x(i_{b+1 \\text{ mod } r})\\rbrace $ .","We have that $S_1$ and $S_2$ are disjoint subsets of $M$ (that is, they form a partition of $M$ ) where the sum of all $S_1$ elements is equal to the sum of all $S_2$ elements, that is, if the cyclic graph constructed from $G$ admits an embedding on the line (which means that its solution to EQ-CDGP is YES), then $M$ has a YES solution for Partition and vice-versa.","This reduction can be made in $O(r)$ time, thus, EQ-CDGP $\\in $ NP-hard.","To illustrate the reasoning from Theorem REF , let $M$ be an instance of Partition such that $M =\\lbrace 1, 4, 5, 6, 7, 9\\rbrace $ .","Figure REF shows its corresponding EQ-CDGP solution.","Figure: Partition instances and corresponding transformations to EQ-CDGP.Since most graph coloring problems in the literature and in real world applications are optimization problems, we define an optimization version of this basic distance geometry graph coloring problem, as shown below.","Definition 2 Minimum Equal Coloring Distance Geometry Problem (MinEQ-CDGP): Given a simple weighted undirected graph $G = (V, E)$ , where, for each $(i, j) \\in E$ , there is a weight $d_{i,j} \\in \\mathbb {N}$ , find an embedding $x: V \\rightarrow \\mathbb {N}$ such that $|x(i) - x(j)| = d_{i,j}$ for each $(i, j) \\in E$ whose span $S$ , defined as $S = \\max _{i \\in V} x(i)$ , that is, the maximum used color, is the minimum possible.","Figure REF shows an example of this model and its corresponding 0-sphere visualization.","Figure: MinEQ-CDGP instance with solution and its 0-sphere representation.On the other hand, instead of equalities, we can consider inequalities, such that the weight $d_{i,j}$ on an edge $(i, j)$ is actually a lower bound for the distance to be respected between the color points $x(i)$ and $x(j)$ , that is, $|x(i) - x(j)| \\ge d_{ij}$ .","Thus, we can modify MinEQ-CDGP to deal with this scenario, which becomes the following model.","Definition 3 Minimum Greater than or Equal Coloring Distance Geometry Problem(MinGEQ-CDGP): Given a simple weighted undirected graph $G = (V, E)$ , where, for each $(i, j) \\in E$ , there is a weight $d_{i,j} \\in \\mathbb {N}$ , find an embedding $x: V \\rightarrow \\mathbb {N}$ such that $|x(i) - x(j)| \\ge d_{i,j}$ for each $(i, j) \\in E$ whose span ($\\max \\limits _{i \\in V} x(i)$ ) is the minimum possible.","MinGEQ-CDGP is equivalent to the bandwidth coloring problem (BCP) [20], which itself is equivalent to the minimum span frequency assignment problem (MS-FAP) [14], [3].","Figure REF In Figure REF , this model, along with its 0-sphere representation, is exemplified.","Figure: MinGEQ-CDGP instance with solution and its 0-sphere representation."],["Special cases","For the models previously stated, we can identify some specific scenarios for which additional properties can be identified.","The first special case is for EQ-CDGP, the decision distance coloring problem, where all distances are the same, that is, the input is a graph with uniform edge weights, as stated below.","Definition 4 Coloring Distance Geometry Problem with Uniform Distances (EQ-CDGP-Unif): Given a simple weighted undirected graph $G = (V, E)$ , and a nonnegative integer $\\varphi $ , find an embedding $x: V \\rightarrow \\mathbb {N}$ such that $|x(i) - x(j)| = \\varphi $ for each $(i, j) \\in E$ .","For the optimization version, we can also define this special case, as shown below.","Definition 5 Minimum Equal Coloring Distance Geometry Problem with Uniform Distances (MinEQ-CDGP-Unif): Given a simple weighted undirected graph $G = (V, E)$ , and a nonnegative integer $\\varphi $ , find an embedding $x: V \\rightarrow \\mathbb {N}$ such that $|x(i) - x(j)| = \\varphi $ for each $(i, j) \\in E$ whose span ($\\max _{i \\in V} x(i)$ ) is the minimum possible.","In this model, an input graph can be defined by its sets of vertices and edges and the $\\varphi $ value, instead of a set of weights for each edge.","A similar special case exists for MinGEQ-CDGP, as stated in the following definition.","Definition 6 Minimum Greater than or Equal Coloring Distance Geometry Problem with Uniform Distances (MinGEQ-CDGP-Unif): Given a simple weighted undirected graph $G = (V, E)$ , and a nonnegative integer $\\varphi $ , find an embedding $x: V \\rightarrow \\mathbb {N}$ such that $|x(i) - x(j)| \\ge \\varphi $ for each $(i, j) \\in E$ whose span ($\\max _{i \\in V} x(i)$ ) is the minimum possible.","When $\\varphi =1$ , MinGEQ-CDGP-Unif is equivalent to the classic graph coloring problem (Figure REF ).","A summary of all distance coloring models, including special cases, is given in Table REF .","Table: Summary of distance coloring models.Figure: Examples of instances for the special cases of distance coloring models withconstant edge weights and feasible solutions for them."],["Feasibility conditions of distance graph coloring problems","In this section, we discuss feasibility conditions related to our proposed EQ-CDGP problems.","Clearly, the problems involving inequality constraints are always feasible.","This is the case for the MinGEQ-CDGP and MinGEQ-CDGP problems (and the special cases with uniform distances, MinGEQ-CDGP-Unif and MinGEQ-CDGP-Unif).","However, this is not so for versions that involve only equality constraints, EQ-CDGP and its special case with uniform distances, the EQ-CDGP-Unif problem."],["Feasibility conditions for EQ-CDGP-Unif","Graphs that admit a solution for the EQ-CDGP-Unif problem are characterized by the following theorem.","Theorem 2 A graph $G$ has solution YES for EQ-CDGP-Unif problem if and only if $G$ is bipartite.","Let $G$ be a graph, input to a EQ-CDGP-Unif problem, where for each edge $v_iv_j$ of $G$ , the distance required is $d_{ij}=\\varphi $ , $\\varphi \\in \\mathbb {N}$ , constant.","Suppose $G$ has a YES solution for the problem such that $x: V \\rightarrow \\mathbb {N}$ is a certificate for that solution.","Let $x(i)$ be the color assigned to $v_i \\in V$ .","Choose an arbitrary path $v_1,v_2,...,v_k$ of $G$ , not necessarily simple.","Then $|x(i)-x(j)|=\\varphi $ , for $|i-j|=1$ .","The latter implies $x(i)=x(i+2)$ , $i=1,2,...,k-2$ .","Consequently, if the path contains the same vector $v_i$ twice, their corresponding indices are the same.","That is, all edges of $G$ are necessarily even, and $G$ is bipartite.","Conversely, if $G$ is bipartite, its vertices admit a proper coloring with two distinct colors.","Assign the value $x(i)$ to the vertices of the first color, and the value $\\varphi + 1$ to the second one.","Then $|x(i)-x(j)|=\\varphi $ , for each edge $v_iv_j$ of $G$ , and EQ-CDGP-Unif has a YES solution.","As an alternative way of proving that if a graph is bipartite then it has a YES solution for EQ-CDGP, observe that, since the input graph is bipartite, it is also 2-colorable (considering the classic graph coloring problem), that is, the entire graph can be colored using only two different colors, which can be determined by considering a single edge from the graph.","All edges $(i, j)$ have the same distance constraint, that is, $|x(i) - x(j)| = \\varphi $ , so the two colors that will be used are {1, 1+$\\varphi $ }, which form the solution for the EQ-CDGP-Unif instance.","In order to prove the converse statement, that is, if a graph has a YES solution for EQ-CDGP, it is bipartite, we will use a proof by contrapositive, which states that if a graph is not bipartite, then it has a NO solution for EQ-CDGP.","This will be done by mathematical induction on odd cycles, since a graph is not bipartite if, and only if, it contains an odd cycle.","Let $|V| = 2z+1$ .","The proof will be by induction on $z$ (the number of vertices).","Base case: $z=1$ .","We have the cycle $C_3$ , with three vertices ($V = \\lbrace 1, 2, 3\\rbrace $ ) and three edges ($\\lbrace (1,2), (1,3), (2,3)\\rbrace $ ), with $|x(i)-x(j)| = \\varphi $ for all of them.","Without loss of generality, let $x(1) = 1$ and $x(2) = 1+\\varphi $ .","Then we have that: Since $(1,3) \\in E$ and $x(1) = 1$ , then $|x(3)-1| = \\varphi $ .","All colors must be positive integers, so $x(3) = 1+\\varphi $ .","Since $(2,3) \\in E$ and $x(2) = 1+\\varphi $ , then $|x(3)-(1+\\varphi )| = \\varphi \\ \\ \\Rightarrow \\ \\ |x(3)-1-\\varphi | = \\varphi $ .","By this inequation, $x(3) = 1$ or $x(3) = 1+ 2\\varphi $ .","From this result, we have that $x(3) = 1+\\varphi $ and ($x(3) = 1$ or $x(3) = 1+ 2\\varphi $ ) at the same time, which is impossible.","Then $C_3$ has a NO solution for EQ-CDGP, as seen in Figure REF .","Induction hypothesis: The cycle $C_{2z+1}$ has a NO solution for EQ-CDGP.","Inductive step: By the inductive hypothesis, the cycle $C_{2z+1}$ is infeasible for EQ-CDGP.","If we consider the cycle $C_{2(z+1)+1} = C_{2z+3}$ , we have that the size of the cycle increases by two vertices, but it will still be an odd cycle.","If we add only one vertex, that is, we consider the cycle $C_{2z+1+1} = C_{2z+2}$ , we will have an even cycle.","Since all even cycles are bipartite, they are feasible in EQ-CDGP according to Theorem REF .","Now, consider that another vertex is added to $C_{2z+2}$ , becoming $C_{2z+3}$ .","Without loss of generality, consider that the new vertex $2z+3$ is adjacent to vertices $2z+2$ and 1, that is, we have $\\lbrace (2z+2, 2z+3), (2z+3, 1)\\rbrace \\subseteq E$ , and $x(2z+2) = 1+\\varphi $ and $x(1) = 1$ (these colors can be seen as having been assigned when we added only one vertex, generating an even cycle).","Then we have that: Since $(2z+2, 2z+3) \\in E$ and $x(2z+2) = 1+\\varphi $ , then $|x(2z+3)-(1+\\varphi )| = \\varphi \\ \\ \\Rightarrow \\ \\ |x(2z+3)-1-\\varphi | = \\varphi $ .","By this inequation, $x(2z+3) = 1$ or $x(2z+3) = 1+ 2\\varphi $ .","Since $(2z+3, 1) \\in E$ and $x(1) = 1$ , then $|x(2z+3)-1| = \\varphi $ .","All colors must be positive integers, so $x(2z+3) = 1+\\varphi $ .","From this result, we have that $x(2z+3) = 1+\\varphi $ and ($x(2z+3) = 1$ or $x(2z+3) = 1+ 2\\varphi $ ) at the same time, which is impossible.","Therefore $C_{2z+3}$ has a NO solution for EQ-CDGP, as can be seen in Figure REF .","Figure: C 3 C_3 graph that has a NO solution for EQ-CDGP-Const when all distances are the same.Figure: Odd cycle C 2z+3 C_{2z+3} that has a NO solution for EQ-CDGP-Const when all distances are the same.As a complementary result, graphs which have odd-length cycles as induced subgraphs will always have a NO solution for EQ-CDGP-Unif, because a graph is bipartite if, and only if, it contains no odd-length cycles.","Since the recognition of bipartite graphs can be done in linear time using a graph search algorithm such as depth-first search (DFS), the EQ-CDGP-Unif problem can be solved in linear time."],["Feasibility conditions for EQ-CDGP","Clearly, Theorem REF does not apply when the distances are arbitrarily defined.","Depending on the edge weights, bipartite graphs may have NO solutions for EQ-CDGP, and graphs which include odd-length cycles may have YES solutions.","Figure REF shows examples of instances considering each case.","However, this decision problem can be easily solved for trees, as shown below.","Figure: Examples of instances for the special cases of distance coloring models withconstant edge weights and feasible solutions for them.Theorem 3 Let $G = (V, E, d)$ , be a tree, where $\\forall (i, j) \\in E\\ \\ d_{i, j}$ is an arbitrary positive integer.","Then $G$ always has a YES solution for EQ-CDGP.","We describe a simple algorithm for assigning colors that satisfy the EQ-CDGP problem.","Initially unmark all vertices.","Choose an arbitrary vertex $v_i$ , assign any positive integer value $x(v_i)$ to $v_i$ , and mark $v_i$ .","Iteratively, choose an unmarked vertex $v_j$ , adjacent to some marked vertex $v_k$ .","Assign the value $x(v_j)=d_{jk}+x(v_j)$ and mark $v_j$ .","Repeat the iteration until all vertices become marked.","The algorithm described in Theorem REF has linear time complexity.","It is important to note that when this procedure is applied to a MinEQ-CDGP instance, that is, the optimization problem with equality constraints, it only guarantees that a feasible solution is found for a tree, not the optimal one."],["Algorithmic techniques and methods to solve EQ-CDGP models","In this section, we show some algorithmic strategies to solve our distance geometry graph coloring models, and discuss some algorithmic strategies considering the EQ-CDGP models proposed in the previous section."],["Branch-prune-and-bound methods","For solving the three distance geometry graph coloring models shown in Section , we developed three algorithms that combine concepts from constraint propagation and optimization techniques.","A branch-and-prune (BP) algorithm was proposed by [16] for the Discretizable Molecular Distance Geometry Problem (DMDGP), based on a previous version for the MDGP by [18].","The algorithm proceeds by enumerating possible positions for the vertices that must be located in three-dimensional space ($\\mathbb {R}^3$ ), by manipulating the set of available distances.","The position for a vertex $i$ , where $i \\in [4, n]$ and $n$ is the number of vertices that must be placed in $\\mathbb {R}^3$ , is determined with respect to the last three vertices that have already been positioned, following the ordering and sphere intersection cited in Section .","However, a distance between the currently positioned vertex and a previous one that was placed before the last three can be violated, which requires feasibility tests to guarantee that the solution being built is valid.","The authors applied the Direct Distance Feasibility (DDF) pruning test, where $\\forall (i, j) \\in E\\ \\ |||x(i) - x(j)|| - d_{i, j}| < \\epsilon $ , and where $\\epsilon $ is a given tolerance.","In this work, we adapted these concepts to study and solve our proposed distance geometry coloring models.","One of the first reflections that can be made is that for the distance geometry coloring models, there are no initial assumptions to be respected, and thus, there is no explicit vertex ordering to be considered, so we build the ordering by an implicit enumerating process.","We mix concepts from branch-and-prune for DMDGP and branch-and-bound procedures to obtain partial solutions (sequences of vertices that have already been colored) that cannot improve on the current best solution.","Our branch-prune-and-bound (BPB) method works as follows.","First, a vertex $i$ that has not been colored yet is selected as a starting point.","This vertex receives the color 1, which is the lowest available (since all colors are positive integers).","Then a neighbor $j$ of $i$ that has not been colored yet is selected.","A color selection algorithm is used for setting a color to $j$ and the process is repeated recursively for neighbors of $j$ that have not been colored yet.","When an uncolored neighbor of the current vertex cannot be found, a uncolored vertex of the graph is used.","Pseudocode for this general procedure is given in Algorithm REF .","[t] Branch-prune-and-bound general algorithm.","0.9 [1] graph $G$ (with set $V$ of vertices and set $E$ of edges), function $d: E \\rightarrow \\mathbb {N}$ of distances for each edge, previous vertex $i$ , current vertex $j$ to be colored, current partial coloring $x$ , best complete coloring found $x_{best}$ , upper bound $ub$ , array $pred$ of predecessors from each vertex (initially all set to -1) and enumeration tree depth $dpt$ .","Branch-Prune-And-Bound$G = (V, E), d, i, j, x, x_{best}, ub, pred, dpt$ each neighbor $k$ of $j$ $predec[k] = -1$ $predec[k] \\leftarrow i$ Set current vertex as predecessor of neighbors $i = -1$ $i \\leftarrow predec[j]$ If this call did not come from a neighbor, use predecessor information $colorsAvail \\leftarrow $ SelectColors$G, d, i, j, x, ub$ $colorsAvail \\ne \\emptyset $ $color \\leftarrow $ element of $colorsAvail$ $colorsAvail \\leftarrow colorsAvail - \\lbrace color\\rbrace $ $x(j) \\leftarrow color$ $\\max \\limits _{v \\in V\\ |\\ v \\text{ is colored}} x(v) \\ge ub$ Remove color from $i$ continue Discard this possible partial solution by bounding FeasibilityTest$G, d, f, x, i$ = false Remove color from $i$ Distance violation, discard partial solution by pruning $dpt = |V|$ If true, then all vertices are colored $\\max \\limits _{v \\in V} x(v)$ $<$ $\\max \\limits _{v \\in V} x_{best}(v)$ $x_{best} \\leftarrow x$ $ub \\leftarrow \\max \\limits _{v \\in V} x(v)$ $hasNeighbor \\leftarrow $ false each neighbor $k$ of $j$ $k$ is not colored $hasNeighbor \\leftarrow $ true Branch-Prune-And-Bound$G, d, f, j, k, x, x_{best}, ub, dpt+1$ $hasNeighbor =$ false each vertex $k$ of $G$ such that $predec[k] \\ne -1$ Only from vertices with predecessors $k$ is not colored Branch-Prune-And-Bound$G, d, f, -1, k, x, x_{best}, ub, dpt+1$ Remove color from $i$ $x_{best}$ We propose different strategies for selecting a color for a vertex and illustrate how the feasibility checking can be done in different levels of the procedure.","Each of these cases are discussed below.","Color selection for a vertex There are two possibilities for determining which colors a vertex can use, determined by the call to SelectColors()), which returns a set of possible colors for a vertex.","The first one, denoted by BPB-Prev, is based on the original BP algorithm by [16].","When a vertex $i$ has to be colored, the single previously colored vertex $j$ is taken into account.","If $j$ is an invalid vertex, which means that $i$ is not an uncolored neighbor of $j$ , then the only color that $i$ can receive is 1.","Otherwise, the function returns a set of cardinality at most 2, whose elements are: $x(j) + d_{i, j}$ .","$x(j) - d_{i, j}$ (returned only if $x(j) > d_{i, j}$ ).","This means that this criterion uses only information from the previous vertex to determine colors, which makes the BPB that uses it an inexact algorithm, something that the original BP for DMDGP also is [16].","However, to counter this in our BPB, when a vertex is colored, its neighbors are marked so that they can use the current vertex as a predecessor in case the search restarts from one of such neighbors.","Since we assume the input graph is connected and the algorithm essentially walks through the graph, this information helps to find the true optimal solution.","This procedure is done in $O(1)$ time, since only two arithmetic operations are made to determine the colors.","An example of this color selection possibility is given in Figure REF .","When using this criterion, we apply the feasibility checking at each colored vertex.","However, an alternative is to prune only infeasible solutions where all vertices have colors, that is, we apply the feasibility test only at the last level of the enumeration tree.","An example of this alternative is shown in Figure REF , where it is possible to see that this strategy makes the tree grow very large.","The second selection criterion is undertaken using information from all colored neighbors to determine the color for the current vertex $i$ .","This is done by solving a system of absolute value inequalities (or equalities, in the case of MinEQ).","Those inequalities arise from the distance constraints for the edges.","Let $i$ be the vertex that must be colored.","The color $x(i)$ must be the solution of a system of absolute value (in)equalities where there is one for each colored neighbor $j$ and each one is as follows: $|x(j) - x(i)|\\ \\ \\text{OP}\\ \\ d_{i, j}$ Where OP is either “=” (for MinEQ-CDGP type problems) or “$\\ge $ ” (for MinGEQ-CDGP type problems).","The color that will be assigned to $j$ is the smallest value that satisfies all (in)equalities.","We note that this procedure always returns a set of cardinality 1, that is, only one color (since only the lowest index is returned) which is also feasible for the partial solution and eventually leads to the optimal solution, although it requires more work per vertex.","This selection strategy runs in $O(ub)$ time, where $ub$ is an upper bound for the span, since, to solve the system, we have to mark each possible solution in the interval $[1, ub]$ and select the smallest value.","Figure REF shows an example of an enumeration tree using this color selection strategy.","Figure: Partial enumeration of solutions starting from vertex 2 for the MinEQ-CDGP instance defined by Figures and using BPB-Prev, with color selection based only on the previous vertex and feasibility checking at each partial solution.Figure: Partial enumeration of solutions starting from vertex 2 for the MinEQ-CDGP instance defined by Figures and using BPB-Prev-FeasCheckFull with color selection based only on the previous vertex and feasibility checking only when all vertices are colored.The backtracking points are indicated when the solution is pruned.Figure: Partial enumeration of solutions starting from vertex 2 for the MinEQ-CDGP instance defined by Figures and using BPB-Select, where a color is determined using a system of absolute value expressions (equalities or inequalities).Feasibility checking When building a partial solution we must verify if it is feasible when not all distances are taken into account at the same time, especially on BPB-Prev.","We used a similar feasibility test to the Direct Distance Feasibility (DDF) used on the BP algorithm for the DMDGP.","Let $i$ be the vertex that has just been colored.","Then we must check, for each neighbor $j$ that has already been colored, if the condition $|x(i)-x(j)| \\ge d_{i, j}$ (if $f((i, j)) = 0$ ) or $|x(i)-x(j)| = d_{i, j}$ (if $f((i, j)) = 1$ ).","This test can be seen as a variation of DDF setting $\\epsilon $ to zero and allowing inequalities in the test.","We denote this procedure as Direct Distance Coloring Feasibility (DDCF) and its pseudocode is given in Algorithm REF .","Direct Distance Coloring Feasibility (DDCF) check 0.9 [1] graph $G$ (with set $V$ of vertices and set $E$ of edges), problem type $t$ (MinGEQ-CDGP or MinEQ-CDGP), matrix $d$ of distances for each edge, current coloring $x$ and vertex $i$ .","DDCF-Check$G, d, f, x, i$ each neighbor $k$ of $i$ $k$ is colored $t$ = MinGEQ-CDGP Inequality constraint $|x(k)-x(i)| > d_{i,j}$ false Equality constraint $|x(k)-x(i)| \\ne d_{i,j}$ false true We note that when selecting a color using the first criterion (only taking into account the previously colored vertex) the feasibility check can be made at each colored vertex or only when all vertices have been colored (which will require that the function DDF-Check() is called for each vertex).","Each check (for only one vertex) runs in $O(|V|)$ time, and if the entire coloring is checked (that is, for all vertices), it runs in $O(|V|^2)$ time.","We also note that, when using the second criterion (using a system of absolute value (in)equalities), the feasibility check can be skipped, since the color that it returns is always feasible.","The combination of these selection criteria and the corresponding feasibility checks result in three possible BPB algorithms, which are summarized in Table REF .","Table: Summary of branch-prune-and-bound methods."],["Computational experiments","In order to analyze the behavior of the proposed distance geometry coloring problems and the branch-prune-and-bound algorithms, we made two main sets of experiments: the first one involved generating many random graphs with different numbers of vertices according to some configurations and counting how many include even or odd cycles (while the rest are trees), since some of the properties of distance geometry coloring are related to these types of graphs.","All algorithms used in these experiments were implemented in C language (compiled with gcc 4.8.4 using options -Ofast -march=native -mtune=native) and executed on a computer equipped with an Intel Core i7-3770 (3,4GHz), 8GB of memory and Linux Mint 17 operating system.","We describe each set of experiments below."],["Counting members of graph classes in random instances","Using Theorems REF and REF , we have information about some types of graphs which always have feasible embeddings for EQ-CDGP and EQ-CDGP-Unif.","Based on this, we generated a large amount of random graphs with different number of vertices and counted how many were cyclic (and based on that, how many there were for each possibility of having even or odd cycles) and how many were trees.","Each random graph always starts as a random spanning tree, that is, a connected undirected graph $G = (V, E)$ , where $|E| =|V|-1$ .","To generate this initial tree, we used a random walk algorithm proposed independently by [5] and [2].","The procedure works by using a set $V^*$ of the vertices outside the tree and a set $W$ of edges of the spanning tree.","Then, whenever the random walk reaches a vertex $j$ outside the tree, the edge $(i, j)$ is added to $E$ and $j$ is removed from $V^*$ .","This continues until $V^* = \\varnothing $ .","We note that this amounts to making a random walk in a complete graph of $|V|$ vertices and it generates trees in a uniform manner, that is, for all possible spanning trees of a given complete graph, each one has the same probability of being generated by the algorithm.","After the initial tree is generated, we add random new edges to it until the graph has the desired number of edges.","This parameter is also randomly set, sampled from interval $\\left[|V|-1, \\frac{|V| (|V|-1)}{2}\\right]$ .","This interval ensures that the generated graph is always connected and is, at least, a tree and, at most, a complete graph.","In Table REF , we outline statistics obtained from using the described procedure to generate 1,000,000 (one million) random graphs for each $|V| \\in \\lbrace 50, 100, 150, 200, 250, 300, 350, 400, 450, 500\\rbrace $ .","As we can observe, most of the graphs (more than 99%) generated have odd cycles, which translates into a very small set of possible EQ-CDGP-Unif instances with feasible embeddings for this configuration of random graphs.","By increasing the number of vertices, more possibilities for generating edges appear, but the number of possible connections which will lead to trees or graphs with even cycles is very small.","In fact, we can deduce that this configuration generated very few bipartite graphs.","For EQ-CDGP (with arbitrary distances), the space of instances with guaranteed feasible embeddings is even smaller, since only trees are certain to have them.","However, as shown in Section , odd and even cycles can have embeddings depending on how the edges are weighted.","In Figure REF , we can observe the growth of the average number of edges between all generated graphs for each number of vertices.","Since the number of edges in a graph is proportional to the square of the number of edges (since $\\frac{|V| (|V|-1)}{2}\\ \\in \\ O(|V|^2)$ , the curve follows a similar pattern, being a half parabola.","Table: Number of random graphs with even, odd or no cycles (trees) and bipartite graphs generated for each number of vertices.","For each size,1,000,000 graphs were generated.Figure: Growth of the average number of edges generated when the number of vertices increases.Figure: Total number of bipartite graphs generated from 1,000,000 random graphs of each number of vertices."],["Results for branch-prune-and-bound algorithms","In order to use some of the random graphs in experiments with the BPB algorithms, we selected four graphs of each type (with even cycle, with odd cycle and trees) for each number of vertices and randomly weighted the edges with a uniform distribution in the interval $[1, 30]$ .","We made two subsets of experiments: the first one involved only the EQ-CDGP and EQ-CDGP-Const models, in order to find a feasible solution for each of its instances that were generated (that is, the algorithms use the pruning procedure, but not bounding - equivalently, stopping the search as soon as the first feasible solution is found), and the second one involved the optimization models for each discussed model (MinEQ-CDGP, MinEQ-CDGP-Const, MinGEQ-CDGP and MinGEQ-CDGP-Const).","Tables REF , REF and REF give detailed results with 4 to 8, 9 to 16 and 18 to 20 vertices, respectively, considering each BPB algorithm applied to the decision versions.","We can see that BPB-Prev reaches a feasible solution faster than the other methods (that is, it solves the decision problem in less time), but it also returns the first feasible solution with a worst span than BPB-Select (noting that BPB-Prev-FeasCheckFull always returns the same span from BPB-Prev because only the pruning point is changed).","However, it is much slower to prove that an instance is infeasible (that is, the answer to the decision problem is NO).","This is explained by the fact that the enumeration tree is smaller in BPB-Select, since instead of two color possibilities for each vertex, there is only one.","Although the time complexity of determining the color for a vertex in BPB-Select is higher (as shown in Table REF ), this is compensated by generating a smaller tree and that the feasibility check is not explicitly needed, since it is guaranteed by the color selection algorithm.","We also note that BPB-Prev-FeasCheckFull has the worst CPU times for infeasible instances, because the method will keep branching the enumeration tree to find a feasible solution, but since feasibility checking is only done at the leaf nodes, the tree will tend to become the full enumeration tree.","In the same manner, Table REF shows the results for the BPB algorithms considering the optimization versions and applied only to feasible MinEQ-CDGP instances.","For almost all of these instances, BPB-Prev is the best method, BPB-Prev-FeasCheckFull shows similar performance and BPB-Select has worse CPU times.","The ties between BPB-Prev and BPB-Prev-FeasCheckFull are explained by the fact that although, in the latter method, the feasibility checking at the leaf nodes increases the time required to find a feasible solution, we keep using the bounding procedures at each node, which reduces the amount of work needed to find the optimal solution.","We also note that, for the 4th Tree instance with 20 vertices, BPB-Select does not find the true optimum for the problem.","This happens because the method is recursively applied only to vertices which have recorded neighbors, in the same manner as the other two BPBs, but the system of absolute value expressions returns only one color, which may not be the one for the optimal solution when applying the recursion only on some vertices.","On some experiments, we detected that, if we generate all vertice orders and apply the color selection of BPB-Select, on them, the optimal solution is found, but the CPU times become very high, since this procedure does not take advantage of the 0-sphere intersection characteristic.","The last experiments were made by applying the BPB algorithms on all instances considered for the algorithms, but by transforming them to MinGEQ-CDGP (changing the $=$ in constraints to $\\ge $ ).","Since they are always feasible for MinGEQ-CDGP because of its equivalence to BCP, we only have to consider optimization problems, as was already done in Section .","The same pattern of previous experiments occur here, with BPB-Prev being the best method of the three, however, the CPU time difference between it and BPB-Prev-FeasCheckFull becomes much more apparent here, since there are many more feasible solutions for MinGEQ-CDGP than MinEQ-CDGP.","Table: Results for BPB algorithms (optimization) applied to MinEQ-CDGP instances - 4 to 20 vertices.Table: Results for BPB algorithms (optimization) applied to MinGEQ-CDGP instances - 4 to 10 vertices.Table: Results for BPB algorithms (optimization) applied to MinEQ-CDGP instances - 12 to 20 vertices."],["Concluding remarks","In this paper, we proposed some distance geometry models for graph coloring problems with distance constraints that can be applied to important, modern real world applications, such as in telecommunications for channel assignment in wireless networks.","In these proposed coloring distance geometry problems (CDGPs), the vertices of the graph are embedded on the real line and the coloring is treated as an assignment of natural numbers to the vertices, while the distances correspond to line segments, whose objective is to determine a feasible intersection of them.","We tackle such problems under the graph theory approach, to establish conditions of feasibility through behavior analysis of the problems in certain classes of graphs, identifying prohibited structures for which the occurrence indicates that it can not admit a valid solution, as well as identifying classes graphs that always admit valid solution.","We also developed exact and approximate enumeration algorithms, based on the Branch-and-Prune (BP) algorithm proposed for the Discretizable Molecular Distance Geometry Problem (DMDGP), combining concepts from constraint propagation and optimization techniques, resulting in Branch-Prune-and-Bound algorithms (BPB), that handle the set of distances in different ways in order to get feasible and optimal solutions.","The computational experiments involved equality and inequality constraints and both uniform and arbitrary sets of distances, where we measure the number of prunes and bounds and the CPU time needed to reach the best solution.","The main contribution of this paper consists of a distance geometry approach for special cases of T-coloring problems with distance constraints, involving a study of graph classes for which some of these distance coloring problems are unfeasible, and branch-prune-and-bound algorithms, combining concepts from the branch-and-bound method and constraint propagation, for the considered problems.","Ongoing and future works include improving the BPB formulations by domain reduction and more specific constraints; developing hybrid methods, involving integer and constraint programming; and applying heuristics, in order to solve the proposed distance coloring models more efficiently.","Studying problems posed to specific classes of graphs, in order to establish other characterizations of feasibility conditions for more general CDGP problems, is also a subject of research in progress.","Acknowledgement We would like to thank Dr. Leo Liberti, professor at LIX - Ãcole Polytechnique, France, for his valuable contribution to this work, having identified a relationship between coloring problems in graphs with additional constraints on the edges and distance geometry problems, and for discussing some of the ideas presented here."]],"string":"[\n [\n \"Distance geometry approach for special graph coloring problems\"\n ],\n [\n \"Abstract One of the most important combinatorial optimization problems is graph coloring.\",\n \"There are several variations of this problem involving additional constraints either on vertices or edges.\",\n \"They constitute models for real applications, such as channel assignment in mobile wireless networks.\",\n \"In this work, we consider some coloring problems involving distance constraints as weighted edges, modeling them as distance geometry problems.\",\n \"Thus, the vertices of the graph are considered as embedded on the real line and the coloring is treated as an assignment of positive integers to the vertices, while the distances correspond to line segments, where the goal is to find a feasible intersection of them.\",\n \"We formulate different such coloring problems and show feasibility conditions for some problems.\",\n \"We also propose implicit enumeration methods for some of the optimization problems based on branch-and-prune methods proposed for distance geometry problems in the literature.\",\n \"An empirical analysis was undertaken, considering equality and inequality constraints, uniform and arbitrary set of distances, and the performance of each variant of the method considering the handling and propagation of the set of distances involved.\"\n ],\n [\n \"Introduction\",\n \"Let $G = (V, E)$ be an undirected graph.\",\n \"A $k$ -coloring of $G$ is an assignment of colors $\\\\lbrace 1, 2, \\\\dots , k\\\\rbrace $ to the vertices of $G$ so that no two adjacent vertices share the same color.\",\n \"The chromatic number $\\\\chi _G$ of a graph is the minimum value of $k$ for which $G$ is $k$ -colorable.\",\n \"The classic graph coloring problem, which consists in finding the chromatic number of a graph, is one of the most important combinatorial optimization problems and it is known to be NP-hard [13].\",\n \"There are several versions of this classic vertex coloring problem, involving additional constraints, in both edges as vertices of the graph, with a number of practical applications as well as theoretical challenges.\",\n \"One of the main applications of such problems consists of assigning channels to transmitters in a mobile wireless network.\",\n \"Each transmitter is responsible for the calls made in the area it covers and the communication among devices is made through a channel consisting of a discrete slice of the electromagnetic spectrum.\",\n \"However, the channels cannot be assigned to calls in an arbitrary way, since there is the problem of interference among devices located near each other using approximate channels.\",\n \"There are three main types of interferences: co-channel, among calls of two transmitters using the same channels; adjacent channel, among calls of two transmitters using adjacent channels and co-site, among calls on the same cell that do not respect a minimal separation.\",\n \"It is necessary to assign channels to the calls such that interference is avoided and the spectrum usage is minimized [3], [15], [14].\",\n \"Thus, the channel assignment scenario is modeled as a graph coloring problem by considering each transmitter as a vertex in a undirected graph and the channels to be assigned as the colors that the vertices will receive.\",\n \"Some more general graph coloring problems were proposed in the literature in order to take the separation among channels into account, such as the T-coloring problem, also known as the Generalized Coloring Problem (GCP) where, for each edge, the absolute difference between colors assigned to each vertex must not be in a given forbidden set [12].\",\n \"The Bandwidth Coloring Problem, a special case of T-coloring where the absolute difference between colors assigned to each vertex must be greater or equal a certain value [20], and the coloring problem with restrictions of adjacent colors (COLRAC), where there is a restriction graph for which adjacent colors in it cannot be assigned to adjacent vertices [1].\",\n \"The separation among channels is a type of distance constraint, so we can see the channel assignment as a type of distance geometry (DG) problem [19] since we have to place the channels in the transmitters respecting some distances imposed in the edges, as can be seen in Figure REF .\",\n \"One method to solve DG problems is the branch-and-prune approach proposed by [16], [17], where a solution is built and if at some point a distance constraint is violated, then we stop this construction (prune) and try another option for the current solution in the search space.\",\n \"See also: [21], [16], [9], [10], [6], [8], [7].\",\n \"For graph theoretic concepts and terminology, see the book by [4].\",\n \"The main contribution of this paper consists of a distance geometry approach for special cases of T-coloring problems with distance constraints, involving a study of graph classes for which some of these distance coloring problems are unfeasible, and branch-prune-and-bound algorithms, combining concepts from the branch-and-bound method and constraint propagation, for the considered problems.\",\n \"The remainder of this paper is organized as follows.\",\n \"Section defines the distance geometry models for some special graph coloring problems.\",\n \"Section shows some properties regarding the structure of those distance geometry graph coloring problems, including the determination of feasibility for some graphs classes.\",\n \"Section formulates the branch-prune-and-bound (BPB) algorithms proposed for the problems and shows properties regarding optimality results.\",\n \"Section shows results of some experiments done with the BPB algorithms using randomly generated graphs for each proposed model.\",\n \"Finally, Section concludes the paper and states the next steps for ongoing research.\",\n \"Figure: Example of channel assignment with distance constraints, wherethe separation is given by the weight in each edge.\",\n \"Theimage on the right shows the network as an undirected graph and the projection of vertices on the real number line,but considering only natural numbers.\"\n ],\n [\n \"Distance geometry and graph colorings\",\n \"We propose an approach in distance geometry for special vertex coloring problems with distance constraints, based on the Discretizable Molecular Distance Geometry Problem (DMDGP), which is a special case of the Molecular Distance Geometry Problem, where the set $V$ of vertices from the input graph $G$ are ordered such that the set $E$ of edges contain all cliques on quadruplets of consecutive vertices, that is, any four consecutive vertices induce a complete graph ($\\\\forall i \\\\in \\\\lbrace 4, \\\\dots , n\\\\rbrace \\\\ \\\\forall j, k \\\\in \\\\lbrace i-3, \\\\dots , i\\\\rbrace \\\\ (\\\\lbrace j, k\\\\rbrace \\\\in E)$ ) [16].\",\n \"Furthermore, a strict triangular inequality holds on weights of edges between consecutive vertices in such ordering ($\\\\forall i \\\\in \\\\lbrace 2, \\\\dots , n-1\\\\rbrace \\\\ d_{i-1, i+1} < d_{i-1, i}\\\\ +\\\\ d_{i, i+1}$ ).\",\n \"All coordinates are given in $\\\\mathbb {R}^3$ space.\",\n \"The position for a point $i$ (where $i \\\\ge 4$ ) can be determined using the positions of the previous three points $i-1, i-2$ and $i-3$ by intersecting three spheres with radii $d_{i-3,i}, d_{i-2, i}$ and $d_{i-1, i}$ , obtaining two possible points that are checked for feasibility.\",\n \"A similar reasoning can be used in vertex coloring problems with distance constraints, where the distances that must be respected involve the absolute difference between two values $x(i)$ and $x(j)$ (respectively, the color points attributed to $i$ and $j$ ), but for these problems the space considered is actually unidimensional.\",\n \"The positioning of a vertex $i$ can be determined by using a neighbor $j$ that is already positioned.\",\n \"Thus, we have a 0-sphere, consisting of a projection of a 1-sphere (a circle), which itself is a projection of a 2-sphere (the three-dimensional sphere), as shown in Figure REF .\",\n \"The 0-sphere is a line segment with a radius $d_{i,j}$ , and feasible colorings consist of treating the intersections of these 0-spheres.\",\n \"Figure REF exemplifies the correlations between these types of spheres and shows the example from Figure REF as the positioning of these line segments.\",\n \"Figure: Some types of nn-spheres.\",\n \"A (n-1n-1)-sphere is a projection of a nn-sphere on a lower dimension.Figure: Example from Figure using 0-spheres (line segments).Figure: Specific order of 0-spheres that leads to the optimal solution for Figure .In this work we focus on problems with exact distances between colors, and also in the analysis of different types of BPB algorithms and integer programming models.\",\n \"Based on DMDGP, which is a decision problem involving equality distance constraints, the basic distance graph coloring model we consider also involves equality constraints between colors of two neighbor vertices $i$ and $j$ .\",\n \"That is, the absolute difference between them must be exactly equal to an arbitrary weight imposed on the edge $(i, j)$ , and the solution candidate must satisfy all given constraints.\",\n \"We can formally define as follows.\",\n \"Given a graph $G=(V,E)$ , we define $d_{i,j}$ as a positive integer weight associated to an edge $(i,j) \\\\in E(G)$ .\",\n \"In distance coloring, for each vertex $i$ , a color must be determined for it (denoted by $x(i)$ ) such that the constraints imposed on the edges between $i$ and its neighbors are satisfied.\",\n \"A variation of the classic graph coloring problem consists in finding the minimum span of $G$ , that is, in determining that the maximum $x(i)$ , or color used, be the minimum possible.\",\n \"Based on these preliminary definitions, we describe the following distance geometry vertex coloring problems.\",\n \"Definition 1 Coloring Distance Geometry Problem (CDGP): Given a simple weighted undirected graph $G = (V, E)$ , where, for each $(i, j) \\\\in E$ , there is a weight $d_{i,j} \\\\in \\\\mathbb {N}$ , find an embedding $x: V \\\\rightarrow \\\\mathbb {N}$ (that is, an embedding of $G$ on the real number line, but considering only the natural number points) such that $|x(i) - x(j)| = d_{i,j}$ for each $(i, j) \\\\in E$ .\",\n \"CDGP involves equality constraints, and thus is named as Equal Coloring Distance Geometry Problem and labeled as EQ-CDGP.\",\n \"A solution for this problem consists of a tree, whose vertices are colored with colors that respect the equality constraints involving the weighted edges (see Figure REF ).\",\n \"Since CDGP (or EQ-CDGP) is a decision problem, only a feasible solution is required.\",\n \"This problem is NP-complete, as shown below.\",\n \"Theorem 1 EQ-CDGP is NP-complete.\",\n \"To prove that EQ-CDGP $\\\\in $ NP-complete, we must show that EQ-CDGP $\\\\in $ NP and EQ-CDGP $\\\\in $ NP-hard.\",\n \"1.\",\n \"EQ-CDGP $\\\\in $ NP.\",\n \"Given, for a graph $G = (V, E)$ , an embedding $x: V \\\\rightarrow \\\\mathbb {N}$ , its feasibility can be checked by taking each edge $(i, j) \\\\in E$ and examining if its endpoints do not violate the corresponding distance constraint, that is, if $|x(i) - x(j)| = d_{i, j}$ .\",\n \"If all distance constraints are valid, then $x$ is a certificate for a positive answer to the EQ-CDGP instance, meaning that a certificate for a YES answer can be verified in $O(|E|)$ time, which is linear.\",\n \"Thus, EQ-CDGP $\\\\in $ NP.\",\n \"2.\",\n \"EQ-CDGP $\\\\in $ NP-hard.\",\n \"Since EQ-CDGP is equivalent to 1-Embeddability with integer weights, which is NP-hard [22], we can use the same proof for the latter problem to show that EQ-CDGP is also NP-hard.\",\n \"The proof is made by reducing the Partition problem, which is known to be NP-complete [11] to EQ-CDGP.\",\n \"Consider a Partition instance, consisting of a set $I$ of $r$ integers, that is, $M = \\\\lbrace m_1, m_2, \\\\dots , m_r\\\\rbrace $ .\",\n \"Let $G$ be a weighted graph $G = (V, E)$ , where $G$ is a cycle such that $|V| = |E| = r$ and, for each edge $(i, j)$ , its weight is a natural number denoted by $d_{i, j}$ .\",\n \"This graph is constructed from $M$ by considering: $V = \\\\lbrace i_0, i_1, \\\\dots , i_{r-1}\\\\rbrace .$ $E = \\\\lbrace (i_b, i_{b+1 \\\\text{ mod } r})\\\\ |\\\\ 0 \\\\le b \\\\le r\\\\rbrace .$ $d_{i_b, i_{b+1 \\\\text{ mod } r}} = m_b\\\\ (\\\\forall 0 \\\\le b \\\\le r).$ Now, let $x: V \\\\rightarrow \\\\mathbb {N}$ be an embedding of the vertices on the number line.\",\n \"If it is a valid embedding, then we can define two sets: $S_1 = \\\\lbrace m_b\\\\ |\\\\ x(i_b) < x(i_{b+1 \\\\text{ mod } r})\\\\rbrace $ .\",\n \"$S_2 = \\\\lbrace m_b\\\\ |\\\\ x(i_b) > x(i_{b+1 \\\\text{ mod } r})\\\\rbrace $ .\",\n \"We have that $S_1$ and $S_2$ are disjoint subsets of $M$ (that is, they form a partition of $M$ ) where the sum of all $S_1$ elements is equal to the sum of all $S_2$ elements, that is, if the cyclic graph constructed from $G$ admits an embedding on the line (which means that its solution to EQ-CDGP is YES), then $M$ has a YES solution for Partition and vice-versa.\",\n \"This reduction can be made in $O(r)$ time, thus, EQ-CDGP $\\\\in $ NP-hard.\",\n \"To illustrate the reasoning from Theorem REF , let $M$ be an instance of Partition such that $M =\\\\lbrace 1, 4, 5, 6, 7, 9\\\\rbrace $ .\",\n \"Figure REF shows its corresponding EQ-CDGP solution.\",\n \"Figure: Partition instances and corresponding transformations to EQ-CDGP.Since most graph coloring problems in the literature and in real world applications are optimization problems, we define an optimization version of this basic distance geometry graph coloring problem, as shown below.\",\n \"Definition 2 Minimum Equal Coloring Distance Geometry Problem (MinEQ-CDGP): Given a simple weighted undirected graph $G = (V, E)$ , where, for each $(i, j) \\\\in E$ , there is a weight $d_{i,j} \\\\in \\\\mathbb {N}$ , find an embedding $x: V \\\\rightarrow \\\\mathbb {N}$ such that $|x(i) - x(j)| = d_{i,j}$ for each $(i, j) \\\\in E$ whose span $S$ , defined as $S = \\\\max _{i \\\\in V} x(i)$ , that is, the maximum used color, is the minimum possible.\",\n \"Figure REF shows an example of this model and its corresponding 0-sphere visualization.\",\n \"Figure: MinEQ-CDGP instance with solution and its 0-sphere representation.On the other hand, instead of equalities, we can consider inequalities, such that the weight $d_{i,j}$ on an edge $(i, j)$ is actually a lower bound for the distance to be respected between the color points $x(i)$ and $x(j)$ , that is, $|x(i) - x(j)| \\\\ge d_{ij}$ .\",\n \"Thus, we can modify MinEQ-CDGP to deal with this scenario, which becomes the following model.\",\n \"Definition 3 Minimum Greater than or Equal Coloring Distance Geometry Problem(MinGEQ-CDGP): Given a simple weighted undirected graph $G = (V, E)$ , where, for each $(i, j) \\\\in E$ , there is a weight $d_{i,j} \\\\in \\\\mathbb {N}$ , find an embedding $x: V \\\\rightarrow \\\\mathbb {N}$ such that $|x(i) - x(j)| \\\\ge d_{i,j}$ for each $(i, j) \\\\in E$ whose span ($\\\\max \\\\limits _{i \\\\in V} x(i)$ ) is the minimum possible.\",\n \"MinGEQ-CDGP is equivalent to the bandwidth coloring problem (BCP) [20], which itself is equivalent to the minimum span frequency assignment problem (MS-FAP) [14], [3].\",\n \"Figure REF In Figure REF , this model, along with its 0-sphere representation, is exemplified.\",\n \"Figure: MinGEQ-CDGP instance with solution and its 0-sphere representation.\"\n ],\n [\n \"Special cases\",\n \"For the models previously stated, we can identify some specific scenarios for which additional properties can be identified.\",\n \"The first special case is for EQ-CDGP, the decision distance coloring problem, where all distances are the same, that is, the input is a graph with uniform edge weights, as stated below.\",\n \"Definition 4 Coloring Distance Geometry Problem with Uniform Distances (EQ-CDGP-Unif): Given a simple weighted undirected graph $G = (V, E)$ , and a nonnegative integer $\\\\varphi $ , find an embedding $x: V \\\\rightarrow \\\\mathbb {N}$ such that $|x(i) - x(j)| = \\\\varphi $ for each $(i, j) \\\\in E$ .\",\n \"For the optimization version, we can also define this special case, as shown below.\",\n \"Definition 5 Minimum Equal Coloring Distance Geometry Problem with Uniform Distances (MinEQ-CDGP-Unif): Given a simple weighted undirected graph $G = (V, E)$ , and a nonnegative integer $\\\\varphi $ , find an embedding $x: V \\\\rightarrow \\\\mathbb {N}$ such that $|x(i) - x(j)| = \\\\varphi $ for each $(i, j) \\\\in E$ whose span ($\\\\max _{i \\\\in V} x(i)$ ) is the minimum possible.\",\n \"In this model, an input graph can be defined by its sets of vertices and edges and the $\\\\varphi $ value, instead of a set of weights for each edge.\",\n \"A similar special case exists for MinGEQ-CDGP, as stated in the following definition.\",\n \"Definition 6 Minimum Greater than or Equal Coloring Distance Geometry Problem with Uniform Distances (MinGEQ-CDGP-Unif): Given a simple weighted undirected graph $G = (V, E)$ , and a nonnegative integer $\\\\varphi $ , find an embedding $x: V \\\\rightarrow \\\\mathbb {N}$ such that $|x(i) - x(j)| \\\\ge \\\\varphi $ for each $(i, j) \\\\in E$ whose span ($\\\\max _{i \\\\in V} x(i)$ ) is the minimum possible.\",\n \"When $\\\\varphi =1$ , MinGEQ-CDGP-Unif is equivalent to the classic graph coloring problem (Figure REF ).\",\n \"A summary of all distance coloring models, including special cases, is given in Table REF .\",\n \"Table: Summary of distance coloring models.Figure: Examples of instances for the special cases of distance coloring models withconstant edge weights and feasible solutions for them.\"\n ],\n [\n \"Feasibility conditions of distance graph coloring problems\",\n \"In this section, we discuss feasibility conditions related to our proposed EQ-CDGP problems.\",\n \"Clearly, the problems involving inequality constraints are always feasible.\",\n \"This is the case for the MinGEQ-CDGP and MinGEQ-CDGP problems (and the special cases with uniform distances, MinGEQ-CDGP-Unif and MinGEQ-CDGP-Unif).\",\n \"However, this is not so for versions that involve only equality constraints, EQ-CDGP and its special case with uniform distances, the EQ-CDGP-Unif problem.\"\n ],\n [\n \"Feasibility conditions for EQ-CDGP-Unif\",\n \"Graphs that admit a solution for the EQ-CDGP-Unif problem are characterized by the following theorem.\",\n \"Theorem 2 A graph $G$ has solution YES for EQ-CDGP-Unif problem if and only if $G$ is bipartite.\",\n \"Let $G$ be a graph, input to a EQ-CDGP-Unif problem, where for each edge $v_iv_j$ of $G$ , the distance required is $d_{ij}=\\\\varphi $ , $\\\\varphi \\\\in \\\\mathbb {N}$ , constant.\",\n \"Suppose $G$ has a YES solution for the problem such that $x: V \\\\rightarrow \\\\mathbb {N}$ is a certificate for that solution.\",\n \"Let $x(i)$ be the color assigned to $v_i \\\\in V$ .\",\n \"Choose an arbitrary path $v_1,v_2,...,v_k$ of $G$ , not necessarily simple.\",\n \"Then $|x(i)-x(j)|=\\\\varphi $ , for $|i-j|=1$ .\",\n \"The latter implies $x(i)=x(i+2)$ , $i=1,2,...,k-2$ .\",\n \"Consequently, if the path contains the same vector $v_i$ twice, their corresponding indices are the same.\",\n \"That is, all edges of $G$ are necessarily even, and $G$ is bipartite.\",\n \"Conversely, if $G$ is bipartite, its vertices admit a proper coloring with two distinct colors.\",\n \"Assign the value $x(i)$ to the vertices of the first color, and the value $\\\\varphi + 1$ to the second one.\",\n \"Then $|x(i)-x(j)|=\\\\varphi $ , for each edge $v_iv_j$ of $G$ , and EQ-CDGP-Unif has a YES solution.\",\n \"As an alternative way of proving that if a graph is bipartite then it has a YES solution for EQ-CDGP, observe that, since the input graph is bipartite, it is also 2-colorable (considering the classic graph coloring problem), that is, the entire graph can be colored using only two different colors, which can be determined by considering a single edge from the graph.\",\n \"All edges $(i, j)$ have the same distance constraint, that is, $|x(i) - x(j)| = \\\\varphi $ , so the two colors that will be used are {1, 1+$\\\\varphi $ }, which form the solution for the EQ-CDGP-Unif instance.\",\n \"In order to prove the converse statement, that is, if a graph has a YES solution for EQ-CDGP, it is bipartite, we will use a proof by contrapositive, which states that if a graph is not bipartite, then it has a NO solution for EQ-CDGP.\",\n \"This will be done by mathematical induction on odd cycles, since a graph is not bipartite if, and only if, it contains an odd cycle.\",\n \"Let $|V| = 2z+1$ .\",\n \"The proof will be by induction on $z$ (the number of vertices).\",\n \"Base case: $z=1$ .\",\n \"We have the cycle $C_3$ , with three vertices ($V = \\\\lbrace 1, 2, 3\\\\rbrace $ ) and three edges ($\\\\lbrace (1,2), (1,3), (2,3)\\\\rbrace $ ), with $|x(i)-x(j)| = \\\\varphi $ for all of them.\",\n \"Without loss of generality, let $x(1) = 1$ and $x(2) = 1+\\\\varphi $ .\",\n \"Then we have that: Since $(1,3) \\\\in E$ and $x(1) = 1$ , then $|x(3)-1| = \\\\varphi $ .\",\n \"All colors must be positive integers, so $x(3) = 1+\\\\varphi $ .\",\n \"Since $(2,3) \\\\in E$ and $x(2) = 1+\\\\varphi $ , then $|x(3)-(1+\\\\varphi )| = \\\\varphi \\\\ \\\\ \\\\Rightarrow \\\\ \\\\ |x(3)-1-\\\\varphi | = \\\\varphi $ .\",\n \"By this inequation, $x(3) = 1$ or $x(3) = 1+ 2\\\\varphi $ .\",\n \"From this result, we have that $x(3) = 1+\\\\varphi $ and ($x(3) = 1$ or $x(3) = 1+ 2\\\\varphi $ ) at the same time, which is impossible.\",\n \"Then $C_3$ has a NO solution for EQ-CDGP, as seen in Figure REF .\",\n \"Induction hypothesis: The cycle $C_{2z+1}$ has a NO solution for EQ-CDGP.\",\n \"Inductive step: By the inductive hypothesis, the cycle $C_{2z+1}$ is infeasible for EQ-CDGP.\",\n \"If we consider the cycle $C_{2(z+1)+1} = C_{2z+3}$ , we have that the size of the cycle increases by two vertices, but it will still be an odd cycle.\",\n \"If we add only one vertex, that is, we consider the cycle $C_{2z+1+1} = C_{2z+2}$ , we will have an even cycle.\",\n \"Since all even cycles are bipartite, they are feasible in EQ-CDGP according to Theorem REF .\",\n \"Now, consider that another vertex is added to $C_{2z+2}$ , becoming $C_{2z+3}$ .\",\n \"Without loss of generality, consider that the new vertex $2z+3$ is adjacent to vertices $2z+2$ and 1, that is, we have $\\\\lbrace (2z+2, 2z+3), (2z+3, 1)\\\\rbrace \\\\subseteq E$ , and $x(2z+2) = 1+\\\\varphi $ and $x(1) = 1$ (these colors can be seen as having been assigned when we added only one vertex, generating an even cycle).\",\n \"Then we have that: Since $(2z+2, 2z+3) \\\\in E$ and $x(2z+2) = 1+\\\\varphi $ , then $|x(2z+3)-(1+\\\\varphi )| = \\\\varphi \\\\ \\\\ \\\\Rightarrow \\\\ \\\\ |x(2z+3)-1-\\\\varphi | = \\\\varphi $ .\",\n \"By this inequation, $x(2z+3) = 1$ or $x(2z+3) = 1+ 2\\\\varphi $ .\",\n \"Since $(2z+3, 1) \\\\in E$ and $x(1) = 1$ , then $|x(2z+3)-1| = \\\\varphi $ .\",\n \"All colors must be positive integers, so $x(2z+3) = 1+\\\\varphi $ .\",\n \"From this result, we have that $x(2z+3) = 1+\\\\varphi $ and ($x(2z+3) = 1$ or $x(2z+3) = 1+ 2\\\\varphi $ ) at the same time, which is impossible.\",\n \"Therefore $C_{2z+3}$ has a NO solution for EQ-CDGP, as can be seen in Figure REF .\",\n \"Figure: C 3 C_3 graph that has a NO solution for EQ-CDGP-Const when all distances are the same.Figure: Odd cycle C 2z+3 C_{2z+3} that has a NO solution for EQ-CDGP-Const when all distances are the same.As a complementary result, graphs which have odd-length cycles as induced subgraphs will always have a NO solution for EQ-CDGP-Unif, because a graph is bipartite if, and only if, it contains no odd-length cycles.\",\n \"Since the recognition of bipartite graphs can be done in linear time using a graph search algorithm such as depth-first search (DFS), the EQ-CDGP-Unif problem can be solved in linear time.\"\n ],\n [\n \"Feasibility conditions for EQ-CDGP\",\n \"Clearly, Theorem REF does not apply when the distances are arbitrarily defined.\",\n \"Depending on the edge weights, bipartite graphs may have NO solutions for EQ-CDGP, and graphs which include odd-length cycles may have YES solutions.\",\n \"Figure REF shows examples of instances considering each case.\",\n \"However, this decision problem can be easily solved for trees, as shown below.\",\n \"Figure: Examples of instances for the special cases of distance coloring models withconstant edge weights and feasible solutions for them.Theorem 3 Let $G = (V, E, d)$ , be a tree, where $\\\\forall (i, j) \\\\in E\\\\ \\\\ d_{i, j}$ is an arbitrary positive integer.\",\n \"Then $G$ always has a YES solution for EQ-CDGP.\",\n \"We describe a simple algorithm for assigning colors that satisfy the EQ-CDGP problem.\",\n \"Initially unmark all vertices.\",\n \"Choose an arbitrary vertex $v_i$ , assign any positive integer value $x(v_i)$ to $v_i$ , and mark $v_i$ .\",\n \"Iteratively, choose an unmarked vertex $v_j$ , adjacent to some marked vertex $v_k$ .\",\n \"Assign the value $x(v_j)=d_{jk}+x(v_j)$ and mark $v_j$ .\",\n \"Repeat the iteration until all vertices become marked.\",\n \"The algorithm described in Theorem REF has linear time complexity.\",\n \"It is important to note that when this procedure is applied to a MinEQ-CDGP instance, that is, the optimization problem with equality constraints, it only guarantees that a feasible solution is found for a tree, not the optimal one.\"\n ],\n [\n \"Algorithmic techniques and methods to solve EQ-CDGP models\",\n \"In this section, we show some algorithmic strategies to solve our distance geometry graph coloring models, and discuss some algorithmic strategies considering the EQ-CDGP models proposed in the previous section.\"\n ],\n [\n \"Branch-prune-and-bound methods\",\n \"For solving the three distance geometry graph coloring models shown in Section , we developed three algorithms that combine concepts from constraint propagation and optimization techniques.\",\n \"A branch-and-prune (BP) algorithm was proposed by [16] for the Discretizable Molecular Distance Geometry Problem (DMDGP), based on a previous version for the MDGP by [18].\",\n \"The algorithm proceeds by enumerating possible positions for the vertices that must be located in three-dimensional space ($\\\\mathbb {R}^3$ ), by manipulating the set of available distances.\",\n \"The position for a vertex $i$ , where $i \\\\in [4, n]$ and $n$ is the number of vertices that must be placed in $\\\\mathbb {R}^3$ , is determined with respect to the last three vertices that have already been positioned, following the ordering and sphere intersection cited in Section .\",\n \"However, a distance between the currently positioned vertex and a previous one that was placed before the last three can be violated, which requires feasibility tests to guarantee that the solution being built is valid.\",\n \"The authors applied the Direct Distance Feasibility (DDF) pruning test, where $\\\\forall (i, j) \\\\in E\\\\ \\\\ |||x(i) - x(j)|| - d_{i, j}| < \\\\epsilon $ , and where $\\\\epsilon $ is a given tolerance.\",\n \"In this work, we adapted these concepts to study and solve our proposed distance geometry coloring models.\",\n \"One of the first reflections that can be made is that for the distance geometry coloring models, there are no initial assumptions to be respected, and thus, there is no explicit vertex ordering to be considered, so we build the ordering by an implicit enumerating process.\",\n \"We mix concepts from branch-and-prune for DMDGP and branch-and-bound procedures to obtain partial solutions (sequences of vertices that have already been colored) that cannot improve on the current best solution.\",\n \"Our branch-prune-and-bound (BPB) method works as follows.\",\n \"First, a vertex $i$ that has not been colored yet is selected as a starting point.\",\n \"This vertex receives the color 1, which is the lowest available (since all colors are positive integers).\",\n \"Then a neighbor $j$ of $i$ that has not been colored yet is selected.\",\n \"A color selection algorithm is used for setting a color to $j$ and the process is repeated recursively for neighbors of $j$ that have not been colored yet.\",\n \"When an uncolored neighbor of the current vertex cannot be found, a uncolored vertex of the graph is used.\",\n \"Pseudocode for this general procedure is given in Algorithm REF .\",\n \"[t] Branch-prune-and-bound general algorithm.\",\n \"0.9 [1] graph $G$ (with set $V$ of vertices and set $E$ of edges), function $d: E \\\\rightarrow \\\\mathbb {N}$ of distances for each edge, previous vertex $i$ , current vertex $j$ to be colored, current partial coloring $x$ , best complete coloring found $x_{best}$ , upper bound $ub$ , array $pred$ of predecessors from each vertex (initially all set to -1) and enumeration tree depth $dpt$ .\",\n \"Branch-Prune-And-Bound$G = (V, E), d, i, j, x, x_{best}, ub, pred, dpt$ each neighbor $k$ of $j$ $predec[k] = -1$ $predec[k] \\\\leftarrow i$ Set current vertex as predecessor of neighbors $i = -1$ $i \\\\leftarrow predec[j]$ If this call did not come from a neighbor, use predecessor information $colorsAvail \\\\leftarrow $ SelectColors$G, d, i, j, x, ub$ $colorsAvail \\\\ne \\\\emptyset $ $color \\\\leftarrow $ element of $colorsAvail$ $colorsAvail \\\\leftarrow colorsAvail - \\\\lbrace color\\\\rbrace $ $x(j) \\\\leftarrow color$ $\\\\max \\\\limits _{v \\\\in V\\\\ |\\\\ v \\\\text{ is colored}} x(v) \\\\ge ub$ Remove color from $i$ continue Discard this possible partial solution by bounding FeasibilityTest$G, d, f, x, i$ = false Remove color from $i$ Distance violation, discard partial solution by pruning $dpt = |V|$ If true, then all vertices are colored $\\\\max \\\\limits _{v \\\\in V} x(v)$ $<$ $\\\\max \\\\limits _{v \\\\in V} x_{best}(v)$ $x_{best} \\\\leftarrow x$ $ub \\\\leftarrow \\\\max \\\\limits _{v \\\\in V} x(v)$ $hasNeighbor \\\\leftarrow $ false each neighbor $k$ of $j$ $k$ is not colored $hasNeighbor \\\\leftarrow $ true Branch-Prune-And-Bound$G, d, f, j, k, x, x_{best}, ub, dpt+1$ $hasNeighbor =$ false each vertex $k$ of $G$ such that $predec[k] \\\\ne -1$ Only from vertices with predecessors $k$ is not colored Branch-Prune-And-Bound$G, d, f, -1, k, x, x_{best}, ub, dpt+1$ Remove color from $i$ $x_{best}$ We propose different strategies for selecting a color for a vertex and illustrate how the feasibility checking can be done in different levels of the procedure.\",\n \"Each of these cases are discussed below.\",\n \"Color selection for a vertex There are two possibilities for determining which colors a vertex can use, determined by the call to SelectColors()), which returns a set of possible colors for a vertex.\",\n \"The first one, denoted by BPB-Prev, is based on the original BP algorithm by [16].\",\n \"When a vertex $i$ has to be colored, the single previously colored vertex $j$ is taken into account.\",\n \"If $j$ is an invalid vertex, which means that $i$ is not an uncolored neighbor of $j$ , then the only color that $i$ can receive is 1.\",\n \"Otherwise, the function returns a set of cardinality at most 2, whose elements are: $x(j) + d_{i, j}$ .\",\n \"$x(j) - d_{i, j}$ (returned only if $x(j) > d_{i, j}$ ).\",\n \"This means that this criterion uses only information from the previous vertex to determine colors, which makes the BPB that uses it an inexact algorithm, something that the original BP for DMDGP also is [16].\",\n \"However, to counter this in our BPB, when a vertex is colored, its neighbors are marked so that they can use the current vertex as a predecessor in case the search restarts from one of such neighbors.\",\n \"Since we assume the input graph is connected and the algorithm essentially walks through the graph, this information helps to find the true optimal solution.\",\n \"This procedure is done in $O(1)$ time, since only two arithmetic operations are made to determine the colors.\",\n \"An example of this color selection possibility is given in Figure REF .\",\n \"When using this criterion, we apply the feasibility checking at each colored vertex.\",\n \"However, an alternative is to prune only infeasible solutions where all vertices have colors, that is, we apply the feasibility test only at the last level of the enumeration tree.\",\n \"An example of this alternative is shown in Figure REF , where it is possible to see that this strategy makes the tree grow very large.\",\n \"The second selection criterion is undertaken using information from all colored neighbors to determine the color for the current vertex $i$ .\",\n \"This is done by solving a system of absolute value inequalities (or equalities, in the case of MinEQ).\",\n \"Those inequalities arise from the distance constraints for the edges.\",\n \"Let $i$ be the vertex that must be colored.\",\n \"The color $x(i)$ must be the solution of a system of absolute value (in)equalities where there is one for each colored neighbor $j$ and each one is as follows: $|x(j) - x(i)|\\\\ \\\\ \\\\text{OP}\\\\ \\\\ d_{i, j}$ Where OP is either “=” (for MinEQ-CDGP type problems) or “$\\\\ge $ ” (for MinGEQ-CDGP type problems).\",\n \"The color that will be assigned to $j$ is the smallest value that satisfies all (in)equalities.\",\n \"We note that this procedure always returns a set of cardinality 1, that is, only one color (since only the lowest index is returned) which is also feasible for the partial solution and eventually leads to the optimal solution, although it requires more work per vertex.\",\n \"This selection strategy runs in $O(ub)$ time, where $ub$ is an upper bound for the span, since, to solve the system, we have to mark each possible solution in the interval $[1, ub]$ and select the smallest value.\",\n \"Figure REF shows an example of an enumeration tree using this color selection strategy.\",\n \"Figure: Partial enumeration of solutions starting from vertex 2 for the MinEQ-CDGP instance defined by Figures and using BPB-Prev, with color selection based only on the previous vertex and feasibility checking at each partial solution.Figure: Partial enumeration of solutions starting from vertex 2 for the MinEQ-CDGP instance defined by Figures and using BPB-Prev-FeasCheckFull with color selection based only on the previous vertex and feasibility checking only when all vertices are colored.The backtracking points are indicated when the solution is pruned.Figure: Partial enumeration of solutions starting from vertex 2 for the MinEQ-CDGP instance defined by Figures and using BPB-Select, where a color is determined using a system of absolute value expressions (equalities or inequalities).Feasibility checking When building a partial solution we must verify if it is feasible when not all distances are taken into account at the same time, especially on BPB-Prev.\",\n \"We used a similar feasibility test to the Direct Distance Feasibility (DDF) used on the BP algorithm for the DMDGP.\",\n \"Let $i$ be the vertex that has just been colored.\",\n \"Then we must check, for each neighbor $j$ that has already been colored, if the condition $|x(i)-x(j)| \\\\ge d_{i, j}$ (if $f((i, j)) = 0$ ) or $|x(i)-x(j)| = d_{i, j}$ (if $f((i, j)) = 1$ ).\",\n \"This test can be seen as a variation of DDF setting $\\\\epsilon $ to zero and allowing inequalities in the test.\",\n \"We denote this procedure as Direct Distance Coloring Feasibility (DDCF) and its pseudocode is given in Algorithm REF .\",\n \"Direct Distance Coloring Feasibility (DDCF) check 0.9 [1] graph $G$ (with set $V$ of vertices and set $E$ of edges), problem type $t$ (MinGEQ-CDGP or MinEQ-CDGP), matrix $d$ of distances for each edge, current coloring $x$ and vertex $i$ .\",\n \"DDCF-Check$G, d, f, x, i$ each neighbor $k$ of $i$ $k$ is colored $t$ = MinGEQ-CDGP Inequality constraint $|x(k)-x(i)| > d_{i,j}$ false Equality constraint $|x(k)-x(i)| \\\\ne d_{i,j}$ false true We note that when selecting a color using the first criterion (only taking into account the previously colored vertex) the feasibility check can be made at each colored vertex or only when all vertices have been colored (which will require that the function DDF-Check() is called for each vertex).\",\n \"Each check (for only one vertex) runs in $O(|V|)$ time, and if the entire coloring is checked (that is, for all vertices), it runs in $O(|V|^2)$ time.\",\n \"We also note that, when using the second criterion (using a system of absolute value (in)equalities), the feasibility check can be skipped, since the color that it returns is always feasible.\",\n \"The combination of these selection criteria and the corresponding feasibility checks result in three possible BPB algorithms, which are summarized in Table REF .\",\n \"Table: Summary of branch-prune-and-bound methods.\"\n ],\n [\n \"Computational experiments\",\n \"In order to analyze the behavior of the proposed distance geometry coloring problems and the branch-prune-and-bound algorithms, we made two main sets of experiments: the first one involved generating many random graphs with different numbers of vertices according to some configurations and counting how many include even or odd cycles (while the rest are trees), since some of the properties of distance geometry coloring are related to these types of graphs.\",\n \"All algorithms used in these experiments were implemented in C language (compiled with gcc 4.8.4 using options -Ofast -march=native -mtune=native) and executed on a computer equipped with an Intel Core i7-3770 (3,4GHz), 8GB of memory and Linux Mint 17 operating system.\",\n \"We describe each set of experiments below.\"\n ],\n [\n \"Counting members of graph classes in random instances\",\n \"Using Theorems REF and REF , we have information about some types of graphs which always have feasible embeddings for EQ-CDGP and EQ-CDGP-Unif.\",\n \"Based on this, we generated a large amount of random graphs with different number of vertices and counted how many were cyclic (and based on that, how many there were for each possibility of having even or odd cycles) and how many were trees.\",\n \"Each random graph always starts as a random spanning tree, that is, a connected undirected graph $G = (V, E)$ , where $|E| =|V|-1$ .\",\n \"To generate this initial tree, we used a random walk algorithm proposed independently by [5] and [2].\",\n \"The procedure works by using a set $V^*$ of the vertices outside the tree and a set $W$ of edges of the spanning tree.\",\n \"Then, whenever the random walk reaches a vertex $j$ outside the tree, the edge $(i, j)$ is added to $E$ and $j$ is removed from $V^*$ .\",\n \"This continues until $V^* = \\\\varnothing $ .\",\n \"We note that this amounts to making a random walk in a complete graph of $|V|$ vertices and it generates trees in a uniform manner, that is, for all possible spanning trees of a given complete graph, each one has the same probability of being generated by the algorithm.\",\n \"After the initial tree is generated, we add random new edges to it until the graph has the desired number of edges.\",\n \"This parameter is also randomly set, sampled from interval $\\\\left[|V|-1, \\\\frac{|V| (|V|-1)}{2}\\\\right]$ .\",\n \"This interval ensures that the generated graph is always connected and is, at least, a tree and, at most, a complete graph.\",\n \"In Table REF , we outline statistics obtained from using the described procedure to generate 1,000,000 (one million) random graphs for each $|V| \\\\in \\\\lbrace 50, 100, 150, 200, 250, 300, 350, 400, 450, 500\\\\rbrace $ .\",\n \"As we can observe, most of the graphs (more than 99%) generated have odd cycles, which translates into a very small set of possible EQ-CDGP-Unif instances with feasible embeddings for this configuration of random graphs.\",\n \"By increasing the number of vertices, more possibilities for generating edges appear, but the number of possible connections which will lead to trees or graphs with even cycles is very small.\",\n \"In fact, we can deduce that this configuration generated very few bipartite graphs.\",\n \"For EQ-CDGP (with arbitrary distances), the space of instances with guaranteed feasible embeddings is even smaller, since only trees are certain to have them.\",\n \"However, as shown in Section , odd and even cycles can have embeddings depending on how the edges are weighted.\",\n \"In Figure REF , we can observe the growth of the average number of edges between all generated graphs for each number of vertices.\",\n \"Since the number of edges in a graph is proportional to the square of the number of edges (since $\\\\frac{|V| (|V|-1)}{2}\\\\ \\\\in \\\\ O(|V|^2)$ , the curve follows a similar pattern, being a half parabola.\",\n \"Table: Number of random graphs with even, odd or no cycles (trees) and bipartite graphs generated for each number of vertices.\",\n \"For each size,1,000,000 graphs were generated.Figure: Growth of the average number of edges generated when the number of vertices increases.Figure: Total number of bipartite graphs generated from 1,000,000 random graphs of each number of vertices.\"\n ],\n [\n \"Results for branch-prune-and-bound algorithms\",\n \"In order to use some of the random graphs in experiments with the BPB algorithms, we selected four graphs of each type (with even cycle, with odd cycle and trees) for each number of vertices and randomly weighted the edges with a uniform distribution in the interval $[1, 30]$ .\",\n \"We made two subsets of experiments: the first one involved only the EQ-CDGP and EQ-CDGP-Const models, in order to find a feasible solution for each of its instances that were generated (that is, the algorithms use the pruning procedure, but not bounding - equivalently, stopping the search as soon as the first feasible solution is found), and the second one involved the optimization models for each discussed model (MinEQ-CDGP, MinEQ-CDGP-Const, MinGEQ-CDGP and MinGEQ-CDGP-Const).\",\n \"Tables REF , REF and REF give detailed results with 4 to 8, 9 to 16 and 18 to 20 vertices, respectively, considering each BPB algorithm applied to the decision versions.\",\n \"We can see that BPB-Prev reaches a feasible solution faster than the other methods (that is, it solves the decision problem in less time), but it also returns the first feasible solution with a worst span than BPB-Select (noting that BPB-Prev-FeasCheckFull always returns the same span from BPB-Prev because only the pruning point is changed).\",\n \"However, it is much slower to prove that an instance is infeasible (that is, the answer to the decision problem is NO).\",\n \"This is explained by the fact that the enumeration tree is smaller in BPB-Select, since instead of two color possibilities for each vertex, there is only one.\",\n \"Although the time complexity of determining the color for a vertex in BPB-Select is higher (as shown in Table REF ), this is compensated by generating a smaller tree and that the feasibility check is not explicitly needed, since it is guaranteed by the color selection algorithm.\",\n \"We also note that BPB-Prev-FeasCheckFull has the worst CPU times for infeasible instances, because the method will keep branching the enumeration tree to find a feasible solution, but since feasibility checking is only done at the leaf nodes, the tree will tend to become the full enumeration tree.\",\n \"In the same manner, Table REF shows the results for the BPB algorithms considering the optimization versions and applied only to feasible MinEQ-CDGP instances.\",\n \"For almost all of these instances, BPB-Prev is the best method, BPB-Prev-FeasCheckFull shows similar performance and BPB-Select has worse CPU times.\",\n \"The ties between BPB-Prev and BPB-Prev-FeasCheckFull are explained by the fact that although, in the latter method, the feasibility checking at the leaf nodes increases the time required to find a feasible solution, we keep using the bounding procedures at each node, which reduces the amount of work needed to find the optimal solution.\",\n \"We also note that, for the 4th Tree instance with 20 vertices, BPB-Select does not find the true optimum for the problem.\",\n \"This happens because the method is recursively applied only to vertices which have recorded neighbors, in the same manner as the other two BPBs, but the system of absolute value expressions returns only one color, which may not be the one for the optimal solution when applying the recursion only on some vertices.\",\n \"On some experiments, we detected that, if we generate all vertice orders and apply the color selection of BPB-Select, on them, the optimal solution is found, but the CPU times become very high, since this procedure does not take advantage of the 0-sphere intersection characteristic.\",\n \"The last experiments were made by applying the BPB algorithms on all instances considered for the algorithms, but by transforming them to MinGEQ-CDGP (changing the $=$ in constraints to $\\\\ge $ ).\",\n \"Since they are always feasible for MinGEQ-CDGP because of its equivalence to BCP, we only have to consider optimization problems, as was already done in Section .\",\n \"The same pattern of previous experiments occur here, with BPB-Prev being the best method of the three, however, the CPU time difference between it and BPB-Prev-FeasCheckFull becomes much more apparent here, since there are many more feasible solutions for MinGEQ-CDGP than MinEQ-CDGP.\",\n \"Table: Results for BPB algorithms (optimization) applied to MinEQ-CDGP instances - 4 to 20 vertices.Table: Results for BPB algorithms (optimization) applied to MinGEQ-CDGP instances - 4 to 10 vertices.Table: Results for BPB algorithms (optimization) applied to MinEQ-CDGP instances - 12 to 20 vertices.\"\n ],\n [\n \"Concluding remarks\",\n \"In this paper, we proposed some distance geometry models for graph coloring problems with distance constraints that can be applied to important, modern real world applications, such as in telecommunications for channel assignment in wireless networks.\",\n \"In these proposed coloring distance geometry problems (CDGPs), the vertices of the graph are embedded on the real line and the coloring is treated as an assignment of natural numbers to the vertices, while the distances correspond to line segments, whose objective is to determine a feasible intersection of them.\",\n \"We tackle such problems under the graph theory approach, to establish conditions of feasibility through behavior analysis of the problems in certain classes of graphs, identifying prohibited structures for which the occurrence indicates that it can not admit a valid solution, as well as identifying classes graphs that always admit valid solution.\",\n \"We also developed exact and approximate enumeration algorithms, based on the Branch-and-Prune (BP) algorithm proposed for the Discretizable Molecular Distance Geometry Problem (DMDGP), combining concepts from constraint propagation and optimization techniques, resulting in Branch-Prune-and-Bound algorithms (BPB), that handle the set of distances in different ways in order to get feasible and optimal solutions.\",\n \"The computational experiments involved equality and inequality constraints and both uniform and arbitrary sets of distances, where we measure the number of prunes and bounds and the CPU time needed to reach the best solution.\",\n \"The main contribution of this paper consists of a distance geometry approach for special cases of T-coloring problems with distance constraints, involving a study of graph classes for which some of these distance coloring problems are unfeasible, and branch-prune-and-bound algorithms, combining concepts from the branch-and-bound method and constraint propagation, for the considered problems.\",\n \"Ongoing and future works include improving the BPB formulations by domain reduction and more specific constraints; developing hybrid methods, involving integer and constraint programming; and applying heuristics, in order to solve the proposed distance coloring models more efficiently.\",\n \"Studying problems posed to specific classes of graphs, in order to establish other characterizations of feasibility conditions for more general CDGP problems, is also a subject of research in progress.\",\n \"Acknowledgement We would like to thank Dr. Leo Liberti, professor at LIX - Ãcole Polytechnique, France, for his valuable contribution to this work, having identified a relationship between coloring problems in graphs with additional constraints on the edges and distance geometry problems, and for discussing some of the ideas presented here.\"\n ]\n]"},"id":{"kind":"string","value":"1606.04978"}}},{"rowIdx":1198,"cells":{"text":{"kind":"list like","value":[["Global adiabaticity and non-Gaussianity consistency condition"],["Abstract In the context of single-field inflation, the conservation of the curvature perturbation on comoving slices, $\\R_c$, on super-horizon scales is one of the assumptions necessary to derive the consistency condition between the squeezed limit of the bispectrum and the spectrum of the primordial curvature perturbation.","However, the conservation of $\\R_c$ holds only after the perturbation has reached the adiabatic limit where the constant mode of $\\R_c$ dominates over the other (usually decaying) mode.","In this case, the non-adiabatic pressure perturbation defined in the thermodynamic sense, $\\delta P_{nad}\\equiv\\delta P-c_w^2\\delta\\rho$ where $c_w^2=\\dot P/\\dot\\rho$, usually becomes also negligible on superhorizon scales.","Therefore one might think that the adiabatic limit is the same as thermodynamic adiabaticity.","This is in fact not true.","In other words, thermodynamic adiabaticity is not a sufficient condition for the conservation of $\\R_c$ on super-horizon scales.","In this paper, we consider models that satisfy $\\delta P_{nad}=0$ on all scales, which we call global adiabaticity (GA), which is guaranteed if $c_w^2=c_s^2$, where $c_s$ is the phase velocity of the propagation of the perturbation.","A known example is the case of ultra-slow-roll(USR) inflation in which $c_w^2=c_s^2=1$.","In order to generalize USR we develop a method to find the Lagrangian of GA K-inflation models from the behavior of background quantities as functions of the scale factor.","Applying this method we show that there indeed exists a wide class of GA models with $c_w^2=c_s^2$, which allows $\\R_c$ to grow on superhorizon scales, and hence violates the non-Gaussianity consistency condition."],["Introduction","A period of accelerated expansion during the early stages of the evolution of the Universe, called inflation [1], [2], [3], is able to account for several otherwise difficult to explain features of the observed Universe such the high level of isotropy of the cosmic microwave background (CMB) [4] radiation and the small value of the curvature.","Some of the simplest inflationary models are based on a single slowly-rolling scalar field, and they are in good agreement with observations.","It is commonly assumed in slow-roll models that adiabaticity in the thermodynamic sense, $\\delta P_{nad}\\equiv \\delta P-c_w^2\\delta \\rho =0$ where $c_w^2=\\dot{P}/\\dot{\\rho }$ , implies the conservation of the curvature perturbation on uniform density slices $\\zeta $ , and hence the conservation of the curvature perturbation on comoving slices ${\\mathcal {R}}_c$ , on super-horizon scales.","In [5] it was shown that there can be important exceptions, i.e.","in some cases thermodynamic adiabaticity does not necessarily imply the super-horizon conservation of ${\\mathcal {R}}_c$ and $\\zeta $ , and that they can differ from each other.","This can happen even for models in which $c_w^2=c_s^2$ .","An example is ultra-slow-roll (USR) inflation [6], [7], which has exact adiabaticity $\\delta P_{nad}=0$ on all scales.","In USR inflation, both ${\\mathcal {R}}_c$ and $\\zeta $ exhibit super-horizon growth but their behavior is very different from each other.","As has been stressed in [8], the non-freezing of ${\\mathcal {R}}_c$ has important phenomenological consequences.","Since the freezing of ${\\mathcal {R}}_c$ on superhorizon scales is a necessary ingredient [9] for Maldacena's consistency relation [10] to hold, models that do not conserve ${\\mathcal {R}}_c$ can actually violate that consistency condition.","In this paper focusing on K-inflation, i.e., Einstein-scalar models with a general kinetic term, we explore in a general way other single field models which have $c_w^2=c_s^2$ , hence satisfy $\\delta P_{nad}=0$ on all scales which we call globally adiabatic (GA), but which may not conserve ${\\mathcal {R}}_c$ .","We find a generalization of the USR model.","A different generalization without imposing the condition $c_w^2=c_s^2$ was discussed in [11], [12].","The method we adopt is based on establishing a general condition for the non-conservation of ${\\mathcal {R}}_c$ in terms of the dependence of the background quantities, in particular the slow-roll parameter $\\epsilon \\equiv -\\dot{H}/H^2$ and the sound velocity $c_s$ , on the scale factor $a$ .","We first derive the necessary condition for the comoving curvature perturbation ${\\mathcal {R}}_c$ to grow on superhorizon scales.","Next we determine $\\rho (a)$ and $P(a)$ by solving the continuity equation.","Then using the equivalence between barotropic fluids and $K$ -inflationary models which satisfy the condition $c_w^2=c_s^2$ [13], [14], we determine the corresponding Lagrangian for the equivalent scalar field model.","Using this method we obtain a new class of GA scalar field models which do not conserve ${\\mathcal {R}}_c$ .","Throughout the paper we denote the proper-time derivative by a dot ($\\dot{~}=d/dt$ ), the conformal-time derivative by a prime ($\\prime {~}=d/d\\eta =a\\,d/dt$ ) and the Hubble expansion rates in proper and conformal times by $H=\\dot{a}/a$ and ${\\cal H}=a^{\\prime }/a$ , respectively.","We also use the terminology “adiabaticity” for thermodynamic adiabaticity $\\delta P_{nad}=0$ throughout the paper."],["Conservation of ${\\mathcal {R}}_c$ and global adiabaticity","We set the perturbed metric as $ds^2&=&a^2\\Bigl [-(1+2A)d\\eta ^2+2\\partial _jB dx^jd\\eta \\nonumber \\\\&& \\qquad +\\left\\lbrace \\delta _{ij}(1+2{\\cal R})+2\\partial _i\\partial _j E\\rbrace dx^idx^j\\right\\rbrace \\Bigr ]\\,.$ In [5] it was shown that independently of the gravity theory and for generic matter the energy-momentum conservation equations imply $\\delta P_{nad}=\\left[\\left(\\frac{c_w}{c_s}\\right)^2-1\\right](\\rho +P) A_c \\,,$ where the subscript $c$ means a quantity evaluated on comoving slices defined by $\\delta T^i_0=0$ (or equivalently slices on which the scalar field is homogeneous).","In the case of general relativity, the additional relation $A_c=\\dot{{\\mathcal {R}}}_c/H$ gives an important relation for the time derivative of ${\\mathcal {R}}_c$ $\\delta P_{ nad}=\\left[\\left(\\frac{c_w}{c_s}\\right)^2-1\\right](\\rho +P)\\frac{\\dot{{\\mathcal {R}}_c}}{H}.","$ The non-adiabatic pressure perturbation is given according to its thermodynamics definition $\\delta P_{nad}\\equiv \\delta P - c_w^2 \\delta \\rho .$ This definition of $\\delta P_{nad}$ is important because it is gauge invariant and $\\delta P_{nad}=\\delta P_{ud}$ , where $\\delta P_{ud}$ is the pressure perturbation on uniform density slices.","It appears in the equation for the curvature perturbation on uniform density slices $\\zeta \\equiv {\\mathcal {R}}_{ud}$ obtained from the energy conservation law [15], $\\zeta ^{\\prime }=-\\frac{{\\cal H}\\delta P_{nad}}{(\\rho +P)}+\\frac{1}{3}\\mathop \\Delta ^{(3)}\\left(v-E^{\\prime }\\right)_{ud}\\,$ where $v$ is the 3-velocity potential ($v=\\delta \\phi /\\phi ^{\\prime }$ for a scalar field).","In general, the curvature perturbations on uniform density and comoving slices are related as $\\zeta ={\\mathcal {R}}_c+\\frac{\\delta P_{ nad}}{3(\\rho +P)(c^2_s-c^2_w)} \\,.$ A common interpretation of these equations (see for example [16], [17]) is that when $\\delta P_{nad}=0$ with $c_w^2 \\ne c_s^2$ , $\\zeta $ and ${\\mathcal {R}}_c$ are equal because of eq.","(REF ), and they are both conserved on super-horizon scales because of eq.","(REF ).","The equation (REF ) is the key relation to understand how ${\\mathcal {R}}_c$ depends on the non-adiabatic pressure $\\delta P_{nad}$ .","First of all let us note that this equation is valid on any scale.","The advantage of it with respect to eq.","(REF ) is that it does not involve gradient terms, so it allows us to directly relate $\\delta P_{\\rm nad}$ to $\\dot{{\\mathcal {R}}}_{c}$ if $c_w^2\\ne c_s^2$ , while in eq.","(REF ) $\\dot{\\zeta }$ depends on spatial gradients, which in the case of USR are not negligible on super-horizon scales [5].","This explains while in USR in which $c_w^2=c_s^2=1$ , both ${\\mathcal {R}}_c$ and $\\zeta $ are not conserved despite $\\delta P_{nad}=0$ .","It should be noted here that for slow-roll attractor models $c_w^2\\ne c_s^2$ in general, and ${\\mathcal {R}}_c$ is time-varying on sub-horizon scales.","This implies that the non-adiabatic pressure perturbation $\\delta P_{nad}$ on sub-horizon scales is not zero.","In other words, the attractor models are adiabatic only on super-horizon scales, and we call these models super-horizon adiabatic (SHA).","From eq.","(REF ) we can immediately deduce that in general relativity there are two possible scenarios for the non-conservation of ${\\mathcal {R}}_c$ , $\\mbox{(1)}\\quad &c_s^2=c_w^2\\,,\\quad \\delta P_{nad}=0\\,,\\nonumber \\\\\\mbox{(2)}\\quad &c_s^2\\ne c_w^2\\,,\\quad \\delta P_{nad}\\ne 0 \\, .$ The second case was studied in [11], [12].","Here we focus on the first case.","It is trivial to see that because of the gauge invariance of $\\delta P_{nad}$ the condition $c_w^2=c_s^2$ automatically implies $\\delta P_{nad}=0$ .","The models satisfying the condition $c_s^2-c_w^2=\\delta P_{nad}=0$ are adiabatic on any scale, and because of this we call them globally adiabatic (GA).","In GA models an explicit calculation can reveal the super-horizon behavior of ${\\mathcal {R}}_c$ , and $\\zeta $ , as was shown in [5] in the case of USR.","Below, we develop an inversion method to find a new class of models that violate the conservation of ${\\mathcal {R}}_c$ without solving the perturbations equations."],["Globally adiabatic K-essence models","The condition $c_w^2=c_s^2$ has been studied in the context of K-inflation [13] described by the action $S=\\frac{1}{2}\\int d^4x\\sqrt{-g}\\left[M^2_{Pl}R+2P(X,\\phi )\\right] \\,,$ and it was shown that it is satisfied by scalar field models with the Lagrangian of the form, $P(X,\\phi )=u(X g(\\phi ))=u(Y)\\,.","$ These models are equivalent to a barotropic perfect fluid, i.e.","a fluid with equation of state $P(\\rho )$ .","See also [18], [19], [20], [21].","We note again that these models are adiabatic on any scale (GA), contrary to the slow-roll attractor models, which are adiabatic only on super-horizon scales (SHA).","The fact that they are mutually exclusive can be readily seen by considering the hypothetical case of $\\delta P_{nad}=0$ and $c_w^2\\ne c_s^2$ .","In this case Eq.","(REF ) which is valid on any scale would mean ${\\mathcal {R}}_c$ should be frozen on all scales.","In contrast, the condition $c_w^2=c_s^2$ allows for the curvature perturbation to evolve both on sub-horizon and super-horizon scales.","In [13] it was shown that is possible to associate any barotropic perfect fluid with an equivalent K-inflation model according to $2\\int ^P\\frac{du}{F(u)}=\\log (Y) \\,, $ where $F(P)=\\rho (P)+P$ and $Y=g(\\phi )X$ with $X=-g^{\\mu \\nu }\\partial _\\mu \\phi \\partial _\\nu \\phi /2$ .","These models are the ones which could violate the conservation of ${\\mathcal {R}}_c$ for adiabatic perturbations, since they satisfy $c_w^2=c_s^2$ .","It is noted of course that the global adiabaticity is not the sufficient condition for the non-conservation of ${\\mathcal {R}}_c$ .","Not all GA models violate the conservation of ${\\mathcal {R}}_c$ on super-horizon scales."],["General conditions for super-horizon growth of ${\\mathcal {R}}_c$","From the equation for the curvature perturbation on comoving slices, $\\frac{\\partial }{\\partial t}\\left(\\frac{a^3\\epsilon }{c_s^2}\\frac{\\partial }{\\partial t}{\\mathcal {R}}_c\\right)-a\\epsilon \\Delta {\\mathcal {R}}_c=0 \\,,$ we can deduce, after re-expressing the time derivative in terms of the derivative respect to the scale factor $a$ , that on superhorizon scales there is (apart from a constant solution) a solution of the form, ${\\mathcal {R}}_c &\\propto & \\int ^a\\frac{da}{a} f(a)\\,;\\quad f(a)\\equiv {\\frac{c_s^2(a)}{Ha^3 \\epsilon (a)}} \\,, $ where we have introduced the function $f(a)$ for later convenience.","In conventional slow-roll inflation $c_s^2$ and $\\epsilon $ are both slowly varying, hence the integral rapidly approaches a constant, rendering ${\\mathcal {R}}_c$ conserved.","The time dependent part of the above solution corresponds to the decaying mode.","The necessary and sufficient condition for super-horizon freezing is that there exists some $\\delta >0$ for which $\\lim _{a\\rightarrow \\infty }a^\\delta f(a)=0 .","$ By definition of inflation, $H$ must be sufficiently slowly varying; $\\epsilon =-\\dot{H}/H^2\\ll 1$ .","So we may neglect the time dependence of $H$ in (REF ) at leading order, while $\\epsilon $ and $c_s^2$ may vary rapidly in time.","For models for which $\\epsilon \\approx a^{-n}$ and $c_s^2\\approx a^q$ we get $f \\propto a^{q+n-3} \\,,$ hence the condition for freezing is $q+n-3<0 \\,.$ If this condition is violated, i.e.","$q+n-3\\ge 0$ , then the solution (REF ) will grow on super-horizon scales.","This happens for example in USR., which corresponds to $c_s^2=1$ and $\\epsilon \\propto a^{-6}$ , i.e.","$q=0$ , and $n=6$ .","(The super-horizon growth of ${\\mathcal {R}}_c$ in USR can also be understood as a direct consequence of the non-attractor nature of USR [22].)","In general, we expect that $q$ would not become very large.","This implies $\\epsilon $ should decrease sufficiently rapidly.","Conversely, if $\\epsilon $ decreases sufficiently rapidly, then the growth of ${\\mathcal {R}}_c$ on superhorizon scales will follow."],["Barotropic model","We have shown that GA models could violate the super-horizon conservation of ${\\mathcal {R}}_c$ , so now we will look for GA K-essence models which do indeed violate it, based on the freezing condition in eq.","(REF ).","Inspired by the equivalence between barotropic fluids and GA K-essence models [13] we will first look for barotropic fluids that can give the growing curvature perturbation on superhorizon scales.","From the very beginning we will set $c_w^2=c_s^2$ .","Using the Friedmann equation we can write the slow-roll parameter $\\epsilon $ as $\\epsilon = -\\frac{\\dot{H}}{H^2}=\\frac{3}{2}\\frac{\\rho +P}{\\rho } \\,.$ In terms of the scale factor and $\\epsilon $ the energy conservation equation reads $\\frac{d\\rho }{da}+\\frac{3}{a}(\\rho +p)=\\frac{d\\rho }{da}+\\frac{2\\epsilon \\rho }{a}=0 .$ We may now define the quantity $b(a)=2\\epsilon \\rho $ .","It appears naturally in the continuity equation and plays a crucial role in regards to the super-horizon behavior of curvature perturbations because the function $f(a)$ can be re-written in terms of it as $f(a)\\propto \\frac{H c_s^2}{a^3 b(a)} \\,.$ Integrating the energy conservation equation we get $\\rho (a)=\\rho _0\\exp \\left[-2\\int _{a_0}^a{\\frac{\\epsilon }{a}da}\\right]=\\int {-\\frac{b(a)}{a} da} \\,.$ Using eq.","(REF ), we then obtain $P(a)=\\left(\\frac{2}{3}\\epsilon -1\\right)\\rho \\,.$ The sound velocity is given by $c_w^2=c_s^2=\\frac{dP}{d\\rho }&=-1+\\frac{1}{3}\\frac{db(a)}{d\\rho }\\nonumber \\\\&=-1+\\frac{1}{3}\\frac{db(a)}{da}\\Big /\\left(\\frac{d\\rho }{da}\\right)\\nonumber \\\\&=-1-\\frac{a}{3b(a)}\\frac{db(a)}{da}\\,.$ We now consider the behavior of $f(a)$ introduced in (REF ).","As mentioned before, we consider the case when $\\epsilon $ decreases sufficiently rapidly.","In this case, $\\rho =3H^2M_P^2$ approaches a constant rapidly.","Hence the time dependence of $\\rho $ may be neglected compared to that of other quantities that vary far more rapidly.","With this approximation, assuming $\\epsilon \\propto a^{-n}$ , we find $c_s^2\\approx \\frac{n-3}{3}\\,,$ which means $q\\approx 0$ , and $f(a)={\\frac{c_s^2(a)}{Ha^3 \\epsilon (a)}}\\propto a^{n-3}\\,,$ which satisfies the condition for the growth if $n>3$ , in accordance with the original anticipation.","In passing, it is interesting to note that the condition $n>3$ implies $c_s^2>0$ , a necessary condition to avoid the gradient instability of the perturbation.","Thus virtually all GA models that are free from the gradient instability exhibit superhorizon growth of the comoving curvature perturbation ${\\mathcal {R}}_c$ ."],["Scalar field model","Let us now find a scalar field model that corresponds to the barotropic model discussed in the previous section.","As a warm-up, let us consider the USR case, whose fluid interpretation has already been studied in [23].","In this case, we exactly have $c_s^2=1$ .","From eq.","(REF ), this implies $b/2=\\epsilon \\rho \\bigl (=3(\\rho +P)/2\\bigr )\\propto a^{-6}$ .","Also $c_s^2=1$ implies $\\rho =P+const.$ Inserting this into eq.","(REF ) gives $\\frac{2dP}{2P+const.","}=\\frac{dY}{Y}\\,.$ Thus up to a constant term $P$ and $Y$ are the same, $P=Y+const..$ Absorbing $g(\\phi )$ in $Y$ into the definition of the scalar field by $g^{1/2}d\\phi \\rightarrow d\\phi $ , this is indeed the Lagrangian for a minimally coupled massless scalar with a cosmological constant: $L=P(\\phi ,X)=X-V_0\\,.","$ This is consistent with $\\rho +P=2X\\propto \\epsilon \\rho \\propto a^{-6}$ .","Let us generalize the USR case.","As in the previous section, we consider models that have the behavior of $\\epsilon \\rho $ as $2\\epsilon \\rho =b(a)\\,,$ where $b(a)$ should decrease faster than $a^{-3}$ asymptotically at $a\\rightarrow \\infty $ but otherwise is an arbitrary function.","Then we have $F(P)\\equiv \\rho +P=2H^2\\epsilon =\\frac{2\\epsilon \\rho }{3}=\\frac{b(a)}{3}\\,,$ which gives $\\frac{dY}{Y}=2\\frac{dP}{F(P)}=6\\frac{dP}{2\\epsilon \\rho }=6\\frac{dP}{b(a)}\\,.$ For $dP$ , using the energy conservation law, we may rewrite it as $dP&=d\\left(-\\rho +F(P)\\right)=-d\\rho +\\frac{db(a)}{3}\\nonumber \\\\&=3\\frac{da}{a}(\\rho +P)+\\frac{db(a)}{3}=b(a)\\frac{da}{a}+\\frac{db(a)}{3}\\,.$ Therefore we have $\\frac{dY}{Y}=6\\frac{dP}{b(a)}=6\\frac{da}{a}+2\\frac{db}{b}\\,.$ Hence $Y\\propto a^6b^2\\,.","$ This is consistent with the USR case in which $b(a) \\propto a^{-6}$ and $Y=X \\propto a^{-6}$ .","This relation is quite useful since it allows to rewrite the freezing function $f(a)$ as $f(a)\\propto \\frac{H c_s^2}{\\sqrt{Y}} \\,,$ from which we can deduce that $Y(a)$ determines the super-horizon behavior of ${\\mathcal {R}}_c$ .","In particular, for the models we are considering in which $c_s$ is constant, we infer that super-horizon growth can happen in the limit $Y\\rightarrow 0$ .","For a given choice of $b(a)$ , eq.","(REF ) can be inverted to give the scale factor as a function of $Y$ , $a=a(Y)$ .","Also eq.","(REF ) can be integrated to give $P=P(a)$ .","Combining these two, one can obtain the Lagrangian for the scalar field, $L=P=P(Y)$ .","Note that in GA models there is a one-to-one correspondence between the scale factor and state variables such as $P(a)$ and $\\rho (a)$ , which is the reason why we can also write a barotropic equation of state $P(\\rho )=P(a(\\rho ))$ .","Once any of the functions $P(a),\\rho (a),b(a),\\epsilon (a),Y(a)$ is specified, all the others are specified too, as well as the equation of state $P(\\rho )$ or its scalar field equivalent Lagrangian $P(Y)$ , which is in fact the basis of the inversion method that we are developing in this paper."],["examples","Here we give a couple of specific K-inflation models that are globally adiabatic and violate the convervation of ${\\mathcal {R}}_c$ .","Given the parametric behaviour of $b\\equiv 2 \\epsilon \\rho $ , our inversion method allows us to deduce the Lagrangian."],["Ex 1: Generalized USR","Let us consider a specific case where $b(a)$ is a power-law function, $2\\epsilon \\rho =b(a)=c a^{-n} \\,.$ where $c$ is a constant.","We assume $n>3$ in order to have the growth on superhorizon scales.","From eq.","(REF ) we have $a\\propto Y^{1/(6-2n)}\\,.$ Now eq.","(REF ) gives $P&=\\int ^a\\left(b(a)\\frac{da}{a}+\\frac{db(a)}{3}\\right)\\nonumber \\\\&=-\\frac{c}{n}a^{-n}+\\frac{c}{3}a^{-n}+const.\\nonumber \\\\&=\\frac{n-3}{3n}b(a)+const..$ Plugging eq.","(REF ) into this, we finally obtain $L=P(Y)= Y^{n/(2n-6)}-V_0 \\,.","$ Since this may be regarded as a natural generalization of the USR case, which corresponds to the case $n=6$ , we call it the generalized USR (GUSR) model.","Lagrangians involving $Y^{\\alpha }$ terms have already been studied in [11], [24], [25], but those models are either not exactly globally adiabatic because of the presence of a not constant potential or they satisfy the relation $\\epsilon \\propto a^{-n}$ only approximately and during a limited time range, while for GUSR $\\epsilon \\propto a^{-n}$ is an exact relation and is valid at any time.","As the Lagrangian is of the type described in eqs.","(REF ) and (REF ) (remember that after a field transformation $Y$ can be made equal to $X$ ), we understand that this scalar field model is indeed equivalent to a barotropic fluid.","Hence we have $c_w^2=c_s^2$ , and therefore $\\delta P_{ nad}=0$ .","Indeed the second condition for super-horizon growth of ${\\mathcal {R}}_c$ given in eq.","(REF ) is satisfied.","More precisely, we note that for the GUSR model, the sound velocity is exactly constant, $c_w^2=c_s^2=\\frac{n-3}{3} \\,.$ The power spectrum of the comoving curvature perturbation can be explicitly computed for this model.","One finds [26] that the spectral index is a function of $n$ : $n_s-1= 6-n$ , in agreement with the scale invariant spectrum of the original ultra slow-roll inflation in which one has $n=6$ .","Hence, the model can be constrained by the observational value.","Note as well, from eq.","(REF ), that to have a slightly red-tilted spectrum, we need a slightly superluminal speed of sound."],["Ex 2: Lambert Inflation","As another example, let us consider the case when $\\epsilon $ is a power-law function, $\\epsilon (a)=\\epsilon _0{a}^{-n}\\,.$ As before, we assume $n>3$ .","In this case, since $d\\log \\rho /d\\log a=-2\\epsilon \\propto a^{-n}$ , we find $\\rho (a)=\\rho _0\\exp \\left[\\frac{2\\epsilon }{n}\\right] \\,.$ It is clear that $\\rho $ approaches a constant $\\rho _0$ asymptotically at $a\\rightarrow \\infty $ .","Inserting eq.","(REF ) and eq.","(REF ) into eq.","(REF ), the sound velocity is given by $c_w^2=c_s^2=-1-\\frac{1}{3}\\left(\\frac{d\\log \\epsilon }{d\\log a}+\\frac{d\\log \\rho }{d\\log a}\\right)=\\frac{n-3+2\\epsilon }{3}\\,.$ Thus $c_s^2$ is time dependent, but it rapidly approaches a constant as $\\epsilon $ decays out.","Also from eq.","(REF ) and eq.","(REF ), we find $b(a)=2\\epsilon \\rho =2\\epsilon \\rho _0\\exp \\left[\\frac{2\\epsilon }{n}\\right]\\,.$ Thus we have $Y\\propto a^6 b^2\\propto a^{6-2n}\\exp \\left[\\frac{4\\epsilon }{n}\\right]\\propto \\epsilon ^{(2n-6)/n}\\exp \\left[\\frac{4\\epsilon }{n}\\right] \\,,$ which implies $Y^{n/(2n-6)}\\propto \\frac{4\\epsilon }{2n-6}\\exp \\left[\\frac{4\\epsilon }{2n-6}\\right]\\,.$ To find the Lagrangian, we manipulate eq.","(REF ) as $dP&=b\\frac{da}{a}+\\frac{db}{3}=-\\frac{b}{n}\\frac{d\\epsilon }{\\epsilon }+\\frac{db}{3}\\nonumber \\\\&=-\\frac{2}{n}\\rho _0e^{2\\epsilon /n}d\\epsilon +\\frac{db}{3}\\,.$ Therefore, integrating this we obtain $P=\\rho _0e^{2\\epsilon /n}\\left(-1+\\frac{2\\epsilon }{3}\\right)+const..$ One can invert eq.","(REF ) to find $\\epsilon $ as a function of $Y$ , and then insert it into the above to obtain the Lagrangian.","Specifically, we introduce the Lambert function $W(x)$ defined by the inverse function of $X(z)=ze^z$ , $z=X^{-1}(ze^z)\\equiv W(ze^z)\\,.$ Setting ${Y}^{n/(2n-6)}=ze^z\\,;\\quad z=\\frac{4\\epsilon }{2n-6}\\,,$ we have $\\frac{4\\epsilon }{2n-6}=W(y)\\,;\\quad y\\equiv Y^{n/(2n-6)}\\,.$ Inserting this into eq.","(REF ), we finally obtain $L&=P(Y)\\nonumber \\\\&=\\rho _0\\left(\\frac{n-3}{3}W(y)-1\\right)\\exp \\left[\\frac{n-3}{n}W(y)\\right]-V_0\\,,$ where $y=y(Y)$ is given in eq.","(REF ).","Note that this model has been derived without making any approximation, and it gives exactly $\\epsilon \\propto a^{-n}$ .","However, as we mentioned before, in the late time limit, there is no difference between $\\epsilon \\propto a^{-n}$ and $\\rho \\epsilon \\propto a^{-n}$ .","Thus the two models discussed above are essentially the same at late times.","This can be easily checked by expanding $W(y)$ around $y=0$ , $W(y)=y-y^2+\\cdots .$ At leading order in $y=Y^{n/(2n-6)}$ , this gives $P(Y)=\\frac{n-3}{3}\\rho _0Y^{n/(2n-6)}-\\rho _0-V_0\\,.$ By absorbing the constant coefficient into $g(\\phi )$ in the definition of $Y$ , $Y=g(\\phi )X$ , and absorbing $\\rho _0$ into the constant $V_0$ , eq.","(REF ) reduces to $P=Y^{n/(2n-6)}-V_0\\,,$ which indeed coincides with the GUSR model, see eq.","(REF ).","Higher order terms in the expansion give an infinite class of models of the type $u(Y)&=&\\sum _i \\beta _i Y^{n_i} \\,, $ where $\\beta _i$ are appropriate coefficients.","Finally, note that in USR and as well in the two examples considered here, the shift symmetry in the potential ($V(\\phi )=V_0$ ) is a direct consequence of the demand $c_w^2=c_s^2$ , which in turn follows from the global adiabaticity of the model.","That is in line with the general statement [27], [28] that for a $k$ -essence theory to describe a fluid, one needs a shift symmetry (i.e., there is no physical clock, the model is of the non-attractor type)."],["Conclusions","By introducing the notion of global adiabaticity, namely, $c_w^2=c_s^2$ and $\\delta P_{nad}=\\delta P-c_w^2\\delta \\rho =0$ , where $c_w^2=\\dot{P}/\\dot{\\rho }$ and $c_s$ is the propagation (phase) speed of the perturbation, we have determined the general conditions for the non-conservation of the curvature perturbations on comoving slices ${\\mathcal {R}}_c$ on super-horizon scales.","We have found that globally adiabatic K-essence models can exhibit this behavior.","We have then developed a method to construct the Lagrangian of a K-essence globally adiabatic (GA) model by specifying the behavior of background quantities such as $\\epsilon \\rho $ where $\\epsilon $ is the slow-roll parameter, using the equivalence between barotropic fluids and GA K-essence models.","We have applied the method to find the equations of state of the fluids and derive the Lagrangian of the equivalent single scalar field models.","Interestingly, we have found that the requirement to avoid the gradient instability, ie, $c_s^2>0$ is almost identical to the condition for the non-conservation on superhorizon scales.","The advantage of our approach is that we have not solved any perturbation equation explicitly, since we have proceeded in the opposite way solving the inversion problem consisting of requiring certain properties to the behavior of the perturbation equation we are interested in.","In other words, instead of starting from a Lagrangian and then solve the perturbations equations we have determined the equation of state or equivalently the Lagrangian which admits a solution of the perturbation equation with the particular behavior we are interested in.","We have shown that the main difference between attractor models and GA models is that the latter are adiabatic on all scales, while attractor models are approximately adiabatic in the sense of $\\delta P_{nad}=0$ only on super-horizon scales and $c_w^2\\ne c_s^2$ .","The detailed study of the new models found in this paper will be done in a separate upcoming work [26] but we can already predict that they can be compatible with observational constraints on the spectral index thanks to the extra parameter $n$ which is not present in USR.","Furthermore they can violate the Maldacena's consistency condition and consequently produce large local shape non-Gaussianity.","In the future it will be interesting to apply the inversion method we have developed to other problems related to primordial curvature perturbations, or to develop a similar method for the adiabatic sound speed as function of the scale factor.","The work of MS was supported by MEXT KAKENHI No. 15H05888.","SM is funded by the Fondecyt 2015 Postdoctoral Grant 3150126.","This work was supported by the Dedicacion exclusica and Sostenibilidad programs at UDEA, the UDEA CODI project IN10219CE and 2015-4044, and Colciencias mobility project COSOMOLOGY AFTER BICEP."]],"string":"[\n [\n \"Global adiabaticity and non-Gaussianity consistency condition\"\n ],\n [\n \"Abstract In the context of single-field inflation, the conservation of the curvature perturbation on comoving slices, $\\\\R_c$, on super-horizon scales is one of the assumptions necessary to derive the consistency condition between the squeezed limit of the bispectrum and the spectrum of the primordial curvature perturbation.\",\n \"However, the conservation of $\\\\R_c$ holds only after the perturbation has reached the adiabatic limit where the constant mode of $\\\\R_c$ dominates over the other (usually decaying) mode.\",\n \"In this case, the non-adiabatic pressure perturbation defined in the thermodynamic sense, $\\\\delta P_{nad}\\\\equiv\\\\delta P-c_w^2\\\\delta\\\\rho$ where $c_w^2=\\\\dot P/\\\\dot\\\\rho$, usually becomes also negligible on superhorizon scales.\",\n \"Therefore one might think that the adiabatic limit is the same as thermodynamic adiabaticity.\",\n \"This is in fact not true.\",\n \"In other words, thermodynamic adiabaticity is not a sufficient condition for the conservation of $\\\\R_c$ on super-horizon scales.\",\n \"In this paper, we consider models that satisfy $\\\\delta P_{nad}=0$ on all scales, which we call global adiabaticity (GA), which is guaranteed if $c_w^2=c_s^2$, where $c_s$ is the phase velocity of the propagation of the perturbation.\",\n \"A known example is the case of ultra-slow-roll(USR) inflation in which $c_w^2=c_s^2=1$.\",\n \"In order to generalize USR we develop a method to find the Lagrangian of GA K-inflation models from the behavior of background quantities as functions of the scale factor.\",\n \"Applying this method we show that there indeed exists a wide class of GA models with $c_w^2=c_s^2$, which allows $\\\\R_c$ to grow on superhorizon scales, and hence violates the non-Gaussianity consistency condition.\"\n ],\n [\n \"Introduction\",\n \"A period of accelerated expansion during the early stages of the evolution of the Universe, called inflation [1], [2], [3], is able to account for several otherwise difficult to explain features of the observed Universe such the high level of isotropy of the cosmic microwave background (CMB) [4] radiation and the small value of the curvature.\",\n \"Some of the simplest inflationary models are based on a single slowly-rolling scalar field, and they are in good agreement with observations.\",\n \"It is commonly assumed in slow-roll models that adiabaticity in the thermodynamic sense, $\\\\delta P_{nad}\\\\equiv \\\\delta P-c_w^2\\\\delta \\\\rho =0$ where $c_w^2=\\\\dot{P}/\\\\dot{\\\\rho }$ , implies the conservation of the curvature perturbation on uniform density slices $\\\\zeta $ , and hence the conservation of the curvature perturbation on comoving slices ${\\\\mathcal {R}}_c$ , on super-horizon scales.\",\n \"In [5] it was shown that there can be important exceptions, i.e.\",\n \"in some cases thermodynamic adiabaticity does not necessarily imply the super-horizon conservation of ${\\\\mathcal {R}}_c$ and $\\\\zeta $ , and that they can differ from each other.\",\n \"This can happen even for models in which $c_w^2=c_s^2$ .\",\n \"An example is ultra-slow-roll (USR) inflation [6], [7], which has exact adiabaticity $\\\\delta P_{nad}=0$ on all scales.\",\n \"In USR inflation, both ${\\\\mathcal {R}}_c$ and $\\\\zeta $ exhibit super-horizon growth but their behavior is very different from each other.\",\n \"As has been stressed in [8], the non-freezing of ${\\\\mathcal {R}}_c$ has important phenomenological consequences.\",\n \"Since the freezing of ${\\\\mathcal {R}}_c$ on superhorizon scales is a necessary ingredient [9] for Maldacena's consistency relation [10] to hold, models that do not conserve ${\\\\mathcal {R}}_c$ can actually violate that consistency condition.\",\n \"In this paper focusing on K-inflation, i.e., Einstein-scalar models with a general kinetic term, we explore in a general way other single field models which have $c_w^2=c_s^2$ , hence satisfy $\\\\delta P_{nad}=0$ on all scales which we call globally adiabatic (GA), but which may not conserve ${\\\\mathcal {R}}_c$ .\",\n \"We find a generalization of the USR model.\",\n \"A different generalization without imposing the condition $c_w^2=c_s^2$ was discussed in [11], [12].\",\n \"The method we adopt is based on establishing a general condition for the non-conservation of ${\\\\mathcal {R}}_c$ in terms of the dependence of the background quantities, in particular the slow-roll parameter $\\\\epsilon \\\\equiv -\\\\dot{H}/H^2$ and the sound velocity $c_s$ , on the scale factor $a$ .\",\n \"We first derive the necessary condition for the comoving curvature perturbation ${\\\\mathcal {R}}_c$ to grow on superhorizon scales.\",\n \"Next we determine $\\\\rho (a)$ and $P(a)$ by solving the continuity equation.\",\n \"Then using the equivalence between barotropic fluids and $K$ -inflationary models which satisfy the condition $c_w^2=c_s^2$ [13], [14], we determine the corresponding Lagrangian for the equivalent scalar field model.\",\n \"Using this method we obtain a new class of GA scalar field models which do not conserve ${\\\\mathcal {R}}_c$ .\",\n \"Throughout the paper we denote the proper-time derivative by a dot ($\\\\dot{~}=d/dt$ ), the conformal-time derivative by a prime ($\\\\prime {~}=d/d\\\\eta =a\\\\,d/dt$ ) and the Hubble expansion rates in proper and conformal times by $H=\\\\dot{a}/a$ and ${\\\\cal H}=a^{\\\\prime }/a$ , respectively.\",\n \"We also use the terminology “adiabaticity” for thermodynamic adiabaticity $\\\\delta P_{nad}=0$ throughout the paper.\"\n ],\n [\n \"Conservation of ${\\\\mathcal {R}}_c$ and global adiabaticity\",\n \"We set the perturbed metric as $ds^2&=&a^2\\\\Bigl [-(1+2A)d\\\\eta ^2+2\\\\partial _jB dx^jd\\\\eta \\\\nonumber \\\\\\\\&& \\\\qquad +\\\\left\\\\lbrace \\\\delta _{ij}(1+2{\\\\cal R})+2\\\\partial _i\\\\partial _j E\\\\rbrace dx^idx^j\\\\right\\\\rbrace \\\\Bigr ]\\\\,.$ In [5] it was shown that independently of the gravity theory and for generic matter the energy-momentum conservation equations imply $\\\\delta P_{nad}=\\\\left[\\\\left(\\\\frac{c_w}{c_s}\\\\right)^2-1\\\\right](\\\\rho +P) A_c \\\\,,$ where the subscript $c$ means a quantity evaluated on comoving slices defined by $\\\\delta T^i_0=0$ (or equivalently slices on which the scalar field is homogeneous).\",\n \"In the case of general relativity, the additional relation $A_c=\\\\dot{{\\\\mathcal {R}}}_c/H$ gives an important relation for the time derivative of ${\\\\mathcal {R}}_c$ $\\\\delta P_{ nad}=\\\\left[\\\\left(\\\\frac{c_w}{c_s}\\\\right)^2-1\\\\right](\\\\rho +P)\\\\frac{\\\\dot{{\\\\mathcal {R}}_c}}{H}.\",\n \"$ The non-adiabatic pressure perturbation is given according to its thermodynamics definition $\\\\delta P_{nad}\\\\equiv \\\\delta P - c_w^2 \\\\delta \\\\rho .$ This definition of $\\\\delta P_{nad}$ is important because it is gauge invariant and $\\\\delta P_{nad}=\\\\delta P_{ud}$ , where $\\\\delta P_{ud}$ is the pressure perturbation on uniform density slices.\",\n \"It appears in the equation for the curvature perturbation on uniform density slices $\\\\zeta \\\\equiv {\\\\mathcal {R}}_{ud}$ obtained from the energy conservation law [15], $\\\\zeta ^{\\\\prime }=-\\\\frac{{\\\\cal H}\\\\delta P_{nad}}{(\\\\rho +P)}+\\\\frac{1}{3}\\\\mathop \\\\Delta ^{(3)}\\\\left(v-E^{\\\\prime }\\\\right)_{ud}\\\\,$ where $v$ is the 3-velocity potential ($v=\\\\delta \\\\phi /\\\\phi ^{\\\\prime }$ for a scalar field).\",\n \"In general, the curvature perturbations on uniform density and comoving slices are related as $\\\\zeta ={\\\\mathcal {R}}_c+\\\\frac{\\\\delta P_{ nad}}{3(\\\\rho +P)(c^2_s-c^2_w)} \\\\,.$ A common interpretation of these equations (see for example [16], [17]) is that when $\\\\delta P_{nad}=0$ with $c_w^2 \\\\ne c_s^2$ , $\\\\zeta $ and ${\\\\mathcal {R}}_c$ are equal because of eq.\",\n \"(REF ), and they are both conserved on super-horizon scales because of eq.\",\n \"(REF ).\",\n \"The equation (REF ) is the key relation to understand how ${\\\\mathcal {R}}_c$ depends on the non-adiabatic pressure $\\\\delta P_{nad}$ .\",\n \"First of all let us note that this equation is valid on any scale.\",\n \"The advantage of it with respect to eq.\",\n \"(REF ) is that it does not involve gradient terms, so it allows us to directly relate $\\\\delta P_{\\\\rm nad}$ to $\\\\dot{{\\\\mathcal {R}}}_{c}$ if $c_w^2\\\\ne c_s^2$ , while in eq.\",\n \"(REF ) $\\\\dot{\\\\zeta }$ depends on spatial gradients, which in the case of USR are not negligible on super-horizon scales [5].\",\n \"This explains while in USR in which $c_w^2=c_s^2=1$ , both ${\\\\mathcal {R}}_c$ and $\\\\zeta $ are not conserved despite $\\\\delta P_{nad}=0$ .\",\n \"It should be noted here that for slow-roll attractor models $c_w^2\\\\ne c_s^2$ in general, and ${\\\\mathcal {R}}_c$ is time-varying on sub-horizon scales.\",\n \"This implies that the non-adiabatic pressure perturbation $\\\\delta P_{nad}$ on sub-horizon scales is not zero.\",\n \"In other words, the attractor models are adiabatic only on super-horizon scales, and we call these models super-horizon adiabatic (SHA).\",\n \"From eq.\",\n \"(REF ) we can immediately deduce that in general relativity there are two possible scenarios for the non-conservation of ${\\\\mathcal {R}}_c$ , $\\\\mbox{(1)}\\\\quad &c_s^2=c_w^2\\\\,,\\\\quad \\\\delta P_{nad}=0\\\\,,\\\\nonumber \\\\\\\\\\\\mbox{(2)}\\\\quad &c_s^2\\\\ne c_w^2\\\\,,\\\\quad \\\\delta P_{nad}\\\\ne 0 \\\\, .$ The second case was studied in [11], [12].\",\n \"Here we focus on the first case.\",\n \"It is trivial to see that because of the gauge invariance of $\\\\delta P_{nad}$ the condition $c_w^2=c_s^2$ automatically implies $\\\\delta P_{nad}=0$ .\",\n \"The models satisfying the condition $c_s^2-c_w^2=\\\\delta P_{nad}=0$ are adiabatic on any scale, and because of this we call them globally adiabatic (GA).\",\n \"In GA models an explicit calculation can reveal the super-horizon behavior of ${\\\\mathcal {R}}_c$ , and $\\\\zeta $ , as was shown in [5] in the case of USR.\",\n \"Below, we develop an inversion method to find a new class of models that violate the conservation of ${\\\\mathcal {R}}_c$ without solving the perturbations equations.\"\n ],\n [\n \"Globally adiabatic K-essence models\",\n \"The condition $c_w^2=c_s^2$ has been studied in the context of K-inflation [13] described by the action $S=\\\\frac{1}{2}\\\\int d^4x\\\\sqrt{-g}\\\\left[M^2_{Pl}R+2P(X,\\\\phi )\\\\right] \\\\,,$ and it was shown that it is satisfied by scalar field models with the Lagrangian of the form, $P(X,\\\\phi )=u(X g(\\\\phi ))=u(Y)\\\\,.\",\n \"$ These models are equivalent to a barotropic perfect fluid, i.e.\",\n \"a fluid with equation of state $P(\\\\rho )$ .\",\n \"See also [18], [19], [20], [21].\",\n \"We note again that these models are adiabatic on any scale (GA), contrary to the slow-roll attractor models, which are adiabatic only on super-horizon scales (SHA).\",\n \"The fact that they are mutually exclusive can be readily seen by considering the hypothetical case of $\\\\delta P_{nad}=0$ and $c_w^2\\\\ne c_s^2$ .\",\n \"In this case Eq.\",\n \"(REF ) which is valid on any scale would mean ${\\\\mathcal {R}}_c$ should be frozen on all scales.\",\n \"In contrast, the condition $c_w^2=c_s^2$ allows for the curvature perturbation to evolve both on sub-horizon and super-horizon scales.\",\n \"In [13] it was shown that is possible to associate any barotropic perfect fluid with an equivalent K-inflation model according to $2\\\\int ^P\\\\frac{du}{F(u)}=\\\\log (Y) \\\\,, $ where $F(P)=\\\\rho (P)+P$ and $Y=g(\\\\phi )X$ with $X=-g^{\\\\mu \\\\nu }\\\\partial _\\\\mu \\\\phi \\\\partial _\\\\nu \\\\phi /2$ .\",\n \"These models are the ones which could violate the conservation of ${\\\\mathcal {R}}_c$ for adiabatic perturbations, since they satisfy $c_w^2=c_s^2$ .\",\n \"It is noted of course that the global adiabaticity is not the sufficient condition for the non-conservation of ${\\\\mathcal {R}}_c$ .\",\n \"Not all GA models violate the conservation of ${\\\\mathcal {R}}_c$ on super-horizon scales.\"\n ],\n [\n \"General conditions for super-horizon growth of ${\\\\mathcal {R}}_c$\",\n \"From the equation for the curvature perturbation on comoving slices, $\\\\frac{\\\\partial }{\\\\partial t}\\\\left(\\\\frac{a^3\\\\epsilon }{c_s^2}\\\\frac{\\\\partial }{\\\\partial t}{\\\\mathcal {R}}_c\\\\right)-a\\\\epsilon \\\\Delta {\\\\mathcal {R}}_c=0 \\\\,,$ we can deduce, after re-expressing the time derivative in terms of the derivative respect to the scale factor $a$ , that on superhorizon scales there is (apart from a constant solution) a solution of the form, ${\\\\mathcal {R}}_c &\\\\propto & \\\\int ^a\\\\frac{da}{a} f(a)\\\\,;\\\\quad f(a)\\\\equiv {\\\\frac{c_s^2(a)}{Ha^3 \\\\epsilon (a)}} \\\\,, $ where we have introduced the function $f(a)$ for later convenience.\",\n \"In conventional slow-roll inflation $c_s^2$ and $\\\\epsilon $ are both slowly varying, hence the integral rapidly approaches a constant, rendering ${\\\\mathcal {R}}_c$ conserved.\",\n \"The time dependent part of the above solution corresponds to the decaying mode.\",\n \"The necessary and sufficient condition for super-horizon freezing is that there exists some $\\\\delta >0$ for which $\\\\lim _{a\\\\rightarrow \\\\infty }a^\\\\delta f(a)=0 .\",\n \"$ By definition of inflation, $H$ must be sufficiently slowly varying; $\\\\epsilon =-\\\\dot{H}/H^2\\\\ll 1$ .\",\n \"So we may neglect the time dependence of $H$ in (REF ) at leading order, while $\\\\epsilon $ and $c_s^2$ may vary rapidly in time.\",\n \"For models for which $\\\\epsilon \\\\approx a^{-n}$ and $c_s^2\\\\approx a^q$ we get $f \\\\propto a^{q+n-3} \\\\,,$ hence the condition for freezing is $q+n-3<0 \\\\,.$ If this condition is violated, i.e.\",\n \"$q+n-3\\\\ge 0$ , then the solution (REF ) will grow on super-horizon scales.\",\n \"This happens for example in USR., which corresponds to $c_s^2=1$ and $\\\\epsilon \\\\propto a^{-6}$ , i.e.\",\n \"$q=0$ , and $n=6$ .\",\n \"(The super-horizon growth of ${\\\\mathcal {R}}_c$ in USR can also be understood as a direct consequence of the non-attractor nature of USR [22].)\",\n \"In general, we expect that $q$ would not become very large.\",\n \"This implies $\\\\epsilon $ should decrease sufficiently rapidly.\",\n \"Conversely, if $\\\\epsilon $ decreases sufficiently rapidly, then the growth of ${\\\\mathcal {R}}_c$ on superhorizon scales will follow.\"\n ],\n [\n \"Barotropic model\",\n \"We have shown that GA models could violate the super-horizon conservation of ${\\\\mathcal {R}}_c$ , so now we will look for GA K-essence models which do indeed violate it, based on the freezing condition in eq.\",\n \"(REF ).\",\n \"Inspired by the equivalence between barotropic fluids and GA K-essence models [13] we will first look for barotropic fluids that can give the growing curvature perturbation on superhorizon scales.\",\n \"From the very beginning we will set $c_w^2=c_s^2$ .\",\n \"Using the Friedmann equation we can write the slow-roll parameter $\\\\epsilon $ as $\\\\epsilon = -\\\\frac{\\\\dot{H}}{H^2}=\\\\frac{3}{2}\\\\frac{\\\\rho +P}{\\\\rho } \\\\,.$ In terms of the scale factor and $\\\\epsilon $ the energy conservation equation reads $\\\\frac{d\\\\rho }{da}+\\\\frac{3}{a}(\\\\rho +p)=\\\\frac{d\\\\rho }{da}+\\\\frac{2\\\\epsilon \\\\rho }{a}=0 .$ We may now define the quantity $b(a)=2\\\\epsilon \\\\rho $ .\",\n \"It appears naturally in the continuity equation and plays a crucial role in regards to the super-horizon behavior of curvature perturbations because the function $f(a)$ can be re-written in terms of it as $f(a)\\\\propto \\\\frac{H c_s^2}{a^3 b(a)} \\\\,.$ Integrating the energy conservation equation we get $\\\\rho (a)=\\\\rho _0\\\\exp \\\\left[-2\\\\int _{a_0}^a{\\\\frac{\\\\epsilon }{a}da}\\\\right]=\\\\int {-\\\\frac{b(a)}{a} da} \\\\,.$ Using eq.\",\n \"(REF ), we then obtain $P(a)=\\\\left(\\\\frac{2}{3}\\\\epsilon -1\\\\right)\\\\rho \\\\,.$ The sound velocity is given by $c_w^2=c_s^2=\\\\frac{dP}{d\\\\rho }&=-1+\\\\frac{1}{3}\\\\frac{db(a)}{d\\\\rho }\\\\nonumber \\\\\\\\&=-1+\\\\frac{1}{3}\\\\frac{db(a)}{da}\\\\Big /\\\\left(\\\\frac{d\\\\rho }{da}\\\\right)\\\\nonumber \\\\\\\\&=-1-\\\\frac{a}{3b(a)}\\\\frac{db(a)}{da}\\\\,.$ We now consider the behavior of $f(a)$ introduced in (REF ).\",\n \"As mentioned before, we consider the case when $\\\\epsilon $ decreases sufficiently rapidly.\",\n \"In this case, $\\\\rho =3H^2M_P^2$ approaches a constant rapidly.\",\n \"Hence the time dependence of $\\\\rho $ may be neglected compared to that of other quantities that vary far more rapidly.\",\n \"With this approximation, assuming $\\\\epsilon \\\\propto a^{-n}$ , we find $c_s^2\\\\approx \\\\frac{n-3}{3}\\\\,,$ which means $q\\\\approx 0$ , and $f(a)={\\\\frac{c_s^2(a)}{Ha^3 \\\\epsilon (a)}}\\\\propto a^{n-3}\\\\,,$ which satisfies the condition for the growth if $n>3$ , in accordance with the original anticipation.\",\n \"In passing, it is interesting to note that the condition $n>3$ implies $c_s^2>0$ , a necessary condition to avoid the gradient instability of the perturbation.\",\n \"Thus virtually all GA models that are free from the gradient instability exhibit superhorizon growth of the comoving curvature perturbation ${\\\\mathcal {R}}_c$ .\"\n ],\n [\n \"Scalar field model\",\n \"Let us now find a scalar field model that corresponds to the barotropic model discussed in the previous section.\",\n \"As a warm-up, let us consider the USR case, whose fluid interpretation has already been studied in [23].\",\n \"In this case, we exactly have $c_s^2=1$ .\",\n \"From eq.\",\n \"(REF ), this implies $b/2=\\\\epsilon \\\\rho \\\\bigl (=3(\\\\rho +P)/2\\\\bigr )\\\\propto a^{-6}$ .\",\n \"Also $c_s^2=1$ implies $\\\\rho =P+const.$ Inserting this into eq.\",\n \"(REF ) gives $\\\\frac{2dP}{2P+const.\",\n \"}=\\\\frac{dY}{Y}\\\\,.$ Thus up to a constant term $P$ and $Y$ are the same, $P=Y+const..$ Absorbing $g(\\\\phi )$ in $Y$ into the definition of the scalar field by $g^{1/2}d\\\\phi \\\\rightarrow d\\\\phi $ , this is indeed the Lagrangian for a minimally coupled massless scalar with a cosmological constant: $L=P(\\\\phi ,X)=X-V_0\\\\,.\",\n \"$ This is consistent with $\\\\rho +P=2X\\\\propto \\\\epsilon \\\\rho \\\\propto a^{-6}$ .\",\n \"Let us generalize the USR case.\",\n \"As in the previous section, we consider models that have the behavior of $\\\\epsilon \\\\rho $ as $2\\\\epsilon \\\\rho =b(a)\\\\,,$ where $b(a)$ should decrease faster than $a^{-3}$ asymptotically at $a\\\\rightarrow \\\\infty $ but otherwise is an arbitrary function.\",\n \"Then we have $F(P)\\\\equiv \\\\rho +P=2H^2\\\\epsilon =\\\\frac{2\\\\epsilon \\\\rho }{3}=\\\\frac{b(a)}{3}\\\\,,$ which gives $\\\\frac{dY}{Y}=2\\\\frac{dP}{F(P)}=6\\\\frac{dP}{2\\\\epsilon \\\\rho }=6\\\\frac{dP}{b(a)}\\\\,.$ For $dP$ , using the energy conservation law, we may rewrite it as $dP&=d\\\\left(-\\\\rho +F(P)\\\\right)=-d\\\\rho +\\\\frac{db(a)}{3}\\\\nonumber \\\\\\\\&=3\\\\frac{da}{a}(\\\\rho +P)+\\\\frac{db(a)}{3}=b(a)\\\\frac{da}{a}+\\\\frac{db(a)}{3}\\\\,.$ Therefore we have $\\\\frac{dY}{Y}=6\\\\frac{dP}{b(a)}=6\\\\frac{da}{a}+2\\\\frac{db}{b}\\\\,.$ Hence $Y\\\\propto a^6b^2\\\\,.\",\n \"$ This is consistent with the USR case in which $b(a) \\\\propto a^{-6}$ and $Y=X \\\\propto a^{-6}$ .\",\n \"This relation is quite useful since it allows to rewrite the freezing function $f(a)$ as $f(a)\\\\propto \\\\frac{H c_s^2}{\\\\sqrt{Y}} \\\\,,$ from which we can deduce that $Y(a)$ determines the super-horizon behavior of ${\\\\mathcal {R}}_c$ .\",\n \"In particular, for the models we are considering in which $c_s$ is constant, we infer that super-horizon growth can happen in the limit $Y\\\\rightarrow 0$ .\",\n \"For a given choice of $b(a)$ , eq.\",\n \"(REF ) can be inverted to give the scale factor as a function of $Y$ , $a=a(Y)$ .\",\n \"Also eq.\",\n \"(REF ) can be integrated to give $P=P(a)$ .\",\n \"Combining these two, one can obtain the Lagrangian for the scalar field, $L=P=P(Y)$ .\",\n \"Note that in GA models there is a one-to-one correspondence between the scale factor and state variables such as $P(a)$ and $\\\\rho (a)$ , which is the reason why we can also write a barotropic equation of state $P(\\\\rho )=P(a(\\\\rho ))$ .\",\n \"Once any of the functions $P(a),\\\\rho (a),b(a),\\\\epsilon (a),Y(a)$ is specified, all the others are specified too, as well as the equation of state $P(\\\\rho )$ or its scalar field equivalent Lagrangian $P(Y)$ , which is in fact the basis of the inversion method that we are developing in this paper.\"\n ],\n [\n \"examples\",\n \"Here we give a couple of specific K-inflation models that are globally adiabatic and violate the convervation of ${\\\\mathcal {R}}_c$ .\",\n \"Given the parametric behaviour of $b\\\\equiv 2 \\\\epsilon \\\\rho $ , our inversion method allows us to deduce the Lagrangian.\"\n ],\n [\n \"Ex 1: Generalized USR\",\n \"Let us consider a specific case where $b(a)$ is a power-law function, $2\\\\epsilon \\\\rho =b(a)=c a^{-n} \\\\,.$ where $c$ is a constant.\",\n \"We assume $n>3$ in order to have the growth on superhorizon scales.\",\n \"From eq.\",\n \"(REF ) we have $a\\\\propto Y^{1/(6-2n)}\\\\,.$ Now eq.\",\n \"(REF ) gives $P&=\\\\int ^a\\\\left(b(a)\\\\frac{da}{a}+\\\\frac{db(a)}{3}\\\\right)\\\\nonumber \\\\\\\\&=-\\\\frac{c}{n}a^{-n}+\\\\frac{c}{3}a^{-n}+const.\\\\nonumber \\\\\\\\&=\\\\frac{n-3}{3n}b(a)+const..$ Plugging eq.\",\n \"(REF ) into this, we finally obtain $L=P(Y)= Y^{n/(2n-6)}-V_0 \\\\,.\",\n \"$ Since this may be regarded as a natural generalization of the USR case, which corresponds to the case $n=6$ , we call it the generalized USR (GUSR) model.\",\n \"Lagrangians involving $Y^{\\\\alpha }$ terms have already been studied in [11], [24], [25], but those models are either not exactly globally adiabatic because of the presence of a not constant potential or they satisfy the relation $\\\\epsilon \\\\propto a^{-n}$ only approximately and during a limited time range, while for GUSR $\\\\epsilon \\\\propto a^{-n}$ is an exact relation and is valid at any time.\",\n \"As the Lagrangian is of the type described in eqs.\",\n \"(REF ) and (REF ) (remember that after a field transformation $Y$ can be made equal to $X$ ), we understand that this scalar field model is indeed equivalent to a barotropic fluid.\",\n \"Hence we have $c_w^2=c_s^2$ , and therefore $\\\\delta P_{ nad}=0$ .\",\n \"Indeed the second condition for super-horizon growth of ${\\\\mathcal {R}}_c$ given in eq.\",\n \"(REF ) is satisfied.\",\n \"More precisely, we note that for the GUSR model, the sound velocity is exactly constant, $c_w^2=c_s^2=\\\\frac{n-3}{3} \\\\,.$ The power spectrum of the comoving curvature perturbation can be explicitly computed for this model.\",\n \"One finds [26] that the spectral index is a function of $n$ : $n_s-1= 6-n$ , in agreement with the scale invariant spectrum of the original ultra slow-roll inflation in which one has $n=6$ .\",\n \"Hence, the model can be constrained by the observational value.\",\n \"Note as well, from eq.\",\n \"(REF ), that to have a slightly red-tilted spectrum, we need a slightly superluminal speed of sound.\"\n ],\n [\n \"Ex 2: Lambert Inflation\",\n \"As another example, let us consider the case when $\\\\epsilon $ is a power-law function, $\\\\epsilon (a)=\\\\epsilon _0{a}^{-n}\\\\,.$ As before, we assume $n>3$ .\",\n \"In this case, since $d\\\\log \\\\rho /d\\\\log a=-2\\\\epsilon \\\\propto a^{-n}$ , we find $\\\\rho (a)=\\\\rho _0\\\\exp \\\\left[\\\\frac{2\\\\epsilon }{n}\\\\right] \\\\,.$ It is clear that $\\\\rho $ approaches a constant $\\\\rho _0$ asymptotically at $a\\\\rightarrow \\\\infty $ .\",\n \"Inserting eq.\",\n \"(REF ) and eq.\",\n \"(REF ) into eq.\",\n \"(REF ), the sound velocity is given by $c_w^2=c_s^2=-1-\\\\frac{1}{3}\\\\left(\\\\frac{d\\\\log \\\\epsilon }{d\\\\log a}+\\\\frac{d\\\\log \\\\rho }{d\\\\log a}\\\\right)=\\\\frac{n-3+2\\\\epsilon }{3}\\\\,.$ Thus $c_s^2$ is time dependent, but it rapidly approaches a constant as $\\\\epsilon $ decays out.\",\n \"Also from eq.\",\n \"(REF ) and eq.\",\n \"(REF ), we find $b(a)=2\\\\epsilon \\\\rho =2\\\\epsilon \\\\rho _0\\\\exp \\\\left[\\\\frac{2\\\\epsilon }{n}\\\\right]\\\\,.$ Thus we have $Y\\\\propto a^6 b^2\\\\propto a^{6-2n}\\\\exp \\\\left[\\\\frac{4\\\\epsilon }{n}\\\\right]\\\\propto \\\\epsilon ^{(2n-6)/n}\\\\exp \\\\left[\\\\frac{4\\\\epsilon }{n}\\\\right] \\\\,,$ which implies $Y^{n/(2n-6)}\\\\propto \\\\frac{4\\\\epsilon }{2n-6}\\\\exp \\\\left[\\\\frac{4\\\\epsilon }{2n-6}\\\\right]\\\\,.$ To find the Lagrangian, we manipulate eq.\",\n \"(REF ) as $dP&=b\\\\frac{da}{a}+\\\\frac{db}{3}=-\\\\frac{b}{n}\\\\frac{d\\\\epsilon }{\\\\epsilon }+\\\\frac{db}{3}\\\\nonumber \\\\\\\\&=-\\\\frac{2}{n}\\\\rho _0e^{2\\\\epsilon /n}d\\\\epsilon +\\\\frac{db}{3}\\\\,.$ Therefore, integrating this we obtain $P=\\\\rho _0e^{2\\\\epsilon /n}\\\\left(-1+\\\\frac{2\\\\epsilon }{3}\\\\right)+const..$ One can invert eq.\",\n \"(REF ) to find $\\\\epsilon $ as a function of $Y$ , and then insert it into the above to obtain the Lagrangian.\",\n \"Specifically, we introduce the Lambert function $W(x)$ defined by the inverse function of $X(z)=ze^z$ , $z=X^{-1}(ze^z)\\\\equiv W(ze^z)\\\\,.$ Setting ${Y}^{n/(2n-6)}=ze^z\\\\,;\\\\quad z=\\\\frac{4\\\\epsilon }{2n-6}\\\\,,$ we have $\\\\frac{4\\\\epsilon }{2n-6}=W(y)\\\\,;\\\\quad y\\\\equiv Y^{n/(2n-6)}\\\\,.$ Inserting this into eq.\",\n \"(REF ), we finally obtain $L&=P(Y)\\\\nonumber \\\\\\\\&=\\\\rho _0\\\\left(\\\\frac{n-3}{3}W(y)-1\\\\right)\\\\exp \\\\left[\\\\frac{n-3}{n}W(y)\\\\right]-V_0\\\\,,$ where $y=y(Y)$ is given in eq.\",\n \"(REF ).\",\n \"Note that this model has been derived without making any approximation, and it gives exactly $\\\\epsilon \\\\propto a^{-n}$ .\",\n \"However, as we mentioned before, in the late time limit, there is no difference between $\\\\epsilon \\\\propto a^{-n}$ and $\\\\rho \\\\epsilon \\\\propto a^{-n}$ .\",\n \"Thus the two models discussed above are essentially the same at late times.\",\n \"This can be easily checked by expanding $W(y)$ around $y=0$ , $W(y)=y-y^2+\\\\cdots .$ At leading order in $y=Y^{n/(2n-6)}$ , this gives $P(Y)=\\\\frac{n-3}{3}\\\\rho _0Y^{n/(2n-6)}-\\\\rho _0-V_0\\\\,.$ By absorbing the constant coefficient into $g(\\\\phi )$ in the definition of $Y$ , $Y=g(\\\\phi )X$ , and absorbing $\\\\rho _0$ into the constant $V_0$ , eq.\",\n \"(REF ) reduces to $P=Y^{n/(2n-6)}-V_0\\\\,,$ which indeed coincides with the GUSR model, see eq.\",\n \"(REF ).\",\n \"Higher order terms in the expansion give an infinite class of models of the type $u(Y)&=&\\\\sum _i \\\\beta _i Y^{n_i} \\\\,, $ where $\\\\beta _i$ are appropriate coefficients.\",\n \"Finally, note that in USR and as well in the two examples considered here, the shift symmetry in the potential ($V(\\\\phi )=V_0$ ) is a direct consequence of the demand $c_w^2=c_s^2$ , which in turn follows from the global adiabaticity of the model.\",\n \"That is in line with the general statement [27], [28] that for a $k$ -essence theory to describe a fluid, one needs a shift symmetry (i.e., there is no physical clock, the model is of the non-attractor type).\"\n ],\n [\n \"Conclusions\",\n \"By introducing the notion of global adiabaticity, namely, $c_w^2=c_s^2$ and $\\\\delta P_{nad}=\\\\delta P-c_w^2\\\\delta \\\\rho =0$ , where $c_w^2=\\\\dot{P}/\\\\dot{\\\\rho }$ and $c_s$ is the propagation (phase) speed of the perturbation, we have determined the general conditions for the non-conservation of the curvature perturbations on comoving slices ${\\\\mathcal {R}}_c$ on super-horizon scales.\",\n \"We have found that globally adiabatic K-essence models can exhibit this behavior.\",\n \"We have then developed a method to construct the Lagrangian of a K-essence globally adiabatic (GA) model by specifying the behavior of background quantities such as $\\\\epsilon \\\\rho $ where $\\\\epsilon $ is the slow-roll parameter, using the equivalence between barotropic fluids and GA K-essence models.\",\n \"We have applied the method to find the equations of state of the fluids and derive the Lagrangian of the equivalent single scalar field models.\",\n \"Interestingly, we have found that the requirement to avoid the gradient instability, ie, $c_s^2>0$ is almost identical to the condition for the non-conservation on superhorizon scales.\",\n \"The advantage of our approach is that we have not solved any perturbation equation explicitly, since we have proceeded in the opposite way solving the inversion problem consisting of requiring certain properties to the behavior of the perturbation equation we are interested in.\",\n \"In other words, instead of starting from a Lagrangian and then solve the perturbations equations we have determined the equation of state or equivalently the Lagrangian which admits a solution of the perturbation equation with the particular behavior we are interested in.\",\n \"We have shown that the main difference between attractor models and GA models is that the latter are adiabatic on all scales, while attractor models are approximately adiabatic in the sense of $\\\\delta P_{nad}=0$ only on super-horizon scales and $c_w^2\\\\ne c_s^2$ .\",\n \"The detailed study of the new models found in this paper will be done in a separate upcoming work [26] but we can already predict that they can be compatible with observational constraints on the spectral index thanks to the extra parameter $n$ which is not present in USR.\",\n \"Furthermore they can violate the Maldacena's consistency condition and consequently produce large local shape non-Gaussianity.\",\n \"In the future it will be interesting to apply the inversion method we have developed to other problems related to primordial curvature perturbations, or to develop a similar method for the adiabatic sound speed as function of the scale factor.\",\n \"The work of MS was supported by MEXT KAKENHI No. 15H05888.\",\n \"SM is funded by the Fondecyt 2015 Postdoctoral Grant 3150126.\",\n \"This work was supported by the Dedicacion exclusica and Sostenibilidad programs at UDEA, the UDEA CODI project IN10219CE and 2015-4044, and Colciencias mobility project COSOMOLOGY AFTER BICEP.\"\n ]\n]"},"id":{"kind":"string","value":"1606.04906"}}},{"rowIdx":1199,"cells":{"text":{"kind":"list like","value":[["Perfect Embezzlement of Entanglement"],["Abstract Van Dam and Hayden introduced a concept commonly referred to as embezzlement, where, for any entangled quantum state $\\phi$, there is an entangled catalyst state $\\psi$, from which a high fidelity approximation of $\\phi \\otimes \\psi$ can be produced using only local operations.","We investigate a version of this where the embezzlement is perfect (i.e., the fidelity is 1).","We prove that perfect embezzlement is impossible in a tensor product framework, even with infinite-dimensional Hilbert spaces and infinite entanglement entropy.","Then we prove that perfect embezzlement is possible in a commuting operator framework.","We prove this using the theory of C*-algebras and we also provide an explicit construction.","Next, we apply our results to analyze perfect versions of a nonlocal game introduced by Regev and Vidick.","Finally, we analyze the structure of perfect embezzlement protocols in the commuting operator model, showing that they require infinite-dimensional Hilbert spaces."],["Introduction","It is well known that an entangled quantum state cannot be produced by local operations alone.","Van Dam and Hayden [2] proposed a method that, in a certain sense, appears to produce additional entanglement by local operations.","They showed that, for any entangled state $\\phi $ and $\\epsilon > 0$ , starting with a special entangled catalyst state $\\psi $ , applying local operations, can produce a state that approximates $\\phi \\otimes \\psi $ within fidelity $1-\\epsilon $ .","Although the entanglement entropy of the state produced cannot exceed that of $\\psi $ , when $\\epsilon $ is small, it is difficult to distinguish between the state produced and $\\phi \\otimes \\psi $ .","The name embezzlement reflects the fact that the protocol “steals\" entanglement from $\\psi $ in order to produce entanglement elsewhere, but in a manner that is difficult to detect.","In the method of [2], fidelity $1-\\epsilon $ can be attained for any $\\epsilon > 0$ , using a catalyst $\\psi $ with entanglement entropy $O(\\log (1/\\epsilon ))$ .","Moreover, it is shown in [2] that the entanglement entropy of the catalyst must be $\\Omega (\\log (1/\\epsilon ))$ to attain this fidelity.","Thus, high fidelity embezzlement requires a large amount of entanglement to begin with.","We consider the question: what kinds of embezzlement are possible when the amount of entanglement in $\\psi $ is allowed to be infinite?","The aforementioned results do not rule out perfect (i.e., fidelity 1) embezzlement in such cases.","On the other hand, the catalytic states $\\psi _{\\epsilon }$ in [2] do not converge to a valid quantum vector state as $\\epsilon $ approaches 0.","This question provides a setting in which the consequences of notions of infinite entanglement can be explored.","We first show that in the tensor product framework, where catalytic states are in the tensor product of two Hilbert spaces, perfect embezzlement is impossible, even if the spaces are infinite dimensional and the entanglement entropy is infinite.","Next, we consider a commuting operator framework, where the notion of “local\" is formalized differently: there is one joint Hilbert space, accessible to both Alice and Bob; however, the operations that Alice performs on this space must commute with those of Bob.","This formalism is used in quantum field theory (see [12], [9], [11], [4], [6], [5] for more discussion about this framework and its relationship with the tensor product framework).","A natural adaptation of the commuting operator framework to the setting of embezzlement is the following.","The catalytic state $\\psi $ is in a jointly accessibe Hilbert space, that we refer to as the resource space $\\mathcal {R}$ .","There are also two additional Hilbert spaces: $\\mathcal {H}_A$ , accessible to Alice only; and $\\mathcal {H}_B$ , accessible to Bob only.","The goal of the protocol is to transform a product state to an entangled state in $\\mathcal {H}_A \\otimes \\mathcal {H}_B$ while using $\\psi $ catalytically, and using operators that are commuting in the following sense.","Alice can apply a unitary operator on $\\mathcal {H}_A \\otimes \\mathcal {R}$ and Bob can apply a unitary operator on $\\mathcal {H}_B \\otimes \\mathcal {R}$ ; however, $U_A \\otimes I_{\\mathcal {H}_B}$ and $I_{\\mathcal {H}_A} \\otimes U_B$ must commute on $\\mathcal {H}_A \\otimes \\mathcal {R}\\otimes \\mathcal {H}_B$ , as illustrated in Figure REF .","Figure: Commuting operator framework as a circuit diagram.We focus on the problem of embezzling a Bell state of the form ${\\textstyle {\\frac{1}{\\sqrt{2}}}}|0\\rangle \\otimes |0\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|1\\rangle \\otimes |1\\rangle $ (though our methodology adapts to more general states).","In this case, $\\mathcal {H}_A = \\mathcal {H}_B = \\mathbb {C}^{2}$ .","A perfect embezzlement protocol consists of a resource space $\\mathcal {R}$ , a catalytic state $\\psi \\in \\mathcal {\\mathcal {R}}$ , and commuting unitary operators $U_A$ and $U_B$ , such that $(U_A \\otimes I_{\\mathcal {H}_B})&(I_{\\mathcal {H}_A} \\otimes U_B) |0\\rangle \\otimes \\psi \\otimes |0\\rangle \\\\&= \\textstyle {\\frac{1}{\\sqrt{2}}}|0\\rangle \\otimes \\psi \\otimes |0\\rangle +\\textstyle {\\frac{1}{\\sqrt{2}}}|1\\rangle \\otimes \\psi \\otimes |1\\rangle .","\\nonumber $ We show that, in this commuting operator framework, a perfect embezzlement protocol exists, where the resource space is a countably infinite dimensional (i.e., separable) Hilbert space.","We show this in two ways: one is a simple existence proof, based on the theory of C*-algebras, which does not yield explicit unitary operations; the other is by an explicit construction.","Next, we consider coherent embezzlement, which was introduced in [10] (where it is referred to as $T_2$ ) and is a refinement of coherent state exchange, introduced in [8].","Coherent embezzlement is related to embezzlement but has the property that it is operationally testable in a sense similar to that of nonlocal games (whereas embezzlement itself does not have this property).","We give reductions between perfect embezzlement and perfect coherent embezzlement to prove that perfect coherent embezzlement is impossible in the tensor product framework; whereas it is possible in the commuting operator framework.","Finally, we prove a theorem concerning the structure of pairs of unitaries that achieve perfect embezzlement in terms of properties of their constituent operators.","We show that at least one of these operators must contain a non-unitary isometry, a term that we will define later.","Since non-unitary isometries do not exist in finite dimensions, this implies that perfect embezzlement in the commuting-operator model cannot be achieved with a finite dimensional resource space."],["Perfect embezzlement is impossible in a tensor product framework","In [2], it is proved that, for any protocol that embezzles within fidelity $1 - \\epsilon $ , the entanglement entropy of the catalyst must be $\\Omega (\\log (1/\\epsilon ))$ .","It follows that perfect embezzlement is impossible with finite-dimensional entanglement in the tensor product framework.","Here, we extend this impossibility result to tensor products of arbitrary Hilbert spaces (where the dimension of the spaces and entanglement entropy can be infinite).","In the tensor product framework, the resource space is of the form $\\mathcal {R} = \\mathcal {R}_A \\otimes \\mathcal {R}_B$ , where $\\mathcal {R}_A$ and $\\mathcal {R}_B$ are arbitrary Hilbert spaces.","Alice has access to $\\mathcal {H}_A \\otimes \\mathcal {R}_A$ and Bob has access to $\\mathcal {H}_B \\otimes \\mathcal {R}_B$ .","Alice and Bob can each apply any unitary operation to the registers that they have access to, as illustrated in Figure REF (left), where the input state is $|0\\rangle \\otimes \\psi \\otimes |0\\rangle $ , for some state $\\psi \\in \\mathcal {R}_A \\otimes \\mathcal {R}_B$ .","The protocol performs perfect embezzlement if its output state is $\\textstyle {\\frac{1}{\\sqrt{2}}}|0\\rangle \\otimes \\psi \\otimes |0\\rangle +\\textstyle {\\frac{1}{\\sqrt{2}}}|1\\rangle \\otimes \\psi \\otimes |1\\rangle $ .","We also define a potentially stronger model, that we refer to as embezzlement with ancillas, which includes the possibility of Alice and Bob employing additional registers as part of their protocol, as illustrated in Figure REF (right).","Figure: Circuit diagram for embezzlement (left) and embezzlement with ancillas (right) in the tensor product framework.Registers ℛ A \\mathcal {R}_A and ℛ B \\mathcal {R}_B contain a bipartite resource state that must be used catalytically.Registers ℋ A =ℋ B =ℂ 2 \\mathcal {H}_A = \\mathcal {H}_B = \\mathbb {C}^{2} are intialized to state |00〉|00\\rangle and are transformed to state 1 2|00〉+1 2|11〉{\\textstyle {\\frac{1}{\\sqrt{2}}}}|00\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|11\\rangle .Registers 𝒢 A \\mathcal {G}_A and 𝒢 B \\mathcal {G}_B are ancillas, whose initial state is unentangled, but they need not be used catalytically.The input to the circuit is of the form $\\gamma _A \\otimes |0\\rangle \\otimes \\psi \\otimes |0\\rangle \\otimes \\gamma _B$ , where $\\psi \\in \\mathcal {R}_A \\otimes \\mathcal {R}_B$ is the catalyst state, and $\\gamma _A \\in \\mathcal {G}_A$ and $\\gamma _B \\in \\mathcal {G}_B$ are the initial states of Alice and Bob's respective ancilla registers, $\\mathcal {G}_A$ and $\\mathcal {G}_B$ (which can be infinite dimensional).","If we express the Hilbert space as $(\\mathcal {H}_A \\otimes \\mathcal {H}_B) \\otimes (\\mathcal {R}_A \\otimes \\mathcal {R}_B) \\otimes (\\mathcal {G}_A \\otimes \\mathcal {G}_B)$ then the input state can be written as $|00\\rangle \\otimes \\psi \\otimes (\\gamma _A \\otimes \\gamma _B)$ .","The protocol performs perfect embezzlement if and only if the output state is of the form $\\bigl ({\\textstyle {\\frac{1}{\\sqrt{2}}}}|00\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|11\\rangle \\bigr )\\otimes \\psi \\otimes \\gamma _{AB},$ for some state $\\gamma _{AB} \\in \\mathcal {G}_A \\otimes \\mathcal {G}_B$ .","Theorem 2.1 Perfect embezzlement is impossible in the tensor product framework, even if Alice and Bob are allowed to use ancillas.","The proof is a straightforward application of the Schmidt decomposition for vectors in tensor products of arbitrary Hilbert spaces.","For arbitrary (not necessarily separable) Hilbert spaces $\\mathcal {H}$ and $\\mathcal {K}$ and any $\\phi \\in \\mathcal {H} \\otimes \\mathcal {K}$ , it is possible to express $\\phi = \\sum _{j = 0}^{\\infty } \\alpha _j\\, u_j\\otimes v_j,$ where $\\alpha _j \\ge 0$ , $\\sum _{j = 0}^{\\infty } |\\alpha _j|^2 = 1$ , $ \\alpha _j \\ge \\alpha _{j+1}$ , $u_0, u_1, \\dots $ are orthonormal vectors in $\\mathcal {H}$ , and $v_0, v_1, \\dots $ are orthonormal vectors in $\\mathcal {K}$ .","Moreover, given these conditions, the coefficients $\\alpha _j$ are unique.","For the convenience of the reader, we include a proof of this in Appendix .","Now taking a Schmidt decomposition of $\\gamma _A \\otimes |0\\rangle \\otimes \\psi \\otimes |0\\rangle \\otimes \\gamma _B$ , with respect to $\\big (\\mathcal {G}_A \\otimes \\mathcal {H}_A \\otimes \\mathcal {R}_A \\big ) \\otimes \\big (\\mathcal {G}_B \\otimes \\mathcal {H}_B \\otimes \\mathcal {R}_B \\big )$ , we obtain Schmidt coefficients $\\alpha _j$ .","Suppose that a perfect embezzlement protocol exists.","Then, since $U_A$ and $U_B$ are local unitaries, the Schmidt coefficients of the initial state $|00\\rangle \\otimes \\psi \\otimes \\gamma _A \\otimes \\gamma _B$ must be the same as those of the final state $({\\textstyle {\\frac{1}{\\sqrt{2}}}}|00\\rangle + {\\textstyle {\\frac{1}{\\sqrt{2}}}}|11\\rangle )\\otimes \\psi \\otimes \\gamma _{AB}$ .","But this is a contradiction, since the largest Schmidt coefficient of the input state is $\\alpha _0$ (which is nonzero) and the largest Schmidt coefficient of the output state is at most ${\\textstyle {\\frac{1}{\\sqrt{2}}}}\\alpha _0$ .","Therefore, there is no perfect embezzlement protocol in the tensor product framework."],["Perfect embezzlement is possible in a commuting operator framework","In this section we show that, since one can approximately embezzle a Bell state to any level of precision in finite dimensions (by the results of [2]), one can perfectly embezzle in infinite dimensions in the commuting operator framework.","Readers unfamiliar with the theory of C*-algebras might prefer to read our primer on C*-algebras in Appendix before tackling this section.","At the end of the section, we explain how to generalize the technique to more general entangled states.","We begin by showing that each commuting operator framework, where $\\mathcal {H}_A = \\mathcal {H}_B = \\mathbb {C}^2$ , yields a set of eight operators on the resource space.","To study the most general commuting framework, it is natural to consider the relations that any such set of operators must satisfy and look for a “universal\" model for such sets of operators.","We will show that the eight operators arising from a commuting operator framework are always a representation of a certain C*-algebra and that the catalyst vector yields a state on this C*-algebra.","We will show that the commuting operator framework together with the catalyst vector achieves perfect embezzlement of a Bell state if and only if the state on this C*-algebra induced by the catalyst vector satisfies a set of four equations.","In this manner the question of whether or not one can perfectly embezzle a Bell state is reduced to a question about the existence of a state on this C*-algebra that satisfies our four equations.","Finally, we show that perfect embezzlement of a Bell state is possible in the commuting operator framework by showing the existence of such a state.","The “universal\" C*-algebra that one needs was first introduced by L.G.","Brown [1], who referred to it as the universal C*-algebra of a non-commutative unitary for reasons that will, hopefully, be clear.","Our viewpoint shows that in a certain sense questions about embezzlement can be interpreted as questions about states on these particular quantum group C*-algebras.","We think that this perspective is new and should lead to interesting links between these two areas.","Let's return to the scenario of Figure REF .","Alice's unitary operation, $U_A: \\mathbb {C}^2 \\otimes \\mathcal {R} \\rightarrow \\mathbb {C}^2 \\otimes \\mathcal {R}$ can be represented by a $2 \\times 2$ matrix of operators on $\\mathcal {R}$ , $U_A= \\begin{pmatrix} U_{00} & U_{01} \\\\ U_{10} & U_{11} \\end{pmatrix}= \\sum _{i,j=0}^1 |i\\rangle \\langle j| \\otimes U_{ij} $ where $U_A( |j\\rangle \\otimes h) = \\sum _{i=0}^1 |i\\rangle \\otimes U_{ij} h$ .","In this case, $ U_A^* = \\begin{pmatrix} U_{00}^* & U_{10}^* \\\\ U_{01}^* & U_{11}^* \\end{pmatrix} $ and the fact that $U_A$ is unitary can be expressed by eight equations involving these operators that are best expressed as $U_A^*U_A = \\begin{pmatrix} I_{\\mathcal {R}} & 0\\\\ 0 & I_{\\mathcal {R}} \\end{pmatrix} = U_AU_A^*,$ where we apply the usual rules of matrix multiplication, being careful to remember that since the entries of $U_A$ are operators, not numbers, they need not commute.","We also recall that when $\\mathcal {R}$ is infinite dimensional, then it is necessary that both $U_A^*U_A$ and $U_AU_A^*$ be the identity to guarantee that $U_A$ is unitary.","Finally, in the special case that $\\dim (\\mathcal {R}) =1$ so that these entries are numbers, then we are back to the usual case of a $2 \\times 2$ complex unitary matrix.","Conversely, if we let $U_A$ be any $2 \\times 2$ matrix of operators on $\\mathcal {R}$ that satisfies Eq.","(REF ) then $U_A$ will define a unitary on $\\mathbb {C}^2 \\otimes \\mathcal {R}$ .","Similarly, Bob's unitary $U_B: \\mathcal {R} \\otimes \\mathbb {C}^2 \\rightarrow \\mathcal {R} \\otimes \\mathbb {C}^2$ is represented by a $2 \\times 2$ matrix of operators on $\\mathcal {R}$ , $U_B= ( V_{ij})$ , whose entries satisfy the same eight equations.","Finally, to have a commuting operator framework as in Figure REF , we need $(U_A \\otimes I_2)(I_2 \\otimes U_B) = (I_2 \\otimes U_B)(U_A \\otimes I_2)$ .","The following proposition translates this condition into equations involving the operator entries.","Proposition 3.1 Let $U_{ij}, V_{kl}, 0 \\le i,j,k,l \\le 1$ be operators on the Hilbert space $\\mathcal {R}$ such that $U_A=( U_{ij}),$ and $U_B= (V_{kl})$ are unitaries.","Then $(U_A \\otimes I_2)(I_2 \\otimes U_B)= (I_2 \\otimes U_B)(I_2 \\otimes U_A)$ if and only if $U_{ij}V_{kl} = V_{kl}U_{ij}$ and $U_{ij}^*V_{kl} = V_{kl}U_{ij}^*$ for all $i,j,k,l$ .","We have that $(U_A \\otimes I_2)(I_2 \\otimes U_B)( |j\\rangle \\otimes h \\otimes |l\\rangle ) = \\sum _{i,k=0}^1 |i\\rangle \\otimes U_{ij}V_{kl}h \\otimes |k\\rangle ,$ and similarly, $(I_2 \\otimes U_B)(U_A \\otimes I_2) ( |j\\rangle \\otimes h \\otimes |l\\rangle ) = \\sum _{i,k=0}^1 |i\\rangle \\otimes V_{kl}U_{ij}h \\otimes |k\\rangle .$ Thus, we see that $(U_A \\otimes I_2)(I_2 \\otimes U_B) = (I_2 \\otimes U_B)(U_A \\otimes I_2)$ is equivalent to $U_{ij}V_{kl} = V_{kl}U_{ij}$ , for all $i,j,k,l.$ However, if an invertible operator commutes with another operator, then its inverse also commutes with that operator.","Hence, $(U_A \\otimes I_2)^{-1} = (U_A^* \\otimes I_2)$ commutes with $(I_2 \\otimes U_B)$ and this is equivalent to $U_{ij}^*V_{kl}= V_{kl}U_{ij}^*$ , for all $i,j,k,l.$ The above equations are generally summarized by saying that the set of operators $\\lbrace U_{ij} \\rbrace $ *-commutes with the set $\\lbrace V_{kl} \\rbrace $ .","Thus, having a commuting operator framework is equivalent to having two unitaries $U_A=(U_{ij})$ , and $U_B=(V_{kl})$ whose entries *-commute.","We wish to study “universal\" properties of $2 \\times 2$ matrices of operators $(U_{ij})$ that give rise to a unitary.","To do this we begin with a unital *-algebra $\\mathcal {U}_2$ with generators, denoted 1 and $u_{ij}, \\, 0 \\le i,j \\le 1$ , subject to the eight equations, $\\begin{pmatrix} u_{00} & u_{01} \\\\ u_{10} & u_{11} \\end{pmatrix}\\begin{pmatrix} u_{00}^* & u_{10}^* \\\\ u_{01}^* & u_{11}^* \\end{pmatrix}=\\begin{pmatrix} u_{00}^* & u_{10}^* \\\\ u_{01}^* & u_{11}^* \\end{pmatrix}\\begin{pmatrix} u_{00} & u_{01} \\\\ u_{10} & u_{11} \\end{pmatrix}= \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix}.$ Thus, whenever there is a Hilbert space $\\mathcal {H}$ and four operators, $U_{ij}$ on that space such that the operator-matrix $U=(U_{ij})$ defines a unitary operator on $\\mathcal {H} \\otimes \\mathbb {C}^2$ , then there is a *-homomorphism, $ \\pi : \\mathcal {U}_2 \\rightarrow B(\\mathcal {H}) \\text{ with } \\pi (u_{ij}) = U_{ij}.$ For $x \\in \\mathcal {U}_2$ he sets $\\Vert x\\Vert = \\sup \\lbrace \\Vert \\pi (x)\\Vert \\colon \\pi \\text{ a *-homomorphism} \\rbrace $ , where the supremum is taken over all Hilbert spaces and all $\\pi $ 's as above.","This defines a norm on $\\mathcal {U}_2$ and that the completion is a C*-algebra, we shall denote $U_{\\operatorname{nc}}(2)$ .","The subscript $\\operatorname{nc}$ stands for “non-commuting\" and is intended to remind us that the generators $u_{ij}$ do not commute.","(This approach generalizes naturally to $d \\times d$ matrices of operators, for $d > 2$ , where the C*-algebra is denoted as $U_{\\operatorname{nc}}(d)$ .)","Note that in the commuting operator framework, the set $U_{ij}$ and the set $V_{kl}$ each gives rise to a *-homomorphism and that these two *-homomorphisms commute.","Thus, it is not hard to see that we have a one-to-one correspondence between commuting operator frameworks and *-homomorphisms of $U_{\\operatorname{nc}}(2) \\otimes U_{\\operatorname{nc}}(2)$ into $B(\\mathcal {R})$ .","Since we want to consider all commuting operator frameworks, we are lead to study $U_{\\operatorname{nc}}(2) \\otimes _{\\max } U_{\\operatorname{nc}}(2)$ .","The study of states on this algebra turns out to be closely related to embezzlement constructions as the following result shows.","Theorem 3.2 There exists a perfect embezzlement protocol in the commuting operator framework if and only if there exists a state $s: U_{\\operatorname{nc}}(2) \\otimes _{\\max } U_{\\operatorname{nc}}(2) \\rightarrow \\mathbb {C}$ such that $s(u_{00} \\otimes u_{00}) = \\frac{1}{\\sqrt{2}}$ , $s(u_{10} \\otimes u_{00}) = \\,0$ , $s(u_{00} \\otimes u_{10}) = \\,0$ , $s(u_{10} \\otimes u_{10}) = \\frac{1}{\\sqrt{2}}$ .","First assume that a perfect embezzlement protocol exists in a commuting operator framework.","Let $\\mathcal {R}$ be a Hilbert space, let $\\psi \\in \\mathcal {R}$ be a unit vector, let $U_A= \\big ( U_{ij} \\big )$ and $U_B= \\big ( V_{kl} \\big )$ be unitaries on $\\mathbb {C}^2 \\otimes \\mathcal {H}$ and $\\mathcal {H} \\otimes \\mathbb {C}^2$ , respectively, such that $U_A \\otimes I_2$ commutes with $I_2 \\otimes U_B$ and let $\\psi $ be a catalyst vector for perfect embezzlement of a Bell state, Define $\\pi : U_{\\operatorname{nc}}(2) \\otimes _{\\max } U_{\\operatorname{nc}}(2) \\rightarrow B(\\mathcal {H})$ to be the *-homomorphism defined by $\\pi (u_{ij} \\otimes 1) = U_{i,j}, \\, \\pi (1 \\otimes u_{kl}) = V_{kl}$ .","Since $\\psi $ is a catalyst vector, $ (U_A \\otimes I_2)(I_2 \\otimes U_B) ( |0\\rangle \\otimes \\psi \\otimes |0\\rangle ) = \\textstyle {\\frac{1}{\\sqrt{2}}}( |0\\rangle \\otimes \\psi \\otimes |0\\rangle + |1\\rangle \\otimes \\psi \\otimes |1\\rangle ).","$ Now define a state on $U_{\\operatorname{nc}}(2) \\otimes _{\\max } U_{\\operatorname{nc}}(2)$ by $s(x) = \\langle \\pi (x) \\psi , \\psi \\rangle $ .","We have that $\\big ( I_2 \\otimes U_B\\big ) \\big ( U_A \\otimes I_2 \\big ) |0\\rangle \\otimes \\psi \\otimes |0\\rangle & = \\sum _{i,j=0}^1 |i\\rangle \\otimes U_{i,0}V_{j,0} \\psi \\otimes |j\\rangle \\\\ & = \\textstyle {\\frac{1}{\\sqrt{2}}}|0\\rangle \\otimes \\psi \\otimes |0\\rangle + \\textstyle {\\frac{1}{\\sqrt{2}}}|1\\rangle \\otimes \\psi \\otimes |1\\rangle ,$ which is equivalent to $ U_{00}V_{00} \\psi = U_{10}V_{10} \\psi = \\textstyle {\\frac{1}{\\sqrt{2}}}\\psi \\text{\\ \\ and \\ \\ } U_{00} V_{10} \\psi = U_{10}V_{00} \\psi = 0 .$ From these equations, it follows that the state $s$ satisfies the four conditions.","Conversely, assume that $s: U_{\\operatorname{nc}}(2) \\otimes _{\\max } U_{\\operatorname{nc}}(2) \\rightarrow \\mathbb {C}$ is a state that satisfies the 4 conditions.","Let $\\pi _s: U_{\\operatorname{nc}}(2) \\otimes _{\\max } U_{\\operatorname{nc}}(2) \\rightarrow B(\\mathcal {H}_s)$ and $\\psi \\in \\mathcal {H}_s$ be the GNS representation of the state so that $s(x) = \\langle \\pi _s(x) \\psi , \\psi \\rangle $ .","If we define $U_A: \\mathbb {C}^2 \\otimes \\mathcal {H} \\rightarrow \\mathbb {C}^2 \\otimes \\mathcal {H}$ by $U_A= \\big ( \\pi (u_{ij} \\otimes 1) \\big )$ and $U_B: \\mathcal {H} \\otimes \\mathbb {C}^2 \\rightarrow \\mathcal {H} \\otimes \\mathbb {C}^2$ by $U_B = \\big ( \\pi (1 \\otimes u_{i,j}) \\big )$ , then $U_A \\otimes I_2$ commutes with $I_2 \\otimes U_B$ .","The operator on the direct sum of four copies of $\\mathcal {H}$ given by $ \\begin{pmatrix} U_{00}V_{00} & U_{01}V_{00} & U_{00}V_{01} & U_{01}V_{01} \\\\ U_{10}V_{00} & U_{11}V_{00} & U_{10}V_{01} & U_{11}V_{01}\\\\ U_{00} V_{10} & U_{01}V_{10} & U_{00}V_{11} & U_{01}V_{11}\\\\ U_{10}V_{10} & U_{11}V_{10} & U_{10} V_{11} & U_{11}V_{11} \\end{pmatrix} $ is unitary.","Hence, $ 1 &= |\\langle U_{00}V_{00} \\psi , \\psi \\rangle |^2 + |\\langle U_{10}V_{10} \\psi , \\psi \\rangle |^2 \\le \\Vert U_{00} V_{00} \\psi \\Vert ^2 + \\Vert U_{10}V_{10}\\psi \\Vert ^2 \\\\ &\\le \\Vert U_{00} V_{00} \\psi \\Vert ^2 + \\Vert U_{10}V_{10}\\psi \\Vert ^2 + \\Vert U_{00}V_{10}\\psi \\Vert ^2 + \\Vert U_{10} V_{00} \\psi \\Vert ^2 =1, $ from which it follows that $ U_{00}V_{00} \\psi = U_{10}V_{10} \\psi = \\textstyle {\\frac{1}{\\sqrt{2}}}\\psi \\text{\\ \\ and \\ \\,} U_{00} V_{10} \\psi = U_{10}V_{00} \\psi = 0 .$ Thus, $\\big ( I_2 \\otimes U_B\\big ) \\big ( U_A \\otimes I_2 \\big ) |0\\rangle \\otimes \\psi \\otimes |0\\rangle & = \\sum _{i,j=0}^1 |i\\rangle \\otimes U_{i,0}V_{j,0} \\psi \\otimes |j\\rangle \\\\ & = \\textstyle {\\frac{1}{\\sqrt{2}}}|0\\rangle \\otimes \\psi \\otimes |0\\rangle + \\textstyle {\\frac{1}{\\sqrt{2}}}|1\\rangle \\otimes \\psi \\otimes |1\\rangle $ and we have a perfect embezzlement protocol.","Thus, we have proven that perfect embezzlement in the commuting operator framework is equivalent to the existence of a state on $U_{\\operatorname{nc}}(2) \\otimes _{\\max } U_{\\operatorname{nc}}(2)$ that satisfies the four equations above.","We now prove that such a state exists.","Theorem 3.3 There exists a state $s: U_{\\operatorname{nc}}(2) \\otimes _{\\max } U_{\\operatorname{nc}}(2) \\rightarrow \\mathbb {C}$ that satisfies the four equations of the previous theorem and consequently perfect embezzlement is possible in the commuting operator framework.","By the results of [2], we have finite dimensional Hilbert spaces $H_n$ unit vectors $h_n\\in H_n$ and unitary operators $U_n, V_n$ on $H_n \\otimes \\mathbb {C}^2$ , such that $(U_n \\otimes I_2) (I_2 \\otimes V_n)(|0\\rangle \\otimes h_n \\otimes |0\\rangle ) -\\frac{1}{\\sqrt{2}}(|0\\rangle \\otimes h_n \\otimes |0\\rangle + |1\\rangle \\otimes h_n \\otimes |1\\rangle )$ has norm less than $1/n$ .","These operators induce *-homomorphisms, $\\pi _n: U_{\\operatorname{nc}}(2) \\otimes _{\\max } U_{\\operatorname{nc}}(2) \\rightarrow B(H_n)$ and states $s_n: U_{\\operatorname{nc}}(2) \\otimes _{\\max } U_{\\operatorname{nc}}(2) \\rightarrow \\mathbb {C}$ defined by $s_n(x) = \\langle \\pi _n(x) h_n , h_n \\rangle $ .","These states satisfy: $\\bigl | s_n( u_{00} \\otimes u_{00}) - \\frac{1}{\\sqrt{2}} \\bigr | < \\frac{1}{n}$ , $| s_n(u_{10} \\otimes u_{00}) | < \\frac{1}{n}$ , $| s_n( u_{00} \\otimes u_{10}) | < \\frac{1}{n}$ , $ \\bigl | s_n(u_{10} \\otimes u_{10}) - \\frac{1}{\\sqrt{2}} \\bigr | < \\frac{1}{n}$ .","Now by the fact that the state space of any unital C*-algebra is compact in the weak*-topology, we may take a limit point $s$ of this sequence of states.","Since the value of $s(u_{i,j} \\otimes u_{k,l})$ must be a limit of the values of $s_n$ on these same elements, $s$ will be a state on $U_{\\operatorname{nc}}(2) \\otimes _{\\max } U_{\\operatorname{nc}}(2)$ that satisfies the 4 conditions exactly.","Remark 3.4 From [2], each of the states, $s_n$ appearing in the above proof is actually a state on $U_{\\operatorname{nc}}(2) \\otimes _{\\min } U_{\\operatorname{nc}}(2)$ .","Hence, by taking a limit point, we obtain a state $s: U_{\\operatorname{nc}}(2) \\otimes _{\\min } U_{\\operatorname{nc}}(2) \\rightarrow \\mathbb {C}$ that satisfies the 4 equations of Theorem REF .","If we apply the GNS construction or any other method to represent it as $s(x) = \\langle \\pi (x) \\psi , \\psi \\rangle $ on some Hilbert space $\\mathcal {H}$ , where $\\pi : U_{\\operatorname{nc}}(2) \\otimes _{\\min } U_{\\operatorname{nc}}(2) \\rightarrow B(\\mathcal {H})$ , then the representation $\\pi $ and the Hilbert space cannot decompose as a tensor product.","Otherwise we would achieve perfect embezzlement in a tensor product framework.","Hence, we obtain an example of a state on a minimal tensor product, such that it cannot be represented using a *-homomorphism that is a spatial tensor product.","In fact, no state on $U_{\\operatorname{nc}}(2) \\otimes _{\\min } U_{\\operatorname{nc}}(2)$ that satisfies just those 4 equations can have a spatial tensor product representation.","Remark 3.5 The coefficients that appear in Theorem REF are a consequence of the fact that we are embezzling a Bell state.","If we wish instead for a perfect embezzlement protocol of a more general vector state, say, of the form $\\sum _{i,j=0}^{d-1} \\alpha _{ij} |i\\rangle \\otimes |j\\rangle $ , then this is equivalent to the existence of a state $s$ on $U_{\\operatorname{nc}}(d) \\otimes _{\\max } U_{\\operatorname{nc}}(d)$ satisfying $s(u_{i0} \\otimes u_{j0}) = \\alpha _{ij}$ , for $0 \\le i,j < d$ .","Moreover, it is shown in [2] that every vector in $\\mathbb {C}^d \\otimes \\mathbb {C}^d$ can be approximately embezzled in a finite dimensional scenario.","Therefore, arguing as above, there is always a state $s$ on $U_{\\operatorname{nc}}(d) \\otimes _{\\min } U_{\\operatorname{nc}}(d)$ satisfying the $d^2$ equations."],["Explicit construction of a perfect embezzlement protocol in a commuting-operator framework","The previous section proves the existence of a perfect embezzlement protocol, but without constructing one explicitly.","Some of the steps of the proof are nonconstructive.","In Theorem REF , an abstract state is obtained by invoking an existence theorem using the weak*-compactness of the set of all states; moreover, in Theorem REF , the Hilbert space is obtained by applying the GNS representation of the state, which is based on the completion of an abstract C*-algebra.","In this section, we give an explicit commuting-operator protocol for perfect embezzlement.","We explain the technique for Bell states, and, at the end of the section, explain how to it extends to more general entangled states."],["The resource space $\\mathcal {R}$ and shift operations on this space","The resource space is the Hilbert space $\\ell ^2$ , whose orthonormal basis is countably infinite.","In order to define the operations used in the protocol, it is useful to think of this space in terms of countably infinite tensor products of states, where all but finitely many of them are fixed.","First, consider the set of infinite tensor products of 2-qubit computational basis states, where all but finitely many of them are in state $|00\\rangle $ .","We can express these states as $|\\dots x_2 x_1 x_0\\,,\\, \\dots y_2 y_1 y_0\\rangle $ , or as $|x,y\\rangle $ (where $x, y \\in \\mathbb {N}_0$ , and $x_j$ and $y_j$ are the binary digits of $x$ and $y$ , respectively, in position $j$ ).","Next, consider the set of infinite tensor products of 2-qubit Bell basis states where all but finitely many of them are $\\frac{1}{\\sqrt{2}}|00\\rangle + \\frac{1}{\\sqrt{2}}|11\\rangle $ .","Let us denote these states as $|0.x_{-1}x_{-2}x_{-3}\\dots \\,,\\, 0.y_{-1}y_{-2}y_{-3}\\dots \\rangle $ , with the convention that the qubits in position $-j$ (i.e., the $j$th qubit pair, corresponding to bits $x_{-j}$ and $y_{-j}$ ) are in the Bell basis state $\\textstyle {\\frac{1}{\\sqrt{2}}}|0y_{-j}\\rangle + \\textstyle {\\frac{1}{\\sqrt{2}}}(-1)^{x_{-j}}|1\\overline{y_{-j}}\\rangle .$ A convenient way of denoting an orthonormal basis for the (spatial) tensor product of the Hilbert spaces generated by the two aforementioned sets is as the set of all $|\\dots x_2 x_1 x_0\\,\\mbox{\\Large .","}\\, x_{-1} x_{-2} \\dots \\,,\\ \\dots y_2 y_1 y_0\\,\\mbox{\\Large .","}\\, y_{-1} y_{-2}\\dots \\rangle ,$ with all but finitely many $x_j$ and $y_{j}$ set to 0.","Equivalently, each basis state can be written as $|x,y\\rangle $ , where $x$ and $y$ are dyadic rational numbersA dyadic rational number is of the form $x = a/2^b$ , where $a, b \\in \\mathbb {N}_0$ .","Each dyadic $x$ can be written in binary as $x = x_{\\ell } \\dots x_2 x_1 x_0 \\hspace*{0.85358pt}\\mbox{\\Large .","}\\hspace*{0.85358pt} x_{-1} x_{-2} \\dots x_{-r}$ .","Formally, for all $j \\in \\mathbb {Z}$ , bit $j$ of $x$ is defined as $x_j = \\lfloor x 2^{-j} \\rfloor \\bmod 2$ .","Note that $2x$ is $x$ with all the binary digits shifted left by 1..","Intuitively, these basis states can be thought of as two-way infinite tensor products, as illustrated in Figure REF .","Figure: Schematic picture of the tensor product structure of the basis states.","Each circle represents a qubit.","In positions 0,1,2,⋯0, 1, 2, \\dots the qubits are in computational basis states.","In positions -1,-2,⋯-1, -2, \\dots the qubits are in Bell basis states.On the left are computational basis states (with all but finitely many in state $|00\\rangle $ ).","On the right are Bell basis states (with all but finitely many in state $\\frac{1}{\\sqrt{2}}|00\\rangle + \\frac{1}{\\sqrt{2}}|11\\rangle )$ .","We now define a left shift $L$ on the Hilbert space spanned by these basis states.","Intuitively, $L$ shifts the two-way infinite tensor product to the left by one.","Formally, we define $L$ as the product of two unitaries.","First, define $L_1$ as the left shift of the digits of $x$ and $y$ $L_1 & |\\dots x_2 x_1 x_0 \\,.\\, x_{-1} x_{-2} \\dots \\,,\\ \\dots y_2 y_1 y_0\\,.\\, y_{-1} y_{-2} \\dots \\rangle \\\\&=|\\dots x_1 x_0 x_{-1}\\,.\\, x_{-2} y_{-3} \\dots \\,,\\ \\dots y_1 y_0 y_{-1} \\,.\\, y_{-2} y_{-3} \\dots \\rangle ,$ or, equivalently, as $L_1 |x,y\\rangle = |2x,2y\\rangle $ .","$L_1$ is unitary because it is a permutation of the basis states.","Note that $L_1$ does not implement the desired left shift $L$ because the qubits in position $-1$ are in the Bell basis whereas the qubits in position 0 are in the computational basis.","A basis conversion is needed when position $-1$ is shifted to position 0.","Define $L_2$ to perform this basis conversion in position 0 as $L_2 & |\\dots x_2 x_1 x_0 \\,.\\, x_{-1} x_{-2} \\dots \\,,\\ \\dots y_2 y_1 y_0 \\,.\\, y_{-1} y_{-2}\\dots \\rangle \\\\&={\\textstyle {\\frac{1}{\\sqrt{2}}}}|\\dots x_2 x_1 0 \\,.\\, x_{-1} x_{-2} \\dots \\,,\\ \\dots y_2 y_1 y_0 \\,.\\, y_{-1} y_{-2}\\dots \\rangle \\\\&\\ \\ + {\\textstyle {\\frac{1}{\\sqrt{2}}}}(-1)^{x_0}|\\dots x_2 x_1 1 \\,.\\, x_{-1} x_{-2} \\dots \\,,\\ \\dots y_2 y_1 \\overline{y_0} \\,.\\, y_{-1} y_{-2}\\dots \\rangle .$ This can be equivalently expressed in terms of arithmetic operations on dyadic rationals as $L_2 |x,y\\rangle ={\\textstyle {\\frac{1}{\\sqrt{2}}}}|x - x_0,y\\rangle + {\\textstyle {\\frac{1}{\\sqrt{2}}}}(-1)^{x_0}|x - x_0 + 1, y - 2y_0 + 1\\rangle .$ $L_2$ is unitary because it is a direct sum of $4 \\times 4$ unitaries.","Finally, define $L = L_2 L_1$ , which is unitary because $L_1$ and $L_2$ are unitary.","An interesting property of $L$ is that applying this operation to the state $|0.0,0.0\\rangle $ yields ${\\textstyle {\\frac{1}{\\sqrt{2}}}}|0.0,0.0\\rangle + {\\textstyle {\\frac{1}{\\sqrt{2}}}}|1.0,1.0\\rangle $ .","In the tensor product picture, $L$ leaves the state of all the qubits intact except for the qubits in position 0, whose state changes from $|00\\rangle $ to $\\frac{1}{\\sqrt{2}}|00\\rangle + \\frac{1}{\\sqrt{2}}|11\\rangle $ .","This is performing something like an embezzlement transformation (in a manner reminiscent of the imaginary “Hilbert hotel\"); however, this $L$ does not decompose into two commuting operations that have the structure illustrated in Figure REF .","In order to obtain such a decomposition, we need to enlarge our Hilbert space.","We begin with some intuition.","In Figure REF , assume that Alice possesses the qubits in the first row and Bob possesses the qubits in the second row.","When Alice's qubits are shifted to the left by one, the picture changes to that of Figure REF .","Figure: Schematic picture of the tensor product of basis states when Alice's qubits (top row) are shifted left by one.Pairs of circles connected by lines are in the Bell basis.Such states are orthogonal to all states of the form of Figure .This can be equivalently expressed by shifting the labels of Alice's qubits as in Figure REF .","Figure: An alternative schematic picture of the tensor product of Figure , where the labels of the qubits on the top row are adjusted to reflect the shift in the top row.More generally, for an arbitrary $r \\in \\mathbb {Z}$ , a left shift of the first row by $r$ is illustrated in Figure REF .","Figure: Schematic picture of a left shift of Alice's qubits (top row) by r∈ℤr \\in \\mathbb {Z}.With the picture of Figure REF in mind, define the Hilbert space $\\mathcal {R}$ as having orthonormal basis states of the form $|r,x,y\\rangle $ , where $x, y$ are dyadic rationals and $r \\in \\mathbb {Z}$ represents the leftward shift of Alice's qubits.","We can interpret $|r,x,y\\rangle $ as an encoding of the following logical state.","For all $j \\in \\mathbb {Z}$ , Alice's logical qubit in position $j+r$ and Bob's logical qubit in position $j$ are in the joint state ${\\left\\lbrace \\begin{array}{ll}|x_{j}y_{j}\\rangle & \\text{if $j \\ge 0$} \\\\[2mm]{\\textstyle {\\frac{1}{\\sqrt{2}}}}|0y_{j}\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}(-1)^{x_{j}}|1\\overline{y_{j}}\\rangle & \\text{if $j < 0$.}\\end{array}\\right.","}$ Now we define the Alice left shift $L_A$ as simply $L_A |r,x,y\\rangle = |r+1,x,y\\rangle .$ $L_A$ is obviously unitary and commutes with $L$ , since they act on different components of $|r,x,y\\rangle $ .","Next, define the Bob left shift $L_B$ as $L_B = L_{A}^{*}L$ (a left shift of both Alice and Bob's qubits followed by a right shift of Alice's qubits).","Note that $L_A$ and $L_B$ commute, since $L_A$ and $L$ commute.","Also, since $L$ is a left shift by both Alice and Bob and $L_{A}^{*}$ is a right shift by Alice, $L_B$ has no net effect on Alice's logical qubits."],["Swap operations between $\\mathcal {H}_A$ , {{formula:5ce0fc52-fcb6-4283-ae7b-84f3bd08f3dc}} and {{formula:3ba76a35-7dd7-4b48-878e-0858d3c7d485}}","Prior to defining our embezzlement protocol, we define swap operations between $\\mathcal {H}_A$ and the logical qubit of Alice in position 0 of $\\mathcal {R}$ , and between $\\mathcal {H}_B$ and the logical qubit of Bob in position 0 of $\\mathcal {R}$ .","The Bob swap $S_B$ is defined simply as the unitary operation that acts on $\\mathcal {H}_B \\otimes \\mathcal {R}$ as $S_B |t\\rangle \\otimes |r,\\, x,\\, \\dots y_1 y_0 \\,.\\, y_{-1} \\dots \\rangle =|y_0\\rangle \\otimes |r,\\, x,\\, \\dots y_1 t \\,.\\, y_{-1} \\dots \\rangle ,$ or, equivalently, as $S_B|t\\rangle \\otimes |r,x,y\\rangle = |y_0\\rangle \\otimes |r,x,y-y_0+t\\rangle $ .","$S_B$ is clearly unitary and commutes with $L_A$ since they act on different components of each basis state $|r,x,y\\rangle \\in \\mathcal {R}$ .","The corresponding Alice swap, acting on $\\mathcal {H}_A \\otimes \\mathcal {R}$ , is more complicated than $S_B$ .","First define $\\tilde{S}_A$ (a naïve Alice swap) as $\\tilde{S}_A |s\\rangle \\otimes |r,\\, \\dots x_1 x_0 \\,.\\, x_{-1} \\dots ,\\, y\\rangle =|x_0\\rangle \\otimes |r,\\, \\dots x_1 s \\,.\\, x_{-1} \\dots ,\\, y\\rangle ,$ or, equivalently, as $\\tilde{S}_A|s\\rangle \\otimes |r,x,y\\rangle = |x_0\\rangle \\otimes |r,x-x_0+s,y\\rangle $ .","$\\tilde{S}_A$ does not swap with Alice's logical qubit in position 0—moreover, $\\tilde{S}_A$ does not commute with $L_B$ .","To swap with Alice's logical qubit in position 0, it is convenient to first define the controlled-$L$, denoted as $C$ , acting on $\\mathcal {R}$ as $C |r,x,y\\rangle = L^{r}|r,x,y\\rangle ,$ which makes sense because $L^{r}$ acts only on the second and third component of $|r,x,y\\rangle $ .","$C$ is unitary because each $L^{r}$ is unitary and $C$ is a direct sum of all $L^{r}$ .","Intuitively, $C^*|r,x,y\\rangle $ is a state in which Alice's literal qubit in position 0 corresponds to Alice's logical qubit in position 0 in $|r,x,y\\rangle $ .","Now we define the actual Alice swap as $S_A = C \\tilde{S}_A C^{*}.$ Clearly $S_A$ is unitary and, for each $|r,x,y\\rangle \\in \\mathcal {R}$ , its effect is localized to Alice's logical qubit in position 0.","$S_A$ and $S_B$ commute because $S_B$ is localized to Bob's logical qubit in position 0.","Moreover, $S_A$ and $L_B$ commute because, for $|r,x,y\\rangle \\in \\mathcal {R}$ , $L_B$ is localized to Bob's logical qubits."],["The embezzlement protocol","The idea is to start with state $|0,\\,0.0,\\,0.0\\rangle $ , perform $L_A$ and $L_B$ , and then swap the two qubits in position 0 of $\\mathcal {R}$ into $\\mathcal {H}_A$ and $\\mathcal {H}_B$ .","Alice performs $U_A = S_A L_A$ and Bob performs $U_B = S_B L_B$ .","Clearly $U_A$ and $U_B$ commute.","Recall that $L_A L_B = L$ .","The state evolves as: 0. initial state $|0\\rangle \\otimes |0\\rangle \\otimes |0,\\,0.0,\\,0.0\\rangle $ 1. after $L_A L_B$ $|0\\rangle \\otimes |0\\rangle \\otimes \\bigl ({\\textstyle {\\frac{1}{\\sqrt{2}}}}|0,\\,0.0,\\,0.0\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|0,\\,1.0,\\,1.0\\rangle \\bigr )$ 2. after $S_A S_B$ $\\bigl ({\\textstyle {\\frac{1}{\\sqrt{2}}}}|0\\rangle \\otimes |0\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|1\\rangle \\otimes |1\\rangle \\bigr )\\otimes |0,\\,0.0,\\,0.0\\rangle $ This completes the protocol for perfect embezzlement in the commuting operator framework.","Remark 4.1 It is easy to adapt the above method to embezzle a more general entangled state, say, of the form $\\phi = \\sum _{i,j=0}^{d-1} \\alpha _{ij} |i\\rangle \\otimes |j\\rangle $ .","First, redefine $\\mathcal {R}$ to be in terms of basis states of the form $|r,x,y\\rangle $ , where $x$ and $y$ are $d$ -adic rational numbers (i.e., with digits in $\\mathbb {Z}_d$ ).","Then it suffices to set the operation $L_2$ (which is the basis change part of the left shift operation) to be any unitary operation on $\\mathbb {C}^d \\otimes \\mathbb {C}^d$ that maps $|0\\rangle \\otimes |0\\rangle $ to $\\phi $ .","The other parts of the protocol are essentially the same."],["Coherent embezzlement games","A purported protocol for embezzlement cannot be tested in the way that nonlocal games can, because Alice and Bob can perform local operations that perfectly map $|0\\rangle \\otimes |0\\rangle $ to $\\frac{1}{\\sqrt{2}}|0\\rangle \\otimes |0\\rangle +\\frac{1}{\\sqrt{2}}|1\\rangle \\otimes |1\\rangle $ using the resource of only a single (concealed) Bell state.","Leung, Toner and Watrous [8] proposed a coherent state exchange game that is related to embezzlement but is operationally testable.","In this game, Alice and Bob each receive a qutrit from a referee as input and they each return a qubit to the referee, who performs a measurement on the returned state to determine whether they win or lose.","There is no perfect strategy for this game using finite entanglement.","It is shown in [8] that, for all $\\epsilon > 0$ : there exists a strategy that succeeds with probability $1 - \\epsilon $ using $O(\\log (1/\\epsilon ))$ -entropy entanglement; moreover, to succeed with probability $1 - \\epsilon $ requires entanglement with entropy $\\Omega (\\log (1/\\epsilon ))$ .","Regev and Vidick [10] presented a simplification of the coherent state exchange game that has the above properties, but where the outputs are classical bits instead of qubits (the inputs are still qutrits).","In [10], this is called the $T_2$ game.","We refer to this as the coherent embezzlement game, to highlight its close relationship with embezzlement.","In this section, we begin by reviewing the definition of the coherent embezzlement (a.k.a.","$T_2$ ) game.","Then we show that a perfect strategy for embezzlement can be converted to a perfect strategy for coherent embezzlement—and vice versa.","By such reductions, we prove that there is a perfect strategy for coherent embezzlement in the commuting operator framework (Theorem REF ), but there is no such perfect strategy in the tensor product framework (Theorem REF ).","We now define the coherent embezzlement game [10].","Alice and Bob each receive two qutrits as input and they each produce a classical bit as output.","The input state that Alice and Bob jointly receive is either $\\phi _0$ or $\\phi _1$ , where $\\phi _0 &= {\\textstyle {\\frac{1}{\\sqrt{2}}}}|0\\rangle \\otimes |0\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}\\bigl ({\\textstyle {\\frac{1}{\\sqrt{2}}}}|1\\rangle \\otimes |1\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|2\\rangle \\otimes |2\\rangle \\bigr ) \\\\\\phi _1 &= {\\textstyle {\\frac{1}{\\sqrt{2}}}}|0\\rangle \\otimes |0\\rangle -{\\textstyle {\\frac{1}{\\sqrt{2}}}}\\bigl ({\\textstyle {\\frac{1}{\\sqrt{2}}}}|1\\rangle \\otimes |1\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|2\\rangle \\otimes |2\\rangle \\bigr ).$ Call Alice and Bob's output bits $a$ and $b$ respectively.","The winning condition is that: when the input is $\\phi _0$ , $a \\oplus b = 0$ ; when the input is $\\phi _1$ , $a \\oplus b = 1$ .","Note that the winning condition does not require Alice and Bob's resource state to be used in a catalytic manner; Alice and Bob are free to destroy this state in their strategy.","In this sense, the coherent embezzlement game is simpler than embezzlement, whose definition depends critically on restoring the resource state.","In the remainder of this section, for technical convenience, we represent qutrits as pairs of qubits using the encoding $0 \\equiv 00$ , $1 \\equiv 10$ (for Alice) or 01 (for Bob), and $2 \\equiv 11$ .","With this encoding, the input states that Alice and Bob jointly receive can be written as, for $c \\in \\lbrace 0,1\\rbrace $ , $\\phi _c &= {\\textstyle {\\frac{1}{\\sqrt{2}}}}|00\\rangle \\otimes |00\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}(-1)^c\\bigl ({\\textstyle {\\frac{1}{\\sqrt{2}}}}|10\\rangle \\otimes |01\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|11\\rangle \\otimes |11\\rangle \\bigr ) \\\\&= {\\textstyle {\\frac{1}{\\sqrt{2}}}}|0\\rangle |00\\rangle |0\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}(-1)^c|1\\rangle \\bigl ({\\textstyle {\\frac{1}{\\sqrt{2}}}}|00\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|11\\rangle \\bigr )|1\\rangle $ (where the first two qubits are Alice's input and the last two qubits are Bob's input).","Theorem 5.1 There is a perfect strategy for the coherent embezzlement game in the commuting operator framework.","Let $\\mathcal {R}$ , $U_A$ , and $U_B$ be as defined in the unitary embezzlement protocol of section .","For coherent embezzlement, the input Hilbert spaces are $\\mathcal {H}_{A_1} \\otimes \\mathcal {H}_{A_2} = \\mathcal {H}_{B_1} \\otimes \\mathcal {H}_{B_2} = \\mathbb {C}^2\\otimes \\mathbb {C}^2$ .","We will show that the protocol in Figure REF performs coherent embezzlement.","Figure: Circuit diagram of a protocol for coherent embezzlement from a protocol for embezzlement based on U A U_A, U B U_B, and ψ∈ℛ\\psi \\in \\mathcal {R}.First note that, since the controlled $U^*_A$ and $U^*_B$ perform the inverse of embezzlement when their control qubits are in state $|1\\rangle $ , they perform a mapping on $\\mathcal {H}_{A_1} \\otimes \\mathcal {H}_{A_2} \\otimes \\mathcal {H}_{B_2} \\otimes \\mathcal {H}_{B_1}$ such that $\\phi _c =&{\\textstyle {\\frac{1}{\\sqrt{2}}}}|0\\rangle |00\\rangle |0\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}(-1)^c|1\\rangle \\bigl ({\\textstyle {\\frac{1}{\\sqrt{2}}}}|00\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|11\\rangle \\bigr )|1\\rangle \\\\[1mm]&\\mapsto \\ \\ {\\textstyle {\\frac{1}{\\sqrt{2}}}}|0\\rangle |00\\rangle |0\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}(-1)^c|1\\rangle |00\\rangle |1\\rangle .$ Note that, on the Hilbert space $\\mathcal {H}_{A_1} \\otimes \\mathcal {H}_{B_1}$ , this is the pure state ${\\textstyle {\\frac{1}{\\sqrt{2}}}}|00\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}(-1)^c|11\\rangle $ .","Finally, since the Hadamard gates perform the mapping ${\\textstyle {\\frac{1}{\\sqrt{2}}}}|00\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}(-1)^c|11\\rangle \\mapsto {\\textstyle {\\frac{1}{\\sqrt{2}}}}|0c\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|1\\overline{c}\\rangle ,$ the result follows.","Theorem 5.2 There is no perfect strategy for the coherent embezzlement game in the tensor product framework (where Alice and Bob are allowed to use ancillas).","The idea is that a perfect strategy for coherent embezzlement can be converted into a perfect strategy for embezzlement.","Suppose that there is a perfect strategy for coherent embezzlement.","Without loss of generality, it can be assumed that this strategy is of the form depicted in Figure REF , where $\\psi \\in \\mathcal {R}_A \\otimes \\mathcal {R}_B$ , Alice and Bob each perform a local unitary operation followed by measurement of one of their qubits.","Figure: Circuit diagram of an arbitrary protocol for coherent embezzlement.ψ∈ℛ A ⊗ℛ B \\psi \\in \\mathcal {R}_A \\otimes \\mathcal {R}_B is the resource state.Without loss of generality, two local unitaries are performed, followed by measurements of two specific qubits.Since we are assuming that the strategy is perfect, for input state $\\phi _0$ , the output bits satisfy $a\\oplus b = 0$ , and for input state $\\phi _1$ , the output bits satisfy $a\\oplus b = 1$ .","Now, consider the protocol in Figure REF .","Figure: Modification of circuit in Figure that performs embezzlement using the resource state ψ∈ℛ A ⊗ℛ B \\psi \\in \\mathcal {R}_A \\otimes \\mathcal {R}_B.We begin by showing that, for each $c \\in \\lbrace 0,1\\rbrace $ , the effect of the first three steps of the protocol in Figure REF is to map $\\phi _c\\otimes \\psi \\otimes \\gamma _A\\otimes \\gamma _B$ to $(-1)^c \\phi _c\\otimes \\psi \\otimes \\gamma _A\\otimes \\gamma _B$ .","Since $a \\oplus b = c$ , After the $U_A \\otimes U_B$ operations have been performed, the state must be of the form $\\alpha |0\\rangle \\otimes |c\\rangle \\otimes \\xi _0 + \\beta |1\\rangle \\otimes |\\overline{c}\\rangle \\otimes \\xi _1,$ when expressed the state in the Hilbert space $\\mathcal {H}_{A_1} \\otimes \\mathcal {H}_{B_1} \\otimes (\\mathcal {H}_{A_2} \\otimes \\mathcal {H}_{B_2} \\otimes \\mathcal {R}_A \\otimes \\mathcal {R}_B \\otimes \\mathcal {G}_A \\otimes \\mathcal {G}_B).$ Therefore, after the two $Z$ gates, the state is $(-1)^c\\bigl (\\alpha |0\\rangle \\otimes |c\\rangle \\otimes \\xi _0 + \\beta |1\\rangle \\otimes |\\overline{c}\\rangle \\otimes \\xi _1\\bigr ),$ and this is mapped by $U^{*}_{A} \\otimes U^{*}_{B}$ to $(-1)^c \\phi _c\\otimes \\psi \\otimes \\gamma _A\\otimes \\gamma _B$ .","This implies that the first three steps of the modified protocol maps $|00\\rangle \\otimes |00\\rangle = {\\textstyle {\\frac{1}{\\sqrt{2}}}}\\phi _0 + {\\textstyle {\\frac{1}{\\sqrt{2}}}}\\phi _1$ on $\\mathcal {H}_{A}\\otimes \\mathcal {H}_B$ (with the other registers in state $\\psi \\otimes \\gamma _A\\otimes \\gamma _B$ ) to ${\\textstyle {\\frac{1}{\\sqrt{2}}}}|10\\rangle \\otimes |01\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|11\\rangle \\otimes |11\\rangle = {\\textstyle {\\frac{1}{\\sqrt{2}}}}\\phi _0 - {\\textstyle {\\frac{1}{\\sqrt{2}}}}\\phi _1.$ Finally, the two $X$ gates of the protocol in Figure REF map the state ${\\textstyle {\\frac{1}{\\sqrt{2}}}}|10\\rangle \\otimes |01\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|11\\rangle \\otimes |11\\rangle $ to ${\\textstyle {\\frac{1}{\\sqrt{2}}}}|00\\rangle \\otimes |00\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|01\\rangle \\otimes |10\\rangle $ .","Therefore, the entire protocol maps $|00\\rangle \\otimes |00\\rangle \\otimes \\psi \\otimes \\gamma _A\\otimes \\gamma _B$ to $\\bigl ({\\textstyle {\\frac{1}{\\sqrt{2}}}}|00\\rangle \\otimes |00\\rangle &+{\\textstyle {\\frac{1}{\\sqrt{2}}}}|01\\rangle \\otimes |10\\rangle \\bigr )\\otimes \\psi \\otimes \\gamma _A\\otimes \\gamma _B \\\\& =|0\\rangle ({\\textstyle {\\frac{1}{\\sqrt{2}}}}|00\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|11\\rangle )|0\\rangle \\otimes \\psi \\otimes \\gamma _A\\otimes \\gamma _B.$ This is perfect embezzlement in the registers $\\mathcal {H}_{A_2} \\otimes \\mathcal {H}_{B_2}$ .","Since this protocol violates Theorem REF , there cannot exist a perfect strategy for coherent embezzlement using entanglement of the form $\\psi \\in \\mathcal {R}_A \\otimes \\mathcal {R}_B$ ."],["Perfect embezzlement requires non-unitary isometries","In this section we obtain further information about the nature of the unitary operators that appear in perfect embezzlement protocols.","We show that some of the operators that occur in such protocols must contain non-unitary isometries.","This result justifies the need for the register shifts that appear in the explicit protocol of the earlier section.","Also, since non-unitary isometries only exist in infinite dimensions, this result implies that perfect embezzlement is impossible with a finite dimensional resource space.","We begin by reviewing the key facts about unitaries and isometries.","Given a Hilbert space $\\mathcal {H}$ a linear map $V: \\mathcal {H} \\rightarrow \\mathcal {H}$ is an isometry provided that $\\Vert Vh\\Vert =\\Vert h\\Vert $ for all $h \\in \\mathcal {H}$ .","Note that isometries are necessarily one-to-one.","The map $V$ is a unitary provided that it is an isometry and it is onto.","A simple dimension count shows that in finite dimensions every isometry is necessarily a unitary.","An example of an isometry that is not a unitary is the unilateral shift S. This is the operator on the Hilbert space $\\ell ^2(\\mathbb {N})$ which has an orthonormal basis $\\lbrace |j\\rangle : j \\in \\mathbb {N} \\rbrace $ , defined by $S(|j\\rangle ) = |j+1\\rangle $ .","It is easily seen that this operator is an isometry and the $|1\\rangle $ is orthogonal to the range of $S$ , so that $S$ is not onto.","Note that the kernel of $S^*$ is equal to the span of $|1\\rangle $ .","We call a linear map $V: \\mathcal {H} \\rightarrow \\mathcal {H}$ a non-unitary isometry provided that it is an isometry that is not onto.","It is not hard to show that if $V$ is a non-unitary isometry, $e_1$ is any unit vector orthogonal to the range of $V$ , and we set $e_n = V^n e_1,$ then this sequence is orthonormal and $V$ acts as a unilateral shift on this subspace.","So, in this sense, non-unitary isometries always contain a space on which they act like unilateral shifts.","Recall that a subspace $\\mathcal {M}$ is called invariant for an operator $C$ provided that $C(\\mathcal {M}) \\subseteq \\mathcal {M}$ .","Lemma 6.1 Let $\\mathcal {H}$ be a Hilbert space, $C: \\mathcal {H} \\rightarrow \\mathcal {H}$ be a contraction (i.e., $\\Vert Ch\\Vert \\le \\Vert h \\Vert , \\, \\forall h \\in \\mathcal {H}$ ), and $\\mathcal {M} \\subseteq \\mathcal {H}$ be a subspace that is invariant for $C$ .","If the restriction of $C$ to $\\mathcal {M}$ acts as a unitary $U$ on $\\mathcal {M}$ , then $\\mathcal {M}$ is also invariant for $C^*$ and the restriction of $C^*$ to $\\mathcal {M}$ is $U^*$ .","This is easiest to see using operator matrices.","Decomposing $\\mathcal {H} = \\mathcal {M} \\oplus \\mathcal {M}^{\\perp }$ we may write $C= \\begin{pmatrix} U &X \\\\Y & Z \\end{pmatrix}$ where $U: \\mathcal {M} \\rightarrow \\mathcal {M},$ $X: \\mathcal {M}^{\\perp } \\rightarrow \\mathcal {M}$ , $Y: \\mathcal {M} \\rightarrow \\mathcal {M}^{\\perp }$ , and $Z: \\mathcal {M}^{\\perp } \\rightarrow \\mathcal {M}^{\\perp }$ .","The fact that $\\mathcal {M}$ is invariant implies that $Y=0$ .","The hypothesis that the restriction of $C$ to $\\mathcal {M}$ is a unitary, means that $U$ is a unitary.","But $C^*= \\begin{pmatrix} U^* & X^*\\\\ 0 & Z^* \\end{pmatrix}$ is a contraction with $U^*$ a unitary.","This forces that for every $h \\in \\mathcal {M},$ $X^*h =0$ .","Thus, $X^*=0$ and $\\mathcal {M}$ is invariant for $C^*$ .","Note that if $U$ was a non-unitary isometry, then $U^*$ would have a kernel and we could no longer conclude that $X^*=0$ .","The next concept that we shall need is the polar decomposition of an operator.","Given an operator $A$ on a Hilbert space $\\mathcal {H}$ , we set $|A|= (A^*A)^{1/2}$ .","Let $\\mathcal {R}(A)$ denote the closure of the range of $A$ and let $\\mathcal {R}(|A|)$ denote the closure of the range of $|A|$ .","Note that for any $x \\in \\mathbb {\\mathcal {\\unknown.","}}H,$ we have $ \\Vert Ax\\Vert ^2 = \\langle Ax, Ax \\rangle = \\langle x, A^*Ax \\rangle =\\langle |A|x, |A|x \\rangle = \\Vert |A|x\\Vert ^2.$ From this equality, it follows that there is a well-defined linear isometry $W: \\mathcal {R}(|A|) \\rightarrow \\mathcal {R}(A)$ defined by setting $W(|A|h) = Ah$ , so that $A= W|A|.$ If we extend $W$ to a map $U: \\mathcal {H} \\rightarrow \\mathcal {H}$ by sitting $U= WQ$ where $Q$ is the orthogonal projection onto $\\mathcal {R}(|A|)$ then we still have $A= U|A|$ .","The map $U$ is called a partial isometry.","This representation $A=U|A|$ is called the polar decomposition of $A$ .","Lemma 6.2 Let $C$ be a contraction and let $C=U|C|$ be its polar decomposition.","If $h$ is a unit vector such that $||C^nh||=1, \\forall n \\in \\mathbb {N}$ , then the closed subspace $\\mathcal {M}$ generated by $\\lbrace C^nh: n \\ge 0 \\rbrace $ is an invariant subspace for $C$ and for any $v \\in \\mathcal {M},$ $\\Vert Cv\\Vert =\\Vert v\\Vert $ .","Clearly $\\mathcal {M}$ is invariant for $C$ .","Set $P = |C|$ so that $C=UP$ with $0 \\le P \\le I$ and $||Ph||=||UPh||= ||Ch||=1$ .","This implies that $Ph=h$ .","Thus, $Ch= Uh$ .","Since $1= ||C^2h|| = ||UPUPh|| = ||UP(Uh)|| = ||P(UH)|| $ we have that $P(Uh) = Uh$ and so $C^2h = UPUPh =UPUh=U^2h.$ Inductively, $P(U^nh) = U^nh$ so that $C^nh = U^n h$ .","Thus, $Cv= Uv$ for any $v \\in \\mathcal {M}$ .","Since $P(U^nh) = U^n h$ for all $n \\ge 0$ , we have that $\\mathcal {M} \\subseteq \\mathcal {R}(|C|)$ .","But $U$ acts isometrically on $\\mathcal {R}(|C|)$ and since $U$ and $C$ are equal on this subspace, $C$ acts isometrically on $\\mathcal {M}$ .","Theorem 6.3 Let $\\mathcal {H}_R, \\psi , U_A=(U_{ij}),$ and $U_B= (V_{kl})$ be a perfect embezzlement protocol.","If $\\mathcal {M}$ is the closed subspace of $\\mathcal {H}$ spanned by $\\lbrace U_{00}^{*n} \\psi : n \\ge 0 \\rbrace $ , then the restriction of $U_{00}^*$ to this invariant subspace is a non-unitary isometry.","Recall that the $U_{ij}$ 's must *-commute with the $V_{kl}$ 's.","The embezzlement relations tell us that $ U_{00}V_{00}\\psi = \\psi /\\sqrt{2}, \\ \\ U_{10}V_{00}\\psi =0, \\ \\ U_{10}V_{10}\\psi = \\psi /\\sqrt{2}, \\ \\ U_{00}V_{10}\\psi =0.$ The fact that $U_A$ and $U_B$ are unitaries implies that $ I= V^*_{00}V_{00}+ V^*_{10}V_{10}= U_{00}^*U_{00} + U_{10}^*U_{10} $ so that $U_{00} \\psi &= U_{00}(V_{00}^*V_{00} + V_{10}^*V_{10})\\psi = V_{00}^*U_{00}V_{00}\\psi = V_{00}^*\\psi /\\sqrt{2}, \\\\V_{00}\\psi &= V_{00}(U_{00}^*U_{00} +U_{10}^*U_{10})\\psi = U_{00}^*\\psi / \\sqrt{2}.$ Iterating, yields $(V_{00}^*)^n\\psi = (\\sqrt{2}U_{00})^n \\psi \\mbox{\\ \\ and \\ \\ } V_{00}^n \\psi = (U_{00}^*/\\sqrt{2})^n \\psi .$ Since $(1/\\sqrt{2})^n \\psi = (U_{00}V_{00})^n\\psi = U_{00}^nV_{00}^n \\psi =U_{00}^n(U_{00}^*/\\sqrt{2})^n \\psi $ , we have that $U_{00}^n U_{00}^{*n}\\psi =\\psi .$ Because $||U_{00}|| \\le 1$ we have that $||U_{00}^{*n} \\psi || =1$ for all $n$ .","Hence, by the previous lemma, $U_{00}^*$ acts isometrically on $\\mathcal {M}$ .","Now if $U_{00}^*$ acted unitarily, then by the earlier lemma, for vectors in this space $U_{00}$ would be the inverse and in particular would act isometrically on $\\mathcal {M}$ .","But then we would have that $\\Vert V_{00}^*\\psi \\Vert = \\sqrt{2} \\Vert U_{00}\\psi \\Vert = \\sqrt{2}$ .","This is impossible, because $U_B$ is a unitary and so $\\Vert V_{00}^*\\Vert \\le 1$ .","This contradiction shows that $U_{00}^*$ must be a non-unitary isometry on $\\mathcal {M}$ .","This yields the following fact.","Recall that for a perfect embezzlement protocol, we are only assuming that the operators commute, not that the resource space has a bipartite tensor structure.","Corollary 6.4 Perfect embezzlement is impossible in the commuting-operator framework if the resource space is finite-dimensional.","If $\\mathcal {H}_R$ is finite dimensional, then $\\mathcal {M}$ is also finite dimensional.","But every isometry on a finite dimensional space is necessarily a unitary, contradicting the fact that $U_{00}^*$ is a non-unitary isometry."],["Acknowledgements","We would like to thank Marius Junge, Debbie Leung, Volkher Scholz, and John Watrous for helpful discussions.","This research was supported in part by Canada's NSERC, a David R. Cheriton Scholarship, and a Mike and Ophelia Lazaridis Fellowship."],["The Schmidt and polar decompositions in infinite dimensions","In this section, for the convenience of the reader, we gather together a few useful results from operator theory that are not well known within the QIT community.","We are claiming no originality.","Definition 1.1 Let $W: H_1 \\rightarrow H_2$ .","Then $W$ is called an isometry if $\\Vert Wh_1\\Vert _2 = \\Vert h_1\\Vert _1$ for every $h_1 \\in H_1$ .","$W$ is called a coisometry iff $W^*:H_2 \\rightarrow H_1$ is an isometry.","$W$ is called a partial isometry if the restriction of $W$ to $ker(W)^{\\perp }$ is an isometry.","In this case the space $ker(W)^{\\perp }$ is called the initial space of $W$ and $ran(W)^- = ker(W^*)^{\\perp }$ is called the final space of $W$ .","Proposition 1.2 Let $H$ and $K$ be Hilbert spaces of arbitrary dimension, let $\\lbrace e_{\\alpha }: \\alpha \\in A \\rbrace $ and $\\lbrace f_{\\beta } : \\beta \\in B \\rbrace $ be o.n.","bases for $H$ and $K$ , respectively.","Then $\\lbrace e_{\\alpha } \\otimes f_{\\beta }: \\alpha \\in A, \\beta \\in B \\rbrace $ is an o.n.","basis for $H \\otimes K.$ Proposition 1.3 (The Polar Decomposition) Let $X: H_1 \\rightarrow H_2$ and let $|X|= (X^*X)^{1/2}.$ Then there is a unique partial isometry $W:H_1 \\rightarrow H_2$ with initial space $ker(X)^{\\perp }=ker(|X|)^{\\perp }= ran(|X|)^-$ and final space $ran(X)^-= ker(X^*)^{\\perp }$ such that $X= W|X|.$ To prove, one simply sets $W(|X|h) = Xh$ and shows that this is well-defined and satisfies the properties.","Note that $W$ is an isometry iff $ker(X) = (0)$ and is a coisometry iff $ran(X)^- = H_2.$ Proposition 1.4 Let $\\lbrace e_k \\rbrace $ be an o.n.","sequence in a Hilbert space, set $u_i = \\sum _k u_{i,k} e_k$ and let $U=(u_{i,j})$ .","Then $\\lbrace u_i \\rbrace $ is o.n.","iff $UU^*= I,$ i.e., $U$ is a coisometry.","Theorem 1.5 (The Infinite Dimensional Schmidt Decomposition) Let $H$ and $K$ be Hilbert spaces of arbitrary dimension and let $x \\in H \\otimes K$ .","Then there are countable orthonormal sets $ u_k \\in H$ and $v_k \\in K$ and $d_k \\ge 0$ and $d_k \\ge d_{k+1}, \\forall k$ , such that $x= \\sum _k d_k u_k \\otimes v_k,$ and so $\\Vert x\\Vert ^2= \\sum _k d_k^2.$ Moreover, if $x = \\sum _k c_k w_k \\otimes z_k$ is another such representation of $x$ , then $c_k=d_k$ for all $k$ .","Pick any orthonormal bases $\\lbrace e_{\\alpha } : \\alpha \\in A \\rbrace $ and $\\lbrace f_{\\beta }: \\beta \\in B \\rbrace $ .","By Proposition REF , we can expand $x = \\sum _{\\alpha , \\beta } z_{\\alpha , \\beta } e_{\\alpha } \\otimes f_{\\beta }$ .","We know that only countably many of the coefficients are non-zero, so we only need countably many $\\alpha $ 's and countably many $\\beta $ 's.","So we can write $x = \\sum _{i,j} z_{i,j} e_i \\otimes f_j.$ and $\\sum _{i,j} |z_{i,j}|^2 = \\Vert x\\Vert ^2$ .","Let $h_j = \\sum _i z_{i, j} e_i$ , so that $x = \\sum _j h_j\\otimes f_j$ , and let $H = \\text{span}\\lbrace h_j\\rbrace $ .","Let $\\lbrace |i\\rangle \\rbrace $ be an orthonormal basis of $H$ , and write $h_j = \\sum _{i} x_{i, j} |i\\rangle $ .","This gives us $x = \\sum _{i, j} x_{i, j}|i\\rangle \\otimes f_j$ .","Let $X = \\sum _{i, j} x_{i, j} |i\\rangle \\langle j|$ be the matrix of a map from $H$ to $H$ .","Note that $X$ is Hilbert-Schmidt and so compact and also has dense range because $\\text{span}\\lbrace |i\\rangle \\rbrace = \\text{span} \\lbrace h_j\\rbrace = \\text{span}\\lbrace \\sum _{i} x_{i, j} |i\\rangle \\rbrace $ .","By Proposition REF , performing polar decomposition on $X$ yields $X = W |X|$ where $W$ is a partial isometry, and $|X| = (X^* X)^{1/2}$ .","Since $|X|$ is compact and positive, it has an orthonormal basis of eigenvectors.","This defines a unitary $V$ such that $V |X| V^* = D$ , where $D =\\sum _{k} d_k |k\\rangle \\langle k|$ is a diagonal matrix and the $d_k$ 's are the singular values of $X$ arranged in decreasing order.","Conjugating $D$ by $V$ , we get $|X| = V^* D V$ .","Combining this with the polar decomposition, we get $X = W|X| = WV^* D V = U D V$ where $U = WV^*$ is a partial isometry.","Moreover, since $X$ has dense range, $U$ is a coisometry.","Let $U = \\sum _{i, j} u_{i,j} |i\\rangle \\langle j|$ , and $V = \\sum _{i, j} v_{i, j}|i\\rangle \\langle j|$ .","Then, $\\sum _{i,j} |x_{i,j}|^2 &=\\mbox{Tr}(X^*X)= \\mbox{Tr}(V^*D^*U^*UDV) \\\\&= \\mbox{Tr}(V^*D^2V) = \\mbox{Tr}(D^2) = \\sum _k d_k^2.$ Now let $u_k = \\sum _{i} u_{i, k} |i\\rangle $ and $v_k = \\sum _j v_{k, j} f_j$ .","Since $U$ is a coisometry, by Proposition REF , $\\lbrace u_k\\rbrace $ is an orthonormal set.","Now we have $x = \\sum _{i, j} x_{i, j} |i\\rangle \\otimes f_j = \\sum _{i, j, k} u_{i, k} d_k v_{k, j} |i\\rangle \\otimes f_j = \\sum _{k} d_k u_k\\otimes v_k.$ The statement about the uniqueness of the sequence $d_k$ follows from the fact that these numbers are the singular values of the Hilbert-Schmidt matrix $X$ and that any other choice of basis for representing $X$ would give rise to a matrix that is obtained from $X$ by pre and post multiplying by unitaries, which does not alter the singular values."],["A primer on C*-algebras","For readers unfamiliar with C*-algebras, we briefly mention the definitions and tools that we shall use.","For very readable general references we recommend [3] or [7].","Given a Hilbert space $\\mathcal {H}$ we let $B(\\mathcal {H})$ denote the set of bounded linear operators from $\\mathcal {H}$ to $\\mathcal {H}$ .","By a C*-algebra of operators we mean a subset $\\mathcal {A} \\subseteq B(\\mathcal {H})$ for some Hilbert space $\\mathcal {H}$ satisfying: $X, Y \\in \\mathcal {A}, \\lambda \\in \\mathbb {C} \\Rightarrow (\\lambda X+Y) \\in \\mathcal {A}$ and $XY \\in \\mathcal {A}$ , $X \\in \\mathcal {A} \\Rightarrow X^* \\in \\mathcal {A}$ , where $X^*$ denotes the adjoint of the operator $X$ (sometimes denoted by $X^{\\dag }$ in the physics literature), $\\mathcal {A}$ is closed in the operator norm, i.e., if $X_n \\in \\mathcal {A}$ , $X \\in B(\\mathcal {H})$ and $\\Vert X_n - X \\Vert \\rightarrow 0,$ then $X \\in \\mathcal {A}$ .","The first condition is the definition of what it means to say that $\\mathcal {A}$ is an algebra over the complex field.","The second condition is that $\\mathcal {A}$ be invariant under the taking of operator adjoints and the third is that it be a closed subset of $B(\\mathcal {H})$ in a certain topology.","C*-algebras of operators also have an abstract characterization.","An algebra $\\mathcal {A}$ over the complex numbers that is equipped with a norm $\\Vert \\cdot \\Vert $ that satisfies $\\Vert xy\\Vert \\le \\Vert x\\Vert \\cdot \\Vert y\\Vert $ is called a normed algebra.","If a normed algebra is complete, i.e., if every Cauchy sequence converges, then it is called a Banach algebra.","Given an algebra $\\mathcal {A}$ over the complex numbers, a *-map is a map from $\\mathcal {A}$ to $\\mathcal {A}$ , $^*: \\mathcal {A} \\rightarrow \\mathcal {A}$ satisfying $(x+y)^*= x^*+y^*$ , $(\\lambda x)^* = \\overline{\\lambda } x^*,$ $(xy)^*= y^*x^*$ and $(x^*)^*=x$ .","An algebra equipped with a *-map is called a *-algebra.","A map $\\pi : \\mathcal {A} \\rightarrow \\mathcal {B}$ between two algebras that is linear and satisfies $\\pi (xy) = \\pi (x) \\pi (y)$ is called a homomorphism.","If both algebras are also *-algebras and the map satisfies $\\pi (x^*) = \\pi (x)^*$ then $\\pi $ is called a *-homomorphism.","Finally, an (abstract) C*-algebra $\\mathcal {A}$ is a Banach *-algebra that satisfies $\\Vert x^*x\\Vert = \\Vert x\\Vert ^2, \\, \\forall x \\in \\mathcal {A}$ .","Note that $B(\\mathcal {H})$ is an abstract C*-algebra and so is every C*-algebra of operators.","The celebrated Gelfand-Naimark-Segal theorem shows that every abstract C*-algebra is in an appropriate sense a C*-algebra of operators.","Theorem 2.1 (Gelfand-Naimark-Segal) Let $\\mathcal {A}$ be an abstract C*-algebra.","Then there is a Hilbert space $\\mathcal {H}$ and a map $\\pi : \\mathcal {A} \\rightarrow B(\\mathcal {H})$ such that: $\\pi $ is a *-homomorphism, $\\Vert \\pi (x)\\Vert = \\Vert x\\Vert $ for all $x \\in \\mathcal {A}$ .","Moreover, if $\\mathcal {A}$ has a unit element $1 \\in \\mathcal {A}$ then, in addition, one can arrange that $\\pi (1) = I_{\\mathcal {H}}$ .","A map satisfying the second condition is called an isometry.","Clearly, an isometry is one-to-one.","Conversely, it is a theorem that every one-to-one *-homomorphism is an isometry.","A one-to-one, onto *-homomorphism is called a *-isomorphism.","The two conditions in the above theorem also guarantee that the range of $\\pi ,$ $\\mathcal {B}= \\pi (\\mathcal {A})$ is a C*-algebra of operators.","Thus, $\\pi $ is a *-isomorphism from the abstract C*-algebra onto a C*-algebra of operators.","A key element of the proof of the above theorem is their theorem on representations of states.","A state on an abstract unital C*-algebra $\\mathcal {A}$ is any linear functional, $s: \\mathcal {A} \\rightarrow \\mathbb {C}$ such that $s(1) =1$ and $s(x^*x) \\ge 0$ for every $x \\in \\mathcal {A}$ .","Theorem 2.2 (GNS state representation theorem) Let $s:\\mathcal {A} \\rightarrow \\mathbb {C}$ be a state on a unital C*-algebra.","Then there exists a Hilbert space $\\mathcal {H}_s$ , a unit vector $\\xi \\in \\mathcal {H}_s$ and a unital *-homomorphism, $\\pi _s: \\mathcal {A} \\rightarrow B(\\mathcal {H}_s)$ such that $s(x) = \\langle \\xi | \\pi _s(x) \\xi \\rangle $ for all $x \\in \\mathcal {A}$ and such that the subspace of vectors of the form $\\lbrace \\pi _s(x) \\xi : x \\in \\mathcal {A} \\rbrace $ is dense in $\\mathcal {H}_s$ .","Thus, the theorem says that for each abstract state there is a way to realize the C*-algebra as a C*-algebra of operators such that the state becomes a vector state."],["The State Space","Given a unital C*-algebra $\\mathcal {A}$ , the set of all states on $\\mathcal {A}$ , denoted $S(\\mathcal {A})$ is a convex set.","Moreover, it is endowed with a topology, called the weak*-topology and in this topology it is a compact set.","A net of states $\\lbrace s_{\\lambda } \\rbrace $ converges to a state $s$ in this topology if and only if $\\lim _{\\lambda } | s_{\\lambda }(a) - s(a)| =0$ for every $a \\in \\mathcal {A}$ ."],["Tensor Products of C*-algebras","Let $\\mathcal {A}$ and $\\mathcal {B}$ be two unital C*-algebras, and let $\\mathcal {A}\\otimes \\mathcal {B}$ be their algebraic tensor product.","Given $x= \\sum _i a_i \\otimes b_i$ and $y = \\sum _j c_j \\otimes d_j$ in $\\mathcal {A} \\otimes \\mathcal {B}$ we define their product by $ xy = \\sum _{i,j} a_i c_j \\otimes b_i \\otimes d_j,$ and a *-map by $x^* = \\sum _i a_i^* \\otimes b_i^*.$ Endowed with these two operations, $\\mathcal {A} \\otimes \\mathcal {B}$ becomes a *-algebra.","Note that the *-subalgebra $\\lbrace a \\otimes 1: a \\in \\mathcal {A} \\rbrace $ can be identified with $\\mathcal {A}$ and similarly, $\\lbrace 1 \\otimes b: b \\in \\mathcal {B} \\rbrace $ can be identified with $\\mathcal {B}$ .","Also $(a \\otimes 1)(1 \\otimes b) = a \\otimes b = (1 \\otimes b)(1 \\otimes a)$ so that these “copies\" of $\\mathcal {A}$ and $\\mathcal {B}$ commute.","There are two important ways to give this *-algebra a norm so that it can be completed to become a C*-algebra.","Given $x \\in \\mathcal {A} \\otimes \\mathcal {B}$ we set $\\Vert x\\Vert _{\\max } = \\sup \\lbrace \\Vert \\pi (x) \\Vert : \\ \\pi : \\mathcal {A} \\otimes \\mathcal {B} \\rightarrow B(\\mathcal {H}) \\text{ is a unital *-homomorphism}\\rbrace , $ where the supremum is taken over all Hilbert spaces $\\mathcal {H}$ and all unital *-homomorphisms.","The completion of $\\mathcal {A} \\otimes \\mathcal {B}$ in this norm is a C*-algebra denoted $\\mathcal {A} \\otimes _{\\max } \\mathcal {B}$ .","Alternatively, if $\\pi _1: \\mathcal {A} \\rightarrow B(\\mathcal {H}_1)$ and $\\pi _2: \\mathcal {B} \\rightarrow B(\\mathcal {H}_2)$ are unital *-homomorphisms, then setting $\\pi (a \\otimes b) = \\pi _1(a) \\otimes \\pi _2(b) \\in B(\\mathcal {H}_1 \\otimes \\mathcal {H}_2)$ and extending linearly, defines a unital *-homomorphism from $\\mathcal {A} \\otimes \\mathcal {B}$ into $B(\\mathcal {H}_1 \\otimes \\mathcal {H}_2)$ denoted by $\\pi = \\pi _1 \\otimes \\pi _2$ .","Given $x \\in \\mathcal {A} \\otimes \\mathcal {B}$ we set $ \\Vert x\\Vert _{\\min } = \\sup \\lbrace \\Vert \\pi _1 \\otimes \\pi _2(x) \\Vert : \\,\\, \\pi _1: \\mathcal {A} \\rightarrow B(\\mathcal {H}_1), \\, \\, \\pi _2: \\mathcal {B} \\rightarrow B(\\mathcal {H}_2) \\\\ \\text{ are unital *-homomorphisms}\\rbrace .$ The completion of $\\mathcal {A} \\otimes \\mathcal {B}$ in this norm is a C*-algebra denoted $\\mathcal {A} \\otimes _{\\min } \\mathcal {B}$ ."]],"string":"[\n [\n \"Perfect Embezzlement of Entanglement\"\n ],\n [\n \"Abstract Van Dam and Hayden introduced a concept commonly referred to as embezzlement, where, for any entangled quantum state $\\\\phi$, there is an entangled catalyst state $\\\\psi$, from which a high fidelity approximation of $\\\\phi \\\\otimes \\\\psi$ can be produced using only local operations.\",\n \"We investigate a version of this where the embezzlement is perfect (i.e., the fidelity is 1).\",\n \"We prove that perfect embezzlement is impossible in a tensor product framework, even with infinite-dimensional Hilbert spaces and infinite entanglement entropy.\",\n \"Then we prove that perfect embezzlement is possible in a commuting operator framework.\",\n \"We prove this using the theory of C*-algebras and we also provide an explicit construction.\",\n \"Next, we apply our results to analyze perfect versions of a nonlocal game introduced by Regev and Vidick.\",\n \"Finally, we analyze the structure of perfect embezzlement protocols in the commuting operator model, showing that they require infinite-dimensional Hilbert spaces.\"\n ],\n [\n \"Introduction\",\n \"It is well known that an entangled quantum state cannot be produced by local operations alone.\",\n \"Van Dam and Hayden [2] proposed a method that, in a certain sense, appears to produce additional entanglement by local operations.\",\n \"They showed that, for any entangled state $\\\\phi $ and $\\\\epsilon > 0$ , starting with a special entangled catalyst state $\\\\psi $ , applying local operations, can produce a state that approximates $\\\\phi \\\\otimes \\\\psi $ within fidelity $1-\\\\epsilon $ .\",\n \"Although the entanglement entropy of the state produced cannot exceed that of $\\\\psi $ , when $\\\\epsilon $ is small, it is difficult to distinguish between the state produced and $\\\\phi \\\\otimes \\\\psi $ .\",\n \"The name embezzlement reflects the fact that the protocol “steals\\\" entanglement from $\\\\psi $ in order to produce entanglement elsewhere, but in a manner that is difficult to detect.\",\n \"In the method of [2], fidelity $1-\\\\epsilon $ can be attained for any $\\\\epsilon > 0$ , using a catalyst $\\\\psi $ with entanglement entropy $O(\\\\log (1/\\\\epsilon ))$ .\",\n \"Moreover, it is shown in [2] that the entanglement entropy of the catalyst must be $\\\\Omega (\\\\log (1/\\\\epsilon ))$ to attain this fidelity.\",\n \"Thus, high fidelity embezzlement requires a large amount of entanglement to begin with.\",\n \"We consider the question: what kinds of embezzlement are possible when the amount of entanglement in $\\\\psi $ is allowed to be infinite?\",\n \"The aforementioned results do not rule out perfect (i.e., fidelity 1) embezzlement in such cases.\",\n \"On the other hand, the catalytic states $\\\\psi _{\\\\epsilon }$ in [2] do not converge to a valid quantum vector state as $\\\\epsilon $ approaches 0.\",\n \"This question provides a setting in which the consequences of notions of infinite entanglement can be explored.\",\n \"We first show that in the tensor product framework, where catalytic states are in the tensor product of two Hilbert spaces, perfect embezzlement is impossible, even if the spaces are infinite dimensional and the entanglement entropy is infinite.\",\n \"Next, we consider a commuting operator framework, where the notion of “local\\\" is formalized differently: there is one joint Hilbert space, accessible to both Alice and Bob; however, the operations that Alice performs on this space must commute with those of Bob.\",\n \"This formalism is used in quantum field theory (see [12], [9], [11], [4], [6], [5] for more discussion about this framework and its relationship with the tensor product framework).\",\n \"A natural adaptation of the commuting operator framework to the setting of embezzlement is the following.\",\n \"The catalytic state $\\\\psi $ is in a jointly accessibe Hilbert space, that we refer to as the resource space $\\\\mathcal {R}$ .\",\n \"There are also two additional Hilbert spaces: $\\\\mathcal {H}_A$ , accessible to Alice only; and $\\\\mathcal {H}_B$ , accessible to Bob only.\",\n \"The goal of the protocol is to transform a product state to an entangled state in $\\\\mathcal {H}_A \\\\otimes \\\\mathcal {H}_B$ while using $\\\\psi $ catalytically, and using operators that are commuting in the following sense.\",\n \"Alice can apply a unitary operator on $\\\\mathcal {H}_A \\\\otimes \\\\mathcal {R}$ and Bob can apply a unitary operator on $\\\\mathcal {H}_B \\\\otimes \\\\mathcal {R}$ ; however, $U_A \\\\otimes I_{\\\\mathcal {H}_B}$ and $I_{\\\\mathcal {H}_A} \\\\otimes U_B$ must commute on $\\\\mathcal {H}_A \\\\otimes \\\\mathcal {R}\\\\otimes \\\\mathcal {H}_B$ , as illustrated in Figure REF .\",\n \"Figure: Commuting operator framework as a circuit diagram.We focus on the problem of embezzling a Bell state of the form ${\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|0\\\\rangle \\\\otimes |0\\\\rangle +{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|1\\\\rangle \\\\otimes |1\\\\rangle $ (though our methodology adapts to more general states).\",\n \"In this case, $\\\\mathcal {H}_A = \\\\mathcal {H}_B = \\\\mathbb {C}^{2}$ .\",\n \"A perfect embezzlement protocol consists of a resource space $\\\\mathcal {R}$ , a catalytic state $\\\\psi \\\\in \\\\mathcal {\\\\mathcal {R}}$ , and commuting unitary operators $U_A$ and $U_B$ , such that $(U_A \\\\otimes I_{\\\\mathcal {H}_B})&(I_{\\\\mathcal {H}_A} \\\\otimes U_B) |0\\\\rangle \\\\otimes \\\\psi \\\\otimes |0\\\\rangle \\\\\\\\&= \\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}|0\\\\rangle \\\\otimes \\\\psi \\\\otimes |0\\\\rangle +\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}|1\\\\rangle \\\\otimes \\\\psi \\\\otimes |1\\\\rangle .\",\n \"\\\\nonumber $ We show that, in this commuting operator framework, a perfect embezzlement protocol exists, where the resource space is a countably infinite dimensional (i.e., separable) Hilbert space.\",\n \"We show this in two ways: one is a simple existence proof, based on the theory of C*-algebras, which does not yield explicit unitary operations; the other is by an explicit construction.\",\n \"Next, we consider coherent embezzlement, which was introduced in [10] (where it is referred to as $T_2$ ) and is a refinement of coherent state exchange, introduced in [8].\",\n \"Coherent embezzlement is related to embezzlement but has the property that it is operationally testable in a sense similar to that of nonlocal games (whereas embezzlement itself does not have this property).\",\n \"We give reductions between perfect embezzlement and perfect coherent embezzlement to prove that perfect coherent embezzlement is impossible in the tensor product framework; whereas it is possible in the commuting operator framework.\",\n \"Finally, we prove a theorem concerning the structure of pairs of unitaries that achieve perfect embezzlement in terms of properties of their constituent operators.\",\n \"We show that at least one of these operators must contain a non-unitary isometry, a term that we will define later.\",\n \"Since non-unitary isometries do not exist in finite dimensions, this implies that perfect embezzlement in the commuting-operator model cannot be achieved with a finite dimensional resource space.\"\n ],\n [\n \"Perfect embezzlement is impossible in a tensor product framework\",\n \"In [2], it is proved that, for any protocol that embezzles within fidelity $1 - \\\\epsilon $ , the entanglement entropy of the catalyst must be $\\\\Omega (\\\\log (1/\\\\epsilon ))$ .\",\n \"It follows that perfect embezzlement is impossible with finite-dimensional entanglement in the tensor product framework.\",\n \"Here, we extend this impossibility result to tensor products of arbitrary Hilbert spaces (where the dimension of the spaces and entanglement entropy can be infinite).\",\n \"In the tensor product framework, the resource space is of the form $\\\\mathcal {R} = \\\\mathcal {R}_A \\\\otimes \\\\mathcal {R}_B$ , where $\\\\mathcal {R}_A$ and $\\\\mathcal {R}_B$ are arbitrary Hilbert spaces.\",\n \"Alice has access to $\\\\mathcal {H}_A \\\\otimes \\\\mathcal {R}_A$ and Bob has access to $\\\\mathcal {H}_B \\\\otimes \\\\mathcal {R}_B$ .\",\n \"Alice and Bob can each apply any unitary operation to the registers that they have access to, as illustrated in Figure REF (left), where the input state is $|0\\\\rangle \\\\otimes \\\\psi \\\\otimes |0\\\\rangle $ , for some state $\\\\psi \\\\in \\\\mathcal {R}_A \\\\otimes \\\\mathcal {R}_B$ .\",\n \"The protocol performs perfect embezzlement if its output state is $\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}|0\\\\rangle \\\\otimes \\\\psi \\\\otimes |0\\\\rangle +\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}|1\\\\rangle \\\\otimes \\\\psi \\\\otimes |1\\\\rangle $ .\",\n \"We also define a potentially stronger model, that we refer to as embezzlement with ancillas, which includes the possibility of Alice and Bob employing additional registers as part of their protocol, as illustrated in Figure REF (right).\",\n \"Figure: Circuit diagram for embezzlement (left) and embezzlement with ancillas (right) in the tensor product framework.Registers ℛ A \\\\mathcal {R}_A and ℛ B \\\\mathcal {R}_B contain a bipartite resource state that must be used catalytically.Registers ℋ A =ℋ B =ℂ 2 \\\\mathcal {H}_A = \\\\mathcal {H}_B = \\\\mathbb {C}^{2} are intialized to state |00〉|00\\\\rangle and are transformed to state 1 2|00〉+1 2|11〉{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|00\\\\rangle +{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|11\\\\rangle .Registers 𝒢 A \\\\mathcal {G}_A and 𝒢 B \\\\mathcal {G}_B are ancillas, whose initial state is unentangled, but they need not be used catalytically.The input to the circuit is of the form $\\\\gamma _A \\\\otimes |0\\\\rangle \\\\otimes \\\\psi \\\\otimes |0\\\\rangle \\\\otimes \\\\gamma _B$ , where $\\\\psi \\\\in \\\\mathcal {R}_A \\\\otimes \\\\mathcal {R}_B$ is the catalyst state, and $\\\\gamma _A \\\\in \\\\mathcal {G}_A$ and $\\\\gamma _B \\\\in \\\\mathcal {G}_B$ are the initial states of Alice and Bob's respective ancilla registers, $\\\\mathcal {G}_A$ and $\\\\mathcal {G}_B$ (which can be infinite dimensional).\",\n \"If we express the Hilbert space as $(\\\\mathcal {H}_A \\\\otimes \\\\mathcal {H}_B) \\\\otimes (\\\\mathcal {R}_A \\\\otimes \\\\mathcal {R}_B) \\\\otimes (\\\\mathcal {G}_A \\\\otimes \\\\mathcal {G}_B)$ then the input state can be written as $|00\\\\rangle \\\\otimes \\\\psi \\\\otimes (\\\\gamma _A \\\\otimes \\\\gamma _B)$ .\",\n \"The protocol performs perfect embezzlement if and only if the output state is of the form $\\\\bigl ({\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|00\\\\rangle +{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|11\\\\rangle \\\\bigr )\\\\otimes \\\\psi \\\\otimes \\\\gamma _{AB},$ for some state $\\\\gamma _{AB} \\\\in \\\\mathcal {G}_A \\\\otimes \\\\mathcal {G}_B$ .\",\n \"Theorem 2.1 Perfect embezzlement is impossible in the tensor product framework, even if Alice and Bob are allowed to use ancillas.\",\n \"The proof is a straightforward application of the Schmidt decomposition for vectors in tensor products of arbitrary Hilbert spaces.\",\n \"For arbitrary (not necessarily separable) Hilbert spaces $\\\\mathcal {H}$ and $\\\\mathcal {K}$ and any $\\\\phi \\\\in \\\\mathcal {H} \\\\otimes \\\\mathcal {K}$ , it is possible to express $\\\\phi = \\\\sum _{j = 0}^{\\\\infty } \\\\alpha _j\\\\, u_j\\\\otimes v_j,$ where $\\\\alpha _j \\\\ge 0$ , $\\\\sum _{j = 0}^{\\\\infty } |\\\\alpha _j|^2 = 1$ , $ \\\\alpha _j \\\\ge \\\\alpha _{j+1}$ , $u_0, u_1, \\\\dots $ are orthonormal vectors in $\\\\mathcal {H}$ , and $v_0, v_1, \\\\dots $ are orthonormal vectors in $\\\\mathcal {K}$ .\",\n \"Moreover, given these conditions, the coefficients $\\\\alpha _j$ are unique.\",\n \"For the convenience of the reader, we include a proof of this in Appendix .\",\n \"Now taking a Schmidt decomposition of $\\\\gamma _A \\\\otimes |0\\\\rangle \\\\otimes \\\\psi \\\\otimes |0\\\\rangle \\\\otimes \\\\gamma _B$ , with respect to $\\\\big (\\\\mathcal {G}_A \\\\otimes \\\\mathcal {H}_A \\\\otimes \\\\mathcal {R}_A \\\\big ) \\\\otimes \\\\big (\\\\mathcal {G}_B \\\\otimes \\\\mathcal {H}_B \\\\otimes \\\\mathcal {R}_B \\\\big )$ , we obtain Schmidt coefficients $\\\\alpha _j$ .\",\n \"Suppose that a perfect embezzlement protocol exists.\",\n \"Then, since $U_A$ and $U_B$ are local unitaries, the Schmidt coefficients of the initial state $|00\\\\rangle \\\\otimes \\\\psi \\\\otimes \\\\gamma _A \\\\otimes \\\\gamma _B$ must be the same as those of the final state $({\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|00\\\\rangle + {\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|11\\\\rangle )\\\\otimes \\\\psi \\\\otimes \\\\gamma _{AB}$ .\",\n \"But this is a contradiction, since the largest Schmidt coefficient of the input state is $\\\\alpha _0$ (which is nonzero) and the largest Schmidt coefficient of the output state is at most ${\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}\\\\alpha _0$ .\",\n \"Therefore, there is no perfect embezzlement protocol in the tensor product framework.\"\n ],\n [\n \"Perfect embezzlement is possible in a commuting operator framework\",\n \"In this section we show that, since one can approximately embezzle a Bell state to any level of precision in finite dimensions (by the results of [2]), one can perfectly embezzle in infinite dimensions in the commuting operator framework.\",\n \"Readers unfamiliar with the theory of C*-algebras might prefer to read our primer on C*-algebras in Appendix before tackling this section.\",\n \"At the end of the section, we explain how to generalize the technique to more general entangled states.\",\n \"We begin by showing that each commuting operator framework, where $\\\\mathcal {H}_A = \\\\mathcal {H}_B = \\\\mathbb {C}^2$ , yields a set of eight operators on the resource space.\",\n \"To study the most general commuting framework, it is natural to consider the relations that any such set of operators must satisfy and look for a “universal\\\" model for such sets of operators.\",\n \"We will show that the eight operators arising from a commuting operator framework are always a representation of a certain C*-algebra and that the catalyst vector yields a state on this C*-algebra.\",\n \"We will show that the commuting operator framework together with the catalyst vector achieves perfect embezzlement of a Bell state if and only if the state on this C*-algebra induced by the catalyst vector satisfies a set of four equations.\",\n \"In this manner the question of whether or not one can perfectly embezzle a Bell state is reduced to a question about the existence of a state on this C*-algebra that satisfies our four equations.\",\n \"Finally, we show that perfect embezzlement of a Bell state is possible in the commuting operator framework by showing the existence of such a state.\",\n \"The “universal\\\" C*-algebra that one needs was first introduced by L.G.\",\n \"Brown [1], who referred to it as the universal C*-algebra of a non-commutative unitary for reasons that will, hopefully, be clear.\",\n \"Our viewpoint shows that in a certain sense questions about embezzlement can be interpreted as questions about states on these particular quantum group C*-algebras.\",\n \"We think that this perspective is new and should lead to interesting links between these two areas.\",\n \"Let's return to the scenario of Figure REF .\",\n \"Alice's unitary operation, $U_A: \\\\mathbb {C}^2 \\\\otimes \\\\mathcal {R} \\\\rightarrow \\\\mathbb {C}^2 \\\\otimes \\\\mathcal {R}$ can be represented by a $2 \\\\times 2$ matrix of operators on $\\\\mathcal {R}$ , $U_A= \\\\begin{pmatrix} U_{00} & U_{01} \\\\\\\\ U_{10} & U_{11} \\\\end{pmatrix}= \\\\sum _{i,j=0}^1 |i\\\\rangle \\\\langle j| \\\\otimes U_{ij} $ where $U_A( |j\\\\rangle \\\\otimes h) = \\\\sum _{i=0}^1 |i\\\\rangle \\\\otimes U_{ij} h$ .\",\n \"In this case, $ U_A^* = \\\\begin{pmatrix} U_{00}^* & U_{10}^* \\\\\\\\ U_{01}^* & U_{11}^* \\\\end{pmatrix} $ and the fact that $U_A$ is unitary can be expressed by eight equations involving these operators that are best expressed as $U_A^*U_A = \\\\begin{pmatrix} I_{\\\\mathcal {R}} & 0\\\\\\\\ 0 & I_{\\\\mathcal {R}} \\\\end{pmatrix} = U_AU_A^*,$ where we apply the usual rules of matrix multiplication, being careful to remember that since the entries of $U_A$ are operators, not numbers, they need not commute.\",\n \"We also recall that when $\\\\mathcal {R}$ is infinite dimensional, then it is necessary that both $U_A^*U_A$ and $U_AU_A^*$ be the identity to guarantee that $U_A$ is unitary.\",\n \"Finally, in the special case that $\\\\dim (\\\\mathcal {R}) =1$ so that these entries are numbers, then we are back to the usual case of a $2 \\\\times 2$ complex unitary matrix.\",\n \"Conversely, if we let $U_A$ be any $2 \\\\times 2$ matrix of operators on $\\\\mathcal {R}$ that satisfies Eq.\",\n \"(REF ) then $U_A$ will define a unitary on $\\\\mathbb {C}^2 \\\\otimes \\\\mathcal {R}$ .\",\n \"Similarly, Bob's unitary $U_B: \\\\mathcal {R} \\\\otimes \\\\mathbb {C}^2 \\\\rightarrow \\\\mathcal {R} \\\\otimes \\\\mathbb {C}^2$ is represented by a $2 \\\\times 2$ matrix of operators on $\\\\mathcal {R}$ , $U_B= ( V_{ij})$ , whose entries satisfy the same eight equations.\",\n \"Finally, to have a commuting operator framework as in Figure REF , we need $(U_A \\\\otimes I_2)(I_2 \\\\otimes U_B) = (I_2 \\\\otimes U_B)(U_A \\\\otimes I_2)$ .\",\n \"The following proposition translates this condition into equations involving the operator entries.\",\n \"Proposition 3.1 Let $U_{ij}, V_{kl}, 0 \\\\le i,j,k,l \\\\le 1$ be operators on the Hilbert space $\\\\mathcal {R}$ such that $U_A=( U_{ij}),$ and $U_B= (V_{kl})$ are unitaries.\",\n \"Then $(U_A \\\\otimes I_2)(I_2 \\\\otimes U_B)= (I_2 \\\\otimes U_B)(I_2 \\\\otimes U_A)$ if and only if $U_{ij}V_{kl} = V_{kl}U_{ij}$ and $U_{ij}^*V_{kl} = V_{kl}U_{ij}^*$ for all $i,j,k,l$ .\",\n \"We have that $(U_A \\\\otimes I_2)(I_2 \\\\otimes U_B)( |j\\\\rangle \\\\otimes h \\\\otimes |l\\\\rangle ) = \\\\sum _{i,k=0}^1 |i\\\\rangle \\\\otimes U_{ij}V_{kl}h \\\\otimes |k\\\\rangle ,$ and similarly, $(I_2 \\\\otimes U_B)(U_A \\\\otimes I_2) ( |j\\\\rangle \\\\otimes h \\\\otimes |l\\\\rangle ) = \\\\sum _{i,k=0}^1 |i\\\\rangle \\\\otimes V_{kl}U_{ij}h \\\\otimes |k\\\\rangle .$ Thus, we see that $(U_A \\\\otimes I_2)(I_2 \\\\otimes U_B) = (I_2 \\\\otimes U_B)(U_A \\\\otimes I_2)$ is equivalent to $U_{ij}V_{kl} = V_{kl}U_{ij}$ , for all $i,j,k,l.$ However, if an invertible operator commutes with another operator, then its inverse also commutes with that operator.\",\n \"Hence, $(U_A \\\\otimes I_2)^{-1} = (U_A^* \\\\otimes I_2)$ commutes with $(I_2 \\\\otimes U_B)$ and this is equivalent to $U_{ij}^*V_{kl}= V_{kl}U_{ij}^*$ , for all $i,j,k,l.$ The above equations are generally summarized by saying that the set of operators $\\\\lbrace U_{ij} \\\\rbrace $ *-commutes with the set $\\\\lbrace V_{kl} \\\\rbrace $ .\",\n \"Thus, having a commuting operator framework is equivalent to having two unitaries $U_A=(U_{ij})$ , and $U_B=(V_{kl})$ whose entries *-commute.\",\n \"We wish to study “universal\\\" properties of $2 \\\\times 2$ matrices of operators $(U_{ij})$ that give rise to a unitary.\",\n \"To do this we begin with a unital *-algebra $\\\\mathcal {U}_2$ with generators, denoted 1 and $u_{ij}, \\\\, 0 \\\\le i,j \\\\le 1$ , subject to the eight equations, $\\\\begin{pmatrix} u_{00} & u_{01} \\\\\\\\ u_{10} & u_{11} \\\\end{pmatrix}\\\\begin{pmatrix} u_{00}^* & u_{10}^* \\\\\\\\ u_{01}^* & u_{11}^* \\\\end{pmatrix}=\\\\begin{pmatrix} u_{00}^* & u_{10}^* \\\\\\\\ u_{01}^* & u_{11}^* \\\\end{pmatrix}\\\\begin{pmatrix} u_{00} & u_{01} \\\\\\\\ u_{10} & u_{11} \\\\end{pmatrix}= \\\\begin{pmatrix} 1 & 0 \\\\\\\\ 0 & 1 \\\\end{pmatrix}.$ Thus, whenever there is a Hilbert space $\\\\mathcal {H}$ and four operators, $U_{ij}$ on that space such that the operator-matrix $U=(U_{ij})$ defines a unitary operator on $\\\\mathcal {H} \\\\otimes \\\\mathbb {C}^2$ , then there is a *-homomorphism, $ \\\\pi : \\\\mathcal {U}_2 \\\\rightarrow B(\\\\mathcal {H}) \\\\text{ with } \\\\pi (u_{ij}) = U_{ij}.$ For $x \\\\in \\\\mathcal {U}_2$ he sets $\\\\Vert x\\\\Vert = \\\\sup \\\\lbrace \\\\Vert \\\\pi (x)\\\\Vert \\\\colon \\\\pi \\\\text{ a *-homomorphism} \\\\rbrace $ , where the supremum is taken over all Hilbert spaces and all $\\\\pi $ 's as above.\",\n \"This defines a norm on $\\\\mathcal {U}_2$ and that the completion is a C*-algebra, we shall denote $U_{\\\\operatorname{nc}}(2)$ .\",\n \"The subscript $\\\\operatorname{nc}$ stands for “non-commuting\\\" and is intended to remind us that the generators $u_{ij}$ do not commute.\",\n \"(This approach generalizes naturally to $d \\\\times d$ matrices of operators, for $d > 2$ , where the C*-algebra is denoted as $U_{\\\\operatorname{nc}}(d)$ .)\",\n \"Note that in the commuting operator framework, the set $U_{ij}$ and the set $V_{kl}$ each gives rise to a *-homomorphism and that these two *-homomorphisms commute.\",\n \"Thus, it is not hard to see that we have a one-to-one correspondence between commuting operator frameworks and *-homomorphisms of $U_{\\\\operatorname{nc}}(2) \\\\otimes U_{\\\\operatorname{nc}}(2)$ into $B(\\\\mathcal {R})$ .\",\n \"Since we want to consider all commuting operator frameworks, we are lead to study $U_{\\\\operatorname{nc}}(2) \\\\otimes _{\\\\max } U_{\\\\operatorname{nc}}(2)$ .\",\n \"The study of states on this algebra turns out to be closely related to embezzlement constructions as the following result shows.\",\n \"Theorem 3.2 There exists a perfect embezzlement protocol in the commuting operator framework if and only if there exists a state $s: U_{\\\\operatorname{nc}}(2) \\\\otimes _{\\\\max } U_{\\\\operatorname{nc}}(2) \\\\rightarrow \\\\mathbb {C}$ such that $s(u_{00} \\\\otimes u_{00}) = \\\\frac{1}{\\\\sqrt{2}}$ , $s(u_{10} \\\\otimes u_{00}) = \\\\,0$ , $s(u_{00} \\\\otimes u_{10}) = \\\\,0$ , $s(u_{10} \\\\otimes u_{10}) = \\\\frac{1}{\\\\sqrt{2}}$ .\",\n \"First assume that a perfect embezzlement protocol exists in a commuting operator framework.\",\n \"Let $\\\\mathcal {R}$ be a Hilbert space, let $\\\\psi \\\\in \\\\mathcal {R}$ be a unit vector, let $U_A= \\\\big ( U_{ij} \\\\big )$ and $U_B= \\\\big ( V_{kl} \\\\big )$ be unitaries on $\\\\mathbb {C}^2 \\\\otimes \\\\mathcal {H}$ and $\\\\mathcal {H} \\\\otimes \\\\mathbb {C}^2$ , respectively, such that $U_A \\\\otimes I_2$ commutes with $I_2 \\\\otimes U_B$ and let $\\\\psi $ be a catalyst vector for perfect embezzlement of a Bell state, Define $\\\\pi : U_{\\\\operatorname{nc}}(2) \\\\otimes _{\\\\max } U_{\\\\operatorname{nc}}(2) \\\\rightarrow B(\\\\mathcal {H})$ to be the *-homomorphism defined by $\\\\pi (u_{ij} \\\\otimes 1) = U_{i,j}, \\\\, \\\\pi (1 \\\\otimes u_{kl}) = V_{kl}$ .\",\n \"Since $\\\\psi $ is a catalyst vector, $ (U_A \\\\otimes I_2)(I_2 \\\\otimes U_B) ( |0\\\\rangle \\\\otimes \\\\psi \\\\otimes |0\\\\rangle ) = \\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}( |0\\\\rangle \\\\otimes \\\\psi \\\\otimes |0\\\\rangle + |1\\\\rangle \\\\otimes \\\\psi \\\\otimes |1\\\\rangle ).\",\n \"$ Now define a state on $U_{\\\\operatorname{nc}}(2) \\\\otimes _{\\\\max } U_{\\\\operatorname{nc}}(2)$ by $s(x) = \\\\langle \\\\pi (x) \\\\psi , \\\\psi \\\\rangle $ .\",\n \"We have that $\\\\big ( I_2 \\\\otimes U_B\\\\big ) \\\\big ( U_A \\\\otimes I_2 \\\\big ) |0\\\\rangle \\\\otimes \\\\psi \\\\otimes |0\\\\rangle & = \\\\sum _{i,j=0}^1 |i\\\\rangle \\\\otimes U_{i,0}V_{j,0} \\\\psi \\\\otimes |j\\\\rangle \\\\\\\\ & = \\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}|0\\\\rangle \\\\otimes \\\\psi \\\\otimes |0\\\\rangle + \\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}|1\\\\rangle \\\\otimes \\\\psi \\\\otimes |1\\\\rangle ,$ which is equivalent to $ U_{00}V_{00} \\\\psi = U_{10}V_{10} \\\\psi = \\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}\\\\psi \\\\text{\\\\ \\\\ and \\\\ \\\\ } U_{00} V_{10} \\\\psi = U_{10}V_{00} \\\\psi = 0 .$ From these equations, it follows that the state $s$ satisfies the four conditions.\",\n \"Conversely, assume that $s: U_{\\\\operatorname{nc}}(2) \\\\otimes _{\\\\max } U_{\\\\operatorname{nc}}(2) \\\\rightarrow \\\\mathbb {C}$ is a state that satisfies the 4 conditions.\",\n \"Let $\\\\pi _s: U_{\\\\operatorname{nc}}(2) \\\\otimes _{\\\\max } U_{\\\\operatorname{nc}}(2) \\\\rightarrow B(\\\\mathcal {H}_s)$ and $\\\\psi \\\\in \\\\mathcal {H}_s$ be the GNS representation of the state so that $s(x) = \\\\langle \\\\pi _s(x) \\\\psi , \\\\psi \\\\rangle $ .\",\n \"If we define $U_A: \\\\mathbb {C}^2 \\\\otimes \\\\mathcal {H} \\\\rightarrow \\\\mathbb {C}^2 \\\\otimes \\\\mathcal {H}$ by $U_A= \\\\big ( \\\\pi (u_{ij} \\\\otimes 1) \\\\big )$ and $U_B: \\\\mathcal {H} \\\\otimes \\\\mathbb {C}^2 \\\\rightarrow \\\\mathcal {H} \\\\otimes \\\\mathbb {C}^2$ by $U_B = \\\\big ( \\\\pi (1 \\\\otimes u_{i,j}) \\\\big )$ , then $U_A \\\\otimes I_2$ commutes with $I_2 \\\\otimes U_B$ .\",\n \"The operator on the direct sum of four copies of $\\\\mathcal {H}$ given by $ \\\\begin{pmatrix} U_{00}V_{00} & U_{01}V_{00} & U_{00}V_{01} & U_{01}V_{01} \\\\\\\\ U_{10}V_{00} & U_{11}V_{00} & U_{10}V_{01} & U_{11}V_{01}\\\\\\\\ U_{00} V_{10} & U_{01}V_{10} & U_{00}V_{11} & U_{01}V_{11}\\\\\\\\ U_{10}V_{10} & U_{11}V_{10} & U_{10} V_{11} & U_{11}V_{11} \\\\end{pmatrix} $ is unitary.\",\n \"Hence, $ 1 &= |\\\\langle U_{00}V_{00} \\\\psi , \\\\psi \\\\rangle |^2 + |\\\\langle U_{10}V_{10} \\\\psi , \\\\psi \\\\rangle |^2 \\\\le \\\\Vert U_{00} V_{00} \\\\psi \\\\Vert ^2 + \\\\Vert U_{10}V_{10}\\\\psi \\\\Vert ^2 \\\\\\\\ &\\\\le \\\\Vert U_{00} V_{00} \\\\psi \\\\Vert ^2 + \\\\Vert U_{10}V_{10}\\\\psi \\\\Vert ^2 + \\\\Vert U_{00}V_{10}\\\\psi \\\\Vert ^2 + \\\\Vert U_{10} V_{00} \\\\psi \\\\Vert ^2 =1, $ from which it follows that $ U_{00}V_{00} \\\\psi = U_{10}V_{10} \\\\psi = \\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}\\\\psi \\\\text{\\\\ \\\\ and \\\\ \\\\,} U_{00} V_{10} \\\\psi = U_{10}V_{00} \\\\psi = 0 .$ Thus, $\\\\big ( I_2 \\\\otimes U_B\\\\big ) \\\\big ( U_A \\\\otimes I_2 \\\\big ) |0\\\\rangle \\\\otimes \\\\psi \\\\otimes |0\\\\rangle & = \\\\sum _{i,j=0}^1 |i\\\\rangle \\\\otimes U_{i,0}V_{j,0} \\\\psi \\\\otimes |j\\\\rangle \\\\\\\\ & = \\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}|0\\\\rangle \\\\otimes \\\\psi \\\\otimes |0\\\\rangle + \\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}|1\\\\rangle \\\\otimes \\\\psi \\\\otimes |1\\\\rangle $ and we have a perfect embezzlement protocol.\",\n \"Thus, we have proven that perfect embezzlement in the commuting operator framework is equivalent to the existence of a state on $U_{\\\\operatorname{nc}}(2) \\\\otimes _{\\\\max } U_{\\\\operatorname{nc}}(2)$ that satisfies the four equations above.\",\n \"We now prove that such a state exists.\",\n \"Theorem 3.3 There exists a state $s: U_{\\\\operatorname{nc}}(2) \\\\otimes _{\\\\max } U_{\\\\operatorname{nc}}(2) \\\\rightarrow \\\\mathbb {C}$ that satisfies the four equations of the previous theorem and consequently perfect embezzlement is possible in the commuting operator framework.\",\n \"By the results of [2], we have finite dimensional Hilbert spaces $H_n$ unit vectors $h_n\\\\in H_n$ and unitary operators $U_n, V_n$ on $H_n \\\\otimes \\\\mathbb {C}^2$ , such that $(U_n \\\\otimes I_2) (I_2 \\\\otimes V_n)(|0\\\\rangle \\\\otimes h_n \\\\otimes |0\\\\rangle ) -\\\\frac{1}{\\\\sqrt{2}}(|0\\\\rangle \\\\otimes h_n \\\\otimes |0\\\\rangle + |1\\\\rangle \\\\otimes h_n \\\\otimes |1\\\\rangle )$ has norm less than $1/n$ .\",\n \"These operators induce *-homomorphisms, $\\\\pi _n: U_{\\\\operatorname{nc}}(2) \\\\otimes _{\\\\max } U_{\\\\operatorname{nc}}(2) \\\\rightarrow B(H_n)$ and states $s_n: U_{\\\\operatorname{nc}}(2) \\\\otimes _{\\\\max } U_{\\\\operatorname{nc}}(2) \\\\rightarrow \\\\mathbb {C}$ defined by $s_n(x) = \\\\langle \\\\pi _n(x) h_n , h_n \\\\rangle $ .\",\n \"These states satisfy: $\\\\bigl | s_n( u_{00} \\\\otimes u_{00}) - \\\\frac{1}{\\\\sqrt{2}} \\\\bigr | < \\\\frac{1}{n}$ , $| s_n(u_{10} \\\\otimes u_{00}) | < \\\\frac{1}{n}$ , $| s_n( u_{00} \\\\otimes u_{10}) | < \\\\frac{1}{n}$ , $ \\\\bigl | s_n(u_{10} \\\\otimes u_{10}) - \\\\frac{1}{\\\\sqrt{2}} \\\\bigr | < \\\\frac{1}{n}$ .\",\n \"Now by the fact that the state space of any unital C*-algebra is compact in the weak*-topology, we may take a limit point $s$ of this sequence of states.\",\n \"Since the value of $s(u_{i,j} \\\\otimes u_{k,l})$ must be a limit of the values of $s_n$ on these same elements, $s$ will be a state on $U_{\\\\operatorname{nc}}(2) \\\\otimes _{\\\\max } U_{\\\\operatorname{nc}}(2)$ that satisfies the 4 conditions exactly.\",\n \"Remark 3.4 From [2], each of the states, $s_n$ appearing in the above proof is actually a state on $U_{\\\\operatorname{nc}}(2) \\\\otimes _{\\\\min } U_{\\\\operatorname{nc}}(2)$ .\",\n \"Hence, by taking a limit point, we obtain a state $s: U_{\\\\operatorname{nc}}(2) \\\\otimes _{\\\\min } U_{\\\\operatorname{nc}}(2) \\\\rightarrow \\\\mathbb {C}$ that satisfies the 4 equations of Theorem REF .\",\n \"If we apply the GNS construction or any other method to represent it as $s(x) = \\\\langle \\\\pi (x) \\\\psi , \\\\psi \\\\rangle $ on some Hilbert space $\\\\mathcal {H}$ , where $\\\\pi : U_{\\\\operatorname{nc}}(2) \\\\otimes _{\\\\min } U_{\\\\operatorname{nc}}(2) \\\\rightarrow B(\\\\mathcal {H})$ , then the representation $\\\\pi $ and the Hilbert space cannot decompose as a tensor product.\",\n \"Otherwise we would achieve perfect embezzlement in a tensor product framework.\",\n \"Hence, we obtain an example of a state on a minimal tensor product, such that it cannot be represented using a *-homomorphism that is a spatial tensor product.\",\n \"In fact, no state on $U_{\\\\operatorname{nc}}(2) \\\\otimes _{\\\\min } U_{\\\\operatorname{nc}}(2)$ that satisfies just those 4 equations can have a spatial tensor product representation.\",\n \"Remark 3.5 The coefficients that appear in Theorem REF are a consequence of the fact that we are embezzling a Bell state.\",\n \"If we wish instead for a perfect embezzlement protocol of a more general vector state, say, of the form $\\\\sum _{i,j=0}^{d-1} \\\\alpha _{ij} |i\\\\rangle \\\\otimes |j\\\\rangle $ , then this is equivalent to the existence of a state $s$ on $U_{\\\\operatorname{nc}}(d) \\\\otimes _{\\\\max } U_{\\\\operatorname{nc}}(d)$ satisfying $s(u_{i0} \\\\otimes u_{j0}) = \\\\alpha _{ij}$ , for $0 \\\\le i,j < d$ .\",\n \"Moreover, it is shown in [2] that every vector in $\\\\mathbb {C}^d \\\\otimes \\\\mathbb {C}^d$ can be approximately embezzled in a finite dimensional scenario.\",\n \"Therefore, arguing as above, there is always a state $s$ on $U_{\\\\operatorname{nc}}(d) \\\\otimes _{\\\\min } U_{\\\\operatorname{nc}}(d)$ satisfying the $d^2$ equations.\"\n ],\n [\n \"Explicit construction of a perfect embezzlement protocol in a commuting-operator framework\",\n \"The previous section proves the existence of a perfect embezzlement protocol, but without constructing one explicitly.\",\n \"Some of the steps of the proof are nonconstructive.\",\n \"In Theorem REF , an abstract state is obtained by invoking an existence theorem using the weak*-compactness of the set of all states; moreover, in Theorem REF , the Hilbert space is obtained by applying the GNS representation of the state, which is based on the completion of an abstract C*-algebra.\",\n \"In this section, we give an explicit commuting-operator protocol for perfect embezzlement.\",\n \"We explain the technique for Bell states, and, at the end of the section, explain how to it extends to more general entangled states.\"\n ],\n [\n \"The resource space $\\\\mathcal {R}$ and shift operations on this space\",\n \"The resource space is the Hilbert space $\\\\ell ^2$ , whose orthonormal basis is countably infinite.\",\n \"In order to define the operations used in the protocol, it is useful to think of this space in terms of countably infinite tensor products of states, where all but finitely many of them are fixed.\",\n \"First, consider the set of infinite tensor products of 2-qubit computational basis states, where all but finitely many of them are in state $|00\\\\rangle $ .\",\n \"We can express these states as $|\\\\dots x_2 x_1 x_0\\\\,,\\\\, \\\\dots y_2 y_1 y_0\\\\rangle $ , or as $|x,y\\\\rangle $ (where $x, y \\\\in \\\\mathbb {N}_0$ , and $x_j$ and $y_j$ are the binary digits of $x$ and $y$ , respectively, in position $j$ ).\",\n \"Next, consider the set of infinite tensor products of 2-qubit Bell basis states where all but finitely many of them are $\\\\frac{1}{\\\\sqrt{2}}|00\\\\rangle + \\\\frac{1}{\\\\sqrt{2}}|11\\\\rangle $ .\",\n \"Let us denote these states as $|0.x_{-1}x_{-2}x_{-3}\\\\dots \\\\,,\\\\, 0.y_{-1}y_{-2}y_{-3}\\\\dots \\\\rangle $ , with the convention that the qubits in position $-j$ (i.e., the $j$th qubit pair, corresponding to bits $x_{-j}$ and $y_{-j}$ ) are in the Bell basis state $\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}|0y_{-j}\\\\rangle + \\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}(-1)^{x_{-j}}|1\\\\overline{y_{-j}}\\\\rangle .$ A convenient way of denoting an orthonormal basis for the (spatial) tensor product of the Hilbert spaces generated by the two aforementioned sets is as the set of all $|\\\\dots x_2 x_1 x_0\\\\,\\\\mbox{\\\\Large .\",\n \"}\\\\, x_{-1} x_{-2} \\\\dots \\\\,,\\\\ \\\\dots y_2 y_1 y_0\\\\,\\\\mbox{\\\\Large .\",\n \"}\\\\, y_{-1} y_{-2}\\\\dots \\\\rangle ,$ with all but finitely many $x_j$ and $y_{j}$ set to 0.\",\n \"Equivalently, each basis state can be written as $|x,y\\\\rangle $ , where $x$ and $y$ are dyadic rational numbersA dyadic rational number is of the form $x = a/2^b$ , where $a, b \\\\in \\\\mathbb {N}_0$ .\",\n \"Each dyadic $x$ can be written in binary as $x = x_{\\\\ell } \\\\dots x_2 x_1 x_0 \\\\hspace*{0.85358pt}\\\\mbox{\\\\Large .\",\n \"}\\\\hspace*{0.85358pt} x_{-1} x_{-2} \\\\dots x_{-r}$ .\",\n \"Formally, for all $j \\\\in \\\\mathbb {Z}$ , bit $j$ of $x$ is defined as $x_j = \\\\lfloor x 2^{-j} \\\\rfloor \\\\bmod 2$ .\",\n \"Note that $2x$ is $x$ with all the binary digits shifted left by 1..\",\n \"Intuitively, these basis states can be thought of as two-way infinite tensor products, as illustrated in Figure REF .\",\n \"Figure: Schematic picture of the tensor product structure of the basis states.\",\n \"Each circle represents a qubit.\",\n \"In positions 0,1,2,⋯0, 1, 2, \\\\dots the qubits are in computational basis states.\",\n \"In positions -1,-2,⋯-1, -2, \\\\dots the qubits are in Bell basis states.On the left are computational basis states (with all but finitely many in state $|00\\\\rangle $ ).\",\n \"On the right are Bell basis states (with all but finitely many in state $\\\\frac{1}{\\\\sqrt{2}}|00\\\\rangle + \\\\frac{1}{\\\\sqrt{2}}|11\\\\rangle )$ .\",\n \"We now define a left shift $L$ on the Hilbert space spanned by these basis states.\",\n \"Intuitively, $L$ shifts the two-way infinite tensor product to the left by one.\",\n \"Formally, we define $L$ as the product of two unitaries.\",\n \"First, define $L_1$ as the left shift of the digits of $x$ and $y$ $L_1 & |\\\\dots x_2 x_1 x_0 \\\\,.\\\\, x_{-1} x_{-2} \\\\dots \\\\,,\\\\ \\\\dots y_2 y_1 y_0\\\\,.\\\\, y_{-1} y_{-2} \\\\dots \\\\rangle \\\\\\\\&=|\\\\dots x_1 x_0 x_{-1}\\\\,.\\\\, x_{-2} y_{-3} \\\\dots \\\\,,\\\\ \\\\dots y_1 y_0 y_{-1} \\\\,.\\\\, y_{-2} y_{-3} \\\\dots \\\\rangle ,$ or, equivalently, as $L_1 |x,y\\\\rangle = |2x,2y\\\\rangle $ .\",\n \"$L_1$ is unitary because it is a permutation of the basis states.\",\n \"Note that $L_1$ does not implement the desired left shift $L$ because the qubits in position $-1$ are in the Bell basis whereas the qubits in position 0 are in the computational basis.\",\n \"A basis conversion is needed when position $-1$ is shifted to position 0.\",\n \"Define $L_2$ to perform this basis conversion in position 0 as $L_2 & |\\\\dots x_2 x_1 x_0 \\\\,.\\\\, x_{-1} x_{-2} \\\\dots \\\\,,\\\\ \\\\dots y_2 y_1 y_0 \\\\,.\\\\, y_{-1} y_{-2}\\\\dots \\\\rangle \\\\\\\\&={\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|\\\\dots x_2 x_1 0 \\\\,.\\\\, x_{-1} x_{-2} \\\\dots \\\\,,\\\\ \\\\dots y_2 y_1 y_0 \\\\,.\\\\, y_{-1} y_{-2}\\\\dots \\\\rangle \\\\\\\\&\\\\ \\\\ + {\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}(-1)^{x_0}|\\\\dots x_2 x_1 1 \\\\,.\\\\, x_{-1} x_{-2} \\\\dots \\\\,,\\\\ \\\\dots y_2 y_1 \\\\overline{y_0} \\\\,.\\\\, y_{-1} y_{-2}\\\\dots \\\\rangle .$ This can be equivalently expressed in terms of arithmetic operations on dyadic rationals as $L_2 |x,y\\\\rangle ={\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|x - x_0,y\\\\rangle + {\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}(-1)^{x_0}|x - x_0 + 1, y - 2y_0 + 1\\\\rangle .$ $L_2$ is unitary because it is a direct sum of $4 \\\\times 4$ unitaries.\",\n \"Finally, define $L = L_2 L_1$ , which is unitary because $L_1$ and $L_2$ are unitary.\",\n \"An interesting property of $L$ is that applying this operation to the state $|0.0,0.0\\\\rangle $ yields ${\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|0.0,0.0\\\\rangle + {\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|1.0,1.0\\\\rangle $ .\",\n \"In the tensor product picture, $L$ leaves the state of all the qubits intact except for the qubits in position 0, whose state changes from $|00\\\\rangle $ to $\\\\frac{1}{\\\\sqrt{2}}|00\\\\rangle + \\\\frac{1}{\\\\sqrt{2}}|11\\\\rangle $ .\",\n \"This is performing something like an embezzlement transformation (in a manner reminiscent of the imaginary “Hilbert hotel\\\"); however, this $L$ does not decompose into two commuting operations that have the structure illustrated in Figure REF .\",\n \"In order to obtain such a decomposition, we need to enlarge our Hilbert space.\",\n \"We begin with some intuition.\",\n \"In Figure REF , assume that Alice possesses the qubits in the first row and Bob possesses the qubits in the second row.\",\n \"When Alice's qubits are shifted to the left by one, the picture changes to that of Figure REF .\",\n \"Figure: Schematic picture of the tensor product of basis states when Alice's qubits (top row) are shifted left by one.Pairs of circles connected by lines are in the Bell basis.Such states are orthogonal to all states of the form of Figure .This can be equivalently expressed by shifting the labels of Alice's qubits as in Figure REF .\",\n \"Figure: An alternative schematic picture of the tensor product of Figure , where the labels of the qubits on the top row are adjusted to reflect the shift in the top row.More generally, for an arbitrary $r \\\\in \\\\mathbb {Z}$ , a left shift of the first row by $r$ is illustrated in Figure REF .\",\n \"Figure: Schematic picture of a left shift of Alice's qubits (top row) by r∈ℤr \\\\in \\\\mathbb {Z}.With the picture of Figure REF in mind, define the Hilbert space $\\\\mathcal {R}$ as having orthonormal basis states of the form $|r,x,y\\\\rangle $ , where $x, y$ are dyadic rationals and $r \\\\in \\\\mathbb {Z}$ represents the leftward shift of Alice's qubits.\",\n \"We can interpret $|r,x,y\\\\rangle $ as an encoding of the following logical state.\",\n \"For all $j \\\\in \\\\mathbb {Z}$ , Alice's logical qubit in position $j+r$ and Bob's logical qubit in position $j$ are in the joint state ${\\\\left\\\\lbrace \\\\begin{array}{ll}|x_{j}y_{j}\\\\rangle & \\\\text{if $j \\\\ge 0$} \\\\\\\\[2mm]{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|0y_{j}\\\\rangle +{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}(-1)^{x_{j}}|1\\\\overline{y_{j}}\\\\rangle & \\\\text{if $j < 0$.}\\\\end{array}\\\\right.\",\n \"}$ Now we define the Alice left shift $L_A$ as simply $L_A |r,x,y\\\\rangle = |r+1,x,y\\\\rangle .$ $L_A$ is obviously unitary and commutes with $L$ , since they act on different components of $|r,x,y\\\\rangle $ .\",\n \"Next, define the Bob left shift $L_B$ as $L_B = L_{A}^{*}L$ (a left shift of both Alice and Bob's qubits followed by a right shift of Alice's qubits).\",\n \"Note that $L_A$ and $L_B$ commute, since $L_A$ and $L$ commute.\",\n \"Also, since $L$ is a left shift by both Alice and Bob and $L_{A}^{*}$ is a right shift by Alice, $L_B$ has no net effect on Alice's logical qubits.\"\n ],\n [\n \"Swap operations between $\\\\mathcal {H}_A$ , {{formula:5ce0fc52-fcb6-4283-ae7b-84f3bd08f3dc}} and {{formula:3ba76a35-7dd7-4b48-878e-0858d3c7d485}}\",\n \"Prior to defining our embezzlement protocol, we define swap operations between $\\\\mathcal {H}_A$ and the logical qubit of Alice in position 0 of $\\\\mathcal {R}$ , and between $\\\\mathcal {H}_B$ and the logical qubit of Bob in position 0 of $\\\\mathcal {R}$ .\",\n \"The Bob swap $S_B$ is defined simply as the unitary operation that acts on $\\\\mathcal {H}_B \\\\otimes \\\\mathcal {R}$ as $S_B |t\\\\rangle \\\\otimes |r,\\\\, x,\\\\, \\\\dots y_1 y_0 \\\\,.\\\\, y_{-1} \\\\dots \\\\rangle =|y_0\\\\rangle \\\\otimes |r,\\\\, x,\\\\, \\\\dots y_1 t \\\\,.\\\\, y_{-1} \\\\dots \\\\rangle ,$ or, equivalently, as $S_B|t\\\\rangle \\\\otimes |r,x,y\\\\rangle = |y_0\\\\rangle \\\\otimes |r,x,y-y_0+t\\\\rangle $ .\",\n \"$S_B$ is clearly unitary and commutes with $L_A$ since they act on different components of each basis state $|r,x,y\\\\rangle \\\\in \\\\mathcal {R}$ .\",\n \"The corresponding Alice swap, acting on $\\\\mathcal {H}_A \\\\otimes \\\\mathcal {R}$ , is more complicated than $S_B$ .\",\n \"First define $\\\\tilde{S}_A$ (a naïve Alice swap) as $\\\\tilde{S}_A |s\\\\rangle \\\\otimes |r,\\\\, \\\\dots x_1 x_0 \\\\,.\\\\, x_{-1} \\\\dots ,\\\\, y\\\\rangle =|x_0\\\\rangle \\\\otimes |r,\\\\, \\\\dots x_1 s \\\\,.\\\\, x_{-1} \\\\dots ,\\\\, y\\\\rangle ,$ or, equivalently, as $\\\\tilde{S}_A|s\\\\rangle \\\\otimes |r,x,y\\\\rangle = |x_0\\\\rangle \\\\otimes |r,x-x_0+s,y\\\\rangle $ .\",\n \"$\\\\tilde{S}_A$ does not swap with Alice's logical qubit in position 0—moreover, $\\\\tilde{S}_A$ does not commute with $L_B$ .\",\n \"To swap with Alice's logical qubit in position 0, it is convenient to first define the controlled-$L$, denoted as $C$ , acting on $\\\\mathcal {R}$ as $C |r,x,y\\\\rangle = L^{r}|r,x,y\\\\rangle ,$ which makes sense because $L^{r}$ acts only on the second and third component of $|r,x,y\\\\rangle $ .\",\n \"$C$ is unitary because each $L^{r}$ is unitary and $C$ is a direct sum of all $L^{r}$ .\",\n \"Intuitively, $C^*|r,x,y\\\\rangle $ is a state in which Alice's literal qubit in position 0 corresponds to Alice's logical qubit in position 0 in $|r,x,y\\\\rangle $ .\",\n \"Now we define the actual Alice swap as $S_A = C \\\\tilde{S}_A C^{*}.$ Clearly $S_A$ is unitary and, for each $|r,x,y\\\\rangle \\\\in \\\\mathcal {R}$ , its effect is localized to Alice's logical qubit in position 0.\",\n \"$S_A$ and $S_B$ commute because $S_B$ is localized to Bob's logical qubit in position 0.\",\n \"Moreover, $S_A$ and $L_B$ commute because, for $|r,x,y\\\\rangle \\\\in \\\\mathcal {R}$ , $L_B$ is localized to Bob's logical qubits.\"\n ],\n [\n \"The embezzlement protocol\",\n \"The idea is to start with state $|0,\\\\,0.0,\\\\,0.0\\\\rangle $ , perform $L_A$ and $L_B$ , and then swap the two qubits in position 0 of $\\\\mathcal {R}$ into $\\\\mathcal {H}_A$ and $\\\\mathcal {H}_B$ .\",\n \"Alice performs $U_A = S_A L_A$ and Bob performs $U_B = S_B L_B$ .\",\n \"Clearly $U_A$ and $U_B$ commute.\",\n \"Recall that $L_A L_B = L$ .\",\n \"The state evolves as: 0. initial state $|0\\\\rangle \\\\otimes |0\\\\rangle \\\\otimes |0,\\\\,0.0,\\\\,0.0\\\\rangle $ 1. after $L_A L_B$ $|0\\\\rangle \\\\otimes |0\\\\rangle \\\\otimes \\\\bigl ({\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|0,\\\\,0.0,\\\\,0.0\\\\rangle +{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|0,\\\\,1.0,\\\\,1.0\\\\rangle \\\\bigr )$ 2. after $S_A S_B$ $\\\\bigl ({\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|0\\\\rangle \\\\otimes |0\\\\rangle +{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|1\\\\rangle \\\\otimes |1\\\\rangle \\\\bigr )\\\\otimes |0,\\\\,0.0,\\\\,0.0\\\\rangle $ This completes the protocol for perfect embezzlement in the commuting operator framework.\",\n \"Remark 4.1 It is easy to adapt the above method to embezzle a more general entangled state, say, of the form $\\\\phi = \\\\sum _{i,j=0}^{d-1} \\\\alpha _{ij} |i\\\\rangle \\\\otimes |j\\\\rangle $ .\",\n \"First, redefine $\\\\mathcal {R}$ to be in terms of basis states of the form $|r,x,y\\\\rangle $ , where $x$ and $y$ are $d$ -adic rational numbers (i.e., with digits in $\\\\mathbb {Z}_d$ ).\",\n \"Then it suffices to set the operation $L_2$ (which is the basis change part of the left shift operation) to be any unitary operation on $\\\\mathbb {C}^d \\\\otimes \\\\mathbb {C}^d$ that maps $|0\\\\rangle \\\\otimes |0\\\\rangle $ to $\\\\phi $ .\",\n \"The other parts of the protocol are essentially the same.\"\n ],\n [\n \"Coherent embezzlement games\",\n \"A purported protocol for embezzlement cannot be tested in the way that nonlocal games can, because Alice and Bob can perform local operations that perfectly map $|0\\\\rangle \\\\otimes |0\\\\rangle $ to $\\\\frac{1}{\\\\sqrt{2}}|0\\\\rangle \\\\otimes |0\\\\rangle +\\\\frac{1}{\\\\sqrt{2}}|1\\\\rangle \\\\otimes |1\\\\rangle $ using the resource of only a single (concealed) Bell state.\",\n \"Leung, Toner and Watrous [8] proposed a coherent state exchange game that is related to embezzlement but is operationally testable.\",\n \"In this game, Alice and Bob each receive a qutrit from a referee as input and they each return a qubit to the referee, who performs a measurement on the returned state to determine whether they win or lose.\",\n \"There is no perfect strategy for this game using finite entanglement.\",\n \"It is shown in [8] that, for all $\\\\epsilon > 0$ : there exists a strategy that succeeds with probability $1 - \\\\epsilon $ using $O(\\\\log (1/\\\\epsilon ))$ -entropy entanglement; moreover, to succeed with probability $1 - \\\\epsilon $ requires entanglement with entropy $\\\\Omega (\\\\log (1/\\\\epsilon ))$ .\",\n \"Regev and Vidick [10] presented a simplification of the coherent state exchange game that has the above properties, but where the outputs are classical bits instead of qubits (the inputs are still qutrits).\",\n \"In [10], this is called the $T_2$ game.\",\n \"We refer to this as the coherent embezzlement game, to highlight its close relationship with embezzlement.\",\n \"In this section, we begin by reviewing the definition of the coherent embezzlement (a.k.a.\",\n \"$T_2$ ) game.\",\n \"Then we show that a perfect strategy for embezzlement can be converted to a perfect strategy for coherent embezzlement—and vice versa.\",\n \"By such reductions, we prove that there is a perfect strategy for coherent embezzlement in the commuting operator framework (Theorem REF ), but there is no such perfect strategy in the tensor product framework (Theorem REF ).\",\n \"We now define the coherent embezzlement game [10].\",\n \"Alice and Bob each receive two qutrits as input and they each produce a classical bit as output.\",\n \"The input state that Alice and Bob jointly receive is either $\\\\phi _0$ or $\\\\phi _1$ , where $\\\\phi _0 &= {\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|0\\\\rangle \\\\otimes |0\\\\rangle +{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}\\\\bigl ({\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|1\\\\rangle \\\\otimes |1\\\\rangle +{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|2\\\\rangle \\\\otimes |2\\\\rangle \\\\bigr ) \\\\\\\\\\\\phi _1 &= {\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|0\\\\rangle \\\\otimes |0\\\\rangle -{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}\\\\bigl ({\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|1\\\\rangle \\\\otimes |1\\\\rangle +{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|2\\\\rangle \\\\otimes |2\\\\rangle \\\\bigr ).$ Call Alice and Bob's output bits $a$ and $b$ respectively.\",\n \"The winning condition is that: when the input is $\\\\phi _0$ , $a \\\\oplus b = 0$ ; when the input is $\\\\phi _1$ , $a \\\\oplus b = 1$ .\",\n \"Note that the winning condition does not require Alice and Bob's resource state to be used in a catalytic manner; Alice and Bob are free to destroy this state in their strategy.\",\n \"In this sense, the coherent embezzlement game is simpler than embezzlement, whose definition depends critically on restoring the resource state.\",\n \"In the remainder of this section, for technical convenience, we represent qutrits as pairs of qubits using the encoding $0 \\\\equiv 00$ , $1 \\\\equiv 10$ (for Alice) or 01 (for Bob), and $2 \\\\equiv 11$ .\",\n \"With this encoding, the input states that Alice and Bob jointly receive can be written as, for $c \\\\in \\\\lbrace 0,1\\\\rbrace $ , $\\\\phi _c &= {\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|00\\\\rangle \\\\otimes |00\\\\rangle +{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}(-1)^c\\\\bigl ({\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|10\\\\rangle \\\\otimes |01\\\\rangle +{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|11\\\\rangle \\\\otimes |11\\\\rangle \\\\bigr ) \\\\\\\\&= {\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|0\\\\rangle |00\\\\rangle |0\\\\rangle +{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}(-1)^c|1\\\\rangle \\\\bigl ({\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|00\\\\rangle +{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|11\\\\rangle \\\\bigr )|1\\\\rangle $ (where the first two qubits are Alice's input and the last two qubits are Bob's input).\",\n \"Theorem 5.1 There is a perfect strategy for the coherent embezzlement game in the commuting operator framework.\",\n \"Let $\\\\mathcal {R}$ , $U_A$ , and $U_B$ be as defined in the unitary embezzlement protocol of section .\",\n \"For coherent embezzlement, the input Hilbert spaces are $\\\\mathcal {H}_{A_1} \\\\otimes \\\\mathcal {H}_{A_2} = \\\\mathcal {H}_{B_1} \\\\otimes \\\\mathcal {H}_{B_2} = \\\\mathbb {C}^2\\\\otimes \\\\mathbb {C}^2$ .\",\n \"We will show that the protocol in Figure REF performs coherent embezzlement.\",\n \"Figure: Circuit diagram of a protocol for coherent embezzlement from a protocol for embezzlement based on U A U_A, U B U_B, and ψ∈ℛ\\\\psi \\\\in \\\\mathcal {R}.First note that, since the controlled $U^*_A$ and $U^*_B$ perform the inverse of embezzlement when their control qubits are in state $|1\\\\rangle $ , they perform a mapping on $\\\\mathcal {H}_{A_1} \\\\otimes \\\\mathcal {H}_{A_2} \\\\otimes \\\\mathcal {H}_{B_2} \\\\otimes \\\\mathcal {H}_{B_1}$ such that $\\\\phi _c =&{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|0\\\\rangle |00\\\\rangle |0\\\\rangle +{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}(-1)^c|1\\\\rangle \\\\bigl ({\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|00\\\\rangle +{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|11\\\\rangle \\\\bigr )|1\\\\rangle \\\\\\\\[1mm]&\\\\mapsto \\\\ \\\\ {\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|0\\\\rangle |00\\\\rangle |0\\\\rangle +{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}(-1)^c|1\\\\rangle |00\\\\rangle |1\\\\rangle .$ Note that, on the Hilbert space $\\\\mathcal {H}_{A_1} \\\\otimes \\\\mathcal {H}_{B_1}$ , this is the pure state ${\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|00\\\\rangle +{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}(-1)^c|11\\\\rangle $ .\",\n \"Finally, since the Hadamard gates perform the mapping ${\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|00\\\\rangle +{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}(-1)^c|11\\\\rangle \\\\mapsto {\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|0c\\\\rangle +{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|1\\\\overline{c}\\\\rangle ,$ the result follows.\",\n \"Theorem 5.2 There is no perfect strategy for the coherent embezzlement game in the tensor product framework (where Alice and Bob are allowed to use ancillas).\",\n \"The idea is that a perfect strategy for coherent embezzlement can be converted into a perfect strategy for embezzlement.\",\n \"Suppose that there is a perfect strategy for coherent embezzlement.\",\n \"Without loss of generality, it can be assumed that this strategy is of the form depicted in Figure REF , where $\\\\psi \\\\in \\\\mathcal {R}_A \\\\otimes \\\\mathcal {R}_B$ , Alice and Bob each perform a local unitary operation followed by measurement of one of their qubits.\",\n \"Figure: Circuit diagram of an arbitrary protocol for coherent embezzlement.ψ∈ℛ A ⊗ℛ B \\\\psi \\\\in \\\\mathcal {R}_A \\\\otimes \\\\mathcal {R}_B is the resource state.Without loss of generality, two local unitaries are performed, followed by measurements of two specific qubits.Since we are assuming that the strategy is perfect, for input state $\\\\phi _0$ , the output bits satisfy $a\\\\oplus b = 0$ , and for input state $\\\\phi _1$ , the output bits satisfy $a\\\\oplus b = 1$ .\",\n \"Now, consider the protocol in Figure REF .\",\n \"Figure: Modification of circuit in Figure that performs embezzlement using the resource state ψ∈ℛ A ⊗ℛ B \\\\psi \\\\in \\\\mathcal {R}_A \\\\otimes \\\\mathcal {R}_B.We begin by showing that, for each $c \\\\in \\\\lbrace 0,1\\\\rbrace $ , the effect of the first three steps of the protocol in Figure REF is to map $\\\\phi _c\\\\otimes \\\\psi \\\\otimes \\\\gamma _A\\\\otimes \\\\gamma _B$ to $(-1)^c \\\\phi _c\\\\otimes \\\\psi \\\\otimes \\\\gamma _A\\\\otimes \\\\gamma _B$ .\",\n \"Since $a \\\\oplus b = c$ , After the $U_A \\\\otimes U_B$ operations have been performed, the state must be of the form $\\\\alpha |0\\\\rangle \\\\otimes |c\\\\rangle \\\\otimes \\\\xi _0 + \\\\beta |1\\\\rangle \\\\otimes |\\\\overline{c}\\\\rangle \\\\otimes \\\\xi _1,$ when expressed the state in the Hilbert space $\\\\mathcal {H}_{A_1} \\\\otimes \\\\mathcal {H}_{B_1} \\\\otimes (\\\\mathcal {H}_{A_2} \\\\otimes \\\\mathcal {H}_{B_2} \\\\otimes \\\\mathcal {R}_A \\\\otimes \\\\mathcal {R}_B \\\\otimes \\\\mathcal {G}_A \\\\otimes \\\\mathcal {G}_B).$ Therefore, after the two $Z$ gates, the state is $(-1)^c\\\\bigl (\\\\alpha |0\\\\rangle \\\\otimes |c\\\\rangle \\\\otimes \\\\xi _0 + \\\\beta |1\\\\rangle \\\\otimes |\\\\overline{c}\\\\rangle \\\\otimes \\\\xi _1\\\\bigr ),$ and this is mapped by $U^{*}_{A} \\\\otimes U^{*}_{B}$ to $(-1)^c \\\\phi _c\\\\otimes \\\\psi \\\\otimes \\\\gamma _A\\\\otimes \\\\gamma _B$ .\",\n \"This implies that the first three steps of the modified protocol maps $|00\\\\rangle \\\\otimes |00\\\\rangle = {\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}\\\\phi _0 + {\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}\\\\phi _1$ on $\\\\mathcal {H}_{A}\\\\otimes \\\\mathcal {H}_B$ (with the other registers in state $\\\\psi \\\\otimes \\\\gamma _A\\\\otimes \\\\gamma _B$ ) to ${\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|10\\\\rangle \\\\otimes |01\\\\rangle +{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|11\\\\rangle \\\\otimes |11\\\\rangle = {\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}\\\\phi _0 - {\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}\\\\phi _1.$ Finally, the two $X$ gates of the protocol in Figure REF map the state ${\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|10\\\\rangle \\\\otimes |01\\\\rangle +{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|11\\\\rangle \\\\otimes |11\\\\rangle $ to ${\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|00\\\\rangle \\\\otimes |00\\\\rangle +{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|01\\\\rangle \\\\otimes |10\\\\rangle $ .\",\n \"Therefore, the entire protocol maps $|00\\\\rangle \\\\otimes |00\\\\rangle \\\\otimes \\\\psi \\\\otimes \\\\gamma _A\\\\otimes \\\\gamma _B$ to $\\\\bigl ({\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|00\\\\rangle \\\\otimes |00\\\\rangle &+{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|01\\\\rangle \\\\otimes |10\\\\rangle \\\\bigr )\\\\otimes \\\\psi \\\\otimes \\\\gamma _A\\\\otimes \\\\gamma _B \\\\\\\\& =|0\\\\rangle ({\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|00\\\\rangle +{\\\\textstyle {\\\\frac{1}{\\\\sqrt{2}}}}|11\\\\rangle )|0\\\\rangle \\\\otimes \\\\psi \\\\otimes \\\\gamma _A\\\\otimes \\\\gamma _B.$ This is perfect embezzlement in the registers $\\\\mathcal {H}_{A_2} \\\\otimes \\\\mathcal {H}_{B_2}$ .\",\n \"Since this protocol violates Theorem REF , there cannot exist a perfect strategy for coherent embezzlement using entanglement of the form $\\\\psi \\\\in \\\\mathcal {R}_A \\\\otimes \\\\mathcal {R}_B$ .\"\n ],\n [\n \"Perfect embezzlement requires non-unitary isometries\",\n \"In this section we obtain further information about the nature of the unitary operators that appear in perfect embezzlement protocols.\",\n \"We show that some of the operators that occur in such protocols must contain non-unitary isometries.\",\n \"This result justifies the need for the register shifts that appear in the explicit protocol of the earlier section.\",\n \"Also, since non-unitary isometries only exist in infinite dimensions, this result implies that perfect embezzlement is impossible with a finite dimensional resource space.\",\n \"We begin by reviewing the key facts about unitaries and isometries.\",\n \"Given a Hilbert space $\\\\mathcal {H}$ a linear map $V: \\\\mathcal {H} \\\\rightarrow \\\\mathcal {H}$ is an isometry provided that $\\\\Vert Vh\\\\Vert =\\\\Vert h\\\\Vert $ for all $h \\\\in \\\\mathcal {H}$ .\",\n \"Note that isometries are necessarily one-to-one.\",\n \"The map $V$ is a unitary provided that it is an isometry and it is onto.\",\n \"A simple dimension count shows that in finite dimensions every isometry is necessarily a unitary.\",\n \"An example of an isometry that is not a unitary is the unilateral shift S. This is the operator on the Hilbert space $\\\\ell ^2(\\\\mathbb {N})$ which has an orthonormal basis $\\\\lbrace |j\\\\rangle : j \\\\in \\\\mathbb {N} \\\\rbrace $ , defined by $S(|j\\\\rangle ) = |j+1\\\\rangle $ .\",\n \"It is easily seen that this operator is an isometry and the $|1\\\\rangle $ is orthogonal to the range of $S$ , so that $S$ is not onto.\",\n \"Note that the kernel of $S^*$ is equal to the span of $|1\\\\rangle $ .\",\n \"We call a linear map $V: \\\\mathcal {H} \\\\rightarrow \\\\mathcal {H}$ a non-unitary isometry provided that it is an isometry that is not onto.\",\n \"It is not hard to show that if $V$ is a non-unitary isometry, $e_1$ is any unit vector orthogonal to the range of $V$ , and we set $e_n = V^n e_1,$ then this sequence is orthonormal and $V$ acts as a unilateral shift on this subspace.\",\n \"So, in this sense, non-unitary isometries always contain a space on which they act like unilateral shifts.\",\n \"Recall that a subspace $\\\\mathcal {M}$ is called invariant for an operator $C$ provided that $C(\\\\mathcal {M}) \\\\subseteq \\\\mathcal {M}$ .\",\n \"Lemma 6.1 Let $\\\\mathcal {H}$ be a Hilbert space, $C: \\\\mathcal {H} \\\\rightarrow \\\\mathcal {H}$ be a contraction (i.e., $\\\\Vert Ch\\\\Vert \\\\le \\\\Vert h \\\\Vert , \\\\, \\\\forall h \\\\in \\\\mathcal {H}$ ), and $\\\\mathcal {M} \\\\subseteq \\\\mathcal {H}$ be a subspace that is invariant for $C$ .\",\n \"If the restriction of $C$ to $\\\\mathcal {M}$ acts as a unitary $U$ on $\\\\mathcal {M}$ , then $\\\\mathcal {M}$ is also invariant for $C^*$ and the restriction of $C^*$ to $\\\\mathcal {M}$ is $U^*$ .\",\n \"This is easiest to see using operator matrices.\",\n \"Decomposing $\\\\mathcal {H} = \\\\mathcal {M} \\\\oplus \\\\mathcal {M}^{\\\\perp }$ we may write $C= \\\\begin{pmatrix} U &X \\\\\\\\Y & Z \\\\end{pmatrix}$ where $U: \\\\mathcal {M} \\\\rightarrow \\\\mathcal {M},$ $X: \\\\mathcal {M}^{\\\\perp } \\\\rightarrow \\\\mathcal {M}$ , $Y: \\\\mathcal {M} \\\\rightarrow \\\\mathcal {M}^{\\\\perp }$ , and $Z: \\\\mathcal {M}^{\\\\perp } \\\\rightarrow \\\\mathcal {M}^{\\\\perp }$ .\",\n \"The fact that $\\\\mathcal {M}$ is invariant implies that $Y=0$ .\",\n \"The hypothesis that the restriction of $C$ to $\\\\mathcal {M}$ is a unitary, means that $U$ is a unitary.\",\n \"But $C^*= \\\\begin{pmatrix} U^* & X^*\\\\\\\\ 0 & Z^* \\\\end{pmatrix}$ is a contraction with $U^*$ a unitary.\",\n \"This forces that for every $h \\\\in \\\\mathcal {M},$ $X^*h =0$ .\",\n \"Thus, $X^*=0$ and $\\\\mathcal {M}$ is invariant for $C^*$ .\",\n \"Note that if $U$ was a non-unitary isometry, then $U^*$ would have a kernel and we could no longer conclude that $X^*=0$ .\",\n \"The next concept that we shall need is the polar decomposition of an operator.\",\n \"Given an operator $A$ on a Hilbert space $\\\\mathcal {H}$ , we set $|A|= (A^*A)^{1/2}$ .\",\n \"Let $\\\\mathcal {R}(A)$ denote the closure of the range of $A$ and let $\\\\mathcal {R}(|A|)$ denote the closure of the range of $|A|$ .\",\n \"Note that for any $x \\\\in \\\\mathbb {\\\\mathcal {\\\\unknown.\",\n \"}}H,$ we have $ \\\\Vert Ax\\\\Vert ^2 = \\\\langle Ax, Ax \\\\rangle = \\\\langle x, A^*Ax \\\\rangle =\\\\langle |A|x, |A|x \\\\rangle = \\\\Vert |A|x\\\\Vert ^2.$ From this equality, it follows that there is a well-defined linear isometry $W: \\\\mathcal {R}(|A|) \\\\rightarrow \\\\mathcal {R}(A)$ defined by setting $W(|A|h) = Ah$ , so that $A= W|A|.$ If we extend $W$ to a map $U: \\\\mathcal {H} \\\\rightarrow \\\\mathcal {H}$ by sitting $U= WQ$ where $Q$ is the orthogonal projection onto $\\\\mathcal {R}(|A|)$ then we still have $A= U|A|$ .\",\n \"The map $U$ is called a partial isometry.\",\n \"This representation $A=U|A|$ is called the polar decomposition of $A$ .\",\n \"Lemma 6.2 Let $C$ be a contraction and let $C=U|C|$ be its polar decomposition.\",\n \"If $h$ is a unit vector such that $||C^nh||=1, \\\\forall n \\\\in \\\\mathbb {N}$ , then the closed subspace $\\\\mathcal {M}$ generated by $\\\\lbrace C^nh: n \\\\ge 0 \\\\rbrace $ is an invariant subspace for $C$ and for any $v \\\\in \\\\mathcal {M},$ $\\\\Vert Cv\\\\Vert =\\\\Vert v\\\\Vert $ .\",\n \"Clearly $\\\\mathcal {M}$ is invariant for $C$ .\",\n \"Set $P = |C|$ so that $C=UP$ with $0 \\\\le P \\\\le I$ and $||Ph||=||UPh||= ||Ch||=1$ .\",\n \"This implies that $Ph=h$ .\",\n \"Thus, $Ch= Uh$ .\",\n \"Since $1= ||C^2h|| = ||UPUPh|| = ||UP(Uh)|| = ||P(UH)|| $ we have that $P(Uh) = Uh$ and so $C^2h = UPUPh =UPUh=U^2h.$ Inductively, $P(U^nh) = U^nh$ so that $C^nh = U^n h$ .\",\n \"Thus, $Cv= Uv$ for any $v \\\\in \\\\mathcal {M}$ .\",\n \"Since $P(U^nh) = U^n h$ for all $n \\\\ge 0$ , we have that $\\\\mathcal {M} \\\\subseteq \\\\mathcal {R}(|C|)$ .\",\n \"But $U$ acts isometrically on $\\\\mathcal {R}(|C|)$ and since $U$ and $C$ are equal on this subspace, $C$ acts isometrically on $\\\\mathcal {M}$ .\",\n \"Theorem 6.3 Let $\\\\mathcal {H}_R, \\\\psi , U_A=(U_{ij}),$ and $U_B= (V_{kl})$ be a perfect embezzlement protocol.\",\n \"If $\\\\mathcal {M}$ is the closed subspace of $\\\\mathcal {H}$ spanned by $\\\\lbrace U_{00}^{*n} \\\\psi : n \\\\ge 0 \\\\rbrace $ , then the restriction of $U_{00}^*$ to this invariant subspace is a non-unitary isometry.\",\n \"Recall that the $U_{ij}$ 's must *-commute with the $V_{kl}$ 's.\",\n \"The embezzlement relations tell us that $ U_{00}V_{00}\\\\psi = \\\\psi /\\\\sqrt{2}, \\\\ \\\\ U_{10}V_{00}\\\\psi =0, \\\\ \\\\ U_{10}V_{10}\\\\psi = \\\\psi /\\\\sqrt{2}, \\\\ \\\\ U_{00}V_{10}\\\\psi =0.$ The fact that $U_A$ and $U_B$ are unitaries implies that $ I= V^*_{00}V_{00}+ V^*_{10}V_{10}= U_{00}^*U_{00} + U_{10}^*U_{10} $ so that $U_{00} \\\\psi &= U_{00}(V_{00}^*V_{00} + V_{10}^*V_{10})\\\\psi = V_{00}^*U_{00}V_{00}\\\\psi = V_{00}^*\\\\psi /\\\\sqrt{2}, \\\\\\\\V_{00}\\\\psi &= V_{00}(U_{00}^*U_{00} +U_{10}^*U_{10})\\\\psi = U_{00}^*\\\\psi / \\\\sqrt{2}.$ Iterating, yields $(V_{00}^*)^n\\\\psi = (\\\\sqrt{2}U_{00})^n \\\\psi \\\\mbox{\\\\ \\\\ and \\\\ \\\\ } V_{00}^n \\\\psi = (U_{00}^*/\\\\sqrt{2})^n \\\\psi .$ Since $(1/\\\\sqrt{2})^n \\\\psi = (U_{00}V_{00})^n\\\\psi = U_{00}^nV_{00}^n \\\\psi =U_{00}^n(U_{00}^*/\\\\sqrt{2})^n \\\\psi $ , we have that $U_{00}^n U_{00}^{*n}\\\\psi =\\\\psi .$ Because $||U_{00}|| \\\\le 1$ we have that $||U_{00}^{*n} \\\\psi || =1$ for all $n$ .\",\n \"Hence, by the previous lemma, $U_{00}^*$ acts isometrically on $\\\\mathcal {M}$ .\",\n \"Now if $U_{00}^*$ acted unitarily, then by the earlier lemma, for vectors in this space $U_{00}$ would be the inverse and in particular would act isometrically on $\\\\mathcal {M}$ .\",\n \"But then we would have that $\\\\Vert V_{00}^*\\\\psi \\\\Vert = \\\\sqrt{2} \\\\Vert U_{00}\\\\psi \\\\Vert = \\\\sqrt{2}$ .\",\n \"This is impossible, because $U_B$ is a unitary and so $\\\\Vert V_{00}^*\\\\Vert \\\\le 1$ .\",\n \"This contradiction shows that $U_{00}^*$ must be a non-unitary isometry on $\\\\mathcal {M}$ .\",\n \"This yields the following fact.\",\n \"Recall that for a perfect embezzlement protocol, we are only assuming that the operators commute, not that the resource space has a bipartite tensor structure.\",\n \"Corollary 6.4 Perfect embezzlement is impossible in the commuting-operator framework if the resource space is finite-dimensional.\",\n \"If $\\\\mathcal {H}_R$ is finite dimensional, then $\\\\mathcal {M}$ is also finite dimensional.\",\n \"But every isometry on a finite dimensional space is necessarily a unitary, contradicting the fact that $U_{00}^*$ is a non-unitary isometry.\"\n ],\n [\n \"Acknowledgements\",\n \"We would like to thank Marius Junge, Debbie Leung, Volkher Scholz, and John Watrous for helpful discussions.\",\n \"This research was supported in part by Canada's NSERC, a David R. Cheriton Scholarship, and a Mike and Ophelia Lazaridis Fellowship.\"\n ],\n [\n \"The Schmidt and polar decompositions in infinite dimensions\",\n \"In this section, for the convenience of the reader, we gather together a few useful results from operator theory that are not well known within the QIT community.\",\n \"We are claiming no originality.\",\n \"Definition 1.1 Let $W: H_1 \\\\rightarrow H_2$ .\",\n \"Then $W$ is called an isometry if $\\\\Vert Wh_1\\\\Vert _2 = \\\\Vert h_1\\\\Vert _1$ for every $h_1 \\\\in H_1$ .\",\n \"$W$ is called a coisometry iff $W^*:H_2 \\\\rightarrow H_1$ is an isometry.\",\n \"$W$ is called a partial isometry if the restriction of $W$ to $ker(W)^{\\\\perp }$ is an isometry.\",\n \"In this case the space $ker(W)^{\\\\perp }$ is called the initial space of $W$ and $ran(W)^- = ker(W^*)^{\\\\perp }$ is called the final space of $W$ .\",\n \"Proposition 1.2 Let $H$ and $K$ be Hilbert spaces of arbitrary dimension, let $\\\\lbrace e_{\\\\alpha }: \\\\alpha \\\\in A \\\\rbrace $ and $\\\\lbrace f_{\\\\beta } : \\\\beta \\\\in B \\\\rbrace $ be o.n.\",\n \"bases for $H$ and $K$ , respectively.\",\n \"Then $\\\\lbrace e_{\\\\alpha } \\\\otimes f_{\\\\beta }: \\\\alpha \\\\in A, \\\\beta \\\\in B \\\\rbrace $ is an o.n.\",\n \"basis for $H \\\\otimes K.$ Proposition 1.3 (The Polar Decomposition) Let $X: H_1 \\\\rightarrow H_2$ and let $|X|= (X^*X)^{1/2}.$ Then there is a unique partial isometry $W:H_1 \\\\rightarrow H_2$ with initial space $ker(X)^{\\\\perp }=ker(|X|)^{\\\\perp }= ran(|X|)^-$ and final space $ran(X)^-= ker(X^*)^{\\\\perp }$ such that $X= W|X|.$ To prove, one simply sets $W(|X|h) = Xh$ and shows that this is well-defined and satisfies the properties.\",\n \"Note that $W$ is an isometry iff $ker(X) = (0)$ and is a coisometry iff $ran(X)^- = H_2.$ Proposition 1.4 Let $\\\\lbrace e_k \\\\rbrace $ be an o.n.\",\n \"sequence in a Hilbert space, set $u_i = \\\\sum _k u_{i,k} e_k$ and let $U=(u_{i,j})$ .\",\n \"Then $\\\\lbrace u_i \\\\rbrace $ is o.n.\",\n \"iff $UU^*= I,$ i.e., $U$ is a coisometry.\",\n \"Theorem 1.5 (The Infinite Dimensional Schmidt Decomposition) Let $H$ and $K$ be Hilbert spaces of arbitrary dimension and let $x \\\\in H \\\\otimes K$ .\",\n \"Then there are countable orthonormal sets $ u_k \\\\in H$ and $v_k \\\\in K$ and $d_k \\\\ge 0$ and $d_k \\\\ge d_{k+1}, \\\\forall k$ , such that $x= \\\\sum _k d_k u_k \\\\otimes v_k,$ and so $\\\\Vert x\\\\Vert ^2= \\\\sum _k d_k^2.$ Moreover, if $x = \\\\sum _k c_k w_k \\\\otimes z_k$ is another such representation of $x$ , then $c_k=d_k$ for all $k$ .\",\n \"Pick any orthonormal bases $\\\\lbrace e_{\\\\alpha } : \\\\alpha \\\\in A \\\\rbrace $ and $\\\\lbrace f_{\\\\beta }: \\\\beta \\\\in B \\\\rbrace $ .\",\n \"By Proposition REF , we can expand $x = \\\\sum _{\\\\alpha , \\\\beta } z_{\\\\alpha , \\\\beta } e_{\\\\alpha } \\\\otimes f_{\\\\beta }$ .\",\n \"We know that only countably many of the coefficients are non-zero, so we only need countably many $\\\\alpha $ 's and countably many $\\\\beta $ 's.\",\n \"So we can write $x = \\\\sum _{i,j} z_{i,j} e_i \\\\otimes f_j.$ and $\\\\sum _{i,j} |z_{i,j}|^2 = \\\\Vert x\\\\Vert ^2$ .\",\n \"Let $h_j = \\\\sum _i z_{i, j} e_i$ , so that $x = \\\\sum _j h_j\\\\otimes f_j$ , and let $H = \\\\text{span}\\\\lbrace h_j\\\\rbrace $ .\",\n \"Let $\\\\lbrace |i\\\\rangle \\\\rbrace $ be an orthonormal basis of $H$ , and write $h_j = \\\\sum _{i} x_{i, j} |i\\\\rangle $ .\",\n \"This gives us $x = \\\\sum _{i, j} x_{i, j}|i\\\\rangle \\\\otimes f_j$ .\",\n \"Let $X = \\\\sum _{i, j} x_{i, j} |i\\\\rangle \\\\langle j|$ be the matrix of a map from $H$ to $H$ .\",\n \"Note that $X$ is Hilbert-Schmidt and so compact and also has dense range because $\\\\text{span}\\\\lbrace |i\\\\rangle \\\\rbrace = \\\\text{span} \\\\lbrace h_j\\\\rbrace = \\\\text{span}\\\\lbrace \\\\sum _{i} x_{i, j} |i\\\\rangle \\\\rbrace $ .\",\n \"By Proposition REF , performing polar decomposition on $X$ yields $X = W |X|$ where $W$ is a partial isometry, and $|X| = (X^* X)^{1/2}$ .\",\n \"Since $|X|$ is compact and positive, it has an orthonormal basis of eigenvectors.\",\n \"This defines a unitary $V$ such that $V |X| V^* = D$ , where $D =\\\\sum _{k} d_k |k\\\\rangle \\\\langle k|$ is a diagonal matrix and the $d_k$ 's are the singular values of $X$ arranged in decreasing order.\",\n \"Conjugating $D$ by $V$ , we get $|X| = V^* D V$ .\",\n \"Combining this with the polar decomposition, we get $X = W|X| = WV^* D V = U D V$ where $U = WV^*$ is a partial isometry.\",\n \"Moreover, since $X$ has dense range, $U$ is a coisometry.\",\n \"Let $U = \\\\sum _{i, j} u_{i,j} |i\\\\rangle \\\\langle j|$ , and $V = \\\\sum _{i, j} v_{i, j}|i\\\\rangle \\\\langle j|$ .\",\n \"Then, $\\\\sum _{i,j} |x_{i,j}|^2 &=\\\\mbox{Tr}(X^*X)= \\\\mbox{Tr}(V^*D^*U^*UDV) \\\\\\\\&= \\\\mbox{Tr}(V^*D^2V) = \\\\mbox{Tr}(D^2) = \\\\sum _k d_k^2.$ Now let $u_k = \\\\sum _{i} u_{i, k} |i\\\\rangle $ and $v_k = \\\\sum _j v_{k, j} f_j$ .\",\n \"Since $U$ is a coisometry, by Proposition REF , $\\\\lbrace u_k\\\\rbrace $ is an orthonormal set.\",\n \"Now we have $x = \\\\sum _{i, j} x_{i, j} |i\\\\rangle \\\\otimes f_j = \\\\sum _{i, j, k} u_{i, k} d_k v_{k, j} |i\\\\rangle \\\\otimes f_j = \\\\sum _{k} d_k u_k\\\\otimes v_k.$ The statement about the uniqueness of the sequence $d_k$ follows from the fact that these numbers are the singular values of the Hilbert-Schmidt matrix $X$ and that any other choice of basis for representing $X$ would give rise to a matrix that is obtained from $X$ by pre and post multiplying by unitaries, which does not alter the singular values.\"\n ],\n [\n \"A primer on C*-algebras\",\n \"For readers unfamiliar with C*-algebras, we briefly mention the definitions and tools that we shall use.\",\n \"For very readable general references we recommend [3] or [7].\",\n \"Given a Hilbert space $\\\\mathcal {H}$ we let $B(\\\\mathcal {H})$ denote the set of bounded linear operators from $\\\\mathcal {H}$ to $\\\\mathcal {H}$ .\",\n \"By a C*-algebra of operators we mean a subset $\\\\mathcal {A} \\\\subseteq B(\\\\mathcal {H})$ for some Hilbert space $\\\\mathcal {H}$ satisfying: $X, Y \\\\in \\\\mathcal {A}, \\\\lambda \\\\in \\\\mathbb {C} \\\\Rightarrow (\\\\lambda X+Y) \\\\in \\\\mathcal {A}$ and $XY \\\\in \\\\mathcal {A}$ , $X \\\\in \\\\mathcal {A} \\\\Rightarrow X^* \\\\in \\\\mathcal {A}$ , where $X^*$ denotes the adjoint of the operator $X$ (sometimes denoted by $X^{\\\\dag }$ in the physics literature), $\\\\mathcal {A}$ is closed in the operator norm, i.e., if $X_n \\\\in \\\\mathcal {A}$ , $X \\\\in B(\\\\mathcal {H})$ and $\\\\Vert X_n - X \\\\Vert \\\\rightarrow 0,$ then $X \\\\in \\\\mathcal {A}$ .\",\n \"The first condition is the definition of what it means to say that $\\\\mathcal {A}$ is an algebra over the complex field.\",\n \"The second condition is that $\\\\mathcal {A}$ be invariant under the taking of operator adjoints and the third is that it be a closed subset of $B(\\\\mathcal {H})$ in a certain topology.\",\n \"C*-algebras of operators also have an abstract characterization.\",\n \"An algebra $\\\\mathcal {A}$ over the complex numbers that is equipped with a norm $\\\\Vert \\\\cdot \\\\Vert $ that satisfies $\\\\Vert xy\\\\Vert \\\\le \\\\Vert x\\\\Vert \\\\cdot \\\\Vert y\\\\Vert $ is called a normed algebra.\",\n \"If a normed algebra is complete, i.e., if every Cauchy sequence converges, then it is called a Banach algebra.\",\n \"Given an algebra $\\\\mathcal {A}$ over the complex numbers, a *-map is a map from $\\\\mathcal {A}$ to $\\\\mathcal {A}$ , $^*: \\\\mathcal {A} \\\\rightarrow \\\\mathcal {A}$ satisfying $(x+y)^*= x^*+y^*$ , $(\\\\lambda x)^* = \\\\overline{\\\\lambda } x^*,$ $(xy)^*= y^*x^*$ and $(x^*)^*=x$ .\",\n \"An algebra equipped with a *-map is called a *-algebra.\",\n \"A map $\\\\pi : \\\\mathcal {A} \\\\rightarrow \\\\mathcal {B}$ between two algebras that is linear and satisfies $\\\\pi (xy) = \\\\pi (x) \\\\pi (y)$ is called a homomorphism.\",\n \"If both algebras are also *-algebras and the map satisfies $\\\\pi (x^*) = \\\\pi (x)^*$ then $\\\\pi $ is called a *-homomorphism.\",\n \"Finally, an (abstract) C*-algebra $\\\\mathcal {A}$ is a Banach *-algebra that satisfies $\\\\Vert x^*x\\\\Vert = \\\\Vert x\\\\Vert ^2, \\\\, \\\\forall x \\\\in \\\\mathcal {A}$ .\",\n \"Note that $B(\\\\mathcal {H})$ is an abstract C*-algebra and so is every C*-algebra of operators.\",\n \"The celebrated Gelfand-Naimark-Segal theorem shows that every abstract C*-algebra is in an appropriate sense a C*-algebra of operators.\",\n \"Theorem 2.1 (Gelfand-Naimark-Segal) Let $\\\\mathcal {A}$ be an abstract C*-algebra.\",\n \"Then there is a Hilbert space $\\\\mathcal {H}$ and a map $\\\\pi : \\\\mathcal {A} \\\\rightarrow B(\\\\mathcal {H})$ such that: $\\\\pi $ is a *-homomorphism, $\\\\Vert \\\\pi (x)\\\\Vert = \\\\Vert x\\\\Vert $ for all $x \\\\in \\\\mathcal {A}$ .\",\n \"Moreover, if $\\\\mathcal {A}$ has a unit element $1 \\\\in \\\\mathcal {A}$ then, in addition, one can arrange that $\\\\pi (1) = I_{\\\\mathcal {H}}$ .\",\n \"A map satisfying the second condition is called an isometry.\",\n \"Clearly, an isometry is one-to-one.\",\n \"Conversely, it is a theorem that every one-to-one *-homomorphism is an isometry.\",\n \"A one-to-one, onto *-homomorphism is called a *-isomorphism.\",\n \"The two conditions in the above theorem also guarantee that the range of $\\\\pi ,$ $\\\\mathcal {B}= \\\\pi (\\\\mathcal {A})$ is a C*-algebra of operators.\",\n \"Thus, $\\\\pi $ is a *-isomorphism from the abstract C*-algebra onto a C*-algebra of operators.\",\n \"A key element of the proof of the above theorem is their theorem on representations of states.\",\n \"A state on an abstract unital C*-algebra $\\\\mathcal {A}$ is any linear functional, $s: \\\\mathcal {A} \\\\rightarrow \\\\mathbb {C}$ such that $s(1) =1$ and $s(x^*x) \\\\ge 0$ for every $x \\\\in \\\\mathcal {A}$ .\",\n \"Theorem 2.2 (GNS state representation theorem) Let $s:\\\\mathcal {A} \\\\rightarrow \\\\mathbb {C}$ be a state on a unital C*-algebra.\",\n \"Then there exists a Hilbert space $\\\\mathcal {H}_s$ , a unit vector $\\\\xi \\\\in \\\\mathcal {H}_s$ and a unital *-homomorphism, $\\\\pi _s: \\\\mathcal {A} \\\\rightarrow B(\\\\mathcal {H}_s)$ such that $s(x) = \\\\langle \\\\xi | \\\\pi _s(x) \\\\xi \\\\rangle $ for all $x \\\\in \\\\mathcal {A}$ and such that the subspace of vectors of the form $\\\\lbrace \\\\pi _s(x) \\\\xi : x \\\\in \\\\mathcal {A} \\\\rbrace $ is dense in $\\\\mathcal {H}_s$ .\",\n \"Thus, the theorem says that for each abstract state there is a way to realize the C*-algebra as a C*-algebra of operators such that the state becomes a vector state.\"\n ],\n [\n \"The State Space\",\n \"Given a unital C*-algebra $\\\\mathcal {A}$ , the set of all states on $\\\\mathcal {A}$ , denoted $S(\\\\mathcal {A})$ is a convex set.\",\n \"Moreover, it is endowed with a topology, called the weak*-topology and in this topology it is a compact set.\",\n \"A net of states $\\\\lbrace s_{\\\\lambda } \\\\rbrace $ converges to a state $s$ in this topology if and only if $\\\\lim _{\\\\lambda } | s_{\\\\lambda }(a) - s(a)| =0$ for every $a \\\\in \\\\mathcal {A}$ .\"\n ],\n [\n \"Tensor Products of C*-algebras\",\n \"Let $\\\\mathcal {A}$ and $\\\\mathcal {B}$ be two unital C*-algebras, and let $\\\\mathcal {A}\\\\otimes \\\\mathcal {B}$ be their algebraic tensor product.\",\n \"Given $x= \\\\sum _i a_i \\\\otimes b_i$ and $y = \\\\sum _j c_j \\\\otimes d_j$ in $\\\\mathcal {A} \\\\otimes \\\\mathcal {B}$ we define their product by $ xy = \\\\sum _{i,j} a_i c_j \\\\otimes b_i \\\\otimes d_j,$ and a *-map by $x^* = \\\\sum _i a_i^* \\\\otimes b_i^*.$ Endowed with these two operations, $\\\\mathcal {A} \\\\otimes \\\\mathcal {B}$ becomes a *-algebra.\",\n \"Note that the *-subalgebra $\\\\lbrace a \\\\otimes 1: a \\\\in \\\\mathcal {A} \\\\rbrace $ can be identified with $\\\\mathcal {A}$ and similarly, $\\\\lbrace 1 \\\\otimes b: b \\\\in \\\\mathcal {B} \\\\rbrace $ can be identified with $\\\\mathcal {B}$ .\",\n \"Also $(a \\\\otimes 1)(1 \\\\otimes b) = a \\\\otimes b = (1 \\\\otimes b)(1 \\\\otimes a)$ so that these “copies\\\" of $\\\\mathcal {A}$ and $\\\\mathcal {B}$ commute.\",\n \"There are two important ways to give this *-algebra a norm so that it can be completed to become a C*-algebra.\",\n \"Given $x \\\\in \\\\mathcal {A} \\\\otimes \\\\mathcal {B}$ we set $\\\\Vert x\\\\Vert _{\\\\max } = \\\\sup \\\\lbrace \\\\Vert \\\\pi (x) \\\\Vert : \\\\ \\\\pi : \\\\mathcal {A} \\\\otimes \\\\mathcal {B} \\\\rightarrow B(\\\\mathcal {H}) \\\\text{ is a unital *-homomorphism}\\\\rbrace , $ where the supremum is taken over all Hilbert spaces $\\\\mathcal {H}$ and all unital *-homomorphisms.\",\n \"The completion of $\\\\mathcal {A} \\\\otimes \\\\mathcal {B}$ in this norm is a C*-algebra denoted $\\\\mathcal {A} \\\\otimes _{\\\\max } \\\\mathcal {B}$ .\",\n \"Alternatively, if $\\\\pi _1: \\\\mathcal {A} \\\\rightarrow B(\\\\mathcal {H}_1)$ and $\\\\pi _2: \\\\mathcal {B} \\\\rightarrow B(\\\\mathcal {H}_2)$ are unital *-homomorphisms, then setting $\\\\pi (a \\\\otimes b) = \\\\pi _1(a) \\\\otimes \\\\pi _2(b) \\\\in B(\\\\mathcal {H}_1 \\\\otimes \\\\mathcal {H}_2)$ and extending linearly, defines a unital *-homomorphism from $\\\\mathcal {A} \\\\otimes \\\\mathcal {B}$ into $B(\\\\mathcal {H}_1 \\\\otimes \\\\mathcal {H}_2)$ denoted by $\\\\pi = \\\\pi _1 \\\\otimes \\\\pi _2$ .\",\n \"Given $x \\\\in \\\\mathcal {A} \\\\otimes \\\\mathcal {B}$ we set $ \\\\Vert x\\\\Vert _{\\\\min } = \\\\sup \\\\lbrace \\\\Vert \\\\pi _1 \\\\otimes \\\\pi _2(x) \\\\Vert : \\\\,\\\\, \\\\pi _1: \\\\mathcal {A} \\\\rightarrow B(\\\\mathcal {H}_1), \\\\, \\\\, \\\\pi _2: \\\\mathcal {B} \\\\rightarrow B(\\\\mathcal {H}_2) \\\\\\\\ \\\\text{ are unital *-homomorphisms}\\\\rbrace .$ The completion of $\\\\mathcal {A} \\\\otimes \\\\mathcal {B}$ in this norm is a C*-algebra denoted $\\\\mathcal {A} \\\\otimes _{\\\\min } \\\\mathcal {B}$ .\"\n ]\n]"},"id":{"kind":"string","value":"1606.05061"}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":11,"numItemsPerPage":100,"numTotalItems":1867602,"offset":1100,"length":100}},"jwt":"eyJhbGciOiJFZERTQSJ9.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc1MTAxNTA1MCwic3ViIjoiL2RhdGFzZXRzL2hvd2V5L3VuYXJYaXZlIiwiZXhwIjoxNzUxMDE4NjUwLCJpc3MiOiJodHRwczovL2h1Z2dpbmdmYWNlLmNvIn0.dEIVhheb13GsKf54r8JzpDZwejk3sBacZG89kyiqOd9jh4uZOasPGrdPalUUHC0seqQcwJeIRhuma_PKO9pTBw","displayUrls":true},"discussionsStats":{"closed":0,"open":1,"total":1},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
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[
"A Maximum Principle for hypersurfaces in $\\mathbb{R}^{n+1}$ with an\n Ideal Contact at Infinity and Bounded Mean Curvature"
],
[
"Abstract We will generalize a Maximum Principle at Infinity in the parabolic case given by De Lima [Ann.",
"Global Anal.",
"Geom.",
"${\\bf 20}$, 325-343 2001] and De Lima and Meeks [Indiana Univ.",
"Math.",
"Journal ${\\bf 53}$ 5, 1211-1223 2004], for disjoints hypersurfaces of $\\mathbb{R}^{n+1}$ with bounded mean curvature without restrictions on the Gaussian curvature.",
"We will also extend for hypersurfaces in $\\mathbb{R}^{n+1}$ a generalization of Hopf's Maximum Principle for hypersurfaces that get close asymptotically."
],
[
"Introduction",
"A classical result in Differential Geometry is the Hopf's Maximum Principle for Hypersurfaces in $\\mathbb {R}^{n+1}$ , which states that under certain conditions related to the Mean Curvature, if two hypersurfaces $M_1$ and $M_2$ are tangent at an interior point $p\\in M_1\\cap M_2$ and this point is an Ideal Contact at $p$ (see Definition REF ), then they coincide in a neighbourhood of $p$ (Theorem REF ).",
"Thinking about this type of contact between two hypersurfaces, De Lima, [1], [2] , and Meeks, [2], established an ideal contact between two disjoints surfaces $M_{1}$ and $M_{2}$ in $\\mathbb {R}^{3}$ , which generalizes the Ideal Contact at $p$ for disjoints surfaces that get asymptotically close to each other.",
"This approximation was name Ideal Contact at infinity (see Definition REF ).",
"Assuming an Ideal Contact at Infinity between the surfaces $M_{1}$ and $M_{2}$ in $\\mathbb {R}^{3}$ , De Lima demonstrates the following Maximum Principle at Infinity for surfaces with bounded Gaussian curvature and constant Mean Curvature $H\\ne 0$ , [1].",
"Theorem 1.1 Let $M_1$ and $M_2$ be two disjoints, complete and properly embedded $H$ -surfaces in $\\mathbb {R}^3$ , with bounded Gaussian Curvature and non-empty boundaries $\\partial M_{1}$ and $\\partial M_{2}$ .",
"If $M_{1}$ and $M_{2}$ have an ideal contact at infinity and either $M_{1}$ or $M_{2}$ is parabolic, then $\\min \\lbrace dist(M_{1},\\partial M_{2}),dist(M_{2},\\partial M_{1})\\rbrace =0.$ In 2004, De Lima, [1], [2], along with Meeks managed to prove in [2] the following non-parabolic version with bounded Gaussian and mean curvatures of Theorem REF .",
"Theorem 1.2 Let $M_1$ be a surface with boundary $\\partial M_1$ and bounded Gaussian curvature, which is properly embedded in $\\mathbb {R}^3$ and whose mean curvature satisfies $b_{0}\\le H_{M_1}\\le b_{1}$ , $b_{0},b_{1}>0$ .",
"Assume $M_2$ is a surface with boundary $\\partial M_2$ , which is properly immersed in $\\mathbb {R}^3$ and such that $\\vert H_{M_2}\\vert \\le b_0$ .",
"Then, if $M_2$ has an contact ideal at infinity wich $M_1$ , one has $\\min \\lbrace dist(M_1,\\partial M_2),dist(M_2,\\partial M_1)\\rbrace =0.$ As an application of Theorem REF De Lima and Meeks also proved the following theorem, which generalizes Hopf's Maximum Principle for surfaces in $\\mathbb {R}^{3}$ with an ideal contact at infinity and bounded Gaussian and mean curvatures: Theorem 1.3 Suppose $M_{1}$ is a properly embedded surface in $\\mathbb {R}^3$ without boundary and of bounded Gaussian Curvature.",
"If the mean curvature function of $M_{1}$ satifies $b_{0}\\le H_{M_1}\\le b_{1}$ , $b_{0},b_{1}>0$ , the surface $M_{2}$ without boundary, which is properly immersed in $\\mathbb {R}^3$ and whose mean curvature satisfies $\\vert H_{M_2}\\vert \\le b_{0}$ , cannot lie on the mean convex side of $M_{1}$ .",
"In [1] De Lima proved the parabolic version of Theorem REF assuming that $M_{1}$ and $M_{2}$ are $H$ -surfaces, $H\\ne 0$ .",
"Our objective in this article is to extend to hypersurfaces of $\\mathbb {R}^{n+1}$ Theorems REF and REF above.",
"It will be done in Theorem REF where we prove the Maximum Principle at Infinity for properly embedded and disjoints hypersurfaces $M_{1}$ and $M_{2}$ in $\\mathbb {R}^{n+1}$ with nonempty boundaries.",
"To do that, we will suppose that $M_{2}$ is complete and that $\\sup _{M_2}\\vert {\\bf H}_{M_2}\\vert \\le \\inf _{M_1} H_{M_1}$ , where $H_{M_1}$ is the mean curvature of $M_{1}$ and ${\\bf H}_{M_2}$ is the mean curvature vector of $M_2$ .",
"We will also assume that $M_{2}$ have an ideal contact at infinity with $M_1$ and that $M_{2}$ is parabolic.",
"However, we will not consider any additional hypothesis about the Gaussian curvature of any hypersurface.",
"We will need two lemmas that will be proved in section , lemmas REF and REF .",
"As an application of Theorem REF we will prove Theorem REF , which is a generalization of Hopf's Maximum Principle for hypersurfaces with an ideal contact, which extends Theorem REF .",
"Such theorem states that under certain conditions, like the ones in Theorem REF , if $M_{2}$ has an empty boundary then it cannot be on the convex side of $M_{1}$ .",
"In Theorem REF we will extend Theorem REF to the case where $M_{1} \\subset \\mathbb {R}^{n+k}$ is a hypersurface and $M_{2} \\subset \\mathbb {R}^{n+k}$ a parabolic $n$ -submanifold, which generalizes Theorem 4 in [3] for asymptotic hypersurfaces.",
"In Theorem REF , as in its corollaries, we used the Omory-Yau's Maximum Principle, see [11], [12], and prove an analogous result to Theorem REF without the hypothesis of $M_2\\subset \\mathbb {R}^{n+1}$ be a parabolic hypersurface, obtaining another generalization of Hopf's Maximum Principle for asymptotics hypersurfaces of $\\mathbb {R}^{n+1}$ ."
],
[
"Preliminaries",
"Given a smooth and oriented hypersurface $M\\subset \\mathbb {R}^{n+1}$ , we denote the mean curvature function and the mean curvature vector of $M$ as $H_{M}$ and $\\textbf {H}_{M}$ , respectively.",
"We will also denote by $\\nabla ^{M}$ and $\\Delta ^{M}$ the gradient and Laplacian of $M$ , respectively."
],
[
"Parabolic Riemannian manifold ",
"Let $M^{n}$ be an $n$ -dimensional Riemannian manifold with smooth (possibly empty) boundary $\\partial M$ and $\\Omega \\subset M$ an open set of $M$ .",
"A function $h\\in C^2(\\Omega )$ is said subharmonic if $\\Delta ^Mh\\ge 0.$ Subharmonic functions will play an important role in this section in that we will address a class of Riemannian manifolds that are characterized by these functions.",
"When a Riemannian manifold $M$ has an empty boundary $\\partial M$ we will say that $M$ is parabolic if it cannot exist a non-constant upper bounded subharmonic function, namely, $M$ is parabolic if $\\Delta ^Mh\\ge 0$ and $\\sup _Mh<+\\infty $ we have $h$ cosntant.",
"As an example of such Riemannian manifolds Cheng e Yau showed that $M=(\\mathbb {R}^2,\\langle ,\\rangle _{can})$ is a parabolic Riemannian manifold, [22].",
"In the case when $\\partial M$ is nonempty will use the following definition of parabolic manifold given by De Lima in [1].",
"Definition 2.1 An $n$ -dimensional complete Riemannian manifold $M^{n}$ with nonempty smooth boundary $\\partial M$ is called parabolic if for any upper bounded subharmonic function $h$ in $M$ , we have $\\sup _{M}h=\\sup _{\\partial M}h.$ The next result, whose proof can be found in [1], gives a sufficient condition for a Riemannian manifold with nonempty boundary be parabolic.",
"Proposition 2.1 (Proposition 2 of [1]) Let $M$ be a complete Riemannian manifold with nonempty, smooth boundary $\\partial M$ .",
"If there exist a proper, positive harmonic funcition defined on $M$ , then $M$ is parabolic.",
"Example 2.1 Let $C$ be a cylinder in $\\mathbb {R}^3$ given by $C=\\lbrace (x_1,x_2,x_3):x_1^2+x_2^2=1,\\;x_3\\ge 1\\rbrace $ .",
"Consider the parametrization given by $X(\\theta ,r)=(\\cos \\theta ,\\sin \\theta ,r)$ , $ 0\\le \\theta < 2\\pi $ , $r\\ge 1$ .",
"It is easy to see that the function $h(\\theta ,r)=r$ on $C$ is positive, proper and harmonic.",
"Thus, Proposition REF gives $C$ parabolic.",
"Next, we will state a proposition of fundamental importance in the demonstration of our main result, whose proof can be found in [1].",
"In the following, $M^{n}$ is a complete, $n$ -dimensional Riemannian manifold, with a (possibly empty) smooth boundary $\\partial M$ and $M^{\\prime }\\subset M$ a complete and embedded $n$ -dimensional Riemannian submanifold of $M$ with nonempty smooth boundary $\\partial M^{\\prime }$ .",
"Proposition 2.2 (Prposition 3 of [1]) Let $M$ and $M^{\\prime }$ be like above.",
"Then $M^{\\prime }$ is parabolic if $M$ is parabolic.",
"We will also use the following lemma, found in [5].",
"Lemma 2.1 (Lemma 2.3 of [5]) Let $A$ be a quadratic form in an $n$ -dimensional Euclidean vectorial space with eigenvalues $\\lambda _ {1} \\le \\cdots \\le \\lambda _{k} \\le \\cdots \\le \\lambda _{n}$ .",
"Then for any $k$ -dimensional subspace $W \\subset V$ we have $\\mathrm {tr} A \\mid _{W}\\ge \\lambda _{1} + \\cdots + \\lambda _{k} .$ Comparing to the maximum principle, we will define next the ideal contact at a point (Definition REF ) and show Hopf's Maximum Principle (Theorem REF ) which, inspired the concept of an ideal contact at infinity (see Definition REF ) for hypersurfaces, [1], [2].",
"Definition 2.2 (Ideal Contact at $p$ ) Let $M_{1}$ and $M_{2}$ be two oriented hypersurfaces in $\\mathbb {R}^{n+1}$ .",
"If $M_{1}$ and $M_{2}$ are tangent at an interior point $p$ and has the same unit normal $\\eta _{0}$ at $p$ , we will say that they have an Ideal contact at $p$.",
"We also say that $M_{1}$ lies above $M_{2}$ near $p$ with respect to $\\eta _{0}$ , if when we express $M_{1}$ and $M_{2}$ as graphics of function $\\phi _{1}$ and $\\phi _{2}$ over the tangent hyperplan in $p$ we have $\\phi _{1} \\ge \\phi _{2}$ in a neighbourghood of $p$ .",
"Theorem 2.1 (Hopf's Maximum Principle, [7]) Let $M_1$ and $M_2$ be oriented hypersurfaces in $\\mathbb {R}^{n+1}$ which have a contact at a point p. Let $H_{M_1}$ and $H_{M_2}$ be their mean curvature function, respectively.",
"If $H_{M_1} \\le H_{M_2}$ at $p$ then $M_1$ cannot lie above $M_2$ , unless they coincide in a neighborhood of $p$ ."
],
[
"The Maximum Principle at Infinity for hypersurfaces in $\\mathbb {R}^{n+1}$",
"We introduze now, and we will use it in the course of this work, the definition of Ideal Contact at Infinity used by De Lima and Meeks, [2].",
"Definition 3.1 (Ideal contact at infinity) Let $M_{1}$ be a propperly embedded hypersurface in $\\mathbb {R}^{n+k}$ with a positive mean curvature function.",
"We say that an $n$ -dimensional submanifold $M_{2} \\subset \\mathbb {R}^{n+k}$ has an Ideal Contact at Infinity with $M_{1}$ if $M_{1}$ and $M_{2}$ are disjoints and there exist sequences of interior points $y_i\\in M_1$ , $x_i\\in M_2$ and $\\lambda _i >0$ , $i\\in \\mathbb {N}$ , with $\\vert y_i - x_i\\vert \\rightarrow 0$ and $x_i - y_i = \\lambda _i{\\bf H}_{M_1}(y_i)$ always when $i\\rightarrow +\\infty $ .",
", Figure REF .",
"Here, as in [2], we say that two disjoints and properly imersed hypersurfaces $M_1$ and $M_2$ , with nonempty boundaries $\\partial M_1$ and $\\partial M_2$ satisfy the Maximum Principles at Infinity if $dist(M_1,M_2)=\\min \\lbrace dist (M_1,\\partial M_2), dist(M_2,\\partial M_1)\\rbrace ,$ were $dist$ is the distance in $\\mathbb {R}^{n+1}$ .",
"Figure: Ideal Contact at InfinityIn this section we state the main result of this paper.",
"Here we suppose $M_{1} \\subset \\mathbb {R}^{n+1}$ to be an oriented smooth hypersurface with positive mean curvature $H_{M_{1}}$ .",
"Theorem 3.1 (Maximum Principles at Infinity) Let $M_{1}$ and $M_{2}$ two propperly embedded and disjoints hypersurfaces in $\\mathbb {R}^{n+1}$ with nonempty boundaries $\\partial M_{1}$ and $\\partial M_{2}$ .",
"Suppose that $M_{2}$ is complete and that $\\sup _{M_2}\\vert \\mathbf {H}_{M_2}\\vert \\le b_0\\le \\inf _{M_1}H_{M_1},\\ \\ b_0>0.$ If $M_{2}$ has an ideal contact at infinity with $M_{1}$ and $M_2$ is parabolic, then $\\min \\lbrace dist(M_1,\\partial M_2),dist(M_2,\\partial M_1)\\rbrace =0.$ Such theorem is a generalization of Theorems REF and REF .",
"Its demonstration leads us to Theorem REF which is a generalization of Hopf's Maximum Principle in $\\mathbb {R}^{n+1}$ for hypersurfaces with an ideal contact at infinity (see Definition REF ) in a way that they are disjoints hypersurfaces that approach each other asymptotically.",
"Exemple REF bellow shows that Theorem REF may be false without the hypothesis of Ideal Contact at Infinity.",
"Example 3.1 Let $M_1$ be the surface of revolution obtained by rotating the curve $\\alpha (t)=(t,0,\\dfrac{1}{1-t^2})$ , $0<t_0< t<1$ , about the $z$ axis and $M_2$ the cylinder $M_2=\\lbrace (x,y,z):x^2+y^2=1,\\; z>z_0>0$ }.",
"By Exemple REF $M_2$ is parabolic, is disjoint of $M_1$ and we have $\\sup _{M_2} \\vert H_{M_2}\\vert =\\dfrac{1}{2}$ .",
"We also have $\\inf _{M_1}H_{M_1}=\\dfrac{1}{2}$ , but $0=dist(M_1,M_1)<\\min \\lbrace dist(M_1,\\partial M_2),dist(M_2,\\partial M_1)\\rbrace .$ This happens because there is no $y\\in M_1$ , $x\\in M_2$ and $\\lambda >0$ such that, $x-y=\\lambda \\mathbf {H}_{M_1}(y)$ , that is, $M_2$ does not have an ideal contact at infinity with $M_1$ ."
],
[
"Preliminary results",
"In the demonstration of Theorem REF we will use Lemmas REF and REF , that demand the following assumptions: Let $M_{1} \\subset \\mathbb {R}^{n+k}$ be a hypersurface, at least $C^{2}$ , with mean curvature $H_{M_{1}}$ according to the unit normal $\\eta $ .",
"We denote by $_n:=\\dfrac{1}{n}(\\lambda _1 + \\cdots + \\lambda _n)$ the $n$ -th mean curvature with respect to $\\eta $ , where $\\lambda _{1} \\le \\lambda _{2}\\le \\cdots \\le \\lambda _{n+k-1} $ are the principal curvatures of $M_{1}$ with respect to $\\eta $ .",
"Also let $M_2\\subset \\mathbb {R}^{n+k}$ be a $n$ -dimensional $C^{2}$ submanifold, $n\\ge 1$ , with mean curvature vector ${\\bf H}_{M_2}=-\\dfrac{1}{n}\\sum _{r=1}^{k}(\\textmd {div}\\;\\eta ^r)\\eta ^r,$ where $\\eta ^{1}, \\cdots , \\eta ^{k}$ are orthonormal vector fields normal to $M_{2}$ .",
"Let $d$ be the distance function $d(x):=dist(x,M_{1})$ .",
"Such function is of class $C^{2}$ in a neighborhood of $M_{1}$ , Lipschitz with constant 1 and oriented by the choice of $\\eta $ , i.e., $\\eta (y)=Dd(x)$ , where $y\\in M_{1}$ is such that $\\vert x-y\\vert =d(x)$ and $D=\\left(\\dfrac{\\partial }{\\partial x^1}, \\dots , \\dfrac{\\partial }{\\partial x^{n+k}}\\right)$ is the gradient of $\\mathbb {R}^{n+k}$ , see [3], [6] .",
"This way, the point $x$ is such as $x=y+d(x)\\eta (y)$ .",
"For each $x$ close to $M_{1}$ we consider a parallel hypersurface $M_{1d(x)}=\\lbrace p\\in \\mathbb {R}^{n+k}:d(p)=d(x)\\rbrace =d^{-1}(d(x)).$ Such hypersurfaces are of class $C^{2}$ and have principal curvatures at $x_{0}$ given by $\\dfrac{\\lambda _1(y_0)}{1-\\lambda _1d(x_0)}\\le \\dfrac{\\lambda _2(y_0)}{1-\\lambda _2d(x_0)}\\le \\cdots \\le \\dfrac{\\lambda _{n+k-1}(y_0)}{1-\\lambda _{n+k-1}d(x_0)},$ where $y_{0} \\in M_{1}$ is such that $\\vert x_{0} - y_{0}\\vert = d(x_0)$ and $\\lambda _1(y_0)\\le \\lambda _2(y_0)\\le \\cdots \\le \\lambda _{n+k-1}(y_0)$ are the principal curvatures of $M_{1}$ at $y_{0}$ if $\\vert d\\vert \\ll 1$ , [3], [6].",
"Observe that this give us $\\dfrac{1}{n}\\left(\\dfrac{\\lambda _1(y_0)}{1-\\lambda _1d(x_0)}+ \\dfrac{\\lambda _2(y_0)}{1-\\lambda _2d(x_0)}+\\cdots + \\dfrac{\\lambda _{n}(y_0)}{1-\\lambda _{n}d(x_0)}\\right)\\ge \\dfrac{1}{n}(\\lambda _1 + \\cdots + \\lambda _n)$ for any hypersurface $M_{1d}$ .",
"In other words, the $n$ -mean curvature $_{n}(x_{0})$ of $M_{1d}$ at $x_{0}$ is not smaller than the $n$ -mean curvature of $M_1$ at $y_0$ .",
"In the following let $\\lbrace e_{1}, e_{2}, \\cdots , e_{n}\\rbrace $ be an orthonormal basis of $T_xM_2$ and denote by $e_i^{T}:=e_i-\\langle e_i, \\eta \\rangle \\eta $ the orthogonal projection of $e_{i}$ over the tangent space $T_xM_{1d}$ .",
"Also let $T_xM_2^{T}$ be the orthogonal projection space of $T_xM_2$ over $T_xM_{1d}$ .",
"Finally, let $II_d$ and $A_d$ be the second fundamental form and the shape operator of $M_{1d}$ , with respect to $\\eta $ , respectively.",
"The following lemma is an adaptation of Lemma 1 in [3], with an analogue demonstration.",
"Lemma 3.1 Let $M_{1}$ and $M_{2}$ be like above and $d$ be the distance function $d(x)=dist(x,M_{1})$ .",
"Suppose in addition that $M_{2}$ has an ideal contact at infinity with $M_{1}$ .",
"So, we have $\\Delta ^{M_2}d - \\displaystyle \\sum _{i,j}^{n}_{ij}II_d(e_i^T,e_j^T) - n\\langle {\\bf H}_{M_2},Dd\\rangle + \\mathrm {tr}A_d\\mid _{T_xM_2^T}=0,$ where $_{ij}=\\dfrac{\\nabla _{e_i}^{M_2}d\\nabla _{e_j}^{M_2}d}{1-\\vert \\nabla ^{M_2}d\\vert ^2}$ and $\\nabla _{e_i}^{M_2}$ is the derivative in the direction of $e_{i}$ .",
"Let $x \\in M_{2}$ be such that $d(x)\\ll 1$ .",
"Such point exist because $M_{2}$ has an ideal contact at infinity with $M_{1}$ .",
"Then, if $\\eta ^{1}, \\cdots ,\\eta ^{k}$ form an orthonormal basis of $T_xM_2^{\\perp }$ and $Dd=\\eta $ is the Euclidian gradient of $d$ , we have that $\\nabla ^{M_2}d= Dd - \\langle Dd,\\eta ^1\\rangle \\eta ^1 - \\cdots - \\langle Dd,\\eta ^k\\rangle \\eta ^k$ and $\\begin{array}{rlll}\\Delta ^{M_2}d & =& \\textmd {div}\\nabla ^{M_2}d&= \\textmd {div}\\; Dd - \\displaystyle \\sum _{r=1}^{k} \\langle Dd,\\eta ^r\\rangle \\textmd {div}\\;\\eta ^r\\\\& & & = \\textmd {div}\\; Dd + n\\langle {\\bf H}_{M_2},Dd^{\\perp }\\rangle \\end{array}$ where $Dd^{\\perp }=\\displaystyle \\sum _{r=1}^k\\langle Dd,\\eta ^r\\rangle \\eta ^r$ is the normal component of $\\eta =Dd$ relative to $M_{2}$ and ${\\bf H}_{M_2}=-\\displaystyle \\dfrac{1}{n}\\sum _{r=1}^{k}(\\textmd {div}\\;\\eta ^r)\\eta ^r$ is the mean curvature vector of $M_{2}$ .",
"Denote by $\\overline{\\nabla }_{e_{i}}$ the Euclidian directional derivative in the direction of $e_{i}$ .",
"Then, as $e_i=e_i^T + \\langle e_i,\\eta \\rangle \\eta $ and $\\vert \\eta \\vert ^2=1,$ it follows from (REF ) that $\\begin{array}{rll}\\Delta ^{M_2}d &=&\\displaystyle \\sum _{i=1}^{n}\\langle e_i, \\overline{\\nabla }_{e_i}\\eta \\rangle + n\\langle {\\bf H}_{M_2},Dd^{\\perp }\\rangle \\\\&=& \\displaystyle \\sum _{i=1}^{n}\\langle e_i^T, \\overline{\\nabla }_{e_i^T}\\eta \\rangle + n\\langle {\\bf H}_{M_2},Dd^{\\perp }\\rangle \\\\&=& -\\displaystyle \\sum _{i=1}^{n}II_d( e_i^T, e_i^T) + n\\langle {\\bf H}_{M_2},Dd^{\\perp }\\rangle \\end{array}$ and as $Dd=\\nabla ^{M_2}d + Dd^{\\perp }$ , we have that $\\Delta ^{M_2}d - n\\langle {\\bf H}_{M_2},Dd\\rangle + \\displaystyle \\sum _{i=1}^{n}II_d( e_i^T, e_i^T)=0.",
"$ As $d(x)\\ll 1$ , because $M_2$ have a contact ideal at infinity with $M_1$ , we have that $\\vert \\nabla ^{M_2}d\\vert (x)< 1$ .",
"That is, $\\vert \\nabla ^{M_2}d\\vert (x)< 1$ if $dist(x,M_1)$ is smoll enough.",
"Then, if we put $\\begin{array}{rllll}g_{ij}&:=&\\langle e_i^T, e_j^T\\rangle =\\langle e_i - \\langle e_i,\\eta \\rangle \\eta ,e_j - \\langle e_j,\\eta \\rangle \\eta \\rangle \\\\&=&\\delta _{ij}-\\langle e_i,\\eta \\rangle \\langle e_j,\\eta \\rangle \\end{array}$ and as $\\vert \\nabla ^{M_2}d\\vert <1$ , we have that $g^{ij}$ , the inverse of $g_{ij}$ is given by $g^{ij}=\\delta _{ij} + \\dfrac{\\langle e_i,\\eta \\rangle \\langle e_j,\\eta \\rangle }{1-\\sum _{i=1}^{n}\\langle e_i, \\eta \\rangle ^2}=:\\delta _{ij}+_{ij}.$ And then, the trace of $A_d$ in $T_x{M_2}^T$ is given by $\\begin{array}{rll}\\mbox{tr}A_d\\mid _{T_x{M_2}^T}&=&\\displaystyle \\sum _{i,j}^{n}g^{ij}\\displaystyle II_d( e_i^T, e_j^T)\\\\&=&\\displaystyle \\sum _{i=1}^{n}II_d( e_i^T, e_i^T) + \\displaystyle \\sum _{i,j}^{n}_{ij}II_d( e_i^T, e_j^T).\\end{array}$ From (REF ) we have $\\Delta ^{M_2}d - n\\langle {\\bf H}_{M_2},Dd\\rangle + \\mbox{tr}A_d\\mid _{T_x{M_2}^T} - \\displaystyle \\sum _{i,j}^{n}_{ij}II_d( e_i^T, e_j^T)=0.",
"$ And the lemma goes on observing that $\\langle e_i,\\eta \\rangle = \\langle e_i,Dd\\rangle =\\langle e_i,\\nabla ^{M_2}d\\rangle + \\langle e_i,Dd^{\\perp }\\rangle = \\langle e_i,\\nabla ^{M_2}d\\rangle = \\nabla _{e_i}^{M_2}d$ and $\\nabla ^{M_2}d =\\sum _{i=1}^{n}(\\nabla _{e_i}^{M_2}d)e_i.$ Then $_{ij}= \\displaystyle \\dfrac{\\langle e_i,\\eta \\rangle \\langle e_j,\\eta \\rangle }{1-\\sum _{i=1}^{n}\\langle e_i, \\eta \\rangle ^2}= \\dfrac{\\nabla _{e_i}^{M_2}d\\nabla _{e_j}^{M_2}d}{1-\\vert \\nabla ^{M_2}d\\vert ^2} .$ Lemma 3.2 Let $M_{1}$ and $M_{2}$ be like in Lemma REF and $d(x)=dist(x,M_{1})$ .",
"Suppose that $M_{2}$ has an ideal contact at infinity with $M_{1}$ and that $\\sup _{M_2}\\vert {\\bf H}_{M_2}\\vert \\le \\inf _{M_1}_n.$ Then we have that $\\Delta ^{M_2}d -C_0 \\vert \\nabla ^{M_2}d\\vert ^2\\le 0,$ for some positive constant $C_0$ .",
"First we will prove that $- n\\langle {\\bf H}_{M_2},Dd\\rangle + \\mbox{tr}A_d\\mid _{T_xM_2^T}\\ge 0.",
"$ By Lemma REF we have that $\\begin{array}{rll}\\dfrac{1}{n}\\mbox{tr}A_d\\mid _{T_xM_2^T}&\\ge &\\dfrac{1}{n}\\left(\\dfrac{\\lambda _1(y)}{1-\\lambda _1d(x)}+ \\dfrac{\\lambda _2(y)}{1-\\lambda _2d(x)}+\\cdots + \\dfrac{\\lambda _{n}(y)}{1-\\lambda _{n}d(x)}\\right)\\\\&\\ge & \\dfrac{1}{n}(\\lambda _1(y) + \\cdots + \\lambda _n(y))\\end{array}$ where $y \\in M_{1}$ is such that $\\vert x-y\\vert = d(x)\\ll 1$ and $\\lambda _1(y), \\cdots , \\lambda _n(y)$ are the $n$ first principal curvatures of $M_{1}$ .",
"As we supposed that $\\sup _{M_2}\\vert {\\bf H}_{M_2}\\vert \\le \\inf _{M_1}_n, $ we conclude that $\\dfrac{1}{n}\\mbox{tr}A_d\\mid _{T_xM_2^T}\\ge \\vert {\\bf H}_{M_2}\\vert $ and by the Schwarz inequality, it follows that $\\langle {\\bf H}_{M_2}, Dd\\rangle \\le \\vert {\\bf H}_{M_2}\\vert \\vert Dd\\vert = \\vert {\\bf H}_{M_2}\\vert ,$ since $Dd =\\eta $ .",
"From this inequality we derive (REF ).",
"For the sake of simplicity of notation denote $\\nabla _{e_i}^{M_2}d=d_i$ .",
"Let $_{ij}$ be given by Lemma REF , then $\\begin{array}{rll}\\displaystyle \\sum _{i,j}^{n}_{ij}II_d( e_i^T, e_j^T)&=&\\displaystyle \\sum _{i,j}^{n}\\dfrac{II_d(e_i^T, e_j^T)d_id_j}{1-\\vert \\nabla ^{M_2}d\\vert ^2}\\\\&=&\\displaystyle \\sum _{i,j}^{n}\\dfrac{II_d( d_ie_i^T, d_je_j^T)}{1-\\vert \\nabla ^{M_2}d\\vert ^2}\\\\&=&\\displaystyle \\dfrac{II_d((\\nabla ^{M_2}d)^T,(\\nabla ^{M_2}d)^T)}{1-\\vert \\nabla ^{M_2}d\\vert ^2}\\\\&\\le & C_0\\vert \\nabla ^{M_2}d\\vert ^2, \\end{array}$ for a positive constant $C_{0}$ .",
"From $\\vert \\nabla ^{M_2}d\\vert \\ll 1$ we conclude the inequality of (REF ), because $M_{2}$ has an ideal contact at infinity with $M_{1}$ .",
"Using now (REF ), (REF ) and (REF ), we conclude the lemma."
],
[
"Proof of the main theorem",
"The demonstration of Theorem REF is similar to the demonstration in the case where we have $H$ -surfaces in $\\mathbb {R}^{3}$ given in Theorem 1 in [1].",
"The difference between them lies in the demonstrations of Lemmas REF and REF above, which equivalents in [1] are the Lemmas 3 and 4, respectively.",
"These will allow us to construct in a convenient set a subharmonic function and assuming by absurd that (REF ) is not true we reach a contradiction related to the parabolicity of $M_{2}$ .",
"Let's suppose that (REF ) is false, i.e., $m_0=\\min \\lbrace dist(M_1,\\partial M_2),dist(M_2,\\partial M_1)\\rbrace >0.$ For $\\epsilon >0$ sufficient small, let $M_2(\\epsilon )=\\lbrace x\\in M_2 : dist(x,M_1)\\le \\epsilon \\rbrace .$ For each $x\\in M_2(\\epsilon )$ , consider the set $S_x=\\lbrace y\\in M_1: \\vert y - x\\vert =dist(x,M_1) \\ \\ and \\ \\ x-y=\\lambda {\\bf H}_{M_1}(y), \\ \\ \\lambda > 0\\rbrace $ and finally define the set $M^{\\prime }_2(\\epsilon )\\subset M_2(\\epsilon )$ as $M^{\\prime }_2(\\epsilon )= \\lbrace x\\in M_2(\\epsilon ): S_x\\ne \\emptyset \\rbrace .$ Note that $M^{\\prime }_2(\\epsilon )$ is non-empty because $M_{1}$ and $M_{2}$ are propperly embedded in $\\mathbb {R}^{n+1}$ and have an ideal contact at infinity.",
"Let $C_2(\\epsilon )\\subset M^{\\prime }_2(\\epsilon )$ be a conex component of $M^{\\prime }_2(\\epsilon )$ .",
"Now take $\\epsilon >0$ such that $m_o>\\epsilon $ .",
"From that last assumption we have that $\\partial M_2\\cap C_2(\\epsilon )=\\emptyset $ .",
"In fact, if otherwise we had $x\\in C_2(\\epsilon )\\subset M^{\\prime }_2(\\epsilon )$ we would have $dist(x,M_1)\\le \\epsilon $ and $x\\in \\partial M_2$ we would have $dist(x,M_1)>\\epsilon $ , because $dist(M_1,\\partial M_2)>\\epsilon $ , which give us $\\partial C_2(\\epsilon )= \\lbrace x\\in C_2(\\epsilon ): dist(x,M_1)=\\epsilon \\rbrace .$ Consider now the distance function $d(x)=dist(x,M_1)$ .",
"By the lemma REF we have that $\\Delta ^{M_2}d - C_0\\vert \\nabla ^{M_2}d\\vert ^2\\le 0 $ for a positive constant $C_{0}$ .",
"Observe that $d\\mid _{\\partial C_2(\\epsilon )}\\equiv \\epsilon $ .",
"We also have that $C_2(\\epsilon )$ is not compact, otherwise we would have a $x^{\\prime }$ in the interior of $C_{2}(\\epsilon )$ such that $d(x^{\\prime })$ would be minimum, and in this case $\\nabla ^{M_2}d(x^{\\prime })=0$ and by (REF ) we would have $\\Delta ^{M_2}d(x^{\\prime })\\le 0$ , contrary to the fact that $x^{\\prime }$ is a inferior minimum point.",
"So $C_2(\\epsilon )$ is not compact and $\\sup _{C_2(\\epsilon )}d=\\epsilon $ .",
"Consider now a function $\\phi $ in $C_2(\\epsilon )$ given by $\\phi (x)=e^{-C_0d(x)}.$ Calculating $\\Delta ^{M_2}\\phi $ using (REF ), we will have that $\\Delta ^{M_2}\\phi = -C_0e^{-C_0d}(\\Delta ^{M_2}d - C_0\\vert \\nabla ^{M_2}d\\vert ^2)\\ge 0$ from which we can conclude that $\\phi $ is subharmonic in $C_{2}(\\epsilon )$ .",
"As we are assuming that $M_{2}$ is parabolic, we have by the Proposition REF that $C_2(\\epsilon )$ is parabolic.",
"So we should have $\\sup _{C_2(\\epsilon )}\\phi = \\sup _{\\partial C_2(\\epsilon )}\\phi = e^{-C_0 \\epsilon }$ which is a contradition because $\\sup _{C_2(\\epsilon )}\\phi = 1 > e^{-C_0 \\epsilon }=\\sup _{\\partial C_2(\\epsilon )}\\phi $ .",
"As this contradition came from the assumption that $m_0>0$ , we have that $\\min \\lbrace dist(M_1,\\partial M_2),dist(M_2,\\partial M_1)\\rbrace =0$ and the proof is complete.",
"In [14] Impera-Pigola-Setti they use the following definition of parabolic Riemannian manifol when $\\partial M\\ne \\emptyset $ (see also [15]).",
"Definition 3.2 Let $M$ an oriented Riemannian manifold with smooth boundary $\\partial M\\ne \\emptyset $ and exterior unit normal $\\nu $ .",
"$M$ is said to be $\\mathcal {N}$ -parabolic if the only solutions to the problem $\\left\\lbrace \\begin{array}{lll}\\Delta ^{M}h\\ge 0 & em& M\\\\\\dfrac{\\partial h}{\\partial \\nu }\\le 0 & em& \\partial M\\\\\\sup _{M}h<+\\infty \\end{array}\\right.$ is the constant function $h\\equiv \\sup _{M}h$ .",
"Getting the following proposition (Appendix A in [14]).",
"Proposition 3.1 Assume that $M$ is an $\\mathcal {N}$ -parabolic manifold with boundary $\\partial M \\ne \\emptyset $ and let $h$ be a solution of the problem $\\left\\lbrace \\begin{array}{lll}\\Delta ^{M}h\\ge 0 & em& M\\\\\\sup _{M}h<+\\infty .\\end{array}\\right.$ Then $\\sup _Mh=\\sup _{\\partial M}h.$ Proving thus that Definition REF implies the Definiton REF given by De Lima em [1].",
"Naturally if $\\partial M = \\emptyset $ , then the $\\mathcal {N}$ -parabolicity is equivalent the parabolicity.",
"In [20], [21] Grigor'yan proved the following theorem Theorem 3.2 Let $M$ a complete Riamannian manifold.",
"If for some point $o\\in M$ $\\dfrac{R}{Vol B_R^M(o)}\\notin L^1(+\\infty )$ or $\\dfrac{1}{Area(\\partial _0 B_R^M(o))}\\notin L^1(+\\infty )$ then $M$ is $\\mathcal {N}$ -parabolic.",
"Were, following the notation of [14], [15] for a non-necessarily connected open set $\\Omega \\subseteq M$ , we defined $\\partial _0\\Omega =\\partial \\Omega \\cap int M$ and $\\partial _1\\Omega =\\partial M\\cap \\Omega .$ Which gives us $\\partial _0 B_R^M(o)=\\partial B_R^M(o)\\cap int M.$ The Theorem REF has as corollary the next result proved by Cheng and Yau, telling us that $(\\mathbb {R}^2,\\langle ,\\rangle _{can})$ is an $\\mathcal {N}$ -parabolic Riemannian manifold, [22].",
"Corrollary 3.1 (Cheng-Yau) Let $M$ a complete Riemannian manifold.",
"If for some point $o\\in M$ and for some sequence $R_k\\rightarrow +\\infty $ $VolB^M_{R_k}(o)\\le cte.R^2_k,$ then $M$ is $\\mathcal {N}$ -parabolic.",
"Observin now that in the PMIProof of Theorem REF , we have that $\\partial _0C_2(\\epsilon )=\\partial C_2(\\epsilon )$ .",
"Therefore, the Ahlfors Maximum Principles, Theorem 7 in [14] (see also [15]), tells us that if $M_2$ is $\\mathcal {N}$ -parabolic in Theorem REF , that is, does not admit non constant function satisfying (REF ), then the function $\\phi (x)=e^{-C_0d(x)}$ that satisfies $\\Delta ^{M_2}\\phi \\ge 0$ on $C_2(\\epsilon )$ is such that $\\sup _{C_2(\\epsilon )}\\phi = \\sup _{\\partial _0 C_2(\\epsilon )}\\phi .$ Allowing us to reach the same contradiction PMIProof of Theorem REF .",
"Guaranteeing the validity of Maximum Principles at Infinity for $\\mathcal {N}$ -parabolic Riemannian manifold.",
"In [16], see also Appendix A in [14], Pessoa-Pigola-Setti they extend the notion of $\\mathcal {N}$ -parabolic Riemannian manifold with the next definition Definition 3.3 We say that a Riemannian manifold $M$ with nonempty boundary $\\partial M$ is $\\mathcal {D}$ -parabolic if every bounded function $h\\in C^{\\infty }(int\\; M)\\cap C^0(M)$ satisfiyng $\\left\\lbrace \\begin{array}{lll}\\Delta ^{M}h= 0 & em& int\\;M\\\\h=0&em&\\partial M,\\end{array}\\right.$ vanishes identically.",
"Not that of Proposition REF every $\\mathcal {N}$ -parabolic Riemannian manifold is $\\mathcal {D}$ -parabolic, but the converse is not true, see Example 4 of [16].",
"When $M$ is a $\\mathcal {D}$ -parabolic Riemannian manifold we have the following proposition.",
"Proposition 3.2 (Proposition 10 of [16]) Let $M$ be a manifold with boundary $\\partial M$ .",
"Then the following are equivalent: $M$ is $\\mathcal {D}$ -parabolic; For every domain $\\Omega \\subset M$ and every bounded function $h\\in C^{\\infty }(int\\; M)\\cap C^0(M)$ satisfying $\\Delta ^{M}h\\ge 0$ on $int\\Omega $ we have $\\sup _{\\Omega }h=\\sup _{\\partial \\Omega }h;$ For every bounded function $h$ satisfying $\\Delta ^{M}h\\ge 0$ on $int\\;M$ we have $\\sup _Mh=\\sup _{\\partial M}h.$ Now with the Proposition REF and with previous discussion we can suppose in Theorem REF $M_2$ an $\\mathcal {N}$ -parabolic or $\\mathcal {D}$ -parabolic hypersurface and guarantee the validity of Maximum Principles at Infinity, namely we have the next theorem Theorem 3.3 Let $M_1$ and $M_2$ two propperly embedded and disjoints hypersurfaces in $\\mathbb {R}^{n+1}$ with nonempty boundaries $\\partial M_1$ and $\\partial M_2$ .",
"Suppose that $M_2$ it is complete and that $\\sup _{M_2}\\vert \\mathbf {H}_{M_2}\\vert \\le b_0\\le \\inf _{M_1}H_{M_1},\\ \\ b_0>0.$ If $M_2$ have an ideal contac at infinity with $M_1$ and $M_2$ is $\\mathcal {N}$ -parabolic (or $\\mathcal {D}$ -parabolic) then $\\min \\lbrace dist(M_1,\\partial M_2),dist(M_2,\\partial M_1)\\rbrace =0.$ For $\\mathcal {D}$ -parabolic hypersurfaces we have the following corollary of Theorem REF .",
"Corrollary 3.2 Let $M_1$ and $M_2$ has in Theorem REF .",
"Assume that $M_2$ it is complete and $\\sup _{M_2}\\vert \\mathbf {H}_{M_2}\\vert \\le b_0\\le \\inf _{M_1}H_{M_1},\\ \\ b_0>0.$ Suppose that there exist relatively compact sets $\\Omega _1\\subset M_1$ and $\\Omega _2\\subset M_2$ such that $M_1\\diagdown \\Omega _1$ is isometric to $M_2\\diagdown \\Omega _2$ .",
"If $M_2$ have an ideal contact at infinity with $M_1$ and $M_1$ or $M_2$ is $\\mathcal {D}$ -parabolic then $\\min \\lbrace dist(M_1,\\partial M_2),dist(M_2,\\partial M_1)\\rbrace =0.$ By Corollary 13 of [16] we have that if $M_1\\diagdown \\Omega _1$ is isometric to $M_2\\diagdown \\Omega _2$ , then $M_1$ is $\\mathcal {D}$ -parabolic if and only if so is $M_2$ .",
"Therefore, if $M_2$ is $\\mathcal {D}$ -parabolic by Theorem REF the lemma is true.",
"If $M_1$ is $\\mathcal {D}$ -parabolic then $M_2$ so is, and again the lemma is true.",
"Remark 3.1 Since the Proposition REF and the Ahlfors Maximum Principles, for $\\mathcal {N}$ -parabolicity, is valid when $\\partial M_2=\\emptyset $ we have that the Theorems REF and REF remains valid if $\\partial M_2=\\emptyset $ , in this case we have that (REF ) remain $dist(M_2,\\partial M_1)=0$ .",
"Remark 3.2 If we suppose that $M_{2}$ is a compact hypersurface, we can withdraw the hypothesis of parabolicity in Theorem REF , using now the Divergence Theorem in its demonstration."
],
[
"Geometric Applications",
"Let $M\\subset \\mathbb {R}^{n+1}$ be a complete hypersurface, propperly embedded and empty boundary.",
"We say that $M$ is convex with respect to the unit normal $\\eta $ if its mean curvature function is positive.",
"Observe that $M$ splits $\\mathbb {R}^{n+1}$ in two connected component.",
"We will define as convex side of $M$ the component of $\\mathbb {R}^{n+1}$ to which the mean curvature vector points at."
],
[
"Applications of the Maximum Principles at Infinty",
"As a consequence of the PMIproof of the Theorem REF we have the following theorem that extends to $\\mathbb {R}^{n+1}$ , without assumptions about the Gaussian curvature, the Corollary 1 in [1] and the Theorem 3.4 in [2], with a similar demonstration.",
"It also extends, in the parabolic case, the Theorem 1 in [9].",
"Theorem 4.1 Let $M_{1}$ and $M_{2}$ be two propperly embedded and disjoints hypersurfaces in $\\mathbb {R}^{n+1}$ with empty boundaries.",
"Suppose that $M_{2}$ is complete and that $ \\sup _{M_2}\\vert {\\bf H}_{M_2}\\vert \\le b_0\\le \\inf _{M_1}H_{M_1}, \\; b_0>0.",
"$ If $M_{2}$ is parabolic, it cannot lie in the convex side of $M_{1}$ .",
"Suppose that $M_{2}$ is in the convex side of $M_{1}$ .",
"If $dist(M_{1},M_{2})=0$ then $M_{1}$ and $M_{2}$ have an ideal contact at infinty.",
"In this case we can proceed with the demonstration of Theorem REF defining for an $\\epsilon >0$ sufficiently close to zero, the sets $M_2(\\epsilon )=\\lbrace x\\in M_2 : dist(x,M_1)\\le \\epsilon \\rbrace .$ For each $x\\in M_2(\\epsilon )$ , consider the sets $S_x=\\lbrace y\\in M_1: \\vert y - x\\vert =dist(x,M_1) \\ \\ e \\ \\ x-y=\\lambda {\\bf H}_{M_1}(y), \\ \\ \\lambda > 0 \\rbrace $ and $M^{\\prime }_2(\\epsilon )\\subset M_2(\\epsilon )$ given by $M^{\\prime }_2(\\epsilon )= \\lbrace x\\in M_2(\\epsilon ): S_x\\ne \\emptyset \\rbrace .$ If $C_2(\\epsilon )\\subset M^{\\prime }_2(\\epsilon )$ is a connected component of $M^{\\prime }_2(\\epsilon )$ , we will have that $\\partial C_2(\\epsilon )= \\lbrace x\\in C_2(\\epsilon ): dist(x,M_1)=\\epsilon \\rbrace .$ because we are assuming that $\\partial M_2=\\emptyset $ .",
"And finally, defining in $C_2(\\epsilon )$ the function $\\phi (x)=e^{-C_0d(x)}$ , where $d(x)=dist(x,M_{1})$ , we will get to the same contradition of Theorem REF , because we are assuming that $M_{2}$ is parabolic.",
"So, $M_{2}$ cannot lie in the convex side of $M_{1}$ if $dist(M_1,M_2)=0$ .",
"If $dist(M_1,M_2)>0$ , there are sequences $y_n\\in M_1$ and $x_n\\in M_2$ in a way that the sequence $y_n - x_n$ have a subsequence that converges to a vector $v\\in \\mathbb {R}^{n+1}$ with $\\vert v\\vert =dist(M_1,M_2)$ .",
"So let $\\overline{M}_2=M_2+v$ .",
"In this way, we have that $dist(M_1,\\overline{M}_2)=0$ and that $M_1\\cap \\overline{M}_2\\ne \\emptyset $ , otherwise $\\overline{M}_2$ have an ideal contact at infinity with $M_1$ , what cannot happen by previous paragraph.",
"Let $p\\in M_1\\cap \\overline{M}_2$ , as $\\vert v\\vert =dist(M_1,M_2)$ and $M_2$ is in the convex side of $M_{1}$ , we have that $M_{1}$ and $\\overline{M}_2$ have an ideal contact at $p$ .",
"By Hopf'S Maximum Principle, $M_{1}$ and $\\overline{M}_2$ coincide in a neighbourhood of $p$ .",
"That means that $M_{1}$ differs from $M_{2}$ by a translation in $\\mathbb {R}^{n+1}$ of length $\\vert v\\vert =dist(M_1,M_2)$ .",
"As the line segment that starts in $M_{1}$ and ends in $M_{2}$ , whose length is $dist(M_1,M_2)$ , is orthogonal to both $M_{1}$ and $M_{2}$ , we have that in this neighborhood of $p$ , $M_{1}$ and $M_{2}$ are parallels hyperplans, contradicting the hyptothesis that the mean curvature function of $M_{1}$ is postive.",
"Then, if $dist(M_{1}, M_{2})>0$ , $M_{2}$ cannot lie in the convex side of $M_{1}$ , which proves the Theorem.",
"Remark 4.1 Generally, an hypersurface $M_1\\subset \\mathbb {R}^{n+k}$ with principal curvatures $\\lambda _1\\le \\cdots \\le \\lambda _n\\le \\cdots \\le \\lambda _{n+k-1}$ with respect to the unit normal $\\eta $ is called $n$ -convex mean with respect to $\\eta $ if $\\lambda _1+\\cdots +\\lambda _n\\ge 0$ .",
"Then the demonstration of Theorem REF above actually give as the following theorem, with a similar demonstration.",
"Theorem 4.2 Let $M_{1}$ be an hypersurface of $\\mathbb {R}^{n+k}$ whose mean curvature function is postive and $M_{2}$ an $n$ -dimensional $C^{2}$ -submanifold of $\\mathbb {R}^{n+k}$ .",
"Also let $M_{1}$ and $M_{2}$ be disjoints, propperly embedded in $\\mathbb {R}^{n+k}$ with empty boundaries.",
"Suppose that $M_{2}$ is complete and that $\\sup _{M_2}\\vert \\mathbf {H}_{M_2}\\vert \\le \\inf _{M_1} _n.$ If $M_{2}$ is parabolic, it cannot lie in the convex side of $M_{1}$ .",
"This last theorem is the version with an ideal contact at infinity of the Theorem 4 in [3] in the case when $M_{2}$ is parabolic and it is called there A barrier principle for submanifolds of arbitrary codimension and bounded mean curvature, see also in [5] the case for minimal submanifolds."
],
[
"Applications of Omori-Yau Maximum Principles",
"Adding an appropriate hypothesis about the Ricci curvature of $M_2$ hypersurface, we can withdraw its condition of parabolicity on Theorem REF and obtain an analog result given in Theorem REF soon.",
"Before, we will demonstrate the Lemma REF below.",
"Lemma 4.1 Let $M\\subset \\mathbb {R}^{n+k}$ be a properly embedded hypersurface and $A$ its shape operator.",
"We denote $d_j=d(x_j)=dist(x_j,M)$ , where $x_j\\in \\mathbb {R}^{n+k}$ it is a sequence of points such that $\\lim _{j\\rightarrow +\\infty }d_j=0$ , and $A_j$ as the shape operator of $M$ in $y_j\\in M$ , such that $\\vert x_j-y_j\\vert =d_j$ .",
"Suppose that $\\limsup _{j\\rightarrow +\\infty }\\Vert A_j\\Vert <+\\infty .$ If $II_{d_j}$ is the second fundamental form of the parallel hypersurface $M_{d_j}=d^{-1}(d_j)$ in $T_{x_j}M_{d_j}$ , then $\\lim _{j\\rightarrow +\\infty }\\Vert II_{d_j}\\Vert <+\\infty .$ If $A_{d_j}$ is the shape operator of $M_{d_j}$ then we have the Riccati's equation, see [17], $\\overline{\\nabla }_{Dd}A_{d_j}+A^2_{d_j}+\\overline{R}(\\cdot \\; ,Dd)Dd=0$ where $\\overline{\\nabla }$ and $\\overline{R}$ are the Riemannian connection and the tensor curvature of the $\\mathbb {R}^{n+k}$ , respectively.",
"For $x_j\\in \\mathbb {R}^{n+k}$ such that $d_j\\ll 1$ consider the normalized geodesic minimizer $\\beta : [0,d_j]\\rightarrow \\mathbb {R}^{n+k}$ with $\\beta (0)=y_j\\in M$ and $\\beta (d_j)=x_j\\in M_{d_j}$ .",
"As $\\beta $ is a line segment joining $y_j$ to $x_j$ and $d_j\\ll 1$ , then $\\beta ^{\\prime }(d)=Dd$ if $d\\in [0,d_j]$ .",
"Let $v\\in T_{x_j}M_{d_j}$ and $V$ its parallel transport alongside $\\beta $ with $V(0)=v_0\\in T_{y_j}M$ .",
"Denoting by $^{\\prime }$ the derivative alongside $\\beta $ we have $\\langle A_{d}(V),V\\rangle ^{\\prime }=\\langle \\overline{\\nabla }_{\\gamma ^{\\prime }}A_{d}(V),V\\rangle =\\langle \\overline{\\nabla }_{Dd}A_{d}(V),V\\rangle .$ Thus, by (REF ) $\\langle A_{d}(V),V\\rangle ^{\\prime }=-\\langle A^2_{d}(V),V\\rangle -\\langle \\overline{R}(V,Dd)Dd,V\\rangle .$ Therefore in $d=0$ we have $\\langle A_{d}(V),V\\rangle ^{\\prime }\\mid _{d=0}=-\\langle A^2_{j}(v_0),v_0\\rangle -\\langle \\overline{R}(v_0,\\eta )\\eta ,v_0\\rangle .$ Thus, the Taylor expansion of $II_{d_j}(v)=\\langle A_{d_j}(v),v\\rangle $ around second fundamental form $II_{j}=II_{d_j}\\mid _{d=0}$ of $M$ it is given by $\\langle A_{d_j}(v),v\\rangle =\\langle A_{j}(v_0),v_0\\rangle -[\\langle A^2_{j}(v_0),v_0\\rangle +\\langle \\overline{R}(v_0,\\eta )\\eta ,v_0\\rangle ]d +O(d^2)$ Being the sectional curvature of $\\mathbb {R}^{n+k}$ null, we have from (REF ) that $\\vert II_{d_j}(v)\\vert \\le \\Vert A_j\\Vert \\vert v_0\\vert ^2+\\Vert A_j\\Vert ^2\\vert v_0\\vert ^2d + O(d^2).$ Therefore from (REF ) $\\Vert II_{d_j}\\Vert \\le \\Vert A_j\\Vert +\\Vert A_j\\Vert ^2d + O(d^2).$ As $j\\rightarrow +\\infty $ in (REF ) we have what we wanted.",
"Theorem 4.3 Let $M_1$ and $M_2$ be two disjoints, without boundary and properly embedded hypersurfaces in $\\mathbb {R}^{n+1}$ .",
"Assume that $M_2$ is complete with Ricci curvature satisfying $Ric_{M_2}\\ge -(n-1)R_0$ , $R_0>0.$ Let $d(x)=dist(x,M_1)$ and $A_j$ as in Lemma REF and suppose that $\\limsup _{j\\rightarrow +\\infty }\\Vert A_j\\Vert <+\\infty $ for every sequence $x_j\\in M_2$ such that $\\lim _{j\\rightarrow +\\infty }d(x_j)=\\inf _{M_2}dist(x,M_1)$ .",
"If $\\sup _{M_2}\\vert {\\bf H}_{M_2}\\vert <\\inf _{M_1}H_{M_1},$ then $M_2$ it connot lie in the convex side of $M_1$ .",
"By Omori-Yau's Maximum Principle, with the version for the minimum, there is a sequence $x_j\\in M_2$ such that $\\lim _{j\\rightarrow \\infty }d(x_j)=\\inf _{M_2}dist(x,M_1)$ and we have that $\\vert \\nabla ^{M_2}d\\vert (x_j) < \\dfrac{1}{j}\\ \\ \\ \\ \\ \\ e \\ \\ \\ \\ \\ \\ \\Delta ^{M_2}d(x_j) > -\\dfrac{1}{j}, \\ \\ \\ \\ \\forall j\\in \\mathbb {N}.$ Suppose that $M_2$ lies on the mean convex side of $M_1$ and that $dist(M_1,M_2)=0$ .",
"Then $M_2$ have an ideal contact at infinity with $M_1$ .",
"This way, for $j\\in \\mathbb {N}$ sufficiently larg, the Lemma REF give us that $\\Delta ^{M_2}d(x_j) - n\\vert {\\bf H}_{M_2}\\vert + \\displaystyle \\sum _{i}^{n}\\lambda _i(d_j) - \\displaystyle \\dfrac{II_{d_j}((\\nabla ^{M_2}d)^T,(\\nabla ^{M_2}d)^T)}{1-\\vert \\nabla ^{M_2}d\\vert ^2}(x_j)\\le 0$ where $\\lambda _i(d_j)$ are the main curvatures of the parallel hypersurface $d^{-1}(d_j)$ in the orientation given by $\\eta $ and $d_j=d(x_j)$ .",
"As we have that $\\Delta ^{M_2}d(x_j) > -\\dfrac{1}{j}$ then (REF ) give us that $\\dfrac{1}{j}> - n\\vert {\\bf H}_{M_2}\\vert + \\displaystyle \\sum _{i}^{n}\\lambda _i(d_j) - \\displaystyle \\dfrac{II_{d_j}((\\nabla ^{M_2}d)^T,(\\nabla ^{M_2}d)^T)}{1-\\vert \\nabla ^{M_2}d\\vert ^2}.$ And from $\\vert \\nabla ^{M_2}d\\vert (x_j) < \\dfrac{1}{j}$ we have that $\\displaystyle \\dfrac{\\vert II_{d_j}((\\nabla ^{M_2}d)^T,(\\nabla ^{M_2}d)^T)\\vert }{1-\\vert \\nabla ^{M_2}d\\vert ^2} \\le \\Vert II_{d_j}\\Vert \\dfrac{\\vert \\nabla ^{M_2}d\\vert ^2}{1-\\vert \\nabla ^{M_2}d\\vert ^2}<\\Vert II_{d_j}\\Vert \\dfrac{1}{j^2-1}.$ Then, from (REF ) $\\begin{array}{rll}\\Vert II_{d_j}\\Vert \\dfrac{1}{j^2-1}+\\dfrac{1}{j}&>& \\displaystyle \\sum _{i}^{n}\\lambda _i(d_j)-n\\vert {\\bf H}_{M_2}\\vert \\\\&\\ge &\\displaystyle \\sum _{i}^{n}\\lambda _i -n\\vert {\\bf H}_{M_2}\\vert .\\end{array}$ As $j\\rightarrow +\\infty $ in (REF ), we have, by Lemma REF , that $n\\vert {\\bf H}_{M_2}\\vert \\ge \\lambda _1 + \\cdots +\\lambda _n$ contradicting (REF ).",
"Thus, if $dist(M_1,M_2)=0$ , $M_2$ cannot lie in the convex side of $M_1$ .",
"If $dist (M_1,M_2)>0$ , let $\\overline{M}_2$ has in second paragraph in the demonstration of Theorem REF .",
"Again $M_1\\cap \\overline{M}_2\\ne \\emptyset $ , otherwise $\\overline{M}_2$ have an ideal contact at infinity with $M_1$ , what cannot happen by previous paragraph.",
"Let $p_0\\in M_1\\cap \\overline{M}_2$ , then $d$ would attain a minimum at $p_0$ .",
"In this case $\\nabla ^{\\overline{M}_2}d(p_0)=0$ and by Lemma REF we have $\\Delta ^{\\overline{M}_2}d(p_0)<0$ , which contradicts that $p_0$ is a minimum point for $d$ .",
"Therefore, $M_2$ cannot lie in the convex side of $M_1$ if $dist(M_1,M_2)>0$ and ends the proof of theorem.",
"Remark 4.2 Let $M$ be an oriented hypersurface isometrically imersed in Riemannian space form $\\mathbb {M}^{n+1}_c$ .",
"If $R_M$ is the normalized scalar curvature of $M$ and $A$ its shape operator, then from Gauss equation we have the following relationship, see for instance [18], $\\Vert A\\Vert ^2=n^2H_M^2-n(n-1)(R_M-c).$ Therefore, if $R_M\\ge c$ we have to $\\Vert A\\Vert ^2\\le n^2H_M^2$ , and so $\\Vert A\\Vert \\le nH_M$ , if $H_M>0$ .",
"Then if the mean curvature of $M$ is bounded will $\\sup _M\\Vert A\\Vert \\le n\\sup _MH_M<+\\infty .$ With the Remark REF , the Theorem REF give us the following corollary.",
"Corrollary 4.1 Let $M_1$ and $M_2$ as in Theorem REF .",
"Assume that $M_2$ is complete with Ricci curvature bounded below.",
"Let $R_{M_1}$ the scalar curvature (normalized) of $M_1$ and suppose that $R_{M_1}\\ge 0$ .",
"Suppose also that $H_{M_1}$ is bounded and that $ \\sup _{M_2}\\vert {\\bf H}_{M_2}\\vert <\\inf _{M_1}H_{M_1}.$ Then $M_2$ cannot lie in the convex side of $M_1$ .",
"Suppose $M_2$ lies in the convex side of $M_1$ and that $dist(M_1,M_2)=0$ .",
"Let $II_{d_j}$ be the second fundamental form of the parallel hypersurface $d^{-1}(d_j)$ , where $d_j=d(x_j)=dist(x_j,M_1)$ and $x_j\\in M_2$ is a sequence of points given by Omori-Yau, i.e., with $d_j\\rightarrow 0$ satisfying (REF ).",
"Then accordingly with the Remark REF and by Lemma REF , we have $\\lim _{j\\rightarrow +\\infty } \\Vert II_{d_j}\\Vert <+\\infty $ .",
"Proceeding like the demonstration of Theorem REF we have the desired.",
"Notice that in the demonstration of Theorem REF the hypotheses about $M_2$ are essentially so that we can use the Omori-Yau maximum principle on function $d(x)=dist(x,M_1)$ .",
"Thus, we can suppose others hypotheses on $M_2$ allowing us to use this principle.",
"This can be done using the Pigola-Rigoli-Setti's Theorem, see [19], which extends the class of Riemannian manifolds for which hold the Omori-Yau maximum principle.",
"Before, however, we give the following definition given by Pigola-Rigoli-Setti also in [19].",
"Definition 4.1 The Omori-Yau maximum principles is said to hold on Riemannian manifold $M$ if for any given $u\\in C^2(M)$ with $u^*=\\sup _{M}u<+\\infty $ , there exist a sequence of points $x_j\\in M$ , depending on $M$ and on $u$ , such that $\\lim _{j\\rightarrow +\\infty }u(x_j)=u^*, \\ \\ \\vert \\nabla ^M u\\vert (x_j)<\\dfrac{1}{j}, \\ \\ \\Delta ^M u(x_j)<\\dfrac{1}{j}.$ Theorem 4.4 (Pigola-Rigoli-Setti) Let $M$ a Riemannian manifold and assume there exists a non-negative function $\\gamma $ satisfying the following: C1) $\\gamma (x)\\rightarrow +\\infty $ as $x \\rightarrow +\\infty $ ; C2) $\\exists B>0$ such that $\\vert \\nabla ^M\\gamma \\vert \\le B\\sqrt{\\gamma }$ off a compact set; C3) $\\exists C>0$ such that $\\Delta ^M\\gamma \\le C\\sqrt{\\gamma G(\\sqrt{\\gamma })}$ off a compact set, were $G:[0,+\\infty )\\rightarrow [0,+\\infty )$ is a smooth function satisfying $G(0)>0, \\ \\ G^{\\prime }(0)\\ge 0, \\ \\ \\displaystyle \\int _{0}^{+\\infty }\\dfrac{ds}{\\sqrt{G(s)}}=+\\infty , \\ \\ \\displaystyle \\limsup _{t\\rightarrow +\\infty }\\dfrac{tG(\\sqrt{t})}{G(t)}<+\\infty .$ Then the Omori-Yau maximum principle holds on $M$ .",
"Because of the Theorem of Pigola-Rigoli-Setti, Theorem REF above, the Corollary REF can be extended to a more general class of Riemannian manifold that satisfy the Omori-Yau Maximum Principle.",
"Then, we have the Theorem 4.5 Let $M_1$ be without boundary and properly embedded hypersurface in $\\mathbb {R}^{n+k}$ with bounded scalar curvature and whose mean curvature satisfies $b_0\\le H_{M_1}\\le b_1$ , where $b_0,b_1>0$ .",
"Let also $M_2$ be without boundary, disjoint of $M_1$ and properly embedded $n$ -dimensional submanifold of $\\mathbb {R}^{n+k}$ .",
"Suppose there exist a non-negative function in $M_2$ satisfying the conditions $C1)$ , $C2)$ and $C3)$ of Theorem REF and $\\sup _{M_2}\\vert \\mathbf {H}_{M_2}\\vert <\\inf _{M_1}_n.$ Then $M_2$ it cannot lie in the convex side of $M_1$ .",
"With analogous proof of Theorem REF and extend in the case of Ideal Contact at Infinity Theorem 4 in [3].",
"Extend also to submanifolds of arbitrary codimension $M_2$ the Theorem 3.4 in [2], Theorem 1 in [9] and Corollary 1 in [1]."
]
] | 1606.04899 |
[
[
"A counterexample to the extension space conjecture for realizable\n oriented matroids"
],
[
"Abstract The extension space conjecture of oriented matroid theory states that the space of all one-element, non-loop, non-coloop extensions of a realizable oriented matroid of rank $d$ has the homotopy type of a sphere of dimension $d-1$.",
"We disprove this conjecture by showing the existence of a realizable uniform oriented matroid of high rank and corank 3 with disconnected extension space."
],
[
"Introduction",
"Given an oriented matroid $M$ , the set of all one-element, non-loop, non-coloop extensions of $M$ has a natural poset structure, and the order complex of this poset is called the extension space $\\mathcal {E}(M)$ of $M$ .",
"A long-standing and central open question in oriented matroid theory is the extension space conjecture, which states that if $M$ is realizable, then $\\mathcal {E}(M)$ is homotopy equivalent to a sphere of dimension $\\operatorname{rank}(M) - 1$ .",
"Sturmfels and Ziegler [18] proved this conjecture for a class of oriented matroids which they called strongly Euclidean oriented matroids, which includes all oriented matroids of rank at most 3 or corank at most 2.",
"However, Santos [16] showed that realizable oriented matroids which are not strongly Euclidean exist both in rank 4 and corank 3.",
"Mnëv and Richter-Gebert [10] showed that the conjecture is false if one removes the realizability assumption on $M$ ; they constructed non-realizable oriented matroids of rank 4 with disconnected extension spaces.",
"In this paper, we disprove the extension space conjecture by showing that there exists a realizable uniform oriented matroid of high rank (possibly around $10^5$ ) and corank 3 with disconnected extension space.",
"Geometrically, one can view the extension space conjecture as an attempt to understand how the space of realizable oriented matroids is embedded in the space of all oriented matroids.",
"In particular, if one fixes a realization $X$ of $M$ , the oriented matroids which can be realized as one-element extensions of $X$ form a poset which is isomorphic to the face lattice of the boundary of a polytope, and hence has spherical order complex $\\mathcal {E}(X)$ .",
"It was hoped that $\\mathcal {E}(X)$ might be a good representation of the topology of the entire extension space of $M$ ; in fact, the extension space conjecture is equivalent to the claim that $\\mathcal {E}(X)$ is a strong deformation retract of $\\mathcal {E}(M)$ [2].",
"However, our result shows that for very large oriented matroids things can be more complicated.",
"It would be interesting to better understand under what conditions and by how much the topologies of $\\mathcal {E}(X)$ and $\\mathcal {E}(M)$ can differ.",
"Before describing the counterexample, we discuss two problems which are also resolved as a result of this counterexample."
],
[
"Combinatorial Grassmannians",
"The extension space conjecture is a special case of a far-reaching conjecture by MacPherson, Mnëv, and Ziegler [11] on “combinatorial Grassmannians.” Any set $\\mathcal {S}$ of oriented matroids on the same ground set can be turned into a topological space by considering the order complex of the usual poset structure on $\\mathcal {S}$ , as above.",
"If $M$ is an oriented matroid and $\\mathcal {S}$ is the set of all rank $r$ strong images [5] of $M$ , then the resulting space is known as the combinatorial Grassmannian $\\mathcal {G}(r, M)$ .",
"These spaces (especially the case where $M$ is the rank $n$ oriented matroid on $n$ elements, in which case this space is known as the MacPhersonian) can be thought of as combinatorial models for the real Grassmannian and play important roles in the theories of combinatorial differential manifolds and matroid bundles; see [9] and [1].",
"The basic problem surrounding these objects is whether or not they are topologically similar to their real counterpart, the real Grassmannian $G(r, \\mathbb {R}^n)$ .",
"MacPherson, Mnëv, and Ziegler conjectured that if $M$ is realizable of rank $n$ , then $\\mathcal {G}(r, M)$ is homotopy equivalent to $G(r, \\mathbb {R}^n)$ .",
"The special case $r = n-1$ of this conjecture is equivalent, through some minor additional arguments, to the extension space conjecture for $M$ .",
"Our result is thus also a counterexample to the combinatorial Grassmannian conjecture.",
"An important remaining open question is whether the MacPhersonian is homotopy equivalent to the appropriate Grassmannian.",
"A positive answer would have ramifications in the application of combinatorial differential manifolds to the study of smooth manifolds.",
"Our result, however, could be taken as evidence against the conjecture, and at the very least demonstrates the combinatorial subtleties underlying the problem."
],
[
"The generalized Baues problem",
"The extension space conjecture is also a special case of the generalized Baues problem in the theory of fiber polytopes [3].",
"This problem studies general classes of polytopal subdivisions which are “induced” by some projection of polytopes; these classes of subdivisions include triangulations of polytopes, zonotopal tilings, and monotone paths on polytopes.",
"Given a polytope and a class of subdivisions, the set of such subdivisions of this polytope form a poset and associated order complex, and thus can be studied as a topological space; this is the goal of the generalized Baues problem.",
"The problem is motivated by the fact that if one restricts this poset to a certain set of subdivisions called coherent subdivisions, one obtains the face lattice of a polytope known as the fiber polytope [4].",
"The conjecture is that the homotopy type of the space does not change when one includes the non-coherent subdivisions.",
"This area of research proved very fruitful and led to major developments in the understanding of flip graphs, which are certain graphs connecting the finest subdivisions of a polytope.",
"See the survey [13] for an overview and the paper [17] and book [6] for more recent results.",
"The connection to the extension space conjecture is as follows: Via the Bohne-Dress theorem [19], the extension space of a realizable oriented matroid $M$ is isomorphic to the space of all non-trivial zonotopal tilings of the zonotope associated to the dual of $M$ .",
"The extension space conjecture is equivalent to the “generalized Baues conjecture for cubes,” which states that for any zonotope, the aforementioned space is homotopy equivalent to a sphere.",
"After Rambau and Ziegler disproved the most general form of the generalized Baues conjecture [12] and Santos disproved the more particular “generalized Baues conjecture for simplices” [17], the generalized Baues conjecture for cubes remained as (possibly) the last unresolved case of interest for the problem.",
"Our result disproves this case by giving a three-dimensional zonotope whose space of non-trivial zonotopal tilings is disconnected.",
"The counterexample in this paper is based on a vector configuration used by the author in [8] to construct a zonotope whose flip graph of zonotopal tilings is not connected.",
"This vector configuration is formed by taking the set $\\lbrace e_i - e_j : 1 \\le i < j \\le 4\\rbrace $ , where $e_i$ is the $i$ -th standard basis vector, and repeating each vector in the set a large number of times.",
"Call this configuration $E_N$ , where $N$ is the number of times each vector is repeated.",
"Let $\\widetilde{E}_N$ be a configuration obtained by perturbing each vector in $E_N$ by a small random displacement in the span of $E_N$ .",
"Our result is the following.",
"Theorem 1.1 For large enough $N$ , with probability greater than 0, $\\widetilde{E}_N$ contains a subconfiguration $E$ such that the oriented matroid dual to the oriented matroid of $E$ has disconnected extension space.",
"The strategy of the proof is to show that the flip graph (Section REF ) of all uniform one-element extensions of the dual oriented matroid of $\\widetilde{E}_N$ is disconnected.",
"A feature of the proof is that it uses probabilistic arguments to show the existence of certain elements in the flip graph; the value of $N$ required for these arguments to work is roughly $10^5$ .",
"We then use a known trick (Proposition REF ) to convert disconnectedness of flip graphs to disconnectedness of entire posets.",
"Unfortunately, the trick only tells us that there is some subconfiguration $E \\subseteq \\widetilde{E}_N$ whose dual oriented matroid has disconnected extension space, and does not tell us what $E$ is.",
"Section 2 gives the relevant background on oriented matroids.",
"Section 3 is the main proof."
],
[
"Oriented matroids",
"We will give a brief overview of oriented matroids.",
"While this overview is self-contained, some familiarity with the basic concepts is helpful.",
"We refer to Björner et al.",
"[5] or Richter-Gebert and Ziegler [14] for a more comprehensive treatment."
],
[
"Basic definitions",
"Throughout Section , let $E$ be a finite set.",
"Let $\\lbrace +,-,0\\rbrace $ be the set of signs, and let $\\lbrace +,-,0\\rbrace ^E$ be the set of sign vectors on $E$ .",
"For $\\alpha \\in \\lbrace +,-,0\\rbrace $ , define $-\\alpha \\in \\lbrace +,-,0\\rbrace $ in the obvious way.",
"For $X \\in \\lbrace +,-,0\\rbrace ^E$ , define $-X \\in \\lbrace +,-,0\\rbrace ^E$ such that $(-X)(e) = -X(e)$ for all $e \\in E$ .",
"Define a partial order on $\\lbrace +,-,0\\rbrace $ by $0 < +$ and $0 < -$ , and extend this to the product order on $\\lbrace +,-,0\\rbrace ^E$ .",
"An oriented matroid is a pair $(E, \\mathcal {L})$ where $\\mathcal {L}$ is a set of sign vectors on $E$ satisfying certain axioms.",
"We will not use this axiomatic description in this paper, but we include it for completeness: Definition 2.1 An oriented matroid is a pair $M = (E, \\mathcal {L})$ where $\\mathcal {L} \\subseteq \\lbrace +,-,0\\rbrace ^E$ such that $(0,\\cdots ,0) \\in \\mathcal {L}$ If $X \\in \\mathcal {L}$ , then $-X \\in \\mathcal {L}$ .",
"If $X$ , $Y \\in \\mathcal {L}$ , then $X \\circ Y \\in \\mathcal {L}$ , where $(X \\circ Y)(e) =\\begin{dcases*}X(e) & if X(e) \\ne 0 \\\\Y(e) & otherwise.\\end{dcases*}$ If $X$ , $Y \\in \\mathcal {L}$ and $e \\in E$ such that $\\lbrace X(e),Y(e)\\rbrace = \\lbrace +,-\\rbrace $ , then there exists $Z \\in \\mathcal {L}$ such that $Z(e) = 0$ and $Z(f) = (X \\circ Y)(f)$ whenever $\\lbrace X(f),Y(f)\\rbrace \\ne \\lbrace +,-\\rbrace $ .",
"The set $\\mathcal {L}$ is called the set of covectors of $M$ .",
"A minimal element of $\\mathcal {L} \\,\\backslash \\,\\lbrace 0\\rbrace $ (with respect to the above product order) is called a cocircuit of $M$ , and the set of cocircuits is denoted $\\mathcal {C}^\\ast (M)$ .",
"Every covector of $M$ can be written as $X_1 \\circ \\cdots \\circ X_k$ where $X_1$ , ..., $X_k \\in \\mathcal {C}^\\ast (M)$ .",
"Given an oriented matroid $M = (E, \\mathcal {L})$ , an element $e \\in E$ is a loop of $M$ if $X(e) = 0$ for all $X \\in \\mathcal {L}$ .",
"An element $e \\in E$ is a coloop of $M$ if there is some $X \\in \\mathcal {L}$ with $X(e) = +$ and $X(f) = 0$ for all $f \\in E \\,\\backslash \\,\\lbrace e\\rbrace $ .",
"An independent set of $M$ is a set $\\lbrace e_1,\\cdots ,e_k\\rbrace \\subseteq E$ such that there exist $X_1$ , ..., $X_k \\in \\mathcal {L}$ with $X_i(e_j) =\\begin{dcases*}+ & if i=j \\\\0 & otherwise.\\end{dcases*}$ All maximal independent sets of $M$ have the same size, and this size is the rank of $M$ .",
"The corank of $M$ is $\\vert E \\vert - \\operatorname{rank}(M)$ .",
"An oriented matroid of rank $d$ is uniform if all $d$ -element subsets of $E$ are independent sets.",
"For $X$ , $Y \\in \\lbrace +,-,0\\rbrace ^E$ , we write $X \\perp Y$ if the set $\\lbrace X(e) \\cdot Y(e) : e \\in E \\rbrace $ is either $\\lbrace 0\\rbrace $ or contains both $+$ and $-$ .",
"The next theorem defines duality of oriented matroids.",
"Theorem 2.2 For any oriented matroid $M = (E,\\mathcal {L})$ of rank $d$ , the pair $M^\\ast = (E,\\mathcal {L}^\\ast )$ where $\\mathcal {L}^\\ast = \\lbrace X \\in \\lbrace +,-,0\\rbrace : X \\perp Y \\text{ for all } Y \\in \\mathcal {L} \\rbrace $ is an oriented matroid of rank $\\vert E \\vert - d$ , called the dual of $M$ .",
"We have $M^{\\ast \\ast } = M$ .",
"Finally, if $M = (E, \\mathcal {L})$ is an oriented matroid and $A \\subseteq E$ , the pair $M \\vert _A = (A, \\mathcal {L}_A)$ where $\\mathcal {L}_A := \\lbrace X \\vert _A : X \\in \\mathcal {L} \\rbrace $ is an oriented matroid called the restriction of $M$ to $A$ ."
],
[
"Topological representation",
"For each $e \\in E$ , let $v_e \\in \\mathbb {R}^d$ be a vector.",
"For each point $x \\in \\mathbb {R}^d$ , we obtain a sign vector $X \\in \\lbrace +,-,0\\rbrace ^E$ by letting $X(e)$ be the sign of the inner product $\\langle x , v_e \\rangle $ .",
"The set of all such sign vectors is the set of covectors of an oriented matroid, which we call the oriented matroid of the vector configuration $\\lbrace v_e\\rbrace _{e \\in E}$ .",
"An oriented matroid is realizable if it is the oriented matroid of some vector configuration.",
"Now assume that all of the $v_e$ above are nonzero, and let $S_e$ be the intersection of the hyperplane normal to $v_e$ with the unit sphere $S^{d-1}$ .",
"The $S_e$ form a sphere arrangement of $(d-2)$ -dimensional spheres in $S^{d-1}$ , and each $S_e$ is oriented in the following way: $S_e$ separates $S^{d-1}$ into two hemispheres, exactly one of which has points with positive inner product with $v_e$ .",
"The topological representation theorem says that all oriented matroids arise as topological deformations of such arrangements; we now describe this more precisely.",
"A pseudosphere in $S^{d-1}$ is an image of $\\lbrace x \\in S^{d-1} : x_d = 0\\rbrace $ under a homeomorphism $\\phi : S^{d-1} \\rightarrow S^{d-1}$ .",
"A psudosphere $S$ separates $S^{d-1}$ into two regions called sides; if we choose one side to be $S^+$ and the other to be $S^-$ , then we say that $S$ is oriented.",
"A pseudosphere arrangement is a collection $\\mathcal {A} = \\lbrace S_e\\rbrace _{e \\in E}$ of oriented pseudospheres in $S^{d-1}$ such that For all $A \\subseteq E$ , the set $S_A := \\bigcap _{e \\in A} S_e$ is homeomorphic to a sphere or empty.",
"If $A \\subseteq E$ and $e \\in E$ such that $S_A \\lnot \\subseteq S_e$ , then $S_A \\cap S_e$ is a pseudosphere in $S_A$ with sides $S_A \\cap S_e^+$ and $S_A \\cap S_e^-$ .",
"Let $\\mathcal {A} = \\lbrace S_e\\rbrace _{e \\in E}$ be a pseudosphere arrangement in $S^{d-1}$ .",
"For each $x \\in S^{d-1}$ , we obtain a sign vector $X \\in \\lbrace +,-,0\\rbrace ^E$ by setting $X(e) =\\begin{dcases*}+ & if x \\in S_e^+ \\\\- & if x \\in S_e^- \\\\0 & if x \\in S_e\\end{dcases*}$ Let $\\mathcal {L}(\\mathcal {A})$ be the set of all sign vectors obtained this way along with the 0 sign vector.",
"Call $\\mathcal {A}$ essential if $\\bigcap _{e \\in E} S_e = \\emptyset $ .",
"We can now state the topological representation theorem.",
"Theorem 2.3 (Folkman-Lawrence [7]) For any essential pseudosphere arrangement $\\mathcal {A}$ in $S^{d-1}$ , $(E, \\mathcal {L}(\\mathcal {A}))$ is an oriented matroid of rank $d$ .",
"Conversely, every oriented matroid without loops is $(E, \\mathcal {L}(\\mathcal {A}))$ for some essential pseudosphere arrangement $\\mathcal {A}$ , and $\\mathcal {A}$ is unique up to homeomorphisms $\\phi : S^{d-1} \\rightarrow S^{d-1}$ .",
"For an oriented matroid $M$ , we call an essential pseudosphere arrangement $\\mathcal {A}$ such that $M = (E, \\mathcal {L}(\\mathcal {A}))$ a topological representation of $M$ .",
"If $M$ has rank $d$ and $\\mathcal {A} = \\lbrace S_e\\rbrace _{e \\in E}$ is a topological representation of $M$ , we call any nonempty $S_A$ (where $A \\subseteq E$ ) for which $\\dim S_A > d-1-\\vert A \\vert $ a special pseudosphere of $\\mathcal {A}$ .",
"$M$ is uniform if and only if $\\mathcal {A}$ has no special pseudospheres.",
"The cocircuits of $M$ are given by points of $S_A$ where $\\dim S_A = 0$ ."
],
[
"Extensions, liftings, and weak maps",
"Let $M = (E, \\mathcal {L})$ be an oriented matroid.",
"Let $M^{\\prime } = (E^{\\prime }, \\mathcal {L}^{\\prime })$ be another oriented matroid such that $E^{\\prime } = E \\cup \\lbrace f\\rbrace $ for some $f \\notin E$ .",
"We say that $M^{\\prime }$ is a one-element extension, or extension, of $M$ if $M = M^{\\prime } \\vert _E$ ; that is, $\\mathcal {L} = \\lbrace X \\vert _E : X \\in \\mathcal {L}^{\\prime } \\rbrace .$ We say that $M^{\\prime }$ is a one-element lifting, or lifting, of $M$ if $\\mathcal {L} = \\lbrace X \\vert _E : X \\in \\mathcal {L}^{\\prime }, X(f) = 0 \\rbrace .$ If $M^{\\prime }$ is an extension (or lifting) of $M$ , we call it trivial if $f$ is a coloop (resp., a loop) of $M^{\\prime }$ .",
"The notions of extension and lifting are dual to each other: $M^{\\prime }$ is a (non-trivial) extension of $M$ if and only if $(M^{\\prime })^\\ast $ is a (non-trivial) lifting of $M^\\ast $ .",
"Finally, if $M^{\\prime }$ is a non-trivial extension of $M$ , then $\\operatorname{rank}(M^{\\prime }) = \\operatorname{rank}(M)$ , and if $M^{\\prime }$ is a non-trivial lifting of $M$ , then $\\operatorname{rank}(M^{\\prime }) = \\operatorname{rank}(M) + 1$ .",
"We can understand liftings better using topological representation.",
"Suppose $M^{\\prime }$ is a lifting of $M$ , and assume $\\operatorname{rank}(M) = d$ and $M$ and $M^{\\prime }$ have no loops.",
"Let $\\mathcal {A} = \\lbrace S_e\\rbrace _{e \\in E^{\\prime }}$ be a topological representation of $M^{\\prime }$ in $S^d$ ; by applying an appropriate homeomorphism $\\phi : S^d \\rightarrow S^d$ , we may assume $S_f = \\lbrace x \\in S^d : x_{d+1} = 0\\rbrace $ and $S_f^+ = \\lbrace x \\in S^d : x_{d+1} > 0\\rbrace $ .",
"Let $\\mathcal {A}^+ = \\lbrace S_e \\cap S_f^+\\rbrace _{e \\in E}$ .Note that $M^{\\prime }$ is determined by $\\mathcal {A}^+$ .",
"In addition, we have $S_e \\ne S_f$ for all $e \\ne f$ , because otherwise, by the definition of a lifting, $e$ would be a loop of $M$ .",
"Consider the “gnomonic projection” which maps $S_f^+$ to $\\mathbb {R}^d$ .",
"The image of $\\mathcal {A}^+$ under this map is a (not necessarily central) arrangement $\\mathcal {B}$ of oriented pseudohyperplanes in $\\mathbb {R}^d$ such that the intersection of $\\mathcal {B}$ with the “sphere at infinity” (that is, $S_f$ ) is a pseudosphere arrangement representing $M$ .",
"Conversely, given such a pseudohyperplane arrangement $\\mathcal {B}$ (with the appropriate definition of “pseudohyperplane arrangment”), we can uniquely construct a lifting $M^{\\prime }$ of $M$ such that the set of covectors of $M^{\\prime }$ which are positive on $f$ is topologically represented by $\\mathcal {B}$ .",
"Hence, liftings of $M$ are given by pseudohyperplane arrangements in $\\mathbb {R}^d$ whose intersection with the sphere at infinity are topological representations of $M$ .",
"Given two oriented matroids $M_1 = (E, \\mathcal {L}_1)$ and $M_2 = (E, \\mathcal {L}_2)$ on the same ground set $E$ , we say that there is a weak map $M_1 \\rightsquigarrow M_2$ if for every $X_2 \\in \\mathcal {L}_2$ , there exists $X_1 \\in \\mathcal {L}_1$ such that $X_1 \\ge X_2$ .",
"We say that this weak map is rank-preserving if $M_1$ and $M_2$ have the same rank.",
"If $M_1 \\rightsquigarrow M_2$ is a rank-preserving weak map, then $M_1^\\ast \\rightsquigarrow M_2^\\ast $ is also a (rank-preserving) weak map [5].",
"For any set $\\mathcal {S}$ of oriented matroids on the same ground set, we obtain a partial order on $\\mathcal {S}$ by letting $M_1 \\ge M_2$ if there is a weak map $M_1 \\rightsquigarrow M_2$ .",
"We call the set of all non-trivial extensions of an oriented matroid $M$ partially ordered this way the extension poset $\\mathcal {E}(M)$ of $M$ .",
"Similarly, we call the poset of all non-trivial liftings of $M$ the lifting poset $\\mathcal {F}(M)$ of $M$ .",
"Since all non-trivial extensions of an oriented matroid $M$ have the same rank, we have $\\mathcal {E}(M) \\cong \\mathcal {F}(M^\\ast )$ .",
"The extension poset (or lifting poset) has a unique minimal element $\\hat{0}$ , corresponding to extension by a loop (resp., lifting by a coloop).",
"Every poset has an associated order complex, which is the simplicial complex whose simplices are finite chains of the poset.",
"The extension space conjecture claims that for any realizable oriented matroid $M$ , the order complex of $\\mathcal {E}(M) \\,\\backslash \\,\\hat{0}$ is homotopy equivalent to a sphere of dimension $\\operatorname{rank}(M)-1$ .",
"Since an oriented matroid is realizable if and only if its dual is, this is equivalent to saying that the order complex of $\\mathcal {F}(M) \\,\\backslash \\,\\hat{0}$ is homotopy equivalent to a sphere of dimension $\\operatorname{corank}(M) - 1$ for any realizable $M$ .",
"We will find a realizable $M$ (with corank greater than 1) such that $\\mathcal {F}(M) \\,\\backslash \\,\\hat{0}$ is disconnected."
],
[
"Flips",
"To prove disconnectedness of some $\\mathcal {F}(M) \\,\\backslash \\,\\hat{0}$ , we will actually only need to look at the maximal elements of $\\mathcal {F}(M)$ .",
"If $M$ is uniform, the maximal elements of $\\mathcal {F}(M)$ are precisely the uniform liftings of $M$ .",
"We will study these uniform liftings through flips.These are called mutations in [5].",
"An equivalent discussion can be found there.",
"The following propositions define a flip and its basic properties.",
"They are easy to see from topological representation; we leave their proofs to the reader.",
"Proposition 2.4 Let $M = (E, \\mathcal {L})$ be a uniform oriented matroid of rank $d$ .",
"Let $D = \\lbrace e_1,\\cdots ,e_d\\rbrace $ be a $d$ -element subset of $E$ .",
"Let $X_1$ , ..., $X_d \\in \\mathcal {L}$ be cocircuits such that $X_i(e_j) = 0$ for all $i \\ne j$ .",
"Suppose that $X_1 \\vert _{E \\,\\backslash \\,D} = X_2 \\vert _{E \\,\\backslash \\,D} = \\cdots = X_d \\vert _{E \\,\\backslash \\,D}.$ Let $X^0 \\in \\lbrace +,-,0\\rbrace ^E$ be the sign vector with $X^0(e) =\\begin{dcases*}0 & if e \\in D \\\\X_1(e) & otherwise\\end{dcases*}$ and let $\\overline{X}_1$ , ..., $\\overline{X}_d$ be the sign vectors with $\\overline{X}_i(e) =\\begin{dcases*}-X_i(e) & if e \\in D \\\\X_i(e) & otherwise.\\end{dcases*}$ Then there are oriented matroids $M^0 = (E,\\mathcal {L}^0)$ and $\\overline{M} = (E,\\overline{\\mathcal {L}})$ such that $\\mathcal {C}^\\ast (M^0) &= \\mathcal {C}^\\ast (M) \\,\\backslash \\,\\lbrace \\pm X_1, \\cdots , \\pm X_d\\rbrace \\cup \\lbrace \\pm X^0\\rbrace \\\\\\mathcal {C}^\\ast (\\overline{M}) &= \\mathcal {C}^\\ast (M) \\,\\backslash \\,\\lbrace \\pm X_1, \\cdots , \\pm X_d\\rbrace \\cup \\lbrace \\pm \\overline{X}_1, \\cdots , \\pm \\overline{X}_d\\rbrace .$ We call $M^0$ a flip of $M$ , and say that $M^0$ is a flip between $M$ and $\\overline{M}$ .",
"We say that that the cocircuits $X_1$ , ..., $X_d$ are involved in this flip.",
"Proposition 2.5 In the situation of Proposition REF , the oriented matroid $\\overline{M}$ is uniform, and there are weak maps $M \\rightsquigarrow M^0$ and $\\overline{M} \\rightsquigarrow M^0$ .",
"Moreover, $M$ , $\\overline{M}$ , and $M^0$ are the only oriented matroids $N$ such that $N \\rightsquigarrow M^0$ .",
"Proposition 2.6 In the situation of Proposition REF , if $X \\in \\mathcal {L}$ is a cocircuit such that for all $1 \\le i \\le d$ , either $X(e_i) = X_i(e_i)$ or $X(e_i) = 0$ , then $X \\in \\lbrace X_1,\\cdots ,X_d\\rbrace $ .",
"Let $\\mathcal {A} = \\lbrace S_e\\rbrace _{e \\in E}$ be a topological representation of $M$ .",
"The conditions on $X_1$ , ..., $X_d$ imply that $S_{e_1}^{X_1(e_1)}$ , ..., $S_{e_d}^{X_d(e_d)}$ bound a simplicial region of $\\mathcal {A}$ .",
"The only cocircuits which correspond to points in the closure of this region are $X_1$ , ..., $X_d$ .",
"Now suppose that $M = (E,\\mathcal {L})$ is a uniform lifting of a uniform oriented matroid $M_0 = (E_0, \\mathcal {L}_0)$ , where $E = E_0 \\cup \\lbrace f\\rbrace $ .",
"Let $D$ , $X_1$ , ..., $X_d$ , and $\\overline{M}$ be as in Proposition REF , and suppose that $\\overline{M}$ is also a lifting of $M_0$ .",
"This implies that $f \\notin D$ and $X_1(f) = X_2(f) = \\cdots = X_d(f) \\ne 0$ .",
"Since replacing all of the $X_1$ , ..., $X_d$ with their negatives does not change $\\overline{M}$ , we may assume $X_1(f) = X_2(f) = \\cdots = X_d(f) = +$ .",
"Along with the original assumptions on the $X_i$ , this completely determines the $X_i$ .",
"In this case, $\\overline{M}$ is determined by $D$ , and we say $D$ is the support of the flip between $M$ and $\\overline{M}$ .",
"Given a uniform oriented matroid $M = (E, \\mathcal {L})$ , let $G(M)$ denote the graph whose vertices are all uniform liftings of $M$ and whose edges are the flips between them.",
"The following is a version of [13], [15] applied to $\\mathcal {F}(M)$ .",
"Proposition 2.7 If $G(M)$ is disconnected, then there is some subset $A \\subseteq E$ such that $\\mathcal {F}(M \\vert _A) \\,\\backslash \\,\\hat{0}$ is disconnected and $\\operatorname{corank}(M \\vert _A) > 1$ .",
"For any poset $P$ , an upper ideal of $P$ is a subposet $I \\subseteq P$ such that $x \\in I$ and $y > x$ implies $y \\in I$ .",
"For any $x \\in P$ , define the upper ideals $I_{\\ge x} = \\lbrace y \\in P : y \\ge x\\rbrace $ and $I_{> x} = \\lbrace y \\in P : y > x\\rbrace $ .",
"The following is an easy exercise.",
"Proposition 2.8 Let $P$ be a finite connected poset, and let $G$ be an upper ideal of $P$ containing all the maximal elements of $P$ .",
"Suppose that $I_{> x}$ is connected for any $x \\in P \\,\\backslash \\,G$ .",
"Then $G$ is connected.",
"Let $G$ be the subposet of $\\mathcal {F}(M)$ consisting of all uniform liftings of $M$ and the flips between them.",
"By Proposition REF , $G$ is an upper ideal of $\\mathcal {F}(M)$ .",
"If $G$ is disconnected, by Proposition REF there is some $M^{\\prime } \\in \\mathcal {F}(M) \\,\\backslash \\,G$ such that $I_{> M^{\\prime }}$ is disconnected.",
"We now use the following.",
"Proposition 2.9 For any non-maximal $M^{\\prime } \\in \\mathcal {F}(M)$ , there exist $A_1$ , ..., $A_k \\subseteq E$ such that $I_{\\ge M^{\\prime }} \\cong \\mathcal {F}(M \\vert _{A_1}) \\times \\cdots \\mathcal {\\times }\\mathcal {F}(M \\vert _{A_k})$ .",
"Let $\\mathcal {A}$ be a topological representation of $M^{\\prime }$ .",
"Let $S_{A_1}$ , ..., $S_{A_k}$ be all of the special pseudospheres of $\\mathcal {A}$ , where $A_i$ is the maximal set $A \\subseteq E \\cup \\lbrace f\\rbrace $ such that $S_A = S_{A_i}$ .",
"Since $M$ is uniform, none of these special pseudospheres intersects $S_f$ , and hence they are all 0-dimensional and $A_i \\subseteq E$ for all $i$ .",
"Thus, moving the arrangement $\\mathcal {A}$ into a more general position (while still representing a lift of $M$ ) is equivalent to moving each of the subarrangements $\\lbrace S_e\\rbrace _{e \\in A_i \\cup \\lbrace f\\rbrace }$ into more general position; in other words, the map $I_{\\ge M^{\\prime }} \\rightarrow \\mathcal {F}(M \\vert _{A_1}) \\times \\cdots \\times \\mathcal {F}(M \\vert _{A_k})$ given by $M^{\\prime \\prime } \\mapsto (M^{\\prime \\prime } \\vert _{A_1 \\cup \\lbrace f\\rbrace }, \\cdots , M^{\\prime \\prime } \\vert _{A_k \\cup \\lbrace f\\rbrace })$ is an isomorphism of posets.",
"Now, let $M^{\\prime } \\in \\mathcal {F}(M) \\,\\backslash \\,G$ and let $I_{\\ge M^{\\prime }} \\cong \\mathcal {F}(M \\vert _{A_1}) \\times \\cdots \\mathcal {\\times }\\mathcal {F}(M \\vert _{A_k})$ as in the previous Proposition.",
"We may assume each $\\mathcal {F}(M \\vert _{A_i})$ is non-trivial.",
"Then $I_{> M^{\\prime }}$ is disconnected only if $k = 1$ and $\\mathcal {F}(M \\vert _{A_1}) \\,\\backslash \\,\\hat{0}$ is disconnected.",
"In this case, if $\\operatorname{corank}(M \\vert _{A_1}) = 1$ , then $M^{\\prime }$ is a flip, which contradicts $M^{\\prime } \\notin G$ .",
"This completes the proof.",
"Thus, to disprove the extension space conjecture, it suffices to show the following.",
"Theorem 2.10 There is a realizable uniform oriented matroid $M$ for which $G(M)$ is disconnected."
],
[
"Main proof",
"The main idea will be to define a large realizable uniform oriented matroid of rank 3 and show that one of its liftings is highly “entangled.” This entanglement will be achieved by what can be thought of as “local reorientations” which are applied on a random subset of its ground set $E$ .",
"Formally, we will build this lifting up from many smaller liftings of the braid arrangement of dimension 3.",
"Many of the ideas here originally appeared in the author's paper [8], and some of the exposition is rewritten from there."
],
[
"Liftings of the 3-dimensional braid arrangement",
"We will work in the tropical projective space $\\mathbb {TP}^3 := \\mathbb {R}^4 / (1,1,1,1)\\mathbb {R}$ , whose points we write as points of $\\mathbb {R}^4$ modulo the relation $x \\sim x + (c,c,c,c)$ for any $c \\in \\mathbb {R}$ .",
"We define an inner product on $\\mathbb {TP}^3$ by $\\langle x,y \\rangle = \\langle x^{\\prime },y^{\\prime } \\rangle _{\\mathbb {R}^4}$ , where $x^{\\prime } \\in \\mathbb {R}^4$ satsifies $x^{\\prime }_1 + \\cdots + x^{\\prime }_4 = 0$ and the residue of $x^{\\prime }$ in $\\mathbb {TP}^3$ is $x$ , and $y^{\\prime }$ is defined similarly.",
"While $\\mathbb {TP}^3$ is isomorphic to $\\mathbb {R}^3$ , it has a more convenient coordinate system for our purposes.",
"Let $\\Gamma _n^k$ denote the set of all ordered $k$ -tuples $(i_1,\\cdots ,i_k)$ of distinct $i_1$ , ..., $i_k \\in [n]$ under the equivalence relation $(i_1,\\cdots ,i_k) \\sim (i_2,\\cdots ,i_k,i_1)$ .",
"We will use $(i_1 \\cdots i_k)$ to denote the equivalence class of $(i_1,\\cdots ,i_k)$ in $\\Gamma _n^k$ .",
"We write $-(i_1 \\cdots i_k)$ to denote $(i_k \\cdots i_1)$ .",
"Let $E_0 := \\Gamma _4^2$ .",
"Let $e_i$ be the $i$ -th standard basis vector of $\\mathbb {R}^4$ mapped to $\\mathbb {TP}^3$ , and let $e_{ij} := e_i - e_j$ .",
"Let $M_0 = (E_0, \\mathcal {L}_0)$ be the oriented matroid of the vector configuration $\\lbrace e_{ij} : 1 \\le i < j \\le 4 \\rbrace $ , where $e_{ij}$ is indexed by $(ij) \\in E_0$ .",
"This oriented matroid is topologically represented by the intersection of the unit 2-sphere in $\\mathbb {TP}^3$ with the braid arrangement $\\mathcal {B}_0 := \\lbrace H_{ij} : 1 \\le i < j \\le 4 \\rbrace $ , where $H_{ij}$ is the oriented hyperplane $\\lbrace x \\in \\mathbb {TP}^3 : x_i - x_j = 0 \\rbrace $ with positive direction $e_{ij}$ .",
"We construct eight specific liftings of $M_0$ .",
"For each $\\gamma = (ijk) \\in \\Gamma _4^3$ , let $\\mathcal {B}_\\gamma $ be the hyperplane arrangement in $\\mathbb {TP}^3$ with hyperplanes $H_{(ij)}^\\gamma &= \\lbrace x \\in \\mathbb {TP}^3 : x_i - x_j = 1 \\rbrace \\qquad & H_{(il)}^\\gamma &= \\lbrace x \\in \\mathbb {TP}^3 : x_i - x_l = 0 \\rbrace \\\\H_{(jk)}^\\gamma &= \\lbrace x \\in \\mathbb {TP}^3 : x_j - x_k = 1 \\rbrace \\qquad & H_{(jl)}^\\gamma &= \\lbrace x \\in \\mathbb {TP}^3 : x_j - x_l = 0 \\rbrace \\\\H_{(ki)}^\\gamma &= \\lbrace x \\in \\mathbb {TP}^3 : x_k - x_i = 1 \\rbrace \\qquad & H_{(kl)}^\\gamma &= \\lbrace x \\in \\mathbb {TP}^3 : x_k - x_l = 0 \\rbrace $ where $\\lbrace l\\rbrace = [4] \\,\\backslash \\,\\lbrace i,j,k\\rbrace $ .",
"To orient these hyperplanes, for any distinct $1 \\le p,q \\le 4$ , define $\\alpha _{pq} =\\begin{dcases*}+ & if p < q \\\\- & if p > q\\end{dcases*}$ and orient each $H_{(pq)}^\\gamma $ so that the $\\alpha _{pq}$ side of $H_{(pq)}^\\gamma $ is in the $e_{pq}$ direction.",
"With this orientation, the intersection of $\\mathcal {B}_\\gamma $ with the sphere at infinity is a topological representation of $M_0$ .",
"Hence, as discussed in Section REF , there is a unique lifting $M_\\gamma = (E_0 \\cup \\lbrace f\\rbrace , \\mathcal {L}_\\gamma )$ of $M_0$ such that $\\mathcal {L}_\\gamma ^+ := \\lbrace X \\in \\mathcal {L}_\\gamma : X(f) = + \\rbrace $ is topologically represented by $\\mathcal {B}_\\gamma $ .",
"It is easily checked that $M_\\gamma $ is a maximal element of $\\mathcal {F}(M_0)$ .",
"Proposition 3.1 Let $\\gamma = (ijk)$ be as above.",
"For each $p \\in \\lbrace i,j,k\\rbrace $ , there are cocircuits $X_1$ , $X_2$ , and $X_3 \\in \\mathcal {L}_\\gamma ^+$ satisfying $X_1((jk)) &= \\alpha _{kj} \\qquad & X_1((ki)) &= 0 \\qquad & X_1((ij)) &= 0 \\qquad & X_1((pl)) &= 0 \\\\X_2((jk)) &= 0 \\qquad & X_2((ki)) &= \\alpha _{ik} \\qquad & X_2((ij)) &= 0 \\qquad & X_2((pl)) &= 0 \\\\X_3((jk)) &= 0 \\qquad & X_3((ki)) &= 0 \\qquad & X_3((ij)) &= \\alpha _{ji} \\qquad & X_3((pl)) &= 0$ There is a cocircuit $X \\in \\mathcal {L}_\\gamma ^+$ satisfying $X((jk)) &= \\alpha _{kj} \\qquad & X((il)) &= 0 \\\\X((ki)) &= \\alpha _{ik} \\qquad & X((jl)) &= 0 \\\\X((ij)) &= \\alpha _{ji} \\qquad & X((kl)) &= 0$ In the arrangement $\\mathcal {B}_\\gamma $ , the covector $X_1$ corresponds to the point $x \\in \\mathbb {TP}^3$ with $x_i = 0$ , $x_j = -1$ , $x_k = 1$ , and $x_l = x_p$ .",
"$X_2$ and $X_3$ can be found similarly.",
"The covector $X$ corresponds to the point $(0,0,0,0)$ ."
],
[
"A group action on $\\Gamma _4^3$",
"We will use many copies of the liftings in the previous section to construct a lifting of a larger oriented matroid.",
"To help in doing so, we define a certain group action on $\\Gamma _4^3$ .",
"For each $\\gamma = (ijk) \\in \\Gamma _4^3$ , we define a function $o_\\gamma : \\binom{[4]}{3} \\rightarrow \\Gamma _4^3$ by $o_\\gamma (\\lbrace i,j,k\\rbrace ) &= (ijk) \\\\o_\\gamma (\\lbrace i,j,l\\rbrace ) &= (ijl) \\\\o_\\gamma (\\lbrace j,k,l\\rbrace ) &= (jkl) \\\\o_\\gamma (\\lbrace k,i,l\\rbrace ) &= (kil)$ where $\\lbrace l\\rbrace = [4] \\,\\backslash \\,\\lbrace i,j,k\\rbrace $ .",
"It is easy to check that $\\gamma $ is determined by $o_\\gamma $ .",
"The relationship of $o_\\gamma $ to $M_\\gamma $ is as follows.",
"Suppose that $\\gamma \\in \\Gamma _4^3$ and $o_\\gamma (\\lbrace i,j,k\\rbrace ) = (ijk)$ .",
"Then the restriction of $\\mathcal {B}_\\gamma $ to the hyperplanes $H_{(ij)}^\\gamma $ , $H_{(jk)}^\\gamma $ , $H_{(ki)}^\\gamma $ is isomorphic to the hyperplane arrangement $H_{ij}^{\\prime } &= \\lbrace x \\in \\mathbb {TP}^3 : x_i - x_j = 1 \\rbrace \\nonumber \\\\H_{jk}^{\\prime } &= \\lbrace x \\in \\mathbb {TP}^3 : x_j - x_k = 1 \\rbrace \\\\H_{ki}^{\\prime } &= \\lbrace x \\in \\mathbb {TP}^3 : x_k - x_i = 1 \\rbrace \\nonumber $ where $H_{(ij)}^\\gamma $ maps to $H_{ij}^{\\prime }$ , etc., and each $H_{pq}^{\\prime }$ is oriented in the same way that $H_{(pq)}^\\gamma $ is.",
"Now, we will map each $\\alpha \\in \\Gamma _4^2$ to a permutation $\\pi _\\alpha : \\Gamma _4^3 \\rightarrow \\Gamma _4^3$ .",
"This map is completely determined by the following rules: For any distinct $i$ , $j$ , $k$ , $l \\in [4]$ , we have $\\pi _{(ij)}(ijk) &= (jil) \\\\\\pi _{(kl)}(ijk) &= (ijl).$ Let $G_{\\Gamma _4^3}$ be the permutation group of $\\Gamma _4^3$ generated by all the $\\pi _\\alpha $ .",
"Proposition 3.2 The following are true.",
"Every element of $G_{\\Gamma _4^3}$ is an involution, and $G_{\\Gamma _4^3}$ is abelian and transitive on $\\Gamma _4^3$ .",
"For $l \\in [4]$ , let $H_l$ be the subgroup of $G_{\\Gamma _4^3}$ generated by $\\pi _{(il)}$ for all $i \\in [4] \\,\\backslash \\,\\lbrace l\\rbrace $ .",
"Let $i$ ,$j$ ,$k \\in [4] \\,\\backslash \\,\\lbrace l\\rbrace $ be distinct, and let $\\Gamma _4^3(ijk)$ be the set of all $\\gamma \\in \\Gamma _4^3$ such that $o_\\gamma (\\lbrace i,j,k\\rbrace ) = (ijk)$ .",
"Then $\\Gamma _4^3(ijk)$ is an orbit of $H_l$ .",
"Since each $\\gamma $ is determined by $o_\\gamma $ , we can view $G_{\\Gamma _4^3}$ as an action on the set of functions $o_\\gamma $ .",
"We check that for all distinct $i$ , $j$ , $k$ , $l \\in [4]$ and $\\gamma \\in \\Gamma _4^3$ , we have $o_{\\pi _{(ij)}\\gamma }(\\lbrace i,j,k\\rbrace ) &= -o_\\gamma (\\lbrace i,j,k\\rbrace ) \\\\o_{\\pi _{(ij)}\\gamma }(\\lbrace i,j,l\\rbrace ) &= -o_\\gamma (\\lbrace i,j,l\\rbrace ) \\\\o_{\\pi _{(ij)}\\gamma }(\\lbrace j,k,l\\rbrace ) &= o_\\gamma (\\lbrace j,k,l\\rbrace ) \\\\o_{\\pi _{(ij)}\\gamma }(\\lbrace k,i,l\\rbrace ) &= o_\\gamma (\\lbrace k,i,l\\rbrace ).$ It follows that we can embed $G_{\\Gamma _4^3}$ as a subgroup of $\\mathbb {Z}_2^4$ .",
"This implies that every element of $G_{\\Gamma _4^3}$ is an involution and $G_{\\Gamma _4^3}$ is abelian.",
"It is also easy to check from the above action on the $o_\\gamma $ that every element of $\\Gamma _4^3$ has orbit of size 8, and hence $G_{\\Gamma _4^3}$ is transitive.",
"From the above action on $o_\\gamma $ , we see that $H_l$ maps $\\Gamma _4^3(ijk)$ to itself and every element of $\\Gamma _4^3(ijk)$ has orbit of size 4 under $H_l$ .",
"Since $\\vert \\Gamma _4^3(ijk) \\vert = 4$ , $\\Gamma _4^3(ijk)$ is an orbit of $H_l$ ."
],
[
"A non-uniform realizable oriented matroid and a lifting",
"We will now construct a non-uniform realizable oriented matroid and one of its liftings.",
"Our desired uniform oriented matroid will be obtained by perturbing this matroid.",
"Let $N$ be a positive integer to be determined later.",
"Let $E = \\lbrace (i,j,r) : 1 \\le i, j \\le 4, i \\ne j, -N \\le r \\le N \\rbrace \\ / \\ (i,j,r) \\sim (j,i,-r).$ That is, the element $(i,j,r) \\in E$ is identified with $(j,i,-r) \\in E$ .",
"Let $M = (E,\\mathcal {L})$ be the oriented matroid of the vector configuration $\\lbrace v_e \\rbrace _{e \\in E}$ , where $v_{(i,j,r)} = e_{ij} \\text{ if } i < j.$ We construct a lifting of $M$ .",
"First, let $\\mathcal {B}$ be the hyperplane arrangement $\\lbrace H_e\\rbrace _{e \\in E}$ where $H_{(i,j,r)} = \\lbrace x \\in \\mathbb {TP}^3 : x_i - x_j = r \\rbrace $ and $H_{(i,j,r)}$ is oriented so that the $\\alpha _{ij}$ side of $H_{(i,j,r)}$ is in the $e_{ij}$ direction.",
"The intersection of $\\mathcal {B}$ with the sphere at infinity is a topological representation of $M$ , and hence $\\mathcal {B}$ defines a lifting $M_{\\mathcal {B}} = (E \\cup \\lbrace f\\rbrace , \\mathcal {L}_{\\mathcal {B}})$ of $M$ .",
"Let $Q$ be the set of $x \\in \\mathbb {TP}^3$ such that if $ijkl$ is a permutation of $[4]$ such that $x_i \\ge x_j \\ge x_k \\ge x_l$ , then $x_i - x_j$ , $x_j - x_k$ , and $x_k - x_l$ are integers at most $N$ .",
"Let $Q^\\star $ be the set of $x \\in Q$ such that $\\vert x_i - x_j \\vert \\le N$ for all $i$ , $j \\in [n]$ .",
"For each $x \\in Q$ , the set of hyperplanes $\\mathcal {B}(x) := \\lbrace H_{(i,j,x_i-x_j)} : 1 \\le i < j \\le 4, \\vert x_i - x_j \\vert \\le N\\rbrace $ intersect at $x$ .",
"If $x \\in Q^\\star $ , then $\\mathcal {B}(x)$ is isomorphic to the braid arrangement $\\mathcal {B}_0$ .",
"We now construct a maximal element of $\\mathcal {F}(M)$ by deforming the arrangement $\\mathcal {B}$ .",
"For each $e = (i,j,r) \\in E$ , let $g_e$ be an independent random element of $G_{\\Gamma _4^3}$ which is 1 with probability $1/2$ and $\\pi _{(ij)}$ with probability $1/2$ .",
"For each $x \\in Q$ , define $\\gamma (x) := \\left( \\prod _{1 \\le i < j \\le 4} g_{(i,j,x_i-x_j)} \\right) (123) \\in \\Gamma _4^3.$ For each $x \\in Q$ , we have an injective map $\\mathcal {B}(x) \\rightarrow \\mathcal {B}_{\\gamma (x)}$ where $H_{(i,j,x_i-x_j)} \\mapsto H_{(ij)}^{\\gamma (x)}$ .",
"The image of this map is a subarrangement of $\\mathcal {B}_{\\gamma (x)}$ , and there is a canonical way to shift the hyperplanes of $\\mathcal {B}(x)$ to obtain an arrangement $\\mathcal {B}(x)^{\\prime }$ which is isomorphic to this image.",
"Now, we construct a pseudohyperplane arrangement $\\mathcal {B}^{\\prime }$ from $\\mathcal {B}$ as follows: For all $x \\in Q$ , we deform $\\mathcal {B}(x)$ so that in a small (i.e.",
"radius $\\ll 1$ ) open neighborhood $U$ around $x$ we have the arrangement $\\mathcal {B}(x)^{\\prime }$ , and at infinity the arrangement is unchanged.",
"This deformation of $\\mathcal {B}(x)$ is not necessarily local to $U$ because if the hyperplanes $H_{(i,j,r)}$ , $H_{(j,k,s)}$ , $H_{(k,i,t)}$ are in $\\mathcal {B}(x)$ , then they intersect at a line in $\\mathcal {B}(x)$ , but their deformations do not mutually intersect in $\\mathcal {B}(x)^{\\prime }$ .",
"To show that $\\mathcal {B}^{\\prime }$ is well-defined, we need to show that for any three such hyperplanes, the restriction of $\\mathcal {B}(x)^{\\prime }$ to the deformations of these hyperplanes is the same for any $\\mathcal {B}(x)$ which contains these hyperplanes.",
"Suppose that $x^1$ , $x^2 \\in Q$ and $H_{(i,j,r)}$ , $H_{(j,k,s)}$ , $H_{(k,i,t)}$ are hyperplanes contained in both $\\mathcal {B}(x^1)$ and $\\mathcal {B}(x^2)$ .",
"Hence, $x_i^1 - x_j^1 = x_i^2 - x_j^2 &= r \\\\x_j^1 - x_k^1 = x_j^2 - x_k^2 &= s \\\\x_k^1 - x_i^1 = x_k^2 - x_i^2 &= t.$ By Proposition REF (b), for any $x \\in Q$ , $o_{\\gamma (x)}(\\lbrace i,j,k\\rbrace )$ depends only on $g_{(i,j,x_i-x_j)}$ , $g_{(j,k,x_i-x_j)}$ , and $g_{(k,i,x_k-x_i)}$ .",
"Hence, $o_{\\gamma (x^1)}(\\lbrace i,j,k\\rbrace ) = o_{\\gamma (x^2)}(\\lbrace i,j,k\\rbrace )$ .",
"From the discussion in Section REF , this implies that the restrictions of $\\mathcal {B}(x^1)^{\\prime }$ and $\\mathcal {B}(x^2)^{\\prime }$ to the deformations of $H_{(i,j,r)}$ , $H_{(j,k,s)}$ , $H_{(k,i,t)}$ are isomorphic arrangements, with the canonical isomorphism.",
"Thus, $\\mathcal {B}^{\\prime }$ is a well-defined pseudohyperplane arrangement.A proof which does not use topological arguments can be found in [8].",
"The intersection of $\\mathcal {B}^{\\prime }$ with the sphere at infinity is a topological representation of $M$ , so we obtain a lifting $M^{\\prime } = (E \\cup \\lbrace f\\rbrace , \\mathcal {L}^{\\prime })$ of $M$ such that $(\\mathcal {L}^{\\prime })^+ := \\lbrace X \\in \\mathcal {L}^{\\prime } : X(f) = +\\rbrace $ is topologically represented by $\\mathcal {B}^{\\prime }$ .",
"By construction, $M^{\\prime }$ is maximal in $\\mathcal {F}(M)$ and $M^{\\prime } \\rightsquigarrow M_{\\mathcal {B}}$ .",
"The following proposition follows from Proposition REF and the properties of $\\mathcal {B}$ .",
"Proposition 3.3 Let $x \\in Q^\\star $ , and assume $\\gamma (x) = (ijk)$ .",
"Let $\\lbrace l\\rbrace = [4] \\,\\backslash \\,\\lbrace i,j,k\\rbrace $ .",
"For all $-N \\le r \\le N$ with $r \\ge x_j - x_k$ , and for all $p \\in \\lbrace i,j,k\\rbrace $ , there are cocircuits $X_1$ , $X_2$ , and $X_3 \\in (\\mathcal {L}^{\\prime })^+$ satisfying $X_1(j,k,r) &= \\alpha _{kj} \\qquad & X_1(k,i,x_k-x_i) &= 0 \\qquad & X_1(i,j,x_i-x_j) &= 0 \\\\ X_2(j,k,r) &= 0 \\qquad & X_2(k,i,x_k-x_i) &= \\alpha _{ik} \\qquad & X_2(i,j,x_i-x_j) &= 0 \\\\ X_3(j,k,r) &= 0 \\qquad & X_3(k,i,x_k-x_i) &= 0 \\qquad & X_3(i,j,x_i-x_j) &= \\alpha _{ji}$ and $X_1(p,l,x_p-x_l) = 0 \\qquad X_2(p,l,x_p-x_l) = 0 \\qquad X_3(p,l,x_p-x_l) = 0.$ There is a cocircuit $X \\in (\\mathcal {L}^{\\prime })^+$ satisfying $X(j,k,x_j-x_k) &= \\alpha _{kj} \\qquad & X(i,l,x_i-x_l) &= 0 \\\\X(k,i,x_k-x_i) &= \\alpha _{ik} \\qquad & X(j,l,x_j-x_l) &= 0 \\\\X(i,j,x_i-x_j) &= \\alpha _{ji} \\qquad & X(k,l,x_k-x_l) &= 0.$ and for any distinct $p$ , $q \\in [4]$ and any $-N \\le u \\le N$ with $u \\ne x_p - x_q$ , $X(p,q,u) = \\operatorname{sign}(x_p-x_q-u)\\alpha _{pq}.$"
],
[
"A uniform realizable oriented matroid",
"We now perturb $M$ and $M^{\\prime }$ .",
"Let $0 < \\delta \\ll 1$ be a small real number.",
"Let $\\Delta := \\lbrace \\delta (e_i + e_j - e_k - e_l) : i, j, k, l \\in [4] \\text{ are distinct} \\rbrace .$ For each $u \\in \\Delta $ and $i \\in [4]$ , define $u^i \\in \\lbrace \\pm 1\\rbrace $ as follows: If $u = \\delta (e_i + e_j - e_k - e_l)$ , then $u^i = u^j = 1 \\qquad u^k = u^l = -1.$ For each $e \\in E$ , let $\\eta _e := u_e + \\epsilon _e$ , where $\\epsilon _e$ is a generic element of $\\mathbb {TP}^3$ with $\\vert \\vert \\epsilon _e \\vert \\vert \\ll \\delta $ , and $u_e$ is chosen independently and uniformly at random from $\\Delta $ .",
"Let $\\widetilde{v}_e := v_e + \\eta _e$ .",
"Let $\\widetilde{M} = (E, \\widetilde{\\mathcal {L}})$ be the oriented matroid of the configuration $\\lbrace \\widetilde{v}_e\\rbrace _{e \\in E}$ .",
"Since the $\\epsilon _e$ are generic, $\\widetilde{M}$ is uniform.",
"Let $H_e^{\\prime }$ be the deformation of $H_e$ in $\\mathcal {B}^{\\prime }$ .",
"We can tilt each pseudohyperplane $H_e^{\\prime }$ “near infinity” to obtain a pseudohyperplane $\\widetilde{H_e^{\\prime }}$ whose normal vector far away from the origin is $\\widetilde{v}_e$ .",
"This gives an arrangement $\\widetilde{\\mathcal {B}^{\\prime }} = \\lbrace \\widetilde{H_e^{\\prime }}\\rbrace _{e \\in E}$ whose intersection with the sphere at infinity is a topological representation of $\\widetilde{M}$ .",
"If $\\delta $ is small enough, this can be done so that within a sphere $S$ of radius $100N$ around the origin, the arrangement $\\widetilde{\\mathcal {B}^{\\prime }}$ is the same as $\\mathcal {B}^{\\prime }$ .",
"In other words, $\\widetilde{\\mathcal {B}^{\\prime }}$ defines a lifting $\\widetilde{M^{\\prime }} = (E \\cup \\lbrace f\\rbrace , \\widetilde{\\mathcal {L}^{\\prime }})$ of $\\widetilde{M}$ such that $\\widetilde{M^{\\prime }} \\rightsquigarrow M^{\\prime }$ .",
"In particular, we have $(\\widetilde{\\mathcal {L}^{\\prime }})^+ \\supseteq (\\mathcal {L}^{\\prime })^+$ , where $(\\widetilde{\\mathcal {L}^{\\prime }})^+ := \\lbrace X \\in \\widetilde{\\mathcal {L}^{\\prime }} : X(f) = +\\rbrace $ .",
"For any distinct $i$ , $j$ , $k \\in [4]$ and any $-N \\le r,s,t \\le N$ , the pseudohyperplanes $\\widetilde{H^{\\prime }}_{(i,j,r)}$ , $\\widetilde{H^{\\prime }}_{(j,k,s)}$ , and $\\widetilde{H^{\\prime }}_{(k,i,t)}$ in $\\widetilde{\\mathcal {B}^{\\prime }}$ intersect at a point which is far from the origin; i.e., outside of $S$ .",
"This point will either be far in the $e_l$ direction or far in the $-e_l$ direction, where $\\lbrace l\\rbrace = [4] \\,\\backslash \\,\\lbrace i,j,k\\rbrace $ .",
"The correct direction depends on the arrangement of $H_{(i,j,r)}^{\\prime }$ , $H_{(j,k,s)}^{\\prime }$ , $H_{(k,i,t)}^{\\prime }$ in $\\mathcal {B}^{\\prime }$ and the sign $\\beta _{ijk}(r,s,t) := \\operatorname{sign}\\left( \\alpha _{ij}u_{(i,j,r)}^l + \\alpha _{jk}u_{(j,k,s)}^l + \\alpha _{ki}u_{(k,i,t)}^l \\right).$ More precisely, we have the following.",
"Proposition 3.4 Let $i$ , $j$ , $k$ , $l \\in [4]$ be distinct and $-N \\le r,s,t \\le N$ .",
"Suppose that there exists $X_0 \\in (\\mathcal {L}^{\\prime })^+$ such that $X_0(i,j,r) = \\alpha _{ji} \\qquad X_0(j,k,s) = \\alpha _{kj} \\qquad X_0(k,i,t) = \\alpha _{ik}.$ Then there exists a cocircuit $X \\in (\\widetilde{\\mathcal {L}}^{\\prime })^+$ such that $X(i,j,r) = 0 \\qquad X(j,k,s) = 0 \\qquad X(k,i,t) = 0$ and for all $p \\in \\lbrace i,j,k\\rbrace $ and $-N \\le u \\le N$ , $X(l,p,u) = \\alpha _{lp} \\cdot \\beta _{ijk}(r,s,t).$ The existence of $X_0$ implies that the restriction of $\\mathcal {B}^{\\prime }$ to $\\lbrace H_{(i,j,r)}^{\\prime }, H_{(j,k,s)}^{\\prime }, H_{(k,i,t)}^{\\prime }\\rbrace $ is isomorphic (with the usual isomorphism) to the arrangement (REF ).",
"Thus, the restriction of $\\widetilde{\\mathcal {B}}^{\\prime }$ to $\\widetilde{H^{\\prime }}_{(i,j,r)}$ , $\\widetilde{H^{\\prime }}_{(j,k,s)}$ , and $\\widetilde{H^{\\prime }}_{(k,i,t)}$ is an arrangement whose behavior away from the origin is the same as the arrangement $\\widetilde{H_{ij}^{\\prime }} &= \\lbrace x \\in \\mathbb {TP}^3 : \\langle x, e_{ij} + \\alpha _{ij}\\eta _{(i,j,r)} \\rangle = 1 \\rbrace \\\\\\widetilde{H_{jk}^{\\prime }} &= \\lbrace x \\in \\mathbb {TP}^3 : \\langle x, e_{jk} + \\alpha _{jk}\\eta _{(j,k,s)} \\rangle = 1 \\rbrace \\\\\\widetilde{H_{ki}^{\\prime }} &= \\lbrace x \\in \\mathbb {TP}^3 : \\langle x, e_{ki} + \\alpha _{ki}\\eta _{(k,i,t)} \\rangle = 1 \\rbrace $ obtained by tilting the hyperplanes in arrangement (REF ).",
"Let $x$ be the intersection of $\\widetilde{H_{ij}^{\\prime }}$ , $\\widetilde{H_{jk}^{\\prime }}$ , and $\\widetilde{H_{ki}^{\\prime }}$ .",
"Let $\\eta _{(i,j,r)} = \\eta _{ij}^l + \\eta _{ij}^\\perp $ , where $\\eta _{ij}^l$ is parallel to $e_l$ and $\\eta _{ij}^\\perp $ is orthogonal to $e_l$ .",
"Similarly let $\\eta _{(j,k,s)} = \\eta _{jk}^l + \\eta _{jk}^\\perp $ and $\\eta _{(k,i,t)} = \\eta _{ki}^l + \\eta _{ki}^\\perp $ .",
"The vectors $e_{ij} + \\alpha _{ij}\\eta _{ij}^\\perp ,\\, e_{jk} + \\alpha _{jk}\\eta _{jk}^\\perp ,\\, e_{ki} + \\alpha _{ki}\\eta _{ki}^\\perp $ lie in the 2-dimensional subspace of $\\mathbb {TP}^3$ orthogonal to $e_l$ .",
"Hence, there are $c_1$ , $c_2$ , $c_3 \\in \\mathbb {R}$ such that $c_1(e_{ij} + \\alpha _{ij}\\eta _{ij}^\\perp ) + c_2(e_{jk} + \\alpha _{ij}\\eta _{jk}^\\perp ) + c_3(e_{ki} + \\alpha _{ki}\\eta _{ki}^\\perp ) = 0.$ Since $e_{ij} + e_{jk} + e_{ki} = 0$ and $\\vert \\vert \\eta _{ij}^\\perp \\vert \\vert $ , $\\vert \\vert \\eta _{jk}^\\perp \\vert \\vert $ , $\\vert \\vert \\eta _{ki}^\\perp \\vert \\vert < 3\\delta $ , we can choose $c_1$ , $c_2$ , and $c_3$ so that $\\vert c_i-1 \\vert < C\\delta $ for all $i$ and some constant $C$ independent of $\\delta $ .",
"Now, we have $c_1 \\langle x, e_{ij} + \\alpha _{ij}\\eta _{(i,j,r)} \\rangle + c_2 \\langle x, e_{jk} + \\alpha _{jk}\\eta _{(j,k,s)} \\rangle + c_3 \\langle x, e_{ki} + \\alpha _{ki}\\eta _{(k,i,t)} \\rangle &= c_1 + c_2 + c_3 \\nonumber \\\\\\Rightarrow \\ \\langle x, c_1 \\alpha _{ij} \\eta _{ij}^l + c_2 \\alpha _{jk} \\eta _{jk}^l + c_3 \\alpha _{ki} \\eta _{ki}^l \\rangle &= c_1 + c_2 + c_3 \\nonumber \\\\\\Rightarrow \\ \\langle x, c_1 \\alpha _{ij} \\eta _{ij}^l + c_2 \\alpha _{jk} \\eta _{jk}^l + c_3 \\alpha _{ki} \\eta _{ki}^l \\rangle &> 0 $ where the last inequality holds for small enough $\\delta $ since $\\vert c_i-1 \\vert < C\\delta $ .",
"By the definition of $\\eta _{ij}$ , we have $\\eta _{ij}^l = \\delta u_{(i,j,r)}^l e_l + o(\\delta ) e_l$ where $o(\\delta ) \\ll \\delta $ , and similarly for $\\eta _{jk}^l$ and $\\eta _{ki}^l$ .",
"Thus, $c_1 \\alpha _{ij} \\eta _{ij}^l + c_2 \\alpha _{jk} \\eta _{jk}^l + c_3 \\alpha _{ki} \\eta _{ki}^l = \\delta ( c_1 \\alpha _{ij} u_{(i,j,r)}^l + c_2 \\alpha _{jk} u_{(j,k,s)}^l + c_3 \\alpha _{ki} u_{(k,i,t)}^l ) e_l + o(\\delta )e_l.$ Since $\\vert c_i-1 \\vert < C\\delta $ for all $i$ , this becomes $c_1 \\alpha _{ij} \\eta _{ij}^l + c_2 \\alpha _{jk} \\eta _{jk}^l + c_3 \\alpha _{ki} \\eta _{ki}^l &= \\delta ( \\alpha _{ij} u_{(i,j,r)}^l + \\alpha _{jk} u_{(j,k,s)}^l + \\alpha _{ki} u_{(k,i,t)}^l ) e_l + O(\\delta ^2) e_l + o(\\delta ) e_l \\\\&= \\delta ( \\alpha _{ij} u_{(i,j,r)}^l + \\alpha _{jk} u_{(j,k,s)}^l + \\alpha _{ki} u_{(k,i,t)}^l ) e_l + o(\\delta ) e_l.$ Hence, $\\operatorname{sign}\\langle x, c_1 \\alpha _{ij} \\eta _{ij}^l + c_2 \\alpha _{jk} \\eta _{jk}^l &+ c_3 \\alpha _{ki} \\eta _{ki}^l \\rangle \\\\&= \\operatorname{sign}\\left( \\delta ( \\alpha _{ij} u_{(i,j,r)}^l + \\alpha _{jk} u_{(j,k,s)}^l + \\alpha _{ki} u_{(k,i,t)}^l ) + o(\\delta ) \\right) \\operatorname{sign}\\langle x, e_l \\rangle \\\\&= \\beta _{ijk}(r,s,t) \\cdot \\operatorname{sign}\\langle x, e_l \\rangle $ since $\\vert \\alpha _{ij} u_{(i,j,r)}^l + \\alpha _{jk} u_{(j,k,s)}^l + \\alpha _{ki} u_{(k,i,t)}^l \\vert \\ge 1$ .",
"With (REF ), we thus have $\\operatorname{sign}\\langle x, e_l \\rangle = \\beta _{ijk}(r,s,t)$ .",
"Returning to $\\widetilde{\\mathcal {B}^{\\prime }}$ , we conclude that $\\widetilde{H^{\\prime }}_{(i,j,r)}$ , $\\widetilde{H^{\\prime }}_{(j,k,s)}$ , and $\\widetilde{H^{\\prime }}_{(k,i,t)}$ intersect at a point which is outside of $S$ and far in the $\\beta _{ijk}(r,s,t) e_l$ direction.",
"The cocircuit $X$ corresponding to this point is the desired cocircuit.",
"We make some final definitions before proceeding.",
"For each $x \\in Q^\\star $ and $l \\in [4]$ , define $\\beta _l(x) := \\beta _{ijk}(x_i - x_j, x_j - x_k, x_k - x_i) \\text{ where } o_{\\gamma (x)}([4] \\,\\backslash \\,\\lbrace l\\rbrace ) = (ijk).$ For each $x \\in Q^\\star $ and $i \\in [4]$ , define $R_{i,+}(x) &:= \\lbrace x + ke_i \\in Q^\\star : k \\in \\mathbb {Z}_+ \\rbrace \\\\R_{i,-}(x) &:= \\lbrace x + ke_i \\in Q^\\star : k \\in \\mathbb {Z}_- \\rbrace .$ Note that at least one of $R_{i,+}(x)$ , $R_{i,-}(x)$ has size at least $N/2$ .",
"Indeed, if $\\vert R_{i,+}(x) \\vert = m$ , then there is some $j \\in [4] \\,\\backslash \\,\\lbrace i\\rbrace $ such that $x_i + m + 1 - x_j \\ge N + 1$ , and hence $x_i - x_j \\ge N - m$ .",
"Similarly if $\\vert R_{i,+}(x) \\vert = n$ , then there is some $k \\in [4] \\,\\backslash \\,\\lbrace i\\rbrace $ such that $x_k - x_i \\ge N - n$ .",
"Thus $x_k - x_j \\ge 2N - m - n$ , and since $x_k - x_j \\le N$ , we obtain $m+n \\ge N$ ."
],
[
"A special set $\\Omega $",
"We now define a special set $\\Omega \\subseteq Q^\\star $ .",
"We show that with positive probability, it satisfies certain density conditions.",
"This will be used to show disconnectedness of $G(\\widetilde{M})$ .",
"Let $\\Omega $ be the set of all $x \\in Q^\\star $ such that if $\\gamma (x) = (ijk)$ and $\\lbrace l\\rbrace = [4] \\,\\backslash \\,\\lbrace i,j,k\\rbrace $ , then $\\vert R_{i,\\beta _i(x)}(x) \\vert , \\vert R_{j,\\beta _j(x)}(x) \\vert , \\vert R_{k,\\beta _k(x)}(x) \\vert \\ge N/2$ and $\\operatorname{sign}(\\alpha _{il}u_{(i,l,x_i-x_l)}^j) &= \\beta _j(x) \\qquad & \\operatorname{sign}(\\alpha _{li}u_{(l,i,x_l-x_i)}^k) &= \\beta _k(x) \\\\\\operatorname{sign}(\\alpha _{jl}u_{(j,l,x_j-x_l)}^k) &= \\beta _k(x) \\qquad & \\operatorname{sign}(\\alpha _{lj}u_{(l,j,x_l-x_j)}^i) &= \\beta _i(x) \\\\\\operatorname{sign}(\\alpha _{kl}u_{(k,l,x_k-x_l)}^i) &= \\beta _i(x) \\qquad & \\operatorname{sign}(\\alpha _{lk}u_{(l,k,x_l-x_k)}^j) &= \\beta _j(x).$ Proposition 3.5 Let $i$ , $j$ , $k \\in [4]$ be distinct and $-N \\le r,s,t \\le N$ be integers with $r+s+t = 0$ .",
"Let $L = \\lbrace x \\in Q^\\star : x_i - x_j = r, x_j - x_k = s, x_k - x_i = t\\rbrace $ and suppose that $o_{\\gamma (y)}(\\lbrace i,j,k\\rbrace ) = (ijk)$ for some $y \\in L$ .",
"Then for each $x \\in L$ , the probability that $\\gamma (x) = (ijk)$ and $x \\in \\Omega $ is at least $1/864$ , and these probabilities are mutually independent over $L$ .",
"By Proposition REF (b), the value of $o_{\\gamma (x)}(\\lbrace i,j,k\\rbrace )$ is the same for all $x \\in L$ .",
"By assumption, this value is $(ijk)$ .",
"Now, for each $x \\in L$ , let $A(x)$ be the event that $\\gamma (x) = (ijk)$ and let $B(x)$ be the event that $x \\in \\Omega $ .",
"By Proposition REF (b), whether $A(x)$ happens depends only on $g_{(i,l,x_i-x_l)}$ , $g_{(j,l,x_j-x_l)}$ , and $g_{(k,l,x_k-x_l)}$ , where $l = [4] \\,\\backslash \\,\\lbrace i,j,k\\rbrace $ .",
"By Proposition REF (a)(b), the probability $A(x)$ happens is $1/4$ .",
"Now fix $x \\in L$ .",
"For each $p \\in \\lbrace i,j,k\\rbrace $ , there is some $\\beta _p \\in \\lbrace +,-\\rbrace $ such that $\\vert R_{p,\\beta _p} \\vert \\ge N/2$ .",
"Now, the event that $\\operatorname{sign}(\\alpha _{il}u_{(i,l,x_i-x_l)}^j) = \\beta _j \\qquad \\text{and}\\qquad \\operatorname{sign}(\\alpha _{li}u_{(l,i,x_l-x_i)}^k) = \\beta _k$ depends only on $u_{(i,l,x_i-x_l)}$ , and from the definition of $\\Delta $ , the probability this event occurs is at least $1/6$ for any choice of $\\beta _j$ and $\\beta _k$ .",
"Hence, the probability that all the events $\\operatorname{sign}(\\alpha _{il}u_{(i,l,x_i-x_l)}^j) &= \\beta _j \\qquad & \\operatorname{sign}(\\alpha _{li}u_{(l,i,x_l-x_i)}^k) &= \\beta _k \\\\\\operatorname{sign}(\\alpha _{jl}u_{(j,l,x_j-x_l)}^k) &= \\beta _k \\qquad & \\operatorname{sign}(\\alpha _{lj}u_{(l,j,x_l-x_j)}^i) &= \\beta _i \\\\\\operatorname{sign}(\\alpha _{kl}u_{(k,l,x_k-x_l)}^i) &= \\beta _i \\qquad & \\operatorname{sign}(\\alpha _{lk}u_{(l,k,x_l-x_k)}^j) &= \\beta _j$ occur is at least $1/216$ .",
"Now, assume that all these events occur.",
"Since $A(x)$ is independent of these events, we can assume that $A(x)$ occurs as well.",
"Then $o_{\\gamma (x)}(\\lbrace j,k,l\\rbrace ) = (jkl)$ , and $\\beta _i(x) &= \\beta _{jkl}(x_j-x_k,x_k-x_l,x_l-x_i) \\\\&= \\operatorname{sign}\\left( \\alpha _{jk} u_{(j,k,s)}^i + \\alpha _{kl}u_{(k,l,x_k-x_l)}^i + \\alpha _{lj}u_{(l,j,x_l-x_j)}^i \\right).$ Since $\\operatorname{sign}(\\alpha _{kl}u_{(k,l,x_k-x_l)}^i) = \\operatorname{sign}(\\alpha _{lj}u_{(l,j,x_l-x_j)}^i) = \\beta _i$ , the right hand side of this expression is $\\beta _i$ no matter what $\\alpha _{jk} u_{(j,k,s)}^i$ is.",
"Hence $\\beta _i(x) = \\beta _i$ .",
"Similarly $\\beta _j(x) = \\beta _j$ and $\\beta _k(x) = \\beta _k$ .",
"It follows that $x \\in \\Omega $ .",
"This occurs with probability at least $(1/4)(1/216) = 1/864$ .",
"Finally, the event $A(x) \\cap B(x)$ depends only on the independent variables $g_{(i,l,x_i-x_l)}$ , $g_{(j,l,x_j-x_l)}$ , $g_{(k,l,x_k-x_l)}$ , $u_{(i,l,x_i-x_l)}$ , $u_{(j,l,x_j-x_l)}$ , and $u_{(k,l,x_k-x_l)}$ , and these variables are different for every $x \\in L$ .",
"So these events are mutually independent over $L$ .",
"Proposition 3.6 For large enough $N$ , with probability greater than 0, we have the following: $\\Omega $ is nonempty, and for every $x \\in \\Omega $ , if $\\gamma (x) = (ijk)$ , then the sets $S_i(x) &:= \\lbrace y \\in R_{i,\\beta _i(x)}(x) \\cap \\Omega : \\gamma (y) = (jkl) \\rbrace \\\\S_j(x) &:= \\lbrace y \\in R_{j,\\beta _j(x)}(x) \\cap \\Omega : \\gamma (y) = (kil) \\rbrace \\\\S_k(x) &:= \\lbrace y \\in R_{k,\\beta _k(x)}(x) \\cap \\Omega : \\gamma (y) = (ijl) \\rbrace $ are all nonempty.",
"Suppose $x \\in \\Omega $ .",
"By Proposition REF , the probability that $S_i(x)$ is empty is at most $\\left( \\frac{863}{864} \\right)^{\\vert R_{i,\\beta _i(x)}(x) \\vert } \\le \\left( \\frac{863}{864} \\right)^{N/2}$ and similarly for $S_j(x)$ and $S_k(x)$ .",
"By the union bound, the probability that at least one of these sets is empty is at most $3(863/864)^{N/2}$ .",
"Since $\\vert \\Omega \\vert \\le \\vert Q^\\star \\vert \\le (2N+1)^3$ , the probability that this happens for at least one $x \\in \\Omega $ is at most $3 (2N+1)^3 \\left( \\frac{863}{864} \\right)^{N/2}.$ Finally, by Proposition REF , the probability that $\\Omega $ is empty is at most $(863/864)^N$ .",
"For large enough $N$ , the number $3 (2N+1)^3 \\left( \\frac{863}{864} \\right)^{N/2} + \\left( \\frac{863}{864} \\right)^N$ is less than 1.",
"So with positive probability, the desired property is satisfied.",
"From now on, we will assume that the conclusion of Proposition REF is satisfied."
],
[
"Proof that $G(\\widetilde{M})$ is disconnected",
"We conclude by showing that $G(\\widetilde{M})$ is disconnected.",
"This proves Theorem REF and disproves the extension space conjecture.",
"In the following, we will reuse the variables $M$ and $\\mathcal {L}$ for convenience.",
"Definition 3.7 Let $G^\\dagger $ denote the set of all uniform liftings $M = (E \\cup \\lbrace f\\rbrace , \\mathcal {L})$ of $\\widetilde{M}$ which satisfy the following.",
"(As usual, $\\mathcal {L}^+ := \\lbrace X \\in \\mathcal {L} : X(f) = + \\rbrace $ and $\\lbrace l\\rbrace = [4] \\,\\backslash \\,\\lbrace i,j,k\\rbrace $ .)",
"For all $x \\in \\Omega $ and $i \\in [4]$ with $\\gamma (x) = (ijk)$ , for all $y \\in \\Omega $ with $y_j - y_k \\ge x_j - x_k$ , and for all $p \\in \\lbrace i,j,k\\rbrace $ , there exist cocircuits $X_1$ , $X_2$ , $X_3 \\in \\mathcal {L}^+$ satisfying $X_1(j,k,y_j-y_k) &= \\alpha _{kj} \\qquad & X_1(k,i,x_k-x_i) &= 0 \\qquad & X_1(i,j,x_i-x_j) &= 0 \\\\X_2(j,k,y_j-y_k) &= 0 \\qquad & X_2(k,i,x_k-x_i) &= \\alpha _{ik} \\qquad & X_2(i,j,x_i-x_j) &= 0 \\\\X_3(j,k,y_j-y_k) &= 0 \\qquad & X_3(k,i,x_k-x_i) &= 0 \\qquad & X_3(i,j,x_i-x_j) &= \\alpha _{ji}$ and $X_1(p,l,x_p-x_l) = 0 \\qquad X_2(p,l,x_p-x_l) = 0 \\qquad X_3(p,l,x_p-x_l) = 0.$ For all $x \\in \\Omega $ with $\\gamma (x) = (ijk)$ , there exists a cocircuit $X \\in \\mathcal {L}^+$ satisfying $X(i,l,x_i-x_l) = 0 \\qquad X(j,l,x_j-x_l) = 0 \\qquad X(k,l,x_k-x_l) = 0$ in addition to the following: For all $y \\in \\Omega $ with $y_j - y_k \\ge x_j - x_k$ , $X(j,k,y_j-y_k) = \\alpha _{kj}$ .",
"For all $y \\in \\Omega $ with $y_k - y_i \\ge x_k - x_i$ , $X(k,i,y_k-y_i) = \\alpha _{ik}$ .",
"For all $y \\in \\Omega $ with $y_i - y_j \\ge x_i - x_j$ , $X(i,j,y_i-y_j) = \\alpha _{ji}$ .",
"For all $x \\in \\Omega $ and $i \\in [4]$ with $\\gamma (x) = (ijk)$ , and for all $y \\in \\Omega $ with $y_j - y_k \\ge x_j - x_k$ , there exists a cocircuit $X \\in \\mathcal {L}^+$ satisfying $X(j,k,y_j-y_k) = 0 \\qquad X(k,l,x_k-x_l) = 0 \\qquad X(l,j,x_l-x_j) = 0$ and for all $z \\in S_i(x)$ and $p \\in \\lbrace j,k,l\\rbrace $ , $X(i,p,z_i-z_p) = \\alpha _{ip} \\cdot \\beta _i(x).$ Proposition 3.8 $\\widetilde{M^{\\prime }} \\in G^\\dagger $ .",
"Property (a) follows from Proposition REF (a) and the fact that $(\\widetilde{\\mathcal {L}^{\\prime }})^+ \\supseteq (\\mathcal {L}^{\\prime })^+$ .",
"Property (b) follows from Proposition REF (b).",
"We now prove (c).",
"Suppose $x \\in \\Omega $ , $i \\in [4]$ , and $y \\in \\Omega $ such that $\\gamma (x) = (ijk)$ and $y_j - y_k \\ge x_j - x_k$ .",
"First, note there is a covector $X_0 \\in (\\widetilde{\\mathcal {L}^{\\prime }})^+$ such that $X_0(j,k,y_j-y_k) = \\alpha _{kj} \\qquad X_0(k,l,x_k-x_l) = \\alpha _{lk} \\qquad X_0(l,j,x_l-x_j) = \\alpha _{jl}.$ Indeed, if $y = x$ , then this holds because $o_{\\gamma (x)}(\\lbrace j,k,l\\rbrace ) = (jkl)$ , and if $y \\ne x$ , this holds because $y_j - y_k > x_j - x_k$ and the properties of $\\mathcal {B}$ .",
"Thus, by Proposition REF , there exists $X \\in (\\widetilde{\\mathcal {L}^{\\prime }})^+$ such that $X(j,k,y_j-y_k) = 0 \\qquad X(k,l,x_k-x_l) = 0 \\qquad X(l,j,x_l-x_j) = 0$ and for all $z \\in S_i(x)$ and $p \\in \\lbrace j,k,l\\rbrace $ , $X(i,p,z_i-z_p) = \\alpha _{ip} \\cdot \\beta _{jkl}(y_j-y_k,x_k-x_l,x_l-x_j).$ We have $\\beta _{jkl}(y_j-y_k,x_k-x_l,x_l-x_j) = \\operatorname{sign}\\left( \\alpha _{jk}u_{(j,k,y_j-y_k)}^i + \\alpha _{kl}u_{(k,l,x_k-x_l)}^i + \\alpha _{lj}u_{(l,j,x_l-x_j)}^i \\right).$ By the definition of $\\Omega $ , we have $\\operatorname{sign}(\\alpha _{kl}u_{(k,l,x_k-x_l)}^i) = \\operatorname{sign}(\\alpha _{lj}u_{(l,j,x_l-x_j)}^i) = \\beta _i(x)$ .",
"So the right hand side of this expression is $\\beta _i(x)$ .",
"Hence, $X(i,p,z_i-z_p) = \\alpha _{ip} \\cdot \\beta _i(x)$ as desired.",
"Proposition 3.9 Suppose $M = (E \\cup \\lbrace f\\rbrace , \\mathcal {L})$ and $\\overline{M} = (E \\cup \\lbrace f\\rbrace , \\overline{\\mathcal {L}})$ are uniform liftings of $\\widetilde{M}$ and there is a flip $M^0$ between $M$ and $\\overline{M}$ .",
"If $M \\in G^\\dagger $ , then $\\overline{M} \\in G^\\dagger $ .",
"We need to show that Definition REF (a)-(c) hold for $\\overline{M}$ .",
"Proof of (a).",
"Suppose there is some $x \\in \\Omega $ , $i \\in [4]$ , $y \\in \\Omega $ , and $p \\in \\lbrace i,j,k\\rbrace $ such that $\\gamma (x) = (ijk)$ , $y_j-y_k \\ge x_j-x_k$ , and $\\overline{\\mathcal {L}}^+$ does not contain three cocircuits satisfying (a).",
"Since $\\mathcal {L}^+$ does contain such cocircuits $X_1$ , $X_2$ , and $X_3$ , the flip $M^0$ must have support $\\lbrace (j,k,y_j-y_k), (k,i,x_k-x_i), (i,j,x_i-x_j), (p,l,x_p-x_l)\\rbrace $ and this flip involves $X_1$ , $X_2$ , and $X_3$ .",
"However, by (b), there exists $X \\in \\mathcal {L}^+$ with $X(j,k,y_j-y_k) = \\alpha _{kj}$ , $X(k,i,x_k-x_i) = \\alpha _{ik}$ ,This equality is obtained by taking $y = x$ in (b)(ii).",
"$X(i,j,x_i-x_j) = \\alpha _{ji}$ , and $X(p,l,x_p-x_l) = 0$ .",
"This contradicts Proposition REF , proving (a).",
"Proof of (b).",
"Suppose there is some $x \\in \\Omega $ with $\\gamma (x) = (ijk)$ such that $\\overline{\\mathcal {L}}^+$ does not contain a cocircuit satisfying (b).",
"Since $\\mathcal {L}^+$ does contain such a cocircuit $X$ , we must have one of the following: For some $y \\in \\Omega $ with $y_j - y_k \\ge x_j - x_k$ , $M^0$ has support $\\lbrace (j,k,y_j-y_k), (i,l,x_i-x_l), (j,l,x_j-x_l), (k,l,x_k-x_l) \\rbrace .$ For some $y \\in \\Omega $ with $y_k - y_i \\ge x_k - x_i$ , $M^0$ has support $\\lbrace (k,i,y_k-y_i), (i,l,x_i-x_l), (j,l,x_j-x_l), (k,l,x_k-x_l) \\rbrace .$ For some $y \\in \\Omega $ with $y_i - y_j \\ge x_i - x_j$ , $M^0$ has support $\\lbrace (i,j,y_i-y_j), (i,l,x_i-x_l), (j,l,x_j-x_l), (k,l,x_k-x_l) \\rbrace .$ We will assume case (i); the other cases are analogous.",
"The cocircuit $X$ must be involved in the flip $M^0$ .",
"By (c), there is a cocircuit $Y \\in \\mathcal {L}^+$ satisfying $Y(j,k,y_j-y_k) = 0$ , $Y(k,l,x_k-x_l) = 0$ , and $Y(j,l,x_j-x_l) = 0$ , and hence it is also involved in this flip.",
"Suppose $z \\in S_i(x)$ .",
"If $\\beta _i(x) = +$ , then $z_i - z_j \\ge x_i - x_j$ by definition of $S_i(x)$ .",
"Hence $X(i,j,z_i-z_j) = \\alpha _{ji}$ by definition of $X$ .",
"But $Y(i,j,z_i-z_j) = \\alpha _{ij}$ by definition of $Y$ .",
"This contradicts the fact that the cocircuits involved in a flip agree outside of the flip's support.",
"Similarly, if $\\beta _i(x) = -$ , then $X(k,i,z_k-z_i) = \\alpha _{ik}$ and $Y(k,i,z_k-z_i) = \\alpha _{ki}$ , a contradiction.",
"Since $S_i(x)$ is nonempty, we have a contradiction, proving (b).",
"Proof of (c).",
"Suppose there is some $x \\in \\Omega $ , $i \\in [4]$ , and $y \\in \\Omega $ such that $\\gamma (x) = (ijk)$ , $y_j - y_k \\ge x_j - x_k$ , and $\\overline{\\mathcal {L}}^+$ does not contain a cocircuit satisfying (c).",
"Since $\\mathcal {L}^+$ does contain such a cocircuit $X$ , there must be some $z \\in S_i(x)$ and $p \\in \\lbrace j,k,l\\rbrace $ such that $M^0$ has support $\\lbrace (j,k,y_j-y_k), (k,l,x_k-x_l), (l,j,x_l-x_j), (i,p,z_i-z_p) \\rbrace .$ Since $z \\in S_i(x)$ , we have $x_k - x_l = z_k - z_l$ and $x_l - x_j = z_l - z_j$ .",
"So we can rewrite the support of the flip as $\\lbrace (j,k,y_j-y_k), (k,l,z_k-z_l), (l,j,z_l-z_j), (i,p,z_i-z_p) \\rbrace .$ Moreover, we have $y_j - y_k \\ge x_j - x_k = z_j - z_k$ , and since $z \\in S_i(x)$ , we have $\\gamma (z) = (ljk)$ .",
"Thus, by (a) with $x$ replaced by $z$ and $i$ replaced by $l$ , there are cocircuits $X_1$ , $X_2$ , $X_3 \\in \\mathcal {L}^+$ such that $X_1(j,k,y_j-y_k) &= \\alpha _{kj} \\qquad & X_1(k,l,z_k-z_l) &= 0 \\qquad & X_1(l,j,z_l-z_j) &= 0 \\\\X_2(j,k,y_j-y_k) &= 0 \\qquad & X_2(k,l,z_k-z_l) &= \\alpha _{lk} \\qquad & X_2(l,j,z_l-z_j) &= 0 \\\\X_3(j,k,y_j-y_k) &= 0 \\qquad & X_3(k,l,z_k-z_l) &= 0 \\qquad & X_3(l,j,z_l-z_j) &= \\alpha _{jl}$ and $X_1(i,p,z_i-z_p) = 0 \\qquad X_2(i,p,z_i-z_p) = 0 \\qquad X_3(i,p,z_i-z_p) = 0.$ Thus, the flip $M_0$ must involve these three circuits.",
"But we showed in the proof of (a) that there cannot be a flip involving these circuits.",
"This proves (c).",
"Thus, $G^\\dagger $ is a nonempty connected component of $G(\\widetilde{M})$ .",
"Since $\\Omega $ is nonempty, it is easy to see that $G^\\dagger $ is not all of $G(\\widetilde{M})$ .",
"Thus $G(\\widetilde{M})$ is disconnected."
]
] | 1606.05033 |
[
[
"Bowen-York Type Initial Data for Binaries with Neutron Stars"
],
[
"Abstract A new approach to construct initial data for binary systems with neutron star components is introduced.",
"The approach is a generalization of the puncture initial data method for binary black holes based on Bowen-York solutions to the momentum constraint.",
"As with binary black holes, the method allows setting orbital configurations with direct input from post-Newtonian approximations and involves solving only the Hamiltonian constraint.",
"The effectiveness of the method is demonstrated with evolutions of double neutron star and black hole -- neutron star binaries in quasi-circular orbits."
],
[
"Introduction",
"Compact object binaries with black hole (BH) and neutron star (NS) components are main targets of gravitational wave (GW) observations.",
"GWs from binary black holes (BBHs) have been recently detected by the Laser Interferometer Gravitational Wave Observatory (LIGO), first detection in the transient event GW150914 [1] and second detection in the transient event GW151226 [2].",
"As advanced LIGO reaches designed sensitivity, GWs from double neutron star (DNS) and black hole - neutron star (BH-NS) binaries will very likely also be detected.",
"Not surprisingly, numerical relativity (NR) simulations played an important role in the analysis of the GW150914 and GW151226 events.",
"Specifically, best fits of a NR waveform to the data were included in the detection paper [1].",
"The papers on parameter estimation [3] and tests of general relativity [4] mentioned that results from BBH simulations were involved in the construction of the phenomenological and effective-one-body waveform models used in the analysis.",
"The same applies to the paper on the burst-type analysis of GW150914 [5].",
"As with GW150914 and GW151226, our ability to distinguish in future GW observations whether a signal originated from a BBH, a DNS, or a BH-NS binary will rely on waveform templates with input from NR.",
"This would be particularly important during the last orbits and coalescence of the binary, where strong dynamical gravity is the most relevant.",
"In this regard, NR simulations of binary systems with NS companions have experience a boost in accuracy and sophistication.",
"These days the simulations routinely include realistic equations of state, magnetic fields, and radiation.",
"But the predicting power of simulations not only hinges on the multi-physics included.",
"The degree to which the initial data represent an accurate astrophysical setting is also crucial.",
"Another important aspect connected to the initial data is the capability to explore a vast range of scenarios.",
"And for this to happen, one needs initial data methodologies that are flexible and computationally inexpensive.",
"In BBH simulations, low-cost and efficient methods to construct astrophysically relevant initial data have been available for some time [6], [7], [8], which is not exactly the case for binaries with NSs.",
"A popular method to construct initial data representing a binary system in a quasi-circular orbit is the conformal thin sandwich approach.",
"The method has been used for BBHs by [9], for DNS by [10], and for BH-NS binaries by [11].",
"The key in those studies was the identification of a helical Killing vector field, so the initial data are approximately time-symmetric, ensuring that the compact objects are in a quasi-circular orbit.",
"The conformal thin sandwich approach requires solving a set of five elliptic equations for the conformal factor, lapse function and shift vector [12], [13], [14].",
"Many groups have used the Lorene code from the Meudon group [15], [16] for this purpose, and other groups have developed their own infrastructure [17], [18], [19], [20], [21], [22], [23].",
"This paper introduces a new approach to construct initial data for binary systems with NS components.",
"The method is simpler than the thin sandwich one, and it has a computational cost and flexibility similar to that of the BBH puncture method.",
"In the BBH puncture approach [24], one only solves the Hamiltonian constraint for the conformal factor.",
"The solution to the momentum constraint is given by the Bowen-York extrinsic curvature [25].",
"Each initial data set is then fully specified by the masses, spins and momenta of the BHs, and their separation.",
"All of these parameters are obtained from integrating the post-Newtonian (PN) equations of motion.",
"The integration starts at large separations and ends at the separation where the NR initial data are constructed.",
"This method is known to yield initial data suitable for stitching together NR and PN evolutions.",
"The new initial data proposal in this paper recycles most of the elements of the puncture BBH initial data.",
"The key step is constructing an extrinsic curvature for NSs similar to the Bowen-York for BHs.",
"The paper is organized as follows: In Section , we provide a quick review of York's initial data formulation.",
"Section reintroduces the Bowen-York extrinsic curvature for arbitrary, spherically symmetric momentum sources.",
"Section discusses an approach to specifying the matter source functions for the initial data equations.",
"Section summarizes the steps to construct initial data.",
"Section reviews the stellar model we will use to represent NSs.",
"Section presents tests with an isolated NS.",
"Results of simulations of DNS and BH-NS binaries are presented in Section .",
"Paper ends with conclusions in Section .",
"The numerical simulations in the present work were carried out with our Maya code [26], [27], [28], [29], [30], [31].",
"The code is based on the BSSN formulation of the Einstein equations [32] and the moving puncture gauge condition [33], [34].",
"Maya is very similar to the Einstein code in the Einstein Toolkit [35].",
"That is, it operates under the Cactus infrastructure [36], with Carpet providing mesh refinements [37] and thorns (modules) generated by the package Kranc [38]."
],
[
"Initial Data at a Glance",
"When the Einstein equations of general relativity are viewed as an initial value problem, the initial data are not completely freely specifiable.",
"They must satisfy the Hamiltonian and momentum constraints: $R + K^2 - K_{ij}K^{ij} &=& 16\\,\\pi \\,\\rho _{\\rm H} \\\\\\nabla _j (K^{ij} - \\gamma ^{ij}K) &=& 8\\,\\pi \\,S^i \\,.$ Above, $\\gamma _{ij}$ and $K_{ij}$ are the metric and extrinsic curvature of the space-like hypersurfaces in the foliation.",
"In addition, $R$ is the Ricci scalar, and $\\nabla _i$ denotes covariant differentiation associated with $\\gamma _{ij}$ .",
"The sources $\\rho _{\\rm H}$ and $S^i$ are obtained from the stress-energy tensor $T_{ab}$ as follows: $\\rho _{\\rm H} &=& n^an^b T_{ab}\\\\S^i &=& -\\gamma ^{ib}n^cT_{bc} \\,,$ where $n^a$ is the unit normal to the space-like hypersurfaces.",
"We are using units in which $G=c=1$ .",
"Latin indices from the beginning of the alphabet denote spacetime indices and from the middle of the alphabet spatial indices.",
"For a perfect fluid, the stress-energy tensor reads $T_{ab} &=& (\\rho +p)\\,u_au_b + p\\,g_{ab}\\nonumber \\\\&=&\\rho _0\\,h\\,u_au_b + p\\,g_{ab}\\,,$ where $h= 1+\\epsilon +p/\\rho _0$ is the enthalpy, $p$ is the pressure, $u^a$ is the 4-velocity of the fluid, $\\rho _0$ is the rest-mass density, $\\epsilon $ is the specific internal energy density, and $\\rho = \\rho _0(1+\\epsilon )$ is the total mass-energy density.",
"In terms of these quantities, the sources in the Hamiltonian and momentum constraints read: $\\rho _{\\rm H} &=& (\\rho +p)\\,W^2 - p = \\rho _0\\,h\\,W^2 - p\\\\S^i &=& (\\rho + p) W u^i = \\rho _0\\,h\\, W u^i$ where $W = -n_au^a$ is the Lorentz factor between normal and fluid observers.",
"Since the initial data consist of the set $\\lbrace \\gamma _{ij}, K_{ij}, \\rho _{\\rm H}, S^i \\rbrace $ , the pressing issue is to identify which “pieces” in these data are to be fixed by the constraint Eqs.",
"(REF ) and (), and which data are indeed freely specifiable.",
"Motivated by the work of Lichnerowicz [12], York and collaborators [39] developed an elegant way of achieving this task.",
"The basis of this approach is using conformal transformations and transverse-traceless decompositions to single out the four quantities fixed by the constraint equations.",
"The transformations and decompositions are: $\\gamma _{ij} &=& \\Phi ^4\\bar{\\gamma }_{ij}\\\\K_{ij} &=& A_{ij} + \\frac{1}{3}\\gamma _{ij} K\\\\A_{ij} &=& \\Phi ^{-2}\\bar{A}_{ij}\\\\\\bar{A}_{ij} &=& \\bar{A}^{\\rm TT}_{ij} + \\bar{A}^{\\rm L}_{ij}\\,.$ With them, Eqs.",
"(REF ) and () reduce to $8\\bar{\\Delta }\\Phi - \\Phi \\,\\bar{R} - \\frac{2}{3} \\Phi ^5 K^2&+& \\Phi ^{-7} \\bar{A}_{ij} \\bar{A}^{ij} \\nonumber \\\\&=& - 16\\,\\pi \\Phi ^5\\,\\rho _{\\rm H} \\\\(\\bar{\\Delta }_{\\rm L} {\\cal W})^i - \\frac{2}{3}\\Phi ^6\\, \\bar{\\nabla }^i K &=& 8\\,\\pi \\,\\Phi ^{10}S^i$ respectively, with $\\bar{A}_{\\rm L}^{ij} &=& (\\bar{L} {\\cal W})^{ij} \\\\\\bar{\\nabla }_i \\bar{A}^{ij}_{\\rm TT} &=& 0\\\\(\\bar{L} {\\cal W})^{ij} &\\equiv & \\bar{\\nabla }^i{\\cal W}^j + \\bar{\\nabla }^j {\\cal W}^i- \\frac{2}{3} \\bar{\\gamma }^{ij} \\bar{\\nabla }_k{\\cal W}^k\\\\(\\bar{\\Delta }_L {\\cal W})^i &\\equiv & \\bar{\\nabla }_j(\\bar{L} {\\cal W})^{ij} \\,.$ Given Eqs.",
"(REF ) and (), constructing initial data translates into specifying the quantities $\\lbrace \\hat{\\gamma }_{ij}, K, \\bar{A}^{\\rm TT}_{ij}, \\rho _{\\rm H}, S^i \\rbrace $ , and solving for the conformal factor $\\Phi $ and vector ${\\cal W}^i$ .",
"A common choice, which we adopt, is to assume conformal flatness ($\\bar{\\gamma }_{ij} = \\eta _{ij}$ ), maximal slicing ($K=0$ ), and $\\bar{A}^{\\rm TT}_{ij} = 0$ .",
"Under these assumptions, the constraints (REF ) and () assume the form $&&\\bar{\\Delta }\\Phi + \\frac{1}{8}\\Phi ^{-7} \\bar{A}_{ij} \\bar{A}^{ij} = - 2\\,\\pi \\Phi ^5\\,\\rho _{\\rm H} \\\\&&(\\bar{\\Delta }_L {\\cal W})^i = 8\\,\\pi \\,\\Phi ^{10}S^i$ with $\\bar{A}^{ij} = \\bar{A}_{\\rm L}^{ij} = (\\bar{L} {\\cal W})^{ij}$ .",
"We exploit the freedom to conformally transform $\\rho _{\\rm H}$ and $S^i$ and set $\\bar{\\rho }_{\\rm H} &=& \\rho _{\\rm H}\\,\\Phi ^{8} \\,,\\\\\\bar{S}^i &=& S^i \\,\\Phi ^{10} \\,,$ and thus Eqs.",
"(REF ) and () read $&&\\bar{\\Delta }\\Phi + \\frac{1}{8}\\Phi ^{-7} \\bar{A}_{ij} \\bar{A}^{ij} = - 2\\,\\pi \\Phi ^{-3}\\,\\bar{\\rho }_{\\rm H} \\\\&&(\\bar{\\Delta }_L {\\cal W})^i = 8\\,\\pi \\,\\bar{S}^i\\,.$ The transformations (REF ) and (), and the expressions (REF ) and () suggest setting in the stress-energy tensor $\\bar{\\rho }= \\Phi ^{8}\\rho $ , $\\bar{p} = \\Phi ^{8}\\,p$ and $\\bar{u}^i = \\Phi ^2u^i$ , and therefore $\\bar{\\rho }_{\\rm H} &=& (\\bar{\\rho }+\\bar{p})\\,W^2 - \\bar{p}\\,,\\\\\\bar{S}^i &=& (\\bar{\\rho }+\\bar{p}) W\\,\\bar{u}^i \\,,$ Notice from $u^au_a = -1$ that $W^2 -1 = \\gamma _{ij} u^iu^j = \\bar{\\gamma }_{ij}\\bar{u}^i\\bar{u}^j = \\bar{W}^2-1$ .",
"Then, with the help of Eq.",
"(), $W^2 -1 =\\bar{\\gamma }_{ij}\\bar{u}^i\\bar{u}^j = \\frac{\\bar{S}^2}{W^2(\\bar{\\rho }+ \\bar{p})^2}\\,,$ and thus $W^2 = \\frac{1}{2}\\left(1 + \\sqrt{1 + \\frac{4\\,\\bar{S}^2}{(\\bar{\\rho }+ \\bar{p})^2}}\\right) \\, ,$ where $\\bar{S}^2 = \\bar{\\gamma }_{ij}\\bar{S}^i\\bar{S}^j$ .",
"In summary, constructing initial data reduces to first specifying $\\bar{\\rho }_{\\rm H}$ and $\\bar{S}^i$ , next solving Eq.",
"() for ${\\cal W}^i$ to construct $\\bar{A}^{ij}$ , and finally solving for $\\Phi $ from Eq.",
"(REF )."
],
[
"Extrinsic Curvature",
"We now consider solutions to the momentum constraint equation $(\\bar{\\Delta }_L {\\cal W})^i = 8\\,\\pi \\,\\bar{S}^i$ .",
"We will first recall the solution that represents BHs and next reintroduce the one suitable to model NSs.",
"For BHs ($\\bar{S}^i = 0$ ), Bowen and York [25] found that point-source solutions to $(\\bar{\\Delta }_L {\\cal W})^i =0$ are given by ${\\cal W}^i &=& -\\frac{1}{4\\,r}\\left[7\\,P^i + l^i(P\\cdot l)\\right]\\\\{\\cal W}^i &=& \\frac{1}{r^2}\\epsilon ^{ijk}l_jJ_k\\,,$ with $l^i = x^i/r$ a unit radial vector and $P\\cdot l = P^i l_i$ .",
"In these solutions, the constant vectors $P^i$ and $J_i$ are respectively interpreted as the linear and angular momentum of the BH.",
"From $\\bar{A}^{ij} = (\\bar{L} {\\cal W})^{ij}$ , the extrinsic curvature associated with these solutions are: $\\bar{A}^{ij} &=& \\frac{3}{2\\,r^2}\\left[ P^i l^j + P^jl^i - ( \\eta ^{ij}-l^il^j) (P\\cdot l)\\right]\\\\\\bar{A}^{ij} &=& \\frac{6}{r^3}l^{(i}\\epsilon ^{j)kl}J_kl_l $ Next is to consider solutions to $(\\bar{\\Delta }_L {\\cal W})^i = 8\\,\\pi \\,\\bar{S}^i$ that can be used to build the extrinsic curvature of a NS.",
"Following Bowen [40], we assume sources of the form $\\bar{S}^i &=& P^i\\,\\sigma (r)\\\\\\bar{S}_i &=& \\epsilon _{ijk}\\,J^jx^k\\,\\kappa (r)\\,.$ At this point, $P^i$ and $J^i$ arbitrary constant vectors, and $\\sigma $ and $\\kappa $ radial functions with compact support on $r \\le r_0$ .",
"The specific form of these functions will be determined in the next section using the following conditions.",
"From the definition of ADM linear momentum [41], one has that $P^i _{\\rm ADM}&=& \\frac{1}{8\\pi }\\int _{\\partial \\Sigma _\\infty }A^{ij} \\,dS_j \\nonumber \\\\&=&\\frac{1}{8\\pi } \\int _\\Sigma \\bar{\\nabla }_j \\bar{A}^{ij} \\sqrt{\\eta } \\,d^3x \\nonumber \\\\&=& \\int _\\Sigma \\bar{S}^{i} \\sqrt{\\eta }\\, d^3x \\nonumber \\\\&=&P^i \\int _\\Sigma \\sigma \\sqrt{\\eta }\\, d^3x \\,.$ Thus, for $P^i_{\\rm ADM} = P^i$ to hold, $\\sigma $ must satisfy the following normalization condition: $\\int _\\Sigma \\sigma \\sqrt{\\eta }\\, d^3x= 4\\,\\pi \\int _{0}^{r_0} \\sigma \\,r^2\\, dr = 1\\,.$ Similarly, from the definition of ADM angular momentum [12], we have that $J^{\\rm ADM}_i &=& \\frac{1}{8\\pi }\\epsilon _{ijk}\\int _{\\partial \\Sigma _\\infty }x^jA^{km} \\,dS_m \\nonumber \\\\&=&\\frac{1}{8\\pi }\\epsilon _{ijk} \\int _\\Sigma x^j \\bar{\\nabla }_m \\bar{A}^{km} \\sqrt{\\eta } \\,d^3x \\nonumber \\\\&=&\\epsilon _{ijk} \\int _\\Sigma x^j \\bar{S}^{k} \\sqrt{\\eta } \\,d^3x \\nonumber \\\\&=&\\epsilon _{ijk}\\epsilon ^{klm} \\int _\\Sigma x^j J_l x_m\\,\\kappa \\sqrt{\\eta }\\,d^3x \\nonumber \\\\&=& \\int _\\Sigma r^2 \\left(J_i - l_i l^j J_j \\right)\\kappa \\sqrt{\\eta } \\,d^3x \\,.$ Adopting Cartesian coordinates and aligning the angular momentum with the $z$ -axis, one gets that $J^{\\rm ADM}_i &=& J_i \\int _\\Sigma r^2\\,\\sin ^2\\theta \\,\\kappa \\sqrt{\\eta } \\,d^3x \\,.$ Thus, in order to have $J^{\\rm ADM}_i = J_i$ , the following normalization condition must hold $2\\,\\pi \\int _0^{r_0}\\int _0^{\\pi } \\sin ^3\\theta \\,r^4\\kappa \\,d\\theta \\,dr = \\frac{8\\,\\pi }{3} \\int _{0}^{r_0} \\kappa \\,r^4\\, dr = 1\\,.$ Given the normalization condition Eq.",
"(REF ) for $\\sigma $ , the solution to $(\\bar{\\Delta }_L {\\cal W})^i = 8\\,\\pi \\, P^i\\,\\sigma $ reads [40] ${\\cal W}^i = -2\\,P^iF + \\frac{1}{2}P^iH +\\frac{1}{2} l^i (P\\cdot l)\\,r H^{\\prime } \\,.$ The functions $F$ and $H$ are given respectively by $F &=& \\frac{1}{r}\\int ^r_0 \\,4\\,\\pi \\,\\sigma \\,r^{\\prime 2}\\,dr^{\\prime }+ \\int ^{r_0}_r 4\\,\\pi \\,\\sigma \\,r^{\\prime }\\,dr^{\\prime }\\,,\\\\H &=& \\frac{1}{r^3}\\int ^r_0 F\\,r^{\\prime 2}\\,dr^{\\prime }\\hspace{36.135pt}\\,.$ With the help of $\\bar{\\nabla }^i r= l^i$ and $\\bar{\\nabla }^i l^j = (\\eta ^{ij}-l^il^j)/r$ , substitution of Eq.",
"(REF ) into $\\bar{A}^{ij} = (\\bar{L} {\\cal W})^{ij}$ yields $\\bar{A}^{ij}&=& (-2\\, F^{\\prime } + H^{\\prime })(P^i l^j + P^jl^i)\\nonumber \\\\&+& (r H^{\\prime \\prime }-H^{\\prime }) (P \\cdot l) l^il^j \\nonumber \\\\&+&\\frac{1}{3}(4\\,F^{\\prime } - rH^{\\prime \\prime } - H^{\\prime }) (P\\cdot l)\\eta ^{ij}\\,.$ With the help of Q = 0r 4r'2 dr' J = rr0 4r' dr' C = 0r 23 r'4 dr' = 0r 12 Q' r'2 + 13 J' r'3 dr' , and F = Q/r + J H = Q/2r + J/3 - C/r3 F' = -Q/r2 H' = -Q/2r2 + 3C/r4 H” = Q/r3 - 12C/r5 , the expression (REF ) for the extrinsic curvature can be rewritten as $\\bar{A}^{ij}&=& \\frac{3Q}{2\\,r^2}\\left[ P^i l^j + P^jl^i - ( \\eta ^{ij}-l^il^j) (P\\cdot l)\\right] \\nonumber \\\\&+& \\frac{3 C}{r^4}\\left[ P^i l^j + P^jl^i + ( \\eta ^{ij}-5\\,l^il^j) (P\\cdot l)\\right] \\,.$ For $r > r_0$ (exterior solution), $Q = 1$ , thus the first term in Eq.",
"(REF ) becomes the Bowen-York curvature for a point mass (REF ).",
"Furthermore, Eq.",
"(REF ) has the correct point mass limit since $Q=1$ and $C=0$ for $r_0 = 0$ .",
"For a spherically symmetric source function $\\kappa $ with angular momentum $J^i$ , the solution to $(\\bar{\\Delta }_L {\\cal W})_i = 8\\,\\pi \\, \\epsilon _{ijk} J^jx^k\\,\\kappa $ is given by [42] ${\\cal W}_i = \\epsilon _{ijk} \\,x^jJ^k\\,G $ where $G &=& \\frac{1}{r^3}\\int ^r_0 \\frac{8\\,\\pi }{3}\\,r^{\\prime 4}\\,\\kappa \\,dr^{\\prime }+ \\int ^{r_0}_r\\frac{8\\,\\pi }{3}\\kappa \\,r^{\\prime }\\,dr^{\\prime }\\,.$ Notice that $G = r^{-3}$ for $r \\ge r_0$ .",
"Substitution of Eq.",
"(REF ) into $\\bar{A}^{ij} = (\\bar{L} {\\cal W})^{ij}$ yields $\\bar{A}^{ij} &=& \\frac{6}{r^3}l^{(i}\\epsilon ^{j)kl}J_kl_l N$ where $N = \\int ^r_0 \\frac{8\\,\\pi }{3}\\,r^{\\prime 4}\\,\\kappa \\,dr^{\\prime }$ Exterior to the source, $N=1$ , and the extrinsic curvature reduces to the point-like solution ().",
"In summary, Eqs.",
"(REF ) and () are the extrinsic curvatures for a point-like source with linear and angular momentum, respectively.",
"In addition, Eqs.",
"() and () are the extrinsic curvatures for a spherically symmetric source with linear and angular momentum, respectively.",
"To construct initial data for compact object binaries, the extrinsic curvature for the binary system will be simply given by a superposition of these solutions, point-like for the BH and spherically symmetric source for the NS.",
"The only input needed are the locations of the compact objects, their linear and angular momenta, and the source functions $\\sigma $ and $\\kappa $ .",
"As with BBHs, the linear and angular momenta of the sources, and their binary separation will be provided by the outcome of integrating the PN equations of motion.",
"It is very important to keep in mind that, because of the spherical symmetry assumption in the source functions $\\sigma $ and $\\kappa $ , the extrinsic curvature will not be able to account for tidal deformations of the star.",
"We are currently considering a generalization that relaxes the spherical symmetry assumption."
],
[
"Source Functions",
"The next step is to specify the source functions $\\sigma $ and $\\kappa $ , as well as the source $\\bar{\\rho }_{\\rm H} = (\\bar{\\rho }+\\bar{p})\\,W^2 - \\bar{p}$ in the Hamiltonian constraint.",
"The starting point is the density $\\bar{\\rho }$ and pressure $\\bar{p}$ from the stellar model of our choice, Recall from Eq.",
"() that $\\bar{S}^i = (\\bar{\\rho }+\\bar{p}) W\\,\\bar{u}^i \\,.$ Thus, for the case of linear momentum, we have that $\\bar{S}^i = (\\bar{\\rho }+ \\bar{p})W\\, \\bar{u}^i= P^i\\,\\sigma \\,.$ We then set $\\sigma = (\\bar{\\rho }+\\bar{p})/\\mathcal {M}\\,, $ with $\\mathcal {M}$ a constant determined by the normalization condition Eq.",
"(REF ) for $\\sigma $ .",
"That is, $1 = 4\\,\\pi \\int _{0}^{r_0} \\sigma \\,r^2\\, dr = \\frac{4\\,\\pi }{\\mathcal {M}} \\int _{0}^{r_0} (\\bar{\\rho }+\\bar{p})\\,r^2\\, dr \\,,$ and thus ${\\mathcal {M}} = 4\\,\\pi \\int _{0}^{r_0} (\\bar{\\rho }+\\bar{p})\\,r^2\\, dr \\,,$ Notice that Eq.",
"(REF ) restricts our choice for $\\bar{\\rho }$ and $\\bar{p}$ to be spherically symmetry solutions since by assumption $\\sigma (r)$ .",
"With this choice for $\\sigma $ , the linear momentum satisfies $P^i = W\\,\\mathcal {M}\\,\\bar{u}^i$ .",
"Since by construction $P^i$ and $\\mathcal {M}$ are constants, $W\\,\\bar{u}^i$ must also be constant within the source distribution.",
"Finally, notice also from Eqs.",
"(REF ), (REF ) and (REF ) that the Lorentz factor is then given by $W^2 = \\frac{1}{2}\\left(1 + \\sqrt{1 + \\frac{4\\,P^2}{\\mathcal {M}^2}}\\right) \\, .$ where $P^2 = \\eta _{ij} P^iP^j$ .",
"For a source with angular momentum, $\\bar{S}_i = \\epsilon _{ijk}\\,J^jx^k\\,\\kappa = (\\bar{\\rho }+ \\bar{p})W\\, \\bar{u}_i\\,.$ As with the previous case, we set $\\kappa = (\\bar{\\rho }+\\bar{p})/\\mathcal {N}\\,, \\,.$ From the normalization condition Eq.",
"(REF ), one has that $1= \\frac{8\\,\\pi }{3} \\int _{0}^{r_0} \\kappa \\,r^4\\, dr = \\frac{8\\,\\pi }{3\\,\\mathcal {N}} \\int _{0}^{r_0} (\\bar{\\rho }+\\bar{p}) \\,r^4\\, dr\\,,$ and thus the constant $\\mathcal {N}$ is given by ${\\mathcal {N}}= \\frac{8\\,\\pi }{3} \\int _{0}^{r_0} (\\bar{\\rho }+\\bar{p}) \\,r^4\\, dr\\,,$ The Lorentz factor in this case reads $W^2 = \\frac{1}{2}\\left(1 + \\sqrt{1 + \\frac{4\\,J^2r^2\\sin ^2\\theta }{\\mathcal {N}^2}}\\right) \\, .$ where $J$ is the magnitude of the angular momentum and $\\theta $ the angle between $J^i$ and $l^i$ .",
"It is important to notice that in this case the Lorentz boost factor is not constant within the star."
],
[
"Initial Data Procedure",
"The centerpiece of our method is solving Eq.",
"(REF ), or equivalently $\\bar{\\Delta }\\Phi + \\frac{1}{8}\\Phi ^{-7} \\bar{A}_{ij} \\bar{A}^{ij} = - 2\\,\\pi \\Phi ^{-3}[(\\bar{\\rho }+\\bar{p})\\,W^2 - \\bar{p}] \\,.$ In this equation, the boost factor $W$ for the stellar model is given by Eq.",
"(REF ) for linear momentum or Eq.",
"(REF ) for angular momentum.",
"In the same equation, $\\bar{A}^{ij}$ is given by the Bowen-York extrinsic curvatures.",
"For point masses, Eq.",
"(REF ) provides the extrinsic curvature with linear momentum and Eq.",
"() the corresponding extrinsic curvature with angular momentum.",
"Similarly, the extrinsic curvature associated with the stellar model is given by Eq.",
"() for linear momentum and Eq.",
"() for angular momentum.",
"In general terms, the sequence of steps to construct initial data for binaries with BHs and NSs components under the proposed method is as follows: Choose masses $M_{1,2}$ of the compact objects and their initial separation $d_0$ deep in the PN regime, with $M = M_1+M_2$ the total mass of the binary and $q = M_1/M_2$ its mass ratio.",
"Integrate the PN equations of motion at the highest order available and stop at a separation $d$ where the NR evolution will begin.",
"Read off the linear momentum $\\vec{P}_{1,2}$ and spin $\\vec{S}_{1,2}$ for each of the binary components.",
"Identify the mass $M_{1(2)}$ with the ADM mass $M_{1(2)}^\\text{ADM}$ of a star in isolation if a NS and with the irreducible mass $M_{1(2)}^\\text{irr}$ if a BH, where $M_\\text{ADM} = -\\frac{1}{2\\pi } \\int _{\\partial \\Sigma _\\infty } \\bar{\\nabla }^i \\Phi \\, dS_i\\,,$ and $M_\\text{irr} \\equiv \\sqrt{{\\cal A}/16\\,\\pi }$ for a BH with apparent horizon area ${\\cal A}$ [12].",
"If object 1(2) is a BH, set its puncture bare mass $m_{1(2)} = M_{1(2)}$ .",
"If object 1(2) is a NS, construct a spherically symmetric stellar model with ADM mass $M_{1(2)}^\\text{ADM}$ .",
"Compute also its rest mass $M^0_{1(2)}$ from $M_0 = \\int _{\\Sigma } \\rho _0 \\,W\\,\\sqrt{\\gamma }\\, d^3x\\,,$ and save the ratio $\\xi _{1(2)} \\equiv M_{1(2)}^\\text{ADM}/M^0_{1(2)}$ .",
"If the compact object is a NS, calculate the functions $\\sigma $ and $\\kappa $ from Eqs.",
"(REF ) and (REF ), respectively.",
"Use the $\\vec{P}$ and $\\vec{S}$ vectors to construct the extrinsic curvature using Eqs.",
"(REF ) and () if a BH, and Eqs.",
"() and () if a NS.",
"The functions $\\sigma $ and $\\kappa $ will also be needed if a NS.",
"The total extrinsic curvature is $\\bar{A}^{ij} = \\bar{A}^{ij}_1 +\\bar{A}^{ij}_2$ .",
"Construct the term $[(\\bar{\\rho }+\\bar{p})\\,W^2 - \\bar{p}]$ in the r.h.s.",
"of Eq.",
"(REF ) for each NS.",
"Superpose the terms if the binary involves a DNS.",
"Solve the Hamiltonian constraint in the form given by Eq.",
"(REF ).",
"If a BH, compute the new irreducible $\\hat{M}^\\text{irr}_{1(2)}$ , and if a NS calculate the new rest mass $\\hat{M}^0_{1(2)}$ .",
"Using $\\xi _{1(2)}$ from Step 3, estimate the new ADM mass $\\hat{M}_{1(2)}^\\text{ADM} = \\xi _{1(2)} \\hat{M}^0_{1(2)}$ .",
"Notice that we are assuming that the ratio $\\xi _{1(2)}$ does not change significantly from iteration to iteration.",
"Next, identify the new mass $\\hat{M}_{1(2)}$ with $\\hat{M}_{1(2)}^\\text{ADM}$ if a NS and $\\hat{M}_{1(2)}$ with $\\hat{M}_{1(2)}^\\text{irr}$ if a BH.",
"Calculate the new total mass $\\hat{M} = \\hat{M}_1+\\hat{M}_2$ and mass ratio $\\hat{q} = \\hat{M}_1/\\hat{M}_2$ .",
"If the new values differ from the values in Step 1 by more than a specified tolerance, adjust the bare masses of the BH or central densities of the NS according to a 2D secant algorithm [43], and return to step 3.",
"For the present work, we solve Eq.",
"(REF ) using a modified version of the 2Punctures spectral code.",
"2Punctures was originally developed by Ansorg [7] to construct BBH initial data; that is, to solve Eq.",
"(REF ) with vanishing r.h.s.",
"and $A_{ij}$ given by Eqs.",
"(REF ) and/or ().",
"Once the conformal factor $\\Phi $ is found from solving Eq.",
"(REF ), the spatial metric and extrinsic curvature are obtained from $\\gamma _{ij} = \\Phi ^4\\eta _{ij}$ and $K_{ij} = \\Phi ^{-2}\\bar{A}_{ij}$ , respectively.",
"The last step is constructing the hydrodynamical fields $\\rho $ , $p$ , $W$ and $u^i$ .",
"Given $\\Phi $ , $\\bar{\\rho }_H$ and $\\bar{S}^i$ , we have that $\\rho _H$ and $S^i$ are considered as known since $\\rho _{\\rm H} = \\Phi ^{-8}\\bar{\\rho }_H$ and $S^i = \\Phi ^{-10}\\bar{S}^i$ .",
"On the other hand, $\\rho _{\\rm H} &=& (\\rho +p)\\,W^2 - p \\\\S^i &=& (\\rho + p) W u^i \\,,$ and from the second equation, $\\gamma _{ij}S^iS^j &=& (\\rho + p)^2 W^2\\gamma _{ij} u^i u^j\\nonumber \\\\&=& (\\rho + p)^2 W^2(W^2 -1)\\,,$ where in the last equality we used that $ \\gamma _{ij} u^iu^j = W^2 -1$ as implied by $u^au_a = -1$ .",
"If we view that $p$ is given by an equation of state, Eqs.",
"(REF ) and (REF ) can be used to solve for $\\rho $ and $W$ .",
"And the last step is to construct $u^i$ from Eq.",
"()."
],
[
"Tolman-Oppenheimer-Volkoff model in isotropic coordinates",
"For the present work, we use a Tolman-Oppenheimer-Volkoff (TOV) stellar model to represent a NS, with a polytropic equation of state $p= K\\,\\rho _0^\\Gamma $ setting $\\Gamma = 2$ and $K = 123.641\\,M_\\odot ^2$ .",
"Since we assume conformal flatness, it is natural to recast the TOV model in isotropic coordinates.",
"TOV models are commonly constructed in coordinates in which the metric takes the form $ds^2 = -\\alpha ^2(\\hat{r})\\,dt^2 + \\left[1-\\frac{2\\,m(\\hat{r})}{\\hat{r}}\\right]^{-1}d\\hat{r}^2 + \\hat{r}^2\\,d\\Omega \\,.$ On the other hand, the form of the metric (isotropic) compatible with our conformal flatness assumption is $ds^2 = -\\alpha ^2(r)\\,dt^2 + \\Phi (r)^4(dr^2 + r^2\\,d\\Omega )\\,.$ In these coordinates, the equations that one needs to solve are the so called “conformal thin sandwich” equations [12].",
"i i = - 18-7 Aij Aij - 2 5 H jj i + 13 ijj = 2 Aijj( -6) + 16 4 Si ii ( ) = [78 -8 Aij Aij + 2 4(H+2 S)] where $\\beta ^i$ is the shift vector, $\\rho _H$ is given by Eq.",
"(REF ), $S^i$ by Eq.",
"() and $S = S^i\\,_i$ with $S_{ij} = \\gamma _i^{a}\\gamma _j^{b}T_{ab}$ .",
"For the metric (REF ), the conformal thin sandwich equations reduce to $\\frac{1}{r^2}(r^2\\,\\Phi ^{\\prime })^{\\prime } &=& - 2\\,\\pi \\Phi ^5\\,\\rho \\\\\\frac{1}{r^2}(r^2\\,\\Theta ^{\\prime })^{\\prime } &=& 2\\,\\pi \\,\\Theta \\Phi ^4(\\rho +6\\,p)$ where primes denote differentiation with respect to $r$ and $\\Theta \\equiv \\alpha \\,\\Phi $ .",
"Notice also that in this case $\\beta ^i = 0$ , $A^{ij} = 0$ , $S^i = 0$ and $\\rho _H = \\rho $ .",
"Finally, from $\\nabla _bT^{ab} = 0$ , one obtains $p^{\\prime } = -(\\rho +p)\\frac{\\alpha ^{\\prime }}{\\alpha }= -(\\rho +p)\\left( \\frac{\\Theta ^{\\prime }}{\\Theta } - \\frac{\\Phi ^{\\prime }}{\\Phi } \\right)$ Therefore, together with an equation of state, constructing TOV stellar models in isotropic coordinates involves solving Eqs.",
"(REF ), () and (REF ).",
"Integration constants are chosen such that in the exterior of the star $\\Phi &=& 1+ \\frac{M}{2\\,r} \\\\\\Theta &=& 1- \\frac{M}{2\\,r}\\,,$ with $M = 2\\pi \\int _0^{r_0} {r}^2 \\Phi ^{5}\\rho \\; dr$ the total mass of the star.",
"Notice that $M = M_{\\rm ADM}$ the ADM mass since Eq.",
"(REF ) can be rewritten as Eq.",
"(REF ).",
"If we denote by $\\Phi _{\\rm tov}$ , $\\rho _{\\rm tov}$ and $p_{\\rm tov}$ the TOV solutions in isotropic coordinates, we then set $\\bar{\\rho }&=& \\Phi _{\\rm tov}^8 \\rho _{\\rm tov}\\\\\\bar{p} &=& \\Phi _{\\rm tov}^8 p_{\\rm tov}\\,,$ and rewrite the Hamiltonian constraint Eq.",
"(REF ) as $&&\\bar{\\Delta }\\Phi + \\frac{1}{8}\\Phi ^{-7} \\bar{A}_{ij} \\bar{A}^{ij} = \\nonumber \\\\&&- 2\\,\\pi \\Phi ^{-3}\\,\\Phi ^8_{\\rm tov}[(\\rho _{\\rm tov}+p_{\\rm tov})\\,W^2 - p_{\\rm tov}] $ Notice that for an isolated TOV stellar model without linear or angular momentum ($\\bar{A}_{ij} = 0$ , $W = 1$ and $\\Phi = \\Phi _{\\rm tov}$ ), Eq.",
"(REF ) reduces to Eq.",
"(REF ), namely $\\bar{\\Delta }\\Phi _{\\rm tov} = - 2\\,\\pi \\Phi ^5_{\\rm tov}\\,\\rho _{\\rm tov}\\,.$"
],
[
"Single Neutron Star with Linear Momentum",
"As a first test of the proposed method, we will consider an isolated NS with linear momentum.",
"We use a TOV stellar model with mass $M_* = 1.543 \\,M_\\odot $ , radius $R_* = [13.4]{km}$ , and central density $\\rho _c = 6.235\\times 10^{14}{gr\\,cm^{-3}}$ .",
"We endow the star with linear momentum within the range $0 \\le P/M_* \\le 0.4$ .",
"Figure REF depicts with dots the ADM mass $M_{\\rm ADM}$ as a function of $P/M_*$ , and with triangles the rest mass $M_0$ .",
"In the same figure, squares denote the quantity $M_*\\,W$ , where the Lorentz boost factor $W$ is calculated from Eq.",
"(REF ).",
"Notice that for small values of the linear momentum $M_{\\rm ADM} \\approx M_*\\,W$ .",
"Also, it is not difficult to show from Eq.",
"(REF ) and the Hamiltonian constraint (REF ) that $M_{\\rm ADM} = M_* + O(P^2)$ , consistent with the growth observed in Fig.",
"REF .",
"Figure: ADM mass M ADM M_{\\rm ADM} (dots), rest mass M 0 M_0 (triangles) and M * WM_*W (squares) as a function of P/M * P/M_* for a single NS.",
"Solid line represents a fit to M ADM =M * +cP 2 M_{\\rm ADM} = M_* + cP^2.To further understand the changes that the momentum introduces to the TOV solution, we plot in Fig.",
"REF the relative differences with respect to the TOV solution of the total mass-energy density $\\rho $ (top panel) and conformal factor $\\Phi $ (bottom panel) along the $x$ -axis, after solving the Hamiltonian constraint for a star with a linear momentum $P/M_* = 0.1$ .",
"The relative differences are computed as follows: $\\delta \\rho &=& \\frac{\\rho -\\rho _{\\rm tov}}{\\rho _{\\rm tov}} \\\\\\delta \\Phi &=& \\frac{\\Phi -\\Phi _{\\rm tov}}{\\Phi _{\\rm tov}}$ The differences in the mass-energy density are entirely due to the conformal factor.",
"From $\\rho = \\Phi ^{-8}\\,\\bar{\\rho }$ and $\\bar{\\rho }= \\Phi _{\\rm tov}^8\\,\\rho _{\\rm tov}$ , one has that $\\rho = ( \\Phi /\\Phi _{\\rm tov})^{-8}\\rho _{\\rm tov}$ , and thus from (REF ) $\\delta \\rho = (\\Phi ^{-8}-\\Phi _{\\rm tov}^{-8})/\\Phi _{\\rm tov}^{-8}\\,.$ Figure: Relative differences along the xx-axis between the TOV solution and the corresponding solution for a TOV star with momentum P/M * =0.1P/M_* = 0.1.",
"Top panel shows the relative differences δρ\\delta \\rho in total mass-energy and bottom panel those in the conformal factor δΦ\\delta \\Phi .Figure: Density ρ\\rho profiles along the xx-axis for a TOV star with P/M * =0.1P/ M_* = 0.1 at various times throughout the evolution.",
"The profiles have been normalized to the initial central density ρ c \\rho _c and shifted to be centered at x=0x=0.Figure: Evolution of the central density of the star in Fig.",
"normalized to the initial central value ρ c \\rho _c.In general terms, the evolutions of the initial data for a single neutron star with linear momentum were satisfactory.",
"The evolutions were carried out with the same gauge conditions used for puncture BH evolutions [33], [34].",
"We noticed, however, few percent variations in the size and internal structure in the star during the course of the evolution.",
"The changes in the size of the star are shown in Fig.",
"REF , where we superimpose density profiles from different times for the case of a star with $P/M_* = 0.1$ .",
"Notice that the deformations are more prominent in the leading edge of the star (i.e.",
"positive axis).",
"Oscillations reveal themselves also in the central density of the star.",
"Fig.",
"REF shows the evolution of the central density in the star for the same case."
],
[
"Compact Object Binary Evolutions",
"Next, we test the performance of our prescription to construct initial data with evolutions of DNS and BH-NS binary systems."
],
[
"Non-spinning Double Neutron Star Binary",
"We consider first an equal-mass DNS system.",
"The NSs have a mass of $1.568\\, M_\\odot $ , coordinate radius $13.1$ km, and they are initially separated by [54.6]km.",
"The configuration is similar to the model 1.62-45 in [44].",
"In their case, the stars have a mass of $1.62 \\,M_\\odot $ , and their initial coordinate separation is [45]km.",
"The results of this simulation were obtained using 7 levels of mesh refinement.",
"The finest mesh had resolution of $0.150\\,M_\\odot = 0.221$ km and extent of $26.6$ km.",
"The wave-zone grid resolution was $9.58\\,M_\\odot = 14.1$ km.",
"Figure REF shows the coordinate trajectory of one of the NS stars and Fig.",
"REF the corresponding coordinate separation of the binary.",
"The data in both figures end at the “point-of-contact\" (PoC), which occurs at approximately $[18]{ms}$ after the start of the simulation or at a separation of approximately 25 km.",
"A hypermassive neutron star (HMNS) forms $[4]{ms}$ after the PoC, which collapses to a BH in approximately $[8]{ms}$ .",
"The collapse of the HMNS in [44] is $[10]{ms}$ , a difference that we attribute primarily to resolution effects.",
"Figure: Amplitude (left panel) and phase (right panel) differences of the Weyl scalar Ψ 4 \\Psi _4 for three different resolutions of non-spinning DNS system simulations.",
"The resolutions in the finest grid are: [0.45]km[0.45]{km} (Low), [0.315]km[0.315]{km} (Medium), and [0.225]km[0.225]{km} (High).",
"The (Medium–High) resolution is also presented in black re-scaled with a factor of 2.49, corresponding to 2nd order convergence.Figure REF shows the evolution of the central density normalized to its initial value.",
"For comparison, see Fig.",
"12 in [44].",
"The oscillations in Fig.",
"REF for times earlier than $[18]{ms}$ are similar, and likely due to the same reasons, to those seen in the case of a single NS with linear momentum (see Fig.",
"REF ).",
"Since the amplitude of the oscillations decrease by increasing the initial separation of the binary, we suspect that the origin of the oscillations is because the TOV star has not been able to adjust to the linear momentum added and to the gravitational field by its companion.",
"Similar oscillations have been observed in other initial data methods, for instance, in the work by [19].",
"We are currently investigating whether the prescription introduced by [19] to attenuate the oscillations will work in our case.",
"Figure REF shows the 2,2 mode of the Weyl scalar $\\Psi _4$ , extracted at $462\\,M_\\odot $ from the binary, as a function of retarded time.",
"At the beginning of the waveform, there is a small burst.",
"This is the characteristic unphysical burst of radiation observed in NR simulations that start with conformally flat initial data.",
"After the burst, $\\Psi _4$ shows the expected chirp-like structure, the ringing of the HMNS during the time interval $[18]{ms} \\le t \\le [24]{ms}$ , and the quasi-normal-mode (QNM) ring-down of the final BH.",
"Figure: Rest-mass density snapshots from the non-spinning DNS binary evolution.",
"Panels (a), (b) and (c) show the xyxy-plane and panel (d) the xzxz-plane All densities are in units of gcm -3 {g \\, cm^{-3}} and distances in units of M=3.14M ⊙ M = 3.14\\,M_\\odot .Next, we analyze the convergence properties of the Weyl scalar $\\Psi _4$ , focusing only in the time segment before merger.",
"We were unable to get “clean\" convergence estimates during the HMNS phase since numerical dissipation due to resolution effects leads to significant differences in the longevity of the resulting HMNS [45].",
"Figure REF shows differences of amplitude and phase from three simulations with resolutions in the finest grid of $[0.45]{km}$ (Low), $[0.315]{km}$ (Medium), and $[0.225]{km}$ (High).",
"The red line shows the difference (Medium–Low) and the blue line (High–Medium).",
"Assuming 2nd order convergence, the three resolutions imply that (Medium–Low) $\\approx 2.49\\times $ (High–Medium).",
"The black line in Fig.",
"REF depicts $2.49\\times $ (High–Medium) and thus consistency with 2nd order convergence.",
"For reference, the sector of the Maya code handling the geometrical fields is by design 6nd order convergent.",
"The hydrodynamical sector however is at best 3rd order, but near shocks and local extrema can deteriorate to 1st order, as seen in codes similar to ours where convergence order could be as low as 1.8 [46].",
"Finally, Fig.",
"REF depicts snapshots of the rest-mass density during the evolution.",
"Panels (a), (b) and (c) show the $xy$ -plane and panel (d) the $xz$ -plane.",
"All densities are in units of ${g \\, cm^{-3}}$ and distances in units of $M = 3.14\\,M_\\odot $"
],
[
"Spinning Double Neutron Star Binary",
"The second example of evolution of initial data with the proposed scheme is again an equal-mass binary but now with spinning NSs.",
"Both stars have identical spins, anti-aligned to the orbital axis.",
"The NSs have a mass of $1.57\\, M_\\odot $ , coordinate radius $13.1$ km, and dimensionless spin parameter $\\chi _s = -0.05$ .",
"At the beginning of the simulation, the NSs are separated by [61.2]km.",
"With this choice of parameters, the binary system is similar to the case $\\Gamma ^{--}_{050}$ in [47].",
"The grid structure is as follows: the finest mesh has resolution $0.299 M_\\odot = [.442]{km}$ and extent $[26.6]{km}$ .",
"The radiation zone has resolution $19.2 M_\\odot = [28.3]{km}$ .",
"Figure REF shows the coordinate trajectory of one of the NS stars and Fig.",
"REF the corresponding coordinate separation of the binary.",
"Notice from Fig.",
"REF that the system performs 6 full orbits before merger.",
"Also noticeable is the slight kink or sudden drop in separation observed in Fig.",
"REF at the beginning of the evolution.",
"After the drop, the inspiral proceeds very smoothly, with minimal spurious eccentricity.",
"As with the previous case, the data in both figures are depicted up to the PoC, which occurs at approximately $[25]{ms}$ after the start of the simulation or at a separation of $[26]{km}$ .",
"Figure REF shows the evolution of the central density normalized to its initial value.",
"Here again, we observe oscillations in the central density before merger.",
"The HMNS forms at $[26.2]{ms}$ and lasts for $[1.3]{ms}$ before it collapses.",
"From the waveform in Fig.",
"REF , we notice that the HMNS undergoes two bursts.",
"Also, the collapse to BH is faster than in the non-spinning case.",
"This is expected since the spins of NS are anti-aligned with the orbital angular momentum and thus the HMNS is rotating slower than the HMNS in the non-spinning DNS.",
"The energy radiated is estimated to be approximately 0.7% of total mass-energy, and the angular momentum radiated is 16% of total angular momentum.",
"These values are slightly different form those reported by [47]—which are 1.2% and 18% respectively.",
"Finally, Fig.",
"REF depicts snapshots of the rest-mass density during the evolution.",
"Panels (a), (b) and (c) show the $xy$ -plane and panel (d) the $xz$ -plane.",
"All densities are in units of ${g \\, cm^{-3}}$ and distances in units of $M = 3.14\\,M_\\odot $ .",
"Figure: Rest-mass density snapshots from the spinning DNS binary evolution.",
"Panels (a), (b) and (c) show the xyxy-plane and panel (d) the xzxz-plane All densities are in units of gcm -3 {g \\, cm^{-3}} and distances in units of M=3.14M ⊙ M = 3.14\\,M_\\odot ."
],
[
"BH-NS Binary",
"The final example of evolution of initial data is for the case of a BH-NS binary system.",
"The NS has a mass of $1.54 M_\\odot $ and a coordinate radius of $[13.0]{km}$ , and the BH has a mass of $7.7\\,M_\\odot $ (i.e.",
"5:1 mass ratio binary).",
"Both compact objects are non-spinning.",
"The coordinate separation between the BH and the NS is $[117]{km}$ .",
"With these parameters, the BH-NS binary is similar to the M50.145b system in [48].",
"As with the DNS system, we cover the star with a single mesh whose side length is the diameter of the star.",
"The grid structure has 8 levels of refinement, with finest resolution of $0.303 M_\\odot = [0.448]{km}$ .",
"The finest mesh around the BH has extent $9.10 M_\\odot = [13.4]{km}$ .",
"The radiation zone has resolution of $38.8 M_\\odot = [57.3]{km}$ .",
"Figure REF shows the trajectories of the BH (solid line) and NS (dashed line).",
"The orbital separation of the binary is shown in Fig.",
"REF .",
"There is clear indication of spurious eccentricity.",
"We attribute this eccentricity to the relatively small initial separation.",
"Figure REF shows the maximum rest mass density during the course of the evolution.",
"The central density fluctuates as in the previous two cases, with the oscillations decaying at later times.",
"The point at which the central density drops signals the time when the star is disrupted and swallowed by the BH.",
"This is also clear in the 2,2 mode of the Weyl scalar $\\Psi _4$ (see Fig.",
"REF ).",
"At approximately $[36]{ms}$ , $\\Psi _4$ shows the characteristic QNM ringing of a BH.",
"Figure: Rest-mass density snapshots from the bhns binary evolution.",
"Panels (a), (b) and (c) show the xyxy-plane and panel (d) the xzxz-plane All densities are in units of gcm -3 {g \\, cm^{-3}} and distances in units of M=3.14M ⊙ M = 3.14\\,M_\\odot .Figure REF depicts snapshots of the rest-mass density during the BH-NS binary evolution.",
"Panels (a), (b) and (c) show the $xy$ -plane and panel (d) the $xz$ -plane.",
"All densities are in units of ${g \\, cm^{-3}}$ and distances in units of $M = 3.14\\,M_\\odot $"
],
[
"Conclusions",
"We have introduced a new scheme to construct initial data for compact object binaries with NS companions.",
"The method is a generalization of the approach to construct initial data for BBHs in which the BHs are modeled as punctures and the extrinsic curvature is given by the Bowen-York solution to the momentum constraint [25].",
"In the method introduced in the present work, the extrinsic curvature for the NSs is given by the solution derived by Bowen for spherically symmetric sources with linear momentum [40] and angular momentum [42].",
"Given these extrinsic curvature solutions, we developed an iterative prescription to construct compact object binary initial data of DNSs or BH-NSs.",
"The prescription has a relatively low computational cost since it only requires solving the Hamiltonian constraint.",
"As with the BBH case, the method also allows one to specify the intrinsic and orbital parameters of the binary with direct input from PN approximations.",
"The quality of the initial data method was demonstrated with a few examples of evolutions: an isolated NS with linear momentum, DNS binaries, including spinning NSs, and a BH-NS system.",
"The evolutions showed general agreement with similar cases found in the literature [44], [47], [48].",
"In this initial incarnation, the method was not devoid of defects.",
"The NSs showed spurious breathing that translated into oscillations in their density structure.",
"We are currently investigating applying the suggestion by [19] to mitigate the oscillations.",
"In addition, for BH-NS binaries and DNS binaries with unequal masses, there is slight drift of the coordinate center-of-mass.",
"In extreme cases, the drift complicates waveform extraction.",
"We thank P. Marronetti for helpful suggestions.",
"This work was supported by NSF grants 1333360 and 1505824.",
"Computations at XSEDE TG-PHY120016 and the Cygnus cluster at Georgia Tech."
]
] | 1606.04881 |
[
[
"Precision lattice test of the gauge/gravity duality at large-$N$"
],
[
"Abstract We pioneer a systematic, large-scale lattice simulation of D0-brane quantum mechanics.",
"The large-$N$ and continuum limits of the gauge theory are taken for the first time at various temperatures $0.4 \\leq T \\leq 1.0$.",
"As a way to directly test the gauge/gravity duality conjecture we compute the internal energy of the black hole directly from the gauge theory and reproduce the coefficient of the supergravity result $E/N^2=7.41T^{14/5}$.",
"This is the first confirmation of the supergravity prediction for the internal energy of a black hole at finite temperature coming directly from the dual gauge theory.",
"We also constrain stringy corrections to the internal energy."
],
[
"Introduction",
"The gauge/gravity duality conjecture claims that superstring theories and certain supersymmetric gauge theories are equivalent , , .",
"This duality implies that gauge theories provide us with a non-perturbative formulation of superstring theories, which will be essential in understanding the nature of quantum gravity.",
"However, this duality between gauge theories and gravity is still a conjecture.",
"With the aim to establish a non-perturbative formulation of superstring theories based on the duality relation, we must vigorously try to falsify the duality.",
"Gauge/gravity duality can be intuitively understood as a relation between two different descriptions of a system with some D-branes in a string theory.",
"One description of D-branes is given by the low energy effective theory of open strings, where the D-branes are described by a supersymmetric Yang-Mills theory defined on the world-volume of the D-branes.",
"On the other hand, D-branes can also be thought of as solitonic objects in theories of closed strings, which couple to gravity in the bulk.",
"In this picture, the D-branes are described as a source of gravity.",
"This leads to another description of the D-branes in terms of the bulk gravitational theory.",
"Though the equivalence between these two descriptions is naturally expected from the physical viewpoint, no rigorous proof has been given so far.",
"A major obstacle is the fact that, in the duality, the perturbative semi-classical regime of superstring theory is mapped to the non-perturbative regime of the gauge theory, which is very hard to deal with in an analytical way.",
"In order to study the duality, one needs a method of analyzing supersymmetric gauge theories in the strong coupling regime.",
"Numerical simulations of gauge theories, based on lattice discretization, for example, are a powerful tool to study such a regime.",
"By using a discretized lattice theory, one has a robust framework to work with in order to extract information about non-perturbative physics.",
"This is what makes it possible to test the gauge/gravity duality from first principles.",
"For the duality based on D0-branes a lot of positive evidence has been obtained through numerical simulations of a supersymmetric gauge theory known as D0-brane quantum mechanics.",
"In this case, the gravity dual geometry is given by the black 0-brane solution in type IIA supergravity (SUGRA) .",
"At finite temperature, the black 0-brane is characterized by thermodynamic quantities such as entropy and internal energy.",
"In particular, at large-$N$ and low temperature, where the SUGRA approximation becomes valid, the internal energy is given by $E= 7.41 N^2T^{14/5},$ where, $E$ and $T$ are dimensionless internal energy and temperature normalized by appropriate powers of the 't Hooft coupling of D0-brane quantum mechanics.",
"In this paper, we test the duality for D0-branes by performing a systematic, large-scale lattice study of D0-brane quantum mechanics.",
"In particular, we take both the continuum limit, by sending the lattice spacing to zero, and the large-$N$ limit for the first time.",
"This makes possible precise comparison with the result (REF ) in the SUGRA approximation.",
"We calculate the internal energy of D0-brane quantum mechanics and confirm that the internal energy of the black 0-brane (REF ) is reproduced from the D0-brane quantum mechanics — our value is $E=(7.4\\pm 0.5)N^2T^{14/5}$ .",
"We also give predictions for the stringy corrections directly from the gauge theory side.",
"The rest of this paper is organized as follows.",
"In Section , we review D0-brane quantum mechanics in more details and describe the existing literature.",
"Section contains the setup of our lattice simulations and the observables used to test the gauge/gravity duality.",
"In Section we discuss our lattice results and their extrapolation to the continuum and large-$N$ limits, before comparing them to the SUGRA expectations in Section ."
],
[
"D0-brane Quantum Mechanics",
"We consider D0-brane quantum mechanics , which is the low energy effective theory of open strings ending on $N$ D0-branes in 10-dimensional flat space.",
"The Lagrangian in the Euclidean signature is $\\mathcal {L}=\\frac{1}{g_{YM}^2}\\text{Tr}\\Bigg \\lbrace \\frac{1}{2}(D_t X_M)^2- \\frac{1}{4}[X_M,X_{M^{\\prime }}]^2+ i\\bar{\\psi }^\\alpha D_t\\psi ^\\beta + \\bar{\\psi }^\\alpha \\gamma ^M_{\\alpha \\beta }[X_M,\\psi ^\\beta ]\\Bigg \\rbrace .$ Here, $X_M$ $(M=1,2,\\cdots ,9)$ and $\\psi _\\alpha $ $(\\alpha =1,2,\\cdots ,16)$ are $N\\times N$ bosonic and fermionic Hermitian matrices, the covariant derivative $D_t$ is given by $D_t=\\partial _t +i[A_t,\\ \\cdot \\ ]$ where $A_t$ is the $U(N)$ gauge field, and $\\gamma ^M_{\\alpha \\beta }$ are the left-handed part of the gamma matrices in (9+1)-dimensions, which are $16\\times 16$ matrices.",
"This action can be obtained by dimensionally reducing the $\\mathcal {N}=1$ 10D super Yang–Mills or $\\mathcal {N}=4$ 4D SYM to $(0+1)$ -dimension.",
"Historically, this model was also obtained by applying the matrix regularization to the theory of a single supermembrane in 11-dimensional flat space in the light-cone frame .",
"From this perspective, it was conjectured that the model in Equation REF describes second quantized M-theory on 11-dimensional flat space .",
"The coupling constant $g_{YM}$ and the matrix size $N$ are related to parameters of the M-theory as $g_{YM}^2N \\sim R^3 $ and $N \\sim p^+ R$ , where $R$ is the radius of the M-circle and $p^+$ is momentum along the light-cone direction.",
"In order to realize the decompactified limit $R\\rightarrow \\infty $ with $p^+$ fixed, one needs to take a very strong coupling limit of the matrix model.",
"On the other hand, in this paper, we mainly consider the 't Hooft limit of the model, where $\\lambda =g_{YM}^2N$ is fixed and $N \\rightarrow \\infty $ .",
"Therefore we focus on the gauge/gravity duality to type IIA superstring theory .",
"The coupling constant $\\lambda $ has mass dimension 3 and sets the scale of the theory.",
"In the following we fix $\\lambda =1$ without loss of generality, because it amounts to a rescaling of the fields.",
"Intuitively, the off-diagonal elements of the matrices are open strings that connect the D0-branes whose locations are given by the diagonal elements , as sketched in Fig.",
"REF .",
"Black 0-branes are states where all the D0-branes form a single bound bunch, which corresponds to generic non-commuting matrices.",
"Strictly speaking, such bound state and a black 0-brane in SUGRA can be equivalent only at large-$N$ and in the strong coupling limit (low temperatureLow temperature means that the temperature $T$ is much smaller than the typical energy scale, $\\lambda ^{1/3}$ .",
"Hence this implies a strong coupling $\\lambda ^{-1/3}T\\ll 1$ .",
").",
"However the bound state at generic $N$ and temperature is connected smoothly to the black 0-brane at large-$N$ and strong coupling.",
"Hence it can be regarded as the stringy generalization of the black hole.",
"When there is no risk of confusion, we call such bound state simply as the black 0-brane or black hole.",
"Figure: An intuitive interpretation of the matrices X M X_M.The diagonal elements correspond to positions of D0-branes and the off-diagonal elements correspond to the open strings connecting them.This figure is taken from Ref.",
".D0-brane quantum mechanics was first investigated with Monte Carlo methods in Ref. .",
"(Earlier numerical work with the same motivation can be found in Ref. .)",
"Previously, there have been attempts to study the internal energy , , , , , the supersymmetric Polyakov loop and two-point correlation functions , .",
"However, the existing literature claiming to provide strong evidence supporting the gauge/gravity duality could be invalidated, because the numerical results obtained so far were not extrapolated to the continuum limit $L\\rightarrow \\infty $ and the $N\\rightarrow \\infty $ limit.",
"This lack of controlled extrapolations would obstruct a meaningful test of the conjecture.",
"Moreover, the existing results did not have enough accuracy to confirm the supergravity prediction of the internal energy, $E/N^2=7.41T^{14/5}$ .",
"In order to achieve high precision, it is of paramount importance to correctly estimate the discretization errors and corrections due to finite $N$ .",
"We accomplish this for the first time in our study."
],
[
"Lattice Setup",
"In order to study the thermodynamic properties the D0-brane quantum mechanics non-perturbatively, we discretize the theory in a 0+1 dimensional Euclidean spacetime.",
"We then use the discretized action to calculate the theory's partition function by importance sampling field configurations via the rational hybrid Monte Carlo algorithm.",
"By measuring observables on this ensemble of configurations, we get an estimate for the observable's expectation value with an associated statistical uncertainty.",
"Finally, by measuring on ensembles with different lattice spacings, we can extrapolate to the continuum limit, removing the lattice regulator, and get a fully non-perturbative result.",
"As we will show, achieving a reliable continuum extrapolation requires a careful study."
],
[
"Discretized Action and Simulations",
"Consider D0-brane quantum mechanics (REF ) on a Euclidean circle with circumference $\\beta $ .",
"With antiperiodic boundary conditions for the fermions and periodic boundary conditions for the bosons, $\\beta $ is identified with the inverse temperature $1/T$ .",
"This model consists of nine $N\\times N$ bosonic hermitian matrices $X_M$ ($M=1,2,\\cdots ,9$ ), sixteen fermionic matrices $\\psi _\\alpha $ ($\\alpha =1,2,\\cdots ,16$ ) and the gauge field $A_t$ .",
"Both $X_M$ and $\\psi _\\alpha $ are in the adjoint representation of $U(N)$ gauge group, and the covariant derivative $D_t$ acts on them as $D_tX_M = \\partial _t X_M +i[A_t,X_M]$ and $D_t\\psi _\\alpha = \\partial _t\\psi _\\alpha +i[A_t,\\psi _\\alpha ]$ .",
"The 't Hooft coupling $\\lambda =g_{YM}^2N$ has a dimension of $({\\rm mass})^3$ , and can be set to 1 by rescaling time $t$ and the fields.",
"In other words, all dimensionful quantities can be made dimensionless by multiplying appropriate powers of $\\lambda $ .",
"As mentioned before, we choose $\\lambda =1$ .",
"The action is given by $S_{BFSS}=S_b+S_f,$ where $S_b$ the bosonic part and $S_f$ the fermionic part are given by $S_b &= N\\int _0^\\beta dt\\ \\text{Tr}\\left\\lbrace \\frac{1}{2}(D_t X_M)^2 - \\frac{1}{4}[X_M,X_N]^2 \\right\\rbrace , \\\\S_f &= N\\int _0^\\beta dt\\ \\text{Tr}\\left\\lbrace i\\bar{\\psi }\\gamma ^{10}D_t\\psi - \\bar{\\psi }\\gamma ^M[X_M,\\psi ] \\right\\rbrace .$ while $\\gamma ^M$ ($M=1,\\cdots ,10$ ) represent the $16\\times 16$ left-handed part of the 10D gamma matrices $\\Gamma ^{M}$ .",
"Formally, this model is obtained by dimensionally reducing the ten-dimensional ${\\cal N}=1$ super Yang-Mills theory to one dimension.",
"The index $\\alpha $ of the fermionic matrices $\\psi _\\alpha $ corresponds to the spinor index in ten dimensions, and $\\psi _\\alpha $ is Majorana-Weyl in the ten-dimensional sense.",
"For numerical efficiency, we adopt the static diagonal gauge , $A_t=\\frac{1}{\\beta }\\cdot {\\rm diag}(\\alpha _1,\\cdots ,\\alpha _N),\\qquad -\\pi <\\alpha _i\\le \\pi .$ Associated with this gauge fixing, we add to the action the corresponding Faddeev-Popov term $S_{F.P.",
"}=-\\sum _{i<j}^{N}2\\log \\left|\\sin \\left(\\frac{\\alpha _i-\\alpha _j}{2}\\right)\\right|.$ We regularize the theory by discretizing the Euclidean time direction over $L$ lattice sites.",
"Our lattice action is $S_b&=\\frac{N}{2a}\\sum _{t,M}\\text{Tr}\\left\\lbrace \\left(D_+X_M(t)\\right)^2\\right\\rbrace -\\frac{Na}{4}\\sum _{t,M,N}\\text{Tr}\\left\\lbrace [X_M(t),X_N(t)]^2 \\right\\rbrace ,\\\\S_f&=\\sum _{t}\\text{Tr}\\left\\lbrace iN \\bar{\\psi }(t)\\left(\\begin{array}{cc}0 & D_+\\\\D_- & 0\\end{array}\\right)\\psi (t)-aN\\sum _{t,M}\\bar{\\psi }(t)\\gamma ^M[X_M(t),\\psi (t)] \\right\\rbrace ,\\\\S_{F.P.",
"}&=-\\sum _{i<j}^{N}2\\log \\left|\\sin \\left(\\frac{\\alpha _i-\\alpha _j}{2}\\right)\\right|,$ where the gauge links $U=\\exp (iaA_t)$ with $-\\pi \\le \\alpha _i<\\pi $ .",
"The covariant derivative $D_{\\pm }$ can be discretized in different ways which in turn will have different discretization errors.",
"A first discretization that we call “unimproved” defines $D_{\\pm }$ as follows: $D_+f(t) &= Uf(t+a)U^\\dagger -f(t), \\nonumber \\\\D_-f(t) &= f(t)-U^\\dagger f(t-a)U ,$ where $f(t)$ can be a bosonic or a fermionic field defined at site $t$ and the gauge link $U$ is $t$ -independent due to our gauge fixing choice (REF ).",
"This discretized derivative is related to the continuum one $D_t$ by $D_\\pm f(t)= aD_t f(t) +\\mathcal {O}\\left( a^2 \\right)$ .",
"The discretization of $D_{\\pm }$ that we will use in our main results has smaller discretization effects, $\\mathcal {O}\\left( a^3 \\right)$ , and we call it “improved” to reflect this feature.",
"The exact lattice definition is $D_+f(t) &= -\\frac{1}{2}U^2 f(t+ 2a)U^{\\dagger 2} + 2U f(t+ a)U^\\dagger -\\frac{3}{2} f(t),\\nonumber \\\\D_-f(t) &= +\\frac{1}{2}U^{\\dagger 2} f(t- 2a)U^{2} - 2U^\\dagger f(t- a)U +\\frac{3}{2} f(t).$ We calculate with the unimproved and improved lattice actions with the RHMC algorithm, tuning the integration step and trajectory length to attain an acceptance rate of order $80\\%$ .",
"We take advantage of MPI parallelization, where each MPI process takes care $l$ lattice sites and $n\\times n$ sub-blocks of matrices.",
"The number of total processes for a lattice of size $L$ and matrices of size $N \\times N$ is $(L/l)\\cdot (N/n)^2$ .",
"Typically we take $n=l=4$ and, for example, the number of processes is $8^3=512$ for $N=L=32$ .",
"This setup is very advantageous on large parallel machines and allows us to simulate very large values of $N$ and $L$ by scaling our code to greater numbers of MPI processes.",
"The simulation code is publicly available and well documented .",
"An important remark for numerical simulations of the D0-brane quantum mechanics is that the system has flat directions, $[X_M,X_{M^{\\prime }}]=0$ .",
"At large $N$ , the flat directions are lifted dynamically, around the black hole phase.",
"However, at finite $N$ , the black hole is metastable, and the D0-branes (the eigenvalues of the matrices) can be emitted and propagate to infinity.",
"This phenomenon produces an instability in the Monte Carlo evolution which become more and more severe at smaller $N$ and at lower temperatures.",
"In order to obtain meaningful statistical results from simulations, it is of crucial importance to control these flat directions and correctly single out the phase under consideration , , .",
"If this control is missing, wrong answers might be obtained, as happened countlessly many times in the early literature on lattice supersymmetry.",
"In this study, we overcome the instability by taking $N$ sufficiently large that our observables do not show signs of eigenvalue instability over long Monte Carlo histories."
],
[
"Observables",
"On each configuration we measure different observables.",
"The most crucial for this work is the internal energy $E/N^2$ , $E/N^2 = \\frac{3}{2N^2\\beta }\\left( 9(N^2L-1) - 2 \\langle S_b\\rangle \\right).$ We also measure the absolute value of the Polyakov loop, $|P| = \\left|\\frac{1}{N}\\sum _{j=1}^N e^{i\\alpha _j}\\right|$ where $\\alpha _j$ belong to the gauge-fixed link variables, the average size of the eigenvalue bunch (0-brane) $R^2$ , $R^2 \\equiv \\frac{1}{NL} \\sum _{M,t} \\text{Tr}\\left\\lbrace X_M^2\\right\\rbrace \\qquad \\longrightarrow \\qquad \\frac{1}{N\\beta }\\int dt \\text{Tr}X_M^2$ and the potential term $F^2$ (analogous to the square of the field strength), $F^2 = -\\frac{1}{NL}\\sum _{M,M^{\\prime },t} \\text{Tr}\\left\\lbrace \\left[X_M,X_{M^{\\prime }}\\right]^2\\right\\rbrace \\qquad \\longrightarrow \\qquad -\\frac{1}{N\\beta }\\int dt \\text{Tr}[X_M,X_{M^{\\prime }}]^2.$"
],
[
"Phase Quenching",
"One potential issue in simulating this theory is the infamous sign problem — the Pfaffian that results from integrating out the fermions can have an oscillating phase, undermining the probabilistic interpretation of the Euclidean action in the path integral.",
"In our calculation, we follow the usual practice , , , , of simply taking the absolute value of the Pfaffian, quenching the phase.",
"Several studies have found that the phase of the Pfaffian remains close to zero in the temperature region we consider, and the most recent one is Ref. .",
"This means that the sign problem is mild and quenching the phase does not distort the results.",
"Previous results were obtained for relatively small values of $N$ and of the cutoff, but in the same temperature regime we study here.",
"In principle, the effect of the phase can be taken into account by phase reweighting, $\\langle {\\cal O}\\rangle _{\\rm F}=\\frac{\\langle {\\cal O}\\cdot e^{i\\theta }\\rangle _{\\rm PQ}}{\\langle e^{i\\theta }\\rangle _{\\rm PQ}},$ where $\\langle \\ \\cdot \\ \\rangle _{\\rm F}$ and $\\langle \\ \\cdot \\ \\rangle _{\\rm PQ}$ represent the expectation values with the full and phase-quenched theories, and $e^{i\\theta }={\\rm Pfaffian}/|{\\rm Pfaffian}|$ .",
"Interestingly, even when the phase fluctuations become large, it has been observed that the phase quenching does not affect the expectation values of various observables.",
"Figure: The correlation between E/N 2 E/N^2 and |P||P| at N=16N=16, L=16L=16 and T=0.5T=0.5 is shown as 2D histogram where a darker color corresponds to a higher count.The blue and red histograms (three per panel) represent the normalized distribution of E/N 2 E/N^2 and |P||P|, respectively, within the slices on the two-dimensional plot bounded by dashed lines.The histograms coming from different slices are almost indistinguishable.A possible mechanism is suggested in Ref. .",
"Let $\\rho (x)$ be the distribution of the observable ${\\cal O}$ in the phase-quenched simulation, and let $w_x$ be the average of $e^{i\\theta }$ when the value of ${\\cal O}$ is fixed to $x$ .",
"Then $\\langle {\\cal O}\\rangle _{\\rm PQ} &=\\int dx\\ x \\rho (x)\\\\\\langle {\\cal O}\\cdot e^{i\\theta }\\rangle _{\\rm PQ} &=\\int dx\\ x \\rho (x) w_x\\\\\\langle e^{i\\theta }\\rangle _{\\rm PQ} &=\\int dx\\ \\rho (x) w_x\\ .$ Typically $\\rho (x)$ peaks around the average value, $x=\\langle {\\cal O}\\rangle _{\\rm PQ}$ .",
"If $w_x$ is constant around this peak, then $\\langle {\\cal O}\\cdot e^{i\\theta }\\rangle _{\\rm PQ}\\simeq \\langle {\\cal O}\\rangle _{\\rm PQ}\\cdot \\langle e^{i\\theta }\\rangle _{\\rm PQ}$ , and then (REF ) becomes $\\langle {\\cal O}\\rangle _{\\rm F}\\simeq \\langle {\\cal O}\\rangle _{\\rm PQ}$ .",
"Because the calculation of the Pfaffian is very costly, it is difficult to test this scenario directly at large values of $N$ .",
"However, it is possible to indirectly infer the magnitude of the phase fluctuations and their impact on the other observables.",
"In fact, the Polyakov loop has a strong correlation with the phase factor—the phase disappears when $|P|=1$ (up to discretization effects) and the phase fluctuations become larger as $|P|$ decreases.",
"In Fig.",
"REF , we show the correlation between $E/N^2$ and $|P|$ at $N=16$ , $L=16$ and $T=0.5$ .",
"The blue and red histograms represent the distribution of $E/N^2$ and $|P|$ , respectively, with the other quantity restricted within small bins highlighted in the two-dimensional plot.",
"The areas of the histograms are normalized.",
"The $E$ -independence of the distribution of $|P|$ , at least away from the tails, strongly suggests the $E$ -independence of the distribution of the phase, which justifies the phase quenching via the scenario explained above.",
"A more detailed study of the distribution of $|P|$ for various values of the energy, near and away from its average, is reported in Appendix .",
"An explicit calculation of the Pfaffian phase is worthwhile, but we leave it for a future study.",
"In the rest of the paper we assume that the phase-quenched approximation does not influence the internal energy results of our simulations."
],
[
"Results",
"In this section we discuss the statistical needs of our analysis, continuum extrapolations at fixed $N$ (comparing with other calculations when available) and simultaneous continuum and large-$N$ extrapolations.",
"In the end we report a continuum large-$N$ data set that we will use in Section for a direct comparison to supergravity predictions.",
"We also collect our measurements for each ensemble in Appendix ."
],
[
"Statistical Requirements",
"To ensure a faithful estimation of an observable, one must ensure a large number of independent (decorrelated) Monte Carlo samples are taken.",
"In Fig.",
"REF we show an example Monte Carlo history for the ensemble with $N=24$ , $L=32$ , and $T=0.5$ .",
"It is apparent that there are long-lived autocorrelations.",
"Therefore, to achieve many independent samples, lengthy Monte Carlo ensembles are required.",
"Figure: The Monte Carlo history and a corresponding histogram for the energy E/N 2 E/N^2, the Polyakov loop |P||P|, R 2 R^2, and F 2 F^2 of the T=0.5T=0.5 N=24N=24 L=32L=32 ensemble.For each observable, one can see fluctuations that span many Monte Carlo steps.Figure: A study of the statistical stability of E/N 2 E/N^2 for the ensemble N=16N=16, L=32L=32, T=0.5T=0.5.In the left panel we show different thermalization cuts, measuring E/N 2 E/N^2 on the rest of the configurations.In the right panel we show the importance of large statistical samples by measuring on consecutive disjoint sets of trajectories.As the statistical sample grows from 1000 configurations (red squares) to 6000 configurations (blue circles), the central values and uncertainties between sets of configurations become more and more stable and compatible.In both panels we perform the analysis with bins of 50 configurations.For comparison, we also show our final analysis and its uncertainty as a black star in both panels, with its error bar displayed as dashed lines on the right panel.Moreover, accounting for autocorrelations is essential for an accurate estimate of the statistical uncertainty on a given measurement.",
"For each observable on each ensemble, we measure the autocorrelation time $\\tau _\\text{corr}$ using the Madras-Sokal algorithm and form bins of width 3$\\tau _\\text{corr}$ .",
"With those binned measurements we perform a jackknife analysis to estimate the statistical uncertainty.",
"We also independently test that the statistical error associated with our final average is robust by performing different analysis with smaller and larger jackknife bins and making sure that the final uncertainty does not change.",
"In Fig.",
"REF we study the statistical stability of $E/N^2$ for a low-temperature ensemble, $N=24$ , $L=32$ , and $T=0.5$ , with bins 50 trajectories wide.",
"In the left panel we show the residual effects of keeping measurements from too early in the Monte Carlo history by using the whole ensemble and only adjusting the thermalization cut.",
"From the compatibility with later cuts, it is clear that this ensemble has no memory of its initially chosen configuration after 500 trajectories.",
"On each ensemble, we discard 1000 trajectories as a thermalization cut.",
"In the right panel of Fig.",
"REF we show how many configurations are necessary for a stable estimate.",
"We start at trajectory 500 and take the next 1000, 2000, 4000, and 6000 trajectories and perform an independent analysis, and then slide that window to the next disjoint set of trajectories.",
"One can see that for this ensemble, 1000 thermalized trajectories is not enough to achieve a stable statistical estimate, indicating that there can be sizable fluctuations over Monte Carlo time that can dramatically shift the measured value.",
"However, 2000 trajectories seem to be enough to reliably get the eventual central value within the uncertainty.",
"Increasing the window size correctly washes out the effect of lengthy fluctuations and makes each successive analysis agree more reliably.",
"We are therefore confident that most of our statistical samples are large enough to correctly estimate the energy $E/N^2$ .",
"Some ensembles at $T=0.4$ are not very lengthy—though all are longer than 1000 trajectories after the thermalization cut.",
"To compensate for this shortcoming, we inflated their statistical uncertainty by 50% and reperformed all the following analyses.",
"We find very little difference between the two cases.",
"In what follows, we therefore use the uninflated errors."
],
[
"Continuum Extrapolation at Fixed $N$",
"To study the continuum theory, one must measure at a variety of lattice discretizations and extrapolate to the continuum.",
"In this section, we discuss continuum extrapolation at fixed $N$ using our unimproved and improved actions (cfr.",
"equations (REF ) and (REF )).",
"As the lattice spacing $L^{-1}$ gets smaller, one expects an expansion around $L^{-1}=0$ to get better.",
"So, at fixed $N$ the energy should follow $\\frac{E}{N^2} = e_0 + \\frac{e_1}{L} + \\frac{e_2}{L^2} + \\mathcal {O}\\left( L^{-3} \\right)$ where $e_0$ is the continuum-extrapolated value and the other $e_i$ characterize the lattice artifacts.",
"Based on the naïve scaling of the action with the lattice spacing, we expect results with the unimproved action to have larger discretization effects and we check this explicitly in the following for the first time.",
"In Figure REF we show a fixed-$N$ continuum extrapolation for $T=0.7$ $N=16$ so that we can directly compare to the continuum extrapolations of Ref. .",
"One immediately sees that the region where only the leading $L^{-1}$ corrections matter is $L\\gtrsim 16$ —with smaller $L$ the subleading correction is not negligible, so linear fits to lattice data from such small $L$ will be systematically biased towards larger $E/N^2$ .",
"We have checked this rule of thumb for all $T$ and $N$ , and find broad consistency with this observation, which means Refs.",
", may suffer from premature extrapolation.",
"Figure: A comparison between taking the continuum limit for the unimproved and improved actions for T=1.0T=1.0, N=16N=16.Error bars and the error bands on the extrapolated curves represent 1σ\\sigma errors.Knowing that to successfully fit down to $L=8$ with the improved action requires a quadratic fit, we expect additional lattice artifacts to contaminate $L=8$ with the unimproved action, suggesting an additional term is needed to fit the unimproved action to incorporate that point into the continuum limit.",
"Indeed, fitting a quadratic to that point pushes the fit upwards, while fitting a cubic gives perfect agreement with the improved continuum limit.",
"Using the improved action allows us to extrapolate to the continuum in a more controlled manner, because a successful extrapolation requires fitting fewer parameters."
],
[
"Simultaneous Large $N$ and Continuum Extrapolation",
"In order to test the gauge/gravity duality precisely, it is important to take the large-$N$ limit.",
"However, taking the continuum limit at large-$N$ becomes costly even with a quadratic fit, because at small $N$ the physical instability may ruin the Monte Carlo history, while numerical cost grows with $L$ and $N$ .",
"Large-$N$ corrections appear in powers of $N^{-2}$ , at each fixed $L$ , because the 't Hooft counting holds even for the discretized theory.",
"Thus, at a fixed temperature we expect $E/N^2$ to be described by a series like the following $\\frac{E}{N^2} = \\sum _{i,j\\ge 0} \\frac{e_{ij}}{N^{2i}L^j}$ so that $e_{i0}$ are physical, continuum-limit quantities at finite $N$ , $e_{00}$ is the continuum, large-$N$ value, and all other coefficients, $e_{ij}$ with $j>0$ , characterize lattice artifacts.",
"Importantly, by extrapolating in $1/N^2$ and $1/L$ simultaneously, we can take advantage of significantly more data points without increasing the number of fit parameters dramatically.",
"We can truncate (REF ) in various ways and attempt to fit a finite set of $e_{ij}$ .",
"We attempted a six-parameter fit with $i+j\\le 2$ and found our data insufficient to characterize $e_{11}$ or $e_{20}$ without 100% uncertainties, and strong correlation with the other coefficients.",
"We also performed five-parameter fits, omitting either $e_{11}$ and $e_{20}$ and still found the other to be very poorly constrained by our data and highly correlated with the remaining coefficients.",
"Thus, we settled on a four-parameter fit—next to leading order (NLO) in $N^{-2}$ and NNLO in $L^{-1}$ , with no mixed term $\\frac{E}{N^2} \\approx e_{00} + \\frac{e_{01}}{L} + \\frac{e_{02}}{L^2} + \\frac{e_{10}}{N^2}.$ We fit this form to all of our measurements at a given temperature, and find extremely good fit quality together with a very mild dependence on $N$ and—just as in the fixed-$N$ case—important dependence on $L$ .",
"The strong $L$ dependence, which we observe to get stronger at low temperature, raises the possibility that Ref.",
", which at low temperature works only at $L=16$ and has no continuum limit, and Ref.",
", which at $N=16$ extrapolates from the momentum cutoff $\\Lambda \\le 8$ , may be systematically contaminated by discretization artifacts.",
"However, because those references use different discretized actions from that used in this work, their discretization effects may be substantially smaller than in our approach.",
"For example, a direct comparison at $T=0.4$ $N=32$ $L=16$ shows that Ref.",
"'s central value is substantially closer to our continuum limit 0.40(7) than our data point at those parameters 0.835(7).",
"In Figure REF we show the result of the simultaneous continuum- and large-$N$ extrapolation of the measurements of the $T=0.5$ improved action measurements.",
"We also show three fixed-$N$ continuum extrapolations and their subsequent large-$N$ extrapolation.",
"For that ensemble, we fit 13 data points to the four-parameter fit in (REF ) and find a reduced chi-squared (the usual $\\chi ^2$ divided by DOF, the degrees of freedom in the fit) of 7.2/9 and good compatibility with the sequential extrapolation.",
"In Table REF we show the simultaneous continuum and large-$N$ extrapolation by the four-parameter fit in (REF ) of data taken with the improved action at various temperatures.",
"A more complete data set is provided in Appendix .",
"Table: Simultaneous Continuum Large-NN Extrapolations"
]
] | 1606.04951 |
[
[
"Orthogonality of divisorial Zariski decompositions for classes with\n volume zero"
],
[
"Abstract We show that the orthogonality conjecture for divisorial Zariski decompositions on compact Kahler manifolds holds for pseudoeffective (1,1) classes with volume zero."
],
[
"Introduction",
"The orthogonality conjecture of divisorial Zariski decompositions [4] states the following: Conjecture 1.1 Let $(X^n,\\omega )$ be a compact Kähler manifold, and $\\alpha $ a pseudoeffective $(1,1)$ class.",
"Then $\\langle \\alpha ^{n-1}\\rangle \\cdot \\alpha =\\mathrm {Vol}(\\alpha ).$ Here $\\mathrm {Vol}(\\alpha )$ denotes the volume of the class $\\alpha $ [1] and $\\langle \\cdot \\rangle $ is the moving intersection product of classes as introduced by Boucksom [2], [4], [5], [6].",
"The name of this problem comes from the following observation.",
"If we choose an approximate Zariski decomposition of the class $\\alpha $ (see section for the precise details of the construction), given by suitable modifications $\\mu _\\delta :X_\\delta \\rightarrow X$ , for all small $\\delta >0$ , with $\\mu _\\delta ^*(\\alpha +\\delta \\omega )=\\theta _\\delta +[E_\\delta ]$ with $\\theta _\\delta $ a semipositive class and $E_\\delta $ an effective $\\mathbb {R}$ -divisor, then (REF ) is equivalent to $\\lim _{\\delta \\downarrow 0}\\int _{X_\\delta }\\theta _\\delta ^{n-1}\\wedge [E_\\delta ]=0,$ which explains the name.",
"This conjecture was first raised by Nakamaye [14] when $X$ is projective and $\\alpha =c_1(L)$ for some line bundle $L$ .",
"This was solved by Boucksom-Demailly-Păun-Peternell [4] who more generally proved Conjecture REF when $X$ is projective and $\\alpha $ belongs to the real Néron-Severi group, and posed it in the general case.",
"It was observed in [4], [6] that Conjecture REF is equivalent to several other powerful statements, including the fact that the dual cone of the pseudoeffective cone $\\mathcal {E}$ is the movable cone $\\mathcal {M}$ , the weak transcendental Morse inequalities, and the $C^1$ differentiability of the volume function on the big cone.",
"Very recently, Witt-Nyström [19] has proved Conjecture REF when $X$ is projective.",
"Our main result is the following: Theorem 1.2 Conjecture REF holds if $\\mathrm {Vol}(\\alpha )=0$ .",
"The strategy of proof is similar to the one in [4] with one crucial difference.",
"Since the weak transcendental Morse inequality $\\mathrm {Vol}(\\alpha -\\beta )\\geqslant \\int _X\\alpha ^n-n\\int _X\\alpha ^{n-1}\\wedge \\beta ,$ for the difference of two nef classes $\\alpha ,\\beta $ remains conjectural, we employ instead the weaker version $\\mathrm {Vol}(\\alpha -\\beta )\\geqslant \\frac{\\left(\\int _X\\alpha ^n-n\\int _X\\alpha ^{n-1}\\wedge \\beta \\right)^n}{\\left(\\int _X\\alpha ^n\\right)^{n-1}},$ which was proved independently in [18], [17], using the mass concentration technique of Demailly-Păun [10] and its recent improvements by Chiose [7], Xiao [20] and Popovici [16].",
"Inequality (REF ) is too weak to prove Conjecture REF in general, but the fact that the numerator on the RHS has the correct form turns out to be enough to prove Theorem REF .",
"If we define the “difference function” $\\mathcal {D}:\\mathcal {E}\\rightarrow \\mathbb {R}$ by $\\mathcal {D}(\\alpha ):=\\langle \\alpha ^{n-1}\\rangle \\cdot \\alpha -\\mathrm {Vol}(\\alpha ),$ then Conjecture REF simply states that $\\mathcal {D}$ vanishes identically on $\\mathcal {E}$ .",
"As a corollary of Theorem REF we have: Corollary 1.3 The function $\\mathcal {D}:\\mathcal {E}\\rightarrow \\mathbb {R}$ is nonnegative, continuous on $\\mathcal {E}$ , and vanishes on its boundary.",
"Theorem REF and Corollary REF are proved in section .",
"In section we will make some further remarks on the function $\\mathcal {D}$ , and on the relation between Theorem REF and the “cone duality” conjecture."
],
[
"The main theorem",
"In this section we give the proof of Theorem REF and Corollary REF .",
"Let $\\alpha $ be any pseudoeffective $(1,1)$ class on $X$ .",
"By definition of moving intersection products, for any $1\\leqslant p\\leqslant n$ the real $(p,p)$ cohomology class $\\langle \\alpha ^{p}\\rangle $ is defined to be $\\langle \\alpha ^{p}\\rangle =\\lim _{\\delta \\downarrow 0}\\langle (\\alpha +\\delta \\omega )^{p}\\rangle ,$ where for $\\delta >0$ the class $\\alpha +\\delta \\omega $ is big, and in this case we define $\\langle (\\alpha +\\delta \\omega )^{p}\\rangle =[\\langle T_{min,\\delta }^{p}\\rangle ],$ where $T_{min,\\delta }$ is any positive current with minimal singularities in the class $\\alpha +\\delta \\omega $ , and $\\langle T_{min,\\delta }^{p}\\rangle $ denotes the non-pluripolar product [5].",
"The volume of $\\alpha $ (see [1]) is in fact equal to the moving self-intersection product $\\mathrm {Vol}(\\alpha )=\\langle \\alpha ^n\\rangle =\\lim _{\\delta \\downarrow 0}\\langle (\\alpha +\\delta \\omega )^n\\rangle .$ We now review the well-known construction of approximate Zariski decompositions [2], [4], [5], [6], following roughly the argument in [5].",
"Applying Demailly's regularization [9] to $T_{min,\\frac{\\delta }{2}}$ we obtain a sequence of currents $T_{\\delta ,\\varepsilon }, \\varepsilon >0,$ in the big class $\\alpha +\\frac{\\delta }{2}\\omega $ , with analytic singularities, with $T_{\\delta ,\\varepsilon }\\geqslant -\\varepsilon \\omega $ , and with their potentials decreasing to that of $T_{min,\\frac{\\delta }{2}}$ as $\\varepsilon \\rightarrow 0$ .",
"As long as $\\varepsilon \\leqslant \\frac{\\delta }{2}$ , we have that $T_{\\delta ,\\varepsilon }+\\frac{\\delta }{2}\\omega $ and $T_{min,\\frac{\\delta }{2}}+\\frac{\\delta }{2}\\omega $ are closed positive currents in the class $\\alpha +\\delta \\omega $ whose potentials are locally bounded away from the proper analytic subvariety $A:=E_{nK}(\\alpha +\\frac{\\delta }{2}\\omega )$ , and so by weak continuity of the Bedford-Taylor Monge-Ampère operator along decreasing sequences we have $\\left(T_{\\delta ,\\varepsilon }+\\frac{\\delta }{2}\\omega \\right)^{p}\\rightarrow \\left(T_{min,\\frac{\\delta }{2}}+\\frac{\\delta }{2}\\omega \\right)^{p},$ weakly on $X\\backslash A$ as $\\varepsilon \\rightarrow 0$ , for all $1\\leqslant p\\leqslant n$ .",
"It follows that $\\begin{split}\\int _{X\\backslash A}\\left(T_{min,\\frac{\\delta }{2}}+\\frac{\\delta }{2}\\omega \\right)^{p}\\wedge \\omega ^{n-p}&\\leqslant \\liminf _{\\varepsilon \\rightarrow 0}\\int _{X\\backslash A}\\left(T_{\\delta ,\\varepsilon }+\\frac{\\delta }{2}\\omega \\right)^{p}\\wedge \\omega ^{n-p}\\\\&\\leqslant \\int _{X\\backslash A}(T_{min,\\delta })^{p}\\wedge \\omega ^{n-p},\\end{split}$ where the last inequality follows from [5], since all the currents involved have small unbounded locus.",
"But as $\\delta \\rightarrow 0$ both the LHS and the RHS converge to $\\int _X\\langle \\alpha ^p\\rangle \\wedge \\omega ^{n-p},$ and so we may choose a sequence $\\varepsilon (\\delta )\\rightarrow 0$ such that the currents $T_\\delta :=T_{\\delta ,\\varepsilon (\\delta )}-\\frac{\\delta }{2}\\omega $ in the class $\\alpha $ have analytic singularities, satisfy $T_\\delta \\geqslant -\\delta \\omega $ , and are such that $\\lim _{\\delta \\downarrow 0}\\langle (T_{\\delta }+\\delta \\omega )^p\\rangle =\\langle \\alpha ^p\\rangle ,$ for all $1\\leqslant p\\leqslant n$ .",
"Let $\\mu _\\delta :X_\\delta \\rightarrow X$ be a resolution of the singularities of $T_\\delta +\\delta \\omega $ , so that $\\mu _\\delta ^*(T_\\delta +\\delta \\omega )=\\theta _\\delta +[E_\\delta ],$ so that $\\theta _\\delta $ is a smooth semipositive form, $E_\\delta $ is an effective $\\mathbb {R}$ -divisor and $[E_\\delta ]$ denotes the current of integration.",
"We refer to this construction as an approximate Zariski decomposition for the class $\\alpha $ .",
"As discussed above, we have $\\mathrm {Vol}(\\alpha )=\\lim _{\\delta \\downarrow 0}\\int _X \\langle (T_\\delta +\\delta \\omega )^{n}\\rangle =\\lim _{\\delta \\downarrow 0}\\int _{X_\\delta }\\langle (\\mu _\\delta ^*(T_\\delta +\\delta \\omega ))^{n}\\rangle =\\lim _{\\delta \\downarrow 0}\\int _{X_\\delta }\\theta _\\delta ^n,$ $\\begin{split}\\langle \\alpha ^{n-1}\\rangle \\cdot \\alpha &=\\lim _{\\delta \\downarrow 0}\\int _X \\langle (T_\\delta +\\delta \\omega )^{n-1}\\rangle \\wedge \\alpha =\\lim _{\\delta \\downarrow 0}\\int _X \\langle (T_\\delta +\\delta \\omega )^{n-1}\\rangle \\wedge (\\alpha +\\delta \\omega )\\\\&=\\lim _{\\delta \\downarrow 0}\\int _{X_\\delta }\\langle (\\mu _\\delta ^*(T_\\delta +\\delta \\omega ))^{n-1}\\rangle \\wedge \\mu _\\delta ^*(\\alpha +\\delta \\omega )\\\\&=\\lim _{\\delta \\downarrow 0}\\int _{X_\\delta }(\\theta _\\delta ^n+\\theta _\\delta ^{n-1}\\wedge [E_\\delta ]),\\end{split}$ so that the orthogonality relation (REF ) is in general equivalent to the statement that $\\lim _{\\delta \\downarrow 0}\\int _{X_\\delta }\\theta _\\delta ^{n-1}\\wedge [E_\\delta ]=0.$ We now follow [4], and fix a constant $C_0$ such that $C_0\\omega \\pm (\\alpha +\\delta \\omega )$ is nef.",
"We write $E_\\delta =\\mu _\\delta ^*(\\alpha +\\delta \\omega +C_0\\omega )-(\\theta _\\delta +C_0\\mu _\\delta ^*\\omega ),$ as the difference of two nef classes.",
"For $t\\in [0,1]$ write $\\theta _\\delta +tE_\\delta =A-B,$ where $A=\\theta _\\delta +t\\mu _\\delta ^*(\\alpha +\\delta \\omega +C_0\\omega ),$ $B=t(\\theta _\\delta +C_0\\mu _\\delta ^*\\omega ),$ and $A,B$ are nef.",
"Then $\\mathrm {Vol}(\\alpha +\\delta \\omega )=\\mathrm {Vol}(\\theta _\\delta +E_\\delta )\\geqslant \\mathrm {Vol}(\\theta _\\delta +tE_\\delta )=\\mathrm {Vol}(A-B).$ We use [18] (also independently obtained in [17]) and obtain $\\mathrm {Vol}(A-B)\\geqslant \\frac{\\left(\\int _{X_\\delta } A^n -n\\int _{X_\\delta } A^{n-1}\\wedge B\\right)^n}{\\left(\\int _{X_\\delta }A^n\\right)^{n-1}}.$ We follow the same argument as in [4] (see also [12]) and estimate $\\int _{X_\\delta } A^n -n\\int _{X_\\delta } A^{n-1}\\wedge B\\geqslant \\int _{X_\\delta }\\theta _\\delta ^n+nt\\int _{X_\\delta }\\theta _\\delta ^{n-1}\\wedge [E_\\delta ] - 5n^2t^2C_0^n\\int _X\\omega ^n,$ as long as $t\\leqslant \\frac{1}{10n}$ .",
"We choose $t=\\frac{\\int _{X_\\delta }\\theta _\\delta ^{n-1}\\wedge [E_\\delta ]}{10n C_0^n\\int _X\\omega ^n},$ which is easily seen to be less than $\\frac{1}{10n}$ , and so we obtain $\\int _{X_\\delta } A^n -n\\int _{X_\\delta } A^{n-1}\\wedge B\\geqslant \\int _{X_\\delta }\\theta _\\delta ^n+\\frac{1}{20}\\frac{\\left(\\int _{X_\\delta }\\theta _\\delta ^{n-1}\\wedge [E_\\delta ]\\right)^2}{C_0^n\\int _X\\omega ^n},$ and plugging this into (REF ) we obtain $\\mathrm {Vol}(\\alpha +\\delta \\omega )\\geqslant \\frac{\\left(\\int _{X_\\delta }\\theta _\\delta ^n+\\frac{1}{20}\\frac{\\left(\\int _{X_\\delta }\\theta _\\delta ^{n-1}\\wedge [E_\\delta ]\\right)^2}{C_0^n\\int _X\\omega ^n}\\right)^n}{\\left(\\int _{X_\\delta }A^n\\right)^{n-1}}.$ We also have $\\int _{X_\\delta }A^n=\\int _{X_\\delta }(\\theta _\\delta +t\\mu _\\delta ^*(\\alpha +\\delta \\omega +C_0\\omega ))^n\\leqslant \\int _{X_\\delta }\\theta _\\delta ^n + Ct\\leqslant C,$ and so $\\int _{X_\\delta }\\theta _\\delta ^n+\\frac{1}{20}\\frac{\\left(\\int _{X_\\delta }\\theta _\\delta ^{n-1}\\wedge [E_\\delta ]\\right)^2}{C_0^n\\int _X\\omega ^n}\\leqslant C\\mathrm {Vol}(\\alpha +\\delta \\omega )^{\\frac{1}{n}}\\rightarrow C\\mathrm {Vol}(\\alpha )^{\\frac{1}{n}}=0,$ as $\\delta \\rightarrow 0$ , and we conclude that $\\int _{X_\\delta }\\theta _\\delta ^{n-1}\\wedge [E_\\delta ]\\rightarrow 0,$ which proves (REF ).",
"Remark 2.1 Some of the estimates in this proof, such as for example (REF ), are far from being sharp.",
"It is easy to make them sharp, but this does not appear to give any useful improvement.",
"Using the notation as in the proof of Theorem REF , we have that $\\mathcal {D}(\\alpha )=\\lim _{\\delta \\downarrow 0}\\int _{X_\\delta }\\theta _\\delta ^{n-1}\\wedge [E_\\delta ]\\geqslant 0.$ It follows easily from the definitions that moving intersection products are upper-semicontinuous on the pseudoeffective cone and continuous in its interior (the big cone), while it was proved in [1] that the volume function is continuous on the whole pseudoeffective cone.",
"It follows that $\\mathcal {D}$ is continuous on the big cone and upper-semicontinuous on the pseudoeffective cone.",
"By Theorem REF , $\\mathcal {D}$ vanishes on its boundary, and hence it is continuous on all of $\\mathcal {E}$ ."
],
[
"Further Remarks",
"In this section we collect some further remarks on the function $\\mathcal {D}$ and on the cone duality conjecture of [4].",
"The function $\\mathcal {D}$ defined in (REF ) clearly vanishes on the nef cone, where moving intersection products equal usual intersection products (see e.g.",
"[5]).",
"More generally we have the following result, which is a simple consequence of the author's work with Collins [8] and was also observed by Deng [11]: Proposition 3.1 Let $\\alpha $ be a pseudoeffective $(1,1)$ class, and write $\\alpha =P+N$ for its divisorial Zariski decomposition.",
"If the class $P$ is nef then $\\mathcal {D}(\\alpha )=0$ .",
"Recall here that the positive part is given by $P=\\langle \\alpha \\rangle $ , which is in general only nef in codimension 1 [3], [15].",
"First, we show that $\\langle \\alpha ^{n-1}\\rangle \\cdot N=0.$ Since moving products are unchanged if we replace a class by its positive part (see [4], [6]), this is equivalent to showing that $\\langle P^{n-1}\\rangle \\cdot N=0,$ and since by assumption $P$ is nef, this is also equivalent to showing that $P^{n-1}\\cdot N=0.$ But by definition the irreducible components of $N$ are contained in $E_{nK}(\\alpha )=E_{nK}(P)$ (see e.g.",
"[13]), and so are also irreducible components of $E_{nK}(P)$ .",
"By the main theorem of [8] we have therefore $P^{n-1}\\cdot N=0$ .",
"On the other hand since $P$ is nef we also have $\\langle \\alpha ^{n-1}\\rangle \\cdot \\alpha =\\langle \\alpha ^{n-1}\\rangle \\cdot P=\\langle P^{n-1}\\rangle \\cdot P=P^n=\\mathrm {Vol}(P)=\\mathrm {Vol}(\\alpha ),$ as claimed.",
"Remark 3.2 In fact, it is not hard to see (cf.",
"[4]) that in general (REF ) is equivalent to the following two relations both holding $\\langle \\alpha ^{n-1}\\rangle \\cdot P=\\mathrm {Vol}(\\alpha ),$ $\\langle \\alpha ^{n-1}\\rangle \\cdot N=0.$ It was proved in [4] that Conjecture REF implies (and is in fact equivalent to) the “cone duality” conjecture, which states that the dual cone of the pseudoeffective cone $\\mathcal {E}$ of a compact Kähler manifold equals the movable cone $\\mathcal {M}\\subset H^{n-1,n-1}(X,\\mathbb {R})$ .",
"It is easy to see that $\\mathcal {E}\\subset \\mathcal {M}^\\vee $ , so the point is to prove the reverse inclusion.",
"The proof given there starts by assuming that there is a class $\\alpha \\in \\partial \\mathcal {E}\\cap (\\mathcal {M}^\\vee )^\\circ $ , and derives a contradiction, assuming that $X$ is projective and that the class belongs to the real Néron-Severi group.",
"In general we have the following: Proposition 3.3 Let $\\alpha \\in \\partial \\mathcal {E}\\cap (\\mathcal {M}^\\vee )^\\circ $ .",
"Then we must have $\\langle \\alpha ^{n-1}\\rangle =0$ .",
"Assume for a contradiction that $\\langle \\alpha ^{n-1}\\rangle \\ne 0$ in $H^{n-1,n-1}(X,\\mathbb {R})$ .",
"Since this class is represented by a closed nonnegative $(n-1,n-1)$ current, it follows that $\\int _X\\langle \\alpha ^{n-1}\\rangle \\wedge \\omega >0,$ where $\\omega $ is a fixed Kähler metric.",
"We choose an approximate Zariski decomposition $T_\\delta $ with resolutions $\\mu _\\delta :X_\\delta \\rightarrow X$ as before, so that we have also $0<2\\eta \\leqslant \\int _X\\langle \\alpha ^{n-1}\\rangle \\wedge \\omega =\\lim _{\\delta \\downarrow 0} \\int _{X_\\delta }\\theta _\\delta ^{n-1}\\wedge \\mu _\\delta ^*\\omega =\\lim _{\\delta \\downarrow 0} \\int _X (\\mu _\\delta )_*(\\theta _\\delta ^{n-1})\\wedge \\omega ,$ for some fixed $\\eta >0$ .",
"Up to modifying the classes $\\theta _\\delta $ (and $[E_\\delta ]$ ) by subtracting a small multiple of $\\mathrm {Exc}(\\mu _\\delta )$ , we may also assume that the classes $\\theta _\\delta $ are Kähler on $X_\\delta $ , so that the pushforwards $(\\mu _\\delta )_*(\\theta _\\delta ^{n-1})$ are movable classes on $X$ .",
"Let $\\varepsilon >0$ be such that $\\alpha -\\varepsilon \\omega \\in \\mathcal {M}^\\vee $ .",
"Integrating $\\alpha -\\varepsilon \\omega $ against the class $(\\mu _\\delta )_*(\\theta _\\delta ^{n-1})\\in \\mathcal {M}$ we obtain $\\int _X\\alpha \\wedge (\\mu _\\delta )_*(\\theta _\\delta ^{n-1})\\geqslant \\varepsilon \\int _X\\omega \\wedge (\\mu _\\delta )_*(\\theta _\\delta ^{n-1})\\geqslant \\varepsilon \\eta >0,$ for all $\\delta >0$ small.",
"But we also have $\\int _X\\alpha \\wedge (\\mu _\\delta )_*(\\theta _\\delta ^{n-1})=\\int _{X_\\delta }\\mu _\\delta ^*\\alpha \\wedge \\theta _\\delta ^{n-1}\\leqslant \\int _{X_\\delta }\\mu _\\delta ^*(\\alpha +\\delta \\omega )\\wedge \\theta _\\delta ^{n-1}=\\int _{X_\\delta }(\\theta _\\delta ^n+\\theta _\\delta ^{n-1}\\wedge [E_\\delta ]),$ and putting these together we have $\\int _{X_\\delta }(\\theta _\\delta ^n+\\theta _\\delta ^{n-1}\\wedge [E_\\delta ])\\geqslant \\varepsilon \\eta ,$ for all $\\delta >0$ small.",
"Since $\\mathrm {Vol}(\\alpha )=0$ , this implies $\\int _{X_\\delta }\\theta _\\delta ^{n-1}\\wedge [E_\\delta ]\\geqslant \\frac{\\varepsilon \\eta }{2},$ for all $\\delta >0$ small, which is a contradiction to Theorem REF .",
"Therefore we must have that $\\langle \\alpha ^{n-1}\\rangle =0$ .",
"Remark 3.4 Even though the class $\\alpha \\in \\partial \\mathcal {E}\\cap (\\mathcal {M}^\\vee )^\\circ $ does satisfy the orthogonality conjecture (say as in (REF )) by Theorem REF , because it has volume zero, this is not enough to derive a contradiction in general.",
"Indeed, to make the argument in [4] go through, one would need the following quantitative version of orthogonality $\\left(\\int _{X_\\delta }\\theta _\\delta ^{n-1}\\wedge [E_\\delta ]\\right)^2\\leqslant C\\left(\\mathrm {Vol}(\\alpha +\\delta \\omega )-\\int _{X_\\delta }\\theta _\\delta ^n\\right),$ (cf.",
"[4]) which does not follow from the arguments of Theorem REF ."
]
] | 1606.05243 |
[
[
"Sources of The Slow Solar wind During the Solar Cycle 23/24 Minimum"
],
[
"Abstract We investigate the characteristics and the sources of the slow (< 450 km/s) solar wind during the four years (2006-2009) of low solar activity between Solar Cycles 23 and 24.",
"We use a comprehensive set of in-situ observations in the near-Earth solar wind (Wind and ACE) and remove the periods when large-scale interplanetary coronal mass ejections were present.",
"The investigated period features significant variations in the global coronal structure, including the frequent presence of low-latitude active regions in 2006-2007, long-lived low- and mid-latitude coronal holes in 2006 - mid-2008 and mostly the quiet Sun in 2009.",
"We examine both Carrington Rotation averages of selected solar plasma, charge state and compositional parameters and distributions of these parameters related to Quiet Sun, Active Region Sun and the Coronal Hole Sun.",
"While some of the investigated parameters (e.g., speed, the C^{+6}/C^{+4} and He/H ratio) show clear variations over our study period and with solar wind source type, some (Fe/O) exhibit very little changes.",
"Our results highlight the difficulty in distinguishing between the slow solar wind sources based on the inspection of the solar wind conditions."
],
[
"Introduction",
"The heliosphere is permeated by a continuous stream of charged particles emanating from the Sun's hot outer corona.",
"This so-called “solar wind” carries the magnetic field of the Sun throughout the solar system and it is the medium where coronal mass ejections (CMEs), the key drivers of space weather storms in the near-Earth space environment, propagate.",
"Moreover, the boundary conditions for space-weather modelling and forecasting are entirely dependent upon solar observations and, especially, on the knowledge of the origin of the solar wind.",
"Whereas the regions of open magnetic field lines on the Sun, coronal holes, have been firmly established as the main source of the fast solar wind ($\\sim 700~km~s^{-1}$ [31]), the origin of the highly-structured slow ($\\sim < 450~km~s^{-1}$ ) solar wind is still highly debated.",
"The suggested sources for the slow solar wind are numerous, including: 1) fast converging open magnetic field lines near the boundaries of coronal holes [52]; 2) transient plasma blobs that are released from the helmet streamers or pseudo-streamers [53], [54], [47], [44]; 3) plasma released by reconnection between open and closed field lines at the coronal hole boundaries [37], [15], [35], , [11]; 4) hot outflows with speeds up to $\\sim 100~km~s^{-1}$ from the edges of active regions, first reported by [30] and most recently by [51]; and 5) jets originating from coronal bright points (BPs, small loop-like structures that are omni-present including coronal-holes and the quiet-Sun) , [38], [57], [41].",
"There is still no general consensus on whether all the above-mentioned processes can make a significant contribution to the slow solar wind and what is their relative importance in different phases of the solar activity cycle.",
"However, considering the wide range of suggested sources, it is not surprising that the slow solar wind exhibits such variability of its properties [46].",
"The majority of the above-described sources assume that the plasma is intrinsically trapped in closed magnetic field lines where it is released via magnetic reconnection.",
"Small plasma clouds and narrow jets are indeed detected on a daily basis in extreme ultraviolet (EUV) imaging and in spectroscopic [10], , [38] and in white-light coronagraph observations [53], [56], [8].",
"Furthermore, small-scale transients have been identified in-situ [39], [14], [25], [16], , , most of them embedded in the slow solar wind.",
"There is increasing evidence that the transient activity seen in different wavelengths in remote-sensing observations is linked.",
"For instance, an extensive statistical analysis by par10 confirmed the X-ray jets from BPs as the source of the narrow white-light ejections.",
"In addition, in a few cases coronal jets and plasma blobs have been traced to large distances from the Sun using wide angle white-light heliospheric imaging , [57], even up to their direct detection near the Earth [45].",
"However, the rate of small transients detected in-situ is significantly less (a few events per month) than the rate of jets and plasma blobs seen in remote-sensing observations (several tens per day).",
"This suggests that the majority of small transient plasma blobs emitted from the Sun lose their identity before reaching the orbit of the Earth .",
"One way to establish the linkage is to compare elemental abundances and charge state properties at the Sun and in the solar wind [18], .",
"The charge states of the heavy ions (e.g., C$^{+6}/$ C$^{+5}$ , C$^{+6}/$ C$^{+4}$ , and O$^{+7}/$ O$^{+6}$ ), freeze-in in the corona when the solar wind expansion time-scale exceeds the ionization/recombination time-scales and thus provide an indicator of the coronal electron density and temperature profile in the freeze-in region [61], [59].",
"[33] showed that the carbon charge states are the best indicators of the temperature of the corona and in particular the C$^{+6}/$ C$^{+4}$ ratio is sensitive to the solar wind type.",
"However, using ion charge states to probe the solar wind origin is not straightforward.",
"[34] emphasized that charge states evolve considerably over a solar cycle, and hence, although they might be good proxies of the solar wind sources even across a given year, they cannot be used as an absolute discriminator of the source.",
"In turn, the elemental ratios at the Sun depend in a complex way on the chromospheric temperatures, the magnetic field configuration where the plasma originates and the confinement time of the plasma in closed coronal loops.",
"The spectroscopic remote-sensing observations have shown that newly emerging active regions loops tend to have similar abundances of elements with low (e.g., Fe, Ne, Si, Mg) and high (e.g., Fe, Ne, Si, Mg) first ionization potentials (FIP), while the abundance of high FIP elements increases with increasing confinement times [12], .",
"“FIP bias” refers to the enhancement of elements with low first ionization potentials (FIP) (e.g., Fe, Ne, Si, Mg) to those with higher FIP (e.g., O or S).",
"The FIP fractionation has recently been suggested to be driven by the ponderomotive force of the magnetohydrodynamic waves in the chromosphere and low corona, see e.g., [32] and the references therein.",
"In the slow solar wind the abundances of elements with low FIP are typically larger by a factor of two to four compared to those with higher FIP [60].",
"At the Sun strong FIP biases have been reported in active regions [3].",
"In turn, for coronal streamers the results are more controversial and values vary from the core to the edge of the streamers [43], [42], [5], [50].",
"Hence, it is not straightforward to make the association based on elemental abundances due to large variability in those values even in the same coronal source.",
"Similar to heavy charge states, the solar wind composition exhibits a clear solar cycle trend, being considerably higher at solar maximum than during low solar activity periods [34].",
"In addition, specific entropy $\\ln {T_p/n_p^{\\gamma -1}}$ has been shown to differentiate between the solar wind from different source regions [40].",
"The fast solar wind has high temperatures, but low densities, and hence, generally high specific entropies.",
"As mentioned above, for the slow solar wind variations in plasma and magnetic field parameters are relatively large, and consequently, also the variations in the specific entropy.",
"The particularly low specific entropy structure in the slow solar wind is the high density and low temperature heliospheric plasma sheet (HPS).",
"In this paper we study the properties of the slow solar wind during the extended low solar activity period between Solar Cycles 23 and 24 (2006–2009).",
"We examine a comprehensive set of solar wind parameters, including FIP fractionation, heavy ion charge states and specific entropy.",
"The studied period of low solar activity is excellent for investigating the variations in the slow solar wind and its sources.",
"The contribution from large-scale CMEs was very small at this time and the global structure of the coronal magnetic field experienced drastic changes, featuring periods frequent with low-latitude coronal holes and active regions and an extended period of a very quiet Sun.",
"The paper is organized as follows: In Section 2 we present our data and analysis methods.",
"Section 3 gives the results of the statistical analysis.",
"In Section 4 we discuss and summarize our results."
],
[
"Data and methods",
"Our analysis combines 1-hour averaged solar wind plasma and magnetic field observations from the Near-Earth Heliospheric data base (OMNI; [29]) and the charge state and elemental compositional characteristics (Fe/O and C$^{+6}/$ C$^{+4}$ ) from the SWICS instrument of the ACE spacecraft.",
"ACE was launched in August 1997 and it operates close to the Lagrangian point L1.",
"OMNI is a combination of L1 and near-Earth measurements (during our study period Wind and ACE data has been used) and the data has been shifted in time to the magnetopause.",
"We obtained the OMNI data through the NASA Goddard Space Flight Center Coordinated Data Analysis Web (CDAWeb, http://cdaweb.gsfc.nasa.gov/) and the ACE data from the ACE Science Center (http://www.srl.caltech.edu/ACE/ASC/level2/).",
"The reason behind using solar wind plasma magnetic field data from OMNI instead from ACE is that after 2006 there are significant and persistent data gaps in the ACE solar wind density measurements.",
"We select solar wind C$^{+6}/$ C$^{+4}$ ratio to present the heavy ion charge states according to study by [33] (see Introduction) and Fe/O to present the variations in the FIP bias.",
"We also checked the analysis results using other charge states (C$^{+6}/$ C$^{+5}$ and O$^{+7}/$ O$^{+6}$ ), but no significant differences were obtained to those using the C$^{+6}/$ C$^{+4}$ ratio.",
"To obtain only solar wind periods unperturbed by CMEs we have removed the interplanetary CME-intervals by combing the events from a published list in [27] and from the online Richardson and Cane catalog (http://www.srl.caltech.edu/ACE/ASC/DATA/level3/icmetable2.htm).",
"As discussed in Section 1 during our study period (2006–2009) only a handful of interplanetary CMEs were identified near L1.",
"We used synoptic maps produced from full-disk images taken in the 195 Å channel of the Extreme-ultraviolet Imaging Telescope [9] to calculate the fractional active-region and coronal-hole areas for the CR 2039 starting on 2006 January 18 to CR 2091 ending on 2010 January 3.",
"The preparation of the synoptic maps from CR 2039 to CR 2055 is given in [6] (made by E. Benevolenskaya, hereafter EB) and from CR 2056 to CR 2092 in [21] (by N. Karna, NK).",
"To determine the contour levels that determine the AR and CH areas, we first calculated the background emission as the mean of each individual Carrington map for pixels with Digital Numbers (DN) below 1700 DN/s (the pixels above 1700 DN/s are associated with saturated signal rather than real signal most probably caused by cosmic rays not removed by the standard data reduction procedure).",
"The background emission is then obtained as the mean plus 1.1 times the standard deviation of the data for each Carrington map.",
"Similar approach was successfully used by [36] and [48].",
"For the contours of the AR areas we used a value of 1.3 times the background emission for the Carrington maps produced by EB and 1.5 times the background emission for the maps of NK.",
"The reason for the different thresholds is the slightly different resolution of the two datasets of maps.",
"All steps of the analysis were accompanied by a careful visual inspection of the goodness of the contour levels.",
"The coronal-hole contour levels are 0.1$\\times $ BG for the EB maps and 0.23$\\times $ BG for the NK maps.",
"Examples of the AR and CH contours are given in Figure REF .",
"First, we calculated from the CR EIT synoptic maps the relative fractional ares of the Active Region Sun (ARS) and the Coronal Hole Sun (CHS) using the definitions described above.",
"Since the photospheric footpoints of the field lines that map to the ecliptic can have a wide latitudinal range, we calculated these fractions using a latitude range of $\\pm 30^{\\circ }$ .",
"We also checked the results using various other latitudinal ranges up to $\\pm 60^{\\circ }$ .",
"This resulted in essentially similar solar cycle variations in the ARs and CHS fractions.",
"We also investigated in more detail the properties of the slow ($< 450~km~s^{-1}$ ) solar wind associated with different sources, i.e.",
"QS, ARS and CHS.",
"For this part of the study we estimated the in-situ periods that are affected by the QS, ARS, CHS by defining the boundaries of these periods from the synoptic maps, see an example in Figure REF .",
"The corresponding times in the near-Earth solar wind are found by assuming that the solar wind propagates radially at a constant speed.",
"We use the average slow solar wind speed for our whole dataset, 359 $km~s^{-1}$ (corresponding the Sun to Earth transit time of 166.7 hours).",
"For instance, the dotted lines that bound the left most active region in Figure REF correspond to the period of 2007 January 31, 21 UT – 2007 February 2, 19 UT.",
"The corresponding in-situ interval is 2007 February 5, 13 UT – 2007 February 7, 15 UT and is bounded in Figure REF by red lines.",
"We require that the ARS (bounded by the dotted lines), the CHS (bounded by the dash-dotted lines) and the QS periods (bounded by the solid lines) are separated from each other by at least half a day (i.e., $\\sim 7^{\\circ }$ in Carrington maps).",
"The total hours of different sources are distributed as follows: The QS periods cover 7070 hours (294.6 days), the AR periods – 1886 hours (53.6 days), and the CH periods 395 hours (16.5 days).",
"Figure REF also illustrates that solar wind properties may change considerably in the slow solar wind over relatively short time intervals (about three day interval shown).",
"Figure: A synoptic SOHO/EIT 195 Å map for Carrington Rotation 2052 (January 8, 2007 – February 4, 2007).",
"The dotted vertical lines bound the periods when active regions were present (Active Region Sun: ARS), the dash-dotted lines select the coronal holes periods (Coronal Hole Sun: CHS) and the dashed lines mark the quiet Sun (QS) periods.",
"We have required that these periods are separated by half a day (7 ∘ ^{\\circ }).Figure: The solar wind measurements during 2007 February 5, 13 UT – 2007 February 7, 15 UT, corresponding the active region interval in Figure .",
"The panels show from top to bottom: a) C +6 /^{+6}/C +4 ^{+4} ratio, b) Fe/O ratio, c) specific entropy, d) speed, e) density, f) temperature and g) Helium to proton ratio."
],
[
"Statistical results",
"The monthly sunspot number from the Solar Influences Data Center (http://sidc.oma.be) and the maximum latitudinal extend of the neutral line of the heliospheric current sheet (HCS) from the Wilcox Solar Observatory (http://wso.stanford.edu) are shown in Figure REF .",
"The top panel illustrates that solar activity was relatively low during our whole study period, in particular during 2008–2009 when for most of the time the visible disk of the Sun was completely “spotless”.",
"The bottom panel of Figure REF shows that the HCS neutral line tilt reached low values, i.e.",
"the Sun's magnetic field was dipole-like, only towards the end of 2009.",
"This implies that during most of the extended low activity period between Solar Cycles 23 and 24 the solar magnetic field had significant multipole components .",
"See also Figure 5 in [26], which shows the synoptic maps for the HCS neutral line for the selected Carringon Rotations for 2007–2010 demonstrating that between 2007 and mid 2008 the HCS neutral line experienced significant warps and extended to high latitudes.",
"The flat and low latitude configuration was reached only in mid 2009 and in late 2009 the HCS neutral line suddenly shifted to higher latitudes (seen also in Figure REF ) and become more warped.",
"These may also affect the ecliptic slow solar wind.",
"During the times when the HCS neutral line is flat and confined to low latitudes the ecliptic slow wind is largely influenced by the main streamer belt, while during the times when the HCS neutral line is warped and has high tilt, significant contributions from coronal holes and their boundary regions are expected.",
"Figure: The medians of selected solar wind parameters calculated over Carrington Rotations (27.2753 days) during our four year study period of 2006–2009 for the slow (<450< 450) solar wind.",
"The error bars indicate the range from the lower to upper quartile.",
"The green horizontal lines give the medians calculated using the whole investigated period.",
"The panels show from top to bottom a) the normalized occurrence of the `Active Regions Sun” (ARS; red diamonds), and “Coronal Hole Sun” (CHS; blue triangles), solar wind b) C +6 /^{+6}/C +4 ^{+4} ratio, c) Fe/O ratio, d) specific entropy, e) speed, f) density, g) temperature, h) Helium to proton ratio.The top panel of Figure REF shows the normalized (to the maximum value) CR fractions of the CHS and ARS, see Section 2 for the definitions.",
"The variations in the number of active regions should follow the sunspot number and indeed the fraction of ARS was highest from the beginning of 2006 to early 2008, while the active regions were practically absent during the deepest solar minimum in 2008–2009.",
"During the first half of our study period the identified active regions were primarily low-latitude active region related to the “old” cycle 23, while a few active regions detected in the end of 2009 mark the emergence of the mid-latitude active regions pertaining to the “new” cycle 24.",
"The fraction of CHS was highest from the beginning of 2007 to mid 2008.",
"The decrease in the fraction of low-latitude CHS is consistent with the Sun's magnetic field becoming increasingly dipolar by mid-2009 and having increasing effect from the main streamer belt (see discussion above).",
"The medians of the selected solar wind parameters calculated over each investigated Carrington Rotation (i.e.",
"27.2753 days) are displayed in panels b) - h) of Figure REF .",
"As mentioned above, we have considered only those periods when the solar wind speed was $<450~km~s^{-1}$ , and in addition, we have removed interplanetary CME intervals (see Section 2).",
"The error bars show the ranges from the lower to the upper quartile, i.e.",
"the interquartile range (IQR) describing the spread of the values.",
"Figure: Distributions of solar wind parameters for the periods of a) “Quiet Sun” (thick black line), b) “Active Region Sun” (thick red line), and c) “Coronal Hole Sun” (thick blue line).",
"See Section 2 for the details.",
"The thin curves in each panel show the other two distributions to help the comparison.",
"The solid vertical lines show the medians and the dashed vertical lines mark the lower and upper quartiles.The majority of the investigated solar wind parameters show quite modest and random variations over our four year study period, i.e.",
"there is no clear correspondence with the variations of the CHS and ARS fractions.",
"The solar wind from the mid-2008 to mid-2009 when both active regions and coronal holes were absent is the slowest wind and has the lowest He/H ratio.",
"The year 2006 when the active regions were the most abundant is characterized by the largest C$^{+6}/$ C$^{+4}$ ratio and the highest solar wind densities.",
"The He/H ratio was also relatively high.",
"The period from mid-2007 to mid-2008 had relatively large contribution from low-latitude coronal holes.",
"Note that the complete synoptic maps are mostly missing for the first half of 2008, but the inspection of the few existing EIT images reveals a significant presence of low latitude coronal holes during this period of time.",
"The same signature is also visible from Figure 2 of [1] (which ends at mid-2008).",
"This time period is featured by the lowest Fe/O ratio, the fastest solar wind with the lowest densities and the highest temperature, and the highest He/H ratio.",
"Next, we compare the solar wind properties between the periods when there were prominent low/mid-latitude coronal holes and active regions to periods of predominantly quiet Sun (see Section 2 for the description of our approach).",
"The results are shown in Figure REF .",
"In the left panel of Figure REF the black thick lines show the distributions for the QS related slow solar wind periods.",
"The distributions for the ARS and CHS related slow solar wind periods are shown with light red and light blue lines, respectively, to help the comparison between the different sources.",
"Similar logic is used in the middle and right panels, but now the thick lines denote the ARS (red) and CHS (blue) related solar wind, respectively.",
"The solid vertical lines indicate the medians and the dashed vertical lines the lower and upper quartiles of the given distributions.",
"The medians and the lower and upper quartile ranges (i.e.",
"the difference between the median and lower quartile and the upper quartile and median, respectively) are given in Table 1.",
"Figure REF a and Table 1 show that the QS wind is clearly the slowest, while the fastest speeds are found for the CHS wind.",
"Note also that the speed distribution for the CHS wind has relatively small upper quartile range, indicating that the CHS wind has a strong bias towards our slow solar wind upper threshold of 450 $km~s^{-1}$ .",
"The lower quartile for the CHS wind is $350~km~s^{-1}$ (i.e.",
"only 25% of the data had speeds below this value), which is very close to the upper quartile for the QS wind, 370$~km~s^{-1}$ .",
"The shape of the speed distribution and the median for the ARS wind are more similar to the QS wind than to the CHS wind.",
"However, ARS wind speed distribution is flatter in contrast to QS and CHS wind distributions.",
"The CHS and QS wind have larger mean densities than the ARS wind (Figure REF b).",
"This is mainly due to the lack of the highest densities for the ARS related solar wind.",
"The shapes of the density distributions are rather similar for all source types with strong clustering towards the lowest densities and an extended tail towards larger values, see also Table 1.",
"The temperature distributions (panel REF c) exhibit rather similar profiles as the density distributions.",
"The CHS wind has also considerable higher temperatures than the QS and ARS related slow solar wind periods.",
"The lowest temperatures are found for the QS solar wind.",
"The next three panels (REF d–REF f) examine the solar wind charge state and compositional characteristics.",
"The He/H distributions and their medians differ quite considerably between different solar wind types.",
"The ARS wind has considerably larger He/H median than the QS and CHS wind and its distribution features a long and extended tail.",
"The QS wind shows a strong clustering of the He/H values towards the lowest values.",
"For the QS and CHS wind the upper quartiles are 0.023 and 0.022, respectively, close to the median for the ARS wind, 0.020.",
"The Fe/O ratio distributions are remarkably similar for all three source types and the medians and the lower and upper quartiles are also nearly identical.",
"In turn, the C$^{+6}/$ C$^{+4}$ ratio shows clear differences.",
"This ratio is highest for the ARS wind and lowest for the QS wind.",
"Unlike the Fe/O distribution which shows Gaussian-like shape, the C$^{+6}/$ C$^{+4}$ distributions tend to peak at low values and have an extended tail.",
"Note that while the QS wind lacks high C$^{+6}/$ C$^{+4}$ values, the CHS wind has nearly identical fraction of highest values as the ARS wind.",
"The last panel of Figure REF gives the specific entropy.",
"This parameter shows the largest median for the CHS wind and the lowest median for the QS wind.",
"However, the differences in the medians are not very large, but the distributions have quite different shapes.",
"The distribution peaks at high entropy values for the CHS wind while for the QS wind the distribution shape is more Gaussian-like.",
"Figure: Small transient contributions to the slow solar wind.",
"The panels show: Top) the number of small transients identified in the slow solar wind for each Carrington rotation investigated (green line shows the smoothed number), Middle) the normalized occurrence of the `Active Regions Sun” (ARS; red diamonds), and “Coronal Hole Sun” (CHS; blue triangles), i.e, same as in Figure 4, Bottom) the fraction of slow solar wind for each investigated Carrington rotation.The top panel of Figure REF shows the rate of “small solar wind transients” in the slow solar wind from the statistics published in [58].",
"The green line is the smoothed number.",
"Small transients are “ICME-like” coherent structures (often having a flux rope geometry) in the solar wind that have durations significantly less than large-scale CMEs ($< 12$ hours).",
"As discussed in the Introduction, it has been suggested that the small transients would form a significant part of the slow solar wind.",
"The rate of small transients was the lowest in 2006, i.e.",
"during the year with the highest active region contribution.",
"The rate increased during the first half of 2007, i.e., when the CHS contribution increased.",
"For the deepest solar minimum in 2008–2009 the number of small transients increased quite steadily until around September 2009 when the tilt of the neutral line of the HCS suddenly increased (see Figure REF ).",
"Part of the variations in the number of small transients is presumably related to the variations in the fraction of slow solar (middle panel of Figure REF ), which was only about 30-40% during the first half of 2008 and reached nearly 100% in 2009.",
"However, the observed behaviour could also reflect the association of small transients with the ejections of small plasma blobs from streamers and/or from coronal holes boundaries.",
"The last column of Table 1 gives the medians and the upper and lower quartial ranges for the same set of solar wind parameters that were calculated for the QS, ARS and CHS wind averaged over the small transients.",
"The median C$^{+6}/$ C$^{+4}$ values of small transients is similar to that of the QS wind, while speeds are more similar to ARS wind."
],
[
"Discussion and Conclusions",
"In this paper we have investigated the statistical properties of the slow ($< 450~km~s^{-1}$ ) solar wind associated with different solar sources.",
"We studied both the Carrington Rotation medians and the distribution of various solar wind parameters connected to QS, ARS and CHS periods during the four years of low solar activity between 2006 and 2009.",
"While some of the investigated solar wind parameters showed clear differences depending on the dominant source, others had very little effect.",
"For instance, the Carrington rotation medians for the solar wind FIP bias (here investigated using the Fe/O ratio) showed very little change over our four year study period.",
"In addition, the distributions and medians of Fe/O related to solar wind periods from different solar sources were almost identical.",
"These findings are in agreement with lep13 who also reported relatively little variation of the Fe/O ratio over the solar cycle.",
"Our present study further demonstrates that the Fe/O ratio does not vary significantly with the solar wind source, at least on average.",
"One of the clearest temporal trends were observed for the solar wind charge state ratio C$^{+6}/$ C$^{+4}$ .",
"This ratio peaked when the active regions were present and the C$^{+6}/$ C$^{+4}$ median was also clearly higher for the ARS wind than for the QS and CHS winds.",
"These findings are consistent with [17] who studied annual variations in the solar wind speed and the O$^{+7}/$ O$^{+6}$ ratio of the slow solar wind of different origin (QS, ARS and CHS) using a ballistic two-step mapping procedure.",
"As explained in Section 2, we also checked the results using the O$^{+7}/$ O$^{+6}$ ratio and found them to be very similar to those obtained using the C$^{+6}/$ C$^{+4}$ ratio.",
"However, [17] found very little difference in the annual averages of O$^{+7}/$ O$^{+6}$ for the QS and CHS slow solar wind during solar minimum years 2007–2008 (see their Figure 7), while we found that the QS wind had clearly the lowest C$^{+6}/$ C$^{+4}$ values.",
"Overall, our results are also consistent with lep13 who found a decline in the C$^{+6}$ /C$^{+4}$ ratio with decreasing solar activity.",
"Our study suggests that this decline could be related to the decrease in the active region contribution to the slow solar wind.",
"Both our study and [17] reported that the fastest slow solar wind comes from the CHS regions.",
"This is expected as the CHS slow wind includes the transition from the slow wind towards the fast wind.",
"In contrast, [17] found very similar speeds for the QS and ARS wind, while our study shows that the QS wind was clearly the slowest.",
"The slowest speeds during our study period occurred during the deepest minimum in 2009 when the neutral line of the HCS was flat and had low tilt suggesting a large contribution to the ecliptic slow solar wind from the streamer belt region.",
"This unusually slow solar wind period has been reported in several other works as well [22], [49], [28].",
"We also found that the rate of small solar wind transients peaked at this time.",
"The high density tail (and the tendency for low entropies) for the CHS wind may be due to the contribution of the heliospheric plasma sheet (HPS) [4].",
"The HPS corresponds to the stalks of the helmet streamers [55] and this high-density structure regularly precedes slow-fast stream transition regions [23].",
"Some previous studies suggests a negative correlation between the solar wind speed and the He/H ratio, which arises from the changes in the magnetic field expansion factor and the efficiency of the Coulomb drag [2], [19], [24].",
"Such correlation would imply that the CHS related slow wind had the largest He/H values.",
"In our study the Carrington rotation medians of the He/H ratio indeed peaked during the first half of 2008 when the fastest solar wind occurred.",
"However, the distributions showed a clear tendency for the ARS related wind to have highest He/H values.",
"As discussed in lep13 the correlation between the solar wind speed and He/H values is not at all clear.",
"Low He/H values are in particular reported near solar wind sector boundaries [7].",
"Hence, low He/H values for the CHS and QS wind in our study could be related to the crossings of the HCS.",
"Consistent with [17], our study suggests that the quiet Sun has a major contribution to the slow solar wind throughout the solar minimum.",
"This is an expected result as at this time active regions as well as low-latitude coronal holes and/or near-equatorial coronal hole extensions were absent.",
"The origin of the slow solar wind in the QS has been studied by [13] using elemental abundances and freeze-in temperatures as tracers to locate the source regions.",
"The authors speculated that closed loops with lifetime of 1–2 days can possibly be the contributor to the slow solar wind streams.",
"Their observational analysis identified only larger loops (up to 150 000 $km$ long) as a source region.",
"[20] put forward the idea that the slow solar wind is initiated at the corona formation temperature of Fe xii at 1.6$\\times $ 10$^6$ K. The slow solar wind plasma can be produced from magnetic reconnection between the magnetic loops and open field lines of funnels rooted in the network.",
"The plasma released from the closed loops during the reconnection process will supply to the initial outflow of the slow solar wind.",
"As concluded by [20] “there is still a long way to a full understanding of the ultimate origin of solar wind in the CH and in the QS\".",
"Our results highlight the difficulty in distinguishing between the slow solar wind of different origin based on the inspection of the solar wind conditions.",
"However, some clear tendencies could be drawn.",
"The QS slow solar wind stands out as slowest and coldest with the lowest values of specific entropy, C$^{+6}$ /C$^{+4}$ and He/H.",
"In contrast, the solar wind associated with the ARS was the most tenuous with the largest He/H and C$^{+6}$ /C$^{+4}$ ratio.",
"The fastest slow solar wind with the highest temperatures and entropies was related to CHS.",
"According to our study the FIP bias (Fe/O) is not different, at least in the average sense, for different solar wind sources, and hence, cannot be used a reliable discriminator for the solar wind source.",
"However, we would like to emphasize that our results are statistical and the slow solar wind often exhibits large variations over relatively short periods of time.",
"To establish a more detailed linkage would require a more dedicated connection of the solar wind periods to their solar sources using advanced methods such as the non-linear force free field (NLFFF) reconstructions.",
"Academy of Finland project 1218152 is thanked for financial support.",
"MM and EK acknowledge with gratitude the Royal Society international exchange grant for the project “Coupling transient activity from the Sun to the Heliosphere”.",
"MM is supported by the Leverhulme Trust.",
"The OMNI data were obtained through the NSSDC CDAWEB online facility.",
"We acknowledge ACE Science Center for providing SWICS measurements.",
"SOHO is a project of international cooperation between ESA and NASA."
]
] | 1606.05142 |
[
[
"Molecular dynamics simulation of a binary mixture near the lower\n critical point"
],
[
"Abstract 2,6-lutidine molecules mix with water at high and low temperatures but in a wide intermediate temperature range a 2,6-lutidine/water mixture exhibits a miscibility gap.",
"We constructed and validated an atomistic model for 2,6-lutidine and performed molecular dynamics simulations of 2,6-lutidine/water mixture at different temperatures.",
"We determined the part of demixing curve with the lower critical point.",
"The lower critical point extracted from our data is located close to the experimental one.",
"The estimates for critical exponents obtained from our simulations are in a good agreement with the values corresponding to the $3D$ Ising universality class."
],
[
"Introduction",
"It is well recognised that solutions of water with organic molecules may show closed-looped phase diagrams with a miscibility gap.",
"The occurrence of such phase diagrams with more than one critical point is very different from what is observed in mixtures of simple fluids.",
"In simple fluids, the two fluids form a homogeneous mixed phase at higher temperatures, whereas they phase separate at lower temperatures.",
"The appearance of an upper critical point (UCP) terminating the two-phase coexistence results from a competition between the entropy of mixing and the energy.",
"In the case of mixtures of complex species, such as water and some organic molecules, the mechanism behind the occurrence of the miscibility gap is rather involved.",
"As argued in Refs.",
"[1], [2], the existence of a lower critical point (LCP) can be due to formation of directional bonds between water and the organic molecules, e.g., hydrogen bonding.",
"Below the LCP, the strong hydrogen bonding promotes mixing at the expense of rotational degrees of freedom of molecules, which are effectively \"frozen out\" by the bonding.",
"At higher temperatures, thermal fluctuations “unfreezes” these rotational degrees of freedom so that the hydrogen bonding gets destroyed and the mixed phase separates.",
"Theoretical studies of simple lattice models and within the Landau-Ginzburg approach have demonstrated that the miscibility gap in liquid mixtures may indeed emerge due to angular dependent attractive interactions on top of the spherically symmetric ones [3], [4], [5], [2], [6].",
"Experiments show that the shape of closed-loop phase diagrams of aqueous solutions of organic molecules near the LCP is very flat, i.e., concentrations of the species in the two coexisting phases near LCP vary strongly with temperature.",
"Such behavior indicates that in both coexisting liquid phases some local structures are formed which are determined by hydrogen bonding between water and the organic molecules.",
"The features of the miscibility gap, in particular its sensitivity to changes of the intermolecular interactions, has been studied theoretically [7], by computer simulation (for tetrahydrofuran-water mixtures) [9], [10], and experimentally [7], [8] by several techniques, e.g.",
"by deuteration of water, addition of electrolytic impurities or hydrotrops.",
"As follows from these studies, the details of the interactions affect the critical temperature as well as the detailed shape of the phase diagram.",
"The aqueous solution of 2,6-lutidine (2,6-dimethylpyridine) is a binary liquid mixture, which has gained considerable attention in context of its wetting behaviour at silica walls [11], porous glass [12], and colloids [13], [14].",
"Of particular interest has been the effect of temperature changes on the reversible aggregation of colloidal particles dispersed in a 2,6-lutidine/water mixture [15], [16], [17].",
"More recently, 2,6-lutidine/water mixtures were used to determine critical Casimir interactions in colloidal systems [18], [19], [20], [21].",
"The reason for the popularity of 2,6-lutidine/water mixture in these studies is that it possesses a closed loop phase diagram with a relatively wide temperature miscibility gap, i.e., the difference between the upper and lower critical point is large ($\\approx $ 197$^\\circ \\textrm {C}$ ).",
"Further, the LCP is conveniently located near room temperature.",
"The phase diagram as well as static and dynamic critical properties of 2,6-lutidine/water were studied intensively experimentally [22], [23], [24], [25], [26].",
"The LCP has been reported to occur at $T_c\\approx 307.1~$ K and at the lutidine mole fraction $x_{lut}\\approx 6.1\\% $ [22], [24], [25], [27].",
"The location of the LCP is, however, strongly affected by impurities and, moreover, it is sensitive to the method of determination.",
"This is why the values of the lower critical temperature and the critical mass, volume or mole fraction vary in the literature [28], [29], [30], [26], [31].",
"Such uncertainty is especially troublesome for application of a 2,6-lutidine/water liquid mixture as solvent in colloidal systems tuned by critical Casimir interactions, where the precise knowledge of the deviation in temperature and concentration from the critical values is required.",
"Computer simulations of the 2,6-lutidine/water mixture are thus highly desirable.",
"Moreover, such simulations can help to better understand the molecular mechanisms behind the lower critical point, and are a necessary prerequisite for studies of more complicated phenomena such as, for example, formation of of mesostructures in a mixture of water/organic solvent by adding an antagonistic salt, which is composed of hydrophilic cation and hydrophobic anion.",
"Such mesostructures were observed recently in SANS experiments in a mixture of water/3-methylpyridine/NaBPh$_4$ near the lower critical point [32] and off-critical point [33], [34].",
"A similar observation is reported for 2,6-lutidine/water mixture [35].",
"These phenomena are only partially explained theoretically [37], [36].",
"Another interesting topic, which can be an extension of the present work is critical adsorption of 2,6-lutidine/water mixture containing salt (inorganic one) at a charged and selective wall.",
"This phenomenon is of crucial relevance to recent experiments on critical Casimir interactions [18], [20], [38].",
"The findings of these experiments were not yet clarified in a satisfactory manner and there is some controversy about their origin.",
"Some analytical studies of this problem are available in the literature [40], [41], [39].",
"However, these results were obtained within a mesoscopic model and by using various approximations and thus they need to be verified by means of microscopic simulations.",
"In the present paper we propose an atomistic description of the 2,6-lutidine molecule and apply it to study the bulk 2,6-lutidine liquid as well as the 2,6-lutidine/water mixture near the LCP by molecular dynamics simulations.",
"The goal is to check whether our model of the 2,6-lutidine molecule is able to capture the main features of the bulk fluid as well as those of the aqueous solution.",
"All simulations were performed by the Gromacs/4.6.7 package [42].",
"The Gromos54a7 force field [43] was applied for Lennard-Jones (LJ) pair potential parameters, bond lengths and bonded parameters for angles, and dihedrals.",
"The Particle Mesh Ewald (PME) approach [44] was applied for electrostatic interactions, while a cut-off length $r_c=1.2$ nm was applied to the LJ interactions.",
"Simulations in this work were either performed in the NpT ensemble (constant pressure) or NVT ensemble (constant volume); the type of ensemble is mentioned in the text in each case.",
"In the NpT ensemble, the temperature and the pressure were controlled by a V-rescale thermostat [45] and a Parrinello-Rahman barostat [46] (isotropically to $p=1$ atm), respectively.",
"For the NVT ensemble, the temperature was controlled by V-rescale and no pressure coupling was applied.",
"All bond lengths were constrained with the LINCS algorithm [47].",
"For water the TIP4P/2005 model was used [48].",
"The simulation outcomes were analyzed through our own programs, Gromacs and VMD plugins [49], [50].",
"Figure: The 2,6-lutidine molecule with charges from the final parametrisation used in the current work.",
"The molecule is modelled by 11 atoms where CH 3 _3-groups are considered as united atoms."
],
[
"Parametrisation of the 2,6-lutidine molecule",
"We represent the 2,6-lutidine molecule, C$_7$ H$_9$ N, by 11 atoms wit the two CH$_3$ groups treated as single united atoms as shown in Fig.",
"REF , while the hydrogen atoms that are attached to the ring carbons are explicitly included.",
"The GROMOS force field does not provide partial charges on 2,6-lutidine molecule, therefore, we obtained an initial estimation of these charges from quantum chemistry simulations.",
"Then, we varied the values of these charges until two goals were achieved: (i) an agreement with the experimental results for the density $\\rho $ and the heat of vaporization $ \\Delta H_{vap}$ for liquid lutidine at temperatures around room temperature, and (ii) the existence of the LCP for the mixture of 2,6-lutidine and water in the experimental range (details of 2,6-lutidine/water simulation will be given in Subsec.",
"REF ).",
"More precisely, we gradually scaled the partial charges of 2,6-lutidine molecule by a factor in order to change its Coulomb interaction with water until we obtained the LCP temperature close to experimental value.",
"From experiments, we know that we should have a mixture at $T = 280$ K and a phase-separated system at $T = 320$ K. Therefore, for each rescaled charge distribution, simulations were carried out at these two temperatures.",
"The appearance of mixed and phase-separated phases at these two temperatures ensures that the LCP is located somewhere in between.",
"During the rescaling, we also took advantage of the fact that the molecule is symmetric and that the total charge is zero.",
"Moreover, we kept the charges on the hydrogen atoms and CH$_3$ -groups consistent with the Gromos force field and just vary the remaining 3 parameters."
],
[
"Validation of the lutidine model",
"Simulations were performed for $80~$ ns using a $2~$ s time step in the NpT ensemble for the bulk system and in the NVT ensemble for the surface tension calculation.",
"Results collected after an equilibration of 10 ns, are presented together with the corresponding experimental data taken from Refs.",
"[51], [53], [52], [55], [54] in Table.",
"REF .",
"The simulations provide data for the density $ \\rho $ , the static dielectric constant $\\epsilon $ and the heat of vaporization $\\Delta H_{vap}$ in a fair agreement with the experimental ones.",
"The heat of vaporization was calculated as $\\Delta H_{vap} = \\langle U_{gas}\\rangle - \\langle U_{liquid}\\rangle +RT,$ where $\\langle \\cdot \\rangle $ denotes time average.",
"$U_{gas}$ and $U_{liquid}$ are potential energies of lutidine in the gas and liquid phases.",
"The gas phase is considered as ideal so $U_{gas}$ contains only interactions within the molecules.",
"The surface tension $\\gamma $ was obtained from simulations of the liquid with two flat surfaces.",
"This system was created by increasing the periodic box size of the equilibrated bulk system by one order of magnitude in one direction.",
"The usage of a 1.2 nm cut-off for the LJ interactions has resulted in a too small surface tension.",
"Therefore, we performed simulations of this system using PME treatment of LJ interactions (LJ-PME).",
"This gave the surface tension of $32.5 \\pm 0.2~$ mN/m in a better agreement with experiment.",
"With the fractional charges used in the present model, the dipole moment of lutidine is $2.5~$ D. The experimental gas phase value is $1.7~$ D (no experimental value is available for liquid).",
"Similar differences are encountered for most water models; typically they predict about 30% larger dipole moment than possessed by water in a gas phase, although the difference here is slightly bigger.",
"These differences are caused by the mutual polarization of molecules in the liquid phase.",
"The good agreement between the experimental dielectric constant and the one calculated from fluctuations in the total dipole moment of the entire simulation box is reassuring and indicates that the electrostatic properties of the 2,6 lutidine fluids are properly modelled.",
"Finally, the heat capacities at constant pressure and constant volume, $C_p$ and $C_V$ , were calculated from the enthalpy and energy fluctuations in the simulations as $C_p=\\frac{\\sigma ^2_H}{k_BT^2}\\;\\;\\;\\;\\mbox{and}\\;\\;\\;\\;C_V=\\frac{\\sigma ^2_E}{k_BT^2}.$ The calculated value for $C_p$ given in the last row of the Table.",
"REF shows a substantial difference compared to the experimental value.",
"The reason for this is that the classical treatment of the lutidine molecules allows too much energy to be taken up by degrees of freedom that in reality are quantum mechanical.",
"A quantum-correction could, however, be added to the classical heat capacity assuming that the relevant degrees of freedom could be approximated as coupled harmonic oscillators.",
"To do this we follow Refs.",
"[57]-[58] and determine the normal modes of the system from the velocity auto-correlation functions.",
"Table: Simulation results and available experimental values for various physical quantities characterizing 2,6-lutidine.The normal mode distribution $S(\\nu )$ was obtained from the Fourier transform of the velocity correlation functions and the quantum correction to the heat capacity $C_V^{QM.corr}$ was obtained as the difference between the heat capacity of a quantum oscillator and a classical oscillator ($k_B$ ) integrated over the normal mode distribution [57] $C_{V}^{QM.corr}=k_B \\int _0^\\infty d\\nu S(\\nu ) W_{c_V}(\\nu );\\hspace{28.45274pt}W_{c_V}(\\nu )=\\Bigg ( \\frac{u^2 e^u}{(1-e^u)^2}-1 \\Bigg ),$ with $u\\equiv h \\nu /k_BT$ being the energy in thermal units.",
"The quantum corrections to the heat capacities at three temperatures were obtained according to Eq.",
"REF by using the normal mode distributions obtained from simulation of $N=1104$ molecules of 2,6-lutidine at the different temperatures.",
"The simulations were done in the NVT ensemble for $5~$ ns and velocities were stored every $5~$ fs.",
"Although the quantum corrections were calculated at constant volume, they can still be applied to the constant pressure heat capacities, assuming that the difference between quantum corrections in the two ensembles is negligible.",
"The classical heat capacities $C_{V,p}^{class}$ was calculated from the fluctuations in energy and enthalpy calculated from the last $12~$ ns of $20~$ ns-long simulations at constant volume or constant pressure, respectively.",
"There are a few additional contributions to the heat capacity due to quantum mechanical vibrations in the bond lengths which were treated rigidly in the simulations, and due to the quantum mechanical motion of the absent hydrogen atoms in the CH$_3$ groups.",
"These contributions are negligible.",
"The corrected heat capacity $C_{V,p}^{corr}$ was obtained as the sum of the classical value and the quantum correction $C_{V,p}^{corr}=C_V^{QM.corr}+C_{V,p}^{class}.$ In the last two columns of Table.",
"REF , the obtained quantum corrected heat capacities $C^{corr}_p$ and the experimental ones can be compared.",
"Table: Simulation data for the classical and the quantum corrected heat capacities in J mol -1 ^{-1} K -1 ^{-1}.",
"The corrected heat capacities C V corr C^{corr}_V and C p corr C^{corr}_p are given and C p corr C^{corr}_p can be compared to the available experimental values given in the last column.For most fluids (as well as other condensed matter systems) $(C_p-C_V)/C_V$ is much smaller than 1.",
"For liquid water for instance, this quotient is about 0.01.",
"It is therefore a bit surprising that this quotient is as large as about 0.5 for the present system.",
"As a consistency check, we use the exact thermodynamic relation $C_{p}-C_V= V T \\frac{\\alpha _p^2}{\\kappa _T}$ to calculate $C_p-C_V $ for lutidine from the molar volume, coefficient of thermal expansion $\\alpha _p= \\frac{1}{V} \\frac{\\partial V}{\\partial T }|_p$ and the isothermal compressibility $\\kappa _T=-\\frac{1}{V} \\frac{\\partial V}{\\partial p }|_T$ .",
"The thermal expansion coefficient and the isothermal compressibility were obtained from the fluctuations in the NpT simulations.",
"The isothermal compressibility was obtained from the volume fluctuations, while the thermal expansion coefficient was obtained from the cross correlations between volume and enthalpy fluctuations.",
"The appropriate equations are discussed in Ref.",
"[59] and are $\\kappa _T=\\frac{1}{Vk_BT} \\langle (V-\\langle V\\rangle )^2\\rangle \\;\\;\\;\\mbox{and}\\;\\;\\;\\alpha _p=\\frac{1}{Vk_BT^2} \\langle (V-\\langle V\\rangle )(H-\\langle H\\rangle )\\rangle .$ The data from the simulations shown in Table.",
"REF are consistent with the data in Table.",
"REF .",
"For comparison, the experimental data for liquid water (taken from [60]) are also shown in the table.",
"The table shows that the main reason for the big difference between the two heat capacities for lutidine is the large thermal expansion coefficient (which is squared in the Eq.",
"REF ).",
"It is worth mentioning that the similar big difference between $C_p$ and $C_V$ occurs in liquid benzene [61] and is reported experimentally in Ref. [62].",
"Experimental data for benzene are also given in the table.",
"Table: A comparison between VTα p 2 κ T V T \\frac{\\alpha _p^2}{\\kappa _T} and C p -C V C_p-C_V in J mol -1 ^{-1} K -1 ^{-1}, for lutidine (simulations), water and benzene (experiments).Figure: Snapshots of the initial configuration of the 2,6-lutidine/water simulations (top), simulation result at T=280T=280 K (center) and T=380T=380~K (bottom).",
"The orange and blue colors indicate the 2,6-lutidine and water molecules, respectively."
],
[
"Simulations of the 2,6-lutidine/water mixture",
"In this section we present the simulation results for the 2,6-lutidine/water mixture.",
"In order to simulate a 2,6-lutidine/water mixture, we took as initial configuration a box of the size ($L, L, 7L$ ), with $L\\approx 3.8 ~$ nm, containing $N_{lutidine}=2050$ equilibrated bulk lutidine molecules, placed it at the center of the periodic box ($L, L, 7L$ ) with $L\\approx 5.8 ~$ nm and filled (solvated) with $N_{water}=31325$ equilibrated water molecules described by TIP4P/2005 model.",
"This configuration corresponds to a lutidine mole fraction $x_{lut}= 6.14 \\% $ , which is close to the experimental value at the LCP.",
"This initial configuration, which is neither a mixed phase nor a two-phases mixture, is shown in the top panel of Fig.",
"REF ; this configuration we used for all studied temperatures here.",
"The non-cubic shape of the box with one side much longer than the other two sides has been chosen for two reasons.",
"Firstly, we want ”planar interfaces” separating lutidine-rich and lutidine-poor phases for temperatures as close to the critical temperature $T_c$ as possible.",
"As discussed in details in Refs.",
"[63], [64], [65], the larger the size ratio of the rectangular box, the closer one can approach $T_c$ keeping the slab structure of the lutidine-rich phase and, hence, the planar interface.",
"Figure: (top) left: Time evolution of the lutidine mass density at two different coordinates zz of the simulation box corresponding to two phases, right: Time evolution of the lutidine mass density versus zz coordinate of the box at T=380KT=380 K .",
"(center) the same as in the top panel but at T=320KT=320 K. (bottom) left: Time evolution of the lutidine mass density at several different coordinates zz, right: Time evolution of the lutidine mass density versus zz coordinate of the box at T=280KT=280 K. Mass densities are in unit Da/Å 3 ^3.If the size of the box in the $z$ direction is not big enough, the slab structure is not stable close to $T_c$ .",
"Rather, the lutidine-rich phase forms a cylinder or a sphere.",
"Within the simulation box that we have chosen, all volume fractions at all considered temperatures produce a planar interface.",
"Secondly, we want the finite-size effects to be small in order to be able to determine the near-critical properties of a system as accurately as possible [66], [67], [68], [69].",
"This is achieved by choosing the size of the box to be larger than the bulk correlation length for all studied temperatures.",
"Moreover, due to the enlargement of the periodic box in the $z$ -direction the two interfaces in the slab structure do not influence each other.",
"We simulated the 2,6-lutidine/water mixture in the NpT ensemble for various temperatures; the simulation setting is given in Sec.",
"REF .",
"The time evolution of the local mass density of 2,6-lutidine for several temperatures are presented in Fig.",
"REF .",
"The plots show clearly a phase separated system at $T=320$ and $380~ K$ and a homogeneous mixture at $T=280 K$ .",
"The actual equilibration time is determined with the time at which not only the density profiles remain the same within the statistical errors, but also the energies and all hydrogen bonds become stable.",
"Fig.",
"REF presents the density profiles averaged over different time intervals after the equilibration, for two temperatures.",
"Depending on temperature, stable equilibrium structures were reached after $0.7-3.7~\\mu s$ .",
"The simulations took several months, with average run of $50 ~(ns/day)$ for each temperature.",
"Figure: The lutidine mass fraction versus the zz coordinate of the simulation box for temperatures T=380T= 380 and 320 K. The plots show the initial configuration and three time intervals after the equilibration.The results of the simulation are reported in the next section."
],
[
"Phase behaviours of the 2,6-lutidine/water mixture near the lower critical point",
"Fig.",
"REF shows the snapshots of the initial configuration for all simulations (top) and simulation results for temperatures $T=280$ K (center) and $T=380~$ K (bottom).",
"The snapshots show that the two fluids mix at $T=280~$ K, while they phase separate at the higher temperature $T=380~$ K. In order to assure that the phase separation is not effected by the initial configuration, the simulations at these two temperatures were redone with different initial configurations.",
"Although we used a mixed phase as the initial configuration for the higher temperature $T=380~$ K and a phase-separated initial configuration for the lower temperature $T=280~$ K, the same final configurations as given in Fig.",
"REF (bottom and center respectively) were reproduced.",
"Figure: Mass fraction (left axis) and mole fraction (right axis) of 2,6-lutidine obtained from simulations (symbols) fitted to the analytical expression Eq.",
"(dashed lines).",
"The resulting fitting parameters are given in Table.",
".To quantify the phase separation, the mass fraction of 2,6-lutidine $w_{lut}(z)$ has been calculated as a function of $z$ -coordinate from simulations at different temperatures, see Fig.",
"REF .",
"'Classical` theories for the interface separating two coexisting phases such Cahn and Hilliard [70] or Landau-Ginzburg theory, predict a hyperbolic-tangent shaped interfacial density profiles.",
"This has later been verified in simulations of interfaces [76], [66], [71], [72], [73], [74], [75], [77], [78].",
"Since there are two interfaces in the present system, we fit the density profile to the function $w_{lut}(z)= w_{lut}^p+\\frac{w_{lut}^r-w_{lut}^p}{2} \\bigg [ \\tanh \\bigg (\\frac{z-z_0+c}{\\lambda }\\bigg )- \\tanh \\bigg (\\frac{z-z_0-c}{\\lambda } \\bigg ) \\bigg ],$ with $w_{lut}^r$ and $w_{lut}^p$ being the mass fractions of 2,6-lutidine in the lutidine-rich and lutidine-poor phases.",
"$\\lambda $ is a measure of the width of the interface (softness of the transition between the two regions) and is proportional to the correlation length $\\xi $ , which is defined from the decay of the density-density correlation function.",
"$c $ is half width of the lutidine-rich region and $z_0$ is center of the lutidine-rich phase.",
"Fits Eq.",
"REF to the profiles are shown in Fig.",
"REF as dashed lines.",
"The parameters obtained from the fits are given in Table.",
"REF while the coexistence curve obtained from the data in the table is shown in Fig.",
"REF .",
"In order to compare with experiments [22], [24], [25] one may need to convert between mass fractions and mole fractions.",
"The appropriate equations for this are $x_{i}=\\frac{w_{i}}{w_{i}+(1-w_{i})\\frac{M_{i}}{M_{j}}}, \\hspace{56.9055pt} w_{i}=\\frac{x_{i}}{x_{i}+(1-x_{i})\\frac{M_{j}}{M_{i}}},$ where $w,x$ indicates mass and mole fractions respectively, while $i,j$ refers to water and 2,6-lutidine molecules with molar masses $M_{i,j}$ .",
"The mass and mole fractions of 2,6-lutidine are shown in the left and right vertical axes in Fig.",
"REF for several temperatures, and by circles and squares in Fig.",
"REF .",
"Table: Lutidine mass fraction and the correlation length from fits the simulation results to Eq.",
"versus temperature.Figure: Temperature versus 2,6-lutidine mass fractions (circles) given in Table.",
", and mole fractions (squares) for the poor and rich phases obtained using Eq.",
".When $T_c$ is approached from above, $\\lambda $ increases and will eventually not be much smaller than $c$ .",
"Then, the system will be too small to accommodate a saturated lutidine-rich phase.",
"We note that this starts to occur at $T \\approx 320$ K. This makes it difficult to determine the lutidine mass fraction $w^r_{lut}$ in the lutidine-rich phase.",
"At higher temperatures we could just read off the value of $w^r_{lut}$ from the flat part of the mass fraction profiles.",
"The alternative way to determine $w^r_{lut}$ is by fitting the parameters in Eq.",
"REF to simulation data.",
"At high temperatures, this method gives the same values for $w^r_{lut}$ as the ones extracted from the flat parts of the mass fraction profiles, whereas for 315 and 318 K it gives the values that are higher than the maxima in Fig.",
"REF .",
"The question is now whether this procedure could be trusted or not.",
"First we note that the fit of the mass fraction profile to the functional form of Eq.REF looks very good.",
"We tested the validity of this procedure further by running simulations of a smaller system in which the width of the lutidine-rich region is about one third of the present one, at two temperatures 330 and 380 K. For the smaller system the density profiles do not exhibit flat regions corresponding to the lutidine-rich phases at these temperatures, unlike the original simulations.",
"By fitting the mass fraction profiles to Eq.",
"REF , we do however obtain the same values for $w^r_{lut}$ in the lutidine-rich regions as in the large system at the same temperatures, despite that the maximum of the mass fraction profiles are about $20\\%$ smaller compared to the larger system.",
"Although we do know that it eventually may break down close to $T_c$ , but we are clearly not close enough for that.",
"Now, we turn to the critical properties of the system.",
"We fully realize that an accurate calculation of critical exponents and $T_c$ would require simulations closer to $T_c$ , larger systems and a finite-size scaling analysis [79].",
"This would for the present fairly complicated system call for an unjustifiable amount of computer resources.",
"Therefore, based on our simulations, the calculated exponents are obtained within certain amount of statistical errors.",
"As mentioned earlier, in order to minimize the finite-size effects we limited our simulations to the range of temperatures further away from $T_c$ , for which the bulk correlation length of the mixture is distinctively smaller than the linear dimension of the simulation box (see below) and the order parameter is relatively large.",
"Under such conditions we do not expect the mean field behaviour to occur [80], [81].",
"On the other hand, in this temperature range the corrections to the critical scaling are relevant and therefore we will employ them.",
"The critical point and the shape of the coexistence curve were determined by using well established procedures [82], [83], [84], [85], [86], [87], [88].",
"By employing the Wegner expansion [89], the order parameter (OP) which in this case is the mass fraction difference of lutidine in the rich and poor phases $w_{lut}^r-w_{lut}^p$ , can be written as $w_{lut}^r-w_{lut}^p=B_0 \\tau ^\\beta +B_1 \\tau ^{\\beta +\\Delta },$ where $\\tau = \\frac{T-T_c}{T_c}$.",
"The rectilinear diameter can be similarly expressed as $\\frac{w_{lut}^r+w_{lut}^p}{2}=w_{c}+ A_0 \\tau +A_1 \\tau ^{1-\\Gamma }.$ Figure: The coexistence curve (dash-dotted line) determined by fitting Eq.",
"to the order parameter data (circles) from simulations, and the rectilinear diameterobtained by fitting Eq.",
"(dashed line) to the simulation data (squares).The parameters of the fit are given in the first row of Table. .",
"The green diamond symbols represent the experimental data .In Eqs.",
"REF -REF $w_{c}$ is the mass fraction of lutidine at the critical point, $A_{i}$ , $i=0,1$ and $B_{i}$ , $i=0,1$ are non-universal constants, while $\\beta $ , $\\Delta $ and $\\Gamma $ are universal critical exponents.",
"For the $3D$ Ising universality class relevant for the present study, the exponents are approximately $\\beta =0.326$ , $\\Delta =0.5$ and $\\Gamma =0.1$ [90], [91], [92], [93].",
"We fitted the obtained $w_{lut}^p$ and $w_{lut}^r$ from the simulations (Table.",
"REF ) to Eqs.",
"REF -REF .",
"This resulted in the values of $T_c$ and $A_i$ and $B_i, i=0,1$ in the first line of Table.",
"REF , when the three exponents were fixed to their $3D$ Ising universality class values.",
"The results of the fit corresponding to the first row of the Table.",
"REF are shown in Fig.",
"REF together with the experimental coexistence curve [23].",
"The obtained curve exhibits a slight shift to the right as compared to the experimental one.",
"In Fig.",
"REF we show the OP as a function of the reduced temperature together with the fit to Eq.",
"REF with (left panel) and without (right panel) correction-to-scaling term.",
"One can see that the OP is fitted quite well to the power law with the 3D Ising exponent $\\beta $ (after ignoring two data points furthest away from $T_c$ ); including the correction-to scaling makes this fitting work also for temperatures further away from $T_c$ .",
"As a check for consistency of our estimates, we performed fitting treating the exponent $\\beta $ as a free parameter.",
"This fit provides a value of $\\beta $ that is only slightly different from the $3D$ Ising exponent (see Table.",
"REF ).",
"We also estimated the temperature interval in which the UCP of the mixture is located.",
"This was done by running simulations (with the similar setting as in the original ones) for a smaller system ($L_x= 5nm ,L_y=5nm ,L_z=12 nm$ ) at several higher temperatures.",
"The simulations data show a phase-separated mixture at 450 K, while a mixed liquid phase at $510 K$ .",
"This indicates that UCP is located between 450-510 K, in agreement with experiments [22], [25], [27].",
"Figure: (left) The order parameter obtained from simulations (symbols) versus τ\\tau together with a fit to Eq.",
"(parameters from the first row of Table. ).",
"The inset is in log-log scale.",
"The curvature at the higher reduced temperatures in the inset shows the importance of correction-to-scalingtaken into account by exponent Δ\\Delta .",
"(right) The order parameter versus τ\\tau , together with a fit to Eq.",
"without correction-to-scaling, B 1 =0B_1=0, and β\\beta fixed to the Ising value (0.3260.326), while we ignored two temperatures furthest away from T c T_c (inset is in log-log scale).",
"Table: Parameters obtained when fitting Eqs.",
"and to the mass fractions w lut w_{lut} resulted from simulations.",
"In the firstrow β\\beta , Γ\\Gamma and Δ\\Delta were fixed to the 3D3D Ising values, while β\\beta is treated as a free parameters in the second row."
],
[
"The surface tension and the correlation length of 2,6-lutidine/water mixture",
"We computed the surface tension in the phase-separated system (above the LCT) as a function of temperature from the simulations as an integral of the difference between the normal and tangential components of the pressure (stress) tensor across the interface [94]; the result as a function the reduced temperature is plotted in Fig.",
"REF .",
"The surface tension and the correlation length have similar scaling behaviour as the OP, but with different universal exponents [95], [85], [96], [94] $\\gamma =C_0 \\tau ^{\\mu } +C_1 \\tau ^{\\mu +\\Delta },$ $\\lambda = \\lambda _0 \\tau ^{-\\nu } +\\lambda _1 \\tau ^{-\\nu +\\Delta } ,$ where $C_i$ , $i=0,1$ and $ \\lambda _i$ , $i=0,1$ are non-universal constants.",
"We fitted the simulation data of $\\gamma $ and $\\lambda $ to these expressions using the value of $\\Delta $ fixed to its $3D$ Ising universality class value.",
"In the Tables.",
"REF -REF , the first rows are fits with the exponents $\\mu $ and $\\nu $ fixed to the $3D$ Ising universality class values, while the second rows show the fits with $\\mu $ and $\\nu $ as free fitting parameters.",
"The last rows show fits obtained by imposed $T_c= 310.5 \\pm 1.5$ as estimated from the coexistence curve fit (Table.",
"REF ).",
"All these different way of fitting give similar values of parameters.",
"The fits to Eqs.",
"REF -REF with parameters from first rows of Tables.",
"REF -REF (fixed critical exponents $\\mu $ and $\\nu $ ) are shown in Figs.",
"REF -REF .",
"Table: Parameters obtained when fitting Eq.",
"to the surface tension resulted from simulations.",
"In the first row, the critical exponents μ\\mu and Δ\\Delta are fixed, while the second row shows fit with μ\\mu being a free parameter.",
"In the last row the fit has been done by restricting T c T_c to the interval [309-312][309-312] K as estimated from coexistence curve fits (Table.",
").Table: Parameters obtained when fitting Eq.",
"to the thickness of the interface resulted from simulations.",
"In the first row, the critical exponents ν\\nu and Δ\\Delta are imposed to the fits, while in the second and third rows the fits were done with ν\\nu as a free parameter.",
"In the last row the fit has been done by restricting T c T_c to the interval [309-312][309-312] K as estimated from coexistence curve fits (Table.",
").Table: The critical exponents obtained from the simulations compared to those of the 2- and 3-dimensional Ising model and mean field theory."
],
[
"Interactions between water and 2,6-lutidine in the mixture",
"Fig.",
"REF (left) shows the interaction energy between water and 2,6-lutidine molecules per lutidine molecule for different temperatures.",
"The figure indicates that the attraction between water and lutidine becomes stronger upon decreasing temperature.",
"Fig.",
"REF (right) shows the number of hydrogen bonds between water and 2,6-lutidine molecules per lutidine molecule.",
"From the figure it is seen that upon decreasing temperature the number of hydrogen bonds between water and 2,6-lutidine molecules increases, in line with the behaviour seen in Fig.",
"REF (left)."
],
[
"Conclusion",
"In this study, we have proposed an atomistic description of the 2,6-lutidine molecule which we have shown is able to successfully describe bulk 2,6-lutidine liquid.",
"We then have employed this model together with the TIP4P/2005 water model to study the phase behaviour of 2,6-lutidine/water mixture near the LCP.",
"We conclude that by using these models for molecules it is possible to describe the critical properties of the mixture well.",
"From the density profiles computed in simulations we have obtained phase diagram of the mixture with the lower critical temperature $310.5\\pm 1.5 K$ , which is just a couple of degrees higher than the experimental value [22], [24], [25], [27].",
"We have found that the UCP is located between 450-510 K, in agreement with experiments [22], [25], [27].",
"We have computed the order parameter, the surface tension and the correlation length as a function of temperature.",
"As expected, the order parameter and the surface tension vanish upon approaching the LCP from above, while the correlation length increases.",
"Moreover, we have found that close to $T_c$ the temperature dependence of these quantities is well described by power laws.",
"The calculated exponents deviate less than about $0.02$ from those of the 3D Ising universality class [90], [91], [92], [93] to which the studied system belongs.",
"However, the estimated errors $[0.02-0.18]$ are clearly larger than this.",
"A more accurate calculation of the critical exponents and of Tc would require simulations closer to Tc , larger systems and a finite-size scaling analysis [79]."
],
[
"Acknowledgments",
"The work has been supported by the Swedish National Infrastructure for Computing (SNIC) with computer timed for the Center for High Performance Computing (PDC) and High Performance Computing Center North (HPC2N) and by National Science Center (Harmonia Grant No.",
"2015/18/M/ST3/00403).",
"FP and OE would like to thank J. Lidmar and M. Wallin for useful discussions.",
"FP would like to acknowledge E. Lindahl and B. Hess and their group members for helpful discussions during the meetings in SciLifeLab, and from L. Lundberg for quantum chemistry simulations of lutidine molecule."
]
] | 1606.04883 |
[
[
"Weighted Online Problems with Advice"
],
[
"Abstract Recently, the first online complexity class, AOC, was introduced.",
"The class consists of many online problems where each request must be either accepted or rejected, and the aim is to either minimize or maximize the number of accepted requests, while maintaining a feasible solution.",
"All AOC-complete problems (including Independent Set, Vertex Cover, Dominating Set, and Set Cover) have essentially the same advice complexity.",
"In this paper, we study weighted versions of problems in AOC, i.e., each request comes with a weight and the aim is to either minimize or maximize the total weight of the accepted requests.",
"In contrast to the unweighted versions, we show that there is a significant difference in the advice complexity of complete minimization and maximization problems.",
"We also show that our algorithmic techniques for dealing with weighted requests can be extended to work for non-complete AOC problems such as maximum matching (giving better results than what follow from the general AOC results) and even non-AOC problems such as scheduling."
],
[
"Introduction",
"An online problem is an optimization problem for which the input is divided into small pieces, usually called requests, arriving sequentially.",
"An online algorithm must serve each request, irrevocably, without any knowledge of possible future requests.",
"The quality of online algorithms is traditionally measured using the competitive ratio [12], [16], which is essentially the worst case ratio of the online performance to the performance of an optimal offline algorithm, i.e., an algorithm that knows the whole input sequence from the beginning and has unlimited computational power.",
"For some online problems such as Independent Set or Vertex Cover, the best possible competitive ratio is linear in the sequence length.",
"This gives rise to the question of what would happen, if the algorithm knew something about future requests.",
"Semi-online settings, where it is assumed that the algorithm has some specific knowledge such as the value of an optimal solution, have been studied (see [6] for many relevant references).",
"The extra knowledge may also be more problem specific such as an access graph for paging [4], [8].",
"In contrast to problem specific approaches, advice complexity [3], [9], [11] is a quantitative and standardized way of relaxing the online constraint.",
"The main idea of advice complexity is to provide an online algorithm, $\\textsc {Alg}$ , with some partial knowledge of the future in the form of advice bits provided by a trusted oracle which has unlimited computational power and knows the entire request sequence.",
"Informally, the advice complexity of an algorithm is a function of input sequence length, and for a given $n$ , it is the maximum number of advice bits read for input sequences of length $n$ .",
"The advice complexity of a problem is a function of input sequence length and competitive ratio, and for a given competitive ratio $c$ , it is the best possible advice complexity of any $c$ -competitive algorithm for the problem.",
"Advice complexity is formally defined in Section .",
"Upper bounds on the advice complexity for a problem can sometimes lead to (or come from) semi-online algorithms, and lower bounds can show that such algorithms do not exist.",
"Since its introduction, advice complexity has been a very active area of research.",
"Lower and upper bounds on the advice complexity have been obtained for a large number of online problems; a recent list can be found in [17].",
"For a survey on advice complexity, see [6].",
"Recently in [7], the first complexity class for online problems, $\\mathsf {AOC}$ , was introduced.",
"The class consists of online problems that can be described in the following way: The input is a sequence of requests and each request must either be accepted or rejected.",
"The set of accepted requests is called the solution.",
"For each request sequence, there is at least one feasible solution.",
"The class contains minimization as well as maximization problems.",
"For a minimization problem, the goal is to accept as few requests as possible, while maintaining a feasible solution, and for maximization problems, the aim is to accept as many requests as possible.",
"For minimization problems, any super set of a feasible solution is also a solution, and for maximization problems, any subset of a feasible solution is also a feasible solution.",
"The AOC-complete problems are the hardest problems in the class in terms of their advice complexity.",
"The class $\\mathsf {AOC}$ is formally defined in Section .",
"In this paper, we consider a generalization of the problems in the class $\\mathsf {AOC}$ in which each request comes with a weight.",
"The goal is now to either minimize or maximize the total weight of the accepted requests.",
"We separately consider the classes of maximization and minimization problems.",
"For $\\mathsf {AOC}$ -complete maximization problems, we get advice complexity results quite similar to those for the unweighted versions of the problems.",
"On the other hand, for $\\mathsf {AOC}$ -complete minimization problems, the results are a lot more negative: using less than one advice bit per request leads to unbounded competitive ratios, so this gives a complexity class containing harder problems than $\\mathsf {AOC}$ .",
"This is in contrast to unweighted AOC-complete problems, where minimization and maximization problems are equally hard in terms of advice complexity.",
"Recently, differences between (unweighted) AOC minimization and maximization problems were found with respect to online bounded analysis [5] and min- and max-induced subgraph problems [13].",
"Our upper bound techniques are also useful for non-complete $\\mathsf {AOC}$ problems such as Matching in the edge arrival model, as well as non-$\\mathsf {AOC}$ problems such as Scheduling.",
"For any $\\mathsf {AOC}$ -complete problem, $\\Theta (n/c)$ advice bits are necessary and sufficient to obtain a competitive ratio of $c$ .",
"More specifically, for competitive ratio $c$ , the advice complexity is $B(n,c) \\pm O(\\log n)$ , where $B(n,c)= \\log \\left(1+\\frac{(c-1)^{c-1}}{c^{c}}\\right)n,$ and $an/c \\le B(n,c) \\le n/c$ , $a = 1/(e \\ln (2)) \\approx 0.53$ .",
"This is an upper bound on the advice complexity of all problems in $\\mathsf {AOC}$ .",
"In [7], a list of problems including Independent Set, Vertex Cover, Dominating Set, and Set Cover were proven $\\mathsf {AOC}\\text{-complete}$ .",
"The paper [1] studies a semi-online version of scheduling where it is allowed to keep several parallel schedules and choose the best schedule in the end.",
"The scheduling problem considered is makespan minimization on $m$ identical machines.",
"Using $(1/\\varepsilon )^{O(\\log (1/\\varepsilon ))}$ parallel schedules, a $(4/3+\\varepsilon )$ -competitive algorithm is obtained.",
"Moreover, a $(1+\\varepsilon )$ -competitive algorithm which uses $(m/\\varepsilon )^{O(\\log (1/\\varepsilon )/\\varepsilon )}$ parallel schedules is given along with an almost matching lower bound.",
"Note that keeping $s$ different schedules until the end corresponds to working with $s$ different online algorithms.",
"Thus, this particular semi-online model easily translates to the advice model, the advice being which of the $s$ algorithms to run.",
"In this way, the results of [1] correspond to a $(4/3+\\varepsilon )$ -competitive algorithm using $O(\\log ^2(1/\\varepsilon ))$ advice bits and a $(1+\\varepsilon )$ -competitive algorithm using $O(\\log (m/\\varepsilon ) \\cdot \\log (1/\\varepsilon ) / \\varepsilon )$ advice bits.",
"In particular, note that this algorithm uses constant advice in the size of the input and only logarithmic advice in the number of machines.",
"In [15], scheduling on identical machines with a more general type of objective function (including makespan, minimizing the $\\ell _p$ -norm, and machine covering) was studied.",
"The paper considers the advice-with-request model where a fixed number of advice bits are provided along with each request.",
"The main result is a $(1+\\varepsilon )$ -competitive algorithm that uses $O((1/\\varepsilon ) \\cdot \\log (1/\\varepsilon ))$ advice bits per request, totaling $O((n/\\varepsilon ) \\cdot \\log (1/\\varepsilon ))$ bits of advice for the entire sequence."
],
[
"Our results.",
"We prove that adding arbitrary weights, $\\mathsf {AOC}$ -complete minimization problems become a lot harder than $\\mathsf {AOC}$ -complete maximization problems: For $\\mathsf {AOC}\\text{-complete}$ maximization problems, the weighted version is not significantly harder than the unweighted version: For any maximization problem in $\\mathsf {AOC}$ (this includes, e.g., Independent Set), the $c$ -competitive algorithm given in [7] for the unweighted version of the problem can be converted into a $(1+\\varepsilon )c$ -competitive algorithm for the weighted version using only $O((\\log ^2 n)/\\varepsilon )$ additional advice bits.",
"Thus, a $(1+\\varepsilon )c$ -competitive algorithm using at most $B(n,c)+O((\\log ^2 n)/\\varepsilon )$ bits of advice is obtained.",
"For the weighted version of non-complete $\\mathsf {AOC}$ maximization problems, a better advice complexity than $B(n,c)$ may be obtained: For any $c$ -competitive algorithm for an $\\mathsf {AOC}$ maximization problem, P, using $b$ advice bits can be converted into a $O(c \\cdot \\log n)$ -competitive algorithm for the weighted version of P using $b+O(\\log n)$ advice bits.",
"For Weighted Matching in the edge arrival model, this implies a $O(\\log n)$ -competitive algorithm reading $O(\\log n)$ bits of advice.",
"We show that this is best possible in the following sense: For a set of weighted $\\mathsf {AOC}$ problems including Matching, Independent Set and Clique, no algorithm reading $o(\\log n)$ bits of advice can have a competitive ratio bounded by any function of $n$ .",
"Furthermore, any $O(1)$ -competitive algorithm for Matching must read $\\Omega (n)$ advice bits.",
"For all minimization problems known to be $\\mathsf {AOC}\\text{-complete}$ (this includes, e.g., Vertex Cover, Dominating Set, and Set Cover), $n - O(\\log n)$ bits of advice are required to obtain a competitive ratio bounded by a function of $n$ .",
"This should be contrasted with the fact that $n$ bits of advice trivially yields a strictly 1-competitive algorithm.",
"If the largest weight $w_{\\text{max}}$ cannot be arbitrarily larger than the smallest weight $w_{\\text{min}} $ , the $c$ -competitive algorithm given in [7] for the unweighted version can be converted into a $c(1+\\varepsilon )$ -competitive algorithm for the weighted versions using $B(n,c) + O(\\log ^2 n + \\log (\\log (w_{\\text{max}}/w_{\\text{min}})/\\varepsilon ))$ advice bits in total.",
"Our main upper bound technique is a simple exponential classification scheme that can be used to sparsify the set of possible weights.",
"This technique can also be used for problems outside of $\\mathsf {AOC}$ .",
"For example, for scheduling on related machines, we show that for many important objective functions (including makespan minimization and minimizing the $\\ell _p$ -norm), there exist $(1+\\varepsilon )$ -competitive algorithms reading $O((\\log ^2 n)/\\varepsilon )$ bits of advice.",
"For scheduling on $m$ unrelated machines where $m$ is constant, we get a similar result, but with $O((\\log n)^{m+1}/\\varepsilon ^m)$ advice bits.",
"Finally, for unrelated machines, where the goal is to maximize an objective function, we show that under some mild assumptions on the objective function (satisfied, for example, for machine covering), there is a $(1+\\varepsilon )$ -competitive algorithm reading $O((\\log n)^{m+1}/\\varepsilon ^m)$ bits of advice.",
"For scheduling on related and unrelated machines, our results are the first non-trivial upper bounds on the advice complexity.",
"For the case of makespan minimization on identical machines, the algorithm of [1] is strictly better than ours.",
"However, for minimizing the $\\ell _p$ -norm or maximizing the minimum load on identical machines, we exponentially improve the previous best upper bound [15] (which was linear in $n$ ).",
"Throughout the paper, we let $n$ denote the number of requests in the input.",
"We let $\\mathbb {R}_+$ denote the set containing 0 and all positive real numbers.",
"We let $\\log $ denote the binary logarithm $\\log _2$ .",
"For $k\\ge 1$ , $[k]=\\lbrace 1,2,\\ldots , k\\rbrace $ .",
"For any bit string $y$ , let $\\left|y\\right|_0$ and $\\numero {y}$ denote the number of zeros and the number of ones, respectively, in $y$ .",
"We write $x \\sqsubseteq y$ if for all indices, $i$ , $x_i=1 \\Rightarrow y_i =1$ ."
],
[
"Advice complexity and competitive analysis",
"In this paper, we use the “advice-on-tape” model [3].",
"Before the first request arrives, the oracle, which knows the entire request sequence, prepares an advice tape, an infinite binary string.",
"The algorithm $\\textsc {Alg}$ may, at any point, read some bits from the advice tape.",
"The advice complexity of $\\textsc {Alg}$ is the maximum number of bits read by $\\textsc {Alg}$ for any input sequence of at most a given length.",
"$\\textsc {Opt}$ is an optimal offline algorithm.",
"Advice complexity is combined with competitive analysis to determine how many bits of advice are necessary and sufficient to achieve a given competitive ratio.",
"[Competitive analysis [12], [16] and advice complexity [3]] The input to an online problem, P, is a request sequence $\\sigma =\\langle r_1,\\ldots , r_n \\rangle $ .",
"An online algorithm with advice, $\\textsc {Alg}$ , computes the output $y=\\langle y_1,\\ldots , y_n\\rangle $ , where $y_i$ is computed from $\\varphi , r_1,\\ldots , r_i$ , where $\\varphi $ is the content of the advice tape.",
"Each possible output for P is associated with a cost/profit.",
"For a request sequence $\\sigma $ , $\\textsc {Alg} (\\sigma )$ $(\\textsc {Opt} (\\sigma ))$ denotes the cost/profit of the output computed by $\\textsc {Alg} $ $(\\textsc {Opt})$ when serving $\\sigma $ .",
"If P is a minimization (maximization) problem, then $\\textsc {Alg} $ is $c(n)$ -competitive if there exists a constant, $\\alpha $ , such that, for all $n \\in \\mathbb {N}$ , $\\textsc {Alg} (\\sigma )\\le c(n)\\cdot \\textsc {Opt} (\\sigma )+\\alpha $ , ($\\textsc {Opt} (\\sigma )\\le c(n)\\cdot \\textsc {Alg} (\\sigma )+\\alpha $ ), for all request sequences, $\\sigma $ , of length at most $n$ .",
"If the relevant inequality holds with $\\alpha =0$ , we say that $\\textsc {Alg} $ is strictly $c(n)$ -competitive.",
"The advice complexity, $b(n)$ , of an algorithm, $\\textsc {Alg}$ , is the largest number of bits of $\\varphi $ read by $\\textsc {Alg}$ over all possible request sequences of length at most $n$ .",
"The advice complexity of a problem, P, is a function, $f(n,c)$ , $c\\ge 1$ , such that the smallest possible advice complexity of a strictly $c$ -competitive online algorithm for P is $f(n,c)$ .",
"We only consider deterministic online algorithms (with advice).",
"Note that both $b(n)$ and $c(n)$ in the above definition may depend on $n$ , but, for ease of notation, we often write $b$ and $c$ instead of $b(n)$ and $c(n)$ .",
"Also, with this definition, $c\\ge 1$ , for both minimization and maximization problems."
],
[
"Complexity classes",
"In this paper, we consider the complexity class $\\mathsf {AOC}$ from [7].",
"[$\\mathsf {AOC}$ [7]] A problem, P, is in $\\mathsf {AOC}$ (Asymmetric Online Covering) if it can be defined as follows: The input to an instance of P consists of a sequence of $n$ requests, $\\sigma =\\langle r_1, \\ldots , r_n \\rangle $ , and possibly one final dummy request.",
"An algorithm for P computes a binary output string, $y=y_1 \\ldots y_n\\in \\lbrace 0,1\\rbrace ^n$ , where $y_i=f(r_1, \\ldots , r_i)$ for some function $f$ .",
"For minimization (maximization) problems, the score function, $s$ , maps a pair, $(\\sigma ,y)$ , of input and output to a cost (profit) in $\\mathbb {N} \\cup \\lbrace \\infty \\rbrace $ $(\\mathbb {N} \\cup \\lbrace -\\infty \\rbrace )$ .",
"For an input, $\\sigma $ , and an output, $y$ , $y$ is feasible if $s(\\sigma ,y) \\in \\mathbb {N}$ .",
"Otherwise, $y$ is infeasible.",
"There must exist at least one feasible output.",
"Let $S_{\\min }(\\sigma )$ $(S_{\\max }(\\sigma ))$ be the set of those outputs that minimize (maximize) $s$ for a given input $\\sigma $ .",
"If P is a minimization problem, then for every input, $\\sigma $ , the following must hold: For a feasible output, $y$ , $s(\\sigma ,y)=\\numero {y}$ .",
"An output, $y$ , is feasible if there exists a $y^{\\prime }\\in S_{\\min }(\\sigma )$ such that $y^{\\prime }\\sqsubseteq y$ .",
"If there is no such $y^{\\prime }$ , the output may or may not be feasible.",
"If P is a maximization problem, then for every input, $\\sigma $ , the following must hold: For a feasible output, $y$ , $s(\\sigma ,y)=\\left|y\\right|_0$ .",
"An output, $y$ , is feasible if there exists a $y^{\\prime }\\in S_{\\max }(\\sigma )$ such that $y^{\\prime }\\sqsubseteq y$ .",
"If there is no such $y^{\\prime }$ , the output may or may not be feasible.",
"Recall that no problem in $\\mathsf {AOC}$ requires more than $B(n,c) + O(\\log n)$ bits of advice (see Eq.",
"(REF ) for the definition of $B(n,c)$ ).",
"This result is based on a covering design technique, where the advice indicates a superset of the output bits that are 1 in an optimal solution.",
"The problems in $\\mathsf {AOC}$ requiring the most advice are $\\mathsf {AOC}\\text{-complete}$ [7]: [$\\mathsf {AOC}\\text{-complete}$ [7]] A problem ${\\textsc {P}} \\in \\mathsf {AOC} $ is $\\mathsf {AOC}\\text{-complete}$ if for all $c>1$ , any $c$ -competitive algorithm for P must read at least $B(n,c)-O(\\log n)$ bits of advice.",
"In [7], an abstract guessing game, minASGk (Minimum Asymmetric String Guessing with Known History), was introduced and shown to be $\\mathsf {AOC}\\text{-complete}$ .",
"The minASGk-problem itself is very artificial, but it is well-suited as the starting point of reductions.",
"All minimization problems known to be $\\mathsf {AOC}\\text{-complete}$ have been shown to be so via reductions from minASGk.",
"The input for minASGk is a secret string $x=x_1x_2\\ldots x_n\\in \\lbrace 0,1\\rbrace ^n$ given in $n$ rounds.",
"In round $i\\in [n]$ , the online algorithm must answer $y_i\\in \\lbrace 0,1\\rbrace $ .",
"Immediately after answering, the correct answer $x_i$ for round $i$ is revealed to the algorithm.",
"If the algorithm answers $y_i=1$ , it incurs a cost of 1.",
"If the algorithm answers $y_i=0$ , then it incurs no cost if $x_i=0$ , but if $x_i=1$ , then the output of the algorithm is declared to be infeasible (and the algorithm incurs a cost of $\\infty $ ).",
"The objective is to minimize the total cost incurred.",
"Note that the optimal solution has cost $\\numero {x}$ .",
"See the appendix for a formal definition of minASGk and for definitions of other $\\mathsf {AOC}\\text{-complete}$ problems.",
"The problem minASGk is based on the binary string guessing problem [2], [11].",
"Binary string guessing is similar to asymmetric string guessing, except that any wrong guess (0 instead of 1 or 1 instead of 0) gives a cost of 1.",
"In Theorem , we show a very strong lower bound for a weighted version of minASGk.",
"In Theorem , via reductions, we show that this lower bound implies similar strong lower bounds for the weighted version of other $\\mathsf {AOC}\\text{-complete}$ minimization problems.",
"[Weighted $\\mathsf {AOC}$ ] Let ${\\textsc {P}} $ be a problem in $\\mathsf {AOC}$ .",
"We define the weighted version of ${\\textsc {P}} $, denoted $\\textsc {P}_{\\text{w}} $ , as follows: A $\\textsc {P}_{\\text{w}} $ -input $\\sigma =\\langle \\lbrace r_1, w_1\\rbrace ,$ $\\lbrace r_2, w_2\\rbrace ,$ $\\ldots , \\lbrace r_n, w_n\\rbrace \\rangle $ consists of $n$ ${\\textsc {P}} $ -requests, $r_1,...,r_n$ , each of which has a weight $w_i\\in \\mathbb {R}_+$ .",
"The ${\\textsc {P}} $ -request $r_i$ and its weight $w_i$ are revealed simultaneously.",
"An output $y=y_1\\ldots y_n\\in \\lbrace 0,1\\rbrace ^n$ is feasible for the input $\\sigma $ if and only if $y$ is feasible for the ${\\textsc {P}} $ -input $\\langle r_1,\\ldots , r_n\\rangle $ .",
"The cost (profit) of an infeasible solution is $\\infty $ ($-\\infty $ ).",
"If ${\\textsc {P}} $ is a minimization problem, then the cost of a feasible $\\textsc {P}_{\\text{w}} $ -output $y$ for an input $\\sigma $ is $s(\\sigma ,y)=\\sum _{i=1}^n w_iy_i$ If ${\\textsc {P}} $ is a maximization problem, then the profit of a feasible $\\textsc {P}_{\\text{w}} $ -output $y$ for an input $\\sigma $ is $s(\\sigma ,y)=\\sum _{i=1}^n w_i (1-y_i)$"
],
[
"Weighted Versions of $\\mathsf {AOC}$ -Complete Minimization Problems",
"In the weighted version of minASGk, minASGk$_{\\text{w}}$, each request is a weight for the current request and the value 0 or 1 of the previous request.",
"Producing a feasible solution requires accepting (answering 1 to) all requests with value 1, and the cost of a feasible solution is the sum of all weights for requests which are accepted.",
"We start with a negative result for minASGk$_{\\text{w}}$ and then use it to obtain similar results for the weighted online version of Vertex Cover, Set Cover, Dominating Set, and Cycle Finding.",
"For minASGk$_{\\text{w}}$, no algorithm using less than $n$ bits of advice is $f(n)$ -competitive, for any function $f$ .",
"Let $\\textsc {Alg}$ be any algorithm for minASGk$_{\\text{w}}$ reading at most $n-1$ bits of advice.",
"We show how an adversary can construct input sequences where the cost of $\\textsc {Alg}$ is arbitrarily larger than that of $\\textsc {Opt}$ .",
"We only consider sequences with at least one 1.",
"It is easy to see that for the unweighted version of the binary string guessing problem, $n$ bits of advice are necessary in order to guess correctly each time: If there are fewer than $n$ bits, there are only $2^{n-1}$ possible advice strings, so, even if we only consider the $2^n-1$ possible inputs with at least one 1, there are at least two different request strings, $x$ and $y$ , which get the same advice string.",
"$\\textsc {Alg}$ will make an error on one of the strings when guessing the first bit where $x$ and $y$ differ, since up until that point $\\textsc {Alg}$ has the same information about both strings.",
"We describe a way to assign weights to the requests in minASGk$_{\\text{w}}$ such that if $\\textsc {Alg}$ makes a single mistake (either guessing 0 when the correct answer is 1 or vice versa), its performance ratio is unbounded.",
"We use a large number $a>1$ , which we allow to depend on $n$ .",
"All weights are from the interval $[1,a]$ (note that they are not necessarily integers).",
"We let $x=x_1, \\ldots , x_n$ be the input string and set $w_1=a^{1/2}$ .",
"For $i>1$ , $w_i$ is given by: $w_i ={\\left\\lbrace \\begin{array}{ll}\\hfill w_{i-1}\\cdot a^{(-2^{-i})}, \\hfill & \\text{ if $x_{i-1}=0$} \\\\\\hfill w_{i-1}\\cdot a^{(2^{-i})}, \\hfill & \\text{ if $x_{i-1}=1$} \\\\\\end{array}\\right.",
"}$ Since the weights are only a function of previous requests, they do not reveal any information to $\\textsc {Alg}$ about future requests.",
"Observation 1 For each $i$ , the following hold: If $x_i=0$ , then $w_j \\le w_{i}\\cdot a^{(-2^{-n})}$ for all $j>i$ .",
"If $x_i=1$ , then $w_j \\ge w_{i}\\cdot a^{(2^{-n})}$ for all $j>i$ .",
"We argue for each set of inequalities in the observation: (REF ): If $x_i=0$ , for each $j>i$ , $w_j=w_i\\cdot a^{(-2^{-(i+1)})}\\cdot a^{\\sum _{k=i+2}^j(\\pm 2^{-(i+1)})}$ , where the plus or minus depends on whether $x_k=0$ or $x_k=1$ .",
"The value $w_j$ is largest if all of the $x_k$ values are 1, in which case $w_j=w_i\\cdot a^{(-2^{-j})} \\le w_i\\cdot a^{(-2^{-n})}$ .",
"(REF ): The argument of $x_i=1$ is similar, changing minus to plus and vice versa.",
"We claim that if $\\textsc {Alg}$ makes a single mistake, its performance ratio is not bounded by any function of $n$ .",
"Indeed, if $\\textsc {Alg}$ guesses 0 for a request, but the correct answer is 1, the solution is infeasible and $\\textsc {Alg}$ gets a cost of $\\infty $ .",
"We now consider the case where $\\textsc {Alg}$ guesses 1 for a request $j$ , but the correct answer is 0.",
"This request gives a contribution of $w_j =a^b$ , for some $0<b<1$ , to the cost of the solution produced by $\\textsc {Alg}$ .",
"Define $j^{\\prime }$ such that $w_{j^{\\prime }}=\\max \\lbrace w_i \\mid x_i=1\\rbrace $ .",
"Since $\\textsc {Opt}$ only answers 1 if $x_i =1$ , this is the largest contribution to the cost of $\\textsc {Opt}$ from a single request.",
"If $j^{\\prime }>j$ , Observation REF (REF ) gives that $w_{j^{\\prime }} \\le w_j\\cdot a^{(-2^{-n})}=a^b\\cdot a^{(-2^{-n})}=a^{b-2^{-n}}$ .",
"The cost of $\\textsc {Opt}$ is at most $n \\cdot w_{j^{\\prime }} \\le n \\cdot a^{b-2^{-n}}$ .",
"Thus, $\\frac{\\textsc {Alg} (x)}{\\textsc {Opt} (x)} \\ge \\frac{a^b}{n \\cdot a^{b-2^{-n}}}=\\frac{a^{2^{-n}}}{n}.$ Since $a$ can be arbitrarily large (recall that it can be a function of $n$ ), we see that no algorithm can be $f(n)$ -competitive for any specific function $f$ .",
"If $j^{\\prime }<j$ , Observation REF (REF ) gives us that $w_{j} \\ge w_{j^{\\prime }}\\cdot a^{(2^{-n})}$ .",
"Using $w_j=a^b$ , we get $a^{b-2^{-n}} \\ge w_{j^{\\prime }} $ .",
"We can repeat the argument from the case where $j^{\\prime }>j$ to see that no algorithm can be $f(n)$ -competitive for any specific function $f$ .",
"$\\Box $ In order to show that similar lower bounds apply to all minimization problems known to be complete for $\\mathsf {AOC}$ , we define a simple type of advice preserving reduction for online problems.",
"These are much less general than those defined by Sprock in his PhD dissertation [18], mainly because we do not allow the amount of advice needed to change by a multiplicative factor.",
"Let $\\textsc {Opt} _{{\\textsc {P}}}(\\sigma )$ denote the value of the optimal solution for request sequence $\\sigma $ for problem ${\\textsc {P}} $ , and let $|\\sigma |$ denote the number of requests in $\\sigma $ .",
"Let ${\\textsc {P}} _1$ and ${\\textsc {P}} _2$ be two online minimization problems, and let $\\mathcal {I} _1$ be the set of request sequences for ${\\textsc {P}} _1$ and $\\mathcal {I} _2$ be the set of request sequences for ${\\textsc {P}} _2$ .",
"For a given function $g: \\mathbb {N} \\rightarrow \\mathbb {R} _+$ , we say that there is a length preserving $g$ -reduction from ${\\textsc {P}} _1$ to ${\\textsc {P}} _2$ , if there is a transformation function $f: \\mathcal {I} _1 \\rightarrow \\mathcal {I} _2$ such that for all $\\sigma \\in \\mathcal {I} _1$ , $|\\sigma | = |f(\\sigma )| $ , and for every algorithm $\\textsc {Alg}_2$ for ${\\textsc {P}} _2$ , there is an algorithm $\\textsc {Alg}_1$ for ${\\textsc {P}} _1$ such that for all $\\sigma _1\\in \\mathcal {I} _1$ , the following holds: If $\\textsc {Alg}_2$ produces a feasible solution for $\\sigma _2 = f(\\sigma _1)$ with advice $\\phi (\\sigma _2)$ , then $\\textsc {Alg}_1$ , using at most $|\\phi (\\sigma _2)| + g(|\\sigma _2|)$ advice bits, produces a feasible solution for $\\sigma _1$ such that $\\textsc {Alg}_1 (\\sigma _1) \\le \\textsc {Alg}_2 (\\sigma _2) + \\textsc {Opt} _{{\\textsc {P}} _1}(\\sigma _1)$ and $\\textsc {Opt} _{{\\textsc {P}} _1}(\\sigma _1)\\ge \\textsc {Opt} _{{\\textsc {P}} _2}(\\sigma _2)$ , or $\\textsc {Alg}_1 (\\sigma _1) = \\textsc {Opt} _{{\\textsc {P}} _1}(\\sigma _1)$ Note that the transformation function $f$ is length-preserving in that the lengths of the request sequences for the two problems are identical.",
"This avoids the potential problem that the advice for the two problems could be functions of two different sequence lengths.",
"The amount of advice for the problem being reduced to is allowed to be an additive function, $g(n)$ , longer than for the original problem, because this seems to be necessary for some of the reductions showing that problems are $\\mathsf {AOC}\\text{-complete}$ .",
"Since the reductions are only used here to show that no algorithm is $F(n)$ -competitive for any function $F$ , the increase in the performance ratio that occurs with these reductions is insignificant.",
"The following lemma shows how length-preserving reductions can be used.",
"Let ${\\textsc {P}} _1$ and ${\\textsc {P}} _2$ be online minimization problems.",
"Suppose that at least $b_1(n,c)$ advice bits are required to be $(c+1)$ -competitive for ${\\textsc {P}} _1$ and suppose there is a length preserving $g(n)$ -reduction from ${\\textsc {P}} _1$ to ${\\textsc {P}} _2$ .",
"Then, at least $b_1(n,c)-g(n)$ advice bits are needed for an algorithm for ${\\textsc {P}} _2$ to be $c$ -competitive.",
"Let $f$ be the transformation function associated with $g$ .",
"Suppose for the sake of contradiction that there is a (strictly) $c$ -competitive algorithm $\\textsc {Alg}_2 $ for ${\\textsc {P}} _2$ with advice complexity $b_2(n,c) < b_1(n,c)-g(n)$ .",
"Then there exists a constant $\\alpha $ such that for any request sequence $\\sigma _1 \\in \\mathcal {I} _1$ , either $\\textsc {Alg}_1 (\\sigma _1) =\\textsc {Opt} _{{\\textsc {P}} _1}(\\sigma _1)$ or $\\textsc {Alg}_1 (\\sigma _1)& \\le \\textsc {Alg}_2 (\\sigma _2) + \\textsc {Opt} _{{\\textsc {P}} _1}(\\sigma _1)\\\\& \\le c\\cdot \\textsc {Opt} _{{\\textsc {P}} _2}(\\sigma _2) + \\alpha + \\textsc {Opt} _{{\\textsc {P}} _1}(\\sigma _1)\\\\& \\le (c+1)\\cdot \\textsc {Opt} _{{\\textsc {P}} _1}(\\sigma _1) + \\alpha ,$ where $\\sigma _2 = f(\\sigma _1)$ .",
"Thus, $\\textsc {Alg}_1$ is (strictly) $(c+1)$ -competitive, with less than $b_1(n,c)$ bits of advice, a contradiction.",
"$\\Box $ All known $\\mathsf {AOC}\\text{-complete}$ problems were proven complete using length-preserving reductions from minASGk, so the following holds for the weighted versions of all such problems: For the weighted online versions of Vertex Cover, Cycle Finding, Dominating Set, Set Cover, an algorithm reading less than $n-O(\\log n)$ bits of advice cannot be $f(n)$ -competitive for any function $f$ .",
"The reductions in [7] showing that these problems are $\\mathsf {AOC}\\text{-complete}$ are length preserving $O(\\log n)$ -reductions from minASGk, and hence, the theorem follows from Lemma .",
"For Vertex Cover, the following $O(\\log n)$ -reduction can be used (the other three reductions are given in the Appendix REF ): Each input $\\sigma = \\langle x_1, x_2, \\ldots , x_n \\rangle $ to the problem minASGk, is transformed to $f(\\sigma ) = \\langle v_1, v_2, \\ldots , v_n \\rangle $ , where $V=\\lbrace v_1, v_2, \\ldots , v_n \\rbrace $ is the vertex set of a graph with edge set $E=\\lbrace (v_i,v_j) \\colon x_i=1 \\text{ and } i<j\\rbrace .$ Let $V_1 = \\lbrace v_i \\in V \\colon x_1 = 1\\rbrace $ .",
"Note that $V_1 \\setminus \\lbrace v_n\\rbrace $ is a minimum vertex cover of the graph and that no algorithm can reject more than one vertex from $V_1$ , since $V_1$ induces a clique.",
"The advice used by the minASGk algorithm $\\textsc {Alg}_1$ consists of the advice used by the Vertex Cover algorithm $\\textsc {Alg}_2$ and $O(\\log n)$ bits that are either all 0 or give (an encoding of) an index to a position $i$ in the input sequence, such that $v_i \\in V_1$ and $\\textsc {Alg}_2$ rejects $v_i$ .",
"Let $V_{\\textsc {Alg}_2} \\subseteq V$ be the vertex cover constructed by $\\textsc {Alg}_2$ and let $X_{\\textsc {Alg}_1}$ be the set of requests on which $\\textsc {Alg}_1$ returns a 1.",
"Then either $X_{\\textsc {Alg}_1} = V_{\\textsc {Alg}_1}$ or $X_{\\textsc {Alg}_1} = V_{\\textsc {Alg}_2} \\cup \\lbrace v_i\\rbrace $ , where $\\lbrace v_i\\rbrace = V_1 \\setminus V_{\\textsc {Alg}_1}$ .",
"Thus, $\\textsc {Alg}_1 (\\sigma ) \\le \\textsc {Alg}_2 (f(\\sigma ))+\\textsc {Opt} (\\sigma )$ , since $w_i \\le \\textsc {Opt} (\\sigma )$ .",
"$\\Box $"
],
[
"Exponential Sparsification",
"Assume that we are faced with an online problem for which we know how to obtain a reasonable competitive ratio, possibly using advice, in the unweighted version (or when there are only few possible different weights).",
"We use exponential sparsification, a simple technique which can be of help when designing algorithms with advice for weighted online problems by reducing the number of different possible weights the algorithm has to handle.",
"The first step is to partition the set of possible weights into intervals of exponentially increasing length, i.e., for some small $\\varepsilon $ , $0<\\varepsilon <1$ , $\\mathbb {R}_+=\\bigcup _{k=-\\infty }^{\\infty }\\big [(1+\\varepsilon )^k, (1+\\varepsilon )^{k+1}\\big ).$ How to proceed depends on the problem at hand.",
"We now informally explain the meta-algorithm that we repeatedly use in this paper.",
"Note that if $w_1,w_2\\in \\big [(1+\\varepsilon )^k, (1+\\varepsilon )^{k+1}\\big )$ and $w_1\\le w_2$ , then $w_1\\le w_2\\le (1+\\varepsilon )w_1$ .",
"For many online problems, this means that an algorithm can treat all requests whose weights belong to this interval as if they all had weight $(1+\\varepsilon )^{k+1}$ with only a small loss in competitiveness.",
"Consider now a set of weights and let $w_{\\text{max}} $ denote the largest weight in the set.",
"Let $k_{\\text{max}}$ be the integer for which $w_{\\text{max}} \\in \\big [(1+\\varepsilon )^{k_{\\text{max}}},(1+\\varepsilon )^{k_{\\text{max}} +1}\\big )$ .",
"We say that a request with weight $w\\in \\big [(1+\\varepsilon )^{k},(1+\\varepsilon )^{k+1}\\big )$ is unimportant if $k<k_{\\text{max}}- {\\log _{1+\\varepsilon }(n^2)}$ .",
"Furthermore, we will often categorize the request as important if $k_{\\text{max}}-{\\log _{1+\\varepsilon }(n^2)} \\le k < k_{\\text{max}} +1$ and as huge if $k \\ge k_{\\text{max}} +1$ .",
"Each unimportant request has weight $w \\le (1+\\varepsilon )^{k+1}\\le (1+\\varepsilon )^{k_{\\text{max}}-{\\log _{1+\\varepsilon }(n^2)}-1+1}\\le w_{\\text{max}}/n^2$ , so the total sum of the unimportant weights is $O(w_{\\text{max}}/n)$ .",
"For many weighted online problems, this means that an algorithm can easily serve the requests with unimportant weights, as follows.",
"In maximization problems, this is done by rejecting them.",
"In minimization problems, it is done by accepting them.",
"Thus, exponential sparsification (when applicable) essentially reduces the problem of computing a good approximate solution for a problem with $n$ distinct weights to that of computing a good approximate solution with only $O(\\log _{1+\\varepsilon } n)$ distinct weights.",
"For a concrete problem, several modifications of this meta-algorithm might be necessary.",
"Often, the most tricky part is how the algorithm can learn $k_{\\text{max}} $ without using too much advice.",
"One approach that we often use is the following: The oracle encodes the index $i$ of the first request whose weight is close enough to $(1+\\varepsilon )^{k_{\\text{max}}}$ that the algorithm only needs a little bit of advice to deduce $k_{\\text{max}} $ from the weight of this request.",
"If it is somehow possible for the algorithm to serve all requests prior to $i$ reasonably well, then this approach works well.",
"Our main application of exponential sparsification is to weighted $\\mathsf {AOC}$ problems.",
"We begin by considering maximization problems.",
"Note that no assumptions are made about the weights of $\\textsc {P}_{\\text{w}} $ in Theorem .",
"If ${\\textsc {P}} \\in \\mathsf {AOC} $ is a maximization problem, then for any $c>1$ and $0<\\varepsilon \\le 1$ , $\\textsc {P}_{\\text{w}}$ has a strictly $(1+\\varepsilon )c$ -competitive algorithm using $B(n,c)+O(\\varepsilon ^{-1}\\log ^2 n)$ advice bits.",
"Fix $\\varepsilon >0$ .",
"Let $\\sigma =\\langle \\lbrace r_1,w_1\\rbrace ,\\ldots ,\\lbrace r_n, w_n\\rbrace \\rangle $ be the input and let $x=x_1\\ldots x_n\\in \\lbrace 0,1\\rbrace ^n$ specify an optimal solution for $\\sigma $ , with zeros indicating membership in the optimal solution.",
"Throughout most of this proof, we assume that $n$ is sufficiently large.",
"The necessary conditions are discussed at the end of the proof, along with how to handle small $n$ .",
"Define $s=1+\\varepsilon /2$ .",
"Let $V_{\\textsc {Opt}} =\\lbrace i\\colon x_i=0\\rbrace $ .",
"Note that $V_{\\textsc {Opt}} $ contains exactly those rounds in which $\\textsc {Opt} $ answers 0 and thus accepts.",
"Furthermore, for $k\\in \\mathbb {Z}$ , let $V^k=\\lbrace i\\colon s^k\\le w(i) <s^{k+1}\\rbrace $ and let $V_{\\textsc {Opt}} ^k=V_{\\textsc {Opt}} \\cap V^k$ .",
"Finally, let $i_{\\max }\\in V_{\\textsc {Opt}} $ be such that $w(i_{\\max })\\ge w(i)$ for every $i\\in V_{\\textsc {Opt}} $ .",
"The oracle computes the unique $m\\in \\mathbb {Z}$ such that $i_{\\max }\\in V_{\\textsc {Opt}} ^m$ .",
"We say that a request $r_i$ is unimportant if $w(i)<s^{m- {\\log _{s}(n^2)}}$ , important if $s^{m-{\\log _{s}(n^2)}} \\le w(i) < s^{m+1}$ , and huge if $w(i)\\ge s^{m+1}$ .",
"The oracle computes the index $i^{\\prime }$ of the first important request in the input sequence.",
"Assume that $i^{\\prime } \\in V^{m^{\\prime }}$ .",
"The oracle writes the length $n$ of the input onto the advice tape using a self-delimiting encodingFor example, $\\lceil \\log n\\rceil $ could be written in unary ($\\lceil \\log n\\rceil $ ones, followed by a zero) before writing $n$ itself in binary., and then writes the index $i^{\\prime }$ and the integer $m-m^{\\prime }$ (which is at most ${\\log _{s}(n^2)}$ ) onto the tape, using a total of $O(\\log n)$ bits.",
"This advice allows the algorithm to learn $m$ as soon as the first important request arrives.",
"From there on, the algorithm will know if a request is important, unimportant, or huge.",
"Whenever an unimportant or a huge request arrives, the algorithm answers 1 (rejects the request).",
"We now describe how the algorithm and oracle work for the important requests.",
"For each $0\\le j\\le {\\log _{s}(n^2)}$ , let $n_{m-j}=\\left|V^{m-j}\\right|$ .",
"For the requests (whose indices are) in $V^{m-j}$ , we use the covering design based $c$ -competitive algorithm for unweighted $\\mathsf {AOC}$ -problems.",
"This requires $B(n_{m-j}, c) + O(\\log n_{m-j})$ bits of advice.",
"Since $B(n,c)$ is linear in $n$ , this means that we use a total of $b=\\sum _{j=0}^{{\\log _{s}(n^2)}} \\big (B(n_{m-j}, c) + O(\\log n_{m-j})\\big )\\le B(n,c)+O(\\log _{s} n \\cdot \\log n)$ bits of advice.",
"Note that $\\log _{s}(n)\\le 2\\varepsilon ^{-1}\\log n$ for $\\varepsilon /2\\le 1$ , giving the bound on the advice in the statement of the theorem.",
"We now prove that the algorithm achieves the desired competitiveness.",
"We can ignore the huge requests, since neither $\\textsc {Alg} $ nor $\\textsc {Opt} $ accepts any of them.",
"Let $V_{\\textsc {Alg}}$ be those rounds in which $\\textsc {Alg} $ answers 0 and let $V_{\\textsc {Alg}}^k=V_{\\textsc {Alg}}\\cap V^k$ .",
"We consider the important requests first.",
"Fix $0\\le j\\le {\\log _{s}(n^2)}$ .",
"Let $n^{\\textsc {Opt}}_{m-j}=\\left|V_{\\textsc {Opt}} ^{m-j}\\right|$ , i.e., $n^{\\textsc {Opt}}_{m-j}$ is the number of requests in $V^{m-j}$ which are also in the optimal solution $V_{\\textsc {Opt}} $ .",
"By construction, we have $n^{\\textsc {Opt}}_{m-j}\\le c\\left|V_{\\textsc {Alg}}^{m-j}\\right|$ .",
"Since the largest possible weight of a request in $V^{m-j}$ is at most $s$ times larger than the smallest possible weight of a request in $V^{m-j}$ , this implies that $w(V_{\\textsc {Opt}} ^{m-j})\\le s \\cdot c\\cdot w(V_{\\textsc {Alg}}^{m-j})$ .",
"Thus, we get that $\\sum _{j=0}^{{\\log _{s}(n^2)}}w(V_{\\textsc {Opt}} ^{m-j})\\le \\sum _{j=0}^{{\\log _{s}(n^2)}}s \\cdot c\\cdot w(V_{\\textsc {Alg}}^{m-j})= s\\cdot c\\cdot \\textsc {Alg} (\\sigma ).$ We now consider the unimportant requests.",
"If $r_i$ is unimportant, then $w(i)\\le s^{m^{\\prime }-{\\log _{s}(n^2)}}\\le s^{m^{\\prime }} / n^2 \\le w(x_{i_{\\max }})/n^2\\le \\textsc {Opt} (\\sigma )/n^2$ .",
"This implies that $\\sum _{j={\\log _{s}(n^2)}+1}^{\\infty }w(V_{\\textsc {Opt}}^{m-j})\\le n\\frac{\\textsc {Opt} (\\sigma )}{n^2}=\\frac{\\textsc {Opt} (\\sigma )}{n}.$ We conclude that $\\textsc {Opt} (\\sigma )=w(V_{\\textsc {Opt}})=\\sum _{j=0}^{{\\log _{s}(n^2)}}w(V_{\\textsc {Opt}} ^{m-j})+\\sum _{j={\\log _{s}(n^2)}+1}^{\\infty }w(V_{\\textsc {Opt}} ^{m-j}).$ By Eq.",
"(REF ), $\\left( 1-\\frac{1}{n}\\right)\\textsc {Opt} (\\sigma ) \\le \\sum _{j=0}^{{\\log _{s}(n^2)}}w(V_{\\textsc {Opt}} ^{m-j}),$ so by Eq.",
"(REF ), $\\textsc {Opt} (\\sigma ) \\le \\frac{n}{n-1}\\cdot s\\cdot c\\cdot \\textsc {Alg} (\\sigma )$ .",
"Note that for $n\\ge n_0=\\frac{2+2\\varepsilon }{\\varepsilon }$ , $(\\frac{n}{n-1})(1+\\varepsilon /2)\\le (1+\\varepsilon )$ .",
"For inputs of length less than $n_0$ , the oracle writes an optimal solution onto the advice tape, using at most $n_0$ bits.",
"Since $n_0\\le \\frac{4}{\\varepsilon }$ , $b\\in O(\\varepsilon ^{-1}\\log ^2 n)$ as required.",
"For inputs of length at least $n_0$ , we use the algorithm described above.",
"Thus, for every input $\\sigma $ , it holds that $\\textsc {Opt} (\\sigma )\\le (1+\\varepsilon )c\\textsc {Alg} (\\sigma )$ .",
"Since $\\varepsilon $ was arbitrary, this proves the theorem.",
"$\\Box $ It may be surprising that adding weights to $\\mathsf {AOC}\\text{-complete}$ maximization problems has almost no effect, while adding weights to $\\mathsf {AOC}\\text{-complete}$ minimization problems drastically changes the advice complexity.",
"In particular, one might wonder why the technique used in Theorem does not work for minimization problems.",
"The key difference lies in the beginning of the sequence.",
"Let $w_{\\max }$ be the largest weight of a request accepted by $\\textsc {Opt}$ .",
"For maximization problems, the algorithm can safely reject all requests before the first important one.",
"For minimization problems, this approach does not work, since the algorithm must accept a superset of what $\\textsc {Opt}$ accepts in order to ensure that its output is feasible.",
"Thus, rejecting an unimportant request that $\\textsc {Opt}$ accepts may result in an infeasible solution.",
"This essentially means that the algorithm is forced into accepting all requests before the first important request arrives.",
"Accepting all unimportant requests is no problem, since they will not contribute significantly to the total cost.",
"However, accepting even a single huge request can give an unbounded contribution to the algorithm's cost.",
"As shown in Theorem , it is not possible in general for the algorithm to tell if a request in the beginning of the sequence is unimportant or huge without using a lot of advice.",
"However, if the ratio of the largest to the smallest weight is not too large, exponential sparsification is also useful for minimization problems in $\\mathsf {AOC}$ .",
"Essentially, when this ratio is bounded, it is possible for the algorithm to learn a good approximation of $w_{\\max }$ when the first request arrives.",
"This is formalized in Theorem , the proof of which is very similar to the proof of Theorem .",
"If ${\\textsc {P}} \\in \\mathsf {AOC} $ is a minimization problem and $0<\\varepsilon \\le 1$ , then $\\textsc {P}_{\\text{w}}$ with all weights in $[w_{\\text{min}},w_{\\text{max}} ]$ has a $(1+\\varepsilon )c$ -competitive algorithm with advice complexity at most $B(n,c)+O\\left(\\varepsilon ^{-1} \\log ^2 n+ \\log \\left( \\varepsilon ^{-1} \\log \\frac{w_{\\text{max}}}{w_{\\text{min}}}\\right)\\right)\\,.$ Fix $\\varepsilon >0$ .",
"Let $\\sigma =\\langle \\lbrace r_1,w_1\\rbrace ,\\ldots ,\\lbrace r_n, w_n\\rbrace \\rangle $ be the input and let $x=x_1\\ldots x_n\\in \\lbrace 0,1\\rbrace ^n$ specify an optimal solution for $\\sigma $ , with ones indicating membership in the optimal solution.",
"Define $s=1+\\varepsilon /2$ .",
"Let $V_{\\textsc {Opt}} =\\lbrace i\\colon x_i=1\\rbrace $ .",
"Note that $V_{\\textsc {Opt}} $ contains exactly those rounds in which $\\textsc {Opt} $ answers 1 and thus accepts.",
"Furthermore, for $k\\in \\mathbb {Z}$ , let $V^k=\\lbrace i\\colon s^k\\le w(i) <s^{k+1}\\rbrace $ and let $V_{\\textsc {Opt}} ^k=V_{\\textsc {Opt}} \\cap V^k$ .",
"Finally, let $i_{\\max }\\in V_{\\textsc {Opt}} $ be such that $w(i_{\\max })\\ge w(i)$ for every $i\\in V_{\\textsc {Opt}} $ .",
"The oracle computes the unique $m\\in \\mathbb {Z}$ such that $i_{\\max }\\in V_{\\textsc {Opt}} ^m$ .",
"We say that a request $r_i$ is unimportant if $w(i)<s^{m- {\\log _{s}(n^2)}}$ , important if $s^{m-{\\log _{s}(n^2)}} \\le w(i) < s^{m+1}$ , and huge if $w(i) \\ge s^{m+1}$ .",
"The oracle also computes the unique $m\\in \\mathbb {Z}$ such that $s^{m^{\\prime }}\\le w_1 < s^{m^{\\prime }+1}$ and writes the values $n$ and $m-m^{\\prime }$ on the tape in a self-delimiting encoding.",
"The number of advice bits needed to write $m-m^{\\prime }$ is $O(\\log (m-m^{\\prime }))$ .",
"$\\log {(m-m^{\\prime })}& \\le \\log {(\\log _s{w(i_{\\max })} - \\log _s{w_1})}+1 \\\\& \\le \\log {\\log _s{\\frac{w_{\\text{max}}}{w_{\\text{min}}}}}+1\\\\& \\le \\log \\left( 2\\varepsilon ^{-1} \\log \\frac{w_{\\text{max}}}{w_{\\text{min}}} \\right)+1, \\text{ since } \\log _s n \\le 2 \\varepsilon ^{-1}\\log n, \\text{ for } \\varepsilon /2 \\le 1$ Note that since the length of $m-m^{\\prime }$ is not known, we need to use a self-delimiting encoding, which means that we use $O(\\log n+\\log ( \\varepsilon ^{-1} \\log \\frac{w_{\\text{max}}}{w_{\\text{min}}}))$ advice bits at the beginning.",
"This advice allows the algorithm to learn $m$ as soon as the first request arrives.",
"From there on, the algorithm will know if a request is important, unimportant, or huge.",
"Whenever a huge request arrives, the algorithm answers 0 (rejects the request).",
"When an unimportant request arrives, the algorithm answers 1 (accepts the request).",
"We now describe how the algorithm and oracle work for the important requests.",
"For the important requests (whose indices are) in $V^{m-j}$ , we use the covering design based $c$ -competitive algorithm for unweighted $\\mathsf {AOC}$ -problems.",
"This is similar to what we do in the proof of Theorem .",
"The same calculations yield an upper bound on this advice of $B(n,c)+O(\\log _{s} n \\cdot \\log n)$ .",
"Note that $\\log _{s}(n)\\le 2\\varepsilon ^{-1}\\log n$ for $\\varepsilon /2\\le 1$ , giving the bound on the advice in the statement of the theorem.",
"First, we note that the solution produced is valid, since it is a superset of the solution of $\\textsc {Opt}$ .",
"We now argue that the cost of the solution is at most $(1+\\varepsilon )c$ times the cost of $\\textsc {Opt}$ .",
"Following the proof of Theorem and switching the roles of $\\textsc {Opt}$ and $\\textsc {Alg}$ , we have by construction that the cost of the important requests for the algorithm is at most $sc$ times larger than the cost for $\\textsc {Opt}$ on the important requests.",
"For the huge requests, both this algorithm and $\\textsc {Opt}$ incur a cost of zero.",
"We now consider the unimportant requests.",
"If $r_i$ is unimportant, then $w(i)< s^{m-{\\log _{s}(n^2)}}\\le s^{m} / n^2 \\le w(x_{i_{\\max }})/n^2\\le \\textsc {Opt} (\\sigma )/n^2.$ This implies that $\\sum _{j={\\log _{s}(n^2)}+1}^{\\infty }w(V^{m-j})\\le n\\frac{\\textsc {Opt} (\\sigma )}{n^2}=\\frac{\\textsc {Opt} (\\sigma )}{n}.$ Thus, even if the algorithm accepts all unimportant requests and $\\textsc {Opt}$ accepts none of them, it only accepts an additional $\\frac{\\textsc {Opt} (\\sigma )}{n}$ .",
"In total, the algorithm gets a cost of at most $(1+\\frac{1}{n})(1+\\varepsilon /2)c\\textsc {Opt} (\\sigma )$ .",
"For $n\\ge n_0=\\frac{2+\\varepsilon }{\\varepsilon }$ , this is at most $(1+\\varepsilon )c\\textsc {Opt} (\\sigma )$ .",
"For inputs of length less than $n_0$ , the oracle will write an optimal solution onto the advice tape, using at most $n_0$ bits.",
"Since $n_0\\le \\frac{3}{\\varepsilon }$ , $b\\in O(\\varepsilon ^{-1}\\log ^2 n)$ as required.",
"For inputs of length at least $n_0$ , we use the algorithm described above.",
"Thus, for every input $\\sigma $ , it holds that $\\textsc {Opt} (\\sigma )\\le (1+\\varepsilon )c\\textsc {Alg} (\\sigma )$ .",
"$\\Box $"
],
[
"Matching and Other Non-Complete $\\mathsf {AOC}$ Problems",
"We first provide a general theorem that works for all maximization problems in $\\mathsf {AOC}$ , giving better results in some cases than that in Theorem .",
"Let ${\\textsc {P}} \\in \\mathsf {AOC} $ be a maximization problem.",
"If there exists a $c$ -competitive P-algorithm reading $b$ bits of advice, then there exists a $O(c\\cdot \\log n)$ -competitive $\\textsc {P}_{\\text{w}}$ -algorithm reading $O(b+\\log n)$ bits of advice.",
"Use exponential sparsification on the weights with an arbitrary $\\varepsilon $ , say $\\varepsilon =1/2$ , and let $s=1+\\varepsilon $ .",
"For a given request sequence, $\\sigma $ , let $w_{\\max }$ be the maximum weight that $\\textsc {Opt} _{\\textsc {P}_{\\text{w}}}$ accepts.",
"The oracle computes the unique $m \\in \\mathbb {Z}$ such that $w_{\\max } \\in [s^m,s^{m+1})$ .",
"The important requests are those with weight $w$ , where $s^{m-{\\log _s (n^2)}} \\le w < s^{m+1}$ .",
"We consider only the ${\\log _s(n^2)}+1$ important intervals, i.e., the intervals $[s^{i},s^{i+1})$ , $m-{\\log _s(n^2)} \\le i \\le m$ , and index them by $i$ .",
"Let $k$ be the index of the interval of weights contributing the most weight to $\\textsc {Opt} _{\\textsc {P}_{\\text{w}}}(\\sigma )$ .",
"The advice is a self-delimiting encoding of the index, $j$ , of the first request with weight $w \\in [s^{k},s^{k+1})$ , plus the advice used by the given $c$ -competitive P-algorithm.",
"This requires at most $b+O(\\log (n))$ bits of advice.",
"The algorithm rejects all requests before the $j$ th.",
"From the $j$ th request, the algorithm calculates the index $k$ .",
"The algorithm accepts those requests which would be accepted by the P-algorithm when presented with the subsequence of $\\sigma $ consisting of the requests with weights in $[s^{k},s^{k+1})$ .",
"Since, by exponential sparsification, $\\textsc {Opt} _{\\textsc {P}_{\\text{w}}}$ accepts total weight at most $\\frac{1}{n}\\textsc {Opt} _{\\textsc {P}_{\\text{w}}}(\\sigma )$ from requests with unimportant weights, and it accepts at least as much from interval $k$ as from any of the other ${\\log _s (n^2)}+1$ intervals considered, $\\textsc {Opt} _{\\textsc {P}_{\\text{w}}}$ accepts weight at least $(1-\\frac{1}{n})\\frac{\\textsc {Opt} _{\\textsc {P}_{\\text{w}}}(\\sigma )}{{\\log _s (n^2)}+1}$ from interval $k$ .",
"The algorithm, $\\textsc {Alg}$ , described here accepts at least $\\frac{1}{c}$ as many requests as $\\textsc {Opt} _{\\textsc {P}_{\\text{w}}}$ does in this interval, and each of the requests it accepts is at least a fraction $\\frac{1}{s}$ as large as the largest weight in this interval.",
"Thus, $c(1+\\varepsilon )\\textsc {Alg} (\\sigma ) \\ge \\left( \\frac{1-\\frac{1}{n}}{{\\log _s (n^2)}+1}\\right)\\textsc {Opt} _{\\textsc {P}_{\\text{w}}}(\\sigma )$ , so $\\textsc {Alg}$ is $O(c \\log n)$ -competitive.",
"$\\Box $ In the online matching problem, edges arrive one by one.",
"Each request contains the names of the edge's two endpoints (the set of endpoints is not known from the beginning, but revealed gradually as the edges arrive).",
"The algorithm must irrevocably accept or reject them as they arrive, and the goal is to maximize the number of edges accepted.",
"The natural greedy algorithm for this problem is well known to be 2-competitive.",
"In terms of advice, the problem is known to be in $\\mathsf {AOC}$ , but is not $\\mathsf {AOC}\\text{-complete}$ [7].",
"We remark that a version of unweighted online matching with vertex arrivals (incomparable to our weighted matching with edge arrivals) has been studied with advice in [10].",
"There exists a $O(\\log n)$ -competitive algorithm for Weighted Matching reading $O(\\log n)$ bits of advice.",
"The result follows from Theorem since there exists a 2-competitive algorithm without advice for (unweighted) Matching.",
"$\\Box $"
],
[
"Lower bounds",
"First, we present a result which holds for the weighted versions of many maximization problems in $\\mathsf {AOC}$ .",
"It also holds for the weighted versions of $\\mathsf {AOC}\\text{-complete}$ minimization problems, but Theorem gives a much stronger result.",
"For the weighted online versions of Independent Set, Clique, Disjoint Path Allocation, and Matching, an algorithm reading $o(\\log n)$ bits of advice cannot be $f(n)$ -competitive for any function $f$ .",
"To prove Theorem REF , we start by proving the following lemma from which the theorem easily follows.",
"Let ${\\textsc {P}} \\in \\mathsf {AOC} $ and suppose there exists a family $(\\sigma _n)_{n\\in \\mathbb {N}}$ of ${\\textsc {P}} $ -inputs with the following properties: $\\sigma _n=\\langle r_1, r_2,\\ldots , r_n\\rangle $ consists of $n$ requests.",
"$\\sigma _{n+1}$ is obtained by adding a single request to the end of $\\sigma _n$ .",
"If P is a maximization problem, the feasible solutions are those in which at most one request is accepted.",
"If P is a minimization problem, the feasible solutions are those in which at least one request is accepted.",
"Then, no algorithm for the weighted problem $\\textsc {P}_{\\text{w}} $ reading $o(\\log n)$ bits of advice can be $f(n)$ -competitive for any function $f$ .",
"Let $\\textsc {Alg} $ be a $\\textsc {P}_{\\text{w}} $ -algorithm reading at most $b=o(\\log n)$ bits of advice.",
"Let $f(n)>0$ be an arbitrary non-decreasing function of $n$ .",
"We will show that for all sufficiently large $n$ , there exists an input of length $n$ such that the profit obtained by $\\textsc {Opt} $ is at least $f(n)$ times as large as the profit obtained by $\\textsc {Alg} $ .",
"Since $f(n)$ was arbitrary, it follows that $\\textsc {Alg} $ is not $f(n)$ -competitive for any function $f$ .",
"Since $b=o(\\log n)$ , there exists an $N \\in \\mathbb {Z}$ such that for any $n \\ge N$ , $\\textsc {Alg} $ reads less than $\\log (n)-1$ bits of advice on inputs of length at most $n$ .",
"Fix an $n\\ge N$ .",
"For $1\\le i\\le n$ , define the $\\textsc {P}_{\\text{w}} $ -input $\\widehat{\\sigma }_i=\\langle \\lbrace r_1, f(n)\\rbrace , \\lbrace r_2,f(n)^2\\rbrace ,\\ldots ,$ $\\lbrace r_i, f(n)^i\\rbrace \\rangle $ .",
"Consider the set of inputs $\\lbrace \\widehat{\\sigma }_1,\\ldots , \\widehat{\\sigma }_{n}\\rbrace $ .",
"For every $1\\le i\\le n$ , the number of advice bits read by $\\textsc {Alg} $ on the input $\\widehat{\\sigma }_i$ is at most $\\log (n)-1$ (since the length of the input $\\widehat{\\sigma }_i$ is $i\\le n$ ).",
"Thus, by the pigeonhole principle, there must exist two integers $n_1, n_2$ with $n_1<n_2$ such that $\\textsc {Alg} $ reads the same advice on $\\widehat{\\sigma }_{n_1}$ and $\\widehat{\\sigma }_{n_2}$ .",
"If $\\textsc {Alg} $ rejects all requests in $\\widehat{\\sigma }_{n_1}$ , then it achieves a profit of 0 while $\\textsc {Opt}$ obtains a profit of $f(n)^{n_1}$ .",
"If $\\textsc {Alg} $ accepts a request in $\\widehat{\\sigma }_{n_1}$ , then it obtains a profit of at most $f(n)^{n_1}$ .",
"Since $\\textsc {Alg} $ reads the same advice on $\\widehat{\\sigma }_{n_1}$ and $\\widehat{\\sigma }_{n_2}$ and since the two inputs are indistinguishable for the first $n_1$ requests, this means that $\\textsc {Alg} $ also obtains a profit of at most $f(n)^{n_1}$ on the input $\\widehat{\\sigma }_{n_2}$ .",
"But $\\textsc {Opt} (\\widehat{\\sigma }_{n_2})=f(n)^{n_2}$ , and hence $\\textsc {Opt} (\\widehat{\\sigma }_{n_2})/\\textsc {Alg} (\\widehat{\\sigma }_{n_2})\\ge f(n)^{n_2-n_1}\\ge f(n)$ .",
"For minimization problems, we can use the same arguments and the input sequence $\\widehat{\\sigma }_i=\\langle \\lbrace r_1, f(n)^{-1}\\rbrace , \\lbrace r_2,f(n)^{-2}\\rbrace ,\\ldots , \\lbrace r_i, f(n)^{-n}\\rbrace \\rangle $ .",
"$\\Box $ [Proof of Theorem REF ] For Independent Set, we can use the above lemma with a family of cliques $(K_n)_{n\\in \\mathbb {N}}$ , and for Clique, we can use a family of independent sets.",
"For Matching, we can use a family of stars $(K_{1,n})_{n\\in \\mathbb {N}}$ .",
"For Disjoint Path Allocation, we use a path $P_{2n} = \\langle v_1, v_2, \\ldots , v_{2n} \\rangle $ and $r_i = \\langle v_i, v_{i+1}, \\ldots v_{i+n} \\rangle $ .",
"$\\Box $ Returning to the example of Weighted Matching, we now know that $O(\\log n)$ bits suffice to be $O(\\log n)$ -competitive, and that no algorithm can be $f(n)$ -competitive for any function $f$ with $o(\\log n)$ bits of advice.",
"In order to prove that a linear number of advice bits is necessary to achieve constant competitiveness for Weighted Matching, we use a direct product theorem from [14].",
"This uses the concept defined in [14] of a problem being $\\Sigma $ -repeatable.",
"Informally, this means that it is always possible to combine $r$ (sufficiently profitable) input sequences $I_1,I_2,\\ldots ,I_r$ into a single input $g(I_1,I_2,\\ldots ,I_r)$ such that serving this single input gives profit close to that of serving each of the $I_i$ independently and adding the profits.",
"Let $P$ be an online maximization problem and $I$ be the set of possible input sequences.",
"Assume that for every input in $I$ , there are only a finite number of valid outputs.",
"Let $I^*$ be the set of concatenations of sequences (rounds) from $I$ .",
"$P$ is $\\Sigma $ -repeatable with parameters $(k_1,k_2,k_3)$ if there exists a function $g ~:~ I^*\\rightarrow I$ satisfying the following: For every $\\sigma ^* \\in I^*$ with $r$ rounds, $|g(\\sigma ^*)| \\le |\\sigma ^*| +k_1 r$ For every deterministic algorithm $\\textsc {Alg}$ for $P$ , there is a deterministic algorithm $\\textsc {Alg} ^*$ for sequences from $I^*$ , such that for every $\\sigma ^* \\in I^*$ with $r$ rounds, $\\textsc {Alg} ^*(\\sigma ^*) \\ge \\textsc {Alg} (g(\\sigma ^*)-k_2 r$ , Let $\\textsc {Opt} ^*$ denote an optimal algorithm for sequences from $I^*$ .",
"For every $\\sigma ^* \\in I^*$ with $r$ rounds, $\\textsc {Opt} ^*(\\sigma ^*) \\le \\textsc {Opt} (g(\\sigma ^*))+k_3 r$ .",
"An $O(1)$ -competitive algorithm for Weighted Matching must read $\\Omega (n)$ bits of advice.",
"We prove the lower bound using a direct product theorem [14].",
"According to [14], it suffices to show that: (i) Weighted Matching is $\\Sigma $ -repeatable, and (ii) for every $c$ , there exists a probability distribution $p_c$ with finite support such that for every deterministic algorithm $\\textsc {Det} $ without advice, it holds that $\\operatorname{\\mathbb {E}}_{p_c}[\\textsc {Opt} (\\sigma )]\\ge c\\cdot \\operatorname{\\mathbb {E}}_{p_c}[\\textsc {Det} (\\sigma )]$ .",
"Also, there must be a finite upper bound on the profit an algorithm can obtain on an input in the support of $p_c$ .",
"It is trivial to see that Weighted Matching is $\\Sigma $ -repeatable.",
"Fix $c\\ge 1$ and let $k=2c-1$ .",
"We define the probability distribution $p_c$ by specifying a probabilistic adversary: The input graph will be a star $K_{1,m}$ consisting of $m$ edges for some $1\\le m\\le k$ .",
"In round $i$ , the adversary reveals the edge $e_i=(v,v_i)$ where $v_i$ is a new vertex and $v$ is the center vertex of the star.",
"The edge $e_i$ has weight $2^i$ .",
"If $i<k$ , then with probability $1/2$ the adversary will proceed to round $i+1$ , and with probability $1/2$ the input sequence will end.",
"If the adversary reaches round $k$ , it will always stop after revealing the edge $e_k$ of round $k$ .",
"Note that the support of $p_c$ and the largest profit an algorithm can obtain on any input in the support of $p_c$ are both finite.",
"Let $X$ be the random variable which denotes the number of edges revealed by the adversary.",
"Note that $\\Pr (X=j)=2^{-j}$ if $1\\le j<k$ .",
"Consequently, $\\Pr (X=k)=1-\\Pr (X<k)=1-\\sum _{i=1}^{k-1}2^{-i}=2^{-(k-1)}.$ Let $\\textsc {Det} $ be a deterministic algorithm without advice.",
"We may assume that $\\textsc {Det} $ decides in advance on some $1\\le j\\le k$ and accepts the edge $e_j$ (the only other possible deterministic strategy it to never accept an edge, but this is always strictly worse than following any of the $k$ strategies that accepts an edge).",
"If $X<j$ , then the profit obtained by $\\textsc {Det} $ is zero.",
"If $X\\ge j$ , then $\\textsc {Det} $ obtains a profit of $2^j$ .",
"It follows that $\\operatorname{\\mathbb {E}}[\\textsc {Det} (\\sigma )]=\\Pr (X\\ge j)2^j=(1-\\Pr (X<j))2^j=2^{-(j-1)}2^j=2.$ The optimal algorithm $\\textsc {Opt} $ always accepts the last edge of the input.",
"Thus, if $X=j$ , then the profit of $\\textsc {Opt} $ is $2^j$ .",
"It follows that $\\operatorname{\\mathbb {E}}[\\textsc {Opt} (\\sigma )]=\\sum _{j=1}^{k}\\Pr (X=j)2^j=\\sum _{j=1}^{k-1}\\left(2^{-j}2^j\\right)+2^{-(k-1)}2^k=k+1.$ Thus, we conclude that $\\operatorname{\\mathbb {E}}[\\textsc {Opt} (\\sigma )]\\ge \\frac{k+1}{2}\\operatorname{\\mathbb {E}}[\\textsc {Det} (\\sigma )]= c\\operatorname{\\mathbb {E}}[\\textsc {Det} (\\sigma )]$ .",
"$\\Box $ In particular, we cannot achieve constant competitiveness using $O(\\log n)$ bits of advice for Weighted Matching.",
"We leave it as an open problem to close the gap between $\\omega (1)$ and $O(\\log n)$ on the competitiveness of Weighted Matching algorithms with advice complexity $O(\\log n)$ ."
],
[
"Scheduling with Sublinear Advice",
"For the scheduling problems studied, the requests are jobs, each characterized by its size.",
"Each job must be assigned to one of $m$ available machines.",
"If the machines are identical, the load of a job on any machine is simply its size.",
"If the machines are related, each machine has a speed, and the load of a job, $J$ , assigned to a machine with speed $s$ is the size of $J$ divided by $s$ .",
"If the machines are unrelated, each job arrives with a vector specifying its load on each machine.",
"Consider a sequence $\\sigma =\\langle r_1,\\ldots , r_n\\rangle $ of $n$ jobs that arrive online.",
"Each job $r_i\\in \\sigma $ has an associated weight-function $w_i:[m]\\rightarrow \\mathbb {R}_{+}$ .",
"Upon arrival, a job must irrevocably be assigned to one of the $m$ machines.",
"The load $L_j$ of a machine $j\\in [m]$ is defined as $L_j=\\sum _{i\\in M_j}w_i(j)$ where $M_j$ is the set of (indices of) jobs scheduled on machine $j$ .",
"The total load of a schedule for $\\sigma $ is the vector $\\mathbf {L} =(L_1,\\ldots , L_m)$ .",
"We say that $(L_1,\\ldots , L_m) \\le (L^{\\prime }_1,\\ldots , L^{\\prime }_m)$ if and only if $L_i \\le L^{\\prime }_i$ for $1 \\le i \\le m$ .",
"A scheduling problem of the above type is specified by an objective function $f:\\mathbb {R}_+^m\\rightarrow \\mathbb {R}_+$ and by specifying if the goal is to minimize or maximize $f(\\mathbf {L})=f(L_1,\\ldots , L_m)\\in \\mathbb {R}_+$ .",
"We assume that $f$ is non-decreasing, i.e., $f(\\mathbf {L})\\le f(\\mathbf {L}^{\\prime })$ for all $\\mathbf {L} \\le \\mathbf {L}^{\\prime }$ .",
"Some of the classical choices of objective function include: Minimizing the $\\ell _p$ -norm $f_p(\\mathbf {L})=f_p(L_1,\\ldots , L_m)=\\Vert (L_1,\\ldots , L_m)\\Vert _p$ for some $1\\le p\\le \\infty $ .",
"That is, for $1\\le p<\\infty $ , the goal is to minimize $\\left(\\sum _{j\\in [m]} L_j^p\\right)^{1/p}$ and for $p=\\infty $ , the goal is to minimize the makespan $\\max _{j\\in [m]} L_j$ .",
"Maximizing the minimum load $f(\\mathbf {L})=\\min _{j\\in [m]} L_j$ .",
"This is also known as machine covering.",
"Note that this objective function is not a norm$f$ is a norm if $f(\\alpha \\mathbf {v})=|\\alpha |f(\\mathbf {v})$ , $f(\\mathbf {u} +\\mathbf {v})\\le f(\\mathbf {u})+f(\\mathbf {v})$ , and $f(\\mathbf {v})=0 \\Rightarrow \\mathbf {v} =\\mathbf {0}$ ., but it does satisfy that $f(\\alpha \\mathbf {L})=\\alpha f(\\mathbf {L})$ for every $\\alpha \\ge 0$ and $\\mathbf {L} \\in \\mathbb {R}^m_+$ .",
"We begin with a result for unrelated machines.",
"Let P be a scheduling problem on $m$ unrelated machines where the goal is to minimize an objective function $f$ .",
"Assume that $f$ is a non-decreasing norm.",
"Then, for $0<\\varepsilon \\le 1$ , there exists a $(1+\\varepsilon )$ -competitive ${\\textsc {P}} $ -algorithm reading $O\\big ( (\\frac{4}{\\varepsilon } \\log (n)+2)^m \\log (n)\\big )$ bits of advice.",
"In particular, if $m=O(1)$ and $\\varepsilon =\\Omega (1)$ , then there exists a $(1+\\varepsilon )$ -competitive algorithm reading $O(\\operatorname{polylog}(n))$ bits of advice.",
"Since the objective function $f$ is a norm on $\\mathbb {R}^m$ , we will denote it by $\\Vert \\cdot \\Vert $ .",
"Let $\\mathbf {1}_j$ be the $j$ th unit vector (the vector with a 1 in the $j$ th coordinate and 0 elsewhere).",
"Fix an input sequence $\\sigma $ .",
"The oracle starts by computing an arbitrary optimal schedule for $\\sigma $ .",
"Throughout most of this proof, we assume that $n$ is sufficiently large.",
"The necessary conditions are discussed at the end of the proof, along with how to handle small $n$ .",
"Let $\\mathbf {L} _{\\textsc {Opt}}$ be the load-vector of this schedule.",
"Thus, $\\textsc {Opt} (\\sigma )=\\Vert \\mathbf {L} _\\textsc {Opt} \\Vert $ .",
"Let $s=1+\\varepsilon /2$ and let $k$ be the unique integer such that $s^{k}\\le \\Vert \\mathbf {L} _\\textsc {Opt} \\Vert < s^{k+1}$ .",
"A job $r_i\\in \\sigma $ is said to be unimportant if there exists a machine $j\\in [m]$ such that $\\Vert w_i(j)\\mathbf {1}_j\\Vert <s^{k-{\\log _{s}(n^2)}}$ .",
"A job which is not unimportant is important.",
"The oracle uses $O(\\log n)$ bits to encode $n$ using a self-delimiting encoding.",
"It then writes the index $i^{\\prime }$ of the first important job $r_{i^{\\prime }}$ onto the advice tape (or indicates that $\\sigma $ contains no important jobs) using ${\\log (n+1)}$ bits.",
"Let $j^{\\prime }$ be the machine minimizing $\\Vert w_{i^{\\prime }}(j^{\\prime })\\mathbf {1}_{j^{\\prime }}\\Vert $ , where ties are broken arbitrarily.",
"The oracle also writes $\\Delta _{i^{\\prime }} = k-k_{i^{\\prime }}$ , where $k_{i^{\\prime }}$ is the unique integer such that $s^{k_{i^{\\prime }}}\\le \\Vert w_{i^{\\prime }}(j^{\\prime })\\mathbf {1}_{j^{\\prime }}\\Vert < s^{k_{i^{\\prime }}+1}$ onto the advice tape using $O(\\log \\log _s n)$ bits.",
"Scheduling unimportant jobs.",
"If a job $r_i\\in \\sigma $ is unimportant, then the algorithm schedules the job on the machine $j$ minimizing $\\Vert w_i(j)\\mathbf {1}_j\\Vert $ where ties are broken arbitrarily.",
"We now explain how the algorithm knows if a job is unimportant or not.",
"If $r_i\\in \\sigma $ is a job that arrives before the first important job, i.e., if $i< i^{\\prime }$ , then $r_i$ is unimportant by definition.",
"When job $r_{i^{\\prime }}$ arrives, the algorithm can deduce $k$ since it knows $\\Delta _{i^{\\prime }}$ from the advice and since it can compute $\\min _{j} \\Vert w_{i^{\\prime }}(j)\\mathbf {1}_{j}\\Vert $ without help.",
"Knowing $k$ (and the number of jobs $n$ ), the algorithm is able to tell if a job is unimportant or not.",
"Scheduling important jobs.",
"We now describe how the algorithm schedules the important jobs.",
"To this end, we define the type of an important job.",
"For an important job $r_i$ , let $\\Delta _i(1),\\ldots , \\Delta _i(m)$ be defined as follows: For $1 \\le j \\le m$ , if there exists an integer $k_i(j) \\le k$ such that $s^{k_i(j)}\\le \\Vert w_{i}(j)\\mathbf {1}_{j}\\Vert <s^{k_i(j)+1}$ , then $\\Delta _i(j)=k-k_i(j)$ (since $r_i$ is important, $\\Delta _i(j) \\le {\\log _{s}(n^2)}$ ).",
"If no such integer exists, then it must be the case that $ \\Vert w_{i}(j)\\mathbf {1}_{j}\\Vert \\ge s^{k+1}> \\Vert \\mathbf {L} _\\textsc {Opt} \\Vert $ .",
"In this case, we let $\\Delta _i(j)=\\bot $ be a dummy symbol.",
"The type of $r_i$ is the vector $\\mathbf {\\Delta }_i=(\\Delta _i(1),\\ldots ,\\Delta _i(m))$ .",
"Note that there are only $(\\lceil \\log _{s}(n^2)\\rceil +2)^m$ different types.",
"For each possible type $\\mathbf {\\Delta }=(\\Delta (1),\\ldots , \\Delta (m))$ , the oracle writes the number, $a_{\\mathbf {\\Delta }}$ , of jobs of type $\\mathbf {\\Delta }$ onto the advice tape.",
"This requires at most $(\\lceil \\log _{s}(n^2)\\rceil +2)^m\\lceil \\log (n+1)\\rceil $ bits of advice.",
"Note that since $\\Vert \\cdot \\Vert $ is a norm, if $r_i\\in \\sigma $ is of type $\\mathbf {\\Delta }_i=(\\Delta _i(1),\\ldots , \\Delta _i(m))$ , then $s^{k-\\Delta _i(j)}\\Vert \\mathbf {1}_j\\Vert ^{-1}\\le w_i(j)\\le s^{k-\\Delta _i(j)+1}\\Vert \\mathbf {1}_j\\Vert ^{-1}$ if $\\Delta _i(j)\\ne \\bot $ and $w_i(j)>\\Vert \\mathbf {L} _\\textsc {Opt} \\Vert \\Vert \\mathbf {1}_j\\Vert ^{-1}$ if $\\Delta _i(j)=\\bot $ .",
"The algorithm computes an optimal schedule $\\widehat{S}_{\\text{$\\text{imp}$}}$ for the input $\\widehat{\\sigma }$ which for each possible type $\\mathbf {\\Delta }$ contains $a_{\\mathbf {\\Delta }}$ jobs with weight-function $\\widehat{w}_{\\mathbf {\\Delta }}$ where $\\widehat{w}_{\\mathbf {\\Delta }}(j)=s^{k-\\Delta (j)+1}\\Vert \\mathbf {1}_j\\Vert ^{-1}$ if $k(j)\\ne \\bot $ and $\\widehat{w}_{\\mathbf {\\Delta }}(j)=\\infty $ otherwise.",
"This choice of weight-function ensures that if $r_{i}\\in \\sigma $ is a job of type $\\mathbf {\\Delta }_i$ , then for each $j$ with $\\Delta _i(j)\\ne \\bot $ , $w_{i}(j)< \\widehat{w}_{\\mathbf {\\Delta }_i}(j)\\le s \\cdot w_{i}(j).$ When an important job of $\\sigma $ arrives, the algorithm computes the type of the job.",
"Based solely on this type, the algorithm schedules the important jobs in $\\sigma $ by following the schedule $\\widehat{S}_{\\text{imp}}$ for $\\widehat{\\sigma }$ .",
"Let $\\mathbf {L} _{\\text{imp}}$ be the load-vector of the important jobs of $\\sigma $ scheduled by $\\textsc {Alg} $ .",
"Note that by Eq.",
"(REF ), the weight-function of an important job of $\\sigma $ is strictly smaller (for all machines) than the weight-function of the corresponding job of $\\widehat{\\sigma }$ .",
"Thus, since $f$ is non-decreasing $\\Vert \\mathbf {L} _{\\text{imp}}\\Vert $ is bounded from above by the cost of the schedule $\\widehat{S}_{\\text{imp}}$ for $\\widehat{\\sigma }$ .",
"Putting it all together.",
"The optimal schedule for $\\sigma $ computed by the oracle induces a schedule of $\\widehat{\\sigma }$ .",
"Let $\\widehat{\\mathbf {L}}$ be the load-vector of this schedule.",
"By Eq.",
"(REF ), we get that $\\Vert \\widehat{\\mathbf {L}}\\Vert \\le s \\Vert \\mathbf {L} _\\textsc {Opt} \\Vert $ .",
"Thus, the cost of $\\widehat{S}_{\\text{$\\text{imp}$}}$ (which was an optimal scheduling of $\\widehat{\\sigma }$ ) is at most $\\Vert \\widehat{\\mathbf {L}}\\Vert \\le s\\Vert \\mathbf {L} _\\textsc {Opt} \\Vert $ .",
"Let $\\mathbf {L} _{\\text{unimp}}$ be the load-vector of the unimportant jobs scheduled by $\\textsc {Alg} $ .",
"Furthermore, let $M_j$ be the set of indices of the unimportant jobs scheduled by $\\textsc {Alg} $ on machine $j$ .",
"By subadditivity, $\\Vert \\mathbf {L} _{\\text{unimp}}\\Vert &=\\left\\Vert \\sum _{j=1}^m\\sum _{i\\in M_j}w_i(j)\\mathbf {1}_j\\right\\Vert \\le \\sum _{j=1}^m\\sum _{i\\in M_j} \\Vert w_i(j)\\mathbf {1}_j\\Vert < \\sum _{j=1}^m\\sum _{i\\in M_j} s^{k-\\lceil \\log _{s}(n^2)\\rceil } \\\\&\\le n \\frac{\\Vert \\mathbf {L} _\\textsc {Opt} \\Vert }{n^2}\\le \\frac{\\Vert \\mathbf {L} _\\textsc {Opt} \\Vert }{n}.$ We are finally able to bound the cost of the entire schedule created by $\\textsc {Alg} $ : $\\textsc {Alg} (\\sigma )=\\Vert \\mathbf {L} _{\\text{imp}}+\\mathbf {L} _{\\text{unimp}}\\Vert \\le \\Vert \\mathbf {L} _{\\text{imp}}\\Vert +\\Vert \\mathbf {L} _{\\text{unimp}}\\Vert \\le (s+1/n)\\Vert \\mathbf {L} _\\textsc {Opt} \\Vert $ Recall that $s=1+\\varepsilon / 2$ .",
"Thus, if $n \\ge 2/\\varepsilon $ , then $\\textsc {Alg} (\\sigma )\\le (1+\\varepsilon )\\textsc {Opt} (\\sigma )$ .",
"For inputs of length less than $2/\\varepsilon $ , the oracle can simply encode the optimal solution using at most $\\frac{2}{\\varepsilon }\\lceil \\log m\\rceil $ bits of advice.",
"The total amount of advice used by our algorithm is at most $(\\lceil \\log _{s}(n^2)\\rceil +2)^m\\lceil \\log (n+1)\\rceil +O(\\log n+\\log \\log _s n)=O\\big ( (4\\varepsilon ^{-1} \\log (n)+2)^m \\log (n)\\big ).$ $\\Box $ For the following discussion, assume that $\\varepsilon =\\Theta (1)$ .",
"We remark that the $(1+\\varepsilon )$ -competitive algorithm in Theorem is only of interest if the number of machines $m$ is small compared to the number of jobs $n$ .",
"As already noted, the most interesting aspect of Theorem is that our algorithm uses only $\\operatorname{polylog}(n)$ bits of advice if $m$ is a constant.",
"More generally, if $m=o(\\log n / \\log \\log n)$ , then our algorithm will use $o(n)$ bits of advice.",
"On the other hand, if $m=\\Theta (\\log n)$ , then our algorithm uses $\\Omega (\\log (n)^{\\log (n)})$ bits of advice, which is worse than the trivial 1-competitive algorithm which uses $n\\lceil \\log m\\rceil =O(n\\log \\log n )$ bits of advice when $m=\\Theta (\\log n)$ .",
"The advice complexity of the algorithm in Theorem depends on the number of machines $m$ because we want the result to hold even when the machines are unrelated.",
"We now show that when restricting to related machines, we can obtain a $(1+\\varepsilon )$ -competitive algorithm using $O(\\varepsilon ^{-1}\\log ^2 n)$ bits of advice, independent of the number of machines.",
"The proof resembles that of Theorem .",
"The main difference is that we are able to reduce the number of types to $O(\\log ^2 n)$ .",
"Let P be a scheduling problem on $m$ related machines where the goal is to minimize an objective function $f$ .",
"Assume that $f$ is a non-decreasing norm.",
"Then, for $0<\\varepsilon \\le 1$ , there exists a $(1+\\varepsilon )$ -competitive ${\\textsc {P}} $ -algorithm with advice complexity $O\\big (\\varepsilon ^{-1}\\log ^2 n\\big ).$ Since the objective function $f$ is a norm on $\\mathbb {R}^m$ , we will denote it by $\\Vert \\cdot \\Vert $ .",
"Fix an input sequence $\\sigma $ .",
"The oracle starts by computing an arbitrary optimal schedule for $\\sigma $ .",
"Let $s=1+\\varepsilon /2$ .",
"The oracle uses $O(\\log n)$ bits to encode $n$ using a self-delimiting encoding.",
"Let $C_1,\\ldots , C_m$ be the speeds of the $m$ machines.",
"Assume without loss of generality that $\\Vert \\mathbf {1}_j\\Vert /C_j$ attains its minimum value when $j=1$ .",
"Define $B=\\Vert \\mathbf {1}_1\\Vert /C_1$ .",
"Let $\\mathbf {L} _{\\textsc {Opt}}$ be the load-vector of the fixed optimal schedule.",
"Thus, $\\textsc {Opt} (\\sigma )=\\Vert \\mathbf {L} _\\textsc {Opt} \\Vert $ .",
"Let $k$ be the unique integer such that $s^{k}\\le \\Vert \\mathbf {L} _\\textsc {Opt} \\Vert < s^{k+1}$ .",
"A job $r_i\\in \\sigma $ is said to be unimportant if its weight, $w_i$ , satisfies $w_iB < s^{k-{\\log _{s}(n^2)}}$ .",
"A job which is not unimportant is important.",
"Note that $w_iB$ is always bounded from above by $\\Vert \\mathbf {L} _\\textsc {Opt} \\Vert $ since $r_i$ must be placed on some machine.",
"The oracle writes the index $i^{\\prime }$ of the first important job $r_{i^{\\prime }}$ onto the advice tape (or indicates that $\\sigma $ contains no important jobs) using ${\\log n}+1$ bits.",
"The oracle also writes the unique integer $k^{\\prime }$ such that $s^{k-k^{\\prime }}\\le w_{i^{\\prime }}B < s^{k-k^{\\prime }+1}$ onto the advice tape, using $O(\\log \\log _{s}(n))$ bits.",
"We now explain how the algorithm knows if a job is unimportant or not.",
"If $r_i\\in \\sigma $ is a job that arrives before the first important job, i.e., if $i< i^{\\prime }$ , then $r_i$ is unimportant by definition.",
"When job $r_{i^{\\prime }}$ arrives, the algorithm can deduce $k$ since it knows $k^{\\prime }$ from the advice and since it can compute $w_{i^{\\prime }}B$ without help.",
"Knowing $k$ (and the number of jobs $n$ ), the algorithm is able to tell if a job is unimportant or not.",
"If a job $r_i\\in \\sigma $ is unimportant, then the algorithm schedules the job on machine 1.",
"Scheduling important jobs.",
"We now describe how the algorithm schedules the important jobs.",
"To this end, we define the type of an important job.",
"The type of an important job $r_i$ is the non-negative integer $t_i$ such that $s^{k-t_i}\\le w_iB< s^{k-t_i+1}$ .",
"Note that there are only ${\\log _{s}(n^2)}+1$ different types.",
"For each possible type $0\\le t\\le {\\log _{s}(n^2)}$ , the oracle writes the number of jobs $a_t$ of that type onto the advice tape.",
"This requires at most $O(\\log _{s}(n^2)\\log (n))$ bits of advice.",
"Note that since $\\Vert \\cdot \\Vert $ is a norm, if $r_i\\in \\sigma $ is of type $t_i$ , then $s^{k-t_i}B^{-1} \\le w_i\\le s^{k-t_i+1}B^{-1}$ .",
"The algorithm computes an optimal schedule $\\widehat{S}_{\\text{$\\text{imp}$}}$ for the input $\\widehat{\\sigma }$ which for each possible type $0\\le t\\le {\\log _{s}(n^2)}$ contains $a_{t}$ jobs with weight $\\widehat{w}_t=s^{k-k_i+1}B^{-1}$ .",
"This choice of weight ensures that if $r_{i}\\in \\sigma $ is a job of type $t_{i}$ , then, $w_{i}< \\widehat{w}_{t_i}\\le s\\cdot w_{i}.$ When an important job of $\\sigma $ arrives, the algorithm computes the type of the job.",
"Based solely on this type, the algorithm schedules the important jobs in $\\sigma $ by following the schedule $\\widehat{S}_{\\text{imp}}$ for $\\widehat{\\sigma }$ .",
"Let $\\mathbf {L} _{\\text{imp}}$ be the load-vector of the important jobs of $\\sigma $ scheduled by $\\textsc {Alg} $ .",
"Note that by Eq.",
"(REF ), the weight of an important job of $\\sigma $ is strictly smaller than the weight of the corresponding job of $\\widehat{\\sigma }$ .",
"Thus, $\\Vert \\mathbf {L} _{\\text{imp}}\\Vert $ is bounded from above by the cost of the schedule $\\widehat{S}_{\\text{imp}}$ for $\\widehat{\\sigma }$ .",
"Putting it all together.",
"The fixed optimal schedule for $\\sigma $ induces a scheduling of $\\widehat{\\sigma }$ .",
"Let $\\widehat{\\mathbf {L}}$ be the load-vector of this schedule.",
"By Eq.",
"(REF ), we get that $\\widehat{\\mathbf {L}}\\le s\\mathbf {L} _\\textsc {Opt} $ .",
"Thus, the cost of $\\widehat{S}_{\\text{$\\text{imp}$}}$ (which was an optimal scheduling of $\\widehat{\\sigma }$ ) is at most $\\Vert \\widehat{\\mathbf {L}}\\Vert \\le s\\Vert \\mathbf {L} _\\textsc {Opt} \\Vert $ .",
"Let $W_u$ be the total weight of unimportant jobs scheduled on machine 1 by $\\textsc {Alg} $ .",
"We have that $\\Vert (W_u/C_1)\\mathbf {1}_1\\Vert =W_uB\\le n s^{k-{\\log _{s}(n^2)}}\\le n\\frac{\\Vert \\mathbf {L} _\\textsc {Opt} \\Vert }{n^2}\\le \\frac{\\Vert \\mathbf {L} _\\textsc {Opt} \\Vert }{n}.$ We are finally able to bound the cost of the entire schedule $S_{\\text{imp}}\\cup S_{\\text{unimp}}$ created by $\\textsc {Alg} $ : $\\textsc {Alg} (\\sigma )=\\Vert \\mathbf {L} _{\\text{imp}}+(W_u/C_1)\\mathbf {1}_1\\Vert \\le \\Vert \\mathbf {L} _{\\text{imp}}\\Vert +\\Vert (W_u/C_1)\\mathbf {1}_1\\Vert \\le (s+1/n)\\Vert \\mathbf {L} _\\textsc {Opt} \\Vert $ Recall that $s=1+\\varepsilon / 2$ .",
"Thus, if $n>2/\\varepsilon $ , then $\\textsc {Alg} (\\sigma )\\le (1+\\varepsilon )\\textsc {Opt} (\\sigma )$ .",
"For inputs of length less than $2/\\varepsilon $ , the oracle can simply encode the optimal solution using at most $\\frac{2}{\\varepsilon }\\lceil \\log m\\rceil $ bits of advice.",
"The total amount of advice used by our algorithm is $O(\\varepsilon ^{-1}\\log ^2 n)$ .",
"$\\Box $ We now consider scheduling problems where the goal is to maximize an objective function $f$ .",
"Recall that we assume that the objective function is non-decreasing.",
"The most notable example is when $f$ is the minimum load.",
"In the following theorem, we show how to schedule almost optimally on unrelated machines with only a rather weak constraint on $f$ (weaker than $f$ being a norm).",
"Let ${\\textsc {P}} $ be a scheduling problem on $m$ unrelated machines where the goal is to maximize an objective function $f$ .",
"Assume that $f$ is non-decreasing, that $f(\\alpha \\mathbf {L}) \\le \\alpha f(\\mathbf {L})$ for every $\\alpha \\ge 0$ , and $\\mathbf {L} \\in \\mathbb {R}^m_+$ .",
"Then, for every $0<\\varepsilon \\le 1$ , there exists a $(1+\\varepsilon )$ -competitive ${\\textsc {P}} $ -algorithm with advice complexity $O((\\frac{4}{\\varepsilon } \\log (n)+2)^mm^2\\log n).$ In particular, if $m=O(1)$ and $\\varepsilon = \\Omega (1)$ , the advice complexity is $O(\\operatorname{polylog}(n))$ .",
"Fix an input sequence $\\sigma $ and an arbitrary optimal schedule.",
"Let $s=1+\\varepsilon /2$ .",
"The oracle uses $O(\\log n)$ bits to encode $n$ using a self-delimiting encoding.",
"For $1\\le j\\le m$ , let $L_j$ be the load on machine $j$ in the optimal schedule.",
"Furthermore, let $k_j$ be the unique integer such that $s^{k_j}\\le L_j<s^{k_j+1}$ .",
"We say that a job $r_i$ is unimportant to machine $j$ if $w_i(j)<s^{k_j-\\lceil \\log _{s}(n^2)\\rceil }$ , important to machine $j$ if $s^{k_j-\\lceil \\log _{s}(n^2)\\rceil }\\le w_i(j)<s^{k_j+1}$ and huge to machine $j$ if $w_i(j) \\ge s^{k_j+1}$ .",
"Note that if $r_i$ is huge to machine $j$ , $\\textsc {Opt}$ does not schedule $r_i$ on machine $j$ .",
"A job which is important to at least one machine is called important.",
"All other jobs are called unimportant.",
"Note that, by definition, any unimportant job is unimportant (and not huge) to the machine where it is scheduled by $\\textsc {Opt}$ .",
"We number the machines such that the first job which is important to machine $j$ arrives no later than the first job which is important to machine $j^{\\prime }$ for every $j<j^{\\prime }$ .",
"This numbering is written to the advice tape, using $O(m \\log m)$ advice bits.",
"The algorithm works in $m+1$ phases (some of which might be empty).",
"Phase 0 begins when the first request arrives.",
"For $1\\le j\\le m$ , phase $j-1$ ends and phase $j$ begins when the first important job for machine $j$ arrives.",
"Note that the same job could be the first important job for more than one machine.",
"Phase $m$ ends with the last request of $\\sigma $ .",
"For each phase, $j$ , the oracle writes the index, $i$ , of the request starting the phase and the unique integer $\\Delta _i(j)$ such that $s^{k_j-\\Delta _i(j)} \\le w_i(j) < s^{k_j-\\Delta _i(j)+1}$ .",
"The unimportant jobs are scheduled arbitrarily by our algorithm (it will become clear from the analysis of the algorithm that any choice will do).",
"We now describe how the algorithm schedules the important jobs in phase $j$ for $1\\le j\\le m$ .",
"By definition, at any point in phase $j$ , we have received an important job for machines $1,2,\\ldots , j$ and no important job for machine $j+1$ has yet arrived.",
"The type of a job $r_i$ in phase $j$ is a vector $\\mathbf {\\Delta }_i=(\\Delta _i(1),\\ldots , \\Delta _i(j))$ where $\\Delta _i(j^{\\prime })$ is the interval of $r_i$ on machine $j^{\\prime }$ (so $s^{k_j-\\Delta _i(j^{\\prime })} \\le w_i(j^{\\prime }) < s^{k_j-\\Delta _i(j^{\\prime })+1}$ ) or $\\bot $ if the job is not important to machine $j^{\\prime }$ .",
"Note that there are ${(2+\\log _{s}(n^2))^j}$ possible job types in phase $j$ .",
"The oracle considers how the jobs in phase $j$ are scheduled in the fixed optimal schedule.",
"For each job type $\\mathbf {\\Delta }$ and each machine $1\\le j^{\\prime }\\le j$ , the oracle encodes the number of jobs of that type which are scheduled on machine $j^{\\prime }$ during phase $j$ .",
"This can be done using $O(\\lceil 2+\\log _{s}(n^2)\\rceil ^mm\\log n)$ bits of advice for a single phase, and $O(\\lceil 2+\\log _{s}(n^2)\\rceil ^mm^2\\log n)$ bits of advice for all $m$ phases.",
"Equipped with the advice described above, the algorithm simply schedules the important jobs in the current phase based on their types.",
"This ensures that, for each machine $j$ , the total load of important jobs that $\\textsc {Opt}$ schedules on machine $j$ is at most $s$ times as large as the total load of important jobs scheduled by $\\textsc {Alg}$ on machine $j$ (since if $r_i$ and $r_{i^{\\prime }}$ are important to machine $j$ and of the same type, then $w_i(j)< s\\cdot w_{i^{\\prime }}(j)$ ).",
"In order to finish the proof, we need to show that the contribution of unimportant jobs to $\\textsc {Opt} (\\sigma )$ is negligible (recall that all jobs are either important or unimportant).",
"To this end, let $L_j^{\\text{unimp}}$ (resp.",
"$L_j^{\\text{imp}}$ ) be the load on machine $j$ of the unimportant (resp.",
"important) jobs scheduled on that machine in the optimal schedule.",
"Note that $L_j=L_j^{\\text{unimp}}+L_j^{\\text{imp}}$ .",
"By the definition of an unimportant job (and since there trivially can be no more than $n$ unimportant jobs), we find that for every $1\\le j\\le m$ , $L_j^{\\text{unimp}}<n \\cdot s^{-{\\log _s(n^2)}} \\cdot L_j\\le \\frac{L_j}{n}.$ Thus, $L_j=L_j^{\\text{unimp}}+L_j^{\\text{imp}}\\le L_j/n+L_j^{\\text{imp}}$ from which $L_j\\le \\frac{n}{n-1}\\cdot L_j^{\\text{imp}}$ follows, assuming that $n>1$ .",
"Since this holds for all machines, and since as previously argued $\\mathbf {L} _{\\textsc {Opt}}^{\\text{imp}}\\le s\\cdot \\mathbf {L} _{\\textsc {Alg}}^{\\text{imp}}\\le s\\cdot \\mathbf {L} _{\\textsc {Alg}}$ , we get that $\\mathbf {L} _{\\textsc {Opt}}\\le \\frac{n}{n-1} \\mathbf {L} _{\\textsc {Opt}}^{\\text{imp}}\\le s\\frac{n}{n-1} \\mathbf {L} _{\\textsc {Alg}}.$ By assumption, the objective function $f$ satisfies $f(\\alpha \\mathbf {L})\\le \\alpha f(\\mathbf {L})$ and is non-decreasing.",
"Thus, we conclude that $\\textsc {Opt} (\\sigma )&=f(\\mathbf {L} _{\\textsc {Opt}}) \\le f\\left( s\\frac{n}{n-1}\\mathbf {L} _{\\textsc {Alg}}\\right)\\le s\\frac{n}{n-1}f(\\mathbf {L} _{\\textsc {Alg}})=s\\frac{n}{n-1}\\textsc {Alg} (\\sigma ).$ For $n \\ge 2 + \\frac{2}{\\varepsilon }$ , this gives a ratio of at most $1+\\varepsilon $ .",
"$\\Box $ 1"
],
[
"AOC-Complete Problems",
"For completeness, we state the full definition of minASGk from [7]: [[7]] The minimum asymmetric string guessing problem with known history, minASGk, has input $\\langle ?,x_1,\\ldots , x_n\\rangle $ , where $x = x_1 \\ldots x_n \\in \\lbrace 0,1\\rbrace ^n$ , for some $n\\in \\mathbb {N}$ .",
"For $1\\le i \\le n$ , round $i$ proceeds as follows: If $i>1$ , the algorithm learns the correct answer, $x_{i-1}$ , to the request in the previous round.",
"The algorithm answers $y_i=f(x_1,\\ldots , x_{i-1})\\in \\lbrace 0,1\\rbrace $ , where $f$ is a function defined by the algorithm.",
"The output $y=y_1\\ldots y_n$ computed by the algorithm is feasible, if $x\\sqsubseteq y$ .",
"Otherwise, $y$ is infeasible.",
"The cost of a feasible output is $\\numero {y}$ , and the cost of an infeasible output is $\\infty $ .",
"In addition to minASGk, the class of $\\mathsf {AOC}\\text{-complete}$ problems also contains many graph problems.",
"The following four graph problems are studied in the vertex-arrival model, so the requests are vertices, each presented together with its edges to previous vertices.",
"The first three problems are minimization problems and the last one is a maximization problem.",
"In Vertex Cover, an algorithm must accept a set of vertices which constitute a vertex cover, so for every edge in the requested graph, at least one of its endpoints is accepted.",
"For Dominating Set, the accepted vertices must constitute a dominating set, so every vertex in the requested graph must be accepted, or one its neighbors must be accepted.",
"In Cycle Finding, an algorithm must accept a set of vertices inducing a cyclic graph.",
"For Independent Set, the accepted vertices must form an independent set, i.e., no two accepted vertices share an edge.",
"For Disjoint Path Allocation a path $P$ is given, and the requests are subpaths of $P$ .",
"The aim is to accept as many edge disjoint paths as possible.",
"For Set Cover, the requests are finite subsets from a known universe, and the union of the accepted subsets must be the entire universe.",
"The aim is to accept as few subsets as possible."
],
[
"Reductions for Theorem ",
"In the proof of Theorem , a reduction sketch was given for the weighted online version of Vertex Cover.",
"Here we include sketches for the reductions for the weighted versions of Cycle Finding, Dominating Set and Set Cover."
],
[
"Cycle Finding",
"Each input $\\sigma = \\langle x_1, x_2, \\ldots , x_n \\rangle $ to the problem minASGk, is transformed to $f(\\sigma ) = \\langle v_1, v_2, \\ldots , v_n \\rangle $ , where $V=\\lbrace v_1, v_2, \\ldots , v_n \\rbrace $ is the vertex set of a graph with edge set $E=\\lbrace (v_j,v_i) \\colon f^{\\prime }(x_i)=j\\rbrace \\cup \\lbrace (v_\\textsc {min},v_\\textsc {max})\\rbrace $ , where $f^{\\prime }(x_i)$ is the largest $j<i$ such that $x_j=1$ , $\\textsc {max}$ is the largest $i$ such that $x_i=1$ , and $\\textsc {min}$ is the smallest $i$ such that $x_i=1$ .",
"If $\\numero {\\sigma }>2$ , the vertices corresponding to 1s form the only cycle in the graph.",
"The advice used by the minASGk algorithm $\\textsc {Alg}_1$ consists of the advice used by the Cycle Finding algorithm $\\textsc {Alg}_2$ in combination with 1 bit indicating whether or not $\\numero {\\sigma }\\le 2$ and in this case (an encoding of) one or two indices of 1s in the input sequence.",
"If $\\numero {\\sigma }>2$ , then $\\textsc {Alg}_2$ accepts some vertices, and $\\textsc {Alg}_1$ returns a 1 for the $x_i$ corresponding to each of those vertices.",
"If $\\textsc {Alg}_1$ returns a non-optimal feasible set, $\\textsc {Alg}_2$ does too, and the sets have the same weights, so $\\textsc {Alg}_1 (\\sigma ) \\le \\textsc {Alg}_2 (f(\\sigma ))+\\textsc {Opt} (\\sigma )$ .",
"In this case, the weights of the optimal solutions for $\\sigma $ and $f(\\sigma )$ are both the sum of the weights of the elements corresponding to 1s in $\\sigma $ , so $f$ is a length preserving $O(f(n))$ -reduction."
],
[
"Dominating Set",
"Each input $\\sigma = \\langle x_1, x_2, \\ldots , x_n \\rangle $ to the problem minASGk, is transformed to $f(\\sigma ) = \\langle v_1, v_2, \\ldots , v_n \\rangle $ , where $V=\\lbrace v_1, v_2, \\ldots , v_n \\rbrace $ is the vertex set of a graph with edge set $E=\\lbrace (v_i,v_\\textsc {max})\\rbrace .$ , where $\\textsc {max}$ is the largest $i$ such that $x_i=1$ .",
"The advice used by the minASGk algorithm $\\textsc {Alg}_1$ consists of the advice used by the Dominating Set algorithm $\\textsc {Alg}_2$ in combination with 1 bit indicating whether or not $\\numero {\\sigma }=0$ .",
"If $\\numero {\\sigma }\\ge 1$ , then there is another bit of advice indicating whether or not $\\textsc {Alg}_2$ accepted $v_{\\textsc {max}}$ .",
"If $\\textsc {Alg}_2$ did not accept $v_{\\textsc {max}}$ , the advice also contains an index of a vertex corresponding to a 0 in $\\sigma $ which was accepted, plus the index of $v_{\\textsc {max}}$ .",
"If $\\textsc {Alg}_1$ 's solution is feasible, but not optimal, then $\\numero {\\sigma }>0$ and $\\textsc {Alg}_2$ accepts some vertices, and $\\textsc {Alg}_1$ returns a 1 for the $x_i$ corresponding to each of those vertices (though, in the case where $v_{\\textsc {max}}$ was rejected, it answers 1 for $x_{\\textsc {max}}$ and answers 0 for the earlier request indicated by the advice).",
"If $\\numero {\\sigma }>0$ a minimum weight dominating set for $f(\\sigma )$ consists of exactly those vertices corresponding to 1s in $\\sigma $ , so the weights of the optimal solutions for $\\sigma $ and $f(\\sigma )$ are both the sum of the weights of the elements corresponding to 1s in $\\sigma $ , unless $\\textsc {Alg}_2$ did not accept $v_{\\textsc {max}}$ .",
"However, the weight of $x_{\\textsc {max}}\\le \\textsc {Opt} (\\sigma )$ , so $\\textsc {Alg}_1 (\\sigma ) \\le \\textsc {Alg}_2 (f(\\sigma ))+\\textsc {Opt} (\\sigma )$ .",
"Thus, $f$ is a length preserving $O(\\log (n)$ -reduction."
],
[
"Set Cover",
"This reduction is very similar to that for Dominating Set.",
"Each input $\\sigma = \\langle x_1, x_2, \\ldots , x_n \\rangle $ to the problem minASGk, $\\textsc {max}$ is the largest $i$ such that $x_i=1$ .",
"In the set cover instance, the universe is $\\lbrace 1,\\ldots ,n\\rbrace $ , and $f(\\sigma )$ is a set of $n$ requests, where request $i$ is $\\lbrace i\\rbrace $ , unless $i=\\textsc {max}$ , in which case, the set consists of $\\textsc {max}$ and all of the $j$ where $x_j=0$ .",
"As with the reduction to Dominating Set, this is a length preserving $O(\\log (n)$ -reduction."
]
] | 1606.05210 |
[
[
"A Hierarchical Pose-Based Approach to Complex Action Understanding Using\n Dictionaries of Actionlets and Motion Poselets"
],
[
"Abstract In this paper, we introduce a new hierarchical model for human action recognition using body joint locations.",
"Our model can categorize complex actions in videos, and perform spatio-temporal annotations of the atomic actions that compose the complex action being performed.That is, for each atomic action, the model generates temporal action annotations by estimating its starting and ending times, as well as, spatial annotations by inferring the human body parts that are involved in executing the action.",
"our model includes three key novel properties: (i) it can be trained with no spatial supervision, as it can automatically discover active body parts from temporal action annotations only; (ii) it jointly learns flexible representations for motion poselets and actionlets that encode the visual variability of body parts and atomic actions; (iii) a mechanism to discard idle or non-informative body parts which increases its robustness to common pose estimation errors.",
"We evaluate the performance of our method using multiple action recognition benchmarks.",
"Our model consistently outperforms baselines and state-of-the-art action recognition methods."
],
[
"Introduction",
"Human action recognition in video is a key technology for a wide variety of applications, such as smart surveillance, human-robot interaction, and video search.",
"Consequently, it has received wide attention in the computer vision community with a strong focus on recognition of single actions in short video sequences [1], [21], [29], [36].",
"As this area evolves, there has been an increasing interest to develop more flexible models that can extract useful knowledge from longer video sequences, featuring multiple concurrent or sequential actions, which we refer to as complex actions.",
"Furthermore, to facilitate tasks such as video tagging or retrieval, it is important to design models that can identify the spatial and temporal spans of each relevant action.",
"As an example, Figure REF illustrates a potential usage scenario, where an input video featuring a complex action is automatically annotated by identifying its underlying atomic actions and corresponding spatio-temporal spans.",
"Figure: Sample frames from a video sequence featuring a complex action.Our method is able to identify the global complex action, as well as, the temporal andspatial span of meaningful actions (related to actionlets) and local body partconfigurations (related to motion poselets).A promising research direction for reasoning about complex human actions is to explicitly incorporate body pose representations.",
"In effect, as noticed long ago, body poses are highly informative to discriminate among human actions [13].",
"Similarly, recent works have also demonstrated the relevance of explicitly incorporating body pose information in action recognition models [11], [30].",
"While human body pose estimation from color images remains elusive, the emergence of accurate and cost-effective RGBD cameras has enabled the development of robust techniques to identify body joint locations and to infer body poses [25].",
"In this work, we present a new pose-based approach to recognizing and provide detailed information about complex human actions in RGBD videos.",
"Specifically, given a video featuring a complex action, our model can identify the complex action occurring in the video, as well as, the set of atomic actions that compose this complex action.",
"Furthermore, for each atomic action, the model is also able to generate temporal annotations by estimating its starting and ending times, and spatial annotations by inferring the body parts that are involved in the action execution.",
"To achieve this, we propose a hierarchical compositional model that operates at three levels of abstraction: body poses, atomic actions, and complex actions.",
"At the level of body poses, our model learns a dictionary that captures relevant spatio-temporal configurations of body parts.",
"We refer to the components of this dictionary as motion poselets [2], [26].",
"At the level of atomic actions, our model learns a dictionary that captures the main modes of variation in the execution of each action.",
"We refer to the components of this dictionary as actionlets [32].",
"Atoms in both dictionaries are given by linear classifiers that are jointly learned by minimizing an energy function that constraints compositions among motion poselets and actionlets, as well as, their spatial and temporal relations.",
"While our approach can be extended to more general cases, here we focus on modeling atomic actions that can be characterized by the body motions of a single actor, such as running, drinking, or eating.",
"Our model introduces several contributions with respect to prior work [18], [26], [32], [34].",
"First, it presents a novel formulation based on a structural latent SVM model [39] and an initialization scheme based on self-pace learning [15].",
"These provide an efficient and robust mechanism to infer, at test and training time, action labels for each detected motion poselet, as well as, their temporal and spatial span.",
"Second, it presents a multi-modal approach that trains a group of actionlets for each atomic action.",
"This provides a robust method to capture relevant intra-class variations in action execution.",
"Third, it incorporates a garbage collector mechanism that identifies and discards idle or non-informative spatial areas of the input videos.",
"This provides an effective method to process long video sequences.",
"Finally, we provide empirical evidence indicating that the integration of the previous contributions in a single hierarchical model, generates a highly informative and accurate solution that outperforms state-of-the-art approaches."
],
[
"Related Work",
"There is a large body of work on human activity recognition in the computer vision literature [1], [21], [29], [36].",
"We focus on recognizing human actions and activities from videos using pose-based representations and review in the following some of the most relevant previous work.",
"The idea of using human body poses and configurations as an important cue for recognizing human actions has been explored recurrently, as poses provide strong cues on the actions being performed.",
"Initially, most research focused on pose-based action recognition in color videos [8], [27].",
"But due to the development of pose estimation methods on depth images[25], there has been recent interest in pose-based action recognition from RGBD videos [7], [9], [28].",
"Some methods have tackled the problem of jointly recognizing actions and poses in videos [20] and still images [37], with the hope to create positive feedback by solving both tasks simultaneously.",
"One of the most influential pose-based representations in the literature is Poselets, introduced by Bourdev and Malik [3].",
"Their representation relies on the construction of a large set of frequently occurring poses, which is used to represent the pose space in a quantized, compact and discriminative manner.",
"Their approach has been applied to action recognition in still images [19], as well as in videos [26], [33], [41].",
"Researchers have also explored the idea of fusing pose-based cues with other types of visual descriptors.",
"For example, Cheron [5] introduce P-CNN as a framework for incorporating pose-centered CNN features extracted from optical flow and color.",
"In the case of RGBD videos, researchers have proposed the fusion of depth and color features [9], [14].",
"In general, the use of multiple types of features helps to disambiguate some of the most similar actions.",
"Also relevant to our framework are hierarchical models for action recognition.",
"In particular, the use of latent variables as an intermediary representation in the internal layers of the model can be a powerful tool to build discriminative models and meaningful representations [10], [34].",
"An alternative is to learn hierarchical models based on recurrent neural networks [6], but they tend to lack interpretability in their internal layers and require very large amounts of training data to achieve good generalization.",
"While most of the previous work have focused on recognizing single and isolated simple actions, in this paper we are interested in the recognition of complex, composable and concurrent actions and activities.",
"In this setting, a person may be executing multiple actions simultaneously, or in sequence, instead of performing each action in isolation.",
"An example of these is the earlier work of Ramanan and Forsyth [22], with more recent approaches by Yeung [38] and Wei [35].",
"Another recent trend aims at fine-grained detection of actions performed in sequence such as those in a cooking scenario [24], [16].",
"We build our model upon several of these ideas in the literature.",
"Our method extends the state-of-the-art by introducing a model that can perform detailed annotation of videos during testing time but only requires weak supervision at training time.",
"While learning can be done with reduced labels, the hierarchical structure of poselets and actionlets combined with other key mechanisms enable our model to achieve improved performance over competing methods in several evaluation benchmarks."
],
[
"Model Description",
"In this section, we introduce our model for pose-based recognition of complex human actions.",
"Our goal is to build a model with the capability of annotating input videos with the actions being performed, automatically identifying the parts of the body that are involved in each action (spatial localization) along with the temporal span of each action (temporal localization).",
"As our focus is on concurrent and composable activities, we would also like to encode multiple levels of abstraction, such that we can reason about poses, actions, and their compositions.",
"Therefore, we develop a hierarchical compositional framework for modeling and recognizing complex human actions.",
"One of the key contributions of our model is its capability to spatially localize the body regions that are involved in the execution of each action, both at training and testing time.",
"Our training process does not require careful spatial annotation and localization of actions in the training set; instead, it uses temporal annotations of actions only.",
"At test time, it can discover the spatial and temporal span, as well as, the specific configuration of the main body regions executing each action.",
"We now introduce the components of our model and the training process that achieves this goal.",
"Figure: Skeleton representation used for splitting the human body into a set ofspatial regions."
],
[
"Body regions",
"We divide the body pose into $R$ fixed spatial regions and independently compute a pose feature vector for each region.",
"Figure REF illustrates the case when $R = 4$ that we use in all our experiments.",
"Our body pose feature vector consists of the concatenation of two descriptors.",
"At frame $t$ and region $r$ , a descriptor $x^{g}_{t,r}$ encodes geometric information about the spatial configuration of body joints, and a descriptor $x^{m}_{t,r}$ encodes local motion information around each body joint position.",
"We use the geometric descriptor from [18]: we construct six segments that connect pairs of joints at each regionArm segments: wrist-elbow, elbow-shoulder, shoulder-neck, wrist-shoulder, wrist-head, and neck-torso; Leg segments: ankle-knee, knee-hip, hip-hip center, ankle-hip, ankle-torso and hip center-torso and compute 15 angles between those segments.",
"Also, three angles are calculated between a plane formed by three segmentsArm plane: shoulder-elbow-wrist; Leg plane: hip-knee-ankle and the remaining three non-coplanar segments, totalizing an 18-D geometric descriptor (GEO) for every region.",
"Our motion descriptor is based on tracking motion trajectories of key points [31], which in our case coincide with body joint positions.",
"We extract a HOF descriptor using 32x32 RGB patches centered at the joint location for a temporal window of 15 frames.",
"At each joint location, this produces a 108-D descriptor, which we concatenate across all joints in each a region to obtain our motion descriptor.",
"Finally, we apply PCA to reduce the dimensionality of our concatenated motion descriptor to 20.",
"The final descriptor is the concatenation of the geometric and motion descriptors, $x_{t,r} = [x_{t,r}^g ; x_{t,r}^m]$ ."
],
[
"Hierarchical compositional model",
"We propose a hierarchical compositional model that spans three semantic levels.",
"Figure REF shows a schematic of our model.",
"At the top level, our model assumes that each input video has a single complex action label $y$ .",
"Each complex action is composed of a temporal and spatial arrangement of atomic actions with labels $\\mathbf {u}=[u_1,\\dots ,u_T]$ , $u_i \\in \\lbrace 1,\\dots ,S\\rbrace $ .",
"In turn, each atomic action consists of several non-shared actionlets, which correspond to representative sets of pose configurations for action identification, modeling the multimodality of each atomic action.",
"We capture actionlet assignments in $\\mathbf {v}=[v_1,\\dots ,v_T]$ , $v_i \\in \\lbrace 1,\\dots ,A\\rbrace $ .",
"Each actionlet index $v_i$ corresponds to a unique and known actomic action label $u_i$ , so they are related by a mapping $\\mathbf {u} = \\mathbf {u}(\\mathbf {v})$ .",
"At the intermediate level, our model assumes that each actionlet is composed of a temporal arrangement of a subset from $K$ body poses, encoded in $\\mathbf {z} = [z_1,\\dots ,z_T]$ , $z_i \\in \\lbrace 1,\\dots ,K\\rbrace $ , where $K$ is a hyperparameter of the model.",
"These subsets capture pose geometry and local motion, so we call them motion poselets.",
"Finally, at the bottom level, our model identifies motion poselets using a bank of linear classifiers that are applied to the incoming frame descriptors.",
"We build each layer of our hierarchical model on top of BoW representations of labels.",
"To this end, at the bottom level of our hierarchy, and for each body region, we learn a dictionary of motion poselets.",
"Similarly, at the mid-level of our hierarchy, we learn a dictionary of actionlets, using the BoW representation of motion poselets as inputs.",
"At each of these levels, spatio-temporal activations of the respective dictionary words are used to obtain the corresponding histogram encoding the BoW representation.",
"The next two sections provide details on the process to represent and learn the dictionaries of motion poselets and actionlets.",
"Here we discuss our integrated hierarchical model.",
"We formulate our hierarchical model using an energy function.",
"Given a video of $T$ frames corresponding to complex action $y$ encoded by descriptors $\\mathbf {x}$ , with the label vectors $\\mathbf {z}$ for motion poselets, $\\mathbf {v}$ for actionlets and $\\mathbf {u}$ for atomic actions, we define an energy function for a video as: $E(\\mathbf {x},&\\mathbf {v},\\mathbf {z},y) = E_{\\text{motion poselets}}(\\mathbf {z},\\mathbf {x}) \\nonumber \\\\&+ E_{\\text{motion poselets BoW}}(\\mathbf {v},\\mathbf {z}) +E_{\\text{atomic actions BoW}}(\\mathbf {u}(\\mathbf {v}),y) \\nonumber \\\\& + E_{\\text{motion poselets transition}}(\\mathbf {z}) + E_{\\text{actionletstransition}}(\\mathbf {v}).$ Besides the BoW representations and motion poselet classifiers described above, Equation (REF ) includes two energy potentials that encode information related to temporal transitions between pairs of motion poselets ($E_{\\text{motion poseletstransition}}$ ) and actionlets ($E_{\\text{actionlets transition}}$ ).",
"The energy potentials are given by: $&E_{\\text{mot.",
"poselet}}(\\mathbf {z},\\mathbf {x}) = \\sum _{r,t} \\left[ \\sum _{k} {w^r_k}^\\top x_{t,r}\\delta _{z_{(t,r)}}^{k} + \\theta ^r \\delta _{z_{(t,r)}}^{K+1}\\right] \\\\&E_{\\text{mot.",
"poselet BoW}}(\\mathbf {v},\\mathbf {z}) = \\sum _{r,a,k} {\\beta ^r_{a,k}}\\delta _{v_{(t,r)}}^{a}\\delta _{z_{(t,r)}}^{k}\\\\&E_{\\text{atomic act.",
"BoW}}(\\mathbf {u}(\\mathbf {v}),y) =\\sum _{r,s} {\\alpha ^r_{y,s}}\\delta _{u(v_{(t,r)})}^{s} \\\\&E_{\\text{mot.",
"pos.",
"trans.",
"}}(\\mathbf {z}) =\\sum _{r,k_{+1},k^{\\prime }_{+1}} \\eta ^r_{k,k^{\\prime }}\\sum _{t} \\delta _{z_{(t-1,r)}}^{k}\\delta _{z_{(t,r)}}^{k^{\\prime }} \\\\&E_{\\text{acttionlet trans.",
"}}(\\mathbf {v}) =\\sum _{r,a,a^{\\prime }} \\gamma ^r_{a,a^{\\prime }}\\sum _{t}\\delta _{v_{(t-1,r)}}^{a}\\delta _{v_{(t,r)}}^{a^{\\prime }}$ Our goal is to maximize $E(\\mathbf {x},\\mathbf {v},\\mathbf {z},y)$ , and obtain the spatial and temporal arrangement of motion poselets $\\mathbf {z}$ and actionlets $\\mathbf {v}$ , as well as, the underlying complex action $y$ .",
"In the previous equations, we use $\\delta _a^b$ to indicate the Kronecker delta function $\\delta (a = b)$ , and use indexes $k \\in \\lbrace 1,\\dots ,K\\rbrace $ for motion poselets, $a \\in \\lbrace 1,\\dots ,A\\rbrace $ for actionlets, and $s \\in \\lbrace 1,\\dots ,S\\rbrace $ for atomic actions.",
"In the energy term for motion poselets, $w^r_k$ are a set of $K$ linear pose classifiers applied to frame descriptors $x_{t,r}$ , according to the label of the latent variable $z_{t,r}$ .",
"Note that there is a special label $K+1$ ; the role of this label will be explained in Section REF .",
"In the energy potential associated to the BoW representation for motion poselets, $\\mathbf {\\beta }^r$ denotes a set of $A$ mid-level classifiers, whose inputs are histograms of motion poselet labels at those frame annotated as actionlet $a$ .",
"At the highest level, $\\alpha ^r_{y}$ is a linear classifier associated with complex action $y$ , whose input is the histogram of atomic action labels, which are related to actionlet assignments by the mapping function $\\mathbf {u}(\\mathbf {v})$ .",
"Note that all classifiers and labels here correspond to a single region $r$ .",
"We add the contributions of all regions to compute the global energy of the video.",
"The transition terms act as linear classifiers $\\eta ^r$ and $\\gamma ^r$ over histograms of temporal transitions of motion poselets and temporal transitions of actionlets respectively.",
"As we have a special label $K+1$ for motion poselets, the summation index $k_{+1}$ indicates the interval $[1,\\dots ,K+1 ]$ ."
],
[
"Learning motion poselets",
"In our model, motion poselets are learned by treating them as latent variables during training.",
"Before training, we fix the number of motion poselets per region to $K$ .",
"In every region $r$ , we learn an independent set of pose classifiers $\\lbrace w^r_k\\rbrace _{k=1}^K$ , initializing the motion poselet labels using the $k$ -means algorithm.",
"We learn pose classifiers, actionlets and complex actions classifiers jointly, allowing the model to discover discriminative motion poselets useful to detect and recognize complex actions.",
"As shown in previous work, jointly learning linear classifiers to identify body parts and atomic actions improves recognition rates [18], [34], so here we follow a similar hierarchical approach, and integrate learning of motion poselets with the learning of actionlets."
],
[
"Learning actionlets",
"A single linear classifier does not offer enough flexibility to identify atomic actions that exhibit high visual variability.",
"As an example, the atomic action “open” can be associated with “opening a can” or “opening a book”, displaying high variability in action execution.",
"Consequently, we augment our hierarchical model including multiple classifiers to identify different modes of action execution.",
"Inspired by [23], we use the Cattell's Scree test to find a suitable number of actionlets to model each atomic action.",
"Specifically, using the atomic action labels, we compute a descriptor for every video interval using normalized histograms of initial pose labels obtained with $k$ -means.",
"Then, for a particular atomic action $s$ , we compute the eigenvalues $\\lambda (s)$ of the affinity matrix of the atomic action descriptors, which is build using $\\chi ^2$ distance.",
"For each atomic action $s \\in \\lbrace 1,\\dots ,S\\rbrace $ , we find the number of actionlets $G_s$ as $G_s =\\operatornamewithlimits{argmin}_i {\\lambda (s)}_{i+1}^2 / (\\sum _{j=1}^i {\\lambda (s)}_j) + c\\cdot i$ , with $c=2\\cdot 10^{-3}$ .",
"Finally, we cluster the descriptors from each atomic action $s$ running $k$ -means with $k = G_s$ .",
"This scheme generates a set of non-overlapping actionlets to model each single atomic action.",
"In our experiments, we notice that the number of actionlets used to model each atomic action varies typically from 1 to 8.",
"To transfer the new labels to the model, we define $u(v)$ as a function that maps from actionlet label $v$ to the corresponding atomic action label $u$ .",
"A dictionary of actionlets provides a richer representation for actions, where several actionlets will map to a single atomic action.",
"This behavior resembles a max-pooling operation, where at inference time we will choose the set of actionlets that best describe the performed actions in the video, keeping the semantics of the original atomic action labels."
],
[
"A garbage collector for motion poselets",
"While poses are highly informative for action recognition, an input video might contain irrelevant or idle zones, where the underlying poses are noisy or non-discriminative to identify the actions being performed in the video.",
"As a result, low-scoring motion poselets could degrade the pose classifiers during training, decreasing their performance.",
"To deal with this problem, we include in our model a garbage collector mechanism for motion poselets.",
"This mechanism operates by assigning all low-scoring motion poselets to the $(K+1)$ -th pose dictionary entry.",
"These collected poses are associated with a learned score lower than $\\theta ^r$ , as in Equation (REF ).",
"Our experiments show that this mechanism leads to learning more discriminative motion poselet classifiers."
],
[
"Learning",
"Initial actionlet labels.",
"An important step in the training process is the initialization of latent variables.",
"This is a challenging due to the lack of spatial supervision: at each time instance, the available atomic actions can be associated with any of the $R$ body regions.",
"We adopt the machinery of self-paced learning [15] to provide a suitable solution and formulate the association between actions and body regions as an optimization problem.",
"We constrain this optimization using two structural restrictions: i) atomic actions intervals must not overlap in the same region, and ii) a labeled atomic action must be present at least in one region.",
"We formulate the labeling process as a binary Integer Linear Programming (ILP) problem, where we define $b_{r,q}^m=1$ when action interval $q \\in \\lbrace 1,\\dots ,Q_m\\rbrace $ is active in region $r$ of video $m$ ; and $b_{r,q}^m=0$ otherwise.",
"Each action interval $q$ is associated with a single atomic action.",
"We assume that we have initial motion poselet labels $z_{t,r}$ in each frame and region.",
"We describe the action interval $q$ and region $r$ using the histogram $h_{r,q}^m$ of motion poselet labels.",
"We can find the correspondence between action intervals and regions using a formulation that resembles the operation of$k$ -means, but using the structure of the problem to constraint the labels: $\\begin{split}\\text{P1}) \\quad \\min _{b,\\mu } &\\sum _{m=1}^M \\sum _{r=1}^R \\sum _{q=1}^{Q_m} b_{r,q}^md( h_{r,q}^m - \\mu _{a_q}^r) -\\frac{1}{\\lambda } b_{r,q}^m\\\\\\text{s.t.",
"}\\quad & \\sum _{r=1}^R b_{r,q}^m \\ge 1\\text{, }\\forall q\\text{, }\\forall m \\\\& b_{r,q_1}^m + b_{r,q_2}^m \\le 1 \\text{ if } q_1\\cap q_2 \\ne \\emptyset \\text{,}\\forall r\\text{, }\\forall m\\\\& b_{r,q}^m \\in \\lbrace 0,1\\rbrace \\text{, }\\forall q\\text{, }\\forall {r}\\text{, }\\forall m\\end{split}$ with $d( h_{r,q}^m - \\mu _{a_q}^r) = \\sum _{k=1}^K (h_{r,q}^m[k] -\\mu _{a_q}^r[k])^2/(h_{r,q}^m[k] +\\mu _{a_q}^r[k]).$ Here, $\\mu _{a_q}^r$ are the means of the descriptors with action label $a_q$ within region $r$ .",
"We solve $\\text{P1}$ iteratively using a block coordinate descending scheme, alternating between solving $b_{r,q}^m$ with $\\mu _{a}^r$ fixed, which has a trivial solution; and then fixing $\\mu _{a}^r$ to solve $b_{r,q}^m$ , relaxing $\\text{P1}$ to solve a linear program.",
"Note that the second term of the objective function in $\\text{P1}$ resembles the objective function of self-paced learning [15], managing the balance between assigning a single region to every action or assigning all possible regions to the respective action interval.",
"Learning model parameters.",
"We formulate learning the model parameters as a Latent Structural SVM problem [39], with latent variables for motion poselets $\\mathbf {z}$ and actionlets $\\mathbf {v}$ .",
"We find values for parameters in equations (REF -), slack variables $\\xi _i$ , motion poselet labels $\\mathbf {z}_i$ , and actionlet labels $\\mathbf {v}_i$ , by solving: $\\min _{W,\\xi _i,~i=\\lbrace 1,\\dots ,M\\rbrace } \\frac{1}{2}||W||_2^2 + \\frac{C}{M} \\sum _{i=1}^M\\xi _i ,$ where $W^\\top =[\\alpha ^\\top , \\beta ^\\top , w^\\top , \\gamma ^\\top , \\eta ^\\top , \\theta ^\\top ],$ and $ \\begin{split}\\xi _i = \\max _{\\mathbf {z},\\mathbf {v},y} \\lbrace & E(\\mathbf {x}_i, \\mathbf {z}, \\mathbf {v}, y) + \\Delta ( (y_i,\\mathbf {v}_i), (y, \\mathbf {v})) \\\\& - \\max _{\\mathbf {z}_i}{ E(\\mathbf {x}_i, \\mathbf {z}_i, \\mathbf {v}_i, y_i)} \\rbrace , \\; \\;\\; i\\in [1,...M].",
"\\end{split}$ In Equation (REF ), each slack variable $\\xi _i$ quantifies the error of the inferred labeling for video $i$ .",
"We solve Equation (REF ) iteratively using the CCCP algorithm [40], by solving for latent labels $\\mathbf {z}_i$ and $\\mathbf {v}_i$ given model parameters $W$ , temporal atomic action annotations (when available), and labels of complex actions occurring in training videos (see Section REF ).",
"Then, we solve for $W$ via 1-slack formulation using Cutting Plane algorithm [12].",
"The role of the loss function $\\Delta ((y_i,\\mathbf {v}_i),(y,\\mathbf {v}))$ is to penalize inference errors during training.",
"If the true actionlet labels are known in advance, the loss function is the same as in [18] using the actionlets instead of atomic actions: $\\Delta ((y_i,\\mathbf {v}_i),(y,\\mathbf {v})) = \\lambda _y(y_i \\ne y) + \\lambda _v\\frac{1}{T}\\sum _{t=1}^T\\delta ({v_t}_{i} \\ne v_t),$ where ${v_t}_{i}$ is the true actionlet label.",
"If the spatial ordering of actionlets is unknown (hence the latent actionlet formulation), but the temporal composition is known, we can compute a list $A_t$ of possible actionlets for frame $t$ , and include that information on the loss function as $\\Delta ((y_i,\\mathbf {v}_i),(y,\\mathbf {v})) = \\lambda _y(y_i \\ne y) + \\lambda _v\\frac{1}{T}\\sum _{t=1}^T\\delta (v_t \\notin A_t)$"
],
[
"Inference",
"The input to the inference algorithm is a new video sequence with features $\\mathbf {x}$ .",
"The task is to infer the best complex action label $\\hat{y}$ , and to produce the best labeling of actionlets $\\hat{\\mathbf {v}}$ and motion poselets $\\hat{\\mathbf {z}}$ .",
"$\\hat{y}, \\hat{\\mathbf {v}}, \\hat{\\mathbf {z}} = \\operatornamewithlimits{argmax}_{y, \\mathbf {v},\\mathbf {z}} E(\\mathbf {x}, \\mathbf {v}, \\mathbf {z}, y)$ We can solve this by exhaustively enumerating all values of complex actions $y$ , and solving for $\\hat{\\mathbf {v}}$ and $\\hat{\\mathbf {z}}$ using: $\\begin{split}\\hat{\\mathbf {v}}, \\hat{\\mathbf {z}} | y ~ =~ & \\operatornamewithlimits{argmax}_{\\mathbf {v},\\mathbf {z}} ~ \\sum _{r=1}^R \\sum _{t=1}^T \\left( \\alpha ^r_{y,u(v{(t,r)})}+ \\beta ^r_{v_{(t,r)},z_{(t,r)}}\\right.",
"\\\\&\\quad \\quad \\left.+ {w^r_{z_{(t,r)}}}^\\top x_{t,r} \\delta (z_{(t,r)} \\le K) + \\theta ^r \\delta _{z_{(t,r)}}^{K+1} \\right.",
"\\\\& \\quad \\quad \\left.+ \\gamma ^r_{v_{({t-1},r)},v_{(t,r)}} + \\eta ^r_{z_{({t-1},r)},z_{(t,r)}} \\vphantom{{w^r_{z_{(t,r)}}}^\\top x_{t,r}} \\right).",
"\\\\\\end{split}$"
],
[
"Experiments",
"Our experimental validation focuses on evaluating two properties of our model.",
"First, we measure action classification accuracy on several action recognition benchmarks.",
"Second, we measure the performance of our model to provide detailed information about atomic actions and body regions associated to the execution of a complex action.",
"We evaluate our method on four action recognition benchmarks: the MSR-Action3D dataset [17], Concurrent Actions dataset [35], Composable Activities Dataset [18], and sub-JHMDB [11].",
"Using cross-validation, we set $K=100$ in Composable Activities and Concurrent Actions datasets, $K=150$ in sub-JHMDB, and $K=200$ in MSR-Action3D.",
"In all datasets, we fix $\\lambda _y = 100$ and $\\lambda _u = 25$ .",
"The number of actionlets to model each atomic action is estimated using the method described in Section REF .",
"The garbage collector (GC) label $(K+1)$ is automatically assigned during inference according to the learned model parameters $\\theta ^r$ .",
"We initialize the $20\\%$ most dissimilar frames to the $K+1$ label.",
"In practice, at test time, the number of frames labeled as $(K+1)$ ranges from 14% in MSR-Action3D to 29% in sub-JHMDB.",
"Computation is fast during testing.",
"In the Composable Activities dataset, our CPU implementation runs at 300 fps on a 32-core computer, while training time is 3 days, mostly due to the massive execution of the cutting plane algorithm.",
"Using Dynamic Programming, complexity to estimate labels is linear with the number of frames $T$ and quadratic with the number of actionlets $A$ and motion poselets $K$ .",
"In practice, we filter out the majority of combinations of motion poses and actionlets in each frame, using the 400 best combinations of $(k,a)$ according to the value of non-sequential terms in the dynamic program.",
"Details are provided in the supplementary material."
],
[
"Classification of Simple and Isolated Actions",
"As a first experiment, we evaluate the performance of our model on the task of simple and isolated human action recognition in the MSR-Action3D dataset [17].",
"Although our model is tailored at recognizing complex actions, this experiment verifies the performance of our model in the simpler scenario of isolated atomic action classification.",
"The MSR-Action3D dataset provides pre-trimmed depth videos and estimated body poses for isolated actors performing actions from 20 categories.",
"We use 557 videos in a similar setup to [32], where videos from subjects 1, 3, 5, 7, 9 are used for training and the rest for testing.",
"Table REF shows that in this dataset our model achieves classification accuracies comparable to state-of-the-art methods.",
"Table: Recognition accuracy in the MSR-Action3Ddataset."
],
[
"Detection of Concurrent Actions",
"Our second experiment evaluates the performance of our model in a concurrent action recognition setting.",
"In this scenario, the goal is to predict the temporal localization of actions that may occur concurrently in a long video.",
"We evaluate this task on the Concurrent Actions dataset [35], which provides 61 RGBD videos and pose estimation data annotated with 12 action categories.",
"We use a similar evaluation setup as proposed by the authors.",
"We split the dataset into training and testing sets with a 50%-50% ratio.",
"We evaluate performance by measuring precision-recall: a detected action is declared as a true positive if its temporal overlap with the ground truth action interval is larger than 60% of their union, or if the detected interval is completely covered by the ground truth annotation.",
"Our model is tailored at recognizing complex actions that are composed of atomic components.",
"However, in this scenario, only atomic actions are provided and no compositions are explicitly defined.",
"Therefore, we apply a simple preprocessing step: we cluster training videos into groups by comparing the occurrence of atomic actions within each video.",
"The resulting groups are used as complex actions labels in the training videos of this dataset.",
"At inference time, our model outputs a single labeling per video, which corresponds to the atomic action labeling that maximizes the energy of our model.",
"Since there are no thresholds to adjust, our model produces the single precision-recall measurement reported in Table REF .",
"Our model outperforms the state-of-the-art method in this dataset at that recall level.",
"Table: Recognition accuracy in the Concurrent Actions dataset."
],
[
"Recognition of Composable Activities",
"In this experiment, we evaluate the performance of our model to recognize complex and composable human actions.",
"In the evaluation, we use the Composable Activities dataset [18], which provides 693 videos of 14 subjects performing 16 activities.",
"Each activity is a spatio-temporal composition of atomic actions.",
"The dataset provides a total of 26 atomic actions that are shared across activities.",
"We train our model using two levels of supervision during training: i) spatial annotations that map body regions to the execution of each action are made available ii) spatial supervision is not available, and therefore the labels $\\mathbf {v}$ to assign spatial regions to actionlets are treated as latent variables.",
"Table REF summarizes our results.",
"We observe that under both training conditions, our model achieves comparable performance.",
"This indicates that our weakly supervised model can recover some of the information that is missing while performing well at the activity categorization task.",
"In spite of using less supervision at training time, our method outperforms state-of-the-art methodologies that are trained with full spatial supervision.",
"Table: Recognition accuracy in the Composable Activitiesdataset."
],
[
"Action Recognition in RGB Videos",
"Our experiments so far have evaluated the performance of our model in the task of human action recognition in RGBD videos.",
"In this experiment, we explore the use of our model in the problem of human action recognition in RGB videos.",
"For this purpose, we use the sub-JHMDB dataset [11], which focuses on videos depicting 12 actions and where most of the actor body is visible in the image frames.",
"In our validation, we use the 2D body pose configurations provided by the authors and compare against previous methods that also use them.",
"Given that this dataset only includes 2D image coordinates for each body joint, we obtain the geometric descriptor by adding a depth coordinate with a value $z = d$ to joints corresponding to wrist and knees, $z = -d$ to elbows, and $z = 0$ to other joints, so we can compute angles between segments, using $d = 30$ fixed with cross-validation.",
"We summarize the results in Table REF , which shows that our method outperforms alternative state-of-the-art techniques.",
"Table: Recognition accuracy in the sub-JHMDB dataset."
],
[
"Spatio-temporal Annotation of Atomic Actions",
"In this experiment, we study the ability of our model to provide spatial and temporal annotations of relevant atomic actions.",
"Table REF summarizes our results.",
"We report precision-recall rates for the spatio-temporal annotations predicted by our model in the testing videos (first and second rows).",
"Notice that this is a very challenging task.",
"The testing videos do no provide any label, and the model needs to predict both, the temporal extent of each action and the body regions associated with the execution of each action.",
"Although the difficulty of the task, our model shows satisfactory results being able to infer suitable spatio-temporal annotations.",
"We also study the capability of the model to provide spatial and temporal annotations during training.",
"In our first experiment, each video is provided with the temporal extent of each action, so the model only needs to infer the spatial annotations (third row in Table REF ).",
"In a second experiment, we do not provide any temporal or spatial annotation, but only the global action label of each video (fourth row in Table REF ).",
"In both experiments, we observe that the model is still able to infer suitable spatio-temporal annotations.",
"Table: Atomic action annotation performances in the Composable Activitiesdataset.",
"The results show that our model is able to recover spatio-temporalannotations both at training and testing time."
],
[
"Effect of Model Components",
"In this experiment, we study the contribution of key components of the proposed model.",
"First, using the sub-JHMDB dataset, we measure the impact of three components of our model: garbage collector for motion poselets (GC), multimodal modeling of actionlets, and use of latent variables to infer spatial annotation about body regions (latent $\\mathbf {v}$ ).",
"Table REF summarizes our experimental results.",
"Table REF shows that the full version of our model achieves the best performance, with each of the components mentioned above contributing to the overall success of the method.",
"Table: Analysis of contribution to recognition performance fromeach model component in the sub-JHMDB dataset.Second, using the Composable Activities dataset, we also analyze the contribution of the proposed self-paced learning scheme for initializing and training our model.",
"We summarize our results in Table REF by reporting action recognition accuracy under different initialization schemes: i) Random: random initialization of latent variables $\\mathbf {v}$ , ii) Clustering: initialize $\\mathbf {v}$ by first computing a BoW descriptor for the atomic action intervals and then perform $k$ -means clustering, assigning the action intervals to the closer cluster center, and iii) Ours: initialize $\\mathbf {v}$ using the proposed self-paced learning scheme.",
"Our proposed initialization scheme helps the model to achieve its best performance.",
"Table: Results in Composable Activities dataset, with latent 𝐯\\mathbf {v} and different initializations."
],
[
"Qualitative Results",
"Finally, we provide a qualitative analysis of relevant properties of our model.",
"Figure REF shows examples of moving poselets learned in the Composable Activities dataset.",
"We observe that each moving poselet captures a salient body configuration that helps to discriminate among atomic actions.",
"To further illustrate this, Figure REF indicates the most likely underlying atomic action for each moving poselet.",
"Figure REF presents a similar analysis for moving poselets learned in the MSR-Action3D dataset.",
"We also visualize the action annotations produced by our model.",
"Figure REF (top) shows the action labels associated with each body part in a video from the Composable Activities dataset.",
"Figure REF (bottom) illustrates per-body part action annotations for a video in the Concurrent Actions dataset.",
"These examples illustrate the capabilities of our model to correctly annotate the body parts that are involved in the execution of each action, in spite of not having that information during training.",
"Figure: Moving poselets learned from the Composable Activitiesdataset.Figure: Moving poselets learned from the MSR-Action3Ddataset.Figure: Automatic spatio-temporal annotation of atomic actions.",
"Our methoddetects the temporal span and spatial body regions that are involved inthe performance of atomic actions in videos."
],
[
"Conclusions and Future Work",
"We present a hierarchical model for human action recognition using body joint locations.",
"By using a semisupervised approach to jointly learn dictionaries of motions poselets and actionlets, the model demonstrates to be very flexible and informative, to handle visual variations and to provide spatio-temporal annotations of relevant atomic actions and active body part configurations.",
"In particular, the model demonstrates to be competitive with respect to state-of-the -art approaches for complex action recognition, while also proving highly valuable additional information.",
"As future work, the model can be extended to handle multiple actor situations, to use contextual information such as relevant objects, and to identify novel complex actions not present in the training set.",
"Acknowledgements This work was partially funded by the FONDECYT grant 1151018, from CONICYT, Government of Chile; and by the Stanford AI Lab-Toyota Center for Artificial Intelligence Research.",
"I.L.",
"is supported by a PhD studentship from CONICYT."
]
] | 1606.04992 |
[
[
"SQuAD: 100,000+ Questions for Machine Comprehension of Text"
],
[
"Abstract We present the Stanford Question Answering Dataset (SQuAD), a new reading comprehension dataset consisting of 100,000+ questions posed by crowdworkers on a set of Wikipedia articles, where the answer to each question is a segment of text from the corresponding reading passage.",
"We analyze the dataset to understand the types of reasoning required to answer the questions, leaning heavily on dependency and constituency trees.",
"We build a strong logistic regression model, which achieves an F1 score of 51.0%, a significant improvement over a simple baseline (20%).",
"However, human performance (86.8%) is much higher, indicating that the dataset presents a good challenge problem for future research.",
"The dataset is freely available at https://stanford-qa.com"
],
[
"Introduction",
"Reading Comprehension (RC), or the ability to read text and then answer questions about it, is a challenging task for machines, requiring both understanding of natural language and knowledge about the world.",
"Consider the question “what causes precipitation to fall?” posed on the passage in Figure REF .",
"In order to answer the question, one might first locate the relevant part of the passage “precipitation ... falls under gravity”, then reason that “under” refers to a cause (not location), and thus determine the correct answer: “gravity”.",
"How can we get a machine to make progress on the challenging task of reading comprehension?",
"Historically, large, realistic datasets have played a critical role for driving fields forward—famous examples include ImageNet for object recognition [5] and the Penn Treebank for syntactic parsing [13].",
"Existing datasets for RC have one of two shortcomings: (i) those that are high in quality [17], [1] are too small for training modern data-intensive models, while (ii) those that are large [8], [9] are semi-synthetic and do not share the same characteristics as explicit reading comprehension questions.",
"To address the need for a large and high-quality reading comprehension dataset, we present the Stanford Question Answering Dataset v1.0 (SQuAD), freely available at https://stanford-qa.com, consisting of questions posed by crowdworkers on a set of Wikipedia articles, where the answer to every question is a segment of text, or span, from the corresponding reading passage.",
"SQuAD contains 107,785 question-answer pairs on 536 articles, and is almost two orders of magnitude larger than previous manually labeled RC datasets such as MCTest [17].",
"In contrast to prior datasets, SQuAD does not provide a list of answer choices for each question.",
"Rather, systems must select the answer from all possible spans in the passage, thus needing to cope with a fairly large number of candidates.",
"While questions with span-based answers are more constrained than the more interpretative questions found in more advanced standardized tests, we still find a rich diversity of questions and answer types in SQuAD.",
"We develop automatic techniques based on distances in dependency trees to quantify this diversity and stratify the questions by difficulty.",
"The span constraint also comes with the important benefit that span-based answers are easier to evaluate than free-form answers.",
"To assess the difficulty of SQuAD, we implemented a logistic regression model with a range of features.",
"We find that lexicalized and dependency tree path features are important to the performance of the model.",
"We also find that the model performance worsens with increasing complexity of (i) answer types and (ii) syntactic divergence between the question and the sentence containing the answer; interestingly, there is no such degradation for humans.",
"Our best model achieves an F1 score of 51.0%,All experimental results in this paper are on SQuAD v1.0.",
"which is much better than the sliding window baseline (20%).",
"Over the last four months (since June 2016), we have witnessed significant improvements from more sophisticated neural network-based models.",
"For example, [24] obtained 70.3% F1 on SQuAD v1.1 (results on v1.0 are similar).",
"These results are still well behind human performance, which is 86.8% F1 based on inter-annotator agreement.",
"This suggests that there is plenty of room for advancement in modeling and learning on the SQuAD dataset."
],
[
"Existing Datasets",
"We begin with a survey of existing reading comprehension and question answering (QA) datasets, highlighting a variety of task formulation and creation strategies (see Table REF for an overview)."
],
[
"Reading comprehension.",
"A data-driven approach to reading comprehension goes back to [10], who curated a dataset of 600 real 3rd–6th grade reading comprehension questions.",
"Their pattern matching baseline was subsequently improved by a rule-based system [18] and a logistic regression model [15].",
"More recently, [17] curated MCTest, which contains 660 stories created by crowdworkers, with 4 questions per story and 4 answer choices per question.",
"Because many of the questions require commonsense reasoning and reasoning across multiple sentences, the dataset remains quite challenging, though there has been noticeable progress [14], [19], [25].",
"Both curated datasets, although real and difficult, are too small to support very expressive statistical models.",
"Some datasets focus on deeper reasoning abilities.",
"Algebra word problems require understanding a story well enough to turn it into a system of equations, which can be easily solved to produce the answer [12], [11].",
"BAbI [26], a fully synthetic RC dataset, is stratified by different types of reasoning required to solve each task.",
"[4] describe the task of solving 4th grade science exams, and stress the need to reason with world knowledge."
],
[
"Open-domain question answering.",
"The goal of open-domain QA is to answer a question from a large collection of documents.",
"The annual evaluations at the Text REtreival Conference (TREC) [23] led to many advances in open-domain QA, many of which were used in IBM Watson for Jeopardy!",
"[6].",
"Recently, [27] created the WikiQA dataset, which, like SQuAD, use Wikipedia passages as a source of answers, but their task is sentence selection, while ours requires selecting a specific span in the sentence.",
"Selecting the span of text that answers a question is similar to answer extraction, the final step in the open-domain QA pipeline, methods for which include bootstrapping surface patterns [16], using dependency trees [20], and using a factor graph over multiple sentences [22].",
"One key difference between our RC setting and answer extraction is that answer extraction typically exploits the fact that the answer occurs in multiple documents [2], which is more lenient than in our setting, where a system only has access to a single reading passage."
],
[
"Cloze datasets.",
"Recently, researchers have constructed cloze datasets, in which the goal is to predict the missing word (often a named entity) in a passage.",
"Since these datasets can be automatically generated from naturally occurring data, they can be extremely large.",
"The Children's Book Test (CBT) [9], for example, involves predicting a blanked-out word of a sentence given the 20 previous sentences.",
"[8] constructed a corpus of cloze style questions by blanking out entities in abstractive summaries of CNN / Daily News articles; the goal is to fill in the entity based on the original article.",
"While the size of this dataset is impressive, [3] showed that the dataset requires less reasoning than previously thought, and concluded that performance is almost saturated.",
"One difference between SQuAD questions and cloze-style queries is that answers to cloze queries are single words or entities, while answers in SQuAD often include non-entities and can be much longer phrases.",
"Another difference is that SQuAD focuses on questions whose answers are entailed by the passage, whereas the answers to cloze-style queries are merely suggested by the passage."
],
[
"Dataset Collection",
"We collect our dataset in three stages: curating passages, crowdsourcing question-answers on those passages, and obtaining additional answers."
],
[
"Passage curation.",
"To retrieve high-quality articles, we used Project Nayuki's Wikipedia's internal PageRanks to obtain the top 10000 articles of English Wikipedia, from which we sampled 536 articles uniformly at random.",
"From each of these articles, we extracted individual paragraphs, stripping away images, figures, tables, and discarding paragraphs shorter than 500 characters.",
"The result was 23,215 paragraphs for the 536 articles covering a wide range of topics, from musical celebrities to abstract concepts.",
"We partitioned the articles randomly into a training set (80%), a development set (10%), and a test set (10%)."
],
[
"Question-answer collection.",
"Next, we employed crowdworkers to create questions.",
"We used the Daemo platform [7], with Amazon Mechanical Turk as its backend.",
"Crowdworkers were required to have a 97% HIT acceptance rate, a minimum of 1000 HITs, and be located in the United States or Canada.",
"Workers were asked to spend 4 minutes on every paragraph, and paid $9 per hour for the number of hours required to complete the article.",
"The task was reviewed favorably by crowdworkers, receiving positive comments on Turkopticon.",
"On each paragraph, crowdworkers were tasked with asking and answering up to 5 questions on the content of that paragraph.",
"The questions had to be entered in a text field, and the answers had to be highlighted in the paragraph.",
"To guide the workers, tasks contained a sample paragraph, and examples of good and bad questions and answers on that paragraph along with the reasons they were categorized as such.",
"Additionally, crowdworkers were encouraged to ask questions in their own words, without copying word phrases from the paragraph.",
"On the interface, this was reinforced by a reminder prompt at the beginning of every paragraph, and by disabling copy-paste functionality on the paragraph text."
],
[
"Additional answers collection.",
"To get an indication of human performance on SQuAD and to make our evaluation more robust, we obtained at least 2 additional answers for each question in the development and test sets.",
"In the secondary answer generation task, each crowdworker was shown only the questions along with the paragraphs of an article, and asked to select the shortest span in the paragraph that answered the question.",
"If a question was not answerable by a span in the paragraph, workers were asked to submit the question without marking an answer.",
"Workers were recommended a speed of 5 questions for 2 minutes, and paid at the same rate of $9 per hour for the number of hours required for the entire article.",
"Over the development and test sets, 2.6% of questions were marked unanswerable by at least one of the additional crowdworkers."
],
[
"Dataset Analysis",
"To understand the properties of SQuAD, we analyze the questions and answers in the development set.",
"Specifically, we explore the (i) diversity of answer types, (ii) the difficulty of questions in terms of type of reasoning required to answer them, and (iii) the degree of syntactic divergence between the question and answer sentences."
],
[
"Diversity in answers.",
"We automatically categorize the answers as follows: We first separate the numerical and non-numerical answers.",
"The non-numerical answers are categorized using constituency parses and POS tags generated by Stanford CoreNLP.",
"The proper noun phrases are further split into person, location and other entities using NER tags.",
"In Table REF , we can see dates and other numbers make up 19.8% of the data; 32.6% of the answers are proper nouns of three different types; 31.8% are common noun phrases answers; and the remaining 15.8% are made up of adjective phrases, verb phrases, clauses and other types."
],
[
"Reasoning required to answer questions.",
"To get a better understanding of the reasoning required to answer the questions, we sampled 4 questions from each of the 48 articles in the development set, and then manually labeled the examples with the categories shown in Table REF .",
"The results show that all examples have some sort of lexical or syntactic divergence between the question and the answer in the passage.",
"Note that some examples fall into more than one category.",
"Table: We manually labeled 192 examples into one or more of the above categories.",
"Words relevant to the corresponding reasoning type are bolded, and the crowdsourced answer is underlined.Figure: An example walking through the computation of the syntactic divergence between the question Q and answer sentence S.Figure: We use the edit distance between the unlexicalized dependency paths inthe question and the sentence containing the answer to measure syntactic divergence."
],
[
"Stratification by syntactic divergence.",
"We also develop an automatic method to quantify the syntactic divergence between a question and the sentence containing the answer.",
"This provides another way to measure the difficulty of a question and to stratify the dataset, which we return to in Section REF .",
"We illustrate how we measure the divergence with the example in Figure REF .",
"We first detect anchors (word-lemma pairs common to both the question and answer sentences); in the example, the anchor is “first”.",
"The two unlexicalized paths, one from the anchor “first” in the question to the wh-word “what”, and the other from the anchor in the answer sentence and to the answer span “Bainbridge's”, are then extracted from the dependency parse trees.",
"We measure the edit distance between these two paths, which we define as the minimum number of deletions or insertions to transform one path into the other.",
"The syntactic divergence is then defined as the minimum edit distance over all possible anchors.",
"The histogram in Figure REF shows that there is a wide range of syntactic divergence in our dataset.",
"We also show a concrete example where the edit distance is 0 and another where it is 6.",
"Note that our syntactic divergence ignores lexical variation.",
"Also, small divergence does not mean that a question is easy since there could be other candidates with similarly small divergence."
],
[
"Methods",
"We developed a logistic regression model and compare its accuracy with that of three baseline methods."
],
[
"Candidate answer generation.",
"For all four methods, rather than considering all $O(L^2)$ spans as candidate answers, where $L$ is the number of words in the sentence, we only use spans which are constituents in the constituency parse generated by Stanford CoreNLP.",
"Ignoring punctuation and articles, we find that 77.3% of the correct answers in the development set are constituents.",
"This places an effective ceiling on the accuracy of our methods.",
"During training, when the correct answer of an example is not a constituent, we use the shortest constituent containing the correct answer as the target.",
"Table: Features used in the logistic regression model with examples for thequestion “What causes precipitation to fall?”, sentence “Inmeteorology, precipitation is any product of the condensation of atmosphericwater vapor that falls under gravity.” and answer “gravity”.Q denotes question, A denotes candidate answer, and S denotes sentence containing the candidate answer."
],
[
"Sliding Window Baseline",
"For each candidate answer, we compute the unigram/bigram overlap between the sentence containing it (excluding the candidate itself) and the question.",
"We keep all the candidates that have the maximal overlap.",
"Among these, we select the best one using the sliding-window approach proposed in [17].",
"In addition to the basic sliding window approach, we also implemented the distance-based extension [17].",
"Whereas [17] used the entire passage as the context of an answer, we used only the sentence containing the candidate answer for efficiency."
],
[
"Logistic Regression",
"In our logistic regression model, we extract several types of features for each candidate answer.",
"We discretize each continuous feature into 10 equally-sized buckets, building a total of 180 million features, most of which are lexicalized features or dependency tree path features.",
"The descriptions and examples of the features are summarized in Table REF .",
"The matching word and bigram frequencies as well as the root match features help the model pick the correct sentences.",
"Length features bias the model towards picking common lengths and positions for answer spans, while span word frequencies bias the model against uninformative words.",
"Constituent label and span POS tag features guide the model towards the correct answer types.",
"In addition to these basic features, we resolve lexical variation using lexicalized features, and syntactic variation using dependency tree path features.",
"The multiclass log-likelihood loss is optimized using AdaGrad with an initial learning rate of 0.1.",
"Each update is performed on the batch of all questions in a paragraph for efficiency, since they share the same candidates.",
"$L_2$ regularization is used, with a coefficient of 0.1 divided by the number of batches.",
"The model is trained with three passes over the training data."
],
[
"Model Evaluation",
"We use two different metrics to evaluate model accuracy.",
"Both metrics ignore punctuations and articles (a, an, the)."
],
[
"Exact match.",
"This metric measures the percentage of predictions that match any one of the ground truth answers exactly."
],
[
"(Macro-averaged) F1 score.",
"This metric measures the average overlap between the prediction and ground truth answer.",
"We treat the prediction and ground truth as bags of tokens, and compute their F1.",
"We take the maximum F1 over all of the ground truth answers for a given question, and then average over all of the questions."
],
[
"Human Performance",
"We assess human performance on SQuAD's development and test sets.",
"Recall that each of the questions in these sets has at least three answers.",
"To evaluate human performance, we treat the second answer to each question as the human prediction, and keep the other answers as ground truth answers.",
"The resulting human performance score on the test set is 77.0% for the exact match metric, and 86.8% for F1.",
"Mismatch occurs mostly due to inclusion/exclusion of non-essential phrases (e.g., monsoon trough versus movement of the monsoon trough) rather than fundamental disagreements about the answer."
],
[
"Model Performance",
"Table REF shows the performance of our models alongside human performance on the v1.0 of development and test sets.",
"The logistic regression model significantly outperforms the baselines, but underperforms humans.",
"We note that the model is able to select the sentence containing the answer correctly with 79.3% accuracy; hence, the bulk of the difficulty lies in finding the exact span within the sentence.",
"Table: Performance of various methods and humans.",
"Logistic regression outperforms thebaselines, while there is still a significant gap between humans.Table: Performance with feature ablations.",
"We find that lexicalized and dependency tree path features are most important."
],
[
"Feature ablations.",
"In order to understand the features that are responsible for the performance of the logistic regression model, we perform a feature ablation where we remove one group of features from our model at a time.",
"The results, shown in Table REF , indicate that lexicalized and dependency tree path features are most important.",
"Comparing our analysis to the one in [3], we note that the dependency tree path features play a much bigger role in our dataset.",
"Additionally, we note that with lexicalized features, the model significantly overfits the training set; however, we found that increasing $L_2$ regularization hurts performance on the development set.",
"Table: Performance stratified by answer types.",
"Logistic regression performsbetter on certain types of answers, namely numbers and entities.",
"On the otherhand, human performance is more uniform."
],
[
"Performance stratified by answer type.",
"To gain more insight into the performance of our logistic regression model, we report its performance across the answer types explored in Table REF .",
"The results (shown in Table REF ) show that the model performs best on dates and other numbers, categories for which there are usually only a few plausible candidates, and most answers are single tokens.",
"The model is challenged more on other named entities (i.e., location, person and other entities) because there are many more plausible candidates.",
"However, named entities are still relatively easy to identify by their POS tag features.",
"The model performs worst on other answer types, which together form 47.6% of the dataset.",
"Humans have exceptional performance on dates, numbers and all named entities.",
"Their performance on other answer types degrades only slightly."
],
[
"Performance stratified by syntactic divergence.",
"As discussed in Section , another challenging aspect of the dataset is the syntactic divergence between the question and answer sentence.",
"Figure REF shows that the more divergence there is, the lower the performance of the logistic regression model.",
"Interestingly, humans do not seem to be sensitive to syntactic divergence, suggesting that deep understanding is not distracted by superficial differences.",
"Measuring the degree of degradation could therefore be useful in determining the extent to which a model is generalizing in the right way."
],
[
"Conclusion",
"Towards the end goal of natural language understanding, we introduce the Stanford Question Answering Dataset, a large reading comprehension dataset on Wikipedia articles with crowdsourced question-answer pairs.",
"SQuAD features a diverse range of question and answer types.",
"The performance of our logistic regression model, with 51.0% F1, against the human F1 of 86.8% suggests ample opportunity for improvement.",
"We have made our dataset freely available to encourage exploration of more expressive models.",
"Since the release of our dataset, we have already seen considerable interest in building models on this dataset, and the gap between our logistic regression model and human performance has more than halved [24].",
"We expect that the remaining gap will be harder to close, but that such efforts will result in significant advances in reading comprehension."
],
[
"Reproducibility",
"All code, data, and experiments for this paper are available on the CodaLab platform: https://worksheets.codalab.org/worksheets/0xd53d03a48ef64b329c16b9baf0f99b0c/ .",
"We would like to thank Durim Morina and Professor Michael Bernstein for their help in crowdsourcing the collection of our dataset, both in terms of funding and technical support of the Daemo platform."
]
] | 1606.05250 |
[
[
"Jet activity as a probe of high-mass resonance production"
],
[
"Abstract We explore the method of using the measured jet activity associated with a high mass resonance state to determine the corresponding production modes.",
"To demonstrate the potential of the approach, we consider the case of a resonance of mass $M_R$ decaying to a diphoton final state.",
"We perform a Monte Carlo study, considering three mass points $M_R=0.75,\\,1.5\\,,2.5$ TeV, and show that the $\\gamma\\gamma$, $WW$, $gg$ and light and heavy $q\\overline{q}$ initiated cases lead to distinct predictions for the jet multiplicity distributions.",
"As an example, we apply this result to the ATLAS search for resonances in diphoton events, using the 2015 data set of $3.2\\,{\\rm fb}^{-1}$ at $\\sqrt{s}=13$ TeV.",
"Taking the spin-0 selection, we demonstrate that a dominantly $gg$-initiated signal hypothesis is mildly disfavoured, while the $\\gamma\\gamma$ and light quark cases give good descriptions within the limited statistics, and a dominantly $WW$-initiated hypothesis is found to be in strong tension with the data.",
"We also comment on the $b\\overline{b}$ initial state, which can already be constrained by the measured $b$-jet multiplicity.",
"Finally, we present expected exclusion limits with integrated luminosity, and demonstrate that with just a few 10's of ${\\rm fb}^{-1}$ we can expect to constrain the production modes of such a resonance."
],
[
"Introduction",
"Both the ATLAS and CMS collaborations reported the observation of an excess of events in the diphoton mass distribution around 750 GeV [1], [2] in roughly $3\\,{\\rm fb}^{-1}$ of data recorded in 2015 at $\\sqrt{s}=13$ TeV, which was found to be compatible with the data collected at 8 TeV.",
"The possibility that this corresponded to a new resonance state generated a great deal of theoretical interest, and a wide range of BSM models describing it were proposed, see for example [3] for a review and further references.",
"While no significant excess was observed in the larger data set collected in 2016 [4], [5], it is nonetheless interesting to consider how a potential resonance of this type, which might be present at higher mass and hence be observable at the LHC or a future collider, can be produced.",
"One interesting, and in some sense natural possibility, for an excess that is seen in the $\\gamma \\gamma $ decay channel, is that the resonance may couple dominantly to photons, with the coupling to gluons and other coloured particles being either suppressed or absent entirely.",
"This has been discussed in [6], [7], [8], [9], [10], [11], [12], [13], [14], [15].",
"More generally, we may expect significant couplings to quarks and gluons to be present.",
"A method to distinguish between these different production modes, discussed in for instance [16], [8], [9], [17], [18], is to measure the multiplicity of jets produced in addition to the resonance.",
"In this case we may naturally anticipate that if the resonance is dominantly produced in $\\gamma \\gamma $ collisions, then the average jet multiplicity will be lower compared to the $q\\overline{q}$ and $gg$ cases.",
"While additional jets in this case are not parametrically suppressed by $\\sim \\alpha /\\alpha _s$ – no additional $O(\\alpha )$ suppression is introduced by requiring that the quark produced in the initial–state $q\\rightarrow q\\gamma $ splitting leads to a visible jet in the final–state – nonetheless some suppression is present due to the smaller size of $\\alpha $ and lower photon branching probability.",
"It is in addition well known that the particle multiplicity associated with an initial–state gluon is higher compared to the quark case, and so again some difference in jet activity can be expected here.",
"For the vector boson fusion (VBF) $WW$ –initiated channel the resonance is generally produced in association with at least two additional jets, due to the relatively high $p_\\perp $ recoiling quarks in the final state.",
"With this in mind, in this paper we we will consider the case of a scalar resonance $R$ for three mass points $M_R=0.75,\\,1.5\\,,2.5$ TeV, to demonstrate the viability of the method, although this can be readily extended to other spin–parities.",
"We will present a detailed analysis of the expected jet multiplicities corresponding to the different production scenarios, focussing on the $gg$ , light $q\\overline{q}$ and heavy $b\\overline{b}$ , $\\gamma \\gamma $ and $WW$ cases.",
"We will show that the above expectations are indeed born out by a more precise MC analysis, which accounts for the decay of the resonance and full experimental acceptances and jet selection.",
"To demonstrate how such an approach may be applied to data, we will then compare to the 2015 ATLAS [1] measurement of the jet multiplicity in the spin–0 event selection sample, and demonstrate that this limited statistics data already show some mild tension with a dominantly $gg$ –initiated scenario, while the light quark and in particular $\\gamma \\gamma $ –initiated scenarios give good descriptions.",
"The continuum background–only hypothesis also gives a somewhat worse description than these latter cases, and the $WW$ hypothesis is found to give a particularly poor description, with a dominantly $WW$ –initiated production mechanism in strong tension with the data; a similar conclusion will hold for the $ZZ$ , and to a lesser extent, $Z\\gamma $ –initiated mechanisms.",
"In the $b\\overline{b}$ –initiated case there is expected to be a sizeable fraction of $b$ –jets observed in the final state, rendering such a possibility relatively easy to confirm or constrain, even for this limited data set; indeed the ATLAS measurement of the $b$ –jet fraction in the signal region disfavours any sizeable $bb$ –induced production mode.",
"We also compare in an appendix to the 2015 ATLAS data collected with a spin–2 event selection, but still assuming a scalar resonance signal.",
"While the total number of observed events in the signal region is larger, the $S/B$ ratio is lower, and we find that all hypotheses give acceptable descriptions of the data.",
"Although no excess at 750 GeV was observed in the updated 2016 data set, this nonetheless demonstrates the discriminating power of this observable.",
"For this reason we will also present expected exclusion limits on different production scenarios with integrated luminosity, and will show that with just a few 10's of ${\\rm fb}^{-1}$ it is possible to place strong constraints on the $gg$ production mechanism in favour of the $\\gamma \\gamma $ /$q\\overline{q}$ cases, and vice versa.",
"It is more challenging to distinguish between the $u\\overline{u}$ and $\\gamma \\gamma $ modes via this method, although still possible with enough data.",
"The outline of this paper is as follows.",
"In Section we present the details of our analysis and the simplified production model we consider.",
"In Section we present results for the predicted jet multiplicities corresponding to the different production scenarios.",
"In Section we compare to the ATLAS jet multiplicity measurement, and comment on the implications for the various production modes.",
"In Section we present the expected exclusion limits on different production scenarios as a function of the integrated luminosity collected at the LHC.",
"Finally, in Section we conclude.",
"In Appendix we compare to the ATLAS spin–2 event selection, and show that all initial state hypotheses provide acceptable descriptions."
],
[
"Production model and analysis",
"To model the production of a scalar resonance $X\\rightarrow \\gamma \\gamma $ via $\\gamma \\gamma $ , $gg$ and $q\\overline{q}$ initiated production we use an effective theory approach, with corresponding Lagrangian terms $\\nonumber \\mathcal {L}^{gg}&=g_{g} \\,G^{\\mu \\nu }G_{\\mu \\nu }\\,R\\;,\\\\ \\nonumber \\mathcal {L}^{\\gamma \\gamma }&=g_{\\gamma } \\,F^{\\mu \\nu }\\,F_{\\mu \\nu }R\\;,\\\\ \\nonumber \\mathcal {L}^{q\\overline{q}}&=g_q\\, q\\overline{q}R\\;,\\\\\\mathcal {L}^{WW}&=g_W\\, W^{\\mu \\nu }\\,W_{\\mu \\nu }R\\;,$ where we make no assumptions about the size of the couplings $g$ .",
"We consider resonance masses of $M_R=0.75,\\,1.5,\\,2.5$ TeV, with a uniform width of $\\Gamma _{\\rm tot}=45$ GeV, although the results which follow are largely independent of the size of the width.",
"Parton–level events are then generated at LO with up to 2 additional partons in the final–state using MadGraph 5 [19], which are MLM merged at scale $Q_{\\rm cut}=125,\\,250,\\,350$ GeV for $M_R=0.75,\\,1.5,\\,2.5$ TeV, respectively, to parton shower generated with Pythia 8 [20], [21], including hadronization and multiple parton interactions.",
"Events for the Standard Model continuum $\\gamma \\gamma $ process, which proceeds dominantly via $q\\overline{q}\\rightarrow \\gamma \\gamma $ , are generated in the same way.",
"For the $WW$ initial state only the Born–level process, which generally leads to two additional quark jets in the final state, is generated and passed to Pythia for parton shower and hadronization, with no matching required.",
"For the initial–state photon PDFs we apply the approach described in [22], [9], with the MMHT14 LO [23] PDF set used for all other partons, however the normalized distributions we present below are relatively insensitive to these choices.",
"All the analyses performed below make use of Rivet [24].",
"For concreteness, we apply the event selection: Two reconstructed photons, satisfying $p_\\perp >0.4(0.3) M_{\\gamma \\gamma }$ for the leading (subleading) photon and $|\\eta _\\gamma |<2.7$ .",
"The diphoton invariant mass lies in the range $M_R-25 \\,{\\rm GeV} <M_{\\gamma \\gamma }<M_R+25$ GeV.",
"An isolation requirement for all particles in a cone $\\Delta R=0.4$ around the photons direction $E_\\perp ^{\\rm iso}<0.05 E_\\perp ^\\gamma + 6$ GeV.",
"This is guided by the ATLAS [1] selection for the spin–0 resonance sample, however we note that the results which follow are largely insensitive to these precise choices, for example, if a looser set of cuts on the final–state photons is imposed.",
"We reconstruct jets with the anti–$k_t$ algorithm [25] with distance parameter $R=0.6$ , $p_\\perp ^j>25$ GeV and $|\\eta ^j|< 4.4$ .",
"We choose this somewhat larger value of $R$ compared to that taken in [1] as this allows a greater discrimination between the different production modes, while still being relatively insensitive to the influence of the underlying event generated with Pythia for the corresponding minimum jet $p_\\perp $ ."
],
[
"Jet multiplicities",
"Using the event selection and model choice outlined in the previous section we can then evaluate the cross section for resonance production via the $\\gamma \\gamma $ , $gg$ and $q\\overline{q}$ initial–states.",
"In the latter case we will consider both $u\\overline{u}$ and $b\\overline{b}$ , since as we will see there is a non–negligible difference in the expected distributions for light and heavy quark mediated processes.",
"The expected distributions for the other light $d\\overline{d}$ and $s\\overline{s}$ quark initiated modes are very similar to the $u\\overline{u}$ , and thus while we will for concreteness refer in what follows to the `$u\\overline{u}$ ' production mode, this can be considered as the prediction for any light quark–induced process.",
"In addition, as discussed in [18] this method may also be applied to the case of, for example, $WW$ –initiated VBF.",
"We will consider this below, but it is worth pointing out that, as discussed in [9], due to the relatively large mass of the exchanged $t$ –channel $W$ bosons the transverse momentum of the final–state $\\gamma \\gamma $ system may also be a sensitive observable.",
"In particular within the simple approach of that paper we expect that only roughly $\\sim 20$ % of the cross section has $p_\\perp ^{\\gamma \\gamma } < 60$ GeV.",
"The same effect will also be similarly present in the $ZZ$ and $Z\\gamma $ channels.",
"Figure: Exclusive jet multiplicities for minimum jet p ⊥ >p_\\perp > 25 GeV, forthe uu ¯u\\overline{u}, bb ¯b\\overline{b} and WWWW initial–state resonance production processes.",
"Results shown for resonance masses M R =0.75,1.5,2.5M_R=0.75,\\,1.5,\\,2.5 TeV.Figure: Exclusive jet multiplicities for minimum jet p ⊥ >p_\\perp > 50 GeV, for the uu ¯u\\overline{u}, bb ¯b\\overline{b} and WWWW initial–state resonance production processes.",
"Results shown for resonance masses M R =0.75,1.5,2.5M_R=0.75,\\,1.5,\\,2.5 TeV.In Figs.",
"REF and REF we show the cross section for resonance production accompanied by $N_{\\rm jet}$ jets, for $p_\\perp ^j>$ 25 (50) GeV, respectively.",
"The results are presented with the fixed merging scales described in the previous section, but we have checked that when varying these between a wider range the results presented below do not vary by more than $\\sim 10\\%$ , and usually much lower; this can be considered as an estimate of the theoretical uncertainty in our approach, and all results should be interpreted with this in mind.",
"We can see that the difference between the $gg$ and $\\gamma \\gamma $ initial–states for the lower $p_\\perp ^j>$ 25 GeV cut is dramatic, with over $50\\%$ of the $\\gamma \\gamma $ initiated events having no jets passing the selection, while for the $gg$ –initiated process this is below $20\\%$ .",
"In addition, the overall shape of the jet multiplicity distribution is correspondingly different, with the $\\gamma \\gamma $ distribution peaking at 0 jets, while the $gg$ exhibits a peak at $N_{\\rm jet}=1-2$ .",
"At higher $p_\\perp ^j>$ 50 GeV the difference is less dramatic, although the population of the 0 jet bin is still almost a factor of 2 higher in the $\\gamma \\gamma $ case.",
"The cause of this remarkably different behaviour is discussed in detail in [9], and is generated by the simple fact that the initial–state coloured gluons exhibit a strong preference to radiate further, so that there is a strong Sudakov suppression in the cross section for no further emission within the corresponding acceptance; the initial–state photons, on the other hand, exhibit a much weaker preference to radiate further, and while for example the parent quark in the $q\\rightarrow q\\gamma $ splitting may lead to an observable jet in the final–state, the effect of the jet veto is much less significant.",
"A similar trend is seen in all three mass points, although in general the jet multiplicity is observed to increase with increasing mass for the $gg$ and continuum cases, as expected from the increasing phase space for bremstrahlung gluon emission.",
"Interestingly, for the $\\gamma \\gamma $ initial state the multiplicity in fact decreases as $M_R$ is increases.",
"A possible explanation for this effect may be the fact that to very good approximation the only observed jets here are due to the final $q\\rightarrow q\\gamma $ splitting before the hard process; the $N_j \\ge 3$ population is extremely low.",
"As $M_R$ increases the average momentum fraction $x$ carried by the parent quark, and hence the produced quark jet, increases.",
"This will lead to the jet being produced on average at higher rapidity, making it more like to fail the $|\\eta _j|<4.4$ requirement.",
"As a result of these trends, for a $M_R=2.5$ GeV resonance the $\\gamma \\gamma $ to $gg$ ratio in the 0 jet bin is particularly large, and the discriminating power of this approach improves with increasing resonance mass.",
"Figure: Exclusive bb jet multiplicities for different minimum jet p ⊥ p_\\perp , for the bb ¯→Rb\\overline{b}\\rightarrow R production process and for resonance mass M R =0.75M_R=0.75 TeV.In Figs.",
"REF and REF we show the predicted distribution for the $WW$ –initiated VBF channel for $p_\\perp ^j>$ 25 (50) GeV, respectively: here, the resonance is produced in association with two outgoing quarks recoiling against the exchanged $W$ bosons, and which carry on average a relatively high transverse momentum $p_\\perp \\sim M_W$ .",
"These therefore tend to produce observable jets in the final state, so that the predicted jet multiplicity is very high, larger still than in the $gg$ case, and strongly peaking at $N_{\\rm jet}=2$ .",
"As in the $\\gamma \\gamma $ case, as the resonance mass is increased the average multiplicity is seen to decrease, with the 0 and 1 jet bins become more populated.",
"Figure: Ratio of 0 jet to ≥\\ge 1 jet cross sections as a function of the minimum jet p ⊥ p_\\perp for a range of initial–state resonance production processes, and for the continuum γγ\\gamma \\gamma background.",
"Results shown for resonance masses M R =0.75,1.5,2.5M_R=0.75,\\,1.5,\\,2.5 TeV.Figure: Ratio of 0 jet to ≥\\ge 2 jet cross sections as a function of the minimum jet p ⊥ p_\\perp for a range of initial–state resonance production processes, and for the continuum γγ\\gamma \\gamma background.",
"Results shown for resonance masses M R =0.75,1.5,2.5M_R=0.75,\\,1.5,\\,2.5 TeV.We also show in Figs.",
"REF and REF the predictions for heavy $b\\overline{b}$ and light $u\\overline{u}$ quark mediated production.",
"The results for the three mass points are similar, with some increase in jet multiplicity with increasing $M_R$ , similar to the $gg$ case.",
"We observe a non–negligible difference in the distributions between the heavy and light quark cases, with the $b\\overline{b}$ distribution following the $gg$ closely, while the $u\\overline{u}$ is expected to be accompanied by a lower jet multiplicity.",
"The difference between the $u\\overline{u}$ and $gg$ cases is in line with QCD expectations, for which the average particle multiplicity from an initial–state gluon is higher than for a quark [26].",
"For the $b\\overline{b}$ case, as the $b$ –quark PDF is generated entirely by DGLAP emission above the $b$ –quark threshold, there is a much greater contribution from the $g\\rightarrow b\\overline{b}$ process in the initial–state, where the outgoing $b$ –jet is observed within the jet acceptance.",
"Therefore, by requiring one or more $b$ –jets in the final state, the $b\\overline{b}$ initiated production mode may be identified or constrained.",
"In Fig.",
"REF we show the $b$ –jet multiplicity in this production mode for a range of jet $p_\\perp $ cuts, and we can see that roughly $\\sim 50\\%$ of events are expected to be accompanied by an additional $b$ –jet in the final state (for other initial states this fraction is of course significantly lower).",
"The case of $M_R=0.75$ TeV is shown for concreteness, although similar results hold for the other mass points.",
"Although Figs.",
"REF –REF only consider two choices of jet cut, we can also generalise this, showing in Fig.",
"REF the ratio of 0 to $\\ge 1$ jet cross sections as a function of the minimum jet $p_\\perp $ .",
"The same hierarchy in jet ratios between the different production modes and the same trends with increasing $M_R$ described above are clear.",
"Such distributions have been considered in [17] within a distinct analytic SCET approach, for all cases but the $\\gamma \\gamma $ and $WW$ initial states, and the results are found to be similar.",
"In addition, in [9] a simple analytic approach is taken to calculate the 0–jet cross sections in $gg$ and $\\gamma \\gamma $ –mediated production, and again the results are very similar to those here for these cases.",
"It is interesting to observe that the expected trend with the jet $p_\\perp $ cut in Fig.",
"REF is also in general different between the various production modes.",
"In particular, as the $p_\\perp $ cut is increased the jet ratio in the quark mediated processes increases more rapidly, such that for $p_\\perp \\gtrsim 60 $ GeV the $u\\overline{u}$ ratio is in fact higher than the $\\gamma \\gamma $ .",
"On the other hand, while for lower jet $p_\\perp $ cut, the $\\gamma \\gamma $ continuum ratios are very similar to the $u\\overline{u}$ , as expected from the dominantly light $q\\overline{q}$ initial–state for this process, as the $p_\\perp $ cut is increased, and the jet ratios become more sensitive to the structure of the production process, this is no longer the case, with the jet multiplicity being higher in the continuum case.",
"This indicates that at these higher $p_\\perp $ values the predictions may be sensitive to the precise nature of the resonance production process, which is not included in the effective description considered here.",
"Finally, in Fig.",
"REF we show the 0 to $\\ge 2$ jet ratios.",
"The relative hierarchy and trends with the jet $p_\\perp $ cut are comparable to the 0 to $\\ge 1$ case, but in fact the separation between the $\\gamma \\gamma $ and $gg$ predictions, for example, is increased.",
"This indicates that a greater discrimination can be achieved by considering the full jet multiplicity distribution as in Fig.",
"REF rather than an individual ratio.",
"In the following section we will consider quantitively what limits may be set on the different production modes using such distributions."
],
[
"Comparison to ATLAS data",
"In the ATLAS analysis [1] a measurement is presented of the exclusive jet multiplicity in the signal $700<M_{\\gamma \\gamma }<840$ GeV region for the spin–0 resonance selection described in Section .",
"Although no significant excess is observed in the larger 2016 data set, it is nonetheless instructive to compare the results of our study with these data.",
"It should however be emphasised that such data have limited statistics and moreover as these are not presented in an unfolded form, our comparison can only be approximate, as it omits a full ATLAS detector simulation as well as the inclusion of pile–up (although pile–up jets are largely rejected by cuts based on tracking information).",
"Nonetheless for these data the dominant source of experimental uncertainty is certainly statistical, and moreover the more significant differences predicted between for example the $\\gamma \\gamma $ and $gg$ are visible even in light of this, as we will show.",
"In our comparison we include a contribution from the SM continuum $\\gamma \\gamma $ background, which to be as close to the ATLAS analysis as possible, we simply take from their quoted SHERPA [27] MC predictions for the $\\gamma \\gamma $ continuum (which represent 90% of the total background).",
"We take a $S/B$ ratio of 1, roughly corresponding to that seen in the data, and compare to the $N_{\\rm jet}\\le 3$ bins, where 29 of the 31 observed events lie, and the theoretical predictions are more reliable.",
"Table: The χ 2 \\chi ^2 values for the description of the four N jet ≤3N_{\\rm jet} \\le 3 bins for the ATLAS measurement of the exclusive jet multiplicities, for different initial–state resonance production processes.",
"The contribution from the SM γγ\\gamma \\gamma continuum, taken from , is included, and S/B=1S/B=1 is assumed.",
"These values correspond to the distributions shown in Fig.",
".The predicted distributions for the production processes discussed above are shown in Fig.",
"REF , along with the ATLAS data.",
"From a straightforward visual comparison, it is clear that the data show some preference for the $\\gamma \\gamma $ and $u\\overline{u}$ scenarios, in comparison to the background only and in particular the $gg$ and $WW$ cases.",
"To give a more precise estimate, we can evaluate the corresponding $\\chi ^2$ valuesExcluding the $N_{\\rm jet}=3$ bin from the comparison, for which the observed 2 events is quite low, does not significantly affect the results which follow., as shown in Table REF .",
"These results confirm the above expectation, with the description in the $gg$ case being poor; if we assume $n_{\\rm dof}=n_{\\rm bins}-1=3$ for these normalised distributions, this corresponds to roughly a $\\sim 95\\%$ exclusion.",
"On the other hand, the description is very good in the $\\gamma \\gamma $ and $u\\overline{u}$ casesThe fact that the description of the data is so good in the $\\gamma \\gamma $ case, giving a $\\chi ^2/{\\rm dof}\\ll 1$ , should clearly not be expected, and indeed for other choices of $S/B$ and merging scale $Q_{\\rm cut}$ , for example, the value can be higher, although nonetheless corresponding to a very good description of the data.",
"Another contributing factor is that the errors on the ATLAS data are somewhat overestimated for this normalised observable: see below.. Interestingly, the description for the pure $\\gamma \\gamma $ background hypothesis is also relatively poor.",
"Finally, the description for a purely $WW$ initial state is very poor indeed, corresponding to a $\\sim 99.8\\%$ exclusion; we expect similar results in the $ZZ$ and, to a lesser extent, $\\gamma Z$ modes.",
"The description in the $b\\overline{b}$ case is reasonable, but we note that here by far the most discriminating variable is, as discussed in the previous section, the $b$ –jet multiplicity.",
"Even with the relatively low statistics data sample, a measurement of this observable would lend strong support, or place strong constraints on, such a scenario.",
"Indeed, in [1] it is reported that, with a $b$ –tagging efficiency of about 85%, roughly 8% of events in the signal region are found to contain $b$ –jets, consistent within statistical uncertainties with the sideband regions.",
"Comparing with Fig.",
"REF , we can see that $\\sim 60\\%$ of signal events in the purely $bb$ –initiated scenario are predicted to contain $b$ –jets in the ATLAS event selection.",
"Even accounting for the continuum background such a result is in strong tension with a dominantly $bb$ –induced production mode, although to evaluate the exact constraints would require a more precise comparison accounting for the $b$ –jet efficiency and mis–ID rate.",
"We will therefore not consider the $b\\overline{b}$ case in what follows.",
"While the precise values of the $\\chi ^2$ depend on the uncertain $S/B$ ratio, and merging scale $Q_{\\rm cut}$ , these are only found to vary by $\\sim 10\\%$ for reasonable changes in these parameters.",
"Taking our simulation for the $\\gamma \\gamma $ continuum background leads to somewhat larger values of $\\chi ^2$ , although still with the $\\gamma \\gamma $ and $u\\overline{u}$ cases giving very good descriptions of the data, with the same hierarchy observed for the other scenarios.",
"Before concluding this section, it is worth noting that the transverse momentum $p_\\perp ^{\\gamma \\gamma }$ of the diphoton system has also been measured in [1], where it is found that $\\sim 60\\%$ of the observable cross section has $p_\\perp ^{\\gamma \\gamma }<60$ GeV.",
"Such an observable is also sensitive to the production mode of the resonance, with the $gg$ and in particular $WW$ hypotheses predicting broader distributions.",
"While the description of the measured $p_\\perp ^{\\gamma \\gamma }$ distribution is in fact relatively poor in all cases, including the background only hypothesis, due to the apparent downward fluctuation in the second bin, the same hierarchy in the quality of the data descriptions is found as in the case of the jet multiplicity.",
"While these two observables are evidently correlated, this nonetheless gives a consistency check of the overall approach, and indeed with further data and finer binning, the $p_\\perp ^{\\gamma \\gamma }$ observable may also allow further discrimination between the modes.",
"Finally, it is worth emphasising that the results presented above are from a statistical point of view expected to be conservative.",
"In particular, the statistical errors on the normalised jet multiplicity distribution presented by ATLAS in [1] are simply found by rescaling the errors on the observed event numbers in each bin by the total number of observed events.",
"However this ignores the positive correlation between the measured numbers of events in each bin with the total number of observed events, which will lead in general to a reduction in the error size as well as a correlation between different bins for the normalised distribution; for the relatively low number of measured events this effect will not necessarily be negligible.",
"Indeed, taking a simplified approach and treating all uncertainties on the normalised distribution as Gaussian, we find a that a more complete treatment including the full correlation matrix between the different bins leads to a further deterioration in the quality of the description for the $gg$ , $WW$ and background only scenarios.",
"However, to be conservative we do not present such a comparison in detail here."
],
[
"Expected limits on production modes",
"Having found such promising results when comparing to the limited 2015 ATLAS jet multiplicity data, it is natural to consider what limits we can expect on the production modes.",
"To be more precise, we will perform a confidence limit ($CL$ ) hypothesis test, analogous to those used by the ATLAS [28] and CMS [29] analyses for determining the spin and parity of the SM Higgs.",
"The test statistic is defined as $Q=-2\\ln \\frac{\\mathcal {L}(h_1)}{\\mathcal {L}(h_2)}\\;,$ for the initial–state hypothesis, i.e.",
"$h=\\gamma \\gamma , gg, q\\overline{q}$ or more generally some non–trivial admixture of these.",
"This can be used to discriminate between an initial–state production hypotheses $h_1$ in favour of $h_2$ .",
"For an observed value of $Q_{\\rm obs}$ the exclusion of the $h_2$ in favour of $h_1$ is given in terms of the modified confidence level ${\\rm CL}_s=\\frac{P(Q\\ge Q_{\\rm obs}|h_1)}{P(Q\\ge Q_{\\rm obs}|h_2)}$ where $P(Q\\ge Q_{\\rm obs}|h)$ is the probability that the test statistic is at least as high as $Q_{\\rm obs}$ under a hypothesis $h$ .",
"We can thus use this approach to calculate the expected exclusion limits on a specified production mode $h_2$ , assuming the resonance is produced via $h_1$ .",
"To be more concrete we assume three simplified scenarios where the produced resonance is only produced via two of the three $\\gamma \\gamma $ , $gg$ and $u\\overline{u}$ initial–states, i.e.",
"$\\nonumber \\sigma _{gg}&=f_{g\\gamma }\\, \\sigma _{\\rm R} \\quad \\quad \\sigma _{\\gamma \\gamma }=(1-f_{g\\gamma })\\,\\sigma _{\\rm R}\\;,\\\\ \\nonumber \\sigma _{gg}&=f_{g u}\\, \\sigma _{\\rm R} \\quad \\quad \\sigma _{u\\overline{u}}=(1-f_{g u})\\,\\sigma _{\\rm R}\\;, \\\\ \\sigma _{\\gamma \\gamma }&=f_{\\gamma u}\\, \\sigma _{\\rm R} \\quad \\quad \\sigma _{u\\overline{u}}=(1-f_{\\gamma u})\\,\\sigma _{\\rm R}\\;,$ where $\\sigma _R$ is the signal cross section; we do not consider the $WW$ initial state for simplicity, but will comment on this below.",
"We take the case of $M_R=0.75$ TeV in what follows, but similar results follow for higher mass points.",
"To see how these limits can be expected to extend given a certain amount of luminosity, we will for concreteness consider an observed cross section of 7 fb after cuts in the narrower $725<M_{\\gamma \\gamma }<775$ GeV region, with the SM continuum (taken now from our MC sample) and resonance signal present equally, i.e.",
"$S/B=1$ .",
"We also include a 10% systematic uncertainty on the data, as a fairly conservative assumption.",
"The expected 95% exclusion limits on the cross section fractions in (REF ) are shown in Fig.",
"REF .",
"In each case we consider the limit on the fraction $f$ against the hypotheses that the resonance is purely produced in the corresponding production modes; for example we calculate the exclusion on $f_{g\\gamma }$ under both the purely $gg$ and $\\gamma \\gamma $ hypotheses.",
"We can see that already for the integrated luminosity of $3.2$ ${\\rm fb}^{-1}$ corresponding to the ATLAS data, a 95% exclusion of the purely $gg$ hypotheses, corresponding to $f_{g\\gamma },f_{gu}=1$ , is consistent with the projected limits, and with the results of the previous section.",
"Indeed, if we treat the fractions $f$ in (REF ) as free parameters and perform a $\\chi ^2$ minimisation and profile test with respect to the ATLAS data we find that $f_{g\\gamma } (f_{g u}) < 0.71 (0.76)$ at 95% confidence, with minima at 0, while the $f_{\\gamma u}$ is unconstrained; these results are again completely consistent with these expected confidence limits.",
"We can moreover see that with just a few 10s of ${\\rm fb}^{-1}$ any significant $gg$ –induced component can be excluded in favour of the purely $\\gamma \\gamma $ and $u\\overline{u}$ hypotheses, and conversely any significant $\\gamma \\gamma $ or $u\\overline{u}$ induced component can be excluded in favour of the purely $gg$ .",
"On the other hand, for the $u\\overline{u}$ and $\\gamma \\gamma $ scenario, for which the two available production modes predict closer jet multiplicity distributions, see Fig.",
"REF , the situation is more challenging, and a larger $O(100)$ ${\\rm fb}^{-1}$ sample is required for a sizeable contribution from the $\\gamma \\gamma $ mode to be excluded in favour of the purely $u\\overline{u}$ , and vice versa.",
"In such a situation it is likely that other methods for distinguishing between the different initial–states will be more competitive.",
"For example, as discussed in [6], [9], a measurement of just a few resonance events in the essentially background free exclusive channel, where both protons remain intact after the collision, will provide strong evidence in favour of a significant contribution from the $\\gamma \\gamma $ mode, and conversely any lack of observed events will disfavour this.",
"Alternatively, by selecting events with rapidity gaps in the final state the $\\gamma \\gamma $ –initiated contribution may be isolated: it is worth recalling, in particular, that a significant fraction of $\\gamma \\gamma $ –initiated events are expected to occur due to low–scale coherent emission from the protons, which naturally lead to rapidity gaps in the final–state, see [22].",
"Finally, returning to the VBF channel, if we consider the simplified scenario that the resonance is produced by the $WW$ and $\\gamma \\gamma $ modes, i.e.",
"$\\sigma _{WW}=f_{W\\gamma }\\, \\sigma _{\\rm R} \\quad \\quad \\sigma _{\\gamma \\gamma }=(1-f_{W\\gamma })\\,\\sigma _{\\rm R}\\;,$ then we already find that $f_{W\\gamma }<0.48$ at 95% confidence, with similar limits if the additional mode is $gg$ or $u\\overline{u}$ ."
],
[
"Conclusion",
"The observation by the ATLAS and CMS collaborations, in roughly $3\\,{\\rm fb}^{-1}$ of data at $\\sqrt{s}=13$ TeV recorded in 2015, of an excess of events around 750 GeV in the diphoton mass spectrum provoked a great deal of theoretical interest.",
"Although no significant excess was seen in the increased 2016 data set, this initial observation has motivated us to discuss in detail the possibility of using measurements of the jet multiplicity associated with the production of a high–mass resonance as a means to distinguish between different production mechanisms.",
"Specifically, we have considered the $\\gamma \\gamma $ , $gg$ , $WW$ , light quark $q\\overline{q}$ and heavy $b\\overline{b}$ quark initiated processes and shown that in each case the predicted level of jet activity is quite different.",
"In particular, due to the small size of $\\alpha $ and the lower photon branching probability, the jet multiplicity is expected to be lower in the $\\gamma \\gamma $ case compared to the QCD quark and gluons.",
"As the particle multiplicity associated with an initial–state gluon is higher compared to the quark case, the higher jet activity in the $gg$ case also allows this mechanism to be separated from the $q\\overline{q}$ .",
"For the $WW$ –initiated channel the resonance is generally produced in association with at least two additional jets, due to the relatively high $p_\\perp $ recoiling quarks in the final state.",
"For the heavy $b\\overline{b}$ the most discriminating observable is instead simply the $b$ –jet fraction, which is predicted to be significantly higher than in all other scenarios.",
"To demonstrate how such an approach may be applied to data, in this paper we have compared these results with the 2015 ATLAS measurement of the jet multiplicity in the spin–0 signal region associated with the initial diphoton excess observation.",
"We have found that even with these fairly limited data, a purely $gg$ –initiated scenario is disfavoured, while the light $q\\overline{q}$ and $\\gamma \\gamma $ scenarios provide a good description.",
"The continuum background–only hypothesis also gives a somewhat worse description than these latter cases.",
"In addition, we have found that a dominantly $WW$ –initiated production mechanism is in strong tension with the data, and we have seen that the relatively small observed $b$ –jet fraction disfavours a dominantly $b\\overline{b}$ –initiated production mechanism.",
"We have also presented expected exclusion limits on different production hypotheses with the collected integrated luminosity.",
"With just a few 10's of ${\\rm fb}^{-1}$ , we can expect to rule out any sizeable fraction of $gg$ in favour of light $q\\overline{q}$ and $\\gamma \\gamma $ production, and vice versa.",
"It is more challenging to distinguish between the $u\\overline{u}$ and $\\gamma \\gamma $ modes, although still possible with enough data.",
"If a high mass resonance is observed at the LHC or a future collider, one of the first tasks will be to determine as precisely as possible the nature of such a new state.",
"It has been the goal of this paper to present a method for distinguishing between various production modes, which can be applied to any heavy resonance which we might hope to find in the future."
],
[
"Acknowledgements",
"We thank Michelangelo Mangano, Josh McFayden, Valya Khoze, Frank Krauss and Robert Thorne for useful discussions.",
"VAK thanks the Leverhulme Trust for an Emeritus Fellowship.",
"The work of MGR was supported by the RSCF grant 14-22-00281.",
"LHL thanks the Science and Technology Facilities Council (STFC) for support via the grant award ST/L000377/1.",
"MGR thanks the IPPP at the University of Durham for hospitality.",
"MS is supported in part by the European Commission through the “HiggsTools” Initial Training Network PITN-GA-2012- 316704."
],
[
"Cross check: comparison to spin–2 selection",
"In addition to the ATLAS spin–0 selection data considered in Section , a spin–2 event selection was also taken in [1], identical to the spin–0 case, but with a lower $E_\\perp >55$ GeV cut on the final–state photons.",
"Thus the corresponding number of events in this case is larger, with the sample in the spin–0 case being a subset of the spin–2; in the signal $700 < M_{\\gamma \\gamma }<840$ GeV region ATLAS find 31 (70) events for the spin–0 (2) samples.",
"However, if the excess of events is indeed due to a scalar resonance decaying (isotropically) to photons then the lower $E_\\perp $ cuts of the spin–2 selection, for which the relative contribution from the continuum background will be larger, are expected to lead to an overall decrease in the $S/B$ ratio.",
"Figure: Exclusive jet multiplicities, as in Fig.",
", but compared to data taken with the ATLAS spin–2 event selection.",
"The continuum background is taken from , and is included in all signal distributions, assuming a S/BS/B ratio of 0.5.Nonetheless, it is a useful cross check to compare the expected jet multiplicity distributions with the higher statistics data corresponding to the spin–2 selection, still taking our spin–0 resonance hypothesis.",
"We estimate from [1] that $S/B\\sim 0.5$ , lower than the $S/B\\sim 1$ found for the spin–0 selection, and consistent with the discussion above, and moreover with a $S/\\sqrt{B}$ sensitivity that is comparable in both cases, as is found in the ATLAS analysis.",
"The results are shown in Fig.",
"REF : due to the larger background contribution, we can see that the different signal distributions are less well separated, although the smaller statistical errors on the data might in principle allow an increased differentiation.",
"However, we can see from a rough visual comparison that all hypotheses appear to provide a relatively good description of the data, although with the prediction in the 0–jet bin lying a little above the data in the $\\gamma \\gamma $ case.",
"Nonetheless, in Table REF we show the corresponding $\\chi ^2$ values and we can see that description of the data is good in all cases, with a $\\chi ^2/{\\rm dof}\\lesssim 1$ for all hypotheses; this is driven by the fact that the continuum background, which gives the dominant contribution to the measured sample, provides an excellent description of the data.",
"Table: The χ 2 \\chi ^2 values for the description of the four N jet ≤3N_{\\rm jet} \\le 3 bins for the ATLAS measurement of the exclusive jet multiplicities, for different initial–state resonance production processes.",
"The contribution from the SM γγ\\gamma \\gamma continuum, taken from , is included, and S/B=0.5S/B=0.5 is assumed.",
"These values correspond to the distributions shown in Fig.",
"."
]
] | 1606.04902 |
[
[
"The re-flight of the Colorado high-resolution Echelle stellar\n spectrograph (CHESS): improvements, calibrations, and post-flight results"
],
[
"Abstract In this proceeding, we describe the scientific motivation and technical development of the Colorado High-resolution Echelle Stellar Spectrograph (CHESS), focusing on the hardware advancements and testing supporting the second flight of the payload (CHESS-2).",
"CHESS is a far ultraviolet (FUV) rocket-borne instrument designed to study the atomic-to-molecular transitions within translucent cloud regions in the interstellar medium (ISM).",
"CHESS is an objective f/12.4 echelle spectrograph with resolving power $>$ 100,000 over the band pass 1000 $-$ 1600 {\\AA}.",
"The spectrograph was designed to employ an R2 echelle grating with \"low\" line density.",
"We compare the FUV performance of experimental echelle etching processes (lithographically by LightSmyth, Inc. and etching via electron-beam technology by JPL Microdevices Laboratory) with traditional, mechanically-ruled gratings (Bach Research, Inc. and Richardson Gratings).",
"The cross-dispersing grating, developed and ruled by Horiba Jobin-Yvon, is a holographically-ruled, \"low\" line density, powered optic with a toroidal surface curvature.",
"Both gratings were coated with aluminum and lithium fluoride (Al+LiF) at Goddard Space Flight Center (GSFC).",
"Results from final efficiency and reflectivity measurements for the optical components of CHESS-2 are presented.",
"CHESS-2 utilizes a 40mm-diameter cross-strip anode readout microchannel plate (MCP) detector fabricated by Sensor Sciences, Inc., to achieve high spatial resolution with high count rate capabilities (global rates $>$ 1 MHz).",
"We present pre-flight laboratory spectra and calibration results.",
"CHESS-2 launched on 21 February 2016 aboard NASA/CU sounding rocket mission 36.297 UG.",
"We observed the intervening ISM material along the sightline to $\\epsilon$ Per and present initial characterization of the column densities, temperature, and kinematics of atomic and molecular species in the observation."
],
[
"INTRODUCTION",
"The Colorado High-resolution Echelle Stellar Spectrograph (CHESS)[1], [2], [3], [4], [5] is a rocket-borne astronomical instrument that first launched from White Sands Missile Range (WSMR) aboard NASA/CU mission 36.285 UG on 24 May 2014 (CHESS-1).",
"The second flight of the experiment was aboard the NASA/CU mission 36.297 UG on 21 February 2016 (CHESS-2).",
"This paper presents information on the instrument optics, alignment, calibration, and flight results of CHESS-2.",
"This section will cover the instrumental design and scientific objectives of the CHESS experiment.",
"Figure: The Zemax ray trace of CHESS, including the secondary aspect camera system.",
"The mechanical collimator reduces stray light in the line of sight and feeds starlight to the echelle.",
"The echelle disperses UV light into high-dispersion orders, which are focused by the cross disperser onto the detector plane.",
"The different colored lines represent a series of wavelengths across the 1000 -- 1600 Å bandpass.Translucent clouds reside in the transition between the diffuse (traditionally defined as A$_{V}$ $<$ 1) and dense (A$_{V}$ $>$ 3) phases of the interstellar medium (ISM).",
"It is in this regime where the ultraviolet portion of the average interstellar radiation field plays a critical role in the photochemistry of the gas and dust clouds that pervade the Milky Way galaxy.",
"The most powerful technique for probing the chemical structure of translucent clouds is to combine measurements of H$_{2}$ with knowledge of the full carbon inventory (CI, CII, and CO) along a given line of sight.",
"It has been argued that an analysis of the carbon budget should be the defining criterion for translucent clouds, rather than simple measurements of visual extinction[6].",
"Moderate resolution 1000 $-$ 1120 Å spectra from FUSE and higher-resolution data from HST/STIS have been used to show that many of these sightlines have CO/H$_{2}$ $>$ 10$^{-6}$ and CO/CI $\\sim $ 1, consistent with the existence of translucent material in the framework of current models of photodissociation regions in the ISM[7], [8].",
"The CHESS experiment is designed to study translucent clouds with its combination of bandpass and spectral resolution.",
"The 1000 $-$ 1600 Å bandpass contains absorption lines of H$_{2}$ (1000 $-$ 1120 Å), CII (1036 and 1335 Å; however we note that saturation effects can complicate the interpretation of these lines), CI (several between 1103 $-$ 1130, 1261, 1561 Å), and the A $-$ X, B $-$ X, C $-$ X, and E $-$ X bands of CO ($<$ 1510 Å).",
"High resolution (R) $>$ 100,000 is required to resolve the velocity structure of the CI lines and the rotational structure of CO. High-resolution is therefore essential to the accurate determination of the column density of these species[9].",
"The FUV also provides access to many absorption lines of metals, such as iron, magnesium, silicon, and nickel, allowing for an exploration of the depletion patterns in translucent clouds.",
"CHESS, with its high-resolution and large bandpass, especially including wavelengths shorter than 1150 Å, is well-suited to the study of translucent clouds and will help create an observational base for models of the chemistry and physical conditions in interstellar clouds.",
"For the second flight of CHESS, we combine our program on the LISM (begun with the NASA/CU mission 36.271 UG/SLICE[10], [11], [12] and continued on 36.285 UG/CHESS[1]) with a detailed study of the interstellar material in the line of sight to $\\epsilon $ Per (HD 24760).",
"$\\epsilon $ Per is a B0.5III star at d $\\approx $ 300 pc with low$-$ intermediate reddening (E(B-V) = 0.1; log(H$_2$ ) $\\sim $ 19.5), indicating that the sightline may be sampling cool interstellar material.",
"H$_{2}$ , CI, CO, and CII were all detected by Copernicus; however, higher sensitivity and spectral resolution is required for a complete analysis of these types of sightlines[13].",
"Observations by Copernicus and IUE have been used to measure the velocity structure along the sightline to $\\epsilon $ Per, and have found at least three separate cloud structures described by different kinematic behavior and molecular abundances[14].",
"Resolving the various molecular clouds on the $\\epsilon $ Per sightline is the primary goal of CHESS-2.",
"Overall, the line of sight to $\\epsilon $ Per shows typical abundances of molecular material and ionized metal found in translucent clouds, such as H$_{2}$ , FeII and MgII[14], consistent with the sightline towards recent star-forming sites.",
"CHESS-2 provides the first high$-$ resolution data of $\\epsilon $ Per to observe H$_{2}$ , CI, and ionized metal lines simultaneously, allowing us to constrain the metal content, kinematic structure, and photo-dissociation processes of the nearby molecular cloud material around the early B-type star.",
"Figure: A false-color representation of the full-resolution (8192×\\times 8192 digital pixels) laboratory echellogram of CHESS-2 after instrument alignment.",
"The black/blue represents little to no counts in the binned pixel location, while green/yellow represent emission lines from hydrogen (Lyα\\alpha at 1215.67 Å) and molecular hydrogen.",
"Overlaid are arrows showing the direction of dispersion from the echelle and cross disperser, and the tick marks show the approximate location of different spectral regions.",
"The final laboratory calibration image contains over 73 million photon counts."
],
[
"Instrument Design",
"CHESS is an objective f/12.4 echelle spectrograph.",
"The instrument design[4] included the development of two novel grating technologies and flight-testing of a cross-strip anode microchannel plate (MCP) detector.",
"The high-resolution instrument is capable of achieving resolving powers $\\ge $ 100,000 $\\lambda $ /$\\Delta \\lambda $ across a bandpass of 1000 $-$ 1600 Å[5].",
"The operating principle of the instrument is as follows: A mechanical collimator, consisting of an array of 10.74 mm $\\times $ 10.74 mm $\\times $ 1000 mm anodized aluminum tubes, provides CHESS with a total collecting area of 40 cm$^{2}$ , a field of view (FOV) of 0.67$^{\\circ }$ , and allows on-axis stellar light through to the spectrograph.",
"A square echelle grating (ruled area: 100 mm $\\times $ 100 mm), with a designed groove density of 69 grooves/mm and angle of incidence (AOI) of 67$^{\\circ }$ , intercepts and disperses the FUV stellar light into higher diffraction terms (m = 266 $-$ 166).",
"The custom echelle was used to research new etching technologies, namely the electron-beam (e-beam) etching process.",
"The results of the JPL fabrication effort are discussed in Section REF .",
"Instead of using an off-axis parabolic cross disperser[15], CHESS employs a holographically-ruled cross dispersing grating with a toroidal surface figure and ion-etched grooves, maximizing first-order efficiency[5].",
"The cross disperser is ruled over a square area (100 mm $\\times $ 100 mm) with a groove density of 351 grooves/mm and has a surface radius of curvature (RC) = 2500.25 mm and a rotation curvature ($\\rho $ ) = 2467.96 mm.",
"The grating spectrally disperses the echelle orders and corrects for grating aberrations[16].",
"The cross-strip MCP detector[17], [18] is circular in format, 40 mm in diameter, and capable of total global count rates $\\sim $ 10$^{6}$ counts/second.",
"The cross-strip anode allows for high resolution imaging, with the location of a photoelectron cloud determined by the centroid of current read out from five anode “fingers\" along the x and y axes.",
"The CHESS experiment also includes an optical system (aspect camera), used solely to align the spectrograph to the stellar target during calibrations and flight.",
"The aspect camera system uses the positions of the instrument gratings to direct zeroth-order light to a stand-alone camera.",
"Fig.",
"REF shows a Zemax ray trace of collimated light through the spectrograph, and Fig.",
"REF displays an example of a laboratory spectrum taken with the fully integrated and aligned CHESS instrument."
],
[
"Mechanical Spectrograph Structure",
"The opto-mechanical structure of CHESS-2 is identical to that presented in Ref. Hoadley+14.",
"Below, we outline information and specifications about the structure and optical mount designs for CHESS.",
"A SolidWorks rendering of the spectrograph and electronics sections of CHESS is provided in Fig.",
"REF .",
"Figure: A SolidWorks rendering (all stated quantities in units of centimeters) of the spectrograph and electronics sections of CHESS (skins excluded).",
"Labeled are relevant spectrograph structures and optical components referenced in Section .CHESS is an aft-looking payload that uses 17.26 inch-diameter rocket skins and is split into two sections: a vacuum (spectrograph) section and non-vacuum (electronics) section.",
"The two sections are separated by a hermetic bulkhead.",
"The overall length of the payload is 226.70 cm from mating surface to mating surface and the weight of the payload is 364 lbs.",
"The shutter door is the only moving component during flight in the experiment section.",
"The electronics section is 50.80 cm long with one umbilical pocket for an RJ45 Ethernet port.",
"The detector is mounted with a hermetic seal on the electronics side of the vacuum bulkhead, facing into the spectrograph section.",
"The vacuum section uses two 113.36 cm long rocket skins with hermetic joints.",
"The only mechanical component on CHESS (other than the NASA Sounding Rockets Operations Contact (NSROC)-supplied shutter door) is a manual butterfly valve attached along the 180$^{\\circ }$ line on the aft skin.",
"This allows for the evacuation of the experiment throughout development, integration and pre-flight activities, safeguarding the sensitive optical coatings.",
"A carbon-fiber space frame is attached to the aft side of the hermetic bulkhead and suspends the aspect camera, mechanical collimator, echelle grating and cross-disperser in place.",
"The space frame is comprised of three aluminum disks attached to five 2.54 cm diameter x 182.88 cm long carbon fiber tubes.",
"The collimator is a set of 10.74 mm x 10.74 mm x 1000 mm long black-anodized aluminum square tubes bonded together.",
"An aluminum mounting flange is bonded at the center of the assembly and secured to the aft-most disk of the space frame.",
"The echelle grating in CHESS-2 is a rectangular block made of Zerodur (110 mm x 110 mm x 16 mm thick).",
"To provide support during launch, the grating is bonded to a mount via three Invar brackets.",
"Three titanium flexures secure the Invar brackets to an aluminum mount, which is set to the desired AOI and is attached to the forward most disk of the space frame.",
"The flexures prevent stress transfer into the optic due to a coefficient of thermal expansion (CTE) mismatch between mounting components.",
"The cross disperser is a block of fused silica (100 mm x 100 mm x 30 mm thick) with a toroidal surface.",
"Three Invar pads are bonded to the neutral plane of the optic, nearly 120$^{\\circ }$ apart, and are connected to three titanium flexures.",
"The flexures are attached to an aluminum mounting plate and affixed to the middle disk of the space frame.",
"Again, the flexures prevent the surface of the optic from warping due to stress transfer from a CTE mismatch in mounting components.",
"Figure: Left: An image of one of the sample e-beam etched echelle gratings (fabricated by JPL Microdevices Laboratory) tested at CU.",
"The echelle was etched into a PMGI photo-resist overlaid on a silicon substrate and coated with gold for reflectivity measurements in the UV.",
"The dispersion direction of the echelle can be seen by the optical colors on the grating in the photo.",
"There are small squares over the entire grated area of the echelle, due to the periodicity in the etching procedure.Right: An image showing the optical diffraction off of one of the experimental echelle samples from JPL.",
"The JPL e-beam echelles showed secondary diffraction patterns parallel to the primary diffraction axis (“ghosts\"), which we believe are a consequence of the periodicity in the etching process.",
"In the FUV, ghost orders contained as much as 60% of the total counts as their primary order counterparts.",
"In practice, this would have lead to at least 3 overlapping echellograms (primary + two sets of ghost orders) in CHESS, making the data complicated to decipher."
],
[
"Echelle Gratings",
"The CHESS echelle was designed to be a 100 mm x 100 mm x 0.7 mm silicon wafer with a groove density of 69 grooves/mm and AOI = 67$^{\\circ }$ .",
"The CHESS-1 echelle was the final result of a research and development lithography-etching project undertaken by LightSmyth, Inc., meant to suppress scattered light in the FUV, making the grating more efficient in the peak echelle orders.",
"Unfortunately, the etching process was difficult to perfect, and CHESS-1 flew an echelle grating with peak order efficiencies $\\sim $ 1 $-$ 6% across the bandpass[1].",
"To improve the sensitivity of the CHESS instrument for its second flight, we worked in collaboration with the JPL Microdevices Laboratory on an experimental electron-beam (e-beam) etching process to fabricate an echelle grating with low-scatter and high order efficiency.",
"Theoretically, groove efficiency expectations at the peak order in the FUV were around 85%.",
"Practically, we specified a groove efficiency of 60% for JPL to fabricate to demonstrate the e-beam technology for future UV observatories, while the CHESS-2 primary scientific objectives required a minimum groove efficiency of 20% across the bandpass 1000 $-$ 1600 Å.",
"The e-beam process was noted to have better precision and groove etching control than either the mechanical-ruling or lithographic-etching processes[19].",
"An image of one of the samples from JPL is shown in Figure REF .",
"The first sample was etched directly onto the silicon substrate, which proved difficult for JPL to control and resulted in a jagged saw-tooth grating wall.",
"All other grating profiles were fabricated into a thin layer of polymethylglutarimide (PMGI), which was deposited onto the silicon substrate.",
"PMGI is a photo-resist that is more malleable than silicon, making the etching process more precise and easier to control.",
"To measure the order efficiency of the echelles in the FUV, each PMGI grating sample was gold-coated.",
"Figure: A comparison of the peak order groove efficiency (1.0 = 100% efficient) of each JPL e-beam echelle sample at HI-Lyα\\alpha (1215.67 Å).",
"The first sample (#1) was etched directly into the silicon substrate, which created jagged groove profiles and caused much of the light to be lost to scatter.",
"Samples #2 - #6 were etched into PMGI, which was overlaid on the silicon substrate and coated with gold.",
"The best performance achieved by the samples was ∼\\sim 4.5% peak order efficiency (Samples #4 and #6), which incidentally were the samples etched with 100 grooves/mm instead of the 69 grooves/mm.",
"We include the CHESS-2 threshold order efficiency (right).",
"This threshold represents the minimum order efficiency required of the echelle grating to observe the CHESS-2 target (ϵ\\epsilon Per) at a S/N ∼\\sim 20 over the 250-second exposure time.Figure REF shows the groove efficiency of each CHESS echelle grating sample provided by JPL between February $-$ June 2015.",
"All JPL e-beam gratings failed to meet the specified target groove efficiency for the project.",
"We reiterate that the original groove efficiency specification given to JPL was 60%, while the science goals of the CHESS-2 mission required a groove efficiency of at least 20% from 1000 $-$ 1600 Å.",
"The most efficient gratings (Samples #4 and #6) were only 4.5% efficient at Ly$\\alpha $ , which was comparable to the efficiency of the LightSmyth, Inc. echelle flown in CHESS-1[1].",
"We also saw noticeable ghosting effects $-$ secondary diffraction patterns parallel to the primary diffraction axis $-$ from every echelle sample fabricated with the e-beam technique, both in the optical and in the FUV.",
"We suspect this was due to the periodicity in the e-beam process, which left a grid pattern of small squares over the entire grated surface area.",
"Figure REF shows an image of the grid-like pattern and the resulting ghosting off the grating with a 532 nm laser.",
"We note that the R&D project had difficulty meeting the CHESS echelle parameters throughout the fabrication process.",
"We list the deviations from the optimal AOI (67$^{\\circ }$ ) and groove density (69 grooves/mm) of each sample in Table REF .",
"Figure: A comparison of echelle gratings tested for use in the CHESS instrument.",
"We include the best-performing echelle gratings from the lithography etching R&D project undertaken by LightSmyth, Inc. (flown on CHESS-1, 36.285 UG), the e-beam samples fabricated by JPL, and two mechanically-ruled replica gratings from Bach Research, Inc. and Richardson Gratings, respectively.",
"Both mechanically-ruled gratings out-performed the R&D echelles and met the CHESS minimum order efficiency threshold.Due to ongoing challenges with the e-beam process, we opted to explore traditional, mechanically-ruled echelle gratings, which can have higher efficiency but historically display higher inter-order scatter[20].",
"We tested several echelle gratings from two manufactures with histories of providing UV space missions with gratings: Bach Research Inc. (formerly Hyperfine Inc. $-$ Boulder, CO) and Richardson Gratings (formerly Milton-Roy $-$ Rochester, NY).",
"Figure REF shows a comparison of the best-performing gratings by each manufacturer (LightSmyth, Inc.; JPL; Bach Research, Inc.; and Richardson Gratings).",
"The mechanically-ruled echelle gratings outperformed all of the experimental gratings.",
"Given time constraints on the delivery of the grating to meet the launch date of CHESS-2 and their vicinity to CU, we chose to fly the Bach echelle grating for CHESS-2.",
"We note that we have ordered and currently have in-house a Richardson echelle grating ($\\alpha $ = 63$^{\\circ }$ , g = 87 grooves/mm), which both outperformed all echelle gratings tested at CU in the FUV and has the closest matching echelle solutions to those designed for the CHESS instrument (m = 266 $-$ 166 for $\\lambda $ = 1000 $-$ 1600 Å).",
"It will be flown on CHESS-3 (June 2017) and CHESS-4 (early 2018).",
"Table: Comparison of Echelle Parameters Tested for CHESS-2 and fabricated gratings."
],
[
"Cross Disperser Grating",
"The CHESS cross disperser grating is a 100 mm $\\times $ 100 mm $\\times $ 30 mm fused silica optic with a toroidal surface profile.",
"The toroidal surface shape separates the foci of the spatial and sagittal axes of the dispersed light.",
"The optic first focuses light spatially onto the detector, then spectrally behind the detector, ensuring no foci at the locations of either the ion repeller or quantum efficiency (QE) grids.",
"The cross dispersing optic is a novel type of imaging grating that represents a new family of holographic solutions and was fabricated by Horiba Jobin-Yvon (JY).",
"The line densities are low (351 lines per mm, difficult to achieve with the ion-etching process), and the holographic solution allows for more degrees of freedom than were previously available with off-axis parabolic cross dispersing optics.",
"The holographic ruling corrects for aberrations that otherwise could not be corrected via mechanical ruling.",
"The grating is developed under the formalism of toroidal variable line spacing gratings[16] and corresponds to a holographic grating produced with an aberrated wavefront via deformable mirror technology.",
"This results in a radial change in groove density and a traditional surface of concentric hyperboloids from holography, like those used in ISIS[21] and HST/COS[22].",
"Figure: Measured reflectivity (order efficiency ×\\times reflectivity of Al+LiF) of the cross dispersing grating in CHESS over time, overplotted with simple spline curves to show the resemblance of each trial.",
"We focus on the reflectivity of the m = -1 order, which is the dispersion order used in the CHESS instrument.",
"Because LiF can show efficiency degradations when not stored properly, we measure how the order reflectivity changes between CHESS-1 and CHESS-2 without re-coating the optic.",
"No significant degradation of the coating has been measured between the first assembly of CHESS (November 2013) and the build-up of CHESS-2 (November 2015).The cross disperser was delivered in the summer of 2012, and order efficiencies around both the m = +1 and m = -1 orders were measured to be between 20% $-$ 45% in the FUV (900 $-$ 1700 Å) before and after the Al+LiF optical coating.",
"Figure REF shows the reflectivity (order efficiency $\\times $ reflectivity of Al+LiF) of the cross dispersing optic for order m = -1, which is the dispersion order used in the CHESS instrument, for pre-36.285 field operations, post-36.285 launch, and pre-36.297 field operations.",
"Overall, the performance of the cross disperser exceeded our initial expectations, with reflectivity $\\sim $ 30% at Ly$\\alpha $ .",
"The cross disperser is effective at dispersing most of the on-axis light into the m = $\\pm $ 1 orders and suppressing the m = 0 order because of the characteristic sinusoidal groove profiles created via the ion-etching procedure at JY.",
"Additionally, at optical wavelengths, the reflectivity of the m = 0 order becomes comparable to the m = $\\pm $ 1 orders.",
"This allowed us to build a secondary camera system to track the movements of our optical axis and target acquisition during flight."
],
[
"Cross-Strip Anode Microchannel Plate Detector",
"The cross-strip MCP detector was built and optimized to meet the CHESS spectral resolution specifications at Sensor Sciences[17], [18].",
"The detector has a circular format and a diameter of 40 mm.",
"The microchannel plates are lead silicate glass, containing an array of 10-micron diameter channels.",
"They are coated with an opaque cesium iodide (CsI) photocathode, which provides QE = 15 $-$ 35% at FUV wavelengths.",
"When UV photons strike the photocathode to release photoelectrons, the photoelectrons are accelerated down the channels by an applied high voltage ($\\sim $ 3100 V).",
"Along the way, they collide with the walls of the channels, which produces a large gain over the initial single photoelectron.",
"There are two MCPs arranged in a “chevron\" configuration.",
"During flight, the detector achieved spatial resolution of 25 $\\mu $ m over an 8k x 8k pixel format.",
"The QE estimate across the CHESS bandpass, measured by Sensor Sciences, is plotted in Figure REF against the efficiency measurements of the flight echelle and cross disperser for CHESS-2.",
"Figure: Left: Performance (for each grating: peak order efficiency, and for the detector: detector quantum efficiency) of all optical components of CHESS-2.",
"Right: The CHESS-2 effective area, including throughput loss from baffling, compared to the effective area of CHESS-1.",
"After 36.285, we sent the cross-strip anode MCP back to Sensor Sciences, Inc. to replace the CsI photocathode, which had begun to crystallize during field operations of CHESS-1.",
"The total effective area of CHESS-2 is about an order of magnitude larger than that of CHESS-1, deriving mainly from the large gain in echelle order efficiency across the CHESS bandpass (see Figure for a comparison of the echelle performance).The 2015 NASA Cosmic Origins Program Annual Technology Report emphasized that the technology readiness level (TRL) for large format, high count rate, and high QE MCP detectors needs to improve for future UV space missions.",
"One of the goals of the CHESS instrument is to demonstrate the flight performance of the cross strip anode design to raise the TRL level to 6, which was achieved on 36.285.",
"The cross-strip anode MCP detector was re-flown and performed reliably once more on 36.297, handling count rates of 25,000 photons/second for the entire exposure of CHESS-2.",
"However, in a laboratory setting, we have been able to demonstrate count rates of $\\gtrsim $ 150,000 photons/second, which the MCP handled smoothly.",
"The alignment process for CHESS has been described in great detail by Ref.",
"Hoadley+14; please refer to this document for specifics and pictures of the process.",
"Instead of reiterating the entire procedure in detail, we list the steps taken to align CHESS-2 and focus the echellogram for flight: The CHESS gratings were specifically designed to have grating parameters to allow for optical wavelength solutions.",
"Grating solutions were modeled in Zemax for red (632 nm), green (532 nm), and violet (405 nm) wavelengths.",
"We utilize this tool to first align the echelle to the cross disperser, and then align the cross disperser to the detector.",
"For FUV calibrations, the instrument is aligned to an external vacuum system to feed the spectrograph with UV light.",
"We used the aspect camera system to align the instrument to the vacuum chamber.",
"A small, square mirror was set up along one side of the echelle grating to intercept light, which was directed to the center of the cross disperser.",
"Zeroth-order optical light off the cross disperser intercepted another small, flat mirror at the detector bulkhead, which steered the light to the aspect camera.",
"During alignments, white light was tracked with the aspect camera until the proper alignment position was found.",
"Linear vacuum actuators were positioned behind the cross dispersing optic to control the tip, tilt, and optical axis motions of the grating.",
"Using these stages, we centered the echellogram on the detector and focused the spectrum using small actuator steps.",
"The final focused position of the cross disperser was found to be 11.2 $\\pm $ 0.3 mm from the starting position of the cross disperser.",
"Figure REF shows the focus curves for four different wavelengths that are representative of different portions of the echellogram.",
"Figure: Focus curves for CHESS-2 echellogram orders with λ\\lambda == 1040 Å, 1150 Å, 1350 Å, and 1600 Å.",
"Full width at half maximum (FWHM) measurements were taken as the overall order width, not the spectral width of specific emission features within the order.",
"We focused the echellogram closer to the minima of the 1040 Å and 1150 Å orders, at 11.2 mm from the original starting position of the cross disperser during FUV alignments, because the separation of orders with wavelengths λ\\lambda << 1100 Å was critical in the final data product.Figure: Raw images (with edge effects trimmed out) of the CHESS-2 echellograms from pre-flight (December 2015) and post-flight calibrations (March 2016) using an arc lamp flowing H/Ar gas.",
"The brightest feature in both images is HI-Lyα\\alpha (1215.67 Å), which appears in two adjacent orders in the echellogram.",
"The other broad feature(s) are HI-Lyβ\\beta (1025.72 Å), about 1/4 of the way from the top of the image, and HI-Lyγ\\gamma (97.25 Å), barely visible above the Lyβ\\beta features.",
"Both images are scaled the same.",
"Between field operations, launch, and returning to CU, the instrument echellogram shifted slightly to the left and upward in the images shown, corresponding to total shifts << 5 ' ^{\\prime }.After final alignment and focus positions of the echelle and cross disperser were determined, long exposures with a 65/35% hydrogen/argon (H/Ar) gas mixture fed through a hollow-cathode (“arc\") lamp were taken for a complete sampling of H and H$_2$ emission lines in the CHESS bandpass.",
"Pre- and post-launch deep spectra with the H/Ar lamp are taken to characterize the one-dimensional (1D) extracted spectrum, define the wavelength solutions of the instrument, and determine the line spread functions (LSFs) across the bandpass.",
"Figure REF shows the echellogram of CHESS-2 for pre- and post-launch calibrations.",
"Both echellograms are co-additions of multiple exposures taken under vacuum to accumulate more than 60 million photon counts for a complete sampling of H and H$_2$ emission lines.",
"Each exposure was defined by how long we could run the full instrument configuration in vacuum without over-heating the electronics section, which usually lasted around 30 minutes.",
"Each exposure collected was typically between 10 - 20 million photon counts.",
"Extracting the 1D spectra from the echellogram was accomplished in several steps.",
"First, the echellogram had to be rotated very slightly ($\\theta $ $<$ 1$^{\\circ }$ ) to successfully extract the light in each order without contamination from light in adjacent orders.",
"The location of each order was then determined by summing (collapsing) all photon counts along the x-axis, which added all the light in each order together and created peaks where orders were present and troughs at inter-order pixels.",
"This exercise also determined the width of each order, which ranged from 4 $-$ 12 pixels wide in the CHESS-2 echellogram.",
"Figure: The extracted 1D spectra of six orders from the pre-flight calibration echellogram of CHESS-2.",
"Black represents the CHESS-2 extracted spectrum over the order extent.",
"The blue lines are modeled H 2 _2 emission features, using estimated physical parameters of the conditions within the arc lamp, including the column density of H 2 _2 molecules (N(H 2 _{2}) ∼\\sim 10 19 ^{19} cm -2 ^{-2}), effective temperature (T eff _{eff} = 800 K), and electron energy (E elec _{elec} = 50 eV).Once pixel locations and order widths were extracted, we collapsed a given order along the y-axis, creating the 1D spectrum of each order.",
"We used the composition of air through the arc lamp to map out well-known atomic lines and their corresponding orders.",
"Once wavelengths and order locations were known for these emission lines, we used the H/Ar arc lamp echellogram, in conjunction with modeled H$_2$ fluorescent lines, to extrapolate the pixel-to-wavelength conversion for CHESS-2 over the entire FUV bandpass; examples of the pixel-to-wavelength extractions determined from H$_2$ emission features are shown in Figure REF .",
"Once wavelength solutions were known for 20 $-$ 30 orders, we fit a 6th order polynomial function to extrapolate the wavelength calibration over the entire 130 orders in the CHESS-2 echellogram.",
"Figure REF shows the final wavelength calibration for pre- and post-launch laboratory echellograms.",
"The different colored spectra represent separate orders extracted from the echellograms.",
"Because of the lower line density and AOI of the Bach echelle grating, we used higher-order dispersion solutions than were designed for CHESS.",
"This resulted in overlapping wavelength solutions in adjacent orders, making it easier to stitch together the entire CHESS-2 1D spectrum from all order spectra via correlated spectral features.",
"Despite changes in the echellogram between pre- and post-launch calibration images, there were no significant changes in the functional wavelength solution of the 1D extracted spectra.",
"The only (minor) change to the functional wavelength solution was the shift of the starting pixel-to-wavelength conversion, since more shorter wavelength lines were available for extraction in the post-launch echellogram configuration.",
"The final wavelength solution of the pre- and post-flight calibration spectra will provide more accurate radial velocity estimates of photospheric/ISM material and line identifications for atomic, ionized, and molecular material in the line of sight to $\\epsilon $ Per.",
"Figure: Complete first-order wavelength solution for the pre- and post-flight CHESS-2 calibration spectra from 900 -- 1750 Å.",
"The final wavelength solution was determined using H 2 _2 fluorescence emission features and a functional extrapolation of the wavelength with a 6th-order polynomial fit, determined from pixel-to-wavelength solutions manually fitted for ∼\\sim 30 orders with prominent emission features.",
"Over-plotted in magenta is the model H 2 _2 fluorescence inside the arc lamp (T eff _{eff} = 800 K, N(H 2 _{2}) ∼\\sim 10 19 ^{19} cm -2 ^{-2}, E elec _{elec} = 50 eV).",
"Each spectrum is scaled to the highest total counts of the H 2 _2 features; otherwise, Lyα\\alpha would dominate the spectrum and the H 2 _2 features would be washed out in the images.",
"To show how neighboring order spectra overlap and correlate to form the CHESS-2 1D spectrum, individual order spectra have been plotted using different colors.",
"Overlaid are two vertical lines to show where Lyβ\\beta (dark red) and Lyα\\alpha (green) are located in the spectra.Figure: LSF fits of H 2 _2 emission features in the pre-launch calibration spectrum of CHESS-2 (echelle order m = 286, also featured in Figure ).",
"The order spectrum is shown in black.",
"Red and blue Gaussian line fits are shown for the narrow and board Gaussian fits for each line, respectively.",
"The green line is the sum of all Gaussian components to reproduce the spectrum.",
"A modeled H 2 _2 fluorescence spectrum is shown in magenta.",
"We show how the power placed in the narrow and broad Gaussian components of each emission feature changes over the order spectrum; at shorter wavelengths, the narrow and broad Gaussian components hold similar percentages (50%) of the power in the line, but as the wavelength increases across the order, more power is found in the broad Gaussian component (25%/75% in the narrow/broad components).",
"This behavior is seen in order spectra with strong H 2 _2 features, indicating that the cross disperser may be tilted and focusing shorter wavelength features in each order better than the longer wavelength features.",
"For the three emission features, narrow line velocity widths vary between 11.8, 15.2, and 10.7 km/s respectively, while broad line fits have velocity widths between 34.1, 48.8, and 33.0 km/s, which are offset from the narrow line peaks by 17.9, 18.5, and 15.0 km/s.After the wavelength solution was found for the pre-flight calibration echellogram, we determined the LSF and resolving power of the spectrum produced by CHESS-2.",
"To determine the LSF of the instrument, we created a multi-Gaussian fitting routine to describe the line shapes of the emission features present in each echellogram order.",
"An example of how this routine was applied for all emission lines across a given order is shown in Figure REF .",
"At first glance, the emission lines produced by CHESS-2 are not symmetric; they have a sharp peak towards the left (blue-ward) side of the emission line, and a shallower slope back to the continuum level toward the right (red-ward).",
"We described this line shape with two separate Gaussian functions summed together, which resulted in a narrow- and broad-component to each LSF.",
"Each echelle order spectrum showed very similar behavior in LSFs, specifically that the shorter (longer) wavelength end of each order has more power in the narrow (broad) Gaussian fit.",
"We see that the Area(narrow)/Area(broad) of the emission features for shorter wavelengths in the echelle order is $\\sim $ 1, whereas emission features with longer wavelengths in the same echelle order have Area(narrow)/Area(broad) $\\sim $ 0.33.",
"This indicates that the cross disperser may be tilted, resulting in a better focus at one end of the echellogram.",
"We note that for all emission lines, a peak (narrow) line fit was apparent, and we use this component of the LSF to estimate the resolving power of CHESS-2.",
"We also consistently measured narrow line component velocity widths ($v_{narrow}$ ) in the pre-launch CHESS-2 spectrum to be between 3 $-$ 20 km/s, and broad line velocity widths ($v_{broad}$ ) to range between 20 $-$ 60 km/s.",
"For post-flight calibrations, we measured $v_{narrow}$ $\\sim $ 5 $-$ 25 km/s and $v_{broad}$ $\\sim $ 20 $-$ 60 km/s.",
"All high S/N ($>$ 100) emission LSFs across the CHESS-2 bandpass were saved to later convolve with CHESS-2 flight data to eliminate instrumental effects seen in the final science spectrum.",
"Figure: A scatter plot of the measured resolution (via FWHM measurements) of individual electron-impact H 2 _2 emission features in the pre-flight calibration spectrum of CHESS.",
"Orange spots are measured FWHM values of individual line peak cores, and the dashed lines represent resolution cutoffs, based on the FWHM (in microns) of the emission line as a function of wavelength.",
"Both pre- and post-launch resolving powers seem to be concentrated around 25,000 -- 70,000 over the bandpass of CHESS-2, but there is a noticeable concentration of lines at lower resolving powers (larger FWHM) at λ\\lambda << 1300 Å for the post-launch calibration.",
"Physical shifts in the optical alignment between pre- and post-launch calibrations affected the resolving power of the instrument.",
"This may have been caused by tilting the cross disperser and subsequent echellogram, or by moving away from the instrument focus, or a combination of both.We show the FWHM of the narrow component LSFs of the emission lines observed across the CHESS bandpass in Figure REF .",
"Each orange spot represents a different emission line in the CHESS-2 calibration image, and the different dashed lines show the resulting resolving power as a function of FWHM and wavelength (black: R = 25,000; blue: R = 33,333; green: R = 50,000; red: R = 66,666; magenta: R = 100,000).",
"The pre-launch calibration shows a concentration of resolving powers between R = 25,000 and 66,000, with an average resolution between R $\\sim $ 33,000 $-$ 70,000 (velocity width between 9.0 $-$ 4.5 km/s) across the CHESS-2 bandpass.",
"While this is below our nominal resolving power goal of 100,000, we note that the measured resolving power of CHESS-2 using the laboratory calibration data may only represent a lower limit to the instrument capabilities.",
"As mentioned previously by Ref.",
"Hoadley+14, the ray trace for CHESS requires an input on-axis light source with beam spread $<$ 1$^{\\prime \\prime }$ , but we have only been able to demonstrate constraining the spread to 2$^{\\prime \\prime }$ $-$ 3$^{\\prime \\prime }$ in our laboratory vacuum chamber.",
"Additionally, the laboratory arc lamp may have pressure-broadening effects on the H$_2$ electron-impact emission.",
"This would produce emission lines with FWHMs larger than the resolving power of the instrument, resulting in only being able to measure a lower limit to the resolving power of CHESS-2 across the bandpass.",
"However, post-launch calibration results show a shift from the average R $\\sim $ 33,000 $-$ 70,000 to R $\\sim $ 25,000 $-$ 60,000 (velocity width between 12.0 $-$ 5.0 km/s).",
"This indicates that either the instrument shifted slightly out of focus before and/or during launch operations, or the shift in the echellogram affected the line shapes of the spectra.",
"In either case, post-launch resolution of CHESS-2 is degraded from pre-launch values.",
"New LSFs were measured for the post-launch CHESS-2 calibration echellogram to better correct for instrument affects in the science spectrum."
],
[
"CHESS-2 Launch and Preliminary Flight Results",
"CHESS-2 was brought to White Sands Missile Range (WSMR) in late January 2016 for field operations in preparation for launch.",
"CHESS-2 underwent various tests, including vibration, which required a means of determining alignment shifts before launch.",
"We fitted a Bayard-Alpert tube (ionization gauge)[23] with a small, collimating mirror and pinhole (20 $\\mu $ m) to the shutter door, which produced an echellogram with air spectral features (C, N, O, H).",
"We used these images to measure the centroid pixel location (in both x and y axes) of N I and O I emission features.",
"Comparing the new centroid location of 3 $-$ 4 emission lines to reference pixel locations measured at CU, we use the plate scale of the instrument (plate scale = 206265$^{\\prime \\prime }$ /1236.834 mm = 166.77 $^{\\prime \\prime }$ /mm) to estimate alignment shifts before launch.",
"The largest centroid shift measured pre-launch was 105 pixels (over 2k $\\times $ 2k pixels), which corresponded to a physical shift of 1176 $\\mu $ m (1.176 mm), or 3.27$^{\\prime }$ (196.12$^{\\prime \\prime }$ ).",
"Alignment shifts are apparent between pre- and post-launch calibration images (Figure REF ).",
"Given the large FOV of the instrument (0.67$^{\\circ }$ , or 40.2$^{\\prime }$ ), the success criteria specification for the on-target acquisition (5$^{\\prime }$ ), and the ability to demonstrate that the instrument can still collect a science echellogram, this alignment shift was acceptable to continue with launch of the instrument.",
"CHESS-2 was launched aboard NASA mission 36.297 UG from White Sands Missile Range (WSMR) on 21 February 2016 at 09:15pm MST using a two-stage Terrier/Black Brant IX vehicle.",
"The mission was deemed a comprehensive success.",
"The instrument successfully collected data over the allotted $\\sim $ 400 seconds of observing time, with $>$ 200 seconds without up-link maneuvers.",
"When the instrument centered on-target ($\\epsilon $ Per), the count rate was lower than expected ($\\sim $ 25,000 counts/second, instead of 50,000 counts/second), and as much as half of the signal was geo-coronal scattered light.",
"We steered the payload around the FOV for the first half of the flight, to see if we were off-axis from the star, but we elected to move back to the original acquisition position and integrate for the second half of the flight.",
"Figure: The flight data from 36.297 UG (CHESS-2).",
"Left: The science echellogram of ϵ\\epsilon Per after an exposure time of ∼\\sim 200 seconds, with inter-order scatter subtracted from the raw image.",
"The echelle orders are stacked horizontally in the image, with order spectra easily distinguishable in the bottom half of the echellogram.",
"Because the echelle used in CHESS-2 disperses the starlight into very high orders for λ\\lambda << 1200 Å (m >> 280), shorter wavelength orders are more difficult to distinguish and required scattered light subtraction and echellogram collapsing along the order axis of the image.",
"Progress is still being made to better identify and extract the spectra from these orders.",
"Right: A normalized spectrum extracted from one order identified in the raw fight data to show interstellar absorption features in the data.",
"The order (m = 276) ranges from ∼\\sim 1199 -- 1208 Å and shows warm (Si III) and cool (N I) interstellar features against the stellar continuum.",
"Preliminary line fits are shown in blue, and physical quantities derived from the line fits are listed in the bottom right of the plot.",
"Both N I and Si III column densities and b-values are consistent with line of sight cool and warm ISM diagnostics previously explored in the literature, .From the beginning of the flight, we immediately saw photospheric and interstellar absorption features in the echellogram of $\\epsilon $ Per, the prominent features being Ly$\\alpha $ , O I, and C III, as well as stellar continuum for orders with $\\lambda $ $>$ 1300 Å.",
"Once we integrated on $\\epsilon $ Per during the second half of the flight, interstellar features started to appear, including Si III, N I, Si II, and H$_2$ complexes.",
"Figure REF shows the raw flight data (echellogram; left) after $\\sim $ 200 seconds on-target, including an extraction of one order that shows both the N I and Si III absorption features (right).",
"After post-processing of the data (see Section REF ), a preliminary analysis of the N I and Si III absorption features has been completed.",
"Our analysis assumed a Gaussian line profile LSF convolved with an instrument resolving power R = 50,000.",
"The results of the N I and Si III abundances and b-values, which represents both the turbulent velocity and temperature ISM species, are consistent with cool and warm interstellar diagnostics explored in the local ISM[24], [25].",
"Overall, the quality of the flight was S/N $\\gtrsim $ 5, which is sufficient to identify spectral absorption features, but may make it difficult to quantify the column densities and b-values of the line-of-sight measurement.",
"We are currently working on techniques to better extract the stellar spectrum from the geo-coronal background (discussed in Section REF ), making it easier to identify atomic and molecular contents in the science data."
],
[
"Post-Flight Calibration and Science Data Reduction",
"We verified that all essential components of the instrument were working and completed post-launch activities for CHESS-2 upon return from WSMR after the launch of 36.297 UG.",
"First, because we elected to stay on $\\epsilon $ Per for the entirety of the rocket flight, we needed an understanding of how the geo-coronal scattered light may spread across the detector before beginning post-launch calibrations.",
"Therefore, we took a scattered light spectrum with the field operations Bayard-Alpert tube fitted to a mini-Conflat opening in the instrument shutter door.",
"This spectrum provided a rough “flat-field\" tool to apply to the science echellogram.",
"Then, we integrated the instrument into our vacuum chamber and took a deep H/Ar spectrum, complementary to the one taken pre-launch.",
"Post-launch calibrations are described with pre-launch calibrations in Section .",
"One of the biggest surprises during 36.297 UG was the lower-than-expected count rate while on $\\epsilon $ Per.",
"One possibility is that the efficiencies of the gratings and detector were over-estimated before instrument build-up or degraded between instrument build-up and launch.",
"The former seems more plausible, given that the pre- and post-launch calibration exposures were comparable for similar arc lamp conditions.",
"Another possibility is that we aligned the echellogram to the wrong echelle orders.",
"For example, instead of aligning to the main and next brightest adjacent order for a total of 20% efficiency (at Ly$\\alpha $ ), we aligned the echellogram to the main order and less bright adjacent order, giving a lower$-$ than$-$ expected total order efficiency (likely closer to 10% at Ly$\\alpha $ ).",
"In either case, we will be vigilant in the next iterations of CHESS to be certain we align our instrument properly and keep our optical coating pristine during instrument build-up and alignments.",
"Figure: The full 1D spectrum of ϵ\\epsilon Per, with a preliminary wavelength fit, from the ∼\\sim 200 seconds of data taken on 36.297 UG.",
"Order spectra have been cross-correlated, with photon counts summed to create a 1D spectrum spanning the CHESS bandpass (1000 -- 1600 Å).",
"Spectral features are apparent in the data, including the broad Lyα\\alpha feature (1215.67 Å), Si III (1206.51 Å), C III (1174.26 -- 1176.37 Å, appears blended), and several H 2 _2 complexes (∼\\sim 1020 Å, 1050 Å, and 1070 Å).",
"Continued effort is being made to separate the geo-coronal background from the stellar continuum via extracting scattered light profiles from “dark” regions of the science data and subtracting inter-order scatter from the echelle orders, which will help with order extraction, cross-correlation of order spectra, and refinement the wavelength solution of the science spectrum.Figure REF shows a preliminary extraction of the CHESS-2 science data, with a preliminary wavelength solution determined from post-launch calibrations.",
"Post-processing of the science data is on-going, with a focus on subtracting the geo-coronal background from the stellar spectrum.",
"Elimination of the scattered light profile in the flight data allows for better extraction of the echelle order locations in the echellogram, cross-correlation of absorption features in the science spectrum, and better definition of the wavelength solution for the final science spectrum.",
"One way to eliminate the scattered light is to estimate the noise in the data via extraction of photon counts in echellogram “dark” regions $-$ either regions on the detector where the instrument effective area is very small ($\\lambda $ $<$ 900 Å) or absorption lines we know to be saturated, such as Ly$\\alpha $ and O I - and subtracting the scattered light profiles in these regions from similar areas in the echellogram.",
"The geo-coronal background can also be estimated from the inter-order contamination in the flight echellogram, which is determined from taking the sum of all photon counts along the x-axis and extracting the noise from the minima pixel values between echelle orders.",
"This technique minimizes the inter-order counts and exaggerates order spectra locations in the echellogram, making the stellar continuum-to-absorption line ratios deeper in the 1D extracted spectra."
],
[
"Future Launch and Instrument Changes",
"The CHESS instrument is scheduled to launch two more times $-$ CHESS-3 (36.323 UG) is currently set to launch in June 2017 from WSMR, and CHESS-4 will launch in early 2018, ideally during an Australian sounding rocket campaign.",
"CHESS-3 will implement a new echelle from Richardson Grating (87 grooves/mm, 63$^{\\circ }$ blaze angle), which has demonstrated $>$ 50% peak order efficiency at Ly$\\alpha $ (see Figure REF ) and has order solutions identical to those designed for the CHESS instrument (m = 266 $-$ 166).",
"CHESS-4 will see the implementation of an alternative detector technology, the $\\delta $ -doped charge coupled device (CCD), with new readout technologies to accommodate large focal plane arrays[26].",
"The authors would like to thank the students and staff at CU for their tremendous help in seeing CHESS-2 come to fruition.",
"We would also like to thank the NSROC staff at WFF and WSMR for their tireless efforts that pushed us to a smooth launch.",
"NK also thanks Rocket League and the song “Sister Christian\" by Night Ranger for keeping him sane through field operations in February 2016.",
"This work was supported by NASA grant NNX13AF55G."
]
] | 1606.04979 |
[
[
"Vertical line nodes in the superconducting gap structure of Sr2RuO4"
],
[
"Abstract There is strong experimental evidence that the superconductor Sr2RuO4 has a chiral p-wave order parameter.",
"This symmetry does not require that the associated gap has nodes, yet specific heat, ultrasound and thermal conductivity measurements indicate the presence of nodes in the superconducting gap structure of Sr2RuO4.",
"Theoretical scenarios have been proposed to account for the existence of accidental nodes or deep accidental minima within a p-wave state.",
"To elucidate the nodal structure of the gap, it is essential to know whether the lines of nodes (or minima) are vertical (parallel to the tetragonal c axis) or horizontal (perpendicular to the c axis).",
"Here, we report thermal conductivity measurements on single crystals of Sr2RuO4 down to 50 mK for currents parallel and perpendicular to the c axis.",
"We find that there is substantial quasiparticle transport in the T = 0 limit for both current directions.",
"A magnetic field H immediately excites quasiparticles with velocities both in the basal plane and in the c direction.",
"Our data down to Tc/30 and down to Hc/100 show no evidence that the nodes are in fact deep minima.",
"Relative to the normal state, the thermal conductivity of the superconducting state is found to be very similar for the two current directions, from H = 0 to H = Hc2.",
"These findings show that the gap structure of Sr2RuO4 consists of vertical line nodes.",
"Given that the c-axis dispersion (warping) of the Fermi surface in Sr2RuO4 varies strongly from surface to surface, the small a-c anisotropy suggests that the line nodes are present on all three sheets of the Fermi surface.",
"If imposed by symmetry, vertical line nodes would be inconsistent with a p-wave order parameter for Sr2RuO4.",
"To reconcile the gap structure revealed by our data with a p-wave state, a mechanism must be found that produces accidental line nodes in Sr2RuO4."
],
[
"INTRODUCTION",
"Sr$_{2}$ RuO$_4$ is one of the rare materials in which p-wave superconductivity is thought to be realized.",
"Nuclear magnetic resonance [1], [2] and neutron scattering [3] measurements find no drop in the spin susceptibility below the superconducting transition temperature $T_{\\rm c}$ , strong evidence in favour of spin-triplet pairing.",
"Measurements of muon spin rotation [4], [5] and the polar Kerr angle [6] show that time-reversal symmetry is spontaneously broken below $T_{\\rm c}$ .",
"These results (and others) have led to the view that Sr$_{2}$ RuO$_4$ has a chiral p-wave order parameter, with a $d$ -vector given by ${\\mathbf {d}} = \\Delta _0{\\mathbf {z}}(k_x\\pm ik_y)$ [7], [8], [9].",
"Nevertheless, the symmetry of the superconducting order parameter in Sr$_{2}$ RuO$_4$ is still under debate [8], [9].",
"One of the problems is that although the gap structure of a chiral p-wave order parameter is not required by symmetry to go to zero, i.e.",
"to have nodes, anywhere on a two-dimensional Fermi surface, there are in fact low-energy excitations deep inside the superconducting state of Sr$_{2}$ RuO$_4$ , as detected in the specific heat [10], [11], [12], [13], ultrasound attenuation [14] and penetration depth [15] at very low temperature.",
"Theoretical scenarios have been proposed to account for those excitations in terms of either accidental nodes that are perpendicular to the tetragonal $c$ axis (i.e.",
"`horizontal') [16] or deep minima in the superconducting gap along lines parallel to the $c$ axis (i.e.",
"`vertical') [17], [18], [19], [20].",
"The latter vary in depth from sheet to sheet on the three-sheet Fermi surface of Sr$_{2}$ RuO$_4$ .",
"On the large $\\gamma $ sheet, the gap develops deep minima in the $a$ direction because an odd-parity order parameter must go to zero at the zone boundary.",
"These scenarios are difficult to reconcile with the specific heat and thermal conductivity of Sr$_{2}$ RuO$_4$ .",
"When plotted as $C_e/T$ vs $T$ , the electronic specific heat $C_e$ of Sr$_{2}$ RuO$_4$ is perfectly linear below $\\sim $ $T_{\\rm c}$ /2, down to the lowest temperature [10], [11], [12], [13].",
"Gap minima of various depths inevitably lead to deviations from perfect linearity in $C_e/T$ vs $T$ [17].",
"In the clean limit, a truly linear behaviour can only be obtained if the minima on all three sheets are so deep that they extend to negative values, thereby producing accidental nodes.",
"The in-plane thermal conductivity $\\kappa _{\\text{a}}$$(T)$ of Sr$_{2}$ RuO$_4$ decreases smoothly down to the lowest measured temperature, and it extrapolates to a large residual linear term, $\\kappa _{\\text{0}}/T$ , at $T=0$ [21].",
"This residual linear term is robust against impurity scattering, and virtually unaffected by a 10-fold increase in scattering rate [21].",
"This is the classic behaviour of a nodal superconductor whose nodes are imposed by symmetry [22], [25], [27], [28] (Fig.",
"REF ), as in the $d$ -wave state of cuprate superconductors [29].",
"It comes from the linear energy dependence of the density of states at low energy, which produces a compensation between the growth in the density of quasiparticles and the decrease in their mean free path as a function of impurity scattering [27].",
"Such a compensation does not occur in a nodeless $p$ -wave state [23], [24] (Fig.",
"REF ), nor does it occur for accidental nodes in an $s$ -wave state [30].",
"In summary, the known properties of $\\kappa /T$ and $C_e$ in Sr$_{2}$ RuO$_4$ strongly suggest that the low-energy quasiparticles in the superconducting state come from nodes in the gap, not from deep minima.",
"Because accidental nodes do not occur naturally in the chiral $p$ -wave state that is widely proposed for Sr$_{2}$ RuO$_4$ , it is important to establish the presence of nodes.",
"Moreover, because other proposed states have symmetry-imposed line nodes that are either horizontal (chiral $d$ -wave state [31]) or vertical ($f$ -wave state [32], [33]), we need to determine whether line nodes are vertical or horizontal.",
"Figure: Residual linear term in the thermal conductivity, κ 0 /T\\kappa _{\\text{0}}/T, as a function of impurity scattering rate Γ\\Gamma ,both normalized to unity at Γ=Γ c \\Gamma = \\Gamma _{\\rm c}, the critical scattering rate needed to suppress superconductivity.The blue lines are theoretical calculations for a dd-wave state and a fully-gapped pp-wave state , as indicated (left axis).In the clean limit (Γ→0\\Gamma \\rightarrow 0), κ 0 /T\\kappa _{\\text{0}}/T vanishes in the pp-wave case while it reaches a non-zero value in the dd-wave case (open circle),whose value is given by Eq.",
"1, estimated at κ 0 /T\\kappa _{\\text{0}}/T =15.8= 15.8 mW / K 2 ^2 cm in Sr 2 _{2}RuO 4 _4 (see text).The experimental values of κ 0 /T\\kappa _{\\text{0}}/T measured in Sr 2 _{2}RuO 4 _4 are plotted as red symbols (right axis; circles, ; square, this work),taking ℏΓ c =k B T c0 \\hbar \\Gamma _{\\rm c} = k_{\\rm B} T_{\\rm c0}.The black lines are the zero-energy density of states N 0 N_0, normalized by the normal-state value N F N_{\\rm F}(left axis; solid, full-gap pp-wave ; dashed, dd-wave ).Black triangles show N 0 /N F N_0 / N_{\\rm F} for a pp-wave state with a deep gap minimum (Δ min ≃Δ max /4\\Delta _{\\rm min} \\simeq \\Delta _{\\rm max} / 4) .Existing experimental evidence on the direction of line nodes in Sr$_{2}$ RuO$_4$ is contradictory.",
"Measurements of the heat capacity as a function of the angle made by a magnetic field $H$ applied in the basal plane (normal to the $c$ axis) relative to the $a$ axis ([100] direction) reveal a small four-fold variation below 0.25 K that is consistent with vertical line nodes along the $\\Gamma $ M directions [12], [13].",
"However, no such angular variation was detected in the heat conduction down to 0.3 K [34], [35], [36].",
"Moreover, ultrasound attenuation in Sr$_{2}$ RuO$_4$ is rather isotropic in the plane, unexpected if line nodes are vertical [14].",
"In this Letter, we shed new light on the gap structure of Sr$_{2}$ RuO$_4$ by using the directional power of thermal conductivity to determine whether the line nodes are vertical or horizontal.",
"In particular, we probe nodal quasiparticle motion along the $c$ axis as $T \\rightarrow 0$ , from measurements of $\\kappa _{\\text{c}}$ , the thermal conductivity along the $c$ axis, down to $T_{\\rm c}$ /30 (50 mK).",
"We observe a substantial residual term $\\kappa _{\\text{0}}/T$ in the $c$ direction at $H=0$ .",
"Moreover, $\\kappa _{\\text{0}}/T$ is rapidly enhanced by a magnetic field, even as low as $H_{\\rm c2}$ / 100.",
"This confirms that the line nodes in Sr$_{2}$ RuO$_4$ are not deep minima and it shows they must be vertical.",
"Furthermore, quantitative analysis suggests that the line nodes are present on all three Fermi surfaces.",
"If the vertical line nodes are imposed by symmetry, then, by virtue of Blount's theorem [37], they would rule out a spin-triplet state, such as the proposed $p$ -wave state [38].",
"Conversely, if Sr$_{2}$ RuO$_4$ is indeed a $p$ -wave superconductor, then a reason must be found for the presence of accidental line nodes in its gap function.",
"Note that the obvious spin-singlet state that breaks time-reversal symmetry has symmetry-imposed line nodes that are horizontal, not vertical [31]."
],
[
"METHODS",
"Single crystals of Sr$_{2}$ RuO$_4$ were grown by the floating-zone method [39] and annealed in oxygen flow at 1080 $^{\\circ }$ C for 8 days.",
"Both samples were cut into rectangular platelets from the same crystal rod that contained very few Ru inclusions ($\\sim 3$ inclusions / mm$^2$ ).",
"No $3\\,$ K anomaly was detected in either the susceptibility of the large annealed crystal or the resistivity of the small measured samples.",
"The a-axis sample had a length of 4.0 mm along the $a$ axis, and a cross-section of $0.3 \\times 0.18$ mm$^2$ .",
"The c-axis sample had a length of 1.0 mm along the $c$ axis, and a cross-section of $0.4 \\times 0.42$ mm$^2$ .",
"The geometric factor of the a-axis sample was refined by normalizing the room-temperature resistivity to the well-established literature value of $\\rho _{\\rm a}(300~{\\rm K}) = 121~\\mu \\Omega $ cm [40].",
"The geometric factor of the c-axis sample was calculated from sample dimensions and contact separation.",
"The value we find is $\\rho _{\\rm c}(300~{\\rm K}) = 33$ m$\\Omega $ cm, in the range of reported values [41], [42].",
"Contacts were made with silver epoxy (Epo-Tek H20E) heated at $450\\,^{\\circ }\\mathrm {C}$ for 1 hour in oxygen flow.",
"Silver wires were then glued on with silver paint.",
"From our thermal conductivity measurements, we obtain a superconducting transition temperature $T_c = 1.2$ K, consistent with the measured residual resistivity of our $a$ -axis sample, $\\rho _{\\rm a0} = 0.24~\\mu \\Omega $ cm [40].",
"The thermal conductivity was measured using a one heater-two thermometer method [43], with an applied temperature gradient of 2-5% of the sample temperature.",
"Measurements where carried out for two directions of the magnetic field $H$ : $H \\parallel a$ and $H \\parallel c$ .",
"For $H \\parallel a$ , the field was aligned to within $1^{\\circ }$ of the $a$ axis, and perpendicular to the heat current.",
"(For this field direction, a misalignment of $1^{\\circ }$ can cause a decrease of $H_{\\rm c2}$ by 0.1 T [44].)",
"The field was always changed at $T >$ $T_{\\rm c}$ ."
],
[
"H = 0 : IN-PLANE TRANSPORT",
"Fig.",
"REF a shows the thermal conductivity of Sr$_{2}$ RuO$_4$ in zero field, for the current in the plane ($J \\parallel a$ ).",
"The conductivity $\\kappa _{\\text{a}}$ is completely dominated by the electronic contribution [21], $\\kappa _{\\rm e}$ , so that $\\kappa _{\\rm e} >> \\kappa _{\\rm p}$ up to $\\sim 3$ K, where $\\kappa _{\\rm p}$ is the phonon conductivity.",
"In Fig.",
"REF a, a Fermi-liquid fit to the normal-state data (above $T_{\\rm c}$ ) yields $\\kappa _{\\text{N}}/T$ = $L_0 / (a + bT^2)$ , where $L_0 \\equiv (\\pi ^2/3) (k_{\\mathrm {B}}/e)^2$ , with $a = 0.24~\\mu \\Omega $ cm and $b = 8~{\\rm n}\\Omega $ cm/K$^2$ .",
"We see that the Wiedemann-Franz law is satisfied, with $a = \\rho _{a0}$ .",
"Fig.",
"REF b shows a zoom of the data at low temperature, seen to extrapolate to $\\kappa _{\\text{0}}/T$ = $20 \\pm 2$ $\\text{mW}/\\text{K}^2\\text{cm}$ , a large residual linear term in excellent agreement with the value reported for Sr$_{2}$ RuO$_4$ samples of similar $T_{\\rm c}$ [21] (Fig.",
"REF ).",
"In the limit of a vanishing impurity scattering rate $\\Gamma $ , whence $T_{\\rm c}$ $\\rightarrow 1.5$ K, $\\kappa _{\\text{0}}/T$ = $17 \\pm 2$ $\\text{mW}/\\text{K}^2\\text{cm}$ [21].",
"A ten-fold increase in $\\Gamma $ only yields a modest increase in $\\kappa _{\\text{0}}/T$ (Fig.",
"REF ).",
"Such a weak dependence of $\\kappa _{\\text{0}}/T$ on $\\Gamma $ is precisely the behavior expected of a superconductor with symmetry-imposed line nodes, whereby the impurity-induced growth in the quasiparticle density of states is compensated by a corresponding decrease in mean free path, as in a $d$ -wave superconductor [22], [27], [28] (Fig.",
"REF ).",
"(By contrast, accidental nodes and deep minima in an $s$ -wave superconductor are not robust against impurity scattering, so they will in general be lifted or be made shallower, respectively, causing $\\kappa _{\\text{0}}/T$ to vanish or decrease with impurity concentration, respectively [30].)",
"In other words, the remarkable fact that $\\kappa _{\\text{0}}/T$ remains large [21] even when the zero-energy density of states vanishes [10] as $\\Gamma \\rightarrow 0$ in Sr$_{2}$ RuO$_4$ is the clear signature of a line node.",
"Indeed, while impurities in a $p$ -wave superconductor without nodes do induce a zero-energy density of states [23], [24], [26], the associated $\\kappa _{\\text{0}}/T$ vanishes as $\\Gamma \\rightarrow 0$ [23], [24] (Fig.",
"REF ) because the impurity-induced states are localized.",
"The magnitude of $\\kappa _{\\text{0}}/T$ at $\\Gamma \\rightarrow 0$ can be evaluated theoretically from a knowledge of the Fermi velocity $v_{\\rm F}$ and the gap velocity at the node, $v_{\\Delta }$ .",
"For a $d$ -wave gap on a single 2D Fermi surface [28] : $\\frac{\\kappa _{\\text{0}}}{T} = \\frac{k_{\\rm B}^2}{3\\hbar }\\frac{1}{c} \\left(\\frac{v_{\\rm F}}{v_{\\Delta }} + \\frac{v_{\\Delta }}{v_{\\rm F}}\\right)~,$ where $c$ is the interlayer separation along the $c$ axis and $v_{\\Delta } = 2 \\Delta _0 / \\hbar k_{\\rm F}$ , in terms of the Fermi wavevector $k_{\\rm F}$ and the gap maximum $ \\Delta _0$ , in $\\Delta ({\\rm \\phi }) = \\Delta _0 \\cos {\\rm 2\\phi }$ .",
"This expression works very well for overdoped cuprate superconductors such as YBa$_2$ Cu$_3$ O$_7$ and Tl$_2$ Ba$_2$ CuO$_{6 + \\delta }$ [45], quasi-2D metals where the pairing symmetry is established to be $d$ -wave.",
"Let us use Eq.",
"1 to estimate $\\kappa _{\\text{0}}/T$ in Sr$_{2}$ RuO$_4$ .",
"Figure: a)In-plane (aa-axis) thermal conductivity κ a \\kappa _{\\text{a}}(TT) of Sr 2 _{2}RuO 4 _4 at H=0H = 0(open circles).The black line is a Fermi-liquid fit to the normal-state data, κ N /T=L 0 /(a+bT 2 )\\kappa _{\\rm N}/T = L_0 / (a + bT^2), extended below T c T_{\\rm c}.The arrow marks the location of the superconducting transition temperature, T c T_{\\rm c} = 1.2 K,defined as the temperature below which κ/T\\kappa /T deviates from its normal-state behavior.Note that the contribution of phonons to κ a \\kappa _{\\text{a}}, κ p \\kappa _{\\rm p}, is negligible up to 3 K,so that κ a \\kappa _{\\text{a}} ≃\\simeq κ e \\kappa _{\\rm e}, the electronic contribution.The red dashed line is a calculation for a three-band model of a pp-wave state withdeep minimain the gap structure (see text).It provides a good description of the data at high temperature, but it fails below 0.3 K.b)Zoom at low temperature.Data taken in a magnetic field H=0.25H = 0.25 T (H∥aH \\parallel a) are also shown (blue dots).The solid blacklines are a fit of the data to the form κ/T=κ 0 /T+cT n \\kappa /T = \\kappa _0/T + c T^n.The Fermi surface of Sr$_{2}$ RuO$_4$ is quasi two-dimensional and it has been characterized experimentally in exquisite detail [47].",
"It consists of three cylinders: two at the center of the Brillouin zone ($\\beta $ and $\\gamma $ ) and one ($\\alpha $ ) at the corner.",
"The values of $k_{\\rm F}$ and $v_{\\rm F}$ are known precisely for each.",
"In a $d_{x^2-y^2}$ -wave state, each cylinder would have four vertical line nodes (along the $x = \\pm ~y$ directions).",
"Assuming the same gap on each Fermi surface and using the weak-coupling expression $\\Delta _0 = 2.14~k_{\\rm B}$$T_{\\rm c}$ , we get $\\kappa _{\\text{0}}/T$ = 3.7, 7.3 and 4.8 $\\text{mW}/\\text{K}^2\\text{cm}$ for the $\\alpha $ , $\\beta $ and $\\gamma $ sheets, respectively, giving a total conductivity $\\kappa _{\\text{0}}/T$ = 15.8 $\\text{mW}/\\text{K}^2\\text{cm}$ (open circle on the $y$ axis of Fig.",
"REF ).",
"This theoretical value is in remarkably good agreement with the measured value $\\kappa _{\\rm a0}/T = 17 \\pm 2$ $\\text{mW}/\\text{K}^2\\text{cm}$ [21], consistent with line nodes on all three Fermi surfaces.",
"One may ask whether our data are compatible with deep minima instead of nodes.",
"In Fig.",
"REF , we compare our data with calculations for a model of Sr$_{2}$ RuO$_4$ in the clean limit where the gap has symmetry-related minima along the $a$ axis on the $\\gamma $ sheet and very deep minima along the zone diagonals on the $\\alpha $ and $\\beta $ sheets, that result from the model interaction [17].",
"The deepest minima are on the $\\beta $ sheet, where the gap goes down to a value 30 times smaller than its maximal value (on the $\\gamma $ sheet).",
"We see that while the model works well for $T >$ $T_{\\rm c}$ /4, it fails at lower $T$ , forced as it is to go to zero at $T \\rightarrow 0$ since the gap does not have nodes.",
"This comparison shows that our data are inconsistent even with minima so deep that $\\Delta _{\\rm min} \\simeq \\Delta _{\\rm max}/30$ .",
"Taking into account the perfectly linear $T$ dependence of the specific heat below $\\sim $ $T_{\\rm c}$ /2, the case against deep minima in the gap is compelling, for the combined data require that $\\Delta _{\\rm min} \\simeq \\Delta _{\\rm max}/100$ on each of the three Fermi surfaces – a rather artificial situation.",
"Note that adding impurities to the calculation by Nomura [17] would produce a non-zero $\\kappa _{\\text{0}}/T$ , thereby achieving better agreement with experiment (Fig.",
"REF ).",
"However, that magnitude of $\\kappa _{\\text{0}}/T$ would be expected to decrease rapidly as $\\Gamma _0 \\rightarrow 0$ [23], [24], contrary to what is observed experimentally [21] (Fig.",
"REF ).",
"Figure: Out-of-plane (cc-axis) thermal conductivity of Sr 2 _{2}RuO 4 _4.a)At H=0H = 0.The black line is a linear fit to the normal-state data.The arrow marks the location of T c T_{\\rm c} = 1.2 K.In this direction,κ e <<κ p \\kappa _{\\rm e} << \\kappa _{\\rm p},so that the purely electronic term is obtained as κ c \\kappa _{\\text{c}}(TT)/TT in the T=0T=0 limit.b)Zoom on the data at low temperature, at H=0H = 0 (red dots).The black line is a fit of the data to Eq.",
"3 below 0.35 K,with the phonon conductivity in the T→0T \\rightarrow 0 limit given byκ p =BT α \\kappa _{\\rm p} = B T^{\\alpha }, with α=3.0\\alpha = 3.0 and BB given bysound velocity and sample dimensions (see text).The other two lines are the same fit but with α=2.7\\alpha = 2.7, to take into accountthe effect of specular reflection, and BB whether fixed (red line) or free (blue line) (see text).c)Same data as in b) (red, H=0H=0), compared with data in a magnetic fieldH=25H = 25 mT (blue) and H=0.35H = 0.35 T (burgundy), with H∥aH \\parallel a.d)Increase in κ c /T\\kappa _{\\rm c}/T with field H∥aH \\parallel a, where κ c /T\\kappa _{\\rm c}/T is either measured at T=60T = 60 mK (open circles)or extrapolated to T=0T=0, whether linearly as in panel (c) (crosses) or through a fit as in panel (b), red line (full red dots).",
"The red line is a guide to the eye."
],
[
"H = 0 : C-AXIS TRANSPORT",
"Fig.",
"REF a shows the conductivity out of the plane, $\\kappa _{\\text{c}}$$(T)$ ($J \\parallel c$ ).",
"It is completely dominated by the phonon contribution $\\kappa _{\\rm p}$ , since in this direction $\\kappa _{\\rm e}$ is some 2000 times smaller than in the plane (estimated from the resistivity anisotropy).",
"Because of this, the only way to extract the electronic contribution of interest is to obtain the purely fermionic residual linear term at $T=0$ .",
"A zoom on the $c$ -axis conductivity at low temperature is shown in Figs.",
"REF b and REF c. We see that $\\kappa _{\\text{c}}$$/T$ is linear below 0.2 K. We attribute this linear behavior of $\\kappa _{\\rm p}/T$ , also observed in overdoped cuprate superconductors [52], to the scattering of phonons by nodal quasiparticles, as discussed theoretically in ref. Smith2005PRB.",
"A linear fit to $\\kappa _{\\rm {c}}/T$ extrapolates to $\\kappa _{\\rm {c0}}/T = 0.0 \\pm 3$ $\\mu \\text{W}/\\text{K}^2\\text{cm}$ (Fig.",
"REF c).",
"However, $\\kappa _{\\text{c}}$$(T)/T$ cannot continue linearly all the way down to $T = 0$ , for this would imply a divergent phonon mean free path, since $l_{\\rm p} \\propto \\kappa /T^3$ .",
"The sample boundaries impose an upper bound on $l_{\\rm p}$ .",
"For diffuse (non-specular) scattering, $l_0 = 2\\sqrt{S/\\pi }$ , where $S$ is the sample cross-section normal to the heat flow.",
"In the ballistic regime at low temperature, where phonons are scattered by the (rough) sample boundaries, we have [54]: $\\kappa _{\\rm p} = \\frac{1}{3} C_{\\rm p} v_{\\rm p} l_{\\rm 0} = BT^3~~,$ where $C_{\\rm p} = (2 \\pi ^2 k_{\\rm B} / 5) (k_{\\rm B} T / \\hbar v_{\\rm p})^3$ is the phonon specific heat [55] and $v_{\\rm p}$ is the average sound velocity.",
"$v_{\\rm p}$ can be extracted from the measured phonon specific heat $C_{\\rm p} / T^3 = 0.197$ mJ/K$^4$ mole = 3.44 J/K$^4$ m [12], giving $v_{\\rm p} = 3 284$ m/s, a value which is consistent with the measured sound velocities in Sr$_{2}$ RuO$_4$ [14].",
"Using Eq.",
"2, with $l_0 = 0.46$ mm, we get $B = 17.3$ mW / K$^4$ cm.",
"The total thermal conductivity is given by $\\kappa /T = \\kappa _{\\rm c0}/T + \\kappa _{\\rm p}/T$ , where the first term is electronic and the second term is phononic.",
"At low $T$ , two mechanisms scatter phonons: the sample boundaries, already mentioned, and quasiparticles.",
"In Eq.",
"2, the phonon mean free path $l_{\\rm 0}$ is replaced by $l_{\\rm p} = [1 / l_{\\rm 0} + 1 / l_{\\rm e}]^{-1}$ , where $l_{\\rm e}$ is the mean free path due to electron-phonon scattering, with $1 / l_{\\rm e} \\propto T$ [53].",
"Therefore, in the regime where the latter process dominates, we get $\\kappa _{\\rm p} = A T^2$ , as seen in our data at $T > 0.05$ K (Fig.",
"REF c).",
"In the limit $T \\rightarrow 0$ , we expect $\\kappa _{\\rm p} = B T^3$ .",
"We can therefore fit our data to: $\\kappa / T = \\kappa _{\\rm c0}/T + B T^2 / (1 + BT/A)~~.$ Given that $B$ is known and $A$ is fixed by the slope of $\\kappa / T$ above 50 mK, the only free parameter in the fit is the residual linear term $\\kappa _{\\rm c0}/T$ , due to quasiparticle transport.",
"A fit to the zero-field data of Fig.",
"REF b yields $\\kappa _{\\rm c0}/T = 12 \\pm 5~\\mu $ W / K$^2$ cm (black line).",
"Although the cut side surfaces of our $c$ -axis sample are rougher than the mirror-like cleaved or as-grown surface of crystals, there can still be some degree of specular reflection.",
"This was studied on crystals of the cuprate insulator Nd$_2$ CuO$_4$ , with sample surfaces roughened by sanding [54].",
"At $0.15 <ÊT < 0.3$ K, $\\kappa _{\\rm p} = B T^{3}$ , with the prefactor $B$ correctly given by the sound velocities and sample dimensions (Eq. 2).",
"At $T <Ê0.15$ K, specular reflection becomes important and $\\kappa _{\\rm p} = B^{\\prime } T^{2.68}$ , with $B^{\\prime } = 0.6 B$ .",
"Using the same power law to fit our Sr$_{2}$ RuO$_4$ $c$ -axis data, namely $\\kappa _{\\rm p} = B^{\\prime } T^{\\alpha }$ with $\\alpha = 2.7$ and $B^{\\prime } = 0.6~B = 10~\\mu $ W / K$^{3.7}$ cm, we get the red line in Fig.",
"REF b, with $\\kappa _{\\rm c0}/T = 8.5~\\mu $ W / K$^2$ cm.",
"Leaving $B^{\\prime }$ as a free fit parameter yields $\\kappa _{\\rm c0}/T = 6.5~\\mu $ W / K$^2$ cm (blue line).",
"We arrive at a value for the residual linear term of $\\kappa _{\\rm c0}/T = 10 \\pm 5~\\mu $ W / K$^2$ cm.",
"Nodal quasiparticles in Sr$_{2}$ RuO$_4$ must therefore have a non-zero $c$ -axis velocity.",
"This rules out horizontal line nodes – at least in high-symmetry planes (e.g.",
"$k_z = 0$ ) – and it points immediately to vertical line nodes.",
"What magnitude of $\\kappa _{\\rm c0}/T$ do we expect if the line nodes responsible for the large in-plane $\\kappa _{\\rm a0}/T$ are vertical?",
"Assuming all three Fermi surfaces have line nodes, as would be the case for a $d_{x^2-y^2}$ symmetry, then the $a$ -$c$ anisotropy of nodal quasiparticle transport at $T=0$ , in the superconducting state, should be similar to the $a$ -$c$ anisotropy of transport in the normal state.",
"This is what is observed in the quasi-2D iron-based superconductor KFe$_2$ As$_2$ [45], [46], for example.",
"Explicitly, $(\\kappa _{\\text{c0}}/T)/(\\kappa _{\\text{a0}}/T) \\simeq (\\kappa _{\\text{cN}}/T)/(\\kappa _{\\text{aN}}/T)$ , and we therefore expect $\\kappa _{\\text{c0}}/T \\simeq 0.2~\\kappa _{\\text{cN}}/T = 13 \\pm 1 $ $\\mu \\text{W}/\\text{K}^2\\text{cm}$ , since we have $\\kappa _{\\text{a0}}/T = 0.2~\\kappa _{\\text{aN}}/T$ (Fig.",
"REF ) and $\\kappa _{\\text{cN}}/T = 67 \\pm 7$ $\\mu \\text{W}/\\text{K}^2\\text{cm}$ (see Fig.",
"REF b).",
"This is in good agreement with the experimental value quoted above ($10 \\pm 5~\\mu $ W / K$^2$ cm).",
"We conclude that the line nodes in the gap structure of Sr$_{2}$ RuO$_4$ are vertical.",
"In most theoretical proposals, the gap minima do occur along vertical lines.",
"Presumably, some of these minima could accidentally be so deep as to produce nodes.",
"Let us consider different options.",
"First, a scenario of line nodes present only on the $\\alpha $ surface is unrealistic because the full contribution of this small surface to the total in-plane conductivity in the normal state is only 18% of $\\kappa _{\\text{aN}}/T$ [47], less than the zero-field fraction of 20% (Fig.",
"REF b).",
"In other words, the entire $\\alpha $ Fermi surface would have to be normal already at $H=0$ .",
"Such an extreme multi-band character is ruled out by two facts: 1) the residual linear term in $C_e/T$ at $T \\rightarrow 0$ is too small [13]; 2) an increase in impurity scattering does not cause $\\kappa _{\\text{a0}}/T$ to decrease [21] – unlike in CeCoIn$_5$ , where electrons on part of the Fermi surface are uncondensed and $\\kappa _{\\text{a0}}/T \\propto 1 / \\Gamma $ [48].",
"Secondly, a scenario with nodes only on the $\\beta $ surface is unlikely because the $\\beta $ surface accounts for 80% of the total normal-state conductivity along the $c$ axis, but only 37% along the $a$ axis [47].",
"As a result, if only the $\\beta $ surface had nodes, it would alone be responsible for the ratio $(\\kappa _{\\text{a0}}/T)/(\\kappa _{\\text{aN}}/T) = 0.2$ , and it would then necessarily produce a larger ratio along the $c$ axis (by a factor $\\sim 80/37$ ), giving $(\\kappa _{\\text{c0}}/T)/(\\kappa _{\\text{cN}}/T) \\simeq 0.2~(80/37) = 0.43$ , so that $\\kappa _{\\text{c0}}/T \\simeq 29$ $\\mu \\text{W}/\\text{K}^2\\text{cm}$ .",
"Such a large value is not possible, since it exceeds the full measured conductivity at $T = 50$ mK (including phonons) (Fig.",
"REF c).",
"Invoking line nodes on both $\\alpha $ and $\\beta $ surfaces decreases these estimates to $(\\kappa _{\\text{c0}}/T)/(\\kappa _{\\text{cN}}/T) \\simeq 0.2~(89/55) = 0.32$ and $\\kappa _{\\text{c0}}/T \\simeq 22$ $\\mu \\text{W}/\\text{K}^2\\text{cm}$ – still too large.",
"In summary, quantitative analysis indicates that the vertical line nodes in Sr$_{2}$ RuO$_4$ are present on more than one sheet, including the $\\gamma $ sheet (e.g.",
"on $\\gamma $ and $\\beta $ ), and most likely present on all three sheets of the Fermi surface.",
"This is consistent with the nodal structure of a $d_{x^2-y^2}$ pairing state (with line nodes on all three sheets) and that of a $d_{xy}$ state (with line nodes on $\\gamma $ and $\\beta $ , but not $\\alpha $ ).",
"The data would also be consistent with a $p$ -wave pairing state with minima on $\\gamma $ and $\\beta $ that are so deep that they extend to negative values and hence produce accidental nodes.",
"Figure: Residual linear term κ 0 /T\\kappa _{\\text{0}}/T as a function of a magnetic field HH applied along the aa axis (H∥aH \\parallel a).a)For a current in the plane (J∥aJ \\parallel a).The data points (black dots) are κ a0 /T\\kappa _{\\text{a0}}/T obtained by fitting κ a \\kappa _{\\text{a}}/T/T vsvs TT as in Fig.",
"1b.The black line is a constant fit to the data above H c2 H_{\\rm c2} (negligible magnetoresistance for that current direction).It defines κ aN /T\\kappa _{\\text{aN}}/T vsvs HH, and it is consistent with the H=0H=0 value obtainedbyextrapolating κ N /T\\kappa _{\\text{N}}/T above T c T_{\\rm c} to T→0T \\rightarrow 0 (Fig.",
"a).b)Same as in a), but for a current along the cc axis (J∥cJ \\parallel c).The data points (red dots) are κ c0 /T\\kappa _{\\text{c0}}/T obtained by fitting κ c \\kappa _{\\text{c}}/T/T vsvs TT as in Fig.",
"b.Above H c2 H_{\\rm c2}, κ c0 /T\\kappa _{\\text{c0}}/T decreases slightly due to magnetoresistance(see text).The red line is a fit of the data above H c2 H_{\\rm c2} to κ N /T\\kappa _{\\text{N}}/T =a/(b+cH 2 )= a / (b + c H^{2}),which defines κ cN /T\\kappa _{\\text{cN}}/T vsvs HH for this current direction.The value at H→0H \\rightarrow 0 is κ cN /T=67±7\\kappa _{\\text{cN}}/T = 67 \\pm 7 μW/K 2 cm\\mu \\text{W}/\\text{K}^2\\text{cm}.c)Field dependence of κ a0 /T\\kappa _{\\text{a0}}/T (black dots) and κ c0 /T\\kappa _{\\text{c0}}/T (red dots) normalized to their normal-state value,both plotted as (κ 0 /T\\kappa _{\\text{0}}/T)/(κ N /T\\kappa _{\\text{N}}/T) vsvs H/H/H c2 H_{\\rm c2}, with H c2 H_{\\rm c2} =1.25= 1.25 T.For simplicity, we define κ 0 /κ N ≡\\kappa _0 / \\kappa _{\\rm N} \\equiv (κ 0 /T\\kappa _{\\text{0}}/T)/(κ N /T\\kappa _{\\text{N}}/T).The error bars on κ 0 /κ N \\kappa _0 / \\kappa _{\\rm N} come from the combined uncertainties in extrapolatingκ/T\\kappa /T to T=0T=0 to obtain κ 0 /T\\kappa _{\\text{0}}/T and in extending κ N /T\\kappa _{\\text{N}}/T below H c2 H_{\\rm c2}."
],
[
"FIELD DEPENDENCE",
"Applying a magnetic field is a sensitive way to probe the low-lying excitations in a type-II superconductor [57].",
"In the absence of nodes, the quasiparticle states are localized in the vortex cores, and heat conduction proceeds by tunnelling between adjacent vortices, which depends exponentially on inter-vortex separation.",
"As a result, $\\kappa _{\\text{0}}/T$ grows exponentially with $H$ , as observed in all $s$ -wave superconductors, e.g.",
"LiFeAs [58].",
"In a two-band $s$ -wave superconductor like NbSe$_2$ [51], the exponential increase is seen below $H^\\star <<$ $H_{\\rm c2}$ , the effective critical field of the band with the minimum gap.",
"By contrast, in a nodal superconductor quasiparticle states are delocalized even at $T=0$ and $H=0$ .",
"Increasing the field immediately increases their density of states, causing the specific heat to increase as $\\sqrt{H}$ , the so-called Volovik effect.",
"As a result, $\\kappa _{\\text{0}}/T$ grows rapidly with $H$ at the lowest fields [59], as observed in $d$ -wave superconductors, e.g.",
"YBa$_2$ Cu$_3$ O$_y$ [60].",
"Figure: Residual linear term κ 0 /T\\kappa _{\\text{0}}/T as a function of magnetic field, for a field along the cc axis (H∥cH \\parallel c)and a heat current along the cc axis (J∥cJ \\parallel c).The data are plotted as (κ 0 /T\\kappa _{\\text{0}}/T)/(κ N /T\\kappa _{\\text{N}}/T) vsvs H/H/H c2 H_{\\rm c2}, with H c2 H_{\\rm c2} =0.055= 0.055 T.For this field direction (in the longitudinal configuration with a tiny H c2 H_{\\rm c2}), the magneto-resistancein the normal state is negligible, and so κ N /T\\kappa _{\\text{N}}/T is a constant below H c2 H_{\\rm c2}.The data points are κ c0 /T\\kappa _{\\text{c0}}/T obtained by fitting κ c \\kappa _{\\text{c}}/T/T vsvs TT as in Fig.",
"b(red fit).The solid red line is a theoretical calculation for a single-band dd-wave superconductor .In Fig.",
"REF c, we show $c$ -axis data at $H = 25$ mT.",
"We see that even this tiny field ($H_{\\rm c2}$ /50) induces a substantial increase in $\\kappa _{\\text{c}}$$/T$ at $T \\rightarrow 0$ .",
"This proves the existence of nodal quasiparticles with $c$ -axis velocity.",
"In Fig.",
"REF d, a plot of $\\kappa _{\\text{c}}$$/T$ vs $H$ shows how rapid the rise is, whether $\\kappa _{\\text{c}}$$/T$ is measured at $T = 60$ mK (open circles) or extrapolated to $T=0$ , either linearly as in Fig.",
"REF c (crosses) or through the fit described in sec.",
"IV (full red circles).",
"In Fig.",
"REF , we show the $H$ dependence of $\\kappa _{\\text{0}}/T$ in Sr$_{2}$ RuO$_4$ ($H \\parallel a$ ), for both current directions.",
"Both $\\kappa _{\\text{a0}}/T$ and $\\kappa _{\\text{c0}}/T$ have the dependence expected of nodal superconductors, as calculated for a single-band $d$ -wave superconductor [59].",
"In Fig.",
"REF c, we compare the $H$ dependence of $\\kappa _{\\text{a0}}/T$ and $\\kappa _{\\text{c0}}/T$ in normalized units, both plotted as ($\\kappa _{\\text{0}}/T$ )/($\\kappa _{\\text{N}}/T$ ) $\\equiv \\kappa _0 / \\kappa _{\\rm N}$ $vs$ $H/$$H_{\\rm c2}$ .",
"We obtain the normal-state conductivity $\\kappa _{\\text{N}}/T$ below $H_{\\rm c2}$ by extending a fit of the data above $H_{\\rm c2}$ to lower fields.",
"For $J \\parallel a$ , there is negligible $H$ dependence up to 4 T, and so we take $\\kappa _{\\text{N}}/T$ to be constant (Fig.",
"REF a).",
"For $J \\parallel c$ , Sr$_{2}$ RuO$_4$ exhibits a sizable magneto-resistance, which varies as $H^2$ below 2 T or so [61].",
"By the Wiedemann-Franz law, this implies that $\\kappa _{\\text{N}}/T$ $= a / (b + c H^{2})$ .",
"A fit of the data above $H_{\\rm c2}$ to this formula yields the red line in Fig.",
"REF b.",
"The data in Fig.",
"REF c are striking: the two normalized curves are the same, at all fields, within error bars.",
"This is strong confirmation that line nodes are vertical.",
"Indeed, horizontal line nodes would inevitably produce a qualitative difference between the two current directions, roughly $d$ -wave-like (rapid) for $\\kappa _{\\text{a0}}/T$ and $s$ -wave-like (exponential) for $\\kappa _{\\text{c0}}/T$ .",
"The fact that both curves in Fig.",
"REF c are the same is also consistent with line nodes being present on all of the three Fermi surfaces.",
"Indeed, if line nodes were present only on the $\\beta $ surface, for example, $\\kappa _{\\text{c0}}/T$ would exhibit a $d$ -wave-like $H$ dependence, as it is dominated by that surface, while $\\kappa _{\\text{a0}}/T$ would exhibit an $s$ -wave-like $H$ dependence, since it is dominated by the other Fermi surfaces.",
"The electronic specific heat at low temperature also displays a rapid increase at low field.",
"In the $T=0$ limit, the residual linear term $\\gamma _0(H)$ reaches $\\sim 30$ % of its normal-state value $\\gamma _{\\rm N}$ by $H \\simeq 0.1$ $H_{\\rm c2}$ ($H \\parallel a$ ), and then increases more slowly at higher $H$ [11], [13].",
"We see from Fig.",
"REF c, that the field dependences of $\\kappa _0 / \\kappa _{\\rm N}$ and $\\gamma _0 / \\gamma _{\\rm N}$ are similar.",
"To explain the rapid initial rise in $\\gamma _0 / \\gamma _{\\rm N}$ $vs$ $H$ , it was proposed that the $\\alpha $ and $\\beta $ surfaces become normal at a field $H^\\star \\simeq 0.1$ $H_{\\rm c2}$ [12].",
"But this is inconsistent with our data, since it would imply a much larger increase in $\\kappa _0 / \\kappa _{\\rm N}$ for $J \\parallel c$ than for $J \\parallel a$ , given that the $\\beta $ surface accounts for 80% of $\\kappa _{\\rm cN}/T$ but only 37% of $\\kappa _{\\rm aN}/T$ .",
"This is not observed (Fig.",
"REF c).",
"In Fig.",
"REF , we show the effect of applying a magnetic field parallel to the $c$ axis.",
"This is the field direction for which the Volovik effect is the dominant excitation process, and for which most theoretical calculations on quasi-2D superconductors have been carried out (e.g. [59]).",
"The overall field dependence of $\\kappa _0 / \\kappa _{\\rm N}$ is in good agreement with calculations for a single-band 2D $d$ -wave superconductor [63], as seen in Fig.",
"REF .",
"Also, the specific heat of Sr$_{2}$ RuO$_4$ exhibits a nice $\\sqrt{H}$ dependence and detailed $\\sqrt{H}/T$ scaling [13], consistent with the behavior of a single-band $d$ -wave superconductor [62]."
],
[
"SUMMARY",
"In summary, our thermal conductivity measurements confirm that the gap structure of Sr$_{2}$ RuO$_4$ has nodes rather than deep gap minima and they reveal that those nodes are vertical lines along the $c$ axis.",
"In a nutshell, everything about the thermal conductivity of Sr$_{2}$ RuO$_4$ is consistent with a $d$ -wave pairing state, including its absolute magnitude at $T \\rightarrow 0$ , its dependence on temperature, magnetic field and impurity scattering, and its isotropy relative to current direction.",
"A $d$ -wave gap structure is also consistent with the specific heat of Sr$_{2}$ RuO$_4$ [10], [11], [12], [13], including the magnitude of its jump at $T_{\\rm c}$ and its dependence on temperature, magnetic field and impurity scattering.",
"Given that calculations find $p$ -wave and $d$ -wave solutions for Sr$_{2}$ RuO$_4$ to be very close in energy [64], it is tempting to consider a $d$ -wave state for Sr$_{2}$ RuO$_4$ .",
"However, this comes into conflict with some important properties of the material, in particular the absence of a drop in the NMR Knight shift below $T_{\\rm c}$ [1], [2], a signature of spin-triplet pairing, and the onset of muon and Kerr signals below $T_{\\rm c}$ [4], [6], evidence that time-reversal symmetry is broken.",
"These are the natural properties of a chiral $p$ -wave superconductor.",
"Note that the spin-singlet chiral $d$ -wave state also breaks time-reversal symmetry, but its gap function varies as $k_z (k_x + i k_y)$ and therefore has symmetry-imposed line nodes that are horizontal, not vertical [31].",
"We are therefore faced with a situation where Sr$_{2}$ RuO$_4$ appears to adopt a $p$ -wave state with a $d$ -wave-like gap structure.",
"An intriguing solution to this conundrum has been proposed in the so-called $f$ -wave state [32], [33], a combination of $B_g$ and $E_u$ representations ($B_g \\times E_u$ ), where $B_g$ is either $B_{1g}$ ($d_{x^2-y^2}$ ) or $B_{2g}$ ($d_{xy}$ ), with gap functions that vary either as $(k_x^2 - k_y^2)(k_x + i k_y)$ or as $(k_x k_y)(k_x + i k_y)$ , respectively.",
"Further theoretical and experimental work is needed to resolve the puzzle presented to us by the superconducting state of this exceptionally well characterized and otherwise rather conventional three-band metal."
],
[
"ACKNOWLEDGEMENTS",
"We thank J. Corbin, S. Fortier, A. Juneau-Fecteau, and F. F. Tafti for their assistance with the experiments, and A. Balatsky, M. Graf, A. P. Mackenzie, K. Samokhin, M. Sato, J. Sauls, T. Scaffidi, R. Thomale, and S. Yonezawa for stimulating discussions.",
"L.T.",
"acknowledges support from the Canadian Institute for Advanced Research (CIFAR) and funding from the National Science and Engineering Research Council of Canada (NSERC), the Fonds de recherche du Québec - Nature et Technologies (FRQNT), the Canada Foundation for Innovation (CFI) and a Canada Research Chair.",
"The work in Japan was supported by the JSPS KAKENHI (No. JP15H05852).",
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]
] | 1606.04936 |
[
[
"Merging black hole binaries in galactic nuclei: implications for\n advanced-LIGO detections"
],
[
"Abstract Motivated by the recent detection of gravitational waves from the black hole binary merger GW150914, we study the dynamical evolution of black holes in galactic nuclei where massive star clusters reside.",
"With masses of ~10^7M_Sun and sizes of only a few parsecs, nuclear star clusters are the densest stellar systems observed in the local universe and represent a robust environment where (stellar mass) black hole binaries can dynamically form, harden and merge.",
"We show that due to their large escape speeds, nuclear star clusters can keep a large fraction of their merger remnants while also evolving rapidly enough that the holes can sink back to the central regions where they can swap in new binaries that can subsequently harden and merge.",
"This process can repeat several times and produce black hole mergers of several tens of solar masses similar to GW150914 and up to a few hundreds of solar masses, without the need of invoking extremely low metallicity environments or implausible initial conditions.",
"We use a semi-analytical approach to describe the formation and dynamics of black holes in massive star clusters.",
"We find a black hole binary merger rate per volume from nuclear star clusters of ~1.5 Gpc^-3 yr^-1, implying up to a few tens of possible detections per year with Advanced LIGO.",
"Our models suggest a local merger rate of 0.3- 1 Gpc^-3 yr^-1 for high mass black hole binaries similar to GW150914 (total mass >~ 50 M_Sun, redshift z< 0.3); a merger rate comparable to that of high mass black hole binaries that are dynamically assembled in globular clusters.",
"Finally, we show that if all black holes receive high natal kicks, >~50km s^-1, then nuclear star clusters could dominate the local merger rate of binary black holes compared to the merger rate of similar binaries produced in either globular clusters or through isolated binary evolution."
],
[
"Introduction",
"On September 14, 2015 the Advanced LIGO interferometer (aLIGO) detected the event GW150914, which has been interpreted as the first direct observation of gravitational waves (GWs) from the inspiral and merger of a pair of black holes [3].",
"The event GW150914 was produced by two BHs with masses of $36^{+5}_{-4}\\ M_\\odot $ and $29^{+4}_{-4} \\ M_\\odot $ (in the source frame), at a redshift $z \\approx 0.1$ assuming standard cosmology [4].",
"The detection of the gravitational-wave signal of GW150914 has provided the first direct evidence that black holes with mass $\\gtrsim 30\\ M_\\odot $ exist and that they can reside in binary systems.",
"Assuming that the source-frame binary BH merger rate is constant within the volume in which GW150914 could have been detected, and that GW150914 is representative of the underlying binary BH population, the BH-BH merger rate is inferred to be $2-53 \\rm \\ Gpc^{-3}\\ yr^{-1}$ in the comoving frame [5].",
"[6] reviews various channels for the formation of BH binaries that can coalesce within a Hubble time thus becoming potentially detectable by aLIGO.",
"These include dynamical formation in dense stellar environments [90], [77], [14], [36], [95], and isolated binary evolution [16], [34], [101], [19], [32].",
"While most of the former literature focused on BH binaries forming in globular clusters (GCs), little attention has been devoted to the formation of such binaries in nuclear star clusters (NSCs).",
"Yet NSCs have total stellar masses that are comparable to the whole stellar mass of the GC system for the galaxy, at least in the Milky Way, and are the densest and most massive star clusters observed in the local universe [25], [31], representing therefore a natural environment where dynamical processes can efficiently lead to the formation of BH binaries.",
"In this paper we consider the dynamical formation scenario, and explore the contribution to the BH binary merger rate from NSCs.",
"In a stellar cluster, stellar mass BHs formed from the death of massive stars, quickly segregate to the center through dynamical friction [27], [102].",
"In these high-density environments, BHs can efficiently interact with each other and dynamically form new binaries.",
"Such binaries will subsequently harden through three-body interactions [53].",
"Via such dynamical processes, GCs can produce a significant population of BH binaries that, after being ejected from the cluster, will be able to merge in the local universe [96].",
"Over the last years our understanding of the evolution of BHs in star clusters has improved considerably thanks to numerical efforts [2], [79], [106], [80].",
"However, the role of NSCs and their contribution to the BH binary merger rate in the local universe remains quite obscure.",
"As discussed in what follows, NSCs differ from lower mass GCs in at least three important ways, each of these can significantly enhance the BH merger rate and affect the properties of the merging binaries in NSCs.",
"(i) NSCs retain most of their BHs.",
"While natal kicks can easily eject BHs from GCs, the natal kick magnitudes are unlikely to be large enough to eject a considerable number of BHs from NSCs given the large escape speed in these latter systems.",
"Whether dynamically formed BH binaries will merge, and whether the merger will occur inside the cluster also depends on the cluster escape speed.",
"The low escape speed ($\\lesssim 10\\rm \\ km\\ s^{-1}$ ) from low mass clusters ($M_{\\rm cl}\\lesssim 10^5\\ M_\\odot $ ), implies that most BH binaries are ejected early after their formation with an orbital semi-major axis which is typically too large for GW emission to become efficient and drive the merger of the binary in one Hubble time.",
"The vast majority of dynamically formed BH binaries in GCs are also kicked out before merging, but they are able to merge in the local universe [95].",
"As argued by [77] given that NSCs have escape speeds that are several times those of globulars, they can retain most of their BH binaries.",
"Moreover, as we show below, even when accounting for the recoil kick due to anisotropic emission of GW radiation a large fraction of merger products is likely to be retained inside NSCs.",
"(ii) NSCs contain young stellar populations.",
"The common finding emerging from spectroscopic surveys is that NSCs are characterized by complex star formation histories with a mixture of morphological components and different stellar populations spanning a wide range of characteristic ages (from 10 Myr to 10 Gyr) and metallicities [38], [98], [105], [33].",
"This implies that unlike GCs, NSCs can still form fresh BHs and BH binaries at the present time.",
"The presence of significant additional gas not found in old globular clusters could also result in differences in the black hole mass distribution, as well as the dynamics of the underlying black hole population [65], [66].",
"(iii) NSCs reside at the center of galaxies.",
"Therefore, unlike GCs, NSCs are not isolated.",
"In time, newly formed star clusters could migrate by dynamical friction from the galaxy into the NSC itself replenishing BHs that have been kicked out by three-body processes or by GW recoil kicks.",
"The orbital decay of massive star clusters through dynamical friction constitutes an additional source which can repopulate the BH binary population in the nuclei of galaxies [9].",
"In this paper we study the dynamical formation of BH binary mergers in NSCs, with particular focus on NSCs which do not host a central massive black hole (MBH) which we define here as BHs having a mass of $\\gtrsim 10^6\\rm M_\\odot $ .",
"Our cluster models are based on a semi-analytical approach which describes the formation and evolution of BH binaries in static cluster models.",
"Although necessarily approximated, these models are shown to give reasonable results when compared to recent Monte Carlo models of massive GCs [29], [96] and previous BH binary merger rate estimates from NSCs [90], [77].",
"We stress that although the MBH occupation fraction in NSCs is largely unconstrained observationally, it has been long recognized that some NSCs do not have MBHs [71], [81].",
"We note that NSCs with MBHs are very different, dynamically, than NSCs without.",
"If a MBH is present the velocity dispersion keeps growing towards the MBH, which means that no binary will be hard all the way to the center.",
"Here we make use of the semi-analytical galaxy formation models presented in [11] to predict the occupation fraction of MBHs in NSCs and the NSC initial mass function, which combined with the results of our cluster models allows an estimate of the aLIGO detection rate and properties of BH mergers forming in NSCs.",
"Our results suggest that the BH merger event rate from NSCs is substantial, with several tens of events per year detectable with aLIGO.",
"In addition, we propose a new dynamical pathway to the formation of high mass BH binary mergers similar to GW150914.",
"This merger path is exclusive to NSCs and to the most massive GCs.",
"Due to their large escape speeds, such massive clusters can keep a large fraction of their BH merger remnants while also evolving rapidly enough that the holes can sink back to the central regions where they can form a new binary, which will subsequently harden and merge.",
"We find that this process can repeat several times and produce BH mergers of several tens of solar masses and up to a few hundreds of solar masses, without the need of invoking extremely low metallicity environments.",
"The paper is organized as follows.",
"In Sections we discuss the processes leading to the formation and merger of BH binaries in the high density cores of GCs and NSCs, focusing on the processes that can lead to the full ejection of BHs.",
"In Section we describe our semi-analytical approach and derive the expected merger rate of BH binaries in NSCs.",
"In Section we discuss the implications of our results including the aLIGO detection rate and the contribution to the BH merger rate from NSCs hosting central MBHs.",
"Finally, we summarize the main results of our study in Section .",
"Figure: Distribution of escape velocities from NSCs and GCs(histograms) compared to distributions ofnatal kicks taken from Figure 3 of (blue curves).Blue solid and dashed lines correspond todistributions that are typically used to model the kick velocities of neutron stars.The solid line is the Arzoumanian distribution ,thedashed line is the distribution.The twodot-dashed lines are these two distributions but with kick speeds reduced, assuming that the momentum imparted to the black hole is the same as the momentum imparted to the neutron star.",
"Note that if BHs receive natal kicks as large as those of neutron stars, most of them will be ejected fromGCs but not from NSCs.Due to to asymmetries in the mass ejection or in the neutrino flux during core-collapse supernovae the black holes might receive appreciable natal kicks which could eject them from the cluster.",
"Thus, before we discuss the dynamical processes that can lead to the formation and merger of BH binaries in star clusters it is useful to consider natal kicks as a phenomenon that can fully eject BHs from a star cluster thereby aborting the dynamical formation channel for BH mergers.",
"The histograms in Figure REF show the distribution of escape velocities from NSCs and GCs.",
"The escape velocities of GCs are central escape velocities calculated using the photometric data from the catalog by Harris [52] and using single-mass King models with a constant mass-to-light ratio $M/L_v = 3$ .",
"The NSCs escape velocities (from the cluster half-mass radius) were computed from the expression [41]: $v_{\\rm esc}\\approx f_c \\sqrt{\\frac{M_{\\rm cl}}{M_\\odot }\\frac{\\rm pc}{r_{\\rm h}}} \\ \\rm km\\ s^{-1} ,$ where $r_{\\rm h}$ and $M_{\\rm cl}$ are the cluster half-light radius and mass; the coefficient $f_{\\rm c}$ takes into account the dependence of the escape velocity on the concentration of the cluster (i.e., $c=\\log (r_t/r_c)$ , with $r_t$ and $r_c$ the cluster tidal and core radii).",
"The cluster radii and masses were taken from the sample of late-type galaxies of [42].",
"For more than half of the NSCs in these galaxies [40] find that a King profile with a high concentration index, $c=2$ , provides the best fit.",
"This concentration parameter corresponds to $f_c\\approx 0.1$ [60] – this latter is the value of $f_c$ that we adopted in evaluating Eq.",
"(REF ).",
"Figure REF shows that escape velocities from NSCs are substantially larger than those from GCs although the two distributions somewhat overlap near $M_{\\rm cl}\\sim 10^6\\ M_{\\odot }$ where the two type of systems have similar structural properties [25], [31].",
"The distributions of natal kicks ($v_{\\rm natal}$ ; blue lines) in Figure REF were taken from Figure 3 of [93].",
"These authors consider two different neutron star natal kick distributions.",
"One is the [51] distribution, the other is the bimodal distribution for neutron star kicks proposed by [1] which has a lower peak at $\\approx 1000\\rm \\ km\\ s^{-1}$ and the higher peak at $\\approx 100\\rm \\ km\\ s^{-1}$ .",
"We also show two modified versions of these distributions (blue dot-dashed lines), which were obtained by assuming that the momentum imparted on a BH is the same as the momentum given to a neutron star taken from the two former distributions.",
"Thus the kick velocities are reduced in these latter models by the neutron star to BH mass ratio [93].",
"Figure REF shows that BHs receiving natal kicks as large as $v_{\\rm natal} \\gtrsim 50\\rm \\ km\\ s^{-1}$ will escape from GCs before they can dynamically interact, which will suppress the dynamical formation of BH mergers in these systems.",
"However, from Figure REF we also see that BHs will be easily retained in NSCs even for natal kicks as large as a few $100\\rm \\ km\\ s^{-1}$ .",
"Hence, if BHs receive natal kicks of $\\gtrsim 50\\rm \\ km\\ s^{-1}$ , we expect that this will greatly reduce the BH merger rate from GCs [29] as well as that from isolated binary evolution [19] virtually to zero, but it will not significantly alter the merger rate of BH binaries formed dynamically in NSCs unless the birth kick velocities are $\\gg 100\\rm \\ km\\ s^{-1}$ .",
"As we will show in Section REF these basic predictions are in agreement with the results of our cluster models; for now we note that the obvious consequence of the comparison shown in Figure REF is that the NSC $vs$ GC relative contribution to the BH merger rate will depend on the poorly constrained natal kick velocity distribution.",
"In the following we assume that at least some BHs are retained inside the cluster and consider the subsequent formation and dynamical evolution of binary BHs."
],
[
"Mass-segregation",
"After few million years from the birth of a star cluster, the most massive stars explode in supernovae or collapse directly to form BHs.",
"If the BHs are not ejected by their natal kicks, being more massive than a typical main-sequence star, they will migrate to the cluster center via dynamical friction in a process that is generally referred to as mass segregation.",
"In the dense environment of the cluster core BHs can efficiently form binaries which will then harden and eventually merge.",
"A useful reference time is the two-body relaxation timescale evaluated at the half-mass radius of the star cluster [102]: $ t_{\\rm rh} \\approx 4.2\\times 10^9 \\left(15\\over \\ln \\Lambda \\right)\\left(r_{\\rm h}\\over 4\\rm \\ pc\\right)^{3/2}\\left(M_{\\rm cl}\\over 10^7 \\ M_\\odot \\right)^{1/2} \\rm \\ yr$ with $\\ln \\Lambda $ the Coulomb logarithm.",
"On a time $t_{\\rm rh}$ , two-body gravitational interactions of stars are important in driving the dynamical evolution of the cluster.",
"Figure: Half mass-radius (or effective radius) against total clustermass for NSCs, GCs, UCDs.Data are from and .Systems that lie to the right of the dashed linehave t df >5 Gyr t_{\\rm df}>5\\rm \\ Gyr.Such systems are stillin the process of forming a BH subsystem.",
"For the majority of the systems we considered,including most NSCs and UCDs,the bulk of the stellar mass BHs are likely to have alreadyexperienced significant mass segregation.Systems that lie to the left the solid black linehave v esc <50 km s -1 v_{\\rm esc}<50\\rm \\ km\\ s^{-1}.",
"This is an indicative value ofcluster escapevelocity below which BH binaries will be ejected from the clusterbefore merging.",
"In systems that are within the green hatched regionthe BHs are more likely to merge while still inside their host cluster.While low-mass NSCs have short relaxation times, for some of the most massive NSCs the half-mass relaxation time can exceed the Hubble time.",
"However, even in the most massive NSCs the BHs can still segregate at the center on the much shorter dynamical friction timescale [27].",
"More precisely, the BHs will decay to the cluster core on the timescale [22]: $t_{\\rm df}\\approx 0.42\\times 10^9\\left(10\\frac{m_\\star }{m_\\bullet }\\right) \\left( t_{\\rm rh}\\over 4.2\\times 10^9\\rm \\ yr \\right)\\rm \\ yr,$ with $m_\\star $ and $m_\\bullet $ the mass of a typical cluster star and BH respectively.",
"After a time $t_{\\rm df} $ the BHs will dominate the densities inside the cluster core Note that Eq.",
"(REF ) and Eq.",
"(REF ) are strictly valid only for a singular isothermal sphere model..",
"In Figure REF we plot the half-mass radius (or effective radius) versus the total stellar mass for various types of compact clusters: NSCs, GCs and Ultra Compact Dwarfs (UCDs).",
"The dashed line delineates the region below which the dynamical friction timescale becomes shorter than $\\approx 5\\rm \\ Gyr$ suggesting that the BHs in these systems will sink to the center in much less than a Hubble time.",
"Virtually all systems we considered but the most massive NSCs and UCDs ($M_{\\rm cl}\\gtrsim 10^8\\ M_{\\odot }$ ) can evolve rapidly enough so that the BHs will sink to the center where they can participate in dynamical interactions and swap into hard binaries.",
"The formation of such binaries and their dynamical evolution is discussed in the following."
],
[
"formation of BH binaries, hardening and mergers",
"After the BHs segregate to the cluster core, BH binary formation can efficiently occur through the processes described below.",
"During core-collapse, if the densities of BHs become sufficiently high, BH binaries can be assembled through three-body processes in which a binary is formed with the help of a third BH, which carries away the excess energy needed to bound the pair [63], [80].",
"The timescale for three-body binary formation can be written as [63]: $t_{\\rm 3bb} &\\approx & 4\\times 10^{9} \\left(n\\over 10^6{\\rm \\ pc^{-3}}\\right)^{-2}\\left({\\zeta ^{-1} {\\sigma \\over 30 \\rm \\ km\\ s^{-1}}}\\right)^{9} \\\\&&\\left( {m_\\star \\over m_\\bullet }10\\right)^{9/2}\\left( m_\\bullet \\over 10M_\\odot \\right)^{-5} \\nonumber \\rm \\ yr \\ ,$ with $n$ the number density of black holes near the center.",
"The constant $\\zeta \\le 1$ in the previous expression parametrizes the departure of the cluster from equipartition and we have used the relation $ m_\\bullet (\\zeta \\sigma _{\\rm BH})^2=m_\\star \\sigma ^2$ in order to express $t_{\\rm 3bb}$ in terms of the cluster stellar velocity dispersion.",
"In addition to three-body binary formation, BH binaries can potentially form through exchange interactions involving primordial stellar binaries [77].",
"Exchange interactions can lead to the efficient formation of BH binaries only if the cluster contains a number of hard binaries.",
"Thus, this channel might be somewhat suppressed in NSCs – because of the larger velocity dispersion than in GCs a larger fraction of binaries will be soft and will be quickly ionized in NSCs.",
"However, as argued in [77], the reduction is not going to be by a large fraction given that binaries are typically born with roughly equal probability per logarithmic interval of semi-major axis, $ {\\rm d}P/{\\rm d}\\log (a) = \\rm const.$ , in the range $10^{-2}-10^{3}\\rm \\ AU$ [37].",
"If, for example, we consider a NSC with velocity dispersion $\\approx 30\\rm \\ km\\ s^{-1}$ all binaries with semi-major axis $a\\lesssim 1\\rm \\ AU$ will be hard.",
"If we assume a constant probability per $\\log (a)$ for $0.01 < a < 10^3\\ $ AU, then the probability of finding a binary in the range of $a = 0.01-1$ AU is substantial, $\\approx 40$ percent.",
"When a BH gets within a couple of semi-major axis lengths of a binary, the binary will be broken apart and the BH will tend to acquire a companion.",
"The characteristic timescale on which such exchange interaction occurs is $t_{\\rm 1-2}=(n\\Sigma \\sqrt{3}\\sigma )^{-1}$ , where $\\Sigma =\\pi r_p^2\\left[1+2Gm_{123}/(\\sqrt{3}\\sigma )^2r_p\\right]$ is the interaction cross section for periapsis distances $\\le r_p\\approx 2\\rm \\ AU$ and $m_{123}$ is the total mass of the interacting objects.",
"If the cluster core is dominated by stellar binaries, then the timescale for a BH to be capture into a binary is [77]: $t_{\\rm 1-2} &\\approx & 3\\times 10^9 \\left( f_b \\over 0.01 \\right)^{-1}\\left(n\\over 10^6{\\rm \\ pc^{-3}}\\right)^{-1}\\left({\\sigma \\over 30 \\rm \\ km\\ s^{-1}}\\right) \\nonumber \\\\&& \\left(m_{123}\\over 10M_\\odot \\right)^{-1} \\left(a_{\\rm hard}\\over 1\\rm \\ AU\\right)^{-1}\\rm \\ yr \\ ,$ where $a_{\\rm hard} $ is the typical semi-major axis of hard binaries and $f_b$ is the core binary fraction.",
"By comparing the previous equation to Eq.",
"(REF ) we see that even under quite standard conditions (but even more so during core-collapse), three-body binary formation likely dominates the initial dynamical formation of BH binaries in NSCs [80].",
"Two-body binary formation can also occur through gravitational bremsstrahlung in which two initially unbound BHs become bound after a close encounter in which energy is dissipated through gravitational wave radiation.",
"If a BH binary is formed in this manner it mergers almost immediately, without further interactions.",
"However, [63] showed that for velocity dispersions $\\sigma \\lesssim 100\\rm \\ km\\ s^{-1}$ and numbers of BHs $\\lesssim 10^3$ expected in the most massive star clusters we study here, the rate of binary formation from gravitational bremsstrahlung is much less than that of regular three-body binary formation.",
"Therefore, for our investigation, we do not account for binary formation though gravitational bremsstrahlung, but we caution that this process could become important in the most massive NSCs.",
"In addition, we assume that after BH binaries are formed, binary-single interactions dominate over binary-binary interactions, which will be the case unless the binary fraction is very high [80].",
"After BH-binaries are formed they will dominate the dynamics inside the cluster core.",
"Assuming the interaction is now between three BHs each with mass $10\\ M_{\\odot }$ , the typical timescale on which a three-body interaction occurs is: $ t_{\\rm 1-2} &\\approx & 3\\times 10^8 \\zeta ^{-1} \\left( f_b \\over 0.01 \\right)^{-1}\\left(n\\over 10^6\\ \\rm pc^{-3}\\right)^{-1}\\left({\\sigma \\over 30 \\rm km\\ s^{-1}}\\right)\\nonumber \\\\&&\\left( {m_\\star \\over m_\\bullet }10\\right)^{1/2} \\left(m_{123}\\over 30M_\\odot \\right)^{-1} \\left(a_{\\rm hard}\\over 1\\rm \\ AU\\right)^{-1}\\rm \\ yr \\ .$ Given that three-body encounters tend to pair the most massive BHs participating in the interaction, we expect that after a time $\\lesssim t_{\\rm 1-2}$ the most massive BHs in the cluster will become part of a hard binary.",
"After a hard binary is formed it will tend to harden at a constant rate [92] ${{\\rm d} a\\over {\\rm d}t} \\Big |_{\\rm dyn}=-H \\frac{G\\rho }{\\sigma } a^2\\ .$ In this last expression $\\rho $ is the local density of stars and BHs, $H\\approx 20$ is the binary hardening rate and we have assumed all equal mass interlopers.",
"If after a single interaction with a cluster member of mass $m_\\bullet $ the semi-major axis of the binary decreases from $a$ to $a_{\\rm fin}$ , then a binary with components of mass $m_1$ and $m_2$ will recoil with a velocity $v_{\\rm 2-1}^2=G\\mu {m_\\bullet \\over {m_{123}}} \\left(1/a_{\\rm fin}-1/a\\right) \\approx 0.2 G\\mu {m_\\bullet \\over {m_{123}}} {q_3/a}$ , where $\\mu =m_1m_2/m_{\\rm 12}$ , $m_{\\rm 12}=m_1+m_2$ , $m_{123}=m_1+m_2+m_\\bullet $ and $q_3=m_\\bullet /m_{12}$ .",
"In deriving the previous expression we have assumed that in the interaction the binding energy of the binary increases by a fraction $\\approx 0.2q_3$ [92].",
"The previous expressions can be used to derive the limiting semi-major axis below which a three body interaction will eject the binary from the system: $a_{\\rm ej}&= &0.2G\\mu {m_\\bullet \\over m_{123}} {q_3}/{v_{\\rm esc}^2}\\\\&& = 0.07\\left(\\mu {m_\\bullet \\over m_{123}} {q_3}\\frac{1}{M_{\\odot }}\\right)\\left(\\frac{v_{\\rm esc}}{50 \\rm km\\ s^{-1}} \\right)^{-2} \\rm AU\\nonumber \\ .~~~~~$ The binary keeps hardening at a constant rate until either GW radiation takes over and drives its merger or it is ejected from the cluster.",
"The time evolution of the binary semi-major axis due to GW radiation is described by the orbit averaged evolution equation [86]: ${{\\rm d} a\\over {\\rm d} t} \\Big |_{\\rm GW}&=&-{64\\over 5}\\frac{G^3m_1m_2m_{12}}{c^5a^3(1-e^2)^{7/2}} \\\\&&\\left(1+{73\\over 24}e^2+\\frac{37}{96}e^4\\right) \\nonumber \\ ,$ where $e$ is the binary eccentricity.",
"The merger time for the two BHs is: $t_{\\rm GW}\\approx 2\\times 10^9 \\left({m_1m_2m_{12}\\over 10^3M_{\\odot }^3}\\right)^{-1} \\left(a\\over 0.05\\rm \\ AU\\right)^4(1-e^2)^{7/2}\\rm \\ yr \\ .$ Comparing the above expression with the expression for $a_{\\rm ej}$ demonstrates an important point: since the larger the escape velocity from the cluster the smaller $a_{\\rm ej}$ , BH binaries that are produced in NSCs will have shorter merger time and are therefore more likely to merge within one Hubble time than BH binaries from lower-mass GCs.",
"Let $a_{\\rm GW}$ be the semi-major axis at which GW radiation begins to dominate the energy loss from the binary.",
"A reasonable choice is to set $a_{\\rm GW}$ equal to the semi-major axis at which ${{\\rm d} a/ {\\rm d}t} |_{\\rm dyn}={{\\rm d}a/ {\\rm d}t}|_{\\rm GW}$ .",
"Assuming a circular binary, this leads to the relation [74]: $a_{\\rm GW}&=&0.05 \\left({m_{12}\\over 20\\ M_{\\odot }}\\right)^{3/5} \\left(q\\over {(1+q)^2}\\right)^{1/5}\\\\&& \\left({\\sigma \\over 30{\\rm \\ km\\ s^{-1}}}\\right)^{1/5}\\left( 10^6\\ M_{\\odot } {\\rm \\ pc}^{-3}\\over \\rho \\right)^{1/5}\\rm \\ AU \\nonumber \\ ,$ where $q=m_2/m_1\\lesssim 1$ .",
"If $a_{\\rm GW}>a_{\\rm ej}$ merger happens before ejection.",
"By comparing Eq.",
"(REF ) with the expression for $a_{\\rm ej}$ we see that BH binaries that are produced in NSCs are less likely to be ejected from the cluster.",
"The binary will continue to interact with other cluster members until it reaches a semi-major axis $a_{\\rm crit}=\\max (a_{\\rm GW},a_{\\rm ej})$ .",
"After the binary has decayed to $a_{\\rm crit}$ (where it spends most of its lifetime) the timescale between two consecutive interactions becomes [49] $t_{\\rm 2-1}&\\approx & 2\\times 10^7\\ \\zeta ^{-1} \\left(n\\over 10^6 {\\rm \\ pc^{-3}}\\right)^{-1}\\left({\\sigma \\over 30 \\rm \\ km\\ s^{-1}}\\right)\\\\&& \\left( {m_\\star \\over \\ m_\\bullet }10\\right)^{1/2} \\left(a_{\\rm crit}\\over 0.05\\rm \\ AU\\right)^{-1} \\left( m_{12}\\over 20\\ M_{\\odot }\\right)^{-1}\\rm \\ yr \\ .",
"\\nonumber $ If we assume as before that each interaction removes a fraction $0.2q_3$ of the binary binding energy[92], then the timescale required to decay to $a_{\\rm crit}$ from a much larger separation is of order [76]: $t_{\\rm merge} \\approx 5 q_3^{-1} t_{\\rm 2-1}.$ During the hard interaction the interloper will recoil at the speed $v_3=v_{\\rm 2-1}/q_3$ .",
"Thus the field BHs will start being ejected when $v_3\\gtrsim v_{\\rm esc}$ , at this point the binary semi-major axis is $a_3=a_{\\rm ej}/q_3^2$ .",
"At a fractional hardening of $\\approx 0.2q_3$ per interaction, the mass ejected from the cluster required in order to shrink the binary semi-major axis from $a_3$ to $a_{\\rm crit}$ is approximately: $M_{\\rm ej}\\approx m_{12}\\ln \\left({a_3\\over a_{\\rm crit}}\\right),$ so that for low mass clusters $M_{\\rm ej}\\approx m_{12}\\ln \\left({1/ q_3^2}\\right)$ and $M_{\\rm ej}\\approx m_{12}\\ln \\left(a_{\\rm ej}/ a_{\\rm GW} q_3^2\\right)$ for high mass clusters.",
"Given that $a_{\\rm ej}$ decreases with the cluster escape speed, the previous equation implies that the larger the cluster mass the fewer BH interlopers will be ejected, in addition to fewer binaries being ejected; when $v_{\\rm esc} \\gtrsim 120\\left({\\mu } {m_{12} \\over m_{123}}\\frac{4}{M_{\\odot }}\\right)^{1/2}\\left({{a_{\\rm GW}}\\over 0.05\\ {\\rm AU}}\\right)^{-1/2}{\\rm \\ km\\ s^{-1}}$ the BH binary will merge without ejecting any of the field BHs.",
"From the condition $ a_{\\rm ej}<a_{\\rm GW}$ , we derive the critical cluster escape velocity above which binaries will merge before being ejected through hard scattering with surrounding stars: $\\tilde{v}_{\\rm esc}&\\gtrsim &110{\\sqrt{q}\\over 1+q}\\left(\\frac{m_\\bullet }{10\\ M_{\\odot }}\\right) \\\\&&\\left(\\frac{30\\ M_\\odot }{m_{123}} \\right)^{1/2}\\left(a_{\\rm GW}\\over 0.05\\rm \\ AU\\right)^{-1/2}{\\rm \\ km\\ s^{-1}} \\ , \\nonumber $ so that for $m_1=m_2=m_\\bullet =10\\ M_{\\odot }$ we have $\\tilde{v}_{\\rm esc}\\approx 50\\rm \\ km\\ s^{-1}$ .",
"The solid line in Figure REF shows the locus of points where the escape velocity from the clusters, $v_{\\rm esc}(\\rm \\ km\\ s^{-1})\\approx 0.1\\sqrt{M_{\\rm cl}(M_{\\odot })/r_{\\rm h}(\\rm pc)}$ (see Eq.",
"(REF ) above), is equal to $50\\rm \\ km\\ s^{-1}$ .",
"The BH binaries forming in clusters lying to the left of this line are likely to be ejected before merger.",
"The approximate relation $v_{\\rm esc}\\approx 2\\sqrt{3}\\sigma $ implies that only clusters with velocity dispersion $\\sigma \\gtrsim 15\\rm \\ km\\ s^{-1}$ will be able to retain their binaries.",
"In many NSCs and UCDs stellar mass BH binaries will merge while still inside the cluster, while most BH mergers in GCs are expected to occur outside the cluster unless an initially already massive BH ($M\\gtrsim 100\\ M_{\\odot }$ ) is present in the system [50].",
"This result is consistent with Monte Carlo simulations of GC models where the vast majority of BH binary assembled dynamically through $N$ -body interactions are found to merge after escaping from their host systems [36].",
"Figure REF shows that in many NSCs and UCDs, the BH merger remnants are likely to be retained so they might form new BH-binaries, that will subsequently harden and merge.",
"It is therefore possible that BHs in these massive star clusters will undergo a number of repeated mergers and grow considerably.",
"In addition to the recoil kick due to three-body interactions, as two compact objects merge, asymmetric emission of gravitational radiation will also induce a recoil velocity which can eject the merger product from the system.",
"In the next section we discuss this additional effect.",
"Figure: Left panel: distribution of escape velocities from NSCs and GCs(histograms) compared to the distributions ofGW kick velocities of merging BHs(blue curves).The solid blue line corresponds to a model in which the spinmagnitude was chosen randomly in the range χ=[0,1)\\chi = [0, 1); the dashed blue linecorresponds to a high-spin model in which χ=0.9\\chi =0.9.Right panel: probability of remaining inside the cluster as afunction of initial mass of the dominant BH and for different valuesof the cluster escape velocity.",
"Here we have assumedthat the mass of the secondary BH is 10M ⊙ 10\\ M_\\odot .",
"Solid line is forthe uniform spin model; dashed line is for the high-spin model.These plots show that the recoil velocity imparted by the anisotropic emission of GWradiation will lead to the ejection of most BH merger remnantsformed inside GCs, while in NSCs a fraction of BHs will be retained.For this reason BHs of large masscan naturally grow inside NSCs through repeated accretion of lowermass BHs.",
"Thus, NSCs are a likely host environment for thehighest-massBH mergers that are potentially detectable by aLIGO."
],
[
"Gravitational wave recoil",
"The GW recoil velocity of a merged BH depends on the mass ratio and spins of the progenitor BHs.",
"Hence, in order to make predictions about the distribution of recoil velocities for dynamically formed BHs in star clusters we first define the pre-merger BH spin and mass distributions.",
"The mass distribution of BH binaries is quite uncertain.",
"Here, we use the BH mass distribution of the dynamically formed merging black hole binaries from the Monte Carlo models of massive star clusters presented in [29].",
"These distributions contain no BH binary with mass ratio less than $q\\approx 0.5$ .",
"This is expected given that dynamical encounters in star clusters tend to pair and eject tight BH binaries with similar mass components – binaries in dense stellar environments are prone to exchange components, preferentially ejecting lighter partners in favor of more massive companions [100].",
"The distribution of BH spins is also very uncertain.",
"If the BH inspiral is driven predominantly by random gravitational interactions with other BHs and stars we might expect the spin orientations to be close to random.",
"We note that if there is significant coherent gas accretion, the spins might align in a way that might lead to low recoil kicks [24].",
"But stellar-mass BHs are unlikely to accrete enough mass from the interstellar medium for this process to be effective [77].",
"For these reasons, in our computations the misalignment angle of each BH is chosen at random in cos$(\\theta )$ .",
"For the spin magnitudes of the BHs we consider two choices.",
"The blue solid line in the left panel of Figure REF corresponds to a “uniform” model in which the initial spin magnitudes are drawn uniformly from the range ${\\chi }=[0, 1)$ , where $\\vec{\\chi }$ is the dimensionless spin of the BH ($\\vec{\\chi }=\\vec{S}/m_\\bullet ^2$ , where $\\vec{S}$ is the spin angular momentum in units of $m^2$ ).",
"The blue dashed line corresponds to one additional “high-spin” model in which the spin magnitude is set to a fixed value, $\\chi =0.9$ .",
"We note here that our spin magnitude distributions differ for example from those of [83] who adopted low-spinning BHs, leading to low merger kick velocities.",
"Our choice is motivated by observations: typical estimates of stellar-mass BH spins suggest high values, $\\chi > 0.5$ , in many cases [78].",
"In addition, equal-mass non-spinning binaries produce a rotating (Kerr) BH with final spin magnitude $\\chi \\approx 0.69$ [55], so that BHs undergoing more than one merger inside the cluster will have a finite spin magnitude [20].",
"However, we note that configurations leading to rapidly spinning BHs are rare.",
"The dimensionless spin magnitude tends to decrease for a BH that engages in a series of mergers, if the lighter BHs with which it mergers have a constant mass [75], [23].",
"This will keep the growing BH safely in the cluster after the first few mergers: not only does the mass ratio get farther from unity, which decreases the kick, but the spin of the more massive black hole drops as well.",
"After the pre-merger BH spin and mass distributions have been defined, we compute the recoil kick velocity from the following fitting formula based on the results from numerical relativity simulations of [69]: ${\\vec{v}_{\\rm k}} = v_{\\rm m} {{\\hat{e}}_{\\perp ,1}}+ v_{\\perp } ({\\rm cos} \\,\\xi \\, {{\\hat{e}}_{\\perp ,1}} + {\\rm sin} \\,\\xi \\, {{\\hat{e}}_{\\perp ,2}}) + v_{\\parallel } {{\\hat{e}}_{\\parallel }},$ $v_{\\rm m} = A \\eta ^2 \\sqrt{1 - 4\\eta } \\,(1 + B \\eta ),$ $v_{\\perp } = {H \\eta ^2 \\over (1 + q )} (\\chi _{2\\parallel } - q \\chi _{1\\parallel }),$ $v_{\\parallel } = {16 \\eta ^2 \\over (1+ q)} \\left[ V_{1,1} + V_{\\rm A} \\tilde{S}_{\\parallel } + V_{\\rm B} \\tilde{S}_{\\parallel }^2 + V_{\\rm C}\\tilde{S}_{\\parallel }^3 \\right] \\times \\nonumber \\\\|\\, {\\vec{\\chi }_{2\\perp }} - q {\\vec{\\chi }_{1\\perp }} | \\, {\\rm cos}(\\phi _{\\Delta } - \\phi _1),$ where $\\eta \\equiv q/(1+q)^2$ is the symmetric mass ratio; $\\perp $ and $\\parallel $ refer to vector components perpendicular and parallel to the orbital angular momentum, respectively, ${{\\hat{e}}_{\\perp ,1}}$ and ${{\\hat{e}}_{\\perp ,2}}$ are orthogonal unit vectors in the orbital plane, and ${\\vec{\\tilde{S}}} \\equiv 2({\\vec{\\chi }_2} + q^2 {\\vec{\\chi }_1})/(1+q)^2$ .",
"The values of $A = 1.2\\times 10^4$ km s$^{-1}$ , $B = -0.93$ , $H = 6.9\\times 10^3$ km s$^{-1}$ , and $\\xi = 145^{\\circ }$ are from [46] and [68], and $V_{1,1} =3678$ km s$^{-1}$ , $V_{\\rm A} = 2481$ km s$^{-1}$ , $V_{\\rm B} = 1793$ km s$^{-1}$ , and $V_{\\rm C} = 1507$ km s$^{-1}$ are taken from [69].",
"The angle $\\phi _{\\Delta }$ is that between the in-plane component ${\\vec{\\Delta }_{\\perp }}$ of the vector ${\\vec{\\Delta }} \\equiv M^2({\\vec{\\chi _2}} - q {\\vec{\\chi _1}})/(1+q)$ and the infall direction at merger.",
"We take the phase angle $\\phi _1$ of the binary to be random.",
"The histograms in the left panel of Figure REF show the escape velocities from NSCs and GCs computed as described in Section REF .",
"The blue curves show the recoil velocity distributions for our models computed using Eq.",
"(REF ).",
"The recoil velocity distribution in the uniform spin model is peaked at $v_{\\rm k}\\approx 500$ km s$^{-1}$ , while the high-spin model produces significantly larger kicks with typical velocities $v_{\\rm k}\\approx 1000$ km s$^{-1}$ .",
"Note however that in both models there is a substantial fraction of systems that are accelerated with velocities $\\lesssim 100$ km s$^{-1}$ .",
"The left panel of Figure REF suggests that only the most massive GCs have a finite probability of retaining a BH merger remnant formed inside the cluster.",
"Considering also that BH binaries in GCs are likely to be flung before merger due to three-body encounters, we conclude that the retention probability of BH merger remnants in GCs is small.",
"The left panel of Figure REF shows instead that the escape velocities of many NSCs are high enough that a substantial number of mergers are expected to be retained inside these systems.",
"In the right panel of Figure REF we compute the probability of remaining in the cluster for our spin distributions as a function of the initial BH mass and assuming that the secondary BH mass is $10\\ M_{\\odot }$ .",
"For escape velocities $\\lesssim 50\\rm \\ km\\ s^{-1}$ (typical of massive GCs) the probability of remaining inside the cluster after a merger is essentially zero, unless the cluster contains initially a BH seed of mass $\\gtrsim 100\\ M_{\\odot }$ .",
"For escape velocities $200\\rm \\ km\\ s^{-1}$ , which are more typical of NSCs, the probability of retaining a BH merger remnant of initial mass $50\\ M_\\odot $ is approximately $0.5$ or $0.3$ depending on the assumed spin distribution.",
"This makes NSCs excellent candidates for producing massive BH mergers that are potentially observable by aLIGO, because they can retain their BHs while also evolving rapidly enough that the BHs can sink back to the center and dynamically form new binaries which will subsequently merge.",
"This merger channel is expected to occur quite naturally in massive stellar clusters such as NSCs and UCDs, while it is unlikely to happen in lower mass systems such as open clusters and GCs.",
"In the next section we present a semi-analytical model that we use in order to make predictions about the mass distribution and rates of BH binary mergers forming in NSCs.",
"Figure: Schematic diagram illustrating our semi-analytical algorithmto model the evolution of BH binaries in stellar clusters.After we initialize the star cluster, we divide the BH populationin sub-groups each containing an equal number of BHs.Each sub-group contains a BH binary whose components arealways the two BHs which are currently the two most massive in the sub-group.The binary is evolved for a time interval Δt\\Delta t; if the binary merges or is ejectedfrom the cluster a new binary is formed and evolved.",
"After a time interval Δt\\Delta tall BHs are mixed back together and the procedure repeated until either all BHs have been ejectedfrom the cluster or the integration time becomes longer then the Hubble time (T H T_{\\rm H}).Figure: Comparison between the results of our semi-analytical modelwith the results of Monte Carlo models.Upper panel shows the median number of merging BH binaries as afunction of the total number of stars in the cluster.",
"Open symbols are from theMonte Carlo simulations of .",
"Filled symbolsare from our simplified semi-analytical approach.Lower panel gives the median mass of ejected BH-binariesas a function of time of ejection froma GC Monte Carlo model of (black curve)and the average mass of ejected BH-binariesfrom ten semi-analyticalmodels having similar structural properties (red curve).Dashed curves give the region containing 70%70\\% and 90%90\\% of theejected systems in these models.",
"Solid curve is the median of the mass distribution."
],
[
"semi-analytical modeling",
"As argued above the dynamical evolution of NSCs is of great interest as these systems could represent a important source of inspiraling BHs detectable by aLIGO.",
"Yet a good understanding of the dynamical evolution of massive clusters ($M_{\\rm cl}>>10^6\\ M_{\\odot }$ ) and their implications for aLIGO detections is still elusive.",
"The main difficulty is the large number of particles comprising these systems which makes their treatment extremely challenging even for approximate Monte Carlo methods.",
"Here we adopt a semi-analytical approach which allows us to make predictions about the expected rate and properties of inspiraling BH binaries forming in NSCs."
],
[
"Simplified approach",
"First, we define the structural properties of our star clusters.",
"We assign a total stellar mass $M_{\\rm cl}$ to the cluster.",
"For $M_{\\rm cl}\\le 5\\times 10^6\\ M_\\odot $ the half-mass radius is independent on the cluster mass and it is set to $r_h=3\\rm \\ pc$ .",
"In the NSC mass regime, $M_{\\rm cl}> 5\\times 10^6\\ M_\\odot $ , we adopt the fitted relation to the NSCs in late type galaxies from [42]: $\\log (r_h/c1)=\\alpha \\log (M_{\\rm cl}/c2)+\\beta $ , with $\\alpha =0.321$ , $\\beta =-0.011$ , $c1=3.31\\rm \\ pc$ and $c2=3.6\\times 10^6\\rm \\ M_{\\odot }$ .",
"While sampling from the adopted distributions we also accounted for the scatter of the observed relations.",
"The escape velocity from the cluster is then computed using the approximate Eq.",
"(REF ) above; the cluster velocity dispersion is $\\sigma =v_{\\rm esc}/(2\\sqrt{3})$ .",
"The central number density of stars was computed as $n=4\\times 10^6(\\sigma /100\\rm \\ km\\ s^{-1})^2\\rm \\ pc^{-3}$ .",
"This latter expression gives a central number density of stars for a Milky Way like NSC of $4\\times 10^6 \\rm \\ pc^{-3}$ and $\\approx 10^5 \\rm \\ pc^{-3}$ for a $10^6\\ M_{\\odot }$ GC – this is consistent with observed values [52], [73].",
"Next we define the initial mass distribution and number density of BHs in our cluster models.",
"We take the mass distribution of single BHs from Figure 6 of [96].",
"These authors used the stellar evolution code BSE [57], [58] improved with the stellar remnant prescription from [59] and [28].",
"Our models adopt the update prescriptions for stellar winds and supernova fallback, in order to replicate the BH mass distribution of [35] and [18].",
"We consider two values of metallicity, $Z=0.01\\ Z_\\odot $ and $Z=0.25\\ Z_\\odot $ , defined below as low metallicity and high metallicity models.",
"In our calculations we assume that all clusters formed $12\\rm \\ Gyr$ ago regardless of their mass.",
"While this is a good approximation for GCs, NSCs are known to have complex star formation histories, including recent episodes of star formation.",
"We neglect such complication in the following, noting that the bulk of the stellar population in NSCs is also likely to be in old stars formed many Gyrs ago [87].",
"Initially, our cluster models have a total mass in BHs that is $M_\\bullet =0.01M_{\\rm cl}$ .",
"This is the typical mass fraction in BHs expected for standard initial mass functions [56].",
"The total number of BHs is therefore $N_\\bullet \\approx M_\\bullet /\\langle m_\\bullet \\rangle $ , with $\\langle m_\\bullet \\rangle $ the average BH mass in our models.",
"Then we consider natal kicks.",
"For each BH in our fiducial model we compute a natal kick velocity from a Maxwellian given by $\\sigma _{\\rm natal}=265\\rm \\ km\\ s^{-1}$ as commonly done for neutron stars [54], and assume that the natal velocity of a BH of mass $m_\\bullet $ is lowered by the factor $1.4\\ M_\\odot /m_\\bullet $ .",
"For any sufficiently massive BH progenitor ($>40\\ M_\\odot $ ), the fallback completely damps any natal kick, and the BH is retained in the cluster [39].",
"BHs that receive a kick with velocity larger than the escape velocity from the cluster are removed from our models.",
"However, since we only consider massive clusters with large escape velocities, a large fraction of BHs in our models is retained after experiencing a natal kick.",
"This makes our conclusions less sensitive to the prescription we used for natal kicks, provided that the real kick magnitudes are not much larger than what we have adopted here.",
"We discuss in more details the effect of varying the natal kick magnitudes below in Section REF .",
"We assume that after a time $t_{\\rm df} (\\langle m_\\bullet \\rangle )$ the BHs have segregated to the cluster center.",
"After this time, due to the high densities in the core, BH binaries will efficiently form though 3-body binary formation [80] and possibly through exchange interactions with stellar binaries [77].",
"Therefore we assume that after a time $t_{\\rm df} (\\langle m_\\bullet \\rangle )$ , a fraction $f_{\\rm bin}=0.01$ of the BHs end up in hard BH binaries.",
"Although this fraction is quite uncertain, the value we adopted is typical for Monte Carlo models of massive star clusters with low binary fraction [80].",
"After the BHs have segregated to the center and we have assigned a fraction of them to be in BH binaries we follow the evolution, ejection and formation of new binaries adopting the scheme described in what follows.",
"We divide the BH cluster in $N_{\\rm B}=f_{\\rm bin}N_\\bullet /2$ sub-groups, each containing the same number of BHs.",
"We find the two most massive BHs in each sub-group and assume that after a time $t_{1-2}$ they form a binary.",
"Thus, we assume that each sub-group always contains one binary and that this binary is always composed of the two most massive BHs in the sub-group.",
"While each binary is assumed to evolve in the gravitational potential of the entire cluster the adopted numerical scheme allows us to simulate a scenario in which the number of BH binaries in the cluster is approximately constant with time.",
"This, besides allowing us to greatly simplify our approach, appears to be reasonable when compared to the results of Monte Carlo models of massive clusters [80], [29].",
"Moreover, we assume that the BH binaries are always composed of the two most massive black holes in each sub-group because during exchange encounters lighter partners are more likely to be ejected.",
"This favors the formation of high mass binaries with similar mass components [29].",
"In our calculation we conservatively assume that the interactions occur between BHs and stellar binaries so that the timescale for binary formation is the longer timescale given by Eq.",
"(REF ).",
"The binary fraction in evaluating $t_{1-2}$ was computed taking a primordial binary fraction of $0.2$ and lowering this fraction by the number of soft binaries for a constant probability in $\\log (a)$ [37].",
"Any binary forms with an initial semi-major axis $a_{\\rm hard}=1/(\\sigma /30\\rm \\ km\\ s^{-1})^2 \\rm \\ AU$ .",
"Given the cluster velocity dispersion, its density and the mass of the binary, we compute (i) the semi-major axis, $a_{\\rm GW}$ , below which GW radiation will start to dominate (Eq.",
"[REF ]), and (ii) the semi-major axis, $a_{\\rm ej}$ , at which the binary will be ejected as a consequence of three body scatterings.",
"If $a_{\\rm ej}>a_{\\rm GW}$ the binary will merge outside the cluster and will be ejected with a semi-major axis $\\approx a_{\\rm ej}$ ; in this case we evaluate the timescale from the formation of the binary to its ejection, $t_{\\rm ej}$ , using Eq.",
"(REF ) so that the lifetime of the binary is $T=t_{\\rm ej}+t_{\\rm GW}(a=a_{\\rm ej})$ .",
"If $a_{\\rm ej}<a_{\\rm GW}$ the binary will merge inside the cluster; in this latter case the total lifetime of the binary is $T=t_{\\rm merge}+t_{\\rm GW}(a=a_{\\rm GW})$ .",
"In the previous expressions the GW merger timescale, $t_{\\rm GW}$ , was computed by sampling the binary eccentricity from a thermal distribution $N\\propto e^2$ .",
"If the BH binary is ejected from the cluster, then after a time $t_{1-2}$ we form a new binary and, as before, we take its components to be the next two most massive BHs in the sub-group.",
"Then the hardening timescale of the binary is evaluated as before and it is determined whether the new binary will merge inside the cluster, and, if it does, whether it will be retained inside the cluster after merging.",
"If the binary merges inside the cluster (i.e., $a_{\\rm ej}<a_{\\rm GW}$ ) we assign the two progenitor BHs a spin magnitude and orientation from the spin models described in Section REF and compute the GW recoil speed through Eq.",
"(REF ).",
"In order to account for the recoil kick due to the interaction with a third object we compute a total kick velocity as $v_{\\rm tot}=\\sqrt{v^2_{\\rm k}+v^2_{2-1}}$ , with $v_{2-1}$ computed as in Section REF (note that $v_{\\rm tot}\\approx v_{\\rm k}$ typically).",
"If $v_{\\rm tot}>v_{\\rm esc}$ the BH merger remnant is ejected from the cluster, otherwise it is retained.",
"If the BH merger remnant is ultimately retained inside the cluster it will have another chance of interacting with new binaries and experience additional mergers.",
"In this case, we place the BH remnant at a distance $r_{\\rm h}(v_{\\rm tot}/v_{\\rm esc})^2$ from the center and evaluate the dynamical friction timescale for the BH to reach the cluster core through Eq.",
"(REF ).",
"If $t_{\\rm df}$ is greater than $10\\rm \\ Gyr$ the BH is removed from the computation, otherwise after a time $t_{1-2}$ the BH forms a new binary with the next most massive BH in the sub-group.",
"Then, the hardening timescale of the new binary is evaluated as before and it is determined whether the new binary will merge inside the cluster, and if it does whether it will be ejected from the cluster after the recoil due to anisotropic emission of GW radiation.",
"As the binary hardens we calculate the number of field BHs that are ejected through three-body encounters as $N_{\\rm ej}=M_{\\rm ej}/\\langle m_\\bullet \\rangle $ where $M_{\\rm ej}$ is given by Eq.",
"(REF ).",
"If the number of ejected BHs becomes larger than the total initial number of BHs we stop the integration.",
"The previous steps (i.e., binary formation, hardening, and ejection/merger) are repeated for each sub-group for a time-step $\\Delta t$ .",
"After an interval of time $\\Delta t$ the remaining BHs in all the sub-group are mixed together and the procedure described above is repeated recursively until either all BHs have been ejected from the cluster or the total integration time exceeds the Hubble time.",
"The mixing of the sub-groups every $\\Delta t$ allows us to avoid suppressing exchange interactions between massive BHs that might grow in different sub-groups.",
"In what follows we set $\\Delta t=1.5\\times 10^9\\rm \\ yr$ , but found that values in the range $\\Delta t=1-3\\times 10^9\\rm \\ yr$ all produced similar results.",
"The main steps of our semi-analytical algorithm are also schematically illustrated in Figure REF .",
"We note that our prescriptions are oversimplified in many ways and that more accurate Monte-Carlo simulations will be needed in order to confirm our results.",
"One basic simplifying assumption is that the cluster structural properties (e.g., central density, half-mass radius) remain constant in time.",
"We believe that this assumption is also justified in many cases, and especially in very massive clusters where the relaxation timescale is longer.",
"For example, Monte Carlo simulations of moderately massive GCs find that $r_{\\rm h}$ increases with time, but often only by a factor $\\lesssim 3$ throughout the cluster evolution [29].",
"Additionally, in our models we assume that the binary-single interactions rate is always dominant with respect to that of binary-binary interactions.",
"This latter assumption is also reasonable, unless the cluster has a very large initial binary fraction [80].",
"Finally, we note that we do not follow the evolution of the BH spins through consecutive mergers but assume that the spins are always drawn from the assumed distributions.",
"Figure: Mass of merging BH binaries for a range of clustermasses which could represent typical GCs (upper panels) or NSCs(lower panels) as a function of redshift.",
"Evolution proceeds from right to left.We assume here thatall clusters formed 12 Gyr ago.The uniform spin modeldescribed in Section was adopted.",
"Open blue circles are systems the are retainedinside the cluster after merging.",
"Note how almost all mergersoccurring inside low mass clusters are promptly ejected, while forM cl =0.5-5×10 7 M ⊙ M_{\\rm cl}=0.5-5\\times 10^7M_{\\odot } many of the inspiraling BHs areexpected to be retained inside the cluster.Figure: Same as Figure but for Z=0.25Z ⊙ Z=0.25\\ Z_\\odot ."
],
[
"Results",
"Given that our prescriptions are simplified in many ways, we proceed here by testing the results of our models against the results from the Monte Carlo models of [96].",
"In the upper panel of Figure REF we show the total number of mergers per cluster for systems containing different numbers of stars and having different metallicities.",
"In order to convert $M_{\\rm cl}$ in number of stars we have taken a mean stellar mass of $0.55\\rm \\ M_{\\odot }$ typical of old stellar populations [74].",
"Moreover, we select here the BH spins based on the uniform spin model described in Section REF .",
"Our semi-analytical models predict that the total number of mergers increases with cluster mass and so do the Monte Carlo models.",
"The total number of inspirals over 12 Gyr is nearly linearly proportional to the final cluster mass.",
"This result is also in agreement with previous models of GCs and shows that this statement can likely be extrapolated up to numbers of stars of order of a few $10^7M_{\\odot }$ .",
"Our models also predict an inversion of this simple correlation showing that for $M_{\\rm cl} \\gtrsim 10^7M_{\\odot }$ the number of merging BHs flattens or even declines towards larger cluster masses.",
"This is expected given that for such massive clusters with higher values of $\\sigma $ have a larger binary formation time $t_{1-2}$ .",
"The most massive clusters in our integrations, which could represent NSCs, produce up to a few thousand BH mergers per cluster.",
"The lower panel of Figure REF gives the median mass of the ejected BH binaries formed in 10 cluster models with mass $1.2\\times 10^6\\ M_\\odot $ and half-mass radius $r_h=7\\rm \\ pc$ .",
"These results are directly compared to those from Figure 4 in [96] which corresponds to a cluster model of initial mass $1.2\\times 10^6\\ M_\\odot $ and final half-mass radius $r_h\\approx 7\\rm \\ pc$ .",
"The good agreement between the results of our simplified approach and those of Monte Carlo simulations gives a high level of reliability to our semi-analytical models.",
"Figure REF and Figure REF give the masses for each of the BH inspirals occurring in 10 star cluster models of metallicities $Z=0.01 Z_\\odot $ and $Z=0.25\\ Z_\\odot $ respectively.",
"Masses in the ranges $0.5-5\\times 10^6\\ M_{\\odot }$ (upper panel) and $0.5-5\\times 10^7M_{\\odot }$ (lower panel) were considered.",
"The overall structure of the plots agrees well with our understanding of the dynamics of BHs and their evolution in star clusters, and with the results of previous work [96].",
"After the formation of the cluster at high redshift, the BHs segregate to the center, the most massive BHs form binaries and the majority of them are ejected.",
"The cluster processes through its population of BHs that merge and are ejected from most to least massive, so that only low-mass BHs are retained by the present epoch.",
"More massive clusters, which could represent typical NSCs, produce BH mergers in the local universe that are significantly more massive than mergers occurring in lower mass clusters.",
"As also noted in [96] the plateaus in the chirp mass and total mass distributions in Figure REF are mainly a consequence of the maximum BH mass in the initial models, which is regulated by the wind-driven mass loss from the Vink prescription.",
"For the high metallicity models this produces a large population of $30\\ M_\\odot $ BHs, which leads to the formation of a large population of equal-mass mergers with total mass of $60\\ M_\\odot $ .",
"More interestingly, we find that massive cluster models produce an additional collection of binaries at $90\\ M_\\odot $ and $120\\ M_\\odot $ which can be clearly seen at high redshift in Figure REF .",
"One of the two BHs in these binaries has experienced one and two earlier mergers with lower mass BHs respectively.",
"For the the low metallicity models there is no apparent collection of sources as might be expected [96].",
"The decreased efficiency of the stellar winds in the low metallicity models implies that a lower number of high-mass stars are converted into BHs with the maximum-mass set by the wind-driven mass loss prescription, resulting in a wider range of BH masses.",
"In Figure REF and Figure REF we show the total and chirp mass and uncertainties associated with the recent detection of the BH binary merger GW150914 [3].",
"The reported masses of GW150914 are consistent with the masses of black hole mergers from GCs in the local universe.",
"However, even for the low metallicity models only 5 percent of the total number of mergers in GCs produce a merger at low redshift with a total mass significantly larger than $50\\ M_\\odot $ as required to match the total mass of the GW150914 event.",
"In NSCs this percentage is significantly larger, being $\\approx 20$ percent of the total number of inspiraling binaries.",
"In high metallicity clusters (Figure REF ) a smaller number of high mass BH mergers is produced at low redshift making these clusters less likely progenitors of GW150914-like events.",
"In Table 1 we report the mean number of mergers per cluster obtained from our models.",
"NSCs are defined here as clusters with masses in the range $5\\times 10^6 - 5\\times 10^7 \\ M_\\odot $ , while GCs have masses in the range $10^5 -10^7 \\ M_\\odot $ .",
"In order to obtain the mean rate of mergers we weighted the number of mergers from each of the cluster models by a cluster initial mass function (CIMF).",
"For GCs we assume a power law CIMF: ${\\rm d} M/{\\rm d} N\\propto M^{-2}$ [21].",
"For NSCs the initial mass function is largely unknown.",
"Here we take the IMF of NSCs directly from the mass distribution of NSCs at $z=2$ from the galaxy formation models of [11] (their Figure 10).",
"These models produce a mass distribution at $z=0$ that is consistent with the observed NSC mass distribution from [42].",
"We note that here we might be underestimating the number of massive mergers from NSCs occurring at low redshift because we have assumed that these systems are as old as Galactic GCs.",
"In fact, while most NSCs appear to be dominated by old stellar components they are also known to have a complex star formation history and to contain young stellar populations which can produce high mass mergers also at later times (we will come back to this point below).",
"It is also possible that a large fraction of the NSC stars accumulated gradually in time by infalling globular clusters that decayed to the center through dynamical friction.",
"If this process is the main mechanism for NSC formation, then NSCs and GCs will comprise similar stellar populations [9].",
"Table 1 shows that our models predict a few thousands BH mergers per NSC over 12 Gyr of evolution.",
"This expectation also appears to be consistent with previous estimates [90], [77].",
"In addition, NSCs produce between 50 to $\\approx 500$ BH mergers with high mass $>50M_{\\odot }$ at $z<0.3$ depending on the BH spin magnitudes and assumed metallicities distribution of the underlining stellar population.",
"Our GC models produce only a few mergers per cluster within $z<0.3$ and total mass $>50\\ M_{\\odot }$ .",
"These massive binaries are found to form only in the most massive GCs ($M_{\\rm cl}\\gtrsim 10^6M_{\\odot }$ ) of low metallicity.",
"The number of massive mergers at low redshift is also sensitive to the spin magnitude distribution we assume.",
"For high spin models, a smaller number of BHs are retained in the clusters compared to the uniform spin models.",
"Consequently, high spin models produce fewer high mass BH mergers at low redshift compared to models that assume low spins.",
"However, in either spin models a number of inspiraling BH binaries with mass $\\gtrsim 50\\ M_\\odot $ is found to merge at low redshift.",
"Finally, Table 1 gives the number of BH mergers that are retained inside the cluster.",
"Between 10 and 20 percent of high mass ($>50\\ M_\\odot $ ) mergers occurring in NSCs at $z<1$ are retained inside the cluster enabling the formation of even more massive BH mergers.",
"The results presented in this section suggest that NSCs are a natural environment for producing BH mergers that are observable by aLIGO detectors.",
"In addition to this, NSCs can form high mass BH binaries, and mergers with mass consistent with that of GW150914 also in relatively high metallicity environments.",
"The implications of our results are discussed in more detail in the following section.",
"Table: The mean number of inspirals per cluster over 12 Gyr ofevolution, 〈N〉\\langle N \\rangle , that occur at redshift z<0.3z< 0.3 and z<1z< 1and those that occur at redshift z<0.3z< 0.3 and z<1z< 1and have a total mass >50M ⊙ >50\\ M_\\odot .",
"Below we give the fraction of mergersthat are retained inside the clusters."
],
[
"Implications and Discussions",
"Our study shows that a multitude of BH binary mergers can be produced at the center of galaxies where NSCs reside.",
"In the following we derive an approximate expected detection rate from the results of our models and discuss some implications for possible aLIGO detections of these mergers over the next decade.",
"Finally we discuss the production of BH mergers in NSCs hosting a central MBH, and the possibility of a continues supply of BH-binaries in NSCs through episodic and/or continuous star formation."
],
[
"Detection rate estimates",
"Here we use results from semi-analytical galaxy formation models to derive an expected MBH occupation fraction in NSCs and use this as well as the results of the cluster semi-analytical models presented in this paper to make predictions about the merger rate of BH binaries produced in NSCs.",
"We also consider the merger rates from GCs and compare these to estimates made in former studies.",
"To compute the aLIGO merger rate of BH binaries per unit volume we use the following expression: $\\Gamma ^{\\rm NSC}_{\\rm aLIGO}=n_{\\rm gx}\\Gamma _{\\rm merge}f_{\\rm nucleated}$ where $n_{\\rm gx}$ is the number density of galaxies, $\\Gamma _{\\rm merge}$ is the averaged merger rate of BH binaries per cluster that merge within the observable volume, and $f_{\\rm nucleated}$ is the fraction of galaxies which host a NSC but do not have a MBH.",
"While observations show that NSCs and MBHs coexist in some galaxies, and that NSCs exist in most galaxies, $f_{\\rm nucleated}$ remains largely unconstrained.",
"Here we use the results of semi-analytical galaxy formation models that follow the cosmological evolution of galaxies, their MBHs and NSCs.",
"These galaxy formation models are described in [15], [10] and [11].",
"Figure REF shows the fraction of galaxies in these models that contain a NSC but do not host a MBH.",
"These models predict that the number of galaxies hosting a NSC but without a MBH is quite large, being $f_{\\rm nucleated}\\gtrsim 0.5$ for galaxies with total mass $M_{\\rm Gx}\\lesssim 10^{11}\\ M_\\odot $ regardless of galaxy type.",
"Based on Figure REF we adopt here a conservative value of $f_{\\rm nucleated}= 0.5$ , and adopt a number density of galaxies of $0.02\\rm \\ Mpc^{-3}$ [30], [61].",
"Assuming that BH-BH mergers can be seen by aLIGO out to a redshift of $z\\lesssim 0.3$ [6], which corresponds to an age of the universe of $\\approx 10 \\rm \\ Gyr$ [89], then Eq.",
"(REF ) gives a merger rate of BHs in NSCs of $\\Gamma _{\\rm aLIGO}^{\\rm NSC}\\approx 1.5 \\rm \\ Gpc^{-3}\\ yr^{-1}.$ Thus our calculation predicts a substantial number of detectable BH mergers from NSCs.",
"From Table 1 we see also that between 10 and 20 percent of the total number of merger remnants are retained inside NSCs.",
"Adopting an aLIGO detection-weighted comoving volume of $\\approx \\rm 10{\\rm \\ Gpc^{-3}}$ for full design sensitivity [6], the rate in Eq.",
"(REF ) translates into a detection rate of $\\approx 10 \\rm \\ yr^{-1}$ .",
"The merger rate of Eq.",
"(REF ) can be directly compared to that from GCs: $\\Gamma _{\\rm aLIGO}^{\\rm GC}\\approx 5 \\rm \\ Gpc^{-3}\\ yr^{-1},$ which was obtained from the number of mergers per GCs within $z<0.3$ in Table 1 and assuming a number density of GCs equal to $0.77\\rm Gpc^{-3}$ [95].",
"Note that Eq.",
"(REF ) is very well consistent with the rate previously derived by other authors [96].",
"As before, taking an aLIGO detection-weighted comoving volume of $\\approx 10\\rm \\ Gpc^{-3}$ for full design sensitivity, we obtain an aLIGO detection rate of BH mergers from GCs of $\\approx 50\\rm \\ yr^{-1}$ .",
"Figure: Fraction of galaxies containing a NSC (M cl ≥10 5 M ⊙ M_{\\rm cl} \\ge 10^5 \\ M_\\odot ) but no MBH as a function oftotal galaxy mass.Results are from the galaxy formationmodels of and .Dashed line is for galaxies with bulge-to-total-mass-ratio smaller than 0.7.",
"Solid line is for early type galaxies which are defined here as galaxies with bulge-to-total-mass-ratio larger than 0.7.Now we only consider mergers with mass $\\gtrsim 50M_{\\odot }$ occurring at a redshift $z<0.3$ (see Table 1).",
"We define these as possible progenitors of the event GW150914.",
"The detection rate of high mass mergers from NSCs in the local universe is in the range $\\Gamma _{\\rm aLIGO}^{\\rm NSC}(z<0.3;M>50M_{\\odot })\\approx 0.4 -1 \\rm \\ Gpc^{-3}\\ yr^{-1},$ where the lower limit corresponds to high metallicity clusters and the high spin model, and the upper limit to low metallicity clusters and to a uniform BH spin distribution.",
"Interestingly, we find that between 20 and 50 percent of all massive mergers in NSCs are produced by the consecutive merger channel discussed in this paper with a few percent having a mass $\\gtrsim 100M_{\\odot }$ .",
"The corresponding merger rate of high mass BH binaries mergers in GCs is $\\Gamma _{\\rm aLIGO}^{\\rm GC}(z<0.3;M>50M_{\\odot })\\approx 0.05 -1\\rm \\ Gpc^{-3}\\ yr^{-1} \\ ,$ similar to that corresponding to high mass BH mergers produced in NSCs.",
"Both rates of high mass mergers from GCs and NSCs are marginally consistent with the rate of $2-53\\rm \\ Gpc^{-3}\\ yr^{-1}$ of GW150914-like mergers given by [5].",
"From our rate computation we find that the detection rate of BH-BH binaries from NSCs is substantial, and it is about one tenth of that from GCs.",
"Importantly, we also find that the NSC detection rate of high mass BH mergers similar to GW150914 ($M\\approx 60\\ M_\\odot ,z<0.5$ ) is comparable to that from GCs, with many of the mergers being produced through the consecutive merger scenario discussed in this paper.",
"Finally, our results show that most GW150914-like mergers are more likely to be a product of dynamical interactions occurring in massive clusters of low metallicity, also in agreement with previous findings [97].",
"Figure: Mean merger rate per galaxy at redshift z<1z<1as a function of the dispersionof the Maxwellian distribution of natal kick velocities, σ natal \\sigma _{\\rm natal},applied to all BHs independently of mass.Red curves correspond to ametallicity Z=0.25Z ⊙ Z=0.25\\ Z_{\\rm \\odot } andblack curves to a metallicityZ=0.01Z ⊙ Z=0.01\\ Z_{\\rm \\odot }.We have assumed a number of GCs per galaxy equal to 100and a light travel time of 8 Gyr 8\\rm Gyr at z=1z=1.At σ natal ≳50 km s -1 \\sigma _{\\rm natal}\\gtrsim 50 \\rm \\ km\\ s^{-1}the merger rate of BH binaries from NSCs is dominant compared tothe corresponding merger rate from GCs."
],
[
"Dependence on natal kicks",
"[6] noted that “for both dynamical formation in [globular] clusters and isolated binary evolution, the implication of BH binary existence is that BH natal kicks cannot always be high ($ \\gtrsim 100 \\rm \\ km\\ s^{-1}$ ), in order to avoid disrupting or widening the orbits too much, or ejecting the BHs from clusters before they can interact.” For example, large natal kicks will widen the orbits of massive binary progenitors and quench the formation of binary BH systems that will merge within the age of the universe [17].",
"[6] did not consider the possibility that BH binaries can be dynamically assembled inside NSCs which we discuss here.",
"Motivated by the fact that the distribution of formation kicks for BHs is largely uncertain, even at the qualitative level, we assume here that the kick magnitude distributions are fully unconstrained and explore how the NSC and GC BH merger rates are affected by varying these distributions.",
"In Figure REF we plot the mean merger rate per galaxy at $z<1$ in models where the BHs are given a natal kicks taken from Maxwellian distributions with dispersion $\\sigma _{\\rm natal}$ , and the spin magnitude of the BHs is uniformly distributed in the range $\\chi = [0, 1)$ .",
"Unlike the models discussed above, here we assume that the BH natal kick distributions do not depend on the mass of the BHs.",
"As $\\sigma _{\\rm natal}$ increases above $\\gtrsim 20\\rm \\ km\\ s^{-1}$ the merger rate from GCs decreases and becomes comparable to that of NSCs for $\\sigma _{\\rm natal}\\approx 50\\rm \\ km\\ s^{-1}$ .",
"Above this value the merger rate of NSCs begins to decrease as well because a fraction of BHs starts to be ejected from low mass NSCs as well.",
"However, for $\\sigma _{\\rm natal}\\gtrsim 50\\rm \\ km\\ s^{-1}$ we see that the BH merger rate from NSCs becomes significantly larger than the corresponding merger rate from GCs.",
"We conclude that for high natal kicks, $\\gtrsim 50\\rm \\ km\\ s^{-1}$ , NSCs can dominate the merger rate of dynamically formed BH binaries that are detectable by aLIGO.",
"Recent theoretical studies suggest that the birth kicks may not be directly correlated with the BH mass [94], [84], and that the distribution of black hole kick velocities could be similar to that of neutron stars $\\sigma _{\\rm natal} \\approx 200 \\rm \\ km\\ s^{-1}$ [70].",
"If this is the case, then the merger rate of BH binaries from NSCs will greatly dominate over that from GCs.",
"However, we also note that for $\\sigma _{\\rm natal}\\gtrsim 200\\rm \\ km\\ s^{-1}$ the rate of BH mergers from NSCs becomes significantly smaller than the rate of $2-53\\rm \\ Gpc^{-3}\\ yr^{-1}$ implied by the detection of GW150914 [5].",
"Thus very high values of natal kicks are excluded by our analysis when combined with the estimated aLIGO detection rate."
],
[
"Mergers of stellar remnants near massive black holes",
"So far we have considered the merger of BH binaries in stellar clusters which do not host a central MBH.",
"This is certainly not the case for at least a handful of NSCs which are known to have MBHs at their center [99].",
"One example is our own Galaxy which contains a $\\approx 10^7M_{\\odot }$ NSC and a central MBH of $\\approx 4\\times 10^6\\ M_\\odot $ [44], [45].",
"More generally NSCs and MBHs are known to co-exist in galaxies with masses $\\approx 10^{10}\\ M_\\odot $ [99], [47]; galaxies with masses lower than this value show clear evidence for nucleation but little evidence for a MBH.",
"Conversely galaxies with masses above $10^{11}\\ M_\\odot $ are dominated by MBHs but generally show no evidence for nucleation [81].",
"In NSCs containing a central MBH, the merger rates of BH binaries are expected to be significantly different, although not necessarily smaller, than the rates given in Table 1.",
"If a MBH is present it will inhibit core-collapse, causing the formation of a Bahcall-Wolf cusp instead [13].",
"After the BHs have segregated to the center their densities will dominate over the stellar densities within a radius $\\lesssim 0.1$ times the MBH influence radius [56], [48] – note that this latter statement depends on the formation history of the NSC [9].",
"At such small distances from the MBH all binaries will be effectively soft so that any interaction with a third BH or star will tend to make the binary internal orbit wider.",
"In this situation, three-body interactions between binaries and field objects will not lead to BH mergers but rather to the “evaporation” of binaries.",
"Several mechanisms have been discussed in the literature which can produce BH mergers even in the extreme stellar environment of MBHs.",
"Below we briefly review two of these processes: (i) mergers of BH binaries due to Lidov-Kozai (LK) resonance induced by the central MBH [8]; (ii) single-single captures of compact objects due to gravitational wave energy loss [91], [64], [82]."
],
[
"MBH-mediated BH mergers in NSCs",
"[8] showed that near a MBH the dynamical evolution of binaries is dominated by perturbations from the central MBH.",
"In particular, the LK mechanism [67], [62] exchanges the relative inclination of the inner BH binary orbit to its orbit around the MBH for the eccentricity of the BH binary [7].",
"At the peak eccentricity, during a Lidov-Kozai cycle, GW emission can become efficient leading to a merger of the two BHs.",
"More recently [104] performed $N$ -body simulations of small clusters of stars containing a central MBH, to estimate the rate of BH mergers induced by the LK mechanism.",
"These authors found that this mechanism could produce mergers at a maximum rate of $\\approx 2$ per Myr per Milky Way equivalent galaxy.",
"This rate appears to be somewhat comparable to the upper end of the expected detection rate from stellar clusters [96].",
"[104] showed that this rate could translates into a maximum rate per volume of $\\approx 100\\rm \\ Gpc^{-3}\\ yr^{-1}$ .",
"However, as also noted by [104] their merger rate estimates are likely to be an overestimate of the true merger rate from the LK process as they used optimistic values for both the merger fraction as well as for the BH binary fraction.",
"The efficiency of the LK process in inducing BH mergers in NSCs with MBHs is sensitive to major uncertainties.",
"The major challenge to this being that a continuous supply of BH binaries is needed in order to obtain a finite merger rate.",
"In fact, a continuous formation of binaries is part of the assumptions made in the rate estimates of [8] and [104] .",
"Binaries well inside the influence radius of a MBH will be essentially all soft and will be disrupted over the typical timescale [22]: $t_{\\rm ev}&=&{m_{12}\\sigma \\over 16\\sqrt{\\pi } G m_\\star \\rho a \\ln \\Lambda }\\approx 10^7{\\sigma \\over 100 \\rm \\ km\\ s^{-1}}\\\\&&\\left(0.5 \\frac{m_{12}}{m_\\bullet }\\right)\\left(\\frac{\\ln \\Lambda }{10} \\frac{\\rho }{10^6 \\ M_\\odot {\\rm \\ pc^{-3}}}{a\\over \\rm 1\\ AU} \\right)^{-1}.~~~~\\nonumber $ Because $t_{\\rm ev}$ is much shorter than the lifetime of any NSC, we expect that most primordial binaries will be disrupted by now in these systems.",
"[8] describe various processes which may affect the replenishment rate of compact binaries and/or their progenitors in NSCs with a MBH.",
"[85] suggested that disruption of triple stars could leave behind a binary in a close orbit around the MBH and could serve as a continuous source of replenished binaries close to the MBH.",
"In addition, in-situ star formation can also repopulate the binary population of NSCs.",
"Central star formation bursts may occur continuously (or episodically) throughout the evolution of the stellar cusp and lead to a steady population of massive binaries near the center.",
"Finally, NSCs might result from the merger of stellar clusters in the inner galactic regions [11].",
"Such clusters may harbor an inner core cluster of BHs that formed during the cluster evolution.",
"If these BHs are retained in the cluster this mechanism may also contribute to the BH and BH-binary populations in NSCs [9]."
],
[
"Mergers from BH-BH scattering in NSCs",
"[82] showed that in the dense stellar environments such as those of NSCs BH binaries can efficiently form out of GW radiation during BH-BH (single-single) encounters.",
"Interestingly they show that most of the mergers from this channel will have a finite eccentricity while they enter the $10\\rm \\ Hz$ frequency band of aLIGO.",
"This processes could become important for sufficiently large cluster masses.",
"However, the predicted rate of BH-BH mergers from this channel, although very uncertain, is estimated to be only $\\sim 0.01\\rm \\ Gpc^{-3}\\ yr^{-1}$ [103], and therefore it is sub-dominant with respect to the rate from the other processes discussed in this paper.",
"Finally, we note that [12] argued that even the rate of eccentric mergers from BH-BH scattering in NSCs is likely to be much smaller to that of eccentric mergers from BH triples formed in GCs.",
"In conclusion, the role of NSCs containing MBHs in producing BH mergers is a subject of considerable interest, which will likely require high precision $N$ -body simulations of large number of particles.",
"For now, theoretical models suggest that the merger rate of BHs in these systems might be considerably lower than that from NSCs without MBHs."
],
[
"Continuous and episodic star formation in NSCs",
"As mentioned above, most NSCs are known to have undergone a complex star formation history characterized by recurrent episodes of star formation [105].",
"Thus NSCs might still be forming BHs and BH binaries at the present epoch.",
"This is different from GCs where all stars are old (ages $\\gtrsim 10^{10}\\rm \\ yr$ ) and many (if not all) BHs are expected to have been already ejected by now.",
"As an example, our Galactic center contains a large population of young massive stars, many of which reside in a stellar disk.",
"These stars most likely originated in-situ following the fragmentation of a gaseous disk formed from an infalling gaseous clump [26].",
"Such stars formation bursts are thought to occur episodically throughout the evolution of the central cusp.",
"Eclipsing and close binaries are observed among the Galactic center young stars, suggesting star formation as an additional process which can repopulate the binary population (including BH binaries) in the NSC [88].",
"Observational studies of NSCs in external galaxies, including high resolution spectroscopic surveys, have been used to characterize the star formation history and ages of NSCs [105], [98].",
"The common finding emerging from these studies is that most NSCs are characterized by a mixture of morphological components and different stellar populations spanning a wide range of characteristic ages from 10 Myr to 10 Gyr.",
"Observations also suggest that the ages of NSCs and masses depend on the host galaxy Hubble type, with NSCs in early-type spirals being older and more massive than those of late-type spirals [98].",
"More generally the growth of the nuclei might be a continuous and ongoing process occurring during and after most of the host galaxy was formed.",
"As shown in Figure REF and REF , most massive BH mergers ($\\gtrsim 50 M_{\\odot }$ ) occur at early times in the lifetime of a single stellar population star cluster.",
"This is because the most massive objects segregate earlier and are ejected earlier through dynamical interactions.",
"This will not be the case if the clusters form new stellar populations at later times.",
"New episodes of star formations in NSCs will lead to the formation of new BHs with a substantial contribution to the merger rate in the local universe.",
"In-situ star formation could therefor contribute significantly to the detection rate of high mass mergers we previously derived (c.f., Eq.",
"[REF ]), although it will likely not affect the total detection rate of BH mergers in NSCs."
],
[
"Summary",
"Understanding the distribution of BHs at the centers of galaxies is crucial for making predictions about the expected event rate and source properties for high-frequency gravitational wave detectors.",
"Since the distribution of stellar BHs is not known, and Monte Carlo simulations of star clusters are still limited to a few $10^6$ particles, we opted here for a semi-analytical approach which we used in order to make predictions about the properties and rate of BH binary mergers that are dynamically assembled in NSCs.",
"In the future we plan to explore this topic using more accurate, although computationally more demanding, Monte Carlo simulations.",
"The main conclusions of our work are summarized below.",
"1) NSCs produce BH binary mergers at a realistic rate of $\\approx 1.5\\rm \\ Gpc^{-3}\\ yr^{-1}$ .",
"2) BHs in NSCs can experience a number of mergers and grow to masses up to a few hundred solar masses.",
"Although rare, such high-mass BH mergers at low redshift are unique to NSCs, because these are the only clusters with sufficiently high escape velocities such that they can retain a large fraction of their merging BHs.",
"3) Assuming that BHs receive low natal kicks, with an imparted momentum equal to the momentum imparted to neutron stars, then the NSC detection rate of high mass BH mergers similar to GW150914 ($M \\ge 50\\ M_\\odot ,\\ z \\le 0.3$ ) is $0.4-1\\rm \\ Gpc^{-3}\\ yr^{-1}$ .",
"This rate is comparable or larger tan the corresponding merger rate of dynamically formed BH binaries in GCs.",
"4) If BHs receive natal kicks as large as $\\gtrsim 50\\rm \\ km\\ s^{-1}$ then BH binary mergers produced dynamically in NSCs could dominate over the merger rate of similar sources produced either in GCs or through isolated binary evolution.",
"We thank Enrico Barausse, Sourav Chatterjee, Cole Miller, Carl Rodriguez for useful discussions and the anonymous referee for useful suggestions.",
"FA acknowledges support from a CIERA postdoctoral fellowship at Northwestern University.",
"FAR acknowledges support from NSF Grant AST-1312945 and NASA Grant NNX14AP92G, at Northwestern University, and from NSF Grant PHY-1066293 through the Aspen Center for Physics.."
]
] | 1606.04889 |
[
[
"Towards Characterizing International Routing Detours"
],
[
"Abstract There are currently no requirements (technical or otherwise) that BGP paths must be contained within national boundaries.",
"Indeed, some paths experience international detours, i.e., originate in one country, cross international boundaries and return to the same country.",
"In most cases these are sensible traffic engineering or peering decisions at ISPs that serve multiple countries.",
"In some cases such detours may be suspicious.",
"Characterizing international detours is useful to a number of players: (a) network engineers trying to diagnose persistent problems, (b) policy makers aiming at adhering to certain national communication policies, (c) entrepreneurs looking for opportunities to deploy new networks, or (d) privacy-conscious states trying to minimize the amount of internal communication traversing different jurisdictions.",
"In this paper we characterize international detours in the Internet during the month of January 2016.",
"To detect detours we sample BGP RIBs every 8 hours from 461 RouteViews and RIPE RIS peers spanning 30 countries.",
"Then geolocate visible ASes by geolocating each BGP prefix announced by each AS, mapping its presence at IXPs and geolocation infrastructure IPs.",
"Finally, analyze each global BGP RIB entry looking for detours.",
"Our analysis shows more than 5K unique BGP prefixes experienced a detour.",
"A few ASes cause most detours and a small fraction of prefixes were affected the most.",
"We observe about 544K detours.",
"Detours either last for a few days or persist the entire month.",
"Out of all the detours, more than 90% were transient detours that lasted for 72 hours or less.",
"We also show different countries experience different characteristics of detours."
],
[
"Introduction",
"We define an international detour (detour for short) as a BGP path that originates in an AS located in one country, traverses an AS located in a different country and returns to an AS in the original country.",
"Detours have been observed in the Internet, for example, cities located in the African continent communicating via an external exchange point in Europe[11].",
"Many autonomous systems are also multinational, which means that routes traversing the AS may cross international boundaries.",
"There have also been suspicious cases of detours.",
"In November, 2013, the Internet intelligence company Renesys (now owned by Dyn) published an online article detailing an attack they called Targeted Internet Traffic Misdirection [10].",
"Using Traceroute data they discovered three paths that suffered a man-in-the-middle (MITM) attack.",
"One path originated from and was destined to organizations in Denver, CO, after passing through Iceland, prompting concern and uncomfortable discussions with ISP customers.",
"Each of these anecdotes, while interesting in its own right, does not address the broader question about how prevalent such detours are, their dynamics and impact.",
"Characterizing detours is important to several players: (a) as a tool for network engineers trying to diagnose problems; (b) policy makers aiming at adhering to potential national communication policies mandating that all intra-country communication be confined within national boundaries, (c) entrepreneurs looking for opportunities to deploy new infrastructure in sparsely covered geographical areas such as Africa, or (d) privacy-conscious states trying to minimize the amount of internal communication traversing different jurisdictions.",
"Using the methodology developed to detect detours we also present a tool, Netrahttps://github.com/akshah/netra, to monitor the Internet routing system in near real-time and produce alerts.",
"Network operators can not only appear informed about the incident, but also may be able to take action in peer selection in response to the alerts.",
"Finally, longitudinal analysis of detours can give us insight into how routing and network infrastructure evolve over time.",
"In this paper we first develop methodology to detect detours, validate it on live traffic using our tool Netra and then use it to characterize them at a global scale on historical BGP data of January 2016 from RouteViews and RIPE RIS.",
"The rest of this paper is organized as follows.",
"In Section we present related work and highlight previous efforts in direction similar to ours and point out key areas where our work differs from them.",
"In Section we describe our datasets, corresponding usage and reasoning for the choice of our datasets.",
"Section details the methodology used to perform AS geolocation and analysis of our geolocation results.",
"Section explains in detail detour detection process, corresponding terminologies used throughout the paper.",
"In Section we explain our data plane measurements and present validation results.",
"In Section we characterize detours seen in January 2016.",
"First we present aggregate analysis of entire dataset, then classify detours into different categories and finally focus on transient detours in Sections REF , REF and REF respectively.",
"In Sections and we discuss value additions of our work, summarize and present future work respectively."
],
[
"Related Work",
"Detour detection: In November 2013 Renesys reported a few suspicious paths [10].",
"One went from Guadalajara, Mexico to Washington, D.C. via Belarus; another went from Denver, CO through Reykjavik, Iceland, back to Denver.",
"They used mostly data plane information from traceroute for their analysis.",
"In [11] the authors focus on ISP inter-connectivity in the continent of Africa.",
"They searched for paths that leave Africa only to return back.",
"The goal, however, was to investigate large latencies in Africa and ways to reduce it.",
"The premise was that if a route crosses international boundaries it would exhibit high latency.",
"The work pointed to cases where local ISPs are not present at regional IXPs and IXP participants don't peer with each other.",
"Similar to Renesys, they also use traceroute measurements, this time from the BISmark infrastructure (a deployment of home routers with custom firmware) in South Africa.",
"Our study extends beyond Africa and investigates transient in addition to long-lasting detours.",
"In Boomerang [19], the authors again use traceroute to identify routes from Canada to Canada that detour through the US.",
"In this work the motivation was concerns about potential surveillance by the NSA.",
"This work differs from ours in a number of ways: we characterize detours not just for one but 30 countries using control plane information rather than data plane.",
"We use data plane measurement only for validation purposes.",
"Our goal is to not only detect detours but show characteristics about them which previous work does not present.",
"Data plane vs Control plane Incongruities: In [9] authors focus on routing policies and point out cases where routing decisions taken by ASes do not conform to expected behavior.",
"There are complex AS relationships, such as, hybrid or partial transit which impact routing.",
"Such relationships may lead to false positives in our results.",
"However, the paper points out that most violations of expected routing behavior caused by complex AS relationships are very few and most violations were caused by major content providers.",
"Our work identifies detours for variety of ASes, including both large content providers and small institutions.",
"Moreover, in [24] authors argue that such incongruities are caused due to incorrect IP to AS mappings.",
"About 60% of mismatches occur due to IP sharing between adjacent ASes.",
"Authors here show that 63% to 88% of paths observed in control plane are valid in data plane as well.",
"The work in [14] also analyzes the control plane (RIBs and AS paths) to construct a network topology and then uses traceroute to construct country-level paths.",
"The goal of this work was to understand the role of different countries that act as hubs in cross-country Internet paths.",
"Their results show that western countries are important players in country level internet connectivity.",
"Malicious AS detection: In [16] authors present ASwatch, an AS reputation system to detect bulletproof hosting ASes.",
"Similar to our work ASwatch relies on control plane information to detect malicious ASes (that may host botnet C&C servers, phishing sites, etc).",
"The motivation of this work is different than ours.",
"ASwatch attempts to detect malicious ASes by mining their link stability, IP space fragmentation and prefix reachability.",
"ASwatch will not detect ASes that cause detours.",
"The detour origin ASes that our work detects could complement features that ASwatch uses.",
"As authors in [16] point out malicious ASes rewire their routes more frequently than legitimate ones, transient detours might be particularly useful to improve detection capability of ASwatch.",
"Geolocation Accuracy: In context of MaxMind geolocation accuracy, [12] and [20] have shown MaxMind country geolocation to be 99.8% in consensus with other geolocation DBs.",
"In [21] authors use data from Routing Information Registries (RIRs), RIPE DB and Team Cymru to determine all IP blocks and ASes that geolocate to Germany.",
"To validate their geolocation accuracy, authors query the MaxMind database which allows mapping IP addresses to their country of presence.",
"We adopt a more exhaustive strategy than [21].",
"Control-plane-only for detection: One way to detect detours is to use traceroute, analyze reported hops and use latency as an indication of a detour.",
"This approach was followed by [11] that studied peering relationships in Africa; we too use this approach to validate our results on live data.",
"However, we detect detours using only control plane data.",
"This has a number of advantages: 1) Collecting data plane information at an Internet scale is hard.",
"It needs infrastructure and visibility provided by Atlas probes or Ark monitors is limited.",
"Moreover, running too many traceroutes from own network to others might lead to blacklisting.",
"2) Small footprint of our methodology makes it easily reproducible.",
"Any network operator can pull a RIB dump from his/her border router and run Netra to detect detours for prefixes they own."
],
[
"Data Sources",
"We use variety of data sources to perform AS geolocation, BGP RIBS for detour detection and Traceroutes from RIPE Atlas for detour validation.",
"In Table REF we list different datasets with their usage and relevant information about each.",
"Our sampling rate is 3 RIBs per day (one every eight hours, as provided by RIPE RIS) for a total of 38,688 RIBs from 416 peers.",
"This spans 30 countries, which amounts to about 55GB of compressed MRT data.",
"We acknowledge that 30 countries do not necessarily represent global scale, but our scope is limited by placement of peers that provide BGP feeds.",
"We used all v4 peers in our analysis.",
"For geolocation of IP addresses we use MaxMind GeoLite City DB[17].",
"We treat end user IPs and infrastructure IPs differently since MaxMind is known to be more accurate for eye-ball networks only.",
"To gather the list of infrastructure IPs we used list of routers from CAIDA Ark traceroutes[1], OpenIPMap[8], iPlane[5] and RIPE Atlas built-in measurements and the anchoring measurements.",
"The built-in measurements use all the RIPE Atlas probes and the destinations are root servers.",
"The anchoring measurements are from 400 Atlas probes to other 189 Atlas anchors.",
"These infrastructure IPs are then mapped to AS using IP to AS mappings from CAIDA ITDK[4], iPlane or longest prefix match.",
"In addition to BGP sources, we use AS-to-IXP mapping to estimate presence of an AS in a country.",
"We gather AS to IXP mappings from Packet Clearing House (PCH)[6], PeeringDB[7] and by crawling 368 IXP websites that make their participant list public.",
"Finally, we use CAIDA AS Relationship datasets[3] to eliminate false positives from detours detected.",
"In Section we provide more details on how these datasets are used in AS geolocation along with a flowchart (Figure REF )."
],
[
"AS Geolocation",
"To detect detours we are interested in country level geolocation.",
"We define AS geolocation as presence of an AS in a country.",
"An AS can have presence in multiple countries, especially ASes that belong to large providers.",
"We detect the presence of an AS in country A if it : Announces a prefix that geolocates to A or Has infrastructure IPs that geolocate to A or Has a presence at an IXP in A.",
"In Figure REF we show a flowchart detailing AS geolocation processes.",
"There are 3 main steps as described above.",
"In next sections we elaborate on each.",
"Figure: Flowchart: AS-to-Country mapping creation"
],
[
"Prefix Geolocation",
"We begin by geolocating all advertised BGP prefixes by an AS.",
"It is possible that during our analysis in January 2016 some AS erroneously announced prefixes that it did not own.",
"Therefore we perform a simple filtering; we trust an AS to be owner of a prefix if it announced the prefix for at least 15 days in our dataset.",
"We assume most mistakes or hijacks will be less than this duration.",
"Next, to map a BGP prefix to a country we geolocate each IP in the prefix using MaxMind-free.",
"MaxMind could not geolocate 3.8M IPs.",
"We could successfully geolocate 1.48M of these IPs with MaxMind-paid, we could not use remaining 2.32M IPs.",
"Now we use the union of IP geolocation sets to get the BGP prefix geolocation.",
"Due to the 2.32M IPs not geolocating even with paid version of MaxMind, 614 BGP prefixes could not be geolocated.",
"For the remaining 610,722 BGP prefixesWe use BGP prefixes `as is' from the RIBs and do not perform any prefix aggregation.",
"For example, if both /8 and /9 blocks of a prefix were seen in RIBs of the same or different peers, they are treated as 2 separate prefixes.",
"which were geolocated we observe that more than 99% geolocated to single country.",
"We note that 328,398 BGP prefixes were /24s.",
"When BGP prefixes map to more than one country, the average size of the set was 2.9 countries.",
"Finally, we perform union of geolocation sets of all BGP prefixes that an AS announces to create 1st AS to country set."
],
[
"Infrastructure IP Geolocation",
"As mentioned previously, we treat infrastructure IP addresses separately.",
"Router geolocation is known to be inaccurate [15].",
"Therefore for these IPs we want to create country geolocation set as large as possible.",
"We populate list of router IPs from CAIDA Ark Traceroutes, iPlane IP to PoP mappings, OpenIPMap and RIPE built in measurements.",
"Our list included 3M router IPs.",
"This is the `Read Infrastructure IPs' step shown in flowchart Figure REF .",
"To geolocate each router IP we look at country location provided by iPlane, OpenIPMapOpenIPMap is crowdsourced and may not be very accurate.",
"We use cases where confidence level for router geolocation is higher than 90%.",
"and Maxmind-paid and perform a union to give a set of countries.",
"Next step is to map these routers to ASes.",
"IP to AS is a challenging problem and active area of research.",
"We use the best datasets available to create these mappings.",
"Both CAIDA ITDK and iPlane datasets provide IP to AS mappings using the methodology described in [13].",
"For cases where either of these datasets fail to provide IP to AS mapping, we perform longest prefix match on the global routing table and map the IP to the AS announcing the longest matching prefix.",
"Lastly, we combine IP to Country and IP to AS mappings to give 2nd AS to country set."
],
[
"IXP Presence of an AS",
"We extract presence of ASes at different IXPs and add the geolocation of IXP to the AS geolocation.",
"As shown in `Read IXP data' step in Figure REF , we use 3 sources of AS to IXP mappings.",
"First, we crawl 368 IXP websites and extract their corresponding participants.",
"Next, we use PeeringDB 2.0 API [7] and lastly, we use dataset from Packet Clearing House (PCH) that lists participants at IXPs that PCH is also a part of.",
"We then combine geolocation obtained from these IXP sources to obtain 3rd AS to country set.",
"We acknowledge that IXP mappings from websites, PCH and PeeringDB might not be updated regularly and hence lead to mapping of an AS to a country that it does not have a presence in.",
"Note that this will lead to false negative (not false positives) in detour detection, a trade-off we make to error on safe side."
],
[
"AS to Country Set",
"Finally, we map an AS to a set of countries by taking a union of all the 3 steps above.",
"This is the merge step in Figure REF .",
"The distribution of AS geolocation is shown in Figure REF .",
"Perhaps surprisingly, only about 11.6% ASes out of a total of 52,984 geolocated to multiple countries.",
"We believe that this is the result of a practice where most organizations use a different AS number in different countries.",
"If an AS does geolocate to multiple countries we use the set of all countries in our analysis.",
"We could not geolocate 24 ASes because none of their BGP prefixes could be geolocated, no infrastructure IP from our set mapped to it nor did we find its IXP presence in public datasets.",
"These ASes on an average announced only 2 to 3 BGP prefixes.",
"Figure: CDF: Number of countries in AS geolocationTable: Comparison of CAIDA's AS Rank with number of countries in AS GeolocationComparison with CAIDA's AS Rank: Although our end goal is to detect detours, these geolocation results provide interesting insights.",
"To understand more about which ASes geolocate to more than one country we use CAIDA's AS Rank [2].",
"This dataset gives higher ranks to ASes that have large customer cones.",
"Intuitively, ASes with higher rank should resolve to many countries due to their wider presence.",
"Table REF shows ASes with their CAIDA AS rank and corresponding number of countries the AS geolocated to for top 3 and bottom 3 in the first 1000 ranked ASes.",
"As expected, we see that ASes which have large presence with many customers across the world geolocate to large number of countries and low rank ASes with smaller customer cones geolocate to fewer countries."
],
[
"Detour Detection",
"We define a path as having a detour if the origin and destination is country `A' but the path unambiguously includes some other country `B'.",
"Note that this approach examines paths where the prefix origin AS and the AS where the peer is located are in the same country.",
"To analyze the AS path, we provide the following definitions: Prefix Origin: The AS that announces the BGP prefix.",
"Detour Origin AS: The AS that starts a detour in country `A' and diverts the path to foreign country `B'.",
"Detour Origin Country: The country where we approximate location of Detour Origin AS, country `A'.",
"Detour Destination AS: The AS in foreign country `B'.",
"Detour Return AS: The AS where detour returns back in country `A'.",
"Figure: Example showing direction of BGP announcement and direction of observed detourFigure REF illustrates detours.",
"AS0 announces prefix a.b.c.0/24 to AS1, AS2 and AS3.",
"AS1 geolocates to JP whereas AS3, AS2 and AS0 are in the US.",
"In this case, data traversing from AS3 to AS0 will contain a detour from AS2 (Detour Origin) to AS1 (Detour Destination).",
"We do not include sub-paths in our analysis; other portions of the path that may experience a detour.",
"For example, in path AS1{US}-AS2{IN}-AS3{CN}-AS4{IN}-AS5{US}, we only count the detour US-IN-US.",
"We do not count the detour IN-CN-IN.",
"There are some cases where we need to approximate detour origin and country.",
"In a path such as AS1{US}-AS2{US,BR}-AS3{CN}-AS4{US}.",
"We resolve the uncertainty of the detour origin by assuming that it starts in AS2, since there is a likely path to AS2 from AS1 through the US and AS2 starts the detour from US, not BR.",
"We do not characterize possible detours.",
"For example, a path that geolocates to {US}-{US,IN}-{US} may in fact stay within the US and never visit India.",
"In this work we only focus on paths that contain definite detours, such as {US}-{IN}-{US} or {US}-{IN,CN}-{US}.",
"Again, we re-emphasize that in this work we only look at paths that confidently start and end in the same country; paths like {US,BR}-{IN}-{US} or {US}-{IN}-{BR} are not considered.",
"We discard paths where we see an AS whose geolocation is unknown and a detour is not certain.",
"For example, paths like AS1{US}-AS2{}-AS3{US} are discarded.",
"However, if we see the detour occurring before the AS that could not be geolocated we do count it as a valid detour i.e., in AS1{US}-AS2{BR}-AS3{US}-AS4{}-AS5{US}, AS4 does not have geolocation information but the US-BR-US detour occurred earlier.",
"We treat this path as definite detour.",
"We note that in addition to geolocation accuracy there is also some ambiguity about exact country boundaries.",
"Some territories and relationships are currently disputed between multiple authorities and no worldwide consensus exists.",
"For example, Hong-Kong and the People's Republic of China could be considered one or two entities.",
"Hong-Kong is affiliated with China but it is a charter city and has its own independent constitution and judiciary system.",
"For our analysis, we left the resolution of boundaries and countries to the MaxMind database.",
"With this particular example, Hong-Kong and China are treated as two separate entities.",
"MaxMind follows ISO 3166 country codes.",
"In some cases the geolocation from MaxMind is ambiguous: `A1:Anonymous Proxy', `A2: Satellite Provider', `O1: Other Country', `EU: Europe', `AP: Asia/Pacific'.",
"We discard detours caused by these ambiguous codes, such as {DE}-{EU}-{DE}.",
"Filtering peered AS paths: It is possible that the detour origin and the detour return ASes have a peering relationship and in reality traffic was not detoured at all.",
"This, however, is hard to determine with certainty since peering relations and policies are not public.",
"What we can do is provide an upper bound on how many detours may be eliminated due to peering.",
"To detect such cases we use CAIDA's AS relationship dataset [3].",
"This dataset provides information of provider to provider (p2p) and provider to customer (p2c) relationship between ASes.",
"We count cases where p2p link might be used, i.e., data originates from the peer itself or from a downstream customer.",
"In case of p2c link we assume this link is always chosen.",
"We eliminate such paths from our analysis and revisit this issue in the next section summarizing the peering relationships in Table REF .",
"Multi-Origin Prefixes: Some prefixes are announced by more than one ASes.",
"We do not eliminate such cases.",
"So, if a prefix a.b.c.0/24 is seen in RIBs of 2 peers with AS paths `X Y Z' and `P Q R' then we treat each path as independent and detect detour if it fits above mentioned criteria of starting and ending in the same country.",
"In our geolocation dataset we observed 7,579 prefixes of multi-origin (7,247 originated from 2 ASes).",
"Out of these 6,104 suffered a detour.",
"Motivation to not eliminate these prefixes is as follows: Network operators of such prefixes might want to re-evaluate their decisions especially if the ASes originating the prefix are in different countries.",
"This might be a cause of high latency."
],
[
"Detour Validation",
"In this section we validate detours in near real time using traceroutes from RIPE Atlas probes.",
"Our validation comprises of four steps: Run Netra with live BGP feeds from 416 peers to detect detours.",
"When a detour is detected, run corresponding traceroutes (from same country and same AS) using RIPE Atlas.",
"Check if the traceroute and detour see similar AS path.",
"Validate using traceroute IP hops and RTT.",
"Figure: Data plane measurements: Example showing selection of RIPE Atlas probes and target IPs"
],
[
"Data-plane Measurements",
"We ran Netra from May 2nd 2016 noon to midnight (using BGP feeds from 416 peers).",
"When a detour was detected in control plane we selected RIPE Atlas probes in the same country and same AS which we detected detour from and ran traceroute (ICMP Paris-traceroute[18]) to IP addresses in the detoured prefix.",
"The methodology to run data plane measurements is shown in Figure REF .",
"There are a few cases where more than two Atlas probes are present in selected AS; in this case we selected 2 probes that are geographically farthest from each other.",
"By doing this we aimed to account for cases where routes seen from geographically distant vantage points within the same AS are different.",
"To select target IPs from detoured prefix, we break the prefix into its constituent /24s and randomly select an IP from each /24.",
"For example, in a /23 prefix we select 2 IPs belonging to different /24s.",
"By doing this we account for cases where a large prefix, even though in the same country, has different connectivity via different upstream provider.",
"During this live run we detected 6,175 detours.",
"Out of these 5,787 were unique detours ({peer,prefix,aspath} tuple)."
],
[
"Selecting Congruent Paths",
"Only 72 peers saw the 6,175 detours and the 72 peers belong to only 63 ASes.",
"From these 63 ASes we then select ASes that also have active RIPE Atlas probes; there were only 10 ASes that both saw a detour and host a RIPE Atlas probe.",
"169 detours were seen from these 10 ASes corresponding to 6 countries: {Brazil, Italy, Norway, Russia, United States, South Africa}.",
"From the 169 traceroutes we initiated to detoured prefixes, we discard 6 traceroutes where less than 3 hops responded since drawing detour conclusion from these is not possible.",
"Finally, we are left with 163 traceroutes that can be used for validation.",
"We acknowledge that 163 is not a very large number for validation purposes.",
"However, running Netra for more hours does not necessarily increase the number of usable traceroutes for validation by a lot, we are limited by the number of ASes that have RIPE Atlas probes which also see a detour and detour-origin and detour-destination have no peering.",
"In total we detected 85 prefixes (corresponding to the 163 traceroutes) that suffered a detour that was visible from an AS which has RIPE Atlas probes.",
"Note that some detoured prefixes were larger than /24, so we traceroute multiple IPs within it as explained in Section REF .",
"The validation methodology is stated in Algorithm REF .",
"As previous work [22] has pointed out, we found many cases where AS path seen in control plane and AS path seen in data plane do not match.",
"However, these paths can still show detour if the detour origin AS and the detour destination AS are still present in the traceroute observed AS path.",
"We call such AS paths congruent.",
"More specifically, we consider the detoured AS path congruent only if detour origin AS and detour return AS both are present in the traceroute-observed AS path in the same order (detour origin first).",
"For example, if an AS path `A B C D E' in control plane changed to `A X B C E' in data plane where `B' was detour origin and `C' was detour destination, we consider it as a congruent path.",
"To resolve traceroute path to AS path we used CAIDA ITDK and iPlane IP to AS mappings and in cases where no match was found we use longest prefix match on the global routing table for the hop IP.",
"Then we map the longest prefix match to the AS that originated it.",
"Out of all the IPs we saw in 163 traceroutes, only 44 could be mapped to an AS using the IP to AS datasets.",
"All other IPs were mapped using longest prefix match.",
"We observed 113 congruent AS paths.",
"This includes 3 cases, insertions, deletions and mix of both.",
"We borrow nomenclature of these paths from [22].",
"We saw 73 deletions, 29 insertions, 4 mix of insertion and deletions.",
"The remaining 7 AS paths were exact matches.",
"Note that these insertions and deletions occurred only for ASes that were not involved in the detour.",
"Netra Validation [1] validateASPath $\\textit {aspath}\\leftarrow \\textit {AS Path from Traceroute}$ $\\textit {doas}\\leftarrow \\textit {Detour Origin AS from \\texttt {Netra}}$ $\\textit {ddas}\\leftarrow \\textit {Detour Destination AS from \\texttt {Netra}}$ doas,ddas in aspath doas before ddas in aspath Return True [1] validateIPHops $\\textit {ipHops}\\leftarrow \\textit {IP hops from Traceroute}$ $\\textit {ipHopCountries}\\leftarrow \\textit {MaxMind-paid}$ ipHopCountries show detour $\\textit {detourDestTR}\\leftarrow \\textit {Dest.",
"from traceroute}$ $\\textit {detourDestNetra}\\leftarrow \\textit {Dest.",
"from \\texttt {Netra}}$ detourDestTR in detourDestNetra Return True [1] validateRTTs $\\textit {hopRTTs}\\leftarrow \\textit {RTTs from Traceroute}$ hopRTTs show magnitudeJump Return True [1] main loop: Each Detected Detour validateASPath validateIPHops validateRTTs"
],
[
"Validation",
"Now we validate detours detected by our methodology by comparing it with detours seen in data plane.",
"For the 113 congruent AS paths, we evaluate if a data plane detour was seen.",
"We chose to perform two tests.",
"First, we resolve IPs observed in the hops of traceroute to country level geolocation using Maxmind-paid.",
"We detect data plane detour if a path traversed foreign country and returned.",
"We make sure that country visited (detour destination country) in data plane is present in the set of destination countries expected for this particular detour by Netra.",
"We do this filtering to avoid false positives like: Netra detected detour {US}–{GB,DE}–{US} and traceroute detected detour {US}–{IT}–{US}.",
"Although still a detour, since it was not accurately captured we count it as a miss.",
"However, no such case was found.",
"Second, we validate using RTT measurements.",
"We detect RTT based detour if a hop in the traceroute showed increase in RTT by an order of magnitude (at least 10 times increase).",
"The results of this analysis are shown in Figure REF .",
"We observed accuracy of about 85% (97 out of 113) in country-wise method and 90% (102 out of 113) by RTT measurements.",
"The overlap between these two different tests was also large.",
"88 detours were detected in both (77.8%).",
"We investigate further the 9 detours that were seen in country-wise method but not in RTT.",
"These detours covered small geographic area; 4 from Italy to France, 2 Norway to Sweden, 2 from Brazil to US and 1 from Russia to Sweden.",
"RTTs between these countries have been previously reported to be low.",
"Next we investigate 14 cases which were captured in RTT measurements but not in country-wise method.",
"All of these do cross international boundaries.",
"For 12 of these cases, due to large number of traceroute hops (especially towards the end of the traceroute) not responding we don't see the route returning to the origin country, hence not detected by country-wise method.",
"We attribute remaining 2 cases as false positives due to inaccurate AS geolocation.",
"Figure: Validation Results: Live traceroutes using RIPE AtlasIn Figure REF we provide a visualization of the most common detour we observed from Russia.",
"Only visualization is done using OpenIPMap.",
"Figure: Top Detour on May 2nd 2016: Detected using Netra, visualization using OpenIPMap.",
"Dotted arrow represents multiple hops and solid arrow represents direct hop.Validation Discussion: We show that large percentage of detours seen in control plane are accurately reflected in data plane as well.",
"The main challenge is AS paths in both data plane and control plane don't agree in about 30% cases.",
"We note that this could be an artifact of Atlas probes connected differently than the peers which provide BGP feeds.",
"It is, however, possible to learn common AS insertions and deletions over a period of time and evolve detection capabilities."
],
[
"Results",
"In this section we quantify detours detected in January 2016.",
"First, in Section REF we present an overview of all the detours detected in our dataset.",
"In Section REF we define metrics and classify detours based on their stability and availability.",
"In Section REF we focus on transient detours."
],
[
"Aggregate Results",
"We begin by characterizing aggregate results, namely all detours seen by all peers; in other words, we count an incident every time an AS path appears in a RIB of any peer that contains a detour.",
"Many of these incidents are duplicates.",
"Therefore in addition to the total we also present the number of unique detours.",
"As expected, we observe that detours are not generally common.",
"Also, not all peers see a detour.",
"Only 79 peers, out of 416, saw one or more detours.",
"Table REF details the number of detours seen.",
"We analyzed about 14 billion RIB entries and about 544K entries showed a detour; out of theses only 18.9K were unique (most detours re-appear during the month).",
"Figure REF shows the number of detours for each day in January 2016.",
"On an average we find about 17.5K detoured entries per day.",
"Figure: Total number of definite detours per day in January 2016Table: Aggregate number of detours detectedTable: Routes that may have peering relationsNext we examine the visibility of detours, where we observe an uneven distribution among ASes.",
"Just 9 ASes originate more than 50% of the detours.",
"Similarly, some prefixes experience detours more than others.",
"132 prefixes experienced more than 50% of the total detours.",
"Looking at the average length of a detour, we see that a detour visits 1 to 2 foreign ASes before returning to its origin country.",
"Impact of Peering: We now estimate the effect of peering links on detours.",
"Specifically, we are interested in cases where a peering relationship exists between the Detour Origin AS and the Detour Return AS as described in Section using CAIDA AS relationship dataset.",
"If such a link exists, it is possible that traffic traverses that link instead of the detour.",
"Table REF shows the number of detours between ASes that also have peering relations compared to total number of detours without filtering peered paths.",
"We find that 17.4% of the detours are avoided due to peering relations.",
"We do not count these as detours in our analysis.",
"Top Detour Origins and Prefixes: To understand more about the nature of these detours, we focus on the origin and destination ASes.",
"In Table REF we show the common detour origins and country where the AS was approximated to origin the detour from.",
"Next is the percentage of detours out of the total that started from given origin.",
"Following the percentage, is the most frequent destination that was visited from the origin, and lastly is the percentage of detours that went to most common destination from the said origin.",
"We observe that most commonly these were access provider ASes.",
"Similarly, in Table REF we show top impacted prefixes.",
"Table: Top Detour Origin ASNs for all detoured pathsTable: Top Detoured prefixes and corresponding percentagesCountry-wise analysis: To provide an understanding on number of detours per peer in each country we normalize the data by dividing the number of detours by number of peers in the country.",
"The reason to normalize data is simple, RouteViews and RIPE RIS peers are not evenly distributed among different countries.",
"Therefore it is possible that more detours are seen in countries that have more peers due to more visibility.",
"An average number of detours per peer per country provides better insight.",
"Out of 30 countries, only 12 countries observed a detour.",
"Figure REF shows average number of detours per country.",
"Russia showed most number of average detours.",
"Understanding the total number of detours in different countries is important but it does not reflect if detours seen in different countries have different characteristics.",
"In the next section we focus on characterizing these detours.",
"Figure: Average number of detours per country"
],
[
"Characterizing Detours",
"To characterize detours we define two metrics: Detour Dynamics Flap Rate: Measure of stability of a detour; how many times a detour disappeared and reappeared.",
"Duty Cycle: Measure of uptime of a detour throughout the month measurement period.",
"Persistence: Total number of continuous hours a prefix was seen detoured.",
"Before using the above metrics to characterize the detours, we perform data pruning to avoid skewing of data towards ASes that have more peers that provide BGP feeds to RouteViews and RIPE RIS.",
"Also, ASes with multiple peers and similar views can contribute duplicate detours to our dataset.",
"We follow a simple approach to deal with this problem: if an AS contains more than one peer we select the peer that saw the most detours as the representative of that AS.",
"This may potentially undercount detours since some peers in same AS may see different detours.",
"After selecting a representative we are left with 36 (out of 79) peers.",
"We now continue our characterization of detours by looking at detour dynamics.",
"Specifically we focus on flap rate and duty cycle, defined as follows: $\\hspace{22.76228pt}FlapRate=\\frac{Total Transitions}{TotalTime}\\times 100$ $\\hspace{7.11317pt}DutyCycle=\\frac{Total Uptime}{TotalTime}\\times 100$ These metrics provide insights into the life cycle of detours by measuring route uptime and stability.",
"BGP route flapping is a known problem and has been studied in [23] by looking at BGP updates and RFC 2439 provides methods to dampen these.",
"However, in context of this paper duty cycle and flap rate are calculated from the RIBs.",
"We extract detours from the RIBs and evaluate when they disappear and reappear.",
"To understand if country where detours occur plays a role in detour dynamics, next we drill into country specific detours.",
"Figure REF shows a scatter plot of flap rate vs. duty cycle for various detours in US, Brazil and Russia.",
"We selected these three countries because they show the most detours in our dataset; they account for 93% of detours.",
"We see a triangular pattern with some outliers.",
"Large number of detours show high duty cycle and low flap rate.",
"We divide each figure into 4 quadrants based on average flap rate and average duty cycle of all detours.",
"We name quadrants anti-clockwise starting from top right.",
"US detoured paths appear more stable (lower flap rate and higher duty cycle) in $II^{nd}$ quadrant.",
"On the other hand, Russian and Brazilian detoured paths fall mostly in the $I^{st}$ , $III^{rd}$ and $IV^{th}$ quadrant.",
"Russian detours in general showed lower duty cycle than US and Brazil.",
"We also present a similar scatter plot for all the non US, BR and RU detours in Figure REF .",
"In this case we observed detours mostly in extreme ends on $II^{nd}$ and $III^{rd}$ quadrant indicating two categories of detours, either long lasting or very rare events.",
"A network operator can use information like this and decide which quadrant detours are more interesting to focus on.",
"While all of detours may need attention, we believe detours with low duty cycle and low flap rate may need immediate attention.",
"We talk more about this in Section REF .",
"Figure: Flap Rate vs DC for US, RU and BR prefixesFigure: Flap Rate vs DC for Non US, RU and BR prefixesNext, we examine the persistence of detours.",
"Figure REF shows the number of consecutive days a detour was visible by any peer.",
"Note that persistence is measured in number of consecutive hours hence captures different characteristics than duty cycle which measures uptime throughout the dataset.",
"We see a U-shaped pattern in Figure REF , meaning that many detours are either short lived (one day) or they persist for entire month.",
"We take a different view at persistence in Figure REF by plotting CDF of duration in hours.",
"We see that most detours are short-lived, with about 92% lasting less than 72 hours, defined as transient detours.",
"Finally, we examine a specific case of a transient detour, namely flash detours which appeared only once and never appeared again during the month.",
"Figure: Persistence of definite detoured paths as seen by all peersFigure: Distribution of detour durationIn the following section we focus on transient and flash detours.",
"Due to space limitations we do not characterize persistent detours further.",
"We do note, however, that characterizing persistent detours is important for at least some of the reasons we enumerated earlier.",
"We chose to focus on transient detours as they shed light on misconfigurations or even malicious activities, both aspects of routing we understand less."
],
[
"Transient and Flash Detours",
"We first present an understanding of the transient detours on per-country basis.",
"Since there are more than one peers in some countries and different peers see varying number of transient detours, we calculate an average number of transient detours per country by dividing total number of transient detours in a country by number of peers in the given country.",
"This average value per country is presented in Figure REF .",
"We detected transient detours in only 8 countries where Russia topped the list.",
"In comparison to Figure REF Italy and India showed more average number of transient detours than US.",
"Figures REF and REF show a distribution of ASes that initiate detours and prefixes affected by detours.",
"We observe that 4 ASes originate 50% of the transient detours and only 30 prefixes account for 50% of the transient detours.",
"Similar to Table REF , shown in Table REF are the most common transient detour origins and Table REF shows top impacted prefixes by transient detours.",
"AS9002, RETN-AS, started the most number of transient detours in our dataset.",
"We note that in ASWatch [16] authors gathered ground-truth data from security blogs which enlisted AS9002 as a malicious AS.",
"Another previously know malicious AS that appeared in our findings was AS49934 as a detour destination for 7 Russian prefixes.",
"AS49934 is currently unassigned.",
"It was assigned in Ukraine between 2009-10-14 and 2016-01-03 and was known to announce bogus prefixes and host bots.",
"Figure: Average number of transient detours per countryTable: Prefixes affected the most by transient detoured BGP pathsFigure: Distribution of ASes that originated a transient detour.",
"The top 4 Detour Origin ASes account for 50% of all transient detoursFigure: Distribution of prefixes that experienced a transient detour.",
"About 30 prefixes account for 50% of all transient detoursFinally, we look at flash detours.",
"These are detours that appeared only once and were observed in only one RIB of a peer.",
"Flash detours account for 26% of the transient detours, 328 prefixes (6% of all prefixes that suffered detour) experienced at least one flash detour.",
"Owners of the prefix which suffered flash detours might be interested to know such findings.",
"While 328 prefixes suffered flash detours in our dataset, due to space limitation we point out a few interesting ones in Table REF .",
"Table: Some prefixes affected by flash detoursThe list in Table REF raises serious concerns.",
"Data from government agencies, banks, insurance companies can easily be subject to wiretapping once it leaves national boundaries.",
"Based on our control-plane only data, it is not possible to verify if these institutions were attacked or not.",
"Nevertheless, we believe our findings will motivate network operators to look more closely into why their prefix detoured and if they intended it to happen."
],
[
"Discussion",
"In this paper we present a first attempt to characterize detours in the Internet.",
"We sampled BGP routing tables from 416 peers around the world over the entire month of January 2016 to investigate international detours.",
"We see about 18.9K distinct entries in RIBs that show a detour.",
"More than 90% of the detours last less than 72 hours.",
"We also discover that a few ASes cause most of the detours and detours affect a small fraction of prefixes.",
"Some detours appear only once.",
"Our work is the first to present different types of detours, namely, persistent and transient.",
"We also present novel insights on their characteristics such as detour dynamics in different countries, top impacted prefixes and detour origins.",
"Characterizing detours in the Internet is very useful.",
"Customers gain more insight into how their providers route traffic.",
"There is perhaps an expectation from users that if they send traffic to other users in the same country the packets will not step outside national borders; our work provides evidence to the contrary.",
"Network operators can use our methodology and results for diagnostic purposes.",
"A sudden change in RTT may be traced to a detour, or keeping track of what the routing system does.",
"The latter is important to assure customers that their traffic is not subject to monitoring by other governments.",
"Our work is useful to regulators and state officials responsible for network infrastructure, since our work quantifies information about a practice that may run afoul of state policy.",
"State officials can use such information to assure citizens that their traffic stays within national borders or direct ISPs to alter their practices.",
"State agencies that transmit sensitive information may monitor detours to alert for potential MITM attacks.",
"For example, we did observe cases where prefixes belonging to US Washington state government were detoured through Japan; and for some detours from Russia malicious AS49934 appeared as detour destination.",
"Finally, entrepreneurs may use our results when deciding where to establish new Internet exchange points (IXP) or deploy infrastructure in developing countries.",
"Dataset Contributions: We make the geolocation and detours detection data available to the community via a public RESTful API interface.",
"The motivation to do so is as follows.",
"1) Network operators can easily query our database and check if their prefix suffered a detour.",
"2) Internet measurement researchers can use this information to study various BGP anomalies such as route leaks, detecting malicious ASes, etc.",
"Our results on AS and prefix geolocation are available at http://geoinfo.bgpmon.io and detours results can be accessed at http://detours.bgpmon.io."
],
[
"Conclusions and Future Work",
"There is an increasing need, fueled by new national regulations in Europe and Australia, for ISPs to ensure that personal information belonging to their users does not leave the country.",
"It is unclear whether such regulations cover data in transit as well as storage, but data can certainly be sniffed while in transit, violating the original intent.",
"Such regulations may place a substantial burden on ISPs to prove that such data remains within a country for its entire lifetime, even when it moves.",
"It is still far from clear what the implications are on ISP operations.",
"Currently we do not have the tools to monitor data in transit and state with confidence that data has not left a country, even briefly.",
"Our work does not solve this problem.",
"Rather, it lays the ground for an important conversation about the challenges new regulatory frameworks will pose to researchers, industry and network operators.",
"Our work investigates only a small part of the problem, namely finding the subset of paths where we can detect international detours with some confidence.",
"Our work provides some answers, but also brings attention to the problem and will hopefully stimulate more work in this new direction.",
"The gauntlet was thrown and we expect a lot more research in this area.",
"Within its scope, we believe our work was executed carefully by taking into account measurements from both control and data planes.",
"We show that for the cases were able to study there is agreement between the two planes.",
"This is a significant result.",
"Equally significant, our work has also illuminated the difficulties in expanding the scope within the existing measurement infrastructures.",
"One of the main difficulties we encountered for example, is finding measurement points with both control (BGP peers) and data (RIPE probes) monitors to correlate results.",
"This problem cannot be easily solved, it would take substantial effort to scale the existing infrastructures by an order of magnitude or more.",
"Another important obstacle is lack of knowledge about peering relationships between ASes.",
"This is also a hard problem to solve, since such relationships are not readily disclosed.",
"It is interesting, however, to contemplate the issue if regulatory requirements require such disclosures.",
"Based on our results, we believe that it will be hard to solve this problem without substantial data plane monitor deployment to corroborate control plane measurements.",
"ISPs and IXPs may be required to install sophisticated data plane probe infrastructures and geolocation databases may have to become far more accurate for infrastructure IP addresses in order to detect international detours with some certainty.",
"Control plane monitoring is still very important as it provides efficient global monitoring and can immediately flag potential anomalies where data plane monitoring should be directed.",
"Our work shows that it is effective and should be expanded.",
"In the future we plan to continue to build a system that detects international detours in real time.",
"It is very apparent that we need to include both control and data plane measurements and study algorithms that take input from both.",
"Our first goal is to provide ISPs with a tool to alert when a detour has taken place, followed by information about it (origin and destination AS, duration, source and amount of data in the ISP that followed the detour).",
"We also plan to study emerging regulatory requirements and provide feedback about the challenges they pose.",
"Acknowledgement We would like to thank Randy Bush (IIJ) for helpful discussion on AS geolocation; Emile Aben (RIPE NCC) for helpful insights on the paper and using OpenIPMap; Alessandro Improta and Luca Sani (Isolario Project, Italy) for providing code to crawl IXP websites; Roya Ensafi (Princeton) for help in geolocating infrastructure IP addresses."
]
] | 1606.05047 |
[
[
"The Nature and Frequency of the Gas Outbursts in Comet\n 67P/Churyumov-Gerasimenko observed by the Alice Far-ultraviolet Spectrograph\n on Rosetta"
],
[
"Abstract Alice is a far-ultraviolet imaging spectrograph onboard Rosetta that, amongst multiple objectives, is designed to observe emissions from various atomic and molecular species from within the coma of comet 67P/Churyumov-Gerasimenko.",
"The initial observations, made following orbit insertion in August 2014, showed emissions of atomic hydrogen and oxygen spatially localized close to the nucleus and attributed to photoelectron impact dissociation of H2O vapor.",
"Weaker emissions from atomic carbon were subsequently detected and also attributed to electron impact dissociation, of CO2, the relative H I and C I line intensities reflecting the variation of CO2 to H2O column abundance along the line-of-sight through the coma.",
"Beginning in mid-April 2015, Alice sporadically observed a number of outbursts above the sunward limb characterized by sudden increases in the atomic emissions, particularly the semi-forbidden O I 1356 multiplet, over a period of 10-30 minutes, without a corresponding enhancement in long wavelength solar reflected light characteristic of dust production.",
"A large increase in the brightness ratio O I 1356/O I 1304 suggests O2 as the principal source of the additional gas.",
"These outbursts do not correlate with any of the visible images of outbursts taken with either OSIRIS or the navigation camera.",
"Beginning in June 2015 the nature of the Alice spectrum changed considerably with CO Fourth Positive band emission observed continuously, varying with pointing but otherwise fairly constant in time.",
"However, CO does not appear to be a major driver of any of the observed outbursts."
],
[
"INTRODUCTION",
"We have previously [6] described the initial observations of the near-nucleus coma of comet 67P/Churyumov-Gerasimenko made by the Alice far-ultraviolet imaging spectrograph onboard Rosetta made in the first few months following orbit insertion in August 2014.",
"These observations of the sunward limb, made from distances between 10 and 30 km from the comet's nucleus, showed emissions of atomic hydrogen, oxygen, and carbon, spatially localized close to the nucleus and attributed to photoelectron impact dissociation of H$_{2}$ O and CO$_{2}$ vapor.",
"This interpretation is supported by measurements of suprathermal electrons by the Ion and Electron Sensor (IES) instrument on Rosetta [5].",
"Beginning in February 2015, as the activity of the comet increased, the orbit of Rosetta was adjusted to increasing distance from the nucleus due to concerns for spacecraft safety.",
"At distances $\\ge $ 50 km, when pointed towards the nadir, the spatial extent along the Alice 5.5 long slit [25] allows the coma, both sunward and anti-sunward, to be resolved from the nucleus and observed nearly continuously.",
"This geometry allowed Alice, beginning in mid-April 2015, to observe a number of outbursts above the sunward limb.",
"These outbursts are characterized by sudden increases in the atomic emissions, particularly the semi-forbidden O1 $\\lambda $ 1356 multiplet, over a period of 10-30 minutes, without a corresponding enhancement in long wavelength solar reflected light characteristic of dust production.",
"The corresponding increase in the brightness ratio O1 $\\lambda $ 1356/O1 $\\lambda $ 1304 suggests that O$_{2}$, detected for the first time in a comet by [3], is the primary source of the additional gas.",
"This is the same spectroscopic diagnostic used to determine that O$_{2}$ is the dominant species in the exospheres of Europa and Ganymede [11], [10].",
"As the comet rotates the Alice slit samples different regions above the comet's limb, but the magnitude of the increase on the short time scale as well as the spatial distribution along the slit makes it very unlikely that it is due to a spatial gradient or a collimated “jet”.",
"Although C1 $\\lambda $ 1657 is also seen to increase, the variation in the brightness ratio O1 $\\lambda $ 1356/C1 $\\lambda $ 1657 indicates that it cannot be CO$_{2}$ alone, nor can the effects be due to an increase in photoelectron flux.",
"These outbursts do not correlate with any of the visible images of outbursts taken with either OSIRIS [18], [26] or the navigation camera.",
"We also find that CO does not appear to be driving any of the observed outbursts.",
"Alice is a lightweight, low-power, imaging spectrograph designed for in situ far-ultraviolet imaging spectroscopy of comet 67P in the spectral range 700-2050 Å.",
"The slit is in the shape of a dog bone, 5.5 long, with a width of 0.05 in the central 2.0, while the ends are 0.10 wide, giving a spectral resolution between 8 and 12 Å for extended sources that fill its field-of-view.",
"Each spatial pixel or row along the slit is 0.30 long.",
"Details of the instrument have been given by [25]."
],
[
"Gas Outbursts",
"The strongest of the outbursts, from the period May-July 2015, are listed in Table REF .",
"The table includes the heliocentric distance of the comet, $r_h$ , the distance of Rosetta from the center of the comet, $d$ , the sub-spacecraft longitude and latitude, and the solar phase angle at the time of observation.",
"With the exception of the June 18 and June 23 outbursts, the sub-spacecraft positions of the tabulated outburst are uncorrelated, attesting to their random nature.",
"As an example we will consider in detail the outburst of May 23 since the pointing was constant over two rotations of the comet enabling us to obtain nearly continuous light curves except for gaps in the data due to a lack of observations during spacecraft maintenance activities.",
"Light curves for the strongest coma emissions are shown in Fig.",
"REF .",
"Note that prior to the outburst the relative intensities of H1 Lyman-$\\beta $ , O1 $\\lambda $ 1304, and O1 $\\lambda $ 1356, are consistent with the earlier observations that attributed the emissions primarily to electron dissociative excitation of H$_{2}$ O [6].",
"One rotation period ($\\sim $ 12 h) later both the absolute and relative brightnesses have returned to their pre-outburst values, suggesting that the source of volatile ice driving the outburst is located deeper than the skin depth of the diurnal heat wave.in strong contrast to the persistent diurnal variability seen in mass spectrometer data [12], [20], [7].",
"Figure: Temporal variation of the atomic hydrogen and oxygen emissions above the limb on 2015 May 22–24 (top).",
"Individual error bars are not shown but the 1-σ\\sigma statistical uncertainty for all points is <<5%.",
"The bottom panel shows the variation in relative intensities of these emissions.The orientation of the Alice slit projected onto the comet is shown in Fig.",
"REF (left) in an image from the navigation camera (NAVCAM) taken $\\sim $ 45 minutes after the peak brightness.",
"The Sun is towards the top of the image and the comet's rotation axis is roughly perpendicular to the slit.",
"We attribute the secondary peaks seen in the light curve (top panel of Fig.",
"REF ) to geometric effects of the rotation, as visualized in the three-dimensional coma models of [7].",
"Similar effects are seen in the light curves of the other outbursts listed in Table REF .",
"The spectral image in Fig.",
"REF (right) shows the clearly separated coma emissions together with the reflected solar spectrum from the nucleus in rows 11 to 17 of the slit.",
"Again, the Sun is towards the top of the image.",
"Note that the O1 emissions are seen against the solar reflected radiation from the nucleus and into the anti-sunward coma.",
"lccccccc 0pt 9 Major Gaseous Outbursts Observed by Alice in May - July 2015.",
"Date Peak $r_h$ $d$ 2cSub-spacecraft Phase $B_{max}$ (1356) Time (UT)a (AU) (km) Longitude () Latitude () Angle () (rayleighs)b 2015 May 23 12:42 1.58 143 151.2 –17.1 61.1 223 2015 June 18 03:43 1.42 202 214.8 51.3 89.9 173 2015 June 20 15:26 1.40 181 269.3 19.9 89.9 60 2015 June 23 20:39 1.39 196 215.0 53.3 89.7 113 2015 July 04 08:56 1.34 179 111.6 53.8 89.9 83 2015 July 13 01:16 1.30 155 149.3 15.6 88.8 78 astart time of histogram integration bmaximum O1 $\\lambda $ 1356 brightness above the sunward limb in a 0.3 spatial pixel.",
"Figure: Left: NAVCAM context image obtained 2015 May 23 UT 13:15, shortly after the peak emission was observed.",
"Right: Spectral image beginning UT 12:31, 1589 s exposure, three co-added histograms from the time of peak emission.",
"The sunlit nucleus appears in rows 12–17.",
"The distance to the comet was 143 km, the heliocentric distance was 1.58 AU, and the solar phase angle was 61.1.",
"For both images the direction of the Sun is towards the top of the image."
],
[
"Spectra",
"To study the evolution of the gas content during the outburst we present four successive spectra of the sunward coma taken from 10-minute histograms beginning 2015 May 23 UT12:10:07 in the left panel of Fig.",
"REF .",
"These correspond to rows 18 to 21 of the spectral image in Fig REF .",
"The difference between the spectra at the peak of the outburst and the prior spectra is then indicative of the erupting gas.",
"The difference spectrum, shown in the right panel, is characterized by an enhancement in the atomic emissions, particularly atomic oxygen, without a corresponding enhancement in long wavelength solar reflected light characteristic of dust production.",
"The large increase in O1 $\\lambda $ 1356/O1 $\\lambda $ 1304 (intensity ratio $\\ge $ 1:1) suggests O$_{2}$, now known to be present in the cometary ice of 67P [3], as the primary source of the additional gas.",
"Figure: Left: Sequence of four 10-minute histogram spectra above the sunlit limb.",
"The spectra are offset from one another for clarity.",
"The upturn at long wavelengths is due to scattered solar radiation from dust in the coma.",
"Right: The difference between the average of the final two and the first of the spectra shown at left.",
"The difference represents the spectrum of the material ejected into the coma in a ∼\\sim 30 minute period.However, laboratory data on the electron impact dissociative excitation of O$_{2}$ [14] suggests that this ratio should be $\\sim $ 2, as seen in the exospheric spectra of Europa [11] and Ganymede [10].",
"The observed ratio varies in the other outbursts listed in Table REF and likely reflects additional sources of O1 $\\lambda $ 1304 such as photodissociation of O$_{2}$ [2], [16] or electron impact on O (if present).",
"The dramatic change in relative intensities implies that the outburst cannot be due to a sudden increase in photoelectron flux or change in the electron energy distribution.",
"For May 23 this is borne out by data from RPC/IES [5] that shows only a modest uniform increase in electron flux beginning about two hours before the outburst observed by Alice and lasting for 12 hours (K. Mandt, private communication).",
"A significant amount of H$_{2}$ O is also released as evidenced by the presence of H1 Lyman-$\\alpha $ and Lyman-$\\beta $ in the difference spectrum.",
"If we ignore the blending of Lyman-$\\beta $ with O1 $\\lambda $ 1027, also produced by e+O$_{2}$ [1], and assume that all of the emission at 1026 Å is Lyman-$\\beta $ , then we can use the e+H$_{2}$ O cross section for Lyman-$\\beta $ at 100 eV from [19] relative to the e+O$_{2}$ cross section for O1 $\\lambda $ 1356 from [14] to estimate the relative O$_{2}$/H$_{2}$ O abundance in the outburst.",
"From the relative fluxes in the difference spectrum (Fig.",
"REF ) we find O$_{2}$/H$_{2}$ O $\\ge $ 0.5, considerably greater than the mean quiescent value of 0.038 reported by [3].",
"The presence of H$_{2}$ O in the outburst is confirmed by concurrent sub-mm measurements of the H$_{2}$ O column density by MIRO [17] along a line-of-sight contiguous with the central row of the Alice slit that showed a threefold increase at the same time (P. von Allmen, private communication), consistent with the increase in Lyman-$\\beta $ seen in the top panel of Fig.",
"REF .",
"CO began to appear regularly in Alice coma spectra in June 2015 and continues to be present through early 2016.",
"While not detected in the May 23 spectrum the CO Fourth Positive system is seen in several other spectra listed in Table REF .",
"However, it does not appear in any of the difference spectra and thus is unlikely to be the major driver in any of the outbursts presented here.",
"The origin of the C1 $\\lambda $ 1657 emission in the difference spectrum is not clear.",
"Electron impact on CO$_{2}$ would produce emission at 1561 Å with an intensity about half that of C1 $\\lambda $ 1657 [21] as well as CO Cameron band emission at longer wavelengths which is not observed.",
"Other carbon bearing molecules are not precluded as a source [15].",
"We note that N$_{2}$, also discovered for the first time in 67P [22], also has a rich electron excited spectrum of N$_{2}$ bands and atomic and ionic N multiplets within the Alice spectral range [13].",
"None of these are detected in the outburst spectrum so that N$_{2}$, whose mean abundance relative to CO is found by [22] to be less than 1%, also plays no role in the gas outbursts."
],
[
"Spatial Profiles ",
"Additional information about the outburst can be obtained from the profiles of the O1 emissions along the slit.",
"Fig.",
"REF shows the profiles of O1 $\\lambda $ 1304 and O1 $\\lambda $ 1356 for the four spectra shown in Fig.",
"REF .",
"Both profiles peak just above the sunward limb and decrease radially outward.",
"The brightness increases nearly uniformly in each successive 10 minute integration.",
"With an outflow velocity of 0.5 km s$^{-1}$ the escaping O$_{2}$ molecules will exit the Alice field-of-view in $\\sim $ 10 seconds, so the ejection is continuous.",
"From the light curve we see that one rotation later ($\\sim $ 12 h), at the same sub-spacecraft longitude, the emission has returned to its quiescent level.",
"Emission is seen against the nucleus and off the anti-sunward limb, indicating that the outburst is not in the form of a collimated jet but is rather diffuse.",
"As noted above, any dust produced would have been detected by an increase at long wavelengths of reflected solar radiation."
],
[
"Additional Events ",
"For the other dates in Table REF , the light curves, difference spectra, and spatial profiles are all similar to those of the May 23 event.",
"A search through the Alice database for earlier events reveals multiple outbursts with similar spectra on 2015 April 15 and April 29/30.",
"Prior to April, the geometry for observing outbursts (closer distance to the nucleus) was less favorable.",
"In April the spacecraft was at southern latitudes but the longitudes of the outbursts was also random.",
"The rate of detected gas outburst events decreased after perihelion on 2015 August 13.",
"Post-perihelion observations, through the present, continue to show variable O1 $\\lambda $ 1356/O1 $\\lambda $ 1304 intensity ratios although these measurements are not always confined to the short time scales of the “outburst” events described above.",
"These observations provide a means of monitoring the O$_{2}$/H$_{2}$ O abundance in the coma even when Rosetta is at large distances ($\\ge $ 100 km) from the nucleus, complementing ROSINA measurements closer to the comet."
],
[
"DISCUSSION",
"A likely source of gas outbursts is the warming of sub-surface volatile reservoirs as the comet approaches perihelion.",
"Water ice containing frozen O$_{2}$, as has been proposed for the surface of Ganymede [24], would reside below the dust mantle.",
"Sublimated O$_{2}$, together with some H$_{2}$ O, would then percolate through the porous mantle and diffuse into the coma taking some of the dust with it.",
"The coupling of dust to gas sublimated from sub-surface ice has been studied by many authors [4] For 67P, [9] consider only CO, CO$_{2}$, and H$_{2}$ O ices.",
"O$_{2}$ has a sublimation temperature and pressure comparable to those for CO [8] so that the CO models should similarly apply.",
"The absence of dust in the outbursts observed by Alice suggests a different scenario.",
"[23], seeking to explain a narrow, short-lived dust outburst observed by the OSIRIS imager, has proposed a deepening of a pre-existing fracture that would lead to the exposure of a sub-surface ice layer and a subsequent rapid ejection of gas and dust.",
"Although [23] considered a model with CO ice, their calculations should also be valid for O$_{2}$.",
"A narrow very short-lived dust jet would be missed by the Alice slit, while the high density of the escaping gas would collisionally be distributed throughout the coma."
],
[
"SUMMARY",
"We report here the detection by the Alice far-ultraviolet spectrograph on Rosetta of a number of sporadic gas outbursts above the sunward limb.",
"These outbursts are characterized by sudden increases in the emissions of atomic H and O over a period of 10-30 minutes, without a corresponding enhancement in long wavelength solar reflected light characteristic of dust production.",
"The emissions are seen to decay over a period of several hours, returning to their quiescent level after a complete rotation of the comet.",
"Spectroscopic analysis of the ejected gas suggests O$_{2}$ as the principal driver of the additional gas.",
"These outbursts do not correlate with any of the visible images of outbursts taken with either OSIRIS or the navigation camera.",
"A complete accounting of both pre- and post-perihelion events will be presented in a future publication.",
"Rosetta is an ESA mission with contributions from its member states and NASA.",
"We thank the members of the Rosetta Science Ground System and Mission Operations Center teams, in particular Richard Moissl and Michael Küppers, for their expert and dedicated help in planning and executing the Alice observations.",
"We thank Kathleen Mandt and Paul von Allmen for making available their data concerning the May 23 outburst.",
"The Alice team acknowledges continuing support from NASA's Jet Propulsion Laboratory through contract 1336850 to the Southwest Research Institute.",
"The work at Johns Hopkins University was supported by a subcontract from Southwest Research Institute.",
"Rosetta"
]
] | 1606.05249 |
[
[
"Asymptotic Theory of Channeling in the Field of an Atomic Chain and an\n Atomic Plane"
],
[
"Abstract A rigorous theory of diffraction scattering from extended objects is proposed.",
"The present theory is based on a multiple asymptotic expansion of an integral equation for the exact wave function in terms of the large parameters of the problem, which are the range of the potential and the momentum components of the incident particle.",
"For small angles of incidence the density of positively charged particles on the axis of a chain is always lower than unit y and the density of negatively charged particles has a maximum for certain strength of the potential which can exceed considerably unity.",
"The conditions of validity of the proposed approach are obtained."
],
[
"1. Introduction",
"The present theory of channeling is based on the assumption that the potential of a set of atoms forming a crystal can be replaced by the potential of a system of continuous atomic chains or planes.",
"This approach, first proposed by Lindhard [1], can be regarded as a model approach.",
"Its quantum-mechanical version [2 ,3] predicts a number of effects associated with channeling.",
"In particular, radiation which accompanies channeling has received considerable attention [4 - 6].",
"Although the model approach is widely used, it is not quite satisfactory since, strictly speaking, it is not an approximate method and, therefore, it cannot describe correctly finer effects which require extension beyond the average potential such as discreteness of the potential, boundary effects, interstitial atoms, vacancies, etc.",
"The average potential in the aforementioned model approach is introduced artificially (as the zeroth Fourier component of the real potential) and longitudinal scattering disappears in this approach.",
"It follows that the problem of scattering from a realistic potential bounded in space cannot be treated by this method.",
"It is our aim to study elastic scattering of fast particles from the potentials of an atomic chain and an atomic plane within the general scattering theory.",
"It will be shown quite rigorously that the average potential of an atomic chain introduced by Lindhard [1] is obtained from double asymptotic expansion of the Lippmann-Schwinger equation in terms of the length of the atomic chain $L_z \\rightarrow \\infty $ and in terms of the longitudinal momentum component of the incident particle $p_z \\rightarrow \\infty $ .",
"It is then possible to formulate a general method for calculating corrections to the zeroth approximation corresponding to the average potential since such corrections are simply the next higher-order terms in the asymptotic expansion in terms of large parameters in the problem.",
"In the case of an atomic plane, it is necessary to carry out an asymptotic expansion of higher multiplicity.",
"The zeroth-order wave function is expressed in the present approach in terms of a transverse integral equation with a kernel depending on the longitudinal length of an atomic chain or an atomic plane.",
"An analysis of this equation indicates that the density of the wave function for negatively charged particles depends logarithmically on the length of the atomic chain near the maximum and can be much greater than unity.",
"For real single crystals, this effect is known as flux peaking effect[7]."
],
[
"2. SCATTERING FROM AN EXTENDED ATOMIC CHAIN ",
"We shall consider scattering of fast charged particles from the potential of an atomic chain containing $N_z$ atoms separated by the lattice period $c$ .",
"To ensure that the opposite ends of the chain tend uniformly to $+\\infty $ and $-\\infty $ in the limit $N_z \\rightarrow \\infty $ , we shall choose the origin at $z = N_z c/2$ .",
"The Fourier transform of the potential then remains the same for even and odd $N_z$ , and we shall quote the potential for an odd number of atoms $U({\\bf r})=\\sum _{j_z=-\\frac{N_z-1}{2}}^{\\frac{N_z-1}{2}} U_{at} ({\\bf r}+{\\bf n_z} c j_z ), \\\\U({\\bf k})= U_{at} ({\\bf k}) \\frac{\\sin (N_z k_z \\frac{c}{2})}{\\sin (k_z \\frac{c}{2})}.$ Without loss of generality, we shall consider the scattering of nonrelativistic particles whose wave function satisfies the Lippmann -Schwinger equation ($\\hbar = 1$ ) $\\Psi ({\\bf r})=e^{i {\\bf p r}}+ \\int d {\\bf r^{\\prime }} \\frac{e^{i p |{\\bf r-r^{\\prime }}|}}{-4 \\pi |{\\bf r-r^{\\prime }}|} V({\\bf r^{\\prime } }) \\Psi ({\\bf r^{\\prime }}),$ where $V({\\bf r })= 2 M U({\\bf r})$ , and $M$ is the particle mass.",
"It is well known that the solution of the integral equation (3) can be written formally as an infinite Born series $\\Psi ({\\bf r})=\\sum _{n=0}^{\\infty } \\Psi ^{(n)}({\\bf r})$ with terms $ \\Psi ^{(0)}=e^{i {\\bf p r}}$ , $\\Psi ^{(n)}({\\bf r}) =e^{i {\\bf p r}} \\int d {\\bf R_1...R_n} \\frac{e^{i pR_1- i{\\bf p R_1}|}}{-4 \\pi R_1}\\\\ \\nonumber \\times V({\\bf r -R_1 }) ... \\frac{e^{i pR_1- i{\\bf p R_n}|}}{-4 \\pi R_n} V({\\bf r -R_1- ...-R_n }) .$ An eikonal expression for the wave function is obtained from the expansion of $ \\Psi ^{(n)}({\\bf r})$ in powers of $p_z \\rightarrow \\infty $ followed by summation of the first terms of this expansion [8].",
"An additional expansion in powers of $L_z \\rightarrow \\infty $ was considered in [9] within the eikonal approximation, i.e., a double asymptotic expansion has been used (first expansion in powers of $p_z \\rightarrow \\infty $ and then in powers of $L_z \\rightarrow \\infty $ ).",
"However, the eikonal approximation is not uniform with respect to the length of the potential and holds for $L_z << p_z R^2$ .",
"It follows that an expansion where the limit $L_z \\rightarrow \\infty $ is taken first and then the limit $p_z \\rightarrow \\infty $ should yield a different result.",
"In fact, the pole of the Green function (in the momentum space shifted by an amount equal to the particle momentum p) $\\frac{1}{k^2+ 2 {\\bf p k}- i \\delta }, \\nonumber $ i.e., the point $k_z =- p_z + \\sqrt{p^2_x - k_{\\perp } - 2p^2_x k_{\\perp } - i \\delta }$ merges in the limit $p_z \\rightarrow \\infty $ with a removable singularity of the component $k_z = 0$ of the Fourier potential of the chain defined by Eq.",
"(2) and the resulting nonuniform behavior in the asymptotic expansion of the integrals involved is due to merging of certain critical singularities [10 ,11].",
"To obtain the wave function in the field of a chain with $L_z \\gtrsim p_z R^2$ , it is necessary to derive an asymptotic expansion of $\\Psi ^{(n)}({\\bf r})$ which is uniform over the length of the chain $L_z$ .",
"The parabolic approximation of Leontovich and Fock [12] represents such expansion.",
"For $L_z >> p_z R^2$ , the parabolic approximation can be simplified.",
"The simplification in question can be obtained directly from Eq.",
"(5) by expanding with respect to $L_z \\rightarrow \\infty $ and then with respect to $p_z \\rightarrow \\infty $ , and the results are quoted below.",
"Omitting the intermediate steps, we can write the principal term for $\\Psi ^{(1)}$ in the form $\\Psi ^{(1)}({\\bf r}) =e^{i {\\bf p r}} \\int d {\\bf R_{1 \\perp }} -\\frac{i}{8}[H_0^{(1)}(p_{\\perp } R_{1 \\perp }) \\\\ \\nonumber + H_0( \\ln \\frac{N_z c p_{\\perp }}{2 p_z R_{1 \\perp }}, p_{\\perp } R_{1 \\perp }) ] e^{- i{\\bf p_{\\perp } R_{1 \\perp }}} V_{\\perp }({\\bf r_{\\perp } -R_{1 \\perp } }, 0) ,$ Here, $H_0^{(1)} ( \\rho )$ is a Hankel function of first kind; $H_0(\\beta , \\rho )$ is an incomplete Hankel cylindrical function [13].",
"The quantity $V_{\\perp }({\\bf r_{\\perp } -R_{1 \\perp } }, 0) $ is the average potential of an atomic chain which is usually introduced to describe the channeling [1].",
"Since the integrand in Eq.",
"(6) is independent of $r_z$ , we can introduce quite consistently in Eq.",
"(5) analogous expansions in each of the integrals involved.",
"We obtain $\\Psi ^{(n)}({\\bf r}) =e^{i {\\bf p r}} \\int d {\\bf R_{1 \\perp }}...d {\\bf R_{n \\perp }} \\tilde{G}_{p_{ \\perp }}({\\bf R_{1 \\perp }}) e^{- i{\\bf p R_{1 \\perp }}} \\nonumber \\\\ \\times V_{\\perp } ({\\bf r_{\\perp } -R_{1 \\perp } }, 0)... \\tilde{G}_{p_{ \\perp }}({\\bf R_{n \\perp }}) e^{- i{\\bf p R_{n \\perp }}} \\nonumber \\\\ \\times V_{\\perp } ({\\bf r_{\\perp } -R_{1 \\perp } -... - R_{n \\perp } }, 0 ),$ $\\tilde{G}_{p_{ \\perp }}({\\bf R_{ \\perp }})= -\\frac{i}{8}[H_0^{(1)}(p_{\\perp } R_{\\perp })+ H_0( \\ln \\frac{L_z p_{\\perp }}{2 p_z R_{\\perp }}, p_{\\perp } R_{\\perp }) ] .$ Summation of the terms in Eq.",
"(7) yields the wave function in the field of an atomic chain $\\Psi ({\\bf r})=e^{i p_{z} r_{z} } \\phi _{{\\bf p_\\perp }} ({\\bf r_{\\perp } }) ,$ where $\\phi _{{\\bf p_\\perp }} ({\\bf r_{\\perp } }) $ satisfies the following two-dimensional integral equation: $\\phi _{{\\bf p_\\perp }}=e^{i {\\bf p_{\\perp } r_{\\perp } }}+ \\int d {\\bf r^{\\prime }_{\\perp } } \\tilde{G}_{p_{ \\perp }}({\\bf r_{ \\perp }-r^{\\prime }_{ \\perp }}) V_{\\perp }({\\bf r^{\\prime }_{\\perp }},0) \\phi _{{\\bf p_\\perp }} ({\\bf r^{\\prime }_{\\perp } }).",
"\\;\\;$ The solution of the Fredholm Integral equation (10) of second kind can be written in the form $\\phi _{{\\bf p_\\perp }} ({\\bf r_{\\perp } }) =e^{i {\\bf p_{\\perp } r_{\\perp } }}-\\sum _{k=1}^{\\infty }\\frac{a_k}{1- \\lambda _k} \\phi ^{(k)}_{{\\bf p_\\perp }} ({\\bf r_{\\perp } }),$ where $\\lambda _k$ and $\\phi ^{(k)}$ are the eigenvalues and eigenfunctions of the homogeneous Fredholm equation of second kind $\\phi ^{(k)}_{{\\bf p_\\perp }} ({\\bf r_{\\perp } })=\\lambda _k \\int d {\\bf r^{\\prime }_{\\perp } } \\tilde{G}_{p_{ \\perp }}({\\bf r_{ \\perp }-r^{\\prime }_{ \\perp }}) V_{\\perp }({\\bf r^{\\prime }_{\\perp }},0) \\phi ^{(k)}_{{\\bf p_\\perp }} ({\\bf r^{\\prime }_{\\perp } }) \\nonumber \\\\a_k=\\int d {\\bf r_{\\perp } } e^{i {\\bf p_{\\perp } r_{\\perp } }} V_{\\perp }({\\bf r_{\\perp }},0) \\phi ^{(k)}_{{\\bf p_\\perp }} ({\\bf r_{\\perp }}).$ The spectral equation (12) determines the transverse eigenfunctions and transverse eigenenergies for channeling in the field of an isolated atomic chain.",
"The eigenfunctions and eigenvalues depend not only on the longitudinal momentum of the particle $p_z$ , but also on the longitudinal length of the chain $L_z$ .",
"For an axially symmetric chain, the propagator defined by Eq.",
"(8) can be conveniently written in the form ($\\lambda = L_z/4 p_z$ ) $-\\frac{i}{8}[H_0^{(1)}(p_{\\perp } |{\\bf r_{\\perp }-r^{\\prime }_{\\perp }}|)+ H_0( \\ln \\frac{2 \\lambda p_{\\perp }}{|{\\bf r_{\\perp }-r^{\\prime }_{\\perp }}|}, \\nonumber \\\\ p_{\\perp }|{\\bf r_{\\perp } - r^{\\prime }_{\\perp }}|) ] = \\sum _{m=-\\infty }^{\\infty }e^{i m \\varphi } \\tilde{G}^{m}_{p_{ \\perp }}( r_{ \\perp },r^{\\prime }_{ \\perp }),$ where $ \\varphi $ is the angle between ${\\bf r_{\\perp }}$ and ${\\bf r^{\\prime }_{\\perp }}$ , and $\\tilde{G}^{m}_{p_{ \\perp }}( r_{ \\perp },r^{\\prime }_{ \\perp })=\\frac{1}{2 \\pi } \\int _{0}^{\\infty } k_{\\perp } d k_{\\perp } J_{m}( k_{\\perp } r^{\\prime }_{\\perp }) J_{m}( k_{\\perp } r_{\\perp }) \\nonumber \\\\ \\times \\frac{1-e^{-i \\lambda (k^{2}_{\\perp } -p^{2}_{\\perp } ) } }{p^{2}_{\\perp }-k^{2}_{\\perp }+i \\delta }.$ The wave function $ \\phi _{{\\bf p_\\perp }}$ can be now expanded in terms of the azimuthal momentum ( $ \\varphi _{1}$ is the angle between ${\\bf r_{\\perp }}$ and ${\\bf p_{\\perp }}$ ) $\\phi _{{\\bf p_\\perp }} ({\\bf r_{\\perp } }) =\\sum ^{\\infty }_{-\\infty } i^m e^{i m \\varphi _{1}} \\Psi _m( p_{ \\perp },r_{ \\perp })$ where $ \\Psi _m( p_{ \\perp },r_{ \\perp })$ satisfies the following integral equation: $\\Psi _m( p_{ \\perp },r_{ \\perp })=J_{m}( p_{\\perp } r_{\\perp }) + \\nonumber \\\\ 2 \\pi \\int ^{\\infty }_{0} r^{\\prime }_{ \\perp } d r^{\\prime }_{ \\perp } \\tilde{G}^{m}_{p_{ \\perp }}( r_{ \\perp },r^{\\prime }_{ \\perp }) V(r^{\\prime }_{\\perp }, 0) \\Psi _m( p_{ \\perp },r^{\\prime }_{ \\perp }).$ We shall consider two cases corresponding to two limiting values of the parameter $\\rho sh(\\beta )$ for which the incomplete cylindrical function $H_0( \\beta , \\rho ) $ can be expanded in a series[13].",
"In the first case when the condition $\\rho *sh(\\beta ) >>1$ is satisfied and $H_0( \\beta , \\rho ) = H_0^{(1)}(\\rho )+ O(|\\rho * sh(\\beta ) |^{-1}), \\nonumber $ the kernel defined by Eq.",
"(8) becomes a two-dimensional Green function $-(i/4)H_0^{(1)} ( p_{ \\perp } r_{ \\perp } )$ and Eq.",
"(10) reduces to the usual two-dimensional Lippmann -Schwinger equation independent of $L_z$ .",
"In the second case when the inequality $\\rho *sh(\\beta ) <<1$ holds, we obtain [13] $H_0( \\beta , \\rho ) =\\frac{2}{i \\pi } \\beta + O(|\\rho * sh(\\beta ) |).",
"\\nonumber $ This case corresponds to the condition $| \\frac{p^2_{\\perp } L_z}{4 p_z} - \\frac{ |{\\bf r_{\\perp }-r^{\\prime }_{\\perp }}|^2 p_z}{L_z} | <<1.",
"\\nonumber $ Figure: Dependencies of the probability density of finding particles on theatomic chain axis on the magnitude of the potential U 0 =2Z 1 Z 2 e2/cU_0 = 2 Z_1 Z_2 e2/c (inunits of [MR 2 ] -1 [M R^2]^{-1} ).",
"Curves 1 and 8 correspond to scattering of negatively andpositively charged particles for the following values of the parameter ln(L z /γp z R 2 ) \\ln (L_z/\\gamma p_z R^2): 1, 8 - 6.0; 2, 7 - 5.0; 3, 6 - 4.0; 4,5 - 3.0.i.e., to small angles of incidence $\\theta _0 << 2 / \\sqrt{ p_z L_z}$ and to small distances from the axis of the chain $ |{ \\bf r_{ \\perp } -r^{\\prime }_{ \\perp }}| << \\sqrt{L_z / p_z}$ .",
"It will be shown that Eq.",
"(10) holds for angles of incidence $\\theta _0 \\lesssim 1 / p_z R$ and, therefore, for all points within the range of the potential of the chain $|{ \\bf r_{ \\perp } -r^{\\prime }_{ \\perp }}| \\lesssim R$ the condition $ p_{\\perp }|{ \\bf r_{ \\perp } -r^{\\prime }_{ \\perp }}| << 1$ is satisfied and the Hankel function $H_0^{(1)}(p_{\\perp } |{ \\bf r_{ \\perp } -r^{\\prime }_{ \\perp }}|)$ can be expanded in a series.",
"The function $ \\tilde{G}_{p_{ \\perp }}( { \\bf r_{ \\perp } })$ then satisfies the following equation: $\\tilde{G}_{p_{ \\perp }} \\sim \\frac{1}{4 \\pi } \\ln \\frac{|{ \\bf r_{ \\perp } -r^{\\prime }_{ \\perp }}|^2 \\gamma p_z}{L_z}- \\frac{i}{8} , \\;\\; \\gamma =1.78... \\;$ and Eq.",
"(10) can be written in the form $\\phi _{{\\bf p_\\perp }}=e^{i {\\bf p_{\\perp } r_{\\perp } }}+\\nonumber \\\\ \\int d {\\bf r^{\\prime }_{\\perp } } [\\frac{1}{4 \\pi } \\ln \\frac{|{ \\bf r_{ \\perp } -r^{\\prime }_{ \\perp }}|^2 \\gamma p_z}{L_z}- \\frac{i}{8} ] V_{\\perp }({\\bf r^{\\prime }_{\\perp }},0) \\phi _{{\\bf p_\\perp }} ({\\bf r^{\\prime }_{\\perp } }).$ For a potential with azimuthal symmetry, Eqs.",
"(16) and (17) yield the foUowing equation for the wave function with a momentum $m$ : $\\Psi _m( p_{ \\perp },r_{ \\perp })=J_{m}( p_{\\perp } r_{\\perp }) + \\nonumber \\\\ 2 \\pi \\int ^{\\infty }_{0} r^{\\prime }_{ \\perp } d r^{\\prime }_{ \\perp } \\tilde{G}^{^{\\prime }}_{m}( r_{ \\perp },r^{\\prime }_{ \\perp }) V(r^{\\prime }_{\\perp }, 0) \\Psi _m( p_{ \\perp },r^{\\prime }_{ \\perp }).$ $\\tilde{G}^{^{\\prime }}_{m}( r_{ \\perp },r^{\\prime }_{ \\perp }) =\\left\\lbrace \\begin{array}{ll}- \\frac{1}{4 \\pi m} r^{m}_{\\perp <}\\; r^{-m}_{\\perp >} , & \\; m> 0, \\\\\\frac{1}{4 \\pi } \\ln \\frac{r^{2}_{\\perp >} \\gamma p_z}{L_z} - \\frac{i}{8},& \\; m=0.\\end{array} \\right.$ Figure: Dependencies of the distribution of the particle density on the distance from the axis of an extended chain in the case of the maximum defined by Eq.",
"(23) for ln(L z /γp z R 2 )=3.2\\ln (L_z/\\gamma p_z R^2) = 3.2; 1) negatively charged particles; 2) positively charged particles.We shall now determine the probability density that the partic1e lies on the axis of the chain $r_{\\perp } =0$ .",
"For $m =0$ , Eq.",
"(19) yields $\\Psi _0(0,0)=1 + \\nonumber \\\\ \\frac{1}{2} \\int ^{\\infty }_{0} r^{\\prime }_{ \\perp } d r^{\\prime }_{ \\perp }[ \\ln \\frac{r^{^{\\prime }2}_{\\perp } \\gamma p_z}{L_z} - \\frac{i \\pi }{2}] V(r^{\\prime }_{\\perp }, 0) \\Psi _0(0, r^{\\prime }_{ \\perp }),\\nonumber $ and we obtain $|\\Psi _0(0,0)|^2=\\frac{1}{B+C}, \\\\B=[1 - \\frac{1}{2} \\int ^{\\infty }_{0} r^{\\prime }_{ \\perp } d r^{\\prime }_{ \\perp } \\ln \\frac{r^{^{\\prime }2}_{\\perp } \\gamma p_z}{L_z} V(r^{\\prime }_{\\perp }, 0) ]^2, \\nonumber \\\\C=[ \\frac{\\pi }{4} \\int ^{\\infty }_{0} r^{\\prime }_{ \\perp } d r^{\\prime }_{ \\perp } V(r^{\\prime }_{\\perp }, 0) ]^2.\\nonumber $ For $m > 0$ , we obtain the obvious result $\\Psi _m(0,0)= 0$ .",
"Using the Molière potential [14] as an atomic potential, we can easily evaluate (for the corresponding average potential [15]) the integrals which appear in Eq.",
"(21) (see [16]).",
"We then obtain $|\\Psi _0(0,0)|^2=\\frac{1}{B_1+C_1}, \\\\B_1=[1 - M \\frac{2 Z_1 Z_2 e^2 R^2}{c} \\sum _{i=1}^{3} \\alpha _i \\beta _i^{-2} \\ln \\frac{R^{2} \\gamma p_z}{\\beta _i^{2} L_z} ]^2, \\nonumber \\\\C_1=[ \\frac{\\pi }{2} M \\frac{2 Z_1 Z_2 e^2 R^2}{c} \\sum _{i=1}^{3} \\alpha _i \\beta _i^{-2} ]^2,\\nonumber $ where $Z_1$ is the charge of an atom in the target and $Z_2$ is the charge of the incident partic1e; $\\alpha _i$ and $ \\beta _i$ are the parameters of the Molière potential.",
"It follows from Eq.",
"(22) that the density of positively charged particles on the axis of an extended chain ($L_z >> p_z R^2$ ) is always lower than unity and the density of negatively charged partic1es can exceed unity.",
"We note that the proposed approximation holds for an arbitrary strength of the potential $U_0 = 2 Z_1 Z_2 e^2 /c$ .",
"ln particular, for $U_0 \\equiv 0$ , we obtain $|\\Psi _0(0,0)|^2= 1$ which is the correct result in the absence of scatterers.",
"The dependence of $|\\Psi _0(0,0)|^2$ on $U_0$ is shown in Fig.",
"1.",
"The locations and magnitudes of the maxima of the curves in Fig.",
"1 can be easily determined from Eq.",
"(22) $U_{0 max}= \\nonumber \\\\\\frac{ \\sum _{i=1}^{3} \\alpha _i \\beta _i^{-2} \\ln \\frac{R^{2} \\gamma p_z}{\\beta _i^{2} L_z}}{ M R^2([ \\sum _{i=1}^{3} \\alpha _i \\beta _i^{-2} \\ln \\frac{R^{2} \\gamma p_z}{\\beta _i^{2} L_z}]^2+[ \\frac{\\pi }{2} \\sum _{i=1}^{3} \\alpha _i \\beta _i^{-2} ]^2)} , \\\\|\\Psi _{0 max}(0,0)|^2=\\frac{1}{B_1+C_1}, \\nonumber \\\\B_1=[1 - M R^2 U_{0 max} \\sum _{i=1}^{3} \\alpha _i \\beta _i^{-2} \\ln \\frac{R^{2} \\gamma p_z}{\\beta _i^{2} L_z} ]^2, \\nonumber \\\\C_1=[ \\frac{\\pi }{2} M R^2 U_{0 max} \\sum _{i=1}^{3} \\alpha _i \\beta _i^{-2} ]^2.\\nonumber $ For $| \\ln (\\gamma R^2 p_z/L_z) |>>1$ , the density of negatively charged particles on the axis of the chain is logarithmically large and independent of the magnitude of the potential $|\\Psi _{0 max}(0,0)|^2= (\\frac{\\pi }{2} )^2 \\frac{ \\sum _{i=1}^{3} \\alpha _i \\beta _i^{-2} \\ln \\frac{R^{2} \\gamma p_z}{\\beta _i^{2} L_z} }{ \\sum _{i=1}^{3} \\alpha _i \\beta _i^{-2} }$ The radial distribution of charged partlc1es across the potential of the chain, obtained in this case, is shown in Fig.",
"2.",
"For real crystals, the density on the chain axis can increase only for low-energy electrons.",
"For other negatively charged partic1es ($\\mu ^{-}$ , $\\pi ^{-}$ , $\\bar{p}$ , etc.)",
"which have large masses and for ultrarelativistic electrons $e^{-}$ [when $M$ in Eq.",
"(3) is replaced by $E$ ], the quantity $U_{0 max}$ becomes negligibly small for $\\ln (L_z/\\gamma p_z R^2)>>1$ (see Table 1 where the calculations are presented for $\\ln (L_z/\\gamma p_z R^2)=10$ ).",
"To observe an appreciable increase in the density for $\\mu ^{-}$ , $\\pi ^{-}$ , and $\\bar{p}$ , it would be necessary to use high-index crystallographic directions but the effect of neighboring atomic chains would then be important.",
"Table: table titleWe shall now derive expressions for the amplitude and total scattering cross section.",
"The wave function defined by Eq.",
"(9) is valid only in the region $|{\\bf r_z} |\\lesssim L_z$ (see Sec.",
"3).",
"It follows that it can be used to calculate the amplitude provided Eq.",
"(9) is substituted in the expression $f=-\\frac{1}{4 \\pi } \\int d {\\bf r} e^{-i {\\bf p_{f} r}} V({\\bf r}) \\Psi ({\\bf r})$ which requires the knowledge of the wave function only in the region of action of the potential.",
"We stress that such situation is analogous to the situation in the eikonal approximation.",
"The eikonal wave function is a good approximation only in the region of space $|{\\bf r_z}|<< p_z R^2$ (see [17]) and its asymptotic behavior for $|{\\bf r_z}| \\rightarrow \\infty $ cannot be used to calculate the scattering amplitude.",
"Substituting Eq.",
"(9) in Eq.",
"(25), we obtain $f=-\\frac{1}{4 \\pi } \\frac{\\sin (N_s(p_{iz}-p_{f z})\\frac{c}{2})}{\\sin ((p_{iz}-p_{f z})\\frac{c}{2})} c \\times \\nonumber \\\\ \\int d {\\bf r_{\\perp }} e^{-i {\\bf p_{\\perp f} r_{\\perp }}} V_{\\perp }({\\bf r_{\\perp }}, p_{i z}- p_{f z}) \\psi _{\\bf {p} i \\perp } ({\\bf r_{\\perp }}) ,$ where $V_{\\perp }({\\bf r_{\\perp }}, q)$ is the longitudinal component of the chain Fourier potential $V_{\\perp }({\\bf r_{\\perp }}, q) =V_{at}({\\bf r_{\\perp }}, q) /c$ .",
"Since the integral in Eq.",
"(26) is a slowly varying function of $ p_{i z}- p_{f z}$ compared with the function outside the integral, we can write the scattering amplitude in the form $f=-\\frac{1}{2 \\pi } \\frac{\\sin ((p_{iz}-p_{f z})\\frac{L_z}{2})}{(p_{iz}-p_{f z})} \\times \\nonumber \\\\ \\int d {\\bf r_{\\perp }} e^{-i {\\bf p_{\\perp f} r_{\\perp }}} V({\\bf r_{\\perp }},0) \\psi _{\\bf {p} i \\perp } ({\\bf r_{\\perp }}) .$ For small angles of incidence $\\theta _0 <<1$ and for small angles of partic1es leaving the crystal $\\theta _1 <<1$ measured from the direction of the Oz axis, we obtain $p_{i z}- p_{f z}=2p \\sin (\\frac{\\theta _0+ \\theta _1}{2}) \\sin (\\frac{\\theta _1- \\theta _0}{2}) \\simeq \\frac{1}{2} p_z(\\theta _1^2- \\theta _0^2)\\nonumber $ It follows that the amplitude defined by Eq.",
"(27) and the differential scattering cross section $d \\sigma /d \\Omega = |f|^2$ have sharp maxima at $\\theta _0 =\\theta _1 $ .",
"For angles of incidence $\\theta _0 >> 2/ \\sqrt{p_z L_z}$ , the polar width of the maximum for an extended potential $L_z >> p_z R^2 $ is given by $|\\theta _1- \\theta _0|_{ef} \\sim \\simeq \\frac{2}{p_z L_z \\theta _0} .\\nonumber $ The spatial distribution of the amplitudes defined by Eq.",
"(27) is then either ring- or doughnut-shaped, which is observed in experiments on hyperchanneling [18 ,19].",
"For $ \\theta _0 \\lesssim 1/ \\sqrt{p_z L_z}$ , the ring is compressed and becomes a peak.",
"For $ \\theta _0 << 2/ \\sqrt{p_z L_z}$ , the width of the peak is given by $|\\theta _1- \\theta _0|_{eff} \\sim \\simeq \\frac{2}{p_z L_z }\\nonumber $ Using the optical theorem $ \\sigma = (4 \\pi / p) Im \\;f(0)$ , we obtain the following result for the total scattering cross section: $\\sigma =-\\frac{ L_z }{p} Im \\int d {\\bf r_{\\perp }} e^{-i {\\bf p_{\\perp f} r_{\\perp }}} V({\\bf r_{\\perp }},0) \\psi _{p i \\perp } ({\\bf r_{\\perp }})$ For small angles of incidence $ \\theta _0 << 2/ \\sqrt{p_z L_z}$ , the cross section can be approximated by $\\sigma =\\frac{ L_z }{8 p} \\left[ \\left(\\frac{1-\\frac{1}{2} \\int r_{\\perp } d r_{\\perp } V( r_{\\perp },0) \\ln \\frac{ r^2_{\\perp } \\gamma p_z}{L_z}}{2 \\pi \\int r_{\\perp } d r_{\\perp } V( r_{\\perp },0) }\\right)^2 + \\frac{1}{64} \\right]^{-1}.",
"\\;\\;$ With an accuracy up to a coefficient depending on the form of the potential, Eq.",
"(29) reduces to the results of model calculations [20,21].",
"For angles of incidence $ \\theta _0 << 2/ \\sqrt{p_z L_z}$ , the wave function $ \\psi _{\\bf {p} i \\perp } ({\\bf r_{\\perp }})$ is determined by the two-dimensional Lippmann-Schwinger equation and the cross section defined by Eq.",
"(28) is proportional to the two-dimensional scattering amplitude from a potential $V_{\\perp }({\\bf r_{\\perp }}, 0)$ which may be resonant for negatively charged particles [21]."
],
[
"3. SCATTERING FROM AN EXTENDED ATOMIC PLANE ",
"We shall now consider scattering from the potential of a rectangular atomic plane containing $N_y$ atomic chains separated by a period $b$ and each containing $N_z$ atoms.",
"For simplicity, we shall assume that $N_y$ and $N_z$ are odd numbers $U({\\bf r_{\\perp }})=\\sum _{-(N_z-1/2)}^{N_z-1/2} \\sum _{-(N_y-1/2)}^{N_y-1/2} U_{at} ({\\bf r}+{\\bf n_z} c j_z +{\\bf n_y} b j_y ), \\nonumber \\\\U({\\bf k})= U_{at} ({\\bf k}) \\frac{\\sin (N_z k_z \\frac{c}{2})}{\\sin (k_z \\frac{c}{2})} \\frac{\\sin (N_y k_y \\frac{b}{2})}{\\sin (k_y \\frac{b}{2})}.",
"\\;\\;$ As in the case of the chain potential, the eikonal approximation is not valid for an extended plane potential $L_z >>p_z R^2$ and $L_y >> p_y R^2$ .",
"We shall, therefore, base our discussion on the approach developed in Sec.",
"1 for scattering from a chain and seek the expansion of Eq.",
"(5) for $L_z =N_z c\\rightarrow \\infty $ and $L_y = N_y b\\rightarrow \\infty $ , and then perform expansion for $p_z \\rightarrow \\infty $ and $p_y \\rightarrow \\infty $ .",
"The final result which holds subject to the additional condition $p_y/ p_z \\rightarrow 0$ has the form $\\Psi ^{(1)}({\\bf r}) =e^{i {\\bf p r}} \\int ^{\\infty }_{-\\infty } d R_{x} (-\\frac{i}{2})[\\frac{e^{i p_x |R_x|}}{p_x}- i\\left(\\frac{\\pi |R_x|}{2 p_x}\\right)^{1/2} \\times \\; \\\\\\Phi _{1/2}(\\frac{i \\pi }{2} -\\ln \\frac{L_z p_{x}}{2 p_z |R_{x}|}, p_{x} |R_{x}|) ] e^{- i p_{x} R_{x}} V_{\\perp }({r_{x} -R_{x} }, 0,0).",
"\\nonumber $ Here, $ \\Phi _{1/2}(\\beta , \\rho )$ is the incomplete cylindrical function of fractional order [13] and $ V_{\\perp }(r_{x}, 0,0)= V_{at}(r_{x} , 0,0)/c b$ is the average potential of an atomic plane.",
"Since the integral in Eq.",
"(31) is independent of $r_y$ and $r_z$ , we can perform expansions in the multiple integral which determines the correction $\\Psi ^{(n)}$ as in the case of an atomic chain, which yields $\\Psi ^{(n)}({\\bf r}) =e^{i {\\bf p r}} \\int ^{\\infty }_{-\\infty } d R_{1 x} ...\\int ^{\\infty }_{-\\infty } d R_{n x} \\tilde{G}_{p_{x}} ( R_{x}) e^{- i p_x R_{1x}} \\nonumber \\\\ \\times V_{\\perp } ( r_{x} -R_{1 x }, 0,0)... \\tilde{G}_{p_{ x}}( R_{n x}) e^{- i p_x R_{n x}} \\nonumber \\\\ \\times V_{\\perp } ( r_{x} - R_{1 x} -... - R_{n x} , 0 ,0), \\;\\;$ $\\tilde{G}_{p_{x}}( R_{x})= (-\\frac{i}{2})[\\frac{e^{i p_x |R_x|}}{p_x}- i\\left(\\frac{\\pi |R_x|}{2 p_x}\\right)^{1/2} \\times \\; \\nonumber \\\\\\Phi _{1/2}(\\frac{i \\pi }{2} -\\ln \\frac{L_z p_{x}}{2 p_z |R_{x}|}, p_{x} |R_{x}|) ] .$ Summation of the series in Eq.",
"(4) with the terms defined by Eq.",
"(32) yields the following expression for the wave function in the field of a rectangular atomic plane: $\\Psi ({\\bf r})=e^{i p_z r_z+i p_y r_y} \\phi _{p_x} (r_x),$ where the transverse function $ \\phi _{p_x} (r_x)$ satisfies an integral equation $\\phi _{p_x} (r_x)=e^{i p_x r_x} + \\int ^{\\infty }_{-\\infty } d r^{\\prime }_{ x} \\tilde{G}_{p_{x}}( |r_{x}-r^{\\prime }_x|) \\times \\nonumber \\\\V_{\\perp } ( r^{\\prime }_{x} , 0,0) \\phi _{p_x} (r^{\\prime }_x),$ with the kernel $ \\tilde{G}_{p_{x}}( R_{x}) $ determined by Eq.",
"(33).",
"The distribution of the probability density, the amplitude, and the scattering cross section in the field of an atomic plane can be obtained from Eq.",
"(35) by the method described for the atomic chain."
],
[
"4. DISCUSSION OF RESULTS ",
"The conditions of validity of the present approximation can be obtained from the requirement that the terms which were neglected in the derivation of Eqs.",
"(10) and (35) should be small compared with the terms retained, i.e., $|\\Psi ^{(0)}| >> |\\Psi _{m} ^{(n)}| , \\; \\;|\\Psi _{1}^{(n)}| >> |\\Psi _{m} ^{(n)}| , \\; m \\ge 2,$ where $\\Psi _m^{(n)}$ are the terms in the asymptotic expansion of the wave function $\\Psi ^{(n)}$ of the Born series defined by Eq.",
"(4).",
"We can write $\\Psi ^{(1)}$ with an accuracy up to terms $O(1/p^2_z + 1/p_z L_z + 1/ L_z^2)$ for an atomic chain $\\Psi ^{(1)} =\\Psi _1^{(1)}+\\Psi _2^{(1)}+\\Psi _3^{(1)},\\;\\;\\;\\; \\\\\\Psi _1^{(1)}({\\bf r}) =e^{i {\\bf p r}} \\int d {\\bf R_{ \\perp }} \\left(-\\frac{i}{8}\\right) [H_0^{(1)}(p_{\\perp } R_{ \\perp }) + \\;\\; \\nonumber \\\\H_0( \\ln \\frac{N_z c p_{\\perp }}{2 p_z R_{ \\perp }}, p_{\\perp } R_{ \\perp }) ] e^{- i{\\bf p_{\\perp } R_{ \\perp }}} V_{\\perp }({\\bf r_{\\perp } -R_{ \\perp } }, 0), \\;\\; \\nonumber \\\\\\Psi _2^{(1)}({\\bf r}) =e^{i {\\bf p r}} \\frac{r_z}{2 \\pi L_z} \\int d {\\bf R_{ \\perp }} e^{ i \\frac{p_z}{L_z} ({\\bf R_{ \\perp }}- {\\bf p_{\\perp }} \\frac{L_z}{2 p_z})^2} \\times \\nonumber \\\\ V_{\\perp }({\\bf r_{\\perp } -R_{ \\perp } }, 0), \\;\\; \\nonumber \\\\\\Psi _3^{(1)}({\\bf r}) =e^{i {\\bf p r}} \\sum _{n=\\pm 1,\\pm 2,...} \\frac{(-1)^{(N_z-1)n} c }{4 \\pi p_z n} e^{ i \\frac{2 \\pi n}{c}r_z} \\times \\nonumber \\\\ V_{\\perp }({\\bf r_{\\perp } }, \\frac{2 \\pi n}{c}) .\\;\\; \\nonumber $ (a) We shall now consider small angles of incidence $ \\theta _0 << 2/ \\sqrt{p_z L_z}$ .",
"The integrals in Eq.",
"(37) can be easily estimated $|\\Psi _1^{(1)} | \\simeq | \\bar{V}_{\\perp } R^2 \\ln \\frac{p_z \\gamma R^2}{L_z} |, \\; |\\Psi _2^{(1)} | \\simeq | \\bar{V}_{\\perp } R^2 \\frac{r_z }{2 \\pi L_z} |, \\; \\nonumber \\\\|\\Psi _3^{(1)} | \\simeq | \\bar{V}_{\\perp } \\frac{c }{4 \\pi p_z} |.\\nonumber $ Comparing $|\\Psi _1^{(1)} |$ with $|\\Psi _2^{(1)} |$ , we find that the wave function defined by Eq.",
"(9) is valid in the region $|r_z| << |2 \\pi L_z \\ln \\frac{L_z}{p_z \\gamma R^2} |.$ i.e., for $ L_z >> p_z R^2$ , the wave function defined by Eq.",
"(9) is valid virtually in the whole region of action of the chain potential and can be used in the calculation of the scattering amplitude as described above.",
"Comparing $|\\Psi _1^{(1)} |$ and $|\\Psi _3^{(1)} |$ , we obtain the following important criterion: $|p_z R^2 \\ln \\frac{L_z}{p_z \\gamma R^2} | >> \\frac{c}{4 \\pi }$ which indicates that the dynamic length of longitudinal coherence $p_z R^2$ should exceed the distance $c$ separating the atom under study from the preceding and following atoms.",
"We then obtain physical averaging of the potential of an atomic chain.",
"(b) We shall now assume that the angle of incidence satisfies $ \\theta _0 \\gtrsim 2/ \\sqrt{p_z L_z}$ .",
"We then obtain $|\\Psi _1^{(1)} | \\simeq | \\frac{\\bar{V}_{\\perp } R^2}{\\sqrt{p_{\\perp } R}} |, \\;\\;\\; |\\Psi _2^{(1)} | \\simeq | \\frac{n_z \\bar{V}_{\\perp } R^2}{2 \\pi L_z} |.\\nonumber $ The condition $|\\Psi _1^{(1)} | >> |\\Psi _2^{(1)} | $ yields the following restriction on the angle of incidence: $\\theta _0 <<\\frac{ (2 \\pi )^2 }{p_z R} .$ To derive this inequality, we set $r_z \\sim L_z$ to satisfy the requirement that the wave function defined by Eq.",
"(9) should be valid in the whole range of action of the chain potential.",
"Such restriction can be interpreted as follows.",
"It is well known that angles $ \\lesssim 1/p_z R$ are important in the scattering from an isolated atom.",
"For coherent scattering from the whole chain, we require that the next atom should lie within the diffraction cone of the scattering from the preceding atom.",
"This condition can be satisfied provided the axis of the chain lies within a conical diffraction surface.",
"When it lies outside such a conical surface, noncoherent effects become important.",
"Finally, the additional requirement $|\\Psi _2^{(1)} |,|\\Psi _3^{(1)} |<<1$ which follows from the first condition in Eq.",
"(36) should be satisfied.",
"Combining this requirement with the condition (40), we find that the Lindhard angle $\\theta _L \\sim \\sqrt{U_{\\perp } / E} $ lies in the range defined by Eq.",
"(40), i.e., the average potential is applicable for particles above the barrier.",
"We wish to stress that the condition (40) does not contradict our initial assumption (b) when the condition $ \\theta _0 \\gtrsim 2/ \\sqrt{p_z L_z}$ is satisfied, since, for an extended potential $L_z >> p_z R^2$ , the inequality $1/p_z R >> 1/ \\sqrt{p_z L_z}$ always holds.",
"The mathematical framework of the present approach is similar to the eikonal approximation, but our method is valid also in the opposite limiting case.",
"It can be easily seen that the conditions derived above follow from the conditions of the eikonal approximation [22 ,23] (where the expansion is with respect to $p_z$ rather than with respect to $p$ ) provided $p_z R^2$ is replaced by $L_z$ .",
"The eikonal approximation neglects the diffraction effects [17 ,23] In the opposite limit discussed earlier, the diffraction effects become dominant since the diameter of the chain $R$ is much smaller than the dimensions of the Fresnel zone $ \\sqrt{L_z/p_z}$ and the scattering cross section related to diffraction $\\sim L_z/ p_z$ [see Eq.",
"(29)] exceeds considerably the classical cross section $\\sim R^2$ .",
"Naturally, such effects are observable only if the length determined by multiple scattering from potential fluctuations is much longer than the length of the chain."
]
] | 1606.05194 |
[
[
"Effective Chabauty for symmetric powers of curves"
],
[
"Abstract Faltings' theorem states that curves of genus $g \\geq 2$ have finitely many rational points.",
"Using the ideas of Faltings, Mumford, Parshin and Raynaud, one obtains an upper bound on the number of rational points, but this bound is too large to be used in any reasonable sense.",
"In 1985, Coleman showed that Chabauty's method, which works when the Mordell-Weil rank of the Jacobian of the curve is smaller than $g$, can be used to give a good effective bound on the number of rational points of curves of genus $g > 1$.",
"We draw ideas from nonarchimedean geometry to show that we can also give an effective bound on the number of rational points outside of the special set of $d$-th symmetric power of $X$, where $X$ is a curve of genus $g > d$, when the Mordell-Weil rank of the Jacobian of the curve is at most $g-d$ and the curve further satisfies certain rigid analytic conditions."
],
[
"Introduction",
"Throughout the paper, we assume that $X$ is a nice (smooth, projective, and geometrically integral) curve of genus $g$ defined over ${\\mathbb {Q}}$ that has a rational point $O \\in X({\\mathbb {Q}})$ , and $d \\ge 1$ .",
"We aim to generalize the following theorem of Coleman to $\\operatorname{Sym}^dX$ , the symmetric powers of curves: Theorem 1.1 (, Theorem 4) Let $g > 1$ and $p$ be a prime number.",
"Then there is an effectively computable bound $N(g,p)$ such that for every nice curve $X$ defined over ${\\mathbb {Q}}$ of good reduction at $p$ , such that $X$ is of genus $g$ and $g > \\operatorname{Rank}(\\operatorname{Jac}(X))({\\mathbb {Q}})$ , then $\\#X({\\mathbb {Q}}) \\le N(g,p).$ Although this theorem is weaker than Faltings' theorem for curves, which states that any curve of genus $g \\ge 2$ has finitely many rational points, Coleman's bounds are effective and sometimes sharp, in which case Theorem REF can be used to find all rational points of a given curve.",
"Previously, all known bounds were too large to be practical.",
"Coleman divides the set $X({\\mathbb {Q}}_p)$ into finitely many sets called residue disks; the set of ${\\mathbb {Q}}_p$ -points on each residue disk is in bijection with $p{\\mathbb {Z}}_p$ .",
"On each residue disk, a necessary condition for the ${\\mathbb {Q}}_p$ -points of $X$ (considered as an element of $p{\\mathbb {Z}}_p$ ) to be in $X({\\mathbb {Q}})$ is given as a power series equation; each ${\\mathbb {Q}}$ -point is a solution to the power series equation.",
"The number of such solutions can be estimated by using Newton polygons.",
"More generally, consider $\\operatorname{Sym}^dX$ .",
"While it seems plausible that one could generalize Chabauty's method to $\\operatorname{Sym}^dX$ , several problems exist (see § for more explanation).",
"One major such problem is the fact that $(\\operatorname{Sym}^dX)({\\mathbb {Q}})$ is not necessarily finite.",
"However, in such cases, all but finitely many rational points of $\\operatorname{Sym}^dX$ are contained in the special set: Definition 1.2 () Let $X/{\\mathbb {Q}}$ be a projective variety considered as a variety defined over $\\overline{{\\mathbb {Q}}}$ .",
"The special set of $X$ is the Zariski closure of the union of all images of nonconstant rational maps $f: G \\rightarrow X$ of group varieties $G$ into $X$ ; these rational maps may be defined over finitely generated extensions of ${\\mathbb {Q}}$ .",
"We denote the special set of $X$ by ${\\mathcal {S}}(X)$ (despite the name, this is a geometric object).",
"As in , we will find locally analytic functions (written as power series on residue disks) whose solutions contain $(\\operatorname{Sym}^dX)({\\mathbb {Q}})$ , defined by $p$ -adic integrals.",
"These power series cut out a rigid analytic subvariety of $(\\operatorname{Sym}^dX)^{\\textup {an}}$ , denoted $(\\operatorname{Sym}^dX)^{\\eta =0}$ (Here, $(\\operatorname{Sym}^dX)^{\\textup {an}}$ denotes the analytification of $\\operatorname{Sym}^dX$ , in the sense of rigid analytic geometry).",
"Then $(\\operatorname{Sym}^dX - {\\mathcal {S}}(\\operatorname{Sym}^dX))({\\mathbb {Q}})$ is contained in the set $\\lbrace P \\in (\\operatorname{Sym}^dX)^{\\eta =0} : P \\textup { is the point in a $ $-dimensional component in }(\\operatorname{Sym}^dX)^{\\eta =0}\\rbrace ,$$under the following assumption (which always holds if $ RankJ 1$):$ Assumption 1.3 Every positive-dimensional rigid analytic component of $(\\operatorname{Sym}^dX)^{\\eta =0}$ is contained in ${\\mathcal {S}}(\\operatorname{Sym}^dX)^{\\textup {an}}$ .",
"This assumption always holds if $\\operatorname{rk}J \\le 1$ , since we can always choose the locally analytic functions so that $(\\operatorname{Sym}^dX)^{\\eta = 0} \\subseteq \\overline{J({\\mathbb {Q}})}^{p\\textup {-adic}}$ , where $\\overline{J({\\mathbb {Q}})}^{p\\textup {-adic}}$ is at most 1-dimensional $p$ -adic Lie group.",
"The main result of this paper is the following; under Chabauty-type assumptions, as well as the above assumption, one can get an effective upper bound on the number of rational points of $\\operatorname{Sym}^dX$ outside of the special set: Theorem 1.4 Let $d \\ge 1$ , $p$ a prime, and $g \\ge 2$ .",
"Then there exists a number $N(p,d,g)$ that can be computed effectively, such that for every nice curve $X$ defined over ${\\mathbb {Q}}$ of good reduction at $p$ with $\\operatorname{Rank}J \\le g-d$ satisfying Assumption REF , $\\#\\lbrace Q \\in (\\operatorname{Sym}^dX)({\\mathbb {Q}}) \\mid \\textup {$ $ does not belong to the special set}\\rbrace \\le N(p,d,g).$$$ If we impose extra conditions on the above theorem, we can even get a better bound on $N(p,d,g)$ .",
"For example: Corollary 1.5 We can take $N(2,3,3) = 1539$ for any degree 7 odd hyperelliptic curve $X$ such that $\\operatorname{Rank}J({\\mathbb {Q}}) \\le 1$ and such that $X$ has good reduction at 2.",
"The problem of rational points on symmetric powers of curves have been studied in several papers.",
"One result is that of Debarre and Klassen , which studies the Fermat curves (projective plane curves given by $X^N + Y^N = Z^N, N \\ge 4$ ).",
"By Fermat's Last Theorem, we already know that these curves only have finitely many $K$ -points for any number field $K$ , and no nontrivial ${\\mathbb {Q}}$ -points, uses geometric methods to prove the following theorem: Theorem 1.6 () For $N \\ne 6$ , there are only finitely many number fields $K$ with degree $d = [K:{\\mathbb {Q}}] \\le N-2$ such that $F_N(K) \\ne F_N({\\mathbb {Q}})$ .",
"raises the question of applying Chabauty's method to symmetric powers of curves.",
"Then attempts to generalize to symmetric powers of curves: Theorem 1.7 () Let $1 < d < \\gamma $ , and let $X$ be a nice curve of genus $g >2$ and gonality $\\gamma $ , satisfying $\\operatorname{Rank}J({\\mathbb {Q}}) \\le g-d$ .",
"Then there exists a canonical divisor $M$ on $(\\operatorname{Sym}X^d)_{{\\mathbb {Q}}_p}$ such that the complement $\\operatorname{Sym}^dX \\backslash M$ has only finitely many rational points (here, a canonical divisor is a divisor of a meromorphic $d$ -form).",
"Further, $\\#((\\operatorname{Sym}^dX)({\\mathbb {Q}}) \\backslash \\operatorname{red}_p^{-1}(\\bar{M}({\\mathbb {F}}_p))) \\le \\#((\\operatorname{Sym}^dX)({\\mathbb {F}}_p) \\backslash \\bar{M}({\\mathbb {F}}_p)),$ where $\\operatorname{red}_p$ denotes the reduction modulo $p$ map, and $\\bar{M}$ denotes the reduction of $M$ $\\bmod \\unknown.",
"\\textup { } p$ .",
"Also, refines by removing the gonality from the hypothesis of the above statement, and also giving a sufficient criterion for when a residue disk contains a single rational point.",
"Also, he developed a method that can be used to compute $(\\operatorname{Sym}^2X)({\\mathbb {Q}})$ for some curves (two explicit examples are worked out in , §6).",
"Further, the components of the special set contained in the symmetric powers of curves have been studied, e.g.",
"in , for the case of $d=2$ .",
"As $d$ grows, the geometry becomes increasingly complicated, as in .",
"Theorem 1.8 () If $\\operatorname{Sym}^2X$ contains an elliptic curve, then $X$ is either bielliptic or hyperelliptic.",
"Two main ideas are required to obtain an effective bound for the number of points outside of the special set of $\\operatorname{Sym}^dX$ .",
"The first is the approximation of the shape of the generalized Newton polygons of multivariate power series, which gives an upper bound on the number of zeros of the power series equations on residue disks.",
"In general, approximating the shape of the Newton polygons of multivariate power series is hard, but in our case, we have: Xd(p) @>>[d]U@(->[r] (SymdX)(Qp) @(->[r] @>>[d] (SymdX)(p)[r]@>>[d] J (SymdX)(Fp)@(->[r] (SymdX)(Fp) While Chabauty's method typically deals with residue disks ${\\mathcal {U}}\\subseteq (\\operatorname{Sym}^dX)({\\mathbb {Q}}_p)$ , we will instead look at the preimage of the residue disk ${\\mathcal {U}}\\subseteq (\\operatorname{Sym}^dX)({\\mathbb {Q}}_p) \\subseteq (\\operatorname{Sym}^dX)(p)$ in $X^d(p)$ , where ${\\mathcal {U}}$ decomposes into a product of $d$ one-dimensional residue disks ${\\mathcal {U}}_i \\subseteq X(p)$ above $P_i \\in X(\\overline{{\\mathbb {F}}_p})$ .",
"Pullbacks of residue disks ${\\mathcal {U}}$ to $X^d(p)$ has the effect of change of variables on the local coordinates of ${\\mathcal {U}}$ into the uniformizing parameters of $X^d(p)$ , and this writes the multivariate power series constraints as a sum of $d$ single-variable power series (one variable for each power series) over a degree $d$ extension of ${\\mathbb {Q}}_p$ .",
"Then the Newton polygons are much easier to approximate.",
"However, if $\\#(\\operatorname{Sym}^dX)({\\mathbb {Q}}) = \\infty ,$ we wish to count only the rational points outside of the special set, while the $d$ power series equations in $d$ variables have infinitely many common zeros.",
"Under Assumption REF , all such rational points form a subset of the set of zero-dimensional components of $(\\operatorname{Sym}^dX)^{\\eta =0}$ .",
"And such components correspond to the stable intersections of the multivariate power series constraints, whereas the positive-dimensional components of $(\\operatorname{Sym}^dX)^{\\eta =0}$ do not.",
"Thus, we use deformation theory techniques coming from rigid analytic geometry, to deform the power series away from one another to obtain finite intersection in this case.",
"This finite intersection number corresponds to the upper bound on the points outside of the special set.",
"In §, we provide an outline of Chabauty's method for symmetric powers of curves.",
"However, there is an intrinsic difficulty to Chabauty's method that comes from the incongruity between the algebraic and analytic description of the rational points in $\\operatorname{Sym}^dX$ , necessitating Assumption REF .",
"This problem is explained in §REF .",
"Then § will define generalized Newton polygons and the approximation of the shape the Newton polygons, and combine this idea with § to obtain an upper bound for the number of points outside of the special set, when $(\\operatorname{Sym}^dX)^{\\eta =0}$ consists of only zero-dimensional components.",
"We deal with the general case in §, where we explain the idea of small $p$ -adic deformations.",
"Finally, in §, we obtain some consequences of having an effective bound for the number of rational points outside of the special set of $\\operatorname{Sym}^dX$ ."
],
[
"Acknowledgements",
"I would like to thank my advisor, Bjorn Poonen, for introducing me to Chabauty's method, for many helpful conversations on this project, and for his feedback on the exposition of this article.",
"This project became the topic of my PhD thesis at MIT.",
"I also thank Joseph Rabinoff for patiently explaining many of his results to me; many results in Section of this paper were built on his results.",
"I also benefited from conversations with Matt Baker, Jennifer Balakrishnan, Eric Katz, Samir Siksek, Bernd Sturmfels, and David Zureick-Brown."
],
[
"Chabauty on $\\operatorname{Sym}^dX$",
"In this section, we consider the problem of counting rational points outside of the special set of $\\operatorname{Sym}^dX$ .",
"Using Chabauty's method, we will reduce this problem to analyzing the common zeros of $d$ power series in $d$ variables, and the power series have specific forms."
],
[
"Classical Chabauty",
"The exposition in this subsection outlines the classical method of Chabauty that gives an upper bound on $\\#X({\\mathbb {Q}})$ ; what we have here is a summarized version of .",
"Let $\\iota $ be the ${\\mathbb {Q}}$ -embedding $\\iota : X & \\hookrightarrow J\\\\P & \\mapsto [P-O]$ where $J$ is the Jacobian of $X$ , viewed as the group of linear equivalence classes of degree-zero divisors on $X$ .",
"Then $J$ is an abelian variety of dimension $g$ over ${\\mathbb {Q}}$ .",
"By an abuse of notation, we denote $\\iota (X)$ as $X$ .",
"The inclusion $X({\\mathbb {Q}}) \\subseteq J({\\mathbb {Q}})$ holds, since $O \\in X({\\mathbb {Q}})$ .",
"Since $X({\\mathbb {Q}}) \\subseteq X({\\mathbb {Q}}_p)$ , we have $X({\\mathbb {Q}}) \\subseteq X({\\mathbb {Q}}_p) \\cap \\overline{J({\\mathbb {Q}})}$ , where $\\overline{J({\\mathbb {Q}})}$ denotes the $p$ -adic closure of $J({\\mathbb {Q}})$ inside $J({\\mathbb {Q}}_p)$ .",
"Chabauty's result is: Theorem 2.1 () Keep the notation as above.",
"Let $X$ be a curve of genus $g \\ge 2$ over ${\\mathbb {Q}}$ .",
"If $X$ has good reduction at a prime $p$ and if $\\dim \\overline{J({\\mathbb {Q}})} < g$ , then $X({\\mathbb {Q}}_p) \\cap \\overline{J({\\mathbb {Q}})}$ is finite.",
"To compute $\\#(X({\\mathbb {Q}}_p) \\cap \\overline{J({\\mathbb {Q}})})$ explicitly under the hypothesis of Theorem REF , let $J_{{\\mathbb {Q}}_p}$ and $X_{{\\mathbb {Q}}_p}$ be the base changes of $J$ and $X$ to ${\\mathbb {Q}}_p$ .",
"There is a bilinear pairing $J({\\mathbb {Q}}_p) \\times H^0(J_{{\\mathbb {Q}}_p}, \\Omega ^1) &\\rightarrow {\\mathbb {Q}}_p \\\\(Q, \\omega ) &\\mapsto \\int _{O}^Q \\omega $ which induces the logarithm homomorphism $\\log : J({\\mathbb {Q}}_p) \\rightarrow H^0(J_{{\\mathbb {Q}}_p}, \\Omega ^1)^* = T_OJ_{{\\mathbb {Q}}_p},$ where $T_OJ_{{\\mathbb {Q}}_p}$ denotes the tangent space to $J_{{\\mathbb {Q}}_p}$ at the origin $O$ .",
"Since $\\dim \\log (\\overline{J({\\mathbb {Q}})}) < g$ , there exists a hyperplane $H \\subseteq T_OJ_{{\\mathbb {Q}}_p}$ containing $\\log (\\overline{J({\\mathbb {Q}})})$ .",
"This hyperplane $H$ is defined by the vanishing of some $\\omega _J \\in H^0(J_{{\\mathbb {Q}}_p}, \\Omega ^1) \\cong T_OJ_{{\\mathbb {Q}}_p}^*$ , and the restriction of $\\omega _J$ to $X_{{\\mathbb {Q}}_p}$ can be uniquely identified with an $\\omega _X \\in H^0(X_{{\\mathbb {Q}}_p}, \\Omega ^1)$ via: Proposition 2.2 (, Proposition 2.2) The restriction map $H^0(J_{{\\mathbb {Q}}_p}, \\Omega ^1) \\rightarrow H^0(X_{{\\mathbb {Q}}_p}, \\Omega ^1)$ induced by $X \\hookrightarrow J$ is an isomorphism of ${\\mathbb {Q}}_p$ -vector spaces.",
"Then, the map induced from the above bilinear pairing using the $\\omega _J$ given by $J({\\mathbb {Q}}_p) &\\rightarrow {\\mathbb {Q}}_p \\\\Q &\\mapsto \\int _{O}^Q \\omega _J$ vanishes on $\\overline{J({\\mathbb {Q}})}$ by construction, so $\\# (X({\\mathbb {Q}}_p) \\cap \\overline{J({\\mathbb {Q}})})$ is bounded above by the number of zeros of the restriction $\\eta : X({\\mathbb {Q}}_p) &\\rightarrow {\\mathbb {Q}}_p\\\\Q &\\mapsto \\int _O^Q \\omega _X,$ where $\\omega _X = \\iota ^* \\omega _J$ .",
"From a computational perspective, it is known that $\\omega $ has a well-defined power series expansion in terms of a uniformizer $t$ on small enough open subgroups $U$ of $X({\\mathbb {Q}}_p)$ .",
"In fact, $U$ can be taken to be residue disks of the reduction $\\operatorname{red}_p: X({\\mathbb {Q}}_p) \\twoheadrightarrow X({\\mathbb {F}}_p)$ , which are the preimages of any point $Q \\in X({\\mathbb {F}}_p)$ .",
"Such residue disk $U$ can be parametrized by a uniformizer $t$ , which gives a set bijection $t: U \\rightarrow p{\\mathbb {Z}}_p$ .",
"Then one expresses the locally analytic function $\\eta $ as a power series in terms of $t$ on $U$ ; the local coordinates can be chosen so that $\\omega (t) \\in {\\mathbb {Z}}_p[[t]]$ , and the number of zeros of $\\eta $ on each residue disk can then be estimated using Newton polygons."
],
[
"Chabauty on $\\operatorname{Sym}^d X$",
"In theory, it seems plausible that Chabauty's method could still apply to any higher-dimensional variety $Y$ , where the Albanese variety $\\operatorname{Alb}(Y)$ is used in place of the Jacobian, look for all rational points on the image of the Albanese map using a similar technique.",
"However, there exist several problems in generalizing Chabauty's method to arbitrary higher-dimensional varieties.",
"(1) $\\operatorname{Alb}(Y)$ may be trivial: since $\\dim \\operatorname{Alb}(Y) = h^0(Y, \\Omega _1)$ , if $h^0(Y, \\Omega _1) = 0$ , then Chabauty's method yields nothing.",
"For example, if $Y$ is a K3 surface or an Enriques surface, $h^{0,1} = 0$ , so Chabauty's method cannot apply.",
"(2) To understand $Y({\\mathbb {Q}})$ , we need to understand the (rational points of the) fibres of $j$ as well as the rational points of $j(Y)$ .",
"Understanding the fibres may be complicated.",
"(3) The case of higher-dimensional varieties allows for the possibility that $\\#Y({\\mathbb {Q}}) = \\infty $ : for example, if $Y = \\operatorname{Sym}^2 X$ for a hyperelliptic curve $X: y^2 = f(x)$ , then $\\lbrace (t, \\sqrt{f(t)}),(t, -\\sqrt{f(t)})\\rbrace \\in (\\operatorname{Sym}^2X)({\\mathbb {Q}})$ for all $t \\in {\\mathbb {Q}}$ .",
"The first two problems are taken care of by choosing $Y = \\operatorname{Sym}^d X$ , where $X$ is a nice curve of sufficiently high genus $g$ (to be chosen later); then $\\operatorname{Alb}(Y) = \\operatorname{Jac}(X)=:J$ , so the Albanese variety is nontrivial, and understanding the fibres of the Albanese map $j: (\\operatorname{Sym}^dX)({\\mathbb {Q}}) &\\rightarrow J({\\mathbb {Q}})\\\\\\lbrace P_1, \\ldots , P_d\\rbrace &\\mapsto [P_1 + \\cdots + P_d - d\\cdot O],$ is not too difficult: Lemma 2.3 Suppose that $Q \\in (\\operatorname{Sym}^dX)({\\mathbb {Q}})$ .",
"Then the set of rational points on the fibre of $j$ containing $Q$ is isomorphic to ${\\mathbb {P}}^n({\\mathbb {Q}})$ for some $n \\ge 0$ .",
"As $J$ parametrizes the equivalence classes of degree-0 divisors, $Q$ is identified with an effective divisor on $X$ .",
"By *Theorem II.5.19, the set of points giving rise to the same divisor is isomorphic to a finite-dimensional vector space.",
"In particular, if a fibre contains two distinct rational points, then dimension of this vector space is at least 1.",
"To deal with the last problem, we recall from : Theorem 2.4 () Let $A/{\\mathbb {Q}}$ be an abelian variety, and $X \\subseteq A$ be a closed subvariety.",
"Then there exist finitely many subvarieties $Y_i \\subset X$ such that each $Y_i$ is a coset of an abelian subvariety of $A$ and $X({\\mathbb {Q}}) = \\bigcup Y_i({\\mathbb {Q}}).$ Apply this theorem to the image of $j$ to get $j((\\operatorname{Sym}^dX)({\\mathbb {Q}})) = \\bigcup _{\\textup {finite}} Y_i({\\mathbb {Q}})$ .",
"From Lemma REF and Theorem REF , we see that there are two ways of obtaining $\\#(\\operatorname{Sym}^dX)({\\mathbb {Q}}) = \\infty $ : either at least one of the fibres of $j$ is nontrivial, or there exists some $Y_i$ with $\\dim Y_i > 0$ .",
"Excluding the ${\\mathbb {Q}}$ -points on $\\operatorname{Sym}^dX$ accounted by these two possibilities, we are left with finitely many rational points on $\\operatorname{Sym}^dX$ .",
"However, to avoid ambiguity coming from Faltings' theorem, one excludes the special set (Definition REF ) instead, which includes both of the aforementioned possibilities.",
"For the rest of the section, assume that $X$ satisfies $\\operatorname{rk}J \\le g-d$ .",
"Further, let $p$ be a prime, and let $X$ have good reduction at $p$ .",
"We observe that $(\\operatorname{Sym}^dX)({\\mathbb {Q}}) \\subseteq j^{-1}(j(\\operatorname{Sym}^dX)({\\mathbb {Q}}_p) \\cap \\overline{J({\\mathbb {Q}})}).$ We obtain locally analytic functions come from integrating $\\omega _i \\in H^0(J_{{\\mathbb {Q}}_p}, \\Omega ^1)$ , as in the previous section.",
"However, we use the following stronger definition: Definition 2.5 For any $\\omega _J \\in H^0(J_{K}, \\Omega ^1)$ , define the map $\\eta : J(\\overline{{\\mathbb {Q}}}_p) \\rightarrow \\overline{{\\mathbb {Q}}}_p$ by taking the inverse limit of the maps $\\eta _K: J(K) &\\rightarrow K\\\\Q &\\mapsto \\int _{O}^Q \\omega _J,$ for each $p$ -adic field $K$ .",
"The $p$ -adic integrals also satisfy for $Q_1, Q_2 \\in J(\\overline{{\\mathbb {Q}}}_p)$ and $\\omega _J \\in H^0(J_{{\\mathbb {Q}}_p}, \\Omega ^1)$ , $\\int _{O}^{Q_1+Q_2}\\omega _J = \\int _{O}^{Q_1}\\omega _J + \\int _{Q_1}^{Q_1 + Q_2}\\omega _J = \\int _{O}^{Q_1}\\omega _J + \\int _{O}^{Q_2}\\omega _J,$ where the first equality follows from linearity, and the second equality follows from the translation-invariance of $p$ -adic integrals.",
"Then we may define the integral on $\\operatorname{Sym}^dX$ via the pullback of the integral on $W_d \\subseteq J$ , which can be written using the above as $\\int _{O}^{[P_1 + \\cdots + P_d - d \\cdot O]}\\omega _J = \\int _O^{[P_1-O]} \\omega _J + \\cdots + \\int _O^{[P_d-O]}\\omega _J.$ Therefore, the corresponding locally analytic function on $\\operatorname{Sym}^dX$ can be written as $\\eta \\colon (\\operatorname{Sym}^dX)({\\mathbb {Q}}_p) &\\rightarrow {\\mathbb {Q}}_p\\\\\\lbrace P_1,P_2, \\ldots , P_d\\rbrace &\\mapsto \\eta _J([P_1+P_2+ \\cdots + P_d-d\\infty ])\\\\& = \\int _O^{P_1} \\omega _X + \\int _O^{P_2} \\omega _X +\\cdots + \\int _O^{P_d} \\omega _X,$ where $\\omega _X \\in H^0(X_{{\\mathbb {Q}}_p}, \\Omega ^1)$ is the pullback of $\\omega _J$ via the isomorphism given in Proposition REF , and the $P_i$ are defined over some field $K$ with $[K:{\\mathbb {Q}}] \\le d$ .",
"Since $\\dim \\overline{J({\\mathbb {Q}})} \\le g-d$ , then there exist $(\\omega _J)_1, (\\omega _J)_2, \\ldots , (\\omega _J)_d \\in H^0(J_{{\\mathbb {Q}}_p}, \\Omega ^1)$ that are linearly independent, such that the corresponding locally analytic functions on $J_{{\\mathbb {Q}}_p}$ obtained by integrating the $(\\omega _J)_i$ vanish on $\\overline{J({\\mathbb {Q}})}$ .",
"Let $\\eta _1, \\eta _2, \\ldots , \\eta _d$ be the locally analytic functions on $\\operatorname{Sym}^dX$ corresponding to $(\\omega _J)_1, (\\omega _J)_2, \\ldots , (\\omega _J)_d$ , respectively.",
"Possibly $\\eta _1, \\eta _2, \\ldots , \\eta _d$ have infinitely many common zeros, which contain either the linear system of equivalent divisors parametrized by ${\\mathbb {P}}^n$ for some $n \\ge 1$ , or the (infinitely many) rational points coming from the rational points in the special set $({\\mathcal {S}}(\\operatorname{Sym}^dX))({\\mathbb {Q}})$ ."
],
[
"Explicit parametrization of points on residue disks",
"From §REF , we have (at least) $d$ nontrivial and independent locally analytic functions on $(\\operatorname{Sym}^dX)({\\mathbb {Q}}_p)$ whose common zeros contain the $p$ -adic points $j^{-1}((j(\\operatorname{Sym}^dX))({\\mathbb {Q}}_p) \\cap \\overline{J({\\mathbb {Q}})})$ .",
"We estimate the number of common zeros of these analytic functions away from the special set of $\\operatorname{Sym}^dX$ .",
"In order to do this explicitly, we work locally to get power series expansions.",
"Since $X$ has good reduction at $p$ , $X$ is a smooth proper variety over ${\\mathbb {Z}}_p$ .",
"This implies that $\\operatorname{Sym}^dX$ is smooth and proper over ${\\mathbb {Z}}_p$ .",
"By the valuative criterion for properness, $(\\operatorname{Sym}^dX)({\\mathbb {Q}}_p) = (\\operatorname{Sym}^dX)({\\mathbb {Z}}_p)$ .",
"Definition 2.6 A residue disk ${\\mathcal {U}}$ of $\\operatorname{Sym}^dX$ is the preimage of a point in $(\\operatorname{Sym}^dX)({\\overline{{\\mathbb {F}}}}_p)$ under the reduction modulo-$p$ map $\\operatorname{red}_{p}: (\\operatorname{Sym}^dX)(p) \\twoheadrightarrow (\\operatorname{Sym}^dX)({\\overline{{\\mathbb {F}}}}_p).$ If $K$ is a finite extension of ${\\mathbb {Q}}_p$ , the set of $K$ -points of the residue disk ${\\mathcal {U}}$ are defined to be the set ${\\mathcal {U}}\\cap (\\operatorname{Sym}^dX)(K)$ , and these are denoted ${\\mathcal {U}}(K)$ .",
"Let ${\\mathcal {U}}$ be the residue disk over $O \\in (\\operatorname{Sym}^dX)({\\overline{{\\mathbb {F}}}}_p)$ .",
"Then ${\\mathcal {U}}$ fits into the exact sequence $0 \\rightarrow {\\mathcal {U}}\\rightarrow J(p) = J({\\mathcal {O}}_{p}) \\rightarrow J({\\overline{{\\mathbb {F}}}}_p) \\rightarrow 0,$ where the equality in the middle follows from the valuative criterion for properness.",
"Then for any finite extension $K$ of ${\\mathbb {Q}}_p$ , we have ${\\mathcal {U}}(K) = \\lbrace {\\mathcal {P}}\\in (\\operatorname{Sym}^dX)(K): \\operatorname{red}_{p}({\\mathcal {P}}) = \\lbrace P_1, \\ldots , P_d\\rbrace \\rbrace $ .",
"Thus, by a Hensel-type argument, there is a bijection $(u_1, \\ldots , u_d): {\\mathcal {U}}(K) \\stackrel{\\sim }{\\longrightarrow } (p_K {\\mathcal {O}}_K)^d$ between the set of $K$ -points of the residue disk mapping to $\\lbrace P_1, \\ldots , P_d\\rbrace $ via some local coordinates $u_1, \\ldots , u_d$ , where $p_K$ is the uniformizer of $K$ .",
"In practice, this parametrization is not practical: §REF suggests that we write the higher-dimensional integrals in terms of several 1-dimensional integrals expanded around various ${\\mathbb {F}}_{p^n}$ -points $P_i$ ."
],
[
"The case of $\\operatorname{Sym}^2X$",
"For simplicity, we first consider the case when $d=2$ .",
"First suppose that we consider the residue disk ${\\mathcal {U}}({\\mathbb {Q}}_p)$ which consists of the ${\\mathbb {Q}}_p$ -points in $\\operatorname{Sym}^2X$ reducing to $\\lbrace P, P\\rbrace \\in (\\operatorname{Sym}^2X)({\\mathbb {F}}_p)$ for some $P \\in X({\\mathbb {F}}_p)$ .",
"The completion of the local ring of $(X \\times X)_{{\\mathbb {Q}}_p}$ near any pair of points $(Q_1, Q_2) \\in X \\times X$ reducing to $\\lbrace P, P\\rbrace $ is given by ${\\mathbb {Q}}_p[[t_1, t_2]]$ , where $t_1$ and $t_2$ denote the uniformizers for the set of ${\\mathbb {Q}}_p$ -points of the residue disks ${\\mathcal {U}}_1$ around $Q_1$ and ${\\mathcal {U}}_2$ around $Q_2$ in $X$ , respectively.",
"We further assume that $t_1$ and $t_2$ vanish at $Q_1^{\\prime }$ and $Q_2^{\\prime }$ , respectively.",
"This means that we have two bijections $t_1: {\\mathcal {U}}_1 \\stackrel{\\sim }{\\longrightarrow } p{\\mathbb {Z}}_p, \\quad t_2: {\\mathcal {U}}_2 \\stackrel{\\sim }{\\longrightarrow } p{\\mathbb {Z}}_p.$ Since $t_1(Q_1^{\\prime }) = 0$ , and $t_2(Q_2^{\\prime }) = 0$ , for any $\\omega \\in H^0(J_{{\\mathbb {Q}}_p}, \\Omega ^1)$ , the Coleman integral $\\eta : \\operatorname{Sym}^dX({\\mathbb {Q}}_p) &\\rightarrow {\\mathbb {Q}}_p\\\\\\lbrace Q_1, Q_2\\rbrace &\\mapsto \\int _{O}^{\\lbrace Q_1, Q_2\\rbrace } \\omega $ can be written as the following, in terms of the $t_i$ : $\\eta (\\lbrace Q_1, Q_2\\rbrace ) &= \\int _{O}^{\\lbrace Q_1, Q_2\\rbrace }\\omega = \\int _{O}^{Q_1}\\omega + \\int _{O}^{Q_2}\\omega \\\\&= \\int _{O}^{Q_1^{\\prime }}\\omega + \\int _{Q_1^{\\prime }}^{Q_1}\\omega + \\int _{O}^{Q_2^{\\prime }}\\omega + \\int _{Q_2^{\\prime }}^{Q_2}\\omega \\\\&= C + \\int _0^{t_1(Q_1)}\\omega (t_1) + \\int _0^{t_2(Q_2)}\\omega (t_2),$ where $C = \\int _O^{Q_1^{\\prime }}\\omega +\\int _O^{Q_2^{\\prime }}\\omega $ is a constant in ${\\mathbb {Q}}_p$ (that depends on the choice of $Q_i^{\\prime }$ ).",
"One can relate the $t_i$ to the original local coordinates of ${\\mathcal {U}}$ .",
"If $r(Q_1, Q_2) = (P_1, P_2)$ in $(\\operatorname{Sym}^dX)({\\mathbb {F}}_p)$ , then the completion of the local ring at $\\lbrace Q_1, Q_2\\rbrace \\in (\\operatorname{Sym}^2X)({\\mathbb {Q}}_p)$ is given by ${\\mathbb {Q}}_p[[t_1, t_2]]^{S_2} = {\\mathbb {Q}}_p[[u_1, u_2]],$ so one could choose the $t_i$ and the $u_i$ to satisfy $u_1 = t_1 + t_2$ and $u_2 = t_1t_2$ .",
"This means that for each point $\\lbrace Q_1, Q_2\\rbrace \\in (\\operatorname{Sym}^2X)({\\mathbb {Q}}_p)$ , which corresponds bijectively to a unique pair $(u_1, u_2)$ , there are two pairs $(t_1, t_2)$ that correspond to it.",
"On the other hand, if the residue disk ${\\mathcal {U}}$ were the preimage of $\\lbrace P_1, P_2\\rbrace \\in (\\operatorname{Sym}^2X)({\\mathbb {F}}_p)$ with $P_1 \\ne P_2$ under the reduction by $p$ map, the situation is simpler, as we have the following description of the residue disk.",
"${\\mathcal {U}}&= \\lbrace \\lbrace Q_1, Q_2\\rbrace : Q_i \\in X(W({\\mathbb {F}}_{p^2})) \\textup { reducing to } P_i \\in X({\\mathbb {F}}_{p^2})\\rbrace \\\\&\\subseteq X(W({\\mathbb {F}}_{p^2})) \\times X(W({\\mathbb {F}}_{p^2}))$ where $W({\\mathbb {F}}_{p^2})$ denotes the Witt ring of ${\\mathbb {F}}_{p^2}$ .",
"Thus, there are the bijections $(t_1, t_2): {\\mathcal {U}}\\stackrel{\\sim }{\\longrightarrow } p{\\mathbb {Z}}_p \\times p{\\mathbb {Z}}_p,$ possibly defined over some quadratic extension of ${\\mathbb {Q}}_p$ , where ${\\mathcal {U}}_i$ denotes the residue disk around $Q_i$ .",
"Choose the basepoints of the lifts $t_1(Q_1^{\\prime }) = 0$ , and $t_2(Q_2^{\\prime }) = 0$ .",
"Then, we can write for each $\\omega \\in H^0(X_{{\\mathbb {Q}}_p}, \\Omega ^1)$ $\\eta (\\lbrace Q_1, Q_2\\rbrace ) &= \\int _{O}^{\\lbrace Q_1, Q_2\\rbrace }\\omega = \\int _{O}^{Q_1}\\omega + \\int _{O}^{Q_2}\\omega \\\\&= \\int _{O}^{Q_1^{\\prime }}\\omega + \\int _{Q_1^{\\prime }}^{Q_1}\\omega + \\int _{O}^{Q_1^{\\prime }}\\omega + \\int _{Q_2^{\\prime }}^{Q_2}\\omega \\\\&= C + \\int _0^{t_1(Q_1)}\\omega (t_1) + \\int _0^{t_2(Q_2)}\\omega (t_2),$ where the $\\omega (t_i)$ are defined (under appropriate scaling) over the ring of integers of some quadratic extension of ${\\mathbb {Q}}_p$ , and $C = \\int _O^{Q_1^{\\prime }}\\omega + \\int _O^{Q_2^{\\prime }}\\omega $ is a constant in ${\\mathbb {Z}}_p$ (that depends on the choice of $Q_1^{\\prime }$ and $Q_2^{\\prime }$ ) In this case, each pair $(t_1, t_2)$ represents a point on $\\operatorname{Sym}^2X$ exactly once.",
"In either cases, we are able to express the Coleman integral in the following form: Definition 2.7 A power series $f \\in K[[t_1, \\ldots , t_n]]$ is said to be pure if each of its terms are of the form $Ct_i^N$ , with $C \\in K$ and $N \\in {\\mathbb {Z}}_{\\ge 0}$ .",
"In particular, a pure power series does not contain any term that is a product of more than one variable.",
"Our goal for the rest of the section is to find pure power series that are related to the locally analytic functions in $d$ variables that we obtain from Chabauty's method."
],
[
"The case of $\\operatorname{Sym}^dX$",
"In this section, we generalize §REF .",
"More concretely, our aim is to express each $p$ -adic integral obtained from a $\\omega \\in H^0(X_{{\\mathbb {Q}}_p}, \\Omega ^1)$ (see §REF ), which is a power series of each residue disk of $(\\operatorname{Sym}^dX)({\\mathbb {Q}}_p)$ , as a pure power series over some extension field of ${\\mathbb {Q}}_p$ on the residue disk.",
"We will show that this is possible by doing a change of variables on the local coordinates of each residue disk.",
"We fix a holomorphic differential $\\omega $ from which we get one of the $d$ power series vanishing on $j^{-1}(\\overline{J({\\mathbb {Q}})} \\cap j(\\operatorname{Sym}^dX)({\\mathbb {Q}}_p))$ as in section §REF .",
"We now consider the residue disk given as the preimage of the point $\\lbrace P_1, \\ldots , P_d\\rbrace \\in (\\operatorname{Sym}^dX)({\\mathbb {F}}_p)$ under the reduction map modulo $p$ .",
"The multiset $\\lbrace P_1, \\ldots , P_d\\rbrace $ can be decomposed as the disjoint union of the multisets of the form ${\\mathcal {S}}:= \\lbrace P_{i_1}, \\ldots , P_{i_s}: P_{i_1} = \\cdots = P_{i_s}, P_{i_1} \\ne P_j \\textup { for } j \\in \\lbrace 1, \\ldots , d\\rbrace - \\lbrace i_1, \\ldots , i_s\\rbrace \\rbrace ,$ where $\\lbrace i_1, \\ldots , i_s\\rbrace \\subseteq \\lbrace 1, \\ldots , d\\rbrace $ .",
"Consider the locally analytic function obtained by $p$ -adic integration with respect to an $\\omega \\in H^0(J_{{\\mathbb {Q}}_p}, \\Omega ^1)$ on $(\\operatorname{Sym}^dX)({\\mathbb {Q}}_p)$ given by $\\eta _{\\omega }: \\left(\\prod _{{\\mathcal {S}}}(\\operatorname{Sym}^{\\#{\\mathcal {S}}}X)\\right)({\\mathbb {Q}}_p) &\\rightarrow {\\mathbb {Q}}_p\\\\\\prod _{{\\mathcal {S}}}(\\lbrace P_{i_1}, \\ldots , P_{i_s}\\rbrace ) & \\mapsto \\sum _{{\\mathcal {S}}} \\left(\\sum _{k=1}^{\\#{\\mathcal {S}}} \\int _O^{[P_{i_k} - O]}\\omega \\right).",
"\\\\$ As in §REF , the terms corresponding to the different ${\\mathcal {S}}$ can be separated.",
"So we consider the terms that depend on ${\\mathcal {S}}$ from the above expression; namely the terms $\\sum _{k=1}^{\\#{\\mathcal {S}}} \\int _O^{[P_{i_k} - O]}\\omega $ .",
"When $\\#{\\mathcal {S}}= 1$ , we can expand as in , but since $P_{i_1} \\in X({\\mathbb {F}}_q)$ where $q = p^{\\ell }$ for some $\\ell \\ge 1$ , its expansion with respect to the uniformizer $t_1$ satisfies $\\eta _{{\\mathcal {S}}, \\omega } \\in W({\\mathbb {F}}_q)[\\frac{1}{p}][[t_1]]$ .",
"Here, $W({\\mathbb {F}}_q)$ denotes the Witt ring of ${\\mathbb {F}}_q$ , and $W({\\mathbb {F}}_q)[\\frac{1}{p}]$ is the fraction field of the Witt ring.",
"This is the degree-$p^{\\ell }$ unramified extension of ${\\mathbb {Q}}_p$ , and the points in the residue disk of $P_{i_1}$ are parametrized by $t_1 \\in pW({\\mathbb {F}}_q)$ .",
"Now suppose that $\\#{\\mathcal {S}}\\ge 2$ and $P_{{\\mathcal {S}}} \\in X({\\mathbb {F}}_q)$ .",
"Let ${\\mathcal {U}}$ be the residue disk in $(\\operatorname{Sym}^{\\#{\\mathcal {S}}}X)({\\mathbb {Q}}_p)$ reducing to $\\lbrace P_{{\\mathcal {S}}}, \\ldots , P_{{\\mathcal {S}}}\\rbrace $ (the multiset where $P_{{\\mathcal {S}}}$ is repeated $\\#{\\mathcal {S}}$ times).",
"Let ${\\mathcal {U}}_{i_1}, \\ldots ,{\\mathcal {U}}_{i_s}$ be the residue disks around $Q_{i_1}, \\ldots , Q_{i_s}$ in $X(W({\\mathbb {F}}_q))$ with the set bijections $t_{i_j}: {\\mathcal {U}}_{i_j} \\stackrel{\\sim }{\\longrightarrow } pW({\\mathbb {F}}_q)$ for each $1 \\le j \\le s$ , with $t_{i_j}(0) = Q_{i_j}$ .",
"Then the Coleman integral $\\int _O^{Q_{i_1}, \\ldots , Q_{i_s}} \\omega $ can be written as $C + \\int _0^{t_{i_1}(Q_{i_1})} \\omega + \\cdots + \\int _0^{t_{i_s}(Q_{i_s})} \\omega $ where $C$ is a constant depending on the choice of the $Q_{i_j}$ , and $\\omega $ is scaled to have coefficients in $W({\\mathbb {F}}_q)$ .",
"The $t_{i_j}$ are related to the original local coordinates $(u_{i_1}, \\ldots , u_{i_s})$ of ${\\mathcal {U}}$ by $W({\\mathbb {F}}_q)\\left[\\frac{1}{p}\\right][[t_{i_1}, \\ldots , t_{i_s}]]^{S_s} = W({\\mathbb {F}}_q)\\left[\\frac{1}{p}\\right][[u_{i_1}, \\ldots , u_{i_s}]]$ so one can take the $u_{i_k}$ to be the $k$ -th elementary symmetric polynomial in $t_{i_j}$ .",
"This means that for each point $\\lbrace Q_{i_1}, \\ldots , Q_{i_s}\\rbrace \\in (\\operatorname{Sym}^sX)(W({\\mathbb {F}}_q)\\left[\\frac{1}{p}\\right])$ , which corresponds bijectively to a unique pair $(u_{i_1}, \\ldots , u_{i_s})$ , there are $s!$ choices for the $s$ -tuple $(t_{i_j})_j$ .",
"The above discussion leads to the following proposition: Proposition 2.8 (a) A power series $\\eta _{\\omega } = \\left(\\prod _{{\\mathcal {S}}}(\\operatorname{Sym}^sX)\\right)({\\mathbb {Q}}_p) \\rightarrow {\\mathbb {Q}}_p$ obtained from $p$ -adic integration can be re-written via a change of variables as a pure power series, whose coefficients are contained in some extension of ${\\mathbb {Q}}_p$ of degree at most $d$ .",
"(b) Suppose that one obtains $d$ power series $\\eta _1, \\ldots , \\eta _d$ via Chabauty's method as outlined in the previous section, and that one rewrites these power series as $\\eta ^{\\prime }_1, \\ldots , \\eta ^{\\prime }_d$ , where $\\eta ^{\\prime }_i$ are pure power series obtained from part (a).",
"Then there is a $N$ -to-one correspondence between the common zeros of the $\\eta _i$ and $\\eta ^{\\prime }_i$ , where $N = \\prod _{{\\mathcal {S}}} (\\#{\\mathcal {S}})!$ .",
"Further, the solutions to $\\eta _i^{\\prime }$ that correspond to the points $\\operatorname{Sym}^dX({\\mathbb {Q}}_p)$ have $p$ -adic valuations of at least $1/d$ .",
"Now, it remains to associate Newton polygons to these power series, and apply arguments analogous to to try to count the common zeros."
],
[
"Comparison of the algebraic loci and the analytic loci on $\\operatorname{Sym}^dX$",
"We are interested in comparing different sets that contain $(\\operatorname{Sym}^dX)({\\mathbb {Q}})$ inside $(\\operatorname{Sym}^dX)({\\mathbb {C}}_p)$ .",
"The different subsets of $(\\operatorname{Sym}^dX)({\\mathbb {C}}_p)$ that we consider are described below: (i) Faltings' theorem says that given the natural embedding $j: \\operatorname{Sym}^dX \\hookrightarrow J$ using the basepoint $O \\in X$ , $j(\\operatorname{Sym}^dX)({\\mathbb {Q}}) = \\bigcup _{\\textup {finite}} Y_i({\\mathbb {Q}}),$ where the $Y_i$ are cosets of abelian subvarieties of $J$ with $Y \\subseteq W_d$ .",
"The set of points that we are interested in is the set of $p$ -points of the inverse image $\\bigcup j^{-1}(Y_i)$ , denoted ${\\mathcal {F}}(\\operatorname{Sym}^dX)({\\mathbb {C}}_p)$ .",
"We note that ${\\mathcal {F}}(\\operatorname{Sym}^dX)(p)$ depends on the choice of the $Y_i$ .",
"(ii) The set of ${\\mathbb {C}}_p$ -points of the special set ${\\mathcal {S}}(\\operatorname{Sym}^dX)$ of $\\operatorname{Sym}^dX$ : recall that the special set was defined in Definition REF .",
"This set will be denoted ${\\mathcal {S}}(\\operatorname{Sym}^dX)({\\mathbb {C}}_p)$ .",
"(iii) The set $\\lbrace P \\in (\\operatorname{Sym}^dX)({\\mathbb {C}}_p): \\eta _i(j(P)) = 0 \\textup { for all } 1 \\le i \\le d\\rbrace ,$ where the $\\eta _i$ are $d$ independent locally analytic functions on $J(p)$ that vanish on $\\overline{J({\\mathbb {Q}})}$ arising from Chabauty's method.",
"Since this definition depends on the choice of the $\\eta _i$ ; we will fix one such choice here.",
"We denote this set by $(\\operatorname{Sym}^dX)^{\\eta =0}$ .",
"In this section, we relate these different sets.",
"If $d=1$ , $g \\ge 2$ and $\\operatorname{rk}J < g$ , then all of the above sets are zero-dimensional, which makes the comparison simple: We have $\\emptyset = {\\mathcal {S}}(X)({\\mathbb {C}}_p) \\subseteq {\\mathcal {F}}(X)({\\mathbb {C}}_p) \\subseteq (X)^{\\eta =0}$ .",
"For $d>1$ , we will see that these sets do not obey a linear containment relation; in particular, there does not seem to be any inclusion relation between ${\\mathcal {S}}(\\operatorname{Sym}^dX)$ and $(\\operatorname{Sym}^dX)^{\\eta =0}$ ; this necessitates an extra technical hypothesis of Assumption REF to force such an inclusion.",
"This seems to be an intrinsic limitation of Chabauty's method on higher-dimensional varieties; a new idea seems to be necessary to obtain more precise information on the rational points of $\\operatorname{Sym}^dX$ .",
"We review some basics and terminology of rigid analytic geometry in $§\\ref {S: RAG}$ that will enable the comparison of the sets above in $§\\ref {S: comparison}$ ."
],
[
"$(\\operatorname{Sym}^dX)^{\\eta =0}$ as a rigid analytic space",
"We view $(\\operatorname{Sym}^dX)^{\\eta =0}$ as a rigid analytic space, whose admissible cover by affinoid spaces are given by the vanishing of certain Coleman integrals on residue disks, as in §REF (which are elements of the Tate algebra over ${\\mathbb {Q}}_p$ with $d$ variables).",
"For the basic terminology, we refer the readers to and .",
"We note that there is a notion of irreducible components on rigid analytic spaces.",
"The theory of irreducible components was first suggested in , and simplified in .",
"We summarize here: Definition 3.1 A rigid analytic space $X$ is disconnected if there exists an admissible open covering $\\lbrace U,V\\rbrace $ of $X$ with $U \\cap V = \\emptyset $ , where $U,V \\ne \\emptyset $ .",
"Otherwise, $X$ is said to be connected.",
"Definition 3.2 Let $X$ be a rigid analytic space that admits a cover of affinoid spaces $\\lbrace \\operatorname{Sp}A_{\\lambda }\\rbrace _{\\lambda \\in \\Lambda }$ .",
"A morphism $\\pi : \\widetilde{X} \\rightarrow X$ is said to be a normalization if it is isomorphic to the morphism obtained by gluing $\\operatorname{Sp}(\\widetilde{A}_{\\lambda }) \\rightarrow \\operatorname{Sp}A_{\\lambda }$ , where $\\widetilde{A_{\\lambda }}$ denotes the normalization of $A$ in the usual sense.",
"It is known that for any rigid analytic space $X$ , we can find a normalization $\\pi : \\widetilde{X} \\rightarrow X$ ; for example, see *Theorem 1.2.2.",
"Definition 3.3 (, Definition 2.2.2) The irreducible components of a rigid analytic space $X$ are the images of the connected components $X_i$ of the normalization $\\widetilde{X}$ under the normalization map $\\pi : \\widetilde{X} \\rightarrow X$ .",
"Remark 3.4 When $X = \\operatorname{Sp}(A)$ is affinoid, the irreducible components of $X$ are the analytic sets $\\operatorname{Sp}(A/{\\mathfrak {p}})$ for the finitely many minimal prime ideals ${\\mathfrak {p}}$ of the noetherian ring $A$ ."
],
[
"Comparing algebraic and analytic loci in $\\operatorname{Sym}^dX$",
"Now we determine the containment relations of the three sets mentioned at the beginning of this chapter; namely, ${\\mathcal {F}}(\\operatorname{Sym}^dX), {\\mathcal {S}}(\\operatorname{Sym}^dX)$ , and $(\\operatorname{Sym}^dX)^{\\eta =0}$ , as well as $(\\operatorname{Sym}^dX)({\\mathbb {Q}})$ .",
"Lemma 3.5 For any smooth projective curve $X$ with the choice of a rational point $O \\in X({\\mathbb {Q}})$ with good reduction at $p$ , one has $(\\operatorname{Sym}^dX)({\\mathbb {Q}}) \\subseteq (\\operatorname{Sym}^dX)^{\\eta =0}.$ By construction, each $\\eta _i$ mentioned in part (iii) at the beginning of this section satisfies $\\eta _i(P) = 0$ for all $P \\in J({\\mathbb {Q}})$ .",
"Thus, $(\\operatorname{Sym}^dX)({\\mathbb {Q}}) \\subseteq j^{-1}(J({\\mathbb {Q}})) \\subseteq (\\operatorname{Sym}^dX)(p)^{\\eta =0}$ .",
"Lemma 3.6 We keep the notation of $Y_i$ and ${\\mathcal {S}}(\\operatorname{Sym}^dX)({\\mathbb {C}}_p)$ from the beginning of this section.",
"Let $Y=Y_i$ for some $i$ such that $\\dim Y > 0$ .",
"Then $j^{-1}(Y({\\mathbb {C}}_p)) \\subseteq {\\mathcal {S}}(\\operatorname{Sym}^dX)({\\mathbb {C}}_p)$ .",
"We consider two cases: if the generic point of $Y$ has a positive-dimensional preimage, then each $Q \\in Y({\\mathbb {C}}_p)$ is ${\\mathbb {P}}^n$ for some $n > 0$ , so each fibre is contained in ${\\mathcal {S}}(\\operatorname{Sym}^dX)({\\mathbb {C}}_p)$ .",
"On the other hand, if the preimage of the generic point of $Y$ is 0-dimensional, then any irreducible component of $j^{-1}(Y)$ is either covered by positive-dimensional projective spaces, or is birational to $Y$ via the restriction of $j$ .",
"All of these irreducible components are then in the special set.",
"In particular, we note that ${\\mathcal {F}}(\\operatorname{Sym}^dX) \\subseteq {\\mathcal {S}}(\\operatorname{Sym}^dX)$ .",
"Finally, it remains to relate ${\\mathcal {S}}(\\operatorname{Sym}^dX)({\\mathbb {C}}_p)$ and $(\\operatorname{Sym}^dX)^{\\eta =0}$ .",
"In general, neither set is contained in the other; however, with Assumption REF , we immediately get: Proposition 3.7 Let $R_1, \\ldots , R_n$ be the irreducible components of the rigid analytic space $(\\operatorname{Sym}^dX)^{\\eta =0}$ .",
"Further, suppose that $\\operatorname{Sym}^dX$ satisfies Assumption REF .",
"Then for each $R_i$ with $\\dim R_i \\ge 1$ , we have $R_i({\\mathbb {C}}_p) \\subseteq {\\mathcal {S}}(\\operatorname{Sym}^dX)({\\mathbb {C}}_p)$ , and so $(\\operatorname{Sym}^dX)^{\\eta =0} \\subseteq {\\mathcal {S}}(\\operatorname{Sym}^dX)$ .",
"Then taking complements of the relation obtained in Proposition REF inside $\\operatorname{Sym}^dX$ and looking at the ${\\mathbb {Q}}$ -points, one obtains: Corollary 3.8 Under the hypothesis of Proposition REF , one has $\\lbrace {\\mathbb {Q}}\\textup {-points of } \\operatorname{Sym}^dX \\backslash {\\mathcal {S}}(\\operatorname{Sym}^dX)\\rbrace \\subseteq \\bigcup \\lbrace 0 \\textup {-dimensional } R_i\\rbrace .$ Thus, under the hypothesis of Proposition REF , one is still able to interpret the results given from Chabauty's method for higher-dimensional varieties, as Chabauty's method gives an upper bound on $\\bigcup (\\textup {0-dimensional } R_i)$ .",
"For the rest of the paper, we assume that the conditions of Proposition REF hold for $\\operatorname{Sym}^dX$ ."
],
[
"$p$ -adic geometry",
"The goal of this section is to associate a “generalized Newton polygon\" to each multivariate power series, and to state an approximation theorem for the number of roots of a system of equations given by $d$ power series in $d$ variables in general position – that is, having finitely may common zeros – using these Newton polygons.",
"The classical case of $d=1$ is well-known in the literature.",
"To define the Newton polygons for multivariate power series, we review the language necessary to define tropical objects, and state the results in tropical geometry.",
"For a more detailed treatment of tropical geometry, see and ."
],
[
"Tropicalization of rigid analytic hypersurfaces",
"Tropical geometry generalizes the theory of Newton polygons of single-variable power series to power series of several variables.",
"Most of the exposition from this section is taken from .",
"We discuss the tropicalization of affinoid hypersurfaces cut out by a power series over $p$ that arise via Coleman integration.",
"These power series are necessarily convergent on the domain where each of the coordinates has a positive valuation (this has to do with the fact that we can write any $\\omega \\in H^0(X_{{\\mathbb {Q}}_p}, \\Omega ^1)$ as a power series with coefficients in ${\\mathbb {Z}}_p$ ).",
"The tropicalization should be seen as the dual of a Newton polygon; this notion will be made precise in this section.",
"More generally, everything in this section works for a nontrivially valued field $K$ that is complete with respect to the nontrivial valuation.",
"We further assume that $K$ is algebraically closed.",
"Definition 4.1 Let $m = (m_1, \\ldots , m_d) \\in {\\mathbb {Q}}_{\\ge 0}^d$ , and let $P_m := \\lbrace (x_1, \\ldots , x_n) \\in N_{{\\mathbb {R}}} : x_i \\ge m_i \\textup { for } 1 \\le i \\le d\\rbrace $ .",
"Let $U_{P_m} = \\lbrace (x_1, \\ldots , x_d): v(x_i) \\ge m_i \\textup { for all } i\\rbrace .$ The tropicalization map on $U_{P_m}$ is $\\operatorname{trop}: U_{P_m}^d & \\rightarrow (P_m)^d\\\\(\\xi _1, \\ldots , \\xi _d) &\\mapsto (v(\\xi _1), \\ldots , v(\\xi _d)).$ The above tropicalization makes sense for affinoid hypersurfaces cut out by power series convergent on some $U_{P_m}$ , i.e.",
"the power series that are convergent when evaluated on some $x \\in {\\mathbb {Q}}_p^d$ with $v(x) \\in P_m$ .",
"The set of such power series will be denoted by $K \\langle U_{P_m} \\rangle := \\left\\lbrace \\sum _{u \\in {\\mathbb {Z}}_{>0}^d}a_ux^u : a_u \\in K, v(a_u) + \\langle u,w \\rangle \\rightarrow \\infty \\textup { for all } w \\in P_m\\right\\rbrace ,$ where the convergence $v(a_u) + \\langle u,w \\rangle \\rightarrow \\infty $ holds as $u$ ranges over ${\\mathbb {Z}}_{>0}^d$ in any order.",
"For example, if $m = (0, \\ldots , 0)$ , then $K \\langle U_{P_m} \\rangle = K \\langle x_1, \\ldots , x_d \\rangle ,$ the Tate algebra in $d$ variables.",
"Remark 4.2 More generally, it is known that $K \\langle U_{P_m} \\rangle $ is a $K$ -affinoid algebra (*Lemma 6.9(i)), a Cohen-Macaulay ring (*Lemma 6.9(v)), and that $U_P = \\operatorname{Sp}K \\langle U_P \\rangle $ .",
"Definition 4.3 Let $P$ be a polyhedron, and let $f_1, \\ldots , f_n \\in K \\langle U_P\\rangle $ .",
"Let $(f_1, \\ldots , f_n)$ be the ideal in $K \\langle U_P \\rangle $ generated by $f_1, \\ldots , f_n$ .",
"Then $V(f_1, \\ldots , f_n) := \\operatorname{Sp}K \\langle U_P \\rangle /(f_1, \\ldots , f_n).$ Then $V(f_1, \\ldots , f_n)$ is an affinoid subspace of $\\operatorname{Sp}K \\langle U_P \\rangle $ .",
"In our case, each the power series $f \\in K[[x_1, \\ldots , x_d]]$ that arises from Chabauty's method on $\\operatorname{Sym}^dX$ converges when $v(x_i) > 0$ for $1 \\le i \\le d$ .",
"Thus, $f \\in K\\langle U_{P_m} \\rangle $ for any $P_m$ with $m \\in {\\mathbb {Q}}_{>0}^d$ .",
"Let $f \\in K \\langle U_{P_m} \\rangle $ .",
"Now we define $\\operatorname{Trop}(f)$ , the tropical variety corresponding to $f$ , and then outline the procedure for computing $\\operatorname{Trop}(f)$ in $§$ REF .",
"Definition 4.4 For $f \\in K \\langle U_{P_m} \\rangle $ , $\\operatorname{Trop}(f) := \\overline{\\operatorname{trop}(V(f))} = \\overline{\\lbrace (v(\\xi _1), \\ldots , v(\\xi _d)) : f(\\xi ) = 0, \\xi \\in U_{P_m}\\rbrace },$ where $V(f)$ is the affinoid subspace defined by the ideal ${\\mathfrak {a}}= (f) \\subset K \\langle U_{P_m} \\rangle $ .",
"Here, we take the topological closure in $P_m$ .",
"Often, the easiest way to compute $\\operatorname{Trop}(f)$ is by using Lemma REF , which requires these definitions: Definition 4.5 For $0 \\ne f \\in K \\langle U_{P_m} \\rangle $ , write $f = \\sum _{u \\in S_{\\sigma }}a_ux^u$ .",
"The height graph of $f$ is $H(f) = \\lbrace (u, v(a_u)): u \\in {\\mathbb {Z}}_{\\ge 0}^d, a_u \\ne 0\\rbrace \\subseteq {\\mathbb {Z}}_{\\ge 0}^d \\times {\\mathbb {R}}.$ Given $w \\in {\\mathbb {Q}}_{>0}^d$ , we also define $m_f(w) = m(w) = \\min _{u \\in S_{\\sigma }}\\lbrace (-w, 1) \\cdot H(f)\\rbrace ,$ where $\\cdot $ denotes the usual dot product, and $\\operatorname{vert}_w(f) = \\lbrace (u, v(a_u)) \\in H(f): (-w, 1) \\cdot (u, v(a_u)) = m(w)\\rbrace \\subseteq H(f).$ Intuitively, $m(w)$ denotes the minimum valuation achieved assuming that $v(x) = w$ , among the terms of $f$ .",
"Then $\\operatorname{vert}_w(f)$ records the corresponding terms of $f$ with the minimum valuation, again assuming that $v(x) = w$ .",
"The following is the power-series analogue of a well-known result for polynomials; the original result for polynomials is first recorded in an unpublished manuscript by Kapranov, and a proof of this lemma for power series can be found in , Lemma 8.4; also see, for example , Theorem 3.1.3.",
"This gives a useful method to computing $\\operatorname{Trop}(f)$ .",
"Lemma 4.6 $\\operatorname{Trop}(f) = \\overline{\\lbrace w \\in {\\mathbb {Q}}_{\\ge 0}^d: \\# \\operatorname{vert}_w(f) > 1\\rbrace }.$"
],
[
"Tropical intersection theory and Newton polygons",
"In this section, we take $d$ power series in $d$ variables in $K \\langle U_{P_m} \\rangle $ that have finitely many common zeros.",
"We explain that in order to bound the number of common zeros of the $d$ power series, it suffices to know their tropicalizations and their Newton polygons.",
"Since the tropicalizations and the Newton polygons depend only on finitely many terms of the power series convergent on $U_{P_m}$ , this section shows that one can approximate a power series of several variables by a polynomial for the purposes of intersection theory.",
"In a sense, this is a stronger approximation than what Weierstrass preparation theorem can tell us; Weierstrass preparation for multivariate power series approximates $f \\in K[[t_1, \\ldots , t_d]]$ by $f^{\\prime } \\in K[t_1][[t_2, \\ldots , t_d]]$ , whereas here, we approximate $f$ by $f^{\\prime \\prime } \\in K[t_1, \\ldots , t_d]$ .",
"Let $f \\in K \\langle U_{P_m} \\rangle $ .",
"Write $f = \\sum a_u x^u$ .",
"Define $\\operatorname{vert}_{P_m}(f) := \\bigcup _{w \\in P_m}\\operatorname{vert}_w(f).$ It turns out that $\\operatorname{vert}_{P_m}(f)$ is finite: Lemma 4.7 (, Lemma 8.2) Let $f \\in K \\langle U_{P_m} \\rangle $ be nonzero.",
"Then $\\operatorname{vert}_{P_m}(f)$ is finite.",
"This lemma, combined with Lemma REF , tells us that $\\operatorname{Trop}(f)$ determined by only finitely many terms of $f$ .",
"Now, the following lemma shows that if the coefficients of a power series $f$ are perturbed in a way so that their $v$ -adic valuations do not change, and so that $\\operatorname{vert}_P(f)$ does not change, then the tropicalization also stays the same.",
"Lemma 4.8 Let $f, f^{\\prime } \\in K \\langle U_{P_m} \\rangle $ , with $f = \\sum _{u}a_ux^u$ and $f^{\\prime } = \\sum _u a_u^{\\prime } x^u$ .",
"Suppose that $\\operatorname{vert}_{P_m}(f) = \\operatorname{vert}_{P_m}(f^{\\prime })$ .",
"Then $\\operatorname{Trop}(f) = \\operatorname{Trop}(f^{\\prime })$ .",
"Fix $w \\in P_m$ .",
"We claim that $\\operatorname{vert}_w(f) = \\operatorname{vert}_w(f^{\\prime })$ for each such $w$ .",
"Choose $u_0 \\in {\\mathbb {Z}}_{\\ge 0}^d$ minimizing $v(a_u) + \\langle u,w \\rangle $ .",
"This means $m_f(w) = v(a_{u_0}) + \\langle u_0,w \\rangle .$ Thus, $(u_0, v(a_{u_0})) \\in \\operatorname{vert}_w(f) \\subset \\operatorname{vert}_P(f) = \\operatorname{vert}_P(f^{\\prime })$ .",
"So $(u_0, v(a_{u_0})) \\in \\operatorname{vert}_{w^{\\prime }}(f)$ for some $w^{\\prime } \\in P_m$ .",
"In particular, $(u_0, v(a_{u_0})) = (u_0, v(a_{u_0}^{\\prime }))$ .",
"Thus, $\\min _{u \\in {\\mathbb {Z}}_{\\ge 0}^d}(v(a_u) + \\langle u, w \\rangle ) = v(a_{u_0}) + \\langle u,w \\rangle = v(a_{u_0}^{\\prime }) + \\langle u_0, w \\rangle \\ge \\min _{u \\in {\\mathbb {Z}}_{\\ge 0}^d}(v(a_u^{\\prime }) + \\langle u, w \\rangle ).$ The symmetric argument proves the inequality in the other direction, showing that $\\operatorname{vert}_w(f) = \\operatorname{vert}_w(f^{\\prime })$ for each $w \\in P_m$ .",
"Then by Lemma REF , $\\operatorname{Trop}(f) = \\operatorname{Trop}(f^{\\prime })$ .",
"Given $f \\in K \\langle U_{P_m} \\rangle $ , we can find a polynomial $g \\in K \\langle U_{P_m} \\rangle $ such that $\\operatorname{vert}_{P_m}f = \\operatorname{vert}_{P_m} g$ , in which case Lemma REF implies that $\\operatorname{Trop}(f) = \\operatorname{Trop}(g)$ : Corollary 4.9 Let $f \\in K \\langle U_{P_m} \\rangle $ , with $f = \\sum _{u}a_ux^u$ .",
"Let $\\pi : {\\mathbb {Z}}_{\\ge 0}^d \\times {\\mathbb {R}}\\rightarrow {\\mathbb {Z}}_{\\ge 0}^d$ denote the projection map forgetting the last coordinate.",
"Let $S \\subseteq {\\mathbb {Z}}_{\\ge 0}^d$ be a finite set containing $\\pi (\\operatorname{vert}_{P_m}(f))$ .",
"Define the auxiliary polynomial of the power series $f$ with respect to $S$ by $g_S = \\sum _{u \\in S}a_ux^u \\in K \\langle U_P \\rangle .$ Then $\\operatorname{Trop}(f) = \\operatorname{Trop}(g_S)$ .",
"Since $\\operatorname{vert}_{P_m}(f) = \\operatorname{vert}_{P_m}(g_S)$ , and since $\\operatorname{Trop}(f)$ only depends on $\\operatorname{vert}_P(f)$ , the conclusion follows.",
"We show that the terms of a power series $f \\in K \\langle U_{P_m}\\rangle $ contained in $\\operatorname{vert}_{P_m}(f)$ also contains all the information about the valuations of zeros of $f$ counting multiplicity.",
"That is, the information about zeros of power series depends on only finitely many terms of $f$ too, and hence the information about the intersection theory of the power series also depends on finitely many terms of $f$ .",
"If $S$ is a finite set of points in Euclidean space, its convex hull is denoted $\\operatorname{conv}(S)$ .",
"Definition 4.10 Let $f \\in K \\langle U_{P_m} \\rangle $ .",
"For each $w \\in {\\mathbb {R}}_{\\ge 0}^d$ , define its associated Newton polytope $\\gamma _w(f) = {\\gamma }_w = \\pi (\\operatorname{conv}(\\operatorname{vert}_w(f))).$ Remark 4.11 The $g_S$ from Corollary REF are chosen so that the $\\gamma _w(f) = \\gamma _w(g_S)$ as well.",
"It turns out that as long as $\\bigcap _i V(f_i)$ is finite, the information about the common roots of $f_i \\in K \\langle U_P \\rangle $ having a specified valuation $w$ is encoded in ${\\gamma }_w$ , as explained below.",
"Definition 4.12 Let $f_1, \\ldots , f_d \\in K \\langle U_{P_m} \\rangle $ , $Y_i = V(f_i)$ , and $Y = \\bigcap _{i} Y_i$ .",
"Assume that $Y$ is 0-dimensional.",
"Then the intersection multiplicity of $Y_1, \\ldots , Y_d$ at $w \\in {\\mathbb {Q}}^d$ is defined as $i(w; Y_1 \\cdots Y_d) := \\dim _K H^0(Y \\cap U_{\\lbrace w\\rbrace }, {\\mathcal {O}}_{Y \\cap U_{\\lbrace w\\rbrace }}),$ where $U_{\\lbrace w\\rbrace } := \\operatorname{trop}^{-1}(w)$ .",
"In simpler terms, this intersection multiplicity at $w$ is the number of common zeros of the $f_i$ that have the same coordinate-wise valuation as $w$ , counting with multiplicity.",
"Since the $U_{\\lbrace w\\rbrace }$ are disjoint, the intersection number of $Y_1, \\ldots , Y_d$ is then $i(Y_1, \\ldots , Y_d) := \\dim H^0(Y, {\\mathcal {O}}_Y).$ Definition 4.13 Let $Q_1, \\ldots , Q_d$ be bounded polytopes.",
"Define a function $V_{Q_1, \\ldots , Q_d}(\\lambda _1, \\ldots , \\lambda _d) := \\operatorname{vol}(\\lambda _1Q_1 + \\cdots + \\lambda _d Q_d)$ where $+$ denotes the Minkowski sum.",
"The mixed volume of the $Q_i$ , denoted $MV(Q_1, \\ldots , Q_d)$ , is defined as the coefficient of the $\\lambda _1 \\cdots \\lambda _d$ -term of $V_{Q_1, \\ldots , Q_d}(\\lambda _1, \\ldots , \\lambda _d)$ .",
"Theorem 4.14 *Theorem 11.7 Let $f_1, \\ldots , f_d \\in K \\langle U_{P_m} \\rangle $ have finitely many common zeros, and let $w \\in \\bigcap _{i=1}^d \\operatorname{Trop}(f_i)$ be an isolated point in the interior of $P$ .",
"For $i = 1, \\ldots , d$ let $Y_i = V(f_i)$ and let $\\gamma _i = \\gamma _w(f_i)$ .",
"Then $i(w, Y_1 \\cdots Y_d) = MV(\\gamma _1, \\ldots , \\gamma _d).$ In particular, Theorem REF implies that considering the auxiliary polynomials suffices, as the $\\gamma _i$ are determined by only finitely many terms in the $f_i$ .",
"Theorem 4.15 Suppose that $f_1, \\ldots , f_d \\in K \\langle U_{P_m} \\rangle $ , and let $g_i$ be the auxiliary polynomials of the $f_i$ with respect to some finite set $S \\subseteq M$ containing all $u$ such that $(u, v(a_u)) \\in \\operatorname{vert}_{P_m}(f)$ .",
"Then $\\sum _{w \\in P^{\\circ }} i(w, V(f_1) \\cdots V(f_d)) = \\sum _{w \\in P^{\\circ }} i(w, V(g_1) \\cdots V(g_d)),$ if all the summands on both sides are finite.",
"By the choice of the $g_i$ , $\\operatorname{Trop}(f_i) = \\operatorname{Trop}(g_i)$ for each $1 \\le i \\le d$ , and $\\gamma _w(f_i) = \\gamma _w(g_i)$ for each $w \\in P$ .",
"We note from Theorem REF that $\\operatorname{Trop}(f_i), \\operatorname{Trop}(g_i), \\gamma _w(f_i)$ and $\\gamma _w(g_i)$ are the only information required in computing the intersection multiplicities.",
"The following results for polynomials are useful in estimating the number of zeros of power series.",
"First, a definition: Definition 4.16 Let $f = \\sum _{u \\in \\Lambda }a_ux^u$ be a polynomial, where $\\Lambda \\subset {\\mathbb {Z}}^d$ is a finite set.",
"Then the Newton polygon of $f$ is given by $\\operatorname{New}(f) = \\operatorname{conv}(\\lbrace u: u \\in \\Lambda , a_u \\ne 0\\rbrace ) \\subseteq {\\mathbb {R}}^d.$ We recall Bernstein's theorem: Theorem 4.17 () Let $f_1, \\ldots , f_d \\in K[x_1, \\ldots , x_d]$ be polynomials with finitely many common zeros.",
"Then the number of common zeros of the $f_i$ with multiplicity in $(K^{\\times })^d$ is $\\operatorname{MV}(\\operatorname{New}(f_1), \\ldots , \\operatorname{New}(f_d)).$ Further, suppose that the $f_i$ have finitely many common zeros whose valuations belong to $P$ , and also suppose that the $g_i$ have finitely many common zeros in $K^d$ .",
"By Theorem REF , $&\\textup {number of common zeros of the } f_i \\textup { with valuations in } P \\\\&= \\textup {number of common zeros of the } g_i \\textup { with valuations in }P\\\\&\\le \\textup {number of common zeros of the } g_i \\textup { in } (K^{\\times })^d\\\\ &=\\operatorname{MV}(\\operatorname{New}(g_1), \\ldots , \\operatorname{New}(g_d)),$ where the last inequality follows by Bernstein's theorem.",
"Thus, we conclude: Theorem 4.18 Suppose that $f_1, \\ldots , f_d \\in K \\langle U_{P_m} \\rangle $ , and let $g_i$ be the associated auxiliary polynomials of the $f_i$ with respect to some finite set $S \\subseteq M$ containing all $u$ such that $(u, v(a_u)) \\in \\operatorname{vert}_P(f)$ .",
"Suppose further that $\\bigcap _{i=1}^d V(f_i)< \\infty $ and $\\bigcap _{i=1}^dV(g_i) < \\infty $ .",
"Then $\\# \\left( (K^{\\times })^d \\cap \\bigcap _{i=1}^d V(f_i) \\right) \\le MV(\\operatorname{New}(f_1), \\ldots , \\operatorname{New}(f_d)).$ Again, the proof follows from the fact that by the choice of the $g_i$ , we have $\\operatorname{Trop}(f_i) = \\operatorname{Trop}(g_i)$ and $\\gamma _w(f_i) = \\gamma _w(g_i)$ for each $1 \\le i \\le d$ and $w \\in P$ .",
"Remark 4.19 In order to count all solutions of the form $(x_1, \\ldots , x_d)$ , where some of the $x_i$ may be 0, one needs to apply Theorem REF multiple times, while setting some of the $x_i = 0$ ."
],
[
"Continuity of roots",
"Throughout, $K$ is a complete, algebraically closed valued field with respect to a nontrivial, nonarchimedean valuation $v \\colon K^{\\times } \\rightarrow {\\mathbb {Q}}$ .",
"studies the intersection theory of power series in $K \\langle U_P \\rangle $ that have finite intersection.",
"Our goal in this chapter is to analyze the case of the power series in $K \\langle U_P \\rangle $ have possibly infinite intersection; we will show that these power series have “small” deformations that have finite intersection, and that they preserve information about the number of 0-dimensional components of the original intersection.",
"We make this notion precise, and we obtain an upper bound on the number of 0-dimensional components (counting multiplicity) of the original intersection, using the new power series with finitely many common zeros, obtained via small $p$ -adic deformations."
],
[
"Deformation of power series via rigid analytic geometry and polynomial approximations",
"It is known that small deformations do not affect the multiplicity of 0-dimensional components of intersections in rigid analytic spaces.",
"Theorem 5.1 (*Theorem 10.2, Local continuity of roots) Let $A$ be a $K$ -affinoid algebra that is a Dedekind domain and let $S = \\operatorname{Sp}(A)$ .",
"Let $X = \\operatorname{Sp}(B)$ be a Cohen-Macaulay affinoid space of dimension $d+1$ , let $f_1, \\ldots , f_d \\in B$ , and let $Y \\subset X$ be the subspace defined by the ideal ${\\mathfrak {a}}= (f_1, \\ldots , f_d)$ .",
"Suppose that we are given a morphism $\\alpha : X \\rightarrow S$ and a point $t \\in S$ such that the fibre $Y_t = \\alpha ^{-1}(t) \\cap Y$ has dimension zero.",
"Then there is an affinoid subdomain $U \\subset S$ containing $t$ such that $\\alpha ^{-1}(U) \\rightarrow U$ is finite and flat.",
"Let $B_K^d := \\operatorname{Sp}(K \\langle x_1, \\ldots , x_d\\rangle )$ for $d \\ge 1$ .",
"The following is immediate from the above theorem, and is applicable to our situation arising from Chabauty's method: Corollary 5.2 (*Example 10.3) Let $X = B_K^d \\times B_K^1$ and $S = B_K^1$ , with $\\alpha \\colon X \\rightarrow S$ the projection onto the second factor.",
"Let $f_1, \\ldots , f_d \\in K \\langle x_1, \\ldots , x_d, t \\rangle $ .",
"If the specializations $f_{1,0}, \\ldots , f_{d,0}$ at $t=0$ have only finitely many zeros in $B_K^d$ then there exists ${\\varepsilon }>0$ such that if $|s| < {\\varepsilon }$ , then $f_{1,s}, \\ldots , f_{d,s}$ have the same number of zeros (counted with multiplicity) in $B_{\\kappa (s)}^d$ as $f_{1,0}, \\ldots , f_{d,0}$ .",
"Tate algebra in one variable is a Dedekind domain, and $X$ is Cohen-Macaulay (Remark REF ).",
"Definition 5.3 Suppose that $f_1, \\ldots , f_d \\in K \\langle U_{P_m} \\rangle $ are power series such that $Y:=V(f_1, \\ldots , f_d)$ is possibly infinite.",
"Define $N_0(f_1, \\ldots , f_d)$ to be the number of 0-dimensional components of $Y$ , counting multiplicity.",
"If $Y$ is finite, then we drop the subscript 0 to signal its finiteness: $N(f_1, \\ldots , f_d) := N_0(f_1, \\ldots , f_d) = \\dim H^0(Y, {\\mathcal {O}}_Y).$ Also, let $N_0^{\\times }(f_1, \\ldots , f_d)$ denote the number of 0-dimensional components of $Y$ , whose coordinates are in $\\overline{K}^{\\times }$ .",
"Definition 5.4 Let $f = \\sum _{u}a_ux^u\\in K \\langle U_{P_m}\\rangle $ , and define $M(f) &:= \\textup {the set of monomials with nonzero coefficients appearing in f}\\\\&= \\lbrace x^u: a_u \\ne 0\\rbrace .$ Call $f$ nondegenerate if for every $i$ there exists an integer $n > 0$ with $x_i^n \\in M(f)$ .",
"Remark 5.5 Any pure power series arising from Chabauty's method is nondegenerate.",
"We will need to impose the nondegeneracy conditions in all power series $f$ in order to be able to carry out deformations.",
"Now we prove a series of deformation results for non-stable intersections.",
"Lemma 5.6 Let $f \\in K\\langle U_{P_m} \\rangle $ be a nondegenerate power series, and let $q_1, q_2, \\ldots , q_n \\in U_{P_m}$ such that $q_i \\ne 0$ in $K^d$ .",
"Then there exists a polynomial $h$ such that $h$ does not vanish on any of $q_1, \\ldots , q_{\\ell }$ and $M(h) \\subseteq M(f)$ .",
"We will prove by induction on $\\ell $ that there exists $g$ such that $M(g) \\subseteq M(f)$ and $g(q_1), \\ldots , g(q_{\\ell }) \\ne 0$ .",
"The statement is clear when $\\ell = 0$ , so assume that there is a polynomial $g$ with $M(g) \\subseteq M(f)$ satisfying $g(q_1), \\ldots , g(q_{\\ell }) \\ne 0$ and $g(q_{\\ell +1}) = \\cdots = g(q_n) = 0$ , after possibly reordering the $q_i$ .",
"If $\\ell = n$ , then we are done, so assume otherwise.",
"We will show that there exists another polynomial $g^{\\prime } \\in M(f)$ such that $g^{\\prime }$ does not vanish on at least $\\ell +1$ of the points $q_i$ .",
"Choose a monomial $m \\in M(f)$ such that $m(q_{\\ell +1}) \\ne 0$ ; such $m$ exists due to the nondegeneracy condition on $f$ .",
"We may choose $c \\in K^{\\times }$ such that $v(cm(q_i)) > v(g(q_i))$ for $1 \\le i \\le \\ell $ .",
"Then $g^{\\prime } := g + cm$ satisfies the property that $g^{\\prime }(q_1) \\ne 0, \\ldots , g^{\\prime }(q_{\\ell }) \\ne 0$ .",
"Also, $g^{\\prime }(q_{\\ell +1}) = cm(q_{\\ell +1}) \\ne 0$ .",
"The lemma then follows from an inductive argument on $\\ell $ .",
"Proposition 5.7 (a) Suppose that $f_1, \\ldots , f_d \\in K \\langle U_{P_m} \\rangle $ are nondegenerate power series.",
"Then there exist nondegenerate $g_1, \\ldots , g_d \\in K \\langle U_{P_m} \\rangle $ with $\\operatorname{Trop}(f_i) = \\operatorname{Trop}(g_i)$ and $\\gamma _w(f_i) = \\gamma _w(g_i)$ , for $1 \\le i \\le d$ and $w \\in P_m$ , with $N_0(f_1, \\ldots , f_d) \\le N(g_1, \\ldots , g_d).$ (b) Moreover, if the $f_i$ are polynomials, then the $g_i$ may be chosen to be polynomials.",
"We will deform the $f_i$ to the $g_i$ one by one.",
"Specifically, we will prove by induction on $r$ that there exist $g_1, \\ldots , g_r$ such that $\\operatorname{Trop}(f_i) = \\operatorname{Trop}(g_i)$ and $\\gamma _w(f_i) = \\gamma _w(g_i)$ for $i \\in \\lbrace 1, \\ldots , r\\rbrace $ and $w \\in P$ , satisfying $\\operatorname{codim}\\bigcap _{i=1}^r V(g_i) \\ge r$ for each $r \\in \\lbrace 1, \\ldots , d\\rbrace $ , and $N_0(f_1, \\ldots , f_d) \\le N_0(g_1, \\ldots , g_r, f_{r+1}, \\ldots , f_d).$ When $r=1$ , the statement above is clear, by taking $f_1 = g_1$ .",
"Now we prove the statement for $r+1$ .",
"Let $C_1, \\ldots , C_{\\ell }$ be the codimension $r$ irreducible components of $\\bigcap _{i=1}^r V(g_i)$ .",
"Choose points $P_i \\in C_i$ , such that $P_i \\ne 0$ in $K^d$ .",
"We will deform $f_{r+1}$ to $g_{r+1}$ so that $g_{r+1}(P_{i}) \\ne 0$ for each $i$ , while keeping $\\operatorname{Trop}(f_{r+1}) = \\operatorname{Trop}(g_{r+1})$ and $\\gamma _w(f_{r+1}) = \\gamma _w(g_{r+1})$ .",
"This will guarantee that $\\operatorname{codim}\\left(\\bigcap _{i=1}^{r+1}V(g_{r+1})\\right) \\ge r+1$ .",
"From the nondegeneracy assumption of $f_{r+1}$ , we have that $V(M(f_{r+1})) = \\emptyset $ (if $M(f_{r+1})$ contains 1) or $V(M(f_{r+1})) = \\lbrace 0\\rbrace $ (if $M(f_{r+1})$ does not contain 1).",
"In the first case, we may adjust only the constant term to get the desired $g_{r+1}$ .",
"Thus, we may assume that we are in the second case.",
"Then by Lemma REF , we may pick a polynomial $h$ such that $M(h) \\subseteq M(f_{r+1})$ such that $h$ that does not vanish on any $P_i$ .",
"For small enough nonzero ${\\varepsilon }$ , the deformation $f_{r+1} \\mapsto f_{r+1} + {\\varepsilon }h =: g_{r+1}$ does not vanish on any of the $P_i$ ; in this case, the intersection $\\bigcap _{i=1}^{r+1}V(g_i)$ has codimension $r+1$ , as required.",
"Further, since $h \\in M(f_{r+1})$ , after possibly making ${\\varepsilon }$ even smaller, both the tropicalization and the $\\gamma _w$ of $f_{r+1}$ are identical to those of $g_{r+1}$ .",
"Now we will prove that $N_0(g_1, \\ldots , g_d) \\ge N_0(f_1, \\ldots , f_d)$ .",
"It suffices to show that $N_0(g_1, \\ldots , g_{r+1}, f_{r+2}, \\ldots , f_d) \\ge N_0(g_1, \\ldots , g_r, f_{r+1}, \\ldots , f_d)$ for $r \\in \\lbrace 0, \\ldots , d-1\\rbrace $ ; if $r = 0$ , then the previous inequality will be interpreted as $N_0(g_1, f_2, \\ldots , f_d) \\ge N_0(f_1, \\ldots , f_d).$ Let $I$ be the ideal that cut out the dimension $\\ge 1$ components of $V(g_1, \\ldots , g_r, f_{r+1}, \\ldots , f_d)$ in $K \\langle U_P\\rangle $ and let ${\\mathfrak {p}}_1, \\ldots , {\\mathfrak {p}}_{\\ell }$ denote the maximal ideals corresponding to the 0-dimensional components of $V(g_1, \\ldots , g_r, f_{r+1}, \\ldots , f_d)$ .",
"Choose a $f \\in I$ such that $f \\notin {\\mathfrak {p}}_i$ for $1 \\le i \\le \\ell $ .",
"Such choice is possible by the prime avoidance theorem, see for example *Proposition 1.11.",
"Now we apply Theorem REF on $\\operatorname{Sp}B$ , where $B = K \\langle U_P \\rangle _f$ , which states that a small deformation of $V(g_1, \\ldots , g_r, f_{r+1}, \\ldots , f_d)$ preserves all 0-dimensional components away from the positive-dimensional locus.",
"This proves the inequality at the beginning of this paragraph, and consequently part (a) of the proposition.",
"Part (b) follows, since we deform the the $f_i$ by monomials that already appear in $f_i$ ."
],
[
"Explicit computation of the upper bound",
"Now we consider a residue disk ${\\mathcal {U}}\\subseteq (\\operatorname{Sym}^dX)(p)$ whose points reduce to a given ${\\mathcal {Q}}\\in (\\operatorname{Sym}^dX)({\\mathbb {F}}_p)$ .",
"Recall from Proposition REF that Chabauty's method on ${\\mathcal {U}}$ yields $d$ pure power series $f_1, \\ldots , f_d$ in $d$ variables, whose common zeros in $p^d$ with valuations at least $1/d$ correspond to a set containing the points in $j^{-1}(j(\\operatorname{Sym}^dX)({\\mathbb {Q}}_p)) \\cap \\overline{J({\\mathbb {Q}})})$ .",
"Using the results of the previous section, we will obtain an explicit upper bound on the number of common zeros of the $f_i$ in this section by estimating $\\operatorname{New}(f_i)$ .",
"The methods used in this section are reminiscent of .",
"Definition 5.8 For ${\\varepsilon }\\in (0, \\frac{1}{d})$ , $k \\in {\\mathbb {Z}}_{\\ge 0}$ and $d, \\ell \\in {\\mathbb {Z}}_{\\ge 1}$ with $d \\ge \\ell $ , let $\\delta _{{\\varepsilon }}(k,v, \\ell ) := \\max \\left\\lbrace N \\in {\\mathbb {Z}}_{\\ge 0} : v(k+N) \\ge (\\frac{1}{\\ell }-{\\varepsilon })N + v(k)\\right\\rbrace .$ Remark 5.9 We note that $\\delta _{{\\varepsilon }}(k,v,\\ell )$ is well-defined, independent of the choice of ${\\varepsilon }$ ; $v(k+N) = O(\\log N)$ as $N \\rightarrow \\infty $ , while $(\\frac{1}{\\ell }-{\\varepsilon })(N+1)$ increases linearly with $N$ .",
"Notation 5.10 Given $f \\in W({\\mathbb {F}}_q)[[t]]$ , we mean by $\\bar{f}$ the image of $f$ under the natural reduction map of the coefficients $W({\\mathbb {F}}_q)[[t]] \\rightarrow {\\mathbb {F}}_q[[t]]$ .",
"We will denote $\\operatorname{ord}_0(f) := \\operatorname{ord}_{t = 0}(\\bar{f})$ , the exponent of the first term that does not vanish under the reduction map.",
"Lemma 5.11 For any ${\\varepsilon }\\in {\\mathbb {Q}}$ satisfying $0 < {\\varepsilon }< \\frac{1}{d}$ , the following holds: Let $f \\in W({\\mathbb {F}}_q)\\left[\\frac{1}{p}\\right][[t]]$ be such that its derivative $f^{\\prime }$ is in $W({\\mathbb {F}}_q)[[t]]$ , and $\\operatorname{ord}_0 \\bar{f}^{\\prime } = k-1$ for some $k \\ge 1$ .",
"Let $F(t_1, \\ldots , t_{\\ell }) := f(t_1) + \\cdots + f(t_{\\ell }) = \\sum _{u \\in {\\mathbb {Z}}_{\\ge 0}^{\\ell }} a_ut^u,$ where $t^u$ denotes $t_1^{u_1} \\cdots t_{\\ell }^{u_{\\ell }}$ .",
"Let $w \\in P_{{\\varepsilon }} = [{\\varepsilon }, \\infty )^{\\ell }$ .",
"If $u = (u_1, \\ldots , u_{\\ell }) \\in {\\mathbb {Z}}_{\\ge 0}^{\\ell }$ satisfies $u_i > k + \\delta _{{\\varepsilon }}(k,v, \\ell )$ for some $1 \\le i \\le \\ell $ , then $(u, v(a_u)) \\notin \\operatorname{vert}_w(F)$ .",
"Fix $w \\in [{\\varepsilon }, \\infty )^{\\ell }$ .",
"Since $F$ is pure, it suffices to consider $u \\in {\\mathbb {Z}}_{\\ge 0}^{\\ell }$ such that $u_1 > k + \\delta _{{\\varepsilon }}(k,v,\\ell )$ and $u_2 = u_3 = \\cdots = u_d = 0$ .",
"We will show that there exists $u^{\\prime } \\in {\\mathbb {Z}}_{\\ge 0}^{\\ell }$ such that $v(a_{u^{\\prime }}) + \\langle u^{\\prime }, w \\rangle < v(a_u) + \\langle u,w \\rangle .$ Then by the definition of $\\operatorname{vert}_w(F)$ , the conclusion would follow.",
"Write $f^{\\prime }(t) = \\sum _{i \\ge 0} c_it^i$ , so $f(t) = \\sum _{i \\ge 0}\\frac{c_i}{i+1}t^{i+1}$ .",
"Then $c_i \\in {\\mathbb {Z}}_p$ since $f^{\\prime } \\in {\\mathbb {Z}}_p[[t]]$ , and furthermore, $v(c_{k-1}) = 0$ , with $v(c_j) > 0$ for $1 \\le j \\le k-1$ , since $\\operatorname{ord}_0 \\bar{f}^{\\prime } = k-1$ .",
"Then $a_ut^u = \\frac{c^{m}}{m+1}t_1^{m+1}$ , where $m > k + \\delta _{{\\varepsilon }}(k, v, \\ell )$ .",
"We claim that $u^{\\prime } = (k, 0, \\ldots , 0)$ suffices.",
"For any $w \\in [{\\varepsilon }, \\infty )^{\\ell }$ , consider $m(w)&:= \\min _{u^{\\prime \\prime } \\in S_{\\sigma }}\\lbrace v(a_{u^{\\prime \\prime }})+\\langle u^{\\prime \\prime }, w \\rangle \\rbrace \\\\& \\le v\\left(\\frac{c_{k-1}}{k}\\right)+\\langle (k,0, \\ldots , 0), (w_1, w_2, \\ldots , w_{\\ell })\\rangle \\\\&= v \\left(\\frac{c_{k-1}}{k}\\right) + kw_1$ Since $m > k + \\delta _{{\\varepsilon }}(k, v, \\ell )$ , we have $v(m+1) < (m+1-k)w_1 + v(k),$ which rearranges to $-v(k) + kw_1 < -v(m+1) + (m+1)w_1.$ Using $v(c_{k-1}) = 0$ and $v(c_m) \\ge 0$ , this inequality becomes $v\\left( \\frac{c_{k-1}}{k}\\right) + kw_1 < v \\left(\\frac{c_m}{m+1}\\right) + (m+1)w_1.$ That is, $(u, v(a_u)) \\notin \\operatorname{vert}_w(F)$ , as required.",
"Remark 5.12 Lemma REF shows that any pure power series as in the statement of the lemma can be approximated by polynomials whose terms are pure, and whose degree is less than $k + \\delta _{{\\varepsilon }}(k, v, \\ell )$ .",
"This, in turn, means that the Newton polygons of these polynomials are at worst the convex hull of the points $\\lbrace (0, \\ldots , 0)\\rbrace \\cup \\lbrace (k + \\delta _{{\\varepsilon }}(k, v, \\ell ))e_i : 1 \\le i \\le \\ell \\rbrace $ , where the $e_i$ denotes the $i$ -th standard vector.",
"Thus, the Newton polygon can be approximated by a simplex.",
"Definition 5.13 Let $A = (a_{ij})$ be a $d \\times d$ matrix.",
"The permanent of $A$ is $\\operatorname{Per}(A) = \\sum _{\\sigma \\in S_d} \\prod _{i=1}^d (a_{i \\sigma (i)}).$ Lemma 5.14 Let $A = (a_{ij})$ be a $d \\times d$ matrix of positive real numbers, and define the polytopes $X_i \\subseteq {\\mathbb {R}}^d$ for $1 \\le i \\le d$ by the following: $X_i = \\operatorname{conv}(0, a_{i,1}e_1, \\ldots , a_{i,d}e_d)$ Then $\\operatorname{MV}(X_1, \\ldots , X_d) = \\frac{1}{d!}",
"\\operatorname{Per}(A).$ The mixed volume is $& \\textup {coefficient of \\lambda _1 \\cdots \\lambda _d of }\\operatorname{vol}\\left(\\operatorname{conv}(0, (\\lambda _1 a_{11} + \\cdots + \\lambda _da_{d1})e_1, \\ldots , (\\lambda _1 a_{1d} + \\cdots + \\lambda _da_{dd})e_d)\\right) \\\\&= \\textup {coefficient of \\lambda _1 \\cdots \\lambda _d of } \\frac{1}{d!",
"}(\\lambda _1 a_{11} + \\cdots + \\lambda _da_{d1}) \\cdots (\\lambda _1 a_{1d} + \\cdots + \\lambda _da_{dd})\\\\& = \\frac{1}{d!",
"}\\sum _{\\sigma \\in S_d}\\prod _{i=1}^d a_{\\sigma (i),i} = \\frac{1}{d!}",
"\\operatorname{Per}(A).",
"$ Lemma 5.15 Let $f_i \\in K[[t_i]]$ .",
"Suppose further that $f_i$ converges when $v(t_i) \\ge 1/d_i$ and that $f_i^{\\prime }(t_i) = \\sum _{j=0}^{\\infty } c_{ij}t_i^j \\in R[[t_i]]$ for all $i$ , where $R$ is the ring of integers for $K$ .",
"Suppose also that for each $i$ there exists $k_i \\in {\\mathbb {Z}}_{\\ge 0}$ such that the coefficients $c_{ij}$ satisfy $v(c_{ij}) > 0$ for $j < k_i$ and $v(c_{ik_i}) = 0$ .",
"From these data, define a multivariate pure power series $F(t_1, \\ldots , t_n) := f_1(t_1) + \\cdots + f_d(t_n).$ Then the Newton polygon of the pure power series $F \\in K \\langle U_{P_m} \\rangle $ (where $m = (1/m_1, \\ldots , 1/m_n)$ ) is contained in the $d$ -dimensional simplex defined by the convex hull of the vectors $(k_i + \\delta _{{\\varepsilon }}(k_i, v, m_i))e_i,$ where ${\\varepsilon }\\in {\\mathbb {Q}}$ satisfies ${\\varepsilon }\\le 1/m_i$ for all $i$ , and $e_i$ is the $i$ -th standard vector.",
"Straightforward application of Lemma REF and Remark REF to each $f_i$ that show up in the pure power series.",
"Let ${\\mathcal {Q}}= \\lbrace Q_1, \\ldots , Q_d\\rbrace \\in (\\operatorname{Sym}^dX)({\\mathbb {F}}_p)$ .",
"Let ${\\mathcal {U}}$ be the residue disk of $(\\operatorname{Sym}^dX)(p)$ reducing to $\\lbrace Q_1, \\ldots , Q_d\\rbrace $ .",
"Decompose the multiset $\\lbrace Q_1, \\ldots , Q_d\\rbrace $ into disjoint multisets ${\\mathcal {S}}_1, \\ldots , {\\mathcal {S}}_r$ each consisting of a single point with multiplicity $s_j = \\#{\\mathcal {S}}_j$ .",
"Let $L_j$ be the degree-$s_j$ unramified extension of $K_j$ , the field of definition of the points in ${\\mathcal {S}}_j$ , and let $R_j$ be the ring of integers of the $L_j$ .",
"For $1 \\le i \\le d$ , let $f_{i,j} \\in L_j[[t_j]]$ be the power series obtained from Chabauty's method, applied to the residue disk in $(\\operatorname{Sym}^{s_j}X)(K_j)$ above the point ${\\mathcal {S}}_j$ , such that their derivatives $f_{i,j}^{\\prime }$ are in $R_j[[t_j]]$ .",
"Let $F_i(t_1, \\ldots , t_d) = f_{i,1}(t_1) + \\cdots + f_{i,d}(t_d),$ and let $k_{ij} = \\operatorname{ord}_0(f_{ij})$ .",
"Then define the $d \\times d$ matrix $A_{{\\mathcal {P}}} = (a_{ij})$ by $a_{ij} = k_{i,j} + \\delta _{{\\varepsilon }}(k_{i,j}, v, s_i)$ for each residue disk and suitably small ${\\varepsilon }$ .",
"Theorem 5.16 Keep the notation from the previous paragraph.",
"Then the $F_i$ satisfy $N_0^{\\times }(F_1, \\ldots , F_d) \\le \\frac{1}{d!}",
"\\operatorname{Per}(A).$ By Proposition REF , we may as well assume that the power series that we get from Chabauty's method have finitely many common zeros (that is, a deformation of the power series exists, such that the tropicalizations and the $\\gamma _w$ stay constant).",
"This means that, by Theorem REF , that the number of isolated solutions can be written as the mixed volume of Newton polygons.",
"Now combine Lemma REF and Lemma REF .",
"Recall that $N_0^{\\times }(F_1, \\ldots , F_d)$ counts the 0-dimensional components of the common zeros of the $F_i$ in $(p^{\\times })^d$ .",
"Thus, we need to count the solutions in which some of the coordinates are 0 separately.",
"For example, if we wish to count the solutions that are of the form $(p^{\\times })^{(d-1)} \\times \\lbrace 0\\rbrace $ , it suffices to consider $N_0^{\\times }(F_1(t_1, t_2, \\ldots , t_{d-1}, 0), \\ldots , F_{d-1}(t_1, t_2, \\ldots , t_{d-1}, 0)),$ which is bounded above by $\\frac{1}{(d-1)!",
"}\\operatorname{Per}(B)$ , where $B$ is a $(d-1) \\times (d-1)$ minor of $A$ that takes the first $(d-1)$ rows and columns.",
"Thus, let $\\operatorname{Per}(A)^{\\prime } := \\sum _{0 \\le i \\le d} \\sum _{j \\in \\Lambda _i} \\frac{1}{i!",
"}\\operatorname{Per}(A_{ij}),$ where $A_{ij}$ denotes the $i \\times i$ minor of $A$ that takes the first $i$ columns (and any $i$ rows), and $A_{00}$ is the $0 \\times 0$ matrix whose permanent is understood to be 1 (since if (0, ..., 0) were a solution to the $F_i$ , it would contribute at most 1 to $N_0(F_1, \\ldots , F_d)$ ).",
"Theorem 5.17 Suppose $X$ is a nice curve over ${\\mathbb {Q}}$ with good reduction at $p$ satisfying Assumption REF , and let $\\omega _1, \\ldots , \\omega _d \\in H^0(X_{{\\mathbb {Q}}_p}, \\Omega ^1)$ be independent differential forms that vanish on $\\overline{J({\\mathbb {Q}})}$ such that $\\bar{\\omega }_i \\ne 0$ .",
"Then keeping the notation as above, with the $k_{i,j}$ corresponding to the order of vanishing of $\\omega _i$ at the point $P_j$ , the number of points outside of the special set of $(\\operatorname{Sym}^dX)({\\mathbb {Q}}_p)$ is at most $\\sum _{{\\mathcal {P}}\\in (\\operatorname{Sym}^dX)({\\mathbb {F}}_p)} \\frac{1}{N_{{\\mathcal {P}}}}\\operatorname{Per}(A_{{\\mathcal {P}}})^{\\prime }$ Apply the above theorem to each residue disk of $(\\operatorname{Sym}^dX)({\\mathbb {Q}}_p)$ , and use Corollary REF .",
"The $\\frac{1}{N}$ accounts for the ordering of the solutions, since the order of the points does not matter in $\\operatorname{Sym}^dX$ .",
"The above theorem shows that there is an upper bound on the number of points outside of the special set, depending only on the choice of $g, d$ and $p$ .",
"If we bound $\\# (\\operatorname{Sym}^dX)({\\mathbb {F}}_p)$ in terms of $g, d$ and $p$ , then this would complete the proof of Theorem REF : Proposition 5.18 Given a nice curve $X$ of genus $d$ with good reduction at $p$ and $d \\ge 1$ , $\\#((\\operatorname{Sym}^dX)({\\mathbb {F}}_p)) \\le (1 + 2g p^{d/2} + p^d)^d.$ We use the Hasse-Weil bound on $X$ , along with the fact that if $\\lbrace P_1, \\ldots , P_d\\rbrace \\in (\\operatorname{Sym}^dX)({\\mathbb {F}}_p)$ , then $P_i \\in X({\\mathbb {F}}_{p^d})$ for $1 \\le i \\le d$ .",
"Then the proof of Theorem REF follows by combining the statements of Proposition REF and Theorem REF ."
],
[
"An application",
"In this section, we prove the following corollary: Corollary 6.1 We can take $N(2,3,3) = 1539$ for any $X/{\\mathbb {Q}}$ a hyperelliptic curve whose affine model $y^2 = f(x)$ satisfies $\\deg (f) =7$ (so that $g = 3$ ), such that $\\operatorname{Rank}J \\le 1$ , and such that $X$ has good reduction at $p=2$ .",
"As noted in the introduction, Assumption REF is unnecessary when $\\operatorname{rk}J \\le 1$ .",
"(And we expect that 100% of hyperelliptic curves have ranks 0 or 1, assuming Goldfeld's conjecture!)",
"Lemma 6.2 Let $X$ be a smooth projective odd hyperelliptic curve of genus 3 that has good reduction at 2.",
"Then $\\# (\\operatorname{Sym}^2X)({\\mathbb {F}}_2) \\le 19$ .",
"We first note that a mod-2 reduction of an odd hyperelliptic curve of genus 3 corresponds to an equation of the form $y^2 + g(x)y = h(x)$ , with $g(x), h(x) \\in {\\mathbb {F}}_2[x]$ , with $\\deg g \\le 3, \\deg h = 7$ .",
"Let ${\\mathcal {P}}\\in (\\operatorname{Sym}^2X)({\\mathbb {F}}_2)$ .",
"It can be viewed as a multiset of two points ${\\mathcal {P}}= \\lbrace P_1, P_2\\rbrace $ .",
"We denote by $x(P_i)$ and $y(P_i)$ the $x$ - and $y$ -coordinates of $P_i$ , respectively, for $i = 1,2$ .",
"We have two cases: Case 1: When $P_1, P_2 \\in X({\\mathbb {F}}_2)$ .",
"If $P \\in X({\\mathbb {F}}_2)$ , then $x(P), y(P) \\in {\\mathbb {F}}_2$ or $x(P) = \\infty $ , so in particular, one must have $x(P) \\in \\lbrace 0,1, \\infty \\rbrace $ .",
"There are at most two points above each ${\\mathbb {F}}_2$ -point in the map $X \\rightarrow {\\mathbb {P}}^1$ , so there are at most 5 points in $X({\\mathbb {F}}_2)$ .",
"Let $\\#X({\\mathbb {F}}_2) = a$ .",
"Then there are ${a \\atopwithdelims ()2} + a$ points ${\\mathcal {P}}\\in (\\operatorname{Sym}^2X)({\\mathbb {F}}_2)$ of the form $\\lbrace P_1, P_2\\rbrace $ with $P_i \\in X({\\mathbb {F}}_2)$ ; the first term counts $\\lbrace P_1,P_2\\rbrace $ with $P_1$ and $P_2$ distinct, and the second terms counts $\\lbrace P_1,P_2\\rbrace $ with $P_1=P_2$ .",
"Case 2: When $P_1, P_2 \\in X({\\mathbb {F}}_4) \\backslash X({\\mathbb {F}}_2)$ are Galois conjugates.",
"In this case, there are at most 4 points of $X({\\mathbb {F}}_4)-X({\\mathbb {F}}_2)$ above ${\\mathbb {P}}^1(F_4)-{\\mathbb {P}}^1(F_2)$ .",
"But there could also be points of $X({\\mathbb {F}}_4)-X({\\mathbb {F}}_2)$ above ${\\mathbb {P}}^1({\\mathbb {F}}_2)$ ; the number of these is $5-a$ , since all of the 5 ${\\overline{{\\mathbb {F}}}}_2$ -points of X above ${\\mathbb {P}}^1({\\mathbb {F}}_2)$ are either ${\\mathbb {F}}_2$ -points or ${\\mathbb {F}}_4$ -points.",
"So there are at most $9-a$ such points.",
"Clearly, the choice of $1 \\le a \\le 5$ that maximizes ${a \\atopwithdelims ()2} + 9$ is $a=5$ , which means that there are at most 19 points in $(\\operatorname{Sym}^2X)({\\mathbb {F}}_2)$ .",
"Now we focus on a single residue disk of $(\\operatorname{Sym}^2X)({\\mathbb {F}}_2)$ and compute the possible number of points on each residue disk.",
"Since $g=3$ , the degree of $\\bar{\\omega }$ is $2g-2 = 4$ .",
"We start by computing $\\delta _{{\\varepsilon }}(k, v, \\ell )$ in Definition REF for when $k = 1,2,3,4$ .",
"We take ${\\varepsilon }\\in (0, \\frac{1}{2})$ as small as possible, as that minimizes $\\delta _{{\\varepsilon }}(k,v,\\ell )$ .",
"Then we have $\\delta _{{\\varepsilon }}(4,2,2) = 0, \\quad \\delta _{{\\varepsilon }}(3,2,2) = 5, \\quad \\delta _{{\\varepsilon }}(2,2,2) = 2, \\quad \\delta _{{\\varepsilon }}(1,2,2) = 3.$ Thus, for a residue disk over ${\\mathcal {P}}$ , the largest value of $\\operatorname{Per}A_{{\\mathcal {P}}}$ is given from the $2 \\times 2$ matrix whose entries are all $k + \\delta _{{\\varepsilon }}(k,v, \\ell )$ with $k = 3$ .",
"That is, the maximal value for $\\operatorname{Per}A_{{\\mathcal {P}}}$ is 128.",
"Now, there are two $1 \\times 1$ minors that we need to compute, from the definition of $\\operatorname{Per}(A)^{\\prime }$ in the previous chapter.",
"Again, the maximal values for these are 8, obtained when $k=3$ .",
"This gives $\\operatorname{Per}(A)^{\\prime } \\le \\frac{1}{2} \\cdot 128 + 8 + 8 + 1 = 81$ .",
"Now, we apply Theorem REF on the 19 residue disks with $N_{{\\mathcal {P}}} \\ge 1$ and $\\operatorname{Per}(A_{{\\mathcal {P}}})^{\\prime } \\le 81$ .",
"This gives the upper bound of $81 \\times 19 = 1539.$ This completes the proof of Corollary REF .",
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[
"Acknowledgements",
"I would like to thank my advisor, Bjorn Poonen, for introducing me to Chabauty's method, for many helpful conversations on this project, and for his feedback on the exposition of this article.",
"This project became the topic of my PhD thesis at MIT.",
"I also thank Joseph Rabinoff for patiently explaining many of his results to me; many results in Section of this paper were built on his results.",
"I also benefited from conversations with Matt Baker, Jennifer Balakrishnan, Eric Katz, Samir Siksek, Bernd Sturmfels, and David Zureick-Brown."
],
[
"Chabauty on $\\operatorname{Sym}^dX$",
"In this section, we consider the problem of counting rational points outside of the special set of $\\operatorname{Sym}^dX$ .",
"Using Chabauty's method, we will reduce this problem to analyzing the common zeros of $d$ power series in $d$ variables, and the power series have specific forms."
],
[
"Classical Chabauty",
"The exposition in this subsection outlines the classical method of Chabauty that gives an upper bound on $\\#X({\\mathbb {Q}})$ ; what we have here is a summarized version of .",
"Let $\\iota $ be the ${\\mathbb {Q}}$ -embedding $\\iota : X & \\hookrightarrow J\\\\P & \\mapsto [P-O]$ where $J$ is the Jacobian of $X$ , viewed as the group of linear equivalence classes of degree-zero divisors on $X$ .",
"Then $J$ is an abelian variety of dimension $g$ over ${\\mathbb {Q}}$ .",
"By an abuse of notation, we denote $\\iota (X)$ as $X$ .",
"The inclusion $X({\\mathbb {Q}}) \\subseteq J({\\mathbb {Q}})$ holds, since $O \\in X({\\mathbb {Q}})$ .",
"Since $X({\\mathbb {Q}}) \\subseteq X({\\mathbb {Q}}_p)$ , we have $X({\\mathbb {Q}}) \\subseteq X({\\mathbb {Q}}_p) \\cap \\overline{J({\\mathbb {Q}})}$ , where $\\overline{J({\\mathbb {Q}})}$ denotes the $p$ -adic closure of $J({\\mathbb {Q}})$ inside $J({\\mathbb {Q}}_p)$ .",
"Chabauty's result is: Theorem 2.1 () Keep the notation as above.",
"Let $X$ be a curve of genus $g \\ge 2$ over ${\\mathbb {Q}}$ .",
"If $X$ has good reduction at a prime $p$ and if $\\dim \\overline{J({\\mathbb {Q}})} < g$ , then $X({\\mathbb {Q}}_p) \\cap \\overline{J({\\mathbb {Q}})}$ is finite.",
"To compute $\\#(X({\\mathbb {Q}}_p) \\cap \\overline{J({\\mathbb {Q}})})$ explicitly under the hypothesis of Theorem REF , let $J_{{\\mathbb {Q}}_p}$ and $X_{{\\mathbb {Q}}_p}$ be the base changes of $J$ and $X$ to ${\\mathbb {Q}}_p$ .",
"There is a bilinear pairing $J({\\mathbb {Q}}_p) \\times H^0(J_{{\\mathbb {Q}}_p}, \\Omega ^1) &\\rightarrow {\\mathbb {Q}}_p \\\\(Q, \\omega ) &\\mapsto \\int _{O}^Q \\omega $ which induces the logarithm homomorphism $\\log : J({\\mathbb {Q}}_p) \\rightarrow H^0(J_{{\\mathbb {Q}}_p}, \\Omega ^1)^* = T_OJ_{{\\mathbb {Q}}_p},$ where $T_OJ_{{\\mathbb {Q}}_p}$ denotes the tangent space to $J_{{\\mathbb {Q}}_p}$ at the origin $O$ .",
"Since $\\dim \\log (\\overline{J({\\mathbb {Q}})}) < g$ , there exists a hyperplane $H \\subseteq T_OJ_{{\\mathbb {Q}}_p}$ containing $\\log (\\overline{J({\\mathbb {Q}})})$ .",
"This hyperplane $H$ is defined by the vanishing of some $\\omega _J \\in H^0(J_{{\\mathbb {Q}}_p}, \\Omega ^1) \\cong T_OJ_{{\\mathbb {Q}}_p}^*$ , and the restriction of $\\omega _J$ to $X_{{\\mathbb {Q}}_p}$ can be uniquely identified with an $\\omega _X \\in H^0(X_{{\\mathbb {Q}}_p}, \\Omega ^1)$ via: Proposition 2.2 (, Proposition 2.2) The restriction map $H^0(J_{{\\mathbb {Q}}_p}, \\Omega ^1) \\rightarrow H^0(X_{{\\mathbb {Q}}_p}, \\Omega ^1)$ induced by $X \\hookrightarrow J$ is an isomorphism of ${\\mathbb {Q}}_p$ -vector spaces.",
"Then, the map induced from the above bilinear pairing using the $\\omega _J$ given by $J({\\mathbb {Q}}_p) &\\rightarrow {\\mathbb {Q}}_p \\\\Q &\\mapsto \\int _{O}^Q \\omega _J$ vanishes on $\\overline{J({\\mathbb {Q}})}$ by construction, so $\\# (X({\\mathbb {Q}}_p) \\cap \\overline{J({\\mathbb {Q}})})$ is bounded above by the number of zeros of the restriction $\\eta : X({\\mathbb {Q}}_p) &\\rightarrow {\\mathbb {Q}}_p\\\\Q &\\mapsto \\int _O^Q \\omega _X,$ where $\\omega _X = \\iota ^* \\omega _J$ .",
"From a computational perspective, it is known that $\\omega $ has a well-defined power series expansion in terms of a uniformizer $t$ on small enough open subgroups $U$ of $X({\\mathbb {Q}}_p)$ .",
"In fact, $U$ can be taken to be residue disks of the reduction $\\operatorname{red}_p: X({\\mathbb {Q}}_p) \\twoheadrightarrow X({\\mathbb {F}}_p)$ , which are the preimages of any point $Q \\in X({\\mathbb {F}}_p)$ .",
"Such residue disk $U$ can be parametrized by a uniformizer $t$ , which gives a set bijection $t: U \\rightarrow p{\\mathbb {Z}}_p$ .",
"Then one expresses the locally analytic function $\\eta $ as a power series in terms of $t$ on $U$ ; the local coordinates can be chosen so that $\\omega (t) \\in {\\mathbb {Z}}_p[[t]]$ , and the number of zeros of $\\eta $ on each residue disk can then be estimated using Newton polygons."
],
[
"Chabauty on $\\operatorname{Sym}^d X$",
"In theory, it seems plausible that Chabauty's method could still apply to any higher-dimensional variety $Y$ , where the Albanese variety $\\operatorname{Alb}(Y)$ is used in place of the Jacobian, look for all rational points on the image of the Albanese map using a similar technique.",
"However, there exist several problems in generalizing Chabauty's method to arbitrary higher-dimensional varieties.",
"(1) $\\operatorname{Alb}(Y)$ may be trivial: since $\\dim \\operatorname{Alb}(Y) = h^0(Y, \\Omega _1)$ , if $h^0(Y, \\Omega _1) = 0$ , then Chabauty's method yields nothing.",
"For example, if $Y$ is a K3 surface or an Enriques surface, $h^{0,1} = 0$ , so Chabauty's method cannot apply.",
"(2) To understand $Y({\\mathbb {Q}})$ , we need to understand the (rational points of the) fibres of $j$ as well as the rational points of $j(Y)$ .",
"Understanding the fibres may be complicated.",
"(3) The case of higher-dimensional varieties allows for the possibility that $\\#Y({\\mathbb {Q}}) = \\infty $ : for example, if $Y = \\operatorname{Sym}^2 X$ for a hyperelliptic curve $X: y^2 = f(x)$ , then $\\lbrace (t, \\sqrt{f(t)}),(t, -\\sqrt{f(t)})\\rbrace \\in (\\operatorname{Sym}^2X)({\\mathbb {Q}})$ for all $t \\in {\\mathbb {Q}}$ .",
"The first two problems are taken care of by choosing $Y = \\operatorname{Sym}^d X$ , where $X$ is a nice curve of sufficiently high genus $g$ (to be chosen later); then $\\operatorname{Alb}(Y) = \\operatorname{Jac}(X)=:J$ , so the Albanese variety is nontrivial, and understanding the fibres of the Albanese map $j: (\\operatorname{Sym}^dX)({\\mathbb {Q}}) &\\rightarrow J({\\mathbb {Q}})\\\\\\lbrace P_1, \\ldots , P_d\\rbrace &\\mapsto [P_1 + \\cdots + P_d - d\\cdot O],$ is not too difficult: Lemma 2.3 Suppose that $Q \\in (\\operatorname{Sym}^dX)({\\mathbb {Q}})$ .",
"Then the set of rational points on the fibre of $j$ containing $Q$ is isomorphic to ${\\mathbb {P}}^n({\\mathbb {Q}})$ for some $n \\ge 0$ .",
"As $J$ parametrizes the equivalence classes of degree-0 divisors, $Q$ is identified with an effective divisor on $X$ .",
"By *Theorem II.5.19, the set of points giving rise to the same divisor is isomorphic to a finite-dimensional vector space.",
"In particular, if a fibre contains two distinct rational points, then dimension of this vector space is at least 1.",
"To deal with the last problem, we recall from : Theorem 2.4 () Let $A/{\\mathbb {Q}}$ be an abelian variety, and $X \\subseteq A$ be a closed subvariety.",
"Then there exist finitely many subvarieties $Y_i \\subset X$ such that each $Y_i$ is a coset of an abelian subvariety of $A$ and $X({\\mathbb {Q}}) = \\bigcup Y_i({\\mathbb {Q}}).$ Apply this theorem to the image of $j$ to get $j((\\operatorname{Sym}^dX)({\\mathbb {Q}})) = \\bigcup _{\\textup {finite}} Y_i({\\mathbb {Q}})$ .",
"From Lemma REF and Theorem REF , we see that there are two ways of obtaining $\\#(\\operatorname{Sym}^dX)({\\mathbb {Q}}) = \\infty $ : either at least one of the fibres of $j$ is nontrivial, or there exists some $Y_i$ with $\\dim Y_i > 0$ .",
"Excluding the ${\\mathbb {Q}}$ -points on $\\operatorname{Sym}^dX$ accounted by these two possibilities, we are left with finitely many rational points on $\\operatorname{Sym}^dX$ .",
"However, to avoid ambiguity coming from Faltings' theorem, one excludes the special set (Definition REF ) instead, which includes both of the aforementioned possibilities.",
"For the rest of the section, assume that $X$ satisfies $\\operatorname{rk}J \\le g-d$ .",
"Further, let $p$ be a prime, and let $X$ have good reduction at $p$ .",
"We observe that $(\\operatorname{Sym}^dX)({\\mathbb {Q}}) \\subseteq j^{-1}(j(\\operatorname{Sym}^dX)({\\mathbb {Q}}_p) \\cap \\overline{J({\\mathbb {Q}})}).$ We obtain locally analytic functions come from integrating $\\omega _i \\in H^0(J_{{\\mathbb {Q}}_p}, \\Omega ^1)$ , as in the previous section.",
"However, we use the following stronger definition: Definition 2.5 For any $\\omega _J \\in H^0(J_{K}, \\Omega ^1)$ , define the map $\\eta : J(\\overline{{\\mathbb {Q}}}_p) \\rightarrow \\overline{{\\mathbb {Q}}}_p$ by taking the inverse limit of the maps $\\eta _K: J(K) &\\rightarrow K\\\\Q &\\mapsto \\int _{O}^Q \\omega _J,$ for each $p$ -adic field $K$ .",
"The $p$ -adic integrals also satisfy for $Q_1, Q_2 \\in J(\\overline{{\\mathbb {Q}}}_p)$ and $\\omega _J \\in H^0(J_{{\\mathbb {Q}}_p}, \\Omega ^1)$ , $\\int _{O}^{Q_1+Q_2}\\omega _J = \\int _{O}^{Q_1}\\omega _J + \\int _{Q_1}^{Q_1 + Q_2}\\omega _J = \\int _{O}^{Q_1}\\omega _J + \\int _{O}^{Q_2}\\omega _J,$ where the first equality follows from linearity, and the second equality follows from the translation-invariance of $p$ -adic integrals.",
"Then we may define the integral on $\\operatorname{Sym}^dX$ via the pullback of the integral on $W_d \\subseteq J$ , which can be written using the above as $\\int _{O}^{[P_1 + \\cdots + P_d - d \\cdot O]}\\omega _J = \\int _O^{[P_1-O]} \\omega _J + \\cdots + \\int _O^{[P_d-O]}\\omega _J.$ Therefore, the corresponding locally analytic function on $\\operatorname{Sym}^dX$ can be written as $\\eta \\colon (\\operatorname{Sym}^dX)({\\mathbb {Q}}_p) &\\rightarrow {\\mathbb {Q}}_p\\\\\\lbrace P_1,P_2, \\ldots , P_d\\rbrace &\\mapsto \\eta _J([P_1+P_2+ \\cdots + P_d-d\\infty ])\\\\& = \\int _O^{P_1} \\omega _X + \\int _O^{P_2} \\omega _X +\\cdots + \\int _O^{P_d} \\omega _X,$ where $\\omega _X \\in H^0(X_{{\\mathbb {Q}}_p}, \\Omega ^1)$ is the pullback of $\\omega _J$ via the isomorphism given in Proposition REF , and the $P_i$ are defined over some field $K$ with $[K:{\\mathbb {Q}}] \\le d$ .",
"Since $\\dim \\overline{J({\\mathbb {Q}})} \\le g-d$ , then there exist $(\\omega _J)_1, (\\omega _J)_2, \\ldots , (\\omega _J)_d \\in H^0(J_{{\\mathbb {Q}}_p}, \\Omega ^1)$ that are linearly independent, such that the corresponding locally analytic functions on $J_{{\\mathbb {Q}}_p}$ obtained by integrating the $(\\omega _J)_i$ vanish on $\\overline{J({\\mathbb {Q}})}$ .",
"Let $\\eta _1, \\eta _2, \\ldots , \\eta _d$ be the locally analytic functions on $\\operatorname{Sym}^dX$ corresponding to $(\\omega _J)_1, (\\omega _J)_2, \\ldots , (\\omega _J)_d$ , respectively.",
"Possibly $\\eta _1, \\eta _2, \\ldots , \\eta _d$ have infinitely many common zeros, which contain either the linear system of equivalent divisors parametrized by ${\\mathbb {P}}^n$ for some $n \\ge 1$ , or the (infinitely many) rational points coming from the rational points in the special set $({\\mathcal {S}}(\\operatorname{Sym}^dX))({\\mathbb {Q}})$ ."
],
[
"Explicit parametrization of points on residue disks",
"From §REF , we have (at least) $d$ nontrivial and independent locally analytic functions on $(\\operatorname{Sym}^dX)({\\mathbb {Q}}_p)$ whose common zeros contain the $p$ -adic points $j^{-1}((j(\\operatorname{Sym}^dX))({\\mathbb {Q}}_p) \\cap \\overline{J({\\mathbb {Q}})})$ .",
"We estimate the number of common zeros of these analytic functions away from the special set of $\\operatorname{Sym}^dX$ .",
"In order to do this explicitly, we work locally to get power series expansions.",
"Since $X$ has good reduction at $p$ , $X$ is a smooth proper variety over ${\\mathbb {Z}}_p$ .",
"This implies that $\\operatorname{Sym}^dX$ is smooth and proper over ${\\mathbb {Z}}_p$ .",
"By the valuative criterion for properness, $(\\operatorname{Sym}^dX)({\\mathbb {Q}}_p) = (\\operatorname{Sym}^dX)({\\mathbb {Z}}_p)$ .",
"Definition 2.6 A residue disk ${\\mathcal {U}}$ of $\\operatorname{Sym}^dX$ is the preimage of a point in $(\\operatorname{Sym}^dX)({\\overline{{\\mathbb {F}}}}_p)$ under the reduction modulo-$p$ map $\\operatorname{red}_{p}: (\\operatorname{Sym}^dX)(p) \\twoheadrightarrow (\\operatorname{Sym}^dX)({\\overline{{\\mathbb {F}}}}_p).$ If $K$ is a finite extension of ${\\mathbb {Q}}_p$ , the set of $K$ -points of the residue disk ${\\mathcal {U}}$ are defined to be the set ${\\mathcal {U}}\\cap (\\operatorname{Sym}^dX)(K)$ , and these are denoted ${\\mathcal {U}}(K)$ .",
"Let ${\\mathcal {U}}$ be the residue disk over $O \\in (\\operatorname{Sym}^dX)({\\overline{{\\mathbb {F}}}}_p)$ .",
"Then ${\\mathcal {U}}$ fits into the exact sequence $0 \\rightarrow {\\mathcal {U}}\\rightarrow J(p) = J({\\mathcal {O}}_{p}) \\rightarrow J({\\overline{{\\mathbb {F}}}}_p) \\rightarrow 0,$ where the equality in the middle follows from the valuative criterion for properness.",
"Then for any finite extension $K$ of ${\\mathbb {Q}}_p$ , we have ${\\mathcal {U}}(K) = \\lbrace {\\mathcal {P}}\\in (\\operatorname{Sym}^dX)(K): \\operatorname{red}_{p}({\\mathcal {P}}) = \\lbrace P_1, \\ldots , P_d\\rbrace \\rbrace $ .",
"Thus, by a Hensel-type argument, there is a bijection $(u_1, \\ldots , u_d): {\\mathcal {U}}(K) \\stackrel{\\sim }{\\longrightarrow } (p_K {\\mathcal {O}}_K)^d$ between the set of $K$ -points of the residue disk mapping to $\\lbrace P_1, \\ldots , P_d\\rbrace $ via some local coordinates $u_1, \\ldots , u_d$ , where $p_K$ is the uniformizer of $K$ .",
"In practice, this parametrization is not practical: §REF suggests that we write the higher-dimensional integrals in terms of several 1-dimensional integrals expanded around various ${\\mathbb {F}}_{p^n}$ -points $P_i$ ."
],
[
"The case of $\\operatorname{Sym}^2X$",
"For simplicity, we first consider the case when $d=2$ .",
"First suppose that we consider the residue disk ${\\mathcal {U}}({\\mathbb {Q}}_p)$ which consists of the ${\\mathbb {Q}}_p$ -points in $\\operatorname{Sym}^2X$ reducing to $\\lbrace P, P\\rbrace \\in (\\operatorname{Sym}^2X)({\\mathbb {F}}_p)$ for some $P \\in X({\\mathbb {F}}_p)$ .",
"The completion of the local ring of $(X \\times X)_{{\\mathbb {Q}}_p}$ near any pair of points $(Q_1, Q_2) \\in X \\times X$ reducing to $\\lbrace P, P\\rbrace $ is given by ${\\mathbb {Q}}_p[[t_1, t_2]]$ , where $t_1$ and $t_2$ denote the uniformizers for the set of ${\\mathbb {Q}}_p$ -points of the residue disks ${\\mathcal {U}}_1$ around $Q_1$ and ${\\mathcal {U}}_2$ around $Q_2$ in $X$ , respectively.",
"We further assume that $t_1$ and $t_2$ vanish at $Q_1^{\\prime }$ and $Q_2^{\\prime }$ , respectively.",
"This means that we have two bijections $t_1: {\\mathcal {U}}_1 \\stackrel{\\sim }{\\longrightarrow } p{\\mathbb {Z}}_p, \\quad t_2: {\\mathcal {U}}_2 \\stackrel{\\sim }{\\longrightarrow } p{\\mathbb {Z}}_p.$ Since $t_1(Q_1^{\\prime }) = 0$ , and $t_2(Q_2^{\\prime }) = 0$ , for any $\\omega \\in H^0(J_{{\\mathbb {Q}}_p}, \\Omega ^1)$ , the Coleman integral $\\eta : \\operatorname{Sym}^dX({\\mathbb {Q}}_p) &\\rightarrow {\\mathbb {Q}}_p\\\\\\lbrace Q_1, Q_2\\rbrace &\\mapsto \\int _{O}^{\\lbrace Q_1, Q_2\\rbrace } \\omega $ can be written as the following, in terms of the $t_i$ : $\\eta (\\lbrace Q_1, Q_2\\rbrace ) &= \\int _{O}^{\\lbrace Q_1, Q_2\\rbrace }\\omega = \\int _{O}^{Q_1}\\omega + \\int _{O}^{Q_2}\\omega \\\\&= \\int _{O}^{Q_1^{\\prime }}\\omega + \\int _{Q_1^{\\prime }}^{Q_1}\\omega + \\int _{O}^{Q_2^{\\prime }}\\omega + \\int _{Q_2^{\\prime }}^{Q_2}\\omega \\\\&= C + \\int _0^{t_1(Q_1)}\\omega (t_1) + \\int _0^{t_2(Q_2)}\\omega (t_2),$ where $C = \\int _O^{Q_1^{\\prime }}\\omega +\\int _O^{Q_2^{\\prime }}\\omega $ is a constant in ${\\mathbb {Q}}_p$ (that depends on the choice of $Q_i^{\\prime }$ ).",
"One can relate the $t_i$ to the original local coordinates of ${\\mathcal {U}}$ .",
"If $r(Q_1, Q_2) = (P_1, P_2)$ in $(\\operatorname{Sym}^dX)({\\mathbb {F}}_p)$ , then the completion of the local ring at $\\lbrace Q_1, Q_2\\rbrace \\in (\\operatorname{Sym}^2X)({\\mathbb {Q}}_p)$ is given by ${\\mathbb {Q}}_p[[t_1, t_2]]^{S_2} = {\\mathbb {Q}}_p[[u_1, u_2]],$ so one could choose the $t_i$ and the $u_i$ to satisfy $u_1 = t_1 + t_2$ and $u_2 = t_1t_2$ .",
"This means that for each point $\\lbrace Q_1, Q_2\\rbrace \\in (\\operatorname{Sym}^2X)({\\mathbb {Q}}_p)$ , which corresponds bijectively to a unique pair $(u_1, u_2)$ , there are two pairs $(t_1, t_2)$ that correspond to it.",
"On the other hand, if the residue disk ${\\mathcal {U}}$ were the preimage of $\\lbrace P_1, P_2\\rbrace \\in (\\operatorname{Sym}^2X)({\\mathbb {F}}_p)$ with $P_1 \\ne P_2$ under the reduction by $p$ map, the situation is simpler, as we have the following description of the residue disk.",
"${\\mathcal {U}}&= \\lbrace \\lbrace Q_1, Q_2\\rbrace : Q_i \\in X(W({\\mathbb {F}}_{p^2})) \\textup { reducing to } P_i \\in X({\\mathbb {F}}_{p^2})\\rbrace \\\\&\\subseteq X(W({\\mathbb {F}}_{p^2})) \\times X(W({\\mathbb {F}}_{p^2}))$ where $W({\\mathbb {F}}_{p^2})$ denotes the Witt ring of ${\\mathbb {F}}_{p^2}$ .",
"Thus, there are the bijections $(t_1, t_2): {\\mathcal {U}}\\stackrel{\\sim }{\\longrightarrow } p{\\mathbb {Z}}_p \\times p{\\mathbb {Z}}_p,$ possibly defined over some quadratic extension of ${\\mathbb {Q}}_p$ , where ${\\mathcal {U}}_i$ denotes the residue disk around $Q_i$ .",
"Choose the basepoints of the lifts $t_1(Q_1^{\\prime }) = 0$ , and $t_2(Q_2^{\\prime }) = 0$ .",
"Then, we can write for each $\\omega \\in H^0(X_{{\\mathbb {Q}}_p}, \\Omega ^1)$ $\\eta (\\lbrace Q_1, Q_2\\rbrace ) &= \\int _{O}^{\\lbrace Q_1, Q_2\\rbrace }\\omega = \\int _{O}^{Q_1}\\omega + \\int _{O}^{Q_2}\\omega \\\\&= \\int _{O}^{Q_1^{\\prime }}\\omega + \\int _{Q_1^{\\prime }}^{Q_1}\\omega + \\int _{O}^{Q_1^{\\prime }}\\omega + \\int _{Q_2^{\\prime }}^{Q_2}\\omega \\\\&= C + \\int _0^{t_1(Q_1)}\\omega (t_1) + \\int _0^{t_2(Q_2)}\\omega (t_2),$ where the $\\omega (t_i)$ are defined (under appropriate scaling) over the ring of integers of some quadratic extension of ${\\mathbb {Q}}_p$ , and $C = \\int _O^{Q_1^{\\prime }}\\omega + \\int _O^{Q_2^{\\prime }}\\omega $ is a constant in ${\\mathbb {Z}}_p$ (that depends on the choice of $Q_1^{\\prime }$ and $Q_2^{\\prime }$ ) In this case, each pair $(t_1, t_2)$ represents a point on $\\operatorname{Sym}^2X$ exactly once.",
"In either cases, we are able to express the Coleman integral in the following form: Definition 2.7 A power series $f \\in K[[t_1, \\ldots , t_n]]$ is said to be pure if each of its terms are of the form $Ct_i^N$ , with $C \\in K$ and $N \\in {\\mathbb {Z}}_{\\ge 0}$ .",
"In particular, a pure power series does not contain any term that is a product of more than one variable.",
"Our goal for the rest of the section is to find pure power series that are related to the locally analytic functions in $d$ variables that we obtain from Chabauty's method."
],
[
"The case of $\\operatorname{Sym}^dX$",
"In this section, we generalize §REF .",
"More concretely, our aim is to express each $p$ -adic integral obtained from a $\\omega \\in H^0(X_{{\\mathbb {Q}}_p}, \\Omega ^1)$ (see §REF ), which is a power series of each residue disk of $(\\operatorname{Sym}^dX)({\\mathbb {Q}}_p)$ , as a pure power series over some extension field of ${\\mathbb {Q}}_p$ on the residue disk.",
"We will show that this is possible by doing a change of variables on the local coordinates of each residue disk.",
"We fix a holomorphic differential $\\omega $ from which we get one of the $d$ power series vanishing on $j^{-1}(\\overline{J({\\mathbb {Q}})} \\cap j(\\operatorname{Sym}^dX)({\\mathbb {Q}}_p))$ as in section §REF .",
"We now consider the residue disk given as the preimage of the point $\\lbrace P_1, \\ldots , P_d\\rbrace \\in (\\operatorname{Sym}^dX)({\\mathbb {F}}_p)$ under the reduction map modulo $p$ .",
"The multiset $\\lbrace P_1, \\ldots , P_d\\rbrace $ can be decomposed as the disjoint union of the multisets of the form ${\\mathcal {S}}:= \\lbrace P_{i_1}, \\ldots , P_{i_s}: P_{i_1} = \\cdots = P_{i_s}, P_{i_1} \\ne P_j \\textup { for } j \\in \\lbrace 1, \\ldots , d\\rbrace - \\lbrace i_1, \\ldots , i_s\\rbrace \\rbrace ,$ where $\\lbrace i_1, \\ldots , i_s\\rbrace \\subseteq \\lbrace 1, \\ldots , d\\rbrace $ .",
"Consider the locally analytic function obtained by $p$ -adic integration with respect to an $\\omega \\in H^0(J_{{\\mathbb {Q}}_p}, \\Omega ^1)$ on $(\\operatorname{Sym}^dX)({\\mathbb {Q}}_p)$ given by $\\eta _{\\omega }: \\left(\\prod _{{\\mathcal {S}}}(\\operatorname{Sym}^{\\#{\\mathcal {S}}}X)\\right)({\\mathbb {Q}}_p) &\\rightarrow {\\mathbb {Q}}_p\\\\\\prod _{{\\mathcal {S}}}(\\lbrace P_{i_1}, \\ldots , P_{i_s}\\rbrace ) & \\mapsto \\sum _{{\\mathcal {S}}} \\left(\\sum _{k=1}^{\\#{\\mathcal {S}}} \\int _O^{[P_{i_k} - O]}\\omega \\right).",
"\\\\$ As in §REF , the terms corresponding to the different ${\\mathcal {S}}$ can be separated.",
"So we consider the terms that depend on ${\\mathcal {S}}$ from the above expression; namely the terms $\\sum _{k=1}^{\\#{\\mathcal {S}}} \\int _O^{[P_{i_k} - O]}\\omega $ .",
"When $\\#{\\mathcal {S}}= 1$ , we can expand as in , but since $P_{i_1} \\in X({\\mathbb {F}}_q)$ where $q = p^{\\ell }$ for some $\\ell \\ge 1$ , its expansion with respect to the uniformizer $t_1$ satisfies $\\eta _{{\\mathcal {S}}, \\omega } \\in W({\\mathbb {F}}_q)[\\frac{1}{p}][[t_1]]$ .",
"Here, $W({\\mathbb {F}}_q)$ denotes the Witt ring of ${\\mathbb {F}}_q$ , and $W({\\mathbb {F}}_q)[\\frac{1}{p}]$ is the fraction field of the Witt ring.",
"This is the degree-$p^{\\ell }$ unramified extension of ${\\mathbb {Q}}_p$ , and the points in the residue disk of $P_{i_1}$ are parametrized by $t_1 \\in pW({\\mathbb {F}}_q)$ .",
"Now suppose that $\\#{\\mathcal {S}}\\ge 2$ and $P_{{\\mathcal {S}}} \\in X({\\mathbb {F}}_q)$ .",
"Let ${\\mathcal {U}}$ be the residue disk in $(\\operatorname{Sym}^{\\#{\\mathcal {S}}}X)({\\mathbb {Q}}_p)$ reducing to $\\lbrace P_{{\\mathcal {S}}}, \\ldots , P_{{\\mathcal {S}}}\\rbrace $ (the multiset where $P_{{\\mathcal {S}}}$ is repeated $\\#{\\mathcal {S}}$ times).",
"Let ${\\mathcal {U}}_{i_1}, \\ldots ,{\\mathcal {U}}_{i_s}$ be the residue disks around $Q_{i_1}, \\ldots , Q_{i_s}$ in $X(W({\\mathbb {F}}_q))$ with the set bijections $t_{i_j}: {\\mathcal {U}}_{i_j} \\stackrel{\\sim }{\\longrightarrow } pW({\\mathbb {F}}_q)$ for each $1 \\le j \\le s$ , with $t_{i_j}(0) = Q_{i_j}$ .",
"Then the Coleman integral $\\int _O^{Q_{i_1}, \\ldots , Q_{i_s}} \\omega $ can be written as $C + \\int _0^{t_{i_1}(Q_{i_1})} \\omega + \\cdots + \\int _0^{t_{i_s}(Q_{i_s})} \\omega $ where $C$ is a constant depending on the choice of the $Q_{i_j}$ , and $\\omega $ is scaled to have coefficients in $W({\\mathbb {F}}_q)$ .",
"The $t_{i_j}$ are related to the original local coordinates $(u_{i_1}, \\ldots , u_{i_s})$ of ${\\mathcal {U}}$ by $W({\\mathbb {F}}_q)\\left[\\frac{1}{p}\\right][[t_{i_1}, \\ldots , t_{i_s}]]^{S_s} = W({\\mathbb {F}}_q)\\left[\\frac{1}{p}\\right][[u_{i_1}, \\ldots , u_{i_s}]]$ so one can take the $u_{i_k}$ to be the $k$ -th elementary symmetric polynomial in $t_{i_j}$ .",
"This means that for each point $\\lbrace Q_{i_1}, \\ldots , Q_{i_s}\\rbrace \\in (\\operatorname{Sym}^sX)(W({\\mathbb {F}}_q)\\left[\\frac{1}{p}\\right])$ , which corresponds bijectively to a unique pair $(u_{i_1}, \\ldots , u_{i_s})$ , there are $s!$ choices for the $s$ -tuple $(t_{i_j})_j$ .",
"The above discussion leads to the following proposition: Proposition 2.8 (a) A power series $\\eta _{\\omega } = \\left(\\prod _{{\\mathcal {S}}}(\\operatorname{Sym}^sX)\\right)({\\mathbb {Q}}_p) \\rightarrow {\\mathbb {Q}}_p$ obtained from $p$ -adic integration can be re-written via a change of variables as a pure power series, whose coefficients are contained in some extension of ${\\mathbb {Q}}_p$ of degree at most $d$ .",
"(b) Suppose that one obtains $d$ power series $\\eta _1, \\ldots , \\eta _d$ via Chabauty's method as outlined in the previous section, and that one rewrites these power series as $\\eta ^{\\prime }_1, \\ldots , \\eta ^{\\prime }_d$ , where $\\eta ^{\\prime }_i$ are pure power series obtained from part (a).",
"Then there is a $N$ -to-one correspondence between the common zeros of the $\\eta _i$ and $\\eta ^{\\prime }_i$ , where $N = \\prod _{{\\mathcal {S}}} (\\#{\\mathcal {S}})!$ .",
"Further, the solutions to $\\eta _i^{\\prime }$ that correspond to the points $\\operatorname{Sym}^dX({\\mathbb {Q}}_p)$ have $p$ -adic valuations of at least $1/d$ .",
"Now, it remains to associate Newton polygons to these power series, and apply arguments analogous to to try to count the common zeros."
],
[
"Comparison of the algebraic loci and the analytic loci on $\\operatorname{Sym}^dX$",
"We are interested in comparing different sets that contain $(\\operatorname{Sym}^dX)({\\mathbb {Q}})$ inside $(\\operatorname{Sym}^dX)({\\mathbb {C}}_p)$ .",
"The different subsets of $(\\operatorname{Sym}^dX)({\\mathbb {C}}_p)$ that we consider are described below: (i) Faltings' theorem says that given the natural embedding $j: \\operatorname{Sym}^dX \\hookrightarrow J$ using the basepoint $O \\in X$ , $j(\\operatorname{Sym}^dX)({\\mathbb {Q}}) = \\bigcup _{\\textup {finite}} Y_i({\\mathbb {Q}}),$ where the $Y_i$ are cosets of abelian subvarieties of $J$ with $Y \\subseteq W_d$ .",
"The set of points that we are interested in is the set of $p$ -points of the inverse image $\\bigcup j^{-1}(Y_i)$ , denoted ${\\mathcal {F}}(\\operatorname{Sym}^dX)({\\mathbb {C}}_p)$ .",
"We note that ${\\mathcal {F}}(\\operatorname{Sym}^dX)(p)$ depends on the choice of the $Y_i$ .",
"(ii) The set of ${\\mathbb {C}}_p$ -points of the special set ${\\mathcal {S}}(\\operatorname{Sym}^dX)$ of $\\operatorname{Sym}^dX$ : recall that the special set was defined in Definition REF .",
"This set will be denoted ${\\mathcal {S}}(\\operatorname{Sym}^dX)({\\mathbb {C}}_p)$ .",
"(iii) The set $\\lbrace P \\in (\\operatorname{Sym}^dX)({\\mathbb {C}}_p): \\eta _i(j(P)) = 0 \\textup { for all } 1 \\le i \\le d\\rbrace ,$ where the $\\eta _i$ are $d$ independent locally analytic functions on $J(p)$ that vanish on $\\overline{J({\\mathbb {Q}})}$ arising from Chabauty's method.",
"Since this definition depends on the choice of the $\\eta _i$ ; we will fix one such choice here.",
"We denote this set by $(\\operatorname{Sym}^dX)^{\\eta =0}$ .",
"In this section, we relate these different sets.",
"If $d=1$ , $g \\ge 2$ and $\\operatorname{rk}J < g$ , then all of the above sets are zero-dimensional, which makes the comparison simple: We have $\\emptyset = {\\mathcal {S}}(X)({\\mathbb {C}}_p) \\subseteq {\\mathcal {F}}(X)({\\mathbb {C}}_p) \\subseteq (X)^{\\eta =0}$ .",
"For $d>1$ , we will see that these sets do not obey a linear containment relation; in particular, there does not seem to be any inclusion relation between ${\\mathcal {S}}(\\operatorname{Sym}^dX)$ and $(\\operatorname{Sym}^dX)^{\\eta =0}$ ; this necessitates an extra technical hypothesis of Assumption REF to force such an inclusion.",
"This seems to be an intrinsic limitation of Chabauty's method on higher-dimensional varieties; a new idea seems to be necessary to obtain more precise information on the rational points of $\\operatorname{Sym}^dX$ .",
"We review some basics and terminology of rigid analytic geometry in $§\\ref {S: RAG}$ that will enable the comparison of the sets above in $§\\ref {S: comparison}$ ."
],
[
"$(\\operatorname{Sym}^dX)^{\\eta =0}$ as a rigid analytic space",
"We view $(\\operatorname{Sym}^dX)^{\\eta =0}$ as a rigid analytic space, whose admissible cover by affinoid spaces are given by the vanishing of certain Coleman integrals on residue disks, as in §REF (which are elements of the Tate algebra over ${\\mathbb {Q}}_p$ with $d$ variables).",
"For the basic terminology, we refer the readers to and .",
"We note that there is a notion of irreducible components on rigid analytic spaces.",
"The theory of irreducible components was first suggested in , and simplified in .",
"We summarize here: Definition 3.1 A rigid analytic space $X$ is disconnected if there exists an admissible open covering $\\lbrace U,V\\rbrace $ of $X$ with $U \\cap V = \\emptyset $ , where $U,V \\ne \\emptyset $ .",
"Otherwise, $X$ is said to be connected.",
"Definition 3.2 Let $X$ be a rigid analytic space that admits a cover of affinoid spaces $\\lbrace \\operatorname{Sp}A_{\\lambda }\\rbrace _{\\lambda \\in \\Lambda }$ .",
"A morphism $\\pi : \\widetilde{X} \\rightarrow X$ is said to be a normalization if it is isomorphic to the morphism obtained by gluing $\\operatorname{Sp}(\\widetilde{A}_{\\lambda }) \\rightarrow \\operatorname{Sp}A_{\\lambda }$ , where $\\widetilde{A_{\\lambda }}$ denotes the normalization of $A$ in the usual sense.",
"It is known that for any rigid analytic space $X$ , we can find a normalization $\\pi : \\widetilde{X} \\rightarrow X$ ; for example, see *Theorem 1.2.2.",
"Definition 3.3 (, Definition 2.2.2) The irreducible components of a rigid analytic space $X$ are the images of the connected components $X_i$ of the normalization $\\widetilde{X}$ under the normalization map $\\pi : \\widetilde{X} \\rightarrow X$ .",
"Remark 3.4 When $X = \\operatorname{Sp}(A)$ is affinoid, the irreducible components of $X$ are the analytic sets $\\operatorname{Sp}(A/{\\mathfrak {p}})$ for the finitely many minimal prime ideals ${\\mathfrak {p}}$ of the noetherian ring $A$ ."
],
[
"Comparing algebraic and analytic loci in $\\operatorname{Sym}^dX$",
"Now we determine the containment relations of the three sets mentioned at the beginning of this chapter; namely, ${\\mathcal {F}}(\\operatorname{Sym}^dX), {\\mathcal {S}}(\\operatorname{Sym}^dX)$ , and $(\\operatorname{Sym}^dX)^{\\eta =0}$ , as well as $(\\operatorname{Sym}^dX)({\\mathbb {Q}})$ .",
"Lemma 3.5 For any smooth projective curve $X$ with the choice of a rational point $O \\in X({\\mathbb {Q}})$ with good reduction at $p$ , one has $(\\operatorname{Sym}^dX)({\\mathbb {Q}}) \\subseteq (\\operatorname{Sym}^dX)^{\\eta =0}.$ By construction, each $\\eta _i$ mentioned in part (iii) at the beginning of this section satisfies $\\eta _i(P) = 0$ for all $P \\in J({\\mathbb {Q}})$ .",
"Thus, $(\\operatorname{Sym}^dX)({\\mathbb {Q}}) \\subseteq j^{-1}(J({\\mathbb {Q}})) \\subseteq (\\operatorname{Sym}^dX)(p)^{\\eta =0}$ .",
"Lemma 3.6 We keep the notation of $Y_i$ and ${\\mathcal {S}}(\\operatorname{Sym}^dX)({\\mathbb {C}}_p)$ from the beginning of this section.",
"Let $Y=Y_i$ for some $i$ such that $\\dim Y > 0$ .",
"Then $j^{-1}(Y({\\mathbb {C}}_p)) \\subseteq {\\mathcal {S}}(\\operatorname{Sym}^dX)({\\mathbb {C}}_p)$ .",
"We consider two cases: if the generic point of $Y$ has a positive-dimensional preimage, then each $Q \\in Y({\\mathbb {C}}_p)$ is ${\\mathbb {P}}^n$ for some $n > 0$ , so each fibre is contained in ${\\mathcal {S}}(\\operatorname{Sym}^dX)({\\mathbb {C}}_p)$ .",
"On the other hand, if the preimage of the generic point of $Y$ is 0-dimensional, then any irreducible component of $j^{-1}(Y)$ is either covered by positive-dimensional projective spaces, or is birational to $Y$ via the restriction of $j$ .",
"All of these irreducible components are then in the special set.",
"In particular, we note that ${\\mathcal {F}}(\\operatorname{Sym}^dX) \\subseteq {\\mathcal {S}}(\\operatorname{Sym}^dX)$ .",
"Finally, it remains to relate ${\\mathcal {S}}(\\operatorname{Sym}^dX)({\\mathbb {C}}_p)$ and $(\\operatorname{Sym}^dX)^{\\eta =0}$ .",
"In general, neither set is contained in the other; however, with Assumption REF , we immediately get: Proposition 3.7 Let $R_1, \\ldots , R_n$ be the irreducible components of the rigid analytic space $(\\operatorname{Sym}^dX)^{\\eta =0}$ .",
"Further, suppose that $\\operatorname{Sym}^dX$ satisfies Assumption REF .",
"Then for each $R_i$ with $\\dim R_i \\ge 1$ , we have $R_i({\\mathbb {C}}_p) \\subseteq {\\mathcal {S}}(\\operatorname{Sym}^dX)({\\mathbb {C}}_p)$ , and so $(\\operatorname{Sym}^dX)^{\\eta =0} \\subseteq {\\mathcal {S}}(\\operatorname{Sym}^dX)$ .",
"Then taking complements of the relation obtained in Proposition REF inside $\\operatorname{Sym}^dX$ and looking at the ${\\mathbb {Q}}$ -points, one obtains: Corollary 3.8 Under the hypothesis of Proposition REF , one has $\\lbrace {\\mathbb {Q}}\\textup {-points of } \\operatorname{Sym}^dX \\backslash {\\mathcal {S}}(\\operatorname{Sym}^dX)\\rbrace \\subseteq \\bigcup \\lbrace 0 \\textup {-dimensional } R_i\\rbrace .$ Thus, under the hypothesis of Proposition REF , one is still able to interpret the results given from Chabauty's method for higher-dimensional varieties, as Chabauty's method gives an upper bound on $\\bigcup (\\textup {0-dimensional } R_i)$ .",
"For the rest of the paper, we assume that the conditions of Proposition REF hold for $\\operatorname{Sym}^dX$ ."
],
[
"$p$ -adic geometry",
"The goal of this section is to associate a “generalized Newton polygon\" to each multivariate power series, and to state an approximation theorem for the number of roots of a system of equations given by $d$ power series in $d$ variables in general position – that is, having finitely may common zeros – using these Newton polygons.",
"The classical case of $d=1$ is well-known in the literature.",
"To define the Newton polygons for multivariate power series, we review the language necessary to define tropical objects, and state the results in tropical geometry.",
"For a more detailed treatment of tropical geometry, see and ."
],
[
"Tropicalization of rigid analytic hypersurfaces",
"Tropical geometry generalizes the theory of Newton polygons of single-variable power series to power series of several variables.",
"Most of the exposition from this section is taken from .",
"We discuss the tropicalization of affinoid hypersurfaces cut out by a power series over $p$ that arise via Coleman integration.",
"These power series are necessarily convergent on the domain where each of the coordinates has a positive valuation (this has to do with the fact that we can write any $\\omega \\in H^0(X_{{\\mathbb {Q}}_p}, \\Omega ^1)$ as a power series with coefficients in ${\\mathbb {Z}}_p$ ).",
"The tropicalization should be seen as the dual of a Newton polygon; this notion will be made precise in this section.",
"More generally, everything in this section works for a nontrivially valued field $K$ that is complete with respect to the nontrivial valuation.",
"We further assume that $K$ is algebraically closed.",
"Definition 4.1 Let $m = (m_1, \\ldots , m_d) \\in {\\mathbb {Q}}_{\\ge 0}^d$ , and let $P_m := \\lbrace (x_1, \\ldots , x_n) \\in N_{{\\mathbb {R}}} : x_i \\ge m_i \\textup { for } 1 \\le i \\le d\\rbrace $ .",
"Let $U_{P_m} = \\lbrace (x_1, \\ldots , x_d): v(x_i) \\ge m_i \\textup { for all } i\\rbrace .$ The tropicalization map on $U_{P_m}$ is $\\operatorname{trop}: U_{P_m}^d & \\rightarrow (P_m)^d\\\\(\\xi _1, \\ldots , \\xi _d) &\\mapsto (v(\\xi _1), \\ldots , v(\\xi _d)).$ The above tropicalization makes sense for affinoid hypersurfaces cut out by power series convergent on some $U_{P_m}$ , i.e.",
"the power series that are convergent when evaluated on some $x \\in {\\mathbb {Q}}_p^d$ with $v(x) \\in P_m$ .",
"The set of such power series will be denoted by $K \\langle U_{P_m} \\rangle := \\left\\lbrace \\sum _{u \\in {\\mathbb {Z}}_{>0}^d}a_ux^u : a_u \\in K, v(a_u) + \\langle u,w \\rangle \\rightarrow \\infty \\textup { for all } w \\in P_m\\right\\rbrace ,$ where the convergence $v(a_u) + \\langle u,w \\rangle \\rightarrow \\infty $ holds as $u$ ranges over ${\\mathbb {Z}}_{>0}^d$ in any order.",
"For example, if $m = (0, \\ldots , 0)$ , then $K \\langle U_{P_m} \\rangle = K \\langle x_1, \\ldots , x_d \\rangle ,$ the Tate algebra in $d$ variables.",
"Remark 4.2 More generally, it is known that $K \\langle U_{P_m} \\rangle $ is a $K$ -affinoid algebra (*Lemma 6.9(i)), a Cohen-Macaulay ring (*Lemma 6.9(v)), and that $U_P = \\operatorname{Sp}K \\langle U_P \\rangle $ .",
"Definition 4.3 Let $P$ be a polyhedron, and let $f_1, \\ldots , f_n \\in K \\langle U_P\\rangle $ .",
"Let $(f_1, \\ldots , f_n)$ be the ideal in $K \\langle U_P \\rangle $ generated by $f_1, \\ldots , f_n$ .",
"Then $V(f_1, \\ldots , f_n) := \\operatorname{Sp}K \\langle U_P \\rangle /(f_1, \\ldots , f_n).$ Then $V(f_1, \\ldots , f_n)$ is an affinoid subspace of $\\operatorname{Sp}K \\langle U_P \\rangle $ .",
"In our case, each the power series $f \\in K[[x_1, \\ldots , x_d]]$ that arises from Chabauty's method on $\\operatorname{Sym}^dX$ converges when $v(x_i) > 0$ for $1 \\le i \\le d$ .",
"Thus, $f \\in K\\langle U_{P_m} \\rangle $ for any $P_m$ with $m \\in {\\mathbb {Q}}_{>0}^d$ .",
"Let $f \\in K \\langle U_{P_m} \\rangle $ .",
"Now we define $\\operatorname{Trop}(f)$ , the tropical variety corresponding to $f$ , and then outline the procedure for computing $\\operatorname{Trop}(f)$ in $§$ REF .",
"Definition 4.4 For $f \\in K \\langle U_{P_m} \\rangle $ , $\\operatorname{Trop}(f) := \\overline{\\operatorname{trop}(V(f))} = \\overline{\\lbrace (v(\\xi _1), \\ldots , v(\\xi _d)) : f(\\xi ) = 0, \\xi \\in U_{P_m}\\rbrace },$ where $V(f)$ is the affinoid subspace defined by the ideal ${\\mathfrak {a}}= (f) \\subset K \\langle U_{P_m} \\rangle $ .",
"Here, we take the topological closure in $P_m$ .",
"Often, the easiest way to compute $\\operatorname{Trop}(f)$ is by using Lemma REF , which requires these definitions: Definition 4.5 For $0 \\ne f \\in K \\langle U_{P_m} \\rangle $ , write $f = \\sum _{u \\in S_{\\sigma }}a_ux^u$ .",
"The height graph of $f$ is $H(f) = \\lbrace (u, v(a_u)): u \\in {\\mathbb {Z}}_{\\ge 0}^d, a_u \\ne 0\\rbrace \\subseteq {\\mathbb {Z}}_{\\ge 0}^d \\times {\\mathbb {R}}.$ Given $w \\in {\\mathbb {Q}}_{>0}^d$ , we also define $m_f(w) = m(w) = \\min _{u \\in S_{\\sigma }}\\lbrace (-w, 1) \\cdot H(f)\\rbrace ,$ where $\\cdot $ denotes the usual dot product, and $\\operatorname{vert}_w(f) = \\lbrace (u, v(a_u)) \\in H(f): (-w, 1) \\cdot (u, v(a_u)) = m(w)\\rbrace \\subseteq H(f).$ Intuitively, $m(w)$ denotes the minimum valuation achieved assuming that $v(x) = w$ , among the terms of $f$ .",
"Then $\\operatorname{vert}_w(f)$ records the corresponding terms of $f$ with the minimum valuation, again assuming that $v(x) = w$ .",
"The following is the power-series analogue of a well-known result for polynomials; the original result for polynomials is first recorded in an unpublished manuscript by Kapranov, and a proof of this lemma for power series can be found in , Lemma 8.4; also see, for example , Theorem 3.1.3.",
"This gives a useful method to computing $\\operatorname{Trop}(f)$ .",
"Lemma 4.6 $\\operatorname{Trop}(f) = \\overline{\\lbrace w \\in {\\mathbb {Q}}_{\\ge 0}^d: \\# \\operatorname{vert}_w(f) > 1\\rbrace }.$"
],
[
"Tropical intersection theory and Newton polygons",
"In this section, we take $d$ power series in $d$ variables in $K \\langle U_{P_m} \\rangle $ that have finitely many common zeros.",
"We explain that in order to bound the number of common zeros of the $d$ power series, it suffices to know their tropicalizations and their Newton polygons.",
"Since the tropicalizations and the Newton polygons depend only on finitely many terms of the power series convergent on $U_{P_m}$ , this section shows that one can approximate a power series of several variables by a polynomial for the purposes of intersection theory.",
"In a sense, this is a stronger approximation than what Weierstrass preparation theorem can tell us; Weierstrass preparation for multivariate power series approximates $f \\in K[[t_1, \\ldots , t_d]]$ by $f^{\\prime } \\in K[t_1][[t_2, \\ldots , t_d]]$ , whereas here, we approximate $f$ by $f^{\\prime \\prime } \\in K[t_1, \\ldots , t_d]$ .",
"Let $f \\in K \\langle U_{P_m} \\rangle $ .",
"Write $f = \\sum a_u x^u$ .",
"Define $\\operatorname{vert}_{P_m}(f) := \\bigcup _{w \\in P_m}\\operatorname{vert}_w(f).$ It turns out that $\\operatorname{vert}_{P_m}(f)$ is finite: Lemma 4.7 (, Lemma 8.2) Let $f \\in K \\langle U_{P_m} \\rangle $ be nonzero.",
"Then $\\operatorname{vert}_{P_m}(f)$ is finite.",
"This lemma, combined with Lemma REF , tells us that $\\operatorname{Trop}(f)$ determined by only finitely many terms of $f$ .",
"Now, the following lemma shows that if the coefficients of a power series $f$ are perturbed in a way so that their $v$ -adic valuations do not change, and so that $\\operatorname{vert}_P(f)$ does not change, then the tropicalization also stays the same.",
"Lemma 4.8 Let $f, f^{\\prime } \\in K \\langle U_{P_m} \\rangle $ , with $f = \\sum _{u}a_ux^u$ and $f^{\\prime } = \\sum _u a_u^{\\prime } x^u$ .",
"Suppose that $\\operatorname{vert}_{P_m}(f) = \\operatorname{vert}_{P_m}(f^{\\prime })$ .",
"Then $\\operatorname{Trop}(f) = \\operatorname{Trop}(f^{\\prime })$ .",
"Fix $w \\in P_m$ .",
"We claim that $\\operatorname{vert}_w(f) = \\operatorname{vert}_w(f^{\\prime })$ for each such $w$ .",
"Choose $u_0 \\in {\\mathbb {Z}}_{\\ge 0}^d$ minimizing $v(a_u) + \\langle u,w \\rangle $ .",
"This means $m_f(w) = v(a_{u_0}) + \\langle u_0,w \\rangle .$ Thus, $(u_0, v(a_{u_0})) \\in \\operatorname{vert}_w(f) \\subset \\operatorname{vert}_P(f) = \\operatorname{vert}_P(f^{\\prime })$ .",
"So $(u_0, v(a_{u_0})) \\in \\operatorname{vert}_{w^{\\prime }}(f)$ for some $w^{\\prime } \\in P_m$ .",
"In particular, $(u_0, v(a_{u_0})) = (u_0, v(a_{u_0}^{\\prime }))$ .",
"Thus, $\\min _{u \\in {\\mathbb {Z}}_{\\ge 0}^d}(v(a_u) + \\langle u, w \\rangle ) = v(a_{u_0}) + \\langle u,w \\rangle = v(a_{u_0}^{\\prime }) + \\langle u_0, w \\rangle \\ge \\min _{u \\in {\\mathbb {Z}}_{\\ge 0}^d}(v(a_u^{\\prime }) + \\langle u, w \\rangle ).$ The symmetric argument proves the inequality in the other direction, showing that $\\operatorname{vert}_w(f) = \\operatorname{vert}_w(f^{\\prime })$ for each $w \\in P_m$ .",
"Then by Lemma REF , $\\operatorname{Trop}(f) = \\operatorname{Trop}(f^{\\prime })$ .",
"Given $f \\in K \\langle U_{P_m} \\rangle $ , we can find a polynomial $g \\in K \\langle U_{P_m} \\rangle $ such that $\\operatorname{vert}_{P_m}f = \\operatorname{vert}_{P_m} g$ , in which case Lemma REF implies that $\\operatorname{Trop}(f) = \\operatorname{Trop}(g)$ : Corollary 4.9 Let $f \\in K \\langle U_{P_m} \\rangle $ , with $f = \\sum _{u}a_ux^u$ .",
"Let $\\pi : {\\mathbb {Z}}_{\\ge 0}^d \\times {\\mathbb {R}}\\rightarrow {\\mathbb {Z}}_{\\ge 0}^d$ denote the projection map forgetting the last coordinate.",
"Let $S \\subseteq {\\mathbb {Z}}_{\\ge 0}^d$ be a finite set containing $\\pi (\\operatorname{vert}_{P_m}(f))$ .",
"Define the auxiliary polynomial of the power series $f$ with respect to $S$ by $g_S = \\sum _{u \\in S}a_ux^u \\in K \\langle U_P \\rangle .$ Then $\\operatorname{Trop}(f) = \\operatorname{Trop}(g_S)$ .",
"Since $\\operatorname{vert}_{P_m}(f) = \\operatorname{vert}_{P_m}(g_S)$ , and since $\\operatorname{Trop}(f)$ only depends on $\\operatorname{vert}_P(f)$ , the conclusion follows.",
"We show that the terms of a power series $f \\in K \\langle U_{P_m}\\rangle $ contained in $\\operatorname{vert}_{P_m}(f)$ also contains all the information about the valuations of zeros of $f$ counting multiplicity.",
"That is, the information about zeros of power series depends on only finitely many terms of $f$ too, and hence the information about the intersection theory of the power series also depends on finitely many terms of $f$ .",
"If $S$ is a finite set of points in Euclidean space, its convex hull is denoted $\\operatorname{conv}(S)$ .",
"Definition 4.10 Let $f \\in K \\langle U_{P_m} \\rangle $ .",
"For each $w \\in {\\mathbb {R}}_{\\ge 0}^d$ , define its associated Newton polytope $\\gamma _w(f) = {\\gamma }_w = \\pi (\\operatorname{conv}(\\operatorname{vert}_w(f))).$ Remark 4.11 The $g_S$ from Corollary REF are chosen so that the $\\gamma _w(f) = \\gamma _w(g_S)$ as well.",
"It turns out that as long as $\\bigcap _i V(f_i)$ is finite, the information about the common roots of $f_i \\in K \\langle U_P \\rangle $ having a specified valuation $w$ is encoded in ${\\gamma }_w$ , as explained below.",
"Definition 4.12 Let $f_1, \\ldots , f_d \\in K \\langle U_{P_m} \\rangle $ , $Y_i = V(f_i)$ , and $Y = \\bigcap _{i} Y_i$ .",
"Assume that $Y$ is 0-dimensional.",
"Then the intersection multiplicity of $Y_1, \\ldots , Y_d$ at $w \\in {\\mathbb {Q}}^d$ is defined as $i(w; Y_1 \\cdots Y_d) := \\dim _K H^0(Y \\cap U_{\\lbrace w\\rbrace }, {\\mathcal {O}}_{Y \\cap U_{\\lbrace w\\rbrace }}),$ where $U_{\\lbrace w\\rbrace } := \\operatorname{trop}^{-1}(w)$ .",
"In simpler terms, this intersection multiplicity at $w$ is the number of common zeros of the $f_i$ that have the same coordinate-wise valuation as $w$ , counting with multiplicity.",
"Since the $U_{\\lbrace w\\rbrace }$ are disjoint, the intersection number of $Y_1, \\ldots , Y_d$ is then $i(Y_1, \\ldots , Y_d) := \\dim H^0(Y, {\\mathcal {O}}_Y).$ Definition 4.13 Let $Q_1, \\ldots , Q_d$ be bounded polytopes.",
"Define a function $V_{Q_1, \\ldots , Q_d}(\\lambda _1, \\ldots , \\lambda _d) := \\operatorname{vol}(\\lambda _1Q_1 + \\cdots + \\lambda _d Q_d)$ where $+$ denotes the Minkowski sum.",
"The mixed volume of the $Q_i$ , denoted $MV(Q_1, \\ldots , Q_d)$ , is defined as the coefficient of the $\\lambda _1 \\cdots \\lambda _d$ -term of $V_{Q_1, \\ldots , Q_d}(\\lambda _1, \\ldots , \\lambda _d)$ .",
"Theorem 4.14 *Theorem 11.7 Let $f_1, \\ldots , f_d \\in K \\langle U_{P_m} \\rangle $ have finitely many common zeros, and let $w \\in \\bigcap _{i=1}^d \\operatorname{Trop}(f_i)$ be an isolated point in the interior of $P$ .",
"For $i = 1, \\ldots , d$ let $Y_i = V(f_i)$ and let $\\gamma _i = \\gamma _w(f_i)$ .",
"Then $i(w, Y_1 \\cdots Y_d) = MV(\\gamma _1, \\ldots , \\gamma _d).$ In particular, Theorem REF implies that considering the auxiliary polynomials suffices, as the $\\gamma _i$ are determined by only finitely many terms in the $f_i$ .",
"Theorem 4.15 Suppose that $f_1, \\ldots , f_d \\in K \\langle U_{P_m} \\rangle $ , and let $g_i$ be the auxiliary polynomials of the $f_i$ with respect to some finite set $S \\subseteq M$ containing all $u$ such that $(u, v(a_u)) \\in \\operatorname{vert}_{P_m}(f)$ .",
"Then $\\sum _{w \\in P^{\\circ }} i(w, V(f_1) \\cdots V(f_d)) = \\sum _{w \\in P^{\\circ }} i(w, V(g_1) \\cdots V(g_d)),$ if all the summands on both sides are finite.",
"By the choice of the $g_i$ , $\\operatorname{Trop}(f_i) = \\operatorname{Trop}(g_i)$ for each $1 \\le i \\le d$ , and $\\gamma _w(f_i) = \\gamma _w(g_i)$ for each $w \\in P$ .",
"We note from Theorem REF that $\\operatorname{Trop}(f_i), \\operatorname{Trop}(g_i), \\gamma _w(f_i)$ and $\\gamma _w(g_i)$ are the only information required in computing the intersection multiplicities.",
"The following results for polynomials are useful in estimating the number of zeros of power series.",
"First, a definition: Definition 4.16 Let $f = \\sum _{u \\in \\Lambda }a_ux^u$ be a polynomial, where $\\Lambda \\subset {\\mathbb {Z}}^d$ is a finite set.",
"Then the Newton polygon of $f$ is given by $\\operatorname{New}(f) = \\operatorname{conv}(\\lbrace u: u \\in \\Lambda , a_u \\ne 0\\rbrace ) \\subseteq {\\mathbb {R}}^d.$ We recall Bernstein's theorem: Theorem 4.17 () Let $f_1, \\ldots , f_d \\in K[x_1, \\ldots , x_d]$ be polynomials with finitely many common zeros.",
"Then the number of common zeros of the $f_i$ with multiplicity in $(K^{\\times })^d$ is $\\operatorname{MV}(\\operatorname{New}(f_1), \\ldots , \\operatorname{New}(f_d)).$ Further, suppose that the $f_i$ have finitely many common zeros whose valuations belong to $P$ , and also suppose that the $g_i$ have finitely many common zeros in $K^d$ .",
"By Theorem REF , $&\\textup {number of common zeros of the } f_i \\textup { with valuations in } P \\\\&= \\textup {number of common zeros of the } g_i \\textup { with valuations in }P\\\\&\\le \\textup {number of common zeros of the } g_i \\textup { in } (K^{\\times })^d\\\\ &=\\operatorname{MV}(\\operatorname{New}(g_1), \\ldots , \\operatorname{New}(g_d)),$ where the last inequality follows by Bernstein's theorem.",
"Thus, we conclude: Theorem 4.18 Suppose that $f_1, \\ldots , f_d \\in K \\langle U_{P_m} \\rangle $ , and let $g_i$ be the associated auxiliary polynomials of the $f_i$ with respect to some finite set $S \\subseteq M$ containing all $u$ such that $(u, v(a_u)) \\in \\operatorname{vert}_P(f)$ .",
"Suppose further that $\\bigcap _{i=1}^d V(f_i)< \\infty $ and $\\bigcap _{i=1}^dV(g_i) < \\infty $ .",
"Then $\\# \\left( (K^{\\times })^d \\cap \\bigcap _{i=1}^d V(f_i) \\right) \\le MV(\\operatorname{New}(f_1), \\ldots , \\operatorname{New}(f_d)).$ Again, the proof follows from the fact that by the choice of the $g_i$ , we have $\\operatorname{Trop}(f_i) = \\operatorname{Trop}(g_i)$ and $\\gamma _w(f_i) = \\gamma _w(g_i)$ for each $1 \\le i \\le d$ and $w \\in P$ .",
"Remark 4.19 In order to count all solutions of the form $(x_1, \\ldots , x_d)$ , where some of the $x_i$ may be 0, one needs to apply Theorem REF multiple times, while setting some of the $x_i = 0$ ."
],
[
"Continuity of roots",
"Throughout, $K$ is a complete, algebraically closed valued field with respect to a nontrivial, nonarchimedean valuation $v \\colon K^{\\times } \\rightarrow {\\mathbb {Q}}$ .",
"studies the intersection theory of power series in $K \\langle U_P \\rangle $ that have finite intersection.",
"Our goal in this chapter is to analyze the case of the power series in $K \\langle U_P \\rangle $ have possibly infinite intersection; we will show that these power series have “small” deformations that have finite intersection, and that they preserve information about the number of 0-dimensional components of the original intersection.",
"We make this notion precise, and we obtain an upper bound on the number of 0-dimensional components (counting multiplicity) of the original intersection, using the new power series with finitely many common zeros, obtained via small $p$ -adic deformations."
],
[
"Deformation of power series via rigid analytic geometry and polynomial approximations",
"It is known that small deformations do not affect the multiplicity of 0-dimensional components of intersections in rigid analytic spaces.",
"Theorem 5.1 (*Theorem 10.2, Local continuity of roots) Let $A$ be a $K$ -affinoid algebra that is a Dedekind domain and let $S = \\operatorname{Sp}(A)$ .",
"Let $X = \\operatorname{Sp}(B)$ be a Cohen-Macaulay affinoid space of dimension $d+1$ , let $f_1, \\ldots , f_d \\in B$ , and let $Y \\subset X$ be the subspace defined by the ideal ${\\mathfrak {a}}= (f_1, \\ldots , f_d)$ .",
"Suppose that we are given a morphism $\\alpha : X \\rightarrow S$ and a point $t \\in S$ such that the fibre $Y_t = \\alpha ^{-1}(t) \\cap Y$ has dimension zero.",
"Then there is an affinoid subdomain $U \\subset S$ containing $t$ such that $\\alpha ^{-1}(U) \\rightarrow U$ is finite and flat.",
"Let $B_K^d := \\operatorname{Sp}(K \\langle x_1, \\ldots , x_d\\rangle )$ for $d \\ge 1$ .",
"The following is immediate from the above theorem, and is applicable to our situation arising from Chabauty's method: Corollary 5.2 (*Example 10.3) Let $X = B_K^d \\times B_K^1$ and $S = B_K^1$ , with $\\alpha \\colon X \\rightarrow S$ the projection onto the second factor.",
"Let $f_1, \\ldots , f_d \\in K \\langle x_1, \\ldots , x_d, t \\rangle $ .",
"If the specializations $f_{1,0}, \\ldots , f_{d,0}$ at $t=0$ have only finitely many zeros in $B_K^d$ then there exists ${\\varepsilon }>0$ such that if $|s| < {\\varepsilon }$ , then $f_{1,s}, \\ldots , f_{d,s}$ have the same number of zeros (counted with multiplicity) in $B_{\\kappa (s)}^d$ as $f_{1,0}, \\ldots , f_{d,0}$ .",
"Tate algebra in one variable is a Dedekind domain, and $X$ is Cohen-Macaulay (Remark REF ).",
"Definition 5.3 Suppose that $f_1, \\ldots , f_d \\in K \\langle U_{P_m} \\rangle $ are power series such that $Y:=V(f_1, \\ldots , f_d)$ is possibly infinite.",
"Define $N_0(f_1, \\ldots , f_d)$ to be the number of 0-dimensional components of $Y$ , counting multiplicity.",
"If $Y$ is finite, then we drop the subscript 0 to signal its finiteness: $N(f_1, \\ldots , f_d) := N_0(f_1, \\ldots , f_d) = \\dim H^0(Y, {\\mathcal {O}}_Y).$ Also, let $N_0^{\\times }(f_1, \\ldots , f_d)$ denote the number of 0-dimensional components of $Y$ , whose coordinates are in $\\overline{K}^{\\times }$ .",
"Definition 5.4 Let $f = \\sum _{u}a_ux^u\\in K \\langle U_{P_m}\\rangle $ , and define $M(f) &:= \\textup {the set of monomials with nonzero coefficients appearing in f}\\\\&= \\lbrace x^u: a_u \\ne 0\\rbrace .$ Call $f$ nondegenerate if for every $i$ there exists an integer $n > 0$ with $x_i^n \\in M(f)$ .",
"Remark 5.5 Any pure power series arising from Chabauty's method is nondegenerate.",
"We will need to impose the nondegeneracy conditions in all power series $f$ in order to be able to carry out deformations.",
"Now we prove a series of deformation results for non-stable intersections.",
"Lemma 5.6 Let $f \\in K\\langle U_{P_m} \\rangle $ be a nondegenerate power series, and let $q_1, q_2, \\ldots , q_n \\in U_{P_m}$ such that $q_i \\ne 0$ in $K^d$ .",
"Then there exists a polynomial $h$ such that $h$ does not vanish on any of $q_1, \\ldots , q_{\\ell }$ and $M(h) \\subseteq M(f)$ .",
"We will prove by induction on $\\ell $ that there exists $g$ such that $M(g) \\subseteq M(f)$ and $g(q_1), \\ldots , g(q_{\\ell }) \\ne 0$ .",
"The statement is clear when $\\ell = 0$ , so assume that there is a polynomial $g$ with $M(g) \\subseteq M(f)$ satisfying $g(q_1), \\ldots , g(q_{\\ell }) \\ne 0$ and $g(q_{\\ell +1}) = \\cdots = g(q_n) = 0$ , after possibly reordering the $q_i$ .",
"If $\\ell = n$ , then we are done, so assume otherwise.",
"We will show that there exists another polynomial $g^{\\prime } \\in M(f)$ such that $g^{\\prime }$ does not vanish on at least $\\ell +1$ of the points $q_i$ .",
"Choose a monomial $m \\in M(f)$ such that $m(q_{\\ell +1}) \\ne 0$ ; such $m$ exists due to the nondegeneracy condition on $f$ .",
"We may choose $c \\in K^{\\times }$ such that $v(cm(q_i)) > v(g(q_i))$ for $1 \\le i \\le \\ell $ .",
"Then $g^{\\prime } := g + cm$ satisfies the property that $g^{\\prime }(q_1) \\ne 0, \\ldots , g^{\\prime }(q_{\\ell }) \\ne 0$ .",
"Also, $g^{\\prime }(q_{\\ell +1}) = cm(q_{\\ell +1}) \\ne 0$ .",
"The lemma then follows from an inductive argument on $\\ell $ .",
"Proposition 5.7 (a) Suppose that $f_1, \\ldots , f_d \\in K \\langle U_{P_m} \\rangle $ are nondegenerate power series.",
"Then there exist nondegenerate $g_1, \\ldots , g_d \\in K \\langle U_{P_m} \\rangle $ with $\\operatorname{Trop}(f_i) = \\operatorname{Trop}(g_i)$ and $\\gamma _w(f_i) = \\gamma _w(g_i)$ , for $1 \\le i \\le d$ and $w \\in P_m$ , with $N_0(f_1, \\ldots , f_d) \\le N(g_1, \\ldots , g_d).$ (b) Moreover, if the $f_i$ are polynomials, then the $g_i$ may be chosen to be polynomials.",
"We will deform the $f_i$ to the $g_i$ one by one.",
"Specifically, we will prove by induction on $r$ that there exist $g_1, \\ldots , g_r$ such that $\\operatorname{Trop}(f_i) = \\operatorname{Trop}(g_i)$ and $\\gamma _w(f_i) = \\gamma _w(g_i)$ for $i \\in \\lbrace 1, \\ldots , r\\rbrace $ and $w \\in P$ , satisfying $\\operatorname{codim}\\bigcap _{i=1}^r V(g_i) \\ge r$ for each $r \\in \\lbrace 1, \\ldots , d\\rbrace $ , and $N_0(f_1, \\ldots , f_d) \\le N_0(g_1, \\ldots , g_r, f_{r+1}, \\ldots , f_d).$ When $r=1$ , the statement above is clear, by taking $f_1 = g_1$ .",
"Now we prove the statement for $r+1$ .",
"Let $C_1, \\ldots , C_{\\ell }$ be the codimension $r$ irreducible components of $\\bigcap _{i=1}^r V(g_i)$ .",
"Choose points $P_i \\in C_i$ , such that $P_i \\ne 0$ in $K^d$ .",
"We will deform $f_{r+1}$ to $g_{r+1}$ so that $g_{r+1}(P_{i}) \\ne 0$ for each $i$ , while keeping $\\operatorname{Trop}(f_{r+1}) = \\operatorname{Trop}(g_{r+1})$ and $\\gamma _w(f_{r+1}) = \\gamma _w(g_{r+1})$ .",
"This will guarantee that $\\operatorname{codim}\\left(\\bigcap _{i=1}^{r+1}V(g_{r+1})\\right) \\ge r+1$ .",
"From the nondegeneracy assumption of $f_{r+1}$ , we have that $V(M(f_{r+1})) = \\emptyset $ (if $M(f_{r+1})$ contains 1) or $V(M(f_{r+1})) = \\lbrace 0\\rbrace $ (if $M(f_{r+1})$ does not contain 1).",
"In the first case, we may adjust only the constant term to get the desired $g_{r+1}$ .",
"Thus, we may assume that we are in the second case.",
"Then by Lemma REF , we may pick a polynomial $h$ such that $M(h) \\subseteq M(f_{r+1})$ such that $h$ that does not vanish on any $P_i$ .",
"For small enough nonzero ${\\varepsilon }$ , the deformation $f_{r+1} \\mapsto f_{r+1} + {\\varepsilon }h =: g_{r+1}$ does not vanish on any of the $P_i$ ; in this case, the intersection $\\bigcap _{i=1}^{r+1}V(g_i)$ has codimension $r+1$ , as required.",
"Further, since $h \\in M(f_{r+1})$ , after possibly making ${\\varepsilon }$ even smaller, both the tropicalization and the $\\gamma _w$ of $f_{r+1}$ are identical to those of $g_{r+1}$ .",
"Now we will prove that $N_0(g_1, \\ldots , g_d) \\ge N_0(f_1, \\ldots , f_d)$ .",
"It suffices to show that $N_0(g_1, \\ldots , g_{r+1}, f_{r+2}, \\ldots , f_d) \\ge N_0(g_1, \\ldots , g_r, f_{r+1}, \\ldots , f_d)$ for $r \\in \\lbrace 0, \\ldots , d-1\\rbrace $ ; if $r = 0$ , then the previous inequality will be interpreted as $N_0(g_1, f_2, \\ldots , f_d) \\ge N_0(f_1, \\ldots , f_d).$ Let $I$ be the ideal that cut out the dimension $\\ge 1$ components of $V(g_1, \\ldots , g_r, f_{r+1}, \\ldots , f_d)$ in $K \\langle U_P\\rangle $ and let ${\\mathfrak {p}}_1, \\ldots , {\\mathfrak {p}}_{\\ell }$ denote the maximal ideals corresponding to the 0-dimensional components of $V(g_1, \\ldots , g_r, f_{r+1}, \\ldots , f_d)$ .",
"Choose a $f \\in I$ such that $f \\notin {\\mathfrak {p}}_i$ for $1 \\le i \\le \\ell $ .",
"Such choice is possible by the prime avoidance theorem, see for example *Proposition 1.11.",
"Now we apply Theorem REF on $\\operatorname{Sp}B$ , where $B = K \\langle U_P \\rangle _f$ , which states that a small deformation of $V(g_1, \\ldots , g_r, f_{r+1}, \\ldots , f_d)$ preserves all 0-dimensional components away from the positive-dimensional locus.",
"This proves the inequality at the beginning of this paragraph, and consequently part (a) of the proposition.",
"Part (b) follows, since we deform the the $f_i$ by monomials that already appear in $f_i$ ."
],
[
"Explicit computation of the upper bound",
"Now we consider a residue disk ${\\mathcal {U}}\\subseteq (\\operatorname{Sym}^dX)(p)$ whose points reduce to a given ${\\mathcal {Q}}\\in (\\operatorname{Sym}^dX)({\\mathbb {F}}_p)$ .",
"Recall from Proposition REF that Chabauty's method on ${\\mathcal {U}}$ yields $d$ pure power series $f_1, \\ldots , f_d$ in $d$ variables, whose common zeros in $p^d$ with valuations at least $1/d$ correspond to a set containing the points in $j^{-1}(j(\\operatorname{Sym}^dX)({\\mathbb {Q}}_p)) \\cap \\overline{J({\\mathbb {Q}})})$ .",
"Using the results of the previous section, we will obtain an explicit upper bound on the number of common zeros of the $f_i$ in this section by estimating $\\operatorname{New}(f_i)$ .",
"The methods used in this section are reminiscent of .",
"Definition 5.8 For ${\\varepsilon }\\in (0, \\frac{1}{d})$ , $k \\in {\\mathbb {Z}}_{\\ge 0}$ and $d, \\ell \\in {\\mathbb {Z}}_{\\ge 1}$ with $d \\ge \\ell $ , let $\\delta _{{\\varepsilon }}(k,v, \\ell ) := \\max \\left\\lbrace N \\in {\\mathbb {Z}}_{\\ge 0} : v(k+N) \\ge (\\frac{1}{\\ell }-{\\varepsilon })N + v(k)\\right\\rbrace .$ Remark 5.9 We note that $\\delta _{{\\varepsilon }}(k,v,\\ell )$ is well-defined, independent of the choice of ${\\varepsilon }$ ; $v(k+N) = O(\\log N)$ as $N \\rightarrow \\infty $ , while $(\\frac{1}{\\ell }-{\\varepsilon })(N+1)$ increases linearly with $N$ .",
"Notation 5.10 Given $f \\in W({\\mathbb {F}}_q)[[t]]$ , we mean by $\\bar{f}$ the image of $f$ under the natural reduction map of the coefficients $W({\\mathbb {F}}_q)[[t]] \\rightarrow {\\mathbb {F}}_q[[t]]$ .",
"We will denote $\\operatorname{ord}_0(f) := \\operatorname{ord}_{t = 0}(\\bar{f})$ , the exponent of the first term that does not vanish under the reduction map.",
"Lemma 5.11 For any ${\\varepsilon }\\in {\\mathbb {Q}}$ satisfying $0 < {\\varepsilon }< \\frac{1}{d}$ , the following holds: Let $f \\in W({\\mathbb {F}}_q)\\left[\\frac{1}{p}\\right][[t]]$ be such that its derivative $f^{\\prime }$ is in $W({\\mathbb {F}}_q)[[t]]$ , and $\\operatorname{ord}_0 \\bar{f}^{\\prime } = k-1$ for some $k \\ge 1$ .",
"Let $F(t_1, \\ldots , t_{\\ell }) := f(t_1) + \\cdots + f(t_{\\ell }) = \\sum _{u \\in {\\mathbb {Z}}_{\\ge 0}^{\\ell }} a_ut^u,$ where $t^u$ denotes $t_1^{u_1} \\cdots t_{\\ell }^{u_{\\ell }}$ .",
"Let $w \\in P_{{\\varepsilon }} = [{\\varepsilon }, \\infty )^{\\ell }$ .",
"If $u = (u_1, \\ldots , u_{\\ell }) \\in {\\mathbb {Z}}_{\\ge 0}^{\\ell }$ satisfies $u_i > k + \\delta _{{\\varepsilon }}(k,v, \\ell )$ for some $1 \\le i \\le \\ell $ , then $(u, v(a_u)) \\notin \\operatorname{vert}_w(F)$ .",
"Fix $w \\in [{\\varepsilon }, \\infty )^{\\ell }$ .",
"Since $F$ is pure, it suffices to consider $u \\in {\\mathbb {Z}}_{\\ge 0}^{\\ell }$ such that $u_1 > k + \\delta _{{\\varepsilon }}(k,v,\\ell )$ and $u_2 = u_3 = \\cdots = u_d = 0$ .",
"We will show that there exists $u^{\\prime } \\in {\\mathbb {Z}}_{\\ge 0}^{\\ell }$ such that $v(a_{u^{\\prime }}) + \\langle u^{\\prime }, w \\rangle < v(a_u) + \\langle u,w \\rangle .$ Then by the definition of $\\operatorname{vert}_w(F)$ , the conclusion would follow.",
"Write $f^{\\prime }(t) = \\sum _{i \\ge 0} c_it^i$ , so $f(t) = \\sum _{i \\ge 0}\\frac{c_i}{i+1}t^{i+1}$ .",
"Then $c_i \\in {\\mathbb {Z}}_p$ since $f^{\\prime } \\in {\\mathbb {Z}}_p[[t]]$ , and furthermore, $v(c_{k-1}) = 0$ , with $v(c_j) > 0$ for $1 \\le j \\le k-1$ , since $\\operatorname{ord}_0 \\bar{f}^{\\prime } = k-1$ .",
"Then $a_ut^u = \\frac{c^{m}}{m+1}t_1^{m+1}$ , where $m > k + \\delta _{{\\varepsilon }}(k, v, \\ell )$ .",
"We claim that $u^{\\prime } = (k, 0, \\ldots , 0)$ suffices.",
"For any $w \\in [{\\varepsilon }, \\infty )^{\\ell }$ , consider $m(w)&:= \\min _{u^{\\prime \\prime } \\in S_{\\sigma }}\\lbrace v(a_{u^{\\prime \\prime }})+\\langle u^{\\prime \\prime }, w \\rangle \\rbrace \\\\& \\le v\\left(\\frac{c_{k-1}}{k}\\right)+\\langle (k,0, \\ldots , 0), (w_1, w_2, \\ldots , w_{\\ell })\\rangle \\\\&= v \\left(\\frac{c_{k-1}}{k}\\right) + kw_1$ Since $m > k + \\delta _{{\\varepsilon }}(k, v, \\ell )$ , we have $v(m+1) < (m+1-k)w_1 + v(k),$ which rearranges to $-v(k) + kw_1 < -v(m+1) + (m+1)w_1.$ Using $v(c_{k-1}) = 0$ and $v(c_m) \\ge 0$ , this inequality becomes $v\\left( \\frac{c_{k-1}}{k}\\right) + kw_1 < v \\left(\\frac{c_m}{m+1}\\right) + (m+1)w_1.$ That is, $(u, v(a_u)) \\notin \\operatorname{vert}_w(F)$ , as required.",
"Remark 5.12 Lemma REF shows that any pure power series as in the statement of the lemma can be approximated by polynomials whose terms are pure, and whose degree is less than $k + \\delta _{{\\varepsilon }}(k, v, \\ell )$ .",
"This, in turn, means that the Newton polygons of these polynomials are at worst the convex hull of the points $\\lbrace (0, \\ldots , 0)\\rbrace \\cup \\lbrace (k + \\delta _{{\\varepsilon }}(k, v, \\ell ))e_i : 1 \\le i \\le \\ell \\rbrace $ , where the $e_i$ denotes the $i$ -th standard vector.",
"Thus, the Newton polygon can be approximated by a simplex.",
"Definition 5.13 Let $A = (a_{ij})$ be a $d \\times d$ matrix.",
"The permanent of $A$ is $\\operatorname{Per}(A) = \\sum _{\\sigma \\in S_d} \\prod _{i=1}^d (a_{i \\sigma (i)}).$ Lemma 5.14 Let $A = (a_{ij})$ be a $d \\times d$ matrix of positive real numbers, and define the polytopes $X_i \\subseteq {\\mathbb {R}}^d$ for $1 \\le i \\le d$ by the following: $X_i = \\operatorname{conv}(0, a_{i,1}e_1, \\ldots , a_{i,d}e_d)$ Then $\\operatorname{MV}(X_1, \\ldots , X_d) = \\frac{1}{d!}",
"\\operatorname{Per}(A).$ The mixed volume is $& \\textup {coefficient of \\lambda _1 \\cdots \\lambda _d of }\\operatorname{vol}\\left(\\operatorname{conv}(0, (\\lambda _1 a_{11} + \\cdots + \\lambda _da_{d1})e_1, \\ldots , (\\lambda _1 a_{1d} + \\cdots + \\lambda _da_{dd})e_d)\\right) \\\\&= \\textup {coefficient of \\lambda _1 \\cdots \\lambda _d of } \\frac{1}{d!",
"}(\\lambda _1 a_{11} + \\cdots + \\lambda _da_{d1}) \\cdots (\\lambda _1 a_{1d} + \\cdots + \\lambda _da_{dd})\\\\& = \\frac{1}{d!",
"}\\sum _{\\sigma \\in S_d}\\prod _{i=1}^d a_{\\sigma (i),i} = \\frac{1}{d!}",
"\\operatorname{Per}(A).",
"$ Lemma 5.15 Let $f_i \\in K[[t_i]]$ .",
"Suppose further that $f_i$ converges when $v(t_i) \\ge 1/d_i$ and that $f_i^{\\prime }(t_i) = \\sum _{j=0}^{\\infty } c_{ij}t_i^j \\in R[[t_i]]$ for all $i$ , where $R$ is the ring of integers for $K$ .",
"Suppose also that for each $i$ there exists $k_i \\in {\\mathbb {Z}}_{\\ge 0}$ such that the coefficients $c_{ij}$ satisfy $v(c_{ij}) > 0$ for $j < k_i$ and $v(c_{ik_i}) = 0$ .",
"From these data, define a multivariate pure power series $F(t_1, \\ldots , t_n) := f_1(t_1) + \\cdots + f_d(t_n).$ Then the Newton polygon of the pure power series $F \\in K \\langle U_{P_m} \\rangle $ (where $m = (1/m_1, \\ldots , 1/m_n)$ ) is contained in the $d$ -dimensional simplex defined by the convex hull of the vectors $(k_i + \\delta _{{\\varepsilon }}(k_i, v, m_i))e_i,$ where ${\\varepsilon }\\in {\\mathbb {Q}}$ satisfies ${\\varepsilon }\\le 1/m_i$ for all $i$ , and $e_i$ is the $i$ -th standard vector.",
"Straightforward application of Lemma REF and Remark REF to each $f_i$ that show up in the pure power series.",
"Let ${\\mathcal {Q}}= \\lbrace Q_1, \\ldots , Q_d\\rbrace \\in (\\operatorname{Sym}^dX)({\\mathbb {F}}_p)$ .",
"Let ${\\mathcal {U}}$ be the residue disk of $(\\operatorname{Sym}^dX)(p)$ reducing to $\\lbrace Q_1, \\ldots , Q_d\\rbrace $ .",
"Decompose the multiset $\\lbrace Q_1, \\ldots , Q_d\\rbrace $ into disjoint multisets ${\\mathcal {S}}_1, \\ldots , {\\mathcal {S}}_r$ each consisting of a single point with multiplicity $s_j = \\#{\\mathcal {S}}_j$ .",
"Let $L_j$ be the degree-$s_j$ unramified extension of $K_j$ , the field of definition of the points in ${\\mathcal {S}}_j$ , and let $R_j$ be the ring of integers of the $L_j$ .",
"For $1 \\le i \\le d$ , let $f_{i,j} \\in L_j[[t_j]]$ be the power series obtained from Chabauty's method, applied to the residue disk in $(\\operatorname{Sym}^{s_j}X)(K_j)$ above the point ${\\mathcal {S}}_j$ , such that their derivatives $f_{i,j}^{\\prime }$ are in $R_j[[t_j]]$ .",
"Let $F_i(t_1, \\ldots , t_d) = f_{i,1}(t_1) + \\cdots + f_{i,d}(t_d),$ and let $k_{ij} = \\operatorname{ord}_0(f_{ij})$ .",
"Then define the $d \\times d$ matrix $A_{{\\mathcal {P}}} = (a_{ij})$ by $a_{ij} = k_{i,j} + \\delta _{{\\varepsilon }}(k_{i,j}, v, s_i)$ for each residue disk and suitably small ${\\varepsilon }$ .",
"Theorem 5.16 Keep the notation from the previous paragraph.",
"Then the $F_i$ satisfy $N_0^{\\times }(F_1, \\ldots , F_d) \\le \\frac{1}{d!}",
"\\operatorname{Per}(A).$ By Proposition REF , we may as well assume that the power series that we get from Chabauty's method have finitely many common zeros (that is, a deformation of the power series exists, such that the tropicalizations and the $\\gamma _w$ stay constant).",
"This means that, by Theorem REF , that the number of isolated solutions can be written as the mixed volume of Newton polygons.",
"Now combine Lemma REF and Lemma REF .",
"Recall that $N_0^{\\times }(F_1, \\ldots , F_d)$ counts the 0-dimensional components of the common zeros of the $F_i$ in $(p^{\\times })^d$ .",
"Thus, we need to count the solutions in which some of the coordinates are 0 separately.",
"For example, if we wish to count the solutions that are of the form $(p^{\\times })^{(d-1)} \\times \\lbrace 0\\rbrace $ , it suffices to consider $N_0^{\\times }(F_1(t_1, t_2, \\ldots , t_{d-1}, 0), \\ldots , F_{d-1}(t_1, t_2, \\ldots , t_{d-1}, 0)),$ which is bounded above by $\\frac{1}{(d-1)!",
"}\\operatorname{Per}(B)$ , where $B$ is a $(d-1) \\times (d-1)$ minor of $A$ that takes the first $(d-1)$ rows and columns.",
"Thus, let $\\operatorname{Per}(A)^{\\prime } := \\sum _{0 \\le i \\le d} \\sum _{j \\in \\Lambda _i} \\frac{1}{i!",
"}\\operatorname{Per}(A_{ij}),$ where $A_{ij}$ denotes the $i \\times i$ minor of $A$ that takes the first $i$ columns (and any $i$ rows), and $A_{00}$ is the $0 \\times 0$ matrix whose permanent is understood to be 1 (since if (0, ..., 0) were a solution to the $F_i$ , it would contribute at most 1 to $N_0(F_1, \\ldots , F_d)$ ).",
"Theorem 5.17 Suppose $X$ is a nice curve over ${\\mathbb {Q}}$ with good reduction at $p$ satisfying Assumption REF , and let $\\omega _1, \\ldots , \\omega _d \\in H^0(X_{{\\mathbb {Q}}_p}, \\Omega ^1)$ be independent differential forms that vanish on $\\overline{J({\\mathbb {Q}})}$ such that $\\bar{\\omega }_i \\ne 0$ .",
"Then keeping the notation as above, with the $k_{i,j}$ corresponding to the order of vanishing of $\\omega _i$ at the point $P_j$ , the number of points outside of the special set of $(\\operatorname{Sym}^dX)({\\mathbb {Q}}_p)$ is at most $\\sum _{{\\mathcal {P}}\\in (\\operatorname{Sym}^dX)({\\mathbb {F}}_p)} \\frac{1}{N_{{\\mathcal {P}}}}\\operatorname{Per}(A_{{\\mathcal {P}}})^{\\prime }$ Apply the above theorem to each residue disk of $(\\operatorname{Sym}^dX)({\\mathbb {Q}}_p)$ , and use Corollary REF .",
"The $\\frac{1}{N}$ accounts for the ordering of the solutions, since the order of the points does not matter in $\\operatorname{Sym}^dX$ .",
"The above theorem shows that there is an upper bound on the number of points outside of the special set, depending only on the choice of $g, d$ and $p$ .",
"If we bound $\\# (\\operatorname{Sym}^dX)({\\mathbb {F}}_p)$ in terms of $g, d$ and $p$ , then this would complete the proof of Theorem REF : Proposition 5.18 Given a nice curve $X$ of genus $d$ with good reduction at $p$ and $d \\ge 1$ , $\\#((\\operatorname{Sym}^dX)({\\mathbb {F}}_p)) \\le (1 + 2g p^{d/2} + p^d)^d.$ We use the Hasse-Weil bound on $X$ , along with the fact that if $\\lbrace P_1, \\ldots , P_d\\rbrace \\in (\\operatorname{Sym}^dX)({\\mathbb {F}}_p)$ , then $P_i \\in X({\\mathbb {F}}_{p^d})$ for $1 \\le i \\le d$ .",
"Then the proof of Theorem REF follows by combining the statements of Proposition REF and Theorem REF ."
],
[
"An application",
"In this section, we prove the following corollary: Corollary 6.1 We can take $N(2,3,3) = 1539$ for any $X/{\\mathbb {Q}}$ a hyperelliptic curve whose affine model $y^2 = f(x)$ satisfies $\\deg (f) =7$ (so that $g = 3$ ), such that $\\operatorname{Rank}J \\le 1$ , and such that $X$ has good reduction at $p=2$ .",
"As noted in the introduction, Assumption REF is unnecessary when $\\operatorname{rk}J \\le 1$ .",
"(And we expect that 100% of hyperelliptic curves have ranks 0 or 1, assuming Goldfeld's conjecture!)",
"Lemma 6.2 Let $X$ be a smooth projective odd hyperelliptic curve of genus 3 that has good reduction at 2.",
"Then $\\# (\\operatorname{Sym}^2X)({\\mathbb {F}}_2) \\le 19$ .",
"We first note that a mod-2 reduction of an odd hyperelliptic curve of genus 3 corresponds to an equation of the form $y^2 + g(x)y = h(x)$ , with $g(x), h(x) \\in {\\mathbb {F}}_2[x]$ , with $\\deg g \\le 3, \\deg h = 7$ .",
"Let ${\\mathcal {P}}\\in (\\operatorname{Sym}^2X)({\\mathbb {F}}_2)$ .",
"It can be viewed as a multiset of two points ${\\mathcal {P}}= \\lbrace P_1, P_2\\rbrace $ .",
"We denote by $x(P_i)$ and $y(P_i)$ the $x$ - and $y$ -coordinates of $P_i$ , respectively, for $i = 1,2$ .",
"We have two cases: Case 1: When $P_1, P_2 \\in X({\\mathbb {F}}_2)$ .",
"If $P \\in X({\\mathbb {F}}_2)$ , then $x(P), y(P) \\in {\\mathbb {F}}_2$ or $x(P) = \\infty $ , so in particular, one must have $x(P) \\in \\lbrace 0,1, \\infty \\rbrace $ .",
"There are at most two points above each ${\\mathbb {F}}_2$ -point in the map $X \\rightarrow {\\mathbb {P}}^1$ , so there are at most 5 points in $X({\\mathbb {F}}_2)$ .",
"Let $\\#X({\\mathbb {F}}_2) = a$ .",
"Then there are ${a \\atopwithdelims ()2} + a$ points ${\\mathcal {P}}\\in (\\operatorname{Sym}^2X)({\\mathbb {F}}_2)$ of the form $\\lbrace P_1, P_2\\rbrace $ with $P_i \\in X({\\mathbb {F}}_2)$ ; the first term counts $\\lbrace P_1,P_2\\rbrace $ with $P_1$ and $P_2$ distinct, and the second terms counts $\\lbrace P_1,P_2\\rbrace $ with $P_1=P_2$ .",
"Case 2: When $P_1, P_2 \\in X({\\mathbb {F}}_4) \\backslash X({\\mathbb {F}}_2)$ are Galois conjugates.",
"In this case, there are at most 4 points of $X({\\mathbb {F}}_4)-X({\\mathbb {F}}_2)$ above ${\\mathbb {P}}^1(F_4)-{\\mathbb {P}}^1(F_2)$ .",
"But there could also be points of $X({\\mathbb {F}}_4)-X({\\mathbb {F}}_2)$ above ${\\mathbb {P}}^1({\\mathbb {F}}_2)$ ; the number of these is $5-a$ , since all of the 5 ${\\overline{{\\mathbb {F}}}}_2$ -points of X above ${\\mathbb {P}}^1({\\mathbb {F}}_2)$ are either ${\\mathbb {F}}_2$ -points or ${\\mathbb {F}}_4$ -points.",
"So there are at most $9-a$ such points.",
"Clearly, the choice of $1 \\le a \\le 5$ that maximizes ${a \\atopwithdelims ()2} + 9$ is $a=5$ , which means that there are at most 19 points in $(\\operatorname{Sym}^2X)({\\mathbb {F}}_2)$ .",
"Now we focus on a single residue disk of $(\\operatorname{Sym}^2X)({\\mathbb {F}}_2)$ and compute the possible number of points on each residue disk.",
"Since $g=3$ , the degree of $\\bar{\\omega }$ is $2g-2 = 4$ .",
"We start by computing $\\delta _{{\\varepsilon }}(k, v, \\ell )$ in Definition REF for when $k = 1,2,3,4$ .",
"We take ${\\varepsilon }\\in (0, \\frac{1}{2})$ as small as possible, as that minimizes $\\delta _{{\\varepsilon }}(k,v,\\ell )$ .",
"Then we have $\\delta _{{\\varepsilon }}(4,2,2) = 0, \\quad \\delta _{{\\varepsilon }}(3,2,2) = 5, \\quad \\delta _{{\\varepsilon }}(2,2,2) = 2, \\quad \\delta _{{\\varepsilon }}(1,2,2) = 3.$ Thus, for a residue disk over ${\\mathcal {P}}$ , the largest value of $\\operatorname{Per}A_{{\\mathcal {P}}}$ is given from the $2 \\times 2$ matrix whose entries are all $k + \\delta _{{\\varepsilon }}(k,v, \\ell )$ with $k = 3$ .",
"That is, the maximal value for $\\operatorname{Per}A_{{\\mathcal {P}}}$ is 128.",
"Now, there are two $1 \\times 1$ minors that we need to compute, from the definition of $\\operatorname{Per}(A)^{\\prime }$ in the previous chapter.",
"Again, the maximal values for these are 8, obtained when $k=3$ .",
"This gives $\\operatorname{Per}(A)^{\\prime } \\le \\frac{1}{2} \\cdot 128 + 8 + 8 + 1 = 81$ .",
"Now, we apply Theorem REF on the 19 residue disks with $N_{{\\mathcal {P}}} \\ge 1$ and $\\operatorname{Per}(A_{{\\mathcal {P}}})^{\\prime } \\le 81$ .",
"This gives the upper bound of $81 \\times 19 = 1539.$ This completes the proof of Corollary REF .",
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]
] | 1606.05195 |
[
[
"A Class of Parallel Doubly Stochastic Algorithms for Large-Scale\n Learning"
],
[
"Abstract We consider learning problems over training sets in which both, the number of training examples and the dimension of the feature vectors, are large.",
"To solve these problems we propose the random parallel stochastic algorithm (RAPSA).",
"We call the algorithm random parallel because it utilizes multiple parallel processors to operate on a randomly chosen subset of blocks of the feature vector.",
"We call the algorithm stochastic because processors choose training subsets uniformly at random.",
"Algorithms that are parallel in either of these dimensions exist, but RAPSA is the first attempt at a methodology that is parallel in both the selection of blocks and the selection of elements of the training set.",
"In RAPSA, processors utilize the randomly chosen functions to compute the stochastic gradient component associated with a randomly chosen block.",
"The technical contribution of this paper is to show that this minimally coordinated algorithm converges to the optimal classifier when the training objective is convex.",
"Moreover, we present an accelerated version of RAPSA (ARAPSA) that incorporates the objective function curvature information by premultiplying the descent direction by a Hessian approximation matrix.",
"We further extend the results for asynchronous settings and show that if the processors perform their updates without any coordination the algorithms are still convergent to the optimal argument.",
"RAPSA and its extensions are then numerically evaluated on a linear estimation problem and a binary image classification task using the MNIST handwritten digit dataset."
],
[
"Introduction",
"Learning is often formulated as an optimization problem that finds a vector of parameters ${\\mathbf {x}}^*\\in {\\mathbb {R}}^p$ that minimizes the average of a loss function across the elements of a training set.",
"For a precise definition consider a training set with $N$ elements and let $f_{n}:{\\mathbb {R}}^p\\rightarrow {\\mathbb {R}}$ be a convex loss function associated with the $n$ -th element of the training set.",
"The optimal parameter vector ${\\mathbf {x}}^*\\in {\\mathbb {R}}^p$ is defined as the minimizer of the average cost $F({\\mathbf {x}}) := (1/N)\\sum _{n=1}^N f_{n}({\\mathbf {x}})$ , ${\\mathbf {x}}^* := \\operatornamewithlimits{argmin}_{{\\mathbf {x}}} F({\\mathbf {x}}):= \\operatornamewithlimits{argmin}_{{\\mathbf {x}}}\\frac{1}{N}\\sum _{n=1}^N f_{n}({\\mathbf {x}}).$ Problems such as support vector machine classification, logistic and linear regression, and matrix completion can be put in the form of problem (REF ).",
"In this paper, we are interested in large scale problems where both the number of features $p$ and the number of elements $N$ in the training set are very large – which arise, e.g., in text , image , and genomic processing.",
"When $N$ and $p$ are large, the parallel processing architecture in Figure becomes of interest.",
"In this architecture, the parameter vector ${\\mathbf {x}}$ is divided into $B$ blocks each of which contains $p_b\\ll p$ features and a set of $I\\ll B$ processors work in parallel on randomly chosen parameter blocks while using a stochastic subset of elements of the training set.",
"In the schematic shown, Processor 1 fetches functions $f_1$ and $f_n$ to operate on block ${\\mathbf {x}}_b$ and Processor $i$ fetches functions $f_{n^{\\prime }}$ and $f_{n^{\\prime \\prime }}$ to operate on block ${\\mathbf {x}}_{b^{\\prime }}$ .",
"Other processors select other elements of the training set and other blocks with the majority of blocks remaining unchanged and the majority of functions remaining unused.",
"The blocks chosen for update and the functions fetched for determination of block updates are selected independently at random in subsequent slots.",
"Figure: NO_CAPTION"
]
] | 1606.04991 |
[
[
"Monotone and Convex Stochastic Orders for Processes with Independent\n Increments"
],
[
"Abstract We study monotone and convex stochastic orders for processes with independent increments.",
"Our contributions are twofold: First, we relate stochastic orders of the Poisson component to orders of their (generalized) L\\'evy measures.",
"The relation is proven using an interpolation formula for infinitely divisible laws.",
"Second, we derive explicit conditions on the characteristics of the processes.",
"In this case, we prove the conditions via constructions of couplings."
],
[
"Introduction",
"In this article we study monotone and convex stochastic orders for processes with independent increments (PIIs).",
"The law of a PII can be described by a deterministic triplet, called the characteristics, which has a similar structure as a Lévy-Khinchine triplet corresponding to a Lévy process.",
"The first characteristic represents the drift, the second characteristic encodes the Gaussian component and the third characteristic measures the frequency of jumps.",
"Our goal is to give conditions for stochastic orders in terms of the characteristics of PIIs.",
"Let us explain our main ideas.",
"We start with the observation that PIIs can be decomposed into two independent parts: A quasi-left continuous PII and a sum of independent random variables which represents the fixed times of discontinuity.",
"By the independence of the parts, for monotone and convex stochastic orders it suffices to order both parts individually.",
"The fixed times of discontinuities can be ordered by ordering each summand.",
"Hence, our main focus lies on the quasi-left continuous parts.",
"In this regard, our discussion is divided into two parts.",
"In the first one, we decompose the (quasi-left continuous) PIIs further into a Gaussian and a Poisson component.",
"Again, it suffices to order each of them separately.",
"In the case of the Gaussian parts, conditions for finite-dimensional stochastic orders are well-studied.",
"Thus, we restrict our discussion to the Poisson parts, for which we show that ordering the third characteristics implies finite-dimensional stochastic orders.",
"In the Lévy case, the Poisson parts are ordered if, and only if, the Lévy measures are ordered.",
"The main tool in our proof is an interpolation formula for infinitely divisible laws in the spirit of the formulas studied in [9], [10].",
"In the second part, we are interested in conditions which can be read immediately from the characteristics of the PIIs.",
"For the monotone stochastic order, we first give a majorization condition: The PIIs satisfy a drift condition, have the same Gaussian components and their jump frequencies are ordered in the sense that the negative jumps of the stochastically smaller process dominate the negative jumps of the stochastically larger process and vise versa for the positive jumps.",
"Instead of deducing the result from our previous results, we present an alternative proof.",
"The main idea is to couple the processes via so-called Itô maps, which relate Lévy measures to a reference Lévy measure.",
"The alternative proof brings additional aspects to the table: First, it shows that the conditions imply a pathwise order instead of a finite-dimensional one.",
"Second, the Itô maps imply an easy sufficient and necessary condition for the monotone ordering of the third characteristics, which can be considered as a generalization of the ordering of survival functions.",
"Third, the proof illustrates the relation between the conditions and their intuitive interpretations via the characteristics of the processes.",
"We also give cut criteria, which allow the majorization of the frequencies of jumps to change once.",
"In this case, we present a third alternative proof based on another coupling, which is built using the interpretation of the characteristics.",
"For the convex stochastic order, we also give a majorization condition: The expectations and the covariance functions of the PIIs are ordered and the stochastically larger PII has a higher jump frequency than the stochastically smaller PII.",
"For this condition we present a short proof, which applies to all PIIs with finite first moments.",
"It uses the observation that the stochastically larger PII can be decomposed in law into the stochastically smaller PII and a PII with non-negative expectation such that both are independent.",
"Now, as in Strassen's theorem, the convex order follows by Jensen's inequality.",
"Comparison results for Lévy processes and PIIs with absolutely continuous characteristics were obtained by Bergenthum and Rüschendorf [3], [4] and Bäuerle, Blatter and Müller [2].",
"The main idea in [3] is to start with two compound Poisson processes with the same jump intensity and to observe that these processes can be compared by (stochastically) ordering the jump size distribution.",
"By putting mass into the origin, the case of compound Poisson processes with different jump intensities can be reduced to the case with equal jump intensities.",
"Approximation arguments yield conditions for Lévy processes with infinite activity.",
"We show that the results obtained in [3] for compound Poisson processes with equal jump intensity hold for more general PIIs without modifying the characteristics.",
"Moreover, our explicit conditions improve several results in [4] by showing that parts of the conditions are not necessary.",
"The focus in [2] lies on the supermodular stochastic order, which is not studied in this article.",
"As our first part, the proofs are based on an interpolation formula from [9], which applies to functions in $C^2_b$ .",
"Since the supermodular stochastic order is generated by the supermodular functions in $C^2_b$ , the interpolation formula in [9] can be applied directly.",
"The convex stochastic order, however, is not generated by bounded functions and we have to generalize the interpolation formula to Lipschitz continuous functions.",
"This article is structured as follows.",
"In Section we recall the concepts of PIIs and stochastic orders.",
"In Section we state and prove our general conditions and in Section we present our majorization conditions and cut criteria together with the corresponding coupling arguments.",
"In Section we discuss how to generalize our conditions to semimartingales with conditionally independent increments and we give examples.",
"Let us end the introduction with a short remark on notation: For all non-explained notation we refer the reader to [11]."
],
[
"Stochastic Orders and PIIs",
"In this section we introduce the two main objects in this article: Processes with independent increments and stochastic orders.",
"We fix some $d \\in \\mathbb {N}$ .",
"Definition 2.1 An $\\mathbb {R}^d$ -valued càdlàg adapted stochastic process $X$ on the filtered probability space $(\\Omega , {F}, ({F}_t)_{t \\in [0, \\infty )}, P)$ is called PII, if $X_0 = 0$ and $X_t - X_s$ is independent of ${F}_s$ for all $s \\in [0, t]$ and $t \\in [0, \\infty )$ .",
"Stochastic ordering is a concept depending only on probability measures.",
"Since the law of a PII can be described by a deterministic triplet, the filtration $({F}_t)_{t \\in [0, \\infty )}$ is no active player in this article.",
"We formalizes this: Let $h$ be a fixed truncation function.",
"As stated in [11], laws of PIIs have a one-to-one correspondence to a deterministic triplet $(B, C, \\nu )$ , called the characteristics, consisting of the following: $B\\colon [0, \\infty ) \\rightarrow \\mathbb {R}^d$ is càdlàg with $B(h)_0= 0$ .",
"$C \\colon [0, \\infty ) \\rightarrow \\mathbb {R}^d \\otimes \\mathbb {R}^d$ is continuous with $C_0 = 0$ , such that $C_t - C_s$ is non-negative definite for all $0 \\le s < t$ .",
"$\\nu $ is a $\\sigma $ -finite measure on $([0, \\infty ) \\times \\mathbb {R}^d, {B}([0, \\infty )) \\otimes {B}(\\mathbb {R}^d))$ .",
"Providing an intuition, $B$ represents to the drift, $C$ encodes the Gaussian component and $\\nu $ encodes the Poisson component.",
"We define $\\mathcal {F}^m_{st} &\\triangleq \\lbrace f \\colon \\mathbb {R}^{d\\cdot m} \\rightarrow \\mathbb {R}, f \\textup { Borel and increasing}\\rbrace ,\\\\\\mathcal {F}^m_{cx} &\\triangleq \\lbrace f\\colon \\mathbb {R}^{d\\cdot m} \\rightarrow \\mathbb {R}, f \\textup { convex}\\rbrace ,\\\\\\mathcal {F}^m_{icx} &\\triangleq \\lbrace f \\colon \\mathbb {R}^{d \\cdot m} \\rightarrow \\mathbb {R}, f \\textup { increasing and convex}\\rbrace .$ Here, a function $f\\colon \\mathbb {R}^{d \\cdot m} \\rightarrow \\mathbb {R}$ is increasing if $f(x) \\le f(y)$ whenever $x \\le y$ , which means $x_i \\le y_i$ for all $i \\le d \\cdot m$ .",
"For two PIIs $X$ and $Y$ , we write $X \\preceq _\\bullet Y$ if for all $m \\in \\mathbb {N}$ , $0 \\le t_1 < t_2 < ... < t_m < \\infty $ and $f \\in \\mathcal {F}^m_\\bullet $ it holds that $E[f(X_{t_1}, ..., X_{t_m})] \\le E[f(Y_{t_1}, ..., Y_{t_m})],$ whenever the integrals are well-defined.",
"Needless to say that $X$ and $Y$ may defined on different probability spaces and that (REF ) is a property of the laws of $X$ and $Y$ .",
"We denote by $\\mathbb {D}$ the space of all càdlàg functions $[0, \\infty )\\rightarrow \\mathbb {R}$ and equip it with the Skorokhod topology.",
"In the one-dimensional case $d = 1$ we also consider the following class: $\\mathcal {F}_{pst} \\triangleq \\lbrace f \\colon \\mathbb {D} \\rightarrow \\mathbb {R}, f \\text{ Borel and } f(\\alpha ) \\le f(\\omega ) \\text{ if } \\alpha _t \\le \\omega _t \\text{ for all } t \\in [0, \\infty )\\rbrace .$ We use $p$ as an acronym for pathwise.",
"In this case, we write $X \\preceq _{pst} Y$ if $E[f(X)] \\le E[f(Y)] \\text{ for all } f \\in \\mathcal {F}_{pst}.$"
],
[
"Decomposition of PIIs",
"In this section we show that a PII $X$ with characteristics $(B^X, C^X, \\nu ^X)$ can be decomposed into a quasi-left continuous PII $X^{qlc}$ and a sum of independent random variables $X^{ftd}$ such that $X^{qlc}$ and $X^{ftd}$ are independent.",
"We assume that $|h(x)|\\mathbf {1}_{J^X} \\star \\nu ^X_t < \\infty $ for all $t \\in [0, \\infty )$ , where $J^X \\triangleq \\lbrace t \\in [0, \\infty )\\colon \\nu ^X(\\lbrace t\\rbrace \\times \\mathbb {R}^d) > 0\\rbrace .$ Moreover, we set $\\nu ^{X, qlc} (\\operatorname{d}\\hspace{-1.6502pt}t \\times \\operatorname{d}\\hspace{-1.6502pt}x) &\\triangleq \\mathbf {1}_{\\complement J^X}(t) \\nu ^X(\\operatorname{d}\\hspace{-1.6502pt}t \\times \\operatorname{d}\\hspace{-1.6502pt}x),\\\\B^{X, qlc} &\\triangleq B^X - h(x)\\mathbf {1}_{J^X} \\star \\nu ^X.$ For a Borel function $f\\colon [0, \\infty ) \\times \\mathbb {R}^d \\rightarrow [0, \\infty )$ we write $\\left(f \\cdot \\nu ^X\\right)(\\operatorname{d}\\hspace{-1.6502pt}t \\times \\operatorname{d}\\hspace{-1.6502pt}x) \\triangleq f(t, x) \\nu ^X(\\operatorname{d}\\hspace{-1.6502pt}t \\times \\operatorname{d}\\hspace{-1.6502pt}x).$ Following Jacod and Shiryaev [11], we denote the Dirac measure by $\\varepsilon $ .",
"Lemma 3.1 The process $X^{ftd} \\triangleq x\\mathbf {1}_{J^X} \\star \\mu ^X$ is a.s. well-defined as a sum of independent random variables $(\\Delta X_s)_{s \\in J^X}$ such that $P(\\Delta X_s \\in \\operatorname{d}\\hspace{-1.6502pt}x) = \\nu ^X(\\lbrace s\\rbrace \\times \\operatorname{d}\\hspace{-1.6502pt}x) + \\left(1 - \\nu ^X\\left(\\lbrace s\\rbrace \\times \\mathbb {R}^d\\right) \\right) \\varepsilon _0(\\operatorname{d}\\hspace{-1.6502pt}x).$ Moreover, $X^{qlc} \\triangleq X - X^{ftd}$ is a quasi-left continuous PII with characteristics $(B^{X, qlc}, C^X, \\nu ^{X, qlc})$ and $X^{ftd}$ and $X^{qlc}$ are independent.",
"Proof: The process $X^{ftd}$ is well-defined since $E\\left[|h(x)| \\mathbf {1}_{J^X}\\star \\mu ^X_t\\right] = |h(x)|\\mathbf {1}_{J^X}\\star \\nu ^X_t < \\infty ,$ by assumption, and $|x - h(x)|\\mathbf {1}_{J^X}\\star \\mu ^X_t < \\infty $ , by the càdlàg paths of $X$ .",
"The independence of the sequence $(\\Delta X_s)_{s \\in J}$ follows by the independent increments of $X$ and the fact that independence extends to a.s. limits.",
"The formula (REF ) is due to [11].",
"It follows from [11] that $X^{qlc}$ is a quasi-left continuous PII with characteristics $(B^{X, qlc}, C^X, \\nu ^{X, qlc})$ .",
"The independence of $X^{ftd}$ and $X^{qlc}$ follows from [12] and [11].",
"Let us give a few more details on this point: By [12] it suffices to show that for all sequences $0 \\le t_1 < ... < t_n< \\infty $ the vectors $(X^{ftd}_{t_1}, ..., X^{ftd}_{t_n})$ and $(X^{qlc}_{t_1}, ..., X^{qlc}_{t_n})$ are independent.",
"Moreover, by [12], it even suffices to show that for all $0 \\le s < t < \\infty $ the random variables $X^{ftd}_t - X^{ftd}_s$ and $X^{qlc}_t -X^{qlc}_s$ are independent.",
"We define the (deterministic) processes $\\widehat{B} &\\triangleq \\begin{pmatrix}h(x) \\mathbf {1}_{J^X} \\star \\nu ^X\\\\B^{X, qlc}\\end{pmatrix},\\qquad \\widehat{C} \\triangleq \\begin{pmatrix} 0&0\\\\0&C^X\\end{pmatrix}$ and the measure $\\widehat{\\nu }(\\operatorname{d}\\hspace{-1.6502pt}t \\times \\operatorname{d}\\hspace{-1.6502pt}x \\times \\operatorname{d}\\hspace{-1.6502pt}y) &\\triangleq \\mathbf {1}_{J^X}(t) \\nu ^X(\\operatorname{d}\\hspace{-1.6502pt}t \\times \\operatorname{d}\\hspace{-1.6502pt}x) \\varepsilon _{0} (\\operatorname{d}\\hspace{-1.6502pt}y) \\\\&\\hspace{56.9055pt}+ \\mathbf {1}_{\\complement J^X} (t) \\nu ^X(\\operatorname{d}\\hspace{-1.6502pt}t \\times \\operatorname{d}\\hspace{-1.6502pt}y)\\varepsilon _0 (\\operatorname{d}\\hspace{-1.6502pt}x).$ It is routine to check that the triplet $(\\widehat{B}, \\widehat{C}, \\widehat{\\nu })$ satisfies [11] w.r.t.",
"the truncation function $\\hat{h}(x, y) \\triangleq (h(x), h(y))$ .",
"Thus, using [11], it follows that the $\\mathbb {R}^{2d}$ -valued process $(X^{ftd}, X^{qlc})$ is a PII with characteristics $(\\widehat{B}, \\widehat{C}, \\widehat{\\nu })$ corresponding to the truncation function $\\hat{h}$ .",
"Now, an application of [11] yields that for all $u, v \\in \\mathbb {R}^d$ and $0 \\le s < t < \\infty $ $E&\\left[\\exp \\left(\\sqrt{-1} \\left\\langle u, X^{ftd}_t - X^{ftd}_s\\right\\rangle + \\sqrt{-1} \\left\\langle v, X^{qlc}_t - X^{qlc}_s\\right\\rangle \\right)\\right] \\\\&\\hspace{22.76228pt}= E\\left[\\exp \\left(\\sqrt{-1}\\left\\langle u, X^{ftd}_t - X^{ftd}_s\\right\\rangle \\right)\\right] E\\left[\\exp \\left( \\sqrt{-1}\\left\\langle v, X^{qlc}_t - X^{qlc}_s\\right\\rangle \\right)\\right].$ Thus, the independence of $X^{ftd}_t$ and $X^{qlc}_t$ follows by the uniqueness theorem characteristics functions.",
"$\\Box \\hspace{-1.42262pt}$ By the independence we can consider the fixed times of discontinuity and the quasi-left continuous parts separately.",
"Let $Y$ be a second PII with characteristics $(B^Y, C^Y, \\nu ^Y)$ such that $|h(x)| \\mathbf {1}_{J^Y} \\star \\nu ^Y_t < \\infty $ for all $t \\in [0, \\infty )$ .",
"Proposition 3.2 Let $\\bullet \\in \\lbrace pst, st\\rbrace $ .",
"If $X^{ftd} \\preceq _{\\bullet } Y^{ftd}$ and $X^{qlc} \\preceq _{\\bullet } Y^{qlc}$ , then $X \\preceq _{\\bullet } Y$ .",
"If $|x - h(x)| \\star \\nu ^X_t +|x - h(x)|\\star \\nu ^Y_t < \\infty $ for all $t \\in [0, \\infty )$ , then the statement also holds for $\\bullet \\in \\lbrace cx, icx\\rbrace $ .",
"Proof: Let $f \\in \\mathcal {F}_{pst}$ be bounded.",
"Then, by the independence and Fubini's theorem, $E[f(X)] &= \\iint f\\left(\\omega + \\alpha \\right) P(X^{ftd} \\in \\operatorname{d}\\hspace{-1.6502pt}\\omega ) P(X^{qlc} \\in \\operatorname{d}\\hspace{-1.6502pt}\\alpha ) \\nonumber \\\\&\\le \\iint f\\left(\\omega + \\alpha \\right) P(Y^{ftd} \\in \\operatorname{d}\\hspace{-1.6502pt}\\omega ) P(X^{qlc} \\in \\operatorname{d}\\hspace{-1.6502pt}\\alpha )\\nonumber \\\\&= \\iint f\\left(\\omega + \\alpha \\right) P(X^{qlc} \\in \\operatorname{d}\\hspace{-1.6502pt}\\alpha ) P(Y^{ftd} \\in \\operatorname{d}\\hspace{-1.6502pt}\\omega )\\\\&\\le \\iint f\\left(\\omega + \\alpha \\right) P(Y^{qlc} \\in \\operatorname{d}\\hspace{-1.6502pt}\\alpha ) P(Y^{ftd} \\in \\operatorname{d}\\hspace{-1.6502pt}\\omega )\\nonumber = E[f(Y)].$ Since the stochastic order $\\preceq _{pst}$ is generated by the class of bounded functions in $\\mathcal {F}_{pst}$ , see [15], we can conclude that $X \\preceq _{pst} Y$ .",
"The case $\\preceq _{st}$ follows identically.",
"For the convex cases, we note that each (increasing) convex function can be approximated pointwise in a monotone manner by (increasing) Lipschitz continuous convex functions.",
"More precisely, for a function $f \\colon \\mathbb {R}^d \\rightarrow \\mathbb {R}$ we set $f_n (x) \\triangleq \\inf _{z \\in \\mathbb {R}^d} (f(z) + n |x - z|)$ , which is the inf-convolution of $f$ .",
"It is well-known that in the case where $f$ is convex, the inf-convolution $f_n$ is Lipschitz continuous and convex, $f_n(x) \\le f_{n+1}(x) \\le f(x)$ and $f_n \\rightarrow f$ pointwise as $n \\rightarrow \\infty $ , see, e.g., [5].",
"Moreover, if $f$ is increasing, then $f_n$ is also increasing.",
"To see this, note that for $x \\le y$ we have $\\begin{split}f_n(x) = \\inf _{z \\in \\mathbb {R}^d} (f(z) + n|x - z|) &\\le \\inf _{z \\in \\mathbb {R}^d} (f(z + y - x) + n|y - (z + y - x)|)\\\\&= \\inf _{z \\in \\mathbb {R}^d} (f(z) + n|y - z|) = f_n(y).\\end{split}$ Thus, by the monotone convergence theorem, we may restrict ourselves to (increasing) Lipschitz continuous convex functions $\\mathbb {R}^{d \\cdot n} \\rightarrow \\mathbb {R}$ .",
"Denote one of these by $f$ .",
"To use the same argumentation as in the case $\\preceq _{pst}$ , we only have to verify the application of Fubini's theorem, see (REF ).",
"The assumptions $|x - h(x)| \\star \\nu ^X_t < \\infty $ and $|x - h(x)| \\star \\nu ^Y_t < \\infty $ imply that $E[|Y^{ftd}_t|] < \\infty $ and $E[|X^{qlc}_t|] < \\infty $ .",
"Hence, since all Lipschitz continuous functions are of linear growth, we have $\\iint |f(x + y)|& P\\left(\\left(Y^{ftd}_{t_1}, ..., Y^{ftd}_{t_n}\\right) \\in \\operatorname{d}\\hspace{-1.6502pt}x\\right) P\\left(\\left(X^{qlc}_{t_1}, ..., X^{qlc}_{t_n}\\right) \\in \\operatorname{d}\\hspace{-1.6502pt}y\\right)\\\\&\\le \\textup {const. }",
"\\left(1 + \\sum _{k = 1}^n E\\left[\\left|Y^{ftd}_{t_k}\\right|\\right] + \\sum _{j = 1}^n E\\left[\\left|X^{qlc}_{t_j}\\right|\\right] \\right) < \\infty $ for all $0 \\le t_1 < ... < t_n < \\infty $ .",
"Therefore, we can apply Fubini's theorem and the claim follows similar to the case $\\preceq _{pst}$ .",
"$\\Box \\hspace{-1.42262pt}$"
],
[
"Stochastic Orders for the Fixed Times of Discontinuity",
"For the fixed times of discontinuity it suffices to order each summand separately.",
"Let $X$ and $Y$ be as in the previous section.",
"Proposition 3.3 If for all $t \\in [0, \\infty )$ it holds that $\\Delta X^{ftd}_t \\preceq _{st} \\Delta Y^{ftd}_t$ , then $X^{ftd} \\preceq _{pst} Y^{ftd}$ .",
"Moreover, if for all $t \\in [0, \\infty )$ it holds that $|x| \\mathbf {1}_{J^Y} \\star \\nu ^Y_t < \\infty $ and $\\Delta X^{ftd}_t \\preceq _{(i)cx} \\Delta Y^{ftd}_t$ , then $X^{ftd} \\preceq _{(i)cx} Y^{ftd}$ .",
"Here, $\\Delta X^{ftd}_t \\preceq _{\\bullet } \\Delta Y^{ftd}_t$ refers to stochastic orders of $\\mathbb {R}^d$ -valued random variables.",
"Proof: In the case $\\preceq _{pst}$ the claim follows from Strassen's theorem [14]: We find a probability space which supports two sequences $(\\Delta X_t)_{t \\in J^X \\cup J^Y}$ and $(\\Delta Y_t)_{t \\in J^X \\cup J^Y}$ of independent random variables such that $\\Delta X_t$ has law (REF ), $\\Delta Y_t$ has law (REF ) with $\\nu ^X$ replaced by $\\nu ^Y$ and a.s. $\\Delta X_t \\le \\Delta Y_t$ for all $t \\in J^X \\cup J^Y$ .",
"Set $J_t \\triangleq (J^X \\cup J^Y) \\cap [0, t]$ for $t \\in [0, \\infty )$ .",
"We claim that the sums $\\sum _{s \\in J_t} \\Delta X_s$ and $\\sum _{s \\in J_t} \\Delta Y_s$ converge a.s. To see this, set $Z_s \\triangleq \\Delta X_s \\mathbf {1}_{\\lbrace |\\Delta X_s| \\le 1\\rbrace }$ and note, by [11], that we have $\\sum _{s \\in J_t} P(Z_s \\ne \\Delta X_s) = \\sum _{s \\in J_t} P(|\\Delta X_s| > 1) \\le \\nu ^X([0, t] \\times \\lbrace |x| > 1\\rbrace ) < \\infty .$ Hence, by the Borel-Cantelli lemma, $\\sum _{s \\in J_t} \\Delta X_s$ converges a.s. if, and only if, $\\sum _{s \\in J_t} Z_s$ converges a.s.",
"Since $h$ is a truncation function there exists an $\\epsilon > 0$ such that $h(x) = x$ on $\\lbrace |x| \\le \\epsilon \\rbrace $ .",
"Since $\\sum _{s \\in J_t} E[|Z_s|] &\\le |x - h(x)| \\mathbf {1}_{\\lbrace |x| \\le 1\\rbrace } \\star \\nu ^X_t + |h(x)| \\star \\nu ^X_t\\\\&\\le \\textup {const. }",
"\\nu ^X([0, t] \\times \\lbrace \\epsilon < |x| \\le 1\\rbrace ) + |h(x)| \\star \\nu ^X_t < \\infty ,$ due to [11] and the assumption that $|h(x)| \\star \\nu ^X_t < \\infty $ , we conclude that $\\sum _{s \\in J_t} Z_s$ converges a.s.",
"Thus, $\\sum _{s \\in J_t} \\Delta X_s$ converges a.s. and $\\sum _{s \\in J_t} \\Delta Y_s$ converges a.s. by the same arguments.",
"We also claim that $\\sum _{s \\in J_\\cdot } \\Delta X_s$ has the same law as $X^{ftd}$ and $\\sum _{s \\in J_\\cdot } \\Delta Y_s$ has the same law as $Y^{ftd}$ .",
"By the càdlàg paths, we only have to show that the processes have the same finite dimensional distributions, see, for instance, [11].",
"Now, take $0 = t_0 \\le t_1 < ... < t_n < \\infty $ and let $A\\in \\mathbb {R}^n \\otimes \\mathbb {R}^n$ be the lower triangular matrix with $A_{ij} = 1$ for all $i \\ge j$ .",
"Then, $\\left(X_{t_1}^{ftd}, ..., X^{ftd}_{t_n}\\right)^\\text{tr} &= A \\left(X^{ftd}_{t_1} - X^{ftd}_{t_0}, ..., X^{ftd}_{t_n} - X^{ftd}_{t_{n-1}}\\right)^\\textup {tr},\\\\\\left(\\sum _{s \\in J_{t_1}} \\Delta X_s, ..., \\sum _{s \\in J_{t_n}} \\Delta X_s\\right)^\\textup {tr} &= A\\left(\\sum _{s \\in J_{t_1}\\backslash J_{t_0}} \\Delta X_s, ..., \\sum _{s \\in J_{t_n} \\backslash J_{t_{n-1}}} \\Delta X_s\\right)^\\textup {tr}.$ Now, using the uniqueness theorem for characteristic functions and the fact that the entries of the right hand vectors are independent, it suffices to show that for all $0 \\le s < t < \\infty $ the sum $\\sum _{r \\in J_t \\backslash J_s} \\Delta X_r$ has the same law as $X^{ftd}_t - X^{ftd}_s$ .",
"This, however, follows from the fact that for all $u \\in \\mathbb {R}^d$ $E&\\left[ \\exp \\left( \\sqrt{-1}\\ \\bigg \\langle u, \\sum _{r \\in J_t\\backslash J_s} \\Delta X_r\\bigg \\rangle \\right)\\right]\\\\&\\hspace{45.52458pt}= \\prod _{r \\in (s, t]} \\left(1 + \\int \\left(\\exp \\left(\\sqrt{-1} \\langle u, x\\rangle \\right) - 1 \\right) \\nu ^X(\\lbrace r\\rbrace \\times \\operatorname{d}\\hspace{-1.6502pt}x)\\right),$ [11] and the uniqueness theorem for characteristic functions.",
"Thus, we conclude that $\\sum _{s \\in J_\\cdot } \\Delta X_s$ has the same law as $X^{ftd}$ .",
"The same argument also shows that $\\sum _{s \\in J_\\cdot } \\Delta Y_s$ has the same law as $Y^{ftd}$ .",
"Now, a.s. $\\sum _{s \\in J_\\cdot } \\Delta X_s \\le \\sum _{s \\in J_\\cdot } \\Delta Y_s$ implies the stochastic order $X^{ftd} \\preceq _{pst} Y^{ftd}$ .",
"Let us presume that $|x| \\mathbf {1}_{J^Y} \\star \\nu ^Y_t < \\infty $ for all $t \\in [0, \\infty )$ .",
"If $\\Delta X_t \\preceq _{cx} \\Delta Y_t$ for all $t \\in J^X \\cup J^Y$ , then, by Strassen's theorem [15], we find a probability space which supports two sequences $(\\Delta X_t)_{t \\in J^X \\cup J^Y}$ and $(\\Delta Y_t)_{t \\in J^X \\cup J^Y}$ of independent random variables such that $\\Delta X_t$ has law (REF ) and $\\Delta Y_t$ has law (REF ) with $\\nu ^X$ replaced by $\\nu ^Y$ and a.s. $E[\\Delta Y_t |{H}] = X_t$ for all $t \\in J^X \\cup J^Y$ , where ${H} \\equiv \\sigma (\\Delta X_s, s \\in J^X \\cup J^Y)$ .",
"Fix $t \\in [0, \\infty )$ and set $J_t$ as above.",
"The assumption $|x| \\mathbf {1}_{J^Y} \\star \\nu ^Y_t < \\infty $ implies that $E \\left[\\sum _{s \\in J_t} | \\Delta Y_s|\\right] < \\infty .$ Thus, we have a.s. $E\\left[ \\sum _{s \\in J_t} \\Delta Y_s \\bigg | {H} \\right] &= \\sum _{s \\in J_t} E\\left[\\Delta Y_s | {H}\\right] = \\sum _{s \\in J_t} \\Delta X_s.$ Since $\\sum _{s \\in J_\\cdot } \\Delta X_s$ has the same law as $X^{ftd}$ and $\\sum _{s \\in J_\\cdot } \\Delta Y_s$ has the same law as $Y^{ftd}$ , we conclude $X^{ftd} \\preceq _{cx} Y^{ftd}$ from the conditional Jensen's inequality.",
"The case $\\preceq _{icx}$ follows similarly.",
"$\\Box \\hspace{-1.42262pt}$ Explicit conditions for stochastic orders of $\\mathbb {R}^d$ -valued random variables can be found in [15].",
"Next, we will discuss the quasi-left continuous parts."
],
[
"Stochastic Orders for Quasi-Left Continuous PIIs",
"Let $X$ and $Y$ be quasi-left continuous PIIs with characteristics $(B^X, C^X, \\nu ^X)$ and $(B^Y, C^Y, \\nu ^Y)$ respectively.",
"First, we assume that the discontinuous parts of $X$ and $Y$ are of finite variation and that $X$ and $Y$ have first moments, i.e.",
"for all $t \\in [0, \\infty )$ we assume that $|x| \\star \\nu ^X_t +|x| \\star \\nu ^Y_t < \\infty $ .",
"In this case, the PIIs $X$ and $Y$ have a decomposition $X &= B^X - h(x) \\star \\nu ^X + X^c + x \\star \\mu ^X,\\\\Y &= B^Y + h(x) \\star \\nu ^Y + Y^c + x \\star \\mu ^Y,$ where $X^c$ is a Wiener process with variance function $C^X$ and $Y^c$ is a Wiener process with variance function $C^Y$ in the sense of [11].",
"In particular, $X^c$ is independent of $x \\star \\mu ^X$ and $Y^c$ is independent of $x \\star \\mu ^Y$ , see [12].",
"Proposition 3.4 Let $\\bullet \\in \\lbrace pst, st, cx, icx\\rbrace $ .",
"If $B^X - h(x) \\star \\nu ^X + X^c \\preceq _{\\bullet } B^Y - h(x) \\star \\nu ^Y + Y^c$ and $x \\star \\mu ^X \\preceq _{\\bullet } x \\star \\mu ^Y$ , then $X \\preceq _\\bullet Y$ .",
"Proof: This follows as in the proof of Proposition REF .",
"$\\Box \\hspace{-1.42262pt}$ For all $t \\in [0, \\infty )$ the random variable $B^X_t - h(x) \\star \\nu ^X_t + X^c_t$ is Gaussian with expectation $B^X_t - h(x)\\star \\nu ^X_t$ and covariance matrix $C^X_t$ , and the random variable $B^Y - h(x) \\star \\nu ^Y_t + Y^c_t$ is Gaussian with expectation $B^Y_t - h(x) \\star \\nu ^Y_t$ and covariance matrix $C^Y_t$ .",
"Hence, the question when $B^X - h(x) \\star \\nu ^X + X^c \\preceq _{\\bullet } B^Y - h(x) \\star \\nu ^X + Y^c$ is a question when two Gaussian vectors are stochastically ordered.",
"This question, however, is well-studied, see, e.g., [15], and we restrict ourselves to the Poisson sums by assuming that $B^X - h(x) \\star \\nu ^X = B^Y - h(x) \\star \\nu ^Y = 0,\\quad X^c = Y^c = 0.$ We note the following technical observation: Lemma 3.5 There exist a decomposition $K^\\bullet (t, \\operatorname{d}\\hspace{-1.6502pt}x) \\operatorname{d}\\hspace{-1.6502pt}A_t = \\nu ^{\\bullet } (\\operatorname{d}\\hspace{-1.6502pt}t \\times \\operatorname{d}\\hspace{-1.6502pt}x)$ where $K^{\\bullet }$ is a Borel transition kernel from $[0, \\infty )$ to $\\mathbb {R}$ and $A$ is an increasing continuous function of finite variation.",
"Proof: It is well-known that such a decomposition exists, see [11].",
"That we can take the same $A$ for both decompositions is a consequence of the Radon-Nikodym theorem.",
"$\\Box \\hspace{-1.42262pt}$ We write $K^X \\preceq _{\\bullet } K^Y$ if $\\int f(x) K^X(t, \\operatorname{d}\\hspace{-1.6502pt}x) \\le \\int f(x) K^Y(t, \\operatorname{d}\\hspace{-1.6502pt}x)$ for $\\operatorname{d}\\hspace{-1.6502pt}A_t$ -a.a. $t \\in [0, \\infty )$ and all Lipschitz continuous $f \\in \\mathcal {F}^1_{\\bullet }$ with $|f(x)| \\le \\textup {const. }",
"|x|$ .",
"We stress that for Lipschitz continuous functions $f$ the growth condition $|f(x)| \\le \\textup {const. }",
"|x|$ is equivalent to $f(0) = 0$ .",
"Remark 3.6 If $\\operatorname{d}\\hspace{-1.6502pt}A_t$ -a.e.",
"$K^X(\\cdot , \\mathbb {R}^d) < \\infty $ and $K^X(\\cdot , \\mathbb {R}^d) = K^X(\\cdot , \\mathbb {R}^d)$ , then $K^X$ and $K^Y$ are ordered if, and only if, for $\\operatorname{d}\\hspace{-1.6502pt}A_t$ -a.a. $t \\in [0, \\infty )$ the random variables with laws $K^X(t, \\mathbb {R}^d)^{-1} K^X(t, \\operatorname{d}\\hspace{-1.6502pt}x)$ and $K^Y(t, \\mathbb {R}^d)^{-1} K^Y(t, \\operatorname{d}\\hspace{-1.6502pt}x)$ are ordered.",
"Thus, in the one-dimensional case $d = 1$ , we have the following characterizations: $K^X \\preceq _{st} K^Y \\quad &\\Longleftrightarrow \\quad K^Y(t, (- \\infty , x]) \\le K^X(t, (- \\infty , x]) \\\\&\\hspace{71.13188pt}\\text{ for all } x \\in \\mathbb {R} \\text{ and } \\operatorname{d}\\hspace{-1.6502pt}A_t\\text{-a.a. } t \\in [0, \\infty ),\\\\K^X \\preceq _{icx} K^Y \\quad &\\Longleftrightarrow \\quad \\int (y - x)_+ K^X(t, \\operatorname{d}\\hspace{-1.6502pt}y) \\le \\int (y - x)_+K^Y(t, \\operatorname{d}\\hspace{-1.6502pt}y) \\\\&\\hspace{71.13188pt}\\text{ for all } x \\in \\mathbb {R} \\text{ and } \\operatorname{d}\\hspace{-1.6502pt}A_t\\text{-a.a. } t \\in [0, \\infty ),\\\\K^X \\preceq _{cx} K^Y \\quad &\\Longleftrightarrow \\quad \\int (y - x)_+ K^X(t, \\operatorname{d}\\hspace{-1.6502pt}y) \\le \\int (y - x)_+K^Y(t, \\operatorname{d}\\hspace{-1.6502pt}y) \\\\&\\hspace{30.44466pt}\\text{ and } \\int y K^X(t, \\operatorname{d}\\hspace{-1.6502pt}y) = \\int y K^Y(t, \\operatorname{d}\\hspace{-1.6502pt}y) \\\\&\\hspace{71.13188pt}\\text{ for all } x \\in \\mathbb {R} \\text{ and } \\operatorname{d}\\hspace{-1.6502pt}A_t\\text{-a.a. } t \\in [0, \\infty ).$ These characterizations can be deduced from [15] together with the fact that the stochastic order $\\preceq _{st}$ is generated by the increasing functions of class $C^2_b$ , see [15].",
"In Lemma REF below we will see a generalization of the first equivalence to cases where $K^X(\\cdot , \\operatorname{d}\\hspace{-1.6502pt}x)$ and $K^Y(\\cdot , \\operatorname{d}\\hspace{-1.6502pt}x)$ are not finite nor have the same mass.",
"Theorem 3.7 Let $\\bullet \\in \\lbrace st, cx, icx\\rbrace $ .",
"It holds that $K^X \\preceq _{\\bullet } K^Y\\quad \\Longrightarrow \\quad X \\preceq _{\\bullet } Y.$ If $X$ and $Y$ are Lévy processes, then also $K^X \\preceq _{\\bullet } K^Y\\quad \\Longleftarrow \\quad X \\preceq _\\bullet Y,$ where $K^X$ and $K^Y$ are the Lévy measures.",
"Proof: Let us start with the first claim.",
"The following is a version of [4].",
"Lemma 3.8 If for all $0 \\le s < t< \\infty $ it holds that $X_t - X_s \\preceq _\\bullet Y_t - Y_s,$ then $X \\preceq _{\\bullet } Y$ .",
"Proof: We use an induction argument.",
"Take $0 \\le s < t < \\infty $ .",
"Note that $(X_s, X_t) = (X_s, X_s) + (0, X_t - X_s)$ and $(Y_s, Y_t) = (Y_s, Y_s) + (0, Y_t - Y_s)$ , which are sums of independent random variables due the independent increment property of $X$ and $Y$ .",
"Since the monotone and convex stochastic orders are closed w.r.t.",
"identical concentration, see [15], $X_s \\preceq _{\\bullet } Y_s$ implies $(X_s, X_s) \\preceq _{\\bullet } (Y_s, Y_s)$ .",
"Moreover, since all stochastic orders under consideration are closed w.r.t.",
"independent concentration, see [15], $X_{t}- X_s \\preceq _{\\bullet } Y_t - Y_s$ implies $(0, X_t - X_s) \\preceq _{\\bullet } (0, Y_t- Y_s)$ .",
"Thus, by the convolution property, see again [15], $(X_s, X_t) \\preceq _\\bullet (Y_s, Y_t)$ follows.",
"Now, take $0 \\le t_1 < ..., < t_n < \\infty $ .",
"We have $(X_{t_1}, ..., X_{t_n}) &= (X_{t_1}, ..., X_{t_{n-1}}, X_{t_{n-1}}) + (0, ..., 0, X_{t_n} - X_{t_{n-1}}),\\\\(Y_{t_1}, ..., Y_{t_n}) &= (Y_{t_1}, ..., Y_{t_{n-1}}, Y_{t_{n-1}}) + (0, ..., 0, Y_{t_n} - Y_{t_{n-1}}).$ By the independent increment property, the vectors on the right hand sides are independent.",
"Using the induction hypothesis and the same arguments as above concludes the proof.",
"$\\Box \\hspace{-1.42262pt}$ Thus, it suffices to show (REF ).",
"Our main tool is an interpolation formula in the spirit of [10].",
"A related formula was used in [2] to prove a supermodular stochastic order for Lévy processes.",
"Let the process $Z(\\alpha )$ be a PII with characteristics $(\\alpha h(x) \\star \\nu ^{Y} + (1 - \\alpha ) h(x) \\star \\nu ^{X}, 0, \\alpha \\nu ^{Y} + (1- \\alpha ) \\nu ^{X})$ , see [11] for the existence, and set $\\mathcal {L}^\\bullet _{s, t} f (x) \\triangleq \\int _s^t \\int \\left( f(x + y) - f(x)\\right)K^{\\bullet }(r,\\operatorname{d}\\hspace{-1.6502pt}y)\\operatorname{d}\\hspace{-1.6502pt}A_r$ for Lipschitz continuous functions $f \\colon \\mathbb {R}^d \\rightarrow \\mathbb {R}$ .",
"The assumption $|x| \\star \\nu ^\\bullet _t < \\infty $ implies that $\\mathcal {L}^\\bullet _{s,t} f$ is well-defined.",
"Lemma 3.9 For all Lipschitz continuous $f \\colon \\mathbb {R}^d \\rightarrow \\mathbb {R}$ it holds that $E&\\left[f\\left(Y_t - Y_s\\right)\\right] - E\\left[f\\left(X_t - X_s\\right)\\right]\\\\&\\hspace{28.45274pt}= \\int _0^1 \\int \\left(\\mathcal {L}^Y_{s, t} f (z) - \\mathcal {L}^X_{s, t} f(z)\\right) P(Z_t (\\alpha ) - Z_s(\\alpha ) \\in \\operatorname{d}\\hspace{-1.6502pt}z) \\operatorname{d}\\hspace{-1.6502pt}\\alpha .$ Proof: For $f \\in C^2_b(\\mathbb {R}^d, \\mathbb {R})$ the claim follows from [10].",
"The claim for Lipschitz continuous $f$ follows by approximation: First, we approximate $f$ by bounded Lipschitz continuous functions.",
"Set $f_n \\triangleq n$ on $\\lbrace x \\in \\mathbb {R}^d \\colon f(x) \\ge n\\rbrace $ , $f_n \\triangleq - n$ on $\\lbrace x \\in \\mathbb {R}^d \\colon f(x) \\le -n\\rbrace $ and $f_n \\triangleq f$ otherwise.",
"It is routine to check that $f_n$ is Lipschitz continuous with the same Lipschitz constant as $f$ .",
"Thus, for $U \\in \\lbrace X, Y\\rbrace $ we have $|f_n(U_t - U_s)| \\le \\textup {const. }",
"(1 + |U_t - U_s|)$ and $|f_n(z + y) - f_n(z)| \\le \\textup {const. }",
"|y|$ , where both constants are independent of $n$ .",
"Since, using our assumptions, $|U_t - U_s|$ is integrable w.r.t.",
"$P$ and $|y|$ is integrable w.r.t.",
"$\\mathbf {1}_{[0, 1]}(\\alpha ) \\mathbf {1}_{(s, t]}(r) K^U(r, \\operatorname{d}\\hspace{-1.6502pt}y) \\operatorname{d}\\hspace{-1.6502pt}A_r P(Z_t(\\alpha ) - Z_s(\\alpha ) \\in \\operatorname{d}\\hspace{-1.6502pt}z) \\operatorname{d}\\hspace{-1.6502pt}\\alpha $ , we can apply the dominated convergence theorem to obtain $\\lim _{n \\rightarrow \\infty } E[f_n(U_t - U_s)] = E[f(U_t - U_s)]$ and $\\begin{split}\\lim _{n \\rightarrow \\infty } \\int _0^1 &\\int \\mathcal {L}^U_{s, t} f_n (z)P(Z_t (\\alpha ) - Z_s(\\alpha ) \\in \\operatorname{d}\\hspace{-1.6502pt}z) \\operatorname{d}\\hspace{-1.6502pt}\\alpha \\\\&= \\int _0^1 \\int \\mathcal {L}^U_{s, t} f (z)P(Z_t (\\alpha ) - Z_s(\\alpha ) \\in \\operatorname{d}\\hspace{-1.6502pt}z) \\operatorname{d}\\hspace{-1.6502pt}\\alpha .\\end{split}$ Hence, the claim holds for all bounded Lipschitz continuous function.",
"We approximate a second time.",
"Let $\\phi $ be the standard mollifier, i.e.",
"$\\phi (x) = c \\exp ( (|x|^2 - 1)^{-1}) \\mathbf {1}_{\\lbrace |x| \\le 1\\rbrace }$ , where $c$ is a normalization constant, and set $\\phi _n (x) \\triangleq n^{d} \\phi (n x)$ .",
"Define $f_n \\triangleq f * \\phi _n$ , where $*$ denotes the convolution.",
"It is well-known that $f_n \\in C^\\infty (\\mathbb {R}^d, \\mathbb {R})$ , see, for instance, [17].",
"In particular, since $f$ is bounded and $\\frac{\\operatorname{d}\\hspace{-1.6502pt}}{\\operatorname{d}\\hspace{-1.6502pt}x_i}\\phi _n$ has compact support, we deduce from the formula $\\frac{\\operatorname{d}\\hspace{-1.6502pt}}{\\operatorname{d}\\hspace{-1.6502pt}x_i} (f * \\phi _n) = f * \\frac{\\operatorname{d}\\hspace{-1.6502pt}}{\\operatorname{d}\\hspace{-1.6502pt}x_i}\\phi _n$ , see, for instance, [17], that $f * \\phi _n \\in C^2_b(\\mathbb {R}^d, \\mathbb {R})$ .",
"It is routine to check that $f * \\phi _n$ is Lipschitz continuous with the same Lipschitz constant as $f$ .",
"Moreover, for all $x \\in \\mathbb {R}^d$ , we have $\\left| (f * \\phi _n) (x) - f(x)\\right| &\\le \\int _{|z| \\le 1} \\phi (z) \\left| f(x) - f\\left(x - \\frac{z}{n}\\right)\\right| \\operatorname{d}\\hspace{-1.6502pt}z\\\\&\\le \\textup {const. }",
"\\int _{|z| \\le 1} \\frac{\\phi (z)|z|}{n} \\operatorname{d}\\hspace{-1.6502pt}z\\le \\frac{1}{n} \\xrightarrow{} 0,$ i.e.",
"$f_n \\rightarrow f$ pointwise as $n \\rightarrow \\infty $ .",
"Thus, as above, we conclude from the dominated convergence theorem that (REF ) and (REF ) hold.",
"This finishes the proof.",
"$\\Box \\hspace{-1.42262pt}$ For all $f \\in \\mathcal {F}^1_\\bullet $ and $x \\in \\mathbb {R}^d$ we have $f(x + \\cdot ) - f(x) \\in \\mathcal {F}^1_\\bullet $ .",
"Thus, $K^X \\preceq _{\\bullet } K^Y$ implies $\\mathcal {L}^Y_{s, t} f(z) - \\mathcal {L}^Y_{s, t} f(z) \\ge 0$ for all $0 \\le s < t < \\infty $ , $z \\in \\mathbb {R}^d$ and all Lipschitz continuous $f\\in \\mathcal {F}^1_\\bullet $ .",
"In particular, this yields $E[f(X_t - X_s)] \\le E[f(Y_t - Y_s)]$ by Lemma REF .",
"We note that the stochastic order $\\preceq _{st}$ is generated by the increasing functions in $C^2_b(\\mathbb {R}^d, \\mathbb {R})$ .",
"To see this note first that the stochastic order $\\preceq _{st}$ is generated by all increasing functions in $C_b(\\mathbb {R}^d, \\mathbb {R})$ , see [15].",
"Then, applying a mollification argument to each of these functions yields the claim, see [15] or the proof of Lemma REF .",
"Since all $f \\in C^2_b(\\mathbb {R}^d, \\mathbb {R})$ are Lipschitz continuous, we conclude $X_t - X_s \\preceq _{st} Y_t - Y_s$ .",
"To obtain the claim for $\\preceq _{(i)cx}$ we can use the fact that all $f \\in \\mathcal {F}^1_{(i)cx}$ can be approximated by (increasing) convex Lipschitz continuous functions in a monotone manner, see the proof of Proposition REF .",
"Hence, $X_t - X_s \\preceq _{(i)cx} Y_t - Y_s$ follows from the monotone convergence theorem.",
"For the rest of the proof we assume that $X$ and $Y$ are Lévy processes with Lévy measures $K^X$ , $K^Y$ respectively.",
"Lemma 3.10 For $U \\in \\lbrace X, Y\\rbrace $ and all Lipschitz continuous $f\\colon \\mathbb {R}^d \\rightarrow \\mathbb {R}$ the process $f&(U_\\cdot ) - f(0) - \\int _0^\\cdot \\int \\left(f(U_s + x) - f(U_s)\\right) K^U(\\operatorname{d}\\hspace{-1.6502pt}x) \\operatorname{d}\\hspace{-1.6502pt}s$ is a martingale.",
"Proof: The same approximation arguments as used in the proof of Lemma REF yield that it suffices to show the claim for $f \\in C^2_b(\\mathbb {R}^d, \\mathbb {R})$ .",
"In this case, Itô's formula yields that the process is a local martingale.",
"Using that $|f(x)| \\le \\textup {const.}",
"(1 + |x|)$ and $|f(y + x) - f(y)| \\le \\textup {const.}",
"|x|$ yields $\\sup _{s \\in [0, t]}& \\left| f(U_s) - f(0) - \\int _0^s \\int \\left(f(U_r + x) - f(U_r)\\right) K^U(\\operatorname{d}\\hspace{-1.6502pt}x) \\operatorname{d}\\hspace{-1.6502pt}r\\right|\\\\&\\hspace{136.5733pt}\\le \\text{ const.}",
"\\left(1 + t + \\sup _{s \\in [0, t]} |U_s| \\right),$ where the constant is uniform.",
"Since $U$ is a Lévy process, $\\sup _{s \\in [0, t]} |U_s|$ is integrable if $\\int |x - h(x)| K^U(\\operatorname{d}\\hspace{-1.6502pt}x) < \\infty $ , see [18].",
"Hence, the martingale property follows from the dominated convergence theorem.",
"$\\Box \\hspace{-1.42262pt}$ Thanks to this observation and Fubini's theorem, for all Lipschitz continuous functions $f$ with $f(0) = 0$ it holds that $\\lim _{t \\downarrow 0}\\frac{E[f(X_t)]}{t} &= \\lim _{t \\downarrow 0} \\frac{1}{t} \\int _0^t E \\left[ \\int \\left(f (X_s + x) - f(X_s) \\right)K^X(\\operatorname{d}\\hspace{-1.6502pt}x)\\right] \\operatorname{d}\\hspace{-1.6502pt}s\\\\&= E\\left[ \\int \\left(f(X_0 + x) - f(X_0)\\right) K^X(\\operatorname{d}\\hspace{-1.6502pt}x)\\right]= \\int f(x) K^X(\\operatorname{d}\\hspace{-1.6502pt}x).$ Using the same arguments for $X$ replaced by $Y$ , we obtain for all Lipschitz continuous $f$ with $f(0) = 0$ that $0 \\le \\lim _{t \\downarrow 0} \\frac{E[f(Y_{t})] - E[f(X_{t})]}{t} &= \\int f(x) K^Y(\\operatorname{d}\\hspace{-1.6502pt}x) - \\int f(x) K^X(\\operatorname{d}\\hspace{-1.6502pt}x).$ This concludes the proof of Theorem REF .",
"$\\Box \\hspace{-1.42262pt}$ For compound Poisson processes with equal jump intensity a related result was shown in [3] with a different proof.",
"By a modification of the Lévy measure and approximation arguments, the conditions are generalized more general Lévy processes, see [3] for details.",
"In fact, Theorem REF shows that the claims of [3] hold for all finite variation pure-jump Lévy processes without modifying the Lévy measures.",
"Now, we will relax the integrability assumptions.",
"To be precise, let $X$ and $Y$ be quasi-left continuous PIIs with characteristics $(B^X, C^X, \\nu ^X)$ and $(B^Y, C^Y, \\nu ^Y)$ such that $|x - h(x)| \\star \\nu ^\\bullet _t < \\infty $ for all $t \\in [0, \\infty )$ and suppose that $h$ is continuous.",
"Take a sequence $(G_n)_{n \\in \\mathbb {N}} \\subseteq \\mathbb {R}^d$ of Borel sets such that $\\mathbf {1}_{G_n}$ is vanishing in a neighborhood of the origin and $\\bigcup _{n \\in \\mathbb {N}} G_n \\supseteq \\mathbb {R}^d\\backslash \\lbrace 0\\rbrace $ .",
"Theorem 3.11 Suppose that for all $n \\in \\mathbb {N}$ and $t \\in [0, \\infty )$ it holds that $\\mathbf {1}_{G_n}\\cdot K^X \\preceq _{st} \\mathbf {1}_{G_n} \\cdot K^Y, C^X = C^Y$ and $h(x) \\mathbf {1}_{G_n}(x) \\star \\nu ^{Y}_t - h(x)\\mathbf {1}_{G_n}(x) \\star \\nu ^{X}_t \\le B^{Y}_t - B^{X}_t.$ Then $X \\preceq _{st} Y$ .",
"Suppose that for all $n \\in \\mathbb {N}$ and $t \\in [0, \\infty )$ it holds that $\\mathbf {1}_{G_n}\\cdot K^X \\preceq _{icx} \\mathbf {1}_{G_n}\\cdot K^Y, C^Y_t - C^X_t$ is non-negative definite and (REF ) is satisfied.",
"Then $X \\preceq _{icx} Y$ .",
"If, additionally, for all $t \\in [0, \\infty )$ $B^X_t + |x - h(x)| \\star \\nu ^X_t = B^Y_t + |x - h(x)| \\star \\nu ^Y_t,$ then $X \\preceq _{cx} Y$ .",
"Proof: We start with some general observations.",
"Define the truncated processes $Z^{\\bullet }(n) \\triangleq B^{\\bullet } &+ h \\mathbf {1}_{G_n} \\star (\\mu ^\\bullet - \\nu ^\\bullet )+ (x - h(x)) \\mathbf {1}_{G_n} \\star \\mu ^\\bullet .$ Lemma 3.12 $Z^{\\bullet }(n)$ is a PII with characteristics $(B^{\\bullet }, 0, \\mathbf {1}_{G_n}\\cdot \\nu ^{\\bullet })$ .",
"Moreover, if $Z^c$ is a Wiener process with covariance function $C$ which is independent of $Z^\\bullet (n)$ , then $Z^\\bullet (n) + Z^c$ is a PII with characteristics $(B^\\bullet , C, \\mathbf {1}_{G_n} \\cdot \\nu ^\\bullet )$ .",
"Proof: The first claim follows from [11] and the second claim follows from [11].",
"$\\Box \\hspace{-1.42262pt}$ We pose ourselves in the setting of (i).",
"Let $Z^c$ be as in the previous lemma with $C \\triangleq C^X = C^Y$ and set $\\widehat{Z}^\\bullet (n) \\triangleq Z^{\\bullet }(n) + Z^c$ .",
"By Lemma REF , it suffices to show that $X_t - X_s \\preceq _{st} Y_t - Y_s$ for all $0 \\le s < t < \\infty $ .",
"The same arguments as used in the proof of Theorem REF together with Proposition REF yields that $\\widehat{Z}^X(n)_t - \\widehat{Z}^X(n)_s \\preceq _{st} \\widehat{Z}^Y(n)_t - \\widehat{Z}^Y(n)_s$ for all $n \\in \\mathbb {N}$ .",
"It follows from [11] that $\\widehat{Z}^X(n)$ convergences in law to $X$ and $\\widehat{Z}^Y (n)$ converges in law to $Y$ as $n \\rightarrow \\infty $ .",
"Since the stochastic order $\\preceq _{st}$ for $\\mathbb {R}^d$ -valued random variables is closed under weak convergence, see [15], we conclude that $X_t - X_s \\preceq _{st} Y_t - Y_s$ .",
"This proves $X \\preceq _{st} Y$ by Lemma REF .",
"Next, we prove (ii).",
"First, we do not assume (REF ).",
"Let $Z^{\\bullet , c}$ be a Wiener process with covariance function $C^\\bullet $ independent of $Z^\\bullet (n)$ for all $n \\in \\mathbb {N}$ and set $\\widetilde{Z}^\\bullet (n) \\triangleq Z^\\bullet (n) + Z^{\\bullet , c}$ .",
"The same arguments as used in the proof of Theorem REF together with Proposition REF yields that $\\widetilde{Z}^X(n)_t - \\widetilde{Z}^X(n)_s \\preceq _{icx} \\widetilde{Z}^Y(n)_t - \\widetilde{Z}^Y(n)_s$ for all $n \\in \\mathbb {N}$ .",
"We note that [11] implies that $\\widetilde{Z}^X(n)$ convergences in law to $X$ and $\\widetilde{Z}^Y (n)$ converges in law to $Y$ as $n \\rightarrow \\infty $ .",
"For $U \\in \\lbrace X, Y\\rbrace $ the dominated convergence theorem yields that $E \\left[ Z^U (n)_t - Z^U (n)_s \\right] &= B^U_t - B^U_s + \\int (x - h(x)) \\mathbf {1}_{G_n} \\nu ^U((s, t] \\times \\operatorname{d}\\hspace{-1.6502pt}x)\\\\&\\xrightarrow{} B^U_t - B^U_s + \\int (x - h(x)) \\nu ^U ((s, t] \\times \\operatorname{d}\\hspace{-1.6502pt}x)\\\\&= E[U_t - U_s].$ Hence, we conclude from [15] that $X_t - X_s \\preceq _{icx} Y_t - Y_s$ and, therefore, $X \\preceq _{icx} Y$ by Lemma REF .",
"Finally, suppose that (REF ) holds.",
"In this case, $E[X_t - X_s] = E[Y_t - Y_s]$ and we deduce from [15] that $X_t - X_s \\preceq _{cx} Y_t - Y_s$ .",
"Again due to Lemma REF , this yields $X \\preceq _{cx} Y$ .",
"$\\Box \\hspace{-1.42262pt}$ For the stochastic order $\\preceq _{cx}$ it might be natural to assume that $\\mathbf {1}_{G_n}\\cdot K^X \\preceq _{cx} \\mathbf {1}_{G_n} \\cdot K^Y$ .",
"However, in this case $\\int _{G_n} x K^X(t, \\operatorname{d}\\hspace{-1.6502pt}x) = \\int _{G_n} x K^Y(t, \\operatorname{d}\\hspace{-1.6502pt}x)$ has to hold.",
"This seems to be a restrictive assumption on the sequence $(G_n)_{n \\in \\mathbb {N}}$ .",
"On the contrary, (REF ) is a necessary condition for $X \\preceq _{cx} Y$ such that we consider our conditions for $\\preceq _{cx}$ to be weaker."
],
[
"Explicit Conditions for Quasi-Left Continuous PIIs",
"In this section we give explicit conditions and we present alternative proofs via coupling arguments."
],
[
"A Majorization Condition for the Monotone Stochastic Order",
"We suppose that $d = 1$ .",
"Let $X$ and $Y$ be quasi-left continuous PIIs with characteristics $(B^X, C, \\nu ^X)$ and $(B^Y, C, \\nu ^Y)$ .",
"Our approach is based on a coupling constructed via the Itô map which relates Lévy measures to a reference Lévy measure.",
"The main result of this section is the following Theorem 4.1 Assume that $\\begin{split}K^Y (t, (- \\infty , x]) &\\le K^X(t, (- \\infty , x]),\\quad x < 0,\\\\K^X(t, [x, \\infty )) &\\le K^Y(t, [x, \\infty )),\\quad x > 0,\\end{split}$ for $\\operatorname{d}\\hspace{-1.6502pt}A_t$ -a.a. $t \\in [0, \\infty )$ .",
"Moreover, assume that for all $t \\in [0, \\infty )$ $|h(&x)| \\star \\nu ^{X}_t + |h(x)| \\star \\nu ^{Y}_t < \\infty ,\\\\h(x) &\\star \\nu ^{Y}_t - h(x) \\star \\nu ^{X}_t \\le B^{Y}_t - B^{X}_t.$ Then, $X \\preceq _{pst} Y$ .",
"The stochastic order $\\preceq _{pst}$ is stronger than the stochastic order $\\preceq _{st}$ .",
"In this regard, Theorem REF brings a new condition.",
"Proof: We show that there exists a probability space which supports copies of $X$ and $Y$ such that a.s. $X_t \\le Y_t$ for all $t \\in [0, \\infty )$ .",
"This clearly implies $X \\preceq _{pst}Y$ .",
"Let $F$ be the Lévy measure of a 1-stable Lévy process, i.e.",
"$F(\\operatorname{d}\\hspace{-1.6502pt}x) = \\frac{1}{|x|^{2}}\\operatorname{d}\\hspace{-1.6502pt}x$ , and set $\\rho ^\\bullet (t, x) \\triangleq {\\left\\lbrace \\begin{array}{ll}\\sup \\left( y \\in [0, \\infty )\\colon K^\\bullet (t, [y, \\infty )) \\ge \\frac{1}{|x|}\\right),&x > 0,\\\\0,&x = 0,\\\\- \\sup \\left( y \\in [0, \\infty )\\colon K^\\bullet (t, (- \\infty , -y]) \\ge \\frac{1}{|x|}\\right),&x < 0.\\end{array}\\right.",
"}$ Adapting the terminology in [19], the function $\\rho ^\\bullet $ is called Itô map.",
"Now, $K^\\bullet (t, G) = \\int \\mathbf {1}_G (\\rho ^\\bullet (t, x)) F(\\operatorname{d}\\hspace{-1.6502pt}x),\\quad G \\in {B}(\\mathbb {R}),$ see [19].",
"Set $\\nu ^L(\\operatorname{d}\\hspace{-1.6502pt}t \\times \\operatorname{d}\\hspace{-1.6502pt}x) \\triangleq \\operatorname{d}\\hspace{-1.6502pt}A_t F(\\operatorname{d}\\hspace{-1.6502pt}x)$ .",
"Lemma 4.2 There exists a PII $L$ with characteristics $(0, 0, \\nu ^L)$ .",
"Proof: This follows readily from [11].",
"$\\Box \\hspace{-1.42262pt}$ Let $Z^c$ be a Wiener process with variance function $C$ .",
"We set $Z^\\bullet \\triangleq B^{\\bullet } + Z^c + h(\\rho ^\\bullet ) \\star (\\mu ^L - \\nu ^L) + (\\rho ^\\bullet - h(\\rho ^\\bullet )) \\star \\mu ^L.$ Lemma 4.3 The process $Z^X$ is well-defined and has the same law as $X$ .",
"Moreover, the process $Z^Y$ is well-defined and has the same law as $Y$ .",
"Proof: To establish that $Z^\\bullet $ is well-defined, it suffices to verify that $\\mu ^{Z^{\\bullet }} ([0, t] \\times G) \\triangleq \\int _0^t \\int \\mathbf {1}_G (\\rho ^\\bullet (s, x)) \\mu ^L(\\operatorname{d}\\hspace{-1.6502pt}s \\times \\operatorname{d}\\hspace{-1.6502pt}x)$ , where $G \\in {B}(\\mathbb {R})$ , is a random measure of jumps with compensator $\\nu ^{\\bullet }$ .",
"Since $\\mu ^L$ is an optional random measure, so is $\\mu ^{Z^\\bullet }$ .",
"It remains to show that $\\mu ^{Z^\\bullet }$ is $\\widetilde{{P}}$ -$\\sigma $ -finite.",
"Consider the set $G_n \\triangleq \\lbrace x \\in \\mathbb {R} \\colon |x| > \\frac{1}{n}\\rbrace $ .",
"Now, $E[\\mu ^{Z^\\bullet }([0, t] \\times G_n)] = \\nu ^{\\bullet } ([0, t] \\times G_n) < \\infty $ , see [11].",
"Hence, $Z^\\bullet $ is well-defined.",
"Moreover, it follows readily from [11] that $Z^\\bullet $ is a PII with characteristics $(B^{\\bullet }, 0, \\nu ^{\\bullet })$ .",
"Therefore, the equality of the laws follows from [11].",
"$\\Box \\hspace{-1.42262pt}$ Using (REF ), we compute that for all $t \\in [0, \\infty )$ $Z^Y_t - Z^X_t &= B^{Y}_t - B^{X}_t - h(x) \\star \\nu ^{Y}_t + h(x) \\star \\nu ^{X}_t + \\left(\\rho ^Y - \\rho ^X\\right) \\star \\mu ^L.$ Our assumption (REF ) implies that $\\rho ^X\\le \\rho ^Y$ .",
"Hence, using () we obtain $Z^Y_t \\ge Z^X_t$ for all $t \\in [0, \\infty )$ .",
"This concludes the proof of Theorem REF .",
"$\\Box \\hspace{-1.42262pt}$ In view of [11], the finite variation condition (REF ) and the drift condition () are independent of the choice of $h$ .",
"Let us shortly comment on the assumptions of Theorem REF .",
"Proposition 4.4 If $|x| \\star \\nu ^\\bullet _t < \\infty $ for all $t \\in [0, \\infty )$ , then (REF ) holds for $\\operatorname{d}\\hspace{-1.6502pt}A_t$ -a.a. $t \\in [0, \\infty )$ if, and only if, the order $K^X\\preceq _{st} K^Y$ holds.",
"Proof: Denote $F, \\rho ^X$ and $\\rho ^Y$ as in the proof of Theorem REF , let $f \\in \\mathcal {F}^1_{st}$ such that $|f(x)| \\le \\textup {const. }",
"|x|$ and suppose that (REF ) holds.",
"Now, recalling (REF ), for $\\operatorname{d}\\hspace{-1.6502pt}A_t$ -a.a. $t \\in [0, \\infty )$ $\\int f(x) K^X(t, \\operatorname{d}\\hspace{-1.6502pt}x) &= \\int f(\\rho ^X(t, x)) F(\\operatorname{d}\\hspace{-1.6502pt}x)\\\\&\\le \\int f(\\rho ^Y(t, x)) F(\\operatorname{d}\\hspace{-1.6502pt}x) = \\int f(x) K^Y(t, \\operatorname{d}\\hspace{-1.6502pt}x).$ In other words, $K^X\\preceq _{st}K^Y$ holds.",
"For the converse direction, we approximate.",
"Namely, consider $f(y)\\triangleq - \\mathbf {1}_{(- \\infty , x]}(y)$ for some $x > 0$ and set $f_n$ to be the inf-convolution of $f$ , i.e.",
"$f_n (y) \\triangleq \\inf _{z \\in \\mathbb {R}} (f(z) + n |y - z|)$ .",
"Since $f$ is a bounded lower semi-continuous function, by [7] (see also the proof), the inf-convolution $f_n$ is Lipschitz continuous, $f_n(y) \\le f_{n+1}(y) \\le f(y)$ and $f_n \\rightarrow f$ pointwise as $n \\rightarrow \\infty $ .",
"Moreover, $f_n$ is increasing as $f$ is increasing (see (REF )) and $f_n(0) = \\inf _{z \\in \\mathbb {R}} (f(z) + n |z|) = 0$ for $n\\ge |x|^{-1}$ .",
"Thus, using the monotone convergence theorem, $K^X \\preceq _{st} K^Y$ implies the first part of (REF ).",
"The second part follows in the same manner.",
"$\\Box \\hspace{-1.42262pt}$ So far our conditions for the stochastic order $\\preceq _{pst}$ apply to PIIs with discontinuous parts of finite variation.",
"Similarly to Theorem REF , we can relax this assumption using the fact that $\\preceq _{pst}$ is closed under weak convergence.",
"Take a sequence $(G_n)_{n \\in \\mathbb {N}} \\subseteq \\mathbb {R}$ of Borel sets such that $\\mathbf {1}_{G_n}$ is vanishing in a neighborhood of the origin and $\\bigcup _{n \\in \\mathbb {N}} G_n \\supseteq \\mathbb {R}\\backslash \\lbrace 0\\rbrace $ .",
"Theorem 4.5 Let $h$ be continuous and suppose that for all $n \\in \\mathbb {N}$ and $\\operatorname{d}\\hspace{-1.6502pt}A_t$ -a.a. $t \\in [0, \\infty )$ the condition (REF ) holds with $K^X$ replaced by $\\mathbf {1}_{G_n} \\cdot K^X$ and $K^Y$ replaced by $\\mathbf {1}_{G_n} \\cdot K^Y$ , and that for all $n \\in \\mathbb {N}$ and $t \\in [0, \\infty )$ the inequality (REF ) holds.",
"Then $X \\preceq _{pst} Y$ .",
"Proof: Let $\\widehat{Z}^\\bullet (n) \\triangleq Z^\\bullet (n) + Z^c$ , where $Z^\\bullet (n)$ and $Z^c$ are as in Lemma REF .",
"We deduce from [11] that $\\widehat{Z}^{X}(n)$ convergences in law to $X$ and $\\widehat{Z}^{Y}(n)$ converges in law to $Y^{qlc}$ as $n \\rightarrow \\infty $ .",
"Now, Theorem REF yields $\\widehat{Z}^{X}(n) \\preceq _{pst} \\widehat{Z}^{Y}(n)$ and by [14] we conclude that $X \\preceq _{pst} Y$ .",
"$\\Box \\hspace{-1.42262pt}$"
],
[
"Cut Criteria for the Monotone Stochastic Order",
"In this section we study the case where the frequencies of jumps change once.",
"Together with some integrability conditions, this implies the stochastic order $\\preceq _{pst}$ .",
"We start with a technical observation: Lemma 4.6 There exists a $\\sigma $ -finite measure $\\nu $ on $([0, \\infty ) \\times \\mathbb {R}, {B}([0, \\infty )) \\otimes {B}(\\mathbb {R}))$ such that $\\nu ^{X}\\ll \\nu $ and $\\nu ^{Y} \\ll \\nu $ .",
"Proof: Since $\\nu ^{X}$ and $\\nu ^{Y}$ are $\\sigma $ -finite, by the Radon-Nikodym theorem, $\\nu \\triangleq \\nu ^{X} + \\nu ^{Y}$ has the desired properties.",
"$\\Box \\hspace{-1.42262pt}$ Let $|$ be either $[$ or $]$ and let $|^c$ be the converse, i.e.",
"if $| =\\ ]$ , then $|^c = [$ .",
"Proposition 4.7 Let $k \\in \\mathbb {R}$ and suppose that $\\nu $ -a.e.",
"$\\begin{split}\\frac{\\operatorname{d}\\hspace{-1.6502pt}\\nu ^{X}}{\\operatorname{d}\\hspace{-1.6502pt}\\nu }\\mathbf {1}_{[0, \\infty ) \\times |k, \\infty )} &\\le \\frac{\\operatorname{d}\\hspace{-1.6502pt}\\nu ^{Y}}{\\operatorname{d}\\hspace{-1.6502pt}\\nu }\\mathbf {1}_{[0, \\infty ) \\times |k, \\infty )},\\\\\\ \\frac{\\operatorname{d}\\hspace{-1.6502pt}\\nu ^{X}}{\\operatorname{d}\\hspace{-1.6502pt}\\nu }\\mathbf {1}_{[0, \\infty ) \\times (- \\infty , k|^c}&\\ge \\frac{\\operatorname{d}\\hspace{-1.6502pt}\\nu ^{Y}}{\\operatorname{d}\\hspace{-1.6502pt}\\nu } \\mathbf {1}_{[0, \\infty ) \\times (- \\infty , k|^c}.\\end{split}$ Moreover, assume for $\\operatorname{d}\\hspace{-1.6502pt}A_t$ -a.a. $t \\in [0, \\infty )$ $\\begin{split}\\int _{|k, 0]} \\left(K^Y(t, \\operatorname{d}\\hspace{-1.6502pt}x) - K^X(t, \\operatorname{d}\\hspace{-1.6502pt}x)\\right) \\le &\\int _{(- \\infty , k|^c}\\left(K^X(t, \\operatorname{d}\\hspace{-1.6502pt}x) - K^Y(t, \\operatorname{d}\\hspace{-1.6502pt}x)\\right)\\end{split}$ in the case $k < 0$ , and $\\begin{split}\\int _{[0, k|^c} \\left(K^X(t, \\operatorname{d}\\hspace{-1.6502pt}x) - K^Y(t, \\operatorname{d}\\hspace{-1.6502pt}x)\\right) \\le &\\int _{| k, \\infty )}\\left(K^Y(t, \\operatorname{d}\\hspace{-1.6502pt}x) - K^X(t, \\operatorname{d}\\hspace{-1.6502pt}x)\\right)\\end{split}$ in the case $k > 0$ .",
"Finally, suppose that for all $t \\in [0, \\infty )$ $\\left|h(x)\\left(\\frac{\\operatorname{d}\\hspace{-1.6502pt}\\nu ^{Y}}{\\operatorname{d}\\hspace{-1.6502pt}\\nu } - \\frac{\\operatorname{d}\\hspace{-1.6502pt}\\nu ^{X}}{\\operatorname{d}\\hspace{-1.6502pt}\\nu }\\right)\\right| \\star \\nu _t < \\infty ,$ $B^{Y}_t- B^{X}_t - h(x) \\left(\\frac{\\operatorname{d}\\hspace{-1.6502pt}\\nu ^{Y}}{\\operatorname{d}\\hspace{-1.6502pt}\\nu } - \\frac{\\operatorname{d}\\hspace{-1.6502pt}\\nu ^{X}}{\\operatorname{d}\\hspace{-1.6502pt}\\nu }\\right)\\star \\nu _t \\ge 0.$ Then $X \\preceq _{pst} Y$ .",
"We stress that the r.h.s.",
"of (REF ) and (REF ) are finite due to [11].",
"Remark 4.8 Suppose that $X$ and $Y$ are semimartingales with independent increments and that their laws are locally absolutely continuous.",
"Then, by Girsanov's theorem [11], we find $\\nu $ such that (REF ) holds and (REF ) only depends on the Gaussian parts of $X$ and $Y$ .",
"We provide the intuitions behind the assumptions of Theorem REF .",
"The condition (REF ) means that $Y$ has a higher frequency of jumps with size larger than $k$ compared to $X$ and that $X$ has a higher frequency of jumps with size less than $k$ compared to $Y$ .",
"The conditions (REF ) and (REF ) compensate negative jumps which are done by $Y$ but not by $X$ and positive jumps which are done by $X$ but not by $Y$ .",
"Following this intuition, we can construct explicit couplings of $X$ and $Y$ which are pathwise ordered.",
"We reveal that Proposition REF is (modulo integrability issues) a consequence of Theorem REF .",
"However, we think the alternative proof explains very nicely the origin of the conditions and illustrates the relations of the characteristics of the PIIs and the stochastic order.",
"Proof: We only discuss the cases $k = 0$ and $k > 0$ , since the case $k < 0$ follows similarly to the case $k > 0$ .",
"The case $k = 0$ .",
"We set $\\nu ^{X} \\wedge \\nu ^{Y}\\triangleq \\mathbf {1}_{[0, \\infty )}\\cdot \\nu ^{X}+ \\mathbf {1}_{(- \\infty , 0]}\\cdot \\nu ^{Y}.$ Due to [11] we find a filtered probability space which supports a PII $Z$ with characteristics $(0, C,\\nu ^{X} \\wedge \\nu ^{Y})$ , a PII $Z^X$ with characteristics $(B^{X}, 0, \\mathbf {1}_{(- \\infty , 0]} \\cdot (\\nu ^{X}- \\nu ^{Y}))$ and a PII $Z^Y$ with characteristics $(B^{Y}, 0, \\mathbf {1}_{[0, \\infty )}\\cdot (\\nu ^{Y} -\\nu ^{X}))$ such that $Z, Z^X$ and $Z^Y$ are independent.",
"Then, by independence and [11], $Z + Z^X$ has the same law as $X$ and $Z + Z^Y$ has the same law as $Y$ .",
"Moreover, a.s. $x\\mathbf {1}_{[0, \\infty )} \\star \\mu ^{Z^X}_t = 0$ and $x\\mathbf {1}_{(-\\infty , 0]} \\star \\mu ^{Z^Y}_t = 0$ for all $t \\in [0, \\infty )$ , noting the support of the third characteristics.",
"Thanks to [11] and (REF ), we obtain for all $t \\in [0, \\infty )$ $Z^Y_t - Z^X_t= B^{Y}_t - B^{X}_t &- h(x) \\left(\\frac{\\operatorname{d}\\hspace{-1.6502pt}\\nu ^{Y}}{\\operatorname{d}\\hspace{-1.6502pt}\\nu } - \\frac{\\operatorname{d}\\hspace{-1.6502pt}\\nu ^{X}}{\\operatorname{d}\\hspace{-1.6502pt}\\nu }\\right) \\star \\nu _t\\\\&+ x \\mathbf {1}_{[0, \\infty )} \\star \\mu ^{Z^Y}_t - x\\mathbf {1}_{(- \\infty , 0]} \\star \\mu ^{Z^X}_t.$ Hence, using (REF ), $Z_t + Z^X_t \\le Z_t + Z^Y_t$ for all $t \\in [0, \\infty )$ .",
"This proves $X \\preceq _{pst} Y$ .",
"The case $k > 0$ .",
"We define a Borel measure $\\nu ^+$ on $[0, \\infty ) \\times \\mathbb {R} \\times \\mathbb {R} \\times \\mathbb {R}$ by $&\\frac{\\nu ^+ (\\operatorname{d}\\hspace{-1.6502pt}t \\times \\operatorname{d}\\hspace{-1.6502pt}x \\times \\operatorname{d}\\hspace{-1.6502pt}y \\times \\operatorname{d}\\hspace{-1.6502pt}u)}{\\left[ K^X(t, \\operatorname{d}\\hspace{-1.6502pt}x) - K^Y(t, \\operatorname{d}\\hspace{-1.6502pt}x)\\right] \\left[K^Y(t, \\operatorname{d}\\hspace{-1.6502pt}y) - K^X(t, \\operatorname{d}\\hspace{-1.6502pt}y)\\right]\\operatorname{d}\\hspace{-1.6502pt}u \\operatorname{d}\\hspace{-1.6502pt}A_t}\\\\&\\hspace{142.26378pt} \\triangleq \\frac{\\mathbf {1}_{[0, k|^c}(x) \\mathbf {1}_{|k, \\infty )} (y) \\mathbf {1}_{[0, 1]}(u) }{\\int _{[0, k|^c} (K^X(t, \\operatorname{d}\\hspace{-1.6502pt}z) - K^Y(t, \\operatorname{d}\\hspace{-1.6502pt}z))},$ with the convention that $\\frac{1}{0} = 1$ .",
"Here, we use (REF ), i.e.",
"that the denominator on the r.h.s.",
"is bounded from above for $\\operatorname{d}\\hspace{-1.6502pt}A_t$ -a.a. $t \\in [0, \\infty )$ , see [11].",
"Lemma 4.9 There exists a PII $Z^+$ with characteristics $(0, 0, \\nu ^+)$ .",
"Proof: In view of [11] it suffices to show that $\\nu ^+$ is finite on $[0, t] \\times \\mathbb {R}^3$ for all $t \\in [0, \\infty )$ .",
"It holds that $\\nu ^+([0, t] \\times \\mathbb {R}^3) = (\\nu ^{Y} - \\nu ^{X}) ([0, t] \\times |k, \\infty )) < \\infty ,$ see [11].",
"Hence, the lemma is proven.",
"$\\Box \\hspace{-1.42262pt}$ We find a filtered probability space which supports a PII $Z$ with characteristics $(0, C, \\nu ^X \\wedge \\nu ^Y)$ , where $\\nu ^X \\wedge \\nu ^Y \\triangleq \\mathbf {1}_{|k, \\infty )} \\cdot \\nu ^X + \\mathbf {1}_{(- \\infty , k|^c} \\cdot \\nu ^Y$ , a PII $Z^X$ with characteristics $(B^X, 0, \\mathbf {1}_{(- \\infty , 0]} \\cdot (\\nu ^X - \\nu ^Y))$ and the PII $Z^+$ such that they are independent.",
"We define $m(t, u) \\triangleq \\mathbf {1} \\left\\lbrace u \\le \\frac{\\int _{[0, k|^c} (K^X(t, \\operatorname{d}\\hspace{-1.6502pt}z) - K^Y(t, \\operatorname{d}\\hspace{-1.6502pt}z))}{\\int _{|k, \\infty )} (K^Y(t, \\operatorname{d}\\hspace{-1.6502pt}z) - K^X(t, \\operatorname{d}\\hspace{-1.6502pt}z))}\\right\\rbrace .$ Now, set $Z^{X, -} &\\triangleq x m(t, u) \\mathbf {1}_{[0, k|^c}(x) \\mathbf {1}_{|k, \\infty )}(y)\\star \\mu ^{Z^+}- h \\mathbf {1}_{[0, k|^c} \\star (\\nu ^{X} - \\nu ^{Y}),\\\\Z^{Y} &\\triangleq y \\mathbf {1}_{[0, k|^c}(x) \\mathbf {1}_{|k, \\infty )}(y)\\star \\mu ^{Z^+} + B^{Y}- h \\mathbf {1}_{|k, \\infty )} \\star (\\nu ^{Y} - \\nu ^{X}).$ By assumption (REF ), the processes $Z^{X, -}$ and $Z^Y$ are well-defined.",
"Lemma 4.10 The process $Z^{X, -}$ is a PII with characteristics $(0, 0, \\mathbf {1}_{[0, k|^c} \\cdot (\\nu ^{X}- \\nu ^{Y}))$ and $Z^Y$ is a PII with characteristics $(B^{Y}, 0, \\mathbf {1}_{|k, \\infty )} \\cdot (\\nu ^{Y} - \\nu ^{X}))$ .",
"Proof: Let ${C}^+$ be a set of test functions as defined in [11].",
"All functions in ${C}^+$ are bounded and vanish in a neighborhood of the origin.",
"In view of [11] for the first claim suffices to show that for all $g \\in {C}^+$ the process $g \\star \\mu ^{Z^{X, -}} - g \\mathbf {1}_{[0, k|^c} \\star (\\nu ^{X}- \\nu ^{Y})$ is a local martingale.",
"In fact, since $g$ vanishes in a neighborhood of the origin, we have $g \\star \\mu ^{Z^{X, -}} = g (x) m(t, u) \\mathbf {1}_{[0, k|^c} (x) \\mathbf {1}_{|k, \\infty )}(y) \\star \\mu ^{Z^+},$ which compensator is given by $g (x) m(t, u) \\mathbf {1}_{[0, k|^c} (x) \\mathbf {1}_{|k, \\infty )}(y) \\star \\nu ^{+}= g \\mathbf {1}_{[0, k|^c} \\star \\left(\\nu ^X - \\nu ^Y\\right).$ Here, we use (REF ).",
"This proves the first claim.",
"The second claim follows in the same manner.",
"$\\Box \\hspace{-1.42262pt}$ Now, by independence and [11], $Z + Z^X + Z^{X,-}$ has the same law as $X$ and $Z + Z^Y$ has the same law as $Y$ .",
"Moreover, for all $t \\in [0, \\infty )$ $Z^Y_t - Z^X_t - Z^{X, -}_t = B^{Y}_t &- B^{X}_t - h(x) \\left(\\frac{\\operatorname{d}\\hspace{-1.6502pt}\\nu ^{Y}}{\\operatorname{d}\\hspace{-1.6502pt}\\nu } - \\frac{\\operatorname{d}\\hspace{-1.6502pt}\\nu ^{X}}{\\operatorname{d}\\hspace{-1.6502pt}\\nu }\\right) \\star \\nu _t + x \\mathbf {1}_{[0, \\infty )} \\star \\mu ^{Z^Y}_t\\\\&+ (y - x m(t, u))\\mathbf {1}_{[0, k|^c}(x)\\mathbf {1}_{|k, \\infty )}(y) \\star \\mu ^{Z^+}_t.$ Since $m(t, u) \\le 1$ , we obtain $Z^Y_t - Z^X_t - Z^{X, -}_t \\ge 0$ for all $t \\in [0, \\infty )$ .",
"Again, this proves $X \\preceq _{pst} Y$ .",
"$\\Box \\hspace{-1.42262pt}$"
],
[
"A Majorization Condition for the Convex Stochastic Order",
"In this section we show that if the third characteristic of $Y$ dominates the third characteristics of $X$ , then $X \\preceq _{(i)cx} Y$ .",
"Our proof is based on a coupling.",
"We stress that the proof is very robust in sense that it applies to all PIIs with finite first moments.",
"Theorem 4.11 Suppose that for all $t \\in [0, \\infty )$ the matrix $C^Y_t - C^X_t$ is non-negative definite, $\\nu ^{Y}- \\nu ^{X}$ is a non-negative measure, $|x - h(x)| \\star \\nu ^\\bullet _t < \\infty $ and $B^X_t + |x - h(x)| \\star \\nu ^X_t \\le B^Y_t + |x - h(x)| \\star \\nu ^Y_t,$ Then $X \\preceq _{icx} Y$ .",
"If, additionally, (REF ) holds with equality, then also $X \\preceq _{cx} Y$ .",
"Theorem REF generalizes the majorization criterion [4] for Lévy processes with infinite activity by showing that the conditions [4] are not necessary.",
"Proof: In view of [11], we can extend our probability space such that it supports a PII $Z^{Y}$ with characteristics $(B^{Y} - B^{X}, C^Y - C^X, \\nu ^Y - \\nu ^X)$ , which is independent of $X$ .",
"Now, by independence and [11], the process $X + Z^{Y}$ has the same law as $Y$ .",
"Note that $E \\left[Z^{Y}_t\\right] = B^{Y}_t - B^{X}_t + (x - h(x)) \\star (\\nu ^{Y} -\\nu ^{X})_t$ for all $t \\in [0, \\infty )$ .",
"Hence, we obtain a.s. for all $t \\in [0, \\infty )$ $E \\left[X_t + Z^Y_t |\\sigma (X_s, s \\in [0, \\infty ))\\right] &= X_t + E \\left[Z^Y_t\\right]\\\\&{\\left\\lbrace \\begin{array}{ll}\\ge X_t,&\\textup { if (\\ref {eq: ineq moment}) holds}, \\\\= X_t,&\\textup { if (\\ref {eq: ineq moment}) holds with equality}.\\end{array}\\right.",
"}$ Now, $X \\preceq _{(i)cx} Y$ follows from the conditional Jensen's inequality.",
"$\\Box \\hspace{-1.42262pt}$"
],
[
"Generalizations and Examples",
"In this section we discuss generalizations of our conditions to semimartingales with conditionally independent increments (called ${H}$ -SIIs in the following).",
"Moreover, we give examples."
],
[
"Comparison of ${H}$ -SIIs",
"Let $(\\Omega , {F})$ be a Polish space equipped with its topological Borel $\\sigma $ -field and let $P$ be a probability measure on $(\\Omega , {F})$ .",
"We choose two not necessarily right-continuous filtrations $({F}^1_t)_{t \\in [0, \\infty )}$ and $({F}^2_t)_{t \\in [0, \\infty )}$ on $(\\Omega , {F})$ consisting of countably generated $\\sigma $ -fields.",
"Moreover, let ${H}^1$ and ${H}^2$ be two countably generated sub-$\\sigma $ -fields of ${F}$ .",
"For example, ${H}^i$ could be $\\sigma (Z_s, s \\in [0, \\infty ))$ where $Z$ is right- or left-continuous and takes values in a Polish space.",
"For $i =1, 2$ we define the filtrations $\\mathsf {G}^i = ({G}^i_t)_{t \\in [0, \\infty )}$ on $(\\Omega , {F})$ by ${G}^i_t \\triangleq {G}^{i, o}_{t+},$ where ${G}^{i, o}_t \\triangleq {F}^i_t \\vee {H}^i.",
"$ A process $B \\in {V}^d$ (or $\\in {V}^{d \\times d}$ ) is said to have an ${H}^i$ -measurable version, if for all $t \\in [0, \\infty )$ the random variable $B_t$ has an ${H}^i$ -measurable version.",
"We say that a compensator $\\nu $ of a random measure of jumps has an ${H}^i$ -measurable version, if for all $t \\in [0, \\infty )$ and all Borel functions $g \\colon \\mathbb {R}^d \\rightarrow \\mathbb {R}$ such that $|g(x)| \\le 1 \\wedge |x|^2$ the random variable $g \\star \\nu _t$ has an ${H}^i$ -measurable version.",
"Let $X$ be an $\\mathbb {R}^d$ -valued $\\mathsf {G}^1$ -semimartingale with characteristics $(B^X, C^X, \\nu ^X)$ , and $Y$ be an $\\mathbb {R}^d$ -valued $\\mathsf {G}^2$ -semimartingale with characteristics $(B^Y, C^Y, \\nu ^Y)$ , such that $(B^X, C^X, \\nu ^X)$ has an ${H}^1$ -measurable version and that $(B^Y, C^Y, \\nu ^Y)$ has an ${H}^2$ -measurable version.",
"In this setting there exists a regular conditional probability $P(\\cdot |{H}^i)(\\cdot )$ from $(\\Omega , {H}^i)$ to $(\\Omega , {F})$ , see, e.g., [19].",
"The following is the main observation to transfer our conditions for PIIs to ${H}$ -SIIs.",
"Lemma 5.1 There exists a null set $N \\in {F}$ such that for all $\\omega \\in \\complement N$ the process $X$ is a $P(\\cdot |{H}^1)(\\omega )$ -PII with characteristics $(B^X(\\omega ), C^X(\\omega ), \\nu ^X(\\omega ))$ and $Y$ is a $P(\\cdot |{H}^2)(\\omega )$ -PII with characteristics $(B^Y(\\omega ),$ $C^Y(\\omega ), \\nu ^Y(\\omega ))$ .",
"Proof: The claim follows from [11] and [6].",
"$\\Box \\hspace{-1.42262pt}$ In view of this lemma it is strait forward to transfer our conditions for PIIs to ${H}$ -SIIs.",
"More precisely, assume that for all $\\omega \\in \\complement N$ , where $N$ is as in the previous lemma, the characteristics $(B^X(\\omega ), C^X(\\omega ), \\nu ^X(\\omega ))$ and $(B^Y(\\omega ), C^Y(\\omega ), \\nu ^Y(\\omega ))$ satisfy our conditions for $\\preceq _{\\bullet }$ .",
"Then, with abuse of notation, for all $f \\in \\mathcal {F}_{\\bullet }$ $\\int f(X(\\omega ^*)) P(\\operatorname{d}\\hspace{-1.6502pt}\\omega ^*|{H}^1) (\\omega ) \\le \\int f(Y(\\omega ^*)) P(\\operatorname{d}\\hspace{-1.6502pt}\\omega ^*|{H}^2)(\\omega ).$ Since $N$ is a null set, taking expectation yields $X \\preceq _\\bullet Y$ .",
"We omit to state explicit conditions as these are similar to our previous conditions with an additional almost surely.",
"However, we give examples in the following section."
],
[
"Examples",
"In this section we provide examples.",
"Here, we focus on processes which are not discussed in [3], [4].",
"We start with an example of a PII with infinite variation and fixed times of discontinuity.",
"Example 5.2 (Time-inhomogeneous CGMY Lévy processes with fixed times of discontinuity) For $n \\in \\mathbb {N}$ fix $0 < t_1 < t_2 < ... < t_n < \\infty $ .",
"Moreover, for $k =1 , 2$ and $i = 1, ..., n$ let $F^{k, i}$ be probability measures on $(\\mathbb {R}, {B}(\\mathbb {R}))$ such that $F^{k, i}(\\lbrace 0\\rbrace ) = 0$ and for all $x \\in \\mathbb {R}$ we have $F^{2, i} ((- \\infty , x]) \\le F^{1, i} ((-\\infty , x]).$ Let $X^1$ and $X^2$ be real-valued PIIs such that $C^{X^1} = C^{X^2} = 0$ and $\\nu ^{X^1}(\\operatorname{d}\\hspace{-1.6502pt}t \\times \\operatorname{d}\\hspace{-1.6502pt}x) &= \\frac{C_t}{|x|^{1+ Y_t}} e^{- M^1_t x} \\mathbf {1}_{\\lbrace x > 0\\rbrace } \\operatorname{d}\\hspace{-1.6502pt}t \\operatorname{d}\\hspace{-1.6502pt}x \\\\&\\hspace{28.45274pt}+ \\frac{C_t}{|x|^{1 + Y_t}} e^{G^1_t x} \\mathbf {1}_{\\lbrace x < 0\\rbrace } \\operatorname{d}\\hspace{-1.6502pt}t \\operatorname{d}\\hspace{-1.6502pt}x + \\sum _{i = 1}^n \\varepsilon _{t_i}(\\operatorname{d}\\hspace{-1.6502pt}t) F^{1, i}(\\operatorname{d}\\hspace{-1.6502pt}x),\\\\\\nu ^{X^2}(\\operatorname{d}\\hspace{-1.6502pt}t \\times \\operatorname{d}\\hspace{-1.6502pt}x) &= \\frac{C_t}{|x|^{1 + Y_t}} e^{- M^2_t x} \\mathbf {1}_{\\lbrace x > 0\\rbrace } \\operatorname{d}\\hspace{-1.6502pt}t\\operatorname{d}\\hspace{-1.6502pt}x \\\\&\\hspace{28.45274pt}+ \\frac{C_t}{|x|^{1 + Y_t}} e^{G^2_t x}\\mathbf {1}_{\\lbrace x < 0\\rbrace } \\operatorname{d}\\hspace{-1.6502pt}t\\operatorname{d}\\hspace{-1.6502pt}x + \\sum _{i = 1}^n \\varepsilon _{t_i}(\\operatorname{d}\\hspace{-1.6502pt}t) F^{2, i}(\\operatorname{d}\\hspace{-1.6502pt}x).$ Here, $C, G^1, G^2, M^1, M^2 \\colon [0, \\infty ) \\rightarrow (0, \\infty )$ are continuous and bounded away from 0 on compact sets, and $Y \\colon [0, \\infty ) \\rightarrow (- \\infty , 2)$ is continuous and bounded away from 2 on compact sets.",
"If $\\nu = \\nu ^X + \\nu ^Y$ , then the integrability condition (REF ) holds.",
"Moreover, (REF ) is satisfied if $M^2_t \\le M^1_t$ and $G^1_t \\le G^2_t$ for all $t \\in [0, \\infty )$ except of a $\\operatorname{d}\\hspace{-1.6502pt}t$ -null set.",
"If we choose $B^{X^1}$ and $B^{X^2}$ according to (REF ), the Propositions REF and REF imply $X^1\\preceq _{pst} X^2$ .",
"Next, we compare ${H}$ -SIIs, using the notation from the previous section.",
"Example 5.3 (Comparison of Time-Changed Lévy Processes) For $i = 1, 2$ , let $Z^i$ be a $[0, \\infty )$ -valued càdlàg processes and let $V^i$ be a real-valued Lévy process (w.r.t.",
"its natural filtration) with Lévy-Khinchine triplet $(b^{V^i}, 0, F^{V^i})$ .",
"Assume that the process $Z^i$ is independent of $V^i$ and set $X^i_\\cdot \\triangleq V^i_{\\int _0^\\cdot Z^i_s \\operatorname{d}\\hspace{-1.6502pt}s},$ ${F}^i_t \\triangleq \\sigma (X^i_s, Z^i_s, s \\in [0, t])$ and ${H}^i \\triangleq \\sigma (Z^i_t, t \\in [0, \\infty )).$ The next lemma follows from [13].",
"Lemma 5.4 For $i = 1,2$ the process $X^i$ is a $\\mathsf {G}^i$ -semimartingale and its $\\mathsf {G}^i$ -semimartingale characteristics are given by $B^{X^i} = b^{V^i}\\int _0^\\cdot Z^i_s \\operatorname{d}\\hspace{-1.6502pt}s,\\quad C^{X^i} = 0,\\quad \\nu ^{X^i}(\\operatorname{d}\\hspace{-1.6502pt}t \\times \\operatorname{d}\\hspace{-1.6502pt}x) = Z^i_{t-} \\operatorname{d}\\hspace{-1.6502pt}t F^{V^i}(\\operatorname{d}\\hspace{-1.6502pt}x).$ Using Lemma REF and the Theorems REF and REF , we obtain the following Corollary 5.5 Suppose that $\\int |h(x)| F^{V^i}(\\operatorname{d}\\hspace{-1.6502pt}x) < \\infty $ , that $F^{V^2}((- \\infty , x]) &\\le F^{V^1}((- \\infty , x])\\ x < 0,\\\\ F^{V^1}([x, \\infty )) &\\le F^{V^2}([x, \\infty )),\\ x > 0,$ and that a.s. for all $t \\in [0, \\infty )$ $\\hspace{-28.45274pt}\\left(b^{V^1} - \\int h(x) F^{V^1}(\\operatorname{d}\\hspace{-1.6502pt}x)\\right) \\int _0^t Z^1_s \\operatorname{d}\\hspace{-1.6502pt}s \\le \\left(b^{V^2} - \\int h(x) F^{V^2}(\\operatorname{d}\\hspace{-1.6502pt}x)\\right) \\int _0^t Z^2_s \\operatorname{d}\\hspace{-1.6502pt}s.$ Then $X \\preceq _{pst} Y$ .",
"If a.s. $Z^1_t \\le Z^2_t$ for all $t \\in [0, \\infty )$ , $F^{V^2} - F^{V^1}$ is a non-negative measure, $\\int |x - h(x)| F^{V^i} (\\operatorname{d}\\hspace{-1.6502pt}x) < \\infty ,$ and $b^{V^1} \\le \\int (x- h(x)) F^{V^1}(\\operatorname{d}\\hspace{-1.6502pt}x),\\quad b^{V^2} \\ge \\int (x - h(x)) F^{V^2}(\\operatorname{d}\\hspace{-1.6502pt}x),$ then $X^1 \\preceq _{icx} X^2$ .",
"If, additionally, the inequalities in (REF ) are equalities, then $X^1 \\preceq _{cx} X^2$ .",
"Example 5.6 (Comparison of Integrated Lévy Processes) For $i = 1, 2$ , let $B^i$ be a $d$ -dimensional Brownian motion (w.r.t.",
"its natural filtration), and let $\\mu ^i$ be a left-continuous $\\mathbb {R}^d$ -valued process and $\\sigma ^i$ be a left-continuous $\\mathbb {R}^d \\otimes \\mathbb {R}^d$ -valued process, both of which are independent of $B^i$ .",
"Moreover, we set ${F}^i_t \\triangleq \\sigma (B^i_s, \\mu ^i_s, \\sigma ^i_s, s \\in [0, t])$ and ${H}^i \\triangleq \\sigma (\\mu ^i_t, \\sigma ^i_t, t \\in [0, \\infty )).$ Note that, since $B^i$ is independent of both $\\mu ^i$ and $\\sigma ^i$ , it follows from [16] and [1] that $B^i$ is an $({F}_{t+}^i)_{t \\in [0, \\infty )}$ -Brownian motion.",
"Thus, the process $X^i \\triangleq \\int _0^\\cdot \\mu ^i_s \\operatorname{d}\\hspace{-1.6502pt}s + \\int _0^\\cdot \\sigma ^i_s \\operatorname{d}\\hspace{-1.6502pt}B^i_s$ is well-defined as an $({F}^i_{t+})_{t \\in [0, \\infty )}$ -semimartingale.",
"The following lemma follows from [13].",
"Lemma 5.7 For $i = 1, 2$ the process $X^i$ is a $\\mathsf {G}^i$ -semimartingale and its $\\mathsf {G}^i$ -semimartingale characteristics are given by $B^{X^i} = \\int _0^\\cdot \\mu ^i_s \\operatorname{d}\\hspace{-1.6502pt}s,\\quad C^{X^i} = \\int _0^\\cdot \\sigma ^i_s (\\sigma ^i_s)^* \\operatorname{d}\\hspace{-1.6502pt}s, \\quad \\nu ^{X^i} = 0,$ where $(\\sigma ^i)^*$ denotes the adjoint of $\\sigma ^i$ .",
"Obviously, the $\\mathsf {G}^i$ -semimartingale characteristics of $X^i$ are ${H}^i$ -measurable.",
"We deduce the following comparison result from Lemma REF and Theorem REF .",
"Corollary 5.8 If a.s. $\\mu ^1_t \\le \\mu ^2_t$ and $\\sigma ^2_t (\\sigma ^2_t)^* - \\sigma ^1_t (\\sigma ^1_t)^*$ is a non-negative definite matrix for all $t \\in [0, \\infty )$ , then $X \\preceq _{icx} Y$ .",
"If, additionally, a.s. $\\mu ^1_t = \\mu ^2_t$ for all $t \\in [0, \\infty )$ , then $X \\preceq _{cx} Y$ .",
"Remark 5.9 Results in the spirit of Corollary REF were given by Hajek [8] using a coupling technique different from ours.",
"More precisely, in the one-dimensional case, Hajek shows the first part of Corollary REF in the case where $\\mu ^2$ and $\\sigma ^2$ are constant and $\\mu ^1$ and $\\sigma ^1$ are not necessarily independent of $B^1$ ."
]
] | 1606.04993 |
[
[
"A new asymmetric generalisation of the t-distribution"
],
[
"Abstract A 6-parameter fat-tailed distribution is proposed that generalises the t-distribution and allows asymmetry of scale and also of tail power, whilst avoiding the discontinuity of the second derivative of the split-t (AST) distribution.",
"With the sixth parameter set to unity and no asymmetry, the distribution reduces to a t-distribution, but with the sixth parameter reduced, fatter tails than those of the t-distribution are allowed (the tails start earlier) and the distribution generalises Johnson's $S_U$ distribution.",
"Data fitting is illustrated with examples."
],
[
"Keywords",
"skew-t distribution; arcsinh transformation; AST distribution."
],
[
"Introduction",
"There is a widespread problem of modelling and doing inference when the distribution of the variable of interest has long tails and is skewed.",
"This occurs commonly in finance, e.g.",
"for returns, but also in the life sciences, telecommunications, etc.",
"The t-distribution has polynomial tails, and has been widely used in modelling fat-tailed data.",
"The problem then is to generalise the t-distribution so that it can exhibit skewness.",
"After a brief account of previous attempts to do this, a new distribution that does so is described in the next section.",
"The asymmetric fat-tailed distribution of returns etc in finance has been modelled by the generalised asymmetric t distribution (AST)(e.g.",
"Zhu and Galbraith, 2010, 2011) The pdf is $f(x) \\propto \\lbrace 1+((x-\\mu )/c\\phi )^2/(\\nu /r)\\rbrace ^{-(\\nu /r+1)/2}$ for $X < \\mu $ and $f(x) \\propto \\lbrace 1+(c(x-\\mu )/\\phi )^2/(\\nu r)\\rbrace ^{-(\\nu r+1)/2}$ for $X > \\mu $ (our notation).",
"This form gives continuity of $f(x)$ and its first derivative (zero) at $X=\\mu $ .",
"Here for the random variable $X > \\mu $ the pdf is proportional to the pdf of a t-distribution centred at $\\mu $ with scale $c\\phi $ and degrees of freedom $\\nu r$ , while for $X < \\mu $ the pdf is proportional to that of a t-distribution with scale $\\phi /c$ , degrees of freedom $\\nu /r$ .",
"The two constants of proportionality are chosen so that the total probability is unity, and there is no discontinuity at the join.",
"Data can exhibit two types of skewness: asymmetry in scales so that $c \\ne 1$ is usually seen, and sometimes asymmetry in tail behaviours $f(x) \\propto x^{-(\\nu r+1)}$ as $x \\rightarrow \\infty $ and $f(x) \\propto |x|^{-(\\nu /r+1)}$ as $x \\rightarrow -\\infty $ is also found, so that $r \\ne 1$ .",
"This distribution generalises the earlier skew-t distribution of Hansen (1994) and Fernandez and Steel (1998) where the two tail powers are the same.",
"Other skew-t distributions that include the t-distribution as a special case are those of Branco and Dey (2001) and Azzalini and Capitanio (2003).",
"This distribution has the same tail behaviour in both tails, as does that of Jones and Faddy (2003).",
"Rosco, Jones and Pewsey (2011) apply their sinh-arcsinh transformation (Jones and Pewsey, 2009) to the t-distribution so that the transformed t-variable is skew.",
"However, the only distribution that allows both types of skewness is the AST distribution.",
"The AST distribution has the drawback of the gluing together of two disparate halves, which is unaesthetic.",
"It also seems unlikely that such a construct could reflect actual behaviour of the data, and the discontinuity in the second derivative of the log-likelihood function means that the usual regularity conditions for maximum likelihood estimation are not satisfied, and makes inference for parameter values difficult.",
"Experience reveals no problem in fitting the distribution by likelihood maximisation, but the estimation of standard errors on fitted model parameters is problematical because it relies on the second derivative of the log-likelihood.",
"We therefore sought a more natural fat-tailed distribution that possessed both types of asymmetry, and were fortunate to find one that generalises the t-distribution."
],
[
"A new asymmetric distribution",
"In motivating the new distribution, it is perhaps easiest to start with the type IV generalised logistic distribution (Johnson, Kotz and Balakrishnan, 1995, p142).",
"This has pdf $f(y) \\propto \\frac{\\exp (-qy)}{(1+\\exp (-y))^{p+q}}$ , and it is also the log-F distribution (the logarithm of a random variate from the F distribution).",
"This distribution allows asymmetry, the right tail behaving as $f(y) \\propto \\exp (-qy)$ and the left as $f(x) \\propto \\exp (-p|y|)$ .",
"Tadikamalla and Johnson (1982) and Johnson and Tadikamalla (1992) applied Johnson's arcsinh transformation to the logistic distribution $f(y)=\\exp (-y)/(1+\\exp (-y))^2$ , so converting the exponential tail into a power-law tail and obtaining a fat-tailed distribution, but we apply the arcsinh transformation to the type IV logistic distribution, obtaining a 6-parameter distribution with the required properties.",
"We start then with the type IV rescaled generalised logistic pdf, in our notation $f(y)=\\frac{\\alpha (1+r^2)}{r}\\frac{\\lbrace \\exp (\\alpha r y)+\\exp (-(\\alpha /r)y)\\rbrace ^{-\\nu /\\alpha }}{B(\\frac{\\nu /\\alpha }{1+r^2},\\frac{r^2\\nu /\\alpha }{1+r^2})},$ where $B$ denotes the beta function.",
"The parameter $r$ controls the asymmetry, with $r=1$ for a symmetric distribution.",
"Now we apply the arcsinh transformation, where $y=\\ln c+\\sinh ^{-1}((x-\\mu )/\\phi )\\equiv \\ln c+\\ln \\lbrace (x-\\mu )/\\phi +\\sqrt{1+((x-\\mu )/\\phi )^2}\\rbrace .$ To obtain the pdf of $X$ we use the relation between pdfs $f_x(x)=f_y(y)|\\,\\mbox{d}y/\\,\\mbox{d}x|$ , where the Jacobian $\\,\\mbox{d}y/\\,\\mbox{d}x=(1/\\phi )/\\sqrt{1+((x-\\mu )/\\phi )^2}$ , Writing for brevity $cg((x-\\mu )/\\phi )=(x-\\mu )/\\phi +\\sqrt{1+((x-\\mu )/\\phi )^2},$ the pdf is $f(x)=\\frac{\\alpha (1+r^2)}{r\\phi }\\frac{\\lbrace (cg((x-\\mu )/\\phi ))^{\\alpha r}+(c g((x-\\mu )/\\phi ))^{-\\alpha /r}\\rbrace ^{-\\nu /\\alpha }}{B(\\frac{\\nu /\\alpha }{1+r^2},\\frac{r^2\\nu /\\alpha }{1+r^2})}(1+((x-\\mu )/\\phi )^2)^{-1/2}.$ Here $\\nu > 0$ controls tail power, $\\mu $ is in a sense the centre of location (but not necessarily the mean, which may not exist), $\\phi > 0$ is a measure of scale (but not the variance, which may not exist), $r > 0$ controls tail power asymmetry, $c > 0$ controls the scale asymmetry, and $\\alpha > 0$ controls how early `tail behaviour' is apparent.",
"We propose calling this new distribution the GAT distribution (generalised asymmetric t).",
"A gat is a gun in obsolete American slang, so the idea here is that this distribution, with its 6 parameters, is a powerful weapon.",
"To explore the meaning of these parameters, let $x$ become large, when $g(x) \\approx 2cx/\\phi $ , so that $f(x) \\approx \\frac{\\alpha (1+r^2)}{r\\phi B}(2cx/\\phi )^{-r\\nu }(x/\\phi )^{-1}$ , i.e.",
"$f(x) \\propto x^{-\\nu r-1}$ .",
"Since $g((x-\\mu )/\\phi ))g(-(x-\\mu )/\\phi ))=1$ , when $x$ is large and negative $f(x) \\approx \\frac{\\alpha (1+r^2)}{r\\phi B}(2|x|/c\\phi )^{-\\nu /r}(|x|/\\phi )^{-1}$ , i.e.",
"$f(x) \\propto |x|^{-\\nu /r-1}$ .",
"This shows how $\\nu $ , $r$ and $c$ control tail behaviour; $c$ controls the ratio of probability masses in each tail and $r$ controls the ratio of powers of $x$ in each tail.",
"Setting the asymmetry parameters $r=1, c=1$ and setting $\\alpha =1$ the pdf becomes $f(x)=(2/\\phi )\\lbrace (1+((x-\\mu )/\\phi )^2\\rbrace ^{-{(\\nu +1)/2}}/B(\\nu /2,\\nu /2),$ so that on applying the Legendre duplication formula to regain a more familiar form of the constant, we see that $X$ is a rescaled random variate following the t distribution with $\\nu $ degrees of freedom; specialising to $\\phi =\\sqrt{\\nu }$ and $\\mu =0$ we obtain the standard t distribution.",
"This is a restatement of the fact noted by several authors (Johnson, Kotz and Balakrishnan, p346) that for an $F$ distribution with $\\nu $ and $\\nu $ degrees of freedom, $\\frac{\\sqrt{\\nu }}{2}(F^{1/2}_{\\nu ,\\nu }-F^{-1/2}_{\\nu ,\\nu })$ follows a t distribution with $\\nu $ degrees of freedom.",
"With $\\alpha =1$ we have a 5-parameter distribution that turns out to fit returns data almost identically well to the AST distribution, but which does not have the same inferential problems, as the log-likelihood function has no discontinuities in derivatives.",
"The GAT distribution can fit very skew data, and the only area where it cannot reproduce the behaviour of the AST distribution is the case where $\\nu \\rightarrow \\infty $ .",
"The GAT distribution cannot be skew when both tails are normally distributed, but the AST distribution can.",
"On allowing the sixth parameter $\\alpha $ to vary from unity, we have a more general distribution that sometimes fits the data better.",
"As $\\alpha $ increases, the fatness of the tails decreases, while the power-law behaviour remains the same.",
"Thus this distribution allows us to fit financial and other data with fatter tails even than the t-distribution can cope with, by floating $\\alpha $ below unity.",
"As $\\alpha \\rightarrow 0$ and $\\nu \\rightarrow \\infty $ such that $\\eta =\\nu \\alpha $ remains constant, by expanding the pdf about the mode it is readily seen that $Y$ is normally distributed, with mean zero and variance $\\eta ^{-1}$ .",
"Hence $X$ follows Johnson's $S_U$ distribution.",
"In this case the distribution of $X$ has the tail behaviour of a lognormal distribution.",
"It is possible to ignore the Jacobian in (REF ), and so obtain a slightly simpler distribution, for which the mode is easily found as a transformation of the mode of $Y$ , which is $y_0=\\frac{-2\\ln (r)}{\\alpha (r+1/r)}$ .",
"The mode of the GAT distribution on the other hand must be found iteratively, e.g.",
"by using Newton-Raphson iteration from the corresponding value of $x$ , $x_0=\\mu +(\\phi /2)\\lbrace r^{\\frac{-2}{\\alpha (r+1/r)}}/c-cr^{\\frac{2}{\\alpha (r+1/r)}}\\rbrace .$ This distribution fitted data equally well.",
"However, without the Jacobian, the minimum value of $\\nu $ for which the pdf can be defined is $\\nu > \\text{max}(r,1/r)/\\alpha -1$ , which is messy.",
"Hence we do not develop this distribution further.",
"It does however have broadly similar properties to the GAT distribution, and could also be useful."
],
[
"Moments, distribution function and random numbers",
"The moments and distribution function can be expressed analytically in terms of beta functions.",
"We shall need the (complete) beta function $B(a,b)=\\int _0^1 q^{a-1}(1-q)^{b-1}\\,\\mbox{d}q$ , the incomplete beta function $B_I(a,b;c)=\\int _0^c q^{a-1}(1-q)^{b-1}\\,\\mbox{d}q$ and the regularised incomplete beta function $B_R(a,b;c)=B_I(a,b;c)/B(a,b)$ .",
"It is convenient to start by showing that the pdf integrates to unity.",
"This and the moments are best computed by working with $f_y(y)$ rather than $f_x(x)$ , and then writing $x=\\mu +\\phi \\sinh (y-\\ln (c))$ .",
"In (REF ) we can change variable of integration first to $z=y-\\ln (c)$ so that now $x=\\phi (\\exp (z)- \\exp (-z))/2$ , then to $q(z)=\\lbrace 1+\\exp (-\\alpha (r+1/r)z\\rbrace ^{-1}$ .",
"Clearly $0 < q < 1$ , and we have that $z=-\\frac{\\ln ((1-q)/q)}{\\alpha (r+1/r)}$ , and the Jacobian $\\,\\mbox{d}z/\\,\\mbox{d}q=q^{-1}(1-q)^{-1}/(\\alpha (r+1/r))$ .",
"The normalisation given in (REF ) follows, and the mean is $\\text{E}(X)=\\mu +\\phi \\lbrace \\frac{c^{-1}B(\\frac{\\nu }{\\alpha (1+r^2)}+\\delta ,\\frac{\\nu r^2}{\\alpha (1+r^2)}-\\delta )-cB(\\frac{\\nu }{\\alpha (1+r^2)}-\\delta ,\\frac{\\nu r^2}{\\alpha (1+r^2)}+\\delta )}{2B(\\frac{\\nu }{\\alpha (1+r^2)},\\frac{\\nu r^2}{\\alpha (1+r^2)})}\\rbrace $ when $\\nu > \\max (r,1/r)$ and where $\\delta =r/(\\alpha (1+r^2))$ .",
"Higher moments may be computed similarly, the $n$ th moment existing when $\\nu > n$ for the symmetric distribution, else $\\nu > n\\max (r,1/r)$ .",
"The algebra rapidly becomes tedious but not difficult, for example the variance can be derived as $\\text{E}\\lbrace (X-\\mu )^2\\rbrace -\\text{E}\\lbrace (X-\\mu )\\rbrace ^2$ , where $\\text{E}\\lbrace (X-\\mu )^2\\rbrace =\\phi ^2[c^{-2}B(\\frac{\\frac{\\nu }{\\alpha (1+r^2)}+2\\delta ,\\frac{\\nu r^2}{\\alpha (1+r^2)}-2\\delta )+c^2B(\\frac{\\nu }{\\alpha (1+r^2)}-2\\delta ,\\frac{\\nu r^2}{\\alpha (1+r^2)}+2\\delta )}{4B(\\frac{\\nu }{\\alpha (1+r^2)},\\frac{\\nu r^2}{\\alpha (1+r^2)})}]-\\phi ^2/2.$ In general, $\\text{E}(X-\\mu )^n=(\\phi /2)^n\\frac{\\sum _{m=0}^n (-1)^m {n \\atopwithdelims ()m}c^{n-2m}B\\lbrace \\frac{\\nu }{\\alpha (1+r^2)}-(n-2m)\\delta ,\\frac{\\nu r^2}{\\alpha (1+r^2)}+(n-2m)\\delta \\rbrace }{B(\\frac{\\nu }{\\alpha (1+r^2)},\\frac{\\nu r^2}{\\alpha (1+r^2)})}.$ Using the same algebra, the distribution function $F(x)$ is the regularised incomplete beta function $F(x)=B_R(\\frac{\\nu }{\\alpha (1+r^2)},\\frac{\\nu r^2}{\\alpha (1+r^2)}; q(x))$ where $q(x)=\\frac{1}{1+c^{-\\alpha (1+r^2)/r}\\lbrace \\frac{(x-\\mu )}{\\phi }+\\sqrt{1+\\frac{(x-\\mu )^2}{\\phi ^2}}\\rbrace ^{-\\alpha (1+r^2)/r}}.$ Quantiles may be found by Newton-Raphson iteration starting from $x=\\mu $ .",
"Random numbers may best be generated from the GAT distribution by generating them from the corresponding Beta distribution, and then transforming so that $X=\\mu +(\\phi /2)\\lbrace c^{-1}(q/(1-q))^{\\delta }-c(q/(1-q))^{-\\delta }\\rbrace .$ Distribution functions and random variates are useful for a variety of purposes, e.g.",
"the distribution function is needed for EDF-based goodness of fit tests such as the Anderson-Darling test, and random numbers are needed for MCMC, and also in finding the distribution and p-value of a goodness of fit statistic via the parametric bootstrap.",
"We also have the mean absolute deviation $\\text{E}|X-\\text{E}(X)|=\\mu (1-2F(\\text{E}(X)))+\\text{E}(X-\\mu )-$ $\\phi \\lbrace \\frac{c^{-1}B_I(\\frac{\\nu }{\\alpha (1+r^2)}+\\delta ,\\frac{\\nu r^2}{\\alpha (1+r^2)}-\\delta ; q(\\text{E}(X)))-cB_I(\\frac{\\nu }{\\alpha (1+r^2)}-\\delta ,\\frac{\\nu r^2}{\\alpha (1+r^2)}+\\delta ; q(\\text{E}(X)))}{B(\\frac{\\nu }{\\alpha (1+r^2)},\\frac{\\nu r^2}{\\alpha (1+r^2)})}\\rbrace $ In finance, the Value-at-Risk, VaR, is a quantile such that given a distribution of gains, the probability of exceeding a loss, $-X$ is $\\gamma $ , where e.g.",
"$\\gamma =0.02$ .",
"Then $F(-\\text{VaR})=\\gamma $ , and VaR may be found like any quantile, by Newton-Raphson iteration using the distribution function $F$ .",
"The expected shortfall (ES) is the expected loss given that the loss is at least the VaR, so that $\\text{ES}=-\\text{E}(X|X < -\\text{VaR})$ .",
"We have $\\text{ES}=-\\mu -\\phi \\lbrace \\frac{c^{-1}B_I(\\frac{\\nu }{\\alpha (1+r^2)}+\\delta ,\\frac{\\nu r^2}{\\alpha (1+r^2)}-\\delta ;q(-\\text{VaR}))-cB_I(\\frac{\\nu }{\\alpha (1+r^2)}-\\delta ,\\frac{\\nu r^2}{\\alpha (1+r^2)}+\\delta ;q(-\\text{VaR}))}{2B_I(\\frac{\\nu }{\\alpha (1+r^2)},\\frac{\\nu r^2}{\\alpha (1+r^2)};q(-\\text{VaR}))}\\rbrace $ so that given the VaR, ES can be computed using the incomplete beta function."
],
[
"Inference",
"There are several points to consider regarding statistical use of the new distribution.",
"First, any problems arising with estimating standard errors on model parameters with the AST distribution do not arise here, as the GAT distribution has all derivatives continuous.",
"Next, there is a well-known problem with Azzalini's skew-normal distribution (e.g., Hallin and Ley, 2014).",
"This is that the derivative of the log-likelihood with respect to the skewness parameter is zero when the parameter is zero (the skew-normal reduces to a normal distribution).",
"This problem does not occur here.",
"Another is the sequence of model fitting; with so many parameters, how do we proceed?",
"Experience shows that many skew distributions require only the parameters $\\mu , \\phi , \\nu $ and $c$ , so that skewness is modelled purely by having different probability mass in the two tails.",
"We can set $r=\\alpha =1$ .",
"For some financial data the tail powers are different, so we can then try $r \\ne 1$ .",
"It is also worth then trying $\\alpha \\ne 1$ as this sometimes improves the fit.",
"The 4-parameter distribution with parameters $\\eta =\\nu \\alpha $ , $\\mu , \\phi $ and $c$ is sometimes a good fit, where $\\nu \\rightarrow \\infty $ ; this is Johnson's $S_U$ distribution.",
"In practice, to fit this 4-parameter distribution given the ability to fit the full GAT distribution, one can set for example $\\nu =200$ and vary $\\mu , \\phi ,\\alpha $ and $c$ .",
"It is common in frequentist inference to choose the best model, for example the minimum-AIC model.",
"A Bayesian approach would instead put prior distributions on all the parameters.",
"Another inferential issue is that of carrying out regressions on covariates.",
"With a skew distribution it is often not clear which measure of central tendency should be modelled as a function of covariates, e.g.",
"the mean, median, or mode.",
"Here it is suggested that the parameter $\\mu $ be modelled.",
"From (REF ) the mean is $\\mu $ with an added function of the other five parameters.",
"Thus the regression for $\\mu $ is also a regression for the mean, once the term for $\\text{E}(X)-\\mu $ from (REF ) has been added to constant term of the regression.",
"Similarly it is also a regression for the median and mode.",
"From (REF ) the higher moments do not depend on $\\mu $ , and the variance is proportional to $\\phi ^2$ .",
"Thus if modelling the variance as well as the mean, $\\mu $ and $\\phi $ can be modelled.",
"These attractive results follow because the GAT distribution is a location-scale distribution.",
"R codes to compute the pdf, distribution function, quantiles etc are available by email from the author."
],
[
"Examples",
"Figure REF shows the GAT distribution and the AST distribution fitted to data on the breaking strengths of 63 glass fibres.",
"This dataset, first published by Smith and Naylor (1987), has been widely used by developers of skew distributions, such as Jones and Faddy (2003), Azzalini and Capitanio (2003), Ma and Genton (2004) and Jones and Pewsey (2009).",
"Because it is skew to the left, it cannot be fitted well by lifetime distributions.",
"Fitting the GAT distribution with $\\alpha =r=1$ (4 parameters floated) gave minus log-likelihood $-\\ell =11.7557$ , compared with $11.7921$ for the AST distribution.",
"This shows a (very) slightly better fit, which is typical but not inevitable.",
"Allowing tail power ($r$ ) to float hardly improved the fit for either distribution.",
"However, on allowing $\\alpha $ to vary from unity, $\\alpha $ became tiny and $\\nu $ large, and the fit improved slightly with $\\ell =-11.207$ (shown in the figure).",
"Hence Johnson's $S_U$ distribution, a special case of the GAT distribution, gives the best 4-parameter fit.",
"Figure REF shows fits to the heights in centimetres of 100 female athletes.",
"The data are from Cook and Weisberg (1994) and have been used by Arellano-Valle et al (2004) to illustrate the Azzalini skew-normal distribution.",
"Again, the plot is skew to the left.",
"Here, the split-t and GAT distributions with $\\alpha =r=1$ perform very similarly, with $\\ell =-348.3739$ for the GAT and $-348.4578$ for the split-t. Again the GAT distribution fits (very slightly) better.",
"Here, floating $r$ and $\\alpha $ does not improve the fit, although the best 4-parameter distribution is again Johnson's $S_U$ ($\\ell =-348.2031$ ).",
"Tables REF and REF show the results of maximum-likelihood fits to daily FTSE-100 and Nikkei 225 returns, respectively.",
"Again, in both cases the 4-parameter GAT distribution does slightly better than the split-t distribution.",
"In the case of the Nikkei, Johnson's $S_U$ distribution gives a significantly better fit than the split-t distribution.",
"There are financial data where tail powers differ, and the parameter $r$ is needed, but these simple examples really just demonstrate two points.",
"One is that the GAT distribution can fit at least as well as the split-t distribution.",
"The other is that allowing the parameter $\\alpha $ to vary sometimes improves the fit, an option not available with the split-t distribution."
],
[
"Final comments",
"The GAT distribution generalises the t-distribution in three ways: through the two types of skewness (parameters $c$ and $r$ ) and through how soon `tail behaviour' starts (parameter $\\alpha $ ).",
"Its use in inference has been illustrated and described.",
"It is natural to ask whether the distribution can be made multivariate.",
"To date no attractive way of doing this has presented itself.",
"Many multivariate distributions, such as the multivariate t distribution, have the drawback that when the correlation between variables is zero, they are still not independent.",
"Use of a copula allows multivariate distributions of variables that can be truly independent as well as uncorrelated.",
"Currently this is the only way to generate a multivariate-GAT distribution."
],
[
"Acknowledgements",
"I would like to thank Prof. Ian McHale, Dr Dan Jackson, and Prof. Chris Jones, whose comments have improved this paper.",
"Figures and tables Figure: Glass strength data with fitted GAT distribution and split-t (AST) distribution.Figure: Height of female athlete data with fitted GAT distribution and split-t (AST) distribution.Table: Minus the logarithm of the likelihood function for fits to the FTSE-100 returns (8028 values) and fitted parameter values.",
"Parametervalues with (f) following were fixed at that value.Table: Minus the logarithm of the likelihood function for fits to the Nikkei 225 returns (7581 values) and fitted parameter values.",
"Parametervalues with (f) following were fixed at that value."
]
] | 1606.05203 |
[
[
"Detached eddy simulation of shock unsteadiness in an over-expanded\n planar nozzle"
],
[
"Abstract This work investigates the self-excited shock wave oscillations in a three-dimensional planar over-expanded nozzle turbulent flow by means of Detached Eddy Simulations.",
"Time resolved wall pressure measurements are used as primary diagnostics.",
"The statistical analysis reveals that the shock unsteadiness has common features in terms of the root mean square of the pressure fluctuations with other classical shock wave/boundary layer interactions, like compression ramps and incident shocks on a flat plate.",
"The Fourier transform and the continuous wavelet transform are used to conduct the spectral analysis.",
"The results of the former indicate that the pressure in the shock region is characterized by a broad low-frequency content, without any resonant tone.",
"The wavelet analysis, which is well suited to study non stationary process, reveals that the pressure signal is characterized by an amplitude and a frequency modulation in time."
],
[
"Introduction",
"During the sea-level start-up of a liquid rocket engine the nozzle is highly overexpanded and an internal flow separation takes place, characterized by a shock wave boundary layer interaction (SWBLI), which causes the shedding of vortical structures and unsteadiness in the shock wave position.",
"In the nozzle design community, flow separation is considered dangerous, since it produces dynamic side-loads that reduce the safe life of the engine and can lead to a failure of the nozzle structure.",
"The need to improve nozzle performance under overexpanded conditions and to mitigate the side loads fostered several experimental [1], [2], [3] and numerical investigations [4], [5], [6].",
"All these studies revealed two distinct separation processes: the free shock separation (FSS), in which the boundary layer separates from the nozzle wall and never reattaches, and the restricted shock separation (RSS), characterized by a closed recirculation bubble and the reattachment of the shear layer to the wall.",
"According to Schmucker [7], the main cause of side-loads appearance is an asymmetry of the separation location, which produces a tilted separation line, a momentum imbalance and consequently a lateral force.",
"Nave and Coffey [1] observed that, in optimized nozzles, the maximum value of the side loads takes place during the transition from FSS to RSS condition.",
"A literature survey reporting the various studies on the side loads generation and separation shock configurations can be found in Hadjadj and Onofri [8] and in Reijasse et al. [9].",
"But, in spite of all of these studies, a fundamental knowledge of supersonic flow physics in the presence of a shock separation interaction is still needed.",
"One of the several tasks for future investigations recommended by Hadjadj and Onofri [8] is related to the low frequency oscillations of a shock interacting with a turbulent flow separation.",
"This phenomenon, consisting in fluctuating pressure loads and pulsating separated flows, should be carefully considered by rocket nozzle designers.",
"A lot of experimental work has been carried out to understand the unsteadiness of shocks in internal flows.",
"Bogar et al.",
"[10] investigated the unsteady flow characteristics of a supercritical transonic diffuser as a function of shock Mach number and diffuser length.",
"They observed that in the case of attached flow (or very mild separation), the characteristic frequencies have an acoustic nature, and scale with the distance of the shock from the diffuser exit.",
"While, in the case of separated flows, the characteristic frequencies scale with the length of the inviscid core flow.",
"Zaman et al[11] carried out experiments in a supersonic planar diffuser, and documented a transonic tone with weak harmonics.",
"The presence of harmonics led to the theory that the observed frequency was caused by an acoustic feed-back mechanism.",
"In addition, they found that the instability has the higher sound pressure level if the boundary layer before the shock is laminar, while if the boundary layer is tripped, in some of the tests the tone was suppressed.",
"Both Bogar and Zaman underlined that the mechanism for the shock instability is unclear, although they indicate some dependency of the shock dynamics on the downstream separated region.",
"Also Handa et al.",
"[12], in their experimental investigation of a transonic diffuser, underlined two different mechanisms for the shock oscillation.",
"In the first mechanism, pressure disturbances, generated in the downstream turbulent separated region, force the shock to oscillate, resulting in a broad shape of the power spectral density.",
"In addition, the intensity of this movement is mainly governed by the Mach number in front of the shock.",
"The second mechanism foresees the reflection of a disturbance at the diffuser exit (acoustic feedback), resulting in a narrow-shaped power spectral density.",
"More recently, Johnson and Papamoschou [13] have studied the unsteady shock behavior in an over-expanded planar nozzle.",
"Their results indicate a low frequency piston-like shock motion without any resonant tones.",
"Correlations of Pitot pressure with wall pressure indicated a strong coherence of the shear layer instability with the shock motion.",
"All these different experimental investigations suggest a correlation between the shock movement and the acoustic or fluid dynamic characteristics downstream of the flow separation.",
"As far as Large/Detached Eddy Simulations (LES/DES) of this kind of flows are concerned, very few studies can be found in literature.",
"Deck [14] carried out a Delayed Detached Eddy Simulation (DDES) of the end-effect regime in an axi-symmetric over-expanded rocket nozzle flow.",
"While the experimentally measured main properties of the flow motion were rather well reproduced, the computed main frequency resulted to be higher than in the experiment.",
"Olson and Lele [15] performed large eddy simulations of the experiments of Papamoschou, finding a satisfactory agreement between the experimental data and the computed frequency of the shock displacement.",
"The origin of the unsteadiness was attributed to the confinement of exit area by the separated flow.",
"In the present work, Delayed Detached Eddy Simulations (DDES) reproducing the flowfield of the nozzle geometry of Bogar et al.",
"[10] are carried out to completely characterize the shock unsteadiness at different nozzle pressure ratios.",
"Resorting to this computational strategy, any perturbation coming from the upstream boundary layer is not resolved.",
"Therefore the separation shock can only be perturbed by the turbulent recirculating region.",
"In a recent review of Clemens et al.",
"[16] on the low frequency unsteadiness of the shock wave/turbulent boundary layer interaction, it is argued that both upstream and downstream perturbations are at work in the interaction, whether there is a separation bubble or not.",
"However, the influence of the upstream turbulent boundary layer decreases when the size of the separation bubble increases.",
"In flow separated nozzle, the recirculation bubble is longer than in other classical configuration like the compression ramp.",
"Therefore it seems reasonable to attribute the most important influence on the shock movement to the downstream region.",
"In such a case, the DDES technique can be considered adequate.",
"Following the literature [17], the unsteady behavior of shock flow separation interaction is characterized by analyzing the time series of the wall pressure signals.",
"In particular, the stream-wise distributions of the root mean square as well as the probability density functions (pdf) of the pressure signals are discussed.",
"Then the intermittency of the signals is computed, and from its distribution the length scale of the shock motion is quantified, this quantity being directly linked to the level of the side-loads.",
"The shock flow separation interaction is characterized by the presence of several dominating components in the pressure fluctuations, connected to the development of vortical structures and to the shock movement itself.",
"Fourier analysis is used in order to evaluate the main frequencies along the nozzle, from the shock location to the turbulent recirculating zone.",
"In addition to the global spectral characterization, it is of great interest to understand if the dominating components are simultaneously or alternatively present, and to characterize the time variation of their frequency and amplitude.",
"The wavelet transform is an analysis tool well suited to the study of multi-scale, non-stationary processes occurring over finite temporal domains.",
"In particular, it allows to detect the localized variations of power within a time series.",
"In fact, by decomposing a time series into time-frequency space, it is possible to determine both the dominant modes of variability, as well as their time evolution.",
"Therefore, the continuous complex wavelet transform is directly applied in order to identify the time modulation of frequency and amplitude of turbulent structures [18].",
"This characterization contributes to the basic understanding of the shock wave/turbulent boundary layer interaction physics in internal flows.",
"A better knowledge of this phenomenon can help in predicting and possibly control the level of side loads, thus allowing the design of larger area ratio nozzle, improving the nozzle performance and reducing the costs of the access to space.",
"The paper is organized as follows.",
"In section the numerical method is introduced and the use of DDES is discussed in the frame of over-expanded nozzles.",
"Section presents the experimental test cases and the numerical setup.",
"Section REF presents the mean properties of the fields and of the wall pressure distributions, together with the statistical description of the shock separation iteration.",
"Fourier and the wavelet spectral analysis are discussed in Section REF and Section REF respectively.",
"Finally, in the conclusion section the major findings of this investigation are reported."
],
[
"Computational setup",
"To better understand the unsteadiness of SWBLI in supersonic nozzles and the role played in the generation of side loads, large eddy simulations should be ideally carried out to capture the larger structures of the turbulent flow.",
"Unfortunately, the computational cost of a pure (wall-resolved) LES is still very high for high-Reynolds number wall-bounded flows.",
"To overcome this limitation, hybrid RANS/LES modeling approaches have been proposed to simulate massively separated flows, such as the well-known DES [19].",
"A general feature of this approach is that the whole or at least a major part of the attached boundary layer is treated resorting to RANS, while LES is applied only in the separated flow regions.",
"In the following a brief description of the numerical solver used is given, then the main characteristics of DDES are outlined."
],
[
"Physical model",
"We solve the three-dimensional Navier-Stokes equations for a compressible, viscous, heat-conducting gas $\\left.\\begin{aligned}\\frac{\\partial \\rho }{\\partial t} + \\frac{\\partial (\\rho \\, u_j)}{\\partial x_j}& = 0, \\\\\\frac{\\partial (\\rho \\, u_i)}{\\partial t} + \\frac{\\partial (\\rho \\, u_i u_j)}{\\partial x_j} +\\frac{\\partial p}{\\partial x_i} - \\frac{\\partial \\tau _{ij}}{\\partial x_j} & = 0, \\\\\\frac{\\partial (\\rho \\, E)}{\\partial t} + \\frac{\\partial (\\rho \\, E u_j + p u_j)}{\\partial x_j}- \\frac{\\partial (\\tau _{ij} u_i - q_j)}{\\partial x_j} & = 0, \\end{aligned}\\right.$ where $\\rho $ is the density, $u_i$ is the velocity component in the $i$ -th coordinate direction ($i=1,2,3$ ), $E$ is the total energy per unit mass, $p$ is the thermodynamic pressure.",
"The total stress tensor $\\tau _{ij}$ is the sum of the viscous and the Reynolds stress tensor, $\\tau _{ij} = 2 \\, \\rho \\left( \\nu + \\nu _t \\right) S^*_{ij} \\qquad S^*_{ij} = S_{ij} - \\frac{1}{3} \\, S_{kk} \\, \\delta _{ij},$ where the Boussinesq hypothesis is applied through the introduction of the eddy viscosity $\\nu _t$ , and where $S_{ij} = \\left( u_{i,j} + u_{j,i} \\right) /2 $ is the strain-rate tensor, $\\nu $ the molecular viscosity, depending on temperature $T$ through Sutherland's law.",
"Similarly, the total heat flux $q_j$ is the sum of a molecular and a turbulent contribution $q_j = -\\rho \\, c_p \\left( \\frac{\\nu }{\\mathrm {Pr}} + \\frac{\\nu _t}{\\mathrm {Pr}_t} \\right) \\frac{\\partial T}{\\partial x_j},$ $\\mathrm {Pr}$ , $\\mathrm {Pr}_t$ being the molecular and turbulent Prandtl numbers, assumed to be 0.72 and 0.9, respectively.",
"Hybrid RANS/LES capabilities are provided through the implementation of the delayed detached-eddy simulation (DDES) approach based on the Spalart-Allmaras (SA) model [20], which involves a transport equation for a pseudo eddy viscosity $\\tilde{\\nu }$ $\\frac{\\partial (\\rho \\tilde{\\nu })}{\\partial t} + \\frac{\\partial (\\rho \\, \\tilde{\\nu } \\,u_j)}{\\partial x_j} =c_{b1} \\tilde{S} \\rho \\tilde{\\nu } +\\frac{1}{\\sigma }\\left[\\frac{\\partial }{\\partial x_j} \\left[ \\left( \\rho \\nu + \\rho \\tilde{\\nu } \\right) \\frac{\\partial \\tilde{\\nu }}{\\partial x_j} \\right] +c_{b2} \\, \\rho \\left( \\frac{\\partial \\tilde{\\nu }}{\\partial x_j} \\right)^2\\right]-c_{w1} f_w \\rho \\left( \\frac{\\tilde{\\nu }}{\\tilde{d}} \\right)^2,$ where $\\tilde{d}$ is the model length scale, $f_w$ is a near-wall damping function, $\\tilde{S}$ a modified vorticity magnitude, and $\\sigma , c_{b1}, c_{b2}, c_{w1}$ model constants.",
"The eddy viscosity in Eq.",
"REF is related to $\\tilde{\\nu }$ through $\\nu _t = \\tilde{\\nu } \\, f_{v1}$ , where $f_{v1}$ is a correction function designed to guarantee the correct boundary-layer behavior in the near-wall region.",
"In DDES the destruction term in Eq.",
"REF is designed so that the model reduces to pure RANS in attached boundary layers and to a LES sub-grid scale one in the detached flow regions.",
"This is accomplished by defining the length scale $\\tilde{d}$ as $\\tilde{d} = d_w - f_d \\, \\textrm {max} \\left(0, d_w-C_{DES} \\, \\Delta \\right),$ where $d_w$ is the distance from the closest wall, $\\Delta $ is the subgrid length-scale, controlling the wavelengths resolved in LES mode.",
"The function $f_d$ , designed to be 0 in boundary layers and 1 in LES regions, reads as $f_d = 1-\\tanh {\\left[ \\left( 8 r_d \\right)^3 \\right]}, \\qquad r_d = \\frac{\\tilde{\\nu }}{k^2 \\, d_w^2 \\, \\sqrt{U_{i,j} U_{i,j}}},$ where $U_{i,j}$ is the velocity gradient and $k$ the von Karman constant.",
"The introduction of $f_d$ distinguishes DDES from the original DES approach [21] (usually denoted as DES97), ensuring that boundary layers are treated in RANS mode also in the presence of “ambiguous” grids in the sense defined by Spalart et al.",
"[20], for which the wall-parallel spacings do not exceed the boundary layer thickness.",
"The DDES strategy prevents the phenomenon of model stress depletion, consisting in the excessive reduction of the eddy viscosity in the region of switch (grey area) between RANS and LES, which in turn leads to grid-induced separation.",
"Unlike in the original DDES formulation, the sub-grid length scale in this work is not defined as the largest spacing in all coordinate directions $\\Delta _{\\mathrm {max}} = \\max (\\Delta x, \\Delta y, \\Delta z)$ , but it depends on the flow itself, through $f_d$ as follows $\\Delta = \\frac{1}{2}\\left[\\left(1+\\frac{f_d-f_{d0}}{|f_d-f_{d0}|} \\right) \\, \\Delta _{\\textrm {max}} +\\left(1-\\frac{f_d-f_{d0}}{|f_d-f_{d0}|} \\right) \\, \\Delta _{\\textrm {vol}}\\right],$ where $f_{d0} = 0.8$ , $\\Delta _{\\mathrm {vol}} = (\\Delta x \\cdot \\Delta y \\cdot \\Delta z)^{1/3}$ .",
"The improvement over the classical $\\Delta _{max}$ definition is shown in Deck [22], where the problem of the delay in the formation of flow instabilities encountered in early applications of DES/DDES is solved."
],
[
"Numerical method",
"Numerical simulations are carried out by means of a in-house, fully validated compressible flow solver, that exploits a centered second-order finite volume approach and takes advantage of an energy consistent formulation (away from shocks).",
"Cell-face values of the flow variables are obtained from the cell-centered values through suitable reconstructions.",
"In smooth flow regions, the reconstruction is carried out in such a way that the overall kinetic energy of the fluid is preserved, in the limit of inviscid, incompressible flow [23].",
"This property is particularly beneficial for flow regions treated in LES mode, where the grid is sufficiently fine to support the development of LES content, and where the only relevant dissipation (in addition to the molecular one) should be that provided by the turbulence model.",
"The discretization scheme is made to switch to third-order weighted essentially-non-oscillatory (WENO) near discontinuities, as controlled by a modified Ducros sensor [24].",
"The gradients normal to the cell faces needed for the viscous fluxes, are evaluated through second-order central-difference approximations, obtaining compact stencils and avoiding numerical odd-even decoupling phenomena.",
"Time advancement of the semi-discretized system of ODEs resulting from the spatial discretization is carried out by means of a low-storage third-order Runge-Kutta algorithm [25].",
"The code is written in Fortran 90, it uses domain decomposition and it fully exploits the Message Passing Interface (MPI) paradigm for the parallelism."
],
[
"Test case description",
"The experimental diffuser model numerically reproduced in this work is a convergent-divergent channel with a flat bottom and a contoured top wall, as shown in figure REF .",
"The analytical expression of the contoured wall can be found in Bogar et al.",
"[10] The channel height at the throat is 44 mm, the exit-to-throat area ratio is 1.52, the throat cross-sectional aspect ratio is 4.0, and the divergent length to throat height ratio is 7.2.",
"For $x/H_t$ greater than 7.2 the nozzle is characterized by a constant area section.",
"In the experimental case dry air is supplied to the model from a plenum chamber immediately upstream.",
"The flow from the model is vented to the atmosphere, providing a constant-pressure downstream boundary condition.",
"A two-dimensional schematic of the computational domain adopted in the simulations is presented in figure REF .",
"According to Bogar et al.",
"[10] an arbitrarily selected location within the constant area section upstream of the throat ($x_i/H_t=-4.04$ ) has been chosen as a nominal inlet section.",
"The nominal exit section is at $x_e/H_t=12.63$ .",
"In the inlet station a subsonic flow is prescribed by imposing the Mach number $M=0.46$ , the static pressure and the flow direction.",
"The top and bottom surfaces are treated as adiabatic no-slip walls.",
"In the spanwise direction the extent of the domain is $L_z/H_t=4$ , and periodic boundary conditions are imposed.",
"At the exit section a characteristics based boundary condition prescribing the back pressure is assigned.",
"In order to avoid any acoustic coupling a sponge is imposed from the station at $x/H_t=9$ .",
"As stated in the introduction, in fact, the effect of the acoustic feed back is less important when a large recirculating bubble is present.",
"A logically Cartesian structured mesh is generated using the conformal mapping algorithm of Driscoll and Vavasis[26] and the open-source tool gridgen-c.",
"The computational mesh consists in $N_x\\cdot N_y\\cdot N_z= 512\\cdot 192\\cdot 256$ cells for a total number of $N_{xyz} \\approx 25\\cdot 10^6 $ cells.",
"Figure: Two-dimensional schematic of the computational domain with boundary conditionsFigure: Experimental mean top wall pressure distribution , compared with a 2D RANS simulation for NPR = 1.39The present internal flowfield is characterized by the nozzle pressure ratio $NPR = p_0/p_a$ , where $p_0$ and $p_a$ denote the chamber and ambient pressure respectively.",
"In this work three different values are simulated: 1.39, 1.46 and 1.54.",
"The NPR's values are changed by decreasing the back pressure at the exit section, so that the nozzle Reynolds number based on the chamber values and the throat height remains constant: $Re = \\frac{\\sqrt{\\gamma }}{\\mu }\\frac{p_cH_t}{\\sqrt{R_{air}T_0}}=1.5 \\cdot 10^6,$ where $\\gamma $ is the constant specific heat ratio, $\\mu $ is the molecular viscosity evaluated at the chamber temperature $T_0$ and $R_{air}$ is the air gas constant.",
"A preliminary 2D RANS simulation was performed in order to verify that the mesh resolution in the longitudinal and wall normal direction is sufficient to capture the position of separation line.",
"The comparison of the computed top wall pressure distribution with the experiment [10] for NPR = 1.39 is reported in figure REF , and it shows that both the position of the separation point and the pressure behavior in the separated zone are well reproduced."
],
[
"Time-averaged and instantaneous flowfield",
"The time and spanwise averaged numerical Schlieren like visualizations ($||\\nabla \\rho ||$ ) of the three NPR's are presented in Figure REF .",
"The flowfield is characterized by a lambda shock, a recirculation zone and a shear layer.",
"As the NPR increases, the separation shock moves downstream, while the height of the Mach stem decreases.",
"The evolution of the recirculation bubble with the NPR is shown in figure REF b where an enlargement of the back flow region (indicated by the coloured flood region) with the downstream movement of the separation shock can be observed.",
"Figure: Time-averaged and span-wise averaged field of: a) ||∇ρ||||\\nabla \\rho || and b) density (isolines) with the region with negative velocities (coloured) in the transonic nozzle at different NPR's.Figure: Left: streamwise distributions of time and spanwise averaged wall pressure.",
"Right: streamwise distributions of the wall isentropic Mach number, indicating the shock strengthThe top wall pressure distributions in the streamwise direction are presented in figure REF a for the different NPR's.",
"It can be noted the steep increase of the wall pressure due to the lambda foot of the separation shock, then the flat behavior in the separation bubble and finally a mild increase to the back pressure value in the final part of the diffuser.",
"The isentropic Mach number distribution in the streamwise direction is reported in Figure REF b.",
"This value is computed from the isentropic relation between the chamber pressure and wall pressure $p_0/p_w=\\left(1+\\frac{\\gamma -1}{2} M_i^2 \\right)^{\\frac{\\gamma }{\\gamma -1}}$ and it is used to characterize the shock intensity in nozzle separated flows and to reduce the data from different experiments [27].",
"The isentropic Mach numbers characterizing the shock intensities for the test cases with $NPR$ = 1.39, 1.46, and 1.54 are $M_i$ = 1.34, 1.41, and 1.52.",
"The lower value is close to the experimental value of the work of Bogar et al[10], while the other two are higher.",
"It must be noted that Bogar, in order to characterize the shock intensity, employed the local Mach number at the edge of the top wall boundary layer immediately upstream of the shock, instead of the isentropic Mach number.",
"Nonetheless, the two numerical values do not differ significantly.",
"The main characteristics of the instantaneous flowfield are shown in figure REF for NPR=1.54.",
"The turbulent structures are represented by showing a positive iso-value of the $Q$ -criterion [28].",
"This qualitative criterion defines as vortex tubes the regions where the second invariant of the velocity gradient tensor $Q$ is positive: $Q = \\frac{1}{2} (\\Omega _{ij}\\Omega _{ij}-S_{ij}S_{ij})>0 $ where $S_{ij}$ and $\\Omega _{ij}$ are the symmetric and anti-symmetric components of $\\nabla u$ .",
"A value of $Q/(U/H_t)=60$ has been chosen and the iso-surface are colored by the local value of the streamwise velocity.",
"Is is possible to notice at first the roll-up of almost two-dimensional vortical structures in the shear layer, which are bent toward the direction of motion and are rapidly replaced by three-dimensional structures developing downstream.",
"In addition, a global unsteadiness with fluctuations in the separation shock position characterizes the flowfield, as shown in figure REF , where two different snapshots of the density gradient field, showing the extreme position reached by the shock system, are reported for NPR = 1.39.",
"The figure also highlights the early development of shear layer instabilities downstream the separation line, thanks to the defition given by Eq.",
"REF for the subgrid length-scale $\\Delta $ .",
"Figure: Iso-surface of the Q-criterion (Q/(U/H t )=60Q/(U/H_t)=60), colored by the local value of the streamwise velocity, for NPR = 1.54.",
"The shock is visualized by an iso-surface of ∇·u\\nabla \\cdot u; theslice in the Z-plane shows the field of ||∇ρ||||\\nabla \\rho ||.Figure: Numerical Schlieren at two different time instants for NPR = 1.39."
],
[
"Wall pressure signature",
"In this section the statistical properties of the fluctuating wall pressure are analyzed by evaluating the root mean square and the intermittency factor.",
"Figure REF a shows a set of instantaneous wall pressure distributions and illustrates the entity of the shock excursion.",
"The root mean square value (r.m.s) of the top pressure fluctuations is reported in figure REF b.",
"Within the attached boundary layer the r.m.s.",
"of the top pressure fluctuations is zero, since, according to the DDES approach, this flowfield region is automatically treated in RANS mode.",
"Instead, downstream of the separation point, there is a sharp peak in the r.m.s.",
"value, corresponding to the excursion zone of the shock system.",
"Moving downstream, the first part of the recirculation region is characterized by a decrease of the r.m.s.",
"of approximately one order of magnitude, while a mild increase of the r.m.s.",
"is observable in the last part of the diffuser.",
"The maximum value of the r.m.s.",
"increases almost linearly with the isentropic Mach number, as shown in figure REF b.",
"It can be noted that the distribution of the r.m.s.",
"of the wall pressure fluctuations is qualitatively very similar to the distributions found in other classical shock wave/boundary layer interaction; see for example the experimental findings of Dupont on an incident shock on a flat plate [29] and of Dolling on a supersonic flow over a compression ramp [30].",
"Figure: a): Streamwise distributions of instantaneous spanwise averaged wall pressure; b): streamwise distributions of time and spanwise averaged wall pressure and the pressure root mean square.Figure: a): streamwise distribution of the intermittency factor.",
"b): streamwise length of the shock motion L s L_s and maximum value of the pressure r.m.s.",
"as afunction of the wall isentropic Mach number.In the region where the wall pressure $p_w$ is intermittent, an intermittency factor $\\gamma $ can be defined [30], this representing the fraction of the time that $p_w$ is above the maximum pressure of the attached boundary layer, i.e., $\\gamma = \\textrm {time}[p_w > (\\bar{p_w}+3\\sigma _w)]/\\textrm {total time}$ For the undisturbed boundary layer, the experimental value of $\\gamma $ is equal to 0.0015 (close to the theoretical Gaussian value of 0.0013).",
"In the RANS simulated attached boundary layer the value of $\\gamma $ would be zero.",
"An intermittency equal to 0.5 corresponds to case of having the same probability for the shock to be located on the left and on the right of the probe, and coincide with the maximum value in the r.m.s.",
"distribution.",
"The streamwise evolutions of the intermittency for the various NPR's are shown in figure REF a.",
"All the distributions of the intermittency versus $x/H_t$ have the same shape, with the value of 0.5 occurring at the same abscissa of the maximum r.m.s.",
"value.",
"This shape is similar to the one produced by the shock in a supersonic ramp flow (Dolling et al.",
"[30]).",
"The distribution of $\\gamma $ can be used to evaluate the shock excursion length.",
"In fact, at any instant, the furthest upstream position of the separation shock is where the incoming boundary layer is firstly disturbed.",
"Thus the distance over which $\\gamma $ increases from 0.0015 (from zero in the present simulations) to 1 is the absolute length scale $L_s$ of the shock motion.",
"Figure REF b shows the nondimensional shock excursion length scale $L_s/H_t$ as a function of the isentropic Mach number $M_i$ .",
"It can be seen that the trend is almost linear, with an increase of 11% in the value of $L_s/H_t$ when rising the isentropic Mach number from 1.34 to 1.52.",
"Figure: NPR = 1.39The pre-multiplied spectra $f \\, E(f)$ of the pressure signal at the top wall are shown in figure REF for the different NPR's as a function of the dimensional frequency $f$ and the streamwise coordinate $x/H_t$ .",
"The power spectral densities have been computed using the Welch method, subdividing the overall pressure record into $K$ segments with 50% overlapping, which are individually Fourier-transformed.",
"The frequency spectra are then obtained by averaging the periodograms of the various segments, which allows to minimize the variance of the PSD estimator, and by applying a Konno-Omachi smoothing filter [31], which ensures a constant bandwidth on a logarithmic scale.",
"The number of segments is $K=10$ for the three cases here investigated.",
"The spectral maps are characterized by two different zones, qualitatively similar for the various NPR's.",
"The first region is associated with the dynamics of the shock system and is identified by a peak whose characteristic frequency is of the order $O(200) Hz$ , located in the proximity of the shock foot.",
"We point out that, while previous investigations based on URANS identified the shock motion as tonal [32], the low-frequency activity predicted by our DDES is rather broadband, the energy content encompassing a whole decade of frequencies.",
"This behavior is in agreement with recent experiments and LES carried out for canonical supersonic boundary layer interactions [29], [33], [34], [35].",
"The second extended region in the spectral densities is the signature of the turbulent activity in the separation bubble, whose dynamics is well captured by the LES branch of the simulations.",
"The peak of the frequency spectra in this zone is centered around $f \\approx 2500 Hz$ and its streamwise location approximately correspond to the reattachment point.",
"This qualitative scenario is shared by the various NPR's.",
"The main effect of increasing the NPR is to shift downstream the location of the low-frequency peak and to (slightly) decrease the characteristic frequency of the oscillations (see section REF for a comparison with experiments)."
],
[
"Morlet wavelet transform",
"The continuous wavelet transform is applied to the unsteady wall pressure signals in order to decompose them in the time-frequency space.",
"An extended review of the application of wavelets to study turbulence phenomena can be found in Farge [18], while only the key theoretical aspects are here reported.",
"The continuous wavelet transform of a discrete time sequence $p_n$ , with equal spacing $\\delta t$ and $n=0...N-1$ , is defined as the convolution of $p_n$ with a scaled and translated version of the mother wavelet $\\psi _0$ : $W_n(s) = \\sum _{n^{\\prime }=0}^{N-1} p_n \\cdot \\psi ^{*}\\left[\\frac{(n^{\\prime }-n)\\delta t}{s}\\right]$ where $*$ denotes the complex conjugate.",
"By varying the wavelet scale $s$ and translating along the time index $n$ , one can construct a picture showing both the amplitude of any features versus the scale and how this amplitude varies with time.",
"In this study, the Morlet wavelet has been chosen since higher resolution in frequency can be achieved when compared with other mother functions.",
"It consists of a plane wave modulated by a Gaussian: $\\psi _0 (\\eta ) = \\pi ^{-1/4} e^{i \\omega _0 \\eta } e^{-\\eta ^2/2}$ where $\\eta $ is a nondimensional time parameter and $\\omega _0$ is the nondimensional frequency, here taken equal to 6 to satisfy the admissibility condition [36].",
"This wavelet is shown in figure REF both in the time and frequency domains.",
"Figure: Morlet Wavelet base: a) real part (solid line) and imaginary part (dashed line) in the time domain; b) the corresponding wavelet in the frequency domain.From the definition of the wavelet coefficient one can directly define the wavelet power spectrum (WPS) as $|W_n(s)|^2$ .",
"The total energy is conserved under the wavelet transform and the equivalent of the Parseval's theorem for wavelet analysis is $\\sigma ^2 = \\frac{\\delta j \\delta t}{C_{\\delta }N} \\sum _{n=0}^{N-1}\\sum _{j=0}^{J}\\frac{|W_n(s)|^2}{s_j}$ where $\\sigma ^2$ is the variance, $\\delta j$ is the scale spacing and $C_{\\delta }$ is a factor coming from the reconstruction of a $\\delta $ function from its wavelet transform.",
"For more details the interested reader could see the work of Torrence and Compo [36].",
"The energy density is then determined as: $E(s,t)= \\frac{|W_n(s)|^2}{s_j}$ Once a wavelet function has been chosen, it is necessary to determine a set of scales $s$ to use in the transform.",
"In the case of non orthogonal wavelet analysis, it is possible to use an arbitrary set of scales to build up a more complete picture.",
"Generally, it is convenient to write the scale as a fractional powers of two: $s_j=s_0 2^{j\\, \\delta j},~~ j=0,1,...,J$ where $s_0$ is the smallest resolvable scale and J determines the largest scale.",
"The scale $s_0$ should be chosen so that the equivalent wavelet period is approximately equal to $2\\delta t$ .",
"The relationship between the equivalent Fourier period $\\lambda $ and the wavelet scale $s$ can be found analytically [36].",
"For the Morlet wavelet with $\\omega _0=6$ it is possible to find that $\\lambda = 1.03 s$ , therefore they are almost equal.",
"In the present analysis, the following parameter values have been chosen: $\\delta t = 5\\cdot 10^{-5} s$ , $s_0=\\delta t$ , $\\delta j=0.125$ and $J = 88$ ."
],
[
"Results of the wavelet analysis",
"The time series of the fluctuating wall pressures are presented in figure REF b for NPR = 1.54, being the results for the other NPR's very similar.",
"These signals are taken from the numerical probes displayed in figure REF a.",
"The first probe is located upstream the flow separation and its signal is almost constant in time, since this zone is the URANS domain (attached boundary layer).",
"The second probe is located in the region where there is the maximum value of root mean square of the pressure oscillation, that is the region of the shock excursion.",
"As shown in figure REF c, the probability density function of wall pressures is bimodal, this being a characteristics of an intermittent signal.",
"In facts, the wall pressure alternates between two different ranges: that of the attached boundary layer and that of the turbulent flow downstream of the separation shock, spending less time near the mean value which falls between the two extrema [30].",
"Thus there are two maxima in the probability curve.",
"The first peak is associated with the probability of finding $p_w$ in the narrow range of pressure associated with the attached boundary layer, hence showing a sharp peak.",
"The latter peak has a broader maximum, that reflects the probability of finding $p_w$ in the wider range of pressures that can be found downstream of the shock wave.",
"The probe number 3 collects the signal at the beginning of the recirculating flow, which is characterized by a lower value of the oscillation amplitude with respect to the others.",
"Finally, the probe number 4 is located in the region of the vortex shedding, its signal shows a large oscillation amplitude and the probability density function is Gaussian, as shown in figure REF .",
"Figure: a): Streamwise distribution of time and spanwise averaged wall pressure with the root mean square for NPR = 1.53.",
"The numbers indicate the pressure probes.",
"b): intermittent pressure signals at probes from 1 to 4;c) probability density function at probe 2; d) probability density function at probe 4.The wavelet power spectrum of the wall pressure signals describes how the variance $\\sigma ^2$ of the wall pressure is distributed in frequency, as described by equation (REF ).",
"Figure REF a shows the normalized wavelet power spectrum $|W_n(s)|^2/\\sigma ^2$ in the frequency-time plane for the time series of the wall pressure from the second probe of the case at NPR = 1.39.",
"The normalization by $1/\\sigma ^2$ gives a measure of the power relative to the white noise [36].",
"The first aspect that can be extracted from this plot is that the spectrum appears as a collection in time of events, characterized by a variation of the amplitude of the oscillation energy and a variation of the frequency of the most energetic events.",
"For example, it is possible to see an important event at 0.18 s with a characteristic frequency around 220 Hz, then a second event at 0.21 s with an increase in frequency (around 300 Hz) and a third event at 0.24 s with a frequency of 250 Hz.",
"Therefore it can be inferred from the data, that the shock movement is not continuous in time but rather intermittent.",
"This aspect can be better appreciated in figure REF a which shows an enlargement of the wavelet power spectrum of the pressure signal from probe no.",
"2 between 0.15 s and 0.30 s, together with the pressure signal itself.",
"It is evident from the picture that the pressure oscillation has an amplitude modulation, which is well captured by the wavelet power spectrum.",
"To provide a better qualitative description of the dominant frequency modes within each of the main frequency branches, figure REF b shows the time series of a selection of the wavelet coefficients for different frequency modes.",
"The time series, in fact, can be reconstructed by summing the real parts of the wavelet transform over all the scales: $p_w(n) = \\frac{\\delta j \\delta t^{1/2}}{C_{\\delta }\\psi _0(0)}\\sum _{j=0}^{J} \\frac{\\mathcal {R} [ W_n(s_j)]}{s_j^{1/2}}$ From this picture it can be seen that the more relevant contributions come from frequencies between 50 Hz and 600 Hz, being the component at 278 Hz the most important.",
"In addition, it is also possible to appreciate the amplitude modulation of the various components.",
"These findings highlight the importance of an accurate time-frequency wavelet analysis in addition to the classical Fourier spectral analysis, since the energy and frequency fluctuations are not observable by means of the latter, that presents only time average information.",
"Figure REF b shows the global wavelet power spectrum, that is the WPS integrated in time: $\\overline{W^2}(s) = \\frac{1}{N}\\sum _{n=0}^{N-1} |W_n(s)|^2$ while figure REF c shows the global energy density $E(s)= \\frac{\\overline{W^2}(s)}{s_j}$ as a function of the scale.",
"This last form is equivalent to the compensated spectra in the classical Fourier analysis.",
"In this way, it is possible to identify the scales most contributing to the energy, being possible to write [36]: $\\sigma ^2 = \\frac{\\delta j \\delta t}{C_{\\delta }} \\sum _{j=0}^{J}E(s)$ From figure REF c it can be seen that there is a energy bump at large temporal scales (low frequencies), with a maximum at 278 Hz.",
"Therefore, the shock movement seems to be characterized by a broadband motion rather than by a sinusoidal motion.",
"It may be worth full to recall that the frequency which gives the maximum value should be interpreted in a statistical sense, that is as the most probable frequency.",
"The analysis of the pressure signal from the probe 3 (located at the beginning of the recirculation bubble) is reported in figures REF d, REF e and REF f. Most of the energy is still located at low frequencies (lower than 1000 Hz) with an important bump at 278 Hz and a secondary bump at 2226 Hz.",
"This second bump comes from integration in time of the intermittent events which can be seen in the frequency-time space between 2000 and 2500 Hz and it is linked to the vortex shedding of the shear layer.",
"Figures REF g, REF h and REF i represent the spectral analysis of the fourth probe, located in the vortex shedding region.",
"The energy density now indicates that most of the energy is spread at higher frequencies, with a most probable frequency of 2647 Hz.",
"In this region dominates the three dimensional vortical structures, even if there is still a energy contribution from the lower frequencies (below 1000 Hz).",
"Figure REF a compares the global wavelet power spectrum for all the pressure probes at NPR = 1.39, in order to quantify the shifting of the energy from the lower frequencies, characterizing the shock excursion region, to the higher frequencies which characterize the turbulent recirculating region.",
"This picture is qualitatively the same shown by the Fourier analysis.",
"The comparison of the global WPS for the different NPR's at pressure probes no.",
"2 and 4 are shown in figure REF b.",
"It can be seen that, qualitatively, the behavior of the spectra are very similar.",
"While, from a quantitative point of view, the signals from probe no.",
"2 show that NPR=1.54 has the highest power at low frequencies.",
"This is correlated to the highest shock intensity of this NPR.",
"The signals from probe no.",
"4, instead, are similar also from a quantitative point of view, indicating the same behavior for the turbulent separated region.",
"Figure: Global WPS/scale at probe no.",
"4Figure: a): Wavelet Power Spectrum together with wall pressure signal (probe no.",
"2) for NPR = 1.39.;b): Time series for the wavelet coefficients of the pressure signal (probe no.",
"2) for different frequency modes.",
"The amplitude ofthe coefficients are shifted to facilitate the interpretation.Figure: a): Comparison of the global WPS for the different probes at NPR=1.39;b): Comparison of the global WPS for the different NPR's at probe no.",
"2 and 4.Figure: Peak frequencies related to the shock movement as a function of the isentropic wall Mach number, computed with the wavelet analysis and the Fourier analysis.In the experimental data xvxv is the non dimensional length of the nozzle.Finally, figure REF compares the values of the frequencies characterizing the shock movement obtained with wavelet and Fourier analysis with those taken from the Fourier analysis of the experimental data[10] (open symbols).",
"The test case with the isentropic Mach number equal to 1.34 is the only one which falls in the experimental range of shock Mach numbers [10] $M_s$ ($1.280 \\le M_S \\le 1.347$ ).",
"The computed value result to be in reasonable agreement with the experimental ones, although the characteristic frequencies are sightly overestimated.",
"We can speculate that such discrepancy might be ascribed to some differences between the experiment and the simulations, as the presence of side walls and suction slots employed in the experimental configuration to remove the boundary layer."
],
[
"Conclusions",
"Delayed detached eddy simulations (DDES) of a planar nozzle with flow separation have been carried out for a Reynolds number, based on stagnation chamber properties and throat height, equal to $1.5 \\cdot 10^6$ and for different nozzle pressure ratios (or equivalently different isentropic Mach numbers).",
"The nozzle flow simulated in this study is characterized by a separation shock with a classical lambda shape and by an important recirculation zone, which extends for several nozzle throat heights.",
"All the simulations were able to capture a self-sustained unsteadiness of the shock system.",
"As a first step, a classical statistical description of this unsteadiness has been carried out.",
"The shock region is characterized by a well defined peak in the root mean square distribution of the oscillating wall pressure.",
"The amplitude of this peak increases with increasing Mach number.",
"The evaluation of the intermittency factor has allowed to evaluate the shock excursion length, which can reach the 20% of the throat height.",
"All these findings qualitatively agree with the data of the experimental reference nozzle and with the data of other shock configurations, like compression ramps and incident shock waves on a flat plate.",
"The spectral analysis has been conducted by using Fourier analysis and the Morlet wavelet transform, which is a well suited tool to analyze non stationary time series.",
"The Fourier analysis has allowed to individuate a low frequency region, around 250 Hz, associated with the shock movement, and a higher frequency region (around 2500 Hz) associated with the turbulent separated flow.",
"According to the wavelet analysis, the shock movement and the recirculating region have been recognized to be characterized, in the time-frequency space, by a collection of events with a modulation of the oscillation amplitude and a modulation of the frequency."
],
[
"Acknowledgments",
"The simulations have been performed thanks to computational resources provided by the Italian Computing center CINECA under the ISCRA initiative (grant IscrB_SW-DES-1).",
"MB was supported by the SIR program 2014 (jACOBI project, grant RBSI14TKWU), funded by MIUR (Ministero dell'Istruzione dell'Università e della Ricerca).",
"Wavelet software was provided by C. Torrence and G. Compo, and is available at URL: http://atoc.colorado.edu/research/wavelets/."
]
] | 1606.05114 |
[
[
"On the octonionic Bergman kernel"
],
[
"Abstract By introducing a suitable new definition for the inner product on the octonionic Bergman space, we determine the explicit form of the octonionic Bergman kernel, in the framework of octonionic analysis which is non-commutative and non-associative."
],
[
"Introduction",
"In complex analysis the Szegö kernel and Bergman kernel are well-known, which had also been generalized into Clifford analysis (including quaternionic analysis as a special case, see [2]).",
"But in octonionic analysis the existence of such kernels is still unknown, let alone the explicit expressions.",
"The difficulty arises mainly because the octonion algebra (Cayley algebra) is non-associative.",
"The motivation for us to consider this kind of problem is that we want to unify the formulation of the analytic function theory in the largest normed division algebra over $\\mathbb {R}$ , namely, in octonions $\\mathbb {O}$ (including complex numbers, quaternions as its special cases).",
"Recall that in complex analysis the Bergman space on the unit disc is defined to be the collection of functions that are holomorphic and square integrable on the unit disc.",
"This definition can be naturally generalized to octonionic analytic functions.",
"Since the Cayley algebra is non-commutative, there exist two different but symmetric octonionic analytic function theory.",
"In this paper we focus on the left octonionic analytic functions, and we denote by $\\mathcal {B}^2(B)$ the corresponding octonionic Bergman space, where $B$ is the unit ball in $\\mathbb {R}^8$ centered at origin.",
"A nature problem comes: Does the octonionic Bergman kernel exist?",
"and what is it?",
"Of course this problem is closely related to the definition of the associated inner product.",
"Usually the inner product of two Bergman functions $f$ and $g$ is defined to be the integral of $f\\overline{g}$ on $B$ .",
"Since the octonions is non-associative, the usual definition is no longer valid to guarantee the existence of the kernel.",
"We thus need to give a new definition.",
"Definition 1.1 (inner product on $\\mathcal {B}^2(B)$ ) Let $f, g\\in \\mathcal {B}^2(B)$ , we define $(f,g)_B:=\\frac{1}{\\omega _8}\\int _B\\left(\\overline{g(x)}\\frac{\\overline{x}}{|x|}\\right)\\left(\\frac{x}{|x|}f(x)\\right)dV,$ where $\\omega _8$ is the surface area of the unit sphere in $\\mathbb {R}^8$ , $dV$ is the volume element on $B$ .",
"Note that this modified inner product is real-linear and conjugate symmetric.",
"The induced norm $\\Vert f\\Vert _B^2:=(f,f)_B=\\frac{1}{\\omega _8}\\int _B|f|^2dV$ coincides with the norm induced by the usual inner product.",
"We can now state the main theorem of this paper.",
"Theorem 1.1 Let $B(x,a)=\\frac{\\left(6(1-|a|^2|x|^2)+2(1-\\overline{x}a)\\right)(1-\\overline{x}a)}{|1-\\overline{x}a|^{10}},$ then $B(\\cdot ,a)$ is the desired octonionic Bergman kernel, i.e., $B(\\cdot ,a)\\in \\mathcal {B}^2(B)$ , and for any $f\\in \\mathcal {B}^2(B)$ and any $a\\in B$ , there holds the following reproducing formula $f(a)=(f,B(\\cdot ,a))_B.$ The rest of the paper is organized as follows.",
"In Section 2 we give a brief review on the octonion algebra and octonionic analysis.",
"In Section 3 we will exploit our new idea in defining the structure of the inner product to investigate the octonionic Szegö kernel for the unit ball in $\\mathbb {R}^8$ .",
"Section 4 is then devoted to the proof of our main result Theorem REF .",
"In the last section we point out that the Bergman kernel can be unified in one form in both complex analysis and hyper-complex analysis."
],
[
"The octonions",
"If an algebra $\\mathbb {A}$ is meanwhile a normed vector space, and its norm “$\\Vert \\cdot \\Vert $ ” satisfies $\\Vert ab\\Vert =\\Vert a\\Vert \\Vert b\\Vert $ , then we call $\\mathbb {A}$ a normed algebra.",
"If $ab=0$ ($a, b\\in \\mathbb {A}$ ) implies $a=0$ or $b=0$ , then we call $\\mathbb {A}$ a division algebra.",
"Early in 1898, Hurwitz had proved that the real numbers $\\mathbb {R},$ complex numbers $\\mathbb {C},$ quaternions $\\mathbb {H}$ and octonions $\\mathbb {O}$ are the only normed division algebras over $\\mathbb {R}$ ([4]), with the imbedding relation $\\mathbb {R}\\subseteq \\mathbb {C}\\subseteq \\mathbb {H}\\subseteq \\mathbb {O}$ .",
"As the largest normed division algebra, octonions, which are also called Cayley numbers or the Cayley algebra, were discovered by John T. Graves in 1843, and then by Arthur Cayley in 1845 independently.",
"Octonions are an 8 dimensional algebra over $\\mathbb {R}$ with the basis $e_0,e_1,\\ldots ,e_7$ satisfying $e_0^2=e_0,~e_ie_0=e_0e_i=e_i,~e_i^2=-1,~\\text{for}~i=1,2,\\ldots ,7.$ So $e_0$ is the unit element and can be identified with 1.",
"Denote $W=\\lbrace (1,2,3),(1,4,5),(1,7,6),(2,4,6),(2,5,7),(3,4,7),(3,6,5)\\rbrace .$ For any triple $(\\alpha ,\\beta ,\\gamma )\\in W$ , we set $e_\\alpha e_\\beta =e_\\gamma =-e_\\beta e_\\alpha ,\\quad e_\\beta e_\\gamma =e_\\alpha =-e_\\gamma e_\\beta ,\\quad e_\\gamma e_\\alpha =e_\\beta =- e_\\alpha e_\\gamma .$ Then by distributivity for any $x=\\sum _0^7 x_ie_i$ , $y=\\sum _0^7 y_je_j \\in \\mathbb {O}$ , the multiplication $xy$ is defined to be $xy:=\\sum _{i=0}^7\\sum _{j=0}^7x_iy_je_ie_j.$ For any $x=\\sum _0^7 x_ie_i \\in \\mathbb {O}$ , $\\mbox{Re}\\,x:=x_{0}$ is called the scalar (or real) part of $x$ and $\\overrightarrow{x}:=x-\\mbox{Re}\\,x$ is called its vector part.",
"$\\overline{x}:=\\sum _0^7x_i\\overline{e_i}=x_0-\\overrightarrow{x}$ and $|x|:=(\\sum _0^7x_i^2)^\\frac{1}{2}$ are respectively the conjugate and norm (or modulus) of $x$ , they satisfy: $|xy|=|x||y|,$ $x\\overline{x}=\\overline{x}x=|x|^2,$ $\\overline{xy}=\\overline{y}\\,\\overline{x}$ $(x,y\\in \\mathbb {O}).$ So if $x\\ne 0,$ $x^{-1}=\\overline{x}/{|x|^2}$ gives the inverse of $x.$ Octonionic multiplication is neither commutative nor associative.",
"But the subalgebra generated by any two elements is associative, namely, The octonions are alternative.",
"$[x, y, z]:=(xy)z-x(yz)$ is called the associator of $x, y, z\\in \\mathbb {O},$ it satisfies ([1], [5]) $[x,y,z]=[y,z,x]=-[y,x,z], \\quad [x,x,y]=[\\overline{x},x,y]=0.$"
],
[
"The octonionic analysis",
"As a generalization of complex analysis and quaternionic analysis to higher dimensions, the study of octonionic analysis was originated by Dentoni and Sce in 1973 ([3]), and it was not until 1995 that it began to be systematically investigated by Li et al ([6]).",
"Octonionic analysis is a function theory on octonionic analytic (abbr.",
"$\\mathbb {O}$ -analytic) functions.",
"Suppose $\\Omega $ is an open subset of $\\mathbb {R}^8$ , $f=\\sum _0^7f_je_j\\in C^1(\\Omega ,\\mathbb {O})$ is an octonion-valued function, if $Df=\\sum _{i=0}^{7}e_{i}\\frac{\\partial f}{\\partial x_{i}}=\\sum _{i=0}^{7}\\sum _{j=0}^{7}\\frac{\\partial f_j}{\\partial x_{i}}e_ie_j=0$ $\\left(fD=\\sum _{i=0}^{7} \\frac{\\partial f}{\\partial x_{i}}e_{i}=\\sum _{i=0}^{7}\\sum _{j=0}^{7}\\frac{\\partial f_j}{\\partial x_{i}}e_je_i=0\\right),$ then $f$ is said to be left (right) $\\mathbb {O}$ -analytic in $\\Omega $ , where the generalized Cauchy–Riemann operator $D$ and its conjugate $\\overline{D}$ are defined by $D:=\\sum _{i=0}^7e_i\\frac{\\partial }{\\partial x_i},~~\\overline{D}:=\\sum _{i=0}^7\\overline{e_i}\\frac{\\partial }{\\partial x_i}$ respectively.",
"A function $f$ is $\\mathbb {O}$ -analytic means that $f$ is meanwhile left $\\mathbb {O}$ -analytic and right $\\mathbb {O}$ -analytic.",
"From $\\overline{D}(Df)=(\\overline{D}D)f=\\triangle f=f(D\\overline{D})=(fD)\\overline{D},$ we know that any left (right) $\\mathbb {O}$ -analytic function is always harmonic.",
"In the sequel, unless otherwise specified, we just consider the left $\\mathbb {O}$ -analytic case as the right $\\mathbb {O}$ -analytic case is essentially the same.",
"A Cauchy-type integral formula and a Laurent-type series for this setting are: Lemma 2.1 (Cauchy's integral formula, see [3], [8]) Let $\\mathcal {M}\\subset \\Omega $ be an 8-dimensional, compact differentiable and oriented manifold with boundary.",
"If $f$ is left $\\mathbb {O}$ -analytic in $\\Omega $ , then $f(x)=\\frac{1}{\\omega _8}\\int _{y\\in \\partial \\mathcal {M}}E(y-x)(d\\sigma _yf(y)),\\quad x\\in \\mathcal {M}^o,$ where $E(x)=\\frac{\\overline{x}}{|x|^8}$ is the octonionic Cauchy kernel, $d\\sigma _y=n(y)dS$ , $n(y)$ and $dS$ are respectively the outward-pointing unit normal vector and surface area element on $\\partial \\mathcal {M}$ , $\\mathcal {M}^o$ is the interior of $\\mathcal {M}$ .",
"Lemma 2.2 (Laurent expansion, see [14], [13]) Let $\\mathcal {D}$ be an annular domain in $\\mathbb {R}^8$ .",
"If $f$ is left $\\mathbb {O}$ -analytic in $\\mathcal {D}$ , then $f(x)=\\sum _{k=0}^\\infty P_kf(x)+\\sum _{k=0}^\\infty Q_kf(x),\\quad x\\in \\mathcal {D},$ where $P_kf$ and $Q_kf$ are respectively the inner and outer spherical octonionic-analytics of order $k$ associated to $f$ .",
"Octonionic analytic functions have a close relationship with the Stein–Weiss conjugate harmonic systems.",
"If the components of $F$ consist a Stein–Weiss conjugate harmonic system on $\\Omega \\subset \\mathbb {R}^8$ , then $\\overline{F}$ is $\\mathbb {O}$ -analytic on $\\Omega $ .",
"But conversely this is not true ([7]).",
"For more information and recent progress about octonionic analysis, we refer the reader to [6], [9], [10], [11], [12], [14], [15]."
],
[
"The octonionic Szegö kernel",
"To see how our new definition works, let us check the octonionic Szegö kernel for the unit ball in $\\mathbb {R}^8$ .",
"Recall that on the unit ball the octonionic Hardy space $\\mathcal {H}^2(B)$ consists of the left octonionic analytic functions whose mean square value on the sphere is bounded for radius $r\\in [0,1)$ .",
"For any $f\\in \\mathcal {H}^2(B)$ , according to the Cauchy's integral formula, for all $a\\in B$ there holds $f(a)&=\\frac{1}{\\omega _8}\\int _{x\\in S^7}\\frac{\\overline{x}-\\overline{a}}{|x-a|^8}(xf(x))dS\\\\&=\\frac{1}{\\omega _8}\\int _{x\\in S^7}\\left(\\frac{\\overline{1-\\overline{x}a}}{|1-\\overline{x}a|^8}\\overline{x}\\right)(xf(x))dS,$ where $S^7=\\partial B$ is the unit sphere, $dS$ is the area element on $S^7$ .",
"If we define the inner product for $\\mathcal {H}^2(B)$ to be $(f, g)_{S^7}:=\\frac{1}{\\omega _8}\\int _{S^7}(\\overline{\\eta g(\\eta )})(\\eta f(\\eta ))dS=\\frac{1}{\\omega _8}\\int _{S^7}(\\overline{g(\\eta )}\\overline{\\eta })(\\eta f(\\eta ))dS,$ and let $S(x,a)=\\frac{1-\\overline{x}a}{|1-\\overline{x}a|^8},$ then $S(\\cdot ,a)\\in \\mathcal {H}^2(B)$ , and the Cauchy's integral formula can be rewritten as $f(a)=(f,S(\\cdot ,a))_{S^7}.$ We call $S(\\cdot ,a)$ the octonionic Szegö kernel.",
"Denote by $L^2(S^7)$ the space of square integrable (octonion-valued) functions on the unit sphere, for which we define its inner product to be the same as that in (REF ).",
"We have Proposition 3.1 Let $f, g\\in L^2(S^7)$ be associated with the spherical octonionic-analytics expansions: $f(\\omega )=\\sum _{k=0}^\\infty (P_kf(\\omega )+Q_kf(\\omega )),\\quad g(\\omega )=\\sum _{k=0}^\\infty (P_kg(\\omega )+Q_kg(\\omega )),\\quad \\omega \\in S^7.$ Then $(f, g)_{S^7}=&\\sum _{k=0}^\\infty \\left((P_kf, P_kg)_{S^7}+(Q_kf,Q_kg)_{S^7}\\right)\\\\&+\\sum _{k=0}^\\infty \\left((P_kf,Q_{k+1}g)_{S^7}+(Q_{k+1}f, P_kg)_{S^7}\\right).$ From $\\triangle (xP_kf(x))=x\\triangle (P_kf(x))+2D(P_kf(x))=0,$ we can easily see that the restriction of $xP_kf(x)$ on $S^7$ is a spherical harmonic of order $k+1$ .",
"Similarly, the restriction of $xQ_kf(x)$ on $S^7$ is a spherical harmonic of order $k$ .",
"The proposition immediately follows by the fact that spherical harmonics of different orders are mutually orthogonal.",
"Thus we get Corollary 3.1 Let $f\\in L^2(S^7)$ be associated with the spherical octonionic-analytics expansion $f(\\omega )=\\sum _{k=0}^\\infty (P_kf(\\omega )+Q_kf(\\omega )),\\quad \\omega \\in S^7.$ Then $\\Vert f\\Vert ^2_{S^7}=\\sum _{k=0}^\\infty \\left(\\Vert P_kf\\Vert ^2_{S^7}+\\Vert Q_kf\\Vert ^2_{S^7}\\right)+\\sum _{k=0}^\\infty 2{\\rm Re}\\left((P_kf, Q_{k+1}f)_{S^7}\\right).$ Remark: Proposition REF is similar to the Parseval's theorem.",
"It is worthwhile to note that this version is a bit different from that in Clifford analysis where the second part in the summation vanishes ([2]), here $(P_kf, Q_{k+1}g)_{S^7}$ may not be zero.",
"Below we give a counter-example.",
"Let $f(x)=x_1-x_0e_1,$ $g(x)=\\frac{\\overline{x}}{|x|^{12}}(x_1x_2e_4+x_0x_2e_5+x_0x_1e_6).$ Then $P_1f=f$ , $Q_2g=g$ , but $(P_1f, Q_2g)_{S^7}=\\frac{-2e_6}{\\omega _8}\\int _{S^7}x_0^2x_1^2dS\\ne 0.$"
],
[
"Derivation of the octonionic Bergman kernel",
"In this section we will prove Theorem REF .",
"For the main idea we use in the proof one can also refer to [2].",
"[Proof of Theorem REF ] By definition it is straightforward that $(f,g)_B=\\int _0^1r^7(f_r,g_r)_{S^7}dr,$ where $f_r(\\eta )=f(r\\eta )$ , $\\eta \\in S^7$ .",
"Together with Proposition REF , we get $(f,g)_B&=\\sum _{k=0}^\\infty (P_kf,P_kg)_B\\\\&=\\sum _{k=0}^\\infty \\int _0^1r^{2k+7}(P_kf,P_kg)_{S^7}dr\\\\&=\\sum _{k=0}^\\infty (2k+8)^{-1}(P_kf,P_kg)_{S^7}.$ Therefore, $f\\in \\mathcal {B}^2(B)$ if and only if $f$ is left octonionic analytic in $B$ and $\\Vert f\\Vert _B^2=\\sum _{k=0}^\\infty (2k+8)^{-1}\\Vert P_kf\\Vert _{S^7}^2<\\infty .$ From this viewpoint, if $f\\in \\mathcal {H}^2(B)$ , then $\\sqrt{T}f:=\\sum _{k=0}^\\infty \\sqrt{2k+8}P_kf\\in \\mathcal {B}^2(B).$ Similarly, if $g$ is left octonionic analytic in $B_R$ (the ball centered at the origin of radius $R$ , with $R>1$ ), then $\\sqrt{T}g\\in \\mathcal {B}^2(B_{R^{\\prime }})$ , with $1\\le R^{\\prime }<R$ .",
"Consequently, $Tg:=\\sqrt{T}^2g=\\sum _{k=0}^\\infty (2k+8)P_kg\\in \\mathcal {B}^2(B_{R^{\\prime }}),~1\\le R^{\\prime }<R.$ Now, assume $f\\in \\mathcal {B}^2(B)$ , when $|a|<r$ we have $f(a)&=\\frac{1}{\\omega _8}\\int _{\\partial B_r}\\frac{\\overline{x}-\\overline{a}}{|x-a|^8}d\\mu _xf(x)\\nonumber \\\\&=\\frac{r^7}{\\omega _8}\\int _{S^7}\\frac{r\\overline{\\eta }-\\overline{a}}{|r\\eta -a|^8}(\\eta f(r\\eta ))dS\\nonumber \\\\&=\\lim _{r\\rightarrow 1^-}\\frac{r^7}{\\omega _8}\\int _{S^7}\\frac{r\\overline{\\eta }-\\overline{a}}{|r\\eta -a|^8}(\\eta f(r\\eta ))dS\\nonumber \\\\&=\\lim _{r\\rightarrow 1^-}r^7(f_r, S^r(\\cdot ,a))_{S^7},$ where $S^r(x,a)=\\frac{r-\\overline{x}a}{|r-\\overline{x}a|^8}.$ Since $S^r(x,a)$ is left octonionic analytic in $B_{r/|a|}$ ($r/|a|>1$ ) with respect to $x$ , we have $TS^r(\\cdot ,a)=\\sum _{k=0}^\\infty (2k+8)P_kS^r(\\cdot ,a)\\in \\mathcal {B}^2(B).$ So, $(f_r,TS^r(\\cdot ,a))_B&=\\sum _{k=0}^\\infty (2k+8)^{-1}(P_kf_r,(2k+8)P_kS^r(\\cdot ,a))_{S^7}\\nonumber \\\\&=\\sum _{k=0}^\\infty (P_kf_r,P_kS^r(\\cdot ,a))_{S^7}\\nonumber \\\\&=(f_r,S^r(\\cdot ,a))_{S^7}.$ By (REF ) and (REF ) we get $f(a)=\\lim _{r\\rightarrow 1^-}r^7(f_r,TS^r(\\cdot ,a))_B=(f,TS(\\cdot ,a))_B,$ where $S(\\cdot ,a)$ is the octonionic Szegö kernel.",
"We can now see that the octonionic Bergman kernel $B(x,a)$ is $B(x,a)=TS(x,a)=\\sum _{k=0}^\\infty (2k+8)P_kS(x,a).$ The remaining thing we need to do is to evaluate the above summation.",
"To this end, first note that $S(x,a)=\\mathcal {K}(E(x,\\overline{a})),$ where $E(x,a)=\\frac{\\overline{x}-\\overline{a}}{|x-a|^8}$ ($|x|>1$ ), $\\mathcal {K}f:=E(x,0)f(x^{-1})$ is the Kelvin inversion.",
"So, $P_kS(x,a)=\\mathcal {K}(Q_kE(x,\\overline{a}))=\\overline{x}\\overline{Q_kE(x,a)}|x|^{2k+6}.$ Define the adjoint operator $A$ as follows: $(Af)(x):=\\overline{D}(|x|^{-6}\\overline{f}(x/|x|^2)),$ then it is easy to show that $A(Q_kE(x,a))=(2k+8)\\overline{x}\\overline{Q_kE(x,a)}|x|^{2k+6}.$ Hence, $B(x,a)&=\\sum _{k=0}^\\infty A(Q_kE(x,a))\\\\&=A\\left(\\sum _{k=0}^\\infty Q_kE(x,a)\\right)\\\\&=A(E(x,a))\\\\&=\\overline{D}_x\\left(\\frac{x-a|x|^2}{|1-\\overline{x}a|^8}\\right)\\\\&=\\frac{\\left(6(1-|a|^2|x|^2)+2(1-\\overline{x}a)\\right)(1-\\overline{x}a)}{|1-\\overline{x}a|^{10}}.$ The proof of Theorem REF is complete."
],
[
"Final remarks",
"By direct computation one can show that $\\overline{B(x,a)}\\overline{x}=\\overline{D}_a\\left(\\frac{1-|a|^2|x|^2}{|1-a\\overline{x}|^8}\\right).$ In fact, similar formulas also hold in both complex analysis and Clifford analysis.",
"We therefore can unify the reproducing formulas in complex and hyper-complex contexts.",
"Let ${A}$ denote the complex algebra or hyper-complex algebra, i.e., ${A}$ may refer to complex numbers $\\mathbb {C}$ , quaternions $\\mathbb {H}$ , octonions $\\mathbb {O}$ , or Clifford algebra ${C}$ .",
"Assume that the dimension of ${A}$ is $m$ .",
"Then for any function $f$ which belongs to the Bergman space $\\mathcal {B}^2(B_m)$ and any point $a\\in B_m$ ($B_m$ is the unit ball centered at origin in $\\mathbb {R}^m$ ), there holds $f(a)&=(f,B(\\cdot ,a))_{B_m} \\\\&=\\frac{1}{\\omega _m}\\int _{B_m}\\left(\\overline{B(x,a)}\\frac{\\overline{x}}{|x|}\\right)\\left(\\frac{x}{|x|}f(x)\\right)dV\\\\&=\\frac{1}{\\omega _m}\\int _{B_m}\\overline{D}_a\\frac{1-|a|^2|x|^2}{|1-a\\overline{x}|^m}\\left(\\frac{x}{|x|^2}f(x)\\right)dV\\\\&=\\frac{1}{\\omega _m}\\int _{B_m}\\frac{\\left((m-2)(1-|a|^2|x|^2)+2(1-\\overline{a}x)\\right)(\\overline{x}-|x|^2\\overline{a})}{|1-\\overline{x}a|^{m+2}}\\left(\\frac{x}{|x|^2}f(x)\\right)dV,$ where $\\omega _m$ is the surface area of the unit sphere in $\\mathbb {R}^m$ , $dV$ is the volume element on $B_m$ , and $D$ is the generalized Cauchy–Riemann operator in the respective context.",
"Acknowledgements This work was supported by the Scientific Research Grant of Guangdong University of Foreign Studies for Introduction of Talents (No.",
"299–X5122145), the Research Grant of Guangdong University of Foreign Studies for Young Scholars (No.",
"299–X5122199), and the Foundation for Young Innovative Talents in Higher Education of Guangdong, China (No.",
"2015KQNCX037)."
]
] | 1606.05095 |
[
[
"Quantum Wells in Photovoltaic Cells"
],
[
"Abstract The fundamental efficiency limit of a single bandgap solar cell is about 31% at one sun with a bandgap of about Eg = 1.35 eV (1), determined by the trade-off of maximising current with a smaller bandgap and voltage with a larger bandgap.",
"Multiple bandgaps can be introduced to absorb the broad solar spectrum more efficiently.",
"This can be realised in multi- junction cells, for example, where two or more cells are stacked on top of each other either mechanically or monolithically connected by a tunnel junction.",
"An alternative or complementary (see section 1.4) approach is the quantum well cell (QWC)."
],
[
"Quantum Wells in Photovoltaic Cells",
"QWs in PV C Rohr, P Abbott, I M Ballard, D B Bushnell, J P Connolly, N J Ekins-Daukes, K W J Barnham Experimental Solid State Physics, Imperial College London, U.K.",
"The fundamental efficiency limit of a single bandgap solar cell is about 31% at one sun with a bandgap of about E$_g = 1.35$ eV [1], determined by the trade-off of maximising current with a smaller bandgap and voltage with a larger bandgap.",
"Multiple bandgaps can be introduced to absorb the broad solar spectrum more efficiently.",
"This can be realised in multi-junction cells, for example, where two or more cells are stacked on top of each other either mechanically or monolithically connected by a tunnel junction.",
"An alternative – or complementary (see sec:tandem) – approach is the quantum well cell (QWC)."
],
[
"(QWs) are thin layers of lower bandgap material in a host material with a higher bandgap.",
"Early device designs placed the QWs in the doped regions of a p-n device [2], but superior carrier collection is achieved when an electric field is present across the QWs.",
"More recent QWC designs have employed a p-i-n structure [3] with the QWs located in the intrinsic region; a schematic bandgap diagram is shown in MQW-schematic.",
"The carriers escape from the QWs thermally and by tunnelling [4], [5], [6].",
"Figure: Schematic bandgap diagram of a Quantum Well Cell.",
"Theabsorption threshold is determined by the lowest energy levels inthe quantum wells.",
"Carriers escape thermally assisted and bytunnelling.The photocurrent is enhanced in a QWC compared to a cell made without QWs also known as barrier control, and experimentally it is observed that the voltage is enhanced compared with a bulk cell made of the QW material [7].",
"Hence QWCs can enhance the efficiency if the photocurrent enhancement is greater than the loss in voltage [8].",
"The potential for an efficiency enhancement is also discussed in references [9], [10].",
"The number of QWs is limited by the maximum thickness of the i-region maintaining an electric field across it.",
"QWCs have been investigated quite extensively, both on GaAs as well as on InP substrates, and have been discussed in some detail also in references [11], [12].",
"Historically, the first p-i-n QWCs were in the material system AlGaAs/GaAs (barrier/well) on GaAs [13], [14], [15], [16], [17].",
"AlGaAs is closely lattice-matched to GaAs and the bandgap can be easily varied by changing the Al fraction (see Eg-a) up to about 0.7 where the bandgap becomes indirect.",
"However the material quality particularly that of AlGaAs is relatively poor because of contamination during the epitaxial growth, leading to a high number of recombination centres and hence a high dark current.",
"Figure: Lattice constant versus bandgap for In 1-x Ga x As y P 1-y \\rm In_{1-x}\\-Ga_{x}\\-As_{y}\\-P_{1-y} and AlGaAscompounds.",
"Also indicated is the optimum bandgap for asingle-bandgap PV cell under 1 sun, and the emission peaks ofselective emitters Thulia and Holmia.An alternative material to AlGaAs is InGaP which has better material quality, and an InGaP/GaAs QWC has been demonstrated [18].",
"However, the ideal single bandgap for a 1 sun solar spectrum is E$_g = 1.35$ eV, as indicated in Eg-a, while GaAs has a bandgap of E$_g = 1.42$ eV and that of AlGaAs is still higher.",
"The second material should therefore have a smaller bandgap than GaAs to absorb the longer wavelength light, keeping in mind that the quantum confinement raises the effective bandgap of the QWs.",
"GaAs/InGaAs QWCs fulfil this criterion and they have been studied quite extensively [19], [20], [21], [22], [23], [24].",
"However, because InGaAs has a larger lattice constant than GaAs (see Eg-a) it is strained.",
"If the strain exceeds a critical value relaxation occurs at the top and bottom of the MQW stack, and the dislocations result in an increase in recombination and hence increased dark current [22].",
"This limitation means that strained GaAs/InGaAs QWCs cannot improve the efficiency compared to GaAs control cells [24].",
"Strain compensation techniques can be used to overcome this problem (see sec:sb), and QWCs in the material system GaAsP/InGaAs on GaAs have been investigated [25], [24], [26], [27], [28], [29].",
"These devices are also very suitable as bottom cells in a tandem configuration (see sec:sb).",
"QWCs based on InP are of interest for solar as well as for thermo-photovoltaic (TPV) applications.",
"First, material combinations such as InP/InGaAs were investigated [30], [31], [32], [6], which was then extended to quaternary material (lattice-matched to InP) InP/$\\rm In_{1-x}\\-Ga_{x}\\-As_{y}\\-P_{1-y}$ ($x=0.47y$ ) [33], [34], [35].",
"As in the GaAsP/InGaAs system on GaAs, strain compensation techniques have been employed in $\\rm In_{1-x}\\-Ga_{x}\\-As$ /$\\rm In_{1-z}\\-Ga_{z}\\-As$ QWCs on InP [36], [37], [38], [39], [40], [41].",
"QWCs have practical advantages due to both quantised energy levels and the greater flexibility of choice of materials.",
"In particular, this allows engineering of the bandgap to better match the incident spectrum.",
"The absorption threshold can be varied by changing the width of the QW and/or by changing its material composition.",
"This flexibility can be further increased by employing strain compensation techniques which are explained in more detail in Section sec:sb.",
"In this way, longer wavelengths for absorption can be achieved than what is possible with lattice-matched bulk material, allowing optimisation of the bandgap.",
"The application of QWCs (based on InP) for thermophotovoltaics is discussed in sec:tpv.",
"For TPV applications the same concept of strain compensation can be applied to extend the absorption to longer wavelengths.",
"This is important for relatively low temperature sources combined with appropriate selective emitters for example based on Holmia or Thulia.",
"Several studies indicate that QWCs have a better temperature dependence of efficiency than bulk cells [16], [42], [43], [44]."
],
[
"Strain compensation",
"In order to avoid strain relaxation, strain compensation techniques, first proposed by Matthews and Blakeslee [45], can be used to minimise the stress at the interface between the substrate and a repeat unit of two layers with different natural lattice constants.",
"Layers with larger and smaller natural lattice constant compared to the substrate result in compressive and tensile strain respectively as shown in strainbalance.",
"When these two layers are strained against each other, the strain is compensated and the net force exerted on the adjacent layers is reduced.",
"Therefore the build up of strain in a stack can be reduced, and hence its critical thickness is increased so that more such repeat units can be grown on top of each other without relaxation.",
"If the strain compensation conditions are optimised to give zero stress at the interfaces between the repeat units, an unlimited number of periods can be grown in principle.",
"In addition each individual layer has to remain below its critical thickness which means that this concept can only be used for thin layers such as quantum wells.",
"Figure: Schematic diagram of strain compensation: the naturallattice constant of GaAsP (a 1 _1) is smaller than that of the GaAssubstrate (a 0 _0), and GaAsP barriers are therefore tensilestrained, while the natural lattice constant of InGaAs (a 2 _2) islarger, and hence InGaAs QWs are compressively strained.This technique is very suitable for multi quantum well structures; the barriers and wells are made of different materials with larger and smaller bandgaps but they can also have smaller and larger lattice constants (see Eg-a), i.e.",
"with tensile and compressive strain respectively (see strainbalance and strainbalance-bandgap).",
"That means strained materials can be used for the quantum wells in order to reach lower bandgaps without compromising the quality of the device as dislocations are avoided.",
"This technique, which extends the material range allowing further bandgap engineering, was first applied to photovoltaics in GaAsP/InGaAs QWCs [25].",
"Figure: Bandgap diagram of a strain compensated QWC with InGaAsQWs and GaAsP barriers.For highly strained layers the difference in elastic constants becomes significant and it needs to be taken into account when considering the conditions for zero stress [46].",
"The strain for each layer $i$ is $\\epsilon _i = \\frac{a_0 - a_i}{a_i}$ where $a_0$ is the lattice constant of the substrate and $a_i$ the natural lattice constant of layer $i$ .",
"The zero-stress strain-balance conditions are as follows: $\\epsilon _1 t_1 A_1 a_2 + \\epsilon _2 t_2 A_2 a_1 = 0 \\\\a_0 = \\frac{t_1 A_1 a_1 a_2^2 + t_2 A_2 a_2 a_1^2}{t_1 A_1 a_2^2 + t_2 A_2 a_1^2}$ where $A=C_{11} + C_{12} - \\frac{2 C_{12}^2}{C_{11}}$ with elastic stiffness constants $C_{11}$ and $C_{12}$ , different for each layer, depending on the material.",
"The strain energy must be kept below a critical value, however, to avoid the onset of three-dimensional growth [47].",
"Lateral layer thickness modulations, particularly in the tensile strained material (i.e.",
"barriers), are origins of dislocations and result in isolated highly defected regions if the elastic strain energy density reaches a critical value.",
"In practice the strain balancing puts stringent requirements on growth."
],
[
"QWs in Tandem Cells",
"In tandem cells two photovoltaic cells with two different bandgaps are stacked on top of each other.",
"The bandgaps of the two cells can be optimised for the solar spectrum [48], and a contour plot of efficiency as a function of top and bottom cell bandgaps is shown in fan for an AM0 spectrum.",
"Tandem cells can be grown monolithically on a single substrate, connected with a tunnel junction.",
"The top and bottom cell are connected in series, which means that the lower photocurrent of the two cells determines the current of the tandem device.",
"Monolithic growth requires that the materials are lattice matched in order to avoid relaxation.",
"A tandem with a GaInP top cell lattice matched to a GaAs bottom cell does not have the optimum bandgap combination as one can see in fan, and there is no good quality material available with a smaller bandgap than GaAs having the same lattice constant.",
"Figure: Contour plot of tandem cell efficiency as a function oftop and bottom cell bandgaps.The standard approach to matching the currents in the GaInP/GaAs tandem is by thinning the top cell so that enough light is transmitted to the bottom cell to generate more current there [49].",
"This is not an optimum configuration, however, as the quantum efficiency of the top cell is reduced and in addition there are losses in the bottom cell due to more high-energy carriers relaxing to the bandedge.",
"It is possible to grow a virtual substrate, where the lattice constant is relaxed and the misfit dislocations are largely confined to electrically inactive regions of the device.",
"In this way the lattice constant can be changed before growing the tandem.",
"But the bottom and top cell of the tandem should still be lattice matched with respect to each other and hence the combination of bandgaps is restricted indicated with a line in fan; the optimum where the currents in both cells are matched cannot be reached in general.",
"Several groups have grown samples that fall on that line in fan [50], [51].",
"QWs extend the absorption and therefore increase the photocurrent compared with a barrier control; hence a GaAs/InGaAs QWC generates more current than a GaAs cell and can be better matched to a GaInP top cell in a tandem cell under the solar spectrum [17], [23], [52], [53].",
"However, as mentioned in sec:qwc, the voltage deteriorates due to the formation of misfit dislocations with increasing strain when incorporating strained InGaAs QWs [22].",
"Strain compensated QWCs offer an attractive solution in that the link between lattice matching and bandgap is decoupled [24].",
"The bandgap of the bottom cell, for example, can be reduced with a strain compensated QWC, which means that one can move along a horizontal line in fan improving the efficiency of a tandem cell quite rapidly [25].",
"QWCs for both the top and the bottom cell give an extra degree of freedom and optimisation of the bandgaps to obtain maximum efficiency becomes possible."
],
[
"QWCs with light trapping",
"Not all the light is absorbed in the quantum wells because they are optically thin and their number limited.",
"Hence light trapping techniques to increase the number of light passes through the MQW is desirable, boosting the QW photo-response significantly.",
"The simplest form is a mirror on the back surface resulting in two light passes.",
"Texturing the front or the back surface can further increase the path length of the light, in particular if the light is at a sufficiently large angle with respect to the normal so to obtain internal reflections.",
"Another option is to incorporate gratings into the structure to diffract the light to large angles [54].",
"Light trapping is only desirable for the wavelength range where the optically thin QWs absorb.",
"Other parts of the cell are optically thick and hence light trapping for photons with energies greater than the bandgap of the bulk material has a minimal effect if any.",
"The energy of photons that are trapped but not absorbed (e.g.",
"below bandgap photons) and additional high-energy carriers relaxing to the bandedge contribute to cell heating and are therefore undesirable.",
"A solution to this problem is to use a wavelength specific mirror such as a distributed Bragg reflector (DBR) [29].",
"A DBR consists of alternating layers of high and low refractive index material, each one-quarter of a wavelength thick.",
"Constructive interference occurs for the reflected light of the design wavelength and adding periods to the DBR causes a higher reflection due to the presence of more in-phase reflections from the added interfaces.",
"DBRs maintain a high reflectance within a region around the design wavelength known as the stop-band.",
"As the refractive index contrast between the two materials in the DBR increases, the peak reflection rises and the stop-band widens.",
"This technique is particularly attractive for multi-junction cells where the second cell is a QWC with a DBR on an active Ge substrate.",
"In dbr the quantum efficiency of a typical QWC with 20 QW is shown with and without a 20.5 period DBR, as well as the reflectance of the DBR; the sample description of the QWC with DBR is given in tab:qwc-dbr.",
"The quantum efficieny (QE) of the QWC with DBR in dbr is calculated from the measured QE and the calculated DBR reflectance.",
"The DBR reflects back only the relevant wavelength range that can be absorbed in the MQW, significantly boosting its spectral response.",
"Figure: Quantum efficiency of a 20 QW device with and withoutDBR, and reflectance of the DBR.Table: Sample description of QWC with DBR.As the stop-band of the DBR can be made quite abrupt, most longer wavelength light is allowed through, so that an active Ge substrate as a third junction can be employed, which would still produce more than 200 A/m$^2$ , more than enough photocurrent compared with the other two junctions (see below).",
"The transmission through to a Ge substrate of a tandem cell with a GaInP top cell and a QWC with DBR as bottom cell has been modelled with a multi-layer programme and is shown in mqw-dbr-transmission compared with a tandem having a GaAs p-n junction as a bottom cell.",
"As one can see the transmission of photons of energy below the bandgap of the bottom cell is similarly high in both cases.",
"Figure: Calculated transmission through to a Ge substrate of atandem cell with a GaInP top cell and as bottom cell a QWC withDBR, compared with a tandem with a GaAs p-n bottom cell.In dbr-jsc the short-circuit current density $J_{sc}$ for AM0 illumination is shown for this QWC (20 QWs) with and without a 20.5 period DBR, compared with a p-n control cell [29].",
"In a tandem configuration, assuming a cut-off wavelength of 650 nm, corresponding to the bandgap of a GaInP top cell, the p-n control cell is expected to have a typical $J_{sc}$ of 158 A/m$^2$ .",
"Introducing a QWC with DBR improves $J_{sc}$ by 16% to 183 A/m$^2$ , which is much better current matched to the GaInP top cell.",
"A commercial triple-junction cell with GaInP top cell, a standard p-n GaAs junction and an active Ge substrate has an AM0 efficiency of 26.0% [55]; if the GaAs junction is replaced with a QWC with DBR the efficiency is calculated to improve by about 3.4 percentage points or 13% to 29.4%.",
"Table: Short-circuit current densities J sc J_{sc} and efficiencies forAM0 illumination.",
"The tandem J sc J_{sc} assumes a cut-off wavelength of650 nm.",
"The QWC has 20 QWs and the DBR 20.5 periods.",
"Thetriple-junction efficiencies are based on a device with a GaInPtop cell and an active Ge substrate."
],
[
"QWCs for Thermophotovoltaics",
"QWCs for TPV Thermophotovoltaics (TPV) is the same principle as photovoltaics but the source is at a lower temperature than the sun, typically around 1500-2000 K instead of 6000 K, and much closer.",
"In TPV applications often a combustion process is used as heat source (e.g.",
"using fossil fuels or biomass), but other heat sources such as nuclear, indirect solar or industrial high-grade waste heat can be used too.",
"Because the source has a lower temperature in TPV, lower bandgap materials are required to absorb the (near-)infrared light more efficiently.",
"Often a selective emitter is introduced between the source and the photovoltaic cells to obtain a narrow band as opposed to a broad black or grey-body spectrum, increasing the cell conversion efficiency.",
"TPV is described in more detail for example in references [56], [57].",
"QWCs have advantages for TPV applications, and substantial progress has been made in the development of QWCs for TPV focusing on InP-based materials [58].",
"QWCs were first introduced for TPV applications independently by Griffin et al [32] and Freundlich et al [59].",
"These InGaAs/InP QWCs indicated better performance than InGaAs bulk cells, for example enhancing $V_{oc}$ and improved temperature dependence.",
"These are important parameters in TPV, as the cells are very close to a hot source, and the power densities are high giving rise to higher currents which may pose series resistance problems.",
"Further development of QWCs has been in the quaternary system $\\rm In_{1-x}\\-Ga_{x}\\-As_{y}\\-P_{1-y}$ lattice-matched to InP ($x=0.47y$ ), with very good material quality, and which can be optimised for a rare-earth selective emitter Erbia having a peak emission of about 1.5 $\\mu $ m, for example, but which is also attractive for hybrid solar-TPV applications [34].",
"Many TPV systems operate at temperatures more suitable for longer wavelength selective emitters such as Thulia and Holmia with peak emissions of about 1.7 and 1.95 $\\mu $ m respectively.",
"However, the smallest bandgap achievable with lattice-matched material on InP is that of $\\rm In_{0.53}\\-Ga_{0.47}\\-As$ with an absorption edge of about 1.7 $\\mu $ m (see Eg-a.)",
"Strain-compensation techniques can be employed to extend the absorption, which are unique to QWCs [36].",
"Figure: Experimental external quantum efficiency of TPV QWCdevices (plus a bulk In 0.53 Ga 0.47 As \\rm In_{0.53}\\-Ga_{0.47}\\-As device) with successively longerabsorption, including a Thulia spectrum (arbitrary units).$\\rm In_{1-x}\\-Ga_{x}\\-As$ /$\\rm In_{1-z}\\-Ga_{z}\\-As$ strain-compensated QWCs have been shown to extend the absorption (see tpv-qe) while retaining a dark current similar to an $\\rm In_{0.53}\\-Ga_{0.47}\\-As$ bulk cell , and lower than that of a GaSb cell with similar bandgap (see tpv-div) [37], [39].",
"A 40 QW $\\rm In_{1-x}\\-Ga_{x}\\-As$ /$\\rm In_{1-z}\\-Ga_{z}\\-As$ strain-compensated QWC was designed for a Thulia emitter [41], absorbing out to 1.97 $\\mu $ m as shown in tpv-qe.",
"Absorption beyond 2 $\\mu $ m has been achieved with a 2 QW cell, although in this case the collection efficiency was incomplete except at high temperatures and reverse bias [38].",
"Figure: Dark current densities of TPV cells.A mirror at the back can significantly increase the contribution of the QWs as discussed in sec:dbr.",
"Back reflection is particularly important as a form of spectral control in a TPV system, as below-bandgap photons are reflected back to the source increasing the system efficiency.",
"We observe a back reflectivity of about 65% just from a standard gold back contact [39].",
"As an alternative, Si/SiGe QWCs were grown for TPV applications, but the absorption of the SiGe QWs is very low [60]."
],
[
"Conclusions",
"The primary advantage of incorporating quantum wells in photovoltaic cells is the flexibility offered by bandgap engineering by varying QW width and composition.",
"The use of strain-compensation further increases this flexibility by extending the range of materials and compositions that can be employed to achieve absorption thresholds at lattice constants that do not exist in bulk material.",
"In a tandem or multi-junction configuration QWCs allow current matching and optimising the bandgaps for higher efficiencies.",
"Light trapping schemes are an important technique to boost the quantum efficiency in the QWs.",
"DBRs are particularly suited for QWCs in multi-junction devices, allowing light transmission to the lower bandgap junctions underneath.",
"The flexibility of materials, in particular with strain compensation, combined with the enhanced performance of QWCs make them especially suitable for TPV.",
"QWCs are unique in that the absorption can be extended to wavelengths unattainable for lattice-matched bulk cells, while retaining a similar dark current.",
"Strain-compensated QWCs can be optimised for long-wavelength selective emitters such as Thulia."
]
] | 1606.05128 |
[
[
"Solving reaction-diffusion equations on evolving surfaces defined by\n biological image data"
],
[
"Abstract We present a computational approach for solving reaction-diffusion equations on evolving surfaces which have been obtained from cell image data.",
"It is based on finite element spaces defined on surface triangulations extracted from time series of 3D images.",
"A model for the transport of material between the subsequent surfaces is required where we postulate a velocity in normal direction.",
"We apply the technique to image data obtained from a spreading neutrophil cell.",
"By simulating FRAP experiments we investigate the impact of the evolving geometry on the recovery.",
"We find that for idealised FRAP conditions, changes in membrane geometry, easily account for differences of $\\times 10$ in recovery half-times, which shows that experimentalists must take great care when interpreting membrane photobleaching results.",
"We also numerically solve an activator -- depleted substrate system and report on the effect of the membrane movement on the pattern evolution."
],
[
"Introduction",
"Progress in cell microscopy has led to the availability in large quantities of time dependent high resolution images of cells.",
"These may be processed to yield details of evolving cell shapes.",
"Furthermore via fluorescent tagging constituents of the cell membrane, lipids or proteins, can be visualised.",
"This leads to the possibility of assimilating this data within mathematical models for cell signalling, for example in the context of cell migration, and also to develop, investigate and test new mathematical models.",
"In this article we take the cell to be an evolving three-dimensional domain and we are concerned with partial differential equation (PDE) models for processes which take place on the cell membrane.",
"The general goal is to study the effect of the surface evolution on the solution behaviour.",
"We present a numerical approach to solve the PDEs on surfaces that are obtained from postprocessing the image data.",
"This image-based modelling is in the spirit of [31] but now taking into account time dependent domains obtained from observed data.",
"The observations comprise a time series of images.",
"Each item of the time series has a selection of two dimensional cross-sections (slices) of the three dimensional object.",
"Each slice is segmented in order to identify the cell and its boundary.",
"The upshot is a time series of triangulated genus zero surfaces defined by the three dimensional coordinates of triangle vertices.",
"Spherical parameterisation using an equiareal mapping to the unit sphere [4] together with resampling of the mesh ensure that a mesh of constant topology and good quality is maintained.",
"One can then follow the discrete path of a vertex by the knowledge of the position at discrete times yielding a discrete vertex velocity obtained by time differencing the location of vertices at successive time steps.",
"It is natural to use finite element technology in order to solve the PDE models on the triangulations which describe the evolution.",
"Our computational approach is based on an arbitrary Lagrangian–Eulerian (ALE) evolving surface finite element method (ESFEM) [8], [12], [13].",
"The underlying continuum problem formulation requires a given evolving surface including a material velocity field related to the trajectories of material surface points.",
"Our notion is that it describes how the molecules forming the cell boundary would move without any reactions and lateral transport such as diffusion.",
"The contribution of the material velocity in the normal direction describes the geometric evolution of the surface.",
"In general, it may also have a tangential contribution which is associated with lateral transport of mass.",
"The spatial discretisation of the continuum problem involves approximating the evolving surface by an evolving triangulated surface, and approximating the material velocity by a discrete material velocity which is defined on the evolving triangulated surface.",
"In the present context the evolving triangulated surface is known only at discrete times given by the image data acquisition.",
"No information as to where material points move is provided.",
"In order to define the discrete material velocity we postulate a material velocity in the normal direction.",
"The discrete material velocity is defined on each triangle by taking the normal component of the vertex velocity.",
"This means using the tangential component of the vertex velocity as the arbitrary tangential velocity in the ALE approach.",
"We briefly mention that there are other approaches to solving PDEs on evolving surfaces using fixed grids which include space time finite elements, [28], implicit surface level set methods, [9], a Lagrangian level set and particle approach [3], phase field methods, [14] and closest point methods, [29]."
],
[
"Data accessibility",
"The original image data as well as segmented individual slices can be obtained from: http://www2.warwick.ac.uk/fac/sci/systemsbiology/staff/bretschneider/data/ The triangulation data and the code used for the simulations are provided as part of the supplementary material and can be obtained from: https://github.com/tranner/dune-imagedata"
],
[
"Competing interests",
"We have no competing interests."
],
[
"Author's contributions",
"T Bretschneider made the images data accessible and discussed the biological applications.",
"CJ Du wrote and applied the post-processing software for the image data.",
"CM Elliott contributed background information, was involved in the design of the numerical experiments and drafted the manuscript.",
"T Ranner was involved in the design of the numerical experiments and performed and post-processed the simulations.",
"B Stinner drafted the manuscript, described the method for the surface partial differential equation and was involved in the design of the numerical experiments."
],
[
"Acknowledgements and Funding",
"The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for its hospitality during the programme Coupling Geometric PDEs with Physics for Cell Morphology, Motility and Pattern Formation supported by EPSRC Grant Number EP/K032208/1.",
"This research was partially funded by HEIF5 via the Warwick Impact Fund.",
"The research of TR was funded by the Engineering and Physical Sciences Research Council (EPSRC EP/L504993/1).",
"The research of CME was partially supported by the Royal Society via a Wolfson Research Merit Award.",
"TB was supported by BBSRC grants BBI0082091 and BB/M01150X/1."
],
[
"Image data acquisition and processing",
"We briefly sketch the method and refer to [7] for more details.",
"First a downsampled version of each image is segmented using a graph-based approach with automatic seed detection.",
"Secondly, since we know the cell boundary is a genus zero surface any topological artefact generated via the first process must be fixed.",
"For example, small protrusions which cross at a distance away from the body of the cell may result in holes in the segmented image.",
"A level set approach is used to smooth any protrusions until we have a simply connected surface.",
"From the level set representation a spherical parameterisation is obtained based on the method of [4].",
"This is then used to push forward a fixed nice triangulation of the sphere on to the actual surface."
],
[
"Continuum model",
"In the continuum model underlying the computational approach, the cell boundary is denoted by $\\lbrace \\Gamma (t)\\rbrace _{t \\in [0,T]}$ where $\\Gamma (t)$ is a topological sphere embedded in $\\mathbb {R}^3$ at times $t \\in [0,T]$ with a unit normal vector field, pointing out of the cell, $\\nu (\\cdot , t): \\Gamma (t) \\rightarrow \\mathbb {R}^3$ , $t \\in [0,T]$ .",
"We assume that a material particle $p$ located at $x_p(t) \\in \\Gamma (t)$ at a time $t \\in [0,T]$ has a velocity $\\dot{x}_p(t)$ with both a normal component (which determines the evolution of the shape) and tangential components (which are related to transport of material along the surface).",
"A velocity field $v(\\cdot , t) : \\Gamma (t) \\rightarrow \\mathbb {R}^3$ , $t \\in (0,T)$ is defined by $v(x_p(t),t)=\\dot{x}_p(t).$ For a function $f(t) : \\Gamma (t) \\rightarrow \\mathbb {R}$ , $t \\in [0,T)$ , the material time derivative in a point $(x,t)$ with $x \\in \\Gamma (t)$ is defined by $\\partial _t^{(v)} f (x,t) := \\frac{d}{dt} f(x_p(t),t) = \\frac{\\partial \\tilde{f}}{\\partial t}(x,t) + v(x,t) \\cdot \\nabla \\tilde{f}(x,t)$ where $x = x_p(t)$ for a material particle $p$ located at $x_p(t)$ at time $t$ .",
"Note that although the expressions on the right hand side require a smooth extension $\\tilde{f}$ of $f$ to a neighbourhood of the evolving surface in order to be well-defined, the resulting derivatives are independent of the extension.",
"The tangential or surface gradient is defined as the projection of the standard derivative onto the tangent plane of the surface so that $\\nabla _{\\Gamma (t)} f(x,t) := \\nabla \\tilde{f}(x,t) - (\\nabla \\tilde{f}(x,t) \\cdot \\nu (x,t)) \\nu (x,t).$ Analogously for the surface divergence of a vector valued surface field.",
"We can define the Laplace-Beltrami operator as $\\Delta _{\\Gamma (t)} f := \\nabla _{\\Gamma (t)} \\cdot \\nabla _{\\Gamma (t)} f$ .",
"For calculus on surfaces, integration by parts formulas, and transport identities involving the material time derivative we refer to [10].",
"As a model problem we consider a conserved field on an evolving surface subject to Fickian diffusion.",
"The advection diffusion equation is, see [8], $\\partial _t^{(v)} u + u \\nabla _{\\Gamma (t)} \\cdot v - D \\Delta _{\\Gamma (t)} u & = 0 && \\mbox{ on } \\Gamma (t), \\, t \\in (0,T), \\\\u( \\cdot , 0 ) & = u_0 && \\mbox{ on } \\Gamma (0).$ with initial value $u_0 : \\Gamma (0) \\rightarrow \\mathbb {R}$ and diffusion parameter $D>0$ .",
"We refer to [1] for a weak well-posedness analysis of such equations including suitable time-dependent Sobolev spaces."
],
[
"Numerical scheme",
"Recall that the scanning process creates a representation of the cell boundary at each time in terms of a closed evolving triangulated hypersurface.",
"We assume that by increasing the computational effort, i.e., refining the triangulation and the time step size, the resulting triangulations better approximate a closed evolving smooth hypersurface.",
"For simplicity, we use linear surface finite elements and discretise in time by backward Euler.",
"See [6] for higher order approximations in space and [24] for a discussion of time stepping schemes.",
"The triangulated surfaces given at times $t^{(m)}$ , $m = 0, \\dots , M$ with $t^{(M)} = T$ , are denoted by $\\Gamma _h^{(m)}$ .",
"Here, $h$ stands for the diameter of the largest triangle of the $\\Gamma _h^{(m)}$ .",
"Let $S_h^{(m)}$ denote the space of piecewise linear finite element functions over $\\Gamma _h^{(m)}$ with nodal basis $\\lbrace \\chi _j^{(m)} \\rbrace _{j=1}^J$ .",
"Time steps are denoted by $\\tau ^{(m)} = t^{(m)} - t^{(m-1)}$ , $m = 1, \\dots , M$ .",
"The vertex velocity of the triangulation is denoted by $w_h^{(m)} \\in (S_h^{(m)})^3$ and it is defined by choosing the vertex values to be $w_h^{(m)}(x_j^{(m)}) = (x_j^{(m)} - x_j^{(m-1)}) / \\tau ^{(m)}$ where the $x_j^{(m)}$ are the vertex positions of $\\Gamma _h^{(m)}$ .",
"Given an approximation $u_h^{(0)}$ to the initial values $u_0$ , the finite element discretisation of (REF ) is to subsequently find $u_h^{(m)} \\in S_h^{(m)}$ , $m=1, \\dots , M$ , of the form $u_h^{(m)} = \\sum _{j=1}^J u_j^{(m)} \\chi _j^{(m)}$ as the solution to $ \\frac{1}{\\tau ^{(m)}} \\left( \\int _{\\Gamma _h^{(m)}} u_h^{(m)} \\chi _j^{(m)} \\, \\mathrm {d} \\sigma _h - \\int _{\\Gamma _h^{(m-1)}} u_h^{(m-1)} \\chi _j^{(m-1)} \\, \\mathrm {d} \\sigma _h \\right) \\\\+ D \\int _{\\Gamma _h^{(m)}} \\nabla _{\\Gamma _h^{(m)}} u_h^{(m)} \\cdot \\nabla _{\\Gamma _h^{(m)}} \\chi _j^m \\, \\mathrm {d} \\sigma _h + b_{adv} (u_h^{(m)}, \\chi _j^{(m)}) + b_{sld} (u_h^{(m)}, \\chi _j^{(m)}) = 0$ for $j = 1,2, \\ldots , J$ .",
"Here, $b_{adv}$ and $b_{sld}$ are two terms which account for the material velocity $v$ in (REF ) and for streamline diffusion, respectively.",
"The advection term is $b_{adv} (u_h^{(m)}, \\chi _j^{(m)}) = \\int _{\\Gamma _h^{(m)}} u_h^{(m)} \\, \\big {(} w_h^{(m)} -v_h^{(m)} \\big {)} \\cdot \\nabla _{\\Gamma _h^{(m)}} \\chi _j \\mathrm {d} \\sigma _h.$ Here, $v_h^m$ denotes the discrete material velocity.",
"In ALE methods a non-physical arbitrary velocity for the vertices may be used to ensure a better quality of the evolving mesh [12], [13].",
"In our setting the vertex velocity is dependent on the computational tools used to obtain the parametrisation and hence may not be physical.",
"In our case where the material velocity is purely in the normal direction we set $v_h^{(m)} = w_h^{(m)} \\cdot \\nu _h^{(m)} \\nu _h^{(m)}$ on each triangle where $\\nu _h^{(m)}$ is the (piecewise constant) unit normal of the triangle.",
"In some applications the tangential vertex velocity is quite strong in comparison with transport by diffusion, i.e., the problem is advection dominated which can lead to poor approximations on coarse grids.",
"A standard way to deal with this issue is to add a streamline diffusion term, [23], which is of the form $b_{sld} (u_h^{(m)}, \\chi _j^{(m)}) = D g(h) \\int _{\\Gamma _h^{(m)}}( w_h^{(m)} \\cdot \\nabla _{\\Gamma _h^{(m)}} u_h^{(m)} ) ( w_h^{(m)} \\cdot \\nabla _{\\Gamma _h^{(m)}} \\chi _j^{(m)} ) \\, \\mathrm {d} \\sigma _h$ where we took $g(h) = h^2$ in the computations below.",
"In each time step $m$ , (REF ) yields a linear system of equations for coefficients $u_j^{(m)}$ which can be solved using a biconjugate gradient stabilised method."
],
[
"Applications and simulations",
"In this section we illustrate the use of evolving surface finite element methodology on triangular meshes obtained from image data of neutrophil cells.",
"These were labelled with cell mask orange dye to stain the plasma membrane.",
"The neutrophils are quiescent initially and then stimulated by application of fMLP which prompts cell spreading and movement [22].",
"Cell movements are imaged by spinning disk confocal microscopy [20].",
"The acquisition speed is about 80 ms per slice (4 secs/stack).",
"The data comprises of 166 surface triangulations associated with a subset of 214 successive times where each triangular mesh has 20480 triangles with 10242 vertices.",
"In the computations triangulations needed for intermediate times were obtained by interpolation of the vertex positions."
],
[
"Simulation of FRAP experiments",
"Fluorescence recovery after photobleaching (FRAP) is a standard technique to assess the mobility of fluorescently labelled molecules in cells; for example, cell surface receptors which bind extracellular chemoattractants such as fMLP.",
"Within a small region these molecules are irreversibly photo-bleached by a high intensity laser impulse.",
"This is then followed by a fluorescence recovery phase consisting of the motion of non-affected molecules into the bleached region.",
"Monitoring this gives insight into the type of motion and eventually enables the quantification of kinetic parameters.",
"We refer to [15] for an overview of what type of information can be obtained from FRAP.",
"Previous studies have considered models of this recovery process on a flat stationary surface.",
"For example, the mathematical modelling in [21] uses semi-analytical and computational finite difference methods which do not account for the complex shape of the membrane and its evolution.",
"Here, we show that using an accurate representation of the geometry affects the recovery of the concentration, denoted by $c$ , in the bleached region.",
"Our model for FRAP consists of the diffusion equation of the form (REF ).",
"In particular, we neglect effects such as non-instantaneous bleaching and cytoplasmic attachment/detachment kinetics which sometimes are of importance [21].",
"We assume a model in which there is no tangential, advective material transport so that the material velocity is taken in the normal direction.",
"Hence we have $v_h^{(m)} = w_h^{(m)} \\cdot \\nu _h^{(m)} \\nu _h^{(m)}.$ For the diffusion coefficient we take the values $D = 5 \\cdot 10^{-10} \\, \\mathrm {cm}^2 / \\mathrm {sec}$ which is a typical value for G-protein coupled receptors such as fMLP.",
"The initial photobleaching is modelled by a homogeneous constant value $C_0$ except in a region of interest (ROI) $\\Gamma _b(0)$ , defined by the intersection of $\\Gamma (0)$ with an idealised ball of radius $r$ centred in $m$ .",
"Here $r$ is small with respect to the diameter of the cell enclosed by $\\Gamma (0)$ .",
"We remark that realistic bleach profiles and side effects, such as photobleaching by light scattering, due to choosing this (ROI) are neglected and refer to [26] for a discussion of these issues.",
"In our setting the observation time is short and we do not advect the the bleached material region with the surface velocity.",
"Our region of observation of the concentration is thus $\\Gamma _b(t)$ defined by the intersection of $\\Gamma (t)$ with the (ROI), the fixed ball of radius $r$ centred at $m$ .",
"Our approach mimics that of typical experimental set ups where fluorescence recovery is sampled within a fixed (ROI).",
"We rescale the concentration so that $C_0 = 1$ and show concentration relative to $C_0$ .",
"Precisely, the initial condition is given by $c(0,x) = c_0(x) ={\\left\\lbrace \\begin{array}{ll}0 & \\mbox{if } x \\in \\Gamma _b = \\lbrace x \\in \\Gamma : | x - m | < r \\rbrace \\\\1 & \\mbox{otherwise}.\\end{array}\\right.",
"}$ In the computations, we take a piecewise linear interpolation of $c_0$ for initial condition.",
"The time step is chosen to be $0.04 \\, \\mathrm {s}$ - i.e.",
"100th of the difference between successive images.",
"We present three simulations on different surface evolutions using our single data set starting from frames 0, 25 and 77.",
"In each case we take the (ROI) to be a sphere of radius a quarter the radius of the cell centred at $(0.25, 0.25, 0.25)$ .",
"The areas of bleached regions are, respectively, $8\\%$ , $5\\%$ and $3\\%$ of the total cell areas.",
"In the calculations $\\Gamma _b(t)$ is approximated by the evolution of the union of elements comprising the initial discrete (ROI).",
"Figure: Results of FRAP bleaching simulations.",
"The colour scheme is from red (1 arb.)",
"to blue (0 arb.",
").A black contour shows the boundary of the (ROI) Γ b (t)\\Gamma _b(t) in which we track the concentration over time.Snapshots from each simulation are shown in Figure REF .",
"In the first simulation (Figure REF ), the cell surface undergoes very small changes and has no large changes in curvature.",
"In the other two simulations (Figures REF and REF ), the overall cell surfaces display large deviations in curvature.",
"These changes clearly change the rate at which the concentration recovers in $\\Gamma _b$ .",
"In order to quantify the effects of the change in geometry we compute $\\frac{1}{ | \\Gamma _b (t) | }\\int _{\\Gamma _b (t)} c(t) \\, \\mathrm {d} \\sigma $ at each time point.",
"This quantity is then fitted to a recovery curve $A ( 1 - \\exp ( -t / B ) )$ to the first $12 \\, \\mathrm {s}$ of simulation time (i.e.",
"using four images).",
"We use this short time because of the relatively small evolution of the overall cell boundary.",
"The underlying cell geometry is rather different in each case.",
"It follows that $A$ is the asymptotic mean value of the concentration in the evolved bleached region and $B$ is then the recovery timescale.",
"This is performed using the scipy.optimize function curve_fit.",
"The results are shown in Figure REF .",
"The range of resulting recovery rates show a large variation with differences of an order of magnitude.",
"Figure: Curve fitting the recovery times for FRAP simulations.",
"(Left) The recovery curves with: circles for observations;solid line for fitted curves;blue, green and red colour for geometries (A), (B) and (C), respectively.",
"(Right) Fitted values with one standard deviation of the mean shown in brackets.We also report T 1/2 T_{1/2} the time from the bleach to the timepoint wherethe fluorescence intensity reaches the half of the final recoveredintensity.",
"See Section for details on the simulation parameters."
],
[
"Pattern formation on evolving surfaces",
"Systems of reaction-diffusion equations such as Turing systems which involve local autocatalysis and inhibition on a longer range are well established tools to explain biological pattern formation.",
"Recently, the behaviour of such systems on evolving domains has been increasingly attracting interest [3], [2], [33].",
"Without any specific application in mind but to display the capability of our computational methodology we simulate an activator – depleted substrate system taken from [18], [2] for stripes or spots (in dependence of the parameters).",
"In such systems, the activator, denoted by $u$ , grows by consuming a substrate, denoted by $w$ , which thanks to higher diffusivity acts as the long-range inhibitor.",
"The partial differential equations on a moving surface read $ \\begin{split}\\partial _t^{(v)} u + u \\nabla _{\\Gamma (t)} \\cdot v& = D_u \\Delta _{\\Gamma (t)} u + \\gamma ( a - u + u^2 w ) \\\\\\partial _t^{(v)} w + w \\nabla _{\\Gamma (t)} \\cdot v& = D_w \\Delta _{\\Gamma (t)} w + \\gamma ( b - u^2 w ).\\end{split}$ We use the parameters $D_u = 1, \\quad D_w = 10, \\quad \\gamma = 200, \\quad a = 0.1, \\quad b = 0.9.$ For the initial values we consider $10\\%$ random perturbations of the steady-state solution $(1,0.9)$ .",
"The numerical scheme reads as before except that we have to solve for two fields now and the non-linear terms have been linearised using a first order Taylor expansion, i.e., for a function $f(u,w)$ $f( u^{(m)}, w^{(m)} ) \\approx f( u^{(m-1)}, w^{(m-1)} ) \\\\+ f_u( u^{(m-1)}, w^{(m-1)} ) ( u^{(m)} - u^{(m-1)} ) + f_w( u^{(m-1)}, w^{(m-1)} ) ( w^{(m)} - w^{(m-1)} ).$ To ensure stability, we take a computational time step $10^{-4}$ and interpolate between new meshes which are introduced every 1 time unit until time 70.",
"We take a linear interpolation in time of the position of each node.",
"Snapshots of the solution are shown in Figure REF and a video of all time steps is available at http://www.personal.leeds.ac.uk/~scstr/vid/imagedata-video.html We observe that the number and position of spots is changing over time.",
"On stationary surfaces these are related to the eigenvalues of the Laplace-Beltrami operator which depend on the geometry.",
"A careful assessment of the impact of the surface evolution is left for future studies.",
"Figure: Snapshots from results of Brusselator system on evolving surface at times 0,10,20,30,40,50,60,700,10,20,30,40,50,60,70.",
"See Section for details on the simulation parameters."
],
[
"Conclusion",
"A novel approach to solve reaction-advection-diffusion equations on moving surfaces which are obtained from time series of 3D cell image data was presented.",
"By processing the image data a time series of triangulations of the same topology is obtained.",
"An ALE finite element method then is set up to approximate solutions to the surface PDEs.",
"The approach was used to simulate FRAP experiments with a recovery process based on diffusion along the membrane surface only.",
"By fitting recovery curves a dependence on the geometry of the photobleached region of interest was noticed.",
"For experimentalists it is important to note that even in our idealised FRAP setup where we have assumed an idealised bleach region, which only extends through the dorsal side of the membrane and is perfectly symmetrical unlike real bleach profiles, the computed half-times for recovery differ by factors up to 10.",
"For this purpose, the domain $\\Gamma _b(t)$ to measure the recovery was obtained by simply intersecting the bleach profile (ROI) with the actual surface as this is the standard in the analysis of FRAP data [15], [19].",
"This motivated restricting our numerical experiments to small times in which the cell surface changes only a little.",
"However, over a longer time there is a significant evolution of the cell.",
"This suggests that in a FRAP model the bleached region should be advected with the material velocity in order to get more accurate recovery results.",
"This would motivate a change in the experimental set up so that the bleached region can be tracked.",
"Furthermore the analysis would benefit from a higher resolution data both in time and space which can be achieved by modern technology [16].",
"A constant diffusivity was assumed in our FRAP simulations.",
"However, inhomogeneous diffusivities have been proposed, for instance, in dependence on the local geometry [34] or on the composition of the cell membrane [32].",
"Inverse problems such as identifying the diffusivity of membrane resident proteins will require fluorescence data related to the protein density in addition to position data for the cell membrane.",
"From such data, it may also be possible to infer information on the material velocity which is underlying the continuum approach to the surface PDEs.",
"Of particular interest will be models for the lateral transport due to a gross motion of the membrane which often is modelled as a viscous fluid [30].",
"In a second numerical experiment a pattern-forming reaction diffusion system on the cell surface was simulated.",
"Also here an impact of the changing geometry, in this case in terms of the pattern, was observed.",
"Our aim in this paper is to point out that simulations of PDEs may be carried out on time evolving domains generated from data and that the results of standard models depend on the evolving geometry.",
"We expect that the capabilities of this computational surface finite element methodology may be used in parameter identification.",
"For example by formulating inverse problems to infer reaction rates or diffusivities of membrane resident species.",
"But the results also suggest that more systematic investigations of models and their dependence on geometry can be carried out on realistic domains.",
"For example, target models for investigation might be reaction-diffusion systems for cell polarisation as are employed in models for cell motility [27], [11].",
"See also [5] for a study on the quantification of such a model.",
"Significant extensions might be required to account for processes within and outside of the cell,[25], or raft formation in realistic geometries [17]."
]
] | 1606.05093 |
[
[
"The maximum of the minimal multiplicity of eigenvalues of symmetric\n matrices whose pattern is constrained by a graph"
],
[
"Abstract In this paper we introduce a parameter $Mm(G)$, defined as the maximum over the minimal multiplicities of eigenvalues among all symmetric matrices corresponding to a graph $G$.",
"We compute $Mm(G)$ for several families of graphs."
],
[
"Introduction",
"Given a simple undirected graph $G=(V(G),E(G))$ with vertex set $V(G)=\\lbrace 1,2,\\ldots ,n\\rbrace $ , let $S(G)$ be the set of all real symmetric $n \\times n$ matrices $A=(a_{ij})$ such that, for $i \\ne j$ , $a_{ij} \\ne 0$ if and only if $(i,j) \\in E(G)$ .",
"There is no restriction on the diagonal entries of $A$ .",
"The graph $G$ pozes some conditions on the eigenvalues of $A \\in S(G)$ , and several questions in the literature are trying to better understand those conditions.",
"The most general is the question of characterizing all lists of real numbers $\\lbrace \\lambda _1,\\lambda _2,\\ldots ,\\lambda _n\\rbrace $ that can be the spectrum of a matrix $A \\in S(G),$ and is known as the Inverse eigenvalue problem for $G$.",
"This question and the related question of characterizing all possible multiplicities of eigenvalues of matrices in $S(G)$ have been studied primarily for trees [6], [11], [13], [14].",
"A subproblem to the inverse eigenvalue problem for graphs that has attracted a lot of attention over the years is that of minimizing the rank of all $A \\in S(G)$ .",
"Finding the minimal rank of $G$ , defined as $\\mathrm {mr}(G)=\\min \\lbrace \\mathrm {rk}(A); A \\in S(G)\\rbrace ,$ is equivalent to finding the maximal multiplicity of an eigenvalue of $A \\in S(G)$ , denoted by $M(G).$ The minimum rank problem has been resolved for several families of graphs.",
"We refer the reader to an excellent survey paper on the problem [8] where additional references can be found.",
"A more recent survey paper [7] not only gives an up-to-date on the minimum rank problem, but it also talks about several of its variants that can be found in the literature.",
"For example, the possible inertia of matrices $A \\in S(G)$ has been studied in [2], [3], [4] and the minimum number of distinct eigenvalues in [1].",
"For a matrix $A$ , we let $q(A)$ denote the number of distinct eigenvalues of $A$ .",
"For a graph $G$ , we define $q(G) = \\min \\lbrace q(A); \\; A \\in S(G)\\rbrace .$ In [15] we considered the problem of determining for which graphs $G$ there exists a matrix in $S(G)$ whose characteristic polynomial is a square, i.e.",
"the multiplicities of all its eigenvalues are even.",
"This question is closely related to the question of determining for which graphs $G$ there exists a matrix in $S(G)$ with all the multiplicities of eigenvalues at least 2.",
"In this paper we bring this topic further by defining and studying a new parameter for a graph $G$ denoted by $\\mathrm {Mm}(G)$ .",
"For a matrix $A\\in M_n(\\mathbb {R})$ we denote $\\mathrm {Mm}(A)$ to be the minimal eigenvalue multiplicity of $A$ .",
"Then $\\mathrm {Mm}(G)$ is defined to be: $\\mathrm {Mm}(G)=\\max \\lbrace \\mathrm {Mm}(A); A\\in S(G)\\rbrace .$ In order words, we define $\\mathrm {Mm}(G)$ to be the maximum over the minimal multiplicities of eigenvalues among all $A\\in S(G)$ .",
"Clearly, $\\mathrm {Mm}(G) \\le \\lfloor \\frac{n}{2} \\rfloor $ for all nonempty graphs on $n$ vertices.",
"If $\\mathrm {Mm}(G) =1$ , then all matrices in $S(G)$ must have a simple eigenvalue.",
"Moreover, it is clear that we can determine an upper bound for $\\mathrm {Mm}(G)$ in terms of $M(G)$ and $q(G)$ as follows: $\\mathrm {Mm}(G) \\le M(G), \\; \\mathrm {Mm}(G) \\le \\left\\lfloor \\frac{|G|}{q(G)}\\right\\rfloor .$ As we will see later in the paper both bounds can be achieved for some but not all graphs $G$ .",
"It is clear from (REF ) that graphs that have large $\\mathrm {Mm}$ , will have to have relatively small $q$ , and that the two parameters are related.",
"To determine $\\mathrm {Mm}(G)$ for a given graph $G$ we need to control the multiplicities of all the eigenvalues of $A \\in S(G)$ , and in doing so we further our understanding of the Inverse eigenvalue problem for $G$ .",
"Our paper is organised as follows.",
"In Section we introduce the notation and some preliminary result that give introductory insight into $\\mathrm {Mm}(G)$ and mostly follow from known results.",
"In Section we recall a few constructions from the literature and develop some generalizations that we need later.",
"In Section we look for graphs with large $\\mathrm {Mm}$ .",
"We show that the equality $\\mathrm {Mm}(G) = \\lfloor \\frac{n}{2} \\rfloor $ is achieved for practically all graphs with $\\mathrm {mr}_+(G)=2.$ We also determine $\\mathrm {Mm}(G)$ for complete bipartite graphs.",
"In Section we look at the graphs with small $\\mathrm {Mm}(G)$ .",
"We choose from the graphs $G$ on $n$ vertices with $\\mathrm {mr}(G)=n-2$ , and we show that among those graphs there exist graphs with $\\mathrm {Mm}(G)=1$ and graphs with $\\mathrm {Mm}(G)=2$ ."
],
[
"Notation and preliminary results",
"By $M_n(\\mathbb {R})$ we denote the set of all $n \\times n$ matrices with real entries.",
"By $I_n$ we denote the identity matrix in $M_n(\\mathbb {R})$ and by $0_n$ we denote the zero matrix in $M_n(\\mathbb {R})$ .",
"Our notation concerning graphs is as follows.",
"For a graph $G=(V(G),E(G))$ we denote its order by $|G|=|V(G)|$ .",
"The complement $G^c$ of a graph $G$ is the graph on vertices $V(G)$ such that two vertices are adjacent in $G^c$ if and only if they are not adjacent in $G$ .",
"The join $G \\vee H$ of $G$ and $H$ is the graph union $G \\cup H$ together with all the possible edges joining the vertices in $G$ to the vertices in $H$ .",
"The complete graph on $n$ vertices will be denoted by $K_n$ and complete bipartite graph on disjoint sets of cardinality $m$ and $n$ by $K_{m,n}$ .",
"The Cartesian product $G \\square H$ of graphs $G, H$ is a graph with the vertex set $V(G) \\times V(H)$ and $((u_1,u_2),(v_1,v_2)) \\in E(G \\square H)$ if and only if either $u_1=v_1$ and $(u_2,v_2) \\in E(H)$ or $u_2=v_2$ and $(u_1,v_1) \\in E(G)$ .",
"The tensor product of two graphs is defined as $G \\times H=(V(G \\times H), E(G \\times H))$ .",
"where $V(G \\times H)=V(G) \\times V(H)$ and $((u,u^{\\prime }),(v,v^{\\prime })) \\in E(G \\times H)$ if and only if $ (u,v) \\in E(G)$ and $(u^{\\prime },v^{\\prime }) \\in E(H)$ .",
"The strong product $G \\boxtimes H$ of graphs $G, H$ is a graph with the vertex set $V(G) \\times V(H)$ and $((u_1,u_2),(v_1,v_2)) \\in E(G \\boxtimes H)$ if and only if either $u_1=v_1$ and $(u_2,v_2) \\in E(H)$ or $u_2=v_2$ and $(u_1,v_1) \\in E(G)$ or $(u_1,v_1) \\in E(G)$ and $(u_2,v_2) \\in E(H)$ .",
"We start with some basic observations that follow from the known results.",
"Proposition 2.1 Let $G$ be a graph with $|G|=2n$ , $\\mathrm {mr}(G)=n$ and $q(G)=2$ .",
"Then $\\mathrm {Mm}(G)=n$ .",
"Proof.",
"Since $q(G)=2$ , there exists a symmetric matrix $A \\in S(G)$ with exactly two distinct eigenvalues.",
"Let $k$ and $2n-k$ be the corresponding multiplicities.",
"Since maximal multiplicity of an eigenvalue among matrices in $S(G)$ is $n$ , we have $k \\le n$ and $2n-k\\le n$ , proving that $k=n$ .",
"$\\Box $ Note that the same argument as in the proof above, tells us that there is no graph $G$ , $|G|=2n+1$ , with $\\mathrm {mr}(G)=n+1$ and $q(G)=2$ .",
"In the following proposition we gather observations on $\\mathrm {Mm}(G)$ that follow from results in [1] on $q(G).$ Proposition 2.2 For a connected graph $G$ we have $\\mathrm {Mm}(G \\vee G)=|G|$ .",
"If $|G|=2n$ , $|H|=2m$ with $\\mathrm {Mm}(G)=n$ and $\\mathrm {Mm}(H)=m$ , then $\\mathrm {Mm}(G \\cup H)=n+m$ .",
"Let $G^{\\prime }$ be the corona graph of $G$ , i.e.",
"the graph with $2 |G|$ vertices obtained from $G$ by joining each vertex of $G$ with a pendant vertex.",
"Then $\\mathrm {Mm}(G^{\\prime }) \\ge \\mathrm {Mm}(G).$ Let $G$ be a graph on $n$ vertices with $q(G)=2$ .",
"Then $\\mathrm {Mm}(G \\square K_2)=n$ .",
"Proof.",
"In [1] it is proved that $q(G \\vee G)=2$ for a connected graph $G$ .",
"This is proved by showing that there exists a matrix $Q \\in S(G \\vee G)$ of the form $Q=\\left[\\begin{array}{cc}\\sqrt{P} & \\sqrt{I-P} \\\\ \\sqrt{I-P} & -\\sqrt{P}\\end{array}\\right].$ Since $Q^2=I$ this implies that $q(Q)=2$ .",
"More precisely, 1 and $-1$ are the only eigenvalues of $Q$ .",
"Since the trace of $Q$ is 0, it follows that both 1 and $-1$ must have multiplicity $|G|$ and thus $\\mathrm {Mm}(G \\vee G)=|G|$ .",
"Choose $A\\in S(G)$ with eigenvalues $\\lambda _{1}, \\lambda _{2}$ , each having multiplicity $n$ , and $B \\in S(H)$ with eigenvalues $\\mu _{1}$ , $\\mu _{2}$ , each having multiplicity $m$ .",
"Then the matrix $A\\oplus \\frac{\\lambda _{1}-\\lambda _{2}}{\\mu _{1}-\\mu _{2}} \\left(B+\\frac{\\mu _{1}\\lambda _{2}-\\mu _{2}\\lambda _{1}}{\\lambda _{1}-\\lambda _{2}} I\\right)$ has eigenvalues $\\lambda _{1}$ and $\\lambda _{2}$ , each having multiplicity $n+m$ .",
"It was proved in [1] that if $\\lambda $ is an eigenvalue of $A$ with multiplicity $m$ , then the matrix $B=\\left[\\begin{array}{cc}A & I\\\\ I & 0 \\end{array}\\right] \\in S(G^{\\prime })$ has two distinct eigenvalues $\\mu _1(\\lambda )$ and $\\mu _2(\\lambda )$ both with multiplicity $m$ .",
"By [1] there exists $A \\in S(G)$ such that $q(A)=q(G)=2$ and the eigenvalues of $A$ are 1 and $-1$ .",
"Therefore the minimal polynomial of $A$ is equal to $x^2-1.$ As in the proof of [1], we construct a matrix $B=\\left( \\begin{matrix} A & I_n \\\\ I_n & -A \\end{matrix} \\right) \\in S(G \\square K_2).$ Since $A^2=I_n$ , it follows that $B^2=2I_{2n}$ .",
"This proves that the minimal polynomial of matrix $B$ is $x^2-2$ and thus the only possible eigenvalues of $B$ are $\\sqrt{2}$ and $-\\sqrt{2}$ .",
"Since the trace of $B$ is equal to 0, the multiplicities of the two eigenvalues have to be equal and so $\\mathrm {Mm}(G \\square K_2)=n$ .",
"$\\Box $ Proposition 2.3 Let $Q_s$ be the hypercube, $|Q_s|=2^s$ .",
"Then $\\mathrm {Mm}(Q_s)=2^{s-1}$ .",
"Proof.",
"By [1] we have $q(Q_s)=2$ and by [10] we have $\\mathrm {mr}(Q_s)=2^{s-1}$ .",
"Using Theorem REF it follows that $\\mathrm {Mm}(Q_s)=2^{s-1}$ .",
"$\\Box $ Theorem 2.1 Let $G$ be a graph on $n$ vertices with an induced tree on $n-k$ vertices.",
"Then $\\mathrm {Mm}(G)\\le k+1$ .",
"Proof.",
"Suppose $A \\in S(G)$ with $\\mathrm {Mm}(A)=\\mathrm {Mm}(G)$ and let $A_1 \\in S(T)$ be its principal $(n-k) \\times (n-k)$ submatrix for some induced tree $T$ of $G$ on $n-k$ vertices.",
"If $\\mathrm {Mm}(G)\\ge k+2$ , then all the eigenvalues of $A$ have multiplicity at least $k+2$ .",
"So, let us denote the eigenvalues of $A$ by $\\lambda _1=\\ldots =\\lambda _{k+2}\\ge \\lambda _{k+3}\\ge \\ldots \\ge \\lambda _{n}.$ By interlacing, the submatrix $A_1$ has eigenvalues $\\lambda _1=\\lambda _{2}\\ge \\mu _{3}\\ge \\ldots \\ge \\mu _{n-k},$ which contradicts [11] that states that the largest eigenvalue of a tree must be simple.",
"$\\Box $ Corollary REF will show that even for $k=1$ there exist graphs for which the inequality in Theorem REF is strict.",
"Moreover, Corollary REF will show that the converse of the above theorem is not true.",
"In the next proposition we gather some estimates for $\\mathrm {Mm}$ for different products of graphs.",
"The proofs of those estimates are not difficult and are straightforward modifications of the proofs of Theorem 4.1, Theorem 4.2 and Theorem 4.3 in [15].",
"Proposition 2.4 For any two graphs $G$ and $H$ we have: $\\mathrm {Mm}(G \\times H) \\ge \\mathrm {Mm}(G)\\mathrm {Mm}(H)$ $\\mathrm {Mm}(G \\square H) \\ge \\mathrm {Mm}(G)\\mathrm {Mm}(H)$ $\\mathrm {Mm}(G \\boxtimes H) \\ge \\mathrm {Mm}(G)\\mathrm {Mm}(H)$"
],
[
"Some Constructions",
"In this section we leave our discussion of the parameter $\\mathrm {Mm}$ to develop some technical tools that we will need later in the paper to construct matrices with large $\\mathrm {Mm}.$ First we state a construction introduced by Fiedler [9].",
"Theorem 3.1 ([9]) Let $A$ be a symmetric $m \\times m$ matrix with eigenvalues $\\alpha _1,\\ldots ,\\alpha _m$ and let $u$ , $||u||=1$ , be a unit eigenvector corresponding to $\\alpha _1$ .",
"Let $B$ be a symmetric $n \\times n$ matrix with eigenvalues $\\beta _1, \\ldots , \\beta _n$ and let $v$ , $||v||=1$ , be a unit eigenvector corresponding to $\\beta _1.$ Then for any $\\rho $ , the matrix $C=\\left( \\begin{matrix} A & \\rho uv^T \\\\ \\rho vu^T & B \\end{matrix} \\right)$ has eigenvalues $\\alpha _2,\\ldots ,\\alpha _m, \\beta _2, \\ldots ,\\beta _m, \\gamma _1,\\gamma _2$ , where $\\gamma _1, \\gamma _2$ are eigenvalues of the matrix $\\hat{C}=\\left( \\begin{matrix} \\alpha _1 & \\rho \\\\ \\rho & \\beta _1 \\end{matrix} \\right).$ Next we offer a generalization of Theorem REF .",
"Theorem 3.2 Let $A \\in M_n(\\mathbb {R})$ and $B\\in M_m(\\mathbb {R})$ be symmetric matrices, and let $U=\\left( \\begin{matrix} U_1 & U_2 \\end{matrix} \\right) \\in M_n(\\mathbb {R})$ and $V=\\left( \\begin{matrix} V_1 & V_2 \\end{matrix} \\right) \\in M_m(\\mathbb {R})$ be orthogonal matrices such that $U^T AU=\\left( \\begin{matrix} D_1 & 0 \\\\ 0 & D_2 \\end{matrix} \\right) \\; \\text{ and }\\; V^T BV=\\left( \\begin{matrix} E_1 & 0 \\\\ 0 & E_2 \\end{matrix} \\right),$ where $D_1 \\in M_{k}(\\mathbb {R})$ and $E_1 \\in M_l(\\mathbb {R})$ and the partition of the matrices $U$ and $V$ indicated above is consistent with the orders of $D_i$ and $E_i,$ $i=1,2$ .",
"Let $C=\\left( \\begin{matrix} A &U_1 R V_1^T \\\\ V_1 R^T U_1^T & B \\end{matrix} \\right) \\, \\text{ and } \\, W=\\left( \\begin{matrix} U_1 &0 & U_2 &0 \\\\ 0&V_1 & 0 &V_2 \\end{matrix} \\right)$ for any $R \\in \\mathbb {R}^{k \\times l}$ .",
"Then $W$ is an orthogonal matrix and $W^TCW=\\left( \\begin{matrix} D_1 & R & 0 & 0 \\\\R^T & E_1 & 0 & 0\\\\0&0&D_2&0\\\\0&0&0&E_2 \\end{matrix} \\right).$ Proof.",
"Since $U$ and $V$ are orthogonal matrices, it follows that $\\left( \\begin{matrix} I_k&0\\\\0&I_{n-k} \\end{matrix} \\right)=\\left( \\begin{matrix} U_1^T\\\\U_2^T \\end{matrix} \\right) \\left( \\begin{matrix} U_1 & U_2 \\end{matrix} \\right)=\\left( \\begin{matrix} U_1^T U_1 & U_1^TU_2\\\\ U_2^T U_1 & U_2^T U_2 \\end{matrix} \\right)$ and hence $U_1^T U_1=I_k, \\, U_2^T U_2=I_{n-k}, \\, U_1^TU_2=0, \\, U_2^T U_1=0.$ Similarly we get: $V_1^T V_1=I_l, \\, V_2^T V_2=I_{m-l}, \\, V_1^TV_2=0, \\, V_2^T V_1=0.$ Moreover, from $U^T AU=\\left( \\begin{matrix} D_1 & 0 \\\\ 0 & D_2 \\end{matrix} \\right) \\text{ and } V^T AV=\\left( \\begin{matrix} E_1 & 0 \\\\ 0 & E_2 \\end{matrix} \\right)$ we get: $U_1^T A U_1=D_1,\\, U_2^T A U_2=D_2, \\, U_1^TAU_2=0, \\,U_2^T A U_1=0, \\,$ and $V_1^T B V_1=E_1, \\, V_2^T B V_2=E_2, \\, V_1^T BV_2=0, \\, V_2^TB V_1=0.$ Using equations (REF ), (REF ), (REF ) and (REF ) we can easily show that $W^TW=I_{n+m}$ and $W^TCW=\\left( \\begin{matrix} D_1 & R & 0 & 0 \\\\R^T & E_1 & 0 & 0\\\\0&0&D_2&0\\\\0&0&0&E_2 \\end{matrix} \\right),$ which completes the proof.",
"$\\Box $ In the discussion below, we are assuming the definitions and notations from Theorem REF .",
"In our application, we will take: $D_0=\\left( \\begin{matrix} D_1 & R \\\\ R^T & E_1 \\end{matrix} \\right)=\\left( \\begin{matrix} a_1 & 0 & b \\cos \\alpha & b \\sin \\alpha \\\\0 & a_1 & -b \\sin \\alpha & b \\cos \\alpha \\\\ b \\cos \\alpha & -b \\sin \\alpha & a_2 & 0 \\\\b \\sin \\alpha & b \\cos \\alpha & 0 & a_2 \\end{matrix} \\right),$ and the following lemma deals with this situation.",
"Lemma 3.1 If $b=\\sqrt{t(a_1-a_2+t)}$ , where $a_1 > a_2-t$ and $t>0$ , then the eigenvalues of matrix $D_0$ , defined in (REF ), are equal to $a_1+t,a_1+t, a_2-t,a_2-t$ and the orthogonal matrix that diagonalizes $D_0$ is equal to $U_0=\\frac{1}{\\sqrt{a_1-a_2+2t}}\\left( \\begin{matrix} \\frac{b \\sin \\alpha }{\\sqrt{t}}&\\frac{b \\cos \\alpha }{\\sqrt{t}}& -\\sqrt{t}\\sin \\alpha &-\\sqrt{t}\\cos \\alpha \\\\\\frac{b \\cos \\alpha }{\\sqrt{t}} &-\\frac{b \\sin \\alpha }{\\sqrt{t}}&-\\sqrt{t}\\cos \\alpha &\\sqrt{t}\\sin \\alpha \\\\0 & \\sqrt{t}& 0 &\\frac{b}{ \\sqrt{t}}\\\\\\sqrt{t} & 0 & \\frac{b}{ \\sqrt{t}} & 0 \\end{matrix} \\right).$ Proof.",
"The statements in this lemma can be checked by a straightforward direct calculation.",
"$\\Box $ Let $A \\in M_n(\\mathbb {R})$ and $B\\in M_m(\\mathbb {R})$ be symmetric matrices, such that $U^T AU=\\left( \\begin{matrix} a_1 I_2 & 0 \\\\ 0 & D_2 \\end{matrix} \\right) \\; \\text{, }\\; V^T BV=\\left( \\begin{matrix} a_2 I_2& 0 \\\\ 0 & E_2 \\end{matrix} \\right)$ for some orthogonal matrices $U=\\left( \\begin{matrix} U_1 & U_2 \\end{matrix} \\right) \\in M_n(\\mathbb {R})$ and $V=\\left( \\begin{matrix} V_1 & V_2 \\end{matrix} \\right) \\in M_m(\\mathbb {R})$ as in Theorem REF .",
"Let us define $R=\\left( \\begin{matrix} b \\cos \\alpha & b \\sin \\alpha \\\\ -b \\sin \\alpha & b \\cos \\alpha \\end{matrix} \\right)$ with $b=\\sqrt{t(a_1-a_2+t)}$ , $a_1 > a_2-t$ and $t>0$ .",
"Now Theorem REF tells us that the matrix $C=\\left( \\begin{matrix} A &U_1 R V_1^T \\\\ V_1 R^T U_1^T & B \\end{matrix} \\right)$ is orthogonally similar to $\\left( \\begin{matrix} (a_1+t) I_2& 0 & 0 & 0 \\\\0 & (a_2-t)I_2 & 0 & 0\\\\0&0&D_2&0\\\\0&0&0&E_2 \\end{matrix} \\right)$ with orthogonal similarity $\\hat{W}=\\left( \\begin{matrix} U_1 &0 & U_2 &0 \\\\ 0&V_1 & 0 &V_2 \\end{matrix} \\right)\\left( \\begin{matrix} U_0 & 0 \\\\ 0 & I \\end{matrix} \\right)$ , where $U_0$ is defined in (REF ).",
"If we assume that $U_1$ and $V_1$ have no zero entries, then the first four columns of $\\hat{W}$ also do not have any zero entries, for all but a finite set of $\\alpha $ .",
"Since we may choose $\\alpha $ arbitrarily, we have found a matrix $C$ with two orthogonal eigenvectors corresponding to $a_1+t$ that do not have any zero entries, and two orthogonal eigenvectors corresponding to $a_2-t$ that do not have any zero entries.",
"This discussion yields the following corollary.",
"Corollary 3.1 Let $B_i \\in M_{n_i}(\\mathbb {R}),$ $i=1,2,$ be a symmetric matrix with the spectrum $(\\lambda _i, \\lambda _i,\\sigma _i),$ where $\\sigma _i$ is a list of $n_i-2$ real numbers.",
"Let $t>0$ and assume that $B_i$ has at least two orthogonal eigenvectors corresponding to $\\lambda _i$ that do not contain any zero elements and that $\\lambda _1> \\lambda _2-t$ .",
"Then there exists an $n_1 \\times n_2$ matrix $S$ that does not contain any zero elements, such that $C=\\left( \\begin{matrix} B_1 & S \\\\ S^T & B_2 \\end{matrix} \\right) \\in M_{n_1+n_2}(\\mathbb {R})$ has the spectrum $(\\lambda _1+t, \\lambda _1+t, \\lambda _2-t, \\lambda _2-t, \\sigma _1, \\sigma _2)$ .",
"Moreover, $C$ has at least two orthogonal eigenvectors without zero entries corresponding to $\\lambda _1+t$ and at lest two orthogonal eigenvectors without zero entries corresponding to $\\lambda _2-t$ .",
"We will also make use of the following construction from [16].",
"Lemma 3.2 ([16]) Let $B$ be a symmetric $m \\times m$ matrix with eigenvalues $\\mu _1, \\mu _2,\\ldots ,\\mu _m$ , and let $u$ be an eigenvector corresponding to $\\mu _1$ normalized so that $u^Tu=1$ .",
"Let $A$ be an $n \\times n$ symmetric matrix with a diagonal element $\\mu _1$ $A=\\left( \\begin{matrix} A_1 & b \\\\b^T & \\mu _1 \\end{matrix} \\right)$ and eigenvalues $\\lambda _1, \\ldots , \\lambda _n.$ Then the matrix $C=\\left( \\begin{matrix} A_1 & bu^T \\\\ub^T & B \\end{matrix} \\right)$ has eigenvalues $\\lambda _1,\\ldots ,\\lambda _n,\\mu _2,\\ldots , \\mu _m$ .",
"Remark 3.1 In [16] the eigenvectors of $C$ in terms of the eigenvectors of $A$ and $B$ are given in the following way.",
"Let $\\left( \\begin{matrix} v_i \\\\ \\alpha _i \\end{matrix} \\right), \\, v_i \\in \\mathbb {R}^{n-1}, \\, \\alpha _i \\in \\mathbb {R}$ be the orthonormal set of eigenvectors of $A$ corresponding to $\\lambda _i,$ $i=1,2,\\ldots ,n$ , and let $u_i,$ $i=2,\\ldots ,m,$ together with $u$ be the orthonormal set of eigenvectors corresponding to $\\mu _i$ .",
"Then $C$ has the following eigenvectors: eigenvector of $C$ corresponding to eigenvalue $\\lambda _i,$ $i=1,2,\\ldots ,n$ , is equal to: $\\left( \\begin{matrix} v_i \\\\ \\alpha _i u \\end{matrix} \\right),$ and eigenvector of $C$ corresponding to $\\mu _i,$ $i=2,\\ldots ,m$ , is equal to: $\\left( \\begin{matrix} 0 \\\\ u_i \\end{matrix} \\right).$"
],
[
"Graphs having $\\mathrm {mr}_+ G=2$",
"Graphs whose minimal rank is small will typically have large $\\mathrm {Mm}$ .",
"We denote by ${\\rm mr}_+(G)$ the minimum rank among all positive semidefinite symmetric matrices corresponding to $G$ .",
"A characterisation of graphs with $\\mathrm {mr}_+(G) \\le 2$ given in [5] is restated below.",
"Theorem 4.1 ([5]) Let $G$ be a graph on $n$ vertices.",
"Then ${\\rm mr}_+(G)\\le 2$ if and only if $G^c$ has the form $(K_{p_1,q_1}\\cup K_{p_2,q_2} \\cup \\ldots \\cup K_{p_k,q_k})\\vee K_r$ for appropriate nonnegative integers $k, p_1,q_1,\\ldots ,p_k,q_k,r$ with $p_i+q_i>0$ , $i=1,2,\\ldots ,k.$ In this section we show that essentially all graphs $G$ with $\\mathrm {mr}_+(G)=2$ , have the maximal possible $\\mathrm {Mm}$ .",
"Namely, the main purpose of this section is to show the following theorem.",
"Theorem 4.2 Let $G$ be a graph on $n$ vertices with $G^c$ of the form: $(K_{p_0,0}\\cup K_{p_1,q_1}\\cup K_{p_2,q_2} \\cup \\ldots \\cup K_{p_k,q_k})\\vee K_r$ where $p_1,q_1,\\ldots ,p_k,q_k$ are positive integers, $p_0,k,r\\ge 0$ and $(p_0,k)\\ne (1,1)$ .",
"Then for any two nonnegative integers $n_1$ and $n_2$ that satisfy $4+n_1+n_2=n$ there exists a matrix $A \\in S(G)$ that has precisely two distinct eigenvalues $\\lambda $ and $\\mu $ with multiplicities $2+n_1$ and $2+n_2$ , respectively.",
"Corollary 4.1 If $G$ is a graph on $n$ vertices such that $\\mathrm {mr}_+(G)=2$ and $G$ is not of the form $\\left( (K_{1,0} \\cup K_{p,q})\\vee K_r \\right)^c$ , then $\\mathrm {Mm}(G)= \\left\\lfloor \\frac{n}{2} \\right\\rfloor $ .",
"Note that in the case $k=r=0$ , we have $G=(K_{p_0,0})^c=K_{p_0}$ and in the case $p_0=r=0$ , $k=1$ , we have $G=(K_{p,q})^c=K_{p}\\cup K_q$ .",
"Matrices in $S(K_n)$ will be the building blocks in our construction.",
"It will prove useful that result from [15] considers not only the eigenvalues of these matrices, but also says something about the pattern of some of the associated eigenvectors.",
"Theorem 4.3 ([15]) For any given list of real numbers $\\sigma =(\\lambda _1,\\lambda _2,\\ldots ,\\lambda _n)$ , $\\lambda _1 \\ne \\lambda _2$ , there exists $A_n \\in S(K_n)$ with the spectrum $\\sigma $ .",
"Furthermore, given any zero-nonzero pattern of a vector in $\\mathbb {R}^n$ that contains at least two nonzero elements, $A_n$ can be chosen in such a way that there exist an eigenvector corresponding to $\\lambda _1$ with the given pattern.",
"Corollary 4.2 For any integer $n\\ge 2$ we have $\\mathrm {Mm}(K_n)= \\left\\lfloor \\frac{n}{2} \\right\\rfloor $ .",
"While Theorem REF solves the inverse eigenvalue problem for complete graphs, we note that in order for a matrix $A \\in S(K_p \\cup K_q)$ to have an eigenvector corresponding to an eigenvalue $\\lambda $ that has no zero entries, this eigenvalue will need to be a repeated eigenvalue in $A$ .",
"We will use inductive arguments to gradually construct matrices with the desired pattern and eigenvalues.",
"First we present three lemmas to resolve the cases that will be used as the basis of our construction.",
"Lemma 4.1 There exists a matrix in $S((K_{2,0} \\cup K_{1,1})^c)$ with two distinct eigenvalues both with multiplicity 2, and with an orthogonal basis of eigenvectors that do not contain any zero elements.",
"Proof.",
"The matrix $A=\\left(\\begin{array}{cccc}\\frac{2}{2+3 \\sqrt{3}} & \\frac{1}{23} \\left(9-2 \\sqrt{3}\\right) & \\frac{1}{23} \\sqrt{21+26 \\sqrt{3}} & \\frac{1}{23}\\@root 4 \\of {3} \\left(-11+5 \\sqrt{3}\\right) \\\\\\frac{1}{23} \\left(9-2 \\sqrt{3}\\right) & \\frac{2}{2+3 \\sqrt{3}} & \\frac{1}{23} \\sqrt{21+26 \\sqrt{3}} & \\frac{1}{23}\\sqrt{-330+196 \\sqrt{3}} \\\\\\frac{1}{23} \\sqrt{21+26 \\sqrt{3}} & \\frac{1}{23} \\sqrt{21+26 \\sqrt{3}} & -\\frac{2}{23} \\left(-9+2 \\sqrt{3}\\right)& 0 \\\\\\frac{1}{23} \\@root 4 \\of {3} \\left(-11+5 \\sqrt{3}\\right) & \\frac{1}{23} \\sqrt{-330+196 \\sqrt{3}} & 0 & -\\frac{4}{23}\\left(-9+2 \\sqrt{3}\\right) \\\\\\end{array}\\right)$ has the desired properties.",
"$\\Box $ Lemma 4.2 Let $G$ be a graph on $n \\ge 4$ vertices, such that $G^c=K_{1,1}\\cup \\ldots \\cup K_{1,1}.$ Then for all nonnegative integers $n_0,n_1$ satisfying $4+n_0+n_1=n$ there exists a matrix $A \\in S(G)$ that has two distinct eigenvalues with multiplicities $2+n_0$ and $2+n_1$ .",
"Proof.",
"We are looking for an $n \\times n$ matrix $B \\in S\\left( (K_{1, 1} \\cup \\ldots \\cup K_{1, 1})^c\\right)$ with eigenvalues 0 and 1 with multiplicities $m_0 \\ge 2$ and $m_1 \\ge 2$ , respectively.",
"We will prove this claim by induction.",
"In the first step we distinguish between the cases when $m_0$ and $m_1$ are both even and when $m_0$ and $m_1$ are both odd.",
"In the first case let us write $m_0=2 k_0$ and $m_1=2 k_1$ and in the second case let us write $m_0=2 k_0+1$ and $m_1=2 k_1+1.$ Let $k=k_0+k_1.$ In the even case let $A_1=\\left( \\begin{matrix} 1+(k-1)t & 0 \\\\ 0 & 1+(k-1)t \\end{matrix} \\right) \\in S( (K_{1,1})^c)$ and in the odd case let $A_1=\\left(\\begin{array}{cccc}\\frac{2 a}{3}+1 & 0 & -\\frac{a}{3} & \\frac{1}{3} \\sqrt{a^2+\\frac{3 a}{2}} \\\\0 & \\frac{2 a}{3}+1 & -\\frac{a}{3} & -\\frac{1}{3} \\sqrt{a^2+\\frac{3 a}{2}} \\\\-\\frac{a}{3} & -\\frac{a}{3} & \\frac{a+3}{3} & 0 \\\\\\frac{1}{3} \\sqrt{a^2+\\frac{3 a}{2}} & -\\frac{1}{3} \\sqrt{a^2+\\frac{3 a}{2}} & 0 & \\frac{a}{3}\\\\\\end{array}\\right) \\in S\\left((K_{1,1}\\cup K_{1,1})^c\\right),$ where $a=(k-1) t$ and the matrix $A_1$ has the spectrum $(1,0,1+(k-1)t,1+(k-1)t).$ It is easy to check that in both cases we can find two orthogonal eigenvectors with no zero entries corresponding to $1+(k-1)t.$ Starting with $A_1$ we will apply Corollary REF in two ways.",
"For $j=2,\\ldots , k_1$ we take $A_j=\\left( \\begin{matrix} 1-t & 0 \\\\ 0 & 1-t \\end{matrix} \\right) \\in S( (K_{1,1})^c) \\subseteq M_2(\\mathbb {R})$ and define $C_1=A_1$ .",
"By Corollary REF , we recursively construct matrices $C_j=\\left( \\begin{matrix} C_{j-1} & S_j \\\\ S_j^T & A_j \\end{matrix} \\right) \\in S\\left( (K_{1, 1} \\cup \\ldots \\cup K_{1, 1})^c\\right)\\subseteq M_{2j}(\\mathbb {R}),$ where in the even case $C_j$ has eigenvalues $1+(k-j)t$ with multiplicity 2 and 1 with multiplicity $2(j-1)$ , and in the odd case $C_j$ has eigenvalues $1+(k-j)t$ with multiplicity 2, 1 with multiplicity $2(j-1)+1$ , and 0 with multiplicity 1.",
"In addition, the construction guarantees that $C_{j}$ has two eigenvectors with no zero elements corresponding to $1+(k-j)t.$ For $j=1,\\ldots , k_0$ we take $A_{k_1+j}=\\left( \\begin{matrix} -t & 0 \\\\ 0 & -t \\end{matrix} \\right) \\in S( (K_{1,1})^c) \\subseteq M_2(\\mathbb {R}).$ Using Corollary REF we recursively define $C_{k_1+j}=\\left( \\begin{matrix} C_{k_1+j-1} & S_{k_1+j} \\\\ S_{k_1+j}^T & A_{k_1+j} \\end{matrix} \\right) \\in S\\left( (K_{1, 1} \\cup \\ldots \\cup K_{1,1})^c\\right)\\subseteq M_{2(k_1+j)}(\\mathbb {R})$ such that in the even case $C_{k_1+j}$ has eigenvalues $1+(k_0-j)t$ with multiplicity 2, 1 with multiplicity $2(k_1-1)$ and 0 with multiplicity $2j$ , and in the odd case $C_{k_1+j}$ has eigenvalues $1+(k_0-j)t$ with multiplicity 2, 1 with multiplicity $2(k_1-1)+1$ and 0 with multiplicity $2j+1$ .",
"In this way we obtain a matrix $C_{k}=C_{k_1+k_0} \\in S(K_{1,1} \\cup \\ldots \\cup K_{1, 1})^c$ with eigenvalues 0 repeated $2k_0$ times and 1 repeated $2 k_1$ times in the even case, and with eigenvalues 0 repeated $2k_0+1$ times and 1 repeated $2 k_1+1$ times in the odd case.",
"$\\Box $ Remark 4.1 Note that we can, using the same construction as in the proof above, construct a matrix $A\\in S\\left((K_{1,1}\\cup \\ldots \\cup K_{1,1})^c\\right)$ with eigenvalues $1+t$ , $t>0$ , with multiplicity 2, 0 with multiplicity $n_0$ and 1 with multiplicity $n_1$ and two orthogonal eigenvectors corresponding to $1+t$ that do not contain any zero elements.",
"The only difference that we need to make in the proof is for the initial matrix $A_1$ to have eigenvalues $(1+k t,1+k t)$ in the even case and eigenvalues $(1+k t, 1+k t,1,0)$ in the odd case.",
"Lemma 4.3 Let $G$ be a graph on $n\\ge 5$ vertices such that $G^c=K_{1,0} \\cup K_{1,1}\\cup \\ldots \\cup K_{1,1}.$ Then for all nonnegative integers $n_0,n_1$ such that $4+n_0+n_1=n$ there exists a matrix $A \\in S(G)$ that has two distinct eigenvalues with multiplicities $2+n_0$ and $2+n_1$ .",
"Proof.",
"Let $n=2s+1$ .",
"The matrix $\\small {\\left(\\begin{array}{ccccc}\\frac{1}{7} \\left(3+\\sqrt{3}\\right) & 0 & \\frac{\\sqrt{3}}{7} & \\frac{1}{7} \\left(1+\\sqrt{3}\\right) &\\frac{\\sqrt{6}}{7 \\left(3+\\sqrt{3}\\right)} \\\\0 & \\frac{1}{7} \\left(3-\\sqrt{3}\\right) & \\frac{\\sqrt{3}}{7} & \\frac{1}{7} \\left(1-\\sqrt{3}\\right) &-\\frac{\\sqrt{6} \\left(2+\\sqrt{3}\\right)}{7 \\left(3+\\sqrt{3}\\right)} \\\\\\frac{\\sqrt{3}}{7} & \\frac{\\sqrt{3}}{7} & \\frac{3}{7} & 0 & -\\frac{\\sqrt{6}}{7} \\\\\\frac{1}{7} \\left(1+\\sqrt{3}\\right) & \\frac{1}{7} \\left(1-\\sqrt{3}\\right) & 0 & \\frac{2}{7} & \\frac{\\sqrt{2}}{7}\\\\\\frac{\\sqrt{6}}{7 \\left(3+\\sqrt{3}\\right)} & -\\frac{\\sqrt{6} \\left(2+\\sqrt{3}\\right)}{7 \\left(3+\\sqrt{3}\\right)}& -\\frac{\\sqrt{6}}{7} & \\frac{\\sqrt{2}}{7} & \\frac{3}{7} \\\\\\end{array}\\right)}$ is contained in $ S\\left((K_{1,1}\\cup K_{1,1} \\cup K_{1,0})^c\\right)$ and has eigenvalues 1 and 0 with respective multiplicities 2 and 3, and therefore proves our theorem in the case $s=2$ .",
"For $s \\ge 3$ consider a graph $G=(\\underbrace{K_{1,1}\\cup \\ldots \\cup K_{1,1}}_{s-2} \\cup \\left(K_{1,1}\\cup K_{1,1}\\cup K_{1,0}\\right))^c,$ and take arbitrary integers $\\hat{n}_0, \\hat{n}_1\\ge 0$ such that $\\hat{n}_0+\\hat{n}_1+7=n$ .",
"Let $B_1\\in S\\left((K_{1,1}\\cup K_{1,1} \\cup K_{1,0})^c\\right)$ be a matrix with eigenvalues $-t$ , $-t$ , 1, 1, 1, where $0<t <1$ , and two orthogonal eigenvectors corresponding to $-t$ without any zero elements.",
"By example (REF ) and [1] such a matrix exists.",
"Remark REF gives us a matrix $B_2\\in S\\left((K_{1,1}\\cup \\ldots \\cup K_{1,1})^c\\right)$ on $n-5$ vertices with eigenvalues $1+t$ with multiplicity 2, 0 with multiplicity $\\hat{n}_0$ and 1 with multiplicity $\\hat{n}_1$ and two orthogonal eigenvectors corresponding to $1+t$ that do not contain any zero elements.",
"By Corollary REF there exists a matrix $C \\in S(G)$ with eigenvalues 1 with multiplicity $5+\\hat{n}_1$ and 0 with multiplicity $2+\\hat{n}_0$ .",
"We have now covered all the cases, except multiplicities $3, 4$ in the case $s=3$ .",
"This case is solved by the matrix $\\small {\\left(\\begin{array}{ccccccc}\\frac{4}{7} & \\frac{\\sqrt{5}}{7} & \\frac{3}{7 \\sqrt{5}} & \\frac{2}{7 \\sqrt{7} \\left(1+\\sqrt{3}\\right)} & -\\frac{2\\left(2+\\sqrt{3}\\right)}{7 \\sqrt{7} \\left(1+\\sqrt{3}\\right)} & -\\frac{2 \\sqrt{\\frac{3}{7}}}{7} & \\frac{2}{7 \\sqrt{7}}\\\\\\frac{\\sqrt{5}}{7} & \\frac{5}{7} & 0 & -\\frac{\\sqrt{\\frac{5}{7}}}{7 \\left(1+\\sqrt{3}\\right)} & \\frac{\\sqrt{\\frac{5}{7}}\\left(2+\\sqrt{3}\\right)}{7 \\left(1+\\sqrt{3}\\right)} & \\frac{\\sqrt{\\frac{15}{7}}}{7} & -\\frac{\\sqrt{\\frac{5}{7}}}{7} \\\\\\frac{\\sqrt{5}}{7} & 0 & \\frac{1}{7} & \\frac{\\sqrt{\\frac{5}{7}}}{7+7 \\sqrt{3}} & -\\frac{\\sqrt{\\frac{5}{7}}\\left(2+\\sqrt{3}\\right)}{7 \\left(1+\\sqrt{3}\\right)} & -\\frac{\\sqrt{\\frac{15}{7}}}{7} & \\frac{\\sqrt{\\frac{5}{7}}}{7} \\\\\\frac{2}{7 \\sqrt{7} \\left(1+\\sqrt{3}\\right)} & -\\frac{\\sqrt{\\frac{5}{7}}}{7 \\left(1+\\sqrt{3}\\right)} &\\frac{\\sqrt{\\frac{5}{7}}}{7+7 \\sqrt{3}} & \\frac{1}{7} \\left(3+\\sqrt{3}\\right) & 0 & \\frac{\\sqrt{3}}{7} & \\frac{1}{7}\\left(1+\\sqrt{3}\\right) \\\\-\\frac{2 \\left(2+\\sqrt{3}\\right)}{7 \\sqrt{7} \\left(1+\\sqrt{3}\\right)} & \\frac{\\sqrt{\\frac{5}{7}}\\left(2+\\sqrt{3}\\right)}{7 \\left(1+\\sqrt{3}\\right)} & -\\frac{\\sqrt{\\frac{5}{7}} \\left(2+\\sqrt{3}\\right)}{7\\left(1+\\sqrt{3}\\right)} & 0 & \\frac{1}{7} \\left(3-\\sqrt{3}\\right) & \\frac{\\sqrt{3}}{7} & \\frac{1}{7}\\left(1-\\sqrt{3}\\right) \\\\-\\frac{2 \\sqrt{\\frac{3}{7}}}{7} & \\frac{\\sqrt{\\frac{15}{7}}}{7} & -\\frac{\\sqrt{\\frac{15}{7}}}{7} & \\frac{\\sqrt{3}}{7} &\\frac{\\sqrt{3}}{7} & \\frac{3}{7} & 0 \\\\\\frac{2}{7 \\sqrt{7}} & -\\frac{\\sqrt{\\frac{5}{7}}}{7} & \\frac{\\sqrt{\\frac{5}{7}}}{7} & \\frac{1}{7} \\left(1+\\sqrt{3}\\right)& \\frac{1}{7} \\left(1-\\sqrt{3}\\right) & 0 & \\frac{2}{7} \\\\\\end{array}\\right)}$ in $S\\left((K_{1,0} \\cup K_{1,1}\\cup K_{1,1}\\cup K_{1,1})^c\\right)$ with eigenvalues 1 and 0 with respective multiplicities 3 and 4.",
"$\\Box $ The following technical lemma that is a straightforward application of Lemma REF will enable us to complete the proof of the main theorem of this section.",
"Lemma 4.4 Let $\\hat{G}$ be a graph on $n$ vertices with $\\hat{G}^c$ of the form $K_{{p}_0,0}\\cup K_{{p}_1,{q}_1}\\cup K_{{p}_2,{q}_2} \\cup \\ldots \\cup K_{{p}_k,{q}_k},$ where $k,{p}_1,{q}_1,\\ldots ,{p}_k,{q}_k$ are positive integers.",
"Let $G$ be a graph on $n+r\\ge n$ vertices, that we obtain from a graph $\\hat{G}$ by increasing one of the parameters $p_j$ , $j=0,1,\\ldots ,k$ , by $r$ , i.e.",
"$G^c$ is of the form $K_{p_0,0}\\cup K_{p_1,q_1}\\cup \\ldots \\cup K_{p_{j-1,}q_{j-1}} \\cup K_{p_j+r,q_j} \\cup K_{p_{j+1},q_{j+1}}\\cup \\ldots \\cup K_{p_k,q_k}.$ If there exists a matrix in $ A \\in S(\\hat{G})$ with eigenvalues 0 and 1 with multiplicities $2+m_0$ and $2+m_1$ , where $4+m_0+m_1=n$ , such that $A$ does not have any diagonal elements equal to 0 or 1, then for any two nonnegative integers $t$ and $s$ satisfying $t+s=r$ there exists a matrix in $S(G)$ with eigenvalues 0 and 1 with multiplicities $2+m_0+t$ and $2+m_1+s$ .",
"Proof.",
"Let $A \\in S(\\hat{G})$ have eigenvalues 0 and 1 with multiplicities $2+m_0$ and $2+m_1$ .",
"Let $a_{jj}$ be one of the diagonal element of $A$ from a block corresponding to $K_{p_j}$ .",
"Let $B \\in S(K_{r+1})$ have eigenvalues $a_{jj}$ , 0 with multiplicity $t$ and 1 with multiplicity $s$ , $t+s=r$ .",
"(Here we need the condition on the diagonal elements of $A$ , but only in the case when $s=0$ or $t=0$ .",
"If for example $s=0$ and $a_{jj}=0$ , then matrix $B$ that we need does not exist.)",
"Furthermore, we demand that the eigenvector corresponding to $a_{jj}$ has no zero entries.",
"Now we apply Lemma REF to join matrices $A$ and $B$ through the diagonal element $a_{jj}.$ The resulting matrix belongs to $S(G)$ and has the desired eigenvalues.",
"$\\Box $ Proof of Theorem REF .",
"Note that the join with $K_r$ in the statement of the theorem, adds some unconnected vertices to the graph $G$ , therefore it is sufficient to study the graphs without the join.",
"If $(p_0,k)=(0,1)$ , then $G=(K_{p,q})^c=K_p\\cup K_q$ and the statement follows by Theorem REF .",
"Following the proofs of Lemma REF , Lemma REF and Lemma REF it is easy to see that matrices constructed in those lemmas do not have any of the diagonal elements equal to 1 or to 0.",
"Applying Lemma REF to Lemma REF proves the theorem for all graphs of the form $(K_{p_0,0}\\cup K_{p_1,q_1})^c,$ where $p_0 \\ge 2,$ $p_1 \\ge 1$ and $q_1 \\ge 1.$ Lemma REF together with Lemma REF gives the result for all graphs of the form $G=(K_{p_1,q_1}\\cup \\ldots \\cup K_{p_k,q_k})^c,$ where $p_i \\ge 1$ and $q_i \\ge 1$ .",
"Finally, Lemma REF applied to Lemma REF proves the theorem for graphs of the form: $G=(K_{p_0,0} \\cup K_{p_1,q_1}\\cup \\ldots \\cup K_{p_k,q_k})^c,$ where $p_i \\ge 1$ and $q_i \\ge 1$ and $k \\ge 2.$ $\\Box $ Note that in the case $(p_0,k)= (1,1)$ we have that $q\\left((K_{1,0} \\cup K_{p,q})^c\\right)\\ge 3$ by [1].",
"Hence $q\\left((K_{1,0} \\cup K_{p,q})\\vee K_r\\right)^c \\ge 3 $ and this case is excluded in Theorem REF .",
"It follows from the proposition below that $\\mathrm {Mm}\\left((K_{1,0} \\cup K_{p,q})\\vee K_r\\right)^c =\\left\\lfloor \\frac{n}{3}\\right\\rfloor .$ Proposition 4.1 Let $n=p+q+1,$ and let $n_1, n_2, n_3$ be positive integers such that $n_1+n_2+n_3=n.$ Then there exists $A \\in S((K_{1,0} \\cup K_{p,q})^c)$ with eigenvalues $\\lambda _1,\\lambda _2, \\lambda _3$ with multiplicities $n_1, n_2, n_3$ , respectivelly.",
"Proof.",
"For $n=3$ any matrix in $S((K_{1,0} \\cup K_{p,q})^c)$ satisfies the conditions of the proposition.",
"For example $A=\\left(\\begin{array}{ccc}0 & 2 & 1 \\\\2 & 1 & 0 \\\\1 & 0 & 2 \\\\\\end{array}\\right) \\in S((K_{1,0} \\cup K_{1,1})^c).$ has eigenvalues $(3, -\\sqrt{3}, \\sqrt{3})$ .",
"Let $p_i$ and $q_i$ , $i=1,2,3,$ be nonnegative integers such that $p_1+p_2+p_3=p-1$ and $q_1+q_2+q_3=q-1.$ Let $B_1 \\in S(K_p)$ have eigenvalues 1 with multiplicity 1, 3 with multiplicity $p_1$ , $\\sqrt{3}$ with multiplicity $p_2$ and $-\\sqrt{3}$ with multiplicity $p_3$ , $1+p_1+p_2+p_3=p$ .",
"Moreover, we demand that the eigenvector of $B_1$ corresponding to 1 has no zero elements.",
"Similarly, let $B_2 \\in S(K_p)$ have eigenvalues 2 with multiplicity 1, 3 with multiplicity $q_1$ , $\\sqrt{3}$ with multiplicity $q_2$ and $-\\sqrt{3}$ with multiplicity $q_3$ , $1+q_1+q_2+q_3=q$ , with eigenvector corresponding to 2 having no zero elements.",
"Matrices $B_1$ and $B_2$ exist by Theorem REF .",
"Now we use Lemma REF to first join matrix $A$ with $B_1$ through the diagonal element $1,$ and then to join the resulting matrix with $B_2$ through the diagonal element 2.",
"In this way be obtain a matrix in $S((K_{1,0} \\cup K_{p,q})^c)$ with eigenvalues 3 with multiplicity $1+p_1+q_1$ , $\\sqrt{3}$ with multiplicity $1+p_2+q_2$ and $-\\sqrt{3}$ with multiplicity $1+p_3+q_3$ .",
"Since we didn't pose any conditions on the parameters $p_i$ and $q_i$ this proves or claim.",
"$\\Box $"
],
[
"Complete bipartite graphs",
"Another family of graphs with small minimal rank is the set of complete bipartite graphs.",
"Note that $\\mathrm {mr}(K_{m,n})=2$ for any positive integers $m,n$ .",
"We will prove that $\\mathrm {Mm}(K_{m,n})=\\lfloor \\frac{m+n}{3} \\rfloor $ if $m \\ne n$ .",
"Theorem 4.4 Let $m \\le n.$ Then any list of the form $(\\lambda _1,-\\lambda _1,\\lambda _2,-\\lambda _2,\\ldots ,\\lambda _m,-\\lambda _m,\\underbrace{0,\\ldots ,0}_{n-m}),$ where $\\lambda _1 > 0$ and $\\lambda _i \\ge 0$ for $i=2,\\ldots ,m$ , is the spectrum of a matrix $A \\in S(K_{m,n})$ .",
"Proof.",
"Let $A=\\left( \\begin{matrix} 0_m & B \\\\ B^T & 0_n \\end{matrix} \\right),$ where $B$ is an $m \\times n$ matrix.",
"Then, using Schur complement, we can compute the characteristic polynomial of $A$ as follows: $\\det (xI-A)&=\\det (xI_n)\\det (xI_m-x^{-1}BB^T)\\\\&= x^{n-m}\\det (x^2I_m-BB^T).$ So, it follows that if $\\mu _1, \\ldots , \\mu _m$ are the eigenvalues of $BB^T$ , then the eigenvalues of $A$ are $\\sqrt{\\mu _i},$ $-\\sqrt{\\mu _i}$ for $i=1,2,\\ldots , m,$ together with $n-m$ instances of the eigenvalue 0.",
"It remains to show that we can find a matrix $B$ that does not contain any zero elements so that $BB^T$ has eigenvalues $\\lambda _i$ , $i=1,2, \\ldots , m$ .",
"Let $A_0$ be a symmetric matrix in $S(K_m)$ with eigenvalues $\\lambda _1,\\lambda _2,\\ldots ,\\lambda _m$ and let $U$ be an orthogonal matrix that diagonalises $A_0$ : $U^TA_0U=D$ .",
"Then $A_0=UD^{\\frac{1}{2}}(UD^{\\frac{1}{2}})^T,$ and the matrix $B_0=UD^{1/2}$ satisfies the spectral conditions that we need, however it can have some zero entries.",
"Since we have choosen $A_0 \\in S(K_m)$ we know that $B_0$ has no zero rows, so there exists an orthogonal matrix $V$ such that $B_0V$ has no zero elements.",
"(A generic $n \\times n$ orthogonal matrix will accomplish this.)",
"We take $B=B_0V$ to finish the proof.",
"$\\Box $ Corollary 4.3 For any positive integers $m,n$ , $\\mathrm {Mm}(K_{m,n})={\\left\\lbrace \\begin{array}{ll}m, & m=n,\\\\\\lfloor \\frac{m+n}{3} \\rfloor , &\\text{otherwise.}\\end{array}\\right.",
"}$ Proof.",
"Note that by [1], we have $q(K_{m,n})={\\left\\lbrace \\begin{array}{ll}2, & m=n,\\\\3, & m \\ne n.\\end{array}\\right.",
"}$ From (REF ) it follows that $\\mathrm {Mm}(K_{m,m})\\le m$ and $\\mathrm {Mm}(K_{m,n})\\le \\lfloor \\frac{m+n}{3}\\rfloor $ for $m \\ne n$ .",
"In the case $m=n$ this implies by Theorem REF that $\\mathrm {Mm}(K_{m,m})=m$ .",
"If $m \\ne n$ , let $m+n=3s+k,$ where $k \\in \\lbrace 0,1,2\\rbrace $ .",
"If $k\\in \\lbrace 0,1\\rbrace $ , then choose in Theorem REF a matrix $A \\in K_{m,n}$ with eigenvalues $\\lambda $ , $-\\lambda $ and 0 with multiplicities $s$ , $s$ , and $s+k$ , respectively.",
"In the case $k=2$ , take $A \\in K_{m,n}$ with eigenvalues $\\lambda $ , $-\\lambda $ and 0 with multiplicities $s+1$ , $s+1$ and $s$ , respectively.",
"This shows that $\\mathrm {Mm}(K_{m,n})=s= \\lfloor \\frac{m+n}{3} \\rfloor $ if $m \\ne n$ .",
"$\\Box $"
],
[
"Graphs with small $\\mathrm {Mm}$",
"In this section we look at some cases when $\\mathrm {Mm}$ is small.",
"Since we know that the largest and the smallest eigenvalue of a matrix $A \\in S(T)$ , where $T$ is a tree, both have multiplicities 1, [11], we have $\\mathrm {Mm}(T)= 1$ for all trees $T$ .",
"In [12] a characterisation of graphs with maximal multiplicity of an eigenvalue equal to 2 is given.",
"It is shown that if in the graph $G$ there exist two independent induced paths that cover all the vertices of $G$ and such that any edges between the two paths can be drawn so that they do not cross, then $G$ has maximal multiplicity of an eigenvalue equal to 2.",
"Such graphs are called graphs with two parallel paths.",
"It clear, that if $\\mathrm {mr}(G)=n-2$ , then $\\mathrm {Mm}(G)$ is either equal to 1 or to 2.",
"We will show that both cases can occur.",
"First, we start by Theorem that will imply $\\mathrm {Mm}(G)=1$ for certain unicyclic graphs $G$ having $M(G)=2$ .",
"Lemma 5.1 Let $A=\\left( \\begin{matrix} d & a^T \\\\ a & D \\end{matrix} \\right),$ where $a=\\left( \\begin{matrix} a_1 & a_2 & \\ldots & a_n \\end{matrix} \\right)^T$ and $D$ is a diagonal matrix with diagonal elements $d_1,d_2,\\ldots ,d_n$ .",
"Let $\\lambda $ be an eigenvalue of $A$ with multiplicity $k$ .",
"If the multiplicity of $\\lambda $ in $D$ is either $k-1$ or $k$ , then $d_i=\\lambda $ implies $a_i=0.$ Proof.",
"The characteristic polynomial of $A$ is equal to $p(x)=(x-d)\\prod _{i=1}^n(x-d_i)-\\sum _{i=1}^n a_i^2 \\prod _{j \\ne i}(x-d_j).$ Since $\\lambda $ is an eigenvalue of $A$ of multiplicity $k$ , it follows that $p(x)=(x-\\lambda )^kp_1(x)$ , where $p_1(\\lambda )\\ne 0$ .",
"By interlacing, $\\lambda $ is an eigenvalue of $D$ with multiplicity $l$ , where $k-1\\le l \\le k+1$ .",
"Suppose that $l \\le k$ and without loss of generality we let $d_1=d_2=\\ldots =d_{l}=\\lambda $ and we assume $d_{l+1}, \\ldots ,d_n$ are all different from $\\lambda $ .",
"Then we have: $(x-\\lambda )^{k-l+1}p_1(x)&=(x-d)(x-\\lambda )\\prod _{i=l+1}^n(x-d_i)-\\\\ &-\\sum _{i=1}^l a_i^2\\prod _{j=l+1}^n(x-d_j)-\\sum _{i=l+1}^l a_i^2(x-\\lambda )\\prod _{\\begin{array}{c}j=l+1\\\\j\\ne i\\end{array}}^n(x-d_j).$ Since $k-l+1 \\ge 1$ , all but one therm in the above expression is divisible by $(x-\\lambda )$ .",
"The therm that is not divisible by $(x-\\lambda )$ is: $\\sum _{i=1}^l a_i^2\\prod _{j=l+1}^n(x-d_i)=\\left(\\sum _{i=1}^l a_i^2\\right)\\prod _{j=l+1}^n(x-d_i).$ This implies that $\\sum _{i=1}^l a_i^2=0$ and $a_1=\\ldots =a_l=0$ .",
"$\\Box $ Theorem 5.1 Let $G$ be a graph with $A \\in S(G)$ of the form $A=\\left( \\begin{matrix} d & b_1^T & \\ldots &b_p^T & c^T \\\\b_1 & B_1 & &&& \\\\\\vdots &&\\ddots &&&\\\\b_p & & &B_p& \\\\c & &&& D \\end{matrix} \\right) \\in \\mathbb {R}^{n \\times n},$ where $B_i \\in \\mathbb {R}^{n_i \\times n_i}$ , $n_i \\ge 2$ , are not diagonal matrices, and $D$ is an $m \\times m$ diagonal matrix.",
"Moreover, $b_i \\in \\mathbb {R}^{n_i}$ are not zero vectors and $c \\in \\mathbb {R}^{m}$ has no zero elements.",
"Let matrix $A$ be of the form (REF ) and have $t$ distinct eigenvalues.",
"Then $\\mathrm {Mm}(A) \\le \\frac{2(n-m-p-1)}{t+1}+1.$ In particular: $\\mathrm {Mm}(G) \\le \\frac{n-m-p-1}{2}+1.$ Proof.",
"Let $A$ have $t$ distinct eigenvalues $\\lambda _i$ , $i=1,2,\\ldots ,t$ , with multiplicities $k_1, k_2,\\ldots ,k_t$ .",
"If $\\min _i \\lbrace k_i\\rbrace =1$ , then $\\mathrm {Mm}(A)=1$ and the statement follows.",
"So, suppose $k_i \\ge 2$ for $i=1,2,\\ldots ,t$ .",
"By the interlacing theorem each $\\lambda _i$ is an eigenvalue of the submatrix $B\\oplus D=B_1\\oplus \\ldots \\oplus B_p \\oplus D$ of multiplicity $k_i-1$ , $k_i$ or $k_i+1$ .",
"Without loss of generality we may assume that $\\lambda _i$ , $i=1,2,\\ldots ,s$ , have multiplicities $k_i-1$ or $k_i$ as eigenvalues of $B\\oplus D$ , and eigenvalues $\\lambda _i$ , $i=s+1,\\ldots ,t$ , have multiplicities $k_i+1$ as eigenvalues of $B\\oplus D$ .",
"Denote $s^{\\prime }=t-s$ .",
"It follows that $k_1+k_2+\\ldots +k_s-s+k_{s+1}+\\ldots +k_t+s^{\\prime } \\le \\sum _{i=1}^t {\\rm mult}_{B\\oplus D} (\\lambda _i) \\le n-1,$ and from here $s-s^{\\prime }\\ge 1$ .",
"From $t=s+s^{\\prime }$ it now follows that $s \\ge \\frac{t+1}{2}$ .",
"Let $U_B=U_1\\oplus \\ldots \\oplus U_p$ be an orthogonal matrix that diagonalises $B=B_1\\oplus \\ldots \\oplus B_p$ , i.e.",
"$U_BBU_B^T=D_{1}\\oplus \\ldots \\oplus D_p$ , where $D_i\\in \\mathbb {R}^{n_i \\times n_i}$ are all diagonal matrices.",
"Then $U=1 \\oplus U_B \\oplus I_{m}$ puts $A$ in the form of Lemma REF : $UAU^T=\\left( \\begin{matrix} d & b_1^T U_1^T& \\ldots &b_{p}^TU_p^T & c^T \\\\U_1b_1 & D_1 & && \\\\\\vdots &&\\ddots &&\\\\U_pb_{p} && &D_{p} & \\\\c & &&& D \\end{matrix} \\right).$ If for some $j$ , $j \\in \\lbrace 1,2,\\ldots ,p\\rbrace $ , all the eigenvalues of $B_j$ are among $\\lambda _i$ , $i=1,2,\\ldots ,s$ , then $U_jb_j=0$ by Lemma REF and hence $b_j=0$ , a contradiction.",
"So for each $j$ , $j \\in \\lbrace 1,2,\\ldots ,p\\rbrace $ , at least one of the eigenvalues of $B_j$ has to be different than $\\lambda _i$ , $i=1,2,\\ldots ,s$ .",
"Since we are assuming that $c^T$ has no zero elements, Lemma REF also tells us that $\\lambda _i$ , $i=1,\\ldots , s$ , are not eigenvalues of $D$ .",
"Since $k_i \\ge 2$ , $\\lambda _1,\\ldots ,\\lambda _s$ are the eigenvalues of $B$ with multiplicities at least $k_i-1\\ge 1$ .",
"The sum of multiplicities of $\\lambda _1$ , ..., $\\lambda _s$ in $B$ gives us $k_1+k_2+\\ldots +k_s-s \\le n_1+\\ldots +n_p-p.$ Since $\\mathrm {Mm}(A) \\le k_i$ for $i=1,\\ldots ,s$ , we have $s\\, \\mathrm {Mm}(A) \\le n_1+\\ldots +n_p-p+s=n-m-p-1+s$ and therefore $\\mathrm {Mm}(A) \\le \\frac{n-m-p-1}{s}+1 \\le \\frac{2(n-m-p-1)}{t+1}+1.$ In particular, by [1] the minimal number of distinct eigenvalues of $A$ is $q(G) \\ge 3$ and thus we have $t\\ge 3$ and the result is proved.",
"$\\Box $ Corollary 5.1 Let $G$ be a graph with $A \\in S(G)$ of the form $A=\\left( \\begin{matrix} d & b^T & c^T \\\\b & B & 0 \\\\c & 0 & D \\end{matrix} \\right) \\in \\mathbb {R}^{n \\times n},$ where $B\\in M_{n_1 \\times n_1}(\\mathbb {R})$ , $D$ is an $n_2 \\times n_2$ diagonal matrix, $b \\in \\mathbb {R}^{n_1}$ , b$\\ne 0$ , and $c \\in \\mathbb {R}^{n_2}$ has no zero elements.",
"Then $\\mathrm {Mm}(G) \\le \\frac{n_1+1}{2}.$ Remark 5.1 The inequality between $n_1$ and $\\mathrm {Mm}(G)$ is the best possible, since it is achieved for complete graphs $G$ .",
"(In this case $n_2=0$ .)",
"It is not difficult to find examples of $A \\in S(G)$ for which the inequality is achieved, where $G(B)$ is the complete graph on $n_1$ vertices and $n_2$ is kept general.",
"Remark 5.2 For $n_1=2$ in Corollary REF we get $\\mathrm {Mm}(G)=1$ .",
"If $B$ has nonzero off-diagonal entries, then the theorem gives us an example of a graph $G$ on $n$ vertices with $\\mathrm {Mm}(G)=1$ that is not a tree, and whose maximal multiplicity of an eigenvalue is $n-3$ .",
"Example of such graph where $n_1=2$ and $n_2=7$ : [style=thick] in 0,30,60,90,120,150,180 (0:0) node – (:1) node ; [fill=white] in 0,30,60,90,120,150,180 (:1) circle (1mm) ; (0:0) – (0.5,-1) – (-0.5,-1) – (0,0); [fill=white] in (0,0),(0.5,-1),(-0.5,-1) circle (1mm) ; Example 5.1 Let $a^T=(0,0,\\sqrt{\\frac{1}{m+1}})$ , $c^T=(\\sqrt{\\frac{1}{m+1}},\\sqrt{\\frac{1}{m+1}},\\ldots ,\\sqrt{\\frac{1}{m+1}})\\in \\mathbb {R}^m$ and $A_1=\\left(\\begin{array}{ccc}1 & 0 & 0 \\\\0 & -1 & 0 \\\\0 & 0 & 0 \\\\\\end{array}\\right).$ The matrix: $A=\\left( \\begin{matrix} 0 & a^T & c^T \\\\a & A_1 & 0 \\\\c & 0 & 0 \\end{matrix} \\right)\\in \\mathbb {R}^{(m+4)\\times (m+4)}$ has the spectrum $(1,1,-1,-1,0,0,\\ldots ,0),$ where the multiplicity of 0 is $m.$ The matrix $U=\\left(\\begin{array}{ccc}\\frac{1}{\\sqrt{3}} & \\frac{1}{\\sqrt{3}} & \\frac{1}{\\sqrt{3}} \\\\-\\sqrt{\\frac{2}{3}} & \\frac{1}{\\sqrt{6}} & \\frac{1}{\\sqrt{6}} \\\\0 & -\\frac{1}{\\sqrt{2}} & \\frac{1}{\\sqrt{2}} \\\\\\end{array}\\right).$ is an orthogonal matrix, $B=UA_1U^T=\\left(\\begin{array}{ccc}0 & -\\frac{1}{3 \\sqrt{2}}-\\frac{\\sqrt{2}}{3} & \\frac{1}{\\sqrt{6}} \\\\-\\frac{1}{3 \\sqrt{2}}-\\frac{\\sqrt{2}}{3} & \\frac{1}{2} & \\frac{1}{2 \\sqrt{3}} \\\\\\frac{1}{\\sqrt{6}} & \\frac{1}{2 \\sqrt{3}} & -\\frac{1}{2} \\\\\\end{array}\\right) \\in S(K_3)$ and $Ua=\\left(\\begin{array}{c}\\frac{1}{\\sqrt{3} \\sqrt{m+1}} \\\\\\frac{1}{\\sqrt{6} \\sqrt{m+1}} \\\\\\frac{1}{\\sqrt{2} \\sqrt{m+1}} \\\\\\end{array}\\right).$ The matrix $\\left( \\begin{matrix} 0 & a^TU^T & c^T \\\\Ua & B & 0 \\\\c & 0 & 0 \\end{matrix} \\right)$ is similar to $A$ , hence it has the same spectrum.",
"This example shows that the inequality in Corollary REF can be achieved, and that the multiplicity of one of the eigenvalues can be arbitrarily large.",
"Next we introduce a family of graphs $G$ with $M(G)=2$ and $\\mathrm {Mm}(G)=2$ .",
"Proposition 5.1 Let $M_{2n}=\\left( \\begin{matrix} T_n & D_n \\\\ D_n & T_n \\end{matrix} \\right),$ where $T_n$ is an $n \\times n$ tridiagonal matrix with all its nonzero entries equal to 1, and $D_n$ a diagonal matrix with diagonal entries $d_1,\\ldots ,d_n$ satisfying the condition $d_j=-d_{n-j+1}$ for $j=1,2,\\ldots , \\lceil \\frac{n}{2} \\rceil $ .",
"Then the characteristic polynomial of $M_{2n}$ is of the form $p(x)^2$ for some polynomial $p(x)$ of degree $n$ .",
"Proof.",
"First we observe that $\\left( \\begin{matrix} I_n & 0_n \\\\ I_n & I_n \\end{matrix} \\right)\\left( \\begin{matrix} T_n & D_n \\\\ D_n & T_n \\end{matrix} \\right)\\left( \\begin{matrix} I_n & 0_n \\\\ -I_n & I_n \\end{matrix} \\right)=\\left( \\begin{matrix} T_n-D_n & D_n \\\\ 0_n & T_n+D_n \\end{matrix} \\right).$ Let $P_n$ be a permutation matrix with $p_{j,\\, n-j+1}=1$ for $j=1,2,\\ldots n$ .",
"Notice that $P_nT_nP_n=T_n$ , $P_nD_nP_n=-D_n$ , and $P_n(T_n+D_n)P_n=T_n-D_n.$ This shows that $T_n-D_n$ and $T_n+D_n$ are similar, hence have the same spectrum.",
"$\\Box $ Corollary 5.2 Let $G$ be a graph with $M_{2n}\\in S(G)$ , where $M_{2n}$ is defined in Proposition REF .",
"Then $\\mathrm {Mm}(G)=2$ .",
"Proof.",
"The result follows from the fact that graphs $G$ with $M_{2n} \\in S(G)$ are graphs with two parallel paths and therefore $M(G)=2$ .",
"By Proposition REF it follows that $\\mathrm {Mm}(G)=2$ .",
"$\\Box $ Example 5.2 The four connected graphs on 8 vertices having $\\mathrm {Mm}(G)=2$ that are covered by Corollary REF are: [style=thick,scale=1] in 0,1,2 (,0) node – (,1) node (,0) node – (+1,0) node (,1) node – (+1,1) node ; (3,0) – (3,1); [fill=white] in 0,1,2,3 (,0) circle (1mm) (,1) circle (1mm) ; [style=thick,scale=1] in 0,1,2 (,0) node – (+1,0) node (,1) node – (+1,1) node ; (3,0) – (3,1); (0,0) – (0,1); [fill=white] in 0,1,2,3 (,0) circle (1mm) (,1) circle (1mm) ; [style=thick,scale=1] in 0,1,2 (,0) node – (+1,0) node (,1) node – (+1,1) node ; (1,0) – (1,1); (2,0) – (2,1); [fill=white] in 0,1,2,3 (,0) circle (1mm) (,1) circle (1mm) ; In this work we introduced a parameter $\\mathrm {Mm}$ to be studied in connection with the inverse eigenvalue problem for graphs.",
"We listed some basic properties of $\\mathrm {Mm}$ and derived $\\mathrm {Mm}(G)$ for some families of graphs $G$ .",
"We believe that looking at $\\mathrm {Mm}$ is a good way to expose a deeper insight into the eigenvalue structure that is allowed under pattern constraints imposed by a given graph $G$ ."
]
] | 1606.05214 |
[
[
"LP-Based Robust Algorithms for Noisy Minor-Free and Bounded Treewidth\n Graphs"
],
[
"Abstract We give a general approach for solving optimization problems on noisy minor free graphs, where a \\delta-fraction of edges and vertices are adversarially corrupted.",
"The noisy setting was first considered by Magen and Moharrami and they gave a (1 + \\delta)-estimation algorithm for the independent set problem.",
"Later, Chan and Har-Peled designed a local search algorithm that finds a (1 + O(\\delta))-approximate independent set.",
"However, nothing was known regarding other problems in the noisy setting.",
"Our main contribution is a general LP-based framework that yields a (1 + O(\\delta log m log log m))-approximation algorithm for noisy MAX-k-CSPs on m clauses."
],
[
"Introduction",
"Several hard optimization problems often become substantially easier on special classes of graphs such as planar graphs and bounded treewidth graphs.",
"For example, while the maximum independent set problem is notoriously hard on general graphs [27], it admits an efficient approximation scheme on planar graphs [32], [5] and can be solved exactly in polynomial time on bounded treewidth graphs [6].",
"Similarly, while MAX-$k$ -SAT is APX-hard in general [28], planar instances admit an efficient approximation scheme [29] and bounded treewidth instances can be solved exactly in polynomial time [25], [35].",
"In general, there has been extensive work done on designing better algorithms for special graph classes and several general techniques have been developed for this purpose.",
"For problems on bounded treewidth graphs, several techniques based on dynamic programming and deep results from algorithmic graph minor theory and logic have been developed [12], [6], [14], [11], [17].",
"For problems on planar graphs, many surprising approximation guarantees can be obtained based on decomposition approaches.",
"One of the first decomposition approaches is based on the planar-separator theorem [32].",
"Later, Baker [5] developed a more versatile technique based on decomposition into $O(1)$ -outerplanar graphs (which have bounded treewidth).",
"For example, Khanna and Motwani [29] used Baker's technique to obtain efficient approximation schemes for a wide variety of Constraint Satisfaction Problems (CSPs) such as MAX-SAT.",
"Noisy graph models.",
"In this paper, we consider a natural question that was first studied by Magen and Moharrami [33]: What happens to these special graph classes when they are perturbed adversarially?",
"For the maximum independent set problem, [33] considers the setting where an input graph $G$ on $n$ vertices is obtained from some (hidden) underlying planar graph $G_0$ by adding $\\delta n$ arbitrary edges (these are called noisy edges) for some small number $\\delta >0$ and ask: how well can one approximate the maximum independent set (MIS) problem on $G$ ?",
"More generally, one can consider the same question for other optimization problems that are easy on these special graph classes.",
"In this work, we consider MAX-$k$ -SAT (for constant $k$ ) in the noisy setting.",
"To introduce our noise model and to relate to the noisy graph model of [33], we remind the reader of the definition of a factor graph: given a $k$ -SAT formula $\\Phi $ , the factor graph of $\\Phi $ is a bipartite graph $H=(A,B)$ where $A$ contains a vertex for every variable appearing in $\\Phi $ , $B$ contains a vertex for every clause appearing in $\\Phi $ , and a clause-vertex $\\phi $ is connected to a variable-vertex $x$ if and only if $x$ belongs to the clause $\\phi $ .",
"In the noisy setting, the input formula $\\Phi $ is given by an adversary who takes a planar $k$ -SAT formula $\\Phi _0$ (i.e.",
"$\\Phi _0$ 's factor graph is planar) with $n$ variables and $m$ clauses, and adds $\\delta m$ clauses (each clauses contains exactly $k$ literals) to $\\Phi _0$ (resulting in $\\delta m$ vertices and $k \\delta m = O(\\delta m)$ edges being added to the factor graph of $\\Phi _0$ ).",
"We choose to focus on this noise model as it does not change the arity of the original formula $\\Phi _0$ .",
"Our results carry over without much difficulty to more general noise models where one adds both vertices and edges to the factor graph of $\\Phi _0$ , so long as the total number of edges and vertices added is $\\delta m$ .",
"Previous Work.",
"Several previous works have considered approximation algorithms for MIS in the noisy setting.",
"Magen and Moharrami presented an elegant argument showing that $\\alpha (G)$ can be approximated to within a $(1+\\epsilon )$ factor, for $\\epsilon = \\Omega (\\delta )$ , using $O(1/\\epsilon )$ levels of the Sherali-Adams (SA) Hierarchy.",
"Interestingly, this only yields an efficient estimation algorithm for $\\alpha (G)$ and does not give any way to actually find the corresponding independent set.Self-reducibility techniques such as those used for finding independent set in perfect graphs do not seem to work here as the estimation algorithm only gives an approximate answer.",
"In particular, the SA approach only uses the existence of $G_0$ to argue that $SA(G)\\approx \\alpha (G)$ , without actually detecting the noisy edges.",
"Later, Chan and Har-Peled [8] developed a PTAS for planar independent set based on local search which can be used to obtain a $(1+\\epsilon )$ -approximation for MIS in noisy planar graphs.",
"We are not aware of any previous work that studied approximation algorithms for noisy CSPs.",
"Limitations of Current Approaches.",
"For MAX-$k$ -SAT, classic approaches for planar instances based on computing decompositions of the factor graph break down completely in the presence of noise.",
"In particular, the known algorithms for finding planar separators or decomposition into $O(1)$ -outerplanar graphs inherently use the planar structure in various ways, and are easily tricked even with very few noisy edges.",
"In fact, a key motivation of [33] for studying the noisy setting was to design more “robust\" algorithms that are not specifically tailor-made for particular graph classes.",
"Note that in our noise model the adversary is even allowed to add a bounded degree expander on some subset of $O(\\delta n)$ vertices, completely destroying the planar structure and increasing the treewidth to $\\Omega (n)$ .",
"Another natural approach might be to recover some planar graph $\\tilde{G}$ from $G$ without removing too many edges, and then apply the algorithms for planar graphs to $\\tilde{G}$ .",
"However the best known guarantees for this Minimum Planarization problem are too weak for our purpose.",
"In particular, even for bounded degree graphs these algorithms [10], [9] only achieve a $poly(n,\\operatorname{OPT})$ approximation, where $\\operatorname{OPT}$ is the number of noisy edges, and thus only work in our setting when the noise parameter $\\delta = n^{-\\Omega (1)}$ .",
"Finally, one may attempt to apply the local search algorithm of [8].",
"However, the analysis of the local search algorithm of [8] crucially relies on the existence of a decomposition based on the planar separator theorem known as $r$ -divisions [24] and does not seem to apply to noisy instances of MAX-$k$ -SAT."
],
[
"Our Contributions",
"In this work, we give a general approach for solving problems on noisy planar graphs.",
"Using this approach, we give the first algorithm that is able to handle noisy versions of planar MAX-$k$ -SAT.",
"Theorem 1.1 Let $\\Phi _0$ be a planar $k$ -SAT formula over $n$ variables and $m$ clauses, where $k$ is a constant independent of $n$ and $m$ .",
"Let $\\Phi $ be a $k$ -SAT formula obtained by adding $\\delta m$ clauses to $\\Phi _0$ , for some $\\delta > 0$ .",
"Then there is an algorithm such that given any $\\epsilon > 0$ , finds a $(1 + O(\\epsilon + \\delta \\log m \\log \\log m))$ -approximate assignment in time $m^{O(\\log \\log m)^2/\\epsilon }$ .",
"Remark: Since MAX-$k$ -SAT can be approximated within a constant factor, Theorem REF is of interest when $\\delta =O(1/(\\log m \\log \\log m))$ .",
"Theorem REF can also be extended to arbitrary (binary) $k$ -CSPs.",
"For the maximum independent set problem, our approach gives an LP-based approximation algorithm with much better running time and whose approximation ratio has a better dependence on $\\delta $ .",
"Theorem 1.2 Let $G$ be an $n$ -vertex graph obtained by adding $\\delta n$ arbitrary edges to some planar graph $G_0$ , for some $\\delta >0$ .",
"Then there is an algorithm such that given any $\\epsilon > 0$ , finds an independent set of size within a $(1 + O(\\epsilon + \\delta ))$ factor of $\\alpha (G)$ , and runs in time $n^{O(1/\\epsilon ^4)}$ ."
],
[
"General framework",
"The starting point for our results is the following simple observation: Most algorithms for planar graphs use the planar structure only to find a certain structured decomposition of the graph, and once this is found, apply some simple or brute-force algorithm.",
"For example, the planar separator theorems are used to argue that given any $\\epsilon >0$ , a planar graph can be decomposed into disjoint components of size $O(1/\\epsilon ^2)$ by removing some subset $X$ of at most $\\epsilon n$ vertices.",
"Similarly in Baker's decomposition, an $\\epsilon $ fraction of edges $F$ (or vertices) can be removed from a planar graph to decompose it into $O(1/\\epsilon )$ -treewidth graphs.",
"Now consider a noisy planar graph $G$ .",
"We claim that such nice decompositions also exist for $G$ (although it is unclear how to find them).",
"For example, consider the decomposition of the underlying planar graph $G_0$ into bounded size pieces and for every noisy edge in $G\\setminus G_0$ put one of its endpoint in $X$ .",
"This subset $X$ has size $|X| \\le (\\epsilon + \\delta )n$ and removing $X$ splits $G$ into components of size $O(1/\\epsilon ^2)$ .",
"Similarly, let $F$ be the edges removed in Baker's decomposition of $G_0$ , plus the noisy edges in $G\\setminus G_0$ .",
"Clearly, $|F| \\le (\\epsilon + \\delta ) n$ and removing $F$ decomposes $G$ into $O(1/\\epsilon )$ -treewidth graphs.",
"So this leads to the natural question of whether we can directly find such good decompositions, without relying on the topological or other specific structure of the graph.",
"Our main contribution is to show that this can indeed be done using general LP-based techniques.",
"Specifically, we consider the following problems.",
"Bounded Size Interdiction (weaker form): Suppose $G$ can be decomposed into components of size at most $1/\\epsilon ^2$ by removing some (small) subset $X$ of vertices.",
"Find such a subset $X$ .",
"Bounded Treewidth Interdiction: Suppose $G$ can be decomposed into graphs of treewidth at most $w$ , by removing some subset $F$ of edges.",
"Find such a subset $F$ ."
],
[
"Our Results.",
"We show the following results for these general interdiction problems.",
"Our main technical result is the following theorem.",
"Theorem 1.3 Given a graph $G$ and an integer $w>0$ , let $F$ be some subset of edges such that removing them reduces the treewidth of $G$ to $w$ .",
"Then there is an algorithm that runs in time $n^{O(1)}$ and finds a subset of edges $F^{\\prime }$ such that $|F^{\\prime }| = O(\\log n \\log \\log n) |F|$ and removing $F^{\\prime }$ from $G$ reduces the treewidth to $O(w \\log w)$ .",
"Note that this gives a bicriteria $(\\log n \\log \\log n, \\log w)$ -approximation to the Bounded Treewidth Interdiction problem.",
"We also remark that the approximation factors and the running time of the above algorithm do not depend on $w$ .",
"Let us now see how Theorem REF can be used to obtain Theorem REF .",
"The basic idea is to apply Theorem REF to the factor graph $H$ of the noisy formula $\\Phi $ to obtain a formula $\\Phi ^{\\prime }$ whose factor graph has a smaller treewidth, and then compute an exact solution for $\\Phi ^{\\prime }$ .",
"Since $k$ is constant, the noisy formula $\\Phi $ has at most $O(m)$ variables and thus $H$ has $\\Theta (m)$ vertices.",
"Thus, as discussed above, we know that there exists a set $F$ of $(k\\delta + O(\\epsilon /(\\log m\\log \\log m))) m$ edges that we can delete from $H$ to reduce its treewidth to $O(\\log m \\log \\log m/\\epsilon )$ .",
"So we can apply Theorem REF to $H$ with $w = O(\\log m \\log \\log m / \\epsilon )$ to find an edge set $F^{\\prime }$ of size at most $O((\\epsilon + k\\delta (\\log m \\log \\log m))m)$ such that the residual factor graph $H - F^{\\prime }$ has treewidth $O(\\log m (\\log \\log m)^2/\\epsilon )$ .",
"Now observe that deleting the clauses incident to $F^{\\prime }$ gives us a formula $\\Phi ^{\\prime }$ whose factor graph is a subgraph of $H-F^{\\prime }$ , and thus has treewidth $O(\\log m (\\log \\log m)^2/\\epsilon )$ .",
"Thereafter, one can use an exact algorithm for MAX-$k$ -SAT [29] that runs in time $2^{O(\\ell )}\\cdot \\operatorname{poly}(m)$ when the factor graph has treewidth at most $\\ell $ .",
"Next, we analyze the quality of this solution.",
"Denote by $\\operatorname{OPT}$ and $\\operatorname{OPT}^{\\prime }$ the maximum number of satisfiable clauses in $\\Phi $ and $\\Phi ^{\\prime }$ , respectively.",
"At least a constant fraction of the clauses in $\\Phi $ are satisfiable, so $\\operatorname{OPT}\\ge \\Omega (m)$ .",
"Since the formula $\\Phi ^{\\prime }$ is obtained by deleting $O(\\epsilon + \\delta (\\log m \\log \\log m))m$ clauses from $\\Phi $ (recall that $k$ is a constant), we have $\\operatorname{OPT}^{\\prime } \\ge \\operatorname{OPT}- O(\\epsilon + \\delta (\\log m \\log \\log m))m$ .",
"Thus, $\\operatorname{OPT}^{\\prime } \\ge (1 - O(\\epsilon + \\delta \\log m \\log \\log m))\\operatorname{OPT}$ and so our solution is a $(1 + O(\\epsilon + \\delta \\log m \\log \\log m))$ -approximation.",
"We remark that Bounded Treewidth Interdiction is a fundamental problem of interest beyond its application to noisy planar graphs.",
"For instance, treewidth interdiction algorithms have found applications in designing PTASes in several settings [22], [15], [23].",
"Next, we consider the Bounded Size Interdiction problem in Section .",
"Theorem 1.4 (Weaker version) For the weaker form of the Bounded Size Interdiction problem stated above, given any $\\beta \\le 1$ , we can find in time $n^{O(1/\\epsilon ^2)}$ a subset of vertices $X^{\\prime }$ with $|X^{\\prime }| \\le O(|X|/\\beta + \\beta |E|)$ such that $G[V\\setminus X^{\\prime }]$ has no component larger than $1/\\epsilon ^2$ .",
"Here, $|E|$ is the number of edges in $G$ .",
"For a noisy planar graph $G$ , there exists $X \\subseteq V$ of size $|X| \\le (\\epsilon + \\delta )n$ (as discussed above) and $|E| \\le (3+\\delta )n = O(1)n$ .",
"The latter follows as a planar graph has at most $3n-6$ edges.",
"So setting $\\beta = (\\epsilon +\\delta )^{1/2}$ in Theorem REF already gives $X^{\\prime }$ of size $O((\\epsilon + \\delta )^{1/2} n)$ in time $n^{O(1/\\epsilon ^2)}$ .",
"This can be used to design a $(1 + O((\\epsilon + \\delta )^{1/2}))$ -approximation algorithm for independent set in noisy planar graphs.",
"To get the better approximation factor of $1 + O(\\epsilon + \\delta )$ of Theorem REF above, we will use a more refined result (Theorem REF ) that decouples the dependence of $|X^{\\prime }|$ on $\\epsilon $ and $\\delta $ .",
"To prove Theorem REF , we write a configuration LP based formulation and round it suitably.",
"(See Section .)",
"The proof of Theorem REF is much more challenging and requires several new ideas, and we give a broad overview of the algorithm and the proof below."
],
[
"Overview of Techniques for Theorem ",
"First, observe that if $G$ has treewidth at most $w$ , then $F=\\emptyset $ , and the algorithm must return $F^{\\prime }=\\emptyset $ .",
"Thus, the problem is at least as hard as determining the treewidth of $G$ .",
"This is well known to be NP-Hard, and in fact unlikely to admit a polynomial time $O(1)$ approximation under reasonable complexity assumptions [40].",
"This implies that the bicriteria guarantee is necessary, and that it is unlikely that the approximation with respect to $w$ can be made $O(1)$ .",
"At a high level, our algorithm will try to construct a good tree decomposition of width $w$ , while removing some problematic edges along the way.",
"To this end, let us first see how the known algorithms for finding tree decompositions work.",
"Treewidth is characterized up to $O(1)$ factor by the well-linkedness property of a graph, which allows one to construct a tree decomposition by computing small balanced vertex separators recursively.",
"Either the algorithm succeeds at each step and eventually finds a tree decomposition, or returns a well-linked set as a certificate that $G$ has large treewidth.",
"Finding balanced vertex separators is hard but one can use LP or SDP (resp.)",
"formulations [7], [2], [20] based on spreading constraints [19], [4], and lose an $O(\\log w)$ or $O(\\sqrt{\\log w})$ (resp.)",
"factor in the quality of the treewidth.",
"In the noisy setting, our algorithmic task can thus be viewed as detecting which edges to remove so that the above recursive procedure works.",
"To do this, we formulate an LP with variables for which edges to remove (let us call these the $x_{uv}$ variables) so that in the residual graph every subset $S$ of vertices has a small fractional balanced vertex separator of size at most $w$ .",
"However, as there are exponentially many such sets $S$ , this gives a huge overall LP with (both) exponentially many variables and constraints, and it is unclear how to solve it.",
"In particular, we have exponentially many different vertex separator LPs coupled together with the common $x_{uv}$ variables.",
"We describe the algorithm in two parts.",
"First, we assume that we are given the $x_{uv}$ values from some feasible optimum LP solution.",
"Using these $x$ -values, for any given set $S$ , we can now formulate a balanced edge-and-vertex separator LP, where the $x$ -values give the fractional amount by which edges are removed, and in addition at most $w$ vertices are removed.",
"Using standard region-growing techniques jointly on these edge and vertex values, we decide which edges to delete (this adds to $F^{\\prime }$ ), and which vertices lie in the separator for $S$ (these enter the bags in the tree decomposition).",
"Doing this directly gives an $O(\\log ^2 n)$ approximation (provided we ensure that the separator tree is balanced and has depth $O(\\log n)$ ), due to the loss of an $O(\\log n)$ factor on each level of recursion.",
"To reduce this to $O(\\log n \\log \\log n)$ , we use “Seymour's trick\" of more careful region-growing [38], together with some additional technical steps needed to make it work together with the tree decomposition procedure.",
"Second, we describe how to “solve\" the LP.",
"Perhaps surprisingly, this turns out to be quite challenging and requires some new ideas, which may be be useful in other contexts.",
"We only sketch these here, and details can be found in Section .",
"First, we bypass the need to completely solve this LP, by using the Round-or-Separate framework (as in [3], [31]).",
"In particular, the algorithm starts with some possibly infeasible solution $x$ , and tries to construct the tree decomposition.",
"If it succeeds, we are done.",
"Otherwise, it gets stuck at finding a small balanced vertex separator for some set $S$ .",
"At that point, we try to add a violated inequality.",
"However, a crucial point is that we need to find a violated inequality only involving the $x$ -variables.",
"So, a key step is to reformulate the LP to only have the $x$ -variables.",
"This crucially uses LP duality and the structure of the LP that the variables for different sets $S$ are only loosely coupled via the $x$ -variables.",
"After this reformulation, it is still unclear how to find a violated inequality due to the exponential size of the LPs involved.",
"We get around this issue by using some further properties of the Ellipsoid Method and the LP duality."
],
[
"Other Related Work.",
"The noise model considered here gives an interesting interpolation between easy and general worst-case instances.",
"This is similar in spirit to approaches such as smoothed analysis [39], planted models and semi-random models [21], [34], although unlike these models our noise model is completely adversarial.",
"For the general version of the bounded size interdiction problem—given a graph $G$ and a size parameter $s$ , find the smallest vertex set $X$ to delete so that $G - X$ has components of size at most $s$ —Lee [30] independently gave a $O(\\log s)$ -approximation algorithm that runs in time $2^{O(s)}\\operatorname{poly}(n)$ .",
"However, using this for maximum independent set on noisy planar graphs only yields a $(1 + O((\\delta +\\epsilon )\\log (1/\\epsilon ))$ -approximation.",
"Fomin et al.",
"[22] have considered the vertex deletion variant of our treewidth interdiction problem.",
"They obtain a constant approximation factor for the problem, however their approximation factor depends at least exponentially on $w$ which makes it inapplicable in our settingWe need a sublinear dependence on $w$ .",
"To see why, consider for example the noisy MIS problem with $G_0$ as a grid.",
"If we wish to reduce the treewidth to $w$ , we would need to remove an $\\Omega (1/w)$ fraction of the vertices, so if the interdiction algorithm is not an $o(w)$ approximation, it might end up deleting all the vertices.. Also their algorithm is polynomial time only for $w=O(1)$ .",
"A related model was considered by [26] in the context of property-testing and sublinear time algorithms in the bounded degree model."
],
[
"Notation and Preliminaries",
"We always use $G_0$ for the underlying graph, and $G$ for the noisy graph.",
"The number of vertices of $G$ is always $n$ .",
"For a subset $S \\subseteq V$ and $F \\subseteq E$ , we use the notation $G[S] - F$ to denote the subgraph induced on the vertices $S$ , excluding the edges in $F$ .",
"We use the notation $E(S)$ to denote the subset of $E$ with both endpoints in $S$ , and given another subset $S^{\\prime } \\subseteq V$ , we use $E(S,S^{\\prime })$ to denote the subset of $E$ with one endpoint in $S$ and another in $S^{\\prime }$ ."
],
[
"Planar and Minor-Free graphs.",
"The classic planar separator theorem [32] states that any planar graph has a $2/3$ -balanced vertex separator of size $O(\\sqrt{n})$ and that it can be found efficiently.",
"Applying this recursively gives the following.",
"Lemma 2.1 For any planar graph $G$ and any $\\alpha >0$ , there is subset of vertices $X \\subset V$ with $|X| =O(\\alpha n)$ , such that every component $C_i$ of $G_0[V-X]$ has at most $1/\\alpha ^2$ vertices.",
"A more generally applicable technique (see e.g.",
"[29], [18], [13]) is Baker's decomposition [5], which states that for any integer $k$ , a planar graph can be decomposed into pieces of treewidth $O(k)$ (specifically, $k$ -outerplanar graphs) by removing $O(1/k)$ fraction of edges or vertices.",
"A minor of $G$ is a graph $G^{\\prime }$ obtained by deleting and contracting edges.",
"A graph $G$ is $H$ -free if $G$ does not contain a subgraph $H$ as a minor.",
"Planar graphs are exactly the graphs excluding $K_{3,3}$ and $K_5$ as minors.",
"In fact, Robertson and Seymour proved that every graph family closed under taking minors is characterized by a set of excluded minors.",
"Both the planar separator theorem and Baker's decomposition approach extend more generally to $H$ -free graphs [1], [16], [15]."
],
[
"Treewidth.",
"We review some relevant definitions related to treewidth.",
"Definition 2.2 ($\\alpha $ -separator of $S$ in $G$ ) Given a graph $G=(V,E)$ and a set $S \\subset V$ , a vertex set $X \\subset V$ is an $\\alpha $ -separator of vertex set $S$ in $G$ if every component $C$ of $G[V - X]$ has $|C \\cap S| \\le \\alpha |S|$ .",
"Definition 2.3 (Well-linked sets) A vertex set $S$ is $w$ -linked in $G$ if it does not have a $\\frac{1}{2}$ -separator $X$ with $|X| < w$ .",
"The linkedness of $G$ is defined to be the maximum integer $w$ such that there exists a $w$ -linked set in $G$ , and is denoted as $\\operatorname{link}(G)$ .",
"Definition 2.4 (Tree decomposition) A tree decomposition of $G$ is a tree $T$ whose nodes $t$ correspond to vertex subsets $V_t$ of $G$ (called bags) that satisfies the following properties: (i) for every edge $(u,v) \\in E$ , there exists a bag $V_t$ containing both $u$ and $v$ ; (ii) for every vertex $v$ , the bags that contain $v$ form a non-empty subtree of $T$ .",
"The width of the decomposition is $\\operatorname{width}(T) = \\max _{s \\in T} |V_s| - 1$ .",
"Definition 2.5 (Treewidth) The treewidth of $G$ is the minimum integer $w$ such that it has a tree decomposition of width $w$ , and is denoted as $\\operatorname{tw}(G)$ .",
"The following well-known (see e.g.",
"[36]) approximate characterization of treewidth in terms of linkedness will be useful for us.",
"Lemma 2.6 ([36]) For any graph $G$ , $\\operatorname{link}(G) < \\operatorname{tw}(G) < 4\\operatorname{link}(G)$ ."
],
[
"Bounded Treewidth Interdiction",
"Recall that in the Bounded Treewidth Interdiction problem, we are given a graph $G = (V,E)$ , a target treewidth $w$ , and we want to find the minimum set $F$ of edges to delete from $G$ such that $\\operatorname{tw}(G-F) < w$ .",
"In this section, we describe the exponential-size LP and sketch the rounding algorithm used to prove Theorem REF .",
"In the following, when $X$ is a vertex set and $F$ is an edge set, we use the shorthand $G-X$ to mean $G[V - X]$ , and $G-X-F$ to mean $G[V - X] - F$ ."
],
[
"An Exponential-Sized LP",
"Lemma REF gives us a convenient characterization of feasible solutions $F$ which we can use to write an LP.",
"In particular, it says that if $\\operatorname{tw}(G-F) < w$ , then every vertex set $S \\subseteq V$ has a $\\frac{1}{2}$ -separator $X^S$ in $G - F$ of size less than $w$ .",
"Consider LP (REF ) in Figure REF .",
"It has a variable $x_{uv}$ indicating if edge $(u,v) \\in E$ belongs to $F$ .",
"For every subset $S\\subseteq V$ and vertex $v \\in V$ , variable $y^S_v$ indicates if $v$ belongs to the minimum-size $\\frac{1}{2}$ -separator $X^S$ of $S$ in $G-F$ .",
"For $u,v \\in V$ , let $\\mathcal {P}(u,v)$ denote the set of paths between $u$ and $v$ .",
"For a path $P$ , define $E(P)$ to be the set of edges in $P$ and $V(P)$ to be the set of vertices on $P$ , including the endpoints.",
"Figure: LP relaxation for the treewidth interdiction problem.We interpret the solution as follows: the LP assigns a length $x_{uv}$ to each edge $(u,v) \\in E$ and a weight $y^S_v$ for each vertex set $S$ and vertex $v$ .",
"Consider a fixed set $S$ .",
"Without loss of generality, we can assume that the variable $d^S_{uv}$ denotes the distance between $u$ and $v$ induced by the edge lengths $x_e$ and vertex weights $y^S_t$ .",
"In particular, if we define the length of a path $P \\in \\mathcal {P}(u,v)$ to be the sum of edge lengths and vertex weights on the path, including the weights on $u$ and $v$ , then $d^S_{uv}$ is the length of the shortest path between $u$ and $v$ .",
"The variables $d^S_{uv}$ and the last set of constraints are often called spreading metrics and spreading constraints, respectively.",
"It is also easy to see that without loss of generality any feasible solution satisfies $d_{uv}^S \\le 1$ .",
"Note that there is a potentially different metric $d^S$ for each set $S$ , and that the LP has exponentially many constraints and exponentially many variables.",
"Lemma 3.1 LP (REF ) is a relaxation of the treewidth interdiction problem.",
"We show that for every edge set $F$ such that $\\operatorname{tw}(G-F) < w$ , there exists a feasible solution $(x,y,d)$ to LP (REF ) with $\\sum _{(u,v) \\in E} x_{uv} \\le |F|$ .",
"Let $x$ be the indicator vector for $F$ .",
"As $\\operatorname{tw}(G-F)< w$ , by Lemma REF , for each vertex set $S$ , there exists a set $X^S$ of at most $w$ vertices such that no component of $G-X^S-F$ contains more than half of $S$ .",
"Define $y^S$ to be the indicator vector for $X^S$ and $d^S_{uv}=1$ if either $u$ or $v$ lies in $X^S$ , or if $u$ and $v$ lie in separate components of $G-X^S-F$ .",
"The solution $(x,y,d)$ has $\\sum _{(u,v) \\in E} x_{uv} = |F|$ and satisfies the first two sets of constraints.",
"It remains to show that it satisfies the spreading constraints.",
"Fix some $S$ and consider $U \\subseteq S$ with $|U| > |S|/2$ .",
"Let $u$ be a vertex of $U$ .",
"There are two cases to consider: (i) If $u \\in X^S$ , we have $d^S_{uv} = 1$ for all $v \\in U \\setminus \\lbrace u\\rbrace $ and so $\\sum _{v \\in U}d^S_{uv} \\ge |U| - 1 \\ge |U| - |S|/2$ ; (ii) otherwise if $u \\notin X^S$ , let $C$ be the component of $G-X^S-F$ that contains $u$ .",
"We have $|C \\cap S| \\le |S|/2$ since $X^S$ is a $\\frac{1}{2}$ -separator of $S$ in $G-F$ .",
"We also have $d^S_{uv} = 1$ for every $v \\in U - (C \\cap S)$ since $v$ is either in a different component of $G-X^S-F$ or in $X^S$ .",
"Thus, $\\sum _{v \\in U} d^S_{uv} \\ge \\sum _{v \\in U - (C \\cap S)} d^S_{uv} \\ge |U| - |C \\cap S| \\ge |U| - |S|/2.$ Therefore, the solution $(x,y,d)$ is a feasible solution with $\\sum _{(u,v) \\in E} x_{uv} = |F|$ .",
"In the rest of this section, we describe our rounding algorithm for LP (REF ) and prove the following lemma.",
"Lemma 3.2 Given oracle access to a feasible solution $(x, y, d)$ of LP (REF ), we can find in time $\\operatorname{poly}(n)$ an edge set $F$ such that $\\operatorname{tw}(G-F) \\le O(w \\log w)$ and $|F| \\le O(\\log n \\log \\log n)\\sum _{(u,v) \\in E}x_{uv}$ ."
],
[
"Sketch of the Rounding Algorithm",
"Let $(x,y,d)$ be a feasible solution to LP (REF ).",
"As mentioned in the Introduction, our rounding algorithm is based on a recursive algorithm for constructing tree decompositions with Seymour's recursive graph decomposition trick layered on top.",
"We now describe at a high level how these ideas are combined together and highlight the key issues that arise."
],
[
"Classic Tree Decomposition.",
"We begin by outlining the relevant parts of the classic tree decomposition algorithm [37] (which we call $\\textsc {Decomposition}$ ).",
"It is a recursive algorithm: Given a subgraph $H$ of $G$ and an integer $w$ , it either constructs a tree decomposition of width $O(w)$ or it finds a $w$ -linked set $S$ which certifiesIt certifies $\\operatorname{tw}(H) \\ge w$ , and the treewidth of a graph is at least the treewidth of any subgraph.",
"that $\\operatorname{tw}(G) \\ge w$ .",
"There are two key steps (illustrated in Figure REF ) that are important to us: Separate: find a minimum-size $\\frac{2}{3}$ -separator $X$ of a vertex set $S$ in $H$ (the particular choice of $S$ depends on previous recursive steps).",
"Recurse: for each component $C_i$ of $H-X$ , recurse on the subgraph $H_i$ of $H$ which consists of the edges of $H$ induced by $C_i$ and those between $C_i$ and $X$ .",
"We say that $\\textsc {Decomposition}$ “succeeds” if it does not encounter a $w$ -linked vertex set $S$ .",
"In particular, when $\\textsc {Decomposition}$ succeeds, the separators found during its execution can be used to construct a tree decomposition of width $O(w)$ ; when it fails, it has found a $w$ -linked set $S$ during its recursion.",
"Figure: Decomposition\\textsc {Decomposition} finds a separator XX, and for each component C i C_i of H-XH-X, it recurses on the subgraph H i H_i consisting of edges of HH with at least one endpoint in C i C_i."
],
[
"Our Algorithm.",
"Our algorithm (which we call $\\textsc {Interdict}$ ) largely follows along the lines of $\\textsc {Decomposition}$ .",
"The main difference is that we also want to delete edges to ensure that size of the separators found in the recursion are small enough so that we succeed in constructing a tree decomposition of width $O(w\\log w)$ .",
"We still make the same choices about which set $S$ to separate and how to recurse; this allows us to reuse the analysis of $\\textsc {Decomposition}$ to prove that we have deleted enough edges to reduce the treewidth down to $O(w \\log w)$ .",
"In particular, instead of the Separate step, we want to perform a “Delete and Separate” step instead.",
"Delete and Separate: delete a subset $D$ of edges and find a vertex set $X$ of size $O(w \\log w)$ such that $X$ is a $\\frac{2}{3}$ -separator of $S$ in $H-D$ .",
"Here is where LP (REF ) is useful.",
"It is similar to the spreading metric relaxation for finding minimum balanced vertex separators, except that it gives edge-and-vertex separators.",
"As mentioned in the Introduction, one can apply standard region growing techniques in an almost black box fashion along with other tricks to obtain a $(\\log ^2 n, \\log w)$ -approximation to treewidth interdiction.",
"Figure: Interdict\\textsc {Interdict} uses the spreading metric given by LP solution (left), partitions the metric into regions with cut vertices shown as hollow vertices and cut edges shown as dashed edges (center), and for each region, recurses on the subgraph contained in it (right).",
"Note that the subgraph includes the cut vertices on the boundary of the region as well.Obtaining a $(\\log n \\log \\log n, \\log w)$ -approximation needs some care.",
"At a high level, we want to apply region growing recursively using Seymour's recursion.",
"The basic idea is to ensure that not only is $|D|$ bounded by the cost of the LP solution projected on $H$ , but it is also bounded in some nice way by the cost of the LP solution on the subgraphs $H_i$ we recurse on.",
"In particular, we want to use region growing to partition the spreading metric $d^S$ into “regions” $B_1, \\ldots , B_k$ (illustrated in Figure REF ) with the following properties: (bounded charge) the set of edges $\\delta _x(B_i)$ (vertices $\\delta _y(B_i)$ resp.)",
"cut by $B_i$ can be charged to the $x_{uv}$ variables ($y^S_v$ variables, resp.)",
"“contained” in the region, (containment) each subgraph $H_i$ we recurse on is contained inside a region.",
"We can then choose $D = \\delta _x(B_1) \\cup \\cdots \\cup \\delta _x(B_k)$ and $X=\\delta _y(B_1) \\cup \\cdots \\cup \\delta _y(B_k)$ .",
"Due to the structure of the subproblems in the Recurse step, ensuring the second property requires some care with how we implement region growing.",
"The problem is that the separators involve both edges and vertices, and they play different roles, i.e.",
"edges are deleted globally, while vertices are deleted locally for each $S$ .",
"Normally, the region growing technique proceeds by finding a region $B_i$ in the current graph that satisfies the bounded charge property, removes $B_i \\cup \\delta _y(B_i)$ —the vertices contained in or cut by $B_i$ from the graph—and repeats on the residual graph.",
"However, this would also remove any edge $(u,v)$ with $u \\in \\delta _y(B_i)$ , even though $v$ is still remaining in the residual graph.",
"In this case, no future region can contain or cut the edge $(u,v)$ .",
"This is a problem if $v$ ends up being in some component $C_i$ later, as we would have a subgraph $H_i$ that is not contained in any region.Thus, we need to somehow preserve edges between $\\delta _y(B_i)$ and the residual vertices.",
"We will do this by introducing copies of these edges that we need to preserve, called “zombie edges\".",
"A detailed description of Delete and Separate step as well as the rounding algorithm appears in the next subsection."
],
[
"The Delete and Separate Step",
"We start by describing the Delete and Separate step sketched out in Section REF .",
"The main ingredient is the region growing technique."
],
[
"Region Growing.",
"We begin with some standard definitions.",
"We define the ball $B(s,r)$ centered at vertex $s$ with radius $r$ as $B(s,r) = \\lbrace v : d^S_{sv} \\le r\\rbrace $ .",
"We say that a vertex $v$ is cut by $B(s,r)$ if $d^S_{sv} - y^S_v < r < d^S_{sv}$ , i.e.",
"$v$ is only “partially inside” the ball, and use $\\delta _y(s,r)$ to denote the set of vertices cut by $B(s,r)$ ; likewise, an edge $(u,v)$ is cut by $B(s,r)$ if $d^S_{su} \\le r < d^S_{su} + x_{uv}$ and we use $\\delta _x(s,r)$ to denote the set of edges cut by $B(s,r)$ .",
"We define two quantities for the $x$ -cost and $y$ -cost.",
"The cost $\\operatorname{LP-cost}(s,r)$ of $B(s,r)$ is defined to be the total cost of the LP solution “contained” in $B(s,r)$ : $\\operatorname{LP-cost}(s,r) = \\sum _{u \\in B(s,r), v \\in B(s,r) \\cup \\delta _y(s,r)} x_{uv}\\\\+\\sum _{(u,v) \\in \\delta _x(s,r) : u \\in B(s,r), v \\notin B(s,r)} r - d^S_{su}$ Similarly, its weight is defined to be the total fractional weight that is “contained” in $B(s,r)$ : $\\operatorname{wt}(s,r) = \\sum _{v \\in B(s,r)} y^S_v + \\sum _{v \\in \\delta _y(s,r)} r - (d^S_{sv} -y^S_v).$ See Figure REF for an illustration.",
"The $x$ -volume and $y$ -volume is then defined to be $\\operatorname{vol}_x(s,r) = \\frac{\\operatorname{LP-cost}(G)}{n^2} + \\operatorname{LP-cost}(s,r)$ and $\\operatorname{vol}_y(s,r) = \\frac{1}{w} + \\operatorname{wt}(s,r)$ .",
"These notions also extend to subgraphs.",
"Given a subgraph $H^{\\prime }$ , we define $\\operatorname{LP-cost}(H^{\\prime }) = \\sum _{(u,v) \\in H^{\\prime }} x_{uv}$ and $\\operatorname{wt}(H^{\\prime }) = \\sum _{v \\in H^{\\prime }} y^S_v$ .",
"Similarly, $\\operatorname{vol}_x(H^{\\prime }) = \\frac{\\operatorname{LP-cost}(G)}{n^2} + \\operatorname{LP-cost}(H^{\\prime })$ and $\\operatorname{vol}_y(H^{\\prime }) = \\frac{1}{w} + \\operatorname{wt}(H^{\\prime })$ .",
"Figure: In this figure, the edge lengths represents the xx variables, and vertices are represented as circles whose diameters correspond to their y S y^S weights.",
"The dashed arc is centered at v 1 v_1 and has radius 0.80.8.",
"We have B(v 1 ,0.8)={v 1 ,v 2 ,v 3 }B(v_1,0.8) = \\lbrace v_1,v_2,v_3\\rbrace .",
"Its boundary edges and vertices are δ x (v 1 ,0.8)={(v 2 ,v 4 ),(v 1 ,v 4 )}\\delta _x(v_1,0.8) = \\lbrace (v_2,v_4),(v_1,v_4)\\rbrace and δ y (v 1 ,0.8)={v 3 }\\delta _y(v_1,0.8) = \\lbrace v_3\\rbrace , respectively.",
"Towards vol x (v 1 ,0.8)\\operatorname{vol}_x(v_1,0.8), edge (v 1 ,v 2 )(v_1,v_2) contributes 0.10.1, (v 1 ,v 3 )(v_1,v_3) contributes 0.30.3, (v 1 ,v 4 )(v_1,v_4) contributes 0.70.7, and (v 2 ,v 4 )(v_2,v_4) contributes 0.10.1.",
"Towards vol y (v 1 ,0.8)\\operatorname{vol}_y(v_1,0.8), v 1 v_1 contributes 0.10.1, v 2 v_2 contributes 0.50.5, and v 3 v_3 contributes 0.40.4.The following lemma shows that there exists a good radius $r \\in [0,1/12]$ such that the number of edges cut by $B(s,r)$ can be charged to $\\operatorname{vol}_x(s,r)$ and simultaneously, the number of vertices cut by $B(s,r)$ can be charged to $\\operatorname{vol}_y(s,r)$ .",
"Lemma 3.3 (Region Growing Lemma) For every vertex $s$ , we can efficiently find a good radius $r \\in [0,1/12]$ such that $|\\delta _x(s,r)| &\\le O(\\log \\log n) \\cdot \\ln \\frac{\\operatorname{vol}_x(s,1/12)}{\\operatorname{vol}_x(s,r)} \\cdot \\operatorname{vol}_x(s,r),\\\\|\\delta _y(s,r)| &\\le \\kappa \\cdot \\log w \\cdot \\operatorname{vol}_y(s,r),$ for some constant $\\kappa $ .",
"Say that $r$ is $x$ -good if it satisfies the first inequality and $y$ -good if it satisfies the second inequality.",
"Set $r(n):=\\ln \\ln [e (n + 1)]$ and define the following sets of radii: $A_x = &\\Bigl \\lbrace r \\in [0,1/12] : \\\\ &\\frac{|\\delta _x(s,r)|}{\\operatorname{vol}_x(s,r)} >48\\cdot \\ln \\frac{e \\cdot \\operatorname{vol}_x(s,1/12)}{\\operatorname{vol}_x(s,r)}\\cdot r(n) \\Bigr \\rbrace ,$ $A_y &= \\left\\lbrace r \\in [0,1/12] : \\frac{|\\delta _y(s,r)|}{\\operatorname{vol}_y(s,r)} > 48\\cdot \\ln (w^2+1)\\right\\rbrace .$ In other words, $A_x$ and $A_y$ are the sets of radii that are $x$ -bad and $y$ -bad, respectively.",
"We claim that the measure of both $A_x$ and $A_y$ are small: $\\mu (A_x), \\mu (A_y) \\le 1/48$ .",
"The claim then implies that there exists $r \\in [0, 1/12]$ that is simultaneously $x$ -good and $y$ -good.",
"Observe that $\\frac{\\partial \\operatorname{vol}_x(s,r)}{\\partial r} = |\\delta _x(s,r)|$ and $\\frac{\\partial \\operatorname{vol}_y(s,r)}{\\partial r} = |\\delta _y(s,r)|$ .",
"Suppose, towards a contradiction, that $\\mu (A_x) > 1/48$ .",
"We have $&\\frac{\\partial }{\\partial r} \\left(-\\ln \\ln \\frac{e \\cdot \\operatorname{vol}_x(s,1/12)}{\\operatorname{vol}_x(s,r)}\\right)\\\\&= \\frac{|\\delta _x(s,r)|}{\\operatorname{vol}_x(s,r)\\cdot \\ln \\frac{e \\cdot \\operatorname{vol}_x(s,1/12)}{\\operatorname{vol}_x(s,r)}}$ So $&\\ln \\ln \\frac{e \\cdot \\operatorname{vol}_x(s,1/12)}{\\operatorname{vol}_x(s,0)}\\\\&= -\\ln \\ln \\frac{e \\cdot \\operatorname{vol}_x(s,1/12)}{\\operatorname{vol}_x(s,r)} \\ \\Bigg |^{1/12}_0 \\\\&= \\int _0^{1/12} \\frac{|\\delta _x(s,r)|}{\\operatorname{vol}_x(s,r)\\cdot \\ln \\frac{e \\cdot \\operatorname{vol}_x(s,1/12)}{\\operatorname{vol}_x(s,r)}}\\, \\partial r \\\\&\\ge \\mu (A_x) \\cdot 48\\ln \\ln [e (n + 1)]> r(n).$ But $\\ln \\ln \\frac{e \\cdot \\operatorname{vol}_x(s,1/12)}{\\operatorname{vol}_x(s,0)}\\le \\ln \\ln \\frac{e (\\operatorname{vol}_x+ \\operatorname{vol}_x/n)}{\\operatorname{vol}_x/n}= r(n).$ Thus, we have a contradiction and so $\\mu (A_x) \\le 1/48$ .",
"We now turn to bounding $\\mu (A_y)$ .",
"Suppose, towards a contradiction, that $\\mu (A_y) > 1/48$ .",
"$\\ln \\frac{\\operatorname{vol}_y(s,1/48)}{\\operatorname{vol}_y(s,0)}&= \\int _0^{1/12} \\frac{1}{\\operatorname{vol}_y(s,r)} \\frac{\\partial \\operatorname{vol}_y(s,r)}{\\partial r}\\, \\partial r\\\\&= \\int _0^{1/12} \\frac{|\\delta _y(s,r)|}{\\operatorname{vol}_y(s,r)}\\, \\partial r \\\\&\\ge \\mu (A_y) \\cdot 48\\ln (w^2+1)> \\ln (w^2+1).$ But $\\ln \\frac{\\operatorname{vol}_y(s,1/48)}{\\operatorname{vol}_y(s,0)}\\le \\ln \\frac{w + 1/w}{1/w}= \\ln (w^2 + 1).$ Therefore, both $\\mu (A_x), \\mu (A_y) \\le 1/48$ and so there exists $r \\in [0, 1/12]$ that is simultaneously $x$ -good and $y$ -good.",
"To find such an $r$ efficiently, note that as $r$ grows, $\\operatorname{vol}_x(s,r)$ and $\\operatorname{vol}_y(s,r)$ are non-decreasing.",
"Thus, we only need to check the condition at points $r$ when either $\\delta _x(s,r)$ and $\\delta _y(s,r)$ changes, which happens at most $2|V^{\\prime }|$ and $2|E^{\\prime }|$ times, respectively.",
"This is because as $r$ grows, once a vertex leaves $\\delta _y(s,r)$ , it is inside $B(s,r)$ and will never reappear in $\\delta _y(s,r)$ ; and once an edge leaves $\\delta _x(s,r)$ , both of its endpoints are inside $B(s,r)$ and the edge will never reappear in $\\delta _y(s,r)$ .",
"The next lemma shows that as we grow a ball $B(s,r)$ around vertex $s$ , as long as $r \\le 1/12$ , it cannot contain more than two-thirds of $S$ .",
"Lemma 3.4 Let $U$ be a set of vertices.",
"If $\\max _{u,v \\in U}d^S_{uv} \\le 1/6$ , then $|U \\cap S| \\le 2|S|/3$ .",
"We prove the contrapositive.",
"Suppose, that $|U \\cap S| > 2|S|/3$ .",
"Let $u$ be a vertex in $U \\cap S$ .",
"The spreading constraints imply that $\\sum _{v \\in U \\cap S} d^S_{uv} \\ge |U \\cap S| - \\frac{|S|}{2} > |U \\cap S| - \\frac{3|U \\cap S|}{4} = \\frac{|U \\cap S|}{4}.$ Thus, by averaging, there exists $v \\in U \\cap S$ such that $d^S_{uv} > 1/4 > 1/6$ .",
"We are now ready to describe the $\\textsc {Partition}$ algorithm, which executes the Delete and Separate component of our algorithm.",
"Recall that in the Delete and Separate step, the input is a subgraph $H = (V^{\\prime },E^{\\prime })$ and a vertex set $S \\subseteq V^{\\prime }$ , and our goal is find a set $D$ of edges to delete and a vertex set $X$ of size $O(w \\log w)$ such that $X$ is a $\\frac{2}{3}$ -separator of $S$ in $H-D$ ."
],
[
"The $\\textsc {Partition}$ algorithm.",
" Initialization.",
"Let $\\hat{H}= H$ Region growing.",
"While $\\hat{H}$ contains more than two-thirds of $S$ , Find region.",
"Choose an arbitrary vertex $v \\in S$ that is contained in $\\hat{H}$ and find a good radius $r$ such that $B(v,r)$ satisfies the conditions of Lemma REF .",
"Note that distances, balls and boundaries are defined with respect to $\\hat{H}$ .",
"Removal.",
"Remove all vertices in $B(v,r)$ and their incident edges from $\\hat{H}$ .",
"Add zombies.",
"For each edge $(s,u)$ that was removed, if $s \\in \\delta _y(v,r)$ and $u$ is still in $\\hat{H}$ , add zombie vertex $z_u(s)$ with weight $y^S_{z_u(s)} = 0$ to $\\hat{H}$ and zombie edge $(z_u(s), u)$ with length $x_{z_u(s),u} = x_{su}$ .",
"Let $B(v_i,r_i)$ be the ball found in the $i$ -th iteration.",
"Return the vertex set $X = \\bigcup _i \\delta _y(v_i,r_i)$ and the edge set $D = \\bigcup _i \\delta _x(v_i,r_i)$ , replacing each zombie vertex $z_u(s)$ in $X$ with its original vertex $s$ and each zombie edge $(z_u(s), u)$ in $D$ with its original edge $(s,u)$ .",
"Lemma 3.5 Suppose $\\textsc {Partition}$ took $\\ell $ iterations and let $B(v_1,r_1), \\ldots , B(v_\\ell ,r_\\ell )$ be the regions it found.",
"Let $C_1, \\ldots , C_k$ be the components of $H-X-D$ .",
"For every $j \\in [k]$ , define the subgraph $H_j = (V_j,E_j)$ where $E_j$ is the subset of $E^{\\prime }-D$ with at least one endpoint in $C_j$ , and $V_j$ is the set of endpoints of $E_j$ .",
"We have the following: $|D| \\le O(\\log \\log n) \\cdot \\sum _{i=1}^\\ell \\ln \\left(\\frac{\\operatorname{vol}_x(H)}{\\operatorname{vol}_x(v_i,r_i)}\\right) \\operatorname{vol}_x(v_i,r_i)$ , $|X| \\le \\gamma \\cdot (w +|S|/w) \\log w$ , for some constant $\\gamma $ , $|C_i \\cap S| \\le 2|S|/3$ , and The edge set $E_j$ of each subgraph $H_j$ is either contained in a region $B(v_i,r_i)$ (and so $\\operatorname{vol}_x(H_j) \\le \\operatorname{vol}_x(v_i,r_i)$ ) or it is contained in the residual graph $\\hat{H}$ at the end of the execution of $\\textsc {Partition}$ .",
"The first statement follows from the fact that we chose a good radius $r_i$ for each region $B(v_i,r_i)$ and that the $x$ -volume of any region can be at most $\\operatorname{vol}_x(H)$ .",
"Let us now consider the second statement.",
"The fact that we chose a good radius for each region implies that $|X| = \\sum _{i=1}^\\ell |\\delta _y(v_i,r_i)| \\le \\kappa \\cdot \\log w \\sum _{i=1}^\\ell \\operatorname{vol}_y(v_i,r_i)$ for some constant $\\kappa $ .",
"Recall that $\\operatorname{vol}_y(v_i,r_i) = \\operatorname{wt}(v_i,r_i) + 1/w$ .",
"Since each vertex can only contribute towards the weight of at most one region and zombie vertices have zero weight, the total weight of any region is at most $\\sum _{v \\in V^{\\prime }}y^S_v \\le w$ .",
"So, $\\sum _{i=1}^\\ell \\operatorname{vol}_y(v_i,r_i) \\le (w + \\ell /w)$ .",
"Moreover, since the center of each region is a vertex $v_i$ of $S$ , we remove at least one vertex of $S$ in each iteration.",
"Thus, the number of iterations $\\ell $ can be at most $|S|$ .",
"So, $|X| \\le \\kappa \\cdot \\log w \\cdot (w + \\ell /w) \\le \\kappa \\cdot (w + |S|/w) \\log w$ and this proves the second statement of the lemma.",
"Next, we argue that $|C_i \\cap S| \\le 2|S|/3$ .",
"If $C_i$ was a component that remained at the end of the execution of $\\textsc {Partition}$ , then by definition, $|C_i \\cap S| \\le 2|S|/3$ .",
"Suppose $C_i$ was a component that is contained in $B(v_i,r_i)$ .",
"By Lemma REF , it suffices to check that the distance between any two remaining vertices $u$ and $v$ at the start of iteration $i$ is at least $d^S_{uv}$ .",
"Over the previous iterations, $\\textsc {Partition}$ modifies the graph in two ways: by removing edges and vertices, and by introducing zombie edges.",
"Removing edges and vertices clearly cannot decrease the distance between $u$ and $v$ .",
"Zombie edges do not create a shortcut between $u$ and $v$ either.",
"Thus, the distance between $u$ and $v$ in iteration $i$ is at least $d^S_{uv}$ .",
"Finally, each subgraph $H_j$ is connected (since they are the set of edges with one endpoint in a connected component $C_j$ ), thus for each region $B(v_i,r_i)$ , either all the vertices of $H_j$ are in $B(v_i,r_i)$ or all of them are not in $B(v_i,r_i)$ ."
],
[
"Putting it together",
"We are now ready to describe the $\\textsc {Interdict}$ algorithm, which recursively rounds a feasible solution to LP (REF ).",
"It takes as input a subgraph $H = (V^{\\prime },E^{\\prime })$ and a vertex set $S$ , and deletes a set of edges $F$ such that $\\operatorname{tw}(H-F) \\le O(w \\log w)$ .",
"Let $\\gamma $ be the constant in Lemma REF ."
],
[
"The $\\textsc {Interdict}$ Algorithm.",
" Delete and Separate.",
"Use algorithm $\\textsc {Partition}$ to find a set $D$ of edges to delete and a $\\frac{2}{3}$ -separator $X$ of $S$ in $H-D$ .",
"Let $C_1, \\ldots , C_k$ be the components of $H-D-X$ .",
"Delete $D$ .",
"Define subproblems.",
"For $i \\in [k]$ , define the subgraph $H_i = (V_i,E_i)$ where $E_i$ is the subset of $E^{\\prime }-D$ with an endpoint in $C_i$ , and $V_i$ is the set of endpoints of $E_i$ .",
"Recurse.",
"For $i \\in [k]$ , call $\\textsc {Interdict}(H_i, V_i \\cap (X \\cup S))$ .",
"To round a feasible solution of LP (REF ), we call $\\textsc {Interdict}(G,S_0)$ where $S_0$ is an arbitrary set of at most $O(w \\log w)$ vertices.",
"Let $F$ be the set of edges deleted by $\\textsc {Interdict}(G,S_0)$ .",
"Lemma 3.6 $\\operatorname{tw}(G - F) \\le O(w \\log w)$ .",
"We can apply exactly the same analysis of the classic tree decomposition algorithm [37] to argue that the $O(w \\log w)$ -size separators found in the recursion can be used to construct a tree decomposition $T$ of $G-F$ such that the width of $T$ is at most $O(w \\log w)$ .",
"Recall that $\\operatorname{LP-cost}(G) = \\sum _{(u,v) \\in E} x_{uv}$ , the cost of the LP solution $(x, y, d)$ .",
"Lemma 3.7 $|F| \\le O(\\log n\\log \\log n)\\operatorname{LP-cost}(G)$ .",
"Let $k$ be the recursion depth of $\\textsc {Interdict}(G,S_0)$ .",
"For each depth $j$ , let $\\mathcal {H}_j$ be the collection of subgraphs that $\\textsc {Interdict}$ recursed on at depth $j$ , and for each $H \\in \\mathcal {H}_j$ , let $R(H)$ be the set of regions found by $\\textsc {Partition}$ when $\\textsc {Interdict}$ recursed on $H$ .",
"(Note that at depth 0, we have $\\mathcal {H}_0 = \\lbrace G\\rbrace $ .)",
"By Lemma REF , we have the following bound on $|F|$ : $|F| \\le O(\\log \\log n) \\cdot \\sum _j \\sum _{H \\in \\mathcal {H}_j} \\sum _{B(s,r)\\in R(H)}\\ln \\left(\\frac{\\operatorname{vol}_x(H)}{\\operatorname{vol}_x(s,r)}\\right)\\operatorname{vol}_x(s,r).$ For each edge $(u,v) \\in E$ , define $g(u,v) = x_{uv} + \\frac{\\operatorname{LP-cost}(G)}{n^2}$ .",
"We have that $\\sum _{(u,v) \\in E}g(u,v) \\le 2\\operatorname{LP-cost}(G).$ We now show that $\\sum _j \\sum _{H \\in \\mathcal {H}_j} \\sum _{B(s,r) \\in R(H)}\\ln \\left(\\frac{\\operatorname{vol}_x(H)}{\\operatorname{vol}_x(s,r)}\\right)\\operatorname{vol}_x(s,r) \\le O(\\log n) \\sum _{(u,v) \\in E} g(u,v)$ .",
"We will do this by charging $\\sum _{H \\in \\mathcal {H}_j} \\sum _{B(s,r) \\in R(H)}\\ln \\left(\\frac{\\operatorname{vol}_x(H)}{\\operatorname{vol}_x(s,r)}\\right)\\operatorname{vol}_x(s,r)$ to edges of the subgraphs in $\\mathcal {H}_j$ and proving that every edge $(u,v)$ receives a total charge, across all depths, of at most $O(\\log n) g(u,v)$ .",
"Our charging scheme is as follows: for each subgraph $H \\in \\mathcal {H}_j$ and each region $B(s,r) \\in R(H)$ , charge $\\ln \\left(\\frac{\\operatorname{vol}_x(H)}{\\operatorname{vol}_x(s,r)}\\right) g(u,v)$ to each edge $(u,v)$ with at least one endpoint in $B(s,r)$ .",
"Let us first see why the total amount charged per region $B(s,r) \\in R(H)$ is at least $\\ln \\left(\\frac{\\operatorname{vol}_x(H)}{\\operatorname{vol}_x(s,r)}\\right)\\operatorname{vol}_x(s,r)$ .",
"Let $E_H(s,r)$ be the edges with at least one endpoint in $B(s,r)$ .",
"This is exactly the set of edges that contribute to the region's $x$ -volume $\\operatorname{vol}_x(s,r)$ .",
"Therefore, $\\sum _{(u,v) \\in E_H(s,r)} g(u,v)&\\ge \\sum _{(u,v) \\in E_H(s,r)} x_{uv} + \\frac{\\operatorname{LP-cost}(G)}{n^2}\\\\&\\ge \\operatorname{vol}_x(s,r)$ and so the total amount charged is at least $\\ln \\left(\\frac{\\operatorname{vol}_x(H)}{\\operatorname{vol}_x(s,r)}\\right)\\operatorname{vol}_x(s,r)$ .",
"Overall, we have $\\sum _j \\sum _{H \\in \\mathcal {H}_j} \\sum _{B(s,r) \\in R(H)}\\ln \\left(\\frac{\\operatorname{vol}_x(H)}{\\operatorname{vol}_x(s,r)}\\right)\\operatorname{vol}_x(s,r)\\le \\sum _{(u,v) \\in E} \\phi (u,v),$ where $\\phi (u,v)$ is the total charge received by $(u,v)$ .",
"Fix an edge $(u,v) \\in E$ .",
"Let us now upper bound $\\phi (u,v)$ .",
"Since $\\textsc {Interdict}$ recurses on edge-disjoint subgraphs, $(u,v)$ is contained in at most one subgraph of $\\mathcal {H}_j$ for each depth $j$ , and if it does not belong to a subgraph of $\\mathcal {H}_j$ , then it does not belong to any subgraph of $\\mathcal {H}_{j^{\\prime }}$ at lower depths $j^{\\prime } > j$ .",
"In the worst case, $(u,v)$ is contained in some subgraph $H_j \\in \\mathcal {H}_j$ for every depth $j$ .",
"Let us assume this is so and define $B(s_j,r_j) \\in R(H_j)$ to be the region containing $(u,v)$ .",
"Lemma REF tells us that all of $H_j$ is contained in the region $B(s_{j-1},r_{j-1})$ for each $j$ and so $\\operatorname{vol}_x(s_j, r_j) \\ge \\operatorname{vol}_x(H_{j+1})$ .",
"Thus, $\\phi (u,v)&= \\sum _{j=0}^k \\ln \\left(\\frac{\\operatorname{vol}_x(H_j)}{\\operatorname{vol}_x(s_j,r_j)}\\right) g(u,v)\\\\&\\le \\sum _{j=0}^k \\ln \\left(\\frac{\\operatorname{vol}_x(H_j)}{\\operatorname{vol}_x(H_{j+1})}\\right) g(u,v)\\\\&\\le \\ln \\left(\\frac{\\operatorname{vol}_x(G)}{\\operatorname{vol}_x(H_{k+1})}\\right) g(u,v) \\\\&\\le \\ln (n^2+1) g(u,v),$ where the last inequality follows from the fact that $x$ -volume is always at least $\\frac{\\operatorname{LP-cost}(G)}{n^2}$ .",
"Combining this with Inequality (REF ), we get $&\\sum _j \\sum _{H \\in \\mathcal {H}_j} \\sum _{B(s,r) \\in R(H)}\\ln \\left(\\frac{\\operatorname{vol}_x(H)}{\\operatorname{vol}_x(s,r)}\\right)\\operatorname{vol}_x(s,r)\\\\&\\le \\sum _{(u,v) \\in E} \\ln (n^2+1) g(u,v)\\\\&\\le 2\\ln (n^2+1) \\operatorname{LP-cost}(G),$ where the last inequality follows from Inequality (REF ).",
"Finally, plugging this into the right hand side of (REF ) gives us $|F| \\le O(\\log n \\log \\log n)\\operatorname{LP-cost}(G)$ , as desired.",
"Lemmas REF and REF imply Lemma REF ."
],
[
"Using the LP",
"We now come to the problem of how to handle the LP (REF ) in $n^{O(1)}$ time.",
"As discussed in the Introduction, we bypass the need to completely solve this LP using the Round-or-Separate framework.",
"A crucial ingredient here is that if the rounding step gets stuck (is unable to find a small separator), we need to find a violated inequality for the $x$ -variables, and not just for the LP (REF ).",
"In Section REF we show how to reformulate LP (REF ) to only have the $x$ variables.",
"In this reformulation, the coefficients of the inequalities come from feasible points in a polytope with exponentially many variables, so Section REF deals with how to fix such a violated inequality."
],
[
"Reformulating the LP",
"We first reformulate LP (REF )—the original LP with variables $(x,y,d)$ —to obtain another LP with only $x$ -variables.",
"Call a vector $x$ feasible if there exist vectors $y$ and $d$ such that $(x, y, d)$ is a feasible solution to LP (REF ).",
"Define $\\mathcal {F}$ to be the set of all feasible vectors $x$ , and observe that $\\mathcal {F}$ is simply the feasible region of LP (REF ) with the $y$ and $d$ variables projected out.",
"Next, we show how to describe $\\mathcal {F}$ using linear inequalities.",
"For every vertex set $S$ , we define an LP parameterized by $x$ (see Figure REF ).",
"We call this $\\operatorname{sep-LP}(x, S)$ .",
"Figure: sep-LP(x,S)\\operatorname{sep-LP}(x,S)We emphasize that in this LP, only $y$ and $d$ are variables.",
"Recall that this LP is related to the problem of finding a small balanced vertex separator of $S$ in $G$ , provided the edges are removed fractionally to extent $x_e$ .",
"Definition 4.1 Let $\\lambda (x,S)$ be the value of the optimum solution to the $\\operatorname{sep-LP}(x, S)$ .",
"We say that $x$ is $S$ -feasible if $\\lambda (x, S) \\le w$ , and denote by $\\mathcal {F}(S)$ the set of $S$ -feasible vectors.",
"Lemma 4.2 $\\mathcal {F}= \\bigcap _{S \\subseteq V} \\mathcal {F}(S)$ .",
"Suppose $x\\in \\mathcal {F}$ .",
"Then, for every vertex set $S$ , the vectors $y^S$ and $d^S$ are a feasible solution to $\\operatorname{sep-LP}(x,S)$ and $\\sum _v y^S \\le w$ , so $\\lambda (x,S) \\le w$ .",
"On the other hand, suppose $\\lambda (x, S) \\le w$ for all vertex sets $S$ .",
"Define $y$ and $d$ such that for each $S \\in \\mathcal {S}$ , $(y^S, d^S)$ is the optimal solution to $\\operatorname{sep-LP}(x,S)$ .",
"As $\\lambda (x, S) \\le w$ for all $S \\in \\mathcal {S}$ , we have that $(x, y, d)$ is a feasible solution to LP (REF ).",
"Thus, to describe $\\mathcal {F}$ using linear inequalities, it suffices to describe $\\mathcal {F}(S)$ using linear inequalities.",
"By linear programming duality, we have that $\\lambda (x,S) \\le w$ if and only if for every feasible solution to the dual of $\\operatorname{sep-LP}(x,S)$ has objective value at most $w$ .",
"So we consider the dual LP (REF ) (Figure REF ), and denote it by $\\operatorname{flow-LP}(x,S)$ .",
"(We call this $\\operatorname{flow-LP}$ as it is dual to a “cut-type” LP.)",
"Lemma 4.3 The dual to $\\operatorname{sep-LP}(x,S)$ is given by LP (REF ) below.",
"Let us introduce the variables $f^{uv}_P$ for the first set of constraints in LP (REF ), and the variables $ g_{T,v}$ for the second set of constraints.",
"Again, we emphasize that only $f$ and $g$ are variables here (and that $x$ is not a variable).",
"Let us check that the dual is exactly LP (REF ).",
"Rewrite the first set of primal constraints as $\\sum _{t \\in V(P)} y_t - d_{uv} \\ge -\\sum _{e \\in E(P)} x_e \\quad \\forall u,v \\in V, P \\in \\mathcal {P}(u,v)$ In the objective, the coefficient of $f^{uv}_P$ is $-\\sum _{e \\in E(P)} x_e$ and the coefficient of $g_{T,v}$ is $|T| - |S|/2$ .",
"The dual constraints correspond to primal variables $d_{uv}$ and $y_t$ .",
"In the primal, the variable $d_{uv}$ has a coefficient of $-1$ for the constraints corresponding to the dual variables $f^{uv}_P$ for $P \\in \\mathcal {P}(u,v)$ , and a coefficient of 1 in the constraints corresponding to the dual variables $g_{T,t}$ for $t \\in \\lbrace u,v\\rbrace $ and $T \\ni t$ .",
"Similarly, the variable $y_t$ has a coefficient of 1 for the constraints corresponding to the dual variables $f^{uv}_P$ for $u,v \\in V$ and $P \\in \\mathcal {P}(u,v)$ such that $t \\in V(P)$ .",
"So, LP (REF ) is dual to LP (REF ).",
"Figure: flow-LP(x,S)\\operatorname{flow-LP}(x,S)."
],
[
"Separation oracle for $\\mathcal {F}(S)$ .",
"Given a solution $(f, g)$ to $\\operatorname{flow-LP}(x,S)$ , we denote its objective value as $\\operatorname{flow-LP}(x,S,f,g)$ .",
"LP duality implies the following lemma.",
"Lemma 4.4 $x\\in \\mathcal {F}(S)$ if and only if $\\operatorname{flow-LP}(x,S,f,g) \\le w$ for all feasible dual solutions $(f, g)$ .",
"Let us see what we have achieved so far.",
"The expression $\\operatorname{flow-LP}(x,S,f,g) \\le w$ is a linear constraint on $x$ , whose coefficients $(f, g)$ are feasible points in the polytope given by (REF ).",
"Now we crucially note that the coefficients of the constraints on $(f, g)$ depend only on the topology of the graph $G$ and not on $x$ .",
"Definition 4.5 We say that $(f, g)$ is $S$ -valid if it satisfies the constraints of $\\operatorname{flow-LP}(x,S)$ .",
"Thus, Lemma REF gives a description of $\\mathcal {F}(S)$ in terms of linear inequalities.",
"Together with Lemma REF , we get a description of $\\mathcal {F}$ in terms of linear inequalities and so we can reformulate LP (REF ) as follows.",
"$\\boxed{\\begin{aligned}\\mbox{min} \\quad & \\sum _{(u,v) \\in E} x_{uv}\\\\\\mbox{s.t.",
"}\\quad & \\operatorname{flow-LP}(x,S,f,g)\\le w \\quad \\forall S \\subseteq V, (f, g) \\mbox{ $S$-valid}\\end{aligned}}$"
],
[
"Round or Separate",
"We will work with the reformulated LP (REF ).",
"One immediate obstacle is that it is unclear how to get an efficient separation oracle for $\\mathcal {F}$ .",
"In fact, even for a fixed vertex set $S$ , it is not immediately clear how to find a violated inequality for $\\mathcal {F}(S)$ .",
"The problem is that for any constraint, the coefficients of the $x$ -variables are based on $f,g$ , which need to satisfy LP (REF ), and it is not clear how to generate them; e.g.",
"there are exponentially many $f, g$ variables in $\\operatorname{flow-LP}(x,S)$ .",
"To get over these problems, we do the following.",
"We apply the Round-or-Separate framework with the $\\textsc {Interdict}$ algorithm in Section REF .",
"Roughly speaking, we start with some candidate solution $x$ (possibly infeasible), and try to construct a tree decomposition of width $O(w \\log w)$ using $\\textsc {Interdict}$ .",
"If we succeed, we are done.",
"Otherwise, $\\textsc {Interdict}$ could not find a small balanced separator for some vertex set $S$ , and this can only happen if $x\\notin \\mathcal {F}(S)$ .",
"Later, in Lemma REF , we give an efficient procedure that given $x$ and $S$ , determines whether $x\\in \\mathcal {F}(S)$ and if so, outputs a solution $(y^S, d^S)$ that is feasible to $\\operatorname{sep-LP}(x,S)$ and satisfies $\\sum _v y^S_v \\le w$ ; otherwise, it outputs a separating hyperplane, i.e.",
"an $S$ -valid $(f,g)$ such that $\\operatorname{flow-LP}(x,S,f,g) > w$ .",
"In the first case, $\\textsc {Interdict}$ can use $(y^S, d^S)$ to find a $O(w \\log w)$ -size separator of $S$ and make progress.",
"In the second case, we can add the separating hyperplane to find a new candidate $x$ and repeat the whole tree decomposition procedure.",
"By the standard Ellipsoid method-based, separation-versus-optimization framework, the number of such iterations is polynomially bounded.",
"It remains to show how to efficiently separate $\\mathcal {F}(S)$ for any given subset $S$ .",
"We will do this indirectly by using the Ellipsoid Method and applying duality to $\\operatorname{sep-LP}(x, S)$ .",
"Lemma 4.6 There exists a polynomial-time algorithm that given $x$ and $S$ , determines whether $x\\in \\mathcal {F}(S)$ , and if so, outputs a solution $(y^*, d^*)$ that is feasible to $\\operatorname{sep-LP}(x,S)$ and satisfies $\\sum _v y^*_v \\le w$ ; otherwise, it finds a violated inequality for LP (REF ).",
"We apply the Ellipsoid method to find an optimal solution $(y^*,d^*)$ to $\\operatorname{sep-LP}(x, S)$ .",
"We can do this efficiently by using a separation oracle that uses Dijkstra's algorithm to determine the distances $d$ given $x$ and $y$ .",
"If $\\sum _v y^*_v \\le w$ , then $x\\in \\mathcal {F}(S)$ by definition.",
"On the other hand, if $\\sum _v y^*_v > w +\\epsilon $ (where $\\epsilon $ can be made exponentially small), then we can find a violated inequality for (REF ) as follows.",
"As the Ellipsoid method added only polynomially many constraints we can re-solve this LP only on these added constraints and assume that $y^*$ is a solution to this smaller LP on only polynomially many constraints.",
"By complementary slackness, there exists an optimal dual solution $(f^*, g^*)$ where the only non-zero dual variables are those that correspond to these polynomially many primal constraints.",
"Thus, it suffices to solve $\\operatorname{flow-LP}(x,S)$ restricted to these dual variables which makes $\\operatorname{flow-LP}(x,S)$ polynomial in size.",
"The objective function in $\\operatorname{flow-LP}$ then gives the violated inequality for LP (REF )."
],
[
"Bounded Size Interdiction and MIS in Noisy Planar Graphs",
"We consider the Bounded Size Interdiction problem and show how it implies Theorem REF .",
"Consider the noisy graph $G$ obtained by adding $\\delta n$ edges to some planar $G_0$ .",
"Let us view $G$ as obtained by superimposing the noisy edges on the recursive decomposition of $G_0$ given by Lemma REF .",
"This directly implies the following (noisy) decomposition for $G$ .",
"Lemma 5.1 (Noisy Decomposition) Given a $\\delta $ -noisy planar graph $G$ , for any $\\alpha >0$ , there exists a partition $X,C_1,\\ldots ,C_k$ of $V$ with (i) $|X| \\le c\\alpha n$ for some universal constant $c$ , (ii) $|C_i| \\le 1/\\alpha ^2$ for all $i\\in [k]$ , and (iii) at most $\\delta n$ edges whose endpoints lie in distinct (two different) $C_i$ s. Of course as we do not know $G_0$ , it is not clear how to find such a decomposition.",
"Theorem REF shows that this can be done approximately.",
"Theorem 5.2 (Noisy Bounded Size Interdiction) Let $G = (V,E)$ be any graph that has a vertex partition $X,C_1,\\ldots ,C_k$ with $|C_i| \\le s$ for each $i \\in [k]$ , and let $b$ be the total number of edges whose endpoints lie in distinct $C_i$ s. Then for every $\\beta \\le 1$ , we can find in time $n^{O(s)}$ a vertex partition $X^{\\prime },C^{\\prime }_1,\\ldots ,C^{\\prime }_{k^{\\prime }}$ such that (i) $|X^{\\prime }| = O(|X|/\\beta )$ , (ii) $|C^{\\prime }_i| \\le s$ for each $i\\in [k^{\\prime }]$ , and (iii) at most $O(b + \\beta |E|)$ edges whose endpoints lie in distinct $C_i$ s. This implies the following proper decomposition (where the $C^{\\prime }_i$ are components of $G[V-X]$ ).",
"Corollary 5.3 There is a subset $X^{\\prime } \\subset V$ of size $O(|X|/\\beta + b + \\beta |E|)$ such that the components in $G[V-X^{\\prime }]$ have size at most $s$ .",
"For each edge $(u,v)$ with $u \\in C^{\\prime }_i$ and $v \\in C^{\\prime }_j$ for $i\\ne j\\in [k^{\\prime }]$ , put an endpoint (say $u$ ) in $X^{\\prime }$ and remove $u$ from $C^{\\prime }_i$ .",
"As a result, $X^{\\prime }$ has size $ O(|X|/\\beta + b + \\beta |E|)$ and there are no edges between different $C^{\\prime }_i$ and $C^{\\prime }_j$ .",
"Before we prove Theorem REF , let us first see how it implies Theorem REF .",
"Proof of Theorem REF .",
"Given an $\\epsilon > 0$ , set $\\gamma = O(\\epsilon ^2)$ .",
"Applying Lemma REF with $\\alpha =\\gamma $ , $G$ has a noisy decomposition with $|X| = O(\\gamma n)$ and pieces $C_i$ of size $s=1/\\gamma ^2$ and at most $b=\\delta n$ edges between these $C_i$ 's.",
"Moreover $G$ has at most $(3 + \\delta )n \\le 4n$ edges.",
"Applying Corollary REF with $\\beta = \\gamma ^{1/2}$ , gives an $X^{\\prime }$ of size $O(|X|/\\gamma ^{1/2} + \\delta n +\\gamma ^{1/2} n) = O( (\\epsilon + \\delta )n)$ , such that the components of $G[V-X^{\\prime }]$ have at most $1/\\gamma ^2 = 1/\\epsilon ^4$ vertices.",
"By exhaustive search, find an MIS in each $C_i$ separately and return their union $I$ .",
"Clearly, $I$ is a valid independent set.",
"By Theorem REF , the algorithm has overall running time $n^{O(1/\\gamma ^2)} = n^{O(1/\\epsilon ^4)}$ .",
"As $G_0$ is planar and hence 4-colorable, and $G\\setminus G_0$ has at most $\\delta n$ noisy edges, $\\alpha (G) \\ge \\alpha (G_0) - \\delta n \\ge n/4 - \\delta n.$ Moreover, as $|I| \\ge \\alpha (G) - |X^{\\prime }|$ , this gives $|I| \\ge (1- O(\\epsilon +\\delta )) \\alpha (G)$ .",
"$\\Box $ Remark.",
"Theorem REF extends directly to the minor-free case using the separator theorem for minor-free graphs [1], and the fact that these graphs have bounded average degree and thus $\\alpha (G) = \\Omega (n)$ ."
],
[
"LP formulation.",
"Given $0 < \\beta \\le 1$ and $G$ as input, we first write an integer program to find $X$ and the $C_i$ s. Let $a=|X|$ (we can assume that $a$ is known as the algorithm can try every value).",
"For each vertex $v$ , the variable $y_v$ indicates if $v \\in X$ .",
"For each subset $S\\subset V$ with $|S|\\le s$ , the variable $z_S$ indicates if $S$ is one of the pieces $C_i$ .",
"Let $\\mathcal {S}$ be the collection of all such subsets $S$ .",
"For each $(u,v)\\in E$ , the variable $x_{uv}$ indicates whether the edge $(u,v)$ is such that $u \\in C_i$ and $v \\in C_j$ for some $i \\ne j$ .",
"Consider the integer program IP (REF ) in Figure REF .",
"Figure: IP for the Bounded Size Interdiction Problem.This is easily seen to be a valid formulation for the problem.",
"The first set of constraints ensure that $X$ contains at most $a$ vertices.",
"The second set of constraints ensure that each vertex lies in either $X$ or some $C_i$ .",
"The third set of constraints are more involved and force $x_{uv}$ to be 1 if some edge has endpoints in distinct $C_i$ and $C_j$ .",
"In particular, it says that if $u$ lies in some $C_i$ and $v$ does not lie in that $C_i$ , then either $v$ lies in $X$ (i.e.",
"$y_v=1$ ) or $x_{uv}=1$ .",
"Note that the third set of constraints are asymmetric in $u$ and $v$ , and we will put two such constraints (with $u$ and $v$ swapped) for each edge $(u,v)$ .",
"As the objective function exactly measures the number of edges with endpoints in two different $C_i$ 's, it follows that the IP above has a feasible solution with value at most $\\delta n$ .",
"Consider the LP relaxation of this program.",
"It has $O(n^{s})$ variables, and $O(n^2)$ non-trivial constraints.",
"So in time $n^{O(s)}$ , we can find some basic feasible solution with support size at most $O(n^2)$ .",
"We reuse $x_{uv}$ , $y_v$ and $z_S$ to denote some fixed optimum solution to the LP."
],
[
"The Rounding Algorithm.",
"The algorithm will construct the required $X^{\\prime }$ and the collection $\\mathcal {C}$ of sets $C^{\\prime }_i$ from the LP solution by the following preprocessing and sampling procedure.",
"Initialization.",
"We set $X^{\\prime },\\mathcal {C} = \\emptyset $ and $U =V$ , where $U$ denotes the set of vertices not covered by $\\mathcal {C}$ .",
"Preprocessing.",
"Add every vertex $v$ with $y_v \\ge \\beta $ to $X^{\\prime }$ .",
"Set $U = U \\setminus X^{\\prime }$ Sampling to create $\\mathcal {C}$ .",
"Arbitrarily order the sets $S_1,\\ldots ,S_k$ in the support of the LP solution.",
"Repeat the following (phase) until $U$ is empty: Phase.",
"For $i=1,\\ldots ,k$ , sample the set $S_i$ randomly with probability $z_{S_i}$ .",
"If $S_i$ is picked, add $C^{\\prime }=S_i \\cap U$ to the collection $\\mathcal {C}$ , and update $U = U \\setminus S_i$ ."
],
[
"Analysis.",
"Clearly the sets $C^{\\prime }$ produced by the algorithm have size at most $s$ , and they are disjoint.",
"Lemma 5.4 $|X^{\\prime }| \\le a/\\beta $ .",
"As $X^{\\prime }$ is the set of vertices $v$ with $y_v \\ge \\beta $ , there can be at most $a/\\beta $ such vertices by the LP constraint $\\sum _v y_v \\le a$ .",
"Henceforth, we also assume that $\\beta \\le 1/2$ , otherwise choosing $X^{\\prime } = \\emptyset $ and partitioning $V$ arbitrarily into sets $C_1,\\ldots ,C_k$ of size at most $s$ trivially suffices for Theorem REF .",
"We now show that the algorithm runs in expected polynomial time and does not generate a vertex partition with too many edges between distinct $C^{\\prime }_i$ .",
"Lemma 5.5 Let $F$ be the set of edges with endpoints in two distinct $C^{\\prime }_i$ s. Then $\\mathbb {E}[|F|] \\le O(b + \\beta |E|) $ .",
"Moreover, the algorithm terminates after at most $O(n^2 \\log n)$ sampling steps with high probability.",
"We claim that $U$ is empty after $O(\\log n)$ phases, with high probability.",
"After the preprocessing step, each uncovered vertex in $U$ has $y_v < \\beta \\le 1/2$ .",
"Thus, by the second LP constraint, $p_v:=\\sum _{ S\\ni v} z_S = 1- y_v \\ge 1/2$ .",
"So the probability that a vertex $v$ is not covered after $j$ phases is $ \\left(\\prod _{S \\ni v} (1-z_S)\\right)^j \\le \\exp \\left(-j p_v \\right) \\le \\exp \\left(-j/2\\right) \\le (2/3)^j $ The claim now follows from a union bound over the $n$ vertices.",
"We now bound the size of $F$ .",
"Let us focus on an edge $e=(u,v)$ and bound the probability that it is cut, that is, added to $F$ during the Sampling step.",
"Let $U_j$ denote the vertices in $U$ at the end of phase $j$ .",
"The edge is cut in phase $j$ if and only if both $u$ and $v$ remain in $U$ at the end of phase $j-1$ (i.e.",
"$u,v \\in U_{j-1}$ ) and a set $S$ with $|S \\cap \\lbrace u,v\\rbrace | = 1$ is chosen in phase $j$ .",
"As $\\Pr [u,v \\in U_{j-1}] \\le \\Pr [v \\in U_{j-1}] \\le (2/3)^{j-1}$ , this implies that $\\Pr [(u,v) \\mbox{ cut in phase $j$}]\\le (2/3)^{j-1} \\cdot \\sum _{S : |S \\cap \\lbrace u,v\\rbrace | = 1}z_S.$ Moreover, by the third set of constraints in LP (REF ) $\\sum _{S : |S \\cap \\lbrace u,v\\rbrace | = 1}z_S = \\sum _{S : u \\in S \\wedge v\\notin S}z_S + \\sum _{S : v \\in S \\wedge u \\notin S}z_S\\le 2x_{uv} + y_u + y_v.$ Summing (REF ) over all the phases and using (REF ), we get $ \\Pr [(u,v) \\mbox{ cut}] \\le 3(2x_{uv} + y_u + y_v) \\le 6 x_{uv} + 6\\beta , $ where the second inequality follows as both $u$ and $v$ were not chosen in $X$ during the preprocessing step and hence $y_u,y_v \\le \\beta $ .",
"By linearity of expectation, this implies that $ \\mathbb {E}[|F|] = \\sum _{(u,v) \\in E} (6 x_{uv} + 6 \\beta ) = O(b + \\beta |E|).$"
],
[
"Acknowledgements",
"We would like to thank Marcin Pilipczuk and Fedor Fomin for many useful discussions and pointers, and anonymous reviewers for bringing recent local search-based techniques to our attention.",
"Part of this work was done when all the authors were visiting the Simons Institute for the Theory of Computing."
]
] | 1606.05198 |
[
[
"A New Energy Efficient MAC Protocol based on Redundant Radix for\n Wireless Networks"
],
[
"Abstract In this paper, we first propose a redundant radix based number (RBN) representation for encoding the data to be transmitted in a wireless network.",
"This RBN encoding uses three possible values - 0, 1 and $\\bar 1$, for each digit to be transmitted.",
"We then propose to use silent periods (zero energy transmission) for transmitting the 0's in the RBN encoded data thus obtained.",
"This is in contrast to most conventional communication strategies that utilize energy based transmission (EbT) schemes, where energy expenditure occurs for transmitting both 0 and 1 bit values.",
"The binary to RBN conversion algorithm presented here offers a significant reduction in the number of non-zero bits in the resulting RBN encoded data.",
"As a result, it provides a highly energy-efficient technique for data transmission with silent periods for transmitting 0's.",
"We simulated our proposed technique with ideal radio device characteristics and also with parameters of various commercially available radio devices.",
"Experimental results on various benchmark suites show that with ideal as well as some commercial device characteristics, our proposed transmission scheme requires 69% less energy on an average, compared to the energy based transmission schemes.",
"This makes it very attractive for application scenarios where the devices are highly energy constrained.",
"Finally, based on this transmission strategy, we have designed a MAC protocol that would support the communication of such RBN encoded data frames."
],
[
"Introduction",
"During recent years, wireless ad hoc networks have received considerable attention of researchers for their increasing applications in various fields, e.g, military communications, disaster relief, rescue operations, etc.",
"There exist different schemes for transmitting data in a wireless network.",
"Depending on the situation, either both 0 and 1 are represented by non-zero voltage levels, or one of the bit values is represented by a zero voltage level while a non-zero voltage level is used to distinguish the other bit value.",
"An example of the latter is the polar return-to-zero (polar-RZ) transmission scheme, where a 0 corresponds to a zero voltage level, while a 1 is represented by a non-zero voltage level.",
"However, most existing transmission schemes utilize non-zero voltage levels for both 0 and 1 so as to distinguish between a silent and a busy channel.",
"Communication strategies that require energy expenditure for transmitting both 0 and 1 bit values are known as energy based transmission (EbT) schemes.",
"For example, in order to communicate a value of 278, a node will transmit the bit sequence $<1, 0, 0, 0, 1, 0, 1, 1, 0>$ , consuming energy for every bit it transmits.",
"Thus, if the energy required per bit transmitted is $e_b$ , the total energy consumed to transmit the value 278 would be $9e_b$ .",
"In this paper, we propose a communication technique that first recodes a binary data in redundant radix based number (RBN) representation [10] and then uses silent periods to communicate the bit value of '0'.",
"We show that by using the redundant binary number system (RBNS) that utilizes the digits from the set {-1, 0, 1} to represent a number with radix 2, we can significantly reduce the number of non-zero digits that need to be transmitted.",
"The transmission time remains linear in the number of bits used for data representation, as in the binary number system.",
"We finally propose a MAC protocol that would support the communication of such RBN encoded data frames with a significant amount of energy savings.",
"We have simulated our proposed transmission algorithm with both ideal device characteristics and parameters of several commercially available radio devices.",
"The results of these experiments show that, for ideal device characteristics and even for some commercial device characteristics, the increase in energy savings with our proposed algorithm over the existing energy based transmission schemes is, on an average, equal to 69%."
],
[
"Related Work",
"Recent research efforts on reducing energy consumption have mainly been focussed on the MAC layer design [5], optimizing data transmissions by reducing collisions and retransmissions [6] and through intelligent selection of paths or special architectures for sending data [7].",
"A survey of routing protocols in wireless sensor networks can be found in [5].",
"In all such schemes, the underlying communication strategy of sending a string of binary bits is energy based transmissions (EbT) [3], [4], which implies that the communication of any information between two nodes involves the expenditure of energy for the transmission of data bits.",
"In [3], a new communication strategy called Communication through Silence (CtS) has been proposed that involves the use of silent periods as opposed to energy based transmissions.",
"CtS, however, suffers from the disadvantage of being exponential in time.",
"An alternative strategy, called Variable-base tacit communication (VarBaTaC) has been proposed in [4] that uses a variable radix-based information coding coupled with CtS for communication."
],
[
"Preliminaries and Proposed Communication Scheme",
"The redundant binary number system (RBNS) [10] utilizes the digits from the set {-1, 0, 1} for representing numbers using radix 2.",
"In the rest of the paper, for convenience, we denote the digit '-1' by $\\bar{1}$ .",
"In RBNS, there can be more than one possible representation of a given number.",
"For example, the number 7 can be represented as either 111 or $100\\bar{1}$ in RBNS.",
"In this work, we utilize this property of RBNS to recode a message string so as to reduce the number of 1's in the string while transmitting the message [1].",
"The original binary message can, however, be obtained at the receiver end by reconverting the received message from RBN to binary number system [10].",
"The basic idea of our recoding scheme is as follows : Consider a run of $k$ 1's, $k > 1$ .",
"Let $i$ be the bit position for the first 1 in this run, $i \\ge 0$ (bit position 0 refers to the least significant bit at the rightmost end).",
"Let $v$ represent the value of this run of $k$ 1's.",
"Then, $ v = 2^i + 2^{i + 1} + 2^{i + 2} + \\ldots + 2^{k + i - 1}$ Alternatively, we can rewrite equation REF as, $ v = 2^{k + i} - 2^i$ Equation REF can be represented in RBNS by a '1' at bit position $(k + i)$ and a $\\bar{1}$ at bit position $i$ , while all the intermediate 1's between them are converted to 0's.",
"Thus, a long run of 1's can equivalently be replaced by a run of 0's and only two non-zero digits, 1 and $\\bar{1}$ .",
"Observe that for a run of $k$ 1's, $k > 1$ , the savings in terms of the number of non-zero digits is $k - 2$ .",
"However, the number of non-zero digits remain unchanged for $k = 2$ .",
"Thus, if we keep the transmitter switched-off for 0 bit-values, the power consumption of the transmitter will be less than that in energy based transmission (EbT) schemes.",
"Hence, by combining this approach of silent zero transmission with our RBNS-based recoding strategy, a significant reduction in the energy expenditure during data transmission can be achieved when compared to the energy based transmission (EbT) of binary data.",
"Our proposed low energy transmission strategy involves the execution of the following two steps : Algorithm TransmitRBNData Step 1 : Recode the $n$ -bit binary data frame to its equivalent RBNS data frame using steps 1.1 and 1.2 stated below.",
"Step 1.1 : Starting from the least significant bit (lsb) position, scan the string for a run 1's of length $> 1$ .",
"A run of $k$ 1's ($k > 1$ ) starting from bit position $i$ , is replaced by an equivalent representation consisting of a '1' at bit position $k +i$ and a $\\bar{1}$ at bit position $i$ , with 0's in all intermediate bit positions.",
"Step 1.2 : Every occurrence of the bit pattern $\\bar{1}1$ in a string obtained after step 1.1, is replaced by the equivalent bit pattern $0\\bar{1}$ .",
"Step 2 : Transmit the RBNS data frame obtained from step 1 above.",
"Note that the encoding process of an $n$ -bit binary string to its equivalent RBNS representation can result in a RBNS string of length of either $n$ or $n + 1$ symbols.",
"If a run of 1's of length $> 1$ ends in the most significant bit (msb), then by virtue of step 1.1 of TransmitRBNData algorithm, the symbol 1 is placed at the position $msb + 1$ .",
"Otherwise, if the msb was 0, then the RBNS string also has exactly $n$ symbols.",
"Example 1 Consider in a given binary string, a substring, say 110111, with only one '0' trapped between runs of 1's.",
"Then following step 1.1, we would get the string $10\\bar{1}100\\bar{1}$ .",
"Note the presence of the pattern $\\bar{1}1$ for this trapped '0'.",
"Application of step 1.2 of algorithm TransmitRBNData to the bit pattern $\\bar{1}1$ replaces it by $0\\bar{1}$ , thus resulting in a further reduction in the number of non-zero symbols to be transmitted.",
"We now present the receiver side algorithm to receive a RBNS data frame and convert it back to binary : Algorithm ReceiveRBNData Step 1 : Receive the RBNS data frame in a buffer, say $recv\\_buf$ .",
"Step 2 : Set $runflag \\leftarrow false$ .",
"Now starting from lsb, scan the RBNS string in $recv\\_buf$ sequentially to obtain the binary equivalent using steps 2.1 through 2.3 : Step 2.1 : If the $i^{th}$ bit, $recv\\_buf[i] = \\bar{1}$ then execute steps 2.1.1 and 2.1.2, otherwise execute step 2.2 : Step 2.1.1 : If $runflag = false$ then the corresponding output binary bit is 1.",
"Also set $runflag \\leftarrow true$ .",
"Step 2.1.2 : If $runflag = true$ then the corresponding output binary bit is 0.",
"Step 2.2 : If the $i^{th}$ bit, $recv\\_buf[i] = 1$ then execute steps 2.2.1 and 2.2.2, otherwise execute step 2.3 : Step 2.2.1 : If $runflag = true$ then the corresponding output binary bit is 0.",
"Also set $runflag \\leftarrow false$ .",
"Step 2.2.2 : If $runflag = false$ then the corresponding output binary bit is 1.",
"Step 2.3 : If the $i^{th}$ bit, $recv\\_buf[i] = 0$ then execute steps 2.3.1 and 2.3.2 : Step 2.3.1 : If $runflag = true$ then the corresponding output binary bit is 1.",
"Step 2.3.2 : If $runflag = false$ then the corresponding output binary bit is 0.",
"Step 3 : Set $i \\leftarrow i + 1$ and repeat from step 2 until the entire received RBNS data frame is scanned and converted to the binary equivalent.",
"The equivalent binary data is then passed onto the higher layers of the network stack.",
"Table: Number of occurrences of runlengths forn=8n = 8We note that the application of steps 1.1 and 1.2 of the TransmitRBNData algorithm ensures that the bit patterns $1\\bar{1}$ and $\\bar{1}1$ can not occur in the transmitted data.",
"Hence, there is only a unique way of converting the received RBNS data into its binary equivalent."
],
[
"Analysis of the Energy Savings",
"We denote a run of 1's of length $k$ by $R_k$ .",
"Let us append a zero on left of each such $R_k, 1 \\le k \\le n$ and denote the symbol $0\\ R_k$ by $y_k$ .",
"We also denote a single zero by the symbol $y_0$ .",
"Then each such $y_k, 0 \\le k \\le n$ , will be a string of length $k+1$ .",
"To find out the total number of occurrences of $R_k$ 's, $1\\le k \\le n$ , in all possible $2^n$ strings of length $n$ , we would first compute the total number of occurrences of exactly $i_k$ number of $y_k$ 's.",
"Let this number be denoted by the symbol $N_n^{i_k, k}$ .",
"We use a generating function based approach to derive an expression for $N_n^{i_k, k}$ in all possible binary strings of length $n$ .",
"The detailed analysis is given in [1].",
"We have omitted it here for the sake of brevity and state only the final result as follows : For a given $n$ and $k \\ge 1$ , $N_n^{i_k, k}$ is given by, $N_n^{i_k, k} &= &i_k \\sum _{r=1}^{n+1-(k+1)i_k} {{r + i_k} \\atopwithdelims (){i_k}} \\sum _{q=0}^r \\sum _{j=0}^{q} (-1)^{q+j} \\times \\\\& &{{r+m-1-kq-j} \\atopwithdelims (){m-kq-j}} {r \\atopwithdelims ()q} {q \\atopwithdelims ()j} \\\\$ Example 2 For $n=8$ , $k = 2$ and $i_k = 2$ , we get the number : $N_8^{2,2} & = & 2 {3 \\atopwithdelims ()2} \\biggl [{2 \\atopwithdelims ()2} {1 \\atopwithdelims ()0} {0 \\atopwithdelims ()0} \\\\& &- {0 \\atopwithdelims ()0} {1 \\atopwithdelims ()1} {1 \\atopwithdelims ()0} + {{-1} \\atopwithdelims (){-1}} {1\\atopwithdelims ()1} {1 \\atopwithdelims ()1}\\biggr ] \\\\& &+ 2 {4 \\atopwithdelims ()2} \\biggl [{2 \\atopwithdelims ()1} {2 \\atopwithdelims ()0} {0 \\atopwithdelims ()0} - 0 + 0 \\biggr ] \\\\& & + 2 {5 \\atopwithdelims ()2} \\biggl [{2 \\atopwithdelims ()0} {3 \\atopwithdelims ()0} {0 \\atopwithdelims ()0} - 0 + 0 \\biggr ] \\\\& = & 2(0 + 12 + 10) = 44\\\\$ For a given $k \\ge 1$ , if we now sum the expression $N_n^{i_k, k}$ for all possible values of $i_k$ , $1 \\le i_k \\le \\lfloor (n+1)/(k+1) \\rfloor $ , then we get the total number of occurrences of $R_k$ in all possible strings of length $n$ .",
"Table REF shows the total number of occurrences of all possible runlengths of 1's in all possible binary strings of length 8 bits.",
"It was shown in [1] that considering all possible $2^n$ binary strings of length $n$ each, the total number of 1's and $\\bar{1}$ 's in the RBN coded message after applying both the steps 1.1 and 1.2 of the algorithm TransmitRBNData would be $(n + 2)2^{n-2}$ ."
],
[
"Experimental Results",
"Experimental results demonstrate that algorithm TransmitRBNData significantly reduces the energy consumption required for transmission, for different types of application scenarios.",
"We tested our algorithm on several popular compression benchmark test suites [9], [8].",
"The results for these test suites are presented in figure REF .",
"We have omitted the detailed results for each file of the individual benchmark suites for the sake of brevity.",
"For the purpose of the experiments, we assumed a data frame size of 1024 bits.",
"All the reported values in figure REF are with respect to energy based transmission schemes where the transmission of both '0' and '1' bit values require the expenditure of energy.",
"The column \"SiZe\" mentioned in the tables and figures, refers to a silent zero (SiZe) transmission scheme introduced in [1].",
"Table: Theoretical energy savings results fordifferent radios, n=1024n = 1024Considering the mean of the values reported for the SiZe protocol, we find that binary encoded files consists of 42.5% zeroes on an average which thus translates into an increased energy savings of 42.5%, as compared to an EbT transmission scheme.",
"Application of algorithm TransmitRBNdata on binary encoded files to create RBN encoded files cause on an average, an increased savings in energy from 42.5% to 69%, when averaged over the values reported in figure REF .",
"Experimental results also showed that increasing the data frame size increases the fractional savings in energy as longer runs of ones can then be reduced.",
"It increases steeply with the increase in frame size, when the size of the frames is small (8, 16, 32, 64, $\\ldots $ bits) and plateaus out for larger frame sizes.",
"We observed that in general, for frame sizes larger than 1024 bits, the increase in fractional savings is either very small or none.",
"The results show that the maximum increase in energy savings (34.4%) with our proposed algorithm over the SiZe protocol is obtained for the Maximum Compression test suite [9], while the minimum (21.8%) is for the Large Canterbury suite [8].",
"From the results in figure REF we see that there is an increase of $ 69\\% - 42.5\\% = 26.5\\%$ in transmission energy savings, when averaged over all benchmark suites considered in the figure, by using the proposed TransmitRBNData algorithm over the SiZe protocol.",
"Figure: Comparison of Average Energy Savings for theBenchmark Test Suites"
],
[
"Results Considering Device Characteristics",
"The effect of real life device characteristics on the energy savings was studied in details in [2].",
"It was shown in [2] that the fractional energy savings generated by the TransmitRBNData algorithm over EbT transmission scheme is, $\\gamma _{dev} & = \\Bigl (1 - \\frac{n + 2}{4n}\\Bigr )\\Bigl (1 -\\frac{I_{low}}{I_{high}}\\Bigr ) $ while the fractional energy savings generated by the SiZe protocol compared to an EbT transmission scheme is given by, $\\gamma _{SiZe} & = \\frac{I_{high} - I_{low}}{2I_{high}} = \\frac{1}{2}- \\frac{I_{low}}{2I_{high}} $ where, $I_{high}$ and $I_{low}$ denote the current drawn in the transmit (TX) and the active states, respectively.",
"In order to evaluate the performance of our TransmitRBNData algorithm on real-life devices, we considered some of the commercially available radios for our simulation purpose.",
"The results of our simulation are presented in table REF .",
"In table REF , $\\gamma _{SiZe}$ and $\\gamma _{dev}$ refer to the values obtained by substituting the corresponding device parameter values in equations REF and REF , respectively.",
"Table REF shows that $\\gamma _{dev}$ is higher than $\\gamma _{SiZe}$ by at least 16% for Maxim 2820 and Chipcon CC2510Fx chips, while it is higher by nearly 25% for RFM TR1000 and Maxim 1479.",
"The values of $\\gamma ^{sim}_{SiZe}$ and $\\gamma ^{sim}_{dev}$ in figures REF and REF refer to the energy saving results with the SiZe transmission scheme and our proposed algorithm respectively, obtained by running the simulation on the benchmark suites with the corresponding device parameters.",
"The results show that for the TransmitRBNData algorithm, the performance in energy savings is always much better than the SiZe transmission scheme.",
"However, while the graphs of RFM TR1000 and the Maxim 1479 devices show savings that are nearly equal to those reported in figure REF , the savings are somewhat lower for the CC2510Fx and Maxim 2820, due to the fact that the current drawn in the active state for both of these devices is not negligible compared to the current in the TX state.",
"Figure: Device Specific Energy Savings for the MaximumCompression and Calgary Test SuitesFigure: Device Specific Energy Savings for the Canterbury and LargeCanterbury Test Suites"
],
[
"Medium Access Control",
"We present in this section a medium access control for the TransmitRBNData algorithm.",
"Issues related to the design of a MAC protocol such as representation of RBN encoded numbers in the internal buffers at the MAC layer, consideration of suitable modulation schemes for the TransmitRBNData algorithm and receiver-transmitter synchronization for the duration of transmission of a data packet were addressed in [2]."
],
[
"RBNSiZeMAC - Asynchronous MAC Protocol",
"We now present RBNSiZeMAC - an asynchronous MAC protocol that allows the transmission of RBN encoded data, for single channel wireless networks.",
"In order to transmit a frame successfully, a node must compete with other nodes within its neighborhood to win the channel for the time duration it requires to transmit its frame.",
"It is important to prevent simultaneous transmission to the same receiving node as that would garble the frames beyond recovery.",
"Figure: Data frame of RBNSiZeMAC protocolOur protocol uses two classes of frames : data and control.",
"The format of the data frame is shown in figure REF .",
"The data frame consists of three parts : i) a header ii) payload and, iii) a frame trailer.",
"The frame header and the trailer are in binary while the payload is in RBN.",
"The header part consists of a preamble, destination and source addresses, type, length and sync fields.",
"Each frame starts with a Preamble of 2 bytes, each containing the bit pattern 10101010, similar to that of the IEEE 802.3 MAC header.",
"The receiver synchronizes its clock with that of the sender on this preamble field [11].",
"The next two fields in the frame header are the destination and the source addresses respectively, each 6 bytes long.",
"As in the 802.3 standard, the high order bit of the destination address is a 0 for ordinary addresses and 1 for group addresses.",
"A frame with destination address consisting of all 1s is accepted by all nodes in the neighborhood of the transmitter.",
"Next comes a 2 bit type field which indicates whether this is a data or a control frame.",
"The value '00' is used to indicate a data frame while the remaining 3 possible bit patterns (01, 10 and 11) are reserved for the control frames.",
"The 2 byte length field gives the length of the payload part of the frame.",
"Finally, there are another 2 bytes of sync field, each consisting of the binary bit pattern 10101010 to ensure that the receiver and the sender clocks remain synchronized during the entire period of transmission of the payload.",
"This is necessary as we assume that no signal is transmitted by the sender for the symbol 0 in the RBN encoded payload.",
"As our proposed algorithm TransmitRBNData can lead to the creation of long runs of 0's in the RBN encoded data, it is important to ensure that the sender and receiver clocks remain synchronized for the entire duration of the transmission of the payload.",
"The payload field can be maximum 1500 bytes, again similar to the 802.3 frame.",
"The payload part is followed by the binary encoded frame trailer which consists of a 4 byte checksum field computed only on the binary equivalent of the payload of the frame.",
"The checksum algorithm is the standard cyclic redundancy check (CRC) used by IEEE 802.",
"Figure: Control frame of RBNSiZeMAC protocolThe format of the control frame is shown in figure REF .",
"The control frame is in binary.",
"It consists of a 2 byte preamble, similar to that in a data frame, followed by the destination, source and the type fields respectively.",
"The size and interpretation of these three fields are same as in the data frame.",
"The type field is set according to the type of the intended control message, namely RTS, CTS or ACK.",
"The way these three message types are used is exactly the same as in the 802.11 MAC protocol [11].",
"This is followed by a length field which indicates the length of the payload that a node wishes to send in the data frame.",
"In the case of a CTS or ACK frame from a receiving node, if the receiver too has some data to send to the sender, it sets the length field accordingly, otherwise it is set to zero.",
"The last field is the usual 4 byte checksum computed over the rest of the control frame fields.",
"Each node in the network is assumed to possess channel status sensing capability - i.e., whether the channel is idle or busy.",
"The protocol that we propose here is a CSMA/CA (CSMA with collision avoidance) protocol, very similar to the 802.11 protocol.",
"When a node $A$ wants to transmit, it senses the channel.",
"If it remains idle for a time period of $b$ , where $b$ is the maximum possible duration of a frame transmission in RBNSiZeMAC protocol, $A$ sends an RTS (Request to Send) control message to the receiver (say $B$ ).",
"The reason for waiting for at least $b$ time is to avoid interrupting any ongoing frame transmission.",
"Due to encoding of the payload in RBN, it is possible that there may be a long run of the symbol 0 in the data which may otherwise be wrongly interpreted as the channel being idle without any ongoing transmission.",
"If $B$ receives the RTS, it may decide to grant permission to $A$ to transmit, in which case it sends a CTS frame back.",
"If $A$ does not receive any CTS from $B$ or a collision occurs for the RTS frame, each of the colliding nodes waits for a random time, using the binary exponential back-off algorithm, and then retry.",
"The behavior of other nodes in the vicinity of both $A$ and $B$ on hearing the RTS or CTS frames is the same as in 802.11.",
"After a frame has been sent, there is a certain amount of dead time before any node may send a frame.",
"We define two different intervals, similar to the 802.11 [11], for the RBNSiZeMAC protocol : The shortest interval is SIFS (Short InterFrame Spacing) that is used in exactly the same way as in 802.11.",
"After a SIFS interval, the receiver can send a CTS in response to an RTS or an ACK to indicate a correctly received data frame.",
"If there is no transmission after a SIFS interval has elapsed and a time NIFS (Normal InterFrame Spacing) elapses, any node may attempt to acquire the channel to send a new frame in the manner described previously.",
"We use NIFS in exactly the same way as the 802.11 protocol uses the DIFS (DCF InterFrame Spacing) interval."
],
[
"Conclusion",
"The redundant binary number system can be used instead of the binary number system in order to increase the number of zero bits in the data.",
"Coupled with this, the use of silent periods for communicating the 0's in the bit pattern provides a significant amount of energy savings in data transmissions.",
"The transmission time also remains linear in the number of bits used for data representation, as in the binary number system.",
"Simulation results on various benchmark suites show that with ideal as well as some commercial device characteristics, our proposed algorithm offers a reduction in energy consumption of 69% on an average, when compared to existing energy based transmission schemes.",
"Based on this transmission strategy, we have designed a MAC protocol that would support the communication of such RBN encoded data frames for asynchronous communication in a wireless network."
]
] | 1606.04935 |
[
[
"Di-nucleon structures in homogeneous nuclear matter based on two- and\n three-nucleon interactions"
],
[
"Abstract We investigate homogeneous nuclear matter within the Brueckner-Hartree-Fock (BHF) approach in the limits of isospin-symmetric nuclear matter (SNM) as well as pure neutron matter at zero temperature.",
"The study is based on realistic representations of the internucleon interaction as given by Argonne v18, Paris, Nijmegen I and II potentials, in addition to chiral N$^{3}$LO interactions, including three-nucleon forces up to N$^{2}$LO.",
"Particular attention is paid to the presence of di-nucleon bound states structures in $^1\\textrm{S}_0$ and $^3\\textrm{SD}_1$ channels, whose explicit account becomes crucial for the stability of self-consistent solutions at low densities.",
"A characterization of these solutions and associated bound states is discussed.",
"We confirm that coexisting BHF single-particle solutions in SNM, at Fermi momenta in the range $0.13-0.3$~fm$^{-1}$, is a robust feature under the choice of realistic internucleon potentials."
],
[
"Introduction",
"One of the main goals in theoretical nuclear physics is that of accounting for nuclear structures and processes starting from the basic interaction among their constituents.",
"If sub-hadronic degrees of freedom (i.e.",
"quarks and gluons) are not treated explicitly, then this goal relies on realistic representation of the bare internucleon interaction.",
"Such is the case of modern interactions based on quantum-field models, where strengths and form factors are adjusted to best reproduce nucleon-nucleon (NN) scattering observables as well as properties of the deuteron, the only NN bound state in free space.",
"The inclusion of three-nucleon (3N) forces become subject to constraints from three-body bound state data and/or homogeneous nuclear matter [1], [2].",
"Homogeneous nuclear matter, a hypothetical infinite medium of neutrons and protons, is among the simplest many-body nuclear systems.",
"In principle, all properties of this system should be inferred from the bare interaction among its constituents.",
"In this context the Brueckner-Hartree-Fock (BHF) non-relativistic approach at zero temperature offers a well defined framework which enables the evaluation of the energy of the system as a function of the nucleon density $\\rho =\\rho _p+\\rho _n$ , and isospin asymmetry $\\beta =(\\rho _n-\\rho _p)/(\\rho _n+\\rho _p)$ , with $\\rho _p$ ($\\rho _n$ ) denoting proton (neutron) density [1], [3].",
"Extensive applications of the BHF approach has served to assess the consistency of the model to account for saturation properties of isospin-symmetric ($\\beta =0$ ) nuclear matter [1], [4], [5].",
"Not only that but also the resulting BHF $g$ matrix at positive energies has served as an important tool to construct NN effective interactions, subsequently used in the evaluation of microscopic optical model potentials for nucleon scattering off finite nuclei [6].",
"In this work we investigate homogeneous nuclear matter in the framework of the BHF approach at zero temperature considering realistic representations of the bare NN interactions.",
"Particular attention is paid to the manifestation of two-nucleon bound state structures (di-nucleons), expressed as singularities in the $g$ matrix in the search process for self-consistent solutions of single-particle (sp) spectra.",
"As such, this work represents an extension of the investigation reported in Ref.",
"[7] based on Argonne $v_{18}$ (AV18) [8], where an explicit account for di-nucleon structures in symmetric nuclear matter was first reported.",
"Among the main findings reported in that work we mention: a) Nucleon effective masses at low densities can reach up to four times the bare nucleon mass; b) Large size di-nucleon bound states take place at sub-saturation densities; and c) Coexisting sp spectra are identified at low densities, that is to say two distinct sp fields meet self-consistency at a same density.",
"Here we investigate the robustness of these features in SNM under the choice of the bare NN interaction, in addition to their manifestation in the extreme case of pure neutron matter.",
"The BHF approach for interacting nucleons in nuclear matter can be thought as the lowest-order approximation of Brueckner-Bethe-Goldstone (BBG) theory or Self-Consistent Green's Function theory at zero temperature [3].",
"The former is based on the hole-line expansion for the ground state energy [1], where Goldstone diagrams are grouped according to their number of hole lines, with each group summed up separately.",
"The BHF approximation results from the summation of the two-hole-line diagrams, with the in-medium two-body scattering matrix calculated self-consistently with the sp energy spectrum.",
"Although a sp potential is introduced as an auxiliary quantity, its choice conditions the rate of convergence of the expansion for the binding energy.",
"Studies reported in Ref.",
"[5] lead to conclude that the continuous choice for the auxiliary potential yields better convergence over the so called standard choice, where the sp potential is set to zero above the Fermi energy.",
"Thus, we base this work on the continuous choice for the sp potentials.",
"This article is organized as follows.",
"In Sec.",
"we layout the theoretical framework upon which we base the study of homogeneous nuclear matter at zero temperature.",
"In Sec.",
"we present results for symmetric nuclear matter as well as neutronic matter, discuss associated effective masses and occurring in-medium di-nucleon structures.",
"Additionally, we discuss extent to which Hugenholtz-van Hove theorem [9] for sp energies is met.",
"In Sec.",
"we present a summary and the main conclusions of this work."
],
[
"Framework",
"In BBG theory for homogeneous nuclear matter the $g$ matrix depends on the density of the medium, characterized by the Fermi momentum $k_F$ , and a starting energy $\\omega $ .",
"To lowest order in the BHF approximation for nuclear matter in normal state, when only two-body correlations are taken into account, the Brueckner $G$ matrix satisfies $G(\\omega )=v+v\\,\\frac{Q}{\\omega +i\\eta -\\hat{h}_1-\\hat{h}_2}\\,G(\\omega )\\,,$ with $v$ the bare interaction between nucleons, $\\hat{h}_{i}$ the sp energy of nucleon $i$ ($i=1,2$ ), and $Q$ the Pauli blocking operator which for nuclear matter in normal state takes the form $Q| p\\, k\\rangle =\\Theta (p-k_F)\\Theta (k-k_F) | p\\, k\\rangle \\;.$ The solution to Eq.",
"(REF ) enables the evaluation of the mass operator $M(k;E)=\\sum _{\\mid p\\mid \\le k_F}\\langle \\textstyle {\\frac{1}{2}}(k-p)| g_{K}(E+e_p) |\\textstyle {\\frac{1}{2}}(k-p)\\rangle \\;,$ where the $g$ matrix relates to $G$ through $\\langle {k^{\\prime }} {p^{\\prime }}| G(\\omega ) |{k} {p}\\rangle =\\delta ({K^{\\prime }}-{K})\\langle \\textstyle {\\frac{1}{2}}( {k^{\\prime }}-{p^{\\prime }})|g_K(\\omega ) |\\textstyle {\\frac{1}{2}}({k}-{p})\\rangle \\;.$ Here $K$ ($K^{\\prime })$ denotes the total momentum of the NN pair before (after) interaction, with $K = k+p$ , and $K^{\\prime } = k^{\\prime }+p^{\\prime }$ , so that the Dirac delta functions expresses the momentum conservation.",
"The sp energy becomes defined in terms of an auxiliary field $U$ , $e(p)=\\frac{p^2}{2m} + U(p)\\,,$ with $m$ the nucleon mass taken as the average of proton and neutron masses.",
"In the BHF approximation the sp potential is given by the on-shell mass operator, $U(k)=\\textsf {Re}\\, M[k;e(k)]\\;,$ self-consistency requirement which can be achieved iteratively.",
"We have used the continuous choice for the sp fields, so that this condition is imposed at all momenta $k$ [10]."
],
[
"Results",
"We have proceeded to obtain self-consistent solutions for the sp fields $U(k)$ in infinite nuclear matter at various densities, specified by Fermi momenta $k_F\\lesssim 2.5$ fm$^{-1}$ .",
"These searches comprise isospin-symmetric nuclear matter and pure neutron matter.",
"The internucleon interactions considered in this study are AV18 [8], Paris [11], Nijmegen I and II bare potentials [12].",
"In addition to these potentials we include a chiral effective-field-theory ($\\chi $ EFT) interaction based on chiral perturbation theory.",
"The resulting bare interaction is constructed with nucleons and pions as degrees of freedom, with the two-nucleon (2N) part fit to NN data.",
"We consider the chiral 2N force (2NF) up to next-to-next-to-next-to-leading order (N$^{3}$ LO) given by Entem and Machleidt [2].",
"We also consider chiral 3N forces (3NF) in N$^{2}$ LO, using a density-dependent 2NF at the two-body level [13], [14].",
"This density-dependent contribution does not contain correlation effects [15], and is added to the bare 2NF in the calculation of the $G$ matrix.",
"The corresponding Hartree-Fock contribution is subtracted at each iteration to avoid any double counting.",
"For this chiral 3NF contribution, we use the low energy constants $c_D = -1.11$ and $c_E = -0.66$ , reported in Ref.",
"[16], which describe the $^3$ H and $^4$ He binding energies with unevolved NN interactions.",
"All applications include partial waves up to $J=7$ in the NN total angular momentum.",
"For the numerical methodology to treat di-nucleons during self-consistency search we refer the reader to Ref.",
"[7].",
"Files containing self-consistent solutions for sp potentials can be retreived from Ref.",
"[17]."
],
[
"Symmetric nuclear matter",
"The study of SNM requires to take into account NN states with total isospin $T=0$ , and $T=1$ .",
"As a result, one has to include the attractive ${}^3\\textrm {SD}_{1}$ , ${}^3\\textrm {PF}_{2}$ and ${}^1\\textrm {S}_{0}$ channels.",
"As reported in Ref.",
"[7], the calculation of the on-shell mass operator to obtain $U(k)$ requires the evaluation of $g_{K}(\\omega )$ at various configurations of total momentum $K$ of the NN pair and starting energy $\\omega =e(p)+e(k)$ .",
"In the process the $g$ matrix is sampled over regions where it becomes singular, near or at the occurrence of in-medium bound states in these channels.",
"This feature, investigated in the context of AV18 interaction, has led to unveil coexisting sp solutions at Fermi momenta slightly below 0.3 fm$^{-1}$ , that is to say different solutions that meet self-consistency at the same $k_F$ .",
"Details about how these coexisting solutions are disclosed are given in the same reference.",
"In this work we proceed in the same way.",
"Once a sp solution $U(k)$ is obtained for a given Fermi momentum we can evaluate the energy per nucleon $E/A$ , which in the case of two-body forces is given by $\\frac{E_{2N}}{A} =\\frac{\\sum _{k}n(k)\\left[\\frac{k^2}{2m} +\\textstyle {\\frac{1}{2}}U(k)\\right]}{\\sum _{k} n(k)}\\;.$ In this work we use $n(k) = \\Theta (k_F-k)$ , i.e.",
"nuclear matter in normal state.",
"When 3NFs are included in BHF calculations, these enter at two levels.",
"First, a density-dependent effective two-body interaction is added to the bare 2NF in a standard $G$ -matrix calculation.",
"In addition, the total energy has to be corrected to avoid double counting of the 3NF contribution [14], [18].",
"At the lowest order this can be achieved by subtracting the Hartree-Fock contribution due to 3NFs only: $\\frac{E_{3N}}{A}=\\frac{E_{2N}}{A} &-\\frac{1}{12}\\frac{3}{k_F^3}\\int _0^{k_F} k^2 dk \\, \\Sigma _{HF}^{3NF}(k) \\, .$ We stress that the Hartree-Fock self-energy $\\Sigma ^{3NF}_{HF}$ coming from the 3N force is calculated from an effective 2N potential at the lowest order, in keeping with the procedure established Ref. [14].",
"In Fig.",
"REF we present results for the energy per nucleon $E/A$ as function of $k_F$ for symmetric nuclear matter.",
"Here, solid, long-, medium- and short-dashed curves denote AV18, Paris, Nijmegen I and II solutions, respectively.",
"Dotted and dash-dotted curves represent solutions based on N$^3$ LO and N$^3$ LO+3N chiral interactions, respectively.",
"Labels I and II are used to distinguish the two families of solutions.",
"We are aware that, from a physical point of view, the energy of the system should be uni-valuated.",
"The purpose of this figure in displaying separately $E/A$ for the two phases is that of providing a global characterization of the sp solutions.",
"An actual evaluation of the energy of the system at a given $k_F$ would require a more comprehensive analysis, considering contributions from di-nucleons in the different channels and competing phases I and II.",
"Under such considerations the scope of the BHF approximation would become limited.",
"Figure: Energy per nucleon for isospin-symmetric nuclear matteras function of Fermi momentum k F k_F.Solid, long-, medium- and short-dashed curves correspond toAV18, Paris, Nijmegen I and II, respectively.Dotted and dash-dotted curves represent solutions forN 3 ^3LO and N 3 ^3LO+3N chiral interactions, respectively.The results presented in Fig.",
"REF show that all interactions considered yield nearly identical behavior in the range $0\\le k_F\\le 0.4$ fm$^{-1}$ .",
"Additionally, they all exhibit coexisting sp solutions at $k_F$ in the range between ${\\sim }$ 0.13 and $0.28-0.30$ fm$^{-1}$ .",
"In this regard the feature of coexistence is robust under the bare internucleon bare interaction.",
"In Fig.",
"REF we present results for the energy per nucleon $E/A$ in SNM as a function of $k_F$ for solutions in phase II.",
"We use the same convention of curve patterns as in Fig.",
"REF .",
"In this case the interactions exhibit different behaviors, resulting in different saturation points.",
"As observed, chiral interactions are the ones which yield extreme values for the density and binding energy at saturation.",
"On the one side N$^{3}$ LO saturates at $k_F=1.85$ fm$^{-1}$ , with $E/A=-25.7$ MeV, whereas N$^{3}$ LO+3N does so at $k_F=1.30$ fm$^{-1}$ , with $E/A=-12.1$ MeV.",
"The former becomes much too bound at a density $\\sim $$2.7\\rho _0$ , with $\\rho _0=0.16$ fm$^{-3}$ , the accepted saturation density.",
"The latter, instead, saturates near the correct density but becomes underbound by $\\sim $ 4 MeV relative to the accepted value of $16\\pm 1$ MeV.",
"These results are comparable to those reported in Ref. [15].",
"Furthermore, at $k_F$ below $\\sim \\!1$ fm$^{-1}$ , i.e.",
"matter density below $0.07$ fm$^{-3}$ , the behavior of $E/A$ appears insensitive to the interaction.",
"This feature is in agreement with recent reports based on BHF and Monte Carlo calculations using chiral interactions [19], [20].",
"Another feature we note from Fig.",
"REF is the similarity between AV18 and Paris potentials in their $E/A$ vs $k_F$ behavior.",
"Their resulting saturation energies are $-16.8$ and $-16.3$ MeV, respectively, with both interactions saturating at a density near $1.4\\rho _0$ .",
"In the cases of Nijmegen I and II saturation occurs at $2\\rho _0$ and $1.8\\rho _0$ , respectively, while their respective binding occur at $-20.6$ and $-18.3$ MeV.",
"The results we provide here are in reasonable agreement with those reported elsewhere [21].",
"What is new in these results is the actual account for di-nucleon singularities in the $g$ matrix to obtain self-consistent sp fields within BHF.",
"Unfortunately there is now way to artificially suppress di-nucleon occurrences, without altering the bare interaction, in order to isolate the role of in-medium bound states.",
"Figure: Energy per nucleon for isospin-symmetric nuclear matteras function of Fermi momentum k F k_F.Curves follow the same convention as in Fig.",
".The study of nucleon effective masses $m^{*}$ has been subject of interest in various sub-fields [22], [23].",
"The calculated sp spectra of Eq.",
"(REF ) allows us to evaluate the effective mass $\\frac{m^*}{m} =\\frac{k_F}{m}\\left[\\frac{\\partial e(k)}{\\partial k} \\right]^{-1}_{k=k_F}\\,,$ with $m$ the nucleon mass.",
"In Fig.",
"REF we plot the calculated effective-to-bare mass ratio $m^*/m$ as a function of Fermi momentum based on the six interactions we have discussed.",
"Filled and empty circles correspond to results for AV18 and Paris potentials, respectively.",
"Filled and empty squares correspond to Nijmegen I and II potentials, respectively.",
"Filled and empty diamonds denote solutions based on N$^3$ LO and N$^3$ LO+3N chiral interactions, respectively.",
"Labels I and II refer to solutions in phase I and II, respectively.",
"Figure: Nucleon effective mass in isospin-symmetric nuclearmatter as function of Fermi momentum k F k_F.Filled and empty circles represent solution forAV18 and Paris potentials, respectively.Filled and empty squares correspond to Nijmegen I and II potentials,respectively.Filled and empty diamonds denote solutions based onN 3 ^3LO and N 3 ^3LO+3N chiral interactions, respectively.Dotted lines are used to guide the eye.A peculiar feature observed in Fig.",
"REF is the occurrence of $m^{*}/m>1$ at Fermi momenta below $\\sim $ 1 fm$^{-1}$ .",
"In the case of phase I, which starts at $k_F=0$ , the effective mass grows from the bare mass $m$ up to $\\sim \\!4m$ near the maximum $k_F$ of phase I, consistent with findings reported in Ref. [7].",
"We also note that the trend followed by $m^*/m$ vs $k_F$ is very similar for all the interactions considered, an indication of the robustness of the results under changes of the bare internucleon potential.",
"In the case of phase II, the range where $m^*>m$ is restricted to $\\sim \\!0.2<k_F\\lesssim 1.1$ fm$^{-1}$ , or equivalently $\\sim \\!0.003<\\rho /\\rho _0\\lesssim 0.6$ .",
"As discussed in Refs.",
"[7], [24], this feature is closely related to the occurrence of di-nucleon bound states.",
"For $k_F$ near normal densities, i.e.",
"in the range $1.4-1.5$ fm$^{-1}$ , the ratio $m^*/m$ lies within the interval $0.78-0.85$ for all interactions, feature consistent with the typical values of nucleon effective masses."
],
[
"Pure neutron matter",
"The case of pure neutron matter features full suppression of the deuteron channel.",
"Therefore, singularities of $g_K(\\omega )$ represent in-medium bound states formed by neutron pairs, i..e. di-neutrons.",
"The treatment of these singularities is the same as that applied in SNM reported in Ref. [7].",
"In contrast to the case of SNM, however, no coexisting sp spectra are found.",
"In Fig.",
"REF we present results for the energy per nucleon, $E/A$ , for neutronic matter as a function of $k_F$ .",
"We use the same convention of curve patterns as in Fig.",
"REF .",
"As observed, all interactions yield nearly identical energy per nucleon up to $k_F\\sim 1$ fm$^{-1}$ , departing from each other at Fermi momenta above $1.2$ fm$^{-1}$ .",
"All interactions yield monotonic growing $E/A$ as function of $k_F$ , with N$^{3}$ LO+3N providing the highest slope.",
"As in the case of SNM, AV18 and Paris potentials behave very similarly.",
"The smallest slope in the energy comes from N$^{3}$ LO, although both Nijmegen I and II present similar density dependence.",
"For $k_F$ below $\\sim \\!1.2$ fm$^{-1}$ , i.e.",
"neutron densities below $0.06$ fm$^{-3}$ , all interactions exhibit nearly the same behavior, in agreement with other reports based on chiral interactions [19], [20].",
"Figure: Energy per nucleon for pure neutron matteras function of Fermi momentum k F k_F.Curves follow the same convention as in Fig.",
".Effective masses associated to the sp fields for pure neutron matter are shown in Fig.",
"REF .",
"Here we consider all six interactions included in the previous analysis, following the same symbol convention as in Fig.",
"REF .",
"As in the case of SNM, all interaction follow a very similar behavior as function of $k_F$ , with only the chiral interaction N$^{3}$ LO+3N departing from the rest at $k_F$ above 1.5 fm$^{-1}$ .",
"It is also clear that the neutron effective mass is greater than its bare mass at $k_F$ in the range $0.04-1.1$ fm$^{-1}$ , with a maximum value of $\\sim \\!$$1.2 m$ at $k_F$ in the range $0.25-0.5$ fm$^{-1}$ .",
"In the case of N$^{3}$ LO+3N interaction the ratio $m^{*}/m$ exhibits a growth at $k_F$ above 1.5 fm$^{-1}$ , which could be attributed to the relevance of the (density-dependent) 3N force at such high densities.",
"Apart from this interaction at high densities, the behavior of effective masses featuring $m^{*}/m>1$ is also robust under the choice of bare interaction being considered.",
"Figure: Neutron effective mass in pure neutron matteras function of Fermi momentum k F k_F.Symbols follow the same convention as in Fig.",
"."
],
[
"Di-nucleons within BHF",
"As mentioned above, the occurrence of singularities in the $g$ matrix denotes the presence of bound states.",
"This feature becomes explicit with the use of the Lehmann spectral representation for the $g$ matrix [3] $g_{K}(\\omega ) = v +\\sum _{\\alpha }v\\frac{|\\alpha \\rangle \\langle \\alpha |}{\\omega +i\\eta -\\epsilon _\\alpha }Qv\\;,$ where $\\alpha $ runs over discrete and continuous states, $|\\alpha \\rangle $ is an eigenstate of the Hamiltonian $\\hat{H}=\\hat{h}_1+\\hat{h}_2 + v$ , with eigenenergy $\\epsilon _\\alpha $ .",
"The way to infer the energy of bound states for a given pair momentum $K$ is by imposing [7] $\\det [1-v\\Lambda _K(\\omega )]=0\\;,$ with $\\Lambda _K(\\omega )=Q/(\\omega -\\hat{h}_1-\\hat{h}_2)$ , the BHF particle-particle propagator.",
"The energy of the bound state is obtained from the difference $b\\equiv \\omega -\\omega _{th}\\;,$ where $\\omega _{th}$ corresponds to the lowest (threshold) particle-particle energy allowed by the Pauli blocking operator.",
"We investigate the occurrence of di-nucleon bound states in SNM in the channels $^3\\textrm {SD}_1$ and $^{1}\\textrm {S}_0$ as a function of the Fermi momentum for all six NN interactions under study.",
"The condition given by Eq.",
"REF can be investigated for selected values of $K$ , the momentum of the NN pair.",
"In the following we focus on center-of-mass at rest ($K=0$ ).",
"Fig.",
"REF shows results obtained for the di-nucleons in the deuteron channel, where we follow the same symbol convention as in Fig.",
"REF .",
"Labels I and II indicate solutions for phase I and II, respectively.",
"From these results we observe bound states in channel $^{3}\\textrm {SD}_{1}$ take place in phase I at momenta over the range $0\\le k_F\\lesssim 0.3$ fm$^{-1}$ , featuring increasing binding.",
"The highest binding takes place at the upper edge of phase I, where $b\\approx -\\!4.5$ MeV, nearly twice the binding energy of the deuteron in free space.",
"It is also clear that all six interactions yield nearly the same binding.",
"Phase II, in turn, shows bound states from $k_F\\approx 0.14$ fm$^{-1}$ up to $k_F$ between 1.3 and 1.4 fm$^{-1}$ , close to the accepted Fermi momentum at saturation.",
"The maximum binding takes place at $k_F\\approx 0.5$ fm$^{-1}$ , where $b\\approx -\\!2.6$ MeV.",
"Overall, all interactions display the same behavior for $b$ .",
"Figure: In-medium deuteron binding energy in isospin-symmetric nuclearmatter as function of the Fermi momentum k F k_F.Symbols follow the same convention as in Fig.",
".Results for the $^{1}\\textrm {S}_{0}$ channel in SNM are shown in Fig.",
"REF using the same notation as in the previous figure.",
"Note that in this case the energy scale is expressed in keV units.",
"As a result, differences in the binding energy from the different NN interactions appear enhanced.",
"With the exception of Paris potential (open circles), the trend followed by $b$ in phase I is quite similar among the other five interactions, leading to a maximum binding of about 600 keV at $k_F\\approx 0.28$ fm$^{-1}$ .",
"Note also that for $k_F$ below $\\sim \\!0.06$ fm$^{-1}$ no di-nucleon bound states take place, feature shown by all six interactions.",
"This is consistent with the fact that no bound state takes place at zero density (free space) in the $^{1}\\textrm {S}_{0}$ channel.",
"In the case of phase II, di-nucleons take place from $k_F\\gtrsim 0.2$ fm$^{-1}$ up to about $k_F$ slightly above 1 fm$^{-1}$ .",
"The maximum binding takes place at $k_F\\approx 0.7$ fm$^{-1}$ , with $b$ in the range $500-650$ keV.",
"Figure: In-medium di-nucleon binding energy in channel 1 𝖲 0 ^1\\textsf {S}_0for isospin-symmetric nuclear matteras functions of the Fermi momentum k F k_F.Symbols follow the same convention as in Fig.",
".Results for di-neutrons in pure neutron matter are shown in Fig.",
"REF , where we plot $b_{nn}$ as a function of $k_F$ considering all six interactions in this study, applying the same notation as in the previous case.",
"Here we also use keV units for the energy scale.",
"Note that all interactions display similar behavior over $k_F$ , with appearance of di-neutrons at $k_F$ above 0.06 fm$^{-1}$ and disappearance at $k_F\\approx 1.05$ fm$^{-1}$ in the case of Paris and N$^{3}$ LO+3N interactions, and at $k_F\\approx 1.1$ fm$^{-1}$ for the rest.",
"The maximum binding takes place in the vicinity of $k_F\\approx 0.6$ fm$^{-1}$ , with Paris potential leading to the lowest binding of $\\sim \\!-$ 550 MeV.",
"Note that the behavior of $b_{nn}$ in this case shows some quantitative resemblance to that found for the $^{1}\\textrm {S}_{0}$ channel in SNM (c.f.",
"Fig.",
"REF ).",
"Figure: In-medium di-neutron binding energy in pure neutron matteras function of the Fermi momentum k F k_F.Symbols follow the same convention as in Fig.",
"."
],
[
"Hugenholtz-van Hove theorem",
"The Hugenholtz-van Hove (HvH) theorem [9] states a very general result that relates the mean energy of a bound system, $E/N$ , with its chemical potential $\\mu $ .",
"At zero temperature this relationship establishes that [1] $p = -\\frac{E}{V} + \\frac{N}{V} \\mu \\;,$ where $p$ represents the pressure, $E/V$ the energy density, $N/V$ the particle density, and $\\mu $ the chemical potential.",
"The latter should be extracted from the derivative of energy with respect to the number of particles.",
"In the BHF approximation at zero temperature the chemical potential coincides with the Fermi energy $e_F$ of the system –given by the sp energy at the Fermi momentum $k_F$ – hence relying on the auxiliary potential $U(k)$ .",
"Since at saturation the pressure vanishes, Eq.",
"(REF ) reduces to $\\frac{E}{A} = e_F\\,,$ with $A$ the nucleon number.",
"It has been known for some time that the HvH theorem is not satisfied in the BHF approximation for nuclear matter at zero temperature.",
"This limitation has led to go beyond BHF by including higher order contributions in the hole expansion for the nucleon self-energy [25], [26], [27].",
"Such an extension goes beyond the scope of this work.",
"However, it is still instructive to assess the extent to which HvH theorem is violated within the BHF approach for the interactions considered in this work.",
"In Table REF we list the internucleon potentials together with their respective Fermi momentum and density $\\rho $ at saturation.",
"The sixth column displays the difference between the mean energy $E/A$ and the Fermi energy, $e_F=e(k_F)$ .",
"If HvH was satisfied at the saturation point, then Eq.",
"REF would imply only zeros for this column.",
"Such is not the case, as we observe that the difference $(E/A-e_F)$ is comparable to $-\\!E/A$ .",
"The weakest violation of HvH theorem occurs for N$^{3}$ LO+3N chiral interaction, where $E/A-e_F=8.2$ MeV.",
"Table: HvH theorem check at saturation pointThe inclusion of higher-order correlations to mitigate the violation of the HvH theorem has been investigated in Refs.",
"[25], [26], [27].",
"In the present context, this extension would require a significant amount of work.",
"Implications of these considerations in actual calculations remain to be seen, particularly regarding the coexistence of sp solutions in the case of SNM.",
"We have investigated the role of di-nucleon bound states in homogeneous nuclear matter in the cases of isospin-symmetric matter and pure neutron matter.",
"The study has been based on the BHF approach at zero temperature, considering modern bare internucleon interactions, including the case of 3NFs based on chiral N$^{3}$ LO with three-nucleon forces up to N$^{2}$ LO.",
"Special attention is paid to the occurrence of di-nucleon bound states structures in the $^1\\textrm {S}_0$ and $^3\\textrm {SD}_1$ channels, whose explicit treatment is critical for the stability of self-consistent solutions at sub-saturation densities.",
"An analysis of these solutions is made by comparing their associated energy per nucleon $E/A$ , effective masses and in-medium di-nucleon binding energies.",
"An important result from this work is that coexistence of sp solutions in SNM withing the BHF approximation, occurring at Fermi momenta in the range $0.13-0.3$ fm$^{-1}$ and reported in Ref.",
"[7], is a robust property of the system which does not depend on the choice of realistic internucleon potentials.",
"Additionally, all interactions yield very similar behavior of $E/A$ at fermi momenta $k_F\\lesssim $ fm$^{-1}$ .",
"At higher densities the interactions exhibit their differences, resulting in different saturation points in the case of SNM, or different growth of $E/A$ (i..e. pressure) as function of the density.",
"Additionally, we also obtain effective masses larger than bare masses at sub-saturation densities, feature shared by all interactions.",
"In the case of SNM, effective masses in phase I can reach up to four times the bare mass, while in the case of phase II a maximum ratio of $m^*/m\\approx 1.5$ is found at $k_F\\approx 0.5$ fm$^{-1}$ .",
"In the case of pure neutron matter, the highest effective masses occur in the range $0.25\\lesssim k_F\\lesssim 0.5$ fm$^{-1}$ , where $m^*/m$ can reach up to $\\sim \\!",
"1.2$ .",
"Di-nucleons have also been investigated in both SNM and neutronic matter, identified from singularities in the $g$ matrix at starting energies below particle-particle threshold energy.",
"In this work we obtain results consistent to those reported in Ref.",
"[7], but not restricted anymore to AV18 in SNM.",
"Bound states are identified at sub-saturation densities, with deuterons in phase I bound at energies nearly twice that in free space.",
"Deuterons in phase II reach maximum binding at $k_F\\approx 0.7$ fm$^{-1}$ , with binding energies comparable to that in free space.",
"These in-medium bound states get dissolved for $k_F\\gtrsim 1.3$ fm$^{-1}$ , for all the interactions considered.",
"Di-nucleons in channel $^1\\textrm {S}_0$ are found in both, SNM and neutronic matter.",
"Their binding is much weaker to that for deuterons, reaching deepest values between $-\\!700$ and $-\\!500$ keV.",
"In this particular channel di-nucleons get dissolved at Fermi momenta above 1.1 MeV, feature shared by all interactions considered.",
"Overall, the binding properties of di-nucleons appear quite comparable, pointing also to their robustness under the interaction being considered.",
"The occurrence of di-nucleons in nuclear matter is closely related to nuclear pairing phenomena, mechanism responsible for the formation of Cooper pairs and the emergence of superfluid and superconducting states of matter [28], [29].",
"This aspect, addressed to some extent in Ref.",
"[7], has been omitted here since there would be no substantial new information.",
"On this regard we have checked that all interactions behave very similar to AV18 potential.",
"At a more basic level, in this work we have investigated the degree of fulfillment of Hugenholtz-van Hove theorem, finding that BHF approach alone fails considerably.",
"However, studies reported in Refs.",
"[25], [26], [27].",
"point that inclusion of higher order terms in the series expansion would remedy this limitation.",
"Efforts to include rearrangement corrections are underway.",
"F.I.",
"thanks CONICYT fellowship Beca Nacional, Contract No.",
"221320081.",
"This work was supported in part by STFC through Grants ST/I005528/1, ST/J005743/1 and ST/L005816/1.",
"Partial support comes from “NewCompStar”, COST Action MP1304."
]
] | 1606.04982 |
[
[
"Admissible Banach function spaces for linear dynamics with nonuniform\n behavior on the half-line"
],
[
"Abstract For nonuniform exponentially bounded evolution families on the half-line we introduce a class of Banach function spaces on which we define nonuniform evolution semigroups.",
"We completely characterize nonuniform exponential stability in terms of invertibility of the corresponding generators.",
"We emphasize that in particular our results apply to all linear differential equations with bounded operator and finite Lyapunov exponent."
],
[
"Introduction",
"A linear dynamics is called well-posed if we assume the existence, uniqueness and continuous dependence of solutions on initial data.",
"In the case of the nonautonomous equation $dx/dt=A\\left( t\\right) x$ , well-posedness is equivalent to the existence of an evolution family that solves the equation (see, for instance, Proposition 9.3 in [7], p. 478), that is a family of bounded linear operators $\\mathcal {U}$ $=\\left\\lbrace U(t,s)\\right\\rbrace _{t\\ge s\\ge 0}$ acting on the underlying Banach space $X$ , with properties: $U(t,t)=Id$ , $t\\ge 0$ ; $U(t,\\tau )U(\\tau ,s)=U(t,s)$ , $t\\ge \\tau \\ge s\\ge 0$ ; The map $(t,s)\\mapsto U(t,s)x$ is continuous for each $x\\in X$ .",
"If the linear operators $A\\left( t\\right) $ are bounded, then well-posedness is guaranteed [6].",
"In the autonomous case, the equation $dx/dt=Ax$ is well-posed if and only if the linear operator $A$ generates a $C_{0}$ -semigroup (Corollary 6.9 in [7], p. 151).",
"We recall that a family of bounded linear operators $\\mathcal {T}=\\left\\lbrace T(t)\\right\\rbrace _{t\\ge 0}$ acting on $X$ is said to be a $C_{0}$ -semigroup if $T(0)={Id}$ ; $T(t)T(s)=T(t+s)$ for $t,s\\ge 0$ ; $\\lim \\limits _{t\\rightarrow 0_{+}}T(t)x=x$ for every $x\\in X$ .",
"The (closed and densely defined) linear operator $G:D(G)\\subset X\\rightarrow X$ defined by $Gx=\\lim \\limits _{t\\rightarrow 0_{+}}\\frac{T(t)x-x}{t}$ is called the (infinitesimal) generator of the $C_{0}$ -semigroup $\\mathcal {T}$ .",
"For more details on the theory of $C_{0}$ -semigroups we refer the reader to [7], [10].",
"In the particular case of linear dynamics, the concept of uniform asymptotic stability (in the sense of Lyapunov) translates as uniform exponential stability.",
"The evolution family $\\mathcal {U}$ is called uniform exponentially stable if $\\left\\Vert U(t,s)\\right\\Vert \\le Me^{-\\alpha \\left(t-s\\right) }$ , $t\\ge s\\ge 0$ , for some constants $M$ , $\\alpha >0$ .",
"Under such hypothesis it is possible to introduce a $C_{0}$ -semigroup on some Banach function spaces, basically defined as $(T(t)u)(s)={\\left\\lbrace \\begin{array}{ll}U(s,s-t)u(s-t), & \\text{ if }s>t,\\\\U(s,0)u(0), & \\text{ if }0\\le s\\le t,\\end{array}\\right.",
"}$ and it is called the (uniform) evolution semigroup.",
"This construction reduces the study of a nonautonomous equation to the analysis of an autonomous one in the form $dy/dt=Gy$ , where $G$ is the generator of the evolution semigroup.",
"Thus, asymptotic behavior of an evolution family can be essentially characterized in terms of spectral properties of the generator of the corresponding evolution semigroup.",
"The clue is connecting the generator to a special integral equation.",
"For more details on the theory of (classical) evolution semigroups we mention in particular the complete and self-contained monograph of C. Chicone and Yu.",
"Latushkin [5] and the paper of Nguyen Van Minh, F. Räbiger and R. Schnaubelt [9].",
"The theory of uniform behavior is too restrictive, and it is important to consider a more general view, such as nonuniform asymptotic stability.",
"A serious motivation for weakening the notion of uniform exponential behavior lies in the ergodic theory.",
"In this regard, a consistent contribution is due to Ya.",
"Pesin, L. Barreira and C. Valls (we refer the reader to monographs [1], [3] and the references therein).",
"Roughly speaking, while uniformity relates to the finiteness of the Bohl exponents [6], nonuniformity analyses situations when the Lyapunov exponent is finite [3].",
"This type of argumentation is not at all of a formal type, as illustrated in our examples.",
"The orbits of the evolution family in our Example REF are all asymptotically stable, even if the evolution family is not uniform exponentially bounded.",
"In this case it is impossible to construct the evolution semigroup.",
"From another hand, Example REF (Perron) delivers a uniform exponentially bounded evolution family which is not uniform exponentially stable, with all the orbits being asymptotically stable.",
"In this case the evolution semigroup exists, but it does not furnish any kind of information about the (nonuniform) asymptotic behavior of the orbits.",
"To sum up, in many situation the classical tool either does not exist or it is completely useless.",
"Our main goal is to emphasize an important difference between uniform and nonuniform behavior: while in the uniform case all the evolution families reflect into a unique output function space (of functions vanishing at 0 and infinity, for instance), in the nonuniform case there are infinitely many output function spaces, which are the admissible ones.",
"They depend on each evolution family, and on each particular admissible exponent.",
"Let us point out that the theory of evolution semigroups and that of admissibility are quite similar.",
"In fact, admissibility methods deal with couples of Banach function spaces, while in the case of evolution semigroups the spaces into each couple coincide.",
"In this view, we prefer to use the term “admissible” throughout our paper.",
"For a better understanding, we refer the reader to [8], [11] for uniform behavior, and to [4], [13] for the nonuniform setting.",
"We emphasize that the idea of connecting the nonuniform behavior to $C_{0}$ -semigroups is already present in the recent paper [2], in the case of discrete families.",
"This paper is organized as follows.",
"In Section 2 we introduce the set of admissible exponents associated to a nonuniform exponentially bounded evolution family, and the class of the corresponding Banach function spaces that we also call admissible.",
"We emphasize the connections with the conditions on the finiteness of the corresponding Bohl and Lyapunov exponents.",
"In Section 3 we generalize the notion of evolution semigroup for linear dynamics with nonuniform behavior, by shrinking the output space.",
"We call this mathematical object a nonuniform evolution semigroup.",
"Moreover, we point out the connection with the uniform case.",
"The last section presents an introspective study of nonuniformity via evolution semigroups.",
"We characterize nonuniform exponential stability in terms of invertibility of the corresponding infinitesimal generators.",
"In this regard, we introduce a new notion that we call quasi-negative exponent, specific to nonuniform behavior.",
"As a consequence, we prove a spectral mapping theorem for nonuniform evolution semigroups.",
"We consider as our main result the statement in Theorem REF ."
],
[
"Admissible Banach function spaces",
"Throughout our paper $X$ is a Banach space.",
"We denote $C(\\mathbb {R}_{+},X)$ the space of all continuous, $X$ -valued functions defined on the half-line, and $C_{c,0}(\\mathbb {R}_{+},X)$ is the space of all functions in $C(\\mathbb {R}_{+},X)$ with compact support vanishing at 0.",
"We also make use of the following notation: $C_{00}(\\mathbb {R}_{+},X)=\\left\\lbrace u\\in C(\\mathbb {R}_{+},X):\\,\\lim \\limits _{t\\rightarrow \\infty }u(t)=u(0)=0\\right\\rbrace \\text{.",
"}$ Definition 2.1 For any fixed $\\alpha \\in \\mathbb {R}$ , the evolution family $\\mathcal {U}$ is called $\\alpha $ -nonuniform exponentially bounded, if there exists a continuous map $M_{\\alpha }:\\mathbb {R}_{+}\\rightarrow (0,\\infty )$ such that $\\parallel U(t,s)\\parallel \\le M_{\\alpha }(s)e^{\\alpha (t-s)}\\text{, }t\\ge s\\ge 0\\text{.",
"}$ If the above estimation holds for some $\\alpha <0$ , then $\\mathcal {U}$ is called $\\alpha $ -nonuniform exponentially stable.",
"Each $\\alpha $ satisfying (REF ) is called an admissible exponent, and the set of all admissible exponents is denoted $\\mathcal {A}\\left(\\mathcal {U}\\right) $ .",
"Evidently for each evolution family $\\mathcal {U}$ , the set $\\mathcal {A}\\left(\\mathcal {U}\\right) $ is either a (semi) infinite interval, or empty.",
"If $\\mathcal {A}\\left( \\mathcal {U}\\right) \\ne \\emptyset $ , then the evolution family $\\mathcal {U}$ is called nonuniform exponentially bounded, and if $\\mathcal {A}\\left( \\mathcal {U}\\right) $ contains negative admissible exponents, we say that $\\mathcal {U}$ is nonuniform exponentially stable.",
"We emphasize that our paper is devoted to the study of the nonuniform exponentially bounded evolution families, that is we only consider the case $\\mathcal {A}\\left( \\mathcal {U}\\right) \\ne \\emptyset $ .",
"In the above terminology, whenever there exists a bounded map $M_{\\alpha }(s)$ satisfying (REF ) (which is equivalent to the existence of a constant one), we just replace the term “nonuniform” with “uniform”.",
"Also, in such cases we call $\\alpha $ a strict (admissible) exponent, and we denote $\\mathcal {A}_{s}\\left( \\mathcal {U}\\right) $ the set of all strict exponents.",
"Remark 2.2 Assume that the evolution family $\\mathcal {U}$ is reversible (i.e.",
"$U(t,s)$ is invertible for all $t\\ge s\\ge 0$ , $U(s,t)$ denotes its inverse).",
"If the Lyapunov exponent $K_{L}$ is finite and not attained, then $\\mathcal {A}\\left( \\mathcal {U}\\right) =\\left( K_{L},\\infty \\right) $ (in particular $K_{L}=-\\infty $ whenever $\\mathcal {A}\\left( \\mathcal {U}\\right)=\\mathbb {R}$ ).",
"Also if the Bohl exponent $K_{B}$ is finite and not attained, then $\\mathcal {A}_{s}\\left( \\mathcal {U}\\right) =\\left( K_{B},\\infty \\right)$ .",
"The intervals of admissibility are closed at their left endpoints whenever the Lyapunov or the Bohl exponents are attained.",
"Indeed, as $K_{L}=\\inf \\left\\lbrace \\alpha \\in \\mathbb {R}:\\,\\text{there exists }M_{\\alpha }>0\\text{ with }\\left\\Vert U\\left( t,0\\right) \\right\\Vert \\le M_{\\alpha }e^{\\alpha t}\\text{, }t\\ge 0\\right\\rbrace $ and is not attained, if $\\alpha \\in \\mathcal {A}\\left( \\mathcal {U}\\right) $ , replacing $s=0$ in (REF ), one has $\\parallel U(t,0)\\parallel \\le M_{\\alpha }(0)e^{\\alpha t},$ thus $\\alpha \\in (K_{L},\\infty )$ .",
"For $\\alpha \\in (K_{L},\\infty )$ , assuming that $\\mathcal {U}$ is reversible, we get $\\left\\Vert U(t,s)\\right\\Vert =\\left\\Vert U(t,0)U(0,s)\\right\\Vert \\le M_{\\alpha }e^{\\alpha t}\\left\\Vert U(0,s)\\right\\Vert =M_{\\alpha }e^{\\alpha s}\\left\\Vert U(0,s)\\right\\Vert e^{\\alpha (t-s)},$ that implies $\\alpha \\in \\mathcal {A}\\left( \\mathcal {U}\\right) $ .",
"The second statement can be proved similarly if we notice that $K_{B}=\\inf \\left\\lbrace \\alpha \\in \\mathbb {R}:\\,\\text{there exists }M_{\\alpha }>0\\text{ with }\\left\\Vert U\\left( t,s\\right) \\right\\Vert \\le M_{\\alpha }e^{\\alpha (t-s)},\\;t\\ge s\\ge 0\\right\\rbrace .$ Suppose that $\\mathcal {A}\\left( \\mathcal {U}\\right) \\ne \\emptyset $ , and let $\\alpha \\in \\mathcal {A}\\left( \\mathcal {U}\\right) $ .",
"For each $t\\ge 0$ and $u\\in {C}(\\mathbb {R}_{+},X)$ we set $\\varphi _{\\mathcal {U},\\alpha }(t,u)=\\underset{\\tau \\ge t}{\\sup }\\text{}e^{-\\alpha (\\tau -t)}\\parallel U(\\tau ,t)u(t)\\parallel .",
"$ Inequality (REF ) implies $\\parallel u(t)\\parallel \\le \\varphi _{\\mathcal {U},\\alpha }(t,u)\\le M_{\\alpha }(t)\\parallel u(t)\\parallel ,\\;t\\ge 0\\text{.}",
"$ If in particular $u(t)\\equiv x$ for some $x\\in X$ , we step over the norm on $X$ defined in [4], precisely $\\left\\Vert x\\right\\Vert _{t}=\\underset{\\tau \\ge t}{\\sup }$ $e^{-\\alpha (\\tau -t)}\\parallel U(\\tau ,t)x\\parallel $ .",
"In this regard, we notice that the map $\\varphi _{\\mathcal {U},\\alpha }$ in (REF ) can be (indirectly) defined as $\\varphi _{\\mathcal {U},\\alpha }(t,u)=\\left\\Vert u(t)\\right\\Vert _{t}$ .",
"The following result is essential in the sequel.",
"Proposition 2.3 The map $\\mathbb {R}_{+}\\ni t\\mapsto \\varphi _{\\mathcal {U},\\alpha }(t,u)\\in \\mathbb {R}_{+}$ is continuous for each fixed $\\alpha \\in \\mathcal {A}\\left( \\mathcal {U}\\right) $ and $u\\in {C}(\\mathbb {R}_{+},X)$ .",
"In addition, for each $u\\in {C}(\\mathbb {R}_{+},X)$ for which $\\lim \\limits _{t\\rightarrow \\infty }\\varphi _{\\mathcal {U},\\alpha }(t,u)=0$ , there exists (possibly not unique) $t_{u}\\ge 0$ such that $\\sup \\limits _{t\\ge 0}\\varphi _{\\mathcal {U},\\alpha }(t,u)=\\varphi _{\\mathcal {U},\\alpha }(t_{u},u).$ To prove the first statement we set $V(t,s)=e^{-\\alpha (t-s)}U(t,s)$ .",
"It follows that $\\mathcal {V}=\\left\\lbrace V(t,s)\\right\\rbrace _{t\\ge s\\ge 0}$ is also an evolution family with $\\parallel V(t,s)\\parallel \\le M_{\\alpha }(s)$ , $t\\ge s\\ge 0$ .",
"For fixed $u\\in {C}(\\mathbb {R}_{+},X)$ , $t_{0}\\ge 0$ and $\\varepsilon >0$ , there exists $\\delta _{1}$ , $\\delta _{2}>0$ such that $\\left|t-t_{0}\\right|<\\delta _{1}\\Rightarrow M_{\\alpha }\\left(t\\right) \\left\\Vert u\\left( t\\right) -u\\left( t_{0}\\right) \\right\\Vert <{\\varepsilon }/{3}\\text{,}$ $t_{0}-\\delta _{2}<t_{0}\\le t <t_{0}+\\delta _{2} \\Rightarrow M_{\\alpha }\\left(t\\right) \\left\\Vert u\\left( t_{0}\\right) -V\\left( t,t_{0}\\right) u\\left(t_{0}\\right) \\right\\Vert <{\\varepsilon }/{3}\\text{.",
"}$ Let $\\delta =\\max \\left\\lbrace \\delta _{1},\\delta _{2}\\right\\rbrace $ and choose $t\\ge 0$ with $\\left|t-t_{0}\\right|<\\delta $ .",
"We only analyze the case $t\\ge t_{0}$ .",
"For any $\\tau \\ge t$ we have $\\left\\Vert V(\\tau ,t)u\\left( t\\right) \\right\\Vert & \\le \\left\\Vert V(\\tau ,t)\\left( u\\left( t\\right) -u\\left( t_{0}\\right) \\right)\\right\\Vert +\\left\\Vert V(\\tau ,t)u\\left( t_{0}\\right) -V(\\tau ,t_{0})u\\left(t_{0}\\right) \\right\\Vert \\\\& \\quad +\\left\\Vert V(\\tau ,t_{0})u\\left( t_{0}\\right) \\right\\Vert \\\\& \\le \\left\\Vert V(\\tau ,t)\\right\\Vert \\left\\Vert u\\left( t\\right) -u\\left(t_{0}\\right) \\right\\Vert +\\left\\Vert V(\\tau ,t)\\right\\Vert \\left\\Vert u\\left(t_{0}\\right) -V\\left( t,t_{0}\\right) u\\left( t_{0}\\right) \\right\\Vert \\\\& \\quad +\\left\\Vert V(\\tau ,t_{0})u\\left( t_{0}\\right) \\right\\Vert \\\\& \\le M_{\\alpha }(t) \\left\\Vert u\\left( t\\right) -u\\left( t_{0}\\right)\\right\\Vert +M_{\\alpha }(t) \\left\\Vert u\\left( t_{0}\\right) -V\\left(t,t_{0}\\right) u\\left( t_{0}\\right) \\right\\Vert \\\\& \\quad +\\left\\Vert V(\\tau ,t_{0})u\\left( t_{0}\\right) \\right\\Vert \\\\& \\le (2\\varepsilon )/3+\\varphi _{\\mathcal {U},\\alpha }(t_{0},u).$ When taking the supremum with respect to $\\tau \\ge t$ , we get $\\varphi _{\\mathcal {U},\\alpha }(t,u)-\\varphi _{\\mathcal {U},\\alpha }(t_{0},u)<\\varepsilon .$ Similarly one can prove that $\\varphi _{\\mathcal {U},\\alpha }(t_{0},u)-\\varphi _{\\mathcal {U},\\alpha }(t,u)<\\varepsilon $ , that is the map $t\\mapsto \\varphi _{\\mathcal {U},\\alpha }(t,u)$ is continuous at $t_{0}$ .",
"The second statement follows from the continuity of the map in question, together with condition $\\lim \\limits _{t\\rightarrow \\infty }\\varphi _{\\mathcal {U},\\alpha }(t,u)=0$ .",
"For each $\\alpha \\in \\mathcal {A}\\left( \\mathcal {U}\\right) $ we set $\\mathcal {C}(\\mathcal {U},\\alpha )=\\left\\lbrace u\\in C(\\mathbb {R}_{+},X):\\,\\lim \\limits _{t\\rightarrow \\infty }\\varphi _{\\mathcal {U},\\alpha }(t,u)=\\varphi _{\\mathcal {U},\\alpha }(0,u)=0\\right\\rbrace .$ It is a kind of straightforward argument to verify that $\\mathcal {C}(\\mathcal {U},\\alpha )$ is a Banach (function) space equipped with the norm $\\parallel u\\parallel _{\\mathcal {U},\\alpha }=\\sup \\limits _{t\\ge 0}\\text{ }\\varphi _{\\mathcal {U},\\alpha }(t,u),$ called the admissible Banach function space corresponding to the evolution family $\\mathcal {U}$ and the admissible exponent $\\alpha \\in \\mathcal {A}(\\mathcal {U})$ .",
"Eq.",
"(REF ) also implies $C_{c,0}(\\mathbb {R}_{+},X)\\subset \\mathcal {C}(\\mathcal {U},\\alpha )\\subset C_{00}(\\mathbb {R}_{+},X).",
"$ Let us remark that if $\\alpha \\in \\mathcal {A}\\left( \\mathcal {U}\\right) $ , $u\\in C(\\mathbb {R}_{+},X)$ and $\\beta \\ge \\alpha $ , then $\\beta \\in \\mathcal {A}\\left( \\mathcal {U}\\right) $ , $\\varphi _{\\mathcal {U},\\beta }(t,u)\\le \\varphi _{\\mathcal {U},\\alpha }(t,u)$ .",
"Moreover, $\\mathcal {C}(\\mathcal {U},\\alpha )\\subset \\mathcal {C}\\left( \\mathcal {U},\\beta \\right) \\text{ and }\\parallel u\\parallel _{\\mathcal {U},\\beta }\\le \\parallel u\\parallel _{\\mathcal {U},\\alpha }.$"
],
[
"Nonuniform evolution semigroups",
"In this section we introduce the concept of nonuniform evolution semigroup, emphasizing the connections with the uniform case.",
"Theorem 3.1 Each $\\alpha \\in \\mathcal {A}\\left( \\mathcal {U}\\right) $ defines a $C_{0}$ -semigroup $\\mathcal {T}_{\\alpha }=\\left\\lbrace T_{\\alpha }(t)\\right\\rbrace _{t\\ge 0}$ on $\\mathcal {C}(\\mathcal {U},\\alpha )$ by setting $(T_{\\alpha }(t)u)(s)={\\left\\lbrace \\begin{array}{ll}U(s,s-t)u(s-t)\\text{,} & \\text{if }s>t\\text{,}\\\\0\\text{,} & \\text{if }0\\le s\\le t\\text{.}\\end{array}\\right.",
"}$ Moreover, the following estimation holds $\\parallel T_{\\alpha }(t)u\\parallel _{\\mathcal {U},\\alpha }\\le e^{\\alpha t}\\parallel u\\parallel _{\\mathcal {U},\\alpha }\\text{, }u\\in \\mathcal {C}(\\mathcal {U},\\alpha )\\text{.}",
"$ Evidently $T_{\\alpha }(0)={Id}$ and $T_{\\alpha }(t)T_{\\alpha }(s)=T_{\\alpha }(t+s)$ , for all $t,s\\ge 0$ .",
"It remains to prove that the map $T_{\\alpha }(t)$ in (REF ) is well defined on $\\mathcal {C}(\\mathcal {U},\\alpha )$ , and the norm $\\parallel T_{\\alpha }(t)u-u\\parallel _{\\mathcal {U},\\alpha }\\rightarrow 0$ as $t\\rightarrow 0_{+}$ , for each fixed $u\\in \\mathcal {C}(\\mathcal {U},\\alpha )$ .",
"Pick $u\\in \\mathcal {C}(\\mathcal {U},\\alpha )$ and $t\\ge 0$ .",
"For arbitrary $s\\ge t$ we have $\\varphi _{\\mathcal {U},\\alpha }(s,T_{\\alpha }(t)u) & =\\sup \\limits _{\\tau \\ge s}e^{-\\alpha (\\tau -s)}\\parallel U(\\tau ,s-t)u(s-t)\\parallel \\\\& =e^{\\alpha t}\\sup \\limits _{\\tau \\ge s}e^{-\\alpha [\\tau -(s-t)]}\\parallel U(\\tau ,s-t)u(s-t)\\parallel \\\\& \\le e^{\\alpha t}\\varphi _{\\mathcal {U},\\alpha }(s-t,u),$ therefore $\\varphi _{\\mathcal {U},\\alpha }(s,T_{\\alpha }(t)u)\\rightarrow 0$ as $s\\rightarrow \\infty $ , which leads to $T_{\\alpha }(t)u\\in \\mathcal {C}(\\mathcal {U},\\alpha )$ .",
"Notice that the above estimation also proves inequality (REF ).",
"We now show that the space $C_{c,0}(\\mathbb {R}_{+},X)$ is dense in $\\mathcal {C}(\\mathcal {U},\\alpha )$ with respect to the norm $\\parallel \\cdot \\parallel _{\\mathcal {U},\\alpha }$ .",
"For any fixed $u\\in \\mathcal {C}(\\mathcal {U},\\alpha )$ and any non-negative integer $n\\in \\mathbb {N}$ , let us consider a continuous function $\\alpha _{n}:\\mathbb {R}_{+}\\rightarrow [0,1]$ such that $\\alpha _{n}(t)=1,\\text{ for all }t\\in [0,n],\\text{ and }\\alpha _{n}(t)=0,\\text{ for all }t\\ge n+1.$ Putting $u_{n}=\\alpha _{n}u$ , we notice that $u_{n}\\in C_{c,0}(\\mathbb {R}_{+},X)$ .",
"We claim that the limit $\\underset{n\\rightarrow \\infty }{\\lim }\\parallel u_{n}-u\\parallel _{\\mathcal {U},\\alpha }=0$ .",
"Indeed, as $\\,\\lim \\limits _{t\\rightarrow \\infty }\\varphi _{\\mathcal {U},\\alpha }(t,u)=0$ , it follows that for each $\\varepsilon >0$ there exists $\\delta >0$ such that for $t>\\delta $ we have $\\varphi _{\\mathcal {U},\\alpha }(t,u)<\\varepsilon /2$ .",
"Set $n_{0}=[\\delta ]+1$ and choose $n\\ge n_{0}$ .",
"The definition of map $\\alpha _{n}$ readily implies $\\parallel u_{n}-u\\parallel _{\\mathcal {U},\\alpha }=\\sup \\limits _{t\\ge n}\\varphi _{\\mathcal {U},\\alpha }(t,u_{n}-u)\\le \\sup \\limits _{t\\ge n}\\varphi _{\\mathcal {U},\\alpha }(t,u)<\\varepsilon ,$ which concludes the claim.",
"For the second statement, pick $u\\in C_{c,0}(\\mathbb {R}_{+},X)$ .",
"There exist $a$ , $b\\ge 0$ , $a<b$ such that $supp(T_{\\alpha }(t)u-u)\\subset [a,b]$ , for sufficiently small $t\\ge 0$ .",
"For such $t$ we have $\\parallel T_{\\alpha }(t)u-u\\parallel _{\\mathcal {U},\\alpha } & =\\sup \\limits _{s\\ge 0}\\varphi _{\\mathcal {U},\\alpha }(s,T_{\\alpha }(t)u-u)\\\\& \\le \\sup \\limits _{s\\ge 0}M_{\\alpha }(s)\\parallel (T_{\\alpha }(t)u)(s)-u(s)\\parallel \\\\& =\\sup \\limits _{s\\in supp(T_{\\alpha }(t)u-u)}M_{\\alpha }(s)\\parallel (T_{\\alpha }(t)u)(s)-u(s)\\parallel \\\\& \\le K_{\\alpha }\\sup \\limits _{s\\in supp(T_{\\alpha }(t)u-u)}\\parallel (T_{\\alpha }(t)u)(s)-u(s)\\parallel ,$ where $K_{\\alpha }=\\max \\limits _{s\\in [a,b]}M_{\\alpha }(s)$ .",
"Using standard arguments (ex.",
"[12]), one can easily prove that $\\parallel T_{\\alpha }(t)u-u\\parallel _{\\mathcal {U},\\alpha }\\rightarrow 0$ as $t\\rightarrow 0_{+}$ , and this completes the proof.",
"The $C_{0}$ -semigroup $\\mathcal {T}_{\\alpha }$ defined above is called the nonuniform evolution semigroup associated to the evolution family $\\mathcal {U}$ and the admissible exponent $\\alpha $ .",
"We denote $G_{\\mathcal {U},\\alpha }$ its generator.",
"Remark 3.2 If the Banach function spaces $\\mathcal {C}(\\mathcal {U},\\alpha )$ and $\\mathcal {C}(\\mathcal {U},\\beta )$ coincide for some admissible exponents $\\alpha ,\\beta \\in \\mathcal {A}(\\mathcal {U})$ , then $\\mathcal {T}_{\\alpha }=\\mathcal {T}_{\\beta }$ and $G_{\\mathcal {U},\\alpha }=G_{\\mathcal {U},\\beta }$ .",
"While in the uniform case an evolution family defines a unique evolution semigroup $\\mathcal {T}$ on $C_{00}(\\mathbb {R}_{+},X)$ , in the nonuniform case there are infinitely many nonuniform evolution semigroups $\\mathcal {T}_{\\alpha }$ defined on $\\mathcal {C}(\\mathcal {U},\\alpha )$ , $\\alpha \\in \\mathcal {A}(\\mathcal {U})$ .",
"The next proposition emphasizes the connection between the concepts of uniform and nonuniform evolution semigroup.",
"Proposition 3.3 For $\\alpha \\in \\mathcal {A}\\left( \\mathcal {U}\\right) $ , the Banach spaces $\\mathcal {C}(\\mathcal {U},\\alpha )$ and $C_{00}(\\mathbb {R}_{+},X)$ coincide if and only if $\\alpha $ is a strict exponent.",
"In this case, the nonuniform evolution semigroup $\\mathcal {T}_{\\alpha }$ coincides with the (uniform) evolution semigroup $\\mathcal {T}$ on $C_{00}(\\mathbb {R}_{+},X)$ .",
"Necessity.",
"Let us assume that $\\mathcal {C}(\\mathcal {U},\\alpha )=C_{00}(\\mathbb {R}_{+},X)$ , for some $\\alpha \\in \\mathcal {A}\\left(\\mathcal {U}\\right) $ .",
"If $\\alpha $ is not a strict exponent, then for each positive integer $n\\in \\mathbb {N}^{\\ast }$ there exist $t_{n}\\ge s_{n}\\ge 0$ and $x_{n}\\in X$ with $\\parallel x_{n}\\parallel =1$ , such that $\\parallel U(t_{n},s_{n})x_{n}\\parallel >ne^{\\alpha \\left( t_{n}-s_{n}\\right)}.$ If the sequence $\\left\\lbrace s_{n}\\right\\rbrace _{n\\in \\mathbb {N}^{\\ast }}$ is bounded, say $s_{n}\\le k$ , then inequality $n<e^{-\\alpha \\left( t_{n}-s_{n}\\right) }\\parallel U(t_{n},s_{n})x_{n}\\parallel \\le M_{\\alpha }\\left( s_{n}\\right) \\le \\underset{0\\le s\\le k}{\\sup }M_{\\alpha }\\left( s\\right)$ leads to a contradiction, $n<\\underset{0\\le s\\le k}{\\sup }M_{\\alpha }\\left(s\\right) $ for all $n\\in \\mathbb {N}^{\\ast }$ .",
"Thus, without loss of generality one can always assume that the sequence $\\left\\lbrace s_{n}\\right\\rbrace _{n\\in \\mathbb {N}^{\\ast }}$ is strictly increasing, unbounded and let us put $t_{0}=s_{0}=0$ , $x_{0}=0$ .",
"Setting $y_{n}=\\frac{x_{n}}{\\sqrt{n}}$ , $n\\ge 1$ and $y_{0}=0$ , one gets $e^{-\\alpha \\left( t_{n}-s_{n}\\right) }\\parallel U(t_{n},s_{n})y_{n}\\parallel \\ge \\sqrt{n}\\text{, }n\\in \\mathbb {N}\\text{.",
"}\\ $ Consider $u_{y}:\\mathbb {R}_{+}\\rightarrow X$ by $u_{y}\\left( s\\right) =\\frac{s\\left( y_{n+1}-y_{n}\\right) }{s_{n+1}-s_{n}}+\\frac{s_{n+1}y_{n}-s_{n}y_{n+1}}{s_{n+1}-s_{n}},\\text{ if }s\\in \\left[s_{n},s_{n+1}\\right] ,n\\in \\mathbb {N}\\text{.",
"}$ We notice that $u_{y}\\left( s_{n}\\right) =$ $y_{n}$ for each $n\\in \\mathbb {N}$ , that results in $u_{y}\\in C_{00}(\\mathbb {R}_{+},X)$ .",
"From estimation $\\varphi _{\\mathcal {U},\\alpha }(s_{n},u_{y})\\ge e^{-\\alpha \\left( t_{n}-s_{n}\\right) }\\parallel U(t_{n},s_{n})y_{n}\\parallel \\ge \\sqrt{n},$ we deduce $u_{y}\\notin \\mathcal {C}(\\mathcal {U},\\alpha )$ , that is $\\mathcal {C}(\\mathcal {U},\\alpha )\\ne C_{00}(\\mathbb {R}_{+},X)$ , which is a contradiction.",
"Sufficiency.",
"If $\\alpha $ is a strict exponent, then there exists $M_{\\alpha }>0$ such that $\\parallel U(t,s)\\parallel \\le M_{\\alpha }e^{\\alpha (t-s)}$ for $t\\ge s\\ge 0$ .",
"Assumption $u\\in C_{00}(\\mathbb {R}_{+},X)$ implies $\\varphi _{\\mathcal {U},\\alpha }(t,u)\\le M_{\\alpha }\\parallel u(t)\\parallel \\rightarrow 0$ as $t\\rightarrow \\infty $ , hence $u\\in \\mathcal {C}(\\mathcal {U},\\alpha )$ .",
"The following result is taken from the theory of $C_{0}$ -semigroups (see [10] or [7]): Lemma 3.4 Let $\\mathcal {T}=\\left\\lbrace T(t)\\right\\rbrace _{t\\ge 0}$ be a $C_{0}$ -semigroup on a Banach space $E$ , and $G$ its infinitesimal generator.",
"If $x,y\\in E$ , then $x\\in D(G)$ and $Gx=y$ if and only if $T(t)x-x=\\int _{0}^{t}T(\\xi )yd\\xi ,\\;t\\ge 0.$ Let us substitute $E=\\mathcal {C}(\\mathcal {U},\\alpha )$ , $x=u$ , $y=-f$ and $G=G_{\\mathcal {U},\\alpha }$ .",
"This rewrites as follows: if $u$ , $f\\in \\mathcal {C}(\\mathcal {U},\\alpha )$ , then $u\\in D(G_{\\mathcal {U},\\alpha })$ and $G_{\\mathcal {U},\\alpha }u=-f$ if and only if $u(t)=\\int _{0}^{t}U(t,\\xi )f(\\xi )d\\xi \\text{, }t\\ge 0\\text{.}",
"$ We notice that Lemma 1.1 in [9] uses the same source (Lemma REF ) to present its similar conclusions."
],
[
"Criteria for nonuniform exponential stability",
"Lemma 4.1 If there exists $\\alpha \\in \\mathcal {A}(\\mathcal {U})$ , $\\alpha <0$ (i.e.",
"$\\mathcal {U}$ is $\\alpha $ -nonuniform exponentially stable), then the generator $G_{\\mathcal {U},\\alpha }$ is invertible and $\\left\\Vert G_{\\mathcal {U},\\alpha }^{-1}f\\right\\Vert _{\\mathcal {U},\\alpha }\\le -\\frac{1}{\\alpha }\\left\\Vert f\\right\\Vert _{\\mathcal {U},\\alpha }.$ For each $f\\in \\mathcal {C}(\\mathcal {U},\\alpha )$ we define $u_{f}(t)=\\int _{0}^{t}U(t,\\xi )f(\\xi )d\\xi \\text{, }t\\ge 0.$ From $f=0\\Rightarrow u_{f}=0$ we deduce that $G_{\\mathcal {U},\\alpha }$ is a one-to-one map on $\\mathcal {C}(\\mathcal {U},\\alpha )$ .",
"To prove that $G_{\\mathcal {U},\\alpha }$ is invertible, one needs to show first that $u_{f}\\in \\mathcal {C}(\\mathcal {U},\\alpha )$ for each $f\\in \\mathcal {C}(\\mathcal {U},\\alpha )$ .",
"From estimation $\\varphi _{\\mathcal {U},\\alpha }(t,u_{f}) & \\le \\sup \\limits _{\\tau \\ge t}\\text{}e^{-\\alpha (\\tau -t)}\\int _{0}^{t}\\parallel U(\\tau ,\\xi )f(\\xi )\\parallel d\\xi \\\\& \\le \\sup \\limits _{\\tau \\ge t}\\text{ }e^{-\\alpha (\\tau -t)}\\int _{0}^{t}e^{\\alpha (\\tau -\\xi )}\\varphi _{\\mathcal {U},\\alpha }(\\xi ,f)d\\xi \\\\& =\\int _{0}^{t}e^{\\alpha (t-\\xi )}\\varphi _{\\mathcal {U},\\alpha }(\\xi ,f)d\\xi \\text{,}$ as $\\lim \\limits _{t\\rightarrow \\infty }\\varphi _{\\mathcal {U},\\alpha }(t,f)=0$ , one gets $\\lim \\limits _{t\\rightarrow \\infty }\\varphi _{\\mathcal {U},\\alpha }(t,u_{f})=0$ , therefore $u_{f}\\in \\mathcal {C}(\\mathcal {U},\\alpha )$ .",
"We conclude that $G_{\\mathcal {U},\\alpha }$ is algebraically invertible and $-u_{f}=G_{\\mathcal {U},\\alpha }^{-1}f$ .",
"Furthermore, inequality $\\varphi _{\\mathcal {U},\\alpha }(t,u_{f})\\le \\int _{0}^{t}e^{\\alpha (t-\\xi )}\\varphi _{\\mathcal {U},\\alpha }(\\xi ,f)d\\xi $ yields $\\left\\Vert u_{f}\\right\\Vert _{\\mathcal {U},\\alpha }\\le \\left\\Vert f\\right\\Vert _{\\mathcal {U},\\alpha }\\int _{0}^{t}e^{\\alpha (t-\\xi )}d\\xi \\le -\\frac{1}{\\alpha }\\left\\Vert f\\right\\Vert _{\\mathcal {U},\\alpha }\\text{,}$ that is $G_{\\mathcal {U},\\alpha }^{-1}$ is bounded and estimation (REF ) holds.",
"Next Lemma is crucial in the sequel, and generalizes the implication $(ii)\\Rightarrow (i)$ from Theorem 2.2 in [9], for nonuniform exponentially bounded families.",
"Obviously our techniques, based on induction, are of a completely different type (otherwise the methods and constructions used in the bibliographic source we refer to cannot apply in the nonuniform setting).",
"Lemma 4.2 If the generator $G_{\\mathcal {U},\\alpha }$ is invertible for some $\\alpha \\in \\mathcal {A}\\left( \\mathcal {U}\\right) $ , then $\\mathcal {U}$ is nonuniform exponentially stable.",
"Without loss of generality we may assume that $\\alpha \\ge 0$ .",
"Suppose that the operator $G_{\\mathcal {U},\\alpha }:D(G_{\\mathcal {U},\\alpha })\\subset \\mathcal {C}(\\mathcal {U},\\alpha )\\rightarrow \\mathcal {C}(\\mathcal {U},\\alpha )$ is invertible and put $c=c(\\alpha )=\\parallel G_{\\mathcal {U},\\alpha }^{-1}\\parallel .$ For each positive integer $n\\in \\mathbb {N}^{\\ast }$ we denote $\\theta _{n}=\\ln \\frac{e^{n}}{e^{n}-1}\\rightarrow 0$ .",
"For fixed $t>s\\ge 0$ and $n$ large enough such that $s+\\theta _{n}\\le t\\le n$ , let us consider a continuous function $\\alpha _{n}:\\mathbb {R}_{+}\\rightarrow [0,1]$ with $\\alpha _{n}(\\xi )={\\left\\lbrace \\begin{array}{ll}0, & \\text{ if }0\\le \\xi \\le s,\\\\1, & \\text{ if }s+\\theta _{n}\\le \\xi \\le n,\\\\0, & \\text{ if }\\xi \\ge n+\\theta _{n}\\text{.}\\end{array}\\right.",
"}$ Step 1.",
"We prove that $\\parallel U(t,s)\\parallel \\le (c\\alpha +1)M_{\\alpha }(s),\\text{ for }t\\ge s\\ge 0\\text{.}",
"$ Evidently (REF ) holds for $\\alpha =0$ .",
"Assume now that $\\alpha >0$ and fix $t>s\\ge 0$ , and $x\\in X$ .",
"For $n$ large enough we define $f_{n}(\\xi )={\\left\\lbrace \\begin{array}{ll}\\alpha _{n}(\\xi )e^{-\\alpha (\\xi -s)}U(\\xi ,s)x\\text{,} & \\text{if }\\xi >s,\\\\0\\text{,} & \\text{if }0\\le \\xi \\le s\\text{.}\\end{array}\\right.",
"}$ We have $f_{n}\\in C_{c,0}(\\mathbb {R}_{+},X)$ and thus $f_{n}\\in \\mathcal {C}(\\mathcal {U},\\alpha )$ .",
"We claim that $\\parallel f_{n}\\parallel _{\\mathcal {U},\\alpha }\\le M_{\\alpha }(s)\\parallel x\\parallel .$ Indeed, if $\\xi \\le s$ , as $f_{n}(\\xi )=0$ , one has $\\varphi _{\\mathcal {U},\\alpha }(\\xi ,f_{n})=0$ , meanwhile for $\\xi >s$ the following estimation holds: $\\varphi _{\\mathcal {U},\\alpha }(\\xi ,f_{n}) & =\\underset{\\tau \\ge \\xi }{\\sup }\\text{ }e^{-\\alpha (\\tau -\\xi )}\\parallel U(\\tau ,s)x\\parallel \\alpha _{n}(\\xi )e^{-\\alpha (\\xi -s)}\\\\& \\le \\underset{\\tau \\ge \\xi }{\\sup }\\text{ }e^{-\\alpha (\\tau -s)}\\parallel U(\\tau ,s)x\\parallel \\le M_{\\alpha }(s)\\parallel x\\parallel .$ Putting $u_{n}=G_{\\mathcal {U},\\alpha }^{-1}(-f_{n})$ one gets $u_{n}(t) & =\\int _{0}^{t}U(t,\\xi )f_{n}(\\xi )d\\xi \\\\& =\\int _{s}^{s+\\theta _{n}}\\alpha _{n}(\\xi )e^{-\\alpha (\\xi -s)}d\\xi \\,U(t,s)x+\\int _{s+\\theta _{n}}^{t}\\alpha _{n}(\\xi )e^{-\\alpha (\\xi -s)}d\\xi \\,U(t,s)x\\\\& =I_{n}\\,U(t,s)x+\\frac{1}{\\alpha }\\left[ e^{-\\alpha \\theta _{n}}-e^{-\\alpha (t-s)}\\right] U(t,s)x\\text{.",
"}$ Here $I_{n}=\\int _{s}^{s+\\theta _{n}}\\alpha _{n}(\\xi )e^{-\\alpha (\\xi -s)}d\\xi $ .",
"Inequality $0\\le I_{n}\\le \\frac{1}{\\alpha }\\left( 1-e^{-\\alpha \\theta _{n}}\\right) $ leads to $\\frac{1}{\\alpha }e^{-\\alpha \\theta _{n}} & \\parallel U(t,s)x\\parallel \\\\& \\le \\parallel u_{n}(t)\\parallel +\\frac{1}{\\alpha }\\left( 1-e^{-\\alpha \\theta _{n}}\\right) \\parallel U(t,s)x\\parallel +\\frac{1}{\\alpha }e^{-\\alpha (t-s)}\\parallel U(t,s)x\\parallel \\\\& \\le \\parallel u_{n}\\parallel _{\\mathcal {U},\\alpha }+\\frac{1}{\\alpha }\\left(1-e^{-\\alpha \\theta _{n}}\\right) \\parallel U(t,s)x\\parallel +\\frac{1}{\\alpha }M_{\\alpha }(s)\\parallel x\\parallel \\\\& \\le c\\parallel f_{n}\\parallel _{\\mathcal {U},\\alpha }+\\frac{1}{\\alpha }\\left(1-e^{-\\alpha \\theta _{n}}\\right) \\parallel U(t,s)x\\parallel +\\frac{1}{\\alpha }M_{\\alpha }(s)\\parallel x\\parallel \\\\& \\le \\frac{c\\alpha +1}{\\alpha }M_{\\alpha }(s)\\parallel x\\parallel +\\frac{1}{\\alpha }\\left( 1-e^{-\\alpha \\theta _{n}}\\right) \\parallel U(t,s)x\\parallel \\text{.",
"}$ Now inequality (REF ) results immediately when letting $n\\rightarrow \\infty $ .",
"Step 2.",
"For all $k\\in \\mathbb {N}$ the following holds $\\parallel U(t,s)\\parallel \\le \\frac{c^{k}k!",
"}{(t-s)^{k}}(c\\alpha +1)M_{\\alpha }(s)\\text{, }t>s\\ge 0\\text{.}",
"$ Step 1 implies that inequality (REF ) holds for $k=0$ .",
"Assume that (REF ) holds for some $k\\in \\mathbb {N}$ .",
"For fixed $t>s\\ge 0$ , $x\\in X$ and sufficiently large $n$ we consider $g_{n,k}(\\xi )={\\left\\lbrace \\begin{array}{ll}\\alpha _{n}(\\xi )(\\xi -s)^{k}U(\\xi ,s)x\\text{,} & \\text{if }\\xi >s,\\\\0\\text{,} & \\text{if }0\\le \\xi \\le s\\text{.}\\end{array}\\right.",
"}$ Since $g_{n,k}\\in C_{c,0}(\\mathbb {R}_{+},X)$ , it follows that $g_{n,k}\\in \\mathcal {C}(\\mathcal {U},\\alpha )$ .",
"For $0\\le \\xi \\le s$ we notice that $\\varphi _{\\mathcal {U},\\alpha }(\\xi ,g_{n,k})=0$ , and if $\\xi >s$ we have $\\varphi _{\\mathcal {U},\\alpha }(\\xi ,g_{n,k}) & =\\sup \\limits _{\\tau \\ge \\xi }e^{-\\alpha (\\tau -\\xi )}\\parallel U(\\tau ,\\xi )g_{n,k}(\\xi )\\parallel \\\\& =\\sup \\limits _{\\tau \\ge \\xi }e^{-\\alpha (\\tau -\\xi )}\\alpha _{n}(\\xi )(\\xi -s)^{k}\\parallel U(\\tau ,s)x\\parallel \\\\& \\le \\sup \\limits _{\\tau \\ge \\xi }(\\tau -s)^{k}\\parallel U(\\tau ,s)x\\parallel \\\\& \\le c^{k}k!",
"(c\\alpha +1)M_{\\alpha }(s)\\parallel x\\parallel \\text{,}$ that results in $\\parallel g_{n,k}\\parallel _{\\mathcal {U},\\alpha }\\le c^{k}k!",
"(c\\alpha +1)M_{\\alpha }(s)\\parallel x\\parallel .$ If $u_{n,k}=G^{-1}(-g_{n,k})$ , then $u_{n,k}(t) & =\\int _{0}^{t}U(t,\\xi )g_{n,k}(\\xi )d\\xi \\\\& =I_{n,k}\\,U(t,s)x+\\int _{s+\\theta _{n}}^{t}(\\xi -s)^{k}d\\xi \\,U(t,s)x\\\\& =I_{n,k}\\,U(t,s)x+\\frac{1}{k+1}\\left[ (t-s)^{k+1}-\\theta _{n}^{k+1}\\right]U(t,s)x\\text{,}$ where $I_{n,k}=\\int _{s}^{s+\\theta _{n}}\\alpha _{n}(\\xi )(\\xi -s)^{k}d\\xi \\le \\frac{1}{k+1}\\theta _{n}^{k+1}$ .",
"Let us estimate $\\frac{(t-s)^{k+1}}{k+1} & \\,\\parallel U(t,s)x\\parallel \\\\& \\le \\parallel u_{n,k}(t)\\parallel +\\frac{1}{k+1}\\theta _{n}^{k+1}\\,\\parallel U(t,s)x\\parallel +I_{n,k}\\parallel U(t,s)x\\parallel \\\\& \\le c^{k+1}k!",
"(c\\alpha +1)M_{\\alpha }(s)\\parallel x\\parallel +\\frac{2}{k+1}\\theta _{n}^{k+1}\\,\\parallel U(t,s)x\\parallel \\text{.",
"}$ Letting $n\\rightarrow \\infty $ we deduce that (REF ) works for $k+1$ .",
"Step 3.",
"Pick $\\delta \\in (0,1)$ .",
"Multiplying (REF ) by $\\delta ^{k}$ and summing with respect to $k\\in \\mathbb {N}$ one easily gets $\\parallel U(t,s)\\parallel \\le \\frac{c\\alpha +1}{1-\\delta }M_{\\alpha }(s)e^{-\\frac{\\delta }{c}(t-s)},\\,t\\ge s\\ge 0\\text{,}$ that ends our proof.",
"Definition 4.3 We say that the admissible exponent $\\alpha \\in $$\\mathcal {A}\\left( \\mathcal {U}\\right) $ is quasi-negative if $\\mathcal {C}(\\mathcal {U},\\alpha )=\\mathcal {C}(\\mathcal {U},-\\nu )$ , for some admissible exponent $-\\nu <0$ .",
"It follows that each negative admissible exponent is quasi-negative.",
"We also notice that the set of all quasi-negative exponents is an interval.",
"According to Proposition REF , the evolution family $\\mathcal {U}$ is uniform exponentially stable if and only if it has strict quasi-negative exponents.",
"In the nonuniform case the situation is more complicate, as illustrated in the below examples.",
"Example 4.4 For $t\\ge s\\ge 0$ we set $E\\left( t,s\\right) =s\\left( 2+\\sin s\\right) -t\\left( 2+\\sin t\\right)\\text{,}$ and let $U\\left( t,s\\right) x =e^{E\\left( t,s\\right) }x$ , $x\\in X$ .",
"We claim that the evolution family $\\mathcal {U}$ is nonuniform exponentially stable, but not uniform exponentially bounded.",
"In fact, $\\mathcal {A}\\left( \\mathcal {U}\\right) =\\left[ -1,\\infty \\right) \\text{ and }\\mathcal {C}(\\mathcal {U},\\alpha )=\\mathcal {C}\\left( \\mathcal {U},-1\\right)\\text{ for all } \\alpha \\in \\mathcal {A}\\left( \\mathcal {U}\\right) ,$ that is all admissible exponents are quasi-negative.",
"Indeed, let us notice that $E_{1}\\left( t,s\\right) =E\\left( t,s\\right) +t-s =s(1+\\sin s)-t(1+\\sin t)\\le s+s\\sin s=f_{1}\\left( s\\right) \\text{,}$ and as $E_{1}\\left( 2n\\pi +{3\\pi }/{2},s\\right) =f_{1}\\left( s\\right) $ for $n$ sufficiently large, one obtains $\\underset{t\\ge s}{\\sup }E_{1}\\left( t,s\\right) =f_{1}\\left( s\\right) .$ It follows that $-1\\in \\mathcal {A}\\left( \\mathcal {U}\\right) $ .",
"If $\\varepsilon >0$ , then for $E_{\\varepsilon }\\left( t,s\\right) =E\\left( t,s\\right) +\\left(1+\\varepsilon \\right) \\left( t-s\\right) =s\\left( 1-\\varepsilon +\\sin s\\right) -t\\left( 1-\\varepsilon +\\sin t\\right) ,$ we get $E_{\\varepsilon }\\left( 2n\\pi +{3\\pi }/{2},s\\right) =s\\left( 1-\\varepsilon +\\sin s\\right) +\\varepsilon \\left( 2n\\pi +{3\\pi }/{2}\\right) ,$ which implies that $\\underset{t\\ge s}{\\sup } E_{\\varepsilon }\\left(t,s\\right) =\\infty ,$ and consequently $-1-\\varepsilon \\notin \\mathcal {A}\\left(\\mathcal {U}\\right) $ .",
"Thus, $\\mathcal {A}\\left( \\mathcal {U}\\right) =\\left[-1,\\infty \\right) $ .",
"Since the map $f_{1}$ is unbounded, the evolution family $\\mathcal {U}$ is not uniform exponentially bounded.",
"For any fixed $\\alpha \\in \\mathcal {A}\\left( \\mathcal {U}\\right) $ let us define $E_{\\alpha }\\left( t,s\\right) =E\\left( t,s\\right) -\\alpha \\left( t-s\\right)=s\\left( 2+\\alpha +\\sin s\\right) -t\\left( 2+\\alpha +\\sin t\\right) \\text{,}$ and put $f_{2}\\left( s\\right) =\\underset{t\\ge s}{\\sup }E_{\\alpha }\\left(t,s\\right) $ .",
"For $u\\in C(\\mathbb {R}_{+},X)$ we have $\\varphi _{\\mathcal {U},\\alpha }\\left( s,u\\right) & =\\underset{t\\ge s}{\\sup }e^{-\\alpha \\left( t-s\\right) }\\left\\Vert U\\left( t,s\\right) u\\left(s\\right) \\right\\Vert =\\underset{t\\ge s}{\\sup }e^{E_{\\alpha }\\left(t,s\\right) }\\parallel u(s) \\parallel =e^{f_{2}\\left( s\\right) }\\parallel u\\left( s\\right) \\parallel \\text{,}$ and similarly $\\varphi _{\\mathcal {U},-1}\\left( s,u\\right) =e^{f_{1}\\left(s\\right) }\\parallel u\\left( s\\right) \\parallel .$ For any fixed $s\\ge 0$ we denote $n_{s}=\\left[ \\frac{s}{2\\pi }-\\frac{3}{4}\\right] $ , that implies $t_{s}=2(n_{s}+1)\\pi +{3\\pi }/{2}\\in \\left( s,s+2\\pi \\right] $ .",
"Let us remark that $f_{1}\\left( s\\right) -f_{2}\\left( s\\right) \\le f_{1}\\left( s\\right)-E_{\\alpha }\\left( t_{s},s\\right) =\\left( 1+\\alpha \\right) \\left(t_{s}-s\\right) \\le 2\\pi \\left( 1+\\alpha \\right) .$ Gathering the above identities and estimation, one gets $\\varphi _{\\mathcal {U},-1}\\left( s,u\\right) & =e^{f_{1}\\left( s\\right)}\\parallel u\\left( s\\right) \\parallel =e^{f_{1}\\left( s\\right)-f_{2}\\left( s\\right) }\\varphi _{\\mathcal {U},\\alpha }\\left( s,u\\right) \\le e^{2\\pi \\left( 1+\\alpha \\right) }\\varphi _{\\mathcal {U},\\alpha }\\left(s,u\\right) \\text{.",
"}$ We conclude that $\\mathcal {C}(\\mathcal {U},\\alpha )=\\mathcal {C}(\\mathcal {U},-1)$ .",
"For the evolution family in the next example the situation is completely different.",
"It has no admissible exponents both positive and quasi-negative at the same time.",
"Besides, it is uniform exponentially bounded.",
"Example 4.5 For $t\\ge s\\ge 0$ we set $E\\left( t,s\\right) =s\\left( \\sqrt{2}+\\sin \\ln s\\right) -t\\left( \\sqrt{2}+\\sin \\ln t\\right) \\text{,}$ and let $E_{\\alpha }\\left( t,s\\right) =E\\left( t,s\\right) -\\alpha \\left( t-s\\right)=s\\left( \\alpha +\\sqrt{2}+\\sin \\ln s\\right) -t\\left( \\alpha +\\sqrt{2}+\\sin \\ln t\\right) \\text{,}$ for fixed $\\alpha \\in \\mathbb {R}$ .",
"We consider the evolution family $U\\left(t,s\\right) x =e^{E\\left( t,s\\right) }x$ , $t\\ge s\\ge 0$ , $x\\in X$ .",
"We claim that $\\mathcal {A}\\left( \\mathcal {U}\\right) =\\left[ 1-\\sqrt{2},\\infty \\right) $ , the quasi-negative admissible exponents are only the negative ones, and $\\mathcal {U}$ is uniform exponentially bounded.",
"If $\\alpha <1-\\sqrt{2}$ , then $\\alpha =1-\\sqrt{2}-\\varepsilon $ , for some $\\varepsilon >0$ .",
"Let us notice that $E_{1-\\sqrt{2}-\\varepsilon }\\left( t,s\\right) =s\\left( 1-\\varepsilon +\\sin \\ln s\\right) -t\\left( 1-\\varepsilon +\\sin \\ln t\\right) \\text{.",
"}$ For $t_{n}=e^{2n\\pi +\\frac{3\\pi }{2}}$ , $n\\in \\mathbb {N}$ , we have $E_{1-\\sqrt{2}-\\varepsilon }\\left( t_{n},0\\right) = \\varepsilon t_{n}\\rightarrow \\infty ,$ which implies that $\\alpha \\notin \\mathcal {A}\\left( \\mathcal {U}\\right) $ .",
"If $\\alpha \\in \\left[ 1-\\sqrt{2},0\\right) $ , then there exists $\\varepsilon \\in \\left[ 0,\\sqrt{2}-1\\right) $ such that $\\alpha =1-\\sqrt{2}+\\varepsilon $ .",
"In this case we have $E_{1-\\sqrt{2}+\\varepsilon }\\left( t,s\\right) & =s\\left( 1+\\varepsilon +\\sin \\ln s\\right) -t\\left( 1+\\varepsilon +\\sin \\ln t\\right) \\\\& \\le s\\left( 1+\\sin \\ln s\\right) =f_{1}\\left( s\\right) \\text{.",
"}$ Thus, $\\alpha \\in \\mathcal {A}\\left( \\mathcal {U}\\right) $ .",
"We conclude that $\\mathcal {A}\\left( \\mathcal {U}\\right) =\\left[ 1-\\sqrt{2},\\infty \\right) $ .",
"For $\\alpha \\ge 0$ let us put $f_{\\alpha }\\left( t\\right) =t\\left(\\alpha +\\sqrt{2}+\\sin \\ln t\\right) $ , $t\\ge 0$ .",
"The derivative is $f_{\\alpha }^{\\prime }\\left( t\\right) =\\alpha +\\sqrt{2}\\left[ 1+\\sin \\left(\\frac{\\pi }{4}+\\ln t\\right) \\right] \\ge 0.$ Thus, $E_{\\alpha }\\left( t,s\\right) =f_{\\alpha }\\left( s\\right) -f_{\\alpha }\\left(t\\right) = f_{\\alpha }^{\\prime }\\left( \\theta _{t,s}\\right) \\left(s-t\\right) \\le 0\\text{,}\\text{ for }t>s\\ge 0,$ and so $\\left\\Vert U\\left( t,s\\right) \\right\\Vert \\le e^{\\alpha \\left(t-s\\right) }$ .",
"This means that $\\mathcal {U}$ is uniform exponentially bounded.",
"If $\\alpha \\in \\left[ 1-\\sqrt{2},0\\right) $ , as $f_{1}\\left( s\\right) $ is unbounded, the inclusion $\\mathcal {C}(\\mathcal {U},\\alpha )\\subset C_{00}(\\mathbb {R}_{+},X)$ is strict.",
"In this case the admissible exponent $\\alpha $ is not quasi-negative.",
"We consider the next theorem as the main result of the paper.",
"Theorem 4.6 Let $\\alpha \\in \\mathcal {A}\\left( \\mathcal {U}\\right) $ .",
"The generator $G_{\\mathcal {U},\\alpha }$ is invertible if and only if $\\alpha $ is a quasi-negative admissible exponent.",
"The sufficiency follows directly from Lemma REF and Remark REF .",
"For the necessity, assume that $G_{\\mathcal {U},\\alpha }$ is invertible.",
"According to Lemma REF , $\\mathcal {U}$ is nonuniform exponentially stable.",
"Choose $\\nu >0$ with $-\\nu \\in \\mathcal {A}\\left(\\mathcal {U}\\right) $ , $-\\nu \\ne \\inf \\mathcal {A}\\left( \\mathcal {U}\\right) $ .",
"It suffices to consider the case $\\alpha \\ge 0$ .",
"For each fixed $s\\ge 0$ and $x\\in X\\setminus \\left\\lbrace 0\\right\\rbrace $ we construct the map $\\widetilde{u}_{s,x}\\in C(\\mathbb {R}_{+},X)$ given by $\\widetilde{u}_{s,x}\\left( \\xi \\right) ={\\left\\lbrace \\begin{array}{ll}e^{\\nu \\left( \\xi -s\\right) }U\\left( \\xi ,s\\right) x\\text{,} & \\text{if }\\xi >s;\\\\x\\text{,} & \\text{if }0\\le \\xi \\le s\\text{.}\\end{array}\\right.",
"}$ Pick $\\varepsilon >0$ such that $-\\nu -\\varepsilon \\in \\mathcal {A}\\left(\\mathcal {U}\\right) $ .",
"For $t>s$ we have $\\varphi _{\\mathcal {U},\\alpha }\\left( t,\\widetilde{u}_{s,x}\\right) &=\\underset{\\tau \\ge t}{\\sup }\\text{ }e^{-\\alpha \\left( \\tau -t\\right) }e^{\\nu \\left( t-s\\right) }\\left\\Vert U\\left( \\tau ,s\\right) x\\right\\Vert \\\\& \\le \\underset{\\tau \\ge t}{\\sup }\\text{ }e^{\\nu \\left( t-s\\right)}e^{-\\left( \\nu +\\varepsilon \\right) \\left( \\tau -s\\right) }M_{-\\nu -\\varepsilon }\\left( s\\right) \\left\\Vert x\\right\\Vert \\\\& \\le e^{-\\varepsilon \\left( t-s\\right) }M_{-\\nu -\\varepsilon }\\left(s\\right) \\left\\Vert x\\right\\Vert \\text{,}$ therefore $\\underset{t\\rightarrow \\infty }{\\lim }\\varphi _{\\mathcal {U},\\alpha }\\left( t,\\widetilde{u}_{s,x}\\right) =0$ .",
"Proposition REF implies that the below set is nonempty: $\\Lambda _{s,x}=\\left\\lbrace t\\ge 0:\\sup \\limits _{\\xi \\ge 0}\\text{ }\\varphi _{\\mathcal {U},\\alpha }(\\xi ,\\widetilde{u}_{s,x})=\\varphi _{\\mathcal {U},\\alpha }\\left( t,\\widetilde{u}_{s,x}\\right) \\right\\rbrace .$ As the map $\\xi \\mapsto \\varphi _{\\mathcal {U},\\alpha }(\\xi ,\\widetilde{u}_{s,x})$ is continuous, it follows that $t_{s,x}=\\inf \\Lambda _{s,x}\\in \\Lambda _{s,x}$ .",
"We have the alternative: (A1) There exists $\\nu >0$ , $-\\nu \\in \\mathcal {A}\\left( \\mathcal {U}\\right) $ , $-\\nu \\ne \\inf \\mathcal {A}\\left( \\mathcal {U}\\right) $ such that for each $s\\ge 0$ and $x\\in X\\setminus \\left\\lbrace 0\\right\\rbrace $ , $t_{s,x}\\in \\left[s,s+\\frac{1}{\\nu }\\right] $ .",
"(A2) For any sufficiently small $\\nu >0$ , $-\\nu \\in \\mathcal {A}\\left(\\mathcal {U}\\right) $ , $-\\nu \\ne \\inf \\mathcal {A}\\left( \\mathcal {U}\\right) $ , there exists $s\\ge 0$ and $x\\in X\\setminus \\left\\lbrace 0\\right\\rbrace $ with $t_{s,x}>s+\\frac{1}{\\nu }$ .",
"Assume that (A1) holds.",
"In this case, for all $t\\ge s$ and $x\\in X\\setminus \\left\\lbrace 0\\right\\rbrace $ we have $\\parallel \\widetilde{u}_{s,x}\\left( t\\right) \\parallel & \\le \\varphi _{\\mathcal {U},\\alpha }\\left( t,\\widetilde{u}_{s,x}\\right) \\le \\varphi _{\\mathcal {U},\\alpha }\\left( t_{s,x},\\widetilde{u}_{s,x}\\right) \\\\& =\\underset{\\tau \\ge t_{s,x}}{\\sup }\\text{ }e^{-\\alpha \\left( \\tau -t_{s,x}\\right) }e^{\\nu \\left( t_{s,x}-s\\right) }\\left\\Vert U\\left(\\tau ,s\\right) x\\right\\Vert \\text{.",
"}$ The above estimation yields $\\underset{t\\ge s}{\\sup }\\text{ }e^{\\nu \\left( t-s\\right) }\\left\\Vert U\\left(t,s\\right) x\\right\\Vert & \\le \\underset{\\tau \\ge t_{s,x}}{\\sup }\\text{}e^{-\\alpha \\left( \\tau -t_{s,x}\\right) }e^{\\nu \\left( t_{s,x}-s\\right)}\\left\\Vert U\\left( \\tau ,s\\right) x\\right\\Vert \\\\& =\\underset{\\tau \\ge t_{s,x}}{\\sup }\\text{ }e^{\\left( \\alpha +\\nu \\right)\\left( t_{s,x}-s\\right) }e^{-\\alpha \\left( \\tau -s\\right) }\\left\\Vert U\\left( \\tau ,s\\right) x\\right\\Vert \\\\& \\le e^{\\frac{1}{\\nu }\\left( \\alpha +\\nu \\right) }\\underset{\\tau \\ge s}{\\sup }\\text{ }e^{-\\alpha \\left( \\tau -s\\right) }\\left\\Vert U\\left(\\tau ,s\\right) x\\right\\Vert \\text{.",
"}$ Therefore, for all $s\\ge 0$ and $x\\in X$ we have $\\underset{t\\ge s}{\\sup }\\text{ }e^{-\\alpha \\left( t-s\\right) }\\left\\Vert U\\left( t,s\\right) x\\right\\Vert \\le \\underset{t\\ge s}{\\sup }\\text{ }e^{\\nu \\left( t-s\\right) }\\left\\Vert U\\left( t,s\\right) x\\right\\Vert \\le K\\underset{t\\ge s}{\\sup }\\text{ }e^{-\\alpha \\left( t-s\\right) }\\left\\Vert U\\left( t,s\\right) x\\right\\Vert ,$ where $K=e^{\\frac{1}{\\nu }\\left( \\alpha +\\nu \\right) }$ .",
"It turns out that $\\varphi _{\\mathcal {U},\\alpha }\\left( s,u\\right) \\le \\varphi _{\\mathcal {U},-\\nu }\\left( s,u\\right) \\le K\\varphi _{\\mathcal {U},\\alpha }\\left( s,u\\right)\\text{, }s\\ge 0\\text{, }u\\in C(\\mathbb {R}_{+},X).$ We conclude that $\\mathcal {C}(\\mathcal {U},\\alpha )=\\mathcal {C}(\\mathcal {U},-\\nu )$ , as Banach spaces, thus $\\alpha $ is quasi-negative.",
"Assume that (A2) holds.",
"For sufficiently large $n\\in \\mathbb {N}^{\\ast }$ such that $-\\frac{1}{n}\\in \\mathcal {A}\\left( \\mathcal {U}\\right) $ , $-\\frac{1}{n}\\ne \\inf \\mathcal {A}\\left( \\mathcal {U}\\right) $ , we put $s_{n}\\ge 0$ , $x_{n}\\in X\\setminus \\left\\lbrace 0\\right\\rbrace $ and $t_{n}=t_{s_{n},x_{n}}>s_{n}+n$ , as in the statement (A2).",
"Let us define the $C^{1}$ -map $\\psi _{n}:\\mathbb {R}_{+}\\rightarrow \\mathbb {R}_{+}$ by $\\psi _{n}\\left( t\\right) ={\\left\\lbrace \\begin{array}{ll}e^{\\frac{1}{n}\\left( t-s_{n}\\right) }\\text{,} & \\text{if }t>t_{n}\\text{;}\\\\\\frac{t}{n}e^{\\frac{1}{n}\\left( t_{n}-s_{n}\\right) }+e^{\\frac{1}{n}\\left(t_{n}-s_{n}\\right) }\\left( 1-\\frac{t_{n}}{n}\\right) \\text{,} & \\text{if}s_{n}+\\delta _{n}<t\\le t_{n}\\text{;}\\\\a_{n}t^{2}+b_{n}t+c_{n}\\text{, } & \\text{if }s_{n}<t\\le s_{n}+\\delta _{n}\\text{;}\\\\0\\text{,} & \\text{if }0\\le t\\le s_{n}\\text{.}\\end{array}\\right.",
"}$ The constants $a_{n}$ , $b_{n}$ , $c_{n}\\in \\mathbb {R}$ and $\\delta _{n}\\in \\left(s_{n},t_{n}\\right) $ are determined by condition $\\psi _{n}\\in C^{1}\\left(\\mathbb {R}_{+}\\right) $ .",
"We also define $u_{n}$ , $f_{n}:\\mathbb {R}_{+}\\rightarrow X$ by $u_{n}\\left( t\\right) ={\\left\\lbrace \\begin{array}{ll}\\psi _{n}\\left( t\\right) U\\left( t,s_{n}\\right) x_{n}\\text{,} & \\text{if}t>s_{n}\\text{;}\\\\0\\text{,} & \\text{if }0\\le t\\le s_{n}\\text{,}\\end{array}\\right.",
"}$ and $f_{n}\\left( t\\right) ={\\left\\lbrace \\begin{array}{ll}\\psi _{n}^{\\prime }\\left( t\\right) U\\left( t,s_{n}\\right) x_{n}\\text{,} &\\text{if }t>s_{n}\\text{;}\\\\0\\text{,} & \\text{if }0\\le t\\le s_{n}\\text{.}\\end{array}\\right.",
"}$ Evidently $u_{n},$ $f_{n}\\in \\mathcal {C}(\\mathcal {U},\\alpha )$ and $G_{\\mathcal {U},\\alpha }u_{n}=-f_{n}$ .",
"Notice that for $t\\in \\left[0,t_{n}\\right] $ we have $\\varphi _{\\mathcal {U},\\alpha }\\left( t,u_{n}\\right) & =\\underset{\\tau \\ge t}{\\sup }\\text{ }e^{-\\alpha \\left( \\tau -t\\right) }\\psi _{n}\\left( t\\right)\\left\\Vert U\\left( \\tau ,s_{n}\\right) x_{n}\\right\\Vert \\\\& \\le \\underset{\\tau \\ge t}{\\sup }\\text{ }e^{-\\alpha \\left( \\tau -t\\right)}e^{\\frac{1}{n}\\left( t-s_{n}\\right) }\\left\\Vert U\\left( \\tau ,s_{n}\\right)x_{n}\\right\\Vert \\\\& =\\varphi _{\\mathcal {U},\\alpha }\\left( t,\\widetilde{u}_{s_{n},x_{n}}\\right)\\text{,}$ where $\\widetilde{u}_{s_{n},x_{n}}$ is defined in (REF ).",
"If $t\\ge t_{n\\text{ }}$ , as $\\varphi _{\\mathcal {U},\\alpha }\\left( t,u_{n}\\right)=\\varphi _{\\mathcal {U},\\alpha }\\left( t,\\widetilde{u}_{s_{n},x_{n}}\\right) $ , it follows that $\\parallel u_{n}\\parallel _{\\mathcal {U},\\alpha }=\\varphi _{\\mathcal {U},\\alpha }\\left( t_{n},\\widetilde{u}_{s_{n},x_{n}}\\right) $ , and similarly one gets $\\parallel f_{n}\\parallel _{\\mathcal {U},\\alpha }=\\frac{1}{n}\\varphi _{\\mathcal {U},\\alpha }\\left( t_{n},\\widetilde{u}_{s_{n},x_{n}}\\right) $ .",
"These identities imply $\\frac{\\parallel u_{n}\\parallel _{\\mathcal {U},\\alpha }}{\\parallel f_{n}\\parallel _{\\mathcal {U},\\alpha }}=n\\text{,}$ which leads to $\\underset{f\\in \\mathcal {C}(\\mathcal {U},\\alpha )}{\\sup }$ $\\frac{\\left\\Vert G_{\\mathcal {U},\\alpha }^{-1}f\\right\\Vert _{\\mathcal {U},\\alpha }}{\\left\\Vert f\\right\\Vert _{\\mathcal {U},\\alpha }}\\ge n$ .",
"Thus, $G_{\\mathcal {U},\\alpha }^{-1}$ is not bounded, which is false, and eventually only alternative (A1) holds, that proves the claim.",
"As in [9], the spectral mapping theorem is to be deduced from exponential stability.",
"Corollary 4.7 (The Spectral Mapping Theorem) For each $\\alpha \\in \\mathcal {A}\\left(\\mathcal {U}\\right) $ , the nonuniform evolution semigroup $\\mathcal {T}_{\\alpha }=\\left\\lbrace T_{\\alpha }(t)\\right\\rbrace _{t\\ge 0}$ satisfies the identity $e^{t\\sigma \\left( G_{\\mathcal {U},\\alpha }\\right) }=\\sigma \\left( T_{\\alpha }\\left( t\\right) \\right) \\setminus \\left\\lbrace 0\\right\\rbrace \\text{, }t\\ge 0\\text{.",
"}$ Moreover, $\\sigma \\left( G_{\\mathcal {U},\\alpha }\\right) $ is a left half-plane and $\\sigma \\left( T_{\\alpha }\\left( t\\right) \\right) $ , $t\\ge 0$ , is a disc.",
"If $\\lambda \\in \\mathbb {C}$ , then we consider the rescaled evolution family $U_{\\lambda }(t,s)=e^{-\\lambda (t-s)}U(t,s).$ For arbitrary $\\alpha \\in \\mathcal {A}\\left( \\mathcal {U}\\right) $ , $t\\ge 0$ and $u\\in C(\\mathbb {R}_{+},X)$ one has $\\varphi _{\\mathcal {U}_{\\lambda },\\alpha -\\operatorname{Re}\\lambda }\\left(t,u\\right) & =\\underset{\\tau \\ge t}{\\sup }\\text{ }e^{-\\left( \\alpha -\\operatorname{Re}\\lambda \\right) \\left( \\tau -t\\right) }\\left\\Vert U_{\\lambda }\\left( \\tau ,t\\right) u\\left( t\\right) \\right\\Vert \\\\& =\\underset{\\tau \\ge t}{\\sup }\\text{ }e^{-\\alpha \\left( \\tau -t\\right)}\\left\\Vert U\\left( \\tau ,t\\right) u\\left( t\\right) \\right\\Vert =\\varphi _{\\mathcal {U},\\alpha }\\left( t,u\\right) \\text{.",
"}$ We deduce that the admissible sets $\\mathcal {A}\\left( \\mathcal {U}_{\\lambda }\\right) =\\mathcal {A}\\left( \\mathcal {U}\\right) -\\operatorname{Re}\\lambda $ , and the admissible Banach spaces $\\mathcal {C}(\\mathcal {U}_{\\lambda },\\alpha -\\operatorname{Re}\\lambda )=\\mathcal {C}(\\mathcal {U},\\alpha )$ .",
"From another hand, the definition of the generator implies that $D\\left(G_{\\mathcal {U},\\alpha }\\right) =D\\left( G_{\\mathcal {U}_{\\lambda },\\alpha -\\operatorname{Re}\\lambda }\\right) $ and $G_{\\mathcal {U}_{\\lambda },\\alpha -\\operatorname{Re}\\lambda }=G_{\\mathcal {U},\\alpha }-\\lambda Id\\text{.}",
"$ Let $\\lambda \\in \\rho \\left( G_{\\mathcal {U},\\alpha }\\right) $ , that is $G_{\\mathcal {U}_{\\lambda },\\alpha -\\operatorname{Re}\\lambda }$ is invertible.",
"Choose $\\mu \\in \\mathbb {C}$ with $\\operatorname{Re}\\mu \\ge \\operatorname{Re}\\lambda $ .",
"We only have two options: (B1) $\\operatorname{Re}\\mu >\\alpha $ ; (B2) $\\operatorname{Re}\\lambda \\le \\operatorname{Re}\\mu \\le \\alpha $ .",
"In case (B1), Theorem REF and formula (REF ) imply that $G_{\\mathcal {U}_{\\mu },\\alpha -\\operatorname{Re}\\mu }$ is invertible, consequently $\\mu \\in \\rho \\left( G_{\\mathcal {U},\\alpha }\\right) $ .",
"Assume that (B2) holds.",
"According to the same Theorem REF , as $G_{\\mathcal {U}_{\\lambda },\\alpha -\\operatorname{Re}\\lambda }$ is invertible, there exists $\\nu >0$ with $\\mathcal {C}(\\mathcal {U}_{\\lambda },\\alpha -\\operatorname{Re}\\lambda )=\\mathcal {C}(\\mathcal {U}_{\\lambda },-\\nu )$ , and $G_{\\mathcal {U}_{\\lambda },\\alpha -\\operatorname{Re}\\lambda }=G_{\\mathcal {U}_{\\lambda },-\\nu }$ .",
"Using Eq.",
"(REF ), we successively have $G_{\\mathcal {U}_{\\mu },\\alpha -\\operatorname{Re}\\mu } & =G_{\\mathcal {U}_{\\lambda },\\alpha -\\operatorname{Re}\\lambda }+\\left( \\lambda -\\mu \\right) Id\\\\& =G_{\\mathcal {U}_{\\lambda },-\\nu }+\\left( \\lambda -\\mu \\right) Id\\\\& =G_{\\mathcal {U}_{\\mu },-\\nu +\\operatorname{Re}\\lambda -\\operatorname{Re}\\mu }\\text{.",
"}$ Since $-\\nu +\\operatorname{Re}\\lambda -\\operatorname{Re}\\mu <0$ , then $G_{\\mathcal {U}_{\\mu },-\\nu +\\operatorname{Re}\\lambda -\\operatorname{Re}\\mu }$ is invertible, hence $G_{\\mathcal {U}_{\\mu },\\alpha -\\operatorname{Re}\\mu }$ is also invertible.",
"This shows that $\\mu \\in \\rho \\left( G_{\\mathcal {U},\\alpha }\\right)$ , which implies that the spectrum $\\sigma \\left( G_{\\mathcal {U},\\alpha }\\right) $ is a left half-plane.",
"For the rest of the proof we refer the reader to [9].",
"We now recover the result from Theorem 2.2 in [9], valid in the particular case of uniform exponential stability.",
"Corollary 4.8 Let $\\mathcal {U}$ be a uniform exponentially bounded evolution family.",
"Then $\\mathcal {U}$ is uniform exponentially stable if and only if the generator of the corresponding evolution semigroup is invertible."
]
] | 1606.05159 |
[
[
"Semilinear elliptic equations with the pseudo-relativistic operator on a\n bounded domain"
],
[
"Abstract We study the Dirichlet problem for the semilinear equations involving the pseudo-relativistic operator on a bounded domain, (\\sqrt{-\\Delta + m^2} - m)u =|u|^{p-1}u \\quad \\textrm{in}~\\Omega, with the Dirichlet boundary condition $u=0$ on $\\partial \\Omega$.",
"Here, $p \\in (1,\\infty)$ and the operator $(\\sqrt{-\\Delta + m^2} - m)$ is defined in terms of spectral decomposition.",
"In this paper, we investigate existence and nonexistence of a nontrivial solution, depending on the choice of $p$, $m$ and $\\Omega$.",
"Precisely, we show that $(i)$ if $p$ is not $H^1$ subcritical ($p \\geq \\frac{n+2}{n-2}$) and $\\Omega$ is star-shaped, the equation has no nontrivial solution for all $m > 0$; $(ii)$ if $p$ is not $H^{1/2}$ supercritical ($1 <p \\leq \\frac{n+1}{n-1}$), then there exists a least energy solution for all $m>0$ and any bounded domain $\\Omega$; $(iii)$ finally, in the intermediate range ($\\frac{n+1}{n-1}<p<\\frac{n+2}{n-2}$), the problem has a nontrivial solution, provided that $m$ is sufficiently large and the problem -\\Delta u = |u|^{p-1}u \\quad \\textrm{in}~\\Omega, \\qquad u =0\\quad \\textrm{on}~\\partial \\Omega admits a non-degenerate nontrivial solution, for example, when $\\Omega$ is a ball or an annulus."
],
[
"Introduction",
"Let $n \\ge 3$ and $p > 1$ , and let $\\Omega \\subset \\mathbb {R}^n$ be a smooth bounded domain.",
"This paper is devoted to the study of the Dirichlet problem: $\\left\\lbrace \\begin{aligned}\\mathcal {P}_mu &=|u|^{p-1}u &&\\textrm {in}~\\Omega ,\\\\u&=0&&\\textrm {on}~\\partial \\Omega ,\\end{aligned}\\right.$ where $\\mathcal {P}_m$ is the pseudo-relativistic operator given by $\\sqrt{-\\Delta + m^2} - m$ with a particle mass $m > 0$ .",
"On a bounded domain, there are several ways to define a nonlocal square root operator with the zero Dirichlet condition unlike the whole domain $\\mathbb {R}^n$ case.",
"In this paper, we adopt the definition using the spectral decomposition (see Section 2 below for the precise definition).",
"The pseudo-relativistic operator $\\mathcal {P}_m$ appears in the relativistic theory of the quantum mechanics to describe stellar objects, e.g., boson and fermion stars, in astrophysics.",
"We refer the readers to the fundamental works in [22], [23] on the stability of relativistic matter.",
"Based on this physical motivation, there have been many studies conducted on nonlinear problems involving the pseudo-relativistic operator.",
"We refer the readers to [1], [2], [9], [14], [15], [28] for semilinear elliptic equations with the Hartree nonlinearity; [4], [13], [30] for those with power-type nonlinearity; [26], [27], [32] for time evolution equations.",
"The operator $\\mathcal {P}_m$ on a bounded domain appears in the work of Abou Salem, Chen and Vougalter [1], [2], in which the authors investigated the semi-relativistic Schrödinger-Poisson system on a bounded domain.",
"The operator $\\mathcal {P}_m$ is called pseudo-relativistic, because it interpolates the non-relativistic and the ultra-relativistic schrödinger operators, i.e., $2m\\mathcal {P}_m$ formally converges to $-\\Delta $ as $m\\rightarrow \\infty $ as well as $\\mathcal {P}_m$ converges to $\\sqrt{-\\Delta }$ as $m\\rightarrow 0^+$ .",
"When $m$ is near infinity, the nonlinear problem (REF ) thus can be thought of as a perturbation of the Lane-Emden equation, $\\left\\lbrace \\begin{aligned} -\\Delta u &= |u|^{p-1}u && \\textrm {in}~\\Omega ,\\\\u&=0&&\\textrm {on}~\\partial \\Omega ,\\end{aligned}\\right.$ which arises in astrophysics for describing the pressure and the density of a gas sphere in a hydrostatic equilibrium.",
"Here, the power of nonlinearity $p$ stands for the polytropic index (see [10], [19]).",
"On the other hand, as $m$ goes to $0^{+}$ , the equation (REF ) converges to the nonlocal problem $\\left\\lbrace \\begin{aligned} \\sqrt{-\\Delta }\\,u &= |u|^{p-1}u&&\\textrm {in}~\\Omega ,\\\\u&=0&&\\textrm {on}~\\partial \\Omega ,\\end{aligned}\\right.$ where $\\sqrt{-\\Delta }$ corresponds to the square root of the Laplacian $-\\Delta $ defined in [6].",
"The purpose of this paper is to exploit existence and non-existence of a nontrivial solution to the pseudo-relativistic equation (REF ) and compare them with the results on the two limit equations (REF ) and (REF ) as $m\\rightarrow \\infty $ and $m\\rightarrow 0^+$ respectively.",
"Before stating our main theorems, we briefly recall the existence results for the limit problems.",
"An important remark is that the above two problems have the different critical exponents, and thus the ranges of the exponent $p$ for existence and nonexistence of a nontrivial solution are different.",
"Precisely, it is known that the problem (REF ) has a positive solution if $1 < p < \\frac{n+2}{n-2}$ , but it has no nontrivial solution if $p \\ge \\frac{n+2}{n-2}$ and $\\Omega $ is starshaped (see, e.g., [29]).",
"The same results hold for (REF ) if we replace $\\frac{n+2}{n-2}$ with $\\frac{n+1}{n-1}$ (see [6], [31]).",
"As is well-known, the two critical exponents $\\frac{n+2}{n-2}$ and $\\frac{n+1}{n-1}$ are related to the critical Sobolev embeddings $H_0^1 (\\Omega ) \\hookrightarrow L^{2n/(n-2)}(\\Omega )$ and $H_0^{1/2} (\\Omega ) \\hookrightarrow L^{2n/(n-1)}(\\Omega )$ .",
"Coming back to the problem (REF ), we now introduce the notion of a solution for the equation.",
"In Section , it will be shown that the operator $\\mathcal {P}_m$ is bounded from $H^1_0(\\Omega )$ to $L^2(\\Omega )$ .",
"Thus, it is natural to say that a real valued function $u$ on $\\Omega $ is a (strong) solution of $(\\ref {eq-main})$ if $u \\in H^1_0(\\Omega )$ , $|u|^{p-1}u \\in L^2(\\Omega )$ and $\\mathcal {P}_mu = |u|^{p-1}u$ almost everywhere.",
"Our first main theorem establishes existence of a positive nontrivial solution to (REF ) for all $m > 0$ and $1 < p \\le \\frac{n+1}{n-1}$ .",
"This shall be achieved by searching for a nontrivial critical point of the action integral $I_m(u) := \\int _{\\Omega }\\frac{1}{2}|(-\\Delta +m^2)^{1/4}u|^2-\\frac{m}{2}u^2+\\frac{1}{p+1}|u|^{p+1}\\,dx,$ whose Euler-Lagrange equation is (REF ) (see Section ).",
"Theorem 1.1 Suppose that $1<p\\le \\frac{n+1}{n-1}$ and fix any $m > 0$ .",
"Then, the problem (REF ) admits a positive nontrivial solution.",
"Moreover, it is contained in $C^{2,\\alpha }(\\overline{\\Omega })$ for some $0<\\alpha <1$ and minimizes the action integral $I_m$ among all the nontrivial solutions of (REF ).",
"The main novelty of Theorem REF lies on the $H^{1/2}$ critical case ($p = \\frac{n+1}{n-1}$ ), which is twofold.",
"First of all, a non-trivial solution is constructed without any assumption on the size of $m>0$ .",
"Indeed, from existence and non-existence of a non-trivial solution for the limit equations (REF ) and (REF ), it is natural to speculate that when $p = \\frac{n+1}{n-1}$ , a non-trivial solution exists if $m$ is large enough, while a nontrivial solution does not exists if $m$ is sufficiently small.",
"Such a speculation turns out to be true for a similar problem $\\mathcal {P}_m u + \\mu u = |u|^{p-1}u\\quad \\textrm {in}~\\mathbb {R}^n$ on the whole domain.",
"Precisely, in [12], it is shown that the equation (REF ) with $p = \\frac{n+1}{n-1}$ admits a nontrivial solution when $m$ is sufficiently large, but it does not admit any nontrivial solution in $L^\\infty \\cap H^{1/2}$ when $m \\le \\mu $ .",
"However, Theorem REF shows that unlike the whole domain case, the problem on a bounded domain may admit a nontrivial solution even for small $m$ .",
"Another new feature is that a solution is constructed as a minimizer for the action functional in the $H^{1/2}$ critical case ($p = \\frac{n+1}{n-1}$ ), while for the problem (REF ) on the whole domain, a solution is constructed via a contraction mapping argument with no further information about its variational character.",
"Indeed, being a minimizer is in general crucial to explore interesting properties of a solution like uniqueness, symmetries and asymptotical behaviors.",
"In the forthcoming work, we will investigate such properties of a solution in Theorem REF .",
"The main strategy for proving Theorem REF is to find an energy level $c$ at which a (PS) sequence of $I_n$ , i.e., a sequence $\\lbrace u_n\\rbrace $ such that $I_m^{\\prime }(u_n) \\rightarrow 0$ and $I_m(u_n) \\rightarrow c$ as $n \\rightarrow \\infty $ , is precompact.",
"We shall observe that the energy gain raised by the term $\\sqrt{-\\Delta +m^2}\\,u$ is overwhelmed by the energy loss resulting from the term $-mu$ for any $m > 0$ , which consequently gets rid of the possibility of concentrating bubbles and allows us to obtain a nontrivial solution of (REF ).",
"Next, we state our second main result, which concerns about nonexistence.",
"Theorem 1.2 Assume that $p \\ge \\frac{n+2}{n-2}$ , and $\\Omega $ is star-shaped.",
"Then, there exists no nontrivial $L^\\infty $ bounded solution to (REF ) for any $m > 0$ .",
"In order to obtain this result, we shall devise an anisotropic Pohozaev identity for the problem (REF ).",
"We remark that Theorem REF does not cover the range $p \\in (\\frac{n+1}{n-1},\\, \\frac{n+2}{n-2})$ .",
"In fact, at the limit $m \\rightarrow \\infty $ the critical exponent of the limit problem (REF ) is not $\\frac{n+1}{n-1}$ , but rather $\\frac{n+2}{n-2}$ .",
"Thus, one would expect existence of solutions even for $p \\in (\\frac{n+1}{n-1},\\, \\frac{n+2}{n-2})$ , at least when $m > 0$ is sufficiently large.",
"The last main result of this paper confirms this if we further assume the existence of a non-degenerate nontrivial solution to the limit equation (REF ).",
"We say a solution $u_{\\infty } \\in H_0^1 (\\Omega )$ to (REF ) is non-degenerate if the linearized equation of (REF ) at $u_\\infty $ , i.e., $\\left\\lbrace \\begin{array}{rll}-\\Delta v =&p |u_{\\infty }|^{p-1} v &\\quad \\textrm {in}~\\Omega ,\\\\v=&0&\\quad \\textrm {on}~\\partial \\Omega \\end{array}\\right.$ admits only the trivial solution $v=0$ in $H_0^1 (\\Omega )$ .",
"Theorem 1.3 Let $\\frac{n+1}{n-1} < p < \\frac{n+2}{n-2}$ .",
"Suppose that the equation (REF ) admits a nontrivial non-degenerate solution $u_\\infty $ .",
"Then, there exists a large $m_0 >0$ such that if $m > m_0$ , the problem (REF ) has a nontrivial solution $u_m \\in W^{1,n}_0(\\Omega )$ which satisfies $\\Vert (2m)^{\\frac{1}{p-1}}u_m - u_{\\infty }\\Vert _{W_0^{1,n} (\\Omega )} \\le \\left\\lbrace \\begin{aligned} &\\frac{C}{m^2} \\quad \\text{if } p > 2 \\\\ & \\frac{C}{m} \\quad \\text{if } 1 < p \\le 2,\\end{aligned}\\right.$ where $C$ is a positive constant independent of $m$ .",
"Remark 1.4 The existence of a nontrivial non-degenerate solution to (REF ) is known, for example, if one of the following holds: The dimension is two and the domain is convex and $p \\in (1, \\infty )$ [24].",
"the domain $\\Omega $ is a ball or an annulus; and $p \\in (1,\\, \\frac{n+2}{n-2})$ [3], [5].",
"the domain $\\Omega $ is convex and is symmetric about coordinate axes after a rigid motion; and $p$ is smaller but sufficiently close to $\\frac{n+2}{n-2}$ [18].",
"We remark that the functional $I_m$ is also well defined on $H^1_0(\\Omega )$ for $p \\in (\\frac{n+1}{n-1},\\, \\frac{n+2}{n-2})$ , but the variational approach is not suitable to construct a nontrivial solution, because it does not seem possible to obtain the $H^1$ boundedness of a (PS) sequence, due to the fact that $H^{1/2}$ norm appearing in $I_m$ cannot control $H^1$ norm.",
"We construct a non-trivial solution for (REF ) by the alternative approach proposed in our earlier work [12].",
"The key ingredient is existence of a non-degenerate solution to the limit equation (REF ), which is assumed in the theorem, because it allows us to make use of the contraction mapping principle to find a solution to (REF ) near the non-degenerate solution.",
"In this analysis, the Hörmander-Mikhlin Theorem on Fourier multiplier operators is helpful to cover the full range $1<p<\\frac{n+2}{n-2}$ , because it gives some $L^p$ estimates for the pseudo-relativistic operator.",
"We remark that as mentioned in [12], only a shorter range $1<p\\le \\frac{n}{n-2}$ can be included without the Hörmander-Mikhlin Theorem.",
"In our setting, the domain under consideration is bounded so the standard Hörmander-Mikhlin theorem cannot be applied directly.",
"We resolve this difficulty by invoking the generalized Hörmander-Mikhlin theorem in Duong, Sikora and Yan [16], which includes that on a bounded domain.",
"The rest of this paper is organized as follows.",
"In Section , we present some preliminary results concerning relevant function spaces, a precise definition of the operator $\\mathcal {P}_m$ as well as a localization of nonlocal operator $\\mathcal {P}_m$ .",
"Well-defined notions of solution to (REF ) are also given.",
"In Section , we prove the nonexistence result of Theorem REF by establishing an anisotropic Pohozaev identity.",
"In Section and Section , we prove the existence result of Theorem REF .",
"Finally, Section is devoted to dealing with the $H^{1/2}$ supercritical case with $m$ near infinity.",
"Notations.",
"Here, we list some notations that will be used throughout the paper.",
"- $B_n (x,r) = \\lbrace y \\in \\mathbb {R}^n : |y-x| < r\\rbrace .$ - For a domain $D \\subset \\mathbb {R}^n$ , the map $\\nu = (\\nu _1, \\cdots , \\nu _n): \\partial D \\rightarrow \\mathbb {R}^n$ denotes the outward unit normal vector on $\\partial D$ .",
"- $|S^{n-1}| = 2\\pi ^{n/2}/\\Gamma (n/2)$ denotes the Lebesgue measure of $(n-1)$ -dimensional unit sphere $S^{n-1}$ .",
"- The letter $z$ represents a variable in the $\\mathbb {R}^{n+1}$ .",
"Alternatively, this is written as $z = (x,t)$ with $x \\in \\mathbb {R}^n$ and $t \\in \\mathbb {R}$ .",
"- $\\mathbb {R}^{n+1}_{+} = \\lbrace (x_1, \\cdots , x_{n+1}) \\in \\mathbb {R}^{n+1} : x_{n+1} >0\\rbrace .$ - $dS$ stands for the surface measure.",
"In addition, a subscript attached to $dS$ (such as $dS_x$ or $dS_z$ ) denotes the variable of the surface.",
"- $C > 0$ is a generic constant that may vary from line to line."
],
[
"Preliminary results",
"In this section, we prepare some preliminary results that are relevant to our problem.",
"First, we provide the definition of the pseudo-relativistic operator, and describe the notions of solutions.",
"Next, we recall the extension problem, which localizes the nonlocal equation (REF ), and we provide regularity results for (REF ).",
"In the last of the section, we recall the sharp Sobolev-trace inequality and some related entire problems, and we prove an interesting inequality in Proposition REF , which is essential for proving the existence result of Theorem REF ."
],
[
"pseudo-relativistic operator and notions of solutions",
"Here, we define the square root operator $\\sqrt{-\\Delta +m^2}$ with the zero Dirichlet condition for $m \\ge 0$ .",
"Let $\\lbrace \\lambda _n,\\, \\phi _n\\rbrace _{n=1}^\\infty $ be the complete $L^2$ orthonormal system of eigenvalues and eigenfunctions of $\\left\\lbrace \\begin{aligned}-\\Delta \\phi &= \\lambda \\phi &&\\text{in } \\Omega \\\\\\phi &= 0 &&\\text{on } \\partial \\Omega .\\end{aligned}\\right.$ For any $u \\in L^2(\\Omega )$ , we have the unique decomposition $u = \\sum _{i=1}^\\infty c_i\\phi _i$ in $L^2(\\Omega )$ .",
"It is well known that $H^1_0(\\Omega ) = \\left\\lbrace u \\in L^2(\\Omega ) ~\\Big |~ \\sum _{i=1}^\\infty c^2_i\\lambda _i < \\infty \\right\\rbrace .$ For $m \\ge 0$ , we define $\\sqrt{-\\Delta +m^2}\\,u := \\sum _{i=1}^\\infty c_i\\sqrt{\\lambda _i+m^2}\\,\\phi _i.$ and denote $\\mathcal {P}_m :=\\left(\\sqrt{-\\Delta +m^2} -m\\right)$ .",
"Then, the operator $\\mathcal {P}_m$ is a continuous map from $H^1_0(\\Omega )$ to $L^2(\\Omega )$ .",
"Let $H^{1/2}_0(\\Omega )$ be the set $\\lbrace u\\in L^2(\\Omega ) ~|~ \\sum _{i=1}^\\infty c_i^2\\lambda _i^{1/2} < \\infty ,\\, u = \\sum _{i=1}^\\infty c_i\\phi _i \\rbrace $ .",
"For any $p > 1$ , we say a function $u$ is a (strong) solution of (REF ) if $u$ and $|u|^{p-1}u$ belong to $H^1_0(\\Omega )$ and $L^2(\\Omega )$ respectively, and they satisfy (REF ) almost everwhere in $\\Omega $ .",
"Similarly, one can define a one-fourth power operator of the pseudo-relativistic operator by $(-\\Delta +m^2)^{1/4}u := \\sum _{i=1}^\\infty c_i(\\lambda _i+m^2)^{1/4}\\phi _i$ We say that a function $u$ is a weak solution of (REF ) if $u$ and $|u|^{p-1}u$ belong to $H^{1/2}_0(\\Omega )$ and $L^{\\frac{2n}{n+1}}(\\Omega )$ respectively, and they satisfy $\\int _{\\Omega }(-\\Delta +m^2)^{1/4}u\\cdot (-\\Delta +m^2)^{1/4}v-muv +|u|^{p-1}uv\\,dx = 0, \\quad \\text{for all } v \\in H^{1/2}_0(\\Omega ).$ It is obvious that any strong solution is a weak solution and any weak solution $u$ with the property $u \\in H^1_0(\\Omega ),\\, |u|^{p-1}u \\in L^2(\\Omega )$ is a strong solution.",
"For the $H^{1/2}$ critical or subcritical range of $p$ , i.e., $1 < p \\le \\frac{n+1}{n-1}$ , it is also easy to see that the set of weak solutions for (REF ) is the same as the set of critical points of the functional $I_m(u) := \\int _{\\Omega }\\frac{1}{2}|(-\\Delta +m^2)^{1/4}u|^2-\\frac{m}{2}u^2+\\frac{1}{p+1}|u|^{p+1}\\,dx,$ which is continuously differentiable on $H^{1/2}_0(\\Omega )$ .",
"A weak solution $u$ of (REF ) is said to be of least energy if $u$ satisfies $I_m(u) = \\inf \\lbrace I_m(v) ~|~ v \\lnot \\equiv 0,\\, I_m^{\\prime }(v) = 0\\rbrace .$"
],
[
"Localization",
"One ingredient for Theorem REF is to change the original problem (REF ) to an equivalent problem which contains only local differential operators.",
"This kind of technique was introduced by Caffarelli-Silvestre [7] for the localization of the fractional Laplacians $(-\\Delta )^s$ on the whole domain $\\mathbb {R}^n$ .",
"The localization for the spectral fractional Laplacian on a bounded domain was given by Cabre-Tan [6].",
"The localization of the problem turns out to be powerful when we use various useful tools, such as the Pohozaev type identities, the Kelvin transform, and the moving plane method.",
"In this subsection, we briefly review it.",
"We consider a cylinder $\\mathcal {C}:= \\Omega \\times \\mathbb {R}_+$ .",
"The symbol $\\partial _L\\mathcal {C}$ denotes the lateral boundary of $\\mathcal {C}$ , defined by $\\lbrace (x,t) \\in \\mathcal {C} ~|~ x \\in \\partial \\Omega ,\\, t > 0\\rbrace $ .",
"We shall work in the space $H_{0,L}^1 (\\mathcal {C}) = \\lbrace U \\in H^1 (\\mathcal {C}) ~|~ U(x,t) = 0 \\quad \\textrm {on}~\\partial _{L} \\mathcal {C}\\rbrace ,$ which is equipped with the norm $\\Vert U\\Vert := \\left(\\int _{\\mathcal {C}}|\\nabla U(x,t)|^2\\,dxdt\\right)^{1/2}.$ For a given $u \\in H_0^{1/2}(\\Omega )$ , consider a unique function $U \\in H_{0,L}^1 (\\mathcal {C})$ satisfying $\\left\\lbrace \\begin{array}{lll}(-\\Delta + m^2) U(x,t) =0 &\\quad \\textrm {in}~\\mathcal {C},\\\\U(x,t) = 0&\\quad \\textrm {on}~\\partial _L \\mathcal {C},\\\\U(x,0) = u(x) &\\quad \\textrm {on}~\\Omega \\times \\lbrace 0\\rbrace .\\end{array}\\right.$ Then, as in [6] and [4], the following equality holds in distributional sense: $\\sqrt{-\\Delta +m^2} \\,u (x) = \\frac{\\partial }{\\partial \\nu } U (x,0)\\quad x \\in \\Omega .$ Thus, our study of the problem (REF ) can be transformed into the study of the following localized problem: $\\left\\lbrace \\begin{array}{ll}(-\\Delta + m^2) U(x,t)= 0&\\quad \\textrm {in}~\\mathcal {C},\\\\U(x,t) = 0&\\quad \\textrm {on}~\\partial _{L}\\mathcal {C},\\\\\\partial _{\\nu } U(x,0) = mU(x,0) + |U(x,0)|^{p-1}U(x,0) &\\quad \\textrm {on}~\\Omega \\times \\lbrace 0\\rbrace ,\\end{array}\\right.$ which is the Euler-Lagrange equation of the functional $I_{e,m}(U) = \\frac{1}{2}\\int _{\\mathcal {C}}|\\nabla U(x,t)|^2+m^2U(x,t)^2\\,dxdt -\\int _{\\Omega }\\frac{m}{2}U(x,0)^2+\\frac{1}{p+1}|U(x,0)|^{p+1}\\,dx$ defined on $H^1_{0,L}(\\mathcal {C})$ .",
"In other words, we may find a critical point of $I_{e,m}$ in order to find a weak solution of (REF ).",
"In addition, we note that $I_m(u) = I_{e,m}(U),$ whenever $U$ is an extension of $u$ given by (REF ).",
"Therefore, the least energy critical point of $I_{e,m}$ corresponds to the least energy critical point of $I_m$ ."
],
[
"Regularity of weak solutions",
"Here, we are concerned with the regularity property related to the problem (REF ).",
"Indeed, the required regularity can be demonstrated through minor modifications to the arguments in the proofs of Proposition 3.1 and Theorem 5.2 in [6], where the corresponding results were obtained for the case that $m=0$ .",
"Lemma 2.1 Assume that $v \\in H_{0,L}^1 (\\mathcal {C})$ is a weak solution of $\\left\\lbrace \\begin{array}{rll}-\\Delta v + m^2 v &=0&\\textrm {in}~\\mathcal {C},\\\\v&=0&\\textrm {on}~\\partial \\Omega \\times [0,\\infty ),\\\\\\partial _{\\nu } v&=f&\\textrm {on}~\\Omega \\times \\lbrace 0\\rbrace \\end{array}\\right.$ in the sense that $\\int _{\\mathcal {C}} \\nabla v \\cdot \\nabla \\phi + m^2 v \\,\\phi \\, dx dt = \\int _{\\Omega } f (x) \\phi (x,0) dx\\quad \\forall ~\\phi \\in H_{0,L}^1 (\\mathcal {C}).$ Then, we have the following regularity results.",
"If $f \\in L^{q}(\\Omega )$ and $v \\in L^q([0,R]\\times \\Omega )$ for some $q >n+1$ and $R > 0$ , then $v \\in C^{\\alpha } ([0,R/2]\\times \\overline{\\Omega })$ for some $\\alpha \\in (0,1)$ .",
"If $v \\in C^\\alpha ([0,R]\\times \\overline{\\Omega })$ , $f \\in C^{\\alpha } (\\overline{\\Omega })$ , and $f|_{\\partial \\Omega } \\equiv 0$ for some $R > 0$ and $\\alpha \\in (0,1)$ , then $v \\in C^{1,\\alpha }([0,R/2]\\times \\overline{\\Omega })$ .",
"If $v \\in C^{1,\\alpha }([0,R]\\times \\overline{\\Omega })$ , $f \\in C^{1,\\alpha }(\\overline{\\Omega })$ , and $f|_{\\partial \\Omega } \\equiv 0$ for some $R > 0$ and $\\alpha \\in (0,1)$ , then $v\\in C^{2,\\alpha }([0,R/2]\\times \\overline{\\Omega })$ .",
"The proof of this lemma follows along exactly the same lines as that of [6].",
"The fundamental idea in [6] is to make use of the function $w(x,t) = \\int _0^t v(x,s) ds\\quad \\textrm {for}~ (x,t) \\in \\mathcal {C}.$ This function satisfies $\\partial _t (-\\Delta + m^2) w = (-\\Delta +m^2) \\partial _t w =(-\\Delta +m^2) v =0,$ which means that $(-\\Delta +m^2) w$ is independent of $y$ .",
"Note that $(-\\Delta + m^2) w (x,0) = - \\partial _t v(x,0) = f(x)$ for $x \\in \\Omega $ .",
"Hence, $w$ satisfies $\\left\\lbrace \\begin{aligned}(-\\Delta +m^2) w (x,t) &= f (x) &\\textrm {in}~\\mathcal {C},\\\\w&=0&\\textrm {on}~\\partial \\mathcal {C},\\end{aligned}\\right.$ and if we extend $w$ by odd reflection to the whole cylinder $\\Omega \\times \\mathbb {R}$ as $w_{odd} (x,t) = \\left\\lbrace \\begin{array}{ll} w(x,t) &\\textrm {for}~ t \\ge 0,\\\\-w(x,-t)&\\textrm {for}~t \\le 0,\\end{array}\\right.$ then the function $w_{odd}$ satisfies the same problem (REF ) on $\\Omega \\times (-\\infty , \\infty )$ .",
"At this stage, we may use the classical elliptic regularity of $w_{odd}$ , as in [6], which implies the regularity properties of the function $v$ .",
"We refer the reader to [6] for further details.",
"From Lemma REF , the following corollary easily follows.",
"Corollary 2.2 For $p > 1$ , any $L^\\infty $ bounded weak solution of (REF ) belongs to $C^{2,\\alpha }(\\overline{\\Omega })$ for some $\\alpha \\in (0, 1)$ .",
"Note that for any weak solution $u$ of (REF ), there exists a unique function $U$ such that $U(x,0) = u(x)$ satisfying $\\left\\lbrace \\begin{array}{rll}-\\Delta U + m^2 U &=0&\\textrm {in}~\\mathcal {C},\\\\U&=0&\\textrm {on}~\\partial \\Omega \\times [0,\\infty ),\\\\\\partial _{\\nu } U&= |u|^{p-1}u&\\textrm {on}~\\Omega \\times \\lbrace 0\\rbrace .\\end{array}\\right.$ From the maximum principle, one has $\\sup _{\\Omega \\times [0,\\infty )}|U| \\le \\Vert u\\Vert _{L^\\infty (\\Omega )}$ so Lemma REF applies.",
"The next corollary, an another application of Lemma REF says every weak solution to (REF ) is $C^{2,\\alpha }(\\overline{\\Omega })$ when $p$ is $H^{1/2}$ subcritical or critical.",
"Corollary 2.3 Let $p \\in (1, \\frac{n+1}{n-1}]$ .",
"Then, any weak solution of (REF ) belongs to $C^{2,\\alpha }(\\overline{\\mathcal {C}})$ for some $\\alpha \\in (0,1)$ .",
"In particular, any weak solution of (REF ) belongs to $C^{2,\\alpha }(\\overline{\\Omega })$ so that it is a strong solution.",
"In the spirit of Brezis-Kato estimates, we first need to show that $U \\in L^q (\\mathcal {C})$ and $U(\\cdot ,0) \\in L^q(\\Omega )$ for all $q > 1$ .",
"(See, for example, Theorem 5.2 in [6].)",
"Indeed, if we define $U_T = \\min \\lbrace |U|, T\\rbrace $ for $T >1$ and test $U U_T^{2\\beta }$ for (REF ) for $\\beta \\ge 0$ , then we find that $\\int _{\\mathcal {C}} \\nabla U \\nabla (U U_T^{2\\beta }) dx dt + m^2 \\int _{\\mathcal {C}} U^2 U_T^{2\\beta } dxdt = \\int _{\\Omega \\times \\lbrace 0\\rbrace } |U(x,0)|^{p+1} |U_T (x,0)|^{2\\beta } dxdt.$ This implies that $\\int _{\\mathcal {C}} \\nabla U \\nabla (U U_T^{2\\beta }) dx dt \\le \\int _{\\Omega \\times \\lbrace 0\\rbrace } |U(x,0)|^{p+1} |U_T (x,0)|^{2\\beta } dxdt,$ Then, all of the arguments of [6] can be applied in exactly the same manner, to obtain that $\\int _{\\mathcal {C}}|\\nabla (UU_T^\\beta )|^2\\le C(\\beta +1)\\int _{\\Omega \\times \\lbrace 0\\rbrace } |U(x,0)|^{p+1} |U_T (x,0)|^{2\\beta } dxdt$ and consequently $U(\\cdot ,0) \\in L^q(\\Omega )$ for any $q > 1$ .",
"Then, we may apply the Poincare inequality (REF ) to the left hand side of (REF ) and take $T \\rightarrow \\infty $ , to prove that $U \\in L^q (\\mathcal {C})$ for any $q>1$ .",
"Finally, we apply Lemma REF repeatedly to show that $U \\in C^{2,\\alpha }(\\overline{\\mathcal {C}})$ for some $\\alpha \\in (0,1)$ .",
"Note that Corollary REF gives a proof of regularity part of Theorem REF ."
],
[
"Embeddings and entire problems",
"In this subsection, we review the best Sobolev embedding, along with the related entire problems and maximizing functions.",
"Given any $\\lambda > 0$ and $\\xi \\in \\mathbb {R}^n$ , let $w_{\\lambda , \\xi } (x) = \\mathfrak {c}_{n} \\left( \\frac{\\lambda }{\\lambda ^2+|x-\\xi |^2}\\right)^{\\frac{n-1}{2}} \\quad \\text{for } x \\in \\mathbb {R}^n,$ where $\\mathfrak {c}_{n} = 2^{\\frac{n-1}{2}} \\left( \\frac{\\Gamma \\left( \\frac{n+1}{2}\\right)}{\\Gamma \\left( \\frac{n-1}{2}\\right)}\\right)^{\\frac{n-1}{2}}.$ We recall the sharp fractional Sobolev inequality for $n>1$ , $\\left( \\int _{\\mathbb {R}^n} |f(x)|^{\\frac{2n}{n-1}} dx \\right)^{\\frac{n-1}{2n}} \\le \\mathcal {S}_{n} \\left( \\int _{\\mathbb {R}^n} |(-\\Delta )^{1/4} f(x)|^2 dx \\right)^{1 \\over 2} \\quad \\text{for any } f \\in H^{1/2}(\\mathbb {R}^n)$ with the best constant $\\mathcal {S}_{n} =2^{-\\frac{1}{2}} \\pi ^{-1/4} \\biggl [\\frac{\\Gamma \\left(\\frac{n-1}{2}\\right)}{\\Gamma \\left( \\frac{n+1}{2}\\right)}\\biggr ]^{1 \\over 2} \\biggl [ \\frac{\\Gamma (n)}{\\Gamma (n/2)}\\biggr ]^{1 \\over 2n}.$ The equality holds if and only if $u(x) = c w_{\\lambda ,\\xi }(x)$ for any $c > 0,\\ \\lambda >0$ and $\\xi \\in \\mathbb {R}^n$ (refer to [8], [17], [21]).",
"Furthermore, it was shown in [11], [20], [25] that $\\lbrace w_{\\lambda , \\xi }(x): \\lambda >0, \\xi \\in \\mathbb {R}^n \\rbrace $ is the set of all solutions for the problem $(-\\Delta )^{1/2} u = u^{(n+1)/(n-1)},\\quad u > 0 \\quad \\text{in } \\mathbb {R}^n\\quad \\textrm {and}\\quad \\lim _{|x|\\rightarrow \\infty } u(x) = 0.$ We use $W_{\\lambda ,\\xi } \\in \\mathcal {D}^1(\\mathbb {R}^{n+1}_+)$ to denote the (unique) harmonic extension of $w_{\\lambda ,\\xi }$ , so that $W_{\\lambda , \\xi }$ solves $\\left\\lbrace \\begin{aligned}-\\Delta W_{\\lambda ,\\xi }(x,t) &= 0 &\\quad \\text{in~} \\mathbb {R}^{n+1}_+,\\\\W_{\\lambda ,\\xi }(x,0) &= w_{\\lambda ,\\xi }(x) &\\quad \\text{for } x \\in \\mathbb {R}^n.\\end{aligned}\\right.$ It is easy to check that $W_{\\lambda , \\xi } (x,t) = \\mathfrak {c}_n \\left( \\frac{\\lambda }{|x-\\xi |^2+(t+\\lambda )^2} \\right)^{\\frac{n-1}{2}}\\quad \\textrm {for}~(x,t) \\in \\mathbb {R}^n \\times [0,\\infty ),$ and for any $U \\in D^1(\\mathbb {R}^{n+1}_+)$ , one has the trace Sobolev inequality $\\left( \\int _{\\mathbb {R}^n} |U(x,0)|^{\\frac{2n}{n-1}} dx \\right)^{\\frac{n-1}{2n}}\\le \\mathcal {S}_{n} \\left( \\int _0^{\\infty }\\!\\!\\!\\int _{\\mathbb {R}^n} |\\nabla U(x,t)|^2 dx dt \\right)^{1 \\over 2},$ where the equality is attained by some function $U \\in \\mathcal {D}^1(\\mathbb {R}^{n+1}_+)$ if and only if $U(x,t) = c W_{\\lambda ,\\xi }(x,t)$ for any $c > 0,\\ \\lambda >0$ and $\\xi \\in \\mathbb {R}^n$ .",
"We remark that $W_{\\lambda ,\\xi }(x,t) = \\lambda ^{-\\frac{n-1}{2}}W_{1,0}\\left(\\frac{x-\\xi }{\\lambda },\\,\\frac{t}{\\lambda }\\right).$ After this section we simply denote $w_{1,0}$ and $W_{1,0}$ by $w_1$ and $W_1$ , respectively."
],
[
"Trace inequalities",
"Here, we collect some useful trace inequalities which will be invoked frequently in the rest of the paper.",
"Proposition 2.4 There exists a constant $C > 0$ depending only on $n, m$ and $\\Omega $ such that the following inequality holds for any $U \\in H_{0,L}^{1}(\\mathcal {C})$ : $\\left(\\int _{\\mathcal {C}}|\\nabla U(x,t)|^2 dx dt + m^2 \\int _{\\mathcal {C}} |U(x,t)|^2 dx dt - m \\int _{\\Omega } U(x,0)^2 dx\\right) \\ge C \\int _{\\mathcal {C}} |\\nabla U(x,t)|^2 dx dt.$ For given $A >0$ , we apply integration by parts and Young's inequality to get that $\\begin{split}\\int _{\\Omega } mU(x,0)^2 dx &=-m\\int _0^\\infty \\left\\lbrace \\frac{d}{dt}\\int _\\Omega U(x,t)^2 dx\\right\\rbrace dt\\\\&=- 2m \\int _{\\Omega }\\left( \\int _{0}^{\\infty } (\\partial _t U) (x,t) U(x,t) dt\\right) dx\\\\& \\le A m^2 \\int _{\\mathcal {C}} U(x,t)^2 dx dt + \\frac{1}{A} \\int _{\\mathcal {C}} |\\partial _t U(x,t)|^2 dx dt.\\end{split}$ In addition, by applying the Poincare inequality in the level $\\lbrace (x,h): x \\in \\Omega \\rbrace $ for each $h \\ge 0$ , we obtain the estimate $\\int _{\\mathcal {C}} |U(x,t)|^2 dx dt \\le C(\\Omega ) \\int _{\\mathcal {C}} |\\nabla _x U(x,t)|^2 dx dt.$ Now, we apply this estimate to find that $\\begin{split}&\\int _{\\mathcal {C}} |\\nabla U(x,t)|^2 dx dt + m^2 \\int _{\\mathcal {C}} |U(x,t)|^2 dxdt - m \\int _{\\Omega } U(x,0)^2 dx\\\\&\\quad \\quad \\quad \\ge \\frac{1}{2}\\int _{\\mathcal {C}} |\\partial _x U(x,t)|^2 dx dt + \\int _{\\mathcal {C}} |\\partial _t U(x,t)|^2 dx dt\\\\&\\qquad \\quad \\quad + \\left(m^2 + \\frac{C(\\Omega )}{2}\\right) \\int _{\\mathcal {C}} |U(x,t)|^2 dx dt - m\\int _{\\Omega } U(x,0)^2 dx.\\end{split}$ Then, applying the estimate (REF ) here with $A= \\left(\\frac{1}{m^2 + C(\\Omega )/2}\\right)$ , we get that $\\begin{split}&\\int _{\\mathcal {C}} |\\nabla U(x,t)|^2 dx dt + m^2 \\int _{\\mathcal {C}} |U(x,t)|^2 dxdt - m \\int _{\\Omega } U(x,0)^2 dx\\\\&\\quad \\quad \\quad \\ge \\frac{1}{2}\\int _{\\mathcal {C}} |\\partial _x U(x,t)|^2 dx dt + \\frac{C(\\Omega )/2}{m^2 + C(\\Omega )/2}\\int _{\\mathcal {C}} |\\partial _t U(x,t)|^2 dx dt\\\\&\\quad \\quad \\quad \\ge C_1 \\int _{\\mathcal {C}} |\\nabla U(x,t)|^2 dxdt,\\end{split}$ where we have set $C_1 = \\min \\left\\lbrace \\frac{1}{2}, \\frac{C(\\Omega )/2}{m^2 + C(\\Omega )/2}\\right\\rbrace $ .",
"Thus, the proof is complete.",
"As a byproduct of the above proof, we also obtain the following useful trace inequality.",
"Proposition 2.5 For any $U \\in H^1_{0,L}(\\mathcal {C})$ , it holds that $\\int _{\\Omega } mU^2(x,0)\\,dx \\le m^2 \\int _{\\mathcal {C}} U^2\\,dxdt + \\int _{\\mathcal {C}} |\\partial _t U|^2\\,dxdt.$ The following Sobolev-trace embedding is well known (for example, see [6]).",
"Proposition 2.6 For any $U \\in H^1_{0,L}(\\mathcal {C})$ , its trace $U(\\cdot ,0)$ is continuously embedded in $L^q(\\Omega )$ for every $q \\in (1, \\frac{2n}{n-1})$ , i.e., there exists a constant $C > 0$ depending only on $n, q$ and $\\Omega $ such that $\\Vert U(\\cdot ,0)\\Vert _{L^q(\\Omega )} \\le C\\Vert U\\Vert _{H^1_{0,L}(\\mathcal {C})}.$ Moreover, the embedding is compact for any $q \\in (1, \\frac{2n}{n-1})$ ."
],
[
"Nonexistence result (Proof of Theorem ",
"In this section, we prove non-existence of a nontrivial solution to the Dirichlet problem (REF ) in the $H^1$ -critical and supercritical cases.",
"For this purpose, we shall use the following Pohozaev type identity which is obtained by testing $x\\cdot \\nabla _x U$ to (REF ), not $(x,t)\\cdot \\nabla U$ .",
"Proposition 3.1 Let $U \\in H^1_{0, L}(\\mathcal {C}) \\cap L^{\\infty }(\\overline{\\mathcal {C}})$ be a weak solution of (REF ).",
"Then one has $\\begin{split}&\\left(\\frac{n-2}{2}\\right) \\int _{\\mathcal {C}} |\\nabla _x U|^2 dxdt+\\frac{n}{2} \\int _{\\mathcal {C}} |\\partial _t U|^2 dxdt+\\frac{nm^2}{2}\\int _{\\mathcal {C}} U^2 dx dt \\\\&\\qquad \\qquad -\\frac{n}{p+1}\\int _{\\Omega } U^{p+1} dx-\\frac{nm}{2} \\int _{\\Omega } U^2 dx+\\frac{1}{2} \\int _{\\partial _L \\mathcal {C}} (\\nu _x \\cdot x)\\left( \\frac{\\partial U}{\\partial \\nu _x}\\right)^2 dS = 0.\\end{split}$ By Corollary REF , we see that $U \\in C^2(\\mathcal {C})$ .",
"Multiplying the localized problem (REF ) by $x \\cdot \\nabla _x U$ and then integrating out, we get $m^2 \\int _{\\mathcal {C}} U \\cdot (x \\cdot \\nabla _x U) dxdt +\\int _{\\mathcal {C}} (-\\Delta U) \\cdot (x \\cdot \\nabla _x U) dx dt = 0 $ Integrating by parts, the first integral becomes $m^2 \\int _{\\mathcal {C}} U \\cdot (x \\cdot \\nabla _x U) dxdt=\\frac{m^2}{2}\\int _{\\mathcal {C}} x \\cdot \\nabla _x (U^2) dx dt = - \\frac{m^2}{2}\\int _{\\mathcal {C}}n U^2 dx dt,$ while the second integral becomes $\\begin{split}\\int _{\\mathcal {C}} (-\\Delta U) (x \\cdot \\nabla _x U) dxdt& =\\int _{\\partial \\mathcal {C}} -\\frac{\\partial U}{\\partial \\nu } ( x \\cdot \\nabla _x U) dx dt+ \\int _{\\mathcal {C}} \\nabla U \\cdot \\nabla ( x \\cdot \\nabla _x U) dxdt \\\\& = - \\int _{\\Omega } (U^p + mU) (x \\cdot \\nabla _x U) dx-\\int _{\\partial _L \\mathcal {C}} (\\nu _x\\cdot x) \\left| \\frac{\\partial U}{\\partial \\nu _x}\\right|^2 dS\\\\&\\quad + \\int _{\\mathcal {C}} \\nabla U \\cdot \\nabla ( x \\cdot \\nabla _x U) dxdt\\\\&=: I+II+III.\\end{split}$ By integration by parts as in (REF ), one has $I= -\\frac{n}{p+1} \\int _{\\Omega } U^{p+1} - \\frac{nm}{2} \\int _{\\Omega } U^2 dx.$ Next, we split $III$ as follows.",
"$III = \\int _{\\mathcal {C}}(\\nabla _x U) \\cdot \\nabla _x ( x \\cdot \\nabla _x U) dx dt + \\int _{\\mathcal {C}} \\partial _t U \\, \\partial _t ( x \\cdot \\nabla _x U) dx dt=:III_1+III_2.$ By integration by parts again, we write $\\begin{split}III_1&= \\sum _{i,j=1}^n \\int _{\\mathcal {C}}\\partial _i U \\partial _i (x_j \\partial _j U) dx dt= \\sum _{i,j=1}^n \\int _{\\mathcal {C}} \\partial _i U ( \\delta _{ij} \\partial _j U + x_j \\partial _{ij} U) dx dt \\\\& = \\int _{\\mathcal {C}} |\\nabla _x U|^2 dx dt + \\int _{\\mathcal {C}} \\frac{x}{2} \\nabla _x |\\nabla _x U|^2 dx dt \\\\& = \\left(1- \\frac{n}{2}\\right) \\int _{\\mathcal {C}} |\\nabla _x U|^2 dxdt + \\int _{\\partial _L \\mathcal {C}} \\frac{1}{2} (\\nu _x \\cdot x) \\left| \\frac{\\partial U}{\\partial \\nu _x}\\right|^2 dS.\\end{split}$ On the other hand, since $U=0$ on $\\partial _{L}\\mathcal {C}$ implies $\\partial _tU = 0$ on $\\partial _L\\mathcal {C}$ , we have $\\begin{split}III_2&= \\frac{1}{2} \\int _{\\mathcal {C}} x\\cdot \\nabla _x (\\partial _t U)^2 dx dt \\\\&= \\frac{1}{2} \\int _{\\partial _L \\mathcal {C}} (\\nu _x \\cdot x) (\\partial _t U)^2 dxdt-\\frac{n}{2} \\int _{\\mathcal {C}} (\\partial _t U)^2 dxdt \\\\&= -\\frac{n}{2} \\int _{\\mathcal {C}} (\\partial _t U)^2 dxdt.\\end{split}$ Inserting the above identities (REF ), (REF ) and (REF ) into (REF ), we get $\\begin{split}\\int _{\\mathcal {C}} (-\\Delta U) (x \\cdot \\nabla _x U) dxdt& = \\frac{n}{p+1} \\int _{\\Omega } U^{p+1} dx + \\frac{mn}{2} \\int _{\\Omega } U^2 dx-\\int _{\\partial _L \\mathcal {C}} \\frac{1}{2}(\\nu _x \\cdot x) \\left| \\frac{\\partial U}{\\partial \\nu _x}\\right|^2 dS \\\\&\\quad - \\frac{n}{2} \\int _{\\mathcal {C}} |\\partial _t U|^2 dx dt + \\left(1- \\frac{n}{2}\\right) \\int _{\\mathcal {C}} |\\nabla _x U|^2 dxdt.\\end{split}$ Therefore, we conclude that $\\begin{split}&\\left( \\frac{n}{2} - 1 \\right) \\int _{\\mathcal {C}} |\\nabla _x U|^2 dxdt+\\frac{n}{2} \\int _{\\mathcal {C}} |\\partial _t U|^2 dxdt+\\frac{nm^2}{2}\\int _{\\mathcal {C}} U^2 dx dt \\\\&\\qquad \\qquad -\\frac{n}{p+1}\\int _{\\Omega } U^{p+1} dx-\\frac{mn}{2} \\int _{\\Omega } U^2 dx+\\frac{1}{2} \\int _{\\partial _L \\mathcal {C}} (\\nu _x \\cdot x)\\left| \\frac{\\partial U}{\\partial \\nu _x}\\right|^2 dS = 0,\\end{split}$ which is (REF ).",
"Theorem REF follows from combining the Pohozaev identity (REF ), the Nehari identity and the trace inequality.",
"Let $u$ be a bounded weak solution of (REF ) with $p \\ge (n+2)/(n-2)$ .",
"Let $U \\in H^1_{0,L}(\\mathcal {C})$ be a extension of $u$ given by (REF ).",
"Then $U$ satisfies (REF ) and $\\Vert U\\Vert _{L^{\\infty }(\\mathcal {C})} \\le \\Vert U(x,0)\\Vert _{L^{\\infty }(\\Omega )} <\\infty $ by the maximum principle.",
"Multiplying (REF ) by $U$ and taking an integration by parts, we obtain $\\int _{\\Omega } mU^2 + U^{p+1} dx = m^2 \\int _{\\mathcal {C}} U^2 dx + \\int _{\\mathcal {C}} |\\nabla U|^2\\,dxdt.$ Multiplying (REF ) by $-\\frac{n}{p+1}$ and summing up with (REF ), we get $\\begin{split}\\left(\\frac{n}{2}-\\frac{n}{p+1}\\right)\\int _{\\Omega } mU^2\\,dx & =\\left(\\frac{n}{2}-\\frac{n}{p+1}\\right)m^2\\int _{\\mathcal {C}}U^2\\,dxdt+\\frac{1}{2}\\int _{\\partial _L \\mathcal {C}}(\\nu _x \\cdot x)\\left|\\frac{\\partial U}{\\partial \\nu _x}\\right|^2 dS_z \\\\&\\quad + \\left( \\frac{n}{2} - \\frac{n}{p+1}\\right) \\int _{\\mathcal {C}}|\\partial _t U|^2 dxdt+ \\left(\\frac{n}{2}-1-\\frac{n}{p+1}\\right)\\int _{\\mathcal {C}}|\\nabla _x U|^2\\,dxdt.\\end{split}$ Since $p \\ge \\frac{n+2}{n-2}$ , we have $\\left( \\frac{n}{2} - 1 - \\frac{n}{p+1}\\right) \\int _{\\mathcal {C}} |\\nabla _x U|^2 dx dt \\ge 0.$ Recall the trace inequality (REF ) $m^2 \\int _{\\mathcal {C}} U^2 dx dt + \\int _{\\mathcal {C}} |\\partial _t U|^2 dx dt \\ge \\int _{\\Omega } mU^2(x,0) dx,$ from which we conclude that $\\frac{1}{2} \\int _{\\partial _L \\mathcal {C}} (\\nu _x \\cdot x) \\left| \\frac{\\partial U}{\\partial \\nu _x}\\right|^2 dS_z = 0.$ Then we deduce $U \\equiv 0$ from the unique continuation property.",
"This completes the proof."
],
[
"Existence in $H^{1/2}$ subcritical case",
"In this section, we shall construct a positive least energy solution of (REF ) for all $m > 0$ and $p\\in (1, (n+1)/(n-1))$ .",
"We note that once we find a least energy solution of (REF ), the sign-definiteness of it follows from a standard argument.",
"For example, we refer to Theorem 1.2 in [13].",
"Thus, we may focus ourselves on construction of a least energy solution of (REF ).",
"Let $p \\in (1,\\, (n+1)/(n-1))$ .",
"We recall that $I_{e,m}(U) = \\frac{1}{2} \\int _{\\mathcal {C}} |\\nabla U|^2 +m^2 U^2 dx dt - \\frac{m}{2} \\int _{\\Omega }U^2 (x,0) dx - \\frac{1}{p+1} \\int _{\\Omega }|U(x,0)|^{p+1}dx.$ As is mentioned in Section , we may find a least energy critical point of $I_{e,m}$ to find a least energy critical point of $I_{e,m}$ .",
"We shall search for a minimizer of $I_{e,m}$ on the Nehari manifold $\\mathcal {N}_{e,m} := \\lbrace V \\in H^1_{0,L}(\\mathcal {C}) ~|~ J_{e,m}(V) = 0,\\, V \\lnot \\equiv 0 \\rbrace ,$ where $J_{e,m}(V) := I^{\\prime }_{e,m}(V)V$ , i.e., $J_{e,m}(V) =\\int _{\\mathcal {C}} |\\nabla V (x,t)|^2 + m^2 V(x,t)^2 dxdt -m\\int _{\\Omega } V(x,0)^2 dx - \\int _{\\Omega } |V(x,0)|^{p+1} dx.$ Lemma 4.1 $\\mathcal {N}_{e,m}$ is nonempty.",
"Choose any nonzero $V \\in H^1_{0,L}(\\mathcal {C})$ .",
"Then as a function of $t$ , one can see $I_{e,m}(tV) = \\frac{t^2}{2}\\left(\\int _{\\mathcal {C}}|\\nabla V|^2+m^2 V^2\\,dxdy-m\\int _{\\Omega }V(x,0)^2\\,dx\\right)-\\frac{t^{p+1}}{p+1}\\int _{\\Omega }|V(x,0)|^{p+1}\\,dx$ attains a unique local maximum at some $t_0 \\in (0,\\,\\infty )$ .",
"Differentiating with respect to $t$ , we get $I_{e,m}^{\\prime }(t_0 V)V = 0$ , which says $t_0 V \\in \\mathcal {N}_{e,m}$ .",
"Lemma 4.2 There exists a minimizer of $I_{e,m}$ subject to $\\mathcal {N}_{e,m}$ , i.e., there exists a function $U \\in \\mathcal {N}_{e,m}$ such that $I_{e,m}(U) = M_{e,m}$ .",
"Take a minimizing sequence $\\lbrace V_j\\rbrace $ of $I_{e,m}$ subject to $\\mathcal {N}_{e,m}$ .",
"Then one has $\\left(\\frac{1}{2}-\\frac{1}{p+1}\\right)\\left(\\int _{\\mathcal {C}}|\\nabla V_j|^2+m^2 V_j^2\\,dxdt-m\\int _{\\Omega }V_j(x,0)^2\\,dx\\right)= I_{e,m}(V_j) < C,$ from which and Proposition REF we deduce $\\Vert V_j\\Vert $ is bounded for $j$ .",
"Then there exists some $V_0 \\in H^1_{0,L}(\\mathcal {C})$ such that, after extracting a subsequence, $V_j \\rightharpoonup V_0$ weakly in $H^1_{0,L}(\\mathcal {C})$ and $V_j(x,0) \\rightarrow V_0(x,0)$ strongly in $L^{p+1}(\\Omega )$ by Proposition REF .",
"The weakly lower semi-continuity of the functional $I_{e,m}$ and $J_{e,m}$ , again came from Proposition REF , says $I_{e,m}(V_0) \\le M_{e,m}$ and $J_{e,m}(V_0) \\le 0$ .",
"To show that $V_0\\ne 0$ by contradiction, suppose that $V_0 \\equiv 0$ .",
"Then one has $V_j(x,0) \\rightarrow 0$ in $L^{p+1}(\\Omega )$ but since $J_{e,m}(V_j) = 0$ , we have $\\begin{split}\\int _{\\Omega } |V_j (x,0)|^{p+1}\\,dx&=\\int _{\\mathcal {C}} |\\nabla V_j(x,t)|^2 + m^2V_j(x,t)^2\\,dxdt -m\\int _{\\Omega } V_j(x,0)^2\\,dx\\\\& \\ge C\\int _{\\mathcal {C}} |\\nabla V_j(x,t)|^2\\,dxdt\\\\& \\ge C \\Vert V_j (\\cdot , 0)\\Vert _{L^{p+1}(\\Omega )}^2,\\end{split}$ from the trace inequality (REF ) and trace Sobolev inequality.",
"This shows that $\\Vert V_j(\\cdot ,0)\\Vert _{L^p(\\mathbb {R}^n)}$ is bounded below from $C^{\\frac{1}{p-1}}$ and consequently $V_0 \\lnot \\equiv 0$ .",
"Then it is easy to see there is $t_0 \\in (0, 1]$ such that $I_{e,m}(t_0V_0) \\le M_{e,m}$ and $J_{e,m}(t_0V_0) = 0$ .",
"This completes the proof of Lemma REF .",
"Let $U$ be the minimizer obtained in Lemma REF .",
"Since $U \\in \\mathcal {N}_{e,m}$ , $\\begin{aligned}J_{e,m}^{\\prime }(U)U &= 2\\left(\\int _{\\mathcal {C}}|\\nabla U|^2+m^2 U^2\\,dxdt-m\\int _{\\Omega }U (x,0)^2\\,dx\\right) -(p+1) \\int _{\\Omega } |U(x,0)|^{p+1}\\,dx \\\\&= (1-p)\\int _{\\Omega } |U(x,0)|^{p+1}\\,dx \\ne 0\\end{aligned}$ so $J_{e,m}^{\\prime }(U) \\ne 0$ .",
"Then the Lagrange multiplier rule applies to see that for some $\\lambda \\in \\mathbb {R}$ , $I_{e,m}^{\\prime }(U) = \\lambda J_{e,m}^{\\prime }(U).$ By testing $U$ to (REF ), we see $\\lambda = 0$ so that $U$ is a nontrivial solution of (REF ).",
"We finally see that $U$ is a least energy solution of (REF ) since, for every nontrivial solution $V$ of (REF ), one must have $V \\in \\mathcal {N}_{e,m}$ by testing $V$ to the equation (REF )."
],
[
"Existence in $H^{1/2}$ critical case",
"In this section, we prove Theorem REF in the critical case $p =\\frac{n+1}{n-1}$ by finding a critical point of the functional $I_{e,m} (U) = \\frac{1}{2} \\left[ \\int _{\\mathcal {C}} |\\nabla U(x,t)|^2 + m^2 U(x,t)^2\\,dxdt-m\\int _{\\Omega }U(x,0)^2\\, dx \\right] - \\frac{n-1}{2n} \\int _{\\Omega }|U(x,0)|^{\\frac{2n}{n-1}}\\,dx.$ Due to the loss of compactness of the embedding $H^1_{0,L}(\\mathcal {C})\\big |_{\\Omega } \\hookrightarrow L^{2n/(n-1)}(\\Omega )$ , the first step we have to do would be characterizing the levels of $I_{e,m}$ at which the Palais-Smale condition holds.",
"Lemma 5.1 The Palais-Smale condition holds for $I_{e,m}$ at any level $B < \\frac{1}{2n}\\mathcal {S}_n^{-2n}$ .",
"To verify the lemma, assume that $\\lbrace U_k\\rbrace _{k=1}^\\infty $ is a sequence in $H_{0,L}^{1}(\\mathcal {C})$ such that $\\lim _{k\\rightarrow \\infty }I_{e,m}^{\\prime }(U_k) = 0 \\quad \\textrm {and}\\quad \\lim _{k \\rightarrow \\infty } I_{e,m} (U_k) = B$ for some $B < \\frac{1}{2n}\\mathcal {S}_n^{-2n}$ .",
"We need to show that $\\lbrace U_k\\rbrace _{k=1}^\\infty $ converges in $H_{0,L}^{1} (\\mathcal {C})$ up to a subsequence.",
"We observe from (REF ) that $\\sup _{k\\in \\mathbb {N}} \\int _{\\Omega }|U_k(x,0)|^{\\frac{2n}{n-1}}\\, dx < \\infty \\quad \\textrm {and}\\quad \\sup _{k \\in \\mathbb {N}} \\left(\\int _{\\mathcal {C}} |\\nabla U_k|^2 + m^2 U_k^2\\,dxdt-m\\int _{\\Omega }U_k(x,0)^2\\,dx\\right) < \\infty .$ Hence $\\lbrace U_k\\rbrace _{k=1}^\\infty $ has a weakly convergent subsequence in $H_{0,L}^1 (\\mathcal {C})$ , and its limit $U \\in H_{0,L}^1 (\\mathcal {C})$ satisfies $\\left\\lbrace \\begin{array}{rll} (-\\Delta +m^2) U &=0\\quad &\\textrm {in}~\\mathcal {C},\\\\\\partial _{\\nu } U &= m U + U^{\\frac{n+1}{n-1}}~&\\textrm {on}~\\Omega \\times \\lbrace 0\\rbrace .\\end{array}\\right.$ By the embedding properties, we know that $U_k(\\cdot ,0) \\rightarrow U(\\cdot ,0)$ in $L^2 (\\Omega )$ , $U_k \\rightharpoonup U$ in $L^{\\frac{2n}{n-1}}(\\Omega )$ and $U_k \\rightharpoonup U$ in $L^2 (\\mathcal {C})$ .",
"Hence we have $\\lim _{k \\rightarrow \\infty } \\int _{\\mathcal {C}} |\\nabla (U_k - U)|^2+|U_k - U|^2 \\,dxdt = \\lim _{k \\rightarrow \\infty } \\int _{\\mathcal {C}} |U_k|^2 + |\\nabla U_k|^2 dx dt - \\int _{\\mathcal {C}} U^2 + |\\nabla U|^2 dx,$ and the Brezis-Lieb lemma implies that $\\lim _{k \\rightarrow \\infty } \\int _{\\Omega }|U_k - U|^{\\frac{2n}{n-1}} dx = \\lim _{k \\rightarrow \\infty } \\int _{\\Omega } |U_k|^{\\frac{2n}{n-1}} dx - \\int _{\\Omega } |U|^{\\frac{2n}{n-1}} dx.$ From the fact that $\\lim _{k \\rightarrow \\infty } I_{e,m}^{\\prime }(U_k)U_k =0$ , we have $0 = \\lim _{k \\rightarrow \\infty }\\left( \\int _{\\mathcal {C}} |\\nabla U_k|^2 + m^2|U_k|^2\\,dxdt -\\int _{\\Omega } m|U_k|^2\\,dx -\\int _{\\Omega }|U_k|^{\\frac{2n}{n-1}}\\,dx\\right)$ and from (REF ), $0 = \\int _{\\mathcal {C}} |\\nabla U|^2 + m^2|U|^2\\,dxdt -\\int _{\\Omega } m|U|^2\\,dx -\\int _{\\Omega }|U|^{\\frac{2n}{n-1}}\\,dx.$ Subtracting (REF ) from (REF ), we see from (REF ), (REF ) and $L^2$ convergence $U_k (\\cdot ,0) \\rightarrow U(\\cdot ,0)$ in $\\Omega $ that $0 = \\lim _{k \\rightarrow \\infty } \\left(\\int _{\\mathcal {C}} |\\nabla (U_k -U)|^2 + m^2|U_k -U|^2 \\,dxdt- \\int _{\\Omega } |U_k -U|^{\\frac{2n}{n-1}}\\,dx\\right).$ After extracting a subsequence, we define a nonnegative value $J\\ge 0$ by $J: = \\lim _{k\\rightarrow \\infty } \\int _{\\Omega } |U_k -U|^{\\frac{2n}{n-1}} dx =\\lim _{k \\rightarrow \\infty } \\int _{\\mathcal {C}} |\\nabla (U_k -U)|^2 +m^2|U_k -U|^2\\,dxdt.$ If $J =0$ , we are done.",
"Suppose that $J>0$ .",
"Then, using the trace Sobolev inequality (REF ), we get $\\begin{split}J = \\lim _{k \\rightarrow \\infty } \\int _{\\Omega } |U_k -U|^{\\frac{2n}{n-1}}\\,dx &\\le \\mathcal {S}_n^{\\frac{2n}{n-1}}\\lim _{k\\rightarrow \\infty }\\left( \\int _{\\mathcal {C}} |\\nabla (U_k - U)|^2\\,dxdt \\right)^{\\frac{n}{n-1}}\\\\&\\le \\mathcal {S}_n^{\\frac{2n}{n-1}}\\lim _{k \\rightarrow \\infty } \\left( \\int _{\\mathcal {C}}|\\nabla (U_k - U)|^2 + m^2 |U_k -U|^2\\,dxdt \\right)^{\\frac{n}{n-1}}\\\\&\\le \\mathcal {S}_n^{\\frac{2n}{n-1}}J^{\\frac{n}{n-1}},\\end{split}$ which shows $\\mathcal {S}_n^{-2n} \\le J$ .",
"Combining this with (REF ), we get $\\begin{split}&\\lim _{k \\rightarrow \\infty } \\left[ \\frac{1}{2} \\int _{\\mathcal {C}} |\\nabla (U_k -U)|^2 +|U_k - U|^2\\,dxdt - \\frac{n-1}{2n}\\int _{\\Omega } |U_k -U|^{\\frac{2n}{n-1}} dx \\right]\\\\& = \\lim _{k \\rightarrow \\infty } \\frac{1}{2n} \\int _{\\Omega } |U_k- U|^{\\frac{2n}{n-1}} dx \\ge \\frac{1}{2n} \\mathcal {S}_n^{-2n}.\\end{split}$ On the other hand, we may use again (REF ), (REF ) and the fact that $U_k (\\cdot ,0) \\rightarrow U(\\cdot ,0)$ in $L^2 (\\Omega )$ , to deduce that $\\lim _{k \\rightarrow \\infty } I_{e,m} (U_k) = I_{e,m} (U) + \\lim _{k \\rightarrow \\infty } \\left[ \\frac{1}{2} \\int _{\\Omega } |(U_k - U)|^2 + |\\nabla (U_k -U)|^2\\,dxdt - \\frac{n-1}{2n}\\int _{\\Omega } |U_k -U|^{\\frac{2n}{n-1}} dx \\right]$ Noting that $I_{e,m} (U) \\ge 0$ and inserting (REF ) into (REF ), we obtain $B \\ge \\frac{1}{2n}\\mathcal {S}_n^{-2n}$ , which is a contradiction to the assumption that $B < \\frac{1}{2n}\\mathcal {S}_n^{-2n}$ .",
"Hence $J=0$ holds and thus the lemma is proved.",
"We define the mountain pass level $L_{e,m}$ of $I_{e,m}$ by $L_{e,m} := \\inf _{\\gamma \\in \\Gamma }\\max _{t\\in [0,1]}I_{e,m}(\\gamma (t)),$ where $\\Gamma := \\lbrace \\gamma \\in C([0,1],\\,H^1_{0,L}(\\mathcal {C})) ~|~ \\gamma (0) = 0,\\, I_{e,m}(\\gamma (1)) < 0\\rbrace .$ Lemma 5.2 It holds that $L_{e,m} < \\frac{1}{2n}\\mathcal {S}_n^{-2n}$ In particular, the Palais-Smale condition holds for $I_{e,m}$ at the mountain pass level $L_{e,m}$ .",
"For any given $\\Psi \\in H_{0,L}^1 (\\mathcal {C})$ we have $I_{e,m} (t \\Psi ) = \\frac{1}{2} t^2 \\alpha - \\frac{n-1}{2n} t^{\\frac{2n}{n-1}} \\beta ,$ where $\\alpha = \\int _{\\mathcal {C}} |\\nabla \\Psi (x,t)|^2 + m^2 \\Psi (x,t)^2\\,dxdt - m\\int _{\\Omega }\\Psi (x,0)^2\\,dx \\quad \\textrm {and}\\quad \\beta = \\int _{\\Omega } |\\Psi (x,0)|^{\\frac{2n}{n-1}} dx.$ It is easy to see that the maximum value of the map $t \\in (0,\\infty ) \\mapsto I_{e,m} (t \\Psi )$ is attained at $t_0 = (\\alpha /\\beta )^{\\frac{n-1}{2}}$ and the value is equal to $\\begin{split}I_{e,m} (t_0 \\Psi ) &= \\frac{1}{2n} \\alpha ^n \\beta ^{-(n-1)}\\\\& = \\frac{1}{2n} \\frac{ \\left( \\int _{\\mathcal {C}} |\\nabla \\Psi |^2 + m^2 \\Psi ^2\\, dx dt -m\\int _{\\Omega }\\Psi (x,0)^2 dx\\right)^n}{ \\left( \\int _{\\Omega } |\\Psi (x,0)|^{\\frac{2n}{n-1}} dx \\right)^{n-1}}.\\end{split}$ In order to finish the proof, we need to find a function $\\Psi \\in H_{0,L}^1 (\\mathcal {C})$ such that $\\max _{t>0} I_{e,m}(t \\Psi ) < \\frac{1}{2n} \\mathcal {S}_n^{-2n}.$ We may suppose $0 \\in \\Omega $ .",
"We take $\\Psi _{\\lambda }(x,t) := \\phi (x)W_{\\lambda , 0}(x,t)$ where $W_{\\lambda ,0}$ is defined in (REF ) and $\\phi \\in C^\\infty _c(\\Omega )$ satisfying $\\phi = 1$ on some ball $B_\\rho (0) \\subset \\Omega $ .",
"We are now going to estimate each term of $I_{e,m}(t \\Psi _{\\lambda })$ using the decay $W_{1,0}(z) \\le C/(1+|z|)^{n-1}$ for $z \\in \\mathbb {R}^{n+1}_{+}$ .",
"First we estimate $\\begin{split}\\int _{\\mathcal {C}} |\\nabla \\Psi _{\\lambda }|^2 dx dt & = \\int _{\\mathcal {C}} |\\nabla V_{\\lambda }|^2 \\phi ^2\\,dxdt + \\int _{\\mathcal {C}} 2(\\phi V_{\\lambda })\\nabla V_{\\lambda }\\cdot \\nabla \\phi + V_{\\lambda }^2|\\nabla \\phi |^2\\, dxdt\\\\&= \\int _{\\mathbb {R}^{n+1}_{+}}|\\nabla V_{\\lambda }|^2 dx dt + \\int _{\\mathbb {R}^{n+1}_{+}} (1- \\phi ^2 )|\\nabla V_{\\lambda }|^2 dx dt + O (\\lambda ^{n-1})\\\\& = \\int _{\\mathbb {R}^{n+1}_{+}} |\\nabla V|^2 dx dt + O(\\lambda ^{n-1}),\\end{split}$ and similarly, $\\begin{split}\\int _{\\Omega }|\\Psi _{\\lambda }(x,0)|^{\\frac{2n}{n-1}} dx & = \\int _{\\mathbb {R}^{n}} |W_{\\lambda ,0}|^{\\frac{2n}{n-1}}(x,0) \\phi ^{\\frac{2n}{n-1}}(x,0) dx\\\\& = \\int _{\\mathbb {R}^n} |W_{\\lambda ,0}(x,0)|^{\\frac{2n}{n-1}} dx + \\int _{\\mathbb {R}^n} |W_{\\lambda }(x,0)|^{\\frac{2n}{n-1}} ( \\phi (x,0)^{\\frac{2n}{n-1}} - 1)\\,dx\\\\& = \\int _{\\mathbb {R}^n} |W_{1,0}(x,0)|^{\\frac{2n}{n-1}} dx + O(\\lambda ^{n}).\\end{split}$ Also we easily see that $\\int _{\\mathcal {C}} m^2 \\Psi _{\\lambda }^2\\,dxdt \\le \\int _{\\mathbb {R}^{n+1}_{+}} m^2 W_{\\lambda ,0}^2\\,dxdt = C m^2 \\lambda ^{-(n-1)}\\lambda ^{n+1} \\int _{\\mathbb {R}^{n+1}_{+}} W_{1,0}^2\\,dxdt = O(\\lambda ^{2}),$ and $\\begin{split}\\int _{\\Omega } m \\Psi _{\\lambda }^2 (x,0)\\,dx &= m \\lambda \\int _{\\mathbb {R}^n} W_{1,0}^2 (x,0)\\,dx + m \\int _{\\mathbb {R}^n}(\\phi ^2-1)W_{\\lambda ,0}^2(x,0)\\,dx\\\\&= m \\lambda \\int _{\\mathbb {R}^n} W_{1,0}^2 (x,0)\\,dx + O(\\lambda ^{n-1}).\\end{split}$ Merging the estimates above, we deduce that for small $\\lambda > 0$ , $\\begin{split}\\sup _{t >0} E (t \\psi _{\\lambda })& = \\frac{1}{2n} \\frac{\\left( \\int _{\\mathbb {R}^{n+1}_{+}} |\\nabla W_{1,0}|^2\\,dxdt-m\\lambda \\int _{\\mathbb {R}^n} W_{1,0}^2(x,0)\\, dx +O(\\lambda ^2)\\right)^n}{\\left( \\int _{\\mathbb {R}^n} |W_{1,0}(x,0)|^{\\frac{2n}{n-1}} dx + O (\\lambda ^{n}) \\right)^{n-1}}\\\\&< \\frac{1}{2n} \\frac{\\left( \\int _{\\mathbb {R}^{n+1}_+} |\\nabla W_{1,0}|^2 dxdt \\right)^n}{\\left( \\int _{\\mathbb {R}^n} |W_{1,0}(x,0)|^{\\frac{2n}{n-1}} dx \\right)^{n-1}}\\\\&= \\frac{1}{2n} \\int _{\\mathbb {R}^n} |W_{1,0}(x,0)|^{\\frac{2n}{n-1}} dx = \\frac{1}{2n}\\mathcal {S}_n^{-2n},\\end{split}$ where, in the last equalities, we have used that $\\left( \\int _{\\mathbb {R}^n} |W_{1,0}(x,0)|^{\\frac{2n}{n-1}} dx \\right)^{\\frac{n-1}{2n}} = \\mathcal {S}_n\\left( \\int _{\\mathbb {R}^{n+1}_{+}} |\\nabla W_{1,0}|^2 dx dt \\right)^{1/2}= \\mathcal {S}_n \\left( \\int _{\\mathbb {R}^n} |W_{1,0}(x,0)|^{\\frac{2n}{n-1}}dx \\right)^{1/2},$ which shows that $\\int _{\\mathbb {R}^n} |W_{1,0}(x,0)|^{\\frac{2n}{n-1}} dx = \\mathcal {S}_n^{-2n}.$ The proof is finished.",
"We are ready to complete the whole the proof of Theorem REF .",
"To complete the proof, it remains to show $I_{e,m}$ enjoys the mountain pass geometry.",
"In other words, we show that (MP1) there exists $r_0 > 0$ such that $I_{e,m}(V) \\ge 0$ for any $v \\in H^1_{0,L}(\\mathcal {C})$ with $\\Vert v\\Vert \\le r_0$ and $\\inf \\lbrace I_{e,m} (V) ~|~ V \\in H^1_{0,L}(\\mathcal {C}),\\, \\Vert V\\Vert = r_0\\rbrace > 0$ ; (MP2) there exists $V_0 \\in H^1_{0,L}(\\mathcal {C})$ such that $I_{e,m}(V_0) < 0$ .",
"By the trace Sobolev embedding and Proposition REF , wee see that $I_{e,m} (U) \\ge \\frac{1}{C} \\Vert U\\Vert ^2 - C \\Vert U\\Vert ^{\\frac{2n}{n-1}}$ so (MP1) holds.",
"It is easy to see (MP2) also holds by observing the scaling map $t \\mapsto I_{e,m}(tV)$ .",
"Then the existence of a (PS) sequence of $I_{e,m}$ at the mountain pass level $L_{e,m}$ follows from the standard pseudo gradient flow argument (see [33]) so that we have a critical point of $I_{e,m}$ with the level $L_{e,m}$ .",
"Finally, we note that for any nontrivial solution $U$ to (REF ), one must have $\\frac{d}{dt}\\bigg |_{t=1}I_{e,m}(tU) = I_{e,m}^{\\prime }(U)U = 0,$ which shows that $I_{e,m}(U) = \\max _{t>0}I_{e,m}(tU).$ This means that $L_{e,m}$ is the least energy level.",
"The proof is finished."
],
[
"Existence in $H^{1/2}$ supercritical case: Proof of Theorem ",
"In this section, we show that the problem (REF ) has a nontrivial solution even for the supercritical case $\\frac{n+1}{n-1} < p < \\frac{n+2}{n-2}$ under the assumption on $\\Omega $ that the equation (REF ) admits a non-degenerate solution $u_\\infty \\in H^1_0(\\Omega )$ , i.e., we assume there exists a nontrivial solution $u_{\\infty } \\in H^1_0(\\Omega )$ to $-\\Delta u=|u|^{p-1}u \\quad \\textup {in }\\Omega , \\quad u = 0 \\quad \\text{on } \\partial \\Omega $ such that the linearized equation $-\\Delta v = p|u_\\infty |^{p-1}v \\quad \\text{in } \\Omega , \\quad v = 0 \\quad \\text{on } \\partial \\Omega $ admits only trivial solution.",
"By the scaling $u \\mapsto (2m)^{\\frac{1}{p-1}}u$ , the equation (REF ) is transformed to $2m \\mathcal {P}_mu = |u|^{p-1}u.$ Then, we look for a solution $u_m$ of (REF ) of the form $u_m=u_\\infty +w.$ After inserting (REF ) into the equation (REF ) and doing some algebraic manipulation, we get $\\begin{split}\\left[ 2m\\mathcal {P}_m - p|u_{\\infty }|^{p-1}\\right] w =& \\Big \\lbrace |u_{\\infty } +w|^{p-1}(u_\\infty +w) - |u_{\\infty }|^{p-1}u_\\infty - p|u_{\\infty }|^{p-1}\\Big \\rbrace \\\\&\\quad + \\left[ (-\\Delta ) - 2m\\mathcal {P}_m\\right]u_{\\infty }.\\end{split}$ We now define operators $\\left\\lbrace \\begin{split}D_m(\\phi ) &:= \\left[ (-\\Delta ) - 2m\\mathcal {P}_m\\right]\\phi ,\\\\A_m(\\phi ) &:= \\left[ I -\\left(2m\\mathcal {P}_m\\right)^{-1} pu_{\\infty }^{p-1} \\right]\\phi ,\\\\Q(\\phi ) &:= |u_{\\infty } +\\phi |^{p-1}(u_\\infty +\\phi ) - |u_{\\infty }|^{p-1}u_\\infty - p|u_{\\infty }|^{p-1}\\phi \\end{split}\\right.$ acting on suitable function spaces on $\\Omega $ .",
"We then, in a formal sense, arrive at the equation $w = \\mathcal {L}_m \\left[ Q(w) +D_m(u_{\\infty })\\right]$ from (REF ) where we denote $\\mathcal {L}_m := A_m^{-1}(2m\\mathcal {P}_m)^{-1}$ .",
"We define $\\Phi _m(\\phi ) := \\mathcal {L}_m \\left[ Q(\\phi ) +D_m(u_{\\infty })\\right].$ For Theorem REF , it suffices to show $\\Phi $ is contractive on a small ball in $L^q(\\Omega )$ for every $q \\ge np$ .",
"Indeed, if this is done, then we conclude that $u=u_\\infty +w$ , where $w$ is the (unique) fixed point of $\\Phi $ , is a strong solution of (REF ).",
"We provide the following useful lemmas and propositions in order.",
"Recall that $\\lbrace \\lambda _n,\\, \\phi _n\\rbrace _{n=1}^\\infty $ denotes the complete $L^2$ orthonormal system of eigenvalues and eigenfunctions of $\\left\\lbrace \\begin{aligned}-\\Delta \\phi &= \\lambda \\phi \\quad \\text{ in } \\Omega \\\\\\phi &= 0 \\quad \\text{ on } \\partial \\Omega .\\end{aligned}\\right.$ Lemma 6.1 The inverse map $\\mathcal {P}_m^{-1}$ given by $\\mathcal {P}_m^{-1}\\phi := \\sum _{i=1}^\\infty \\frac{c_i}{\\sqrt{\\lambda _i+m^2}-m}\\phi _i, \\quad \\phi = \\sum _{i=1}^\\infty c_i\\phi _i$ is a bounded map on $L^q(\\Omega )$ to $W^{1,q}_0(\\Omega )$ for any $q \\in [2,\\, \\infty )$ .",
"In addition, there exists a positive constant $C_q$ independent of sufficiently large $m$ such that $\\Vert (2m\\mathcal {P}_m)^{-1}\\phi \\Vert _{W^{1,q}_0} \\le C_q\\Vert \\phi \\Vert _{L^q}.$ This follows from Theorem REF and the fact that $\\Vert \\cdot \\Vert _{W^{1,q}_0}$ is equivalent to $\\Vert \\sqrt{-\\Delta }\\cdot \\Vert _{L^q}$ .",
"From Lemma REF and the fact that $u_\\infty \\in L^\\infty (\\Omega )$ , we see that $A_m$ is well-defined and bounded on $L^q(\\Omega )$ for $q \\ge 2$ .",
"The next lemma shows it is also invertible.",
"Lemma 6.2 Fix an arbitrary $q \\in [2,\\, \\infty )$ .",
"Then for sufficiently large $m > 0$ , the operator $A_m$ is invertible on $L^q (\\Omega )$ .",
"In addition, there exists a constant $C_q >0$ independent of large $m$ such that $\\Vert A_m^{-1} \\phi \\Vert _{L^q} \\le C_q \\Vert \\phi \\Vert _{L^q}.$ We first define an operator $A:L^q(\\Omega )\\rightarrow L^q(\\Omega )$ as follows: $A(\\phi ) := (I - K)(\\phi ), \\quad K(\\phi ) := (-\\Delta )^{-1}(p|u_{\\infty }|^{p-1}\\phi )$ Note that, due to the $W^{2,q}$ elliptic estimate and compact Sobolev embedding, the operator $K$ is compact on $L^q(\\Omega )$ .",
"Since $u_\\infty $ is non-degenerate, the kernel of $A$ is trivial.",
"Then the Fredholm altanative implies $A$ is invertible and its inverse $A^{-1}$ is bounded, i.e., there exists a constant $C > 0$ depending only on $n, p, q, \\Omega $ and $u_{\\infty }$ such that $\\Vert A^{-1}(\\phi )\\Vert _{L^q} \\le C \\Vert \\phi \\Vert _{L^q}.$ We then compute that $\\begin{split}A_m & = \\left( I - (-\\Delta )^{-1} p|u_{\\infty }|^{p-1} + \\left( (-\\Delta )^{-1} -(2m\\mathcal {P}_m)^{-1}\\right) p|u_{\\infty }|^{p-1}\\right)\\\\& = A \\bigg [ I + A^{-1} ((-\\Delta )^{-1} -(2m\\mathcal {P}_m)^{-1}) p|u_{\\infty }|^{p-1}\\bigg ]\\end{split}$ We claim that the operator norm $\\left\\Vert ((-\\Delta )^{-1} -(2m\\mathcal {P}_m)^{-1}) p|u_{\\infty }|^{p-1}\\right\\Vert _{\\mathcal {L}(L^q)}$ can be made arbitrarily small as $m \\rightarrow \\infty $ so that $A_m$ is invertible on $L^q(\\Omega )$ and $\\Vert A_m^{-1}\\Vert _{\\mathcal {L}(L^q)}$ is independent of sufficiently large $m$ .",
"Indeed, by (REF ) in Theorem REF , there exists a constant $C$ independent of large $m$ such that $\\Vert (-\\Delta )^{-1}-(2m\\mathcal {P}_m)^{-1}\\Vert _{\\mathcal {L}(L^q)}\\le \\frac{C}{m^2},$ which proves the claim.",
"This completes the proof.",
"Combining Lemma REF and REF , we get the following proposition.",
"Proposition 6.3 For any $q \\ge 2$ , there exists a constant $C_q >0$ such that $\\Vert \\mathcal {L}_m \\phi \\Vert _{L^q (\\Omega )} \\le C_q \\Vert \\phi \\Vert _{L^n (\\Omega )}\\quad \\text{for every } \\phi \\in L^n (\\Omega ).$ This immediately follows from Lemma REF , Lemma REF and the embedding $W^{1,n}_0(\\Omega ) \\hookrightarrow L^q(\\Omega )$ for every $q \\ge 2$ .",
"Next, we establish the estimates for the nonlinear operator $Q(w)$ .",
"Proposition 6.4 Let $q \\ge np$ .",
"Then there exists a constant $C > 0$ depending only on $n,\\, p,\\, q$ and $\\Omega $ such that the nonlinear operator $Q$ defined in (REF ) satisfies the following estimates: If $1<p\\le 2$ , then $\\Vert Q(w) - Q(v)\\Vert _{L^n} \\le C \\left( \\Vert w\\Vert _{L^q} + \\Vert v\\Vert _{L^q}\\right)^{p-1}\\Vert w-v\\Vert _{L^q}.$ If $2 <p$ , then $\\Vert Q(w) - Q(v)\\Vert _{L^n} \\le C \\left( \\Vert u_{\\infty }\\Vert _{L^q} + \\Vert w\\Vert _{L^q} + \\Vert v\\Vert _{L^q} \\right)^{p-2} \\left( \\Vert w\\Vert _{L^q} + \\Vert v\\Vert _{L^q}\\right) \\Vert w-v\\Vert _{L^q}.$ By the fundamental theorem of calculus, we write $Q(w)-Q(v)&=\\Big \\lbrace |u_\\infty +w|^{p-1}(u_\\infty +w)-|u_\\infty +v|^{p-1}(u_\\infty +v)\\Big \\rbrace \\\\&\\quad \\quad -p |u_\\infty |^{p-1}(w-v)\\\\&=\\int _0^1 \\frac{d}{dt}\\Big [|u_\\infty +(1-t)v+tw|^{p-1}(u_\\infty +(1-t)v+tw)\\Big ]dt\\\\&\\quad \\quad -p |u_\\infty |^{p-1}(w-v)\\\\&=p\\int _0^1\\big (|u_\\infty +(1-t)v+tw|^{p-1}-|u_\\infty |^{p-1}\\big )(w-v)dt.$ Suppose that $1<p\\le 2$ .",
"Then, by the elementary inequality $||a|^\\ell -|b|^\\ell |\\le \\big ||a|-|b|\\big |^\\ell \\le |a-b|^\\ell \\quad \\textup { if }0<\\ell <1,$ we have $|Q(w)-Q(v)|\\le C(|w|+|v|)^{p-1}|w-v|,$ from which and the fact that $q \\ge np$ , we get the estimate $\\begin{aligned}\\Vert Q(w)-Q(v)\\Vert _{L^n} \\le C\\Vert Q(w)-Q(v)\\Vert _{L^{\\frac{q}{p}}}&\\le C\\big (\\Vert w\\Vert _{L^{q}}+\\Vert v\\Vert _{L^{q}}\\big )^{p-1}\\Vert w-v\\Vert _{L^{q}}.\\end{aligned}$ If $p>2$ , using the fundamental theorem of calculus again, we find $|Q(w)-Q(v)|\\le C(|u_\\infty |+|w|+|v|)^{p-2}(|w|+|v|)|w-v|.$ From this and by estimating as above, we see that $\\begin{aligned}\\Vert Q(w)-Q(v)\\Vert _{L^{n}}&\\le C\\Vert Q(w)-Q(v)\\Vert _{L^{\\frac{q}{p}}} \\\\&\\le C\\big (\\Vert u_\\infty \\Vert _{L^{q}}+\\Vert w\\Vert _{L^{q}}+\\Vert v\\Vert _{L^{q}}\\big )^{p-2}\\big (\\Vert w\\Vert _{L^q}+\\Vert v\\Vert _{L^{q}}\\big )\\Vert w-v\\Vert _{L^{q}}.\\end{aligned}$ Thus, the proposition is proved.",
"By inserting $v \\equiv 0$ into the above estimates, we obtain the following.",
"Corollary 6.5 Let $q \\ge np$ .",
"Then operator $Q$ is a map from $L^q(\\Omega )$ to $L^n(\\Omega )$ .",
"In addition, there exists a positive constant $C$ depending only on $n,\\, p,\\, q$ and $\\Omega $ such that $\\left\\lbrace \\begin{aligned}\\Vert Q(w)\\Vert _{L^n} &\\le C\\Vert w\\Vert _{L^q}^p \\quad \\text{if } 1 < p \\le 2; \\\\\\Vert Q(w)\\Vert _{L^n} &\\le C(\\Vert u_\\infty \\Vert _{L^q}+\\Vert w\\Vert _{L^q})^{p-2}\\Vert w\\Vert _{L^q}^{2} \\quad \\text{if } 2 < p.\\end{aligned}\\right.$ Now we estimate the remainder term $D_m(u_\\infty )$ .",
"Proposition 6.6 Let $q \\ge 2$ .",
"Then there exists a constant $C > 0$ depending only on $n, q, p$ and $\\Omega $ such that the following estimate holds: $\\Vert D_m(u_\\infty )\\Vert _{L^q(\\Omega )} \\le \\left\\lbrace \\begin{aligned} &\\frac{C}{m^2}\\Vert u_\\infty \\Vert _{W^{4,q}(\\Omega )} \\quad \\text{if } p > 2; \\\\ &\\frac{C}{m}\\Vert u_\\infty \\Vert _{W^{3,q}(\\Omega )} \\quad \\text{if } 1< p \\le 2.",
"\\end{aligned}\\right.$ Let $p > 2$ .",
"We invoke (REF ) and (REF ) in Theorem REF to see that $\\begin{aligned}\\Vert D_m(u_\\infty )\\Vert _{L^q} & = \\left\\Vert \\left[ (-\\Delta ) - 2m\\mathcal {P}_m\\right]u_\\infty \\right\\Vert _{L^q} = \\left\\Vert \\left(\\frac{1}{(-\\Delta )}-\\frac{1}{2m\\mathcal {P}_m}\\right)(-\\Delta )(2m\\mathcal {P}_m)u_\\infty \\right\\Vert _{L^q} \\\\& \\le \\frac{C}{m^2}\\Vert (-\\Delta )(2m\\mathcal {P}_m)u_\\infty \\Vert _{L^q} \\le \\frac{C}{m^2}\\Vert (-\\Delta )^2u_\\infty \\Vert _{L^q} \\le \\frac{C}{m^2}\\Vert u_\\infty \\Vert _{W^{2,q}}.\\end{aligned}$ Similarly, if $1 < p \\le 2$ , then by invoking (REF ) and (REF ) to get $\\begin{aligned}\\Vert D_m(u_\\infty )\\Vert _{L^q} \\le \\frac{C}{m}\\Vert \\sqrt{-\\Delta }(2m\\mathcal {P}_m)u_\\infty \\Vert _{L^q}\\le \\frac{C}{m}\\Vert (\\sqrt{-\\Delta })^3u_\\infty \\Vert _{L^q} \\le \\frac{C}{m}\\Vert u_\\infty \\Vert _{W^{3,q}}.\\end{aligned}$ By combining Proposition REF , Proposition REF and Corollary REF , we finally deduce that $\\Phi _m$ , given by $\\mathcal {L}_m \\left[ Q(\\cdot ) +D_m(u_{\\infty })\\right]$ , is a contraction mapping on a small ball in $L^q(\\Omega )$ whenever $q \\ge np$ .",
"Now we shall find a fixed point of $\\Phi _m$ for sufficiently large $m > 0$ .",
"Proposition 6.7 Let $q \\ge np$ .",
"Then there exists a constant $m_0 > 1$ such that for $m \\ge m_0$ , the map $\\Phi _m$ has a fixed point $w_m$ , i.e., $w_m = \\Phi _m(w_m)$ .",
"Let $\\delta \\in (0,1)$ be a small value to be chosen later.",
"We consider the ball $B_{\\delta } = \\left\\lbrace u \\in L^q (\\Omega ) ~:~ \\Vert u\\Vert _{L^q} \\le \\delta \\right\\rbrace .$ We aim to show that the map $\\Phi _m$ is contractive on $B_{\\delta }$ for large $m >1$ .",
"First, by combining Proposition REF and Proposition REF , we obtain $\\Vert \\Phi _m (w) - \\Phi _m (v) \\Vert _{L^q} \\le C \\delta ^{\\min \\lbrace (p-1),1\\rbrace } \\Vert w-v\\Vert _{L^q}$ for some large constant $C$ independent of $\\delta \\in (0,1)$ and $m$ .",
"Similarly, we have $\\Vert \\Phi _m (w)\\Vert _{L^q} \\le C (\\delta ^{\\min \\lbrace (p-1),1\\rbrace } \\Vert w\\Vert _{L^q} +\\frac{1}{m^\\alpha }),$ where $\\alpha = 2$ if $p > 2$ and $\\alpha = 1$ if $1 < p \\le 2$ .",
"We first choose $\\delta >0$ small so that $C \\delta ^{\\min \\lbrace (p-1),1\\rbrace } < \\frac{1}{2}$ , and then find $m_0 >1$ such that $\\frac{C}{m_0^{\\alpha }} < \\frac{\\delta }{2}$ .",
"Then, the mapping $\\Phi $ is contractive from $B_{\\delta }$ to itself.",
"Therefore, by the contraction mapping principle, there exists a fixed point $w_m \\in B_{\\delta }$ of $\\Phi _m$ when $m \\ge m_0$ .",
"This completes the proof.",
"To complete the proof of Theorem REF , it remains to show the following.",
"Proposition 6.8 Let $q \\ge np$ and a function $w_m \\in L^q(\\Omega )$ be a fixed point of $\\Phi _m$ , constructed in Proposition REF .",
"Then the following holds.",
"$w_m$ is contained in $W_0^{1,n}(\\Omega )$ and there exists some constant $C > 0$ independent of $m$ such that $\\Vert w_m\\Vert _{W^{1,n}_0} \\le \\frac{C}{m^{\\alpha }},$ where $\\alpha = 2$ if $p > 2$ and $\\alpha = 1$ if $1< p\\le 2$ ; the function $u_m := w_m +u_{\\infty }$ is a strong solution of (REF ).",
"Since $w_m$ is a fixed point of $\\Phi _m$ , we see from Lemma REF , Corollary REF and Proposition REF that $\\begin{aligned}\\Vert w_m\\Vert _{W^{1,n}_0} & = \\left\\Vert (2m\\mathcal {P}_m)^{-1}\\left(p|u_\\infty |^{p-1}w_m +Q(w_m)+D_m(u_\\infty )\\right)\\right\\Vert _{W^{1,n}_0} \\\\&\\le C\\left\\Vert p|u_\\infty |^{p-1}w_m +Q(w_m)+D_m(u_\\infty )\\right\\Vert _{L^n} \\\\&\\le C\\left(\\left\\Vert w_m\\right\\Vert _{L^n} +\\left\\Vert Q(w_m)\\right\\Vert _{L^n}+ \\left\\Vert D_m(u_\\infty )\\right\\Vert _{L^n}\\right) \\\\&\\le C\\left(\\left\\Vert w_m\\right\\Vert _{L^q} +\\left\\Vert w_m\\right\\Vert _{L^q}^{\\min \\lbrace 2,p\\rbrace }+ \\frac{1}{m^\\alpha }\\right).\\end{aligned}$ Also, the estimate (REF ) implies $\\Vert w_m\\Vert _{L^q} = \\Vert \\Phi _m(w_m)\\Vert _{L^q} \\le \\frac{1}{2}\\Vert w\\Vert _{L^q} + \\frac{C}{m^\\alpha }.$ Then, combining these two estimates, we get a proof of the first assertion $(i)$ .",
"To prove the assertion $(ii)$ , we only need to check the function $u_m$ defined by $w_m +u_\\infty $ satisfies the regularity assumptions; $u_m \\in H^1_0(\\Omega )$ and $|u_m|^{p-1}u_m \\in L^2(\\Omega )$ .",
"This easily can be seen from the embeddings $L^n(\\Omega ) \\hookrightarrow L^2(\\Omega )$ and $W^{1,n}_0(\\Omega ) \\hookrightarrow L^{2p}(\\Omega )$ ."
],
[
"Norm estimates",
"In this appendix, we give a proof of the boundedness of the operators in Section employing the theorem of Duong-Sikora-Yan [16], which generalizes the classical Hörmander-Mikhlin theorem.",
"For a bounded Borel function $F:\\mathbb {R}_+\\rightarrow \\mathbb {R}$ , we define the operator $F(-\\Delta )$ on $\\Omega $ by $F(-\\Delta )\\phi = \\sum _{i=1}^\\infty F(\\lambda _i)c_i\\phi _i, \\quad \\phi = \\sum _{i=1}^\\infty c_i\\phi _i,$ where $\\lbrace \\lambda _n,\\, \\phi _n\\rbrace _{n=1}^\\infty $ is the complete $L^2$ orthonormal system of eigenvalues and eigenfunctions of $\\left\\lbrace \\begin{aligned}-\\Delta \\phi &= \\lambda \\phi \\quad \\text{ in } \\Omega \\\\\\phi &= 0 \\quad \\text{ on } \\partial \\Omega .\\end{aligned}\\right.$ Proposition 1.1 ($\\textup {\\cite [Proposition 6.6]{DSY}}$ ) Let $s> n/2$ , $r_0 = \\max (1,n/s)$ and $\\eta \\in C^\\infty _c(\\mathbb {R}_+)$ be a function not identically zero.",
"Then for any bounded Borel function $F$ such that $\\sup _{t>0} \\Vert \\eta F(t\\cdot )\\Vert _{W_s^{\\infty }} < \\infty $ , the operator $F(-\\Delta )$ is bounded on $L^p (\\Omega )$ for all $p$ satisfying $r_0 <p<\\infty $ .",
"In addition, $\\Vert F(-\\Delta )\\Vert _{L^p (\\Omega ) \\rightarrow L^p (\\Omega )} \\le C_s \\left( \\sup _{t>0} \\Vert \\eta F(t\\cdot )\\Vert _{W_s^{\\infty }} + |F(0)|\\right).$ To apply this result to our problem, we choose a smooth bump function $\\chi :[0,+\\infty ) \\rightarrow [0,+\\infty )$ such that $\\chi (s) =1$ for $s \\in [0, \\lambda _1 /2)$ and $\\chi (s) = 0$ for $x \\in (\\lambda _1 , \\infty )$ , where $\\lambda _1>0$ is the smallest positive eigenvalue of $(-\\Delta )$ on $\\Omega $ .",
"Then, we have $\\chi (-\\Delta ) =0$ .",
"Hence, our problem (REF ) is equivalent to $\\left( 2m\\mathcal {P}_m + \\chi (-\\Delta ) \\right) u = |u|^{p-1} u.$ We consider functions $P_m:\\mathbb {R}_+\\rightarrow \\mathbb {R}_+$ and $P_\\infty :\\mathbb {R}_+\\rightarrow \\mathbb {R}_+$ defined by $P_m(\\lambda ):= 2m(\\sqrt{\\lambda +m^2}-m)+ \\chi (\\lambda ), \\quad P_\\infty (\\lambda ):= \\lambda +\\chi (\\lambda )$ so that we have $P_m(-\\Delta ) = 2m\\mathcal {P}_m$ and $P_\\infty (-\\Delta ) = -\\Delta $ .",
"Then the following symbol estimates can be shown.",
"Proposition 1.2 For any positive integer $k$ , there is a constant $C_k > 0$ such that for all $m \\in [2,\\infty )$ , $\\left|\\left(\\frac{d}{d\\lambda }\\right)^k \\left(\\frac{P_m(\\lambda )}{P_\\infty (\\lambda )}\\right)\\right|\\le \\frac{C_k}{\\lambda ^k}.$ For any positive integer $k$ , there is a constant $C_k > 0$ such that for all $m \\in [2,\\infty )$ , $\\left|\\left(\\frac{d}{d\\lambda }\\right)^k \\left(\\frac{1}{P_\\infty (\\lambda )}-\\frac{1}{P_m(\\lambda )}\\right)\\right|\\le \\frac{C_k}{\\lambda ^k}\\min \\left\\lbrace \\frac{1}{m^2}, \\frac{1}{m(\\lambda +1)^{\\frac{1}{2}}}\\right\\rbrace .$ The proposition can be proved exactly as in [12], thus we omit its proof.",
"Indeed, the only difference comes from introducing bump function $\\chi $ for controlling the singularities of $P_m^{-1}(\\lambda )$ and $P_\\infty ^{-1}(\\lambda )$ when we differentiate.",
"As a consequence, as inverse operators, $P_m(-\\Delta )$ converges to $P_\\infty (-\\Delta )$ as $m\\rightarrow \\infty $ in $L^q(\\Omega )$ for any $q \\ge 2$ .",
"Theorem 1.3 (Difference between the inverses of two operators) For $1<q<\\infty $ , there exists a constant $C_{q,n} >0$ such that $\\left\\Vert \\left(\\frac{1}{P_\\infty (-\\Delta )}-\\frac{1}{P_m(-\\Delta )}\\right)f\\right\\Vert _{L^q}\\le \\frac{C_{q,n}}{m^2}\\Vert f\\Vert _{L^q}\\quad \\forall \\, m \\in [2,\\infty ).$ For $1<q<\\infty $ , there exists a constant $C_{q,n} >0$ such that $\\left\\Vert \\left(\\frac{1}{P_\\infty (-\\Delta )}-\\frac{1}{P_m(-\\Delta )}\\right)f\\right\\Vert _{L^q}\\le \\frac{C_{q,n}}{m}\\left\\Vert \\frac{1}{P_\\infty (-\\Delta )^{\\frac{1}{2}}}f\\right\\Vert _{L^q}\\quad \\forall \\, m \\in [2,\\infty ).$ For $1< q < \\infty $ , there exists a constant $C_{q,n} > 0$ such that $\\left\\Vert \\frac{P_m(-\\Delta )}{P_\\infty (-\\Delta )}f\\right\\Vert _{L^q}\\le C_{q,n}\\left\\Vert f\\right\\Vert _{L^q}\\quad \\forall \\, m \\in [2,\\infty ).$ The estimates (REF ) and (REF ) immediately follow from Proposition REF , (REF ) and (REF ).",
"To obtain (REF ), we compute for positive integer $k$ , $&\\left|\\left(\\frac{d}{d\\lambda }\\right)^k\\left(\\left(\\frac{1}{P_\\infty (\\lambda )}-\\frac{1}{P_m(\\lambda )}\\right)P_\\infty (\\lambda )^{\\frac{1}{2}}\\right)\\right|\\\\&\\qquad \\le \\sum _{k_1+k_2=k} \\left|\\left(\\frac{d}{d\\lambda }\\right)^{k_1}\\left(\\frac{1}{P_\\infty (\\lambda )}-\\frac{1}{P_m(\\lambda )}\\right)\\right|\\left|\\left(\\frac{d}{d\\lambda }\\right)^{k_2}\\left(P_\\infty (\\lambda )^{\\frac{1}{2}}\\right)\\right|\\\\&\\qquad \\le \\sum _{k_1+k_2=k}\\frac{C_{k_1}}{m\\lambda ^{k_1+1}}\\frac{C_{k_2}}{\\lambda ^{k_2-1}}= \\sum _{k_1+k_2=k}\\frac{C_{k_1}C_{k_2}}{m\\lambda ^{k}}\\\\&\\qquad \\le \\frac{C_k}{m\\lambda ^{k}}.$ Then the estimate (REF ) again follows from Proposition REF .",
"Theorem 1.4 For each $q\\in (1,\\infty )$ , there exists a constant $C_q >0$ such that the following estimate holds: $\\left\\Vert \\frac{\\sqrt{-\\Delta }}{P_m(-\\Delta )}f\\right\\Vert _{L^q}\\le C_{q}\\left\\Vert f\\right\\Vert _{L^q}$ for any $f \\in L^q (\\Omega )$ .",
"By the triangle inequality and (REF ), we prove that $\\left\\Vert \\frac{\\sqrt{-\\Delta }}{P_m(-\\Delta )}f\\right\\Vert _{L^q}&\\le \\left\\Vert \\frac{\\sqrt{-\\Delta }}{P_\\infty (-\\Delta )}f\\right\\Vert _{L^q}+\\left\\Vert \\left(\\frac{\\sqrt{-\\Delta }}{P_\\infty (-\\Delta )}-\\frac{\\sqrt{-\\Delta }}{P_m(-\\Delta )}\\right)f\\right\\Vert _{L^q}\\\\&\\le \\left\\Vert \\frac{\\sqrt{-\\Delta }}{P_\\infty (-\\Delta )}f\\right\\Vert _{L^q}+\\frac{C_q}{m}\\left\\Vert \\frac{\\sqrt{-\\Delta }}{P_\\infty (-\\Delta )^{\\frac{1}{2}}}f\\right\\Vert _{L^q}\\\\&\\le C_{q}\\left\\Vert f\\right\\Vert _{L^{q}}.$ The proof is done.",
"Acknowledgments This research of the first author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (NRF-2017R1C1B5076348).",
"This research of the second author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (NRF-2017R1C1B1008215).",
"This research of the third author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (NRF-2017R1D1A1A09000768)."
]
] | 1606.04892 |
[
[
"A Framework for Optimal Matching for Causal Inference"
],
[
"Abstract We propose a novel framework for matching estimators for causal effect from observational data that is based on minimizing the dual norm of estimation error when expressed as an operator.",
"We show that many popular matching estimators can be expressed as optimal in this framework, including nearest-neighbor matching, coarsened exact matching, and mean-matched sampling.",
"This reveals their motivation and aptness as structural priors formulated by embedding the effect in a particular functional space.",
"This also gives rise to a range of new, kernel-based matching estimators that arise when one embeds the effect in a reproducing kernel Hilbert space.",
"Depending on the case, these estimators can be found using either quadratic optimization or integer optimization.",
"We show that estimators based on universal kernels are universally consistent without model specification.",
"In empirical results using both synthetic and real data, the new, kernel-based estimators outperform all standard causal estimators in estimation error."
],
[
"=1 pdfinfo= Title=A Framework for Optimal Matching for Causal Inference, Author=Nathan Kallus [pages=1-last]AFrameworkforOptimalMatchingforCausalInference.pdf"
]
] | 1606.05188 |
[
[
"Search and Placement in Tiered Cache Networks"
],
[
"Abstract Content distribution networks have been extremely successful in today's Internet.",
"Despite their success, there are still a number of scalability and performance challenges that motivate clean slate solutions for content dissemination, such as content centric networking.",
"In this paper, we address two of the fundamental problems faced by any content dissemination system: content search and content placement.",
"We consider a multi-tiered, multi-domain hierarchical system wherein random walks are used to cope with the tradeoff between exploitation of known paths towards custodians versus opportunistic exploration of replicas in a given neighborhood.",
"TTL-like mechanisms, referred to as reinforced counters, are used for content placement.",
"We propose an analytical model to study the interplay between search and placement.",
"The model yields closed form expressions for metrics of interest such as the average delay experienced by users and the load placed on custodians.",
"Then, leveraging the model solution we pose a joint placement-search optimization problem.",
"We show that previously proposed strategies for optimal placement, such as the square-root allocation, follow as special cases of ours, and that a bang-bang search policy is optimal if content allocation is given."
],
[
"Introduction",
"Content distribution is in the vogue.",
"Nowadays, virtually everybody can create, distribute and download content through the Internet.",
"It is estimated that video distribution will alone account for up to 80% of global traffic by 2017 [1].",
"Despite the success of the current Internet infrastructure to support user demand, scalability challenges motivate clean slate approaches for content dissemination, such as information centric networking.",
"In information centric networks (ICNs), the focus is on content, rather than on hosts [2], [3].",
"Each content has an identification and is associated to at least one custodian.",
"Once a request for a content is generated it flows towards a custodian through routers equipped with caches, referred to as cache-routers.",
"A request that finds the content stored in a cache-router does not have to access the custodian.",
"This alleviates the load at the custodians, reduces the delay to retrieve the content and the overall traffic in the network.",
"To achieve performance gains with respect to existing architectures, in information centric networks cache-routers must efficiently and distributedly determine how to route content requests and where to place contents.",
"ICN architectures, such as NDN [2], are promising solutions for the future Internet.",
"Still, it is unclear the scope at which the proposed solutions are feasible [4].",
"Incrementally deployable solutions are likely to prevail [5], and identifying the simplest foundational attributes of ICN architectures is essential while envisioning their Internet scale deployment.",
"The efficient management of distributed storage resources in the network coupled with the routing of requests for information retrieval are of fundamental importance [6], [7].",
"However, the interplay between search and placement is still not well understood, and there is a need to study search and placement problems under a holistic perspective.",
"In fact, an adequate framework within which to assess the overall performance gains that ICNs can provide is still missing [6].",
"In this paper, we propose and study a simple ICN architecture comprising of a logical hierarchy of cache-routers divided into tiers, where each tier is subdivided into one or more logical domains (Figure REF ).",
"In-between domains, requests are routed from users towards custodians which are assumed to be placed at the top of the hierarchy.",
"Figure: System diagramTo route content requests from users to custodians, a random lookup search takes place in the vicinity of the logically connected cache-routers (horizontal arrows in Figure REF ).",
"Cache-routers within a domain are assumed to form a logical clique.",
"As such, a request that does not find the searched content in a cache-router is forwarded to one of the remaining cache-routers in the same domain.",
"The goal is to opportunistically explore the presence of content replicas in a given domain.",
"If a copy is found in the domain within a reasonable time interval, the content is served.",
"Otherwise, requests are routed from users towards custodians (vertical arrows in Figure REF ).",
"Custodians as well as the name resolution system (NRS) are supplied by third parties at the publishing area, and we focus our attention on the infrastructure from users to publishing areas.",
"By using random walks to opportunistically explore the presence of content replicas closer to users, we avoid content routing tables and tackle the scalability challenge posed in [8].",
"An alternative would be to adopt scoped-flooding [9].",
"However, scoped-flooding is more complex than random walks and requires some level of synchronization between caches.",
"In addition, random walks have been show to scale well in terms of overhead [10].",
"To efficiently and distributedly place content in the cache network, we consider a flexible content placement mechanism inspired by TTL caches.",
"At each cache, a counter is associated to each content stored there, which we refer to as reinforced counter (RC).",
"Whenever the RC surpasses a given threshold, the corresponding content is stored.",
"The RC is decremented at a given established rate, until reaching zero, when the content is evicted.",
"Focusing on two of the simplest possible mechanisms for search and placement, namely random walks and TTL-like caches, our benefits are twofold.",
"From a practitioners point of view, the proposed architecture is potentially deployable at the Internet scale [4].",
"From the performance evaluation perspective, our architecture is amenable to analytical treatment.",
"Our quantitative analysis provides closed-form expressions for different metrics of interest, such as the average delay experienced by users.",
"Given such an architecture, we pose the following questions, How long should the random-walk based search last at each domain so as to optimize the performance metrics of interest?",
"How should the reinforced counters be tuned so as to tradeoff content retrieval delay with server load at the custodian?",
"What parameters have the greatest impact on the performance metrics of the proposed ICN architecture?",
"To answer these questions, we introduce an analytical model that yields: a) the expected delay to find a content (average search time) and; b) the rate at which requests have to be satisfied by custodians.",
"While the expected delay is directly related to users quality of experience, the rate of accesses towards the custodian is associated with publishing costs.",
"The model yields simple closed-form expressions for the metrics of interest.",
"Using the model, we study different tradeoffs involved in the setting of the parameter values.",
"In particular, we study the tradeoff between the time spent in opportunistic exploration around the vicinity of the user in order to find content and the custodian load.",
"In summary, our key contributions are the following: ICN architecture: we propose a simple ICN multi-tiered architecture based on random walks and TTL-like caches.",
"Simplicity easies deployment and allows for analytical treatment, while capturing essential features of other ICN architectures such as the tension between opportunistic exploration of replicas closer to users and exploitation of known paths towards custodians.",
"Analytical model: we introduce a simple analytical model of the proposed ICN architecture that can be helpful in the performance evaluation of ICNs.",
"In particular, we consider the interplay between content placement and search.",
"Using the model we show that we can achieve performance gains using a simple search strategy (random walks) and a logical hierarchical storage organization.",
"Although our analysis is focused on the proposed architecture, we believe that the insights obtained are more broadly applicable to other architectures as well, such as scoped-flooding [9].",
"Parameter tuning: we formulate an optimization problem that leverages the closed-form expressions obtained with the proposed model to determine optimal search and placement parameters under storage constraints.",
"We show that previously proposed strategies for optimal placement, such as the square-root allocation, follow as special cases of our solution, and that a bang-bang search policy is optimal if content allocation is given.",
"Performance studies: we investigate how different parameters impact system performance under different assumptions regarding the relative rate at which requests are issued and content is replaced in the cache-routers.",
"Our numerical investigations consider scenarios in which the assumptions of the optimization problem posed in this paper do not hold.",
"The remainder of this paper is organized as follows.",
"After introducing background in Section , we describe the system studied in this paper in Section .",
"An analytic model of this system is presented in Section .",
"The joint placement and search optimization problem is posed and analyzed in Section and numerical evaluations are presented in Section .",
"Further discussions are presented in Section and Section concludes."
],
[
"Background and Related Work",
"In this section we introduce the background used in this paper.",
"In Section REF we present previously proposed ICN architectures and in Section REF we indicate some of the challenges they pose."
],
[
"ICN Architectures",
"A survey comprising various architectures considered for ICN can be found in [6].",
"In what follows, we focus on five of the prominent architectures, namely DONA, PSIRP, Netinf, Multicache and NDN, which are most relevant to our work.",
"DONA [3] consists of a hierarchy of domains.",
"Each domain includes a logical Resolution Handler (RH) that tracks the contents published in the domain and in the descendant domains.",
"Therefore, the logical RH placed in the highest level of the hierarchy is aware of all the content published in the entire network.",
"RHs provide a hierarchical name resolution service over the routing infra-structure.",
"DONA supports caching through the RH infrastructure.",
"When a RH aims at storing a content, it replaces the IP address of the requester by its own IP address.",
"Then, the content will be delivered first to the RH before being forwarded to the end users, allowing the RH to cache the content within the domain.",
"PSIRP [11], Netinf [12] and Multicache [13] handle name resolution through a set of Request Nodes (RNs) organized according to a hierarchical Distributed Hash Table (DHT).",
"Content is sent to the user through a set of forward nodes (FNs), under a separate network.",
"FNs can advertise cached information to RNs to enhance the search efficiency and cache hit ratio.",
"Nonetheless, as RNs cannot keep track of all replicas within the network, a key challenge consists of determining what is the relevant information to advertise.",
"The NDN [2] architecture handles name resolution using content routing tables.",
"Users issue Interest messages to request a content.",
"Messages are forwarded hop-by-hop by Content Routers (CRs) until the content is found.",
"Messages leave a trail of bread crumbs and the content follows the reverse path set by the trail.",
"As content flows to requesters, the bread crumbs are removed.",
"Published content is announced through routing protocols, with routing tables supporting name aggregation (names are hierarchical).",
"To enhance the discovery of cached contents, Rosensweig et al.",
"[14] allow bread crumbs not to be consumed on the fly when content traverses the network.",
"This allows trails for previously downloaded contents to be preserved."
],
[
"Challenges",
"Some of the main challenges faced by present ICN architectures are discussed in [8].",
"For Name Resolution Services (NRS) lookup proposals, such as Dona, NetInf and PSIRP, the challenge is to build a scalable resolution system which provides: (i) fast mapping of the name of the content to its locators; (ii) fast update of the location of a content since locations can change frequently; (iii) an efficient scheme to incorporate copies of a content in the cache routers.",
"For proposals based on content routing tables, such as NDN, the number of contents may be around $10^{15}$ to $10^{22}$ .",
"Routing table design becomes a challenge as its size is proportional to the number of contents in the system.",
"Route announcements due to replica updates, and link failures, pose additional challenges.",
"Simple hierarchical tiered topologies, wherein each domain comprises a single node, admit closed-form expressions for the expected time to access content [15], [16].",
"In this paper, we consider the case where each domain comprises multiple nodes, which means that routing is non-trivial.",
"To face the scalability challenge related to content routing tables, [9] proposes the use of flooding in each neighborhood, which simplifies design and reduces complexity.",
"In this paper, in contrast, we propose the use of random walks.",
"Random walks are as simple as flooding, and lead to reduced congestion [17], [10], [18], [19] while still taking advantage of spacial and temporal locality [20], [15].",
"For proposals relying on DHTs there exist many unsolved security vulnerabilities that are able to disrupt the pre-defined operation of DHT nodes [21] and need to be overcome.",
"Note that in a network composed of domains where providers care about administrative autonomy, the use of a global hash table becomes unfeasible [22]."
],
[
"System Architecture",
"In this section we describe the system architecture considered in this paper.",
"We begin with a brief overview."
],
[
"Tiers and Domains",
"The system consists of a set of cache-routers partitioned into several logical domains, which are organized into hierarchically arranged tiers (Figure REF ).",
"Each domain consists of a set of routers or cache-routers that are responsible for forwarding requests and caching copies of contents.",
"In what follows, we assume that all routers are equipped with caches, and use interchangeably the terms router and cache-router.",
"Users generate requests at the lowest level of the hierarchy.",
"These requests flow across domains, following the tier hierarchy towards the publishing areas, at the top of the hierarchy.",
"Figure REF displays routers forwarding requests towards a publishing area (green arrows).",
"We consider $M$ logical hierarchical tiers.",
"Tier 1 is the top level tier and tier $M$ is the bottom level constituted by routers that are “closest” to the users, i.e., which are the first to receive requests from users.",
"The publishing area knows how to forward a request to a publisher in case the content is not found in any of the tiers.",
"We adopt a strategy that allows opportunistic encounters between requests and replicas in a best-effort manner.",
"Each cache maintains a counter (one per content), referred to as a reinforced-counter, to establish thresholds to guide content placement at the caches.",
"Copies of popular contents may be cached in the routers.",
"Whenever a request arrives to a domain, it generates a random walk to explore the domain, so as to allow opportunistic encounters with the desired content, taking advantage of the temporal and geographical correlations encountered by popular requests [20].",
"We rely on random walks in order to avoid the control overhead associated to routing table updates and the drawbacks of DHTs discussed in the previous section."
],
[
"Random Walk Search",
"Random walks are one of the simplest search mechanisms with the flexibility to account for opportunistic encounters between user requests and replicas stored within the domains (purple arrows in Figure REF ).",
"Opportunistic encounters satisfy requests without the need for them to reach the publishing area.",
"A request that reaches the publisher area indicates that the corresponding content was not found in any of the domains traversed by it.",
"When a request arrives to a domain, if the cache-router that receives the request does not have the content, it starts a random walk search.",
"The random walk lasts for at most $T$ units of time, only traversing routers in the domain.",
"A time-to-live (TTL) counter is set to limit the amount of search time for a content within a domain.",
"If the content has not been found by the time the TTL counter expires, the router that holds the request transfers it to the next tier above it in the hierarchy.",
"As a request is forwarded up the hierarchy, backward pointers are deployed.",
"These pointers are named bread crumbs.",
"When content is located in the network, two actions are performed: (a) the content is sent to the requester and (b) the content is possibly stored in the caches of the routers that first received the request in each domain (those that initiated the random walk at a domain).",
"Action (b) is performed if the reinforced counters associated with the given content at the considered cache-routers reach a pre-determined threshold.",
"Note that a cache-router may store contents that were found either in its own domain or in tiers above it.",
"The publisher can perform action (a) by either directly sending the content to the requester, or by following the reverse path of the request (blue arrows in Figure REF ), whichever is more efficient.",
"As the content follows the path of bread crumbs, the trail is erased."
],
[
"Reinforced Counter Based Placement",
"We consider a special class of content placement mechanisms, henceforth referred to as reinforced counters (RC), similar in spirit to TTL-caches [16].",
"Each published content in the network is identified by a unique hash key inf.",
"All cache-routers have a set of RCs, one for each content.",
"Reinforced counters are affected by exogenous requests and interdomain requests, but not by endogenous requests inside a given domain, that is, their values are not altered by the random walk search.",
"At any cache, the reinforced-counter associated to a given content is increment by one at every exogenous or interdomain request to that content, and is decremented by one at every tick of a timer.",
"The timer ticks at a rate of $\\mu $ ticks per second.",
"Associated with each RC is a threshold $K$ .",
"Whenever a request for content inf reaches a router, either (i) an already pre-allocated counter for inf is incremented by one in this router or (ii) a new RC is allocated for inf and set to one.",
"If the value of the RC surpasses $K$ , the content is stored after inf is found.",
"RCs are decremented over time.",
"Whenever the RC for inf is decremented from $K+1$ to $K$ the content is evicted from the cache.",
"The counter is deallocated when it reaches zero.",
"Note that the RC dynamics of different contents are uncoupled and the RC values are independent of each other.",
"Cache storage constraints are taken into account in the model by limiting the average number of replicas in each cache, which corresponds to soft constraints.",
"Since hard constraints on the cache occupancy must be enforced, the RC threshold should be set in such a way that the probability of a cache overflow is small [23].",
"By limiting the fraction of time that each content is cached, reinforced counters take advantage of statistical multiplexing of contents in the system."
],
[
"Stateless and Stateful Searches",
"We consider two variants of random walk searches: stateless and stateful.",
"Under stateless searches, requests do not carry any information about previously visited cache-routers.",
"In other words, when a cache-router is visited, the only information that is known is the content of the cache currently being visited.",
"In a stateful search requests either a) remember the cache-routers that have been visited or b) know ahead of time what routers to visit.",
"We assume that in stateful searches the searcher never revisits cache-routers.",
"The stateless and stateful searches are studied in Sections REF and REF , respectively.",
"In this section we present an analytical model to obtain performance metrics for the ICN architecture described in the previous section, illustrated in Figure REF .",
"The model takes into account the performance impact of content search through random walks and the cache management mechanism based on reinforced counters.",
"In particular, the model allows one to compute the probability of finding a content in a domain and the mean time to find it.",
"Using the model we show the benefit of a hierarchical structure and study the tradeoff between the storage requirements of the cache-routers and the load that reaches the publishing area.",
"When a request reaches a cache-router, the local cache is searched and if the content is locally stored it is immediately retrieved and sent to the user.",
"If the content is not found, a random walk search starts in the domain.",
"We assume that the random search takes $V$ time units per each cache-router visited where $V$ is an exponentially distributed random variable with rate $\\gamma $ .",
"Long search times can have an adverse effect on performance; hence, a timer is set when the random walk starts to limit the search time.",
"The search can last for at most $T$ time units.",
"The search ends when the timer expires or the content is found, whichever occurs first.",
"As described in Section REF , if the timer expires the user request is sent to the next cache-router in the tier hierarchy, and the process restarts.",
"Table summarizes the notation used in the remainder of this paper.",
"l p4.8in Parameter Description $1/\\gamma $ average delay per-hop, i.e., average time for the random walk to check for a content at a cache and possibly forward the request $\\mathcal {C}(\\hat{\\Lambda }_c)$ cost incurred by custodian (measured in delay experienced by users) $C$ number of contents $M$ number of tiers $N$ number of caches in domain under consideration $\\lambda _{c,i}$ arrival rate of exogenous and interdomain requests for content $c$ at typical cache of domain $i$ , $\\lambda =\\sum _{i=1}^M \\sum _{c=1}^C \\lambda _{c,i}$ $\\Lambda _c$ exogenous arrival rate of requests for $c$ at the network (except otherwise noted, exogenous requests are issued at tier $M$ ) Variable Description $L_c$ number of replicas of content $c$ in tagged tier $\\pi _{c,i}$ probability that content $c$ is stored at typical cache at domain $i$ $\\alpha _{c,i}$ $=1/\\mu _{c,i}$ Control variable Description $\\mu _{c,i}$ reinforced counter decrement rate for content $c$ at domain $i$ $T_{c,i}$ TTL for content $c$ at domain $i$ , i.e., maximum time to perform a random search for content $c$ at domain $i$ Metric Description $R_{c,i}(t)$ probability of not finding content $c$ at tier $i$ by time $t$ $D_{c,i}$ delay incurred for finding content $c$ at tier $i$ $D_{c}$ delay incurred for finding content $c$ $D$ delay incurred for finding typical content $\\hat{\\Lambda }_c$ rate of requests for content $c$ at the publisher Table of notation.",
"Note that subscripts are omitted when clear from context."
],
[
" Cache hit and insertion ratios ",
"Consider a given tagged tier and cache-router in this tier.",
"We assume that requests to content $c$ arrive to this cache-router according to a Poisson process with rate $\\lambda _c$ .",
"$\\lambda _c$ is also referred to as the content popularity.",
"Recent work [24] using three months of data collected from the largest VoD provider in Brazil indicates that, during peak hours, the Poisson process is well suited to model the video request arrival process.",
"In our numerical experiments, we rely on the Poisson assumption coupled with the Zipf distribution for popularities to characterize the workload.",
"We recall from Section REF that the reinforced counter associated to a given content $c$ is incremented at every request for $c$ and decremented at constant rate $\\mu _c$ .",
"We assume that the counter is decremented at exponentially distributed times with mean $1/\\mu _c$ .",
"Associated with each counter and content is a threshold $K_c$ such that when the counter exceeds $K_c$ , content $c$ must be stored into cache.",
"Let $\\pi _c$ denote the probability that the cache-router contains content $c$ .",
"Due to the assumption of Poisson arrivals and exponential decrement times, the dynamics of each reinforced counter is characterized by a birth-death process.",
"Hence $\\pi _c$ , which is the probability that the reinforced counter has value greater than $K_c$ , is given by $\\pi _c = \\left(\\frac{\\lambda _c}{\\mu _c}\\right)^{K_c+1}$ If $K_c=0$ we have $\\pi _c = \\lambda _c/\\mu _c$ , which we denote by $\\rho _c$ .",
"Let $\\beta _c$ denote the miss rate for content $c$ .",
"Then, $\\beta _c= \\lambda _c (1-\\pi _c)$ In we consider an additional metric of interest, namely the cache insertion rate, which is the rate at which content is inserted into cache.",
"Note that the cache insertion rate is lower than the cache miss rate, as not all misses lead to content insertions.",
"We show that larger values of $K_c$ yield lower insertion rates, which translate into less overhead due to content churn.",
"Despite the advantages of using $K_c>0$ , without loss of generality, and to facilitate the exposition, in the remainder of this paper we assume $K_c=0$ , except otherwise noted."
],
[
"Publisher Hit Probability",
"We start by considering a single domain in a single tiered hierarchy, wherein $N$ cache-routers are logically fully connected, i.e., any cache-router can exchange messages with any other router in the same domain.",
"Our goal is to compute the probability $R(t)$ that a random walk does not find the requested content by time $t$ , $t>0$ .",
"Note that $R(T_c)$ equals the probability that the request is forwarded to the custodian.",
"We consider two slightly different models.",
"As in the previous section, both models assume that requests for a content arrive according to a Poisson process.",
"In what follows we describe the assumptions associated with each model, and comment on their applicability.",
"In Sections REF and REF the analysis of stateless and stateful searches focuses on a tagged content $c$ ."
],
[
"Model 1: Stateless search",
"Recall that a stateless search is a search in which requests do not carry any information about previously visited cache-routers.",
"We assume that searches are sufficiently fast so that the probability that content placement in a domain changes during the search is negligible.",
"This assumption is reasonable if the expected time it takes for the random walker to check for the presence of content $c$ in a cache and to transit from a cache-router to another, $1/\\gamma $ , is very small compared to the mean time between: (a) two requests for $c$ , $1/\\lambda _c$ , and; (b) decrements of the reinforced counter for $c$ , $1/\\mu _c$ .",
"When an inter-domain request for a given content $c$ arrives at a cache-router and a miss occurs, a random stateless search for $c$ starts.",
"After each visit to a cache-router, if the content is not found another cache-router is selected uniformly at random among the remaining $N-1$ cache-routers.",
"Note that, because the search is stateless, nodes can be revisited during the search.",
"In Section REF we discussed the decoupling between RCs of different contents in a given cache.",
"Next, we argue that RCs for different caches in a domain can also be treated independently.",
"Recall that reinforced counters are not affected by endogenous requests inside a given domain, so we restrict ourselves to the impact of inter-domain requests when studying cache occupancies.",
"Due to symmetry, we assume that the rate of requests from outside of a domain for a given content at different cache-routers in a domain are identical.",
"Due to the Poisson assumption, a request for content $c$ that arrives at a tagged cache-router sees the system in equilibrium (PASTA property).",
"Therefore, arrivals will find the content of interest at a given cache with probability $\\pi _c$ , independent of the state of the neighboring caches in that domain.",
"Let $L_c$ be the random variable equal to the number of replicas of the content $c$ in the domain, excluding the router being visited.",
"We have: $P(L_c=l) = \\binom{N-1}{l} \\pi _c^{l}(1-\\pi _c)^{N-1-l} .$ Let $J_c$ denote the number of hops traversed by the stateless request by time $t$ .",
"Since the time between visits is assumed to be exponentially distributed, $R(t|J_c=j,L_c=l) &=& (1 - \\pi _c) (1-w_l)^{j}$ where $w_l$ is the conditional probability that the random walker selects one router with content $c$ from the remaining $N-1$ routers in the domain when there are $l$ replicas of the content in the domain given that the current router does not have the content.",
"Then, $w_l=l/(N-1)$ .",
"Note that $\\pi _c$ depends on the placement policy and is defined partially by its parameter values.",
"Proposition 4.1 The probability $R_c(t|L_c=l)$ is given by $R_c(t|L_c=l) &=& (1 - \\pi _c) e^{-\\gamma \\omega _l t}$ Proof: From (REF ) we have: $R_c(t|L_c=l) &=&(1 - \\pi _c) \\sum _{n=0}^{\\infty } \\frac{(\\gamma t)^n}{n!",
"}(1-\\omega _l)^n e^{-\\gamma t} \\nonumber \\\\&=& \\frac{1 - \\pi _c}{e^{\\gamma t \\omega _l}}\\sum _{n=0}^{\\infty } \\frac{(\\gamma t (1-\\omega _l))^n}{n!",
"}e^{- \\gamma t (1-\\omega _l)} \\nonumber \\\\&=& (1 - \\pi _c)e^{-\\gamma \\omega _l t}$ $\\square $ Proposition 4.2 (Stateless search) The probability ${R}_c(t)$ that a walker does not find a requested tagged content in a domain by time $t$ is given by: $R_c(t) &=&{ \\left( { e^{-\\gamma t / (N-1)} \\pi _c + (1-\\pi _c) } \\right) }^{(N-1)} (1-\\pi _c)$ Proof: Unconditioning (REF ) on $L_c$ , yields $R(t) &=& \\sum _{l=0}^{N-1} R(t|L_c=l) \\binom{N-1}{l} \\pi _c^{l} (1-\\pi _c)^{(N-1-l)} \\nonumber \\\\&=& (1-\\pi _c) \\sum _{l=0}^{N-1} e^{-\\gamma \\omega _l t} \\binom{N-1}{l} \\pi _c^{l}(1-\\pi _c)^{(N-1-l)} \\nonumber \\\\&=& (1-\\pi _c) \\sum _{l=0}^{N-1} \\binom{N-1}{l} \\left( {e^{-\\gamma t/(N-1)} \\pi _c }\\right)^{l} (1-\\pi _c)^{N-1-l} \\nonumber \\\\&=& (1-\\pi _c) \\left( { \\pi _c e^{-\\gamma t / (N-1) } + (1-\\pi _c) } \\right)^{(N-1)}$ $\\square $ According to (REF ), $R_c(\\infty ) = (1-\\pi _c)^{N}$ .",
"As $t$ increases, the probability that the walker does not find content $c$ approaches the probability that all $N$ caches within the domain do not hold the content."
],
[
"Model 2: Stateful search",
"In this section, we consider stateful searches wherein requests remember the cache-routers that have been visited, i.e., after the search is initiated, the searcher chooses the next router to visit uniformly at random, from those that have not yet been visited before.",
"Alternatively, requests know ahead of time what routers to visit.",
"This latter approach is discussed in .",
"Under a stateful search, the searcher never revisits cache-routers.",
"This is possible because cache-routers are logically fully-connected.",
"As in the stateless model, we assume that arrivals of inter-domain requests for content $c$ at cache-routers are characterized by Poisson processes.",
"Therefore, the random searches for $c$ that are initiated at a tagged router $i$ are characterized by a Poisson process modulated by the RC of router $i$ , whose dynamics is governed by a birth-death Markovian process.",
"It is shown in [25] that the PASTA property holds for Poisson processes modulated by independent Markovian processess.",
"Therefore, a search that starts at router $i$ and arrives at router $k \\ne i$ sees the RC at $k$ in equilibrium, i.e., the request issued at router $i$ finds the desired content at cache $k$ with probability $\\pi _c$ .",
"Conditioning on $J_c=j$ hops being traversed by time $t$ , the probability that content $c$ is not found is given by $\\tilde{R}_c(t | J_c=j) &=& (1-\\pi _c)^{j+1}$ It remains to remove the conditioning on $J_c$ .",
"We assume, as in the stateless model, that the search takes an exponentially distributed random delay at each hop, independent of the system state.",
"Proposition 4.3 (Stateful search) The probability $\\tilde{R}(t)$ that a tagged content is not found by a stateful search by time $t$ is given by $\\tilde{R}_c(t) = (1-\\pi _c)( e^{-\\gamma \\pi _c t} + g(N)) $ where $g(N) = (1-\\pi _c)^{N-1}\\sum _{n=N}^{\\infty } \\frac{(\\gamma t)^n}{n!}",
"e^{-\\gamma t} \\left(1-(1-\\pi _c)^{n+1-N}\\right)$ Proof: The proof is similar to that of Proposition REF .",
"The time between cache visits is an exponential random variable with rate $\\gamma $ .",
"It follows from (REF ) that $\\tilde{R}_c(t) &=& \\sum _{n=0}^{N-1} \\tilde{R}_c(t|J=n) \\frac{(\\gamma t)^n}{n!}",
"e^{-\\gamma t} + \\tilde{R}_c(t|J=N-1) \\sum _{n=N}^{\\infty } \\frac{(\\gamma t)^n}{n!}",
"e^{-\\gamma t}\\\\&=& (1-\\pi _c)\\left( \\sum _{n=0}^{N-1} \\frac{(\\gamma t)^n}{n!}",
"e^{-\\gamma t} (1-\\pi _c)^n + (1-\\pi _c)^{N-1} \\sum _{n=N}^{\\infty } \\frac{(\\gamma t)^n}{n!}",
"e^{-\\gamma t} \\right) \\nonumber \\\\&=& (1-\\pi _c)\\left( \\sum _{n=0}^{\\infty } \\frac{((1-\\pi _c) \\gamma t)^n }{n!}",
"\\frac{e^{-\\gamma (1-\\pi _c)t}}{e^{\\gamma \\pi _c t}} + g(N) \\right) \\nonumber \\\\&=& (1-\\pi _c) \\left( e^{-\\gamma \\pi _c t} + g(N) \\right)$ $\\square $ For large values of $N$ , it follows from Proposition REF that $\\tilde{R}_c(t) \\approx (1-\\pi _c) e^{-\\gamma \\pi _c t} $ The validity of the large $N$ assumption can be checked by using the Normal distribution approximation for the Poisson distribution.",
"For instance, the sum $\\sum _{n=N}^{\\infty } \\frac{(\\gamma t)^n}{n!}",
"e^{-\\gamma t}$ that appears in the expression of $g(N)$ is well approximated by the complementary cumulative distribution of the Normal distribution, $1-\\Phi \\left(\\frac{N-\\gamma t}{\\sqrt{\\gamma t}}\\right)$ , for values of $N > \\gamma t + 4 \\sqrt{\\gamma t}$ , where $\\Phi (x)$ is the cumulative distribution function of the standard Normal distribution.",
"According to (REF ), $\\tilde{R}_c(\\infty ) = 0$ .",
"As the random walk progresses, contents are dynamically inserted and evicted from the caches and the walker eventually finds the desired content."
],
[
"Multi-tier Networks",
"In the previous sections we considered a single tiered network.",
"In what follows we extend these results to the multi-tier case.",
"In Section we discuss the potential performance benefits of a multi-tiered architecture.",
"Refer to Figure REF and let $M$ denote the number of tiers.",
"Let $\\hat{\\Lambda }_c$ denote the publisher load accounting for the requests filtered at the $M$ tiers.",
"Let $R_{c,i}(T_{c,i})$ denote the probability that a search that reaches domain $i$ fails to find content $c$ at that domain.",
"The load for content $c$ that arrives at the publishing area is given by: $\\hat{\\Lambda }_c =\\Lambda _c \\prod _{i=1}^{M} R_{c,i}(T_{c,i})$ where $\\prod _{i=1}^{M} R_{c,i}(T_{c,i})$ is the probability that a request arrives at the publishing area and $\\Lambda _c$ is the load generated by the users for content $c$ which are all placed at tier $M$ .",
"Note that replacing $R_{c,i}(T_{c,i})$ by $\\tilde{R}_{c,i}(T_{c,i})$ corresponds to using the stateful model in place of the stateless one.",
"Let ${D}_{c,i}$ be a random variable denoting the delay experienced by requests for content $c$ at domain $i$ .",
"Recall that $T_{c,i}$ is the maximum time a walker spends searching for content $c$ in domain $i$ .",
"In what follows, we make the dependence of $D_{c,i}$ on $T_{c,i}$ explicit.",
"It follows from [26] that $E[{D}_{c,i}(T_{c,i})] = \\int _{0}^{T_{c,i}} R_{c,i}(t)dt $ Under the stateless model, $E[{D}_{c,i}(T_{c,i})]$ does not admit a simple closed form solution and must be obtained through numerical integration of (REF ).",
"On the other hand, when the stateful model is employed, we obtain, after replacing (REF ) into (REF ), $E[{D}_{c,i}(T_{c,i})] &=& \\int _{0}^{T_{c,i}} (1-\\pi _{c,i}) e^{ -\\gamma \\pi _{c,i} t}dt \\\\&=& (1-\\pi _{c,i})\\frac{1-e^{-\\gamma \\pi _{c,i} T_{c,i}}}{\\pi _{c,i} \\gamma }.",
"$ Let $D_c$ denote the delay to find content $c$ , including the time required for the publishing area to serve the request if needed.",
"Then, $E[D_c]$ is given by: $E[D_c] = \\left(\\sum _{i=1}^{M} E[D_{c,i}(T_{c,i})]\\prod _{j=i+1}^{M} R_{c,j}(T_{c,j}) \\right) +\\mathcal {C}(\\hat{\\Lambda }_{c}) \\prod _{j=1}^{M} R_{c,j}(T_{c,j}),$ where $\\mathcal {C}(\\hat{\\Lambda }_{c})$ is the mean cost (measured in time units) to retrieve a content at the publishing area as a function of the load $\\hat{\\Lambda }_{c}$ .",
"Recall that tier 1 (resp., tier $M$ ) is the closest to the custodians (resp., users).",
"Therefore, $\\prod _{j=i+1}^{M} R_{c,j}(T_{c,j})$ corresponds to the fraction of requests to content $c$ that reach tier $i$ , for $i=1, \\ldots , M-1$ ."
],
[
"Parameter Tuning",
"In this section we consider the problem of minimizing average delay under average storage constraints.",
"To this aim, we use the stateful model that was introduced in the previous section.",
"While in Section the analysis targeted a single tagged content, in this section we account for the limited space available in the caches and for contents that compete for cache space.",
"To simplify presentation, we consider a single tier ($M=1$ ).",
"We also assume that the delays experienced by requests at the custodian are given and fixed, equal to $\\mathcal {C}$ .",
"Let $D_c$ denote the delay experienced by a requester of content $c$ .",
"$E[D_c]$ is obtained by substituting (REF ) into (REF ), $ E[D_c] = (1-\\pi _c)\\left( \\frac{1-e^{-\\gamma \\pi _c T_c }}{\\pi _c \\gamma } + \\mathcal {C} e^{-\\gamma \\pi _c T_c} \\right)$ and $ E[D] = \\sum _{c=1}^C \\frac{\\lambda _c}{\\lambda } E[D_c]$ Let $\\alpha _c = 1/\\mu _c$ , $\\mathbf {\\alpha } = (\\alpha _1, \\alpha _2, \\ldots , \\alpha _C)$ and $\\mathbf {T} = (T_1, T_2, \\ldots , T_C)$ .",
"In light of (REF ) and (REF )-(REF ), we pose the following joint placement and search optimization problem: $\\min _{(\\mathbf {\\alpha },\\mathbf {T})} && E[D]= \\sum _{c=1}^C \\frac{\\lambda _c}{\\lambda } (1-\\lambda _c \\alpha _c) \\left(\\frac{1-e^{-\\gamma \\lambda _c \\alpha _c T_c }}{\\lambda _c \\alpha _c \\gamma } + \\mathcal {C} e^{-\\gamma \\lambda _c \\alpha _c T_c} \\right) \\nonumber \\\\s.t.",
"&& \\sum _{c=1}^{C}\\lambda _c \\alpha _c = B$ Note that we impose a constraint on the expected buffer size, i.e., the number of expected items in the cache cannot exceed the buffer size $B$ .",
"Similar constraint has been considered, for instance, in [27].",
"Moreover, recent work [23] shows that, for TTL caches, we can size the buffer as $B(1 + \\epsilon )$ , where $B$ (resp., $\\epsilon $ ) grows in a sublinear manner (resp., shrinks to zero) with respect to $C$ , and content will not need to be evicted from the cache before their timers expire, with high probability.",
"The reinforced counter vector $\\mathbf {\\alpha }$ impacts content placement, while the random walk vector $\\mathbf {T}$ impacts content search.",
"By jointly optimizing for placement and search parameters, under storage constraints, we obtain insights about the interplay between these two fundamental mechanisms.",
"In what follows, we do not solve the joint optimization problem directly.",
"Instead, to simplify the solution, we solve two problems independently: first, we consider the optimal placement given a search strategy, and then the optimal search given a pre-determined placement.",
"In our case studies we discuss the impact of these simplifications."
],
[
"Optimal Placement Given Search Strategy",
"We first address the optimal placement problem, that is we determine how the buffer space at the cache-routers should be statistically divided among the contents to optimize the overall performance.",
"We begin by considering large time to live values.",
"In the limit when $T_c=\\infty $ , the time spent locally searching for a content is unbounded.",
"Under this assumption, the optimization problem stated in (REF ) reduces to $\\min _{\\mathbf {\\pi }}&& \\sum _{c=1}^C \\frac{\\lambda _c}{\\lambda } ( 1-\\pi _c)\\left(\\frac{1}{\\pi _c \\gamma } \\right) \\nonumber \\\\s.t.",
"&& \\sum _{c=1}^{C}\\pi _c = B $ We construct the Lagrange function, $\\mathcal {L}(\\mathbf {\\pi },\\beta ) =\\sum _{c=1}^C \\frac{\\lambda _c}{\\lambda } \\frac{( 1-\\pi _c)}{\\pi _c \\gamma }+ \\beta \\left( \\sum _{c=1}^C \\pi _c - B \\right)$ where $\\beta $ is a Lagrange multiplier.",
"Setting the derivative of the Lagrangian with respect to $\\pi _c$ equal to zero and using (REF ) yields, $\\beta =\\frac{\\left(\\sum _{c=1}^C \\sqrt{\\lambda _c}\\right)^2 }{ \\gamma \\lambda B^2}.$ Therefore, $\\pi _c = B {\\frac{\\sqrt{\\lambda _c}}{\\left(\\sum _{c=1}^C \\sqrt{\\lambda _c}\\right)}}, c=1, \\ldots , C. $ When $B=1$ , the optimal policy (REF ) is the square-root allocation proposed by Cohen and Shenker [28] in the context of peer-to-peer systems.",
"It is interesting that we obtain a similar result for the ICN system under study.",
"This is because in both cases the optimization problem can be reformulated as to minimize $ \\sum _{c=1}^C (\\lambda _c/\\lambda )/\\pi _c$ under the constraint that $\\sum _{c=1}^C \\pi _c =B$ .",
"In [28] the term $1/\\pi _c$ is the mean time to find content $c$ , which is the average of a geometric random variable with probability of success $\\pi _c$ .",
"In the ICN system under study, the term $1/\\pi _c$ follows from expression (REF )."
],
[
"Special Case: $ T=0$",
"Next, we consider the case $T=0$ .",
"When a request for $c$ arrives at a cache-router and does not find the content, the request is automatically sent to the next level in the hierarchy of tiers.",
"Then, the optimization problem reduces to $\\min _{\\mathbf {\\pi }} && E[D]= \\sum _{c=1}^C \\frac{\\lambda _c}{\\lambda } ( 1-\\pi _c) \\mathcal {C}\\\\s.t.",
"&& \\sum _{c=1}^{C}\\pi _c = B$ In this case, the optimal solution consists of ordering contents based on $\\lambda _c$ and storing the $B$ most popular ones in the cache, i.e., $\\pi _c=1$ for $c=1, \\ldots , B$ and $\\pi _c=0$ otherwise.",
"Note that this rule was shown to be optimal by Liu, Nain, Niclausse and Towsley [29] in the context of Web servers."
],
[
"Special Case: $\\gamma T$ small",
"For $\\gamma T << 1$ , we have $e^{-\\gamma \\pi _c T} \\approx 1-\\gamma \\pi _c T$ .",
"The optimization problem is given by $\\min _{\\mathbf {\\pi }} && E[D]= \\sum _{c=1}^C \\frac{\\lambda _c}{\\lambda } ( 1-\\pi _c)\\left(T+(1-\\gamma \\pi _c T) \\mathcal {C} \\right) \\\\s.t.",
"&& \\sum _{c=1}^{C}\\pi _c = B \\\\&& 0 \\le \\pi _c \\le 1$ Note that the objective function can be rewritten as $E[D]= \\sum _{c=1}^C \\frac{\\lambda _c}{\\lambda } \\left( \\mathcal {C} T \\gamma \\pi _c^2 + \\left(-(\\mathcal {C}+T)- \\mathcal {C} T \\gamma \\right) \\pi _c + (\\mathcal {C}+T)\\right)$ This is a special separable convex quadratic program, known as the economic dispatch problem [30] or continuous quadratic knapsack [31].",
"It can be solved in linear time using techniques presented in [32].",
"Alternatively, in we present the dual of the problem above, which naturally yields a simple interactive gradient descent solution algorithm."
],
[
"Optimal Search Given Placement",
"In this section we address the optimal search problem, that is the choice of the $T_c$ 's, when placement is given (the $\\pi _c$ 's have been determined).",
"Then, the problem reduces to $ \\min _{\\mathbf {T}} && \\sum _{c=1}^C \\frac{\\lambda _c}{\\lambda } \\left(1-\\pi _c\\right)\\left(\\frac{1-e^{- \\gamma \\pi _c T_c }}{\\gamma \\pi _c} + \\mathcal {C} e^{-\\gamma \\pi _c T_c} \\right)\\\\s.t.",
"&& T_c \\ge 0, c=1, \\ldots , C$ For each content $c$ the function to be minimized is $f(T)$ , $f(T)=\\frac{1}{\\gamma \\pi _c} \\left(1-e^{- \\gamma \\pi _c T}\\right) + \\mathcal {C} e^{- \\pi _c \\gamma T} $ and $\\frac{df(T)}{dT}= e^{- \\gamma \\pi _c T} - \\gamma \\pi _c \\mathcal {C} e^{- \\gamma \\pi _c T}$ For a given content $c$ , a random walk search should be issued with $T=\\infty $ whenever ${df(T)}/{dT} < 0$ , i.e., if $1 - \\gamma \\pi _c \\mathcal {C} < 0$ .",
"Otherwise, the request for content $c$ should be sent directly to the publishing area; $T_c =\\left\\lbrace \\begin{array}{ll}\\infty , & \\pi _c > {1}/({\\mathcal {C} \\gamma }) \\\\0, & \\textrm {otherwise}\\end{array} \\right.$ Remarks: Although we do not solve the joint placement and search optimization problem, the special cases considered above provide some guidance for system tuning.",
"The studies we conduct in the following section provide evidence of the usefulness of our model solutions.",
"In addition, we may try different approximation approaches to solve the combined placement and search problem.",
"For instance, one such approach is to first optimize for the $\\pi _c$ 's assuming $T$ is large and set $T_c = \\infty $ for all contents that satisfy $\\pi _c > {1}/({\\mathcal {C} \\gamma })$ (see (REF )).",
"Then, set $T_c=0$ for the contents for which $\\pi _c \\le {1}/({\\mathcal {C} \\gamma })$ , and recompute $\\pi _c$ for such contents using the solution presented in Section REF so as to fill the available buffer space.",
"The performance of this and other heuristics is subject for future research."
],
[
"Evaluation",
"In this section we report numerical results obtained using the proposed model.",
"Our goals are a) to show tradeoffs involved in the choice of the time to live (TTL) parameter, b) to illustrate the interplay between content search and placement, and c) to numerically solve the optimization problems posed in this paper, giving insights about the solutions.",
"In Sections REF and REF we consider the stateless model, and in Section REF we consider the stateful one."
],
[
"Tradeoff in The Choice of TTL: Single Content Scenario",
"In this section we consider a single content that is to be served in the three-tiered topology shown in Figure REF (b).",
"Let $\\Lambda _c=1$ .",
"We assume that the number of replicas of the content remains fixed while the walker traverses each domain (Section REF ).",
"In addition, we assume that $\\pi $ and $T$ are equal at the three considered domains (this assumption will be removed in the other considered scenarios).",
"As requests are filtered towards the custodian, the rate of requests decreases when moving from tier 3 to tier 1.",
"The rate at which reinforced counters are decremented also decreases, in order to keep $\\pi $ constant.",
"Figure: Illustrative topologyFigure REF (a) shows the the expected delay to reach the custodian and the custodian load for different values of $\\pi $ and $T$ .",
"For a given value of $\\pi $ , the dotted lines indicate that as $T$ increases the load at the custodian decreases and the expected delays in the domains increases.",
"In contrast, for a given value of $T$ , as $\\pi $ increases, content becomes more available, which causes a decrease in the load at the custodian and in the expected delay.",
"Figure: Scatter plot indicating the tradeoff in the choice of TTL TT: (a) larger values of TT reduce load in custodian at cost of increased expected delay in domain; (b) expected delay as a function of expected delay in domains, assuming cost at custodian 𝒞(Λ ^ c )=1/(0.9-Λ ^ c )\\mathcal {C}(\\hat{\\Lambda }_c)=1/(0.9-\\hat{\\Lambda }_c).Next, our goal is to evaluate the expected delay.",
"To this aim, we use an M/M/1 queue to model the delay at the custodian.",
"We let the custodian cost be given by $\\mathcal {C}(\\hat{\\Lambda }_{c})=1/(0.9-\\hat{\\Lambda }_c)$ , which corresponds to the delay of an M/M/1 queue with service capacity of 0.9.",
"Figure REF (b) shows how the expected delay (obtained with equation (REF )) varies as a function of $\\pi $ and $T$ .",
"For $\\pi =0.05$ and $\\pi =0.1$ , as $T$ increases, the expected delay $E[D_c]$ first decreases and then increases.",
"The initial decrease occurs due to a decrease in the custodian load.",
"Nonetheless, as $T$ further increases the gains due to decreased load at the custodian are dominated by the increased expected delay before reaching the custodian.",
"The optimal value of $T$ is approximately 1.5 and 0.5 for $\\pi $ equal to 0.05 and 0.1, respectively."
],
[
"Benefits of Load Aggregation",
"While in the previous section we studied the dynamics of a single content, now we consider four content popularities: very low, low, medium and high.",
"In Figures REF and REF we plot expected delay both for the one-tiered architecture (Figure REF (a)) and the three-tiered architecture (Figure REF (b)).",
"The request arrival rate for each type of content was obtained from real data collected from a major Brazilian broadband service provider [24].",
"The content request rates are $\\lambda _1=0.8$ , $\\lambda _2=0.5$ , $\\lambda _3=0.1$ and $\\lambda _4=0.01$ req/sec.",
"The Request Counter (RC) of each content is decremented at constant rate $\\mu =1$ in the three tiers.",
"The value of $\\pi $ varies for each content in each tier due to the fact that content requests are filtered out as they travel towards the custodian.",
"In addition, we assume that $T$ is equal in the three domains, the number of replicas of each content remains fixed while the walker traverses each domain (Section REF ), and the mean time to retrieve a content from the publishing area exponentially increases with respect to the amount of requests hitting the publishing area, $\\mathcal {C}(\\hat{\\Lambda }_c)=e^{\\hat{\\Lambda }_c}$ .",
"Figures REF and REF show the benefits of load aggregation that occurs in the three-tiered architecture: requests that are not satisfied in tier three are aggregated in the second and third tiers.",
"Aggregation increases the probability to find the content in these tiers.",
"We observe that contents with low and medium popularities benefit the most from load aggregation.",
"Note that the expected delay decreases by several orders of magnitude for low popularity contents when we consider a three-tiered architecture.",
"For very low and high popularity contents, a significant reduction is not observed.",
"For highly popular contents, the probability to store the content in at least one of the tiers is high in both architectures, and only a small fraction of the requests is served by the publishing area.",
"For very low popularity contents, the opposite occurs: the majority of requests are served by the publishing area, as the probability that content is stored in one of the tiers is very low.",
"Figure: Expected Delay: very low and low popularity contentsFigure: Expected Delay: medium and high popularity contentsFigures REF and REF show that the three-tiered architecture yields lower delays, for all content popularities.",
"Next, we consider the optimal TTL choice in the three-tiered topology.",
"For very low popularity contents, the best choice is $T=0$ as the majority of requests must be served by the publishing area.",
"For high popularity contents, the best choice is also $T=0$ because the probability to find the content in the first router of the domain is very high.",
"On the other hand, for low popularity contents, Figure REF shows that the mean delay is minimized when $T \\approx 0.1$ ."
],
[
"Validation of the Optimal Solution",
"In this example our goal is to obtain the values of $\\pi _c$ and $T_c$ , $c=1,2,3$ , that minimize expected delay.",
"We consider three contents with high, medium and low popularity sharing a memory that can store, on average, one replica of content, $B=1$ .",
"The publisher cost is $\\mathcal {C}=10$ , the random search time is $1/\\gamma =40$ ms and the content request rates are $\\lambda _1=0.8$ , $\\lambda _2=0.1$ and $\\lambda _3=0.002$ req/sec.",
"As in the previous section, content popularities were inspired by data collected from a major Brazilian broadband service provider [24].",
"Figure: Minimum expected delay for each value of π c \\pi _c and T c T_c.Using (REF ), we compute the expected delay for different values of $\\pi _c$ and $T_c$ , $\\pi _c$ varying from $0.01$ to $0.99$ and $T_c$ varying from 0 to 30s, $i=1,2,3$ .",
"The results of our exhaustive search for the minimum delay are reported in Figure REF .",
"Figure REF (a) shows the minimum average delay attained as a function of $\\pi _1$ , $\\pi _2$ and $\\pi _3$ , considering all possible values of the other parameters.",
"Similarly, Figure REF (b) shows the minimum attainable average delay as a function of $T_1$ , $T_2$ and $T_3$ .",
"For large values of $T$ , it was shown in Section REF that (REF ) yields the optimal values of $\\pi _c$ .",
"For our experimental parameters, (REF ) yields $\\pi _1=0.71$ , $\\pi _2=0.25$ and $\\pi _3=0.04$ .",
"These values are very close to the three points that minimize the expected delay obtained using the exhaustive search, as shown in Figure REF (a), which indicates the usefulness of the closed-form expressions derived in this paper.",
"Even though the solutions we obtained do not account for joint search and placement, they yield relevant guidelines that can be effectively computed in a scalable fashion.",
"The exhaustive search for solutions took us a few hours using a Pentium IV machine, whereas the evaluation of the proposed closed-form expressions takes a fraction of seconds."
],
[
"Joint Placement and Search Optimization",
"In this paper, we introduced a new architecture, followed by a model and its analysis that couples search through random walks with placement through reinforced counters to yield simple expressions for metrics of interest.",
"The model allows us to pose an optimization problem that is amenable to numerical solution.",
"Previous works considered heuristics to solve the joint placement and search problem [33], [34], [35], accounting for the tradeoff between exploration and exploitation of paths towards content replicas [36].",
"To the best of our knowledge, we are the first to account for such a tradeoff using random walks, which have previously been proposed in the context of peer-to-peer systems as an efficient way to search for content [10].",
"We are also not aware of previous works that generalize the cache utility framework [37], [38] from a single cache to a cache network setting."
],
[
"Threats to Validity",
"In this section we discuss some of the limitations and simplifying assumptions, as well as extensions subject for future work."
],
[
"Threats to Internal Validity",
"The parameters used in numerical evaluations serve to illustrate different properties of the proposed model.",
"It remains for one to apply the proposed framework in a realistic setting, showing how to make it scale for hundreds of contents whose popularities vary over time.",
"Section provides a first step towards that goal."
],
[
"Threats to External Validity",
"In this paper, we consider a simple setup which allows us to obtain an analytical model amenable to analysis.",
"The extension to caches with TTL replacement policy, as well as other policies such as LRU, FIFO and Random, is a subject for future research.",
"In Section we focused on a single domain when analyzing the optimal placement and search problem.",
"The extension to multiple domains under the assumption that the workload to each domain is Poisson is straightforward.",
"Nonetheless, validating the extent to which this assumption is valid is subject for future work.",
"Finally, we have focused on the placement and search strategies.",
"We assumed throughout this paper the ZDD assumption (zero delay for downloads).",
"Accounting for the effects of service capacities for download on system performance is out of the scope of this work."
],
[
"Conclusion",
"Content search and placement are two of the most fundamental mechanisms that must be addressed by any content distribution network.",
"In this paper, we have introduced a simple analytical model that couples search through random walks and placement through a TTL-like mechanism.",
"Although the proposed model is simple, it captures the key tradeoffs involved in the choice of parameters.",
"Using the model, we posed an optimization problem which consists of minimizing the expected delay experienced by users subject to expected storage constraints.",
"The solution to the optimization problem indicates for how long should one wait before resorting to custodians in order to download the desired content.",
"We believe that this paper is a first step towards a more foundational understanding of the relationship between search and placement, which is key for the efficient deployment of content centric networks."
],
[
"Acknowledgments",
"Guilherme Domingues, E. de Souza e Silva, Rosa M. M. Leão and Daniel S. Menasché are partially supported by grants from CNPq and FAPERJ.",
"Don Towsley is partially supported by grants from NSF."
],
[
"Cache Insertion Rate",
"In this appendix we study the rate at which content is inserted into cache.",
"Recall that associated with each counter and content there is a threshold $K$ , such that when the reinforced counter exceeds $K$ , the corresponding content must be stored into cache.",
"Next, we consider the impact of $K$ on the cache insertion rate.",
"The cache insertion rate for a given content is the rate at which that content is brought into the cache.",
"Similarly, the cache eviction rate is the rate at which content is evicted from the cache.",
"Due to flow balance, in steady state the cache insertion rate equals the cache eviction rate.",
"Let $\\psi _c$ be the insertion rate.",
"Recall that $\\lambda _c$ and $\\mu _c$ are the request arrival rate for content $c$ and the rate at which the counter associated to content $c$ is decremented, respectively (Table ).",
"Then $\\psi _c = \\lambda _c \\rho _c^K (1-\\rho _c) = { \\pi _c (\\mu _c - \\lambda _c) }$ Recall that the content miss rate is given by $\\lambda _c \\sum _{i=0}^K \\rho _c^i (1-\\rho _c) = \\lambda _c (1-\\pi _c)$ .",
"We note that, except for $K=0$ , the content insertion rate is strictly smaller than the content miss rate.",
"Let us now consider the impact of $K$ on the insertion rate, assuming a constant miss rate.",
"For a given miss rate, $\\pi _c$ is determined.",
"Once $\\pi _c$ is established, it follows from (REF ) that larger values of $K$ yield smaller values of $\\mu _c$ .",
"A decrease in $\\mu _c$ , in turn, causes a reduction in the insertion rate (see eq.",
"(REF )).",
"A smaller insertion rate, for the same hit ratio, has several advantages: (a) first, increasing the number of cache writes slows down servicing the requests for other contents, that is, cache churn increases which reduces throughput [39], [4], [40], [41]; (b) if flash memory is used for the cache, write operations are much slower than reads; (c) writes wear-out the flash memory; and (d) additional writes mean increasing power consumption.",
"Reducing the cache eviction rate might also lead to a reduction in network load.",
"To appreciate this point, consider a scenario similar to the one presented in [42].",
"A custodian is connected to a cache through one route, and to clients through another separate route.",
"The link between the custodian and the cache is used only when a cache insertion is required.",
"The link between the custodian and the clients, in contrast, is used after every cache miss, irrespectively of whether the cache miss resulted in a cache insertion.",
"In this case, reducing the cache insertion rate produces a reduction in the load of the link between the custodian and the cache.",
"In summary, larger values of $K$ favor a reduction in the insertion rate, which benefits system performance.",
"The impact of $K$ is similar in spirit to that of $k$ in $k$ -LRU [41] and $N$ in $N$ -hit caching [40]."
],
[
"Dual Problem For $\\gamma T <<1$",
"Let $K_{2,i}&=&\\frac{\\lambda _i}{\\lambda }\\mathcal {C} T \\gamma \\\\K_{1,i}&=&\\frac{\\lambda _i}{\\lambda }\\left(-(\\mathcal {C}+T)- \\mathcal {C} T \\gamma \\right) \\\\K_{0,i}&=& (\\mathcal {C}+T)$ Let $\\mathbf {1}$ be a row vector of ones.",
"The optimization problem posed in Section REF can be stated as a quadratic program, $\\min && \\frac{1}{2} {\\mbox{$\\pi $}} ^T {\\bf Q} {\\mbox{$\\pi $}} + {\\bf c} ^T {\\mbox{$\\pi $}} \\\\s.t.",
"&& {\\bf A} {\\mbox{$\\pi $}} \\le {\\bf b} \\\\&& \\mathbf {1} {\\mbox{$\\pi $}} = B$ where $ {\\bf Q} $ is a diagonal matrix with $Q(i,i)=2 K_{2,i}$ , $ {\\bf c} $ is a vector with $c(i)=K_{1,i}$ and $\\begin{array}{ll} {\\bf A} =\\underbrace{\\begin{bmatrix}1 & 0 & \\cdots & 0 \\\\0 & \\ddots & \\ddots & \\vdots \\\\\\vdots & \\ddots & \\ddots & 0 \\\\0 & \\cdots & 0 & 1 \\\\-1 & 0 & \\cdots & 0 \\\\0 & \\ddots & \\ddots & \\vdots \\\\\\vdots & \\ddots & \\ddots & 0 \\\\0 & \\cdots & 0 & -1 \\end{bmatrix}}_C& {\\bf b} = {\\begin{bmatrix}1 \\\\1 \\\\\\vdots \\\\1 \\\\0 \\\\0 \\\\\\vdots \\\\0 \\end{bmatrix}}\\end{array}$ Note that because $ {\\bf Q} $ is a positive-definite matrix, there is a unique global minimizer [43].",
"Let $ {\\mbox{$\\delta $}} = ( {\\mbox{$\\nu $}} , {\\mbox{$\\upsilon $}} )$ , where $\\nu _i$ and $\\upsilon _i$ are the Lagrange multipliers associated with the constraints $\\pi _i \\le 1$ and the non-negativity constraint $\\pi _i \\ge 0$ , $i=1, \\ldots , C$ , respectively.",
"The Lagrangian is given by $\\mathcal {L}( {\\mbox{$\\pi $}} , {\\mbox{$\\delta $}} , \\epsilon ) &=& \\frac{1}{2} {\\mbox{$\\pi $}} ^T {\\bf Q} {\\mbox{$\\pi $}} + {\\bf c} ^T {\\mbox{$\\pi $}} + {\\mbox{$\\delta $}} ^T( {\\bf A} {\\mbox{$\\pi $}} - {\\bf b} ) +\\epsilon (\\mathbf {1} {\\mbox{$\\pi $}} - B) \\\\&=& \\frac{1}{2} \\sum _{i=1}^C \\pi _i^2 q_i + \\sum _{i=1}^C c_i \\pi _i + \\sum _{i=1}^C \\nu _i(\\pi _i-1) + \\sum _{i=1}^C \\upsilon _i(-\\pi _i) + \\epsilon \\left(\\sum _{i=1}^C \\pi _i - B\\right) \\nonumber \\\\ \\\\&=& \\sum _{i=1}^C \\pi _i \\left( \\frac{q_i \\pi _i}{2} + c_i + \\nu _i -\\upsilon _i + \\epsilon \\right) - \\left( \\sum _{i=1}^C \\nu _i \\right) - \\epsilon B$ To determine the dual function $g( {\\mbox{$\\delta $}} , \\epsilon )$ , defined as $g( {\\mbox{$\\delta $}} , \\epsilon ) = \\inf _{ {\\mbox{$\\pi $}} } \\mathcal {L}( {\\mbox{$\\pi $}} , {\\mbox{$\\delta $}} , \\epsilon )$ we note that $\\nabla _{ {\\mbox{$\\pi $}} } \\mathcal {L}( {\\mbox{$\\pi $}} , {\\mbox{$\\delta $}} , \\epsilon ) = 0 \\Rightarrow {\\mbox{$\\pi $}} ^{\\star } = - {\\bf Q} ^{-1}( {\\bf A} ^T {\\mbox{$\\delta $}} + {\\bf c} + \\mathbf {1}^T\\epsilon )$ Then, $\\pi ^{\\star }_i = \\frac{-1}{q_i} (c_i + \\nu _i - \\upsilon _i + \\epsilon )$ The dual function is $g( {\\mbox{$\\delta $}} , \\epsilon ) = -\\frac{1}{2} \\sum _{i=1}^C (\\pi ^{\\star }_i)^2{q_i} - \\left(\\sum _{i=1}^C \\nu _i\\right) -\\epsilon B$ The dual problem is also a quadratic program, $\\max _{\\epsilon , {\\mbox{$\\delta $}} }&& -\\frac{1}{2} ( {\\mbox{$\\pi $}} ^{\\star })^T {\\bf Q} {\\mbox{$\\pi $}} ^{\\star } - {\\bf b} ^T {\\mbox{$\\delta $}} - B \\epsilon \\\\s.t.",
"&& {\\mbox{$\\delta $}} \\ge 0$ The dual problem naturally yields an asynchronous distributed solution [44]."
],
[
"Stateful Model",
"Two possible ways to implement stateful searches are: (a) when an inter-domain request arrives at a router and finds that the request cannot be immediately satisfied, a search is initiated and the searcher pre-selects $j$ out of the remaining $N-1$ routers to conduct the search or; (b) after the search is initiated, the searcher chooses the next router to visit uniformly at random, from those that have not yet been visited before.",
"In this Appendix we consider the case in which routers are pre-selected at the beginning of the search.",
"We assume that $\\gamma $ is very large compared to the rate at which RCs are updated.",
"Let $J$ be a random variable denoting the number of routers to be visited by time $t$ excluding the first visited router, and as before, let $L_c$ be the number of replicas of content $c$ in the domain under consideration.",
"Note that as we do not allow revisits, $J \\le N-1$ .",
"Conditioning on $J=j$ visited routers and $L_c=l$ content replicas present in the $N-1$ possible caches to visit, $\\tilde{R}_c(t | J=j, L_c=l) =(1-\\pi _c) \\frac{ {{N-1-l} \\atopwithdelims ()j} }{{{N-1} \\atopwithdelims ()j}}.$ We assume, like in Section REF , that the search is sufficiently fast compared to the rate at which content is replaced.",
"Replacing (REF ) into (REF ), $\\tilde{R}_c(t | J=j)&=& \\sum _{l=0}^{N-1} \\tilde{R}_c(t | J=j, L_c=l) { {N-1} \\atopwithdelims ()l } \\pi _c^l (1-\\pi _c)^{N-1-l} \\\\&=& \\sum _{l=0}^{N-1-j} \\tilde{R}_c(t | J=j, L_c=l) { {N-1} \\atopwithdelims ()l } \\pi _c^l (1-\\pi _c)^{N-1-l} \\\\&=& (1-\\pi _c) \\sum _{l=0}^{N-1-j} { {N-1-j} \\atopwithdelims ()l } \\pi _c^l (1-\\pi _c)^{N-1-l}\\\\&=& (1-\\pi _c)^{j+1}.",
"$ () follows from () since $\\tilde{R}_c(t | J=j)=0$ if $l>N-1-j$ as at least one of the $j$ routers necessarily has the content.",
"It is interesting to observe that () and (REF ) are identical, although derived from two different sets of assumptions."
]
] | 1606.05034 |
[
[
"Limits of stability in supported graphene nanoribbons subject to bending"
],
[
"Abstract Graphene nanoribbons are prone to in-plane bending even when supported on flat substrates.",
"However, the amount of bending that ribbons can stably withstand remains poorly known.",
"Here, by using molecular dynamics simulations, we study the stability limits of 0.5-1.9 nm wide armchair and zigzag graphene nanoribbons subject to bending.",
"We observe that the limits for maximum stable curvatures are below ~10 deg/nm, in case the bending is externally forced and the limit is caused by buckling instability.",
"Furthermore, it turns out that the limits for maximum stable curvatures are also below ~10 deg/nm, in case the bending is not forced and the limit arises only from the corrugated potential energy landscape due to the substrate.",
"Both of the stability limits lower rapidly when ribbons widen.",
"These results agree with recent experiments and can be understood by means of transparent elasticity models."
],
[
"Limits of stability in supported graphene nanoribbons subject to bending Topi Korhonen Pekka Koskinen [email:]pekka.koskinen@iki.fi NanoScience Center, Department of Physics, University of Jyvaskyla, 40014 Jyväskylä, Finland 61.46.-w,62.25.-g,68.65.Pq,68.55.-a Graphene nanoribbons are prone to in-plane bending even when supported on flat substrates.",
"However, the amount of bending that ribbons can stably withstand remains poorly known.",
"Here, by using molecular dynamics simulations, we study the stability limits of $0.5-1.9$ nm wide armchair and zigzag graphene nanoribbons subject to bending.",
"We observe that the limits for maximum stable curvatures are below $\\sim 10$ deg/nm, in case the bending is externally forced and the limit is caused by buckling instability.",
"Furthermore, it turns out that the limits for maximum stable curvatures are also below $\\sim 10$ deg/nm, in case the bending is not forced and the limit arises only from the corrugated potential energy landscape due to the substrate.",
"Both of the stability limits lower rapidly when ribbons widen.",
"These results agree with recent experiments and can be understood by means of transparent elasticity models.",
"Today graphene nanoribbons can be fabricated at atomic precision, but only in the presence of a stabilizing substrate.",
"[1] The substrate stabilizes flimsy ribbons and suppresses their tendency to twist, fold and ripple.",
"[2], [3], [4], [5], [6] However, even substrates cannot fully prevent all deformations, most of which induce mechanical strains that alter ribbons' electronic properties.",
"[7], [8], [9] Actually, such strain engineering of electronic properties is gaining popularity, whereby detailed knowledge of mechanical stability limits is becoming increasingly valuable.",
"[10] Mechanical strain can be created for example by lattice mismatch, by impurities and lattice defects, and by the fabrication process itself.",
"[11] Compressive strain, in particular, is often limited by buckling instability.",
"For uniaxial compression buckling has been observed in experiments at $0.5$ % strain and in simulations at $0.8$ % strain.",
"[12], [13] In graphene nanoribbons, however, the most pertinent deformation is not uniaxial compression but bending.",
"Yet, the mechanical stability limits of supported ribbons subject to bending remain unexplored.",
"In this letter, therefore, we aimed to address two fundamental questions: How much can a graphene nanoribbon of given width bend on a given substrate until it buckles?",
"And, to what extent can it remain bent due to the corrugation potential energy of the substrate alone, without external forcing?",
"As it will turn out, both of these questions could be answered by transparent modeling.",
"Figure: (color online) 7-armchair graphene nanoribbons subject to bending.",
"(a) In experiments bending was controlled by the tip of an atomic force microscope, whose movements are denoted by arrows.",
"Buckling is seen as the bright kink.",
"(b) In simulations ribbons were bent by fixing their front ends and by turning their tail ends.",
"The rightmost geometry shows the buckled geometry.",
"(c) Maximum curvature without external forcing.",
"After manipulation the ribbon remained bent by the substrate corrugations alone.",
"Scale bar, 10 nm.",
"(d) In the simulations one end of the ribbon (green tail) was pinned to (set in registry with) the substrate while the other end was turned to the maximum stable curvature beyond which the entire ribbon started sliding.",
"The experimental figures in panels (a) and (c) are reproduced from Ref.",
"bendingandbucklingiop by Creative Commons Attribution licence; image ordering has been changed.Our simulations were closely related to the recent experiments of van der Lit et al.",
"in Ref.",
"bendingandbucklingiop (Fig.",
"1).",
"There an atomically precise 7-armchair graphene nanoribbon was bent at low temperature on Au(111) surface by an atomic force microscope (AFM) tip.",
"Under forced bending and above certain maximum curvature the ribbon was observed to buckle off the substrate (Figs. 1a).",
"Furthermore, ribbon was observed to withstand certain maximum curvature, presumably due to the lateral energy corrugations arising solely from the substrate interactions (Figs. 1c).",
"To investigate the buckling instability in more detail, we simulated ribbons subject to forced bending (Fig. 1b).",
"We simulated hydrogen-passivated $N$ -armchair ($N=5,\\;7,\\;9,\\;11,$ and 13) and $N$ -zigzag ($N=4,\\;6,\\;8,$ and 10) graphene nanoribbons of widths $w\\approx 0.5-1.9$ nm and lengths given by $1/10$ aspect ratio.",
"The C-C, C-H, and H-H interactions were modeled by the empirical reactive bond-order potential REBO.",
"[15] The ribbons were initially relaxed on a model Au substrate, which assumed an interaction with the ribbon described by a $z$ -dependent potential with 20 meV/Å$^2$ adhesion, $3.4$ Å equilibrium distance, and a functional form suggested by the Lennard-Jones 12-6 potential (Fig.REF ).",
"[16], [17], [18] This substrate model ignores lateral energy corrugation, but it is expected to be a good approximation, because graphene nanoribbons that are out of registry with respect to the Au(111) substrate have been shown not to experience any lateral forces, and thus to exhibit superlibricity.",
"[19], [20] Figure: (color online) Ribbon's adhesion energy per atom as a function of distance from the substrate.",
"Under superlubric conditions surface adhesion is modeled by laterally homogeneous Lennard-Jones (LJ) potential.",
"Under conditions where registry effects are important, the adhesion is modeled by Kolmogorov-Crespi (KC) potential, which models energy corrugations by making the energy minimum registry-dependent (shown with adhesion curves for AA, AB, and saddle (S) point configurations).The supported ribbons were simulated by the LAMMPS code, using 1 fs time step and Langevin thermostat at 10 K temperature and 5 ps damping time.",
"[22] First the ribbons were thermalized on the model substrate.",
"Then they were gradually bent by fixing one end and slowly (quasi-statically) turning the other end while simultaneously allowing its free movement in the plane (Fig. 1b).",
"At a later instant the turning direction was reversed, and the simulation terminated with straight ribbons.",
"At the initial stages of the simulations the bending was smooth and the ribbons remained adhered to the substrate.",
"Here we quantify the amount of bending both by the in-plane curvature $\\kappa =1/R$ , where $R$ is the radius of curvature, and by the dimensionless curvature $\\Theta = \\kappa w/2$ , which also equals the absolute amount of strain at the ribbon edges.",
"Using straightforward continuum elasticity theory, the elastic energy during this initial stage is $E_\\text{bend}(\\Theta ) = (1/6)k w l \\Theta ^2 [1 - 2 \\tau /(k w)]^2,$ where $w$ is ribbon width, $l$ is ribbon length, $k = 19$ eV/Å$^2$ is graphene's in-plane modulus, and $\\tau $ is the stress at the passivated armchair ($\\tau _{ac} = -1.5$ eV/Å) or zigzag edges ($\\tau _{zz} = -0.2$ eV/Å), as given by the REBO potential.",
"[23] Eq.",
"(REF ) gives the elastic energy below $\\Theta \\lesssim 3$ % at fair accuracy (Fig.",
"3a).",
"During this initial stage we observed weak ripples at the inner edges of the ac-ribbons.",
"Ripples were notable up- and down-displacements of alternating armchair units and observable along the entire ribbon.",
"They have been observed also in straight ribbons where they have been attributed to chemically induced edge stress; here the edge stress was created mostly by the bent geometry itself.",
"[24], [25] When curvature increased, the rippling amplitude increased, but wavelength remained fixed.",
"These ripples were observed only for the ac-ribbons as zz-ribbons remained almost completely flat prior to bucling.",
"When the increasing curvature reached a critical limit, the in-plane stress finally became unbearable and the ribbon suddenly buckled (Fig.",
"1c).",
"Buckling allowed two parts of the ribbon to straighten, which released in-plane elastic energy, although at the expense of lost adhesion and increased out-of-plane bending energy.",
"Buckling occurred later for narrow ribbons than for wide ribbons.",
"The events during the bending-straightening simulations are best gauged through the maximum height of the ribbon above the substrate (Fig. 3b).",
"Initially the buckle was formed at $\\Theta _{b^{\\prime }}$ , but upon straightening it remained stable also for curvatures $\\Theta <\\Theta _{b^{\\prime }}$ so that when the ribbon finally unbuckled at $\\Theta _b$ , roughly half the buckling curvature, the result was a notable hysteresis.",
"The buckling-unbuckling process was reversible; plastic deformations did not occur.",
"These observations are in agreement with experiments that also showed the restoring of the initial geometry.",
"In particular, for $N=7$ ac-ribbon the buckling occurred in experiments at curvature of 4 deg/nm, in reasonable agreement with the computational curvature of 6 deg/nm.",
"[14] Note that it is justifiable to compare experiments only to the smaller curvature $\\Theta _b$ , because in macroscopic time scales random perturbations help drive the system toward buckled geometry already at smaller curvatures.",
"Figure: (color online) Trends in buckling instabilities.",
"(a) Simulated elastic energy densities (thin wiggly lines) compared with the elastic model of Eq.",
"() (thick solid lines) for ac-ribbons of different widths.",
"Curves are offset for clarity.",
"(b) The maximum height of the ac-ribbons above the substrate.",
"The bending and straightening simulations show hysteresis in the buckling: buckling requires larger curvature than unbuckling.",
"Dotted line is the buckling threshold.",
"(c) Buckling and unbuckling curvatures for different ribbons and temperatures as defined by the threshold in panel b.To understand the general width-dependence in the buckling (Fig.",
"REF c), let us develop a model that accounts for the in-plain strain, out-of-plain bending, and substrate adhesion energies.",
"In the model the ribbon is treated as two aligned narrow strands that represent the compressed and stretched halves of the ribbon.",
"The aligned strands are next to the neutral line and separated by $w_{eff} = \\alpha w$ , where the width-dependent parameter $\\alpha $ ($\\lesssim 1$ ) is later fitted to account for the averaging.",
"Upon buckling the outer strand remains flat but the height profile of the inner strand acquires the form $y(l) = A \\sin ^2(l/\\lambda \\pi )$ ($0\\le l \\le \\lambda $ ), where $A$ is the buckling amplitude and $l$ is the distance measured along the strand.",
"This profile decreases the strand length by $\\Delta l = \\pi ^2A^2/(4 \\lambda )$ and thereby relieves the compressive strain energy at the inner edge by $wk\\Theta \\Delta l/2$ and the tensile strain energy at the outer edge by the same amount.",
"This approach is similar to that in Ref. bendinginduceddelamcm.",
"Adding this strain energy release to the loss in Lennard-Jones energy ($\\int w/2[V_{LJ}(y)-V_{LJ}(u)]\\text{d}l$ ) and the out of plane bending energy associated with the height profile ($\\int \\frac{w}{4} D y^{\\prime \\prime 2}(l)\\text{d}l$ ), the energy difference between purely bent and buckled ribbon becomes $\\Delta E(\\Theta ) = A^2 \\frac{w}{2} \\left[-k \\frac{\\pi ^2 A^2}{2 \\lambda }\\Theta \\alpha + \\frac{15}{2}\\frac{\\epsilon _{vdw}\\lambda }{\\sigma ^2} + \\frac{D \\pi ^4}{\\lambda ^3} \\right].$ Here $\\epsilon _{vdw}$ is the adhesion energy per unit area, $\\sigma $ is the interlayer distance, and $D = 1.0$ eV is graphene's bending modulus.",
"[26], [27] Buckling occurs when the first term becomes large enough due to the increasing curvature so that $\\Delta E(\\Theta _b) = 0$ .",
"The energy of the buckled geometry is further minimized by $\\partial \\Delta E/\\partial \\lambda |_{\\lambda =\\lambda _b}=0$ .",
"Solving these equations yields $\\lambda _b = 9$ Å and $\\Theta _b(T) = 2/(k \\sigma \\alpha )\\sqrt{30 \\epsilon _{vdw}D} \\approx 0.023 \\times \\alpha ^{-1}.$ Fit to the simulations gives $\\alpha _i = 1/(\\beta _i w^{-1}+1)$ , where $\\beta _{ac}=7$ Å and $\\beta _{zz} = 5$ Å, which provide a good agreement with the simulated buckling curvatures (Fig.",
"REF c).",
"The fit is physically meaningful and obeys the consistency requirement $\\alpha \\lesssim 1$ .",
"The validity of the model is probably limited for ribbon widths below few nanometers, although $\\Theta _b=2.3$ % is a reasonable limit for very wide ribbons, too.",
"While our simulations included ribbons only with hydrogen-passivated zigzag and armchair edges, also other edges with other passivations or edge reconstructions are possible.",
"[28], [29], [30], [31] Especially in free-standing graphene the edges may create sizable corrugations.",
"[32], [25] On substrates these corrugations diminish in magnitude, but do not vanish completely.",
"[33] However, here the edge stresses are small due to hydrogen passivation and the lateral stresses due to bending are so large that the effect of edge stress is fairly small.",
"This is suggested already by the quantitatively similar buckling behavior in zigzag and armchair ribbons (Fig.",
"REF c).",
"For completeness, we repeated buckling simulations for armchair ribbons also at room temperature.",
"As the main result, the effect of temperature was to reduce the hysteresis and initiate buckling at slightly smaller curvatures (Fig.",
"REF c).",
"On average, however, the buckling occurred around the same curvature as described by the model fitted at low temperature.",
"In the next set of simulations, we investigated the limits of maximal curvature in armchair ribbons allowed by the substrate energy corrugation alone.",
"In these simulations we chose to place the ribbons on a graphene substrate modeled by the Kolmogorov-Crespi (KC) registry-dependent interlayer potential.",
"[21] This model substrate was obviously different from the Au(111) substrate in the experiments, but our choice was a necessary compromise for a feasible substrate model with a realistic energy corrugation.",
"Namely, the frequently used Lennard-Jones potential typically yields an order of magnitude too low energy corrugation for sliding, and proper registry-dependent potentials for graphene and Au(111) are missing.",
"[34] Nevertheless, the ribbon adhesions for both Au and graphene substrates are similar, so the KC potential was an attempt to combine a well-defined substrate model with a realistic corrugation energy landscape.",
"In these simulations one end of the ribbon was first appended by a tail of length $L_t$ that was pinned to the substrate by setting it in full registry (Fig. 1f).",
"The other end was then gradually turned until the maximum stable curvature beyond which the pinning was released and the tail started sliding, causing straightening of the ribbon.",
"The ribbon was considered stable at given $L_t$ and $\\kappa $ if it remained in place for 20 ps, although it was evident already within few ps whether the curvature was stable or not.",
"The maximum curvature limits were then searched for each ribbon width with several tail lengths.",
"Figure: (color online) Maximum stable in-plane curvatures for different ribbons as a function of the added tail length L t L_t.",
"Dashed lines are the model estimates from Eq. .",
"Inset: maximum curvature limit as a function of ribbon width at L t =0L_t=0.Simulations show that narrow ribbons withstand higher curvatures than wide ribbons and that maximum curvatures increase when the tail lengths increase (Fig. 4).",
"It is notable that certain finite curvatures can be achieved even in the absence of any added tail (inset of Fig. 4).",
"This occurs because also ribbon's end is close to registry and not yet subject to superlubric behavior.",
"By geometry considerations we therefore approximate that the length $L_{t^{\\prime }} = \\sqrt{2Ra + a^2}$ close to the end of the ribbon is still pinned to the substrate, where $a$ is a length scale for the tolerance in a lateral displacement that is still considered to be in registry.",
"Thus, the total length of the substrate-pinned ribbon at the end equals $L_\\text{pin}=L_{t^{\\prime }}+L_t$ .",
"This assumption serves as a starting point for a model for the maximum curvature limit.",
"In the model we consider the pinned part to be subject to a bending moment $kw^2\\Theta /6$ imposed by the unpinned part.",
"This moment must not exceed a maximum value, lest the pinned part starts to slide.",
"At the maximum the bending moment equals the maximum allowed moment, or $\\frac{1}{6}kw^2\\Theta = \\int _\\text{pinned} r\\times ( f\\text{d}\\mathcal {A}),$ where $f$ is the maximum force per unit area during sliding, averaged over all sliding directions.",
"The integration is over the pinned part of length $L_\\text{pin}$ and $r$ is the distance to its center of mass.",
"Fitting the force parameter $f$ with a chosen tolerance $a=0.7$ Å to simulation data yields $f=0.7$ meV/Å$^3$ .",
"The maximal force per unit area for sliding in an armchair direction is $f_{max}=2.3$ meV/Å$^3$ , which confirms the physical interpretation of the fit ($f\\approx 0.3\\times f_{max}$ ).",
"[35] Upon inserting these parameters into the model Eq.",
"(REF ), the trends in maximum curvatures get reproduced surprisingly well (Fig. 4).",
"The model underestimates the maximum curvatures as compared to simulations, which is however not surprising given the highly discrete nature of the short-tail limit (inset of Fig. 4).",
"The model predicts pinning at roughly constant edge strain of $\\sim 0.9\\%$ , but in simulations the allowed edge strain depends somewhat on ribbon width, changing as ribbons widen from $\\sim 0.9$ % for $N=5$ to $\\sim 1.5$ % for $N=13$ .",
"Such dependence may originate due to thermal fluctuations, which affect narrow and wide ribbons differently due to the different number of pinned atoms.",
"These simulations can be compared to the experimentally observed pinning in Ref.",
"bendingandbucklingiop, although with caution.",
"The energy corrugations for graphene ribbons on Au(111) and on graphene are probably different, but likely of similar magnitude due to the similarity of the adhesion itself.",
"[16] To this end, note that the model in Eq.",
"(REF ) suggests that the substrate affects the trends only through the averaged parameter $f$ .",
"Thus, even though the symmetry in Au(111) differs from that in graphene, it is not unreasonable to expect that the results would correspond also to Au substrate, at least semi-quantitatively.",
"Such correspondence is further supported by the rough agreement between the experimental (2 deg/nm for gold substrate) and simulated ($1.4$ deg/nm for model graphene substrate) maximum curvatures for a 7-armchair ribbon.",
"[14] At any rate, the parameter $f$ allows transferring the results to any other substrate, making the model highly versatile.",
"While in buckling the effect of temperature was clearly small, in pinning its effect is more ambiguous.",
"Although the energy corrugation per atom $\\sim 9$ meV corresponds only to the temperature of $T\\sim 100$ K, the pinning still occurred also at room temperature, at least withing time scales accessible to the simulations (20 ps).",
"The general tendency of an increased temperature was to modestly decrease the maximal pinning curvature, although the results became less clear.",
"While at low temperatures the possible unpinning of the tail was fast ($\\sim 1-2$ ps) and clear-cut, at high temperatures thermal fluctuations brought unambiguity by introducing more variations to the time scale of unpinning.",
"Thus, reliable determination of structure stability would have required simulation times beyond reasonable limits, as also indicated by recently observed sliding phenomena.",
"[36] To conclude, these simulations and the associated models provide transparent understanding for the stability limits in supported graphene nanoribbons subject to bending.",
"Narrow 5-, 7-, 9-, 11-, and 13-armchair ribbons require only minimal pinned parts to maintain curvatures around 1 deg/nm (radius of curvature $R\\approx 60$ nm).",
"Although such curvatures are gentle, other studies have found them to cause predictable modifications in ribbons' electronic and optical properties.",
"In particular, simulations in Ref.",
"Graphenenanoribbonssubjecttogentlebendsprb showed that the energy gap for $N$ -armchair graphene nanoribbons change according to the expression $\\Delta E_g(\\Theta )=\\frac{1}{2}(-1)^q\\gamma \\delta \\Theta ^2,$ where $q=\\mod {(}N,3)$ (restricted to $q=0,\\,1$ ) is the ribbon family, $\\gamma =1.7$ describes bond anharmonicity that is relevant for bending-induced stretching, and $\\delta =12$ eV is an electromechanical coupling constant related to gap changes during the stretching of straight ribbons.",
"Combining Eq.",
"(REF ) with Eq.",
"(REF ), the buckling-limited maximum energy gap change becomes directly $|\\Delta E_g^{max}(w)|=5.4\\times (7\\text{ Å}\\times w^{-1}+1)^2\\text{ meV}.$ For the ribbons studied here this amounts from 23 meV ($N=5$ ) to 10 meV ($N=13$ ) gap changes.",
"For wider ribbons the maximum gap change shrinks.",
"In the case of pinning the maximum curvature depends on the tail length $L_t$ , but it is always limited by Eq.",
"(REF ), so with unconstrained bending Eq.",
"(REF ) gives the upper limit for gap changes.",
"Buckling, however, can modify the electronic properties even more than bending.",
"Simulations showed that narrow ribbons remained flat above 4 deg/nm curvatures ($R\\approx 14$ nm), but stability was strongly width-dependent; ribbons wider than $1.5$ nm remained flat only below 2 deg/nm ($R\\gtrsim 29$ nm).",
"The obtained stability limits thus provide guidelines to design experiments and to choose structures that would be stable enough for reliable device operation.",
"Because the adhesion energies for most van der Waals bound, physisorbed two-dimensional materials are of similar magnitude, we expect the presented elastic models to have applicability for several other ribbon and substrate materials.",
"[17] To this end, we propose that the stabilities of bent ribbons could even be used as a measurement technique to investigate the interaction between different nanoribbons and substrates.",
"Acknowledgements: We thank the Academy of Finland for funding (Projects No.",
"283103 & 251216) and CSC - IT Center for Science in Finland for computer resources."
]
] | 1606.04908 |
[
[
"The Edit Distance Transducer in Action: The University of Cambridge\n English-German System at WMT16"
],
[
"Abstract This paper presents the University of Cambridge submission to WMT16.",
"Motivated by the complementary nature of syntactical machine translation and neural machine translation (NMT), we exploit the synergies of Hiero and NMT in different combination schemes.",
"Starting out with a simple neural lattice rescoring approach, we show that the Hiero lattices are often too narrow for NMT ensembles.",
"Therefore, instead of a hard restriction of the NMT search space to the lattice, we propose to loosely couple NMT and Hiero by composition with a modified version of the edit distance transducer.",
"The loose combination outperforms lattice rescoring, especially when using multiple NMT systems in an ensemble."
],
[
"Introduction",
"Previous work suggests that syntactic machine translation such as Hiero [4] and Neural Machine Translation (NMT) [16], [30], [6], [2] are very different and have complementary strengths and weaknesses [25], [29].",
"Recent attempts to combine syntactic SMT and NMT report large gains over both baselines.",
"Authors in [25] used NMT to rescore $n$ -best lists which were generated with a syntax-based system.",
"They report that even with 1000-best lists, the gains of using the NMT rescorer often do not saturate.",
"Syntactically Guided NMT [29] constrains the NMT search space to Hiero translation lattices which contain significantly more hypotheses than $n$ -best lists.",
"In SGNMT, an NMT beam decoder with a relatively small beam can explore spaces much larger than $n$ -best lists, yielding BLEU score improvements with far fewer expensive NMT evaluations.",
"However, these rescoring approaches enforce an exact match between the NMT and syntactic decoders.",
"In general, this kind of hard restriction is best avoided when combining diverse systems [19], [11].",
"For example, in speech recognition, ROVER [10] is a system combination approach based on a soft voting scheme.",
"In machine translation, minimum Bayes-risk (MBR) decoding [17] can be used to combine multiple systems [8].",
"MBR also does not enforce exact agreement between systems as it distinguishes between the hypothesis space and the evidence space [12], [31].",
"We find that Hiero lattices generated by grammars extracted with the usual heuristics [4] do not provide enough variety to explore the full potential of neural models, especially when using NMT ensembles.",
"Therefore, we present a “soft” lattice-based combination scheme which uses standard operations on finite state transducers such as composition.",
"Our method replaces the hard combination in previous methods with a similarity measure based on the edit distance, and gives the NMT decoder more freedom to diverge from the Hiero translations.",
"We find that this loose coupling scheme is especially useful when using NMT ensembles.",
"Figure: Modified edit distance transducer EE.",
"`a' is an NMT OOV."
],
[
"Combining Hiero and NMT via Edit Distance Transducer",
"In contrast to the strict coupling in SGNMT, we propose to loosely couple Hiero and NMT via an edit distance transducer and shortest distance search.",
"With loose coupling, the NMT decoder is not restricted to the Hiero lattice as in previous work, but runs independently to produce translation lattices on its own, which are then combined with the Hiero lattices.",
"The combination does not require an exact match.",
"Instead, we will describe a procedure for combining NMT and Hiero that captures similarity under the edit distance and both the NMT and Hiero translation system scores.",
"This scheme is implemented efficiently using standard FST operations [1].",
"First, we introduce the FST composition operation and the edit distance transducer.",
"We will describe the whole pipeline in Sec.",
"REF ."
],
[
"Composition of Finite State Transducers",
"The composition of two weighted transducers $T_1$ , $T_2$ (denoted as $T_1 \\circ T_2$ ) over a semiring $(\\mathbb {K},\\oplus ,\\otimes )$ is defined following [24] $[T_1 \\circ T_2](x,y)=\\bigoplus _z T_1(x,z)\\otimes T_2(z,y).$ We will make extensive use of this operation as tool for building complex automata which make use of both the NMT and Hiero translation lattices."
],
[
"The Edit Distance Transducer",
"Composition can be used together with a “flower automaton” to calculate the edit distance between two sequences [23].",
"The edit distance transducer shown in Fig.",
"REF transduces a sequence $x$ to another sequence $y$ over the alphabet $\\lbrace \\text{a},\\text{b}\\rbrace $ and accumulates the number of edit operations via the transitions with cost 1.",
"In our case, $x$ corresponds to an NMT hypothesis which is to be combined with a Hiero hypothesis $y$ .",
"In contrast to SGNMT, where we require an exact match between NMT and Hiero (up to UNKs), our edit-distance-based scheme allows different hypotheses to be combined.",
"We replaced the standard definition of the edit distance transducer [23] by a finer-grained model designed to work well for combining NMT and Hiero.",
"Instead of uniform costs, we lower the cost for UNK substitutions as we want to encourage substituting NMT UNKs by words in the Hiero translation.",
"We distinguish between three types of edit operations.",
"Type I: Substituting UNK with a word outside the NMT vocabulary is free.",
"Type II: For substitutions of UNK with a word inside the NMT vocabulary we add the cost $\\lambda _{sub}$ .",
"Type III: All other edit operations are penalized with cost $\\lambda _{edit}$ (and $\\lambda _{edit}>\\lambda _{sub}$ ).",
"We will refer to the modified edit distance transducer as $E$ .",
"Fig.",
"REF shows $E$ over the alphabet $\\lbrace \\text{a},\\text{b},\\text{UNK}\\rbrace $ , with `a' being an NMT OOV.",
"Figure: Projection of the best path: Π UNK (ShortestPath(C))\\Pi _{UNK}(\\text{ShortestPath}(C)).",
"The final hypothesis is die regionale Politik in Grosswahlstadt darf jedoch nicht leiden.Figure: UNK extension transducer UU."
],
[
"Loose Coupling of Hiero and NMT",
"Our edit-distance-based scheme combines an NMT translation lattice $N$ with a Hiero translation lattice $H$ .",
"Weights in $N$ and $H$ are scaled by $\\lambda _{nmt}$ and $\\lambda _{hiero}$ , respectively.",
"The similarity measure between NMT and Hiero translations is parametrized with $\\lambda _{ins}$ , $\\lambda _{edit}$ , and $\\lambda _{sub}$ .",
"We keep the various costs separated by using transducers with tropical sparse tuple vector semirings [15].",
"Instead of single real-valued arc weights, this semiring uses vectors which can hold multiple features.",
"The inner product of these vectors with a constant parameter vector determines the final weights on the arcsThe ucam-smt tutorial contains details to the tropical sparse tuple vector semiring: http://ucam-smt.github.io/tutorial/basictrans.html#lmert_veclats_tst.",
"The sparse tuple vector semiring enables us to optimize the $\\lambda $ -parameters with LMERT [21] on a development set.",
"Examples for $H$ and $N$ are shown in Fig.",
"REF and Fig.",
"REF .",
"The shortest path in $H$ containing the string nicht erlaubt sein sollte zu has grammatical and stylistic flaws but is complete, whereas there is a better path in $N$ with an UNK.",
"Our goal is to merge these two hypotheses by using the NMT translation in $N$ with the UNK replaced by a word from the Hiero lattice $H$ .",
"Adding UNK insertions.",
"We found that often NMT produces an isolated UNK token, even if multiple tokens are required.",
"Therefore, we allow extending a single UNK token to a sequence of up to three UNK tokens.",
"This is realized by replacing UNK arcs in $N$ with the transducer $U$ shown in Fig.",
"REF using OpenFST's Replace operation.",
"Fig.",
"REF shows the result of the replace operation when applied to the example lattice $N$ in Fig.",
"REF .",
"We denote this operation as follows: $\\text{Replace}(N,\\mathtt {UNK},U)$ Composition with the edit distance transducer.",
"The next step finds the edit distances to the Hiero hypotheses as described in Sec.",
"REF .",
"$C := \\text{Replace}(N,\\mathtt {UNK},U)\\circ E \\circ H$ Shortest path.",
"The above operation generates very large lattices, and dumping all of them is not feasible.",
"We could use disambiguation [15], [22] on the combined transducer $C$ to find the best alignment for each unique NMT hypothesis.",
"However, we only need the single shortest path in order to generate the combined translation.",
"$\\text{ShortestPath}(C)$ Projection.",
"A complete path in the transducer $C$ has an NMT hypothesis on the input labels (marked green in Fig.",
"REF ) and a Hiero hypothesis on the output labels (marked blue in Fig.",
"REF ).",
"Therefore, we can generate different translations from the best path in $C$ .",
"If we project the input labels on the output labels with OpenFST's Project, we obtain a hypothesis $\\widehat{t}_{NMT}$ in the NMT lattice $N$ .",
"$\\widehat{t}_{NMT} = \\Pi _1(\\text{ShortestPath}(C))$ However, $\\widehat{t}_{NMT}$ still contains UNKs.",
"If we project on the input labels, we end up with the aligned Hiero hypothesis without UNKs (blue labels in Fig.",
"REF ) $\\widehat{t}_{Hiero} = \\Pi _2(\\text{ShortestPath}(C))$ but we do not use the NMT translation directly.",
"Therefore, we introduce a new projection function $\\Pi _{UNK}$ which switches between preserving symbols on the input and output tapes: if the input label on an arc is UNK, we write the output label over the input label.",
"Otherwise, we write the input label over the output label.",
"This is equivalent to projecting the output labels to the input labels only if the input label is UNK, and then projecting the input labels to the output labels.",
"As shown in Fig.",
"REF , we obtain the NMT hypothesis, but the UNK is replaced by the matching word Grosswahlstadt from the Hiero lattice.",
"Thus, the final combined translation is described by the following term: $\\widehat{t}_{comb} = \\Pi _{UNK}(\\text{ShortestPath}(C))$ In general, the final hypothesis $\\widehat{t}_{comb}$ is a mix of an NMT and a Hiero hypothesis.",
"We do not search for $\\widehat{t}_{comb}$ directly but for pairs of NMT and Hiero translations which optimize the individual model scores as well as the distance between them.",
"Stated more formally, the shortest path in $C$ yields a pair $(\\widehat{t}_{NMT},\\widehat{t}_{Hiero})$ for which holds $\\begin{aligned}\\widehat{t}_{NMT},\\widehat{t}_{Hiero} =\\operatornamewithlimits{argmin}_{(t_{N},t_{H})\\in N \\times H} \\Big ( d_{edit}(t_{N},t_{H})\\\\+\\lambda _{nmt}\\cdot S_{N}(t_{N}|s)+\\lambda _{hiero}\\cdot S_{H}(t_{H}|s)\\Big )\\end{aligned}$ where $d_{edit}(t_{N},t_{H})$ is the modified edit distance between $t_{N}$ and $t_{H}$ (according $E$ and $U$ ), and $S_{N}(t_{N}|s)$ and $S_{H}(t_{H}|s)$ are the scores NMT and Hiero assign to the translations given source sentence $s$ .",
"If we interpret these scores as negative log-likelihoods, we arrive at a probabilistic interpretation of Eq.",
"REF .",
"$\\begin{aligned}\\widehat{t}_{NMT},\\widehat{t}_{Hiero} &=\\operatornamewithlimits{argmax}_{(t_{N},t_{H})\\in N \\times H} \\Big (\\\\& \\mathrm {e}^{-d_{edit}(t_{N},t_{H})}\\cdot P(t_{N},t_{H}|s) \\Big ) \\\\\\end{aligned}$ with (assuming independence) $P(t_{N},t_{H}|s):=P_{N}(t_{N}|s)^{\\lambda _{nmt}}\\cdot P_{H}(t_{H}|s)^{\\lambda _{hiero}}.$ Eq.",
"REF suggests that we maximize the product of two quantities – the similarity between Hiero and NMT hypotheses and their joint probability.",
"The FST operations allow to optimize over the set $N \\times H$ efficiently.",
"Note that the NMT lattice $N$ is rather small in our case ($|N|\\le 20$ ) due to the small beam size used in NMT decoding.",
"This makes it possible to solve Eq.",
"REF almost always without pruning We limit the Hiero lattices to a maximum of 100,000 nodes with OpenFST's Prune to remove the worst outliers.."
],
[
"Experimental Setup",
"The parallel training data includes Europarl v7, Common Crawl, and News Commentary v10.",
"Sentence pairs with sentences longer than 80 words or length ratios exceeding 2.4:1 were deleted, as were Common Crawl sentences from other languages [28].",
"We use news-test2014 (the filtered version) as a development set, and keep news-test2015 and news-test2016 as test sets.",
"The NMT systems are built using the Blocks framework [32] based on the Theano library [3] with the network architecture and hyper-parameters as in [2]: the encoder and decoder networks consist of 1000 gated recurrent units [6].",
"The decoder uses a single maxout [13] output layer with the feed-forward attention model described in [2].",
"In our final ensemble, we use 8 independently trained NMT systems with vocabulary sizes between 30,000 and 60,000.",
"Rules for our En-De Hiero system were extracted as described in [9].",
"A 5-gram language model for the Hiero system was trained on WMT16 parallel and monolingual data [14].",
"We apply gentle post-processing to the German output for fixing small number and currency formatting issues.",
"The English source sentences in the training corpus are lower-cased.",
"During decoding, we lower case only in-vocabulary words, and pass through OOVs with correct casing.",
"We apply a simple heuristic for recognizing surnames to avoid literal translation of them into GermanWe mark a word as surname if it has occurred after a first name, is on a census list of known surnames, and is written with a capitalized initial letter.. Table: English-German lower-cased BLEU scores calculated with Moses mteval-v13a.pl.Table: Projection methods on news-test2016 with NMT 8-ensemble."
],
[
"Results",
"http://matrix.statmt.org/ Tab.",
"REF reports performance on news-test2014, news-test2015, and news-test2016The code we used for SGNMT and ensembling is available at http://ucam-smt.github.io/sgnmt/html/..",
"Similarly to previous work [29], we observe that rescoring Hiero lattices with NMT (SGNMT) outperforms both NMT and Hiero baselines significantly on all test sets.",
"For SGNMT, we see further improvements of between +0.7 BLEU (news-test2014) and +1.1 BLEU (news-test2015) by using NMT ensembles rather than single NMT.",
"However, these gains are rather small considering the improvements from using ensembles for the (pure) NMT baseline (between +1.9 BLEU and +2.2 BLEU).",
"Our combination scheme makes better use of the ensembles.",
"We report 31.3 BLEU on news-test2016, which in the English-German WMT'16 evaluation is among the best systems (within 0.1 BLEU) which do not use back-translation [26].",
"Back-translation is a technique for making use of monolingual data in NMT training, and we expect our system could benefit from back-translation, although we leave this analysis to future work.",
"The combination procedure we propose is non-trivial.",
"It is not immediately clear how the gains arise as the final scores are mixtures between edit distance costs, NMT scores, and Hiero scores.",
"In the remainder we will try to provide some insight.",
"Unless stated otherwise, we report investigations into the Hiero + NMT 8-system ensemble which yields the best results in Tab.",
"REF .",
"Figure: Percentage of t ^ Hiero \\widehat{t}_{Hiero} hypotheses found in the baseline Hiero nn-best list.First, we focus on the projection function $\\Pi _{UNK}(\\cdot )$ which switches between preserving the input and output label at the UNK symbol to produce the combined translation $\\widehat{t}_{comb}$ (Eq.",
"REF ).",
"As explained in Sec.",
"REF , we can use OpenFST's Project operation to fetch the NMT and Hiero hypotheses $\\widehat{t}_{NMT}$ and $\\widehat{t}_{Hiero}$ which have been used to produce the combined translation (Eq.",
"REF and REF ).",
"Tab.",
"REF shows that the hypotheses that are aligned in the final transducer are often not the 1-best translations of any of the baseline systems.",
"Remarkably, using the $\\widehat{t}_{Hiero}$ translations results in 30.4 BLEU, which is a very substantial improvement over the baseline Hiero system (26.0 BLEU).",
"Note that this BLEU score is achieved with hypotheses from the original Hiero lattice $H$ but weighted in combination with the NMT scores and the edit distance.",
"However, these selected paths are often given very low scores by Hiero: in only 8.6% of the sentences is the Hiero hypothesis left unchanged.",
"If we look for $\\widehat{t}_{Hiero}$ in the Hiero $n$ -best list, we find that even very deep 20,000-best lists contain only 63.5% of the Hiero hypotheses which were selected by the combination scheme (Fig.",
"REF ).",
"This indicates the benefit in using lattice-based approaches over $n$ -best lists.",
"Next, we investigate the distance measure between NMT and Hiero translations, which is realized with the UNK insertion transducer $U$ and the modified edit distance transducer $E$ (Sec.",
"REF ).",
"Tab.",
"REF shows that UNK insertions are relatively rare compared to the edit operations of types II and III allowed by $E$ (Sec.",
"REF ).",
"The average edit distance between NMT and Hiero disregarding UNKs on the best path (type III) is 1.74.",
"In 61.7% of the cases the input and output labels differ not only at UNK – i.e.",
"in only 38.3% of the sentences do we have an exact match between NMT and Hiero.",
"We note that UNK is often replaced with an NMT in-vocabulary word (55.9% of the sentences).",
"It seems that NMT often produces an UNK even if a better word is in the NMT vocabulary.",
"This could be due to the over-representation of UNK in the NMT training corpus.",
"To study the effectiveness of our edit distance transducer based combination scheme in correcting NMT UNKs, we trained individual NMT systems with vocabulary sizes between 10,000 and 60,000.",
"Tab.",
"REF shows that nearly one in six tokens (16.3%) produced by our pure NMT system with a vocabulary size of 30,000 are UNKs.",
"Increasing the NMT vocabulary to 50k or 60k does improve pure NMT very significantly, but results show that these improvements are already captured by the combination scheme with Hiero.",
"As in the literature, we see large variation in performance over individual NMT systems even with the same vocabulary size [27], which could explain the small performance drop when increasing the vocabulary size from 50k to 60k.",
"Table: BLEU scores on news-test2016 for different vocabulary sizes (single NMT).",
"Each individual NMT system is combined with Hiero as described in Sec.",
".Figure: BLEU score over the number of systems in the ensemble on news-test2016.One important practical issue for system building is the number of systems to be ensembled as training each individual NMT system takes a significant amount of time.",
"Fig.",
"REF indicates that even for 8-ensembles the gains for pure NMT do not seem to saturate.",
"The combination with Hiero via edit distance transducer also greatly benefits from using ensembles, but most of the gains are gotten with fewer systems."
],
[
"Conclusion and Future Work",
"We have presented a method based on the edit distance that is effective in combining Hiero SMT systems with NMT ensembles.",
"Our approach makes use of standard WFST operations, and we showed the effectiveness of the approach with a successful WMT'16 submission for English-German.",
"In the future, we are planning to add back-translation [26] and investigate the use of character- or subword-based NMT [27], [5], [18], [7], [20] within our combination framework."
],
[
"Acknowledgements",
"This work was supported by the U.K. Engineering and Physical Sciences Research Council (EPSRC grant EP/L027623/1)."
]
] | 1606.04963 |
[
[
"The Bivariate Lack-of-Memory Distributions"
],
[
"Abstract We treat all the bivariate lack-of-memory (BLM) distributions in a unified approach and develop some new general properties of the BLM distributions, including joint moment generating function, product moments and dependence structure.",
"Necessary and sufficient conditions for the survival functions of BLM distributions to be totally positive of order two are given.",
"Some previous results about specific BLM distributions are improved.",
"In particular, we show that both the Marshall--Olkin survival copula and survival function are totally positive of all orders, regardless of parameters.",
"Besides, we point out that Slepian's inequality also holds true for BLM distributions."
],
[
"The Bivariate Lack-of-Memory Distributions Gwo Dong Lin, Xiaoling Dou and Satoshi Kuriki Academia Sinica, Taiwan, Waseda University, Japan and The Institute of Statistical Mathematics, Japan Abstract.",
"We treat all the bivariate lack-of-memory (BLM) distributions in a unified approach and develop some new general properties of the BLM distributions, including joint moment generating function, product moments and dependence structure.",
"Necessary and sufficient conditions for the survival functions of BLM distributions to be totally positive of order two are given.",
"Some previous results about specific BLM distributions are improved.",
"In particular, we show that both the Marshall–Olkin survival copula and survival function are totally positive of all orders, regardless of parameters.",
"Besides, we point out that Slepian's inequality also holds true for BLM distributions.",
"2010 AMS Mathematics Subject Classifications.",
"Primary: 62N05, 62N86, 62H20, 62H05.",
"Key words and phrases: Lack-of-memory property, bivariate lack-of-memory distributions, Marshall and Olkin's BVE, Block and Basu's BVE, Freund's BVE, likelihood ratio order, usual stochastic order, hazard rate order, survival function, copula, survival copula, positive quadrant dependent, total positivity, Slepian's lemma/inequality.",
"Postal addresses: Gwo Dong Lin, Institute of Statistical Science, Academia Sinica, Taipei 11529, Taiwan, R.O.C.",
"(E-mail: gdlin@stat.sinica.edu.tw) Xiaoling Dou, Waseda University, 3-4-1 Ohkubo, Shinjuku, Tokyo 169-8555, Japan (E-mail: xiaoling@aoni.waseda.jp) Satoshi Kuriki, The Institute of Statistical Mathematics, 10-3 Midoricho, Tachikawa, Tokyo 190-8562, Japan (E-mail: kuriki@ism.ac.jp) 1.",
"Introduction The classical univariate lack-of-memory (LM) property is a remarkable characterization of the exponential distribution which plays a prominent role in reliability theory, queuing theory and other applied fields (Feller 1965, Fortet 1977, Galambos and Kotz 1978).",
"The recent bivariate LM property is, however, shared by the famous Marshall and Olkin's, Block and Basu's as well as Freund's bivariate exponential distributions, among many others; see, e.g., Chapter 10 of Balakrishnan and Lai (2009), Chapter 47 of Kotz et al.",
"(2000) and Kulkarni (2006).",
"These bivariate distributions have been well investigated individually in the literature.",
"Our main purpose in this paper is, however, to develop in a unified approach some new general properties of the bivariate lack-of-memory (BLM) distributions which share the same bivariate LM property.",
"In Section 2, we first review the univariate and bivariate LM properties, and then summarize the important known properties of the BLM distributions.",
"We derive in Section 3 some new general properties of the BLM distributions, including joint moment generating function, product moments and stochastic inequalities.",
"The dependence structures of the BLM distributions are investigated in Section 4.",
"We find necessary and sufficient conditions for the survival functions (and the densities if they exist) of BLM distributions to be totally positive of order two.",
"Some previous results about specific BLM distributions are improved.",
"In particular, we show that both the Marshall–Olkin survival copula and survival function are totally positive of all orders, regardless of parameters.",
"In Section 5, we study the stochastic comparisons in the family of all BLM distributions and point out that Slepian's lemma/inequality for bivariate normal distributions also holds true for BLM distributions.",
"2.",
"Lack-of-Memory Property We first review the well-known univariate lack-of-memory property.",
"Let $X$ be a nonnegative random variable with distribution function $F.$ Then $F$ satisfies (multiplicative) Cauchy's functional equation ${\\overline{F}(x+y)=\\overline{F}(x)\\overline{F}(y), \\ x\\ge 0, y\\ge 0,}$ where ${\\overline{F}(x)=1-F(x)=\\Pr (X>x)},$ if and only if $F(0)=1$ ($X$ degenerates at 0) or $F(x)=1-\\exp ({-\\lambda x}),\\ x\\ge 0,$ for some constant $\\lambda >0,$ denoted by $X\\sim Exp(\\lambda )$ ($X$ has an exponential distribution with positive parameter $\\lambda $ ).",
"If $X$ is the lifetime of a system with positive survival function $\\overline{F},$ then Eq.",
"(1) is equivalent to ${\\Pr (X>x+y|\\ X>y)=\\Pr (X>x),~~x\\ge 0, \\ y\\ge 0.",
"}$ This means that the conditional probability of a system surviving to time $x+y$ given surviving to time $y$ is equal to the unconditional probability of the system surviving to time $x$ .",
"Namely, the failure performance of the system does not depend on the past, given its present condition.",
"In such a case (2), we say that the distribution $F$ lacks memory at each point $y.$ So Eq.",
"(1) is called the LM property or memoryless property of $F.$ For simplicity, we consider only positive random variable $X\\sim F$ from now on.",
"Then, the LM property (1) holds true iff $X\\sim Exp(\\lambda )$ for some $\\lambda >0.$ We next consider the bivariate LM property.",
"Let the positive random variables $X$ and $Y$ have joint distribution $H$ with marginals $F$ and $G$ .",
"Namely, $(X,Y)\\sim H,\\ X\\sim F,\\ Y\\sim G$ .",
"Moreover, denote the survival function of $H$ by $\\overline{H}(x,y)\\equiv \\Pr (X>x,\\,Y>y)=1-F(x)-G(y)+H(x,y),\\ x,y\\ge 0.$ An intuitive extension of the LM property (2) to the bivariate case is the strict BLM property: ${\\Pr (X>x+s,\\,Y>y+t|\\ X>s,Y>t)=\\Pr (X>x,\\,Y>y),\\ x,y,s,t\\ge 0}$ ($H$ lacks memory at each pair $(s,t)$ ), which is equivalent to $\\overline{H}(x+s,\\,y+t)=\\overline{H}(x,y)\\overline{H}(s,t),\\ \\forall \\ x,y,s,t\\ge 0,$ if the survival function $\\overline{H}$ is positive.",
"In a two-component system, this means as before that the conditional probability of two components surviving to times $(x+s,y+t)$ given surviving to times $(s, t)$ is equal to the unconditional probability of these two components surviving to times $(x,y)$ .",
"But Eq.",
"(3) has only one solution (Marshall and Olkin 1967, p. 33), namely, the independent bivariate exponential distribution with survival function ${\\overline{H}(x,y)=\\exp [{-(\\lambda x+\\delta y)}], \\ x,y\\ge 0,}$ for some constants $\\lambda ,\\delta >0;$ in other words, $X$ and $Y$ are independent random variables and $X\\sim Exp(\\lambda ),\\ Y\\sim Exp(\\delta )$ for some positive parameters $\\lambda , \\delta $ .",
"In their pioneering paper, Marshall and Olkin (1967) considered instead the weaker BLM property (with $s=t$ ) $\\Pr (X>x+t,\\,Y>y+t|\\ X>t,\\,Y>t)=\\Pr (X>x,\\,Y>y),\\ x,y,t\\ge 0$ ($H$ lacks memory at each equal pair $(t,t)$ ), and solved the functional equation ${\\overline{H}(x+t,\\,y+t)=\\overline{H}(x,y)\\overline{H}(t,t),\\ \\forall \\ x,y,t\\ge 0.}",
"$ It turns out that for given $(X,Y)\\sim H$ with marginals $F, G$ on $(0,\\infty )$ , $H$ satisfies the BLM property (4) iff its survival function is of the form ${ \\overline{H}(x,y)=\\left\\lbrace \\begin{array}{cc}e^{-\\theta y}\\,\\overline{F}(x-y), & x\\ge y\\ge 0 \\vspace{2.84544pt}\\\\e^{-\\theta x}\\,\\overline{G}(y-x), & y\\ge x\\ge 0,\\end{array}\\right.",
"}$ where $\\theta $ is a positive constant (see also Barlow and Proschan 1981, p. 130).",
"For convenience, denote by $BLM(F,G,\\theta )$ the BLM distribution $H$ with marginals $F,G$ , parameter $\\theta >0$ and survival function $\\overline{H}$ in (5), and denote by ${\\cal BLM}$ the family of all BLM distributions, namely, ${\\cal BLM}=\\lbrace H: H=BLM(F,G,\\theta ),\\ \\hbox{where}\\ \\theta >0,\\ \\hbox{and}\\ F,\\ G\\ \\hbox{are marginal distributions} \\rbrace .$ Theorem 1 below summarizes some important known properties of the BLM distributions; for more details, see Marshall and Olkin (1967), Block and Basu (1974), Block (1977), and Ghurye and Marshall (1984).",
"For convenience, denote $a\\vee b=\\max \\lbrace a,b\\rbrace $ and $a\\wedge b=\\min \\lbrace a,b\\rbrace .$ Theorem 1.",
"Let $(X,Y)\\sim H=BLM(F,G,\\theta )\\in {\\cal BLM}.$ Then the following statements are true.",
"(i) The marginals $F$ , $G$ have densities $f$ , $g$ , respectively.",
"Moreover, the right-hand derivatives $f(x)=\\lim _{\\varepsilon \\rightarrow 0^+}[F(x+\\varepsilon )-F(x)]/{\\varepsilon }$ and $g(x)=\\lim _{\\varepsilon \\rightarrow 0^+}[G(x+\\varepsilon )-G(x)]/{\\varepsilon }$ exist for all $x\\ge 0$ , which are right-continuous and are of bounded variation on $[0,\\infty )$ .",
"(ii) $\\Pr (X-Y>t)=\\overline{F}(t)-f(t)/\\theta $ and $\\Pr (Y-X>t)=\\overline{G}(t)-g(t)/\\theta $ for all $t\\ge 0.$ (iii) Both $e^{\\theta x}f(x)$ and $e^{\\theta x}g(x)$ are increasing (nondecreasing) in $x\\ge 0.$ (iv) $F(x)+G(x)\\ge 1-\\exp ({-\\theta x}),\\ x\\ge 0.$ (v) $X\\wedge Y\\sim Exp(\\theta )$ and is independent of $X-Y.$ (vi) $f(0)\\vee g(0)\\le \\theta \\le f(0)+g(0).$ (vii) $f^{\\prime }(x)+\\theta f(x)\\ge 0,\\ g^{\\prime }(x)+\\theta g(x)\\ge 0,\\ x\\ge 0,$ if $f$ and $g$ are differentiable.",
"Remark 1.",
"Some of the above necessary conditions (i)–(vii) also play as sufficient conditions for $(X,Y)$ to obey a BLM distribution.",
"For example, in addition to the above conditions (vi) and (vii), assume that the marginal densities are absolutely continuous, then the $\\overline{H}$ in (5) is a bona fide survival function.",
"This is a slight modification of Theorem 5.1 of Marshall and Olkin (1967) who required (vi$^{\\prime }$ ) $[f(0)+g(0)]/2\\le \\theta \\le f(0)+g(0)$ instead of (vi) above.",
"Note that conditions (vi) and (vi$^{\\prime }$ ) are different unless $f(0)=g(0)$ , and that (vi) is a consequence of (iii) and (iv) (see Corollary 2(i) below and Ghurye and Marshall 1984, p. 792).",
"On the other hand, the condition (v) together with continuous marginals $F,G$ also implies that $(X,Y)$ has a BLM distribution (Block 1977, p. 810).",
"It is interesting to recall that for independent nondegenerate random variables $X$ and $Y$ , the above independence of $X\\wedge Y$ and $X-Y$ is a characterization of the exponential/geometric distributions under suitable conditions (see Ferguson 1964, 1965, Crawford 1966, and Rao and Shanbhag 1994).",
"Namely, in general, the BLM distributions share the same independence property of $X\\wedge Y$ and $X-Y$ with independent exponential/geometric random variables.",
"Remark 2.",
"There are some more observations: (a) $\\Pr (X=Y)=[f(0)+g(0)]/\\theta -1$ by the above (ii), (b) at least one of $f(0)$ and $g(0)$ is positive, (c) the survival function $\\overline{H}$ in (5) is purely singular (i.e., $X=Y$ almost surely) iff $\\theta =[f(0)+g(0)]/2$ iff $f(0)=g(0)=\\theta $ (because $f(0)\\ne g(0)$ implies $\\theta >[f(0)+g(0)]/2$ by (vi)), and (d) $\\overline{H}$ is absolutely continuous (i.e., $X\\ne Y$ almost surely) iff the marginal densities together satisfy $f(0)+g(0)=\\theta $ (see Ghurye and Marshall 1984, p. 792).",
"In view of the above results, the survival function (5) of $H=BLM(F,G,\\theta )$ can be rewritten as the convex combination of two extreme ones: ${\\overline{H}(x,y)}=\\left(2-\\frac{f(0)+g(0)}{\\theta }\\right){\\overline{H}{\\!",
"}_a(x,y)}+\\left(\\frac{f(0)+g(0)}{\\theta }-1\\right){\\overline{H}{\\!",
"}_s(x,y)} ,\\ x,y\\ge 0,$ where $H_a$ is absolutely continuous and $H_s$ is purely singular with survival function $\\overline{H}{\\!",
"}_s(x,y)=\\exp [-\\theta \\max \\lbrace x,y\\rbrace ],\\,x,y\\ge 0.$ Clearly, the parameter $\\theta $ regulates $H$ between $H_a$ and $H_s.$ On the other hand, Ghurye and Marshall (1984, Section 3) gave an interesting random decomposition of $(X,Y)\\sim H\\in {\\cal BLM}$ and represented $\\overline{H}$ as a Laplace–Stieltjes integral by another bivariate survival function.",
"See also Ghurye (1987) and Marshall and Olkin (2015) for further generalizations of the BLM distributions.",
"Remark 3.",
"Kulkarni (2006) proposed an interesting and useful approach to construct some BLM distributions by starting with marginal failure rate functions.",
"First, choose two real-valued functions $r_1, r_2$ and a constant $\\theta $ satisfying the following (modified) conditions: (a) The functions $r_i, i=1,2$ , are absolutely continuous on $[0,\\infty )$ and $\\theta >0$ .",
"(b) $0\\le r_i(x)\\le \\theta ,\\ x\\ge 0,\\ i=1,2.$ (c) $\\int _0^{\\infty }r_i(x)dx=\\infty ,\\ i=1,2.$ (d) $r_i(x)(\\theta -r_i(x))+r_i^{\\prime }(x)\\ge 0,\\ x\\ge 0,\\ i=1,2.$ (e) $r_1(0)+r_2(0)\\ge \\theta .$ Then set $\\overline{F}(x)=\\exp (-\\int _0^xr_1(t)dt), x\\ge 0,$ and $\\overline{G}(y)=\\exp (-\\int _0^yr_2(t)dt), y\\ge 0.$ In this way, the $H$ defined through (5) is a bona fide BLM distribution because the above conditions (a)–(e) together imply that conditions (vi) and (vii) in Theorem 1 hold true (see Remark 1).",
"Conversely, under the above smoothness conditions on $r_i$ and the setting of $F$ and $G$ , if $H$ in (5) is a BLM distribution, then its marginal failure rate functions $r_i$ should satisfy conditions (b)–(e) from which some properties in Theorem 1 follow immediately (Kulkarni 2006, Proposition 1).",
"We now recall three important BLM distributions in the literature.",
"For more details, see, e.g., Chapter 10 of Balakrishnan and Lai (2009).",
"Example 1.",
"Marshall and Olkin's (1967) bivariate exponential distribution (BVE) If both marginals $F$ and $G$ are exponential, then the $BLM(F,G,\\theta )\\in {\\cal BLM}$ defined in (5) reduces to the Marshall–Olkin BVE with survival function of the form ${\\overline{H}(x,y)}&=&{\\exp [-\\lambda _1x-\\lambda _2 y-\\lambda _{12}\\max \\lbrace x,y\\rbrace ]}\\\\&\\equiv & \\frac{\\lambda _1+\\lambda _2}{\\lambda }{\\overline{H}{\\!",
"}_a(x,y)}+\\frac{\\lambda _{12}}{\\lambda }{\\overline{H}{\\!",
"}_s(x,y)} ,\\ \\ x,y\\ge 0,$ where $\\lambda _1,\\lambda _2, \\lambda _{12}$ are positive constants, $\\lambda =\\lambda _1+\\lambda _2+\\lambda _{12}$ , and $H_a$ , $H_s$ (written explicitly below) are absolutely continuous and singular bivariate distributions, respectively.",
"In practice, the Marshall–Olkin BVE arises from a shock model for a two-component system.",
"Formally, the lifetimes of two components are $(X, Y)=(X_1\\wedge X_3,\\ X_2\\wedge X_3),$ where $X_1\\sim Exp(\\lambda _1),$ $X_2\\sim Exp(\\lambda _2)$ and $X_3\\sim Exp(\\lambda _{12})$ are independent.",
"So they have a joint survival function $\\overline{H}$ defined in (6).",
"The singular part in (7) is identified by the conditional probability: $\\overline{H}{\\!",
"}_s(x,y)=\\Pr (X>x,\\,Y>y|\\ X_3\\le X_1\\wedge X_2)=\\exp [-\\lambda \\max \\lbrace x,y\\rbrace ],$ while the absolutely continuous part $\\overline{H}{\\!",
"}_a$ is calculated from $\\overline{H}$ and $\\overline{H}{\\!",
"}_s$ via (7) (see the next example).",
"Example 2.",
"Block and Basu's (1974) bivariate exponential distribution The Block–Basu BVE is actually the absolute continuous part $H_a$ of Marshall–Olkin BVE in (7) and has a joint density of the form ${ {h}(x,y)=\\left\\lbrace \\begin{array}{cc}\\frac{\\lambda _2\\lambda (\\lambda _1+\\lambda _{12})}{\\lambda _1+\\lambda _2}\\exp [{-(\\lambda _1+\\lambda _{12})x-\\lambda _2y}], &x\\ge y> 0\\vspace{5.69046pt}\\\\\\frac{\\lambda _1\\lambda (\\lambda _2+\\lambda _{12})}{\\lambda _1+\\lambda _2}\\exp [{-\\lambda _1x-(\\lambda _2+\\lambda _{12})y}], &y>x> 0,\\end{array}\\right.",
"}$ where $\\lambda _1,\\lambda _2,\\lambda _{12}>0,$ and $\\lambda =\\lambda _1+\\lambda _2+\\lambda _{12}.$ Its survival function is equal to $& &\\overline{H}(x,y)={\\overline{H}{\\!",
"}_a(x,y)}\\\\&=&\\frac{\\lambda }{\\lambda _1+\\lambda _2}\\exp [-\\lambda _1 x-\\lambda _2y-\\lambda _{12}\\max \\lbrace x,y\\rbrace ]-\\frac{\\lambda _{12}}{\\lambda _1+\\lambda _2}\\exp [-\\lambda \\max \\lbrace x,y\\rbrace ],\\ x,y\\ge 0.$ Note that in this case, the marginals are not exponential but rather negative mixtures of two exponentials.",
"Specifically, $\\overline{F}(x)=\\frac{\\lambda }{\\lambda _1+\\lambda _2}\\exp [-(\\lambda _1+\\lambda _{12})x]-\\frac{\\lambda _{12}}{\\lambda _1+\\lambda _2}\\exp ({-\\lambda x}),\\ x\\ge 0,$ and $\\overline{G}(y)=\\frac{\\lambda }{\\lambda _1+\\lambda _2}\\exp [{-(\\lambda _2+\\lambda _{12})y}]-\\frac{\\lambda _{12}}{\\lambda _1+\\lambda _2}\\exp ({-\\lambda y}),\\ y\\ge 0.$ Example 3.",
"Freund's (1961) bivariate exponential distribution The Freund BVE has a joint density of the form ${ {h}(x,y)=\\left\\lbrace \\begin{array}{cc}\\alpha ^{\\prime }\\beta \\exp [{-(\\alpha +\\beta -\\alpha ^{\\prime })y-\\alpha ^{\\prime }x}], &x\\ge y> 0\\vspace{2.84544pt}\\\\\\alpha \\beta ^{\\prime }\\exp [{-(\\alpha +\\beta -\\beta ^{\\prime })x-\\beta ^{\\prime }y}], & y> x> 0,\\end{array}\\right.",
"}$ where $\\alpha ,\\alpha ^{\\prime },\\beta ,\\beta ^{\\prime }>0.$ If $\\alpha +\\beta > \\alpha ^{\\prime }\\vee \\beta ^{\\prime }$ , its survival function is equal to ${\\overline{H}(x,y)=\\left\\lbrace \\begin{array}{cc}\\frac{\\beta }{\\alpha +\\beta -\\alpha ^{\\prime }}\\exp [{-(\\alpha +\\beta -\\alpha ^{\\prime })y-\\alpha ^{\\prime }x}]+\\frac{\\alpha -\\alpha ^{\\prime }}{\\alpha +\\beta -\\alpha ^{\\prime }}\\exp [{-(\\alpha +\\beta )x}], & x\\ge y\\ge 0\\vspace{5.69046pt}\\\\\\frac{\\alpha }{\\alpha +\\beta -\\beta ^{\\prime }}\\exp [{-(\\alpha +\\beta -\\beta ^{\\prime })x-\\beta ^{\\prime }y}]+\\frac{\\beta -\\beta ^{\\prime }}{\\alpha +\\beta -\\beta ^{\\prime }}\\exp [{-(\\alpha +\\beta )y}], & y\\ge x\\ge 0.\\end{array}\\right.",
"}$ It worths noting that by choosing $\\alpha =\\frac{\\lambda _1\\lambda }{\\lambda _1+\\lambda _2},\\ \\beta =\\frac{\\lambda _{2}\\lambda }{\\lambda _1+\\lambda _2},\\ \\alpha ^{\\prime }=\\lambda _1+\\lambda _{12}$ and $\\beta ^{\\prime }=\\lambda _2+\\lambda _{12},$ Freund's BVE (9) reduces to Block and Basu's BVE (8).",
"3.",
"New General Properties of BLM Distributions Let $(X,Y)\\sim H=BLM(F,G,\\theta )\\in {\\cal BLM}$ with marginals $F$ and $G$ on $(0,\\infty )$ , parameter $\\theta >0$ and survival function (5).",
"Denote the Laplace-Stieltjes transform of $X$ ($Y,$ resp.)",
"by $L_X$ ($L_Y,$ resp.",
"), and that of $(X,Y)$ by ${\\cal L}.$ Then we have Theorem 2.",
"The Laplace-Stieltjes transform of $(X,Y)\\sim H=BLM(F,G,\\theta )\\in {\\cal BLM}$ is ${\\cal L}(s,t)\\equiv E\\left[e^{-sX-tY}\\right]=\\frac{1}{\\theta +s+t}\\left[(\\theta +s)L_X(s)+(\\theta +t)L_Y(t)\\right]-\\frac{\\theta }{\\theta +s+t},\\ s, t>0.$ To prove Theorem 2, we need the following lemma due to Lin et al. (2016).",
"Lemma 1.",
"Let $(X,Y)\\sim H$ defined on ${\\mathbb {R}}_{+}^{2}=[0,\\infty )\\times [0,\\infty )$ .",
"Then the Laplace-Stieltjes transform of $(X,Y)$ is equal to ${\\cal L}(s,t)=st\\int _{0}^{\\infty }\\!\\!\\int _{0}^{\\infty }\\overline{H} (x,y)e^{-sx-ty}dxdy-1+L_X(s)+L_Y(t),~ s,\\,t\\ge 0.$ Proof of Theorem 2.",
"We have to calculate the double integral $\\int _{0}^{\\infty }\\!\\!\\int _{0}^{\\infty }\\overline{H}(x,y)e^{-sx-ty}dxdy=\\int \\!\\!\\int _{x\\ge y}+\\int \\!\\!\\int _{y\\ge x}\\equiv A_1+A_2,$ where, by changing variables and by integration by parts, $A_1&=&\\int _0^{\\infty }\\!\\!e^{-(\\theta +t)y}\\int _{y}^{\\infty }e^{-sx}\\overline{F}(x-y)dxdy=\\int _0^{\\infty }\\!\\!e^{-(\\theta +s+t)y}\\int _{0}^{\\infty }e^{-sz}\\overline{F}(z)dzdy\\\\&=&\\frac{1}{\\theta +s+t}\\left[-\\frac{1}{s}\\int _0^{\\infty }\\overline{F}(z)de^{-sz}\\right]=\\frac{1}{\\theta +s+t}\\left[\\frac{1}{s}(1-L_X(s))\\right],$ and similarly, $A_2=\\frac{1}{\\theta +s+t}\\left[\\frac{1}{t}(1-L_Y(t))\\right].$ Lemma 1 together with the above $A_1$ and $A_2$ completes the proof.",
"Denote the moment generating function (mgf) of $X$ ($Y,$ resp.)",
"by $M_X$ ($M_Y,$ resp.",
"), and that of $(X, Y)$ by ${\\cal M}.$ Then we have the following general result.",
"Theorem 3.",
"Let $(X,Y)\\sim H=BLM(F,G,\\theta )\\in {\\cal BLM}$ and let $r,s$ be real numbers such that $s+t<\\theta .$ Then the mgf of $(X,Y)$ is ${\\cal M}(s,t)\\equiv E\\left[e^{sX+tY}\\right]=\\frac{1}{\\theta -s-t}\\left[(\\theta -s)M_X(s)+(\\theta -t)M_Y(t)\\right]-\\frac{\\theta }{\\theta -s-t},$ provided the expectations (mgfs) exist.",
"To prove Theorem 3, we need instead the following lemma due to Lin et al. (2014).",
"Lemma 2.",
"Let $(X,Y)\\sim H$ defined on ${\\mathbb {R}}_{+}^{2}.$ Let $\\alpha $ and $\\beta $ be two increasing and left-continuous functions on ${\\mathbb {R}}_{+}$ .",
"Then the expectation of the product $\\alpha (X)\\beta (Y)$ is equal to $E[\\alpha (X)\\beta (Y)]=\\int _{0}^{\\infty }\\!\\!\\int _{0}^{\\infty }\\overline{H} (x,y)d\\alpha (x)d\\beta (y) -\\alpha (0)\\beta (0)+\\alpha (0)E[\\beta (Y)]+\\beta (0)E[\\alpha (X)],$ provided the expectations exist.",
"Proof of Theorem 3.",
"Case (i): $s,t\\ge 0.$ Let $\\alpha (x)=e^{sx}$ and $\\beta (y)=e^{t y}$ in Lemma 2, then ${\\cal M}(s,t)=st\\int _{0}^{\\infty }\\!\\!\\int _{0}^{\\infty }\\overline{H} (x,y)e^{sx+ty}dxdy-1+M_X(s)+M_Y(t).$ We have to calculate the double integral $\\int _{0}^{\\infty }\\!\\!\\int _{0}^{\\infty }\\overline{H}(x,y)e^{sx+ty}dxdy=\\int \\!\\!\\int _{x\\ge y}+\\int \\!\\!\\int _{y\\ge x}\\equiv B_1+B_2,$ where, as before, $B_1=\\frac{1}{\\theta -s-t}\\left[\\frac{1}{s}(-1+M_X(s))\\right]~~\\hbox{and}~~B_2=\\frac{1}{\\theta -s-t}\\left[\\frac{1}{t}(-1+M_Y(t))\\right].$ Lemma 2 together with the above $B_1$ and $B_2$ completes the proof of Case (i).",
"Case (ii): $s\\ge 0,\\ t< 0.$ To apply Lemma 2, set $\\alpha (x)=e^{sx}$ and $\\beta (y)=1-e^{ty}.$ Then both $\\alpha $ and $\\beta $ are increasing functions on ${\\mathbb {R}}_+$ and $E[\\alpha (X)\\beta (Y)]=M_X(s)-{\\cal M}(s,t).$ Therefore, ${\\cal M}(s,t)=M_X(s)-E[\\alpha (X)\\beta (Y)]= M_X(s)-1+M_Y(t)+st\\int _{0}^{\\infty }\\!\\!\\int _{0}^{\\infty }\\overline{H} (x,y)e^{sx+ty}dxdy.$ As before, we carry out the above double integral and complete the proof of Case (ii).",
"Case (iii): $s< 0,\\ t\\ge 0.$ Set $\\alpha (x)=1-e^{sx}$ and $\\beta (y)=e^{ty}$ in Lemma 2.",
"The remaining proof is similar to that of Case (ii) and is omitted.",
"Case (iv): $s, t<0.$ This case was treated in Theorem 1.",
"The proof is completed.",
"Next, we consider the product moments of BLM distributions.",
"Theorem 4.",
"For positive integers $i$ and $j$ , the product moment $E[X^iY^j]$ of $(X,Y)\\sim H=BLM(F,G,\\theta )\\in {\\cal BLM}$ is of the form $E[X^iY^j]=i\\,j\\,\\sum _{k=0}^{i-1}\\frac{1}{i-k}{i-1\\atopwithdelims ()k}\\frac{\\Gamma (j+k)}{\\theta ^{j+k}}E[X^{i-k}]+\\,i\\,j\\,\\sum _{k=0}^{j-1}\\frac{1}{j-k}{j-1\\atopwithdelims ()k}\\frac{\\Gamma (i+k)}{\\theta ^{i+k}}E[Y^{j-k}],$ provided the expectations exist.",
"The first product moment has a neat representation in terms of marginal means and the parameter $\\theta $ , from which we can calculate Pearson's correlation of BLM distributions.",
"Corollary 1.",
"$E[XY]=\\frac{1}{\\theta }(E[X]+E[Y])$ provided the expectations exist.",
"To prove Theorem 4 above, we will apply the following lemma due to Lin et al. (2014).",
"Lemma 3.",
"Let $(X,Y)\\sim H$ defined on ${\\mathbb {R}}_+^2$ , and let the expectations $E[X^rY^s]$ , $E[X^r]$ and $E[Y^s]$ be finite for some positive real numbers $r$ and $s$ .",
"Then the product moment $E[X^rY^s]=rs\\int _0^{\\infty }\\!\\!\\int _0^{\\infty }\\overline{H}(x,y)x^{r-1}y^{s-1}dxdy.$ Proof of Theorem 4.",
"We have to calculate the double integral $\\int _{0}^{\\infty }\\!\\!\\int _{0}^{\\infty }\\overline{H}(x,y)x^{i-1}y^{j-1}dxdy=\\int \\!\\!\\int _{x\\ge y}+\\int \\!\\!\\int _{y\\ge x}\\equiv C_1+C_2,$ where, by changing variables and by integration by parts, $C_1&=&\\int _0^{\\infty }\\!\\!e^{-\\theta y}y^{j-1}\\int _{y}^{\\infty }x^{i-1}\\overline{F}(x-y) dxdy=\\int _0^{\\infty }\\!\\!e^{-\\theta y}y^{j-1}\\int _{0}^{\\infty }(y+z)^{i-1}\\overline{F}(z)dzdy\\\\&=&\\sum _{k=0}^{i-1}{i-1\\atopwithdelims ()k}\\int _0^{\\infty }\\!\\!y^{j-1+k}e^{-\\theta y}\\int _{0}^{\\infty }z^{i-1-k}\\overline{F}(z)dzdy\\\\&=&\\sum _{k=0}^{i-1}{i-1\\atopwithdelims ()k}\\frac{\\Gamma (j+k)}{\\theta ^{j+k}}\\left[\\frac{1}{i-k}\\int _0^{\\infty }\\overline{F}(z)dz^{i-k}\\right]=\\sum _{k=0}^{i-1}\\frac{1}{i-k}{i-1\\atopwithdelims ()k}\\frac{\\Gamma (j+k)}{\\theta ^{j+k}}E[X^{i-k}],$ and similarly, $C_2=\\sum _{k=0}^{j-1}\\frac{1}{j-k}{j-1\\atopwithdelims ()k}\\frac{\\Gamma (i+k)}{\\theta ^{i+k}}E[Y^{j-k}].$ Finally, Lemma 3 together with the above $C_1$ and $C_2$ completes the proof.",
"For moment generating functions of some specific BLM distributions, see Chapter 47 of Kotz et al.",
"(2000), while for product moments of such distributions, see Nadarajah (2006).",
"For the next and later results, we need some notations in reliability theory.",
"For random variables $X\\sim F$ and $Y\\sim G,$ we say that $X$ is smaller than $Y$ in the usual stochastic order (denoted by $X\\le _{st} Y$ ) if $\\overline{F}(x)\\le \\overline{G}(x)$ for all $x$ , that $X$ is smaller than $Y$ in the hazard rate order (denoted by $X\\le _{hr} Y$ ) if $\\overline{G}(x)/\\overline{F}(x)$ is increasing in $x$ , and that $X$ is smaller than $Y$ in the reversed hazard rate order (denoted by $X\\le _{rh} Y$ ) if ${G}(x)/{F}(x)$ is increasing in $x$ .",
"Suppose $F$ and $G$ have densities $f$ and $g$ , respectively.",
"Then we say that $X$ is smaller than $Y$ in the likelihood ratio order (denoted by $X\\le _{\\ell r} Y$ ) if $g(x)/f(x)$ is increasing in $x$ .",
"For more definitions of the related stochastic orders, see, e.g., Müller and Stoyan (2002), Shaked and Shanthikumar (2007), Lai and Xie (2006) as well as Kayid et al.",
"(2016).",
"The latter studied stochastic comparisons of the age replacement models.",
"On the other hand, for a distribution $F$ itself we define the notions of increasing failure rate (IFR), decreasing failure rate (DFR), increasing failure rate in average (IFRA), and decreasing failure rate in average (DFRA) as follows.",
"We say that (a) $F$ is IFR (DFR, resp.)",
"if $-\\log \\overline{F}(x)$ is convex (concave, resp.)",
"in $x\\ge 0,$ and (b) $F$ is IFRA (DFRA, resp.)",
"if $-(1/x)\\log \\overline{F}(x)$ is increasing (decreasing, resp.)",
"in $x> 0$ , or, equivalently, $\\overline{F}^{\\alpha }(x)\\le (\\ge ,\\,\\hbox{resp.",
"})\\,\\, \\overline{F}(\\alpha x)$ for all $\\alpha \\in (0,1)$ and $x\\ge 0.$ (See Barlow and Proschan 1981, Chapters 3 and 4.)",
"The bivariate IFRA and bivariate DFRA distributions $H$ can be defined similarly: $H$ is bivariate IFRA (DFRA, resp.)",
"if $\\overline{H}^{\\alpha }(x,y)\\le (\\ge ,\\,\\hbox{resp.",
"})\\,\\,\\overline{H}(\\alpha x, \\alpha y)$ for all $\\alpha \\in (0,1)$ and $x,\\ y\\ge 0$ (see Block and Savits 1976, 1980).",
"It worths mentioning that there are some other definitions of bivariate IFRA distributions that all extend the univariate case (see, e.g., Esary and Marshall 1979 or Shaked and Shanthikumar 1988).",
"Using reliability language, we have the following useful results.",
"Especially, Theorem 5(iii) means that in the ${\\cal BLM}$ family, positive bivariate aging plays in favor of positive univariate aging in the sense of IFRA, and vice versa.",
"This is in general not true even under the condition of positive dependence for lifetimes; see Bassan and Spizzichino (2005, Remark 6.8), which analyzed the relations among univariate and bivariate agings and dependence.",
"Theorem 5.",
"Let $(X, Y)\\sim H=BLM(F,G, \\theta )\\in {\\cal BLM}$ and ${Z\\sim Exp(\\theta )}$ .",
"Then (i) $Z\\le _{\\ell r} X$ and $Z\\le _{\\ell r} Y$ ; (ii) $Z\\le _{st} X$ and $Z\\le _{st} Y$ ; $Z\\le _{hr} X$ and $Z\\le _{hr} Y$ ; $Z\\le _{rh} X$ and $Z\\le _{rh} Y$ ; (iii) $(X,Y)$ has a bivariate IFRA distribution iff both marginals $F$ and $G$ are IFRA; (iv) $(X,Y)$ has a bivariate DFRA distribution iff both marginals $F$ and $G$ are DFRA.",
"Proof.",
"Part (i) follows immediately from Theorem 1(iii) (see Ghurye and Marshall 1984), while part (ii) follows from the fact that the likelihood ratio order is stronger than the usual stochastic order, hazard rate order and reversed hazard rate order (Müller and Stoyan 2002, pp. 12–13).",
"Part (iii) holds true by verifying that $\\overline{H}(\\alpha x,\\alpha y)\\ge \\overline{H}^{\\alpha }(x,y)\\ \\forall \\alpha \\in (0,1),\\,x,y$ $\\ge 0$ , if, and only if, (a) $\\overline{F}(\\alpha x)\\ge \\overline{F}^{\\alpha }(x)\\ \\forall \\alpha \\in (0,1),\\,x\\ge 0,$ and (b) $\\overline{G}(\\alpha y)\\ge \\overline{G}^{\\alpha }(y)\\ \\forall \\alpha \\in (0,1),\\,y\\ge 0.$ The proof of part (iv) is similar.",
"Applying the above stochastic inequalities, we can simplify the proof of some previous known results.",
"For example, we have Corollary 2.",
"Let $(X, Y)\\sim H=BLM(F,G, \\theta )\\in {\\cal BLM}$ .",
"Then the following statements are true.",
"(i) Both hazard rates of marginals $F, G$ are bounded by $\\theta $ and hence $\\theta \\ge f(0)\\vee g(0)$ .",
"(ii) Both the functions $F(-\\frac{1}{\\theta }\\log (1-t))$ and $G(-\\frac{1}{\\theta }\\log (1-t))$ are convex in $t\\in [0,1)$ , and hence $f^{\\prime }(x)+\\theta f(x)\\ge 0,\\ g^{\\prime }(x)+\\theta g(x)\\ge 0,\\ x\\ge 0,$ if $f$ and $g$ are differentiable.",
"(iii) Let $S_F, S_G$ be the supports of marginals $F, G$ with densities $f,g$ , respectively.",
"Then $S_F=[a_F,\\infty ), S_G=[a_G,\\infty )$ for some nonnegative constants $a_F, a_G$ with $a_Fa_G=0,$ and $f, g$ are positive on $(a_F,\\infty ), (a_G,\\infty )$ , respectively.",
"(iv) If $H$ is not absolutely continuous, then $f(0)>0$ , $g(0)>0,$ and hence $a_F=a_G=0$ .",
"Proof.",
"Part (i) follows from the facts $Z\\le _{hr} X$ and $Z\\le _{hr} Y$ , where ${Z\\sim Exp(\\theta )}$ , while part (ii) is due to the probability-probability plot characterization for $Z\\le _{\\ell r} X$ and $Z\\le _{\\ell r} Y$ (see Theorem 1.4.3 of Müller and Stoyan 2002).",
"Part (iii) follows from the facts $Z\\le _{\\ell r} X$ , $Z\\le _{\\ell r} Y$ and Theorem 1(iv), because the latter implies that at least one of the left extremities $a_F$ and $a_G$ of marginal distributions should be zero.",
"Finally, to prove part (iv), we note that $\\Pr (X-Y>0)=1-f(0)/\\theta $ and $\\Pr (Y-X>0)=1-g(0)/\\theta $ by Theorem 1(ii) (see Ghurye and Marshall 1984, p. 789).",
"So if $H$ is not absolutely continuous, $\\Pr (X=Y)>0,$ and hence $f(0)=\\theta \\Pr (X\\le Y)>0$ and $g(0)=\\theta \\Pr (Y\\le X)>0$ .",
"The proof is complete.",
"4.",
"Dependence Structures of BLM Distributions Recall that a bivariate distribution $H$ with marginals $F$ and $G$ is positively quadrant dependent (PQD) if $H(x,y)\\ge F(x)G(y)\\ \\ \\forall \\, x,y\\ge 0,\\ \\ \\hbox{or,equivalently,}\\ \\ \\overline{H}(x,y)\\ge \\overline{F}(x)\\overline{G}(y)\\ \\ \\forall \\, x,y\\ge 0,$ which implies that $H$ has a nonnegative covariance by Hoeffding representation for covariance (see, e.g., Lin et al.",
"2014, p. 2).",
"A stronger (positive dependence) property than the PQD is the total positivity defined below.",
"For a nonnegative function $K$ on the rectangle $(a,b)\\times (c,d)$ (or on the product of two subsets of ${\\mathbb {R}}$ ), we say that $K(x,y)$ is totally positive of order $r$ (TP$_r,\\, r\\ge 2$ ) in $x$ and $y$ if for each fixed $s\\in \\lbrace 2,3,\\ldots ,r\\rbrace $ and for all $a<x_1<x_2<\\cdots <x_s<b\\ \\mbox{and} \\ c<y_1<y_2<\\cdots <y_s<d$ , the determinant of the $s\\times s$ matrix $(K(x_i,y_j))$ is nonnegative.",
"The function $K$ is said to be TP$_{\\infty }$ if it is TP$_r$ for any order $r\\ge 2$ (Karlin 1968).",
"The total positivity plays an important role on various concepts of bivariate dependence (see, e.g., Shaked 1977 and Lee 1985).",
"Moreover, applying total positivity of the bivariate distribution or its survival function, we can derive some useful probability inequalities, among many applications to applied fields including statistics, reliability and economics (see, e.g., Gross and Richards 1998, 2004, and Karlin and Proschan 1960).",
"Especially, the latter studied the totally positive kernels that arise from convolutions of Pólya type distributions.",
"We now characterize the TP$_{2}$ property of the survival functions of BLM distributions.",
"Theorem 6.",
"Let $(X,Y)\\sim H=BLM(F,G,\\theta )\\in {\\cal BLM}.$ Then the survival function $\\overline{H}$ is TP$_{2}$ iff the marginal distributions $F$ and $G$ are IFR and together satisfy $\\overline{F}(x)\\overline{G}(x)\\le \\exp (-\\theta x),\\, x\\ge 0.$ Proof.",
"We define the cross-product ratio of $\\overline{H}$ : $r\\equiv r(x_1,x_2;y_1,y_2)=\\frac{\\overline{H}(x_1,y_1)\\overline{H}(x_2,y_2)}{\\overline{H}(x_1,y_2)\\overline{H}(x_2,y_1)},\\ \\ 0<x_1<x_2,\\ 0<y_1<y_2.$ Then, by definition, $\\overline{H}$ is TP$_{2}$ iff $r(x_1,x_2;y_1,y_2)\\ge 1$ for all $0<x_1<x_2,\\ 0<y_1<y_2$ .",
"(Necessity) Suppose that $\\overline{H}$ is TP$_{2}$ .",
"Then for all $0<x_1=y_1<x_2=y_2$ , we have $r=r(x_1,x_2;x_1,x_2)=\\frac{\\overline{H}(x_1,x_1)\\overline{H}(x_2,x_2)}{\\overline{H}(x_1,x_2)\\overline{H}(x_2,x_1)}=\\frac{\\exp (-\\theta (x_2-x_1)) }{\\overline{F}(x_2-x_1)\\overline{G}(x_2-x_1)}\\ge 1.$ This implies that $\\overline{F}(x)\\overline{G}(x)\\le \\exp (-\\theta x),\\, x\\ge 0.$ Next, we prove that the marginal distribution $G$ is IFR.",
"Note that the following statements are equivalent: (i) $g(y)/\\overline{G}(y)$ is increasing in $y\\ge 0$ , (ii) $\\frac{\\overline{G}(y+t)}{\\overline{G}(t)}$ is decreasing in $t\\in (0,\\infty )$ for each $y\\ge 0$ (Barlow and Proschan 1981, p. 54), (iii) $\\frac{\\overline{G}(t)}{\\overline{G}(y+t)}$ is increasing in $t\\in (0,\\infty )$ for each $y\\ge 0$ , (iv) $\\frac{\\overline{G}(y-x_2)}{\\overline{G}(y-x_1)}$ is increasing in $y> x_2$ for any fixed $0<x_1<x_2$ , (v) the ratio $r^*_{\\overline{G}}\\equiv \\frac{\\overline{G}(y_1-x_1)\\overline{G}(y_2-x_2)}{\\overline{G}(y_2-x_1)\\overline{G}(y_1-x_2)}\\ge 1\\ \\ \\hbox{for all}\\ \\ 0<x_1<x_2< y_1<y_2.$ The latter is true because in this case $r^*_{\\overline{G}}=r(x_1,x_2;y_1,y_2)\\ge 1$ by (5) and the assumption.",
"Similarly, we can prove that $F$ is IFR because the ratio $r^*_{\\overline{F}}\\equiv \\frac{\\overline{F}(x_1-y_1)\\overline{F}(x_2-y_2)}{\\overline{F}(x_2-y_1)\\overline{F}(x_1-y_2)}\\ge 1\\ \\ \\hbox{for all}\\ \\ 0<y_1<y_2< x_1<x_2.$ (Sufficiency) Suppose that the marginal distributions $F$ and $G$ are IFR and together satisfy $\\overline{F}(x)\\overline{G}(x)\\le \\exp (-\\theta x),\\, x\\ge 0.$ Then we want to prove that $\\overline{H}$ is TP$_{2}$ , that is, for all $0<x_1<x_2,\\ 0<y_1<y_2$ , the cross-product ratio $r=r(x_1,x_2;y_1,y_2)\\ge 1.$ Without loss of generality, we consider only three possible cases below, $(a)\\ 0<x_1\\le x_2\\le y_1\\le y_2,\\ \\ (b)\\ 0<x_1\\le y_1\\le x_2\\le y_2,\\ \\ (c)\\ 0<x_1\\le y_1\\le y_2\\le x_2,$ because the remaining cases can be proved by exchanging the roles of $F$ and $G$ .",
"For case (a), we have $r\\ge 1$ by the equivalence relations shown in the necessity part and by the continuity of $H$ when $x_2=y_1$ .",
"For case (b), the cross-product ratio $r=\\frac{\\exp (-\\theta x_2)\\overline{G}(y_1-x_1)\\overline{G}(y_2-x_2)}{\\exp (-\\theta y_1)\\overline{G}(y_2-x_1)\\overline{F}(x_2-y_1)}\\ge \\frac{\\overline{G}(y_1-x_1)\\overline{G}(y_2-x_2)\\overline{G}(x_2-y_1)}{\\overline{G}(y_2-x_1)},$ because $\\overline{F}(x_2-y_1)\\overline{G}(x_2-y_1)\\le \\exp (-\\theta (x_2-y_1))$ by the assumption.",
"Recall that any IFR distribution is new better than used (Barlow and Proschan 1981, p. 159).",
"Therefore, $\\overline{G}(x+y)\\le \\overline{G}(x)\\overline{G}(y)$ for all $x,y\\ge 0,$ and hence the last $r\\ge 1.$ Similarly, for case (c), $r=\\frac{\\exp (-\\theta y_2)\\overline{G}(y_1-x_1)\\overline{F}(x_2-y_2)}{\\exp (-\\theta y_1)\\overline{G}(y_2-x_1)\\overline{F}(x_2-y_1)}\\ge \\frac{\\overline{G}(y_1-x_1)\\overline{G}(y_2-y_1)}{\\overline{G}(y_2-x_1)}\\times \\frac{\\overline{F}(y_2-y_1)\\overline{F}(x_2-y_2)}{\\overline{F}(x_2-y_1)}\\ge 1,$ by the assumptions.",
"This completes the proof.",
"Recall also that for any bivariate distribution $H$ with marginals $F$ and $G$ , there exist a copula $C$ (a bivariate distribution with uniform marginals on $[0,1]$ ) and a survival copula $\\hat{C}$ such that $H(x,y)=C(F(x), G(y))$ and $\\overline{H}(x,y)=\\hat{C}(\\overline{F}(x),\\overline{G}(y))$ for all $x,y\\in {\\mathbb {R}}\\equiv (-\\infty ,\\infty ).$ Namely, $C$ links $H$ and $(F,G)$ , while $\\hat{C}$ links $\\overline{H}$ and $(\\overline{F},\\overline{G})$ .",
"Corollary 3.",
"Let $(X,Y)\\sim H=BLM(F,G,\\theta )\\in {\\cal BLM}.$ Then the survival copula $\\hat{C}$ of $H$ is TP$_{2}$ iff the marginal distributions $F$ and $G$ are IFR and together satisfy $\\overline{F}(x)\\overline{G}(x)\\le \\exp (-\\theta x),\\, x\\ge 0.$ Proof.",
"Since the marginal $F$ is absolutely continuous on the support $[a_F,\\infty )$ with positive density $f$ on $(a_F,\\infty )$ (see Corollary 2(iii) above), $F$ is strictly increasing and continuous on $(a_F,\\infty )$ .",
"Similarly, the marginal $G$ is strictly increasing and continuous on $(a_G,\\infty )$ .",
"By Theorem 6, it suffices to prove that $\\overline{H}$ is TP$_{2}$ on $(a_F,\\infty )\\times (a_G,\\infty )$ iff its survival copula $\\hat{C}$ is TP$_{2}$ on $(0,1)^2$ .",
"Recall the facts (i) $\\overline{H}(x,y)=\\hat{C}(\\overline{F}(x),\\overline{G}(y))$ , $(x,y)\\in (a_F,\\infty )\\times (a_G,\\infty )$ , (ii) $\\hat{C}(u,v)=\\overline{H}(\\overline{F}^{\\,-1}(u),\\overline{G}^{\\,-1}(v))$ , $u,v\\in (0,1)$ , where $\\overline{F}^{\\,-1},\\overline{G}^{\\,-1}$ are inverse functions of $\\overline{F},\\overline{G}$ , respectively, and (iii) all the functions $\\overline{F}, \\overline{G}, \\overline{F}^{\\,-1}$ and $\\overline{G}^{\\,-1}$ are decreasing.",
"The required result then follows immediately (see, e.g., Lemma 5(ii) below).",
"The counterpart of TP$_{2}$ property is the reverse regular of order two (RR$_2$ ).",
"For a nonnegative function $K$ on $(a,b)\\times (c,d)$ , we say that $K$ is RR$_2$ if the determinant of the $2\\times 2$ matrix $(K(x_i,y_j))$ is non-positive for all $a<x_1<x_2<b$ and $c<y_1<y_2<d$ (see, e.g., Esna–Ashari and Asadi 2016 for examples of RR$_2$ joint densities and survival functions).",
"Mimicking the proof of Theorem 6, we conclude that for $H=BLM(F,G,\\theta )\\in {\\cal BLM}$ , the survival function $\\overline{H}$ is RR$_2$ iff the survival copula $\\hat{C}$ of $H$ is RR$_2$ iff the marginal distributions $F$ and $G$ are DFR and satisfy $\\overline{F}(x)\\overline{G}(x)\\ge \\exp (-\\theta x),\\, x\\ge 0.$ To construct such a BLM distribution with RR$_2$ survival function, we first consider the Pareto Type II distribution (or Lomax distribution) $F$ with density function $f(x)=(\\alpha /\\beta ) (1+x/\\beta )^{-(\\alpha +1)},\\ x\\ge 0,$ and survival function $\\overline{F}(x)=(1+x/\\beta )^{-\\alpha },\\ x\\ge 0,$ where $\\alpha ,\\ \\beta >0.$ Then choose the parameters: $\\alpha \\ge 1,\\ \\beta >0$ and $\\theta =(\\alpha +1)/\\beta .$ It can be checked that the $\\overline{H}$ defined in (5) with $G=F$ is a bona fide survival function, and is RR$_2$ if $\\alpha =1.$ It is seen that all the conditions in Theorem 6 are satisfied by the Marshall–Olkin BVE.",
"Therefore, the survival function and survival copula of the Marshall–Olkin BVE are both TP$_{2}$ , regardless of parameters; a more general result will be given in Theorem 8 below.",
"We next characterize, by a different approach, the TP$_{2}$ property of some joint densities of absolutely continuous BLM distributions.",
"Theorem 7.",
"Let $H=BLM(F,G,\\theta )\\in {\\cal BLM}$ be absolutely continuous and have joint density function $h$ .",
"Suppose that the marginal density functions $f$ and $g$ are three times differentiable on $(0,\\infty )$ and that $\\theta f(0^+)+f^{\\prime }(0^+)=\\theta g(0^+)+g^{\\prime }(0^+)$ is finite.",
"Assume further the functions $h_1(x|\\theta )\\equiv \\theta f(x)+f^{\\prime }(x)>0,\\ \\ x>0,\\ \\ \\hbox{and}\\ \\ h_2(y|\\theta )\\equiv \\theta g(y)+g^{\\prime }(y)>0,\\ \\ y>0.$ Then the joint density function $h$ is TP$_{2}$ iff the marginal densities satisfy (i) $\\left(h_i^{\\prime }(x|\\theta )\\right)^2\\ge h_i^{\\prime \\prime }(x|\\theta )h_i(x|\\theta ),\\, x>0,\\,i=1,2,$ and (ii) $h_1(x|\\theta )h_2(x|\\theta )\\le h_1^2(0^+|\\theta )\\exp (-\\theta x),\\,x>0$ .",
"To prove this theorem, we need the concept of local dependence function and the following lemma, in which part (ii) is essentially due to Holland and Wang (1987, p. 872).",
"An alternative (complete) proof of part (ii) is provided below.",
"In their proof, Holland and Wang (1987) assumed implicitly the integrability of the local dependence function, while Kemperman (1977, p. 329) gave without proof the same result under continuity (smoothness) condition (see also Newman 1984).",
"Wang (1993) proved that a positive continuous bivariate density on a Cartesian product $(a,b)\\times (c,d)$ is uniquely determined by its marginal densities and local dependence function when the latter exists and is integrable.",
"On the other hand, Jones (1996, 1998) investigated the bivariate distributions with constant local dependence.",
"Lemma 4.",
"Let $K$ be a positive function on $D=(a,b)\\times (c,d)$ .",
"Then we have (i) $K$ is TP$_{2}$ on $D$ iff $\\log K$ is 2-increasing; (ii) $K$ is TP$_{2}$ on $D$ iff the local dependence function $\\gamma _K(x,y)\\equiv \\frac{\\partial ^2}{\\partial x\\partial y}\\log K(x,y)\\ge 0~~\\hbox{on}\\ D,$ provided the second-order partial derivatives exist.",
"Proof.",
"Part (i) is trivial by the definition of 2-increasing functions (see Nelsen 2006, p. 8), and part (ii) follows from part (i) and the fact that under the smoothness assumption, $\\log K$ is 2-increasing iff the local dependence function $\\gamma _K(x,y)\\ge 0.$ To prove part (ii) directly, note that the following statements are equivalent: (a) $\\frac{\\partial ^2}{\\partial x\\partial y}\\log K(x,y)\\ge 0$ on $D$ , (b) $\\frac{\\partial }{\\partial y}\\log [K(x_2,y)/K(x_1,y)]\\ge 0$ for all $y$ and for all $x_1<x_2$ , (c) $\\log [K(x_2,y)/K(x_1,y)]$ is increasing in $y$ for all $x_1<x_2$ , (d) $K(x_2,y)/K(x_1,y)$ is increasing in $y$ for all $x_1<x_2$ , (e) $K(x_2,y_2)/K(x_1,y_2)\\ge K(x_2,y_1)/K(x_1,y_1)$ for all $y_1<y_2,\\ x_1<x_2$ , (f) the cross-product ratio of $K$ satisfies: $K(x_1,y_1)K(x_2,y_2)/[K(x_1,y_2)K(x_2,y_1)]\\ge 1$ for all $x_1<x_2,\\ y_1<y_2$ , and (g) the function $K$ is TP$_{2}$ on $D$ .",
"The proof is complete.",
"Proof of Theorem 7.",
"By the assumptions, the joint density function of $H$ is of the form $h(x,y)=\\left\\lbrace \\begin{array}{cc}e^{-\\theta y}\\,h_1(x-y|\\theta ), & x\\ge y \\vspace{2.84544pt}\\\\e^{-\\theta x}\\,h_2(y-x|\\theta ), & x\\le y,\\end{array}\\right.$ where $h_i(0|\\theta )\\equiv h_i(0^+|\\theta ),\\,i=1,2.$ For $x\\ne y$ , the local dependence function of $h$ is $\\gamma _h(x,y)=\\frac{\\partial ^2}{\\partial x\\partial y}\\log h(x,y)=\\left\\lbrace \\begin{array}{cc}\\frac{[h_1^{\\prime }(x-y|\\theta )]^2-h_1^{\\prime \\prime }(x-y|\\theta )h_1(x-y|\\theta )}{h_1^2(x-y|\\theta )}, & x> y \\vspace{5.69046pt}\\\\\\frac{[h_2^{\\prime }(y-x|\\theta )]^2-h_2^{\\prime \\prime }(y-x|\\theta )h_2(y-x|\\theta )}{h_2^2(y-x|\\theta )}, & x< y.\\end{array}\\right.$ Therefore, $\\gamma _h(x,y)\\ge 0$ for all $(x,y)$ with $x\\ne y$ iff the property (i) holds true.",
"(Necessity) If $h$ is TP$_{2}$ on $(0,\\infty )^2$ , then it is also TP$_{2}$ on each rectangle (rectangular area) in the region ${\\cal A}_1=\\lbrace (x,y): x>y>0\\rbrace $ or in ${\\cal A}_2=\\lbrace (x,y): y>x>0\\rbrace $ , and hence the property (i) holds true by Lemma 4 and the above observation.",
"Next, the property (ii) follows from the fact that for all $0<x_1=y_1<x_2=y_2$ , the cross-product ratio $r_h$ of $h$ satisfies $1\\le r_h\\equiv r_h(x_1,x_2;y_1,y_2)=\\frac{h(x_1,y_1)h(x_2,y_2)}{h(x_1,y_2)h(x_2,y_1)}=\\frac{\\exp (-\\theta (x_2-x_1))h_1(0|\\theta )h_2(0|\\theta )}{h_1(x_2-x_1|\\theta )h_2(x_2-x_1|\\theta )}.$ This completes the proof of the necessity part.",
"(Sufficiency) Suppose $0<x_1<x_2$ and $0<y_1<y_2$ , then we want to prove the cross-product ratio $r_h\\ge 1$ under the assumptions (i) and (ii).",
"If the rectangle with four vertices $P_i, i=1,2,3,4$ , where $P_1=(x_1,y_1), P_2=(x_2, y_1), P_3=(x_2,y_2), P_4=(x_1,y_2)$ , lies entirely in the region ${\\cal A}_1$ or ${\\cal A}_2$ , then $r_h\\ge 1$ by the assumption (i) and Lemma 4.",
"If $0<x_1=y_1<x_2=y_2$ , then the assumption (ii) implies $r_h\\ge 1$ .",
"For the remaining cases, we apply the technique of factorization of the cross-product ratio if necessary.",
"For example, if $P_*=(x_1,y_*)\\in \\overline{P_1P_4}$ and $P^*=(x^*,y_2)\\in \\overline{P_4P_3}$ denote the intersection of the diagonal line $x=y$ and boundary of the rectangle, where $x_1<x^*<x_2$ and $y_1<y_*<y_2$ , then we split the original rectangle into four sub-rectangles by adding the new point $(x^*,y_*)$ and calculate the ratio $r_h(x_1,x_2;y_1,y_2)=r_h(x_1,x^*;y_1,y_*)r_h(x_1,x^*;y_*,y_2)r_h(x^*,x_2;y_1,y_*)r_h(x^*,x_2;y_*,y_2)\\ge 1,$ each factor being greater than or equal to one by the previous results.",
"The proof is complete.",
"It is known that the Marshall–Olkin BVE (6) is PQD, so are its copula $C$ and survival copula $\\hat{C}$ (see Barlow and Proschan 1981, p. 129).",
"Moreover, $\\overline{H}$ and $\\hat{C}$ are TP$_2$ due to Theorem 6 and its corollary (see also Nelsen 2006, p. 163, for a direct proof) and both are even TP$_{\\infty }$ if $\\lambda _1=\\lambda _2$ (Lin et al. 2016).",
"We are now able to extend these results to the following.",
"Theorem 8.",
"The Marshall–Olkin survival function $\\overline{H}$ and survival copula $\\hat{C}$ are both TP$_{\\infty },$ regardless of parameters.",
"To prove this theorem, we need two more useful lemmas.",
"Lemma 5 is well-known (see, e.g., Marshall et al.",
"2011, p. 758), while Lemma 6 is essentially due to Gantmacher and Krein (2002), pp.",
"78–79 (see also Karlin 1968, p. 112, for an alternative version).",
"Lemma 5.",
"Let $r\\ge 2$ be an integer.",
"(i) If $k(x,y)$ is TP$_r$ in $x$ and $y$ , and if both $u$ and $v$ are nonnegative functions, then the product function $K(x,y)=u(x)\\,v(y)\\,k(x,y)$ is TP$_r$ in $x$ and $y$ .",
"(ii) If $k(x,y)$ is TP$_r$ in $x$ and $y$ , and if $u$ and $v$ are both increasing, or both decreasing, then the composition function $K(x,y)=k(u(x),v(y))$ is TP$_r$ in $x$ and $y$ .",
"Lemma 6.",
"Let $\\phi $ and $\\psi $ be two positive functions on $(a,b).$ Define the symmetric function ${K_s}(x,y)=\\left\\lbrace \\begin{array}{cc}\\psi (x)\\,\\phi (y), &\\,\\, a<y\\le x<b\\vspace{2.84544pt}\\\\\\phi (x)\\,\\psi (y), &\\,\\, a<x\\le y<b.\\end{array}\\right.$ If $\\phi (x)/\\psi (x)$ is nondecreasing in $x\\in (a,b)$ , then the function $K_s(x,y)$ is TP$_{\\infty }$ in $x$ and $y$ .",
"Proof of Theorem 8.",
"We prove first that the Marshall–Olkin survival function $\\overline{H}$ is TP$_{\\infty }.$ Rewrite the survival function (6) as $\\overline{H}(x,y)&=&\\left\\lbrace \\begin{array}{cc}\\exp [-(\\lambda _1+\\lambda _{12})x-\\lambda _{2}y], & x\\ge y \\vspace{2.84544pt}\\\\\\exp [-(\\lambda _2+\\lambda _{12})y-\\lambda _{1}x], & x\\le y\\end{array}\\right.\\vspace{2.84544pt}\\\\&=&\\exp [-\\lambda _1x-\\lambda _2y]K_s(x,y),$ where the symmetric function $K_s(x,y)=\\left\\lbrace \\begin{array}{cc}\\exp (-\\lambda _{12}x), & x\\ge y \\vspace{2.84544pt}\\\\\\exp (-\\lambda _{12}y), & x\\le y.\\end{array}\\right.$ Let $\\phi (x)=1$ and $\\psi (y)=\\exp (-\\lambda _{12}y)$ .",
"Then by Lemma 6, we see that the function $K_s$ in (10) is TP$_{\\infty },$ so is $\\overline{H}$ by Lemma 5(i).",
"Next, recall that the Marshall–Olkin survival copula $\\hat{C}(u,v)=\\overline{H}(\\overline{F}^{\\,-1}(u),\\ \\overline{G}^{\\,-1}(v)),\\ \\ \\ u,v\\in (0,1),$ where $\\overline{F}^{\\,-1}$ and $\\overline{G}^{\\,-1}$ are the inverse (decreasing) functions of $\\overline{F}(x)=\\exp [-(\\lambda _1+\\lambda _{12})x]$ and $\\overline{G}(y)=\\exp [-(\\lambda _2+\\lambda _{12})y]$ , respectively.",
"Therefore, $\\hat{C}$ is TP$_{\\infty }$ by Lemma 5(ii).",
"It is well known that if a bivariate distribution $H$ has TP$_2$ density, then both $H$ and its joint survival function $\\overline{H}$ are TP$_2$ (see, e.g., Balakrishnan and Lai 2009, p. 116).",
"A more general result is given as follows.",
"Theorem 9.",
"If the bivariate distribution $H$ has TP$_r$ density with $r\\ge 2$ , then both $H$ and $\\overline{H}$ are TP$_r$ .",
"Consequently, if $H$ has TP$_{\\infty }$ density, then both $H$ and $\\overline{H}$ are TP$_{\\infty }$ .",
"Proof.",
"Let us consider first the TP$_{\\infty }$ indicator functions $K_1(x, y)=\\hbox{I}_{(-\\infty , x]}(y)$ and $K_2(x,y)=\\hbox{I}_{[x,\\infty )}(y)$ , and then apply Theorem 3.5 of Gross and Richards (1998) restated below.",
"For example, to prove the TP$_r$ property of $H$ , we have to claim that for all $x_1<\\cdots <x_r$ and $y_1<\\cdots <y_r$ , the determinant of each $s\\times s$ sub-matrix $(H(x_i,y_j))$ (with $2\\le s\\le r$ ) is nonnegative.",
"To prove this, let us recall that $H(x_i,y_j)=E[\\hbox{I}_{(-\\infty , x_i]}(X)\\,\\hbox{I}_{(-\\infty ,y_j]}(Y)]=E[\\phi (i,X)\\,\\psi (j,Y)]$ , where $\\phi (i,x)=\\hbox{I}_{(-\\infty , x_i]}(x)$ is TP$_r$ in two variables $i\\in \\lbrace 1,2,\\ldots , r\\rbrace $ and $x\\in {\\mathbb {R}},$ and $\\psi (j,y)=\\hbox{I}_{(-\\infty , y_j]}(y)$ is TP$_r$ in two variables $j\\in \\lbrace 1,2,\\ldots , r\\rbrace $ and $y\\in {\\mathbb {R}}$ .",
"Then Gross and Richards' Theorem applies and hence $H$ is TP$_r$ .",
"Similarly, the survival function $\\overline{H}$ is TP$_r$ .",
"The proof is complete.",
"Gross and Richards' (1998) Theorem.",
"Let $r\\ge 2$ be an integer and let the bivariate $(X,Y)\\sim H$ have TP$_r$ density.",
"Assume further that both the functions $\\phi (i,x)$ and $\\psi (i,x)$ are TP$_r$ in two variables $i\\in \\lbrace 1,2,\\ldots , r\\rbrace $ and $x\\in {\\mathbb {R}}$ .",
"Then the $r\\times r$ matrix $(E[\\phi (i,X)\\,\\psi (j,Y)])$ is totally positive, that is, all its minors (of orders $\\le r$ ) are nonnegative real numbers.",
"As mentioned in Balakrishnan and Lai (2009, p. 124), the Block–Basu BVE (8) is PQD if $\\lambda _1=\\lambda _2.$ We now extend this result to the following.",
"Theorem 10.",
"(i) If $\\lambda _1=\\lambda _2$ in (8), then the Block–Basu BVE has TP$_{\\infty }$ density.",
"(ii) If $\\alpha =\\beta \\le \\alpha ^{\\prime }=\\beta ^{\\prime }$ in (9), then the Freund BVE has TP$_{\\infty }$ density.",
"Proof.",
"Take $\\phi (x)=c_1\\exp ({-\\lambda _1x})$ and $\\psi (y)=c_2\\exp [{-(\\lambda _2+\\lambda _{12})y}]$ for some constants $c_1, c_2>0.$ Then part (i) follows from (8) and Lemma 6.",
"Part (ii) can be proved similarly.",
"Remark 4.",
"The same approach applies to other bivariate (non-BLM) distributions like Li and Pellerey's (2011) generalized Marshall–Olkin bivariate distribution described below.",
"In Marshall and Olkin's (1967) shock model: $(X, Y)= (X_1\\wedge X_3,\\, X_2\\wedge X_3)$ , we assume instead that $X_1, X_2, X_3$ are independent general positive random variables (not limited to exponential ones) and that $X_i\\sim F_i,\\ i=1,2,3.$ Let $R_i=-\\log \\overline{F}{\\!",
"}_i$ be the hazard function of $X_i.$ Then the generalized Marshall–Olkin bivariate distribution $H$ has survival function ${\\overline{H}(x,y)}&=&\\Pr (X>x,\\,Y>y)= \\Pr (X_1>x,\\,X_2>y,\\, X_3>\\max \\lbrace x,y\\rbrace )\\nonumber \\\\&=&\\exp [-R_1(x)-R_2(y)-R_3(\\max \\lbrace x,y\\rbrace )] ,\\ \\ \\ x,y\\ge 0,$ which is PQD (Li and Pellerey 2011).",
"(For other related shock models, see Marshall and Olkin 1967, Ghurye and Marshall 1984 as well as Aven and Jensen 2013, Section 5.3.4.)",
"We now extend this result and Theorem 8 as follows.",
"Theorem 11.",
"Let $H$ be the generalized Marshall–Olkin distribution defined in (11).",
"Then (i) the survival function $\\overline{H}$ is TP$_{\\infty };$ (ii) the survival copula $\\hat{C}$ of $H$ is TP$_{\\infty }$ , provided the functions $\\overline{F}{\\!}_1\\overline{F}{\\!",
"}_3$ and $\\overline{F}{\\!}_2\\overline{F}{\\!",
"}_3$ are both strictly decreasing.",
"Proof.",
"Write the survival function (11) as $\\overline{H}(x,y)&=&\\left\\lbrace \\begin{array}{cc}\\exp [-(R_1(x)+R_3(x))-R_2(y)], & x\\ge y \\vspace{2.84544pt}\\\\\\exp [-(R_2(y)+R_3(y))-R_1(x)], & x\\le y\\end{array}\\right.\\\\&=&\\exp [-R_1(x)-R_2(y)]K_s(x,y),$ where the symmetric function $K_s(x,y)=\\left\\lbrace \\begin{array}{cc}\\exp [-R_3(x)], & x\\ge y \\vspace{2.84544pt}\\\\\\exp [-R_3(y)], & x\\le y.\\end{array}\\right.$ By taking $\\phi (x)=1$ and $\\psi (y)=\\exp [-R_3(y)]$ in Lemma 6, we know that the function $K_s$ in (12) is TP$_{\\infty }$ , and hence the survival function $\\overline{H}$ is TP$_{\\infty }$ by Lemma 5(i).",
"This proves part (i).",
"To prove part (ii), we note that the marginal survival functions of $H$ are $\\overline{F}(x)=\\exp [-\\tilde{R}_1(x)],\\ x\\ge 0,$ and $\\overline{G}(y)=\\exp [-\\tilde{R}_2(y)],\\ y\\ge 0,$ where the two functions $\\tilde{R}_1(x)=R_1(x)+R_3(x),\\ x\\ge 0,$ and $\\tilde{R}_2(y)=R_2(y)+R_3(y),\\ y\\ge 0,$ are strictly increasing by the conditions on $F_i,\\ i=1,2,3.$ This in turn implies that the marginal distribution functions ${F}$ and ${G}$ are strictly increasing and hence the survival copula $\\hat{C}(u,v)=\\overline{H}(F^{-1}(1-u), G^{-1}(1-v)),\\ \\ \\ u,v\\in (0,1),$ because $F^{-1}(F(t))=t,\\ t\\in (0,1),$ where the quantile function $F^{-1}(t)=\\inf \\lbrace x: F(x)\\ge t\\rbrace ,\\, t\\in (0,1)$ (see, e.g., Shorack and Wellner 1986, p. 6).",
"Therefore, $\\hat{C}$ is TP$_{\\infty }$ by part (i) and Lemma 5(ii).",
"The proof is complete.",
"5.",
"Stochastic Comparisons of BLM Distributions To provide more information about BLM distributions, we can study stochastic comparisons in the ${\\cal BLM}$ family.",
"As usual, define the notions of the upper orthant order ($\\le _{uo}$ ), the concordance order ($\\le _{c}$ ) and the Laplace transform order ($\\le _{Lt}$ ) as follows.",
"Let $(X_i,Y_i)\\sim H_i$ with marginals $(F_i,G_i)$ , $i=1,2,$ on ${\\mathbb {R}}_+.$ Then denote (i) $(X_1,Y_1)\\le _{uo} (X_2,Y_2)$ if $\\overline{H}{\\!",
"}_1(x,y)\\le \\overline{H}{\\!",
"}_2(x,y)$ for all $x,y\\ge 0$ , (ii) $(X_1,Y_1)\\le _{c}(X_2,Y_2)$ if $(F_1,G_1)=(F_2,G_2)$ and $(X_1,Y_1)\\le _{uo} (X_2,Y_2)$ , and (iii) $(X_1,Y_1)\\le _{Lt}(X_2,Y_2)$ if ${\\cal L}_1(s,t)\\ge {\\cal L}_2(s,t)$ for all $s, t\\ge 0$ (Müller and Stoyan 2002, Shaked and Shanthikumar 2007).",
"We have, for example, the following results whose proofs are straightforward and are omitted.",
"Theorem 12.",
"Let $(X_i,Y_i)\\sim H_i=BLM(F_i,G_i,\\theta _i)\\in {\\cal BLM},$ $i=1,2.$ Then we have (i) $X_1\\le _{st}X_2,$ $Y_1\\le _{st}Y_2$ and $\\theta _1\\ge \\theta _2,$ iff $(X_1,Y_1)\\le _{uo}(X_2,Y_2)$ , or, equivalently, $E[K(X_1,Y_1)]\\le E[K(X_2,Y_2)]$ for any bivariate distribution $K$ on ${\\mathbb {R}}_+^2$ ; (ii) $F_1=F_2, G_1=G_2$ and $\\theta _1\\ge \\theta _2,$ iff $(X_1,Y_1)\\le _{c}(X_2,Y_2)$ , or, equivalently, $E[k_1(X_1)k_2(Y_1)]\\le E[k_1(X_2)k_2(Y_2)]$ for all increasing functions $k_1,k_2$ , provided the expectations exist; and (iii) if $X_1\\le _{Lt}X_2,$ $Y_1\\le _{Lt}Y_2$ and $\\theta _1=\\theta _2$ , then $(X_1,Y_1)\\le _{Lt}(X_2,Y_2)$ , or, equivalently, $E[k_1(X_1)k_2(Y_1)]\\ge E[k_1(X_2)k_2(Y_2)]$ for all completely monotone functions $k_1,k_2$ , provided the expectations exist.",
"When $H_1$ and $H_2$ have the same pair of marginals $(F,G)$ , Theorem 12(i) reduces, by Corollary 1, to the following interesting result which is related to the famous Slepian's inequality for bivariate normal distributions (see the discussion in Remark 5 below).",
"Corollary 4.",
"Let $(X_i,Y_i)\\sim H_i=BLM(F,G,\\theta _i)\\in {\\cal BLM},$ with correlation $\\rho _i$ , $i=1,2.$ Then $\\rho _1\\le \\rho _2$ iff $\\overline{H}{\\!",
"}_1(x,y)\\le \\overline{H}{\\!",
"}_2(x,y)$ for all $x,y\\ge 0,$ or, equivalently, ${H}_1(x,y)\\le {H}_2(x,y)$ for all $x,y\\ge 0.$ Remark 5.",
"In Corollary 4 above, if we consider standard bivariate normal distributions instead of BLM ones, then the conclusion also holds true and the necessary part is the so-called Slepian's lemma/inequality; see Slepian (1962), Müller and Stoyan (2002), p. 97, and Hoffmann-J$ø$ rgensen (2013) for more general results.",
"In Wikipedia, it was said that while this intuitive-seeming result is true for Gaussian processes, it is not in general true for other random variables.",
"However, as we can see in Corollary 4, there are infinitely many BLM distributions sharing the same Slepian's inequality with bivariate normal ones.",
"Remark 6.",
"We finally compare the effects of the dependence structure of BLM distributions in different coherent systems.",
"Consider a two-component system and let the two components have lifetimes $(X,Y)\\sim H=BLM(F,G,\\theta )$ .",
"Then the lifetime of a series system composed of these two components is $X\\wedge Y\\sim Exp(\\theta )$ , while the lifetime of a parallel system composed of the same components is $X\\vee Y$ obeying the distribution $H_p(z)=e^{-\\theta z}-1+F(z)+G(z),\\ z\\ge 0.$ Therefore the mean times to failure of series and parallel systems are, respectively, $E[X\\wedge Y]=\\int _0^{\\infty }\\overline{H}(x,x)dx=1/\\theta $ (decreasing in $\\theta $ ) and $E[X\\vee Y]=E[X]+E[Y]-\\int _0^{\\infty }\\overline{H}(x,x)dx=E[X]+E[Y]-1/\\theta $ (increasing in $\\theta $ ).",
"The latter further implies that $\\theta \\ge (E[X]+E[Y])^{-1}$ (compare with Theorem 1(vi)) and that $E[XY]\\in [1/\\theta ^2, (E[X]+E[Y])^2]$ by Corollary 1, provided the expectations exist.",
"See also Aven and Jensen (2013, Section 2.3) for special cases with exponential marginals as well as Lai and Lin (2014) for more general results.",
"Acknowledgments.",
"The authors would like to thank the Editor-in-Chief and two Referees for helpful comments and constructive suggestions which improve the presentation of the paper.",
"The paper was presented at the Ibusuki International Seminar, Ibusuki Phoenix Hotel, held from 6th to 8th March 2016 by Waseda University, Japan.",
"The authors thank the organizer Professor Masanobu Taniguchi for his kind invitation and the audiences for their comments and suggestions.",
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"Methodology, 29, 1–9.",
"Marshall, A. W. and Olkin, I.",
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"Statist.",
"Assoc., 62, 30–44.",
"Marshall, A. W. and Olkin, I.",
"(2015).",
"A bivariate Gompertz–Makeham life distribution.",
"J. Multivariate Anal., 139, 219–226.",
"Marshall, A. W., Olkin, I. and Arnold, B. C. (2011).",
"Inequalities: Theory of Majorization and Its Applications, 2nd ed.",
"Springer, New York.",
"Müller, A. and Stoyan, D. (2002).",
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"Wiley, New York.",
"Nadarajah, S. (2006).",
"Exact distributions of $XY$ for some bivariate exponential distributions.",
"Statistics, 40, 307–324.",
"Nelsen, R. B.",
"(2006).",
"An Introduction to Copulas, 2nd ed.",
"Springer, New York.",
"Newman, C. (1984).",
"Asymptotic independence and limit theorems for positively and negatively dependent random variables.",
"In: Inequalities in Statistics and Probability (Y. L. Tong, ed.",
"), IMS Lecture Notes–Monograph Series, Vol.",
"5, 127–140.",
"Rao, C. R. and Shanbhag, D. N. (1994).",
"Choquet–Deny Type Functional Equations with Applications to Stochastic Models.",
"Wiley, New York.",
"Shaked, M. (1977).",
"A family of concepts of positive dependence for bivariate distributions.",
"J. Amer.",
"Statist.",
"Assoc., 72, 642–650.",
"Shaked, M. and Shanthikumar, J. G. (1988).",
"Multivariate conditional hazard rates and the MIFRA and MIFR properties.",
"J. Appl.",
"Probab., 25, 150–168.",
"Shaked, M. and Shanthikumar, J. G. (2007).",
"Stochastic Orders.",
"Springer, New Jersey.",
"Shorack, G. R. and Wellner, J. A.",
"(1986).",
"Empirical Processes with Applications to Statistics.",
"Wiley, New York.",
"Slepian, D. (1962).",
"The one-sided barrier problem for Gaussian noise.",
"Bell System Technical Journal, 41, 463-501.",
"Wang, Y. J.",
"(1993).",
"Construction of continuous bivariate density functions.",
"Statist.",
"Sinica, 3, 173–187.",
"Aven, T. and Jensen, U.",
"(2013).",
"Stochastic Models in Reliability, 2nd ed.",
"Springer, New York.",
"Balakrishnan, N. and Lai, C.-D. (2009).",
"Continuous Bivariate Distributions, 2nd ed.",
"Springer, New York.",
"Barlow, R. E. and Proschan, F. (1981).",
"Statistical Theory of Reliability and Life Testing: Probability Models, To Begin With.",
"Silver Spring, MD.",
"Bassan, B. and Spizzichino, F. (2005).",
"Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes.",
"J. Multivariate Anal., 93, 313–330.",
"Block, H. W. (1977).",
"A characterization of a bivariate exponential distribution.",
"Ann.",
"Statist., 5, 808–812.",
"Block, H. W. and Basu, A. P. (1974).",
"A continuous bivariate exponential extension.",
"J. Amer.",
"Statist.",
"Assoc., 69, 1031–1037.",
"Block, H. W. and Savits, T. H. (1976).",
"The IFRA closure problem.",
"Ann.",
"Probab., 4, 1030–1032.",
"Block, H. W. and Savits, T. H. (1980).",
"Multivariate increasing failure rate average distributions.",
"Ann.",
"Probab., 8, 793–801.",
"Crawford, G. B.",
"(1966).",
"Characterization of geometric and exponential distributions.",
"Ann.",
"Math.",
"Statist., 37, 1790–1795.",
"Esary, J. D. and Marshall, A. W. (1979).",
"Multivariate distributions with increasing hazard rate average.",
"Ann.",
"Probab., 7, 359–370.",
"Esna–Ashari, M. and Asadi, M. (2016).",
"On additive–multiplicative hazards model.",
"Statistics, 50, 1421–1433.",
"Feller, W. (1965).",
"An Introduction to Probability Theory, Vol.",
"I. Wiley, New York.",
"Ferguson, T. S. (1964).",
"A characterization of the exponential distribution.",
"Ann.",
"Math.",
"Statist., 35, 1199–1207.",
"Ferguson, T. S. (1965).",
"A characterization of the geometric distribution.",
"Amer.",
"Math.",
"Monthly, 72, 256–260.",
"Fortet, R. (1977).",
"Elements of Probability Theory.",
"Gordon and Breach, New York.",
"Freund, J. E. (1961).",
"A bivariate extension of the exponential distribution.",
"J. Amer.",
"Statist.",
"Assoc., 56, 971–977.",
"Galambos, J. and Kotz, S. (1978).",
"Characterizations of Probability Distributions.",
"Springer, New York.",
"Gantmacher, F. R. and Krein, M. G. (2002).",
"Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems, Revised edn.",
"Translation based on the 1941 Russian original, Providence, RI.",
"Ghurye, S. G. (1987).",
"Some multivariate lifetime distributions.",
"Adv.",
"Appl.",
"Probab., 19, 138–155.",
"Ghurye, S. G. and Marshall, A. W. (1984).",
"Shock processes with aftereffects and multivariate lack of memory.",
"J. Appl.",
"Probab., 21, 768–801.",
"Gross, K. I. and Richards, D. St. P. (1998).",
"Algebraic methods toward higher-order probability inequalities.",
"In: Stochastic Processes and Related Topics (B. Rajput et al., eds.",
"), 189–211, Birkhäuser, Boston.",
"Gross, K. I. and Richards, D. St. P. (2004).",
"Algebraic methods toward higher-order probability inequalities, II.",
"Ann.",
"Probab., 32, 1509–1544.",
"Hoffmann-J$ø$ rgensen, J.",
"(2013).",
"Slepian's inequality, modularity and integral orderings.",
"High Dimensional Probability VI, 19–53, Progress in Probability, 66, Springer, Basel.",
"Holland, P. W. and Wang, Y. J.",
"(1987).",
"Dependence function for continuous bivariate densities.",
"Comm.",
"Statist.",
"– Theory and Methods, 16, 863–876.",
"Jones, M. C. (1996).",
"The local dependence function.",
"Biometrika, 83, 899–904.",
"Jones, M. C. (1998).",
"Constant local dependence.",
"J. Multivariate Anal., 64, 148–155.",
"Karlin, S. (1968).",
"Total Positivity, Vol. I.",
"Stanford University Press, CA.",
"Karlin, S. and Proschan, F. (1960).",
"Pólya type distributions of convolutions.",
"Ann.",
"Math.",
"Statist., 31, 721–736.",
"Kayid, M., Izadkhah, S. and Alshami, S. (2016).",
"Laplace transform ordering of time to failure in age replacement models.",
"J. Korean Statist.",
"Theory, 45, 101–113.",
"Kemperman, J. H. B.",
"(1977).",
"On the FKG–inequality for measures on a partially ordered space.",
"Indagationes Mathematicae, 80, 313–331.",
"Kotz, S., Balakrishnan, N. and Johnson, N. L. (2000).",
"Continuous Multivariate Distributions, Vol.",
"1: Models and Applications, 2nd ed.",
"Wiley, New York.",
"Kulkarni, H. V. (2006).",
"Characterizations and modelling of multivariate lack of memory property.",
"Metrika, 64, 167–180.",
"Lai, C.-D. and Lin, G. D. (2014).",
"Mean time to failure of systems with dependent components.",
"Appl.",
"Math.",
"Comput., 246, 103–111.",
"Lai, C.-D. and Xie, M. (2006).",
"Stochastic Ageing and Dependence for Reliability.",
"Springer, New York.",
"Lee, M.-L.T.",
"(1985).",
"Dependence by total positivity.",
"Ann.",
"Probab., 13, 572–582.",
"Li, X. and Pellerey, F. (2011).",
"Generalized Marshall–Olkin distributions and related bivariate aging properties.",
"J. Multivariate Anal., 102, 1399–1409.",
"Lin, G. D., Dou, X., Kuriki, S. and Huang, J. S. (2014).",
"Recent developments on the construction of bivariate distributions with fixed marginals.",
"Journal of Statistical Distributions and Applications, 1: 14.",
"Lin, G. D., Lai, C.-D. and Govindaraju, K. (2016).",
"Correlation structure of the Marshall–Olkin bivariate exponential distribution.",
"Statist.",
"Methodology, 29, 1–9.",
"Marshall, A. W. and Olkin, I.",
"(1967).",
"A multivariate exponential distribution.",
"J. Amer.",
"Statist.",
"Assoc., 62, 30–44.",
"Marshall, A. W. and Olkin, I.",
"(2015).",
"A bivariate Gompertz–Makeham life distribution.",
"J. Multivariate Anal., 139, 219–226.",
"Marshall, A. W., Olkin, I. and Arnold, B. C. (2011).",
"Inequalities: Theory of Majorization and Its Applications, 2nd ed.",
"Springer, New York.",
"Müller, A. and Stoyan, D. (2002).",
"Comparison Methods for Stochastic Models and Risks.",
"Wiley, New York.",
"Nadarajah, S. (2006).",
"Exact distributions of $XY$ for some bivariate exponential distributions.",
"Statistics, 40, 307–324.",
"Nelsen, R. B.",
"(2006).",
"An Introduction to Copulas, 2nd ed.",
"Springer, New York.",
"Newman, C. (1984).",
"Asymptotic independence and limit theorems for positively and negatively dependent random variables.",
"In: Inequalities in Statistics and Probability (Y. L. Tong, ed.",
"), IMS Lecture Notes–Monograph Series, Vol.",
"5, 127–140.",
"Rao, C. R. and Shanbhag, D. N. (1994).",
"Choquet–Deny Type Functional Equations with Applications to Stochastic Models.",
"Wiley, New York.",
"Shaked, M. (1977).",
"A family of concepts of positive dependence for bivariate distributions.",
"J. Amer.",
"Statist.",
"Assoc., 72, 642–650.",
"Shaked, M. and Shanthikumar, J. G. (1988).",
"Multivariate conditional hazard rates and the MIFRA and MIFR properties.",
"J. Appl.",
"Probab., 25, 150–168.",
"Shaked, M. and Shanthikumar, J. G. (2007).",
"Stochastic Orders.",
"Springer, New Jersey.",
"Shorack, G. R. and Wellner, J. A.",
"(1986).",
"Empirical Processes with Applications to Statistics.",
"Wiley, New York.",
"Slepian, D. (1962).",
"The one-sided barrier problem for Gaussian noise.",
"Bell System Technical Journal, 41, 463-501.",
"Wang, Y. J.",
"(1993).",
"Construction of continuous bivariate density functions.",
"Statist.",
"Sinica, 3, 173–187."
]
] | 1606.05097 |
[
[
"Recycled Pulsars: Spins, Masses and Ages"
],
[
"Abstract Recycled pulsars are mainly characterized by their spin periods, B-fields and masses.",
"All these quantities are affected by previous interactions with a companion star in a binary system.",
"Therefore, we can use these quantities as fossil records and learn about binary evolution.",
"Here, I briefly review the distribution of these observed quantities and summarize our current understanding of the pulsar recycling process."
],
[
"Introduction",
"Recycled pulsars, or millisecond pulsars (MSPs), represent the advanced phase of stellar evolution in close, interacting binaries.",
"Their observed orbital and stellar properties are fossil records of their evolutionary history and thus one can use these systems as key probes of stellar astrophysics [5], [28], [30].",
"The recycled pulsar is an old neutron star and the first formed compact object in the present-day observed binary system.",
"This neutron star was spun up to a high spin frequency via accretion of mass and angular momentum once the secondary star evolved.",
"In this recycling phase the system is observable as a low-mass X-ray binary [5], [21].",
"Over the last four decades, the number of known recycled pulsars has increased to $>300$ [16], of which $\\sim \\!200$ are in binaries.",
"The remaining $\\sim \\!100$ isolated recycled pulsars have evaporated their companion [12], [22], lost it in a supernova explosion [26] or in an exchange encounter in a globular cluster [23], [10]."
],
[
"Spins of recycled pulsars",
"The observed spin periods of recycled pulsars span between 1.4 ms [13] and about 200 ms [25].",
"There is a clear correlation between these spin periods and the nature of the companion star responsible for the recycling process [30].",
"The more massive and evolved the companion star is at the onset of the mass transfer, the slower is the final spin rate of the recycled pulsar.",
"The reason is that the duration of the mass-transfer (X-ray) phase is much shorter for massive and/or giant stars, which therefore leads to less efficient recycling.",
"[30] also derived a correlation between the final equilibrium spin period of a recycled pulsar and the (minimum) amount of accreted material needed for spin-up.",
"Consider a pulsar with a typical mass of $1.4\\;M_{\\odot }$ and a recycled spin period of either 2, 5, 10 or 50 ms. To obtain such spins requires accretion of (at least) 0.10, 0.03, 0.01 or $10^{-3}\\;M_{\\odot }$ , respectively.",
"Therefore, it is no surprise that observed recycled pulsars with massive companions (CO/ONeMg white dwarfs or neutron stars) are much more slow rotators, in general, compared to MSPs with He white dwarf companions Figure: Observed spin period distribution of MSPs.",
"Where are the sub-ms MSPs?",
".Figure: Mass measurements and 68% uncertainty intervalsfor binary pulsars with white dwarf companions (purple, top) and neutron star companions (blue, bottom).",
"From .An interesting question is why we don't detect any MSPs spinning faster than 1.4 ms (cf.",
"Fig.",
"REF ).",
"Answering this question is not only important for understanding magnetosphere and accretion physics, but could also be relevant for constraining the equation-of-state of neutron stars [18].",
"Previously, the missing sub-ms MSPs were suspected to be related to Doppler smearing of radio pulsations in tight binary orbits, which could cause a selection bias against detection of sub-ms MSPs.",
"However, this issue is less serious in present-day acceleration searches (at least for dispersion measures, $DM<100$ ).",
"Three other ideas have been proposed to explain the saturation of MSP spin rates: i) emission of gravitational waves [6], [7], ii) limited accretion torques in the pulsar magnetosphere [15], and iii) spin-down at Roche lobe decoupling [29].",
"Comparison of spin rates of accreting MSPs and radio MSPs [20] gives some support for the latter hypothesis, although no firm conclusions can be drawn.",
"It is possible that more than one effect is at work.",
"Finally, the concept of a spin-up line in the $P\\dot{P}$ –diagram cannot be uniquely defined, and instead one should consider a broad spin-up valley for the birth location of recycled pulsars [30]."
],
[
"Masses of recycled pulsars",
"In Fig.",
"REF , we have plotted the distribution of rather precisely measured masses of pulsars with white dwarf and neutron star companions.",
"These mass measurements span between $1.17\\pm 0.01\\;M_{\\odot }$ [17] and $2.01\\pm 0.04\\;M_{\\odot }$ [2].",
"The largest spread in pulsar masses is seen in systems with white dwarf companions.",
"For the combined sample of masses, [3] argue in favor of a bimodal distribution with a low- and a high-mass mass peak centered at $\\sim \\!1.39\\;M_{\\odot }$ and $1.81\\;M_{\\odot }$ , respectively.",
"For binary pulsars with He white dwarf companions (the descendants of LMXBs), one can apply the white dwarf mass–orbital period relation [27] to help constrain the mass of the neutron star.",
"In recent years, evidence has accumulated that accreting neutron stars are very inefficient accretors [1], [3].",
"Therefore, we are inclined to believe that the observed mass distribution is closely resembling the birth distribution of neutron stars in these two different classes of pulsar binaries.",
"In particular, for pulsars with neutron star companions, the typical amount of accreted mass obtained from numerical modelling [32] of Case BB Roche-lobe overflow is only $\\sim \\!10^{-3}\\;M_{\\odot }$ ."
],
[
"Ages of recycled pulsars",
"To understand the formation and evolution of recycled pulsars, it is of uttermost importance to determine their true ages.",
"Only when trues ages of MSPs are established, is it possible to gain knowledge of pulsar evolution from the observed distribution of MSPs in the $P\\dot{P}$ –diagram.",
"The characteristic age, $\\tau \\equiv P/(2\\dot{P})$ is in many cases a bad estimate of true age, cf.",
"Fig.",
"REF .",
"This is particularly the case for recycled MSPs with small values of the magnetic field.",
"Since the locations of these pulsars in the $P\\dot{P}$ –diagram are basically frozen on a Hubble time, they could in principle have been recycled just a few years ago – yet they have characteristic ages approaching 100 Gyr.",
"The only way to more accurately estimate the age of a recycled pulsar is by determining the cooling age of its white dwarf companion [34], [14]."
],
[
"Conclusions",
"In this summary, I only very briefly touched on the spins, masses and ages of recycled pulsars.",
"New exciting pulsar discoveries at an ever-increasing rate, e.g.",
"triple MSPs [24], [31], eccentric MSPs [8], [11], [9], and transitional MSPs [4], [19] keep driving this field forward with fruitful results and interesting lessons to be learnt on close binary evolution for the next many years to come.",
"And with an almost certain guarantee for more surprises.",
"I thank the Cosmic-Lab organizer, Francesco R. Ferraro, for the invitation to the MODEST 16 conference.",
"I also thank the International Space Science Institute (ISSI) in Bern, Switzerland, for funding and hosting an international team (ID 319, led by Alessandro Papitto) studying transitional millisecond pulsars."
]
] | 1606.05117 |
[
[
"Division by 2 on hyperelliptic curves and jacobians"
],
[
"Abstract Let $K$ be an algebraically closed field of characteristic different from 2, $g$ a positive integer, $f(x)$ a degree $(2g+1)$ polynomial with coefficients in $K$ and without multiple roots, $C: y^2=f(x)$ the corresponding genus $g$ hyperelliptic curve over $K$ and $J$ the jacobian of $C$.",
"We identify $C$ with the image of its canonical embedding into $J$ (the infinite point of $C$ goes to the zero point of $J$).",
"For each point $P=(a,b)\\in C(K)$ there are $2^{2g}$ points $\\frac{1}{2}P \\in J(K)$.",
"We describe explicitly the Mumford represesentations of all $\\frac{1}{2}P$.",
"The rationality questions for $\\frac{1}{2}P$ are also discussed."
],
[
"Introduction",
"Let $K$ be an algebraically closed field of characteristic different from 2.",
"If $n$ and $i$ are positive integers and $\\mathbf {r}=\\lbrace r_1, \\dots , r_n\\rbrace $ is a sequence of $n$ elements $r_i\\in K$ then we write $\\mathbf {s}_i(\\mathbf {r})=\\mathbf {s}_i(r_1, \\dots , r_n)\\in K$ for the $i$ th basic symmetric function in $r_1, \\dots , r_n$ .",
"If we put $r_{n+1}=0$ then $\\mathbf {s}_i(r_1, \\dots , r_n)=\\mathbf {s}_i(r_1, \\dots , r_n, r_{n+1})$ .",
"Let $g \\ge 1$ be an integer.",
"Let ${\\mathcal {C}}$ be the smooth projective model of the smooth affine plane $K$ -curve $y^2=f(x)=\\prod _{i=1}^{2g+1}(x-\\alpha _i)$ where $\\alpha _1,\\dots , \\alpha _{2g+1}$ are distinct elements of $K$ .",
"It is well known that ${\\mathcal {C}}$ is a genus $g$ hyperelliptic curve over $K$ with precisely one infinite point, which we denote by $\\infty $ .",
"In other words, ${\\mathcal {C}}(K)=\\lbrace (a,b)\\in K^2\\mid b^2=\\prod _{i=1}^{2g+1}(a-\\alpha _i)\\rbrace \\bigsqcup \\lbrace \\infty \\rbrace .$ Clearly, $x$ and $y$ are nonconstant rational functions on ${\\mathcal {C}}$ , whose only pole is $\\infty $ .",
"More precisely, the polar divisor of $x$ is $2 (\\infty )$ and the polar divisor of $y$ is $(2g+1)(\\infty )$ .",
"The zero divisor of $y$ is $\\sum _{i=1}^{2g+1} ({\\mathfrak {W}}_i)$ where ${\\mathfrak {W}}_i=(\\alpha _i,0) \\in {\\mathcal {C}}(K) \\ \\forall i=1, \\dots , 2g+1.$ We write $\\iota $ for the hyperelliptic involution $\\iota : {\\mathcal {C}}\\rightarrow {\\mathcal {C}}, (x,y)\\mapsto (x,-y), \\ \\infty \\mapsto \\infty .$ The set of fixed points of $\\iota $ consists of $\\infty $ and all ${\\mathfrak {W}}_i$ .",
"It is well known that for each $P \\in {\\mathcal {C}}(K)$ the divisor $(P)+\\iota (P)-2(\\infty )$ is principal.",
"More precisely, if $P=(a,b)\\in {\\mathcal {C}}(K)$ then $(P)+\\iota (P)-2(\\infty )$ is the divisor of the rational function $x-a$ on $C$ .",
"If $D$ is a divisor on ${\\mathcal {C}}$ then we write $\\mathrm {supp}(D)$ for its support, which is a finite subset of ${\\mathcal {C}}(K)$ .",
"We write $J$ for the jacobian of ${\\mathcal {C}}$ , which is a $g$ -dimensional abelian variety over $K$ .",
"If $D$ is a degree zero divisor on ${\\mathcal {C}}$ then we write $\\mathrm {cl}(D)$ for its linear equivalence class, which is viewed as an element of $J(K)$ .",
"We will identify ${\\mathcal {C}}$ with its image in $J$ with respect to the canonical regular map ${\\mathcal {C}}\\hookrightarrow J$ under which $\\infty $ goes to the zero of group law on $J$ .",
"In other words, a point $P \\in {\\mathcal {C}}(K)$ is identified with $\\mathrm {cl}((P)-(\\infty ))\\in J(K)$ .",
"Then the action of $\\iota $ on ${\\mathcal {C}}(K)\\subset J(K)$ coincides with multiplication by $-1$ on $J(K)$ .",
"In particular, the list of points of order 2 on ${\\mathcal {C}}$ consists of all ${\\mathfrak {W}}_i$ .",
"Recall [21] that if $D$ is an effective divisor of (nonnegative) degree $m$ , whose support does not contain $\\infty $ , then the degree zero divisor $D-m(\\infty )$ is called semi-reduced if it enjoys the following properties.",
"If ${\\mathfrak {W}}_i$ lies in $\\mathrm {supp}(D)$ then it appears in $D$ with multiplicity 1.",
"If a a point $Q$ of ${\\mathcal {C}}(K)$ lies in $\\mathrm {supp}(D)$ and does not coincide with any of ${\\mathfrak {W}}_i$ then $\\iota (P)$ does not lie in $\\mathrm {supp}(D)$ .",
"If, in addition, $m \\le g$ then $D-m(\\infty )$ is called reduced.",
"It is known ([9], [21]) that for each ${\\mathfrak {a}}\\in J(K)$ there exist exactly one nonnegative $m$ and (effective) degree $m$ divisor $D$ such that the degree zero divisor $D-m(\\infty )$ is reduced and $\\mathrm {cl}(D-m(\\infty ))={\\mathfrak {a}}$ .",
"(E.g.,, the zero divisor with $m=0$ corresponds to ${\\mathfrak {a}}=0$ .)",
"If $m\\ge 1, \\ D=\\sum _{j=1}^m(Q_j)\\ \\mathrm { where } \\ Q_j=(b_j,c_j) \\in {\\mathcal {C}}(K) \\ \\forall \\ j=1, \\dots , m$ (here $Q_j$ do not have to be distinct) then the corresponding ${\\mathfrak {a}}=\\mathrm {cl}(D-m(\\infty ))=\\sum _{j=1}^m Q_j \\in J(K).$ The Mumford's representation ([9], [21] of ${\\mathfrak {a}}\\in J(K)$ is is the pair $(U(x),V(x))$ of polynomials $U(x),V(x)\\in K[x]$ such that $U(x)=\\prod _{j=1}^r(x-a_j)$ is a degree $r$ monic polynomial while $V(x)$ has degree $m<\\deg (U)$ , the polynomial $V(x)^2-f(x)$ is divisible by $U(x)$ , and $D-m(\\infty )$ coincides with the gcd (i.e., with the minimum) of the divisors of rational functions $U(x)$ and $y-V(x)$ on ${\\mathcal {C}}$ .",
"This implies that each $Q_j$ is a zero of $y-V(x)$ , i.e., $b_j=V(a_j), \\ Q_j=(a_j,V(a_j))\\in {\\mathcal {C}}(K) \\ \\forall \\ j=1, \\dots m.$ Such a pair always exists, it is unique, and (as we've just seen) uniquely determines not only ${\\mathfrak {a}}$ but also divisors $D$ and $D-m(\\infty )$ .",
"(The case $\\alpha =0$ corresponds to $m=0, D=0$ and the pair $(U(x)=1, V(x)=0)$ .)",
"Conversely, if $U(x)$ is a monic polynomial of degree $m\\le g$ and $V(x)$ a polynomial such that $\\deg (V)<\\deg (U)$ and $V(x)^2-f(x)$ is divisible by $U(x)$ then there exists exactly one ${\\mathfrak {a}}=\\mathrm {cl}(D-m(\\infty ))$ where $D-m(\\infty )$ is a reduced divisor such that $(U(x),V(x)$ is the Mumford's representation of $\\mathrm {cl}(D-m(\\infty ))$ .",
"Let $P=(a,b)$ be a $K$ -point on ${\\mathcal {C}}$ , i.e., $a,b \\in K, \\ b^2=f(a)=\\prod _{i=1}^n(a-\\alpha _i).$ The aim of this note is to divide explicitly $P$ by 2 in $J(K)$ , i.e., to give explicit formulas for the Mumford's representation of all $2^{2g}$ divisor classes $\\mathrm {cl}(D-g(\\infty ))$ such that $2D+\\iota (P)$ is linearly equivalent to $(2g+1)\\infty $ , i.e., $2\\mathrm {cl}(D-g(\\infty ))=P \\in {\\mathcal {C}}(K)\\subset J(K).$ (It turns out that each such $D$ has degree $g$ and its support does not contain any of ${\\mathfrak {W}}_i$ .)",
"The paper is organized as follows.",
"In Section we obtain auxiliary results about divisors on hyperelliptic curves.",
"In particular, we prove (Theorem REF ) that if $g>1$ then the only point of ${\\mathcal {C}}(K)$ that is divisible by two in ${\\mathcal {C}}(K)$ (rather than in $J(K)$ ) is $\\infty $ (of course, if $g>1$ ).",
"We also prove that ${\\mathcal {C}}(K)$ does not contain points of order $n$ if $2<n\\le 2g$ .",
"In Section we describe explicitly for a given $P=(a,b)\\in {\\mathcal {C}}(K)$ the Mumford's representation of $2^{2g}$ divisor classes $\\mathrm {cl}(D-g(\\infty ))$ such that $D$ is an effective degree $g$ reduced divisor on ${\\mathcal {C}}$ and $2\\mathrm {cl}(D-g(\\infty ))=P \\in {\\mathcal {C}}(K)\\subset J(K).$ The description is given in terms of square roots $\\sqrt{a-\\alpha _i}$ 's ($1\\le i\\le 2g+1$ ), whose product is $-b$ .",
"(There are exactly $2^{2g}$ choices of such square roots.)",
"In Section we discuss the rationality questions, i.e., the case when $f(x),{\\mathcal {C}},J$ and $P$ are defined over a subfield $K_0$ of $K$ and ask when dividing $P$ by 2 we get a point of $J(K_0)$ .",
"Sections and deal with torsion points on certain naturally arised subvarieties of $J$ containg ${\\mathcal {C}}$ .",
"In particular, we discuss the case of a generic hyperelliptic curve in characteristic zero, using as a starting point results of B. Poonen - M. Stoll [11] and of J. Yelton [22].",
"Our approach is based on ideas of J.-P. Serre [17] and F. Bogomolov [4].",
"This paper is a follow up of [24], [3] where the (more elementary) case of elliptic curves is discussed.",
"Acknowledgements.",
"I am deeply grateful to Bjorn Poonen for helpful stimulationg discussions.",
"This work was partially supported by a grant from the Simons Foundation (#246625 to Yuri Zarkhin).",
"I've started to write this paper during my stay in May-June 2016 at the Max-Planck-Institut für Mathematik (Bonn, Germany), whose hospitality and support are gratefully acknowledged."
],
[
"Divisors on hyperelliptic curves",
"Lemma 2.1 (Key Lemma) Let $D$ be an effective divisor on ${\\mathcal {C}}$ of degree $m>0$ such that $m \\le 2g+1$ and $\\mathrm {supp}(D)$ does not contain $\\infty $ .",
"Assume that the divisor $D-m(\\infty )$ is principal.",
"Suppose that $m$ is odd.",
"Then: (i) $m=2g+1$ and there exists exactly one polynomial $v(x)\\in K[x]$ such that the divisor of $y-v(x)$ coincides with $D-(2g+1)(\\infty )$ .",
"In addition, $\\deg (v)\\le g$ .",
"(ii) If ${\\mathfrak {W}}_i$ lies in $\\mathrm {supp}(D)$ then it appears in $D$ with multiplicity 1.",
"(iii) If $b$ is a nonzero element of $K$ and a $K$ -point $P=(a,b) \\in {\\mathcal {C}}(K)$ lies in $\\mathrm {supp}(D)$ then $\\iota (P)=(a,-b)$ does not lie in $\\mathrm {supp}(D)$ .",
"Suppose that $m=2d$ is even.",
"Then there exists exactly one monic degree $d$ polynomial $u(x)\\in K[x]$ such that the divisor of $v(x)$ coincides with $D-m(\\infty )$ .",
"In particular, every point $Q \\in {\\mathcal {C}}(K)$ appears in $D-m(\\infty )$ with the same multiplicity as $\\iota (Q)$ .",
"Leh $h$ be a rational function on ${\\mathcal {C}}$ , whose divisor coincides with $D-m(\\infty )$ .",
"Since $\\infty $ is the only pole of $h$ , the function $h$ is a polynomial in $x,y$ and therefore may be presented as $h=s(x)y-v(x), \\ \\mathrm { with } \\ u,v \\in K[x].$ If $s=0$ then $h$ has at $\\infty $ the pole of even order $2\\deg (v)$ and therefore $m=2\\deg (v)$ .",
"Suppose that $s \\ne 0$ .",
"Clearly, $s(x)y$ has at $\\infty $ the pole of odd order $2\\deg (s)+(2g+1) \\ge (2g+1)$ .",
"So, the orders of the pole for $s(x)y$ and $v(x)$ are distinct, because they have different parity and therefore the order $m$ of the pole of $h=s(x)y-v(x)$ coincides with $\\max (2\\deg (s)+(2g+1), 2\\deg (v))\\ge 2g+1$ .",
"This implies that $m=2g+1$ ; in particular, $m$ is even.",
"It follows that $m$ is even if and only if $s(x)=0$ , i.e., $h=-v(x)$ ; in addition, $\\deg (v)\\le (2g+1)/2$ , i.e., $\\deg (v)\\le g$ .",
"In order to finish the proof of (2), it suffices to divide $-v(x)$ by its leading coefficient and denote the ratio by $u(x)$ .",
"(The uniqueness of monic $u(x)$ is obvious.)",
"Let us prove (1).",
"Since $m$ is odd, $m=2\\deg (s)+(2g+1)>2\\deg (v).$ Since $m \\le 2g+1$ , we obtain that $\\deg (s)=0$ , i.e., $s$ is a nonzero element of $K$ and $2\\deg (v)< 2g+1$ .",
"The latter inequality means that $\\deg (v)\\le g$ .",
"Dividing $h$ by the constant $s$ , we may and will assume that $s=1$ and therefore $h=y-v(x)$ with $v(x)\\in K[x], \\ \\deg (v) \\le g.$ This proves (i).",
"(The uniqueness of $v$ is obvious.)",
"The assertion (ii) is contained in Proposition 13.2(b) on pp.",
"409-10 of [21].",
"In order to prove (iii), we just follow arguments on p. 410 of [21] (where it is actually proven).",
"Notice that our $P=(a,b)$ is a zero of $y-v(x)$ , i.e.",
"$b-v(a)=0$ .",
"Since, $b\\ne 0$ , $v(a)=b \\ne 0$ and $y-v(x)$ takes on at $\\iota (P)=(a,-b)$ the value $-b-v(a)=-2b \\ne 0$ .",
"This implies that $\\iota (P)$ is not a zero of $y-v(x)$ , i.e., $\\iota (P)$ does not lie in $\\mathrm {supp}(D)$ .",
"Remark 2.2 Lemma REF (1)(ii,iii) asserts that if $m$ is odd the divisor $D-m(\\infty )$ is semi-reduced.",
"See [21].",
"Corollary 2.3 Let $P=(a,b)$ be a $K$ -point on ${\\mathcal {C}}$ and $D$ an effective divisor on ${\\mathcal {C}}$ such that $m=\\deg (D)\\le g$ and $\\mathrm {supp}(D)$ does not contain $\\infty $ .",
"Suppose that the degree zero divisor $2D+\\iota (P)-(2m+1)(\\infty )$ is principal.",
"Then: (i) $m=g$ and there exists a polynomial $v_D(x)\\in K[x]$ such that $\\deg (v)\\le g$ and the divisor of $y-v_D(x)$ coincides with $2D+\\iota (P)-(2g+1)(\\infty )$ .",
"In particular, $-b=v(a)$ .",
"(ii) If a point $Q$ lies in $\\mathrm {supp}(D)$ then $\\iota (Q)$ does not lie in $\\mathrm {supp}(D)$ .",
"In particular, none of ${\\mathfrak {W}}_i$ lies in $\\mathrm {supp}(D)$ ; $D-g(\\infty )$ is reduced.",
"(iii) The point $P$ does not lie in $\\mathrm {supp}(D)$ .",
"One has only to apply Lemma REF to the divisor $2D+\\iota (P)$ of odd degree $2m+1\\le 2g+1$ and notice that $\\iota (P)=(a,-b)$ is a zero of $y-v(x)$ while $\\iota ({\\mathfrak {W}}_i)={\\mathfrak {W}}_i$ for all $i=1, \\dots , 2g+1$ .",
"Let $d \\le g$ be a positive integer and $\\Theta _d \\subset J$ be the image of the regular map ${\\mathcal {C}}^{d} \\rightarrow J, \\ (Q_1, \\dots , Q_{d}) \\mapsto \\sum _{i=1}^{d} Q_i\\subset J.$ It is well known that $\\Theta _d$ is a closed $d$ -dimensional subvariety of $J$ that coincides with ${\\mathcal {C}}$ for $d=1$ and with $J$ if $d \\ge g$ ; in addition, $\\Theta _d\\subset \\Theta _{d+1}$ for all $d$ .",
"Clearly, each $\\Theta _d$ is stable under multiplication by $-1$ in $J$ .",
"We write $\\Theta $ for the $(g-1)$ -dimensional theta divisor $\\Theta _{g-1}$ .",
"Theorem 2.4 Suppose that $g>1$ and let ${\\mathcal {C}}_{1/2}:=2^{-1}{\\mathcal {C}}\\subset J$ be the preimage of ${\\mathcal {C}}$ with respect to multiplication by 2 in $J$ .",
"Then the intersection of ${\\mathcal {C}}_{1/2}(K)$ and $\\Theta $ consists of points of order dividing 2 on $J$ .",
"In particular, the intersection of ${\\mathcal {C}}$ and $C_{1/2}$ consists of $\\infty $ and all ${\\mathfrak {W}}_i$ 's.",
"Suppose that $m \\le g-1$ is a positive integer and we have $m$ (not necessarily distinct) points $Q_1, \\dots Q_m$ of ${\\mathcal {C}}(K)$ and a point $P\\in {\\mathcal {C}}(K)$ such that in $J(K)$ $2\\sum _{j=1}^m Q_j=P.$ We need to prove that $P=\\infty $ , i.e., it is the zero of group law in $J$ and therefore $\\sum _{j=1}^m Q_j$ is an element of order 2 (or 1) in $J(K)$ .",
"Suppose that this is not true.",
"Decreasing $m$ if necessary, we may and will assume that none of $Q_j$ is $\\infty $ (but $m$ is still positive and does not exceed $g-1$ ).",
"Let us consider the effective degree $m$ divisor $D=\\sum _{j=1}^m (Q_j)$ on ${\\mathcal {C}}$ .",
"The equality in $J$ means that the divisors $2[D-m(\\infty )]$ and $(P)-(\\infty )$ on ${\\mathcal {C}}$ are linearly equivalent.",
"This means that the divisor $2D+(\\iota (P))-(2m+1)(\\infty )$ is principal.",
"Now Corollary REF tells us that $m=g$ , which is not the case.",
"The obtained contradiction proves that the intersection of ${\\mathcal {C}}_{1/2}$ and $\\Theta $ consists of points of order 2 and 1.",
"Since $g>1$ , ${\\mathcal {C}}\\subset \\Theta $ and therefore the intersection of ${\\mathcal {C}}$ and ${\\mathcal {C}}_{1/2}$ also consists of points of order 2 or 1, i.e., lies in the union of $\\infty $ and all ${\\mathfrak {W}}_i$ 's.",
"Conversely, since each ${\\mathfrak {W}}_i$ has order 2 in $J(K)$ and $\\infty $ has order 1, they all lie in ${\\mathcal {C}}_{1/2}$ (and, of course, in ${\\mathcal {C}}$ ).",
"Remark 2.5 It is known [16] that the curve ${\\mathcal {C}}_{1/2}$ is irreducible.",
"(Its projectiveness and smoothness follow readily from the projectiveness and smoothness of ${\\mathcal {C}}$ and the étaleness of multiplication by 2 in $J$ .)",
"See [7] for an explicit description of equations that cut out ${\\mathcal {C}}_{1/2}$ in a projective space.",
"Corollary 2.6 Suppose that $g>1$ .",
"Let $n$ an integer such that $3 \\le n \\le 2g$ .",
"Then ${\\mathcal {C}}(K)$ does not contain a point of order $n$ in $J(K)$ .",
"In particular, ${\\mathcal {C}}(K)$ does not contain points of order 3 or 4.",
"Suppose that such a point say, $P$ exists.",
"Clearly, $P$ is neither $\\infty $ nor one of ${\\mathfrak {W}}_i$ , i.e., $P \\ne \\iota (P)$ .",
"Suppose that $n$ is odd.",
"Then we have $n=2m+1$ with $1\\le m<g$ .",
"This implies that $mP \\in \\Theta $ and $2(mP)=2mP=-P=\\iota (P) \\in {\\mathcal {C}}(K).$ It follows from Theorem REF that either $mP=0$ in $J(K)$ or $(2m)P=2(mP)=0$ in $J(K)$ .",
"However, the order of $P$ in $J(K)$ is $n=2m+1>m\\ge 1$ and we get a desired contradiction.",
"Assume now that $n$ is even.",
"Then we have $n=2m$ with $1<m\\le g$ .",
"Then $mP$ has order 2 in $J(K)$ .",
"It follows that $mP=-mP=m(-P)=m\\ \\iota (P).$ This means that the degree zero divisors $m(P)-m(\\infty )$ and $m(\\iota (P))-m(\\infty )$ belong to the same linear equivalence class.",
"Since both divisors are reduced, they must coincide (see [21]).",
"This implies that $P=\\iota (P)$ , which is not the case and we get a desired contradiction.",
"Remark 2.7 If $\\mathrm {char}(K)=0$ and $g>1$ then the famous theorem of M. Raynaud (conjectured by Yu.I.",
"Manin and D. Mumford) asserts that an arbitrary genus $g$ smooth projective curve over $K$ embedded into its jacobian contains only finitely many torsion points [12].",
"Using a $p$ -adic approach, B. Poonen [10] developed and implemented an algorithm that finds all complex torsion points on genus 2 hyperelliptic curves ${\\mathcal {C}}:y^2=f(x)$ such that $f(x)$ has rational coefficients.",
"(See also [11].)",
"Theorem 2.8 Suppose that $g>1$ and let $N>1$ be positive integer.",
"Suppose that $N \\le 2g-1$ and let us put $d(N)=\\left[\\frac{2g}{N+1}\\right].$ Let $K_0$ be a subfield of $K$ such that $f(x)\\in K_0[x]$ .",
"Let ${\\mathfrak {a}}$ be a $K$ -point on $\\Theta _{d(N)}$ .",
"Suppose that there is a field automorphism $\\sigma \\in \\mathrm {Aut}(K/K_0)$ such that $\\sigma ({\\mathfrak {a}})=N{\\mathfrak {a}}$ or $-N{\\mathfrak {a}}$ .",
"Then ${\\mathfrak {a}}$ has order 1 or 2 in $J(K)$ .",
"Clearly, $(N+1)\\cdot d(N)<2g+1$ .",
"Let us assume that $2{\\mathfrak {a}}\\ne 0$ in $J(K)$ .",
"We need to arrive to a contradiction.",
"Then there is a positive integer $r\\le d(N)$ and a sequence of points $P_1, \\dots , P_r$ of ${\\mathcal {C}}(K)\\setminus {\\infty }$ such that $\\tilde{D}:=\\sum _{j=1}^r(P_j)-r(\\infty )$ is the Mumford's representation of ${\\mathfrak {a}}$ while (say) $P_1$ does not coincides with any of $W_i$ (here we use the assumption that $2{\\mathfrak {a}}\\ne 0$ ); we may also assume that $P_1$ has the largest multiplicity say, $M$ among $\\lbrace P_1, \\dots , P_r\\rbrace $ .",
"(In particular, none of $P_j$ 's coincides with $\\iota (P_1)$ .)",
"Then $\\sigma (\\tilde{D})=\\sum _{j=1}^r(\\sigma (P_j))-r(\\infty )$ is the Mumford's representation of $\\sigma {\\mathfrak {a}}$ .",
"In particular, the multiplicity of each $\\sigma (P_j)$ in $\\sigma (\\tilde{D})$ does not exceed $M$ ; similarly, the multiplicity of each $\\iota \\sigma (P_j)$ in $\\iota \\sigma (\\tilde{D})$ does not exceed $M$ .",
"Suppose that $\\sigma ({\\mathfrak {a}})=N{\\mathfrak {a}}$ .",
"$N\\tilde{D}+\\iota \\sigma (\\tilde{D}) =N\\left[\\sum _{j=1}^r(P_j)\\right]+\\left[\\sum _{j=1}^r(\\iota \\sigma (P_j))\\right]-r(N+1)(\\infty )$ is a principal divisor on ${\\mathcal {C}}$ .",
"Since $m:=r(N+1) \\le (N+1)\\cdot d(N)<2g+1$ , we are in position to apply Lemma REF , which tells us right away that $m$ is even and there is a monic polynomial $u(x)$ of degree $m/2$ , whose divisor coincides with $N\\tilde{D}+\\iota \\sigma (\\tilde{D})$ .",
"This implies that a point $Q\\in {\\mathcal {C}}(K)$ appears in $N\\tilde{D}+\\iota \\sigma (\\tilde{D})$ with the same multiplicity as $\\iota {Q}$ .",
"It follows that $\\iota {P_1}$ is (at least) one of $\\iota \\sigma (P_j)$ 's.",
"Clearly, the multiplicity of $P_1$ in $N\\tilde{D}+\\iota \\sigma (\\tilde{D})$ is, at least, $NM$ while the multiplicity of $\\iota (P_1)$ is, at most, $M$ .",
"This implies that $NM\\le M$ .",
"Taking into account that $N>1$ , we obtain the desired contradiction.",
"If $\\sigma ({\\mathfrak {a}})=-N{\\mathfrak {a}}$ then literally the same arguments applied to to the principal divisor $N\\tilde{D}+\\sigma (\\tilde{D}) =N\\left[\\sum _{j=1}^r(P_j)\\right]+\\left[\\sum _{j=1}^r(\\sigma (P_j))\\right]-r(N+1)(\\infty )$ also lead to the contradiction."
],
[
"Division by 2",
"Suppose we are given a point $P=(a,b) \\in {\\mathcal {C}}(K) \\subset J(K).$ Since $\\mathrm {dim}(J)=g$ , there are exactly $2^{2g}$ points ${\\mathfrak {a}}\\in J(K)$ such that $P=2{\\mathfrak {a}}\\in J(K).$ Let us choose such an ${\\mathfrak {a}}$ .",
"Then there is exactly one effective divisor $D=D({\\mathfrak {a}})\\qquad \\mathrm {(1)}$ of positive degree $m$ on ${\\mathcal {C}}$ such that $\\mathrm {supp}(D)$ does not contain $\\infty $ , the divisor $D-m(\\infty )$ is reduced, and $m \\le g, \\ \\mathrm {cl}(D-m (\\infty ))={\\mathfrak {a}}.$ It follows that the divisor $2D+(\\iota (P))-(2m+1)(\\infty )$ is principal and, thanks to Corollary REF , $m=g$ and $\\mathrm {supp}(D)$ does not contains any of ${\\mathfrak {W}}_i$ .",
"(In addition, $D-g(\\infty )$ is reduced.)",
"Then the degree $g$ effective divisor $D=D({\\mathfrak {a}})=\\sum _{j=1}^{g}(Q_j)\\qquad \\mathrm {(2)}$ with $Q_i=(c_j,d_j)\\in {\\mathcal {C}}(K)$ .",
"Since none of $Q_j$ coincides with any of ${\\mathfrak {W}}_i$ , $c_j \\ne \\alpha _i \\ \\forall i,j.$ By Corollary REF , there is a polynomial $v_D(x)$ of degree $ \\le g$ such that the degree zero divisor $2D+(\\iota (P))-(2g+1)(\\infty )$ is the divisor of $y-v_D(x)$ .",
"Since the points $\\iota (P)=(a,-b)$ and all $Q_j$ 's are zeros of $y-v_D(x)$ , $b=-v_D(a), \\ d_j=v_D(c_j) \\ \\forall j=1, \\dots , g.$ It follows from Proposition 13.2 on pp.",
"409–410 of [21] that $\\prod _{i=1}^{2g+1}(x-\\alpha _i)-v_D(x)^2=f(x)-v_D(x)^2=(x-a)\\prod _{j=1}^g (x-c_j)^2.\\qquad \\mathrm {(3)}$ In particular, $f(x)-v_D(x)^2$ is divisible by $u_D(x):=\\prod _{j=1}^g (x-c_j).\\qquad \\mathrm {(4)}$ Remark 3.1 Summing up: $D=D({\\mathfrak {a}})=\\sum _{j=1}^{g}(Q_j), \\ Q_j=(c_j,v_D(c_j)) \\ \\forall j=1, \\dots , g$ and the dgree $g$ monic polynomial $u_D(x)=\\prod _{j=1}^g (x-c_j)$ divides $f(x)-v_D(x)^2$ .",
"By Prop.",
"13.4 on p. 412 of [21], this implies that reduced $D-g(\\infty )$ coincides with the gcd of the divisors of $u_D(x)$ and $y-v_D(x)$ .",
"Therefore the pair $(u_D,v_D)$ is the Mumford's representation of ${\\mathfrak {a}}$ if $\\deg (v_D)<g=\\deg (u_D).$ This is not always the case: it may happen that $\\deg (v_D)=g=\\deg (u_D)$ (see below).",
"However, if we replace $v_D(x)$ by its remainder with respect to the division by $u_D(x)$ then we get the Mumford's representation of ${\\mathfrak {a}}$ (see below).",
"If in (3) we put $x=\\alpha _i$ then we get $-v_D(\\alpha _i)^2=(\\alpha _i-a)\\left(\\prod _{j=1}^g(\\alpha _i-c_j)\\right)^2,$ i.e., $v_D(\\alpha _i)^2=(a-\\alpha _i)\\left(\\prod _{j=1}^g(c_j-\\alpha _i)\\right)^2 \\ \\forall \\ i=1, \\dots , 2g+1.$ Since none of $c_j-\\alpha _i$ vanishes, we may define $r_i=r_{i,D}:=\\frac{v_D(\\alpha _i)}{\\prod _{j=1}^g(c_j-\\alpha _i)} \\qquad \\mathrm {(5)}$ with $r_i^2=a-\\alpha _i \\ \\forall \\ i=1, \\dots , 2g+1 \\qquad \\mathrm {(6)}$ and $\\alpha _i= a-r_i^2, \\ c_j-\\alpha _i=r_i^2-a+c_j \\ \\forall \\ i=1, \\dots , 2g+1; j=1, \\dots , g.$ Clearly, all $r_i$ 's are distinct elements of $K$ , because their squares are obviously distinct.",
"(By the same token, $r_{j_1} \\ne \\pm r_{j_2}$ if $j_1\\ne j_2$ .",
"Notice that $\\prod _{i=1}^{2g+1}r_i= \\pm b, \\qquad \\mathrm {(7)}$ because $b^2=\\prod _{i=1}^{2g+1}(a-\\alpha _i)=\\prod _{i=1}^{2g+1} r_i^2.",
")\\qquad \\mathrm {(8)}$ Now we get $r_i=\\frac{v_D(a-r_i^2)}{\\prod _{j=1}^g(r_i^2-a+c_j)},$ i.e., $r_i \\prod _{j=1}^g(r_i^2-a+c_j)-v_D(a-r_i^2)=0 \\ \\forall \\ i=1, \\dots 2g+1.$ This means that the degree $(2g+1)$ monic polynomial (recall that $\\deg (v_D)\\le g$ ) $h_{\\mathbf {r}}(t):=t \\prod _{j=1}^g (t^2-a+c_j) -v(a-t^2)$ has $(2g+1)$ distinct roots $r_1, \\dots , r_{2g+1}$ .",
"This means that $h_{\\mathbf {r}}(t)= \\prod _{i=1}^{2g+1}(t-r_i).$ Clearly, $t \\prod _{j=1}^g (t^2-a+c_i)$ coincides with the odd part of $h_{\\mathbf {r}}(t)$ while $-v_D(a-t^2)$ coincides with the even part of $h_{\\mathbf {r}}(t)$ .",
"In particular, if we put $t=0$ then we get $ (-1)^{2g+1}\\prod _{i=1}^{2g+1}r_i=-v_D(a)=b,$ i.e., $\\prod _{i=1}^{2g+1}r_i=- b.\\qquad \\mathrm {(9)}$ Let us define $\\mathbf {r}=\\mathbf {r}_D:=(r_1, \\dots , r_{2g+1}) \\in K^{2g+1}.$ Since $\\mathbf {s}_i(\\mathbf {r})=\\mathbf {s}_i(r_1, \\dots , r_{2g+1})$ is the $i$ th basic symmetric function in $r_1, \\dots , r_{2g+1}$ , $h_{\\mathbf {r}}(t)=t^{2g+1}+\\sum _{i=1}^{2g+1} (-1)^{i}\\mathbf {s}_i(\\mathbf {r}) t^{2g+1-i}=[t^{2g+1}+\\sum _{i=1}^{2g} (-1)^{i}s_i(\\mathbf {r}) t^{2g+1-i}] +b.$ Then $t\\prod _{j=1}^g (t^2-a+c_j)=t^{2g+1}+\\sum _{j=1}^g \\mathbf {s}_{2j}(\\mathbf {r})t^{2g+1-2j},$ $-v_D(a-t^2)=[-\\sum _{j=1}^{g} \\mathbf {s}_{2j-1}(\\mathbf {r}) t^{2g-2j+2}]+b.$ It follows that $\\prod _{j=1}^g (t-a+c_j)=t^g+\\sum _{j=1}^g \\mathbf {s}_{2j-1}(\\mathbf {r})t^{g-j},$ $v_D(a-t)=\\sum _{j=1}^{g} \\mathbf {s}_{2j-1}(\\mathbf {r}) t^{g-j+1}-b.$ This implies that $v_D(t)=\\left[\\sum _{j=1}^{g} \\mathbf {s}_{2j-1}(\\mathbf {r}) (a-t)^{g-j+1}\\right]-b.\\qquad \\mathrm {(10)}$ It is also clear that if we consider the degree $g$ monic polynomial $U_{\\mathbf {r}}(t):=u_D(t)=\\prod _{j=1}^g (t-c_j)$ then $U_{\\mathbf {r}}(t)=(-1)^g \\left[(a-t)^g+\\sum _{j=1}^g \\mathbf {s}_{2j}(\\mathbf {r})(a-t)^{g-j}\\right].",
"\\qquad \\mathrm {(11)}$ Recall that $\\deg (v_D) \\le g$ and notice that the coefficient of $v(x)$ at $x^g$ is $(-1)^{g}\\mathbf {s}_1(\\mathbf {r})$ .",
"This implies that the polynomial $V_{\\mathbf {r}}(t):=v_D(t)-(-1)^{g}\\mathbf {s}_1(\\mathbf {r}) U_{\\mathbf {r}}(t)=$ $\\left[\\sum _{j=1}^{g} \\mathbf {s}_{2j-1}(\\mathbf {r}) (a-t)^{g-j+1}\\right]-b-\\mathbf {s}_1(\\mathbf {r})\\left[(a-t)^g+\\sum _{j=1}^g \\mathbf {s}_{2j}(\\mathbf {r})(a-t)^{g-j}\\right]\\qquad \\mathrm {(12)}$ has degree $<g$ , i.e., $\\deg (V_{\\mathbf {r}})<\\deg (U_{\\mathbf {r}})=g.$ Clearly, $f(x) - V_{\\mathbf {r}}(x)^2$ is still divisible by $U_{\\mathbf {r}}(x)$ , because $u_D(x)=U_{\\mathbf {r}}(x)$ divides both $f(x)-v_D(x)^2$ and $v_D(x)- V_{\\mathbf {r}}(x)$ .",
"On the other hand, $d_j=v_D(c_j)=V_{\\mathbf {r}}(c_j) \\ \\forall j=1, \\dots g,$ because $U_{\\mathbf {r}}(x)$ divides $v_D(x)- V_{\\mathbf {r}}(x)$ and vanishes at all $b_j$ .",
"Actually, $\\lbrace b_1, \\dots , b_g\\rbrace $ is the list of all roots (with multiplicities) of $U_{\\mathbf {r}}(x)$ .",
"So, $D=D({\\mathfrak {a}})=\\sum _{j=1}^{g}(Q_j), \\ Q_j=(c_j,v_D(c_j))=(c_j,V_{\\mathbf {r}}(c_j)) \\ \\forall j=1, \\dots , g.$ This implies (again via Prop.",
"13.4 on p. 412 of [21]) that reduced $D-g (\\infty )$ coincides with the gcd of the divisors of $U_{\\mathbf {r}}(x)$ and $y-V_{\\mathbf {r}}(x)$ .",
"It follows that the pair $(U_{\\mathbf {r}}(x), V_{\\mathbf {r}}(x))$ is the Mumford's representation of $\\mathrm {cl}(D-g (\\infty ))={\\mathfrak {a}}$ .",
"So, the formulas (11) and (12) give us an explicit construction of ($D({\\mathfrak {a}})$ and) ${\\mathfrak {a}}$ in terms of $\\mathbf {r}=(r_1, \\dots , , r_{2g+1})$ for each of $2^{2g}$ choices of ${\\mathfrak {a}}$ with $2{\\mathfrak {a}}=P\\in J(K)$ .",
"On the other hand, in light of (6)-(8), there is exactly the same number $2^{2g}$ of choices of square roots $\\sqrt{a-\\alpha _i}$ ($1\\le i \\le 2g$ ), whose product is $-b$ .",
"Combining it with (9), we obtain that for each choice of square roots $\\sqrt{a-\\alpha _i}$ 's with $\\prod _{i=1}^{2g+1}\\sqrt{a-\\alpha _i}=-b$ there is precisely one ${\\mathfrak {a}}\\in J(K)$ with $2{\\mathfrak {a}}=P$ such that the corresponding $r_i$ defined by (5) coincides with chosen $\\sqrt{a-\\alpha _i}$ for all $i=1, \\dots , 2g+1$ , and the Mumford's representation $(U_{\\mathbf {r}}(x), V_{\\mathbf {r}}(x))$ for this ${\\mathfrak {a}}$ is given by explicit formulas (11)-(12).",
"This gives us the following assertion.",
"Theorem 3.2 Let $P=(a,b)\\in {\\mathcal {C}}(K)$ .",
"Then the $2^{2g}$ -element set $M_{1/2,P}:=\\lbrace {\\mathfrak {a}}\\in J(K)\\mid 2{\\mathfrak {a}}=P\\in {\\mathcal {C}}(K)\\subset J(K)\\rbrace $ can be described as follows.",
"Let ${\\mathfrak {R}}_{1/2,P}$ be the set of all $(2g+1)$ -tuples ${\\mathfrak {r}}=({\\mathfrak {r}}_1, \\dots , {\\mathfrak {r}}_{2g+1})$ of elements of $K$ such that ${\\mathfrak {r}}_i^2=a-\\alpha _i \\ \\forall \\ i=1, \\dots , 2g+1; \\ \\prod _{i=1}^{2g+1}{\\mathfrak {r}}_i=-b.$ Let $\\mathbf {s}_i({\\mathfrak {r}})$ be the $i$ th basic symmetric function in ${\\mathfrak {r}}_1, \\dots , {\\mathfrak {r}}_{2g+1}$ .",
"Let us put $U_{{\\mathfrak {r}}}(x)=(-1)^g \\left[(a-x)^g+\\sum _{j=1}^g \\mathbf {s}_{2j}({\\mathfrak {r}})(a-x)^{g-j}\\right],$ $V_{{\\mathfrak {r}}}(x)=\\left[\\sum _{j=1}^{g} \\mathbf {s}_{2j-1}({\\mathfrak {r}}) (a-x)^{g-j+1}\\right]-b-\\mathbf {s}_1({\\mathfrak {r}})\\left[(a-x)^g+\\sum _{j=1}^g \\mathbf {s}_{2j}({\\mathfrak {r}})(a-x)^{g-j}\\right].$ Then there is a natural bijection between ${\\mathfrak {R}}_{1/2,P}$ and $M_{1/2,P}$ such that ${\\mathfrak {r}}\\in {\\mathfrak {R}}_{1/2,P}$ corresponds to ${\\mathfrak {a}}_{{\\mathfrak {r}}}\\in M_{1/2,P}$ with Mumford's representation $(U_{{\\mathfrak {r}}},V_{{\\mathfrak {r}}})$ .",
"More explicitly, if $\\lbrace c_1, \\dots , c_g\\rbrace $ is the list of $g$ roots (with multiplicities) of $U_{{\\mathfrak {r}}}(x)$ then ${\\mathfrak {r}}$ corresponds to ${\\mathfrak {a}}_{{\\mathfrak {r}}}=\\mathrm {cl}(D-g(\\infty ))\\in J(K), \\ 2{\\mathfrak {a}}_{{\\mathfrak {r}}}=P$ where the divisor $D=D({\\mathfrak {a}}_{{\\mathfrak {r}}})=\\sum _{j=1}^g (Q_j), \\ Q_j=(b_j,V_{{\\mathfrak {r}}}(b_j))\\in {\\mathcal {C}}(K) \\ \\forall \\ j=1, \\dots , g.$ In addition, none of $\\alpha _i$ is a root of $U_{{\\mathfrak {r}}}(x)$ (i.e., the polynomials $U_{{\\mathfrak {r}}}(x)$ and $f(x)$ are relatively prime) and ${\\mathfrak {r}}_i=\\mathbf {s}_1({\\mathfrak {r}})+(-1)^g\\frac{V_{{\\mathfrak {r}}}(\\alpha _i)}{U_{{\\mathfrak {r}}}(\\alpha _i)}\\ \\forall \\ i=1, \\dots , 2g+1.$ Actually we have already proven all the assertions of Theorem REF except the last formula for ${\\mathfrak {r}}_i$ .",
"It follows from (4) and (5) that ${\\mathfrak {r}}_i=(-1)^g \\frac{v_{D({\\mathfrak {a}}_{{\\mathfrak {r}}})}(\\alpha _i)}{u_{D({\\mathfrak {a}}_{{\\mathfrak {r}}})}(\\alpha _i)}=(-1)^g \\frac{v_{D({\\mathfrak {a}}_{{\\mathfrak {r}}})}(\\alpha _i)}{U_{{\\mathfrak {r}}}(\\alpha _i)}.$ It follows from (12) that $v_{D({\\mathfrak {a}}_{{\\mathfrak {r}}})}(x)=(-1)^{g}\\mathbf {s}_1({\\mathfrak {r}}) U_{{\\mathfrak {r}}}(x)+V_{{\\mathfrak {r}}}(x).$ This implies that ${\\mathfrak {r}}_i=(-1)^g\\frac{(-1)^{g}\\mathbf {s}_1({\\mathfrak {r}}) U_{{\\mathfrak {r}}}(\\alpha _i)+V_{{\\mathfrak {r}}}(\\alpha _i)}{U_{{\\mathfrak {r}}}(\\alpha _i)}=\\mathbf {s}_1({\\mathfrak {r}})+(-1)^g\\frac{V_{{\\mathfrak {r}}}(\\alpha _i)}{U_{{\\mathfrak {r}}}(\\alpha _i)}.$ Example 3.3 Let us take as $P=(a,b)$ the point ${\\mathfrak {W}}_{2g+1}=(\\alpha _{2g+1},0)$ .",
"Then $b=0$ and ${\\mathfrak {r}}_{2g+1}=0$ .",
"We have $2g$ arbitrary independent choices of (nonzero) square roots ${\\mathfrak {r}}_j=\\sqrt{\\alpha _{2g+1}-\\alpha _j}$ with $1 \\le j \\le 2g$ (and always get an element of ${\\mathfrak {R}}_{1/2,P}$ ).",
"Now Theorem REF gives us (if we put $a=\\alpha _{2j+1},b=0$ ) all $2^{2g}$ points ${\\mathfrak {a}}_{{\\mathfrak {r}}}$ of order 4 in $J(K)$ with $2{\\mathfrak {a}}_{{\\mathfrak {r}}}={\\mathfrak {W}}_{2j+1}$ ."
],
[
"Rationality Questions",
"Let $K_0$ be a subfield of $K$ and $K_0^{\\mathrm {sep}}$ its separable algebraic closure in $K$ .",
"Recall that $K_0^{\\mathrm {sep}}$ is separably closed.",
"Clearly, $\\mathrm {char}(K_0)=\\mathrm {char}(K_0^{\\mathrm {sep}})=\\mathrm {char}(K) \\ne 2.$ Let us assume that $f(x)\\in K_0[x]$ , i.e., all the coefficients of $f(x)$ lie in $K_0$ .",
"However, we don't make any additional assumptions about its roots $\\alpha _j$ ; still, all of them lie in $K_0^{\\mathrm {sep}}$ , because $f(x)$ has no multiple roots.",
"Recall that both ${\\mathcal {C}}$ and $J$ are defined over $K_0$ ; the point $\\infty \\in {\\mathcal {C}}(K_0)$ and therefore the embedding ${\\mathcal {C}}\\hookrightarrow J$ is defined over $K_0$ ; in particular, ${\\mathcal {C}}$ is a closed algebraic $K_0$ -subvariety of $J$ .",
"Let us assume that our $K$ -point $P=(a,b)$ of ${\\mathcal {C}}$ lies in ${\\mathcal {C}}(K_0^{\\mathrm {sep}})$ , i.e., $a,b\\in K_0^{\\mathrm {sep}}$ and $P=(a,b)\\in {\\mathcal {C}}(K_0^{\\mathrm {sep}})\\subset J(K_0^{\\mathrm {sep}})\\subset J(K).$ In the notation of Theorem REF , for each ${\\mathfrak {r}}\\in M_{1/2,P}$ all its components ${\\mathfrak {r}}_i$ lie in $K_0^{\\mathrm {sep}}$ , because ${\\mathfrak {r}}_i^2=a-\\alpha _i \\in K_0^{\\mathrm {sep}}$ .",
"This implies that the monic degree $2g+1$ polynomial $h_{{\\mathfrak {r}}}(t)=\\prod _{i=1}^{2g+1}(t-{\\mathfrak {r}}_i)=t^{2g+1}+\\sum _{i=1}^{2g}(-1)^i \\mathbf {s}_i({\\mathfrak {r}})t^{2g+1-i} \\in K_0^{\\mathrm {sep}}[t],$ i.e., all $\\mathbf {s}_i({\\mathfrak {r}}) \\in K_0^{\\mathrm {sep}}$ .",
"It follows immediately from the explicit formulas above that the Mumford representation $(U_{{\\mathfrak {r}}},V_{{\\mathfrak {r}}})$ of ${\\mathfrak {a}}_r=\\mathrm {cl}(D({\\mathfrak {a}}_r)-g(\\infty ))$ consists of polynomials $U_{{\\mathfrak {r}}}$ and $V_{{\\mathfrak {r}}}$ with coefficients in $K_0^{\\mathrm {sep}}$ .",
"In addition, ${\\mathfrak {a}}_{{\\mathfrak {r}}}$ lies in $J(K_0^{\\mathrm {sep}})$ , because $2{\\mathfrak {a}}_{{\\mathfrak {r}}}=P \\in J(K_0^{\\mathrm {sep}})$ , the multiplication by 2 in $J$ is an étale map and $K_0^{\\mathrm {sep}}$ is separably closed.",
"Lemma 4.1 Suppose that either $K_0$ is a perfect field (e.g., $\\mathrm {char}(K)=0$ or $K_0$ is finite) or $\\mathrm {char}(K_0)>g$ .",
"Suppose that $P=(a,b) \\in {\\mathcal {C}}(K_0^{\\mathrm {sep}})\\subset J(K_0^{\\mathrm {sep}}).$ Then for all ${\\mathfrak {r}}\\in {\\mathfrak {R}}_{1/2,P}$ the Mumford representation $(U_{{\\mathfrak {r}}},V_{{\\mathfrak {r}}})$ of ${\\mathfrak {a}}_r=\\mathrm {cl}(D({\\mathfrak {a}}_r)-g(\\infty ))$ enjoys the following properties.",
"(i) The polynomial $U_{{\\mathfrak {r}}}(x)$ splits over $K_0^{\\mathrm {sep}}$ , i.e., all its roots $b_j$ lie in $K_0^{\\mathrm {sep}}$ .",
"(ii) The divisor $D=D({\\mathfrak {a}}_{{\\mathfrak {r}}})=\\sum _{j=1}(Q_j)$ where $Q_j=(c_j,V_{{\\mathfrak {r}}}(c_j))\\in {\\mathcal {C}}(K_0^{\\mathrm {sep}}) \\ \\forall \\ j=1, \\dots , g.$ If $K_0$ is perfect then $K_0^{\\mathrm {sep}}$ is algebraically closed and there is nothing to prove.",
"So, we may assume that $\\mathrm {char}(K_0^{\\mathrm {sep}}) =\\mathrm {char}(K_0)>g$ .",
"In order to prove (i), recall that $\\deg (U_{{\\mathfrak {r}}})=g$ .",
"Every root $c_j$ of $U_{{\\mathfrak {r}}}(x)$ lies in $K$ and the algebraic field extension $K_0^{\\mathrm {sep}}(b_j)/K_0^{\\mathrm {sep}}$ has finite degree that does not exceed $\\deg (U_{{\\mathfrak {r}}})=g<\\mathrm {char}(K_0^{\\mathrm {sep}})$ and therefore this degree is not divisible by $\\mathrm {char}(K_0^{\\mathrm {sep}})$ .",
"This implies that the field extension $K_0^{\\mathrm {sep}}(b_j)/K_0^{\\mathrm {sep}}$ is separable.",
"Since $K_0^{\\mathrm {sep}}$ is separably closed, the overfield $K_0^{\\mathrm {sep}}(c_j)=K_0^{\\mathrm {sep}}$ , i.e., $c_j$ lies in $K_0^{\\mathrm {sep}}$ .",
"This proves (i).",
"As for (ii), since $V_{{\\mathfrak {r}}}(x)\\in K_0^{\\mathrm {sep}}[x]$ and all $c_j\\in K_0^{\\mathrm {sep}}$ , we have $V_{{\\mathfrak {r}}}(c_j)\\in K_0^{\\mathrm {sep}}$ and therefore $Q_j=(c_j,V_{{\\mathfrak {r}}}(c_j))\\in {\\mathcal {C}}(K_0^{\\mathrm {sep}})$ .",
"This proves (ii).",
"Remark 4.2 If $g=2$ then the conditions of Lemma REF do not impose any additional restrictions on $K_0$ .",
"(The case $\\mathrm {char}(K)=2$ was excluded from the very beginning.)",
"Remark 4.3 If $P=(a,b)\\in {\\mathcal {C}}(K_0)$ then for each ${\\mathfrak {r}}\\in {\\mathfrak {R}}_{1/2,P}$ $\\mathbf {s}_{2g+1}({\\mathfrak {r}})=(-1)^{2g+1}\\prod _{i=1}^{2g+1}{\\mathfrak {r}}_i=-\\prod _{i=1}^{2g+1}{\\mathfrak {r}}_i=-(-b)=b\\in K_0.$ This observation (reminder) explains the omission of $i=2g+1$ in the following statement.",
"Theorem 4.4 Suppose that a point $P =(a,b) \\in {\\mathcal {C}}(K_0)\\subset J(K_0),$ i.e., $a,b \\in K_0, \\ b^2=f(a).$ If ${\\mathfrak {r}}$ is an element of ${\\mathfrak {R}}_{1/2,P}$ then ${\\mathfrak {a}}_{{\\mathfrak {r}}}$ lies in $J(K_0)$ if and only if $h_{{\\mathfrak {r}}}(t)$ lies in $K_0[t]$ , i.e., $\\mathbf {s}_i({\\mathfrak {r}}) \\in K_0 \\ \\forall \\ i=1, \\dots , 2g.$ Let $\\bar{K}_0$ be the algebraic closure of $K_0$ .",
"Clearly, $\\bar{K}_0$ is algebraically closed and $K_0\\subset K_0^{\\mathrm {sep}}\\subset \\bar{K}_0\\subset K.$ In the course of the proof we may and will assume that $K=\\bar{K}_0$ .",
"Let ${\\mathfrak {r}}$ be an element of${\\mathfrak {R}}_{1/2,P}$ .",
"We know that ${\\mathfrak {a}}_{{\\mathfrak {r}}} \\in J(K_0^{\\mathrm {sep}})$ and the corresponding polynomials $U_{{\\mathfrak {r}}}(x)$ and $V_{{\\mathfrak {r}}}(x)$ have coefficients in $K_0^{\\mathrm {sep}}$ .",
"This means that there is a finite Galois field extension $E/K_0$ with Galois group $\\mathrm {Gal}(E/K)$ such that $K_0\\subset E \\subset K_0^{\\mathrm {sep}}$ such that ${\\mathfrak {a}}_{rr}\\in J(E); \\ U_{{\\mathfrak {r}}}(x), V_{{\\mathfrak {r}}}(x) \\in E[x].$ Let $\\mathrm {Aut}(K/K_0)$ be the group of all field automorphisms of $K$ that leave invariant every element of $K_0$ .",
"Clearly, the (sub)field $E$ is $\\mathrm {Aut}(K/K_0)$ -stable and the natural (restriction) group homomorphism $\\mathrm {Aut}(K/K_0) \\rightarrow \\mathrm {Gal}(E/K_0)$ is surjective.",
"Since the subfield $E^{\\mathrm {Gal}(E/K_0)}$ of Galois invariants coincides with $K_0$ , we conclude that the subfield of invariants $E^{\\mathrm {Aut}(K/K_0)}$ also coincides with $K_0$ .",
"It follows that $E[x]^{\\mathrm {Aut}(K/K_0)}=K_0[x], \\ J(E)^{\\mathrm {Aut}(K/K_0)}=J(K_0).$ Clearly, for each $\\sigma \\in \\mathrm {Aut}(K/K_0)$ the Mumford representation of $\\sigma {\\mathfrak {a}}_{{\\mathfrak {r}}}$ is $(\\sigma U_{{\\mathfrak {r}}}, \\sigma V_{{\\mathfrak {r}}})$ .",
"Now let us assume that ${\\mathfrak {a}}_{{\\mathfrak {r}}}\\in J(K_0)$ .",
"Then $\\sigma {\\mathfrak {a}}_{{\\mathfrak {r}}}={\\mathfrak {a}}_{{\\mathfrak {r}}} \\ \\forall \\ \\sigma \\in \\mathrm {Aut}(K/K_0).$ The uniqueness of Mumford's representations implies that $\\sigma U_{{\\mathfrak {r}}}(x)=U_{{\\mathfrak {r}}}(x), \\ \\sigma V_{{\\mathfrak {r}}}(x)=V_{{\\mathfrak {r}}}(x) \\ \\forall \\ \\sigma \\in \\mathrm {Aut}(K/K_0).$ It follows that $U_{{\\mathfrak {r}}}(x), V_{{\\mathfrak {r}}}(x) \\in K_0[x].$ Taking into account that $a,b \\in K_0$ , we obtain from the formulas in Theorem REF that $\\mathbf {s}_i({\\mathfrak {r}}) \\in K_0 \\ \\forall \\ i=1, \\dots , 2g.$ Conversely, let us assume that for a certain ${\\mathfrak {r}}\\in {\\mathfrak {R}}_{1/2,P}$ $\\mathbf {s}_i({\\mathfrak {r}}) \\in K_0 \\ \\forall \\ i=1, \\dots , 2g.$ (We know that $\\mathbf {s}_{2g+1}({\\mathfrak {r}})$ also lies in $K_0$ .)",
"This implies that both $U_{{\\mathfrak {r}}}(x)$ and $V_{{\\mathfrak {r}}}(x)$ lie in $K_0[x]$ .",
"In other words, $\\sigma U_{{\\mathfrak {r}}}(x)=U_{{\\mathfrak {r}}}(x), \\ \\sigma V_{{\\mathfrak {r}}}(x)=V_{{\\mathfrak {r}}}(x) \\ \\forall \\ \\sigma \\in \\mathrm {Aut}(K/K_0).$ This means that for every $\\sigma \\in \\mathrm {Aut}(K/K_0)$ both ${\\mathfrak {a}}_{{\\mathfrak {r}}}$ and $\\sigma {\\mathfrak {a}}_{{\\mathfrak {r}}}$ have the same Mumford representation, namely, $(U_{{\\mathfrak {r}}}, V_{{\\mathfrak {r}}})$ .",
"This implies that $\\sigma {\\mathfrak {a}}_{{\\mathfrak {r}}}={\\mathfrak {a}}_{{\\mathfrak {r}}} \\ \\forall \\ \\sigma \\in \\mathrm {Aut}(K/K_0),$ i.e., ${\\mathfrak {a}}_{{\\mathfrak {r}}} \\in J(E)^{\\mathrm {Aut}(K/K_0)}=J(K_0).$ Theorem 4.5 Suppose that a point $P =(a,b) \\in {\\mathcal {C}}(K_0)\\subset J(K_0),$ i.e., $a,b \\in K_0, \\ b^2=f(a).$ Then the following conditions are equivalent.",
"(i) $\\alpha _i \\in K_0$ and $a-\\alpha _i$ is a square in $K_0$ for all $i$ with $1\\le i \\le 2g+1$ .",
"(ii) All $2^{2g}$ elements ${\\mathfrak {a}}\\in J(K)$ with $2{\\mathfrak {a}}=P$ actually lie in $J(K_0)$ .",
"Assume (i).",
"Then ${\\mathfrak {a}}={\\mathfrak {a}}_{{\\mathfrak {r}}}$ for a certain ${\\mathfrak {r}}\\in {\\mathfrak {R}}_{1/2,P}$ .",
"Our assumptions imply that all ${\\mathfrak {r}}_i=\\sqrt{a-\\alpha _i}$ lie in $K_0$ and therefore $\\mathbf {s}_i({\\mathfrak {r}}) \\in K_0 \\ \\forall \\ i=1, \\dots , 2g.$ Now Theorem REF tells us that ${\\mathfrak {a}}_{{\\mathfrak {r}}}\\in J(K_0)$ .",
"This proves (ii).",
"Assume (ii).",
"It follows from Theorem REF that $\\mathbf {s}_i({\\mathfrak {r}})\\in K_0$ for all ${\\mathfrak {r}}\\in {\\mathfrak {R}}_{1/2,P}$ and $i$ with $1 \\le i \\le 2g+1$ .",
"In particular, for $i=1$ $\\sum _{i=1}^{2g+1}{\\mathfrak {r}}_i=\\mathbf {s}_1({\\mathfrak {r}})\\in K_0 \\ \\forall \\ {\\mathfrak {r}}\\in {\\mathfrak {R}}_{1/2,P}.$ Pick any ${\\mathfrak {r}}\\in {\\mathfrak {R}}_{1/2,P}$ and for any index $l$ ($1\\le l\\le 2j+1$ ) consider ${\\mathfrak {r}}^{(l)} \\in {\\mathfrak {R}}_{1/2,P}$ such that ${\\mathfrak {r}}^{(l)}_l={\\mathfrak {r}}_l, \\ {\\mathfrak {r}}^{(l)}_i =-{\\mathfrak {r}}_i \\ \\forall i \\ne l.$ We have $\\mathbf {s}_1({\\mathfrak {r}})\\in K_0, \\ -2\\mathbf {s}_1({\\mathfrak {r}})+2 {\\mathfrak {r}}_l=\\mathbf {s}_1({\\mathfrak {r}}^{(l)})\\in K_0.$ This implies that ${\\mathfrak {r}}_l \\in K_0$ .",
"Since ${\\mathfrak {r}}_{\\ell }^2=a-\\alpha _l$ and $a \\in K_0$ , we conclude that $\\alpha _l$ lies in $K_0$ and $a-\\alpha _l$ is a square in $K_0$ .",
"This proves (i).",
"Remark 4.6 In the case of elliptic curves (i/e., when $g=1$ ) Theorem REF is well known, see, e.g., [5].",
"The following assertion was inspired by results of Schaefer [14].",
"Theorem 4.7 Let us consider the $(2g+1)$ -dimensional commutative semisimple $K_0$ -algebra $L=K_0[x]/f(x)K_0[x]$ .",
"A $K_0$ -point $P=(a,b)$ on ${\\mathcal {C}}$ is divisible by 2 in $J(K_0)$ if and only if $(a-x)+f(x)K_0[x] \\in K_0[x]/f(x)K_0[x]=L$ is a square in $L$ .",
"For each $q(x)\\in K_0[x]$ we write $\\overline{q(x)}$ for its image in $K_0[x]/f(x)K_0[x]$ .",
"For each $i=1, \\dots 2g+1$ there is a homomorphism of $K_0$ -algebras $\\phi _i: L=K_0[x]/f(x)K_0[x] \\rightarrow K_0^{\\mathrm {sep}}, \\ \\overline{q(x)}=q(x)+f(x)K_0[x]\\mapsto q(\\alpha _i);$ the intersection of the kernels of all $\\phi _i$ is $\\lbrace 0\\rbrace $ .",
"Indeed, if $\\overline{q(x)}\\in \\ker (\\phi _i)$ then $q(x)$ is divisible by $x-\\alpha _i$ and therefore if $\\overline{q(x)}$ lies in $\\ker (\\phi _i)$ for all $i$ then $q(x)$ is divisible by $\\prod _{i=1}^{2g+1}(x-\\alpha _i)=f(x)$ , i.e., $\\overline{q(x)}=0$ in $ K_0[x]/f(x)K_0[x]$ .",
"Clearly, $\\phi _i(\\bar{x})=\\alpha _i, \\ \\phi _i(\\overline{a-x})=a-\\alpha _i .$ Since $f(x)$ lies in $K_0[x]$ , the set of its roots $\\lbrace \\alpha _1, \\dots , \\alpha _{2g+1}\\rbrace $ is a Galois-stable subset of $K_0^{\\mathrm {sep}}$ .",
"This implies that for each $q(x)\\in K_0[x]$ and $\\mathcal {Z}=\\overline{q(x)}\\in K_0[x]/f(x)K_0[x]$ the product $\\mathrm {H}_{\\mathcal {Z}}(t)=H_{q(x)}(t) :=\\prod _{i=1}^{2g+1}\\left(t-\\phi _i(\\overline{q(x)}\\right)=\\prod _{i=1}^{2g+1}(t-q(\\alpha _i))$ is a degree $(2g+1)$ monic polynomial with coefficients in $K_0$ .",
"In particular, if $q(x)=a-x$ then $H_{a-x}(t)=\\mathrm {H}_{\\overline{a-x}}(t) =\\prod _{i=1}^{2g+1}[t-(a-\\alpha _i)].$ Assume that $P$ is divisible by 2 in $J(K_0)$ , i.e., there is ${\\mathfrak {a}}\\in J(K_0)$ with $2{\\mathfrak {a}}=P$ .",
"It follows from Theorems REF and REF that there is ${\\mathfrak {r}}\\in {\\mathfrak {R}}_{1/2,P}$ such that ${\\mathfrak {a}}_{{\\mathfrak {r}}}={\\mathfrak {a}}$ and all $\\mathbf {s}_i({\\mathfrak {r}})$ lie in $K_0$ .",
"This implies that both polynomials $U_{{\\mathfrak {r}}}(x)$ and $V_{{\\mathfrak {r}}}(x)$ have coefficients in $K_0[x]$ .",
"Recall (Theorem REF ) that $f(x)$ and $U_{{\\mathfrak {r}}}(x)$ are relatively prime.",
"This means that $\\overline{U_{{\\mathfrak {r}}}(x)}=U_{{\\mathfrak {r}}}(\\bar{x})$ is a unit in $K_0[x]/f(x)K_0[x]$ .",
"Therefore we may define $\\mathcal {R}=\\mathbf {s}_1({\\mathfrak {r}})+(-1)^g\\frac{V_{{\\mathfrak {r}}}(\\bar{x})}{U_{{\\mathfrak {r}}}(\\bar{x})} \\in K_0[x]/f(x)K_0[x].$ The last formula of Theorem REF implies that for all $i$ we have $\\phi _i(\\mathcal {R})={\\mathfrak {r}}_i$ and therefore $\\phi _i(\\mathcal {R}^2)={\\mathfrak {r}}_i^2=a-\\alpha _i=\\phi _i(a-\\bar{x}).$ This implies that $\\mathcal {R}^2=a-\\bar{x}$ .",
"It follows that $a-\\bar{x}=(a -x)+f(t)K_0[t] \\in K_0[x]/f(x)K_0[x]$ is a square in $K_0[x]/f(x)K_0[x]$ .",
"Conversely, assume now that there is an element $\\mathcal {R} \\in L$ such that $\\mathcal {R}^2=a-\\bar{x}=\\overline{a-x}.$ This implies that $\\phi _i(\\mathcal {R})^2=\\phi _i(\\overline{a-x})=a-\\alpha _i,$ i.e., $\\phi _i(\\mathcal {R})=\\sqrt{a-\\alpha _i} \\ \\forall \\ i=1, \\dots , 2g+1.$ This implies that $\\prod _{i=1}^{2g+1}\\phi _i(\\mathcal {R})=\\sqrt{f(a)}=\\pm b.$ Since $(-1)^{2g+1}=-1$ , replacing if necessary, $\\mathcal {R}$ by $-\\mathcal {R}$ , we may and will assume that $\\prod _{i=1}^{2g+1}\\phi _i(\\mathcal {R})=-b.$ Now if we put ${\\mathfrak {r}}_i=\\phi _i(\\mathcal {R}) \\ \\forall \\ i=1, \\dots , 2g+1; \\ {\\mathfrak {r}}=({\\mathfrak {r}}_1, \\dots , {\\mathfrak {r}}_{2g+1})$ then ${\\mathfrak {r}}\\in {\\mathfrak {R}}_{1/2,P}$ and $h_{{\\mathfrak {r}}}(t)=\\prod _{i=1}^{2g+1}(t-{\\mathfrak {r}}_i)=\\prod _{i=1}^{2g+1}(t-\\phi _i(\\mathcal {R}))=\\mathrm {H}_{\\mathcal {R}}(t).$ Since $\\mathfrak {H}_{\\mathcal {R}}(t)$ lies in $K_0[t]$ , the polynomial $h_{{\\mathfrak {r}}}(t)$ also lies in $K_0[t]$ .",
"It follows from Theorem REF that ${\\mathfrak {a}}_{{\\mathfrak {r}}}\\in J(K_0)$ .",
"Since $2{\\mathfrak {a}}_{{\\mathfrak {r}}}=P$ , the point $P$ is divisible by 2 in $J(K_0)$ .",
"Remark 4.8 If one assumes additionally that $\\mathrm {char}(K_0)=0$ and $P$ is none of ${\\mathfrak {W}}_i$ (i.e., $a \\ne \\alpha _i$ for any $i$ ) then the assertion of Theorem REF follows from [14]."
],
[
"Torsion points on $\\Theta _d$",
"We keep the notation of Section .",
"In particular, $K_0$ be a subfield of $K$ such that $f(x) \\in K_0[x].$ Notice that the involution $\\iota $ is also defined over $K_0$ , the absolute Galois group $\\mathrm {Gal}(K_0)$ leaves invariant $\\infty $ and permutes points of ${\\mathcal {C}}(K_0^{\\mathrm {sep}})$ ; in addition, it permutes elements of $J(K_0^{\\mathrm {sep}})$ , respecting the group structure on $J(K_0^{\\mathrm {sep}})$ .",
"If $n$ is a positive integer that is not divisible by $\\mathrm {char}(K)$ then we write $J[n]$ for the kernel of multiplication by $n$ in $J(K)$ .",
"It is well known that $J[n]$ is a free $Z/n{\\mathbb {Z}}$ -module of rank $2g$ that lies in $J(K_0^{\\mathrm {sep}})$ ; in addition, it is a $\\mathrm {Gal}(K_0)$ -stable subgroup of $J(K_0^{\\mathrm {sep}})$ , which gives us the (continuous) group homomorphism $\\rho _{n,,J}:\\mathrm {Gal}(K_0)\\rightarrow \\mathrm {Aut}_{{\\mathbb {Z}}/n{\\mathbb {Z}}}(J[n])$ that defines the Galois action on $J[n]$ .",
"We write $\\tilde{G}_{n,J,K_0}$ for the image $\\rho _{n,J}(\\mathrm {Gal}(K_0))\\subset \\mathrm {Aut}_{{\\mathbb {Z}}/n{\\mathbb {Z}}}(J[n]).$ Let $\\mathrm {Id}_n$ be the identity automorphism of $J[n]$ .",
"The following assertion was inspired by a work of F. Bogomolov [4] (where the $\\ell $ -primary part of the Manin-Mumford conjecture was proven).",
"Theorem 5.1 Suppose that $g>1$ and $n\\ge 3$ is an integer that is not divisible by $\\mathrm {char}(K)$ .",
"Let $N>1$ be an integer that is relatively prime to $n$ and such that $N \\le 2g-1$ and $\\tilde{G}_{n,J,K_0}$ contains either $N\\cdot \\mathrm {Id}_n$ or $-N\\cdot \\mathrm {Id}_n$ .",
"Let us put $d(N):=[2g/(N+1)]$ .",
"Then $\\Theta _{d(N)}(K)$ does not contain nonzero points of order dividing $n$ except points of order 1 or 2.",
"In particular, if $n$ is odd then $\\Theta _{d(N)}(K)$ does not contain nonzero points of order dividing $n$ .",
"Clearly, $(N+1)\\cdot d(N)<2g+1$ .",
"Suppose that ${\\mathfrak {b}}$ is a nonzero point of order dividing $n$ in $\\Theta _{d(N)}(K)$ .",
"We need to prove that $2{\\mathfrak {b}}=0$ .",
"Indeed, ${\\mathfrak {b}}\\in J[n]\\subset J(K_0^{\\mathrm {sep}})$ and therefore ${\\mathfrak {b}}\\in \\Theta _d(K)\\bigcap J(K_0^{\\mathrm {sep}})=\\Theta _d(K_0^{\\mathrm {sep}}).$ By our assumption, there is $\\sigma \\in \\mathrm {Gal}(K)$ such that $\\sigma ({\\mathfrak {a}})=N{\\mathfrak {a}}$ or $-N{\\mathfrak {a}}$ for all ${\\mathfrak {a}}\\in J[n]$ .",
"This implies that $\\sigma ({\\mathfrak {b}})=N{\\mathfrak {b}}$ or $-N{\\mathfrak {b}}$ .",
"It follows from Theorem REF that $2{\\mathfrak {b}}=0$ in $J(K)$ .",
"Example 5.2 Suppose that $K$ is the field ${\\mathbb {C}}$ of complex numbers, $g=2$ and ${\\mathcal {C}}$ is the genus 2 curve $y^2=x^5-x+1.$ Let us put $N=2$ .",
"Then $d(N)=2$ .",
"Let $n=\\ell $ be an odd prime.",
"Then ${\\mathbb {Z}}/n{\\mathbb {Z}}$ is the prime field ${\\mathbb {F}}_{\\ell }$ .",
"Results of L. Dieulefait [6] and Serre's Modularity Conjecture [18] that was proven by C. Khare and J.-P. Wintenberger [8] imply that $\\tilde{G}_{\\ell ,J,K_0}$ is “as large as possible\"; in particular, it contains all the homotheties ${\\mathbb {F}}_{\\ell }^{*}\\cdot \\mathrm {Id}_{\\ell }$ .",
"This implies that $\\tilde{G}_{\\ell ,J,K_0}$ contains $2\\cdot \\mathrm {Id}_{\\ell }$ , since $\\ell $ is odd.",
"It follows from Corollary REF that $\\Theta _1={\\mathcal {C}}({\\mathbb {C}})$ does not contain points of order $\\ell $ for all odd primes $\\ell $ .",
"Actually, using his algorithm mentioned above, B. Poonen had already checked that the only torsion points on this curve are the Weierstrass points ${\\mathfrak {W}}_i$ (of order 2) and $\\infty $ (of order 1) [10].",
"Notice that the Galois group of $x^5-x+1$ over ${\\mathbb {Q}}$ is the full symmetruc group ${\\mathbf {S}}_5$ .",
"This implies that the ring of ${\\mathbb {C}}$ -endomorphisms of $J$ coincides with ${\\mathbb {Z}}$ [23].",
"In particular, $J$ is an absolutely simple abelian surface.",
"Theorem 5.3 Suppose that $g>1$ , $K_0={\\mathbb {Q}}, K={\\mathbb {C}}$ and $\\alpha _1, \\dots , \\alpha _{2g+1} \\in {\\mathbb {C}}$ are algebraically independent (transcendental) elements of ${\\mathbb {C}}$ (i.e., ${\\mathcal {C}}:y^2=\\prod _{i=1}^{2g+1}(x-\\alpha _i)$ is a generic hyperelliptic curve).",
"Then: (i) $\\Theta _{[2g/3]}({\\mathbb {C}})$ does not contain nonzero points of odd order.",
"(ii) All 2-power torsion points in $\\Theta _{[g/2]}({\\mathbb {C}})$ have order 1 or 2.",
"We will prove Theorems REF in Section .",
"Remark 5.4 Let $K_0, K={\\mathbb {C}}$ and ${\\mathcal {C}}$ be as in Theorem REF .",
"(i) B. Poonen and M. Stoll [11] proved that the only torsion points on this generic curve are the Weierstrass points ${\\mathfrak {W}}_i$ (of order 2) and $\\infty $ (of order 1).",
"(ii) Let $s_1, \\dots , s_{2g+1} \\in {\\mathbb {C}}$ be the corresponding basic symmetric functions in $\\alpha _1, \\dots , \\alpha _{2g+1}$ and let us consider the (sub)field $L:={\\mathbb {Q}}(s_1, \\dots , s_{2g+1}\\subset {\\mathbb {Q}}(\\alpha _1, \\dots , \\alpha _{2g+1})=K_0.$ Then $f(x)$ lies in $L[x]$ and its Galois group over $L$ is the full symmetric group ${\\mathbf {S}}_{2g+1}$ .",
"This implies that the ring of ${\\mathbb {C}}$ -endomorphisms of $J$ coincides with ${\\mathbb {Z}}$ [23].",
"In particular, $J$ is an absolutely simple abelian variety.",
"(Of course, this result is well known.)",
"It follows from the generalized Manin-Mumford conjecture (also proven by M. Raynaud [13]) that the set of torsion points on $\\Theta _d({\\mathbb {C}})$ is finite for all $d<g$ ."
],
[
"Abelian varieties with big $\\ell $ -adic Galoid images",
"We need to recall some basic facts about fields of definition of torsion points on abelian varieties.",
"Recall that a positive integer $n$ is not divisible by $\\mathrm {char}(K)$ and the rank $2g$ free ${\\mathbb {Z}}/n{\\mathbb {Z}}$ -module $J[n]$ lies in $J(K^{\\mathrm {sep}})$ .",
"Clearly, all $n$ th roots of unity of $K$ lie in $K^{\\mathrm {sep}}$ .",
"We write $\\mu _n$ for the order $n$ cyclic multiplicative group of $n$ th roots of unity in $K^{\\mathrm {sep}}$ .",
"We write $K(\\mu _n)\\subset K^{\\mathrm {sep}}$ for the $n$ th cyclotomic field extension of $K$ and $\\chi _n: \\mathrm {Gal}(K) \\rightarrow ({\\mathbb {Z}}/n{\\mathbb {Z}})^{*}$ for the $n$ th cyclotomic character that defines the Galois action on all $n$ th roots of unity.",
"The Galois group $\\mathrm {Gal}(K(\\mu _n)/K)$ of the abelian extension $K(\\mu _n/K$ is canonically isomorphic to the image $\\chi _n(\\mathrm {Gal}(K))\\subset ({\\mathbb {Z}}/n{\\mathbb {Z}})^{*}=\\mathrm {Gal}({\\mathbb {Q}}(\\mu _n)/{\\mathbb {Q}});$ the equality holds if and only if the degree $[K(\\mu _n):K]$ coincides with $\\phi (n)$ where $\\phi $ is the Euler function.",
"For example, if $K$ is the field ${\\mathbb {Q}}$ of rational numbers then for all $n$ $[{\\mathbb {Q}}(\\zeta _n):{\\mathbb {Q}}]=\\phi (n), \\ \\chi _n(\\mathrm {Gal}({\\mathbb {Q}}))= ({\\mathbb {Z}}/n{\\mathbb {Z}})^{*}.$ The Jacobian $J$ carries the canonical principal polarization that is defined over $K_0$ and gives rise to a nondegenerate alternating bilinear form (Weil-Riemann pairing) $\\bar{e}_n: J[n] \\times J[n] \\rightarrow {\\mathbb {Z}}/n{\\mathbb {Z}}$ such that for all $\\sigma \\in \\mathrm {Gal}(K)$ and ${\\mathfrak {a}}_1, {\\mathfrak {a}}_2 \\in J[n]$ we have $\\bar{e}_n(\\sigma {\\mathfrak {a}}_1,\\sigma {\\mathfrak {a}}_2)=\\chi _n(\\sigma )\\cdot \\bar{e}_n({\\mathfrak {a}}_1,{\\mathfrak {a}}_2).$ (Such a form is defined uniquely up to multiplication by an element of ${\\mathbb {Z}}/n{\\mathbb {Z}}$ and depends on a choice between of an isomorphism between $\\mu _n$ and ${\\mathbb {Z}}/n{\\mathbb {Z}}$ .)",
"Let $\\mathrm {Gp}(J[n],\\bar{e}_n)\\subset \\mathrm {Aut}_{{\\mathbb {Z}}/n{\\mathbb {Z}}}(J[n])$ be the group of symplectic similitudes of $\\bar{e}_n$ that consists of all automorphisms $u$ of $J[n]$ such that there exists a constant $c = c(u) \\in ({\\mathbb {Z}}/n{\\mathbb {Z}})^{*}$ such that $\\bar{e_n}\\bar{e}_n(u{\\mathfrak {a}}_1,u{\\mathfrak {a}}_2) = c(u)\\cdot \\bar{e_n}\\bar{e}_n({\\mathfrak {a}}_1,{\\mathfrak {a}}_2)\\ \\forall {\\mathfrak {a}}_1,{\\mathfrak {a}}_2 \\in J[n].$ The map $\\mathrm {mult}_n: \\mathrm {Gp}(J[n],\\bar{e}_n) \\rightarrow ({\\mathbb {Z}}/n{\\mathbb {Z}})^{*}, \\ u\\mapsto c(u)$ is a surjective group homomorphism, whose kernel coincides with the symplectic group $\\mathrm {Sp}(J[n],\\bar{e}_n)\\cong \\mathrm {Sp}_{2g}({\\mathbb {F}}_{\\ell })$ of $\\bar{e}_n$ .",
"Both $\\mathrm {Sp}(J[n],\\bar{e}_n)$ and the group of homotheties $({\\mathbb {Z}}/n{\\mathbb {Z}})\\mathrm {Id}_n$ are subgroups of $\\mathrm {Gp}(J[n],\\bar{e}_n)$ .",
"The Galois-equivariance of the Weil-Riemann pairing implies that $\\tilde{G}_{n,J,K_0}\\subset \\mathrm {Gp}(J[n],\\bar{e}_n)\\subset \\mathrm {Aut}_{{\\mathbb {Z}}/n{\\mathbb {Z}}}(J[n]).$ It is also clear that for each $\\sigma \\in \\mathrm {Gal}(K)$ $\\chi _n(\\sigma )=c(\\rho _{n,J.K_0}(\\sigma ))=\\mathrm {mult}_n(\\rho _{n,J.K_0}(\\sigma )) \\in ({\\mathbb {Z}}/n{\\mathbb {Z}})^{*}.$ Since $\\mathrm {Sp}(J[n],\\bar{e}_n)=\\ker (\\mathrm {mult}_n)$ , we obtain the following useful assertion.",
"Lemma 6.1 Let us assume that $\\chi _n(\\mathrm {Gal}(K))= ({\\mathbb {Z}}/n{\\mathbb {Z}})^{*}$ (E.g., $K={\\mathbb {Q}}$ or the field ${\\mathbb {Q}}(t_1, \\dots t_d)$ of rational functions in $d$ independent varibles over $\\mathbb {{\\mathbb {Q}}}$ .)",
"Suppose that $\\tilde{G}_{n,J,K_0}$ contains $\\mathrm {Sp}(J[n],\\bar{e}_n)$ .",
"Then $\\tilde{G}_{n,J,K_0}=\\mathrm {Gp}(J[n],\\bar{e}_n)$ .",
"In particular, $\\tilde{G}_{n,J,K_0}$ contains the whole group of homotheties $({\\mathbb {Z}}/n{\\mathbb {Z}})^{*}\\cdot \\mathrm {Id}_n$ .",
"Example 6.2 Let $K_0$ , $K={\\mathcal {C}}$ and ${\\mathcal {C}}$ be as in Theorem REF , i.e., ${\\mathcal {C}}$ is a generic hyperelliptic curve.",
"(i) B. Poonen and M. Stoll proved [11] that if $n$ is odd then $\\tilde{G}_{n,J,K_0}$ contains $\\mathrm {Sp}(J[n],\\bar{e}_n)$ .",
"It follows from Lemma REF that $\\tilde{G}_{n,J,K_0}=\\mathrm {Gp}(J[n],\\bar{e}_n)$ for all odd $n$ .",
"In particular, it contains $({\\mathbb {Z}}/n{\\mathbb {Z}})^{*}\\cdot \\mathrm {Id}_n$ and therefore contains $2\\cdot \\mathrm {Id}_n$ .",
"(ii) Let us assume that $n=2^e$ is a a power of 2.",
"J. Yelton [22] proved that that $\\tilde{G}_{n,J,K_0}$ contains the level 2 congruence subgroup $\\Gamma (2)$ of $\\mathrm {Sp}(J[n],\\bar{e}_n)$ defined by the condition $\\Gamma (2)=\\lbrace g \\in \\mathrm {Sp}(J[n],\\bar{e}_n)\\mid g\\equiv \\mathrm {Id}_n \\bmod 2\\rbrace \\lhd \\mathrm {Sp}(J[n],\\bar{e}_n).$ Let us consider the level 2 congruence subgroup $G\\Gamma (2)$ of $\\mathrm {Gp}(J[n],\\bar{e}_n)$ defined by the condition $G\\Gamma (2)=\\lbrace g \\in \\mathrm {Gp}(J[n],\\bar{e}_n)\\mid g\\equiv \\mathrm {Id}_n \\bmod 2\\rbrace \\lhd \\mathrm {Gp}(J[n],\\bar{e}_n).$ Clearly, $G\\Gamma (2)$ contains $3\\cdot \\mathrm {Id}_n$ while the intersection of $G\\Gamma (2)$ and $\\mathrm {Sp}(J[n],\\bar{e}_n)$ coincides with $\\Gamma (2)$ .",
"The latter means that $\\Gamma (2)$ coincides with the kernel of the restriction of $\\mathrm {mult}_n$ to $G\\Gamma (2)$ .",
"In addition,one may easily check that $\\mathrm {mult}_n(G\\Gamma (2))=({\\mathbb {Z}}/n{\\mathbb {Z}})^{*}=\\mathrm {mult}_n(\\mathrm {Gp}(J[n],\\bar{e}_n),$ since $({\\mathbb {Z}}/n{\\mathbb {Z}})^{*}=\\lbrace c \\in {\\mathbb {Z}}/n{\\mathbb {Z}})\\mid c \\equiv 1 \\bmod 2\\rbrace .$ This implies that $\\tilde{G}_{n,J,K_0}$ contains $G\\Gamma (2)$ .",
"In particular, $\\tilde{G}_{n,J,K_0}$ contains $3\\cdot \\mathrm {Id}_n$ .",
"(See also [11].)",
"Recall that $d(2)=[2g/3]$ .",
"Combining Theorem REF (with $N=2$ and any odd $n$ ) with Example REF (i), we conclude that $\\Theta _{[2g/3]}({\\mathbb {C}})$ does not contain nonzero points of odd order $n$ .",
"This proves (i).",
"Recall that $d(3)=[2g/4]=[g/2]$ .",
"Combining Theorem REF (with $N=3$ and $n=2^e$ ) with Example REF (ii), we conclude that all 2-power torsion points in $\\Theta _{[g/2]}({\\mathbb {C}})$ are points of order 1 or 2.",
"The rest of this paper is devoted to the proof of the following result.",
"Theorem 6.3 Let $K_0$ be the field ${\\mathbb {Q}}$ of rational numbers, $K={\\mathbb {C}}$ the field of complex numbers.",
"Suppose that $g>1$ .",
"Let $S$ be a non-empty set of odd primes such that for all $\\ell \\in S$ the image $\\tilde{G}_{\\ell ,J,K_0}=\\mathrm {Gp}(J[\\ell ],\\bar{e}_{\\ell })$ .",
"If $n>1$ is a positive odd integer, all whose prime divisors lie in $S$ then $\\Theta _{[2g/3]}({\\mathbb {C}})$ does not contain nonzero points of order dividing $n$ .",
"Let us start with the following elementary observation on Galois properties of torsion points on $J$ .",
"Remark 6.4 Let $\\tilde{G}_n$ be the derived subgroup $[\\tilde{G}_{n,J,K_0}, \\tilde{G}_{n,J,K_0}]$ of $\\tilde{G}_{n,J,K_0}$ .",
"Then $\\tilde{G}_n$ is a normal subgroup of finite index in $\\tilde{G}_{n,J,K_0}$ .",
"Let $K_{0,n}\\subset K_0^{\\mathrm {sep}}$ be the finite Galois extension of $K_0$ such that the absolute Galois (sub)group $\\mathrm {Gal}(K_{0,n})\\subset \\mathrm {Gal}(K_0)$ coincides with the preimage $\\rho _{n,J}^{-1}(\\tilde{G}_n))\\subset \\rho _{n,J}^{-1}(\\tilde{G}_{n,J,K_0})=\\mathrm {Gal}(K).$ We have $\\tilde{G}_{n,J,K_{0,n}}=\\rho _{n,J}(\\mathrm {Gal}(K_{0,n}))=$ $\\tilde{G}_n=[\\tilde{G}_{n,J,K_0}, \\tilde{G}_{n,J,K_0}] \\subset [\\mathrm {Gp}(J[n],\\bar{e}_n),\\mathrm {Gp}(J[n],\\bar{e}_n)]\\subset \\mathrm {Sp}(J[n],\\bar{e}_n).$ This implies that $\\tilde{G}_{n,J,K_{0,n}}\\subset \\mathrm {Sp}(J[n],\\bar{e}_n).$ Let $m>1$ be an integer dividing $n$ .",
"The inclusion of Galois modules $J[m]\\subset J[n]$ induces the surjective group homomorphisms $\\tilde{G}_{n,J,K_0}\\twoheadrightarrow \\tilde{G}_{m,J,K_0}, \\ \\tilde{G}_{n,J,K_{0,n}}\\twoheadrightarrow \\tilde{G}_{m,J,K_{0,n}}\\subset \\tilde{G}_{m,J,K_0};$ the latter homomorphism coincides with the restriction of the former one to the (derived) subgroup $\\tilde{G}_{n,J,K_{0,n}}\\subset \\tilde{G}_{n,J,K_0}$ .",
"This implies that $\\tilde{G}_{m,J,K_{0,n}}=[\\tilde{G}_{m,J,K_0}, \\tilde{G}_{m,J,K_0}]$ is the derived subgroup of $\\tilde{G}_{m,J,K_0}$ .",
"In addition, $\\tilde{G}_{m,J,K_{0,n}}=[\\tilde{G}_{m,J,K_0}, \\tilde{G}_{m,J,K_0}]\\subset [\\mathrm {Gp}(J[m],\\bar{e}_m),\\mathrm {Gp}(J[m],\\bar{e}_m)]\\subset \\mathrm {Sp}(J[m],\\bar{e}_m).$ Recall that $g \\ge 2$ .",
"Now assume that $m=\\ell $ is an odd prime dividing $n$ .",
"Then $\\mathrm {Sp}(J[\\ell ],\\bar{e}_{\\ell })$ is perfect, i.e., coincides with its own derived subgroup.",
"Assume also that $\\tilde{G}_{\\ell ,J,K_0})$ contains $\\mathrm {Sp}(J[\\ell ],\\bar{e}_{\\ell })$ .",
"Then $\\mathrm {Sp}(J[\\ell ],\\bar{e}_{\\ell })\\supset \\tilde{G}_{\\ell ,J,K_{0,n}}=$ $[\\tilde{G}_{\\ell ,J,K_0}),\\tilde{G}_{\\ell ,J,K_0})\\supset [\\mathrm {Sp}(J[\\ell ],\\bar{e}_{\\ell }),\\mathrm {Sp}(J[\\ell ],\\bar{e}_{\\ell })]=\\mathrm {Sp}(J[\\ell ],\\bar{e}_{\\ell })$ and therefore $\\tilde{G}_{\\ell ,J,K_{0,n}}=\\mathrm {Sp}(J[\\ell ],\\bar{e}_{\\ell }).$ We will also need the following result about closed subgroups of symplectic groups over the ring ${\\mathbb {Z}}_{\\ell }$ of $\\ell $ -adic integers ([17], [20]).",
"Lemma 6.5 Let $g \\ge 2$ be an integer and $\\ell $ an odd prime.",
"Let $G$ be a closed subroup of $\\mathrm {Sp}(2g,{\\mathbb {Z}}_{\\ell })$ such that the corresponding reduction map $G \\rightarrow \\mathrm {Sp}(2g,{\\mathbb {Z}}/\\ell {\\mathbb {Z}})$ is surjective.",
"Then $G=\\mathrm {Sp}(2g,{\\mathbb {Z}}_{\\ell })$ .",
"The result follows from [20] applied to $p=q=\\ell , k={\\mathbb {F}}_{\\ell }, W(k)={\\mathbb {Z}}_{\\ell }, G=\\mathrm {Sp}_{2g}.$ Corollary 6.6 Let $g \\ge 2$ be an integer and $\\ell $ an odd prime.",
"Then for each positive integer $i$ the group $\\mathrm {Sp}_{2g}({\\mathbb {Z}}/\\ell ^i{\\mathbb {Z}})$ is perfect.",
"The case $i=1$ is well known.",
"Let $i\\ge 1$ be an integer.",
"It is also well known that the reduction modulo $\\ell ^i$ map $\\mathrm {red}_i: \\mathrm {Sp}_{2g}({\\mathbb {Z}}_{\\ell })\\rightarrow \\mathrm {Sp}_{2g}({\\mathbb {Z}}/\\ell ^i{\\mathbb {Z}})$ is a surjective group homomorphism.",
"This implies that the reduction modulo $\\ell $ map $\\mathrm {\\overline{red}}_{i,1}: \\mathrm {Sp}_{2g}({\\mathbb {Z}}/\\ell ^i{\\mathbb {Z}})\\rightarrow \\mathrm {Sp}_{2g}({\\mathbb {Z}}/\\ell {\\mathbb {Z}})$ is also a surjective group homomorphism.",
"Clearly, $\\mathrm {red}_1$ coincides with the composition $\\mathrm {\\overline{red}}_{i,1}\\circ \\mathrm {red}_i$ .",
"Suppose that $\\mathrm {Sp}_{2g}({\\mathbb {Z}}/\\ell ^i{\\mathbb {Z}})$ is not perfect and let $H:=[\\mathrm {Sp}_{2g}({\\mathbb {Z}}/\\ell ^i{\\mathbb {Z}}), \\mathrm {Sp}_{2g}({\\mathbb {Z}}/\\ell ^i{\\mathbb {Z}})]$ be the derived subgroup of $\\mathrm {Sp}_{2g}({\\mathbb {Z}}/\\ell ^i{\\mathbb {Z}})$ .",
"Since $\\mathrm {Sp}(2g,{\\mathbb {Z}}/\\ell {\\mathbb {Z}})$ is perfect, i.e., coincides with its derived subgroup, $\\mathrm {\\overline{red}}_{i,1}(H) =\\mathrm {Sp}(2g,{\\mathbb {Z}}/\\ell {\\mathbb {Z}}).$ Now the closed subgroup $G:= \\mathrm {red}_i^{-1}(H)\\subset \\mathrm {Sp}_{2g}({\\mathbb {Z}}_{\\ell })$ maps surjectively on $\\mathrm {Sp}_{2g}({\\mathbb {Z}}/\\ell {\\mathbb {Z}})$ but does not coincide with $\\mathrm {Sp}_{2g}({\\mathbb {Z}}_{\\ell })$ , because $H$ is a proper subgroup of $\\mathrm {Sp}_{2g}({\\mathbb {Z}}/\\ell ^i{\\mathbb {Z}})$ and $\\mathrm {\\overline{red}}_{i,1}$ is surjective.",
"This contradicts to Lemma REF , which proves the desired perfectness.",
"The following lemma will be proven at the end of this section.",
"Lemma 6.7 Suppose that $g>1$ .",
"Suppose that $n>1$ is an odd integer that is not divisible by $\\mathrm {char}(K)$ .",
"If for all primes $\\ell $ dividing $n$ the image $\\tilde{G}_{\\ell ,J,K_0}$ contains $\\mathrm {Sp}(J[n],\\bar{e}_{\\ell })$ then $\\tilde{G}_{n,J,K_0}$ contains $\\mathrm {Sp}(J[n],\\bar{e}_n)$ .",
"In addition, if $K_0$ is the field ${\\mathbb {Q}}$ of rational numbers then $\\tilde{G}_{n,J,K_0}= \\mathrm {Gp}(J[n],\\bar{e}_n).$ Remark 6.8 Thanks to Lemma REF , the second assertion of Lemma REF follows from the first one.",
"Recall that $\\mathrm {Gp}(J[\\ell ],\\bar{e}_{\\ell })$ contains $\\mathrm {Sp}(J[\\ell ],\\bar{e}_{\\ell })$ .",
"It follows from Lemma REF that $\\tilde{G}_{n,J,K_0}= \\mathrm {Gp}(J[n],\\bar{e}_n).$ This implies that $\\tilde{G}_{n,J,K_0}$ contains $2\\cdot \\mathrm {Id}_n$ , because it contains the whole $({\\mathbb {Z}}/n{\\mathbb {Z}})^{*}\\mathrm {Id}_n$ .",
"It follows from Corollary REF that ${\\mathcal {C}}(K)$ does not contain points of order $n$ .",
"First, let us do the case when $n$ is a power of an odd prime $\\ell $ .",
"Let $\\ell \\ne \\mathrm {char}(K)$ be a prime.",
"Let $T_{\\ell }(J)$ be the $\\ell $ -adic Tate module of $J$ that is the projective limit of $J[\\ell ^i]$ where the transition maps $J[\\ell ^{i+1}] \\rightarrow J[\\ell ^i]$ are multiplications by $\\ell $ .",
"It is well known that $T_{\\ell }(J)$ is a free ${\\mathbb {Z}}_{\\ell }$ -module of rank $2g$ , the Galois actions on $J[\\ell ^i]$ 's are glued together to the continuous group homomorphism $\\rho _{\\ell ,J,K_0}:\\mathrm {Gal}(K)\\rightarrow \\mathrm {Aut}_{{\\mathbb {Z}}_{\\ell }}(T_{\\ell }(J))$ such that the canonical isomorphisms of ${\\mathbb {Z}}_{\\ell }$ -modules $T_{\\ell }(J)/\\ell ^i T_{\\ell }(J)=J[\\ell ^i]$ become isomorphisms of Galois modules.",
"(Recall that ${\\mathbb {Z}}/\\ell ^i{\\mathbb {Z}}={\\mathbb {Z}}_{\\ell }/\\ell ^i{\\mathbb {Z}}_{\\ell }$ .)",
"The polarization $\\lambda $ gives rise to the alternating perfect/unimodular ${\\mathbb {Z}}_{\\ell }$ -bilinear form $e_{\\ell }: T_{\\ell }(J)\\times T_{\\ell }(J) \\rightarrow {\\mathbb {Z}}_{\\ell }$ such that for each $\\sigma \\in \\mathrm {Gal}(K)$ $e_{\\ell }(\\rho _{\\ell }(\\sigma )(v_1),\\rho _{\\ell }(\\sigma )(v_2))=\\chi _{\\ell }(\\sigma )\\cdot e_{\\ell }(v_1,v_2) \\ \\forall \\ v_1,v_2\\in T_{\\ell }(J).$ Here $\\chi _{\\ell }:\\mathrm {Gal}(K) \\rightarrow {\\mathbb {Z}}_{\\ell }^{*}$ is the (continuous) cyclotomic character of $\\mathrm {Gal}(K)$ characterized by the property $\\chi _{\\ell }(\\sigma )\\bmod \\ell ^i=\\bar{\\chi }_{\\ell ^i}(\\sigma ) \\ \\forall \\ i.$ This implies that $G_{\\ell ,J,K_0}=\\rho _{\\ell ,J,K_0}\\mathrm {Gal}(K)) \\subset \\mathrm {Gp}(T_{\\ell }(J),e_{\\ell }$ where $\\mathrm {Gp}(T_{\\ell }(J),e_{\\ell })\\subset \\mathrm {Aut}_{{\\mathbb {Z}}_{\\ell }}(T_{\\ell }(J))$ is the group of symplectic similitudes of $e_{\\ell }$ .",
"Clearly, $\\mathrm {Gp}(T_{\\ell }(J),e_{\\ell })$ contains the corresponding symplectic group $\\mathrm {Sp}(T_{\\ell }(J),e_{\\ell })\\cong \\mathrm {Sp}_{2g}({\\mathbb {Z}}_{\\ell })$ and the subgroup of homotheties/scalars ${\\mathbb {Z}}_{\\ell }^{*}$ .",
"It is also clear that the derived subgroup $[\\mathrm {Gp}(T_{\\ell }(J),e_{\\ell }), \\mathrm {Gp}(T_{\\ell }(J),e_{\\ell })]$ lies in $\\mathrm {Sp}(T_{\\ell }(J),e_{\\ell })$ .",
"For each $n=\\ell ^i$ the reduction map modulo $\\ell ^i$ sends $\\mathrm {Gp}(T_{\\ell }(J),e_{\\ell })$ onto $\\mathrm {Gp}(J[\\ell ^i],\\bar{e}_{\\ell ^i})$ , $\\mathrm {Sp}(T_{\\ell }(J),e_{\\ell })$ onto $\\mathrm {Sp}(J[\\ell ^i],\\bar{e}_{\\ell ^i})$ and ${\\mathbb {Z}}_{\\ell }^{*}$ onto $({\\mathbb {Z}}/\\ell ^i{\\mathbb {Z}})^{*}\\cdot \\mathrm {Id}_{\\ell ^i}$ .",
"In particular, if $\\ell $ is odd then the scalar $2\\in {\\mathbb {Z}}_{\\ell }^{*}$ goes to $2\\cdot \\mathrm {Id}_{\\ell ^i}\\in \\mathrm {Gp}(J[\\ell ^i],\\bar{e}_{\\ell ^i}).$ As for $G_{\\ell ,J,K_0}$ , its image under the reduction map modulo $\\ell ^i$ coincides with $\\tilde{G}_{\\ell ^i,J,K_0}$ .",
"It is known [15] that $G_{\\ell ,J,K_0}$ is a compact $\\ell $ -adic Lie subgroup in $\\mathrm {Gp}(T_{\\ell }(J),e_{\\ell })$ and therefore is a closed subgroup of $\\mathrm {Gp}(T_{\\ell }(J),e_{\\ell })$ with respect to $\\ell $ -adic topology.",
"Clearly, the intersection $G_{\\ell }:=G_{\\ell ,J,K_0}\\bigcap \\mathrm {Sp}(T_{\\ell }(J),e_{\\ell })$ is a closed subgroup of $\\mathrm {Sp}(T_{\\ell }(J),e_{\\ell })$ .",
"In addition, the derived subgroup of $G_{\\ell ,J,K_0}$ $[G_{\\ell ,J,K_0},G_{\\ell ,J,K_0}]\\subset G_{\\ell ,J,K_0} \\bigcap [\\mathrm {Gp}(T_{\\ell }(J),e_{\\ell }), \\mathrm {Gp}(T_{\\ell }(J),e_{\\ell })]\\subset G_{\\ell ,J,K_0}\\bigcap \\mathrm {Sp}(T_{\\ell }(J),e_{\\ell })=G_{\\ell },$ i.e., $[G_{\\ell ,J,K_0},G_{\\ell ,J,K_0}]\\subset G_{\\ell }.$ Let us assume that $\\ell $ is odd and $\\tilde{G}_{\\ell ,J,K_0}$ contains $\\mathrm {Sp}(J[\\ell ],\\bar{e}_{\\ell })$ .",
"Then the reduction modulo $\\ell $ of $[G_{\\ell ,J,K_0},G_{\\ell ,J,K_0}]$ contains the derived subgroup $[\\mathrm {Sp}(J[\\ell ],\\bar{e}_{\\ell }),\\mathrm {Sp}(J[\\ell ],\\bar{e}_{\\ell })]$ .",
"Since our assumptions on $g$ and $\\ell $ imply that the group $\\mathrm {Sp}(J[\\ell ],\\bar{e}_{\\ell })$ is perfect, i.e., $[\\mathrm {Sp}(J[\\ell ],\\bar{e}_{\\ell }),\\mathrm {Sp}(J[\\ell ],\\bar{e}_{\\ell })]=\\mathrm {Sp}(J[\\ell ],\\bar{e}_{\\ell }),$ the reduction modulo $\\ell $ of $[G_{\\ell ,J,K_0},G_{\\ell ,J,K_0}]$ contains $\\mathrm {Sp}(J[\\ell ],\\bar{e}_{\\ell })$ .",
"This implies that the reduction modulo $\\ell $ of $G_{\\ell }$ also contains $\\mathrm {Sp}(J[\\ell ],\\bar{e}_{\\ell })$ .",
"Since $G_{\\ell }$ is a (closed) subgroup of $\\mathrm {Sp}(T_{\\ell }(J),e_{\\ell })$ , its reduction modulo $\\ell $ actually coincides with $\\mathrm {Sp}(J[\\ell ],\\bar{e}_{\\ell })$ .",
"It follows from Lemma REF that $G_{\\ell }=\\mathrm {Sp}(T_{\\ell }(J),e_{\\ell }).$ In particular, the reduction of $G_{\\ell }$ modulo $\\ell ^i$ coincides with $\\mathrm {Sp}(J[\\ell ^i],\\bar{e}_{\\ell ^i})$ for all positive integers $i$ .",
"Since $G_{\\ell ,J,K_0}$ contains $G_{\\ell }$ , its reduction modulo $\\ell ^i$ contains $\\mathrm {Sp}(J[\\ell ^i],\\bar{e}_{\\ell ^i})$ .",
"This means that $\\tilde{G}_{\\ell ^i,J,K_0}$ contains $\\mathrm {Sp}(J[\\ell ^i],\\bar{e}_{\\ell ^i})$ for all positive $i$ .",
"This proves Lemma REF for all $n$ that are powers of an odd prime $\\ell $ .",
"Now let us consider the general case.",
"So, $n>1$ is an odd integer.",
"Let $S$ be the (finite nonempty) set of prime divisors $\\ell $ of $n$ and $n=\\prod _{\\ell \\in S}\\ell ^{\\mathbf {d}(\\ell )}$ where all $\\mathbf {d}(\\ell )$ are positive integers.",
"Using Remark REF , we may replace if necessary $K_0$ by $K_{0,n}$ and assume that $\\tilde{G}_{\\ell ,J,K_0}=\\mathrm {Sp}(J[\\ell ],\\bar{e}_{\\ell })$ for all $\\ell \\in S$ .",
"The already proven case of prime powers tells us that $\\tilde{G}_{\\ell ^{d(\\ell )},J,K_0}=\\mathrm {Sp}\\left(J\\left[\\ell ^{\\mathbf {d}(\\ell )}\\right],\\bar{e}_{\\ell }\\right)$ for all $\\ell \\in S$ .",
"On the other hand, we have ${\\mathbb {Z}}/n{\\mathbb {Z}}=\\oplus _{\\ell \\in S}{\\mathbb {Z}}/\\ell ^{\\mathbf {d}(\\ell )}{\\mathbb {Z}}, \\ J[n]=\\oplus _{\\ell \\in S} J\\left[\\ell ^{\\mathbf {d}(\\ell )}\\right],$ $\\mathrm {Gp}(J[n], \\bar{e}_n)=\\prod _{\\ell \\in S} \\mathrm {Gp}\\left(J\\left[\\ell ^{\\mathbf {d}(\\ell )}\\right], \\bar{e}_{\\ell ^{\\mathbf {d}(\\ell )}}\\right), \\ \\mathrm {Sp}(J[n], \\bar{e}_n)=\\prod _{\\ell \\in S}\\mathrm {Sp}\\left(J\\left[\\ell ^{\\mathbf {d}(\\ell }\\right], \\bar{e}_{\\ell ^{\\mathbf {d}(\\ell )}}\\right),$ $\\tilde{G}_{n,J,K_0} \\subset \\prod _{\\ell \\in S}\\tilde{G}_{\\ell ^{\\mathbf {d}(\\ell )},J,K_0}=\\prod _{\\ell \\in S}\\mathrm {Sp}(J[\\ell ^{\\mathbf {d}(\\ell }], \\bar{e}_{\\ell ^{\\mathbf {d}(\\ell )}}).$ Recall that the group homomorphisms $\\tilde{G}_{n,J,K_0} \\rightarrow \\tilde{G}_{\\ell ^{\\mathbf {d}(\\ell )},J,K_0}=\\mathrm {Sp}\\left(J\\left[\\ell ^{\\mathbf {d}(\\ell }\\right], \\bar{e}_{\\ell ^{\\mathbf {d}(\\ell )}}\\right)$ (induced by the inclusion of the Galois modules $J\\left[\\ell ^{\\mathbf {d}(\\ell )}\\right]\\subset J[n]$ ) are surjective.",
"We want to use Goursat's Lemma and Ribet's Lemma [19], in order to prove that the subgroup $\\tilde{G}_{n,J,K_0}\\subset \\prod _{\\ell \\in S}\\mathrm {Sp}(J[\\ell ^{\\mathbf {d}(\\ell }], \\bar{e}_{\\ell ^{\\mathbf {d}(\\ell )}})$ coincides with the whole product.",
"In order to do that, we need to check that simple finite groups that are quotients of $\\mathrm {Sp}(J[\\ell ^{\\mathbf {d}(\\ell }]$ 's are mutually nonisomorphic for different $\\ell $ .",
"Recall that $\\mathrm {Sp}\\left(J\\left[\\ell ^{\\mathbf {d}(\\ell }\\right], \\bar{e}_{\\ell ^{\\mathbf {d}(\\ell )}})\\right) \\cong \\mathrm {Sp}_{2g}\\left({\\mathbb {Z}}/\\ell ^{\\mathbf {d}(\\ell )}{\\mathbb {Z}}\\right)$ and therefore is perfect.",
"Therefore, all its simple quotients are also perfect, i.e., are finite simple nonabelian groups.",
"Clearly, the only simple nonabelian quotient of $\\mathrm {Sp}_{2g}\\left({\\mathbb {Z}}/\\ell ^{\\mathbf {d}(\\ell )}{\\mathbb {Z}}\\right)$ is $\\Sigma _{\\ell }:=\\mathrm {Sp}_{2g}({\\mathbb {Z}}/\\ell {\\mathbb {Z}})/\\lbrace \\pm 1\\rbrace .$ However, the groups $\\Sigma _{\\ell }$ are perfect and mutually nonisomorphic for distinct $\\ell $ [1], [2].",
"This ends the proof.",
"Remark 6.9 Remark REF , Lemmas REF and REF , and their proofs remain true if one replaces the jacobian $J$ by any principally polarized $g$ -dimensional abelian variety $A$ over $K_0$ with $g \\ge 2$ ."
]
] | 1606.05252 |
[
[
"Common envelope events with low-mass giants: understanding the\n transition to the slow spiral-in"
],
[
"Abstract We present a three-dimensional (3D) study of common envelope events (CEEs) to provide a foundation for future one-dimensional (1D) methods to model the self-regulated phase of a CEE.",
"The considered CEEs with a low-mass red giant end with one of three different outcomes -- merger, slow spiral-in, or prompt formation of a binary.",
"To understand which physical processes determine different outcomes, and to evaluate how well 1D simulations model the self-regulated phase of a CEE, we introduce tools that map our 3D models to 1D profiles.",
"We discuss the differences in the angular momentum and energy redistribution in 1D and 3D codes.",
"We identified four types of ejection processes: the pre-plunge-in ejection, the outflow during the plunge-in, the outflow driven by recombination, and the ejection triggered by a contraction of the circumbinary envelope.",
"Significant mass is lost in all cases, including the mergers.",
"Therefore a self-regulated spiral-in can start only with a strongly reduced envelope mass.",
"We derive the condition to start a recombination outflow, which can proceed either as a runaway or a stationary outflow.",
"We show that the way the energy of the inspiraling companion is added to the envelope in 1D studies intensifies the envelope's entropy increase, alters the start of the recombination outflow, and leads to different outcomes in 1D and 3D studies.",
"The steady recombination outflow may dispel most of the envelope in all slow spiral-in cases, making the existence of a long-term self-regulated phase debatable, at least for low-mass giant donors."
],
[
"Introduction",
"A common envelope event (CEE) is an episode in the life of a binary system during which the outer layers of one of the stars expand to engulf the companion – thus producing an envelope around both stars, or a common envelope (CE).",
"The concept was originally proposed almost 40 years ago by [38] (this publication cites private communication with Ostriker as well as [48], for the origin of this idea).",
"The concept was later developed into the modern state of the energy formalism in [49] and [26].",
"This brief but crucial episode leads either to a complete merger, or to the expulsion of the CE, leaving a drastically shrunk (short-period) binary – a likely future gravitational wave source, or X-ray source, or SN Ia progenitor, etc.",
"The merged star can form a variety of exotic objects, or produce a long gamma-ray burst, one of the most luminous events known to occur in the Universe [24].",
"It is widely accepted that a CEE is the main mechanism by which an initially wide binary star is converted into a very close binary star, or by which two stars merge.",
"Despite being vital for understanding a vast number of important binary systems, CEE theory is presently very poorly understood.",
"Brute-force numerical simulations of CEEs are hard [44], [46], [39], [45], [36], [35], [37]; often, key elements of the event proceed not only on a dynamical timescale, but also on a thousand to million times longer thermal time-scale.",
"Various physical processes including radiation transfer, convective energy transport and nuclear burning can take place during a CEE [24].",
"Understanding the transition between a fast CEE that can only be modelled using a three-dimensional (3D) hydrodynamical code with simplified physics, and a slow CEE that can only be modelled with a one-dimensional (1D) stellar code, where various mechanisms for the energy transport, nuclear energy generation and other physics are included, was advocated to be the most important step for further progress in CEE studies [24].",
"Several research groups have recently restarted to carry out 1D studies of slow CEEs [31], [14], [29], [30].",
"Determining or constraining the initial conditions for 1D CEEs, starting from the moment when the dynamical phase ended, is crucially needed.",
"In addition, 1D simulations, starting from the pioneers [47], [32], are naturally forced to use either assumptions or prescriptions, which have not yet been verified against 3D simulations.",
"For example, prescriptions for the angular momentum transfer or its distribution, or how the released orbital energy is deposited in the envelope.",
"Understanding the initial conditions at the start of the slow spiral-in, however, can not be done without understanding how the dynamical phases proceeded: how the energy redistribution between the orbit and the envelope takes place in 3D, as compared to what 1D treatments we adopt, whether any envelope material was lost prior the slow spiral-in, and how important quantities (e.g., specific angular momentum and entropy of the material) have evolved by the end of the rapid evolution.",
"In this paper, we study the transition to the slow spiral-in by analyzing 3D simulations of CEEs with a low-mass giant donor and companions of several different masses.",
"The outcomes of those CEEs vary from a rapid and complete envelope ejection to the strong binary shrinkage that can be indicative of a merger (§ ).",
"We are most interested in understanding the start of the intermediate regime, when the binary orbital dissipation is not dynamical anymore, but the envelope has not yet been ejected – the regime that 1D codes are best suited to model.",
"To provide a transition between our results and 1D simulations, we discuss several ways to map the 3D CEE simulations to 1D (§ ).",
"We compare the results of this mapping with the results of the simplified assumptions and prescriptions that 1D codes have used to study slow CEEs.",
"We also use our mapped simulations to analyze the physical processes during a transition to a slow spiral-in, to find clues about what leads to the branching of the outcomes of CEEs, to consider how strong the asymmetry of the CE at the start of a slow CEE is, and to discuss the challenge with density inversion that a 1D approach may have.",
"We provide a detailed technical description of how energies are considered in the 1D and 3D approaches (§ ).",
"This leads to a discussion of how the two kinds of energies can be compared between the approaches and whether the simplified prescription is valid during the transition from fast to the slow spiral-in.",
"We also provide an equation, and its limitations, for how energy conservation should be used in the 1D approach after the start of the slow spiral-in.",
"We analyze the angular momentum redistribution in 3D simulations and compared the results to existing 1D approaches (§ ).",
"We discuss in § how entropy generation is different between the 3D and 1D approaches.",
"We argue that the currently-used 1D treatments of energy conservation via “heating” lead to a different outcome compared to 3D simulations.",
"In particular, the usual 1D treatment leads to a different entropy profile at the start of the slow spiral-in.",
"We discuss next how recombination governs the CE ejecta (§ ).",
"We then advocate that artificially forced entropy generation, coupled with recombination, may produce unrealistic outcomes in 1D codes.",
"Finally, we consider the ejecta, identify several types of ejecta at various CEE phases, and discuss the physics behind each type of ejecta (§ ).",
"We conclude with several recommendations for 1D codes that can improve the modeling of CEEs in 1D (§ ).",
"In this paper, we adopt the definitions of the distinct phases of a CEE as described by [43], see also the thorough details in the review by [24].",
"For a more quantitative differentiation of the phases, we indicate here the typical rates of the orbital dissipation: I.",
"Loss of corotation.",
"During this phase, the change in the orbital separation $a$ is less than one per cent over the orbital period $P_{\\rm orb}$ , $|(\\dot{a} P_{\\rm orb})| /a < 0.01$ .",
"The companion orbits either outside of the future CE, or inside the CE's outer expanded and rarefied layers.",
"II.",
"Plunge-in.",
"This is the fastest phase of the orbital shrinkage, when the rate of change of the orbital separation is large, $|(\\dot{a} P_{\\rm orb})|/a 0.1$ .",
"The companion plunges inside the CE, and, at the end of the phase, most of the CE mass is outside the companion's orbit.",
"During the plunge-in, the concept of a Keplerian binary orbit is equivocal.",
"III.",
"Self-regulating spiral-in.",
"During this stage the change in the orbital energy $ E_{\\rm orb}$ is small, $ | (\\dot{E}_{\\rm orb}P_{\\rm orb}) / E_{\\rm orb}| < 0.01$ .",
"The companion orbits inside the CE.",
"With a 3D code, a proper self-regulating regime is not possible to achieve – a 3D code typically lacks consideration of energy transport that operates on a thermal timescale [37], among other effects.",
"Therefore in this paper we will rather consider the initial phase of the self-regulating spiral-in, to which we will refer here as the “slow spiral-in”.",
"We define that the slow spiral-in starts with a similar criterion as the self-regulating spiral-in, i.e.",
"when $| (\\dot{E}_{\\rm orb} P_{\\rm orb}) / E_{\\rm orb}| < 0.01$ .",
"In our simulations, this stage is modelled for the first few dynamical timescales of the expanded envelope after the plunge-in has ended.",
"On the other hand its duration can be compared to about several thousands orbital periods of the formed close binary.",
"A well-established slow spiral-in – thousands binary orbits after the start of the slow spiral-in – could be quite different from the start of the slow spiral-in for the region around the binary.",
"This is becuase this region's dynamical timescale is comparable to the binary period, but has evolved for many thousands of the binary periods.",
"On the other hand, it is still in the initial phase of the self-regulated spiral-in as the modelled time is smaller than the thermal timescale of the envelope.",
"For the definition of phase III above (as well as for the slow spiral-in), we use the orbital energy dissipation as an indicator, rather than the orbital separation decrease.",
"This is because in our code during this period the orbital energy of the now close binary is nearly constant, as expected, but the binary has a non-zero, albeit small, eccentricity, and the decrease of its average orbital separation is harder to obtain numerically.",
"Note that due to the adopted quantitative classification of the principal phases as described above, there are also transitional phases: between the end of the corotation and the start of the plunge-in, and between the end of the plunge-in and the start of the slow spiral-in."
],
[
"Numerical setup of simulations",
"To study a CEE from before the Roche lobe overflow and to a well-established slow spiral-in, we use STARSMASHER, a Smoothed Particle Hydrodynamics (SPH) 3D code [28], [12], [27].",
"An SPH approach is appropriate to model the interaction between two stars without imposing boundary conditions, and is commonly used in the community for this class of problems [7], [44], [5], [39].",
"The code STARSMASHER was originally developed to treat interactions between two stars, and has been frequently used in studies of this class of problems [12], [22], [2], [27], [18], [36], [35], [42].",
"As an example, one of STARSMASHER's important features, developed specifically for studies of binaries, is the specially designed relaxation setup of a close-to-contact binary [27].",
"This relaxation procedure minimizes spurious effects of artificial viscosity that may affect the start of the spiral-in.",
"STARSMASHER's internal physics has been recently upgraded to take into account recombination processes [35].",
"The modification was done by replacing the code's default equation of state that includes ideal gas and radiation pressure [28] by the tabulated equation of state from MESA that accounts for states of ionization [40].",
"The version of SPH code we use evolves specific internal energy of an SPH particle $u_i$ and density of an SPH particle $\\rho _i$ [12], and pressure then is found from the internal energy, density, and the adopted equation of state.",
"It is this implementation of the more complete equation of state that has enabled modelling of the complete CE ejection , for the first time.",
"For this study of the branching of CEE outcomes, we chose for a donor a low-mass red giant with a mass of $1.8~M_\\odot $ , a core mass of $0.318~M_\\odot $ , and radius $16.3~R\\odot $ (for more details on the ambiguity in the definition of the donor radius in 3D, see the discussion in [36]).",
"To create the initial red giant donor star, we use the TWIN/EV stellar code [8], [9].",
"To obtain different categories of CEE outcomes, we varied the mass of the companion, $M_{\\rm comp}$ , considering $0.36, 0.20, 0.15, 0.10$ and $0.05 M_\\odot $ companions.",
"At the start of the simulations, donors in all binaries are within their Roche Lobes (RLs).",
"We consider the case of non-synchronized donors.",
"While synchronization has a small effect on the outcomes , we chose to start with non-synchronized binaries as it significantly speeds up the start of the interaction.",
"The red giant envelope is modelled with $10^5$ particles.",
"The red giant core and the companions are modelled as point masses, where a point mass only interacts gravitationally with normal SPH particles.",
"Such point masses are also referred to as special particles.",
"In an SPH code, the gravitational potential equation contains an extra smoothing term: the smoothing length $h_i$ [17].",
"In our simulations, for $10^5$ particles, a smoothing length for the red giant core is $0.35 R_\\odot $ [27].",
"In previous studies, a larger number of particles has been used to represent the donor [39].",
"However, the modelled phase of a CEE was substantially shorter.",
"For example, [39], have started their simulations with placing the companion on the surface of the donor, and have finished their simulations at the moment that we define as the start of the slow spiral-in.",
"A similar phase in our simulations takes only between 1 and 2 per cent of the computational time.",
"About 10 per cent of the computational time is spent before the plunge-in starts, and about 90 per cent of the computational time is spent after the plunge-in, until the slow spiral-in is “well-established.” The simulations take on average $5\\times 10^6$ time-step integrations.",
"This allows us to follow about 15,000-35,000 binary periods after the end of the plunge-in.",
"The physical timescale of the simulations is about 1000 days.",
"While it would be great to model a donor at better resolution, it is not computationally feasible yet to get both a long-term evolution of a CEE, and model it with a resolution substantially larger than $10^5$ .",
"The resolution test for STARSMASHER with the original equation of state was performed by [18].",
"As compared to that version of the code, only the equation of state has been changed.",
"[35] carried out STARSMASHER CEE simulations with $10^5$ and $2\\times 10^5$ particles in an attempt to test resolution effects.",
"The two simulations were done for the version of the code that includes the recombination physics.",
"The test has shown that the final orbital separation varies by only a few per cent and that both simulations produce a similar envelope ejection (the ejected mass mass, the timescale of the ejection, etc).",
"While doubling resolution is not sufficient to thoroughly test convergence, this is the most that can be achieved at presentA run with $8\\times 10^5$ particles would require 8-10 GPU years with NVIDIA driver M2070, as was tested.",
"The use of K40, which is currently the fastest NVIDIA driver in the world would reduce GPU time by 25 per cent.",
"Given the low communication speed between the GPU nodes (hardware limitation, and thus the scaling is only effective up to 4 GPUs and 16 CPUs), and the cumbersome queue setup, this run would require more than 2 years of waiting time in the real world, if started at the available Westgrid GPU clusters..",
"This test suggests that most phenomena that are discussed in this manuscript are not likely to be rebutted by a larger resolution run.",
"We warn however that some results presented in this paper should be taken with caution, as future studies made with a substantially larger resolution may negate the phenomenon that is produced by a small number of particles – specifically, the shell-triggered ejection discussed later in § .",
"The orbital energies and the total energies of the CE systems for each companion at the start of the simulations are shown in Table REF .",
"The donor's envelope has an initial total binding energy $-4.4\\times 10^{47}$ ergs, an initial potential energy $-8.8\\times 10^{47}$ ergs, an initial thermal energy $4.4\\times 10^{47}$ ergs (without recombination energy), and additionally $4.7\\times 10^{46}$ ergs is stored as recombination energy (see more details on how the recombination energy is found in § 7).",
"We distinguish the unbound envelope material, $M_{\\rm ej}$ , and the currently bound envelope material, $M_{\\rm env}$ .",
"These masses are found using the technique described in [36].",
"Principally, if an SPH particle has negative total energy, it is bound.",
"If an SPH particle has positive total energy, it is unbound and belongs to the ejecta.",
"In Table REF , we provide the final values at the end of the simulations, but note that the mass of the ejecta can not be simply explained with one number, and more details about the ejecta will be given in § .",
"During a CEE, a CE system can be described in terms of various energies: the binding energy of the envelope $E_{\\rm bind, env}$ , the internal energy of the envelope $U_{\\rm in,env}$ , the thermal energy of the envelope $E_{\\rm th,env}$ , the recombination energy of the envelope $E_{\\rm rec,env}$ , the potential energy of the envelope $E_{\\rm pot,env}$ , the kinetic energy of the envelope, $E_{\\rm kin,env}$ , the orbital energy of the binary $E_{\\rm orb}$ , the total energy of the ejecta $E_{\\rm tot,ej}$ , and the kinetic energy of the ejecta $E_{\\rm kin, ej}$ .",
"We can also trace the angular momentum of the envelope $J_{\\rm env}$ , the orbital angular momentum of the binary $J_{\\rm orb}$ and the angular momentum of the ejecta $J_{\\rm ej}$ .",
"Details of how those quantities are obtained from our 3D simulations can be found in § REF and Appendix .",
"Values of the most important quantities at the start and the end of each simulation can be found in Table REF .",
"Note that in Table REF , energies have the index “3D”, as they are obtained assuming 3D energy definitions (see § REF ), and can be different from those inferred by the definitions of the 1D approach (see § REF ).",
"We clarify that in all our simulations presented in this paper the total angular momentum and the total energy are conserved (the error on energy conservation is less than 0.1% of the initial total energy, and the error on angular momentum conservation is less than 0.001% of the initial total angular momentum)."
],
[
"CEE outcomes",
"We classify the outcomes of our 3D simulations of CEEs as: $\\bullet $ Binary formation – if the CE is ejected and $|(\\dot{E}_{\\rm orb} P_{\\rm orb}) / E_{\\rm orb}| < 0.01$ .",
"$\\bullet $ Slow spiral-in – if the CE has not been fully ejected and no further rapid mass outflow of the envelope material is taking place on a timescale longer than a few dynamical timescales of the expanded CE (although, see § about the shell-triggered ejecta).",
"During this stage, the orbital energy release is decreased to $|(\\dot{E}_{\\rm orb} P_{\\rm orb}) / E_{\\rm orb}| < 0.01$ .",
"$\\bullet $ Merger – if the orbital separation is $<0.15R_\\odot $ (see discussion below on the ambiguity of a merger case).",
"Figure: Evolution of orbital separations in the binaries with 0.05,0.10,0.15,0.200.05,0.10, 0.15, 0.20 and 0.36M ⊙ 0.36 M_\\odot companions.",
"On the left panel, the time axis isshifted to show the relative orbital evolution in more detail for each system.",
"Thetime shifts are 520,550,558,510520, 550, 558, 510 and 270 days for 0.05,0.10,0.15,0.200.05,0.10, 0.15, 0.20 and 0.36M ⊙ 0.36 M_\\odot companions,respectively.",
"On the right panel, the time axis is not shifted.",
"Please note that the time that has passedfrom the start of the simulation to the start of the plunge-in depends strongly on the degree of the initial Roche lobe underflow, and that quantity is slightly different in the simulations.The dashed line indicates where the companion and thered giant core will definitely merge.",
"We note that the apparentnon-periodic pattern of the orbital evolution, especially noticeablein the case of the 0.36M ⊙ 0.36 M_\\odot companion, is because we can only storeevery 10th model when we run a long simulation.",
"We have checked that ifwe store every model for a period of time, theapparent pattern disappears, and during the slow spiral-in eachbinary has constant, albeit slowlydecreasing, orbital period, as expected.",
"The orbital energy does notoscillate, but due to the nonzero eccentricity, the orbital separation, measured at particular times, shows oscillations.The start of the displayed orbital evolution on the left panel corresponds to the moment of time when the orbital decay is increasing to |(a ˙P orb )|/a=0.03|(\\dot{a} P_{\\rm orb})| /a = 0.03 for the binary with 0.05M ⊙ 0.05 M_\\odot companion, |(a ˙P orb )|/a=0.01|(\\dot{a} P_{\\rm orb})| /a = 0.01 for the binaries with 0.10,0.150.10, 0.15 and 0.20M ⊙ 0.20 M_\\odot companions, and |(a ˙P orb )|/a=0.005|(\\dot{a} P_{\\rm orb})| /a = 0.005 for the binary with 0.36M ⊙ 0.36 M_\\odot companion.For each case, the start of the plunge-in, when |(a ˙P orb )|/a|(\\dot{a} P_{\\rm orb})| /a becomes greater than 0.1, is indicated with a star symbol.The end of the plunge-in, when |(a ˙P orb )|/a|(\\dot{a} P_{\\rm orb})| /a becomes less than 0.1, is indicated with a circle symbol.The start of the slow spiral-in, when |(E ˙ orb P orb )/E orb || (\\dot{E}_{\\rm orb} P_{\\rm orb}) / E_{\\rm orb}| becomes less than 0.01, is indicated with a square symbol.A triangle symbols indicates when |(E ˙ orb P orb )/E orb || (\\dot{E}_{\\rm orb} P_{\\rm orb}) / E_{\\rm orb}| becomes less than 10 -4 10^{-4}.In Figure REF we show the evolution of the orbital separation for all cases.",
"We have obtained all three possible outcomes: mergers (with companions of mass $0.05$ and $0.1M_\\odot $ ), slow spiral-ins (with companions of mass $0.15$ and $0.2M_\\odot $ ) and binary formation ($0.36M_\\odot $ companion).",
"The quantities that describe the simulations are shown in TableREF .",
"We provide these values when the separation becomes $0.15 R_\\odot $ , or at 50 days after the end of the plunge-in.",
"For the binary formation case, we also list in TableREF values for when the entire envelope is lost (about 700 days after the plunge-in has ended).",
"We also indicate in Figure REF the start of the plunge-in, the end of the plunge-in, and the start of the slow-spiral-in for all the cases.",
"For example, in the simulation with a $0.15~M_\\odot $ companion, at the end of the plunge-in – when the orbital separation stops changing quickly – the binary separation is about three times larger than during the “well-established” slow spiral-in (50 days after the end of the plunge-in), and is about two times larger than at the start of the slow spiral-in.",
"The transition between the end of the plunge-in and the start of the slow spiral-in lasts for only about 3 days, which is nonetheless about 100 binary orbits.",
"It takes another 400 binary orbits before the orbital decay decreases to $| (\\dot{E}_{\\rm orb} P_{\\rm orb}) / E_{\\rm orb}| \\sim 10^{-4}$ .",
"At about 50 days after the end of the plunge-in, or after about 3000 binary orbits, $| (\\dot{E}_{\\rm orb} P_{\\rm orb}) / E_{\\rm orb}| \\sim 10^{-5}$ .",
"For the case when the “binary formation” occurs in the simulation, half of the envelope is ejected soon after the end of the plunge-in (on day 374 after the start of the simulation), and it takes about 700 more days to steadily eject the rest of the envelope.",
"We note that before the envelope is fully ejected, this “binary formation” case is not substantially different from the other two cases that we designate here as “slow spiral-in” cases, except for the fact that the CEE evolution is faster with the $0.36~M_\\odot $ companion.",
"We call a “merger” all simulated systems for which the geometrical separation between the core and the companion is $0.15 R_\\odot $ .",
"For a non-degenerate companion, this limit on mergers is naturally consistent with the radii of very low-mass stars.",
"For a degenerate companion, technically, the simulations can be forced to run until the separation becomes $0.01~R_\\odot $ .",
"However, there are other reasons to stop the simulation earlier.",
"In an SPH code, if the distance between the two point masses is less than two times their smoothing lengths, then there is an extra smoothing in the gravitational potential equation [17].",
"Then the orbital energy is not the same as would be found from their geometrical separation.",
"In addition, the orbital energy depends also on the presence of the usual SPH particles within the RLs of the special particles (see below § REF for more details).",
"Note that these effects take place while the total energy is conserved; the portions of the energy assigned to the binary, and to the envelope, become dependent on the smoothing length.",
"We find that a noticeable (about a few per cent) mismatch between the geometric orbital separation and the “energy”- derived orbital separation starts to appear when the distance between the special particles is less than $\\sim 0.15 R_\\odot $ .",
"[34].",
"Accordingly, the potential energy (by its absolute value) is not as large as would be calculated purely by the distance between the particles, if a Keplerian orbit were assumed.",
"As we have checked, decreasing the smoothing length by a factor of 2 and increasing the number of particles by a factor of 8 improves the consistency, but leads to an increase of the computational time by a factor of 64.",
"This makes the problem currently computationally unfeasible.",
"The CEEs with the companions of $0.05 M_\\odot $ and $0.1 M_\\odot $ have therefore been assigned to be “mergers”, while in Nature, if the companions are compact, CEEs in these binaries could result in a slow spiral-in with an orbit that is smaller than our cut-off distance $0.15 R_\\odot $ .",
"In the following Sections, we will discuss the differences in the processes that may lead to these three types of the outcomes.",
"We will refer to the binary formation case with $0.36 M_\\odot $ companion as BF36, to the slow spiral-in cases with $0.20 M_\\odot $ and $0.15M_\\odot $ companions as SS20 and SS15, and to the merger cases with $0.10 M_\\odot $ and $0.05 M_\\odot $ companions as M10 and M05.",
"In this Section we introduce the mapping tools that convert the results of 3D simulations to 1D space, both for the analysis of physical processes that will be done in this paper, and for the possible future comparisons with 1D simulations of CEEs.",
"The outcomes of hydrodynamical simulations are not fully symmetrical, while a typical evolutionary stellar code deals with a spherically symmetric 1D star.",
"In the 1D case, the centre of symmetry is naturally located at the centre of the modelled star, in its core.",
"The centre of a 3D object during and after a CEE is not that indisputable.",
"During the Phase I only the outer layers are perturbed, and the donor star mainly keeps its original symmetry around the core.",
"During the dynamical Phase II, no symmetry can be expected.",
"Finally, during the Phase III, symmetry – from the dynamic point of view – is expected to form around the centre of mass of the binary that rotates inside the CE.",
"However, this phase may continue long enough that the energy transport starts to affect the envelope's temperature and density profile (self-regulated spiral-in).",
"On the one hand, the energy transport will be symmetric around the nuclear energy source, if that nuclear energy source (e.g., shell burning) is still active.",
"On the other hand, the binary's orbital energy release may stem from a different point and shift the average centre of symmetry, as the energy generation rate can outpace the nuclear energy source.",
"It is inevitable therefore that no true spherical symmetry can exist, but we can evaluate how the choice of the centre of symmetry can affect the anticipated 1D spherically symmetric stellar structure.",
"The mapping of our 3D object to 1D object is done by using a mass weighted average of a given quantity over a spherical shell of radius, $r$ and thickness, $\\delta r$ : $\\bar{X} (r) = \\frac{ \\sum _{k} x_k m_k }{\\sum _k m_k}$ Here $\\bar{X} (r) $ is a discrete (mesh-type, similar to 1D stellar profiles) profile of the quantity $X$ , the mesh-points spaced by $\\delta r$ .",
"The variable $x_k$ is the value of the quantity $X$ that an SPH particle $k$ has, $m_k$ is the mass of the same SPH particle $k$ , and $k$ is the index of the SPH particle.",
"The summation goes over all particles $k$ such that the distance from each particle to the chosen centre, $r_k$ , is between $r$ and $\\delta r$ .",
"We compare the two choices for the centre of symmetry, the red giant core and the centre of mass of the binary during a well-established spiral-in, i.e.",
"the formed binary has made a few thousand orbits.",
"We find that for the layers outside of the orbit, the choice of the centre of symmetry is less significant, and the differences between the averaged values are within a few per cent (see Fig REF ).",
"The large number of the periods is expected to result in a spin-up of most of the mass that is located within a few binary orbits around the centre of mass of the binary.",
"Indeed, we note that independently on the centre's choice, there is almost no stellar material within a few orbits around the centre of mass of the binary.",
"For example, consider the case of simulation SS15 at the moment when 122 days have passed after the the start of the slow spiral-in, or about 8000 binary periods.",
"The orbital separation at that time is $\\sim 0.2 R_\\odot $ .",
"Except for the only two SPH particles that are within the RL of the companion, and two more particles that are located at about $6 R_\\odot $ , all the other SPH particles are located at least beyond $8 R_\\odot $ from the binary, creating a pronounced density inversion around the binary for about 40 orbital separations (see Fig.",
"REF ).",
"We note that the density inversion was observed in previous 3D CEE simulations, for example see the study by [39].",
"In their work, the CEE simulations were done with the two types of the numerical methods and compared with an SPH type code and with a grid-based code.",
"We note that there, the density inversion was more pronounced in simulations with an SPH code, than with a grid based code, and we attribute it to the reason we described above.",
"We should however stress that in our case, we discuss the state of the CE system when the formed close binary has revolved for about 8000 orbital periods, affecting its neighborhood, while in [39] the density profiles are made for a moment when the formed binary has made less than 50 orbital periods.",
"Using our definitions of the CEE phases as in §2, their binary has just started the slow spiral-in and has had less “relative” time to spin-up its neighboring SPH particles.",
"At the similar stage of the CEE – the start of the spiral-in – our envelope had not yet formed a hollow shell.",
"We note as well that in 1D CEE simulations such a strong density inversion and core stripping have never been reproduced [19].",
"This is likely relevant to the fact that 1D stellar codes, when modelling a CEE, add the energy as a “heat” instead of the mechanical energy (we will discuss how the form of the added energy affects the outcomes in § and § ).",
"Another plausible reason is that the tidal spin-up of the envelope from the binary is not taken into account as it should be, and the spin-up is deposited at an incorrect location.",
"We anticipate that the decoupling between the binary and the envelope might be partially a numerical problem, due to the small number of particles that are left near the binary, and the smoothing length in that region.",
"Unfortunately, decreasing the mass of SPH particles by a factor of two, increasing thereby a simulations's resolution by a factor of two, we run in the same problem when decreasing the smoothing length to obtain a better value for the gravitational potential, as was discussed in § .",
"Nonethless, we believe that the formation of a hollow shell is a physical phenomenon by virtue of a tidal spin-up that operated during many binary timescales.",
"We can also compare profiles of thermodynamic quantities for “polar” SPH particles (those particles that have a polar angle within $\\vartheta <25^0$ around the zenith direction that goes through the red giant core, and for negative $Z$ , $\\vartheta >155^0$ ), and for “equatorial” SPH particles (those that are located close to the equatorial plane and have their polar angle within $75^0<\\vartheta <105^0$ ).",
"To be more specific, the mass-radius relation for the coordinates is the same as for the averaging of the whole 3D object, but the average values of any quantity are found only for the particles that are within these polar angles.",
"We find that, as expected, at the end of the plunge-in the asymmetry is strong – up to an order of magnitude difference in values that can be found in polar and equatorial directions (see Fig REF , the left panel).",
"The most pronounced deviations are for the polar direction, where density and pressure are substantially lower than their averages over all angles in the proximity of the binary orbit and at large radii, while denser in between.",
"This is likely because of the active ongoing “outflows” in the equatorial direction, which are compensated by somewhat slower “inflows” in the polar direction.",
"During a slow spiral-in, when the binary orbit almost does not decay during several dynamical timescales of the expanded envelope (e.g., 40 days in case SS15), non-negligible asymmetry is still present.",
"Qualitatively, the density profiles in opposite directions are consistent with deviations expected from rotation.",
"Relative differences for the density in polar and equatorial directions in the extended envelope, still can reach a factor of a few, with the density in the equatorial direction being smaller (when taken at the same distance from the red giant core)."
],
[
"Energies definitions in a 1D approach",
"In the 1D approache, the gravitational potential, or the specific gravitational potential energy in the envelope during a CEE is $\\phi _{\\rm env 1D}(r)= - \\left( \\frac{Gm(r)}{r} + f_{\\rm ins} \\frac{GM_{\\rm comp}}{r}\\right) \\ .$ Here $m(r)$ is the local mass coordinate within the star (excluding the companion), $r$ is the radial coordinate, $f_{\\rm ins}$ indicates the effect of the companion, and $f_{\\rm ins}=1$ if the companion orbits within $r$ , or $f_{\\rm ins}=0$ if the companion orbits outside $r$ .",
"The origin of the second term in this equation will be explained in more detail in §REF .",
"The potential energy of the envelope $E_{\\rm pot,env1D}$ [20], the internal energy of the envelope $E_{\\rm int,env1D}$ (this energy consists of the thermal energy of the envelope $E_{\\rm th,env1D}$ and the recombination energy of the envelope $E_{\\rm rec,env1D}$ ), the binding energy of the envelope, and the kinetic energy of the envelope $E_{\\rm kin,env1D}$ , are defined as [24] $E_{\\rm pot, env 1D} &=& - \\int _{M_{\\rm core}}^{M} \\left(\\frac{Gm}{r} + f_{\\rm ins} \\frac{GM_{\\rm comp}}{r}\\right) \\, dm\\ ; \\\\E_{\\rm th, env 1D} &=&\\int _{M_{\\rm core}}^{M} e_{\\rm th}\\, dm\\ ; \\\\E_{\\rm rec, env 1D} &=&\\int _{M_{\\rm core}}^{M} \\varepsilon _{\\rm rec}\\, dm ; \\\\U_{\\rm int, env 1D} &=&\\int _{M_{\\rm core}}^{M} u\\, dm = E_{\\rm th, env 1D} + E_{\\rm rec, env 1D}\\ ; \\\\E_{\\rm bind, env 1D} &=& U_{\\rm int, env 1D}+E_{\\rm pot, env 1D}\\ ; \\\\E_{\\rm kin, env 1D} &=&\\int _{M_{\\rm core}}^{M} 0.5 V^2 \\, dm \\ .$ Here $M$ is the total mass of the star, $m$ is the local mass coordinate, $u$ is the specific internal energy and $u=e_{\\rm th}+ \\varepsilon _{\\rm rec}$ , $\\varepsilon _{\\rm rec}$ is the specific recombination energy, and $e_{\\rm th}$ is the specific thermal energy for which no recombination energy is taken into account.",
"$V$ is velocity; note that in 1D, velocities for $E_{\\rm kin, env 1D}(M_{\\rm core})$ do not include the donor's movement on its binary orbit, but only the relative velocities in the corotation frame.",
"The recombination energy $E_{\\rm rec, env 1D}$ is static potential energy, which is not available immediately and is only released as a response of the envelope on its expansion.",
"This energy release may or may not be triggered during a CEE [25]."
],
[
"Energies definitions in a 3D SPH code",
"In a 3D approach, the gravitational potential that a particle $i$ has is defined as $\\phi _i=\\sum _{l\\ne i}^N m_l \\varphi _{i,l}\\ ,$ where $\\varphi _{i,l}$ is the gravitational potential between two SPH particles of unit mass that have a distance between them $|\\mathbf {r}_i-\\mathbf {r}_l|$ [17].",
"The potential, internal, recombination, and kinetic energies of an SPH particle are: $E_{{\\rm pot},k}&=&m_k \\phi _k,\\\\U_{{\\rm int},k}&=&m_k u_k, \\\\E_{{\\rm rec},k}&=&m_k \\varepsilon _{\\rm rec}, \\\\E_{{\\rm kin},k}&=&0.5 m_k (v_{x,k}^2+v_{y,k}^2+v_{z,k}^2) \\ .$ Here $v_{x,k}$ , $v_{y,k}$ , and $v_{z,k}$ are integrated from the equations of motion (i.e., they are calculated with respect to the coordinate system).",
"The variable $u_k$ in our SPH code is integrated over time using the equation of thermal energy change [33], [12], an implementation that guarantees conservation of total energy and entropy in the absence of shocks.",
"$\\varepsilon _{\\rm rec}$ is a component of $u_k$ .",
"The total energy of a CE system that consists of the envelope (the gas that is not yet ejected to infinity), the donor's core and the companion, is: $E_{\\rm tot,CE}&=&\\sum _k^N (0.5E_{{\\rm pot},k}+U_{{\\rm int},k}+E_{{\\rm kin},k}) \\\\ \\nonumber & = &\\sum _k^N 0.5m_k \\sum _{l\\ne k}^N m_l \\varphi _{k,l}+ U_{{\\rm int,CE}}+E_{{\\rm kin,CE}} \\ .$ Here the summations goes over all SPH particles that are still gravitationally bound to the CE system.",
"The total internal energy of the CE system is the same as the energy of the envelope, $U_{{\\rm int,CE}}=U_{{\\rm int,env3D}}$ .",
"The first term in the Equation (REF ) is the total potential energy of the CE system: $E_{\\rm pot,CE} = \\sum _k^N 0.5m_k \\sum _{l\\ne k}^N m_l \\varphi _{k,l} \\ .$ The potential energy of the envelope consists of such components as the potential energy between the envelope and the core $E_{\\rm pot,e-core}$ , the self-gravitating energy of the envelope $E_{\\rm pot,e-sg}$ , and the potential energy between the envelope and the companion $E_{\\rm pot,e-comp}$ : $E_{\\rm pot,env 3D} &= &\\sum _k^{\\rm Env} m_k \\bigg (M_{\\rm core} \\varphi _{k,\\rm core} + 0.5 \\sum _{l\\ne k}^{\\rm Env} m_l \\varphi _{k,l} \\nonumber \\\\& +& M_{\\rm comp} \\varphi _{k,\\rm comp} \\bigg ) \\ ,$ where the summation is only for the normal (non-special) particles belonging to the envelope.",
"The potential energy of the CE system is then: $E_{\\rm pot,CE} = E_{\\rm pot,env 3D} \\nonumber + M_{\\rm comp} M_{\\rm core} \\varphi _{\\rm comp,\\rm core} ,$ and the total energy of the CE system is $E_{\\rm tot,CE} &=& U_{\\rm int,env3D} + E_{\\rm pot,env 3D} \\nonumber \\\\ &+& E_{\\rm kin, CE} + M_{\\rm comp} M_{\\rm core} \\varphi _{\\rm comp,\\rm core} ,$ The “orbital energy” of the binary system is : $E_{\\rm orb, 3D}&=& 0.5\\mu |V_{12}|^2 + 0.5 \\sum _i^{\\rm RL1, RL2} m_i\\phi _i \\\\&-& 0.5 \\sum _j^{\\rm RL1} m_j\\phi _j^{\\rm RL1}- 0.5 \\sum _k^{\\rm RL_2} m_k\\phi _k^{\\rm RL_2} \\ , \\nonumber $ where $\\mu =M_1M_2/(M_1+M_2)$ is the reduced mass, and $\\vec{V}_{12}=\\vec{V}_1-\\vec{V}_2$ is the relative velocity of the two stars.",
"The first term gives the kinetic energy.",
"The second term is the gravitational energy of the binary, with the sum being over all particles $i$ that are inside the two immediate RLs, where each RL depends on the mass of all particles that are bound to the whole mass within that RL, and are not simply functions of the masses of the RG core and the companion.",
"The third and the fourth terms correspond to the removal of the self-gravitational energy of the donor (the sum being over particles $j$ within the RL of the star 1) and of the companion (the sum being over particles $k$ within the RL of the star 2), respectively.",
"The caveat here of course is that during a plunge-in, there is no orbit, and the orbital energy found using Equation (REF ) is not related to the separation between the particles.",
"It can be seen that the orbital energy $E_{\\rm orb, 3D}$ and the potential energy of the envelope $E_{\\rm pot,env 3D}$ have some similar terms – the potential energy between the particles in the RL of the companion and the core, the potential energy between the particles in the RL of the core and the companion, and the potential energy between the particles in the two different RLs.",
"That is because a particle, when it is inside the RL, is both part of the envelope and of the binary.",
"Therefore, it is not possible to decompose the total potential energy of the CE system into the intrinsic potential energy of the binary and the potential energy of the envelope.",
"Instead, we can introduce the reduced orbital energy, where only the core and the companion are considered: $E_{\\rm orb,CC} &=& 0.5 M_{\\rm core} v_{\\rm core}^2 + 0.5 M_{\\rm comp}v_{\\rm comp}^2 \\nonumber \\\\&+& M_{\\rm comp} M_{\\rm core} \\varphi _{\\rm comp,\\rm core} \\ .$ Note that this energy cannot in principle determine the current orbital separation of the binary.",
"We stress that to find the orbital separation $a$ after the plunge using energy, one has to use $E_{\\rm orb, 3D}$ ; during the plunge, there is no orbital separation - energy relation.",
"We also introduce the kinetic energy of the envelope $E_{\\rm kin,env3D}& =& 0.5 \\sum _k^{\\rm Env} m_k v_k^2\\\\ &= &E_{\\rm kin, CE}- 0.5 M_{\\rm core} v_{\\rm core}^2 - 0.5 M_{\\rm comp}v_{\\rm comp}^2 \\nonumber \\ .$ Unlike the 1D case, this energy is non-zero even at the beginning, since the star's envelope rotates with the binary.",
"Finally, we can rewrite the energy equation of the CE system as: $E_{\\rm tot,CE}= E_{\\rm orb,CC} + U_{\\rm int,env3D} + E_{\\rm pot,env3D} + E_{\\rm kin,env3D}\\ .$ Note that in a 3D code, the total energy of all particles (including those that are unbound) is conserved but this is not so for $E_{\\rm tot,CE}$ .",
"Hence this equation can serve as the energy conservation only if no particles have become ejecta."
],
[
"The energy conservation in 1D",
"Let us transfer the energy conservation Equation (REF ) to 1D.",
"As previously, this is for the case when there is no ejecta.",
"1.",
"The internal energy: $U_{\\rm int,env3D} &=& U_{\\rm int,env1D}\\ .$ 1.",
"The orbital energy can be described using binary orbital energy: $E_{\\rm orb,CC} &=& - \\frac{GM_{\\rm core}M_{\\rm comp}}{2a} \\ .$ Note that this is not valid during a plunge, where instead Equation (REF ) should be used.",
"2.",
"The kinetic energy.",
"There is no simple direct link between $E_{\\rm kin,env3D}$ and $E_{\\rm kin,env1D}$ .",
"In the first case, one measures velocities in a stationary reference frame that moves with the centre of mass, and in the second case, velocities are measured in the corotation frame.",
"Before the plunge, $E_{\\rm kin,env3D}\\simeq {GM_{\\rm env} M_{\\rm comp}^2}/{(2a (M+M_{\\rm comp}))}$ , and after the plunge, if the angular velocities in the spun-up envelope are taken into account for 1D, $E_{\\rm kin,env3D}\\simeq E_{\\rm kin,env1D}$ .",
"4.",
"The potential energy.",
"In 1D, an added mass inside the envelope (an orbiting companion) technically produces the same potential as a “thin spherical shell.” A “thin spherical shell” does not create a potential inside of it, nor does it create a gravitational force that would act on an object inside of it (Newton's shell theorem).",
"This is why the potential energy in 1D is written as in Equation (REF ) (in §REF we also will discuss what error is produced by the thin shell approximation in Equation (REF ), i.e., for the CE that is outside of the companion's orbit).",
"The difference between 3D and 1D potential energy should include $E_{\\rm pot,e-comp}^{\\rm out}$ , which is the fraction of $E_{\\rm pot,e-comp}$ that sums only over the SPH particles that are inside the companion's orbit.",
"Note that if thin shell approximation is taken to be valid at any moment of the CEE evolution, it will also imply that $E_{\\rm pot,e-comp}^{\\rm out}=0$ when the companion is still outside of the envelope.",
"However, at the same moment, the true value is well approximated by the point mass expression, $E_{\\rm pot,e-comp}^{\\rm out}\\simeq - GM_{\\rm env} M_{\\rm comp}/a$ .",
"As a result, in 1D, a partially inconsistent approach is usually taken: for the envelope mass outside of the orbit, the companion is treated as a thin shell (this is the second term in Equation (REF )); and for the envelope mass inside of the orbit, $m_{\\rm env}(r<a)$ , the companion is treated as a point mass, $E_{\\rm pot,e-comp}^{\\rm out} = - Gm_{\\rm env} (r<a) M_{\\rm comp}/a$ .",
"We can now rewrite the energy conservation equation (REF ) using “1D values”: $E_{\\rm tot,CE} &=& 0.5 \\left(M_{\\rm core} v_{\\rm core}^2 + M_{\\rm comp} v_{\\rm comp}^2 + \\int _{M_{\\rm core}}^{M} v^2 dm \\right)\\nonumber \\\\& + & \\int _{M_{\\rm core}}^{M} \\left( u - \\frac{Gm}{r} \\right)\\, dm- \\frac{GM_{\\rm core} M_{\\rm comp}}{a}\\\\&-& \\int _{m(r>a)}^{M} \\frac{GM_{\\rm comp}}{r} dm - \\frac{Gm_{\\rm env }(r<a) M_{\\rm comp}}{a} \\nonumber \\ ,$ where all velocities are in the inertial frame, and velocities of the core and the companion are not related in a simple way to the orbital separation, especially during the plunge.",
"After the plunge-in, a simplification can be made: $E_{\\rm tot,CE} &=& - \\frac{GM_{\\rm core} M_{\\rm comp}}{2a} +0.5 \\int _{M_{\\rm core}}^{M} v^2 dm\\nonumber \\\\& + & \\int _{M_{\\rm core}}^{M} \\left( u - \\frac{Gm}{r} \\right)\\, dm\\\\&-& \\int _{M_{\\rm core}}^{M} \\frac{GM_{\\rm comp}}{r} dm \\nonumber \\ .$ In conclusion, given the complications with the kinetic energy, the potential energy, and the non-Keplerian orbit of the companion during the plunge, a 1D code cannot self-consistently conserve energy during the plunge-in."
],
[
"The validity of the thin shell approximation",
"To assess the thin shell approximation for the fraction of the envelope that is outside of the orbit, we can find the “reduced” 3D gravitational potential for each SPH particle in the envelope, $\\phi _{i}^{\\rm red}$ , in a way that will measure the same quantity as does the 1D definition for the gravitational potential: $\\phi _{i}^{\\rm red} (r) =\\sum _{l\\ne i}^{\\le r} m_l \\varphi _{i,l} + M_{\\rm core} \\varphi _{i,\\rm core}+ f_{\\rm ins} M_{\\rm comp} \\varphi _{i,\\rm comp} \\ ,$ where the summation is only for envelope particles that have a distance to the chosen “centre” smaller than $r$ .",
"As previously, for the numerical mapping from 3D to 1D, $\\phi ^{\\rm red}_{\\rm 3D}(r)$ is found by averaging the particles that have $r$ within some small neighborhood $[r;r+\\delta r]$ .",
"Summed over all the envelope particles, this quantity produces the equivalent of $E_{\\rm pot,env1D}$ .",
"As can be expected, at the end of the plunge-in, the difference between the 3D averaged value for $\\phi _{\\rm 3D}^{\\rm red}$ and the 1D value $\\phi _{\\rm env1D}^{\\rm red}$ is small (see Figure REF ).",
"However, during the plunge-in, the 1D approximation often provides a significantly less negative potential well in the envelope near the companion than the 3D model provides for $\\phi _{\\rm 3D}^{\\rm red}$ .",
"The deviation can be as large as 50% of the value over a region of several solar radii.",
"After the envelope had decoupled, the gravitational potential within the inner envelope's “shell” is also not in good agreement between the two approaches.",
"By its design, this comparison only demonstrates the effect of the companion on the difference between the two potentials above the orbit.",
"However, the full potential inside the orbit will also depend on the companion.",
"Typically, the 3D approach provides a deeper potential well in the neighborhood of the orbit, while far above the orbit, the potential well can be more shallow than in the 1D thin shell approximation (see Figure REF ).",
"The value of the specific potential energy does not affect the structure calculations in a 1D model, which are affected only by the value of the local gravitational acceleration.",
"Note that this is a post-processed value.",
"However, it plays a role in what is considered in 1D simulations as the immediate energy budget as well as the rate of the energy transfer from the orbit to the envelope, and therefore may affect the rate of the orbital dissipation."
],
[
"Where and in what form the orbital energy is deposited",
"In both 3D and 1D approaches, the recombination energy is deposited self-consistently where it has been released.",
"This is not the same for the orbital energy release.",
"A 1D approach must come up with a prescription on how to deposit the energy that corresponds to the independently obtained orbital dissipation.",
"We can get a hint from our 3D simulations on where the orbital energy is effectively deposited, and in what form it is deposited.",
"In Figure REF we show where the local specific energies are changing while the orbit is shrinking.",
"It can be seen that the potential energy and the kinetic energy are clearly increasing everywhere above the orbit, while the internal energy is changing at a much smaller pace than that of the potential and kinetic energies.",
"As expected, the change in the real potential above the orbit is similar to that of the reduced potential.",
"In the regions where the orbit had passed between the two moments of time for which the derivatives are found, the absolute value of the time derivative of the reduced potential is a couple of orders of magnitude larger that that of the real potential.",
"Below the orbit, the 3D potential derivative is substantially more noisy than that of the reduced potential, as it is affected by the entire envelope above.",
"The derivatives indicate that we cannot detect the orbital energy deposition inside the binary orbit, but outside of the orbit it affects the entire envelope, primarily changing the mechanical energy of the envelope – the potential energy and the kinetic energy (and kinetic energy is converted to the potential energy with time).",
"This will become important in § , where we will discuss how the form of the energy deposition may affect the outcome.",
"Figure: Accumulated gain in the average specific angular momentum,shown for the case SS15.",
"The black solid line shows the instantaneous orbitalseparation between the core and the companion.",
"The colors of thecircles indicate the Keplerian specific angular momentum that thislocation would have.",
"The dashed purple line indicates the surfaceof the envelope.",
"White color implies that thatthe accumulated gain at this location is below the cut-off minimum value, log 10 Δl cmb <-3\\log _{10} \\Delta l_{\\rm cmb}<-3.Figure: The mass-averaged angular velocity Ω ¯ mass \\bar{\\Omega }_{\\rm mass}(red dots, top panels) and the specific angular momentum ll (blackdots, bottom panel), for the case M10.The blue dots on the top left and middle panels demonstrate the value of theKeplerian velocity that this envelope could have.",
"The left panelshows that case at the end of the plunge-in (t=613 days),the middle panel shows the profiles 50 days later (t=664 days, thisis after the merger took place), and the right top panel shows thecomparison of the polar and equatorial regions for the same moment(t=664 days).",
"The bottom right panel shows the ratio of the polaror equatorial angular velocities to the average.Figure: The mass-averaged angular velocity Ω ¯ mass \\bar{\\Omega }_{\\rm mass}(red dots, top panels) and the specific angular momentum ll (blackdots, bottom panel), for the case SS15.The left panelshows that case at the end of the plunge-in (t=626 days),the middle panel shows the profiles 175 days later (t=801 days), and the right top panel shows thecomparison of the polar and equatorial regions for the same moment(t=801 days).",
"Other information as in Figure ."
],
[
"Average angular momentum in 3D case",
"The $Z$ -component of the angular momentum for each SPH particle is found with respect to a particular point that should be moving without acceleration (for technical details, see Appendix ; below, whenever we talk about an angular momentum, we will imply only its Z-component).",
"However, the core of the red giant – our preferred centre of symmetry for other CE system properties – is not stationary, nor is it moving at a nearly constant velocity.",
"On the contrary, the core is orbiting around a point which, at least during the slow spiral-in, coincides well with the centre of mass of the core-companion binary system.",
"The angular momentum therefore cannot be calculated with respect to the core, and if such a calculation is carried out, the value of the angular momentum oscillates.",
"We choose therefore to calculate the averaged angular momentum with respect to the centre of the core-companion binary.",
"The caveat with this approach is that the centre of mass of the binary can also be moving with an acceleration.",
"Therefore, during the plunge-in phase this averaged quantity is also not entirely meaningful, and becomes most useful only at the end of the spiral-in.",
"In Figure REF we show the accumulated gain in the specific angular momentum in the envelope, computed from the start of the CEE simulation.",
"There are two important features that are present in all the simulations: We do not see that the region inside the orbit gains angular momentum or spins up.",
"While we have said that the value of the average angular momentum is not fully self-consistent before the slow spiral-in, the almost complete absence of any spin-up is nonetheless important.",
"Most of the initial binary angular momentum is absorbed by the outer layers of the envelope.",
"It is those layers that get ejected during the plunge-in, leaving the remaining CE with a smaller fraction of the total initial angular momentum (this can be seen both in Figure REF and from data shown in Table REF )."
],
[
"Angular velocity for 1D star",
"To understand how to relate our 3D star to a 1D star, we will briefly review how the treatment of rotation is done for 1D stars in stellar codes.",
"Generally, angular momentum transport is calculated using a diffusion equation.",
"A diffusion equation is meant to trace and to be written implicitly for the angular momentum, while in practice it is usually written using the angular velocity variable.",
"Currently, there are several ways to write the diffusion equation, however, the basis for all of them is [11]: $\\frac{\\partial \\Omega }{\\partial t} = \\frac{1}{\\rho r^4} \\frac{\\partial }{\\partial r}\\left( \\rho r^4 D \\frac{\\partial \\Omega }{\\partial r}\\right) \\ .$ Here $D$ is some diffusion coefficient, and $\\Omega $ is angular velocity.",
"The differences between several modern modifications are then based on what kinds of instabilities are taken into account to find the diffusion coefficient $D$ .",
"A more complete form of the diffusion equation can also take into account the flux due to non-zero radial velocity, and the temporal loss of the angular momentum [19].",
"It is important that the transition from the angular momentum as a “diffusing quantity” to the angular velocity as the main variable can be done only by choosing and fixing the relation between angular velocity and specific angular momentum.",
"Specifically, the above diffusion equation (REF ) adopts the assumption that the angular velocity is only a function of radius.",
"This allows a simplification in which the star is treated as if it is composed of thin spherical shells, where each shell has constant angular velocity, and has specific moment of inertia $i=2/3 r^2$ .",
"At the same time, it is anticipated that the centrifugal force affects the local effective gravity.",
"Hence mass shells need to correspond to isobars rather than to spherical shells [10], [15], [16], [41].",
"As a result, in stellar codes where isobars are considered, stellar structure variables are taken as being constant on isobars, while the angular momentum diffusion is still based on spherical symmetry.",
"In this paper, we chose to reduce the 3D angular momentum distribution into a 1D angular velocity distribution by adopting the same relation between the specific angular momentum and the angular velocity as was done implicitly for the angular momentum diffusion equation (REF ).",
"Therefore, we provide values of $\\Omega (r)$ as averaged on spherical shells.",
"There are two ways to do the averaging for the angular velocity.",
"First, we can find the total angular momentum in a thin shell that is located at the radius $r$ and has a thickness $\\delta r$ , and then use the relation for a specific moment of inertia of a thin shell: $\\bar{\\Omega }_{\\rm spher}(r_{\\rm shell})) = \\frac{3}{2}\\frac{\\bar{l} (r)}{\\bar{r}^2} = \\frac{3}{2} \\frac{1 }{\\bar{r}^2}\\frac{\\sum _{i\\in \\rm shell} l_{i} m_i}{\\sum _{i\\in \\rm shell} m_i} \\ .$ Here the summation is only done for the particles which belong to the thin shell located at $r$ with a thickness $\\delta r$ .",
"On the other hand, we can also find the mass-averaged angular velocity at each shell: $\\bar{\\Omega }_{\\rm mass}(r_{\\rm shell}) =\\frac{\\sum _{i\\in \\rm shell} \\Omega _{i} m_i}{\\sum _{i\\in \\rm shell} m_i}= \\frac{\\sum _{i\\in \\rm shell} l_i m_i/ r_{z,i}^2}{\\sum _{i\\in \\rm shell} m_i} \\ .$ Here $r_{z,i}$ is the distance from the particle to the rotation axis.",
"We did not find a substantial difference between $\\bar{\\Omega }_{\\rm mass}(r)$ and $\\bar{\\Omega }_{\\rm spher}(r) $ .",
"There is some discrepancy for small values of the angular velocity, where more numerical noise is present in $\\bar{\\Omega }_{\\rm spher}(r)$ .",
"Like the case with the specific angular momentum, we do not observe a spin up of the regions inside the orbit, and the Keplerian (binary) angular velocity always exceeds greatly that of the surrounding matter.",
"There is only a short period of time when the matter around the orbit approaches close to the value of the binary angular velocity.",
"This moment takes place just before the envelope material expands beyond the binary.",
"It is close to the same time when the envelope and the inner binary have effectively decoupled, and the hollow shell structure seen in Figure REF started to form."
],
[
"Angular velocity distribution during the slow spiral-in",
"At the start of the slow spiral-in, the specific angular momentum, unlike the case of a typical rotating star, does not rise monotonically from the centre to the surface.",
"We can separate the envelope in two regions: the inner region, with an almost constant value of specific angular momentum, and the outer region, where the angular momentum was quickly transferred from the companion to the envelope during the plunge-in (see Figures REF and REF ).",
"Between the two regions the value of the specific angular momentum at the start of the spiral-in drops (see Figure REF ).",
"The angular velocity profile and the specific angular momentum profile evolve very quickly to “stable” profiles after the plunge-in ends.",
"In a stable situation, the specific angular momentum increases with the distance from the centre.",
"This is established after about a few dynamical timescales of the expanded CE, where $\\tau _{\\rm d, CE}\\approx 50$ days.",
"The angular velocity profile become flat in most of the envelope (by mass).",
"The outer part of the envelope is small by mass but is more than 80% of the envelope radius.",
"There, the angular momentum transport is not efficient, and the angular velocity drops.",
"We note that either at the start of the slow spiral-in, or during the slow spiral-in, the angular velocity in the envelope is about an order of magnitude less than its local Keplerian velocity (see Figures REF and REF ).",
"In none of our 3D simulations does the angular velocity of the inner part of the envelope approach the binary's angular velocity.",
"Figure: Entropy generation (shown as the ratio of the cumulativeincrease of the entropy from the start of the simulation, over its initial value, left panels) andvelocity divergence (right panels).We show the case SS15.",
"The top panels show long-term evolution,and the bottom panels show a zoom-in of the plunge-in phase.Black solid lines show the location of the companion.Dotted purple lines on the plots for velocity divergence indicate the surface of the envelope.Figure: Evolution of the outer layers of the common envelope.",
"We show the case BF36.",
"The figure shows density (left panels), entropy (middle panels) and velocity divergence (right panels), as functions of the radial coordinate.The top panels show long-term evolution,and the bottom panels show a zoom-in to the plunge-in phase.Black solid lines show the location of the companion.",
"The dotted purple lines indicate the surface of the envelope.Note that the plots show only the material that remains bound, not the ejecta."
],
[
"Entropy",
"In 1D studies, a common feature in the results is the formation of an “entropy bubble” in the envelope [19], [25].",
"An entropy bubble is when part of the material of the envelope acquires a significantly higher entropy than it had before the CEE (note also, in convective envelopes, the pre-CEE entropy profile of a low-mass giant envelope is close to flat, with the exception of the surface layer).",
"The entropy can increase by a factor of a few compared to its initial value.",
"This high entropy region also is often separated from the surface by a region where the entropy did not change strongly from its initial value, forming therefore an internal entropy bubble.",
"The reason for this entropy bubble formation is the way that the energy conservation is implemented in 1D codes.",
"To conform with energy conservation, the energy that is released from the shrinking orbit has to be added to the envelope.",
"And the commonplace way to do that is by adding a new “luminosity” term in the energy equation [47], [32].",
"This consideration, while using the term “luminosity”, is equivalent to adding the net heat $dQ$ to the internal energy of the envelope material, and as a result increases the entropy.",
"Recently, it has been shown that where the energy is added, does matter, and can define the outcome.",
"The same overall energy input to the envelope, but added at different locations, can result either in a slow spiral-in, or in the envelope's ejection [25].",
"In § we will show that it matters in what form the energy is added, and the entropy is the key to understanding the difference.",
"In our 3D simulations, entropy is generated due to shocks and shear friction, where artificial viscosity leads to dissipation of local velocity differences by converting them into heat.",
"As a sanity check, we have checked that our code provides perfectly adiabatic evolution when material recombines, if no shocks or other matter interactions between the particles are involved, and there is no artificial heating term.",
"The velocity divergence is a good indicator of both shocks and a strong shear [6].",
"Therefore one of the best ways to trace the entropy generation is by looking at the velocity divergence (see Figures REF and REF ).",
"The velocity divergence is one of the main variables in the SPH code we use and is calculated using Equation (19) from [28].",
"We can distinguish how the entropy is generated during the three phases.",
"During phase I, prior to the plunge-in, when the companion's orbit is still outside the radius of the donor, we find that the surface layers are “shock heated” and obtain high entropy.",
"Those surface layers “overheated” by shocks are quickly ejected.",
"A similar behavior continues even while the companion is orbiting inside the envelope, which had expanded to several times its initial size.",
"While the companion is already orbiting inside the envelope (but before the plunge-in), very little of the donor's mass ($\\ll 1\\%$ ) is outside the companion's orbit.",
"Both the entropy generation area and the region with high velocity divergence propagate inside the orbit, down to the initial donor's radius, but do not affect much of the envelope's mass (this can be seen best in Figure REF ).",
"The situation changes when more than 1% of the donor's mass surrounds the orbit, and the companion plunges inside the layers that were not shocked previously.",
"Rapid plunge-in takes place, and the entropy is mainly generated outside the orbit.",
"The material with high entropy and high angular momentum becomes quickly unbound, leaves the envelope, and cannot any longer be seen in Figures REF and REF .",
"During the slow spiral-in phase, we detect entropy generation due to two processes: quick expansion provides shocks once more, and some entropy possibly comes from the flattening of the rotational profile.",
"As a result, the entropy is increased in the internal part of the envelope, while in the outer part of the envelope the entropy is almost unchanged (see Figure REF , where this entropy generation can be seen during the two episodes of mass ejection at the top panel showing the long-term evolution of SS15).",
"This entropy generation can appear to be similar to what was described above as an “entropy bubble” observed in the past 1D studies.",
"It is noteworthy, however, that the entropy is increased inside this bubble only by about 30% as compared to its initial value.",
"This is much less than predicted by 1D simulations (a factor of several times).",
"The importance of this lack of entropy generation in 3D as compared to 1D will be shown in §."
],
[
"The role of the recombination in driving the CE ejection",
"The specific energy that is released due to recombination of hydrogen and helium, in the case that they were initially fully ionized, is $\\varepsilon _{\\rm rec} \\approx 1.3\\times 10^{13} {\\rm erg\\ g}^{-1} \\times \\left( X f_{\\rm HI} + Y f_{\\rm HeII} + 1.46 Y f_{\\rm HeI} \\right)$ Here $X$ is the hydrogen mass fraction, $Y$ is the helium mass fraction, $f_{\\rm HI}$ is the fraction of hydrogen that becomes neutral, $f_{\\rm HeI}$ is the fraction of helium that becomes neutral, and $f_{\\rm HeII}$ is the fraction of helium that becomes only singly ionized.",
"In the giant star that we have modelled, the envelope during the giant stage has $X=0.673$ and $Y=0.306$ .",
"In the case of complete recombination, the released energy is $\\sim 1.5\\times 10^{13}$ ergs per gram.",
"The equation of state that we use in our SPH code also takes into account ionization of other elements.",
"In more details, we use the tabulated equation of state incorporated from MESA [40] and implemented as described in [35].",
"This tabulated equation of state includes recombination energy for H, He, C, N, O, Ne, and Mg.",
"The dominant contribution to the total recombination energy comes from hydrogen and helium, and the other elements only provide 3% of the total recombination energy.",
"When averaged over the whole envelope, the available specific recombination energy per gram for our donor is $\\simeq 1.6\\times 10^{13}$ erg/g, providing $4.7\\times 10^{46}$ ergs in total.",
"We can compare the efficiency of this energy release to the specific potential energy of the same matter, assuming that the matter is located at a distance $r$ from a gravitating mass to which it is still bound, $m_{\\rm grav}$ : $\\varepsilon _{\\rm pot} = - \\frac{G m_{\\rm grav}}{r} = - 1.9 \\times 10^{15}{\\rm erg\\ g}^{-1}\\ \\frac{m_{\\rm grav}/M_\\odot }{r/R\\odot }$ Next we use the standard assumption that in a stellar envelope, the binding energy is about half of the potential energy (note that here we use the binding energy in its 1D definition).",
"Comparison of $\\varepsilon _{\\rm rec}$ and $\\varepsilon _{\\rm pot}$ shows that if the material is cooled down to the point when it can start recombination, and the material at the same moment is located at $r_{\\rm rec}65 R_\\odot \\times m_{\\rm grav}/M_\\odot $ , then recombination alone can eject the material with no other energy sources needed.",
"Note that this radius, depending on $m_{\\rm grav}$ , is smaller than the radius of evolved stars (e.g., asymptotic giant branch stars), however, the material in the envelope of an evolved star is usually too hot to start the recombination.",
"For the hydrogen recombination alone to trigger the material outflow, $r_{\\rm rec,H}105 R_\\odot \\times m_{\\rm grav}/M_\\odot $ .",
"As more of the envelope matter is lost, the radius at which the recombination can act as an energy source powering the outflow, gets smaller.",
"When the upper layers of the envelope are lost, the internal layers start their expansion towards a new equilibrium.",
"Together, these two effects produce a recombination runaway of the envelope that can take place once the recombination starts [25].",
"Indeed, let us have a look at the case BF36, where the entire envelope is lost.",
"After the plunge-in, during the slow spiral-in, when the envelope is constantly outflowing, the recombination takes places at about the same radius continuously (see Figure REF ).",
"The inner hollow shell has expanded so much, that there is no double ionized helium left.",
"Hydrogen recombination starts at about $\\sim 155R_\\odot $ (this is the radius where 5% of H is recombined), while $r_{\\rm rec,H}115 R_\\odot $ (and it decreases as the envelope outflows).",
"This case is a clear recombination runaway.",
"Now let us consider the case SS15.",
"Hydrogen recombination starts a bit lower.",
"E.g., on day 720, 5% of hydrogen is recombined at $130R_\\odot $ .",
"Since the post-plunge-in mass below the hydrogen recombination zone is larger than in the case BF36, we find $r_{\\rm rec,H}\\sim 160 R_\\odot $ .",
"Hence, hydrogen recombination is not capable of rapidly ejecting the envelope in this case.",
"We find the case SS20 to be intermediate; similarly to the case BF36, the recombination is established to take place at the radius above $r_{\\rm rec,H}$ .",
"However, the mass loss rate is noticeably lower than the mass loss rate in the case BF36.",
"It is hard to classify this now as a runaway, as the timescale to lose the entire envelope is longer than the dynamical timescale.",
"We classify this as a steady recombination outflow.",
"Why, in the case of a more massive companion, does the recombination start at a larger radius?",
"In the case of a more massive companion, deeper, initially hotter envelope layers gain the energy to expand enough to start recombination.",
"E.g.",
"in the SS15 case, the immediate post plunge-in expansion leads to helium recombination starting at the $0.8 M_\\odot $ mass coordinate, while in the BF36 case, it reached down to $0.45 M_\\odot $ .",
"In the latter case, the disturbed layers were initially hotter, and hence this material had to expand to larger distances before it was cold enough to recombine.",
"We can consider adiabatic expansion of gas that initially had some specific potential energy and specific internal energy, when some (specific) energy is added to the gas.",
"The sum of these energies is the boosted total specific energy of the gas.",
"The layer will start expanding, to find itself at a new “equilibrium” radius $R_{\\rm eq}$ , where its total energy, assuming that the gas is not moving, is the sum of its new potential energy and new internal energy, and is equal to the boosted total specific energy of the gas at its old location.",
"In that new location, the gas will have $T(r)/T_0 \\approx (R_0/R_{\\rm eq})^{2}$ , where $T_0$ is the gas temperature before the expansion, and $R_0$ is the location of the gas before the expansion.",
"Now it can be seen why the difference in the entropy generation between 3D and 1D models, found in § , plays a role in the outcomes.",
"In the case when mechanical energy is added, as in 3D simulations, $T_0$ is the initial temperature of the (almost) unperturbed star.",
"In the case when the energy is added to the internal energy in the form of heat (resulting in a substantially larger entropy increase in the shared envelope), as in 1D simulations, $T_0$ is increased compared to the initial temperature of the same layer in the same unperturbed star.",
"Let us compare the effect of adding the energy mechanically, or as heat, on a specific example – the 1D profile of the star that was used to create our giant in the 3D CEE study presented here.",
"To each mass mesh point in this 1D profile, we add the same amount of energy via the two ways described, and find the temperature when the mass shell reaches its equilibrium radius.",
"Indeed, we find numerically that the second case (where the energy is added as heat) results in a smaller equilibrium radius than the first case (where mechanical energy is added).",
"Adding heat energy also results in matter having a hotter temperature when it reaches the equilibrium radius.",
"When we compare the temperature of two mass shells that have reached the same equilibrium radius after the energy was added, we find that in the added heat energy case, the temperature at the equilibrium radius is also hotter.",
"While no deterministic conclusion can be made on the trend for all possible cases, it is clear that when the energy is added directly to the internal energy, the radius at which the recombinations will start is different."
],
[
"The ejecta",
"In our simulations we distinguish four types of ejection processes: “Initial ejection” of the outer layers of the original envelope, $\\delta M_{\\rm ej}^{\\rm start}$ .",
"This low-mass ejection process takes place before the plunge-in starts, and is seen in all simulated CEEs.",
"A hint of this initial low-mass ejection is suggested from the evolution of the expanding outer layers of the envelope (prior to the start of the plunge-in at about 350 days) shown in Figure REF .",
"[23] have argued, based on the V1309 Sco merger, that the ejecta that accompanies a merger can be estimated by comparing the orbital energy release $\\delta E_{\\rm orb}(r)$ with the local binding energy of the envelope $\\delta E_{\\rm bind, env1D}(r)$ .",
"Indeed, we observe that in all of our simulations the orbital energy does not influence the envelope inside the orbit.",
"This supports the idea that the orbital energy release can be compared with the binding energy of the material during the spiral-in, and not only at the final orbit.",
"The initial low-mass ejection, $\\delta M_{\\rm ej}^{\\rm start}$ , can then be found by using: $\\delta E_{\\rm orb} (r) + \\delta E_{\\rm bind, env 1D}(r) =0.$ From the estimates that use the binding energy profile of the unperturbed star, we find that the expected $\\delta M_{\\rm ej}^{\\rm start}$ is 0.010, 0.022, 0.034, 0.047, and $0.090 \\, M_\\odot $ , in cases for companion masses of 0.05, 0.1, 0.15, 0.2, and $0.36 \\,M_\\odot $ , respectively.",
"In 3D simulations using the same companion masses, we find that $\\delta M_{\\rm ej}^{\\rm start}$ is 0.01, 0.02, 0.03, 0.045, and $0.1 \\, M_\\odot $ .",
"Therefore Equation (REF ) provides a good method for estimating $\\delta M_{\\rm ej}^{\\rm start}$ .",
"However, this is the least massive ejection process, even in cases of mergers.",
"“Plunge-in ejection” of the envelope, $\\delta M_{\\rm ej}^{\\rm plunge}$ .",
"This ejection process takes place at the end of the fast plunge-in, when the circumbinary envelope has just formed and the fast orbit depletion has ended.",
"Like the “Initial ejection”, it takes place in all of our simulated CEEs.",
"An example can be seen in Figure REF : this is when the mass of the envelope is sharply decreasing to $1.4 M_\\odot $ , on the timescale of only a few days.",
"The process of the ejection can also be observed in Figure REF for the case BF36, where it takes place after the plunge-in starts (approximately between 355 days and 365 days).",
"This ejection process is likely powered by the mechanical energy that the envelope has absorbed by the time that the orbit has become decoupled from the envelope.",
"We did not find a good method to predict the associated energy release given the initial energy profile.",
"For $\\delta M_{\\rm ej}^{\\rm plunge} + \\delta M_{\\rm ej}^{\\rm start}$ , we find from the 3D simulations that 0.03, 0.3, 0.43, 0.53, and $0.75\\,M_\\odot $ were ejected, for companion masses of 0.05, 0.1, 0.15, 0.2 and $0.36 \\,M_\\odot $ , respectively.",
"In the case of BF36, half of the initial envelope was ejected during this stage.",
"It is about the same fraction as was found in the simulations that were performed without recombination energy taken into account (about 40% in [35] and about 25% in [45]).",
"We cannot rule out however that the inclusion of the recombination might have enhanced the plunge-in ejection, as in the case BF36 it led to 50% of the envelope being ejected during this stage.",
"“Recombination runaway” is the process that led to the complete envelope ejection in the case BF36 [34].",
"It takes place when the post plunge-in envelope has cooled down sufficiently to start hydrogen recombination, while at the same time the envelope size has expanded beyond $r_{\\rm rec, H}$ (see more details in § ).",
"We find that in SS20 and SS15 models, this process does not start promptly after the plunge-in has occurred, and that the material is ejected at a slower rate during this process than in the case of the BF36 case (see Figure REF ).",
"Therefore, for these cases this process can be thought of as a “steady recombination outflow”.",
"The average mass loss rates of these outflows are 2, 0.25 and 0.15 $M_\\odot $ per year, for BF36, SS20 and SS15 models, respectively (the dynamical timescale of the expanded envelope is about a year).",
"In case of BF36, the recombination driven outflow slowed down to about 0.4 $M_\\odot $ per year at about 200 days after the plunge-in.",
"“Shell-triggered ejection” is the sudden ejection of a substantial part of the envelope during the slow spiral-in.",
"This process takes place about one dynamical timescale of the expanded envelope after the plunge-in process has ended.",
"This phenomenon can be seen for the case SS15 at about 735 days (Figures REF and REF ).",
"This ejection process is partially powered by the energy release from the re-collapsing hollow shell, and partially powered by the triggered recombination.",
"Recall, that this hollow shell is bound after the plunge-in, and while it initially expands, it starts to recollapse on its dynamical timescale.",
"This recollapse results in redistribution of the energy in the envelope, where the contracting shell provides energy for the outer layers, shedding away part of these layers.",
"It is not clear at the moment what total fraction of the envelope this process can eventually remove.",
"First, this process is accompanied with a steady recombination outflow, making it difficult to disentangle the two processes.",
"Second, while this process is continuous, it is very slow (even slower than the outflow in the case of the SS20 model), and thus the model becomes a self-regulated spiral-in.",
"Third, we cannot yet compute the contraction of this shell all the way to the binary orbit; thus we can not see if this process can extract more energy from the binary and produce several similar re-expansions.",
"In the latter case, no self-regulated spiral-in situation can be expected.",
"We find however, that the first episode of the shell-triggered ejection takes away almost all of the angular momentum that was remaining in the CE, leaving the CE with less than 0.02% of its initial angular momentum.",
"In addition, the envelope that remains after the shell-triggered ejection is only marginally bound — its binding energy is $3\\times 10^{44}$ ergs.",
"If there are no bouncing re-expansions, steady recombination outflow can eject the remaining envelope.",
"This shell-triggered process does not always happen even for the same initial binary.",
"We ran the initial SS15 binary twice, with the different Courant numbers (for gas SPH particles only).",
"In particular, we changed the second Courant number $C_{N,2}$ [28].",
"Both simulations conserved energy and angular momentum extremely well.",
"The larger value for the Courant number $C_{N,2}$ was 0.6 (used in the simulation SS15), and the smaller value was 0.3 (used in the simulation SS15sC).",
"Both values that we used for $C_{N,2}$ are in the recommended range, between 0.1 and 0.6, for the version of the code we use (Lombardi, priv.",
"comm.).",
"We find that in the run with the smaller Courant number, SS15sC, the prompt ejection does not occur, and the evolution is similar to SS20, just with a bit slower mass-loss rate of the steady recombination-driven outflows.",
"However, we cannot attribute this divergence between the two runs purely to a numerical error.",
"We traced the difference in the outcomes to the differences in the properties of the SPH particles that are currently in the hollow shell, where those properties were acquired during the plunge-in interaction with the special particles that represent the binary.",
"In the SS15sC case, a fraction of SPH particles in the inner region of the hollow shell have acquired a higher entropy, likely due to more shocks they have experienced.",
"These particles have a higher internal energy, and a higher total energy, if compared to SPH particles located at a similar distance from the binary in the SS15 case.",
"The total energy of SPH particles in the shell is about the same, but the distribution of the energies between the particles is slightly different, and in a wider range than in the case SS15.",
"We cannot determine one of the energy distributions between the shell particles as to be more proper than the other.",
"The natural expectation for SS15sC high-energy particles is that their “fallback” will start when the shell expands more that in the case SS15.",
"However, while the shell was still expanding (beyond the distance it had maximally expanded in the case SS15), the material starts He recombination (from the double ionized state to the single ionized state).",
"This recombination gives an additional energy boost to the hollow shell particles, and prevents its fallback, at least for the duration of the simulation we obtained for SS15sC (for 40 days after the fallback had started in the case SS15).",
"A small deviation in the energy for a fraction of SPH particles in the hollow shell have resulted in a qualitative change of the envelope ejection picture.",
"This indicates that even during the established steady recombination outflow, a CE is recombinationally-dynamically unstable with respect to its inner shell recollapse.",
"The condition to prevent this dynamical instability seems to start the fallback after Helium started its recombination.",
"As a confirmation, we find that a weak version of this instability takes place in all the models at the moment when the inner shell started its fallback.",
"As a weak version, we mean that the envelope changes its mass loss rate at about the same time when the inner shell contracts or re-expands, but this mass loss rate change is not nearly as noticeable as in the case of SS15.",
"In all those “weak” cases, Helium is not doubly ionized in the inner shell at the time of fallback.",
"To answer on whether the strong shell-triggered ejection process is a numerical artifact due to a small number of particles near the core (recalling that the total number of SPH particles in this simulation is 100,000), or is the real physical effect that occurs if the fallback starts before the Helium recombination, a study with significantly more particles is required; this is unfeasible at this time, due to the long computational times needed to model a well estbalished slow spiral-in [34].",
"We stress in this Section that usually under the “CE ejection” term only complete ejection — the one that leads to naked binary formation — is considered.",
"Such complete ejection is an implicit assumption of the energy budget formalism.",
"However, it is important, that even when a CEE fails to eject the entire envelope, a significant fraction of the mass of the envelope is lost during plunge-in ($\\delta M_{\\rm ej}^{\\rm plunge}$ ), and that during the slow spiral-in the CE mass will decrease further."
],
[
"Discussion",
"We discussed how to average different quantities obtained in 3D CEE simulations to produce information useful for 1D studies.",
"As could have been expected, we find that the asymmetry of the profiles is extreme during the plunge-in, and is non-negligible during a slow spiral-in.",
"For example, the angle-dependent deviation of thermodynamic quantities at the start of the slow spiral-in can reach 50%.",
"Second, the typical assumption of spherical symmetry (i.e., the “thin-shell” approximation for the companion) underestimates the depth of the potential well near the orbit.",
"Finally, the decoupling of the binary and the envelope during a slow spiral-in, with the formation of a dense hollow shell around the binary, provides challenges for 1D codes (where it would appear as a sharp density inversion).",
"We reviewed the energy balance and discussed how energy conservation can be treated in 1D codes.",
"We outlined three problems when one tries to keep the energy conserved self-consistently in 1D, and provided the energy conservation equation that can be used in 1D codes during a slow spiral-in, once the post-plunge-in configuration is known.",
"We find that most of the angular momentum is quickly lost from the CE system with the ejecta.",
"In our simulations, there is no corotation between the binary and any part of the CE during any phase of a CEE.",
"In some cases, after the hollow shell formation, only a few particles could have been left within a RL of either the core or the companion; only these particles can be considered as having been in “corotation”.",
"During a slow spiral-in, the angular velocity becomes constant in most (by mass) of the CE, the spherical symmetry approximation works well, and the value of the angular velocity is significantly smaller than the local Keplerian velocity anywhere in the envelope.",
"The flat profile for the angular velocity we derive from our 3D simulations differs strongly from the results of typical 1D CE evolution.",
"In the latter, the region around the binary is always in corotation, as well as a region inside the binary orbit.",
"The angular velocity is then obtained with the diffusion equation, and is found to decrease steadily with the distance from the binary , , .",
"It can be argued that the difference between the 3D and 1D profiles may arise from the different ways that viscosity is treated in the two approaches.",
"It is hard to evaluate how well artificial viscosity in SPH matches the convective viscosity [36].",
"It is clear however that 3D SPH simulations, when compared to 1D studies, at the same time, are less efficient in transferring angular momentum to the regions near the orbit, and are more efficient in the distribution of the angular momentum throughout the envelope.",
"Therefore neither too low, nor too high artificial viscosity in SPH, as compared to what is taken as viscosity in 1D simulations, can work as an explanation of these profile differences.",
"In addition, the significant loss of angular momentum with the ejecta does not lead to as significant a spin-up of the envelope as 1D codes predict.",
"Frictional heating from the differential rotation is therefore not expected for most of the envelope, except during the start of the slow spiral-in, when the angular velocity profile quickly flattens.",
"In addition, the evolution of the entropy of the CE material differs between 3D simulations and 1D results.",
"In 3D, there is no entropy generation within the orbit during the plunge-in.",
"While we can see entropy generation near the surface and a small “entropy bubble” formation, the entropy increase is much smaller than what 1D studies predict — 3D simulations predict a 30% increase, while 1D results predict a factor of a several increase.",
"In 1D, the entropy is generated because the energy was added as heat.",
"We have identified four types of ejection processes: the initial ejection, the plunge-in outflow, the recombination runaway outflow, and the shell-triggered ejection.",
"The initial ejection and the plunge-in ejection take place in all the CEEs we have considered, including those that end up with a merger.",
"We provide a simple way to find the mass of the initial ejecta.",
"The prompt plunge-in ejection carries away substantially more mass, but there is no easy way to estimate the magnitude of this ejection.",
"The shell-triggered ejection takes place during the slow spiral-in and is caused by the hollow shell fallback; this provides another prompt ejection process as part of the CE.",
"The recombination runaway outflow starts during a slow spiral-in, once the expanded envelope is cooled down to start hydrogen recombination above $r_{\\rm rec,H}$ .",
"In this case, the rest of the envelope can be removed within several dynamical timescales of the expanded envelope.",
"Since the radius at which the recombination energy release overcomes the potential well depends on the entropy of the material, the entropy generation observed in 1D codes will likely predict different outcomes of 1D CE evolution.",
"The combination of a difference in the entropy profile and the entropy's effect on recombination runaway was likely the reason why recombination runaway found in recent simplified 1D studies of CEEs [25] have happened in models where either a lot of heat was added to the entire envelope, or less heat was added, but in a region that was more confined to the bottom of the envelope (thus keeping the upper envelope's entropy unchanged).",
"The slow timescales of recombination runaway can be up to several hundreds day, and are getting longer as the mass of the companion decreases.",
"For these cases, 3D simulations are no longer self-consistent, as this is a timescale on which radiative losses can become important, and hence 1D codes must be used.",
"Note that the envelope is not stationary at any moment, and instead exhibits “stationary” outflows.",
"Similar recombination driven outflows were considered for the case of simplified hydrogen envelopes [4], where several possible modes of instabilities were discussed.",
"A recent study of simplified CEE in 1D has shown that the CE is prone to dynamical instabilities [25], and a runaway recombination was found, but a connection to steady recombination outflows was not made then.",
"The energetic consequences for CEE outcomes in the case of a stationary outflow instead of a prompt dynamical ejection were discussed in [21], but, further studies of the instabilities of envelopes with steady outflows powered by recombination are highly needed.",
"Using the tools developed for this paper, we inspected the models presented in [34], where all of the modelled CEEs resulted in binary formation.",
"There, the models were only analyzed for their final states – binaries parameters and energy taken away by the ejected mass.",
"We find that the length of the envelope removal via recombination runaway is increasing with the potential well of the donor.",
"The donor that we consider now had the longest envelope removal time (700 days) out of all the models considered in [34] (the next longest timescale of 250 days occurred in a model that used a $1.6~M_\\odot $ red giant with a $0.32~M_\\odot $ core).",
"In the current study, we find that the ejection timescale increases as the companion mass decreases.",
"This implies that the model that provides the timescales and plateau luminosities of the Luminous Red Novae powered by CEEs [23], in its future developments, should take into account CEE recombination runaway features, such as the initial envelope binding energy and the mass ratio of the companions.",
"For a proper treatment of a self-regulated spiral-in, one needs to know the orbit at which the companion slowed down its fall, the mass that remained in the envelope after the companion's plunge, how much angular momentum remained in the envelope, and how much energy was carried away by the ejecta.",
"This can be done only in conjunction with the preliminary 3D simulations that are performed until at least the end of the plunge-in.",
"For the cases when it is not possible, we outline several important points: The plunge-in takes place on a timescale comparable to a freefall timescale.",
"No energy conservation during the plunge can be properly treated in 1D.",
"Instead a CE structure should be constructed assuming almost adiabatic envelope expansion as a result of the plunge.",
"No heat should be added to the envelope — we stress that it is important to produce an adiabatic envelope expansion.",
"Despite changing the energy of the envelope by the same amount when the heat is added, or when the mechanical energy is added, the change of the CE material after these two ways to add energy alters which envelope layers would start recombination.",
"We strongly recommend the use of kinetic energy injection instead of heat injection.",
"An effort should be given to model a hollow shell outside of the inner binary in 1D.",
"The envelope unavoidably loses a substantial fraction of its mass.",
"While the minimum (initial) ejecta can already be estimated, future studies are needed to determine how to estimate mass involved in the plunge-in ejection.",
"The expanded envelope should be checked for the condition of recombination runaway, as in this case no long-term self-regulated spiral-in can take place.",
"All 1D codes should be prepared to detect and treat steady outflows.",
"Since most of the angular momentum is lost with the ejected material, the magnitude of the angular velocity in the envelope is likely to be very low.",
"Hence it is rather insignificant to affect either internal structure or provide frictional heating.",
"In this study, we only considered the specific case of a CEE where the donor star is a low-mass giant, and all the conclusions are made for this case of the donor.",
"While we may expect that the characteristic behaviors we found will also occur in CEEs with more massive donors, future studies and comparisons of 3D simulations with 1D studies using other donors, especially more massive donors, are highly needed."
],
[
"Acknowledgments",
"NI thanks NSERC Discovery and Canada Research Chairs Program.",
"JLAN acknowledges CONACyT for its support.",
"The authors thank their colleagues, C. Heinke, S. Morsink, E. Rosolowsky, and G. Sivakoff for checking the English in the manuscript.",
"We thank an unknown referee for a number of helpful remarks that helped to improve the manuscript.",
"Computations were made on the supercomputer Guillimin from McGill University, managed by Calcul Québec and Compute Canada.",
"The operation of this supercomputer is funded by the Canada Foundation for Innovation (CFI), Ministére de l'Économie, de l'Innovation et des Exportations du Québec (MEIE), RMGA and the Fonds de recherche du Québec - Nature et technologies (FRQ-NT)."
],
[
"Angular momentum in 3D codes",
"The angular momentum of a particle $k$ in SPH is $\\mathbf {L}_k=\\mathbf {r}_{k}\\times (m_k\\mathbf {v}_{k})=L_{x,k}\\hat{x}+L_{y,k}\\hat{y}+L_{z,k}\\hat{z},$ where the components of the angular momentum for $x$ , $y$ , and $z$ are given by $L_{x,k}&=&m_k(y_kv_{z,k}-z_kv_{y,k}),\\\\L_{y,k}&=&m_k(z_kv_{x,k}-x_kv_{z,k}),\\\\L_{z,k}&=&m_k(x_kv_{y,k}-y_kv_{x,k}),$ respectively.",
"The total angular momentum for the system is, $\\mathbf {L}=\\sum _k^N \\mathbf {L}_k,$ and its magnitude is $L=|\\mathbf {L}|=\\sqrt{L_x^2+L_y^2+L_z^2} \\ ,$ where $L_x&=&\\sum _k^N m_k(y_kv_{z,k}-z_kv_{y,k}),\\\\L_y&=&\\sum _k^N m_k(z_kv_{x,k}-x_kv_{z,k}),\\\\L_z&=&\\sum _k^N m_k(x_kv_{y,k}-y_kv_{x,k}).$ The angular momentum (by its components) around a particular point $p$ can be calculated as, $L_{x,k}^p&=&m_k\\left[(y_k-y^p)(v_{z,k}-v_{z}^p)-(z_k-z^p)(v_{y,k}-v_{y}^p)\\right],\\\\L_{y,k}^p&=&m_k\\left[(z_k-z^p)(v_{x,k}-v_x^p)-(x_k-x^p)(v_{z,k}-v_z^p)\\right],\\\\L_{z,k}^p&=&m_k\\left[(x_k-x^p)(v_{y,k}-v_y^p)-(y_k-y^p)(v_{x,k}-v_x^p)\\right].$"
]
] | 1606.04923 |
[
[
"Improved limit on the $^{225}$Ra electric dipole moment"
],
[
"Abstract Background: Octupole-deformed nuclei, such as that of $^{225}$Ra, are expected to amplify observable atomic electric dipole moments (EDMs) that arise from time-reversal and parity-violating interactions in the nuclear medium.",
"In 2015, we reported the first \"proof-of-principle\" measurement of the $^{225}$Ra atomic EDM.",
"Purpose: This work reports on the first of several experimental upgrades to improve the statistical sensitivity of our $^{225}$Ra EDM measurements by orders of magnitude and evaluates systematic effects that contribute to current and future levels of experimental sensitivity.",
"Method: Laser-cooled and trapped $^{225}$Ra atoms are held between two high voltage electrodes in an ultra high vacuum chamber at the center of a magnetically shielded environment.",
"We observe Larmor precession in a uniform magnetic field using nuclear-spin-dependent laser light scattering and look for a phase shift proportional to the applied electric field, which indicates the existence of an EDM.",
"The main improvement to our measurement technique is an order of magnitude increase in spin precession time, which is enabled by an improved vacuum system and a reduction in trap-induced heating.",
"Results: We have measured the $^{225}$Ra atomic EDM to be less than $1.4\\times10^{-23}$ $e$ cm (95% confidence upper limit), which is a factor of 36 improvement over our previous result.",
"Conclusions: Our evaluation of systematic effects shows that this measurement is completely limited by statistical uncertainty.",
"Combining this measurement technique with planned experimental upgrades we project a statistical sensitivity at the $1\\times10^{-28}$ $e$ cm level and a total systematic uncertainty at the $4\\times10^{-29}$ $e$ cm level."
],
[
"Introduction",
"A permanent electric dipole moment (EDM) in a non-degenerate system would violate both parity inversion (P) and time reversal (T) symmetries.",
"Assuming invariance under the combined charge conjugation (C), P, and T transformations, an EDM also violates CP symmetry.",
"Measurements of EDMs in a wide variety of systems, such as the neutron [1], [2], electron [3], and nuclei [4], [5], place stringent limits on various sources of CP violation.",
"For example, measurements of the neutron EDM [1], [2] have traditionally been used to restrict the CP-violating parameter of quantum chromodynamics to be $\\Theta \\le 10^{-10}$ , whereas one would expect it to be of order unity [6].",
"Identifying additional sources of CP violation is essential to understanding how matter came to dominate over anti-matter in our universe.",
"Although the Standard Model (SM) includes CP violation observed in $K$ meson [7] and $B$ meson [8], [9] decay, these mechanisms are unable to reproduce the observed matter-antimatter asymmetry [10].",
"CP violation included in the SM predicts EDMs that are many orders of magnitude below current experimental limits such that observation of a non-zero EDM in the foreseeable future would be a signature of new physics.",
"The EDM of a diamagnetic atom arises primarily from CP violating interactions in the nuclear medium.",
"Although atomic electrons screen these effects from observation in the lab frame, this screening is imperfect due to the finite size of the nucleus and relativistic electron motion.",
"The portion of the nuclear EDM that survives electron screening is characterized by the Schiff moment [11].",
"While the most precise EDM measurement of a diamagnetic atom was performed on ${}^{199}$ Hg [4], [5], $^{225}$ Ra is a promising isotope for an EDM search because octupole deformation of the $^{225}$ Ra nucleus enhances its EDM by over two orders of magnitude compared to ${}^{199}$ Hg [12], [13], [14], [15].",
"Nuclear octupole deformation also simplifies EDM calculations for $^{225}$ Ra making them more consistent compared to ${}^{199}$ Hg [16].",
"There are two primary obstacles to achieving an EDM measurement in $^{225}$ Ra.",
"First, $^{225}$ Ra is radioactive with a 14.9 day half-life.",
"This makes $^{225}$ Ra atoms scarce and difficult to handle.",
"Second, the vapor pressure of radium is too low to perform spin precession measurements in a vapor cell.",
"Fortunately, the atomic structure of $^{225}$ Ra allows us to circumvent these challenges using laser cooling and trapping.",
"Recently, we reported the first measurement of the $^{225}$ Ra EDM [17], representing both the first EDM measurement using an atom with an octupole deformed nucleus and the first EDM measurement using laser-trapped atoms.",
"Although the $95\\%$ confidence upper limit from this measurement is not yet competitive with that of $^{199}$ Hg for limiting CP violation in the nucleus according to current theoretical calculations [18], $^{225}$ Ra measurements have the potential to achieve rapid improvements by many orders of magnitude through a series of experimental upgrades.",
"In this work, we report the first such upgrade which allows us to improve the $^{225}$ Ra EDM $95\\%$ confidence upper limit by a factor of 36 to $\\left| d(^{225}\\mathrm {Ra}) \\right| \\le 1.4\\times 10^{-23}$ $e$ cm.",
"The primary improvement over our previous measurement is an improved vacuum system and a new optical dipole trap (ODT) [19] geometry which prolongs the atom trap lifetime and allows us to extend spin precession measurements from 2 s to 20 s. EDMs of hadronic systems (neutrons, diamagnetic atoms and molecules) can be described as a linear combination of several potential sources of CP violation.",
"The traditional method used to interpret EDM limits in these systems is to assume that the EDM is generated by only one source.",
"This approach provides a simple way to directly interpret an EDM limit from a single system but fails to account for the scenario where an EDM arises from multiple sources of CP violation.",
"Presently, experimental limits on EDMs of several systems have been reported and it is now possible to combine EDM limits from these systems to simultaneously constrain multiple sources of CP violation [18].",
"Within this framework, when the $^{225}$ Ra EDM limit reaches $10^{-25}$ $e$ cm, it will improve constraints on all CP violating interactions in the nuclear medium.",
"Figure: A diagram of the experimental apparatus with important sections labeled.",
"Atoms exit the ≈\\approx 500 ∘ ^\\circ C oven through a narrow nozzle where they are transversely cooled and collimated, longitudinally slowed, and trapped in a 3D MOT using laser light at 714 nm.",
"One out of every 10 6 10^6 atoms exiting the oven is trapped in the MOT.",
"The MOT is overlapped with the focus of at 50 W 1550 nm laser which forms an ODT.",
"The focusing lens for the ODT is translated by ≈\\approx 1 m to transfer the atoms between copper electrodes in the center of a glass tube with a vacuum below 10 -11 10^{-11} torr.",
"We start with ≈\\approx 10,000 225 ^{225}Ra atoms in the MOT and trap ≈\\approx 1,000 atoms between the electrodes.",
"After the atoms Larmor precess for 20 s in a B-field along z ^\\hat{z} we detect ≈\\approx 500 atoms.",
"An E-field of 65 kV/cm is applied either parallel or anti-parallel to the B-field during spin precession."
],
[
"Experimental Setup",
"We measure the EDM of $^{225}$ Ra using a tabletop apparatus that employs widely used techniques in atomic physics [20], [21], [22], [19].",
"Figure REF shows a diagram of our apparatus.",
"Section REF introduces the relevant atomic structure that allows us to manipulate radium atoms and describes the journey the atoms take through our apparataus from when they are loaded into the oven to when they arrive at the EDM measurement region.",
"Here, we emphasize the changes and improvements since laser cooling and trapping of radium [23] and ODT transport of radium atoms [24] were first demonstrated.",
"Section REF describes the measurement procedure we use to interrogate the atoms after they arrive at the EDM measurement region.",
"Our experiment operates cyclically such that each experimental “cycle” lasts 100 s. For each cycle, 60 s is spent preparing a new sample of atoms for interrogation and 40 s is spent interrogating the atoms using two separate 20 s precession measurements."
],
[
"Atom preparation",
"Figure REF shows the electronic energy levels of radium and the hyperfine structure of $^{225}$ Ra for the three states that are relevant for laser cooling, trapping, and detection.",
"In $^{225}$ Ra, each state with $J\\ne 0$ is split into two hyperfine states with $F=J\\pm I$ .",
"Here $J$ is the total electronic angular momentum and $I=1/2$ is the nuclear spin.",
"The atoms are transversely cooled, slowed and trapped using laser light tuned near the ${}^1$ S${}_0$ $F=1/2$ to ${}^3$ P${}_1$ $F = 3/2$ (red) transition at 714 nm.",
"Here, F is the total angular momentum including nuclear spin.",
"The lifetime of the red transition is 420 ns which limits the photon scattering rate for this transition to 0.38 MHz.",
"This is almost two orders of magnitude weaker than typical primary cooling transitions used in other alkaline-earth atoms with similar electronic structure (e.g.",
"Sr, Yb).",
"While experiments that use these alkaline-earth atoms leverage the dipole-allowed ${}^1$ S${}_0$ to ${}^1$ P${}_1$ (blue) transition, this transition is especially cumbersome in radium because branching ratios from ${}^1$ P${}_1$ to $^1$ D$_2$ and metastable $^3$ D$_{2,1}$ states are so large that a complicated repumping scheme is necessary both for longitudinal slowing [21] and magneto optical trap (MOT) [22] operation.",
"In contrast, the red transition requires a single repump laser at 1428.6 nm which excites the $^3$ D$_1$ to $^1$ P$_1$ transition [23].",
"We obtain $^{225}$ Ra from the National Isotope Development Center at Oak Ridge National Laboratory.",
"Our previous EDM measurement combined two separate data runs which used 3 mCi and 6 mCi shipments of $^{225}$ Ra respectively [17].",
"In the current measurement, we use a single 9 mCi shipment, which equates to 225 ng of $^{225}$ Ra.",
"The $^{225}$ Ra is delivered to our laboratory as radium nitrate salt.",
"We dissolve the salt in a solution of nitric acid along with 4 $\\mathrm {\\mu }$ Ci of $^{226}$ Ra.",
"We can selectively cool and trap either isotope independently and with no isotopic contamination by changing the frequency of the lasers we use to cool and trap atoms.",
"We use $^{226}$ Ra ($\\tau _{1/2}=1,600$ y) to setup and optimize the experimental apparatus since the greater abundance of this isotope allows us to load more atoms into the oven.",
"Unfortunately, $^{226}$ Ra has zero total angular momentum in its ground state since it lacks nuclear spin and thus, cannot have an EDM in the lab frame.",
"The radium nitrate solution is deposited on a 2.5 cm square piece of aluminum foil and allowed to dry.",
"Before loading the foil into the oven, we place two 25 mg granules of metallic barium on the foil and wrap the foil around the barium.",
"The foil and two additional 25 mg granules of barium are then placed in the oven.",
"With a sufficient vapor pressure of barium, the radium nitrate and barium undergo an oxidation-reduction reaction to produce metallic radium.",
"The heart of the oven assembly is a titanium crucible, in which we place the radium-coated foil and barium.",
"Atoms enter the vacuum chamber through a nozzle with a cylindrical opening that is 8.3 cm long and 0.15 cm in diameter.",
"The back side of the crucible attaches to a vacuum flange via a stainless steel tube that is perforated to reduce heat conductance to the vacuum chamber.",
"Two radiative heaters made from tungsten filament coil surround the crucible and nozzle.",
"We heat the oven to between $400^\\circ $ C and $520^\\circ $ C. During an EDM measurement, the $^{225}$ Ra flux degrades due to oven depletion and radioactive decay.",
"We gradually increase the oven temperature to maintain a constant $^{225}$ Ra flux.",
"Once the oven approaches $520^\\circ $ C we notice a decrease in the trap lifetime and further increases in the oven temperature do not result in a useful increase in atom flux.",
"Three layers of in-vacuum passive heat shields surround the heaters inside a fourth layer of heat shielding that is water-cooled.",
"Figure: A level diagram including the lowest 9 energy levels of radium.",
"The frequency axis is plotted to scale for the upper 8 levels.",
"States with orbital angular momentum L=2 (D-states) are offset horizontally for clarity.",
"Colored arrows represent the atomic transitions we drive with laser light in our experiment.",
"Transitions at 714 nm, 1429 nm, and 483 nm are used for cooling/trapping, repumping, and detection, respectively.",
"Insets a), b), and c) show the isotope shift between 226 ^{226}Ra and 225 ^{225}Ra and the hyperfine structure present in 225 ^{225}Ra resulting from the nuclear spin I=1/2I=1/2.",
"Dotted lines are the 226 ^{226}Ra energy levels.Dashed lines are offset from the 226 ^{226}Ra levels by the isotope shift between 225 ^{225}Ra and 226 ^{226}Ra.",
"The isotope shift represents the frequency difference between a 226 ^{226}Ra state and the center of gravity of the corresponding 225 ^{225}Ra hyperfine states, indicated by the solid lines.The upper left corner of each inset is labeled with the energy level it describes.",
"The vertical axes in all three insets have a consistent scale such that the plotted separations accurately represent relative level separation between hyperfine states.",
"Frequency values in a) and c) are taken from and values in b) are taken from .Transverse cooling [20] of the atomic beam is achieved using 150 mW of near-resonant 714 nm laser light that is expanded to a $\\approx $ 2 cm beam diameter ($1/e^2$ ) and split into two orthogonal beams propagating at near normal incidence to the atomic beam propagation direction ($\\hat{x}$ ).",
"The beams make multiple bounces between two pairs of 2.5 cm by 18 cm mirrors to increase their interaction time with atoms and improve collimation by slightly decreasing their angle with respect to the atomic beam normal after each reflection.",
"Each dimension individually gives a gain of $\\approx $ 9 in the MOT loading rate and they combine to give a total gain of $\\approx $ 80.",
"Before we can capture atoms in a MOT, it is necessary to slow their longitudinal velocity.",
"We use a Zeeman slower [21] operating on the red transition.",
"The magnitude of deceleration is determined by the lifetime of the laser-excited state, or equivalently, the line-width of the transition.",
"The lifetime of the red transition limits our 0.9 m slowing region to a capture velocity of 60 m/s.",
"This velocity class represents only $0.5\\%$ of the atoms exiting our oven for a typical operating temperature of 500$^\\circ $ C. Slowed atoms are trapped in a MOT which combines three orthogonal pairs of circularly polarized, counter-propagating laser beams with a quadrupole B-field to create a trapping potential.",
"We operate the MOT in three “phases” which are distinguished by the intensity and detuning of the MOT laser beams and the B-field gradient we apply.",
"The details of each phase have been described previously [24].",
"During the first phase we set the beam intensity, frequency detuning, and B-field gradient to optimize loading atoms into the MOT.",
"After loading atoms for 50 s, we decrease laser intensity and detuning while increasing B-field gradient to compress the atoms and image their fluorescence on a CCD camera.",
"Finally, we further decrease laser intensity and detuning to optimize cooling and loading the atoms into an ODT.",
"The MOT is overlapped with the focus of a 50 W, 1550 nm laser beam (transport beam).",
"Having a $1/e^2$ radius of 50 $\\mu $ m at its focus, the transport beam forms an ODT with a depth of $\\approx $ 400 $\\mu $ K. The focus is formed using a 10 cm diameter lens with a focal length of 2 m. Atoms are transferred to the transport beam with near unity efficiency in the final phase of the MOT.",
"After transfer is complete, we extinguish the MOT laser beams and translate the focusing lens by approximately one meter to transfer the atoms from the MOT chamber to the EDM measurement region.",
"Optimum transfer efficiency occurs for a sinusoidal position versus time motion profile with a transport time of 9 s. Previously, a 6 s transport time was used [17], however, improvements to the vacuum system reduced atom loss due to background gas collisions and created a new optimum transport time.",
"The measurement region is located at the center of a 1.5 m long glass tube which extends from the MOT chamber horizontally along $\\hat{y}$ .",
"Here, the atoms are compressed in a one-dimensional (1D) MOT along $\\hat{y}$ and transferred to a second ODT based on a 20 W, linearly polarized, single mode 1550 nm laser with a 50 $\\mu $ m $1/e^2$ radius at its focus (the holding beam).",
"This transfer procedure has been described in detail previously [24].",
"After the atoms are transferred into the holding beam, the transport beam is extinguished along with all 714 nm laser light and the 1D MOT coil loop circuit is electrically disconnected by a relay.",
"The holding beam propagates along $\\hat{x}$ and is linearly polarized along $\\hat{y}$ .",
"Our previous EDM measurements [17] used a 10 W retro-reflected holding beam.",
"The interference pattern generated by this previous geometry more tightly confines the atoms along $\\hat{x}$ .",
"Tighter confinement reduces sensitivity to some systematic effects, however, this geometry also reduces the trap lifetime to 10 s due to laser induced heating.",
"With a non-retro-reflected 20 W holding beam, we achieve a trap lifetime of $\\approx $ 40 s. The increased sensitivity to systematic effects is not a concern in the current measurement.",
"We have re-evaluated all known systematic effects for this measurement and find them to be well below the current level of statistical sensitivity (See Sec.",
".)",
"Figure: a) A cross-sectional diagram of the copper electrodes used in the apparatus.",
"Measurements are labeled in mm.",
"The solid circle represents a hole for connecting to a 3.2 mm diameter copper lead through which the electrodes are connected to the high voltage power supply.",
"The rounded edges have a 4 mm radius of curvature.",
"b) A picture of one of the copper electrodes used in this work.In the measurement region, button-shaped copper electrodes are located directly above and below the atoms.",
"The electrode surfaces facing the atoms are circular with a 1.6 cm diameter.",
"(See Fig.",
"REF for a detailed diagram.)",
"Atoms are trapped near the center of the $2.3\\pm 0.1$ mm electrode gap.",
"A Macor support structure holds the electrodes in the glass vacuum chamber.",
"Outside the vacuum chamber, coils for the 1D MOT are wrapped around an aluminum spacer that fits snugly around the glass tube.",
"A cosine-theta coil is wound around a larger cylindrical aluminum tube oriented with its symmetry axis along $\\hat{y}$ to provide a uniform B-field along $\\hat{z}$ with a magnitude of 2.6 $\\mu $ T at the location of the atoms.",
"Three layers of mu-metal shielding enclose this measurement region with a shielding factor of 20,000 for B-fields along $\\hat{z}$ ."
],
[
"Field Alignment",
"Fields we apply to the atoms during precession measurements are oriented to minimize sensitivity to systematic effects.",
"Therefore, an accurate evaluation of their relative orientations is critical to estimating the importance of numerous systematic effects.",
"For the current measurement, a rough alignment of the applied fields based on the mechanical constraints of our apparatus is sufficient to suppress systematic effects to below the current level of statistical sensitivity (See Sec.",
").",
"However, it is still necessary to accurately measure the orientation of the applied fields relative to each other to reliably estimate the magnitude of these effects.",
"Furthermore, ongoing upgrades to our measurement technique will enable $^{225}$ Ra EDM measurements with a statistical sensitivity at the $10^{-26}$ $e$ cm level.",
"We show in Sec.",
"that to avoid systematic effects at this level, we must align our fields to within 0.002 rad of their design orientations.",
"The following section describes our evaluation of the current field orientation and the method we will use to achieve the more stringent alignment requirements of future measurements with improved statistical sensitivity.",
"To evaluate the present field orientation, we use gravity ($-\\hat{z}$ ) and the projection of the glass tube in the horizontal plane ($\\hat{y}$ ) as reference dimensions and measure the orientation of the fields relative to reference surfaces using a precision digital level with a National Institute of Standards and Technology (NIST)-traceable calibration and 350 $\\mu $ rad absolute accuracy.",
"The orientation of fields in the horizontal plane is determined using mechanical tolerances of the apparatus.",
"We determine the direction of the following relevant fields: the applied static E-field (the given direction is for E-field applied parallel to the B-field), $\\hat{\\epsilon }_s$ , the applied B-field, $\\hat{B}$ , the holding beam propagation, $\\hat{k}$ , the holding beam polarization, $\\hat{\\epsilon }$ , and the holding beam B-field, $\\hat{b}$ .",
"By definition, the glass tube can only have a misalignment in the $\\hat{z}$ -$\\hat{y}$ plane.",
"We measure this misalignment to be $0.01\\pm 0.01$ rad toward the $\\hat{z}$ axis.",
"The Macor mount for the electrodes is constrained inside the glass tube such that the misalignment of the tube within the $\\hat{z}$ -$\\hat{y}$ plane determines the misalignment of $\\hat{\\epsilon }_s$ .",
"We image the electrodes along the glass tube axis using a CCD camera with a 1 to 1 imaging system to determine the electrode gap and similar images of the electrodes along $\\hat{x}$ determine the misalignment of $\\hat{\\epsilon }_s$ in the $\\hat{z}$ -$\\hat{x}$ plane from the reduced gap size.",
"These measurements determine the direction of $\\hat{\\epsilon }_s$ which is included in Table REF .",
"Fluxgate magnetic field probes are mounted in a rectangular aluminum plate that is mechanically constrained to have an insignificant yaw angle (rotation about $\\hat{z}$ ) relative to the glass tube.",
"We then determine the pitch and roll angles of the plate using a digital level.",
"The probes measure the magnetic field along three orthogonal axes offset 5 cm along the $\\hat{z}$ axis from the location of the atoms.",
"Using the orientation of the mounting plate and the probes within the plate, we determine the direction of $\\hat{B}$ and use the measured gradient of the $\\hat{z}$ component of the B-field to estimate the uncertainty the 5 cm offset creates in our knowledge of the B-field direction at the atoms.",
"The final uncertainty in the direction of $\\hat{B}$ is dominated by the manufacturer-specified uncertainty of the alignment of each probe within its sealed housing.",
"The holding beam is mechanically constrained, by aperture in the magnetic shields, to propagate along $\\hat{x}$ to within $\\pm 0.03$ rad in any one direction.",
"The polarizer that determines the polarization of the holding beam is referenced to a laser table surface which is level to better than 0.001 rad such that $\\hat{\\epsilon }$ is along $\\hat{y}$ with an uncertainty that is determined by the uncertainty in $\\hat{k}$ .",
"This gives that the polarization vector is constrained to the $\\hat{x}$ -$\\hat{y}$ plane to within $\\pm 0.001$ rad since it requires coupled misalignment of the holding beam in two dimensions.",
"In future measurements, it is possible to align the applied fields to within of 0.002 rad of their design orientation using commercial products to evaluate the field alignment and the current capabilities of the experimental apparatus to adjust field orientations.",
"Using an autocollimator to optically evaluate in-vacuum surface orientations, we can determine the orientation of our electrode spacer, and thus, our applied E-field, relative to a level surface.",
"Using trim and gradient coils that are already implemented in our apparatus, we can then measure and adjust the orientation of the applied B-field to be parallel to the E-field using commercial 3-axis fluxgate magnetometers with sufficiently accurate orientation and orthogonality specifications for the three fluxgate magnetic field probes.",
"Finally, we can orient the propagation direction of our holding beam to be parallel to a surface that is normal to the applied E-field using irises with precise heights mounted to this surface.",
"In this scheme, the E-field will define the $\\hat{z}$ axis while the holding beam will define the $\\hat{x}$ axis."
],
[
"Measurement Procedure",
"We manipulate and detect the spin state of $^{225}$ Ra using 483 nm laser light resonant on the ${}^1$ S${}_0$ , $F = 1/2$ to ${}^1$ P${}_1$ , $F = 1/2$ (blue spin-dependent) transition.",
"Our blue laser has circular polarization and co-propagates with the holding beam along $\\hat{x}$ .",
"We adjust the polarization purity to be greater than $99\\%$ at a polarimeter that samples the beam after it has passed through the vacuum chamber.",
"The circularly polarized blue laser is used to both polarize and detect the atoms.",
"To polarize the atoms, we apply a 150 $\\mu $ s laser pulse that optically pumps the atoms such that their nuclear spins point along $\\hat{x}$ .",
"In this state, the atoms no longer scatter photons from the blue laser and so we call this the “dark” state.",
"The applied B-field is along $\\hat{z}$ so, when the polarizing pulse is over, atoms Larmor precess about the B-field.",
"After 1/2 of a precession period, we say that the atoms are in the “bright” state, because their probability to scatter a photon from the blue laser is maximal.",
"An atom excited by the blue laser has a 2/3 probability to decay to the bright state and a 1/3 probability to decay to the dark state, thus, we can scatter a maximum of 3 photons per atom on average.",
"For atom detection, the collimated blue laser and the shadow cast by atoms scattering photons out of the laser are imaged on a CCD camera using a 300 mm focal length lens located 600 mm from the atom cloud.",
"To produce a “detection image” we activate the CCD camera while a 60 $\\mu $ s laser pulse is applied to the atom cloud.",
"For the laser intensity we use, this pulse duration optimizes the signal to noise ratio (SNR) of the atom signal and corresponds to scattering 2.1 photons per bright state atom [26].",
"The laser beam is much larger than the imaged region so in the absence of bright state atoms we get an average of 2,200 counts per pixel on the CCD camera.",
"Bright state atoms scatter photons out of the laser beam and the CCD image contains both the laser beam intensity profile and the shadow that the atoms cast by scattering photons out of the laser beam.",
"The shadow of $^{225}$ Ra atoms depletes $\\approx $ 100 counts from CCD pixels that correspond to the atom cloud location.",
"The atoms precess about the applied B-field with a period of $34.7\\pm 0.3$ ms.",
"The uncertainty in the precession period is derived from our EDM measurement, which is designed to be sensitive to relative phase accumulation and not absolute frequency.",
"For each experimental cycle, we take 5 detection images of the atoms.",
"Figure REF illustrates the timing sequence.",
"The time between the end of the $i^\\mathrm {th}$ polarizing pulse and the start of the $i^\\mathrm {th}$ detection pulse is $\\Delta T_i$ .",
"The first detection image uses $\\Delta T_1=17.4$ ms, corresponding to 1/2 of a precession period and maximal light scattered out of the detection pulse.",
"Images of atoms taken with $\\Delta T_i=17.4$ ms determine the number of atoms in the trap.",
"The second detection image is taken with $\\Delta T_2=20,000+\\delta $ ms where $\\delta $ varies from -10 ms to 40 ms (see Fig.",
"REF ).",
"During precession before the second image, a $\\pm $ 67 kV/cm electric field is applied to the atoms by charging the top electrode to $\\mp $ 15.5 kV and holding the bottom electrode at ground.",
"Each experimental cycle, we alternate between applying an E-field that is parallel to the B-field and applying an E-field that is anti-parallel to the B-field.",
"The E-field on and off ramps are identical to avoid any systematic effects from the B-field that is induced by the ramp (see Sec.",
"REF ).",
"The E-field is on at full strength for 19.2 s out of the total $20,000+\\delta $ ms $\\Delta T_2$ .",
"To avoid shifting and broadening the blue transition via the DC Stark effect in the strong E-field, the ramp on is initiated after the completion of the polarization pulse and the ramp off is timed such that the E-field is less that $1\\%$ of its maximum value at the start of atom detection.",
"Figure: A diagram of the polarizing and detection pulse sequence used in this measurement.",
"Wider blue bars represent polarizing pulses and the narrow blue bars are detection pulses.",
"The red trapezoid designates when the E-field is ramped on.",
"The label ΔT i \\Delta T_i is used to denote the time between the end of the polarization pulse and the start of the detection pulse for image ii.",
"For images 1, 3, and 5, ΔT 1,3,5 =17.4\\Delta T_{1,3,5}=17.4ms and for images 2 and 4 ΔT 2,4 =20,000+δ\\Delta T_{2,4}=20,000+\\delta ms. After an image is taken, we wait300 ms before applying another polarizing pulse to allow enough time for the camera to read out all captured pixels and become available for the next image.The third detection image is taken with $\\Delta T_3=17.4$ ms.",
"This third image is used to normalize the second image since $\\Delta T_3$ is chosen to place all atoms in the bright state.",
"The third image is a good measure of the number of atoms that were present during the second image since the lifetime of atoms in the trap ($\\approx $ 40 s) is much longer than the time between images 2 and 3 (350 ms).",
"Thus, we can determine the nuclear spin population fraction directly by dividing the atom signal we extract from image 2 by that of image 3.",
"The fourth and fifth detection images are taken with $\\Delta T_4=20,000+\\delta $ ms and $\\Delta T_5=17.4$ ms, respectively.",
"These detection images are completely analogous to images 2 and 3 with one notable exception: no E-field is applied during $\\Delta T_4$ .",
"Data taken with no applied E-field tests for changes in precession frequency that are quadratic in E-field.",
"Since the applied parallel and anti-parallel E-fields are known to be identical only at the 0.7$\\%$ level, any such effect could lead to a systematic effect in our EDM measurement resulting from imperfect E-field reversal (see Sec.",
"REF ).",
"Following the detection images, we extinguish the holding beam for 400 ms to remove any remaining atoms and then take several background images (see Sec.",
").",
"Figure: A collection of 4 images that demonstrate the effectiveness of background subtraction.",
"a) An average of 8 226 ^{226}Ra images atoms taken during the first detection image.",
"Periodic fluctuations are visible in the blue laser beam intensity due to interference fringes that are caused by optical elements whose surfaces create reflections that co-propagate with the input beam.",
"b) An average of the same images after background subtraction.",
"c) An average of 63 images of 225 ^{225}Ra atoms taken during the first detection image.",
"d) An average of those same images after background subtraction.",
"We account for the nontrivial shape of the atom cloud image by weighting each 225 ^{225}Ra image with a corresponding 226 ^{226}Ra image before integrating the image.In the preceding section, we described the procedure we use to acquire images of the atoms and the general purpose for each image.",
"This section will describe how we use these images to extract the nuclear spin state of the atom cloud and how, by varying $\\delta $ , we extract the relative Larmor precession phases of the three E-field conditions using a simultaneous fit to the entire data set.",
"Detection images contain the intensity profile of the blue laser beam as well as the shadow of the atom cloud caused by bright state atoms scattering photons out of the laser beam.",
"Before we can extract information about the atoms, we need to remove the laser beam “background” because it contains sources of noise that are large compared to the atom signal and vary over time.",
"In principle, spatial fluctuations of the laser beam intensity are limited only by photon shot noise such that pixels that detect on average N photons will have fluctuations of $\\sqrt{N}$ .",
"However, scattering from dust or other imperfections in the beam path and multiple reflection paths that interfere with the main transmitted beam all create distortions to the intensity profile of our blue laser that can change on timescales of several seconds.",
"In order to properly interpret the images, we remove these distortions using 5 background images taken before the atoms arrive and 20 background images taken after they have been dropped.",
"The time between background images is 333 ms and the distortions change between images, but they change in a regular and repeatable manner.",
"We take 25 background images in each experimental cycle which is sufficient to effectively remove background distortions.",
"We use a least squares fit to determine the linear combination of background images that best subtracts background distortions from each detection image [27], leaving behind only the atom shadow.",
"Figure REF demonstrates the effectiveness of this technique.",
"The noise level we observe in background-subtracted images is only a factor of 1.2 larger than the predicted photon shot noise in a detection pulse.",
"As shown in Fig.",
"REF , the background-subtracted atom shadow is also distorted due to, for example, thermal effects in the imaging optics caused by the holding beam.",
"To compensate, we run the experimental cycle with $^{226}$ Ra and average the images from each cycle to create high SNR atom images for each of the five detection images.",
"We use these $^{226}$ Ra images to weight $^{225}$ Ra images pixel for pixel before summing over all pixels to extract a number proportional to the strength of the atom signal, which we term the “shadow magnitude”.",
"We create new weighting images approximately once every 24 hours to accommodate long-term drifts in the position of the holding beam.",
"Since the SNR of each individual $^{225}$ Ra shadow image is low, we average shadow magnitudes from multiple experimental cycles for each detection image.",
"This procedure involves binning the data and determining an uncertainty for each bin based on the scatter of shadow magnitudes within each bin.",
"In collecting data into bins, we aim to compromise between minimizing the statistical uncertainty and accounting for slow drifts in the experimental conditions.",
"A detailed account of this procedure can be found in Appendix .",
"The probability to scatter photons out of the blue laser is maximal for the bright state and minimal for the dark state, so the shadow magnitude is proportional to the bright state population fraction.",
"However, the shadow magnitude is also proportional to the number of atoms in the trap during the image.",
"To eliminate fluctuations in the signal size due to fluctuations in atom number, we divide the averaged shadow magnitude from image 2(4) by the corresponding average shadow magnitude from image 3(5).",
"This value is equal to the bright state population fraction because the atoms in image 3(5) are all in the bright state.",
"For each E-field condition and $\\delta $ that we measure in the experiment, we extract the bright state population fraction and its associated uncertainty by propagating the uncertainties of the averaged shadow magnitudes.",
"Whenever we make more than one measurement using the same $\\delta $ and E-field condition, we calculate the weighted mean and weighted error of the mean to combine multiple measurements into a single mean value and uncertainty.",
"The measured bright state population fraction versus $\\delta $ is plotted in Fig.",
"REF for all three E-field conditions.",
"We simultaneously fit this data to the three equations below using chi-square minimization.",
"The equations that describe the spin precession of the three E-field conditions are: $\\begin{array}{c}y_\\mathrm {off} = \\frac{A}{1+P}\\left[1-P\\cos {(\\omega \\Delta T_4)}\\right], \\\\\\\\y_\\mathrm {parallel(anti\\mbox{-}parallel)} = \\frac{A}{1+P}\\left[1- P \\cos {(\\omega \\Delta T_2 + \\theta \\pm \\Delta \\phi /2)}\\right].\\end{array}$ Here, $A$ is a normalization constant, $P$ is the signal contrast, $\\omega $ is the angular precession frequency, $\\theta $ is the phase difference between E-field on and E-field off data, and $\\Delta \\phi $ is the phase difference between the E-field parallel and E-field anti-parallel data resulting from an atomic EDM.",
"We allow $A$ , $P$ , $\\omega $ , $\\theta $ , and $\\Delta \\phi $ to fit to the data.",
"Figure REF plots the functions from Eqn.",
"REF using the best fit values for the five fit parameters.",
"Table REF lists the best values for the fit parameters and their 1-$\\sigma $ standard errors.",
"The uncertainties in Table REF include any possible fit parameter correlations, however, the calculated covariance matrix elements show that the correlations of $\\Delta \\phi $ , which corresponds to the EDM-induced phase difference between E-field parallel and E-field anti-parallel, with all other fit parameters are insignificant.",
"For example, a 1-$\\sigma $ change in $A$ from its best fit value creates a 0.01-$\\sigma $ change in the best fit value of $\\Delta \\phi $ .",
"Since we measure population fraction using a $\\Delta T_{2,4}$ ($\\approx $ 20 s) that is long compared to the variation in $\\delta $ across the data ($\\approx $ 50 ms), the fit finds multiple local minima for the fitted value of $\\omega $ .",
"The value we quote for $\\omega $ corresponds to the global best fit and the variance is calculated as the range over which the $\\chi ^2$ of the fit increases by 1 globally.",
"For all other parameters, the variance is calculated as the inverse of the second partial derivative of the chi-square function with respect to the parameter.",
"With 25 degrees of freedom in the fit, we find $\\chi ^2/25=1.4$ .",
"Figure: Nuclear spin precession data for three E-field conditions: E-field parallel to B-field, E-field anti-parallel to B-field, and E-field off.The plot shows population fraction in the bright state versus δ\\delta , where ΔT 2,4 =20,000+δ\\Delta T_{2,4}=20,000+\\delta ms.Lines represent a simultaneous fit of all the data to Eqns.",
"using a chisquare minimization fitting program.Table: Best fit values and uncertainties for fit parameters used to fit experimental data to Eqn.",
".",
"Values for ω\\omega are given in rad//s and values for θ\\theta and Δφ\\Delta \\phi are given in rad.With an E-field applied parallel(anti-parallel) to the B-field, the frequency of Larmor precession is altered in the presence of a permanent electric dipole moment, $d$ , according to: $\\hbar \\omega _{\\pm } = 2\\mu B \\pm 2 d E.$ Here, $\\hbar $ is the Planck constant, $\\omega _\\pm $ is the precession frequency with the E-field applied parallel(anti-parallel), $\\mu $ is the magnetic dipole moment of the $^{225}$ Ra ${}^{1}\\mathrm {S}_{0}$ ground state, and $E$ is the magnitude of the applied E-field.",
"For an E-field of magnitude $E$ applied over a period of time, $\\tau $ , Eqn.",
"REF also relates the phase difference between precession curves measured with the E-field applied parallel and anti-parallel to the permanent electric dipole moment as follows: $d = \\frac{\\hbar \\Delta \\phi }{4E\\tau }.$ We take $\\tau $ to be the total precession time and $E$ to be the average magnitude of the applied E-field to account for the ramping on and off of the applied E-field.",
"Using the best fit value of $\\Delta \\phi $ to experimental data, we get that the electric dipole moment of $^{225}$ Ra is equal to $(4\\pm 6_\\mathrm {stat}\\pm 0.2_\\mathrm {syst})\\times 10^{-24}$ $e$ cm, which gives a $95\\%$ confidence upper limit of $\\left|d({}^{225}\\mathrm {Ra})\\right|<1.4\\times 10^{-23}$ $e$ cm.",
"This measurement improves upon our previous result [17] by a factor of 36."
],
[
"Systematic Effects",
"Systematic effects mimic the signal of a true EDM by causing a true or apparent spin precession phase shift between the two conditions: E-field parallel to B-field and E-field anti-parallel to B-field.",
"We have evaluated known systematic effects for the current work and the results of this analysis are included below.",
"We use the mean value and uncertainty from our evaluation of each systematic effect to calculate a $68.3\\%$ confidence upper limit and use this value as the 1-$\\sigma $ uncertainty for the effect.",
"The uncertainties for all effects are shown in Tab.",
"REF .",
"The total systematic uncertainty of the measurement is determined by adding the uncertainties from all individual effects in quadrature.",
"Certain systematic effects arise from correlations between an experimental parameter and the applied E-field.",
"To limit these effects, we directly measure the parameter in question throughout the data run and compare the average value during each E-field condition to determine if an E-field correlated change occurs.",
"Our measurement of the E-field correlated change is used to calculate the magnitude of the resulting systematic effect.",
"We also estimate the level at which systematic effects will limit future iterations of our experiment under two scenarios.",
"In the first scenario ($\\alpha $ ), we estimate the systematic sensitivity of the current experimental apparatus with the applied fields aligned to within 0.002 rad of their designed orientation.",
"This is possible with autocollimator-assisted installation of improved (commercially-available) fluxgate magnetometers, along with B-field trimming.",
"For the $\\alpha $ scenario, we assume a statistical sensitivity of $10^{-26}$ $e$ cm.",
"In a second scenario ($\\beta $ ), we imagine a measurement with the improved alignment from the $\\alpha $ scenario, with atoms trapped in a retro-reflected holding beam geometry (as in our previous measurement [17]), and with an atomic magnetometer isotope co-trapped with $^{225}$ Ra in the holding beam (a co-magnetometer).",
"Furthermore, we assume a statistical sensitivity of $10^{-28}$ $e$ cm.",
"A statistical sensitivity at this level is possible by implementing the upgrades outlined in section .",
"We find that the total systematic uncertainty of the current measurement is $2\\times 10^{-25}$ $e$ cm.",
"For future measurements, we estimate that in the $\\alpha $ scenario we will achieve a total systematic uncertainty of $5\\times 10^{-27}$ $e$ cm and in the $\\beta $ scenario this estimate reduces to $4\\times 10^{-29}$ $e$ cm."
],
[
"E-squared effects",
"Any effect that produces a phase shift proportional to the square of the applied E-field cannot result from an EDM but still can produce a systematic effect if the E-field reversal is imperfect.",
"In this measurement, we determine that the magnitude of the parallel and anti-parallel E-fields are matched to within $0.7\\%$ using a calibrated high voltage (HV) divider.",
"We make spin precession measurements with no E-field applied to place an experimental limit on any spin precession phase shift proportional to the square of the applied E-field.",
"Using our measurement of $\\theta $ , the differential phase acquired during spin precession between E-field off data and E-field on data, and an E-field imbalance of $0.7\\%$ , we obtain a 1-$\\sigma $ uncertainty of $1\\times 10^{-25}$ $e$ cm for this effect in the current measurement.",
"We do not predict that we will see phase shifts proportional to the square of the E-field even in the $\\beta $ scenario.",
"In the current experimental procedure, we estimate that we would need a statistical sensitivity of $10^{-25}$ $e$ cm to measure known sources of phase shifts proportional to the square of the E-field.",
"The dominant effect arises from E-field gradients that push the atoms in opposite directions for the two E-field polarities combined with a B-field gradient that creates a position dependent precession frequency.",
"Future implementations of the experiment will use a retro-reflected holding beam which confines the atoms more tightly and reduces the statistical sensitivity needed to observe this effect to $10^{-30}$ $e$ cm.",
"It is important to note that as the statistical sensitivity of the experiment improves, measurements of this systematic will improve in lockstep.",
"Measurements of phase shifts proportional to the square of the applied E-field will always have the same statistical sensitivity as the associated EDM measurement.",
"This systematic effect, however, is reduced by the magnitude imbalance of the applied E-fields.",
"For example, the current measurement has a statistical sensitivity at the $10^{-23}$ $e$ cm level and we balance the E-fields to $0.7\\%$ , enabling us to evaluate this effect with an uncertainty of $10^{-25}$ $e$ cm.",
"In the $\\alpha $ and $\\beta $ scenarios, we will be able to measure this effect with a precision of $7\\times 10^{-29}$ and $7\\times 10^{-31}$ $e$ cm, respectively, using this same technique."
],
[
"B-field correlations",
"The phase acquired during spin precession is directly proportional to the applied B-field and any change of the B-field during spin precession that correlates with the applied E-field will produce a false EDM signal.",
"This false EDM signal is given by: $d_\\mathrm {false}=\\frac{\\mu \\Delta B}{E},$ where $\\Delta B$ is the measured difference in B-field between when the E-field applied parallel and when the E-field is applied anti-parallel.",
"We place three orthogonal fluxgate magnetometer probes offset 5 cm along $\\hat{z}$ from the atoms and continuously measure the B-field during our EDM measurement.",
"We have constrained the gradient in the B-field magnitude along $\\hat{z}$ to be less than $0.1\\%$ /cm by measuring the B-field magnitude at three locations along the $\\hat{z}$ -axis within the cosine-theta coil using a rubidium magnetometer.",
"Taking into account that this effect is only sensitive to changes in the B-field that correlate with changes in the applied E-field, the offset of the fluxgate does not significantly contribute to our measurement of this systematic (i.e.",
"for a measured E-field correlated change in B-field at the fluxgate, we need to allow for the possibility that the correlated change at the atoms is greater than the measured value by at most $1\\%$ due to the 5 cm offset of the probe from the atoms).",
"To achieve a precise measurement of $\\Delta B$ we measure the magnetic field during $\\Delta T_2$ throughout the entire data run and take the difference between pairs of adjacent experimental cycles with opposite E-field polarities.",
"We average all of these differential measurement to reduce our sensitivity to magnetic field noise on timescales longer than 200 s and get that $\\Delta B=-0.3\\pm 0.5$ pT.",
"This gives a 1-$\\sigma $ uncertainty of $1\\times 10^{-25}$ $e$ cm for this systematic effect.",
"For future measurements, we have dramatically decreased B-field measurement noise by low-pass filtering our fluxgate signals.",
"We now achieve the manufacturer specified measurement noise of 6 pT for an integration time of 1 s. In the $\\alpha $ scenario, we estimate a 15 day measurement time where for 19.8 s of each 100 s cycle we can compare the measured B-field from the three E-field conditions to determine if an E-field correlated change in B-field occurs.",
"In this scenario we predict an uncertainty of $5\\times 10^{-27}$ $e$ cm for our measurement of this systematic effect In the $\\beta $ scenario, a co-magnetometer can be used to measure the B-field at the atoms and with greater sensitivity.",
"$^{171}$ Yb or $^{199}$ Hg may be attractive co-magnetometer species because they have similar atomic structure to $^{225}$ Ra and are predicted to be much less sensitive to CP violating effects in the nuclear medium [28], [14].",
"The spin precession measurement can proceed exactly as in $^{225}$ Ra and the statistical sensitivity will improve proportional to the square root of the number of atoms.",
"Since the co-magnetometer atom will be a stable isotope, we conservatively predict that we will be able to trap a factor of ten more magnetometer atoms than $^{225}$ Ra atoms.",
"This leads to a measurement of the E-field correlated change in B-field that is a factor of 3 more precise than the EDM measurement or an uncertainty of $3\\times 10^{-29}$ $e$ cm for this systematic effect."
],
[
"Blue laser frequency correlations with E-field",
"Frequency fluctuations of the 483 nm absorption imaging laser that are correlated with the applied E-field could create an EDM-like signal.",
"The current measurement is insensitive to any such correlation since we split the data between parts of the precession curve with opposite slopes.",
"However, this will still lead to a differential amplitude between the two E-field polarities, which can create an EDM systematic through correlations between the fit parameters $A$ and $\\Delta \\phi $ .",
"We measure the E-field correlated frequency fluctuations of our 483 nm laser by continuously monitoring light transmission through a passive optical cavity that serves as a frequency reference using a battery powered phototdiode.",
"We measure the fractional amplitude of E-field correlated transmission fluctuations, $\\Delta A_\\mathrm {cav}$ to be $-75\\pm 80$ ppm which gives a $68.3\\%$ confidence upper limit of 115 ppm.",
"To connect cavity transmission fluctuations to atomic scattering fluctuations, we use a peak normalized Lorentzian function to describe the cavity and atomic resonances as a function of frequency.",
"To proceed with the most conservative estimation of an EDM systematic effect, we assume that the light is exactly centered on the cavity resonance, where a measurement of intensity fluctuations is insensitive to frequency fluctuations and that the laser light is detuned from atomic resonance by $\\Gamma _\\mathrm {Ra}/(2\\sqrt{3})$ , where $\\Gamma _\\mathrm {Ra}$ is the full width at half maximum (FWHM) of the blue transition.",
"At this detuning, the amplitude of light scattered by atoms is most sensitive to laser frequency.",
"For these assumptions, it can be shown that the fractional fluctuations in atomic scattering amplitude, $\\Delta A_\\mathrm {Ra}$ , is related to $\\Delta A_\\mathrm {cav}$ by $\\Delta A_\\mathrm {Ra} = \\frac{3\\sqrt{3}}{8}\\frac{\\Gamma _\\mathrm {cav}}{\\Gamma _\\mathrm {Ra}}\\sqrt{\\Delta A_\\mathrm {cav}},$ where $\\Gamma _\\mathrm {cav}$ is the FWHM of the cavity resonance and we assume $\\left|\\Delta A_\\mathrm {cav}\\right|\\ll 1$ .",
"Then the false phase shift between E-field polarities caused by correlations between $A$ and $\\Delta \\phi $ is given by, $\\Delta \\phi _\\mathrm {false}=\\Delta A_\\mathrm {Ra}\\frac{\\rho _{A,\\Delta \\phi }}{\\rho _{A, A}}.$ Here, $\\rho _{A,\\Delta \\phi }$ is the covariance matrix element connecting fit parameters $A$ and $\\Delta \\phi $ and $\\rho _{A,A}$ is the variance of $A$ .",
"Finally, we can calculate the false EDM signal with the added suppression of 0.09 because we detect atoms at least 390 ms after the E-field has been ramped down with a $1/e$ time of 160 ms. We calculate the systematic effect of blue laser frequency fluctuations to have a 1-$\\sigma $ uncertainty of $4\\times 10^{-28}$ $e$ cm in our current measurement.",
"In both the $\\alpha $ and $\\beta $ scenarios, we can improve our measurement of blue laser frequency fluctuations by locking to the side of the cavity resonance.",
"This changes the dependence of $\\Delta A_\\mathrm {Ra}$ on $\\Delta A_\\mathrm {cav}$ from Eqn.",
"REF to, $\\Delta A_\\mathrm {Ra} = \\frac{\\Gamma _\\mathrm {cav}}{\\Gamma _\\mathrm {Ra}}\\Delta A_\\mathrm {cav}.$ This gives that the upper limit on a false EDM signal due to E-field correlated changes in blue laser frequency would be $8\\times 10^{-30}$ $e$ cm."
],
[
"Blue laser power correlations with E-field",
"As part of our measurement, we image each blue laser detection pulse on a CCD camera.",
"For the EDM measurement, we subtract the “background” signal and look only at the “shadow” of the atoms that they create by scattering photons out of the blue laser pulse.",
"However, the background is a direct measure of the blue laser power used to image the atoms.",
"For adjacent experimental cycles, we compare the backgrounds subtracted from detection image 2 to obtain the difference in blue laser power between images taken directly after the application of a parallel and anti-parallel E-field.",
"Combining all pairwise measurements into a weighted mean and error, we get that the 1-$\\sigma $ uncertainty in correlated power fluctuations is $0.2\\%$ .",
"For the most conservative systematic estimation we assume that we are far below saturation such that fluctuations in laser power are directly proportional to fluctuations in atomic scattering.",
"Then using the analysis of Section REF to connect fluctuations in atomic photon scattering to a false EDM signal, we limit systematic effects from blue laser power correlations with E-field to a 1-$\\sigma $ uncertainty of $7\\times 10^{-28}$ $e$ cm.",
"For this measurement, the standard deviation of fractional pairwise power differences is $3\\%$ .",
"However, the fundamental photon shot noise of the detection pulse is $0.2\\%$ .",
"By implementing intensity feedback on the acousto-optic modulator that creates the blue laser pules for atom detection, we can achieve shot-noise-limited uniformity in laser pulse intensity.",
"The statistical sensitivity of our blue laser power correlation measurement will improve proportional to the improvement in pulse intensity variation.",
"Furthermore, the sensitivity of the EDM measurement can be additionally suppressed by normalizing the atom absorption signal by the laser pulse power.",
"Thus, we will be able to improve our evaluation of this systematic to $1\\times 10^{-31}$ $e$ cm."
],
[
"Holding ODT power correlations",
"A correlation between the holding beam power and the applied E-field could lead to a systematic effect in our measurement.",
"For example, residual circular polarization in the holding beam causes a vector AC stark shift of the ground state that shifts the Larmor precession frequency proportional to the holding beam power.",
"We limit the contribution of this effect using measured correlations between holding beam power and applied E-field along with a calculation of the shift in $^{225}$ Ra that is similar to a calculation reported for $^{199}$ Hg [29].",
"For diamagnetic atoms having nuclear spin, such as $^{225}$ Ra and $^{199}$ Hg, the vector AC Stark shift is due to hyperfine structure in excited states.",
"Since the dominant contributions arise from transitions to the ${}^{3}\\mathrm {P}_{1}$ and ${}^{1}\\mathrm {P}_{1}$ states, we sum over the AC Stark shifts from these transitions to obtain the differential shift that would occur between the ground state Zeeman sub-levels in a circularly polarized laser beam, $\\nu _v$ .",
"Using the parameters of the current 20 W, single pass holding beam, we calculate $\\nu _v = 50$ Hz.",
"Since linearly polarized light cannot cause a vector shift, it is suppressed in our experiment from the linear polarization of the holding beam.",
"The holding beam passes through a calcite polarizer with a 100,000:1 extinction ratio before entering the vacuum chamber and we measure the polarization purity to be $>99.9\\%$ after passing through the vacuum chamber, which implies a polarization at the atoms of better than 99%, with a high level of confidence.",
"Also, since the vector shift is directed along the laser’s propagation direction, which is perpendicular to the electrodes, there is a further suppression.",
"The final shift can be expressed as $\\Delta \\nu _{m_F}=m_F\\nu _V\\left(|\\varepsilon _L|^2-|\\varepsilon _R|^2\\right)\\cos (\\theta _{BH}),$ where $\\theta _{BH}$ is the angle between the holding beam and the applied B-field and the coefficients $\\varepsilon _R$ and $\\varepsilon _L$ represent the right- and left-hand circular components of the holding beam polarization.",
"This effect is suppressed by two orders of magnitude since the holding beam polarization is over $99\\%$ linear.",
"Furthermore, the measured field alignment gives that the $68.3\\%$ confidence upper limit for $\\cos (\\theta _{BH})$ is 0.1.",
"The resulting false EDM signal is given by $d_\\mathrm {false}=\\Delta \\nu _{1/2}\\frac{h}{2E}\\frac{\\Delta P}{P_0},$ where $\\Delta P/P_0$ is the fractional difference in holding beam power between applying an E-field parallel and anti-parallel to the applied B-field.",
"By monitoring the holding beam during E-field applications we measure no correlation between holding beam power and E-field and limit $\\Delta P/P_0$ to a $68.3\\%$ confidence upper limit of of $8\\times 10^{-5}$ .",
"This corresponds to a 1-$\\sigma $ uncertainty of $6\\times 10^{-26}$ $e$ cm for the associated systematic effect on our measurement.",
"As the statistical sensitivity improves to the level where these effects may become detectable, we can limit this systematic in a model-independent way by directly measuring Larmor precession frequency shifts as a function of holding beam power.",
"With the current apparatus we can change the power of the holding beam between 20 W ($P_0$ ), and 10 W ($0.5P_0$ ) without observing a significant change in atom number.",
"In addition to the three field conditions we currently measure, we will add three additional field conditions corresponding to the three E-field conditions at half the holding beam power.",
"To extract the power shift, we will only use data with the E-field off, however, we will see in Section REF that we will need to compare spin precession at two holding beam powers with the E-field on to limit Stark interference.",
"The power shift is related to $\\phi _\\mathrm {H/L}$ , the measured phase shift between precession at $P_0$ and at $0.5P_0$ by $\\Delta \\nu _{1/2}=\\Delta \\phi _\\mathrm {H/L}/(\\pi \\tau ).$ We can use the statistical uncertainty of our EDM measurement to estimate the statistical uncertainty of $\\phi _\\mathrm {H/L}$ .",
"First, we use Eqn.",
"REF to relate the statistical uncertainty in an EDM measurement to that of a phase shift measurement.",
"The EDM measurement is designed to be sensitive to the E-field on conditions and we must increase the statistical uncertainty of $\\phi _\\mathrm {H/L}$ by a factor of $\\sqrt{2}$ since there are twice as many E-field on measurement compared to E-field off measurements.",
"Then we use Eqn.",
"REF to relate an uncertainty in phase to an uncertainty in the false EDM signal caused by this effect.",
"We get that $\\Delta d_\\mathrm {false}=4\\sqrt{2}d_\\mathrm {stat}\\frac{\\Delta P}{P_0},$ where $d_\\mathrm {stat}$ is the statistical uncertainty of the EDM measurement, and $\\Delta $$d_\\mathrm {false}$ is the statistical uncertainty in our evaluation of this systematic effect.",
"Thus, we get that in the $\\alpha $ and $\\beta $ scenarios, we can evaluate this systematic with an uncertainty of $9\\times 10^{-30}$ and $9\\times 10^{-32}$ $e$ cm, respectively."
],
[
"Leakage Current",
"Leakage current creates a false EDM signal from the induced B-field created by leakage electrons as they travel between electrodes.",
"Since leakage current changes sign with the applied E-field and also grows in magnitude with increasing E-field, this can be a particularly troublesome systematic.",
"The two most troublesome paths for leakage electrons are through the vacuum between the electrodes and along the inner surface of the Macor electrode spacer.",
"We use simple models to limit the effect leakage current can have in either path.",
"For the current measurement, the most menacing path would be emission from one electrode such that the electrons pass close by the atom cloud but not through it.",
"Electrons emitted from an electrode will be accelerated by the E-field and travel in the direction of the applied E-field which induces a B-field, $B_\\mathrm {ind}$ , that is perpendicular to the velocity of the electrons.",
"The dominant effect of $B_\\mathrm {ind}$ is to alter the Larmor precession frequency due to misalignment between the applied E- and B-fields.",
"We treat the electrons as an infinite wire and assume they pass within 50 $\\mu $ m of all atoms, which is roughly the radius of the atom cloud.",
"Then electrons traveling along this path give a false EDM signal that is given by $d_\\mathrm {false}=\\frac{\\mu \\mathbf {B}_\\mathrm {ind}}{E}\\cdot \\hat{B} = \\frac{\\mu }{E}\\frac{\\mu _0 I}{2\\pi r}\\sin {\\theta _{EB}}.$ Here, $I$ is the leakage current, $\\mu _0$ is the vacuum permeability, $r$ is the distance of closest approach for the electron beam, and $\\theta _{EB}$ is the angle between the applied E-field and B-field.",
"We calculate the $68.3\\%$ upper limit for $\\theta _{EB}$ using the measured field orientations and their associated uncertainties to get that $\\theta _{EB}\\le 0.1$ rad.",
"The leakage current monitor during this data run measured a leakage current consistent with zero and having a $68.3\\%$ upper limit of 2 pA.",
"Using Eqn.",
"REF , we get a 1-$\\sigma $ uncertainty for this systematic effect of $3\\times 10^{-28}$ $e$ cm.",
"The next most important possible path for the electrons would be if they travel along the inner surface of the Macor electrode spacer, making full or partial loops along the way which induces a B-field having a component along the $\\hat{B}$ direction.",
"This could result from (e.g.)",
"impurities in the spacer or the irregular shape of the spacer.",
"To put a conservative limit on this effect, we determine the loops or partial loops that induce the largest B-field along $\\hat{z}$ and imagine that the leakage current travels along this path.",
"Moreover, we assume that this path reverses perfectly under E-field reversal.",
"Because the spacer has large gaps machined into it for optical access, a solenoidal path is impossible, leaving the most important path to be a loop in the Macor near the surface where the electrode makes contact with the spacer.",
"With a leakage current of 2 pA, each such loop generates a systematic shift at the level of $4.5\\times 10^{-29}$ $e$ cm.",
"Even in the $\\alpha $ scenario, where the electrode alignment has been corrected, this second possibility cannot be excluded, and so we use a two-loop path as an upper limit for that case, giving $9\\times 10^{-29}$ $e$ cm.",
"In the $\\beta $ scenario, direct B-field measurements with a co-magnetometer would eliminate the need to individually consider the effect of leakage current because this systematic effect would be incorporated into the E-field correlated change in B-field systematic discussed in Section REF ."
],
[
"$\\mathbf {E}\\times \\mathbf {v}$ effects",
"Atoms traveling with velocity $\\mathbf {v}$ in a non-zero E-field, $\\mathbf {E}$ , will experience a B-field $\\mathbf {B}_\\mathrm {motion}$ that is equal to $\\mathbf {B}_\\mathrm {motion}=\\gamma \\left(\\frac{\\mathbf {v}}{c^2}\\times \\mathbf {E}\\right).$ Here, $\\gamma $ is approximately equal to one for the non-relativistic velocities of our atoms.",
"We consider atoms at the Doppler cooling limit of 9 $\\mu $ K for the ${}^{1}\\mathrm {S}_{0}$ to ${}^{3}\\mathrm {P}_{1}$ transition which have a root mean square (RMS) velocity of $v_D=0.022$ m/s in the holding beam trap if we assume harmonic motion.",
"Then $\\left|\\mathbf {B}_\\mathrm {motion}\\right| = 1.6\\times 10^{-12}$ T. However, motion of the atoms in any one dimension is periodic with a period given by $\\tau _\\mathrm {trap}=2\\pi /\\omega _\\mathrm {trap}$ , where $\\omega _\\mathrm {trap}$ is the trap frequency.",
"Thus, the effect of the B-field induced during the first half of $\\tau _\\mathrm {trap}$ is exactly canceled by the equal and opposite B-field induced during the second half.",
"To put the most conservative limit possible on this effect, we imagine a synchronization between Larmor precession and atom motion such that the atom motion is identical during every precession measurement such that there is a maximal un-canceled B-field from the final $\\tau _\\mathrm {trap}/2$ of the total precession time $\\tau $ .",
"The induced B-field is perpendicular to $\\hat{\\epsilon }_s$ and it alters the precession frequency due to misalignment between $\\hat{\\epsilon }_s$ and $\\hat{B}$ .",
"The false EDM signal from this effect is then given by $d_\\mathrm {false}=\\frac{\\mu \\left|\\mathbf {B}_\\mathrm {motion}\\right|}{E}\\frac{\\tau _\\mathrm {trap}}{2\\tau }\\sin {\\theta _{EB}},$ We use the trap frequency for the weakly confined dimension of the holding beam ($\\omega _{\\mathrm {trap},\\hat{x}}=2\\pi \\times 4.25$ rad/s) to obtain the maximal possible effect.",
"This gives a 1-$\\sigma $ uncertainty of $4\\times 10^{-28}$ $e$ cm for this systematic effect.",
"In the $\\alpha $ scenario, this reduces to $7\\times 10^{-30}$ $e$ cm due to better field alignment.",
"For the $\\beta $ scenario, this effect is accounted for by the E-field correlated changed in B-field from Section REF because the co-magnetometer atoms measure the field directly at the location of the $^{225}$ Raatoms."
],
[
"E-field Ramping",
"The ramp up and down of the E-field during spin precession will induce magnetic fields, which can in principle lead to a phase shift proportional to the applied E-field.",
"This effect cancels perfectly in the event that the ramp up and down are temporally symmetric.",
"In our experiment, the ramps are computer controlled by an arbitrary waveform generator to ensure the time reversal symmetry of ramping on and off.",
"Nevertheless, we estimate the maximum possible magnitude for this effect by considering the differential phase shift between the two E-field polarities from just the E-field ramp on without considering any cancellation of the effect from the ramp off.",
"There are two effects that lead to induced B-fields that are proportional to the applied E-field: $\\begin{array}{c}\\mathbf {B}_\\mathrm {ind} = \\mathbf {B}_\\mathrm {cur}+\\mathbf {B}_{dE/dt},\\quad \\mathrm {where} \\\\\\\\\\mathbf {B}_\\mathrm {cur}=\\frac{\\mu _0}{4\\pi }\\int _C\\frac{I d\\mathbf {l}\\times \\mathbf {r}^{\\prime }}{\\left|\\mathbf {r}^{\\prime }\\right|^3}, \\quad \\mathrm {and} \\\\\\\\\\oint _{\\partial \\Sigma ^{\\prime }} \\mathbf {B}_\\mathrm {dE/dt}\\cdot d\\mathbf {l}^{\\prime }=\\mu _0\\varepsilon _0\\frac{d}{dt}\\iint _{\\Sigma ^{\\prime }}\\mathbf {E}\\cdot d\\mathbf {S}^{\\prime }.\\end{array}$ Here, $\\mathbf {B}_\\mathrm {cur}$ is the B-field that is induced by the current, $I$ , traveling through the copper lead to the electrode along path $C$ during the field ramp.",
"Also, $\\mathbf {B}_{dE/dt}$ is the B-field that is induced on the edge of a surface, $\\Sigma $ , due to the changing E-field flux through the surface.",
"For simplicity, we assume a linear ramp and model the copper lead as a half infinite wire along $\\hat{y}$ that is offset from the atoms by 13.15 mm (see Fig.",
"REF ).",
"We consider an increasing E-field that is uniform between the electrodes and zero everywhere else and consider atoms displaced horizontally from the electrode center to the edge of the electrode, where the induced B-field is strongest.",
"We imagine a displacement along $\\hat{y}$ such that $\\mathbf {B}_\\mathrm {cur}$ and $\\mathbf {B}_{dE/dt}$ are in the same direction and consider the effect of $\\mathbf {B}_\\mathrm {ind}$ on the precession frequency due to misalignment between the applied B- and E-fields.",
"The observed phase shift between the two E-field polarities due to this effect is then, $\\Delta \\phi _\\mathrm {false}=2\\pi t_0\\Delta \\nu _\\mathrm {false}=2\\pi t_04\\mu |B_\\mathrm {ind}|\\sin {\\left(\\theta _{EB}\\right)}/h,$ where $t_0$ is the duration of the ramp.",
"Using Eqn.",
"REF we calculate the 1-$\\sigma $ uncertainty in the corresponding false EDM signal to be $9\\times 10^{-28}$ $e$ cm.",
"In the $\\alpha $ scenario, this reduces to $2\\times 10^{-29}$ $e$ cm due to better field alignment.",
"Since this effect arises from an E-field correlated change in B-field, in the $\\beta $ scenario the co-magnetometer atoms would directly evaluate this effect as a contribution to the systematic discussed in Section REF ."
],
[
"Stark Interference",
"Stark interference is an effect, third order in perturbation theory, that allows for interaction between an applied DC electric field and the AC electromagnetic field of an intense laser beam [29].",
"It results in a vector shift of the ground state which is linear in the DC electric field and also linear in the intensity of the laser.",
"Since the effect is linear in the electric field, it can mimic an EDM, and so we expect to observe a false EDM signal caused by the holding beam.",
"Fortunately, Stark interference can be distinguished from a true EDM by observing its dependence on the holding beam power, since only a false EDM will change in size as a function of laser power.",
"Here we estimate the expected magnitude of this effect.",
"The vector dependence of Stark interference is most simply stated if we write the frequency shift between ground state Zeeman sub-levels as the sum of two components: $\\Delta \\nu = \\nu _1(\\hat{b}\\cdot \\hat{\\sigma })(\\hat{\\epsilon }\\cdot \\hat{\\epsilon }_s)+\\nu _2(\\hat{b}\\cdot \\hat{\\epsilon }_s)(\\hat{\\epsilon }\\cdot \\hat{\\sigma }),$ where $\\hat{b}$ is the direction of the holding beam B-field, $\\hat{\\sigma }$ is the spin quantization axis, $\\hat{\\epsilon }$ is the holding beam polarization direction, and $\\hat{\\epsilon }_s$ is the direction of the applied static E-field.",
"Expressions for $\\nu _1$ and $\\nu _2$ are given in Appendix .",
"We use the measured orientation of the applied field and their associated uncertainties to determine the $68.3\\%$ confidence upper limits for the vector products in Eqn.",
"REF .",
"We determine that $(\\hat{b}\\cdot \\hat{\\sigma })(\\hat{\\epsilon }\\cdot \\hat{\\epsilon }_s)\\le 0.03$ and $(\\hat{b}\\cdot \\hat{\\epsilon }_s)(\\hat{\\epsilon }\\cdot \\hat{\\sigma })\\le 0.1$ .",
"The false EDM signal is then obtained by calculating the phase shift this effect would cause in our EDM measurement using Eqn.",
"REF .",
"This gives a 1-$\\sigma $ uncertainty of $6\\times 10^{-26}$ $e$ cm for this systematic effect.",
"In the $\\alpha $ scenario, this reduces to $2\\times 10^{-27}$ due to better field alignment.",
"This effect is further suppressed in a retro-reflected holding beam geometry up to the power imbalance of the initial and retro-reflected beams.",
"In the $\\beta $ scenario, we expect this systematic to appear at the $2\\times 10^{-28}$ $e$ cm level assuming a power imbalance of the holding beam at the $10\\%$ level.",
"The size of this effect will decrease proportional to improvements in holding beam power balance.",
"To sufficiently correct for this systematic, we will need to directly evaluate Stark interference by varying the holding beam power as discussed in Section REF .",
"It is possible to evaluate this systematic below the statistical uncertainty of the EDM measurement by first evaluating Stark interference in an intense traveling wave trap for both the co-magnetometer atom and for $^{225}$ Ra.",
"We estimate that Stark interference will cause a phase shift that corresponds to a false EDM signal at the $10^{-25}$ $e$ cm level for this geometry if we use the current maximum laser power of 30 W. This will provide a ratio measurement at the $0.1\\%$ level of the effect of Stark interference in the co-magnetometer to that in $^{225}$ Ra.",
"Then we can measure the EDM-like phase shift of the co-magnetometer atom versus holding beam power in the retro-reflected geometry and use the precisely measured ratio to determine the effect of Stark interference on the EDM measurement with the precision of a co-magnetometer measurement, which is $3\\times 10^{-29}$ $e$ cm."
],
[
"Geometric phase",
"Spins moving in an inhomogeneous magnetic field experience a shift to their Larmor frequency due to the accumulation of geometric phases.",
"Part of this frequency shift can be proportional to the applied E-field and provide a false EDM signal.",
"The worst case scenario occurs when atoms take a peripheral orbit in the trap and thus trace out a large area with their orbit.",
"In our experiment, we have a thermal cloud of atoms and must average over all types of orbits.",
"Using the treatment in Ref.",
"[30], the false EDM signal from geometric phases is given by $d_\\mathrm {false}=\\frac{-F\\hbar }{2B_{0z}^2 c^2}\\left|v_{xy}\\right|^2\\frac{\\partial B_{0z}}{\\partial z}\\frac{1}{1-\\omega _r^2/\\omega _0^2},$ where $F=1/2$ is the total spin, $\\left|v_{xy}\\right|=\\sqrt{2/3}v_\\mathrm {D}$ is the RMS speed in the $\\hat{x}-\\hat{y}$ plane, $B_{0z}$ is the magnitude of the applied B-field, $\\omega _0$ is the Larmor frequency, and $\\omega _r$ is the trap frequency.",
"For the gradient in $B_{0z}$ we use $0.1\\%$ /cm which is the measured upper limit on the gradient of the B-field magnitude for displacement along $\\hat{z}$ .",
"The trap is asymmetric in the $\\hat{x}-\\hat{y}$ plane with trap frequencies $\\omega _x=4$ Hz and $\\omega _y=610$ Hz.",
"We calculate the uncertainty in this effect for $\\omega _x$ because it is closest to $\\omega _0$ and produces the largest effect.",
"The absolute value places an upper bound of $7\\times 10^{-30}$ $e$ cm, which we use as an estimate of the 1-$\\sigma $ uncertainty in this effect.",
"In the $\\beta $ scenario which uses a retro-reflected holding beam geometry, this reduces to $5\\times 10^{-33}$ $e$ cm due to the changes in trap frequencies since cancellation of this effect improves as the trap frequencies and the precession frequency become more different."
],
[
"Summary",
"Table REF summarizes the current and projected 1-$\\sigma $ uncertainties of all the systematic effects discussed above.",
"The total systematic uncertainty of $2\\times 10^{-25}$ $e$ cm is the quadrature sum of uncertainties for all individual effects.",
"The total systematic uncertainty is added in quadrature with the statistical uncertainty to calculate the $95\\%$ confidence upper limit, although the systematic uncertainty is insignificant for this calculation.",
"The systematic uncertainty reported here represents an upper limit for the current measurement.",
"We have shown in the above discussion that for future EDM measurements with improved statistical uncertainty, we can limit the total systematic uncertainty to below the statistical uncertainty of the measurement.",
"Improved field alignment in the $\\alpha $ scenario and the addition of a co-magnetometer in the $\\beta $ scenario allow ever more precise evaluation of systematic effects.",
"It is important to stress that, even in the $\\beta $ scenario, the listed uncertainties do not represent fundamental limitations.",
"Through more careful evaluation of the apparatus, co-magnetometry, and direct evaluation of certain systematic effects, we will be able to further reduce the uncertainty of these effects in an EDM measurement as the statistical sensitivity of the experiment improves.",
"Table: A list of systematic effects and the associated 1-σ\\sigma uncertainties for each effect in units of ee cm.",
"No corrections are added in to the measured EDM value due to the small size of the effects and the dependence on conservative theoretical models.",
"Rather, we calculate the 68.3%68.3\\% confidence upper limit for all effects and take this value as the 1-σ\\sigma systematic uncertainty.",
"Confidence limits are calculated from mean values and uncertainties of physical measurements that are used in the calculations of systematic effects.",
"The third and fourth columns list projected uncertainties for the α\\alpha and β\\beta scenarios, respectively.",
"Projected uncertainties that are labeled “N/A” do not need to be individually considered when using a co-magnetometer.",
"We would observe these effects as an E-field correlated change in B-field and we would directly limit these effects with our limit on the B-field correlations systematic effect."
],
[
"Discussion and Outlook",
"This measurement of the $^{225}$ Ra EDM represents a significant improvement over our previous result [17] and demonstrates the progress that our experiment is poised to achieve.",
"However, the achieved upper limit of $\\left|d\\left(^{225}\\mathrm {Ra}\\right)\\right|\\le 1.4\\times 10^{-23}$ $e$ cm still does not improve current limits on CP violation in the nuclear medium [18] that are derived from measurements of the $^{199}$ Hg EDM [4], [5] and the neutron EDM [1].",
"However, there are several major experimental upgrades under development that can dramatically improve future measurements of the $^{225}$ Ra EDM.",
"One limiting aspect of the current experiment is the small number of photons per atom that scatter out of the detection laser.",
"On average, it is possible to get 3 photons per atom before an atom decays to the dark state.",
"With so few photons scattered, it is unsurprising that the current measurement is limited by photon shot noise in the detection laser.",
"An increase in the number of photons scattered per atom would increase the signal size proportionally.",
"Since laser photon shot noise is the dominant noise source, increasing the signal size does not increase noise until we become limited by quantum projection noise (QPN), a fundamental source of noise resulting from projecting a quantum superposition state onto one of the basis states.",
"In this measurement, the laser light is either scattered out of the beam or not, corresponding to projecting the atom into either the bright or dark state, respectively.",
"We can use the ${}^{1}\\mathrm {S}_{0}$ , $F=1/2$ to ${}^{1}\\mathrm {P}_{1}$ , $F=3/2$ (blue cycling) transition to detect atom number, which can scatter up to 1000 photons per atom on average before the atoms decay to a long-lived metastable state.",
"However, both the dark and bright nuclear spin states scatter light resonant on this transition so although this state is useful for detecting atom number, it cannot directly detect spin precession.",
"In future measurements, we plan to implement a nuclear-spin-dependent coherent transfer to the metastable $^3$ D$_1$ state using stimulated Raman adiabatic passage (STIRAP) [31].",
"This will allow us to detect spin precession using the blue cycling transition and improve the statistical sensitivity of an EDM measurement by over an order of magnitude.",
"With each atom contributing more signal to the measurement, the remaining fundamental limit to the statistical sensitivity of our experiment will then be QPN.",
"Sensitivity to an EDM is greatest if we measure for phase shifts in Larmor precession when the slope of population fraction versus precession time is greatest.",
"This corresponds to interrogating the atoms when they are in an equal superposition of the bright and dark states.",
"The QPN associated with projecting this state onto either the bright or dark state is proportional to the square root of the atom number while the signal is proportional to the atom number.",
"Thus, the statistical sensitivity of our measurement improves as the square root of the number of atoms we interrogate.",
"In future measurements, we can significantly gain in statistical sensitivity by increasing the number of atoms we interrogate by many orders of magnitude.",
"First, we can address the disparity between the velocity of atoms exiting the oven and the capture velocity of the Zeeman slower by developing a longitudinal slower based on the blue cycling transition.",
"Although it is straightforward to design a variable magnetic field for atom deceleration with the blue cycling transition, keeping the three necessary repump lasers on resonance throughout the slower will add additional complexity to the slower operation because their Zeeman shifts are not matched to that of the blue cycling transition.",
"Effective atom slowing based on this transition is currently being developed using chirped frequency ramps rather than a variable magnetic field, a technique first demonstrated with sodium atoms [32].",
"Furthermore, as new isotope production capabilities begin operation, the amount of $^{225}$ Ra that we load into the oven can be substantially increased.",
"In the near future, efforts within the National Isotope Development Center to produce more $^{225}$ Ac for the medical industry will also produce more $^{225}$ Ra.",
"Further in the future, efforts to harvest rare isotopes produced at the upcoming Facility for Rare Isotope Beams [33] have the potential to provide several orders of magnitude more $^{225}$ Ra than current production methods.",
"In addition to improvements in atom detection and increases in atom number, we are also fabricating new electrodes to achieve higher E-fields.",
"The current electrodes initially demonstrated a peak E-field of 100 kV/cm but now routinely only produce 65 kV/cm.",
"In contrast, other groups have demonstrated titanium electrodes that produce 800 kV/cm with no dark current using standard surface preparation techniques and a similar geometry to the current electrodes [34].",
"In the QPN limited regime, a simple estimation for the statistical sensitivity of an EDM measurement is given by $\\sigma _{EDM}=\\frac{\\hbar }{2E\\sqrt{\\tau N T}},$ where $\\sigma _{EDM}$ is the standard error of the EDM measurement, $E$ is the magnitude of the applied E-field, $\\tau $ is the precession time, $N$ is the number of atoms, and $T$ is the total measurement time.",
"An attractive route to achieving a statistical sensitivity of $1\\times 10^{-28}$ $e$ cm is to leverage new cooling and slowing techniques based on the blue transition and increases in the amount of available $^{225}$ Ra such that we increase the number of interrogated atoms from 500 to $5\\times 10^6$ atoms.",
"With improved electrodes, we would then need to achieve a field of 150 kV/cm to reach $1\\times 10^{-28}$ $e$ cm with an integration time of 60 days.",
"The combination of improving atom detection, increasing atom number, and increasing E-field strength form a solid framework for future $^{225}$ Ra EDM measurements that are many orders of magnitude more precise than the current results.",
"Furthermore, there are no known systematic effects that would prohibit a measurement at this level.",
"Taking into account the unique sensitivity $^{225}$ Ra has to CP violation in the nuclear medium, a measurement at the $10^{-28}$ $e$ cm level would probe CP violation with unprecedented precision.",
"This work is supported by U.S. Department of Energy (DOE), Office of Science, Office of Nuclear Physics, under contracts No.",
"DE-AC02-06CH11357 and No.",
"DE-FG02-99ER41101.",
"$^{225}$ Ra used in this research was supplied by DOE, Office of Science, Isotope Program in the Office of Nuclear Physics.",
"M. B. acknowledges support from Argonne Director’s postdoctoral fellowships."
],
[
"Data Averaging",
"We use the term “cycle” to refer to a single implementation of the experiment consisting of MOT loading, atom transfer, and atom detection.",
"For each cycle we produce five shadow magnitudes from the five atom detection images, however, image 1 is currently not used for analysis.",
"We operate the experiment over multiple cycles without changing $\\delta $ .",
"We accumulate shadow magnitudes taken with the same $\\delta $ and E-field condition from multiple cycles (and their associated normalization images) into “bins” and calculate the mean and standard error of the image data in each bin.",
"The sign of the E-field is alternated each cycle, so bins contain data from every other cycle for images 2 and 3 and data from each cycle for images 4 and 5, because no E-field is applied during the precession time before image 4.",
"The number of shadow magnitudes in each bin is known as the “bin size” and we analyze the data using bin sizes from 3 to 64, which is typically the maximum number of cycles that occur before we change $\\delta $ .",
"A “set” of data refers to the data from multiple cycles that are taken during continuous operation of the experiment while no changes to $\\delta $ are made.",
"We never collect data from different sets into the same bin even if both sets are taken with the same values of $\\delta $ .",
"This binning procedure is useful to optimize the statistical sensitivity of our measurement because we have known sources of non-Gaussian noise that occur at different time scales.",
"On one hand, making bins larger reduces sensitivity to power fluctuations in the blue laser.",
"The power is manually adjusted between sets to scatter 2.1 photons per bright state atom, however, within a set the power is observed to fluctuate within $\\pm 10\\%$ of its initial value.",
"Since the time scale for these drifts is similar to the cycle time of 100 s, collecting more cycles together in one bin reduces sensitivity to these fluctuations.",
"On the other hand, drifts in atom number due to oven temperature changes and oven depleting occur at a much longer time scale of several hours to days.",
"By setting the bin size small enough such that the atom signal is normalized before the change in atom number becomes significant, we can reduce sensitivity to non-random fluctuations in atom number.",
"We find that a bin size of 32 is optimal based on the reduced chi-square of the fit to the model (Eqn.",
"REF ).",
"This corresponds to collecting all data with E-field applied into a single bin and separating the data with no E-field applied into two bins for the most common data set size of 64 experimental cycles.",
"This shows that separation into data sets is already sufficient to eliminate sensitivity to non-random fluctuations in atom number."
],
[
"Stark Interference Details",
"We perform a similar calculation of this effect to one for $^{199}$ Hg [29], where the energy shift of the $m_F$ ground state Zeeman sublevel is given by $\\begin{array}{lcl}\\Delta \\mathbb {E}(m_F) & = & \\frac{\\mu _B E_0^2 E_s}{4c\\hbar ^2}\\sum _{S,F^{\\prime },F^{\\prime \\prime },m_{F^{\\prime }},m{F^{\\prime \\prime }}}{^{2S+1}\\mathrm {P}_1,F^{\\prime },m_{F^{\\prime }}}{\\hat{\\mathbf {\\epsilon }}\\cdot \\mathbf {r}}{{}^1\\mathrm {S}_0,F=1/2,m_F}\\times \\\\\\\\& &{^{2S+1}\\mathrm {P}_1,F^{\\prime \\prime },m_{F^{\\prime \\prime }}}{\\hat{\\mathbf {b}}^*\\cdot \\left(\\mathbf {L}+2\\mathbf {S}\\right)}{{}^{2S+1}\\mathrm {P}_1,F^{\\prime },m_{F^{\\prime }}}\\times \\\\\\\\& &{{}^1\\mathrm {S}_0,F=1/2,m_F}{\\hat{\\mathbf {\\epsilon }}_s\\cdot \\mathbf {r}}{^{2S+1}\\mathrm {P}_1,F^{\\prime \\prime },m_{F^{\\prime \\prime }}}\\frac{1}{\\left(\\omega ^{\\prime } - \\omega _L\\right)\\left(\\omega ^{\\prime \\prime } - \\omega _L\\right)}+\\mathrm {perm.",
"}+\\mathrm {c.r.",
"}\\end{array}$ Here, $\\mu _B$ is the Bohr magneton, $E_0$ is the holding beam electric field amplitude, $\\hat{\\epsilon }$ is the polarization vector of the holding beam, $E_s$ is the magnitude of the applied static E-field, $\\hat{\\epsilon }_s$ is the direction of the applied E-field, $\\hat{b}$ is the direction of the holding beam magnetic field, and $\\omega ^{\\prime }(^{\\prime \\prime })$ is the frequency of the ${^1S_0}$ to ${^{2S+1}P_1, F^{\\prime }(^{\\prime \\prime }), m_{F^{\\prime }}(^{\\prime \\prime })}$ transition.",
"In principle, “$\\mathrm {perm.",
"}$ ” represents all six permutations of $\\hat{\\mathbf {\\epsilon }}\\cdot \\mathbf {r}$ , $\\hat{b}\\cdot \\left(\\mathbf {L}+2\\mathbf {S}\\right)$ , and $\\hat{\\epsilon }_s\\cdot \\mathbf {r}$ , however, magnetic dipole amplitudes within the ground state manifold are suppressed by the ratio of $\\mu _B/\\mu _N$ , where $\\mu _N$ is the nuclear magneton, so only the two permutations that have magnetic dipole amplitudes between excited state hyperfine levels are significant.",
"We also need to include counter-rotating terms, designated by “$\\mathrm {c.r.",
"}$ ” in Eqn.",
"REF .",
"The sum over excited states is limited to the ${}^{1}\\mathrm {P}_{1}$ and ${}^{3}\\mathrm {P}_{1}$ states since the electric dipole amplitudes between these states and ${}^{1}\\mathrm {S}_{0}$ dominate those of other states by many orders of magnitude.",
"To simplify Eqn.",
"REF , we use a standard technique to reduce dipole matrix elements by expressing them in terms of rank 1 spherical tensor operators $T_q^{1}$ and employing the Wigner-Eckart theorem to factor out all angular dependence.",
"The energy shift due to a specific combination of tensor operators $T_{q_\\epsilon }^1$ , $T_{q_s}^1$ , and $T_{q_b}^1$ is then given by, $\\begin{array}{c}\\Delta E(m_F,q_\\epsilon ,q_s,q_b) = \\frac{\\mu _B E_0^2 E_s}{4c\\hbar ^2}\\sum _{S,F^{\\prime },F^{\\prime \\prime },m_{F^{\\prime }},m_{F^{\\prime \\prime }}}\\left[\\frac{(-1)^{q_b}G(m_F,F^{\\prime },m_{F^{\\prime }},F^{\\prime \\prime },m_{F^{\\prime \\prime }},q_s,-q_b,q_\\epsilon )}{(\\omega ^{\\prime }-\\omega _L)(\\omega ^{\\prime \\prime }-\\omega _L)} \\right.",
"\\\\\\\\\\left.",
"+ \\frac{(-1)^{q_\\epsilon }G(\\text{``\\quad ''},q_s,q_b,-q_\\epsilon )}{(\\omega ^{\\prime }+\\omega _L)(\\omega ^{\\prime \\prime }+\\omega _L)} + \\frac{(-1)^{q_\\epsilon }G(\\text{``\\quad ''},-q_\\epsilon ,q_b,q_s)}{(\\omega ^{\\prime }-\\omega _L)(\\omega ^{\\prime \\prime }-\\omega _L)} + \\frac{(-1)^{q_b}G(\\text{``\\quad ''},q_\\epsilon ,-q_b,q_s)}{(\\omega ^{\\prime }+\\omega _L)(\\omega ^{\\prime \\prime }+\\omega _L)} \\right],\\end{array}$ where the function G is defined to be $\\begin{array}{c}G(m_F,F^{\\prime },m_{F^{\\prime }},F^{\\prime \\prime },m_{F^{\\prime \\prime }},q_1,q_2,q_3) \\equiv (2F^{\\prime }+1)(2F^{\\prime \\prime }+1)(-1)^{1+3F^{\\prime \\prime }+7I+J^{\\prime \\prime }+2J^{\\prime }-m_F-m_{F^{\\prime }}-m_{F^{\\prime \\prime }}+3F^{\\prime }} \\\\\\\\\\times \\left\\lbrace \\begin{array}{ccc}J^{\\prime \\prime } & 1 & J^{\\prime } \\\\F^{\\prime } & I & F^{\\prime \\prime }\\end{array}\\right\\rbrace \\end{array}\\left(\\begin{array}{ccc}I & 1 & F^{\\prime \\prime } \\\\-m_F & q_1 & m_{F^{\\prime \\prime }}\\end{array}\\right)$ (ccc F” 1 F' -mF” q2 mF' ) (ccc F' 1 I -mF' q3 mF ) |1S0||d||2S+1P1|2 {cc 2/3 if S=0 3/6 if S=1 .",
".",
"Here, curly brackets denote a Wigner 6-j symbol, parentheses denote a Wigner 3-j symbol, and the double vertical lines represent a reduced matrix element.",
"Values of the reduced matrix elements used in this calculation are taken from [35].",
"Finally we group the energy shifts, expressed in the form of Eqn.",
"REF , into a single shift if they have the same vector dependence.",
"Since only two nonzero vector components exist, we can express the total frequency shift between Zeeman sub-levels due to stark interference in terms of the two frequencies, $\\nu _1$ and $\\nu _2$ , from Eqn.",
"REF .",
"These frequencies are given by $\\begin{array}{lcl}\\nu _1 &=& -\\frac{1}{2h}\\left(\\left[\\Delta E(1/2,-1,0,-1)+\\Delta E(1/2,1,0,1)-\\Delta E(1/2,-1,0,1)-\\Delta E(1/2,1,0,-1)\\right]-\\right.\\\\\\\\& &\\left.\\left[\\Delta E(-1/2,-1,0,-1)+\\Delta E(-1/2,1,0,1)-\\Delta E(-1/2,-1,0,1)-\\Delta E(-1/2,1,0,-1)\\right]\\right)\\\\\\\\&& \\mathrm {and} \\\\\\\\\\nu _2 &=&-\\frac{1}{2h}\\left(\\left[\\Delta E(1/2,0,-1,-1)+\\Delta E(1/2,0,1,1)-\\Delta E(1/2,0,-1,1)-\\Delta E(1/2,0,1,-1)\\right]\\right.\\\\\\\\& &\\left.\\left[\\Delta E(-1/2,0,-1,-1)+\\Delta E(-1/2,0,1,1)-\\Delta E(-1/2,0,-1,1)-\\Delta E(-1/2,0,1,-1)\\right]\\right).\\end{array}$"
]
] | 1606.04931 |
[
[
"Deep Reinforcement Learning Discovers Internal Models"
],
[
"Abstract Deep Reinforcement Learning (DRL) is a trending field of research, showing great promise in challenging problems such as playing Atari, solving Go and controlling robots.",
"While DRL agents perform well in practice we are still lacking the tools to analayze their performance.",
"In this work we present the Semi-Aggregated MDP (SAMDP) model.",
"A model best suited to describe policies exhibiting both spatial and temporal hierarchies.",
"We describe its advantages for analyzing trained policies over other modeling approaches, and show that under the right state representation, like that of DQN agents, SAMDP can help to identify skills.",
"We detail the automatic process of creating it from recorded trajectories, up to presenting it on t-SNE maps.",
"We explain how to evaluate its fitness and show surprising results indicating high compatibility with the policy at hand.",
"We conclude by showing how using the SAMDP model, an extra performance gain can be squeezed from the agent."
],
[
"Introduction",
"Deep Q Network (DQN) is an off-policy learning algorithm that uses a Convolutional Neural Network (CNN; [6]) to represent the action-value function.",
"Agents trained using DQN are showing superior performance on a wide range of problems [10].",
"Their success, and that of Deep Neural Network (DNN) in general, is explained by its ability to learn good representations automatically.",
"Unfortunately, its high expressiveness is also the source of its unclarity, making it very hard to analyze.",
"Visualization methods for DNN try to tackle this problem by analyzing and interpreting the learned representations [24], [3], [22].",
"However, these methods were developed for supervised learning tasks, assuming the data is i.i.d, thus overlooking the temporal structure of the learned representation.",
"A major challenge in Reinforcement Learning (RL) is scaling to higher dimensions in order to solve real-world applications.",
"Spatial abstractions such as state aggregation [1], tries to tackle this problem by grouping states with similar characteristics such as policy behaviour, value function or dynamics.",
"On the other hand, temporal abstractions (i.e., options or skills [17]) can help an agent to focus less on lower level details of a task and more on high level planning [2], [11].",
"The problem with these methods is that finding good abstractions is typically done manually which hampers their wide use.",
"The internal model principle [4], \"Every good key must be a model of the lock it opens\", was formulated mathematically for control systems by [15], claiming that if a system is solving a control task, it must necessarily contain a subsystem which is capable of predicting the dynamics of the system.",
"In this work we follow the same line of thought and claim that DQNs are learning an underlying spatio-temporal model of the problem, without implicitly being trained to.",
"We identify this model as an Semi Aggregated Markov Decision Process (SAMDP), an approximation of the true MDP that allows human interpretability.",
"[23] used hand-crafted features in order to interpret policies learned by DQN agents.",
"They revealed that DQNs are automatically learning spatio-temporal representations such as hierarchical state aggregation and skills.",
"The main drawback of their approach is that they used a manual reasoning of a t-Distributed Stochastic Neighbour Embedding (t-SNE) map [20], a tedious process that requires careful inspection as well as an experienced eye.",
"Moreover, their claim to observe skills is not supported with any quantitative evidence.",
"In contrast, we use temporal aware clustering algorithms in order to aggregate the state space, and automatically reveal the underlying spatio-temporal structure of the t-SNE map.",
"The aggregated states uniquely identify skills and allow us to estimate the SAMDP transition probabilities and reward signal empirically.",
"In particular our main contributions are SAMDP: a model that gives a simple explanation on how DRL agents solve a task - by hierarchically decomposing it into a set of sub-problems and learning specific skills at each.",
"Automatic analysis: we suggest quantitative criteria that allows us to select good models and evaluate their consistency.",
"Interpretation: we developed a novel visualization tool that gives a qualitative understanding of the learned policy.",
"Shared autonomy: the SAMDP model allows us to predict situations where the DQN agent is not performing well.",
"In such occasions we suggest to take the control from the agent and ask for expert advice."
],
[
"Background",
"We briefly review the standard reinforcement learning framework of discrete-time, finite Markov decision processes (MDPs).",
"In this framework, the goal of an RL agent is to maximize its expected return by learning a policy $\\pi :S \\rightarrow \\Delta _A$ , a mapping from states $s \\in S$ to probability distribution over actions $A$ .",
"At time $t$ the agent observes a state $s_t \\in S$ , selects an action $a_t \\in A$ , and receives a reward $r_t$ .",
"Following the agents action choice, it transitions to the next state $s_{t+1} \\in S$ .",
"We consider infinite horizon problems where the cumulative return at time $t$ is given by $R_t = \\sum _{t^{\\prime }=t}^\\infty \\gamma ^{t^{\\prime }-t}r_t$ , and $\\gamma \\in [0,1]$ is the discount factor.",
"The action-value function $Q^{\\pi }(s,a) = \\mathbb {E} [R_t|s_t = s, a_t = a, \\pi ]$ represents the expected return after observing state $s$ , taking action $a$ after which following policy $\\pi $ .",
"The optimal action-value function obeys a fundamental recursion known as the optimal Bellman Equation: $Q^* (s_t,a_t)=\\mathbb {E} \\left[r_t+\\gamma \\underset{a^{\\prime }}{\\mathrm {max}}Q^*(s_{t+1},a^{\\prime }) \\right].$ Deep Q Networks: The DQN algorithm approximates the optimal Q function using a CNN.",
"The training objective it to minimize the expected TD error of the optimal Bellman Equation: $\\mathbb {E}_{s_t,a_t,r_t,s_{t+1}}\\left\\Vert Q_{\\theta }\\left(s_{t},a_{t}\\right)-y_{t}\\right\\Vert _{2}^{2}$ [10].",
"DQN is an offline learning algorithm that collects experience tuples $\\left\\lbrace s_{t,}a_{t},r_{t},s_{t+1},\\gamma \\right\\rbrace $ and stores them in the Experience Replay (ER) [8].",
"At each training step, a mini-batch of experience tuples are sampled at random from the ER.",
"The DQN maintains two separate Q-networks.",
"The current Q-network with parameters $\\theta $ , and the target Q-network with parameters $\\theta _{target}$ .",
"The parameters $\\theta _{target}$ are set to $\\theta $ every fixed number of iterations.",
"In order to capture the MDP dynamics, the final DQN representation is a concatenation of several consecutive states.",
"Skills, Options, Macro-actions, [17] are temporally extended control structures, denoted by $\\sigma $ .",
"A skill is defined by a triplet: $\\sigma = <I,\\pi ,\\beta >.$ I defines the set of states where the skill can be initiated.",
"$\\pi $ is the intra-skill policy, and $\\beta $ is the set of termination probabilities determining when a skill will stop executing.",
"$\\beta $ is typically either a function of state $s$ or time $t$ .",
"Any MDP with a fixed set of skills is a Semi-Markov Decision Process (SMDP).",
"Planning with skills can be performed by learning for each state the value of choosing each skill.",
"More formally, an SMDP can be defined by a five-tuple $<S, \\Sigma , P, R, \\gamma >,$ where $S$ is the set of states, $\\Sigma $ is the set of skills, $P$ is the SMDP transition matrix, $\\gamma $ is the discount factor and the SMDP reward is defined by: $R_s^{\\sigma } = \\mathbb {E}[r_s^{\\sigma }] = \\mathbb {E}[r_{t+1} + \\gamma r_{t+2} + \\cdot \\cdot \\cdot + \\gamma ^{k-1} r_{t+k} | s_t=s,\\sigma ]$ The Skill Policy $\\mu : S\\rightarrow \\Delta _\\Sigma $ is a mapping from states to a probability distribution over skills.",
"The action-value function $Q_\\mu (s, \\sigma ) = \\mathbb {E} [\\sum ^\\infty _{t=0} \\gamma ^t R_t |(s, \\sigma ), \\mu ] $ represents the value of choosing skill $\\sigma \\in \\Sigma $ at state $s \\in S$ , and thereafter selecting skills according to policy $\\mu $ .",
"The optimal skill value function is given by: $Q_{\\Sigma }^*(s,\\sigma ) = \\mathbb {E} [R_s^{\\sigma } + \\gamma ^k \\underset{\\sigma ^{\\prime }\\in \\Sigma }{\\mathrm {max}} Q_{\\Sigma }^*(s^{\\prime },\\sigma ^{\\prime })] \\hspace{5.0pt}$ [16]."
],
[
"Semi Aggregated Markov Decision Processes",
"Reinforcement Learning problems are typically modeled using the MDP formulation.",
"The abundant theory developed for MDP throughout the years gave rise to various algorithms for efficiently solving MDPs, and finding good policies.",
"MDP however, is not the optimal modeling choice when one wishes to analyze a given policy.",
"Policy analysis methods typically suffer from the cardinality of the state space and the length of the planning horizon.",
"For example, building a graphical model that explains the policy will be too large (in terms of states), and complex (in terms of planning horizon) for a human to comprehend.",
"If the policy one wishes to analyze is known to be planning using temporally-extended actions (i.e.",
"skills), then one may resort to SMDP modeling.",
"The SMDP model reduces the planning horizon dramatically and simplifies the graphical model.",
"There are two problems however with this approach.",
"First, it requires to identify the set of skills used by the policy, a long-standing challenging problem with no easy solution.",
"Second, one is still facing the high complexity of the state space.",
"Figure: Left: Illustration of state aggregation and skills.",
"Primitive actions (orange arrows) cause transitions between MDP states (black dots) while skills (red arrows) induce transitions between SAMDP states (blue circles).",
"Right: Modeling approaches for analyzing policies.",
"MDP (top-left): a policy is analyzed in the MDP state space SS, with the original set of primitive actions AA.",
"SMDP (top-right): using the set of identified skills (A→Σ)(A \\rightarrow \\Sigma ), the policy is easier to analyze.",
"AMDP (bottom-left): State aggregation allows to reduce state space complexity (S→C)(S \\rightarrow C).",
"SAMDP (bottom-right): identifying skills in the AMDP model reduces the planning horizon (S→C,A→Σ)(S \\rightarrow C ,A \\rightarrow \\Sigma ).A different modeling approach is to aggregate similar states first.",
"This is useful when there is a reason to believe that groups of states share common attributes such as similar policy, value function or dynamics.",
"State aggregation is a well studied problem that can be solved by applying clustering on the MDP state representation.",
"These models are not necessarily Markovian, however they can provide great simplification of the state space.",
"With a slight abuse of notation we denote this model as Aggregated MDP (AMDP).",
"Under the right state-representation, the AMDP can also help to identify skills (if exist).",
"We argue that this is possible if the AMDP dynamics is such that the majority of the transitions are done within the clusters, followed by rare transitions between clusters.",
"As we will show in the experiments section, DQN indeed provides a good state representation that allows skill identification.",
"If the state representation contains both spatial and temporal hierarchies, then the AMDP model can be further simplified into an SAMDP model.",
"Under SAMDP modeling, both the state-space cardinality and the planning horizon are reduced, making policy reasoning more feasible.",
"We summarize our observations about the different modeling approaches in Figure REF .",
"In the remaining of this section we explain the SAMDP modeling in detail and focus on explaining how to empirically build an SAMDP model from experience.",
"To do so we explain how to aggregate states, identify skills and estimate the transition probabilities and reward measures.",
"Finally we discuss how to evaluate the fitness of an empiric SAMDP model to the data."
],
[
"State aggregation",
"We evaluate a DQN agent, by letting it play multiple trajectories with an $\\epsilon $ -greedy policy.",
"During evaluation we record all visited states, neural activations, value estimations, and index them by their visitation order.",
"We treat the neural activations as the state representation that the DQN agent has learned.",
"[23] showed that this state representation captures a spatio-temporal hierarchy and therefore makes a good candidate for state aggregation.",
"We then apply t-SNE on the neural activations data, a non-linear dimensionality reduction method that is particularly good at creating a single map that reveals structure at many different scales.",
"t-SNE reduces the tendency of points to crowd together in the center of the map by using a heavy tailed Student-t distribution in the low dimensional space.",
"The result is a compact, well separated representation, that is easy to visualize and interpret.",
"We represent an MDP state $s_i$ by a feature vector $x_i\\in \\mathbb {R}^3$ , comprised of the two t-SNE coordinates and the DQN value estimate.",
"Using this representation we aggregate the state space by applying clustering algorithms and define the AMDP states $C$ as the resulting clusters.",
"Standard clustering algorithms assume that the data is drawn from an i.i.d distribution, however our data is generated from an MDP which violates this assumption.",
"K-means [9] for state aggregation Input: MDP sates feature representation $(x_1, x_2, \\cdots , x_n).$ Output: SAMDP states $(c_1, c_2, \\cdots , c_k).$ Objective: minimize the within-cluster sum of squares: $\\underset{\\mathbf {C}}{\\operatorname{arg\\,min}} \\sum _{i=1}^{k} \\sum _{\\mathbf {x} \\in C_i} \\left\\Vert \\mathbf {x} - \\mu _i \\right\\Vert ^2$ where $\\mu _i$ is the mean of points in $c_i$ .",
"Repeat until convergence: Assignment step, each observation $x_i$ is assigned to its closest cluster center: $ C_i^{(t)} = \\big \\lbrace x_p : \\big \\Vert x_p - \\mu ^{(t)}_i \\big \\Vert ^2 \\le \\big \\Vert x_p - \\mu ^{(t)}_j \\big \\Vert ^2 \\forall j, 1 \\le j \\le k \\big \\rbrace .",
"$ Update step, each cluster center $\\mu _j$ is updated to be the mean of its constituent instances: $\\mu ^{(t+1)}_i = \\frac{1}{|C^{(t)}_i|} \\sum _{x_j \\in C^{(t)}_i} x_j.$ In order to alleviate this problem, we suggest two versions of K-Means (Algorithm REF ) that take into account the temporal structure of the data.",
"(1) Spatio-Temporal Cluster Assignment that encourages temporal coherency by modifying the assignment step in the following way: $C_i^{(t)} = \\big \\lbrace x_p : \\big \\Vert X_{p-w:p+w} - \\mu ^{(t)}_i \\big \\Vert ^2 \\le \\big \\Vert X_{p-w:p+w} - \\mu ^{(t)}_j \\big \\Vert ^2, \\forall j, 1 \\le j \\le k \\big \\rbrace $ Where $p$ is the time index of observation $x_p$ , $X_{p-w:p+w}$ is the set of $2w$ points before and after $x_p$ along the trajectory.",
"In this way, a point $x_p$ is assigned to a cluster $\\mu _j$ , if its neighbours along the trajectory are also close to $\\mu _j$ .",
"(2) Entropy Regularization Cluster Assignment that creates simpler models by adding an entropy regularization term to the K-mean assignment step: $C_i^{(t)} = \\big \\lbrace x_p : \\big \\Vert x_p - \\mu ^{(t)}_i \\big \\Vert ^2 + d \\cdot e^{t-1}_{x_p \\rightarrow i} \\le \\big \\Vert x_p - \\mu ^{(t)}_j \\big \\Vert ^2 + d \\cdot e^{t-1}_{x_p \\rightarrow j}, \\forall j, 1 \\le j \\le k \\big \\rbrace .$ Where $d$ is a penalty weight, and $e^{t-1}_{x_p \\rightarrow i}$ indicates the entropy (as defined in Section REF ) gain of changing $x_p$ assignment to cluster $i$ in the SMDP obtained at iteration $t-1$ .",
"This is equivalent to minimizing an energy function, the sum of the K-means objective function and an entropy term.",
"We also considered Agglomerative Clustering, a bottom-up hierarchical approach.",
"Starting with a mapping from points to clusters (e.g., each point is a singular cluster), the algorithm advances by merging pairs of clusters such that a linkage criteria is minimized.",
"In order to encourage temporal coherency in cluster assignments we define a new linkage criteria based on [21]: $c(A,B)=(1-\\lambda ) \\cdot mean\\lbrace \\Vert x_a-x_b\\Vert :a \\in A, b \\in B\\rbrace + \\lambda \\cdot e_{\\lbrace A,B\\rbrace \\rightarrow AB}$ where $e_{\\lbrace A,B\\rbrace \\rightarrow AB}$ measures the difference between the entropy of the corresponding SMDP before and after merging clusters $A,B$ ."
],
[
"Temporal abstractions",
"We define the SAMDP skills by their initiation and termination AMDP states $C$ : $\\sigma _{ij} = <\\lbrace c_i \\rbrace ,\\pi _{i,j},\\lbrace c_j \\rbrace >.$ More implicit, once the DQN agent enters an AMDP state $c_i$ at an MDP state $s_t \\in c_i$ , it follows the skill policy $\\pi _{i,j}$ for $k$ steps, until it reaches a state $s_{t+k} \\in c_j$ , s.t $i \\ne j$ .",
"Note that we do not define the skill policy implicitly, but we will observe later that our model successfully captures spatio-temporal defined skill policies.",
"We set the SAMDP discount factor $\\gamma $ same as was used to train the DQN.",
"We now turn to estimate the SAMDP probability matrix and reward signal.",
"For that goal we make the following assumptions: Definition 1.",
"A deterministic probability matrix, is a probability matrix such that each of its rows contains one element that equals to 1 and the others equal to 0.",
"Assumption 1.",
"The MDP transition matrices $P_A: P^{a \\in A}_{i,j}=Pr(x_j|x_i,a)$ are deterministic.",
"This assumption limits our analysis for environments with deterministic dynamics.",
"However, many interesting problems are in fact deterministic, e.g., Atari2600 benchmarks, Go, Chess etc.",
"Assumption 2.",
"The policy played by the DQN agent is deterministic.",
"Although DQN chooses actions deterministically (by selecting the action that corresponds to the maximal Q value in each state), we allow $5\\%$ $\\epsilon $ stochastic exploration.",
"This introduces errors into our model that we will later analyze.",
"Given the DQN policy, the MDP is reduced into a Markov Reward Process (MRP) with probability matrix $P^{\\pi ^{DQN}}_{i,j}=Pr(x_j|x_i,a=\\pi ^{DQN}(x_i))$ .",
"Note that by Assumptions 1 and 2, this is also a deterministic probability matrix.",
"The SAMDP transition probability matrix $P_\\Sigma : P^{\\sigma \\in \\Sigma }_{i,j}=Pr(c_j|c_i,\\sigma )$ , indicates the probability of moving from state $c_i$ to $c_j$ given that skill $\\sigma $ is chosen.",
"It is also a deterministic probability matrix by our definition of skills (Equation REF ).",
"Our goal is to estimate the probability matrix that the DQN policy induces on the SAMDP model: $P^{\\pi ^{DQN}}_{i,j}=Pr(c_j|c_i,\\sigma =\\pi ^{DQN}(c_i))$ .",
"We do not require this policy to be deterministic from two reasons.",
"First, we evaluate the DQN agent with an $\\epsilon $ -greedy policy.",
"While almost deterministic in the view of a single time step, the variance of its behaviour increases as more moves are played.",
"Second, the aggregation process is only an approximation.",
"For example, a given state may contain more than one \"real\" state and therefore hold more than one skill with different transitions.",
"A stochastic policy can solve this disagreement by allowing to choose skills at random.",
"This type of modeling does not guarantee that our SAMDP model is Markovian and we are not claiming it to be.",
"SAMDP is an approximation of the the true dynamics that simplifies it over space and time to and allow human interpretation.",
"Finally, we estimate the skill length $k_\\sigma $ and SAMDP reward for each skill from the data using Equation REF .",
"In the experiments section we show that this model is in fact consistent with the data by evaluating its value function: $V_{SAMDP} = ( I+\\gamma ^{k}P )^{-1}r$ and the greedy policy with respect to it: $\\pi _{greedy}(c_i) = \\underset{j}{\\mbox{argmax}} \\lbrace R_{\\sigma _{i,j}}+\\gamma ^{k_{\\sigma _{i,j}}}v_{SAMDP}(c_j) \\rbrace $"
],
[
"Evaluation criteria",
"We follow the analysis of [5] and define criteria to measure the fitness of a model empirically.",
"We define the Value Mean Square Error(VMSE) as the normalized distance between two value estimations: $\\mbox{VMSE} = \\frac{\\Vert v^{DQN}-v^{SAMDP} \\Vert }{\\Vert v^{DQN}\\Vert }.$ The SAMDP value is given by Equation REF and the DQN value is evaluated by averaging the DQN value estimates over all MDP states in a given cluster (SAMDP state): ${v^{DQN}(c_j)}=\\frac{1}{|C_j|}\\sum _{i: s_i \\in c_j}v^{DQN}(s_i)$ .",
"The Minimum Description Length (MDL; [13]) principle is a formalization of the celebrated Occam’s Razor.",
"It copes with the over-fitting problem for the purpose of model selection.",
"According to this principle, the best hypothesis for a given data set is the one that leads to the best compression of the data.",
"Here, the goal is to find a model that explains the data well, but is also simple in terms of the number of parameters.",
"In our work we follow a similar logic and look for a model that best fits the data but is still “simple”.",
"Instead of considering \"simple\" in terms of the number of parameters, we measure the simplicity of the spatio-temporal state aggregation.",
"For spatial simplicity we define the Inertia: $I = \\sum _{i=0}^{n}\\min _{\\mu _j \\in C}(||x_j - \\mu _i||^2)$ which measures the variance of MDP states inside a cluster (AMDP state).",
"For temporal simplicity we define the entropy: $e= - \\sum _i \\lbrace |C_i| \\cdot \\sum _j{P_{i,j} \\log P_{i,j}} \\rbrace $ , and the Intensity Factor which measures the fraction of in/out cluster transitions: $F = \\sum _j \\frac{P_{jj}}{\\sum _i P_{ji}}.$ To summarize, the stages of building an SAMDP model are: Evaluate : Run the trained (DQN) agent, record visited states, representations and Q-values.",
"Reduce : Apply t-SNE on the state representations to obtain a low dimensional map.",
"Aggregate : Cluster states in the map.",
"Model : Fit an SAMDP model, select the best model.",
"Visualize : Visualize the SAMDP on top of the t-SNE map."
],
[
"Experiments",
"Setup.",
"We evaluate our method on three Atari2600 games, Breakout, Pacman and Seaquest.",
"For each game we collect 120k game states (each represented by 512 features), and Q-values for all actions.",
"We apply PCA to reduce the data to 50 dimensions, then we apply t-SNE using the Barnes Hut approximation to reach the desired low 2 dimension.",
"We run the t-SNE algorithm for 3000 iterations with perplexity of 30.",
"We use Spatio-Temporal K-means clustering (Section REF ) to create the AMDP states (clusters), and evaluate the transition probabilities between them using the trajectory data.",
"We overlook flicker-transitions where a cluster is visited for less than $f$ time steps before transiting out.",
"Finally we truncate transitions with less than 0.1 probability.",
"Figure: Model Selection: Correlation between criteria pairs for the SAMDP model of Breakout.Model Selection.",
"We perform a grid search on two parameters: i) number of clusters $N^c \\in [15,25]$.",
"ii) window size $w \\in [1,7]$.",
"We found that models larger (smaller) than that are too cumbersome (simplistic) to analyze.",
"We select the best model in the following way: Let $e(w,n),i(w,n),v(e,n),f(e,n)$ be the entropy, inertia, VMSE, and intensity factor respective measures of configuration $(w,n)$ in the greed search.",
"Let $E=\\lbrace e(w,n)\\rbrace , I=\\lbrace i(w,n)\\rbrace , V=\\lbrace v(w,n)\\rbrace , F=\\lbrace f(w,n)\\rbrace $ be the corresponding sets grouped over all grid search configurations.",
"We sort each set from good to bad, i.e.",
"from minimum to maximum (except for intensity factor where larger values are considered better).",
"We then iteratively intersect the p-prefix of all sets (i.e.",
"the first p elements of each set) starting with 1-prefix.",
"We stop when the intersection is non empty and choose the configuration at the intersection.",
"Figure REF shows the correlation between pairs of criteria (for Breakout).",
"Overall, we see a tradeoff between spatial and temporal complexity.",
"For example, in the bottom left plot, we observe correlation between the Inertia and the Intensity Factor; a small window size $w$ leads to well-defined clusters in space (low Inertia) at the expense of a complex transition matrix (small intensity factor).",
"A large $w$ causes the clusters to be more spread in space (large Inertia), but has the positive effect of intensifying the in-cluster transitions (high intensity factor).",
"We also measure the p-value of the chosen model with the null hypothesis being the SAMDP model constructed with randomly clustered states.",
"We tested 10000 random SAMDP models, none of which scored better than the chosen model (for any of the evaluation criteria).",
"Qualitative Evaluation.",
"Examining the resulting SAMDP (Figure REF ) it is interesting to note the sparsity of transitions.",
"This indicate that clusters are well located in time.",
"Inspecting the mean image of each cluster also reveal some insights about the nature of the skills hiding within.",
"We also see evidence for the \"tunnel-digging\" option described in [23] in the transitions between clusters 11,12,14 and 4.",
"Figure: SAMDP visualization for Breakout over the t-SNE map colored by value estimates (low values in blue and high in red).Model Evaluation.",
"We evaluate our model using three different methods.",
"First, the VMSE criteria (Figure REF , top): high correlation between the DQN values and the SAMDP values gives a clear indication to the fitness of the model to the data.",
"Second, we evaluate the correlation between the transitions induced by the policy improvement step and the trajectory reward $R^j$ .",
"To do so, we measure $P_i^j:$ the empirical distribution of choosing the greedy policy at state $c_i$ in that trajectory.",
"Finally we present the correlation coefficients at each state: $corr_i = corr(P_i^j,R^j)$ (Figure REF , center).",
"Positive correlation indicates that following the greedy policy leads to high reward.",
"Indeed for most of the states we observe positive correlation, supporting the consistency of the model.",
"The third evaluation is close in spirit to the second one.",
"We create two transition matrices $T^+,T^-$ using k top-rewarded trajectories and k least-rewarded trajectories respectively.",
"We measure the correlation of the greedy policy $T^G$ with each of the transition matrices for different values of k (Figure REF bottom).",
"As clearly seen, the correlation of the greedy policy and the top trajectories is higher than the correlation with the bad trajectories.",
"Figure: Model Evaluation.",
"Top: Value function consistency.",
"Center: greedy policy correlation with trajectory reward.",
"Bottom: top (blue), least (red) rewarded trajectories.Eject Button: Performance improvement.",
"In the following experiment we show how the SAMDP model can help to improve the performance of a trained policy.",
"The motivation for this experiment stems from the idea of shared autonomy [12].",
"There are domains where errors are not permitted and performance must be as high as possible.",
"The idea of shared autonomy is to allow an operator to intervene in the decision loop in critical times.",
"For example, it is known that in 20$\\%$ of commercial flights, the auto-pilot returns the control to the human pilots.",
"For this experiment we first build an SAMDP model and then let the agent to play new (unseen) trajectories.",
"We project the online state visitations onto our model and monitor its transitions along it.",
"We define $T^+,T^-$ as above.",
"If the likelihood of $T^-$ with respect to the online trajectory is greater than the likelihood of $T^+$ , we press the Eject button and terminate this execution (a procedure inspired by option interruption [18]).",
"We're interested to measure the average performance of the un-terminated trajectories with respect to all trajectories.",
"The performance improvement achieved with and without using the Eject button is presented in Table REF .",
"Table: Performance gain using eject button averaged over 60 trajectories.",
"Numbers are reported for DQN agents we train ourselves."
],
[
"Discussion",
"In this work we considered the problem of automatically building an SAMDP model for analyzing trained policies.",
"Starting from a t-SNE map of neural activations, and ending up with a compact model that gives a clear interpretation for complex RL tasks.",
"We showed how SAMDP can help in identifying skills that are well defined in terms of initiation and termination sets.",
"However, the SAMDP doesn't offer much information about the skill policy and we suggest to further investigate it in future work.",
"It would also be interesting to see whether skills of different states actually represent the same behaviour.",
"Most importantly, the skills we find are determined by the state aggregation.",
"Therefore, they are impaired by the artifacts of the clustering method used.",
"In future work we will consider other clustering methods that better relate to the topology (such as spectral-clustering), to see if they lead to better skills.",
"In the Eject experiment we showed how SAMDP model can help to improve the policy at hand without the need to re-train it.",
"It would be even more interesting to use the SAMDP model to improve the training phase itself.",
"The strength of SAMDP in identifying spatio and temporal hierarchies could be used for harnessing DRL hierarchical algorithms [19], [7].",
"For example by automatically detecting sub-goals or skills.",
"Another question we're interested in answering is whether a global control structure exists?",
"Motivated by the success of policy distillation ideas [14], it would be interesting to see how well an SAMDP built for game A, explains game B?",
"Finally we would like to use this model to interpret other DRL agents that are not specifically trained to approximate value such as deep policy gradient methods."
]
] | 1606.05174 |
[
[
"Spectroscopic Identification of Type 2 Quasars at Z < 1 in SDSS-III/BOSS"
],
[
"Abstract The physics and demographics of type 2 quasars remain poorly understood, and new samples of such objects selected in a variety of ways can give insight into their physical properties, evolution, and relationship to their host galaxies.",
"We present a sample of 2758 type 2 quasars at z $\\leq$ 1 from the SDSS-III/BOSS spectroscopic database, selected on the basis of their emission-line properties.",
"We probe the luminous end of the population by requiring the rest-frame equivalent width of [OIII] to be > 100 {\\AA}.",
"We distinguish our objects from star-forming galaxies and type 1 quasars using line widths, standard emission line ratio diagnostic diagrams at z < 0.52 and detection of [Ne V]{\\lambda}3426{\\AA} at z > 0.52.",
"The majority of our objects have [OIII] luminosities in the range 10^8.5-10^10 L$_{\\odot}$ and redshifts between 0.4 and 0.65.",
"Our sample includes over 400 type 2 quasars with incorrectly measured redshifts in the BOSS database; such objects often show kinematic substructure or outflows in the [OIII] line.",
"The majority of the sample has counterparts in the WISE survey, with median infrared luminosity {\\nu}L{\\nu}[12{\\mu}m] = 4.2 x 10^44 erg/sec.",
"Only 34 per cent of the newly identified type 2 quasars would be selected by infrared color cuts designed to identify obscured active nuclei, highlighting the difficulty of identifying complete samples of type 2 quasars.",
"We make public the multi-Gaussian decompositions of all [OIII] profiles for the new sample and for 568 type 2 quasars from SDSS I/II, together with non-parametric measures of line profile shapes and identify over 600 candidate double-peaked [OIII] profiles."
],
[
"Introduction",
"Much, if not most, of the supermassive black hole growth activity in the Universe is hidden by gas and dust [7], [44].",
"The precise accounting of the demographics of active galactic nuclei (AGNs) of different types and at different redshifts is of significant interest because of the growing realization that the growth of supermassive black holes may have had a strong impact on the evolution of massive galaxies [91], [85], [87], especially during the obscured but intrinsically luminous (quasar) phase [81], [36].",
"Circumnuclear gas and dust make obscured (type 2) AGNs faint at optical, ultraviolet and soft X-ray wavelengths.",
"But luminous type 2 quasars ($L_{\\rm bol}10^{45}\\,{\\rm erg\\,s^{-1}}$ ) may be identified using surveys at hard X-ray [68], [12], [33], [13], infrared [46], [88], [55], [45], [19], [89], [21], [24], [47], [44], and radio [62], [56] wavelengths.",
"However, because different selection methods probe somewhat different populations of objects, there is not yet agreement about the obscuration fraction as a function of redshift and luminosity [92], [12], [73], [48], especially in pencil-beam surveys which contain very few objects at the luminous end of the luminosity function.",
"Very large area surveys are important for discovering such rare sources, and thus despite the suppression of the apparent optical flux by obscuration, $\\sim 1000$ type 2 quasars have been selected using their characteristic strong narrow emission lines from the Sloan Digital Sky Survey (SDSS; [101]), both at low ($z<1$ , [40], [30], [103], [73], [63]) and at high ($z2$ , [5], [80]) redshifts.",
"The Baryon Oscillation Spectroscopic Survey (BOSS; [17]) is one of the four major surveys of the third phase of SDSS, SDSS-III (2009-2014; [22]).",
"It collected spectra of over a million galaxies [72] and over 300,000 quasars [79] selected from SDSS imaging data to measure the scale of baryon acoustic oscillations as a function of redshift [9].",
"The BOSS spectrograph [86] covers the range $3600-10400$ Å, with a resolution of 1500-2600, depending on wavelength.",
"The BOSS spectroscopic pipeline [11] fits the resulting spectra with templates of common types of objects to provide redshifts and spectroscopic classifications, and measures the strengths and widths of various emission lines.",
"Spectroscopic targeting in BOSS probes fluxes $\\sim 2$ mag fainter than those accessible to the SDSS-I/II surveys [17], and thus one might expect that the BOSS survey may be able to uncover a previously missed population of optically obscured type 2 quasars.",
"This paper selects type 2 quasars from the BOSS spectroscopic data.",
"In Section we describe the sample selection, using various techniques to select $z < 0.52$ quasars (where standard emission-line ratio selection works well) and those at higher redshift (where the presence of the [Ne$\\,$V]$\\lambda $ 3426Å line allows us to distinguish AGN from star-forming galaxies).",
"We also identify a significant number of type 2 quasars whose redshifts are incorrectly measured by the BOSS pipeline.",
"In Section we discuss optical and multi-wavelength properties of the sample.",
"We summarize in Section .",
"We use a $h$ =0.7, $\\Omega _m$ =0.3, $\\Omega _{\\Lambda }$ =0.7 cosmology throughout this paper.",
"While SDSS uses vacuum wavelengths, we quote emission line wavelengths in air following established convention – for example, [O$\\,$III]$\\lambda $ 5007Å (hereafter [O$\\,$III]) has a vacuum wavelength 5008.3Å.",
"Objects are identified in the figures by their SDSS spectroscopic ID in the order plate - fiber - MJD."
],
[
"Sample selection",
"In this paper we identify type 2 quasar candidates from the complete SDSS-III/BOSS spectroscopic database (Data Release 12; [4]).",
"The first catalog of luminous $z1$ type 2 AGNs in the SDSS data (DR1) [103] was designed to be as inclusive as possible, covering [O$\\,$III] luminosities between $3.8\\times 10^{40}$ and $3.8\\times 10^{43}\\,{\\rm erg\\,s^{-1}}$ , although with completeness and selection efficiency strongly varying with line luminosity (see also [40], [31]).",
"In the catalog by [73], we set a minimal luminosity threshold $L$ [O$\\,$III]$>3.8\\times 10^{41}\\,{\\rm erg\\,s^{-1}}$ , because it was not practical to accurately measure weaker emission lines in moderate-redshift, low signal-to-noise ratio (SNR) spectra and because we were interested in the objects at the quasar (rather than Seyfert) end of the luminosity range.",
"[102] carried out a kinematic analysis of the [O$\\,$III] emission line of this sample, showing evidence for outflows correlated with radio power and infrared luminosity.",
"This analysis required high SNR spectra of the emission lines, and thus the sample was further restricted to luminosities $L$ [O$\\,$III]$>1.2\\times 10^{42}\\,{\\rm erg\\,s^{-1}}$ .",
"In this paper we adopt a somewhat different approach.",
"We rely on the strong empirical relationship between the rest equivalent width (REW) of the [O$\\,$III] emission line and its luminosity (Figure REF ) and aim to select type 2 quasar candidates with REW[O$\\,$III]$>$ 100Å as completely as possible.",
"Because [O$\\,$III] leaves the BOSS spectral coverage at $z \\sim 1$ , our sample is limited in practice to redshifts below unity.",
"The correlation in Figure REF shows that the [O$\\,$III] equivalent width cut should result in a sample which is essentially complete at $L$ [O$\\,$III]$>3.8\\times 10^{42}\\,{\\rm erg\\,s^{-1}}$ ; 92 per cent of the objects in [73] with these luminosities have REW[O$\\,$III]$>$ 100Å.",
"Similarly 56 per cent of the [73] objects with $L$ [O$\\,$III] between $1.2\\times 10^{42}\\,{\\rm erg\\,s^{-1}}$ and $3.8\\times 10^{42}\\,{\\rm erg\\,s^{-1}}$ have REW[O$\\,$III]$>$ 100Å, so we expect to include about half of the type 2 quasars in the BOSS sample within this luminosity range.",
"The optical continuum of type 2 quasars is a poor measure of their intrinsic power since it is suppressed by extinction, but [O$\\,$III] emission is thought to arise outside of the obscuring region, and its luminosity is correlated with bolometric luminosity [34].",
"Empirically, in type 1 quasars $L_{\\rm [OIII]} =1.2\\times 10^{42}\\,{\\rm erg\\,s^{-1}}$ corresponds to an intrinsic (unobscured) absolute AB magnitude of $M_{2500}=-24.0$ mag, and $3.8\\times 10^{42}\\,{\\rm erg\\,s^{-1}}$ corresponds to $M_{2500}=-25.3$ mag [73].",
"Therefore, we estimate that the REW[O$\\,$III]$>100$ Å criterion selects objects that are more luminous than the traditional (though arbitrary) boundary between Seyferts and quasars ($M_B\\sim -23$ mag), at the bright end of the quasar luminosity function at $z<1$ [76], [37].",
"In this paper we define type 1 and type 2 quasars by their classical optical signatures.",
"Specifically, in type 1 quasars, we expect to see both the broad-line region and the narrow-line region, whereas in type 2 quasars the broad-line region is obscured [7].",
"Therefore, the width of the Balmer lines [30] and the ratio of the strengths of the [O$\\,$III] and H$\\beta $ lines [103] play a role in distinguishing type 1 from type 2 quasars.",
"Furthermore, type 2 quasars need to be separated from star-forming galaxies, which also have strong emission lines but with line ratios characterized by underlying ionizing radiation produced solely by stars.",
"We start with objects identifiable using traditional emission line ratio diagnostic diagrams [10], [94] in Section REF .",
"At redshifts $z>0.52$ , the H$\\alpha +$ [N$\\,$II]$\\lambda \\lambda $ 6548,6583ÅÅ emission line complex moves out of the BOSS spectral coverage, and we develop an alternative method that utilizes the [Ne$\\,$V]$\\lambda $ 3426Å emission line to identify type 2 quasar candidates in Section REF .",
"Recovery of type 2 quasars which have been assigned erroneous redshifts by the BOSS pipeline is described in Section REF .",
"We present the final sample in Section REF .",
"Figure: [O\\,III] rest equivalent widths and emission line luminositiesof type 2 quasars from the sample presented in this paper (blue) and (red).",
"While the sample selection is performed usingline measurements from the BOSS pipeline, we remeasure[O\\,III] luminosities and REWs as described inSection , and these are the values shown in thisfigure.",
"The vertical dashed line shows our selection criterionREW[O\\,III]>100>100Å as measured with our fits as described in§ .",
"There is a strong positive correlation between the equivalent widths and luminosities of the [O\\,III] emission line."
],
[
"Selection at $z < 0.52$",
"The majority of extragalactic sources with emission-line optical spectra fall into three categories: (i) star-forming galaxies; (ii) type 1 (broad-line, unobscured) AGNs; and (iii) type 2 (narrow-line, obscured) AGNs.",
"We identify 9454 emission-line objects in BOSS DR12 with REW[O$\\,$III]$>100$ Å and pipeline redshift $z<1$ , both as measured by the BOSS pipeline [11].",
"For the initial selection, we use spZline data productshttp://data.sdss3.org/datamodel/files/BOSS_SPECTRO_REDUX/RUN2D/PLATE4/RUN1D/spZline.html for line flux measurements, but we remeasure the [O$\\,$III] fluxes and equivalent widths more accurately as a final step of our catalog presentation in Section REF .",
"In this section we consider only objects flagged as having confident redshift measurements by the BOSS pipeline (i.e., the ZWARNING flag is set to zero; see [11]), but we return to this issue in Section REF .",
"Most of the type 1 AGNs are removed from this sample by the REW cut, as the typical REW[O$\\,$III] in these objects is $\\sim 13$ Å [93].",
"At $z<0.52$ , for which the [N$\\,$II]$\\lambda \\lambda $ 6548,6583ÅÅ doublet is covered by the BOSS wavelength range, we use diagnostic line-ratio diagrams [10], [94], [41], [30] to separate type 2 AGNs from star-forming galaxies.",
"Our sample of 9454 high REW objects includes 4102 objects with redshift $z<0.52$ and with $>3\\,\\sigma $ detections of the relevant lines as measured by the BOSS pipeline: [N$\\,$II]$\\lambda $ 6583Å, H$\\alpha \\lambda $ 6563Å, [O$\\,$III]$\\lambda $ 5007Å and H$\\beta \\lambda $ 4861Å.",
"The distribution of these objects in the [O$\\,$III]/H$\\beta $ vs [N$\\,$II]/H$\\alpha $ plane is shown in Figure REF .",
"At these high equivalent widths, most of the star-forming galaxies tend to have relatively low metallicities because [O$\\,$III] is one of the few available coolants of the low-metallicity gas.",
"These galaxies separate cleanly from the AGNs [41], [40], [30]; at this equivalent width and luminosity, objects lying between the star-forming and AGN branches would have to have unusually high luminosity in both components, which is quite rare.",
"We select AGNs with the cut shown in Figure REF : $\\frac{\\rm \\hbox{[O$\\,${\\scriptsize III}]}\\lambda 5007Å}{\\rm H\\beta }>10.0-16.7\\times \\frac{\\rm \\hbox{[N$\\,${\\scriptsize II}]}\\lambda 6583Å}{\\rm H\\alpha }.$ This cut further reduces the type 1 AGN contamination of the sample, as [O$\\,$III]/H$\\beta $ tends to be low in type 1 objects, where H$\\beta $ is dominated by the broad component.",
"We find 1693 type 2 quasar candidates at $z<0.52$ which have REW[O$\\,$III]$>100$ Å and which are above the diagnostic cut given by equation (REF ).",
"A visual inspection of these candidate yields a final list of 1606 type 2s at $z<0.52$ , after removing a modest number of objects with weak broad-line components that the BOSS pipeline failed to model.",
"Figure: Emission-line ratio diagnostic diagram for 4102 objects withz<0.52z<0.52 and REW>100>100Å.",
"All measurements are from the BOSSpipeline.",
"Most type 1 AGNs are rejected by thisequivalent width criterion, so the majority of sources in thediagram are type 2 AGNs and low-metallicity, low-redshiftstar-forming galaxies.",
"The dashed line shows our type 2 AGNselection criterion (equation )."
],
[
"Selection at $z>0.52$",
"Various methods have been suggested for separating the spectra of type 2 AGNs from star-forming galaxies at redshifts where the full set of diagnostic emission lines is not available [103], [73], [23].",
"In our case, we need to develop new selection criteria to identify type 2 quasar candidates at $z>0.52$ , when [N$\\,$II]$\\lambda $ 6583Å moves out of the BOSS wavelength range.",
"We use the [Ne$\\,$V]$\\lambda $ 3426Å emission line to distinguish AGN from star-forming galaxies [103], [23].",
"The ionization energy of Ne$^{4+}$ is 97 eV, so the presence of this line implies that the gas has been ionized with intense radiation in the hard UV and soft X-ray range.",
"Star formation produces essentially no emission at these wavelengths, so the mere detection of the [Ne$\\,$V] emission is an unambiguous sign of the presence of an AGN.",
"The BOSS pipeline does not automatically measure [Ne$\\,$V] fluxes, so we measure them using single-Gaussian fits in all candidate spectra.",
"To test how well such [Ne$\\,$V]-based selection performs, we measure [Ne$\\,$V] fluxes in AGN and star-forming galaxies with $0.40 < z < 0.52$ from the sample shown in Figure REF .",
"The classification of these objects is already known from the standard diagnostic diagrams.",
"The distribution of the SNR of the [Ne$\\,$V] line is shown in Figure REF : only 9 per cent of the objects classified as star-forming by the line ratio criterion of equation (REF ) have [Ne$\\,$V] detections; the vast majority of the objects classified as star-forming with [Ne$\\,$V] detections turn out to be Type 1 objects, with broad bases to the H$\\alpha $ lines.",
"The few exceptions all lie close to the boundary between the star-forming and AGN branches in the emission line ratio diagram (Figure REF ), and thus are likely an admixture of the two components.",
"On the other hand, 98 per cent of the BPT-selected AGNs show significant [Ne$\\,$V] emission.",
"Thus the mere detection of the [Ne$\\,$V] line is indeed a good indicator of the presence of an AGN and results in a fairly complete sample.",
"There are 4143 objects in BOSS DR12 with $z>0.52$ and REW[O$\\,$III]$>100$ Å.",
"For a source to be selected as a type 2 quasar candidate, we require that the [Ne$\\,$V] line be detected at SNR${}>3$ in the BOSS spectra for $z>0.52$ objects with REW[O$\\,$III]${}>100$ Å.",
"2191 objects satisfy these criteria.",
"The SNR criterion is of course dependent on the SNR of the BOSS spectra themselves, and we find that the median spectral SNR per pixel of the high equivalent width objects drops steadily from 4 at $z=0.4$ to 2.5 at $z>0.6$ .",
"Thus our [Ne$\\,$V]-based selection likely becomes less complete at high redshifts.",
"Figure: [Ne\\,V] SNR distribution of type 2 quasars at 0.4<z<0.520.4<z<0.52 (redhistogram), type 2 quasar candidates at z>0.52z > 0.52 (blue histogram)and a subset of the star-forming galaxies at z<0.52z<0.52 (greenhistogram).",
"The dashed line marks the SNR>3{}> 3 cutoff that wechoose for our quasar selection for objects with z>0.52z > 0.52.",
"Whileapproximately 98 per cent of confirmed type 2 quasars at 0.4<z<0.520.4< z <0.52show [Ne\\,V] SNR>3{}> 3, over 91 per cent of star forming galaxies showno [Ne\\,V] detection with SNR>3{}> 3.The sample at this stage still includes a substantial number of type 1 AGN.",
"Figure REF shows the full width at half maximum (FWHM) of the H$\\beta $ line against the ratio of [O$\\,$III] to H$\\beta $ for the 2191 objects.",
"The sample is clearly bimodal in H$\\beta $ line width, which is used as a classical distinguishing characteristic between type 1 and type 2 AGNs at low luminosities [42], [30].",
"We choose FWHM(H$\\beta ) = 1000$ km s$^{-1}$ as the cut-off to remove broad-line type 1 AGNs.",
"With this cut, we select 1250 type 2 candidates with $z > 0.52$ , REW[O$\\,$III] $>100$ Å, [Ne$\\,$V] SNR $> 3$ and FWHM(H$\\beta $ ) $<$ 1000 km s$^{-1}$ .",
"A visual inspection yields 796 type 2 quasars.",
"Most of the candidates rejected by this visual inspection showed weak broad H$\\beta $ , broad [Mg$\\,$II]$\\lambda 2800$ Å or a strong blue continuum (the latter often associated with narrow-line Seyfert 1 galaxies; see [97]).",
"In the presence of strong quasar-driven outflows, the kinematics of the forbidden-line region can sometimes result in FWHM of the extended emission line region in excess of 1000 km s$^{-1}$ .",
"Indeed, blueshifted asymmetries of the [O$\\,$III] line have been recognized as the signature of outflows since the 1980's [35], [18], [96], [98], although the relationship between outflows and FWHM significantly larger than the depth of the galaxy potential well became clear only much more recently [67], [66], [27], [26], [95], [29], [102], [106].",
"To avoid missing the sources with the highest FWHM, we visually inspect the 941 objects in Figure REF above the FWHM(H$\\beta $ )$=$ 1000 km s$^{-1}$ cutoff line.",
"We identify an additional 10 type 2 quasars.",
"This is only a small fraction of the strongly kinematically disturbed type 2 quasars in our sample, most of which are identified in the BOSS catalog using another method (Section REF ).",
"Our final sample from [Ne$\\,$V]-based selection at $z>0.52$ includes 806 type 2 quasars with $z > 0.52$ .",
"It is possible that we have been overly aggressive in rejecting objects with weak broad components in H$\\beta $ or [Mg$\\,$II]$\\lambda $ 2800Å.",
"The problem of weeding out type 1 AGNs with weak broad lines from genuine type 2 candidates is inherently difficult.",
"Even when the direct lines of sight to the nucleus are obscured, some quasar emission can escape along other directions, scatter off the interstellar medium of the host galaxy and reach the observer [8], [7], [104].",
"If the scattering is more efficient than a few percent, then this component can make a noticeable enough contribution to the integrated spectrum of the object that we would see weak broad components in the Balmer and [Mg$\\,$II] lines as well as a continuum rising to the blue.",
"Short of conducting polarimetry or spectropolarimetry, we cannot distinguish such objects from type 1 AGN with weak lines and weak continuum.",
"Thus our selection procedure in which we reject all objects with detectable broad components unfortunately biases our sample against type 2 quasars with high scattering efficiency.",
"Another problem is that some genuine type 2 quasars show narrow features near [Mg$\\,$II].",
"In [104], He II $\\lambda $ 2734Å and C II $\\lambda $ 2838Å (vacuum wavelengths) are tentatively identified as possible satellite features to [Mg$\\,$II].",
"In lower SNR data or in an object with high velocities in forbidden lines, these features could be blended together and be erroneously interpreted as a broad Mg component.",
"We keep objects without other indications of being type 1 quasars, if the satellite lines are clearly spectroscopically resolved from [Mg$\\,$II].",
"Finally, in unobscured AGNs the region near Mg has strong emission from multiple lines of [Fe$\\,$II].",
"In particular, narrow-line Seyfert 1 galaxies show a broad Fe complex peaking at 2300-2400Å and two Fe complexes on either side of [Mg$\\,$II] [15].",
"Thus Mg can appear as a narrow core with broad “shoulders” which are actually Fe complexes.",
"Even if such objects show no other signatures of being unobscured, we reject them from our sample.",
"Because we require a high REW of [O$\\,$III], a narrow H$\\beta $ , and low [Fe$\\,$II] during the visual inspection stage we expect little to no contamination of our sample by narrow-line Seyfert 1 galaxies.",
"Figure: The relationship between FWHM(Hβ\\beta ) and [O\\,III]/Hβ\\beta for 2191 AGNs with z>0.52z > 0.52,REW[O\\,III]>>100Å, and >3σ>3\\,\\sigma detection of the[Ne\\,V] line.",
"All measurements are from the BOSS pipeline.",
"The objects inthe bottom right quadrant have narrow Hβ\\beta lines and high[O\\,III]/Hβ\\beta ratios and are likely type 2 AGNs.",
"Theobjects in the top left have broad Hβ\\beta lines and low[O\\,III]/Hβ\\beta ratios (presumably dominated by the broad Hβ\\beta component) and are mostly type 1 AGNs.",
"We choose FWHM(Hβ\\beta )= =1000 km s -1 ^{-1} (dashed line) as our selection cut to removebroad-line type 1 AGNs.",
"Visual inspection found only 10objects with broader Hβ\\beta lines which belong in the type 2category."
],
[
"Selection of type 2s with incorrect/unreliable redshifts",
"Even though 99.8 per cent of the redshifts the BOSS pipeline flags as reliable are correct [1], [11], there are still some objects with incorrect redshifts in the database.",
"Type 2 quasars can be among those mis-classified objects because there is no proper template for them in the BOSS pipeline.",
"To identify such objects, we explore three ways in which the pipeline is known to respond erroneously to a type 2 spectrum.",
"First, the strong [O$\\,$III] emission line of type 2 quasars could be mis-identified as Ly$\\alpha $ , resulting in a mistakenly high redshift.",
"For example, mis-identification of [O$\\,$III]$\\lambda $ 5007Å at redshift $z_{\\rm true}=0.5$ as Ly$\\alpha $ would yield $z_{\\rm wrong}=5.18$ .",
"Second, the [O$\\,$III] emission line could be mis-identified as H$\\alpha $ , which would result in a mistakenly low redshift.",
"For example, mis-identification of [O$\\,$III]$\\lambda $ 5007Å at $z_{\\rm true}=0.5$ as H$\\alpha $ would yield $z_{\\rm wrong}=0.114$ .",
"Finally, the redshift could be measured correctly, but the pipeline could indicate that it has low confidence in the result.",
"In the first possibility, an [O$\\,$III] line at $z=0$ (i.e., at 5007Å) misinterpreted as Ly$\\alpha $ will be assigned a redshift 3.12.",
"We matched the list of 26,489 BOSS objects with $z>3.12$ (as measured by the BOSS pipeline) against the visually inspected (type 1) BOSS quasar catalog [70]; objects that match are presumed to have correct redshifts.",
"Visually inspecting the 1715 objects which remain yields 61 type 2 quasar candidates, whereas the rest are mostly genuine high redshift type 1 AGNs.",
"All 61 type 2 candidates show strong [O$\\,$III] emission.",
"An example of a type 2 quasar selected using this method is shown in the top panel of Figure REF .",
"In the second possibility, [O$\\,$III] is mis-identified as H$\\alpha $ , which is only possible for $z_{\\rm true}>0.31$ .",
"The conversion from observed to rest-frame equivalent width implies that our desired cut of REW[O$\\,$III]$=$ 100Å corresponds to a listed REW for the line, interpreted wrongly as H$\\alpha $ , of 131Å.",
"An object with H$\\alpha $ with such a high equivalent width is likely to also exhibit strong [O$\\,$III].",
"We thus looked for objects with listed H$\\alpha $ REWs greater than 131Å, but with REW[O$\\,$III] less than 5Å.",
"This yields 625 objects, of which 369 (well over half) are in fact type 2 quasars at the wrong redshift.",
"Most of the remaining candidates are artifacts with noisy spectra.",
"An example of a type 2 quasar from this selection method is shown in the bottom panel of Figure REF .",
"Figure: Example type 2 quasars identified assuming that the BOSSpipeline mistook [O\\,III] for another strong emission line: forLyα\\alpha in the top panel (true redshift z true =0.548z_{\\rm true}=0.548) andfor Hα\\alpha in the bottom panel (true redshift z true =0.5803z_{\\rm true}=0.5803).",
"Each quasar is indicated with its plate, fiber, andModified Julian Date (MJD (see § ).",
"Thesespectra have been smoothed with a five-pixel boxcar.For all selection methods presented so far, we require that the BOSS pipeline be confident about the redshift: the ZWARNING flag [11] must be set to 0.",
"We select another interesting subsample of 1050 type 2s, identifying those objects that the BOSS pipeline flags as having problematic redshift measurements, i.e., non-sky fibers with ZWARNING!=0 with measured REW[O$\\,$III]${}>100$ Å.",
"We visually inspect this subsample and identify 78 type 2s.",
"The BOSS pipeline redshift of these type 2s is essentially always right, despite the ZWARNING flag (indeed, had the redshifts been wrong, the REW[O$\\,$III] measurement would have been meaningless).",
"The 508 type 2 quasar candidates selected in this section are particularly interesting because they tend to show strongly disturbed kinematics in their [O$\\,$III] emission lines, which is presumably why their redshifts are either mis-identified by the pipeline or the pipeline is not confident about the redshift.",
"This also explains why we see such a clean separation of FWHM(H$\\beta $ ) in Section REF and Figure REF : the majority of strongly kinematically disturbed objects with broad forbidden lines are placed at a wrong redshift and are therefore not correctly identified using the [Ne$\\,$V] SNR cut employed in that section."
],
[
"BOSS catalog of type 2 quasars",
"We now have 1606 spectroscopic observations of type 2 quasars from Section REF , 806 from Section REF and 508 from Section REF , adding up to a total of 2920 unique spectroscopic observations.",
"Accounting for objects with multiple spectroscopic observations, our sample represents 2758 unique sources.",
"We provide the full catalog as an online FITS table (http://zakamska.johnshopkins.edu/data.htm).",
"The complete data structure of the catalog is described in Table .",
"Our basic identification method is by the BOSS Plate and Fiber Number on which this object was observed spectroscopically, together with the Modified Julian Date (MJD) of the spectroscopic observation.",
"We also provide right ascension and declination, as measured by the SDSS [3], [2].",
"In the cases when the same object has multiple spectra in the database, we flag the first appearance of the source with a `unique' flag and any subsequent appearances are flagged with the spectroscopic identification of the first available spectrum.",
"The catalog includes [Ne$\\,$V] emission line measurements described in Section REF , [O$\\,$III] emission line measurements described in Section REF and infrared luminosity measurements described in Section REF ."
],
[
"Optical colors and target selection",
"We present the redshift distribution of each of our subsamples in Figure REF .",
"The majority ($\\sim 85$ per cent) of the objects in the final sample lie within the redshift range $0.4 < z <0.7$ .",
"This is a reflection of how these objects were selected for spectroscopy in the BOSS survey; 2480 of the type 2 quasars in the final catalog were targeted as CMASS galaxies.",
"These are selected with a series of magnitude and color cuts, as described by [72], to be a roughly stellar mass-limited sample of galaxies (CMASS stands for “Constant Mass”).",
"Figure: The redshift distribution of our type 2 quasars in Δz=0.01\\Delta z=0.01 bins, showing thefive selection criteria.",
"The red histogram corresponds to the 1606type 2 quasars from Section ; the blue histogram isthe 806 type 2 quasars from Section , and the green,black and magenta histograms correspond to the two samples ofwrong-redshift-selected type 2 quasars and the ZWARNINGselected sample in Section .",
"All redshifts are asmeasured by the pipeline described in Section .Figure REF shows the SDSS measured colors (measured using model magnitudes corrected for Galactic extinction following [82]) of the objects in the sample as a function of redshift, following [74].",
"Also shown are median colors for CMASS galaxies in general, as well as for type 1 quasars from SDSS-I/II [83].",
"The type 2 quasars in our sample tend to be appreciably bluer than the bulk of CMASS galaxies in $g-r$ and $i-z$ , but relatively red in $r-i$ .",
"Some of this interesting behaviour is due to the high equivalent widths of the emission lines in type 2 quasar spectra.",
"Specifically, the [O$\\,$III] line falls in the $i$ band from redshift 0.4 to 0.6, making the $r-i$ colors relatively red, close to those of CMASS galaxies, and $i-z$ colors blue, close to or even bluer than those of type 1 quasars.",
"The continuum color of our sample is bluer than that of CMASS galaxies, as seen in $g-r$ colors which are not dominated by emission lines, though they are not as blue as those of type 1 quasars.",
"The dominant contribution to the rest-frame ultraviolet continuum of type 2 quasars is likely to be scattered light [104], [105], [69], though star formation is also possible [107], [100].",
"Type 2 quasar hosts are known [53] to be even more strongly star-forming than type 1 quasar host galaxies which appear with young stellar populations at these redshifts [43], [59], [60].",
"Figure: The observed SDSS colors of objects in our sample as afunction of redshift.",
"The red points correspond to those objectswith the highest [O\\,III] equivalent widths.",
"The blue points give themedian colors of galaxies selected by the CMASS algorithm, while the green points are the median for quasarsfrom SDSS-I/II .The fact that $\\sim 85$ per cent of our type 2 quasars are selected by the BOSS galaxy target selection algorithms means that most of our type 2 quasars are resolved in SDSS imaging.",
"This result is consistent with [73]; $\\sim 50$ per cent of their type 2 quasar sample were selected by the main galaxy target selection algorithm [90].",
"It also suggests that there may be an additional population of unresolved type 2 quasars yet to be identified which reside in compact galaxies; such objects would not be selected by the CMASS targeting algorithm and therefore would not have BOSS spectra."
],
[
"Refitting the [O$\\,$",
"Our sample relies strongly on [O$\\,$III] measurements and is limited in REW[O$\\,$III].",
"We initially use the measurements of this quantity from the BOSS pipeline outputs, but these are not ideal: the pipeline fits a single Gaussian to the line, and forces the width of that Gaussian to be the same for all forbidden lines fit.",
"Following [102], we refit the [O$\\,$III] doublet over the rest wavelength range 4910Å–5058Å for all our objects, assuming a 2.996:1 intensity ratio for the two [O$\\,$III] lines, and forcing the redshifts and profiles of [O$\\,$III]$\\lambda $ 4959Å and [O$\\,$III]$\\lambda $ 5007Å to be the same.",
"Unlike the extreme objects discussed in [106], in no case is the [O$\\,$III] line broad enough to be affected by the H$\\beta $ line, and we thus do not include it in the modeling.",
"The SNR of the spectra in these strong lines is adequate to allow detailed fits.",
"Fits are carried out assuming a linear continuum, and one, two, three or four Gaussians; if adding an extra Gaussian component leads to a decrease in reduced $\\chi ^2$ of $<10$ per cent, we accept the fit with a smaller number of components.",
"The vast majority of the fits require two or three Gaussians; there are only two sources that require four.",
"These fits give accurate measurements of [O$\\,$III] luminosities and equivalent widths; we used these results in Figure REF .",
"Because these fits are sensitive to wings in the profile that are not fit by a single Gaussian, they tend to give equivalent widths which are systematically higher than those measured by the BOSS pipeline.",
"Specifically, the median REW[O$\\,$III] for our type 2 sample as measured by the pipeline is 165Å whereas our refits give a median REW[O$\\,$III] of 186.0Å.",
"This means that our sample is somewhat incomplete close to the REW[O$\\,$III] limit of 100Å; there is presumably a population of objects with pipeline equivalent widths somewhat below 100Å, which would move above this limit with the detailed fits we have described here.",
"We provide the complete multi-Gaussian decomposition for all sources in the catalog, as well as the 568 objects with $L>1.2\\times 10^{42}\\,{\\rm erg\\,s^{-1}}$ in the [73] catalog [102], as online FITS tables (http://zakamska.johnshopkins.edu/data.htm).",
"The names of the columns in the kinematic catalog are listed in Table .",
"In addition to tabulating the Gaussian components, we provide non-parametric measures of the [O$\\,$III] profiles, including the full widths at 25 per cent and 50 per cent of the maximum (in km s$^{-1}$ ) and the non-parametric measures defined by [102] following [96].",
"Specifically, for every emission line profile we measure the velocities $v_x$ at which $x$ per cent of line power accumulates.",
"This allows us to define the widths encompassing 50 per cent, 80 per cent and 90 per cent of line power: $w_{50}\\equiv v_{75}-v_{25}$ , $w_{80} \\equiv v_{90}-v_{10}$ and $w_{90} \\equiv v_{95}-v_{05}$ , measured in km s$^{-1}$ .",
"These widths are more sensitive to weak broad components than the traditional full width at half maximum measures.",
"We also tabulate the dimensionless relative asymmetry $R\\equiv ((v_{95}-v_{50})-(v_{50}-v_{05}))/w_{90}$ , which is negative for profiles with a heavier blue-shifted wing, and dimensionless kurtosis $r_{9050}\\equiv w_{90}/w_{50}$ which is larger for profiles with heavy wings and narrow cores.",
"A comprehensive study of the relationships between these measures and their relevance to quasar winds is presented by [102].",
"In what follows we use $w_{80}$ as a measure of the [O$\\,$III] kinematics.",
"Figure REF shows the profiles and fits for those objects with the objects with the most dramatic outflows, as measured by $w_{80}$ .",
"Figure: [O\\,III] spectra of the ten objects with the broadest[O\\,III] emission, as measured by the width containing 80 per cent of theline power w 80 w_{80}.",
"The spectrum of the[O\\,III]λ\\lambda 4959,5007Å region is shown in grey, and the modelcomponents (continuum and two or three Gaussians) are shown in red for the5007Å line (the fit is performed simultaneously on both lines inthe doublet assuming the same kinematic structure).",
"The summedmodel for both lines of the [O\\,III] doublet is shown inblack.",
"Vertical dashed lines show v 10 v_{10}, v 50 v_{50} and v 90 v_{90},so that w 80 w_{80} is encompassed between the left and rightlines.",
"The ten objects shown in the figure have w 80 w_{80} values inthe range 2558 – 3912 km s -1 ^{-1}.",
"These values are comparable tothe maximal widths found in the sample, but fall short of the extreme values (up to 5409 kms -1 ^{-1}) found in high-luminosity red quasars at high redshifts.",
"Each spectrum is labeled with its plate, fiber, andMJD.In both catalogs, we flag candidate double-peaked [O$\\,$III] profiles (see [52]).",
"While most of these profiles likely result from biconical quasar-driven winds, where each plowed shell may appear as a separate Gaussian component [28], [32], a small fraction of these objects could be due to kpc-scale binary active nuclei [51], [84], [14].",
"Separating these two possibilities is not yet possible without extensive follow-up observations, so we identify possible double-peaked emitters in the catalog exclusively based on the shape of the [O$\\,$III] line.",
"While there is no formal definition of what constitutes a double-peaked profile, for identifying candidate kpc-scale binaries we are interested in profiles with two distinct narrow components in the [O$\\,$III] profile which are kinematically separated from one another by an amount comparable to their velocity dispersion.",
"We identify candidate double-peaked profiles in two ways.",
"First, following [52] we identify sources with a minimum in the fitted [O$\\,$III] profile and examine them visually.",
"Most of these objects are retained as double-peaked candidates.",
"Second, we visually examine all profiles and flag those that do not have a minimum but nonetheless appear to have two distinct kinematic components.",
"Candidates are visually flagged based on the observed profiles, regardless of whether the two distinct components are accurately captured by the multi-Gaussian fits.",
"Examples of objects from both selection methods are shown in Figure REF .",
"In the new BOSS sample, 420 of the 2920 spectra show a profile minimum.",
"Of these, 363 (86 per cent) are retained as double-peaked after visual inspection, and an additional 183 without a model profile minimum are identified visually.",
"In the [102] sample, there are, respectively, 60 and 48 double-peaked candidates identified by the two algorithms.",
"Figure: Six of the double-peaked candidates in the catalog.",
"Thethree objects in the left panels are selected by requiring that thefitted profile has a minimum, whereas the three objects in the rightpanels are selected by visual inspection.",
"Each spectrum is labeledby its plate, fiber, and MJD."
],
[
"Crossmatch with AllWISE",
"The dust within the obscuring medium along the line of sight to a type 2 AGN absorbs much of the energy of optical and ultraviolet photons.",
"This dust emits at mid-infrared wavelengths, to which the dust is more transparent (though obscuration may be significant even at these wavelengths; [65]).",
"Thus observations of type 2 quasars in the mid-infrared may provide a more direct probe of their bolometric emission than the narrow emission lines we have used so far.",
"We matched our sample against the AllWISE [99], [16] catalog of the Wide-field Infrared Survey Explorer (WISE) using a 5 matching radius, picking the nearest match when multiple matches are found.",
"Approximately 97 per cent of the objects in our type 2 quasar sample have a successful match in AllWISE.",
"To estimate the contamination rate, we offset the positions of our sources by 1 and re-match, resulting in a matching rate of 9 per cent within 5.",
"Therefore, a few per cent of our AllWISE matches may be random associations, or have their fluxes contaminated by unrelated objects.",
"We fit piece-wise power-laws between each pair of adjacent WISE bands to determine a flux density, and thus a luminosity ($\\nu L_\\nu $ ) at rest frame 5 and 12 $\\mu $ m. Figure REF shows the distribution of our sources in WISE color space, using the filters at 3.4, 4.6, and 12$\\mu $ m. The median SNRs of the detections of our objects in the 3.4, 4.6, 12 and 22 $\\mu $ m bands are 26.7, 16.9, 6.8, and 4.0 respectively.",
"The figure also includes type 2 quasars from [73]; the two distributions are quite similar.",
"The “wedge\" denoted by the dashed lines is the luminous AGN selection region as defined by [58], [57], analogous to other proposed mid-infrared color cuts used for obscured AGN selection [46], [88], [89].",
"It is striking that only 34 per cent of our sample is encompassed by this wedge; thus, there is a substantial number of sources with strong optical signatures of a type 2 quasar which would not be identified by the standard color-based infrared selection criteria.",
"In Vega magnitudes, $[3.4]-[4.6]\\simeq 0$ corresponds to the color of an old stellar population dominated by the Rayleigh-Jeans tail of the spectral energy distribution of stellar photospheres, thus there are few objects bluer than this cutoff.",
"In the absence of any thermal re-emission by dust, this would also be the typical color of a type 2 quasar whose mid-infrared emission is dominated by the host galaxy.",
"Contribution of warm dust emission moves sources to the right (toward the redder $[3.4]-[12]$ color) and contribution of hot dust emission moves sources upward (toward the redder $[3.4]-[4.6]$ color).",
"Type 2 quasars can be obscured even at mid-infrared wavelengths, which is known both from theoretical models [71] and observations which show that they are significantly redder in the infrared than type 1 quasars [50].",
"Therefore, the hot dust contribution is not strong enough in more than half of the sample to push the objects into the mid-infrared wedge.",
"It has been suggested that the objects that lie outside the wedge have WISE fluxes dominated by starlight [6], [20], [19], [57], but that would be surprising given the high inferred AGN luminosities for our sample.",
"Figure: The distribution of our sample and that of in the WISE ([3.4]-[4.6][3.4] - [4.6] vs [4.6]-[12][4.6] - [12]) colors.",
"The `wedge\" denoted by the dashed lines is the luminous AGN selection region as defined by , .",
"Only 34 per cent of the BOSS type 2 quasars are within this region, indicating that many type 2 quasars would be missed in infrared-selected samples."
],
[
"[O$\\,$",
"Figure REF shows the relationship between the [O$\\,$III] luminosities (based on the model fits described in Section REF ) with the rest-frame 12$\\mu $ m luminosity for all objects with WISE counterparts in our sample and that of [73].",
"The 12$\\mu $ m luminosity is a proxy for a bolometric luminosity, or at least the luminosity associated with hot dust close to the central engine.",
"The two luminosities are strongly correlated, suggesting that [O$\\,$III] is a useful, albeit rough, proxy for bolometric luminosity in obscured AGN [34].",
"We calculate the best fits to the joint sample using two methods: (i) we make the least-squares fit by minimizing perpendicular offsets and (ii) we calculate the best-fit linear relationship (i.e., with slope equal to unity).",
"The resulting best fits are: $& \\log _{10}\\left(\\frac{\\rm L_{[OIII]}}{\\rm erg\\ s^{-1}}\\right) - 42.5 = (0.78\\pm 0.04)\\times \\\\& \\left[\\log _{10}\\left(\\frac{\\rm \\nu L_{\\nu }[12\\mu m]}{\\rm erg\\ s^{-1}}\\right)-44.5\\right]+ (-0.065\\pm 0.017); \\nonumber \\\\& \\log _{10}\\left(\\frac{\\rm L_{[OIII]}}{\\rm erg\\ s^{-1}}\\right) =\\log _{10}\\left(\\frac{\\rm \\nu L_{\\nu }[12\\mu m]}{\\rm erg\\ s^{-1}}\\right)- (2.090\\pm 0.005).$ These should be considered approximate scaling relations, because both our new sample and the [73] sample are affected by their respective [O$\\,$III] luminosity cutoffs (visible in Figure REF ).",
"Furthermore, there are fewer objects at high luminosity, so that our fit is more heavily weighted by the less luminous objects.",
"The correlation between [O$\\,$III] luminosity and 12 luminosity is tighter than the one between [O$\\,$III] luminosity and 5 luminosity [102], presumably because the 5 luminosity is more strongly affected by geometric effects and dust extinction.",
"Figure: The relationship between [O\\,III] and rest-frame 12μ\\mu mluminosity for the quasars in our sample and those of.",
"All objects with WISE counterparts are shownhere, not just those lying within the wedge ofFigure .",
"The solid line is the best-fit power-lawobtained by minimizing perpendicular residuals and the dashed lineis the best-fit linear dependence, with best fits quantified byequations ().",
"[O$\\,$III] kinematics, in turn, can be a useful proxy for the strength of the quasar-driven outflows on host galaxy scales.",
"In low-luminosity AGNs, the kinematics of the forbidden emission lines are strongly correlated with galaxy rotation and/or bulge velocity dispersion [98], [25], indicating that the emission-line gas is in dynamical equilibrium with the galaxy.",
"This is not the case in quasars [102], where the characteristic velocities probed by [O$\\,$III] emission are too high to be contained by the galactic potential.",
"The velocity width asymmetry and kurtosis of [O$\\,$III] are all correlated with one another, suggesting that any of these values can serve as a proxy for outflow strength.",
"[102] found that the strongest correlations are between the velocity width of the [O$\\,$III] line (as measured by the $w_{90}$ parameter, Section REF ), and radio and infrared emission in the [73] sample.",
"In Figure REF we investigate the relationship between $w_{90}$ , the velocity width containing 90 per cent of line power, with the [O$\\,$III] luminosity, the rest-frame 12$\\mu $ m luminosity, and the [O$\\,$III] equivalent width for the objects in our new BOSS sample.",
"There is no correlation with equivalent width, and a weak one with [O$\\,$III] luminosity.",
"The correlation with infrared luminosity, however, is quite strong, suggesting that indeed the velocity width reflects the outflow velocity and that the outflow activity is driven by the bolometric luminosity of the AGN.",
"[106] have found objects at the peak epoch of quasar activity at $z \\sim 2.5$ that lie at the extreme end of this diagram – with extremely high infrared luminosities and extremely broad [O$\\,$III]; $w_{90}$ up to 5000 km s$^{-1}$ .",
"Figure: The relationships between w 90 w_{90} (the velocity width of[O\\,III] containing 90 per cent of line power) and [O\\,III] line luminosity,12μ\\mu m infrared luminosity, and [O\\,III] rest equivalent width.",
"In eachpanel, the red points are median values in bins, with theinterquartile range indicated.",
"[O\\,III] velocity width, which is aproxy for quasar wind activity, is most strongly correlated withquasar infrared luminosity."
],
[
"Extreme [Ne$\\,$",
"Our sample includes objects with very strong [Ne$\\,$V]$\\lambda $ 3426Å emission, including sources with [Ne$\\,$V]/H$\\beta >1$ .",
"Among type 2 quasars in the [73] sample, the mean and standard deviation of the quantity $\\log $ ([Ne$\\,$V]/H$\\beta $ ) are $-0.3$ and 0.2, respectively, with only 6 per cent of objects showing [Ne$\\,$V]/H$\\beta >1$ .",
"In our newly selected BOSS sample, these values are, correspondingly, $-0.3$ dex, 0.3 dex and 13 per cent.",
"The higher fraction of objects with [Ne$\\,$V]/H$\\beta >$ 1 is likely due to our explicit selection requirement that [Ne$\\,$V] be detected in the $z>0.52$ subsample.",
"One example of such an object from the BOSS sample is shown in Figure REF .",
"In addition to the very strong [Ne$\\,$V] ([Ne$\\,$V]/H$\\beta \\simeq 2$ in this source), these objects also show unusual emission features at 5721Å and 6087Å, which we identify as transitions of [FeVII] [78].",
"The ionization potentials of [Ne$\\,$V] and [FeVII] are 97 and 99 eV, respectively, almost identical, and thus it is not surprising that the strength of these features are strongly correlated.",
"[78], [77] deem such objects `coronal line forest AGNs' and explore the hypothesis that these emission lines arise from the inner wall of the obscuring material.",
"It is thus rather unusual to see these features in type 2 quasars, where it is expected that the line of sight to this emitting region should be obscured.",
"[77] argue that even in the classical unification model with a toroidal obscuring region, a small fraction of viewing directions might result in both strong coronal lines and obscured broad-line region (another such example from the [73] sample is analyzed by [95]).",
"This picture is consistent with the distribution of dust temperatures in these objects and in type 2 quasars, in that coronal-line AGNs are warmer as inferred from the infrared colors than are other type 2 quasars.",
"The statistics of the coronal-line AGNs in the type 2 population might offer clues to the geometric structure of the obscuring material and provide constraints on its clumpiness [64].",
"Figure: The BOSS spectrum (smoothed with a 5-pixel boxcar) of anextreme [Ne\\,V] emitter from the newly selected BOSS sample.Prominent emission lines are marked.",
"The [O\\,III] and Hα\\alpha linepeaks are offscale.",
"The stellar continuum is also apparent, showingCalcium K absorption and a strong Balmer break."
],
[
"Extended emission-line region",
"One intriguing object identified using our initial selection is the spectrum of an extended emission-line region associated with the merging pair Mrk 266 (Fig.",
"REF ), photo-ionized by the AGN in one of the merging nuclei.",
"This region of ionized gas, roughly 12 kpc from the central nucleus [38], [39], [61], has essentially no associated starlight, and thus displays [O$\\,$III] with a very high equivalent width.",
"Another well-known example of such an extended emission line nebula with AGN line ratios is Hanny's Voorwerp [49], which is found near a galaxy which no longer hosts an active nucleus but presumably did in the recent past.",
"A systematic search for such objects in the SDSS data is conducted by Sun et al.",
"(in prep.).",
"We removed the off-nuclear spectrum of Mrk 266 from the sample because it is not an integrated spectrum of the entire galaxy.",
"Figure: The SDSS image of Mrk 266.",
"The image is 51.2 '' 51.2^{\\prime \\prime } on aside; North is up and East is to the left.",
"This is a grigri colorcomposite, prepared following the approach of .",
"Thesquares indicate the regions which have SDSS spectra; the SDSS imagedeblender identified the blue region in in the north as a separate“galaxy” .",
"The northernmost spectrum was identified using ourspectroscopic search for objects with high equivalent width, highionization emission lines.",
"The blue color of the nebular emission in thisregion is due to the strong [O\\,III] dominating the g-g-band emission inthis extended nebula."
],
[
"Composite spectrum",
"We construct a composite spectrum from our sample by shifting all spectra to their rest frames (using the adopted redshifts listed in the catalog), rebinning the spectra onto a common rest wavelength grid and calculating the error-weighted average.",
"In Figure REF we further normalize the composite spectrum by a continuum obtained by spline-interpolating between relatively line-free regions.",
"In this presentation the overall continuum shape is lost, but the equivalent widths of the features are preserved.",
"We provide the composite spectrum in an online FITS table (http://zakamska.johnshopkins.edu/data.htm).",
"The high SNR continuum reveals a multitude of emission features due to the quasar-ionized gas and absorption features due to the stellar photospheres and interstellar medium of the host galaxies.",
"Figure: The composite spectrum of all BOSS type 2 quasars normalized by an approximate continuum obtained by spline-interpolating between relatively line-free regions.",
"The top spectrum shows the brightest emission lines, the middle panel shows a zoom in on the same data, and the bottom panel shows the number of objects contributing at each wavelength.",
"Only a small subset of all detected lines are marked, following line identifications by ."
],
[
"Conclusions",
"In this paper we present a sample of 2758 type 2 (obscured) quasars at $z1$ selected from the SDSS-III/BOSS spectroscopic database.",
"We aim to select sources with high equivalent width emission lines, REW([O$\\,$III]$\\lambda $ 5007Å)$>$ 100Å.",
"At low redshifts ($z<0.52$ ) we use standard emission-line diagnostic diagrams to separate type 2 candidates from star-forming galaxies.",
"At higher redshifts ($z>0.52$ ), when H$\\alpha $ and other diagnostics move out of the BOSS spectral coverage, we require a detection of [Ne$\\,$V]$\\lambda $ 3426Å which requires ionization by an AGN and narrow H$\\beta $ to separate type 2 quasars from type 1 quasars.",
"An interesting subsample of 508 objects has erroneous or uncertain redshifts in the SDSS database.",
"We select such sources by assuming that [O$\\,$III] – one of the strongest lines in the type 2 quasar spectra – is mistaken for another strong line, either Ly$\\alpha $ or H$\\alpha $ , by the pipeline.",
"Additionally, we examine sources with high REW of [O$\\,$III] and with redshifts flagged by the SDSS database pipeline as uncertain.",
"These sources tend to have strongly kinematically disturbed emission lines and they are therefore poorly matched against the standard templates.",
"Because our selection algorithms for such sources are not exhaustive, small numbers of interesting type 2 quasars with broad [O$\\,$III] could remain unidentified in the SDSS spectroscopic database.",
"While in low-luminosity AGNs the kinematics of the forbidden emission lines tend to trace the potential of the AGN host galaxy, in powerful quasars [O$\\,$III] profile shapes appear to be strongly related to quasar-driven winds.",
"We conduct multi-Gaussian decomposition of [O$\\,$III] for all objects in this sample and calculate all commonly used non-parametric measures of [O$\\,$III] profile shape.",
"We release complete kinematic decomposition information for both the new catalog of BOSS type 2 quasars and for our previous kinematic analysis of 568 luminous ($L$ [O$\\,$III]$>1.2\\times 10^{42}\\,{\\rm erg\\,s^{-1}}$ ) type 2 quasars selected from SDSS I/II [73], [102].",
"We also identify 654 candidate objects with double-peaked [O$\\,$III] profiles which could be interesting for studies of quasar-driven winds or of binary AGNs.",
"As we determine by matching the sample to the WISE survey, the type 2 quasars presented here have high infrared luminosities, with median $\\nu L_{\\nu }=1.7\\times 10^{44}\\,{\\rm erg\\,s^{-1}}$ at rest-frame 5 and $4.2\\times 10^{44}\\,{\\rm erg\\,s^{-1}}$ at rest-frame 12.",
"If quasars were isotropic emitters at 12, we could apply a typical bolometric correction at this wavelength of $\\sim 9$ [75] to estimate the median bolometric luminosity of our sample to be $\\sim 4\\times 10^{45}\\,{\\rm erg\\,s^{-1}}$ .",
"Intriguingly, despite very high luminosities, fewer than half of the objects in our sample have $[3.6-4.5]$ color red enough for them to be selected using the common infrared color selection methods used to identify AGNs.",
"Thus it is likely that type 2 quasars are obscured even at mid-infrared wavelengths, so that hot dust emission from the inner parts of the obscuring material remains largely invisible to the observer [50].",
"Therefore, the bolometric corrections are likely higher than those for type 1 quasars, and so then is our estimated median bolometric luminosity.",
"The demographics of quasars remain an interesting unsolved problem in astronomy, with obscured quasars now thought to play an important role in galaxy evolution.",
"This work, alongside other approaches, makes it clear that different selection methods result in largely different samples of objects.",
"Samples selected by infrared, optical and X-ray methods overlap at the tens of per cent level [47], but none of the selection methods results in a complete sample.",
"The combination of multi-wavelength approaches and extensive studies of samples selected at different wavelengths will be required to measure the demographics of quasars and to determine the geometry and the spatial structure of the obscuring material.",
"Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the U.S. Department of Energy Office of Science.",
"The SDSS-III web site is http://www.sdss3.org/.",
"SDSS-III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS-III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mellon University, University of Florida, the French Participation Group, the German Participation Group, Harvard University, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, Max Planck Institute for Extraterrestrial Physics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.",
"l|l Entries in the catalog of BOSS type 2 quasars (data model) name of parameter comments plate, fiber, mjd Spectroscopic identification z Adopted redshift based on kinematic fits ra, dec Right ascension and declination in decimal degrees oiiiflux, oiiilum, oiiirew Fluxes (in units of 10$^{-17}\\,{\\rm erg\\,s^{-1}}$ cm$^{-2}$ ), luminosities (in units of 10$^{42}\\,{\\rm erg\\,s^{-1}}$ ) and rest equivalent widths (in Å) of [O III]$\\lambda $ 5007Å from the complete kinematic fits (Section REF ) oiiiw80 Velocity width containing 80 per cent of [O III]$\\lambda $ 5007Å power (in km s$^{-1}$ ) nevflux, nevrew, nevsnr Fluxes, rest equivalent widths and signal-to-noise ratios of [Ne V]$\\lambda $ 3426Å from single-Gaussian fits magu, magg, magr, magi, magz Model SDSS magnitudes (corrected for [82] extinction) emagu, emagg, emagr, emagi, emagz Errors on the model SDSS magnitudes w1, w2, w3, w4 WISE catalog magnitudes (in Vega system) ew1, ew2, ew3, ew4 Errors on the WISE magnitudes lum5, lum12 Rest-frame luminosities $\\nu L_{\\nu }$ at 5 and 12 calculated from piece-wise interpolation between WISE fluxes (in units of 10$^{42}\\,{\\rm erg\\,s^{-1}}$ ) select A string value indicating how the object was selected: possible values are `Lowz' (Section REF ), `Highz' (Section REF ), `Wrongz1', `Wrongz2' and `Zwarning' (Section REF ) unique A string value set to `unique' if the object is making its first or only appearance in the catalog; if not, the flag is set to the spectroscopic ID of the first appearance of the source in the format `pppp-ffff-mmmmm' l|l Kinematic parameters of [O$\\,$III]$\\lambda $ 5007Å (data model) name of parameter comments plate, fiber, mjd Spectroscopic identification z Adopted redshift amp1, vel1, sig1 Amplitude (in units of $10^{-17}\\,{\\rm erg\\,s^{-1}}$ cm$^{-2}$ Å$^{-1}$ ), velocity offset (in km s$^{-1}$ ) and velocity dispersion (in km s$^{-1}$ ) of the first Gaussian component as measured in the frame placed at the adopted redshift amp2$-$ 4, vel2$-$ 4, sig2$-$ 4 Same as above for additional Gaussian components; 0 amplitude indicates the component is not required by the fit fwhm, fwqm Full width at half maximum and at quarter maximum of the line profile, in km s$^{-1}$ w50, w80, w90 Velocity widths containing 50 per cent, 80 per cent and 90 per cent of the line profile power, in km s$^{-1}$ relasym, r9050 Dimensionless relative asymmetry $R$ and kurtosis parameter $r_{9050}$ dp Double-peaked candidate flag: 0 – no, 1 – visual inspection, 2 – profile minimum and visual inspection These data are provided in two on-line FITS tables: one for the new catalog of SDSS-III type 2 quasars presented here and one for the 568 objects with [O$\\,$III] kinematics calculated by [102].",
"The files can be found at http://zakamska.johnshopkins.edu/data.htm."
]
] | 1606.04976 |
[
[
"Holographic Construction of Quantum Field Theory using Wavelets"
],
[
"Abstract Wavelets encode data at multiple resolutions, which in a wavelet description of a quantum field theory, allows for fields to carry, in addition to space-time coordinates, an extra dimension: scale.",
"A recently introduced Exact Holographic Mapping [C.H.",
"Lee and X.-L. Qi, Phys.",
"Rev.",
"B 93, 035112 (2016)] uses the Haar wavelet basis to represent the free Dirac fermionic quantum field theory (QFT) at multiple renormalization scales thereby inducing an emergent bulk geometry in one higher dimension.",
"This construction is, in fact, generic and we show how higher families of Daubechies wavelet transforms of 1+1 dimensional scalar bosonic QFT generate a bulk description with a variable rate of renormalization flow.",
"In the massless case, where the boundary is described by conformal field theory, the bulk correlations decay with distance consistent with an Anti-de-Sitter space (AdS3) metric whose radius of curvature depends on the wavelet family used.",
"We propose an experimental demonstration of the bulk/boundary correspondence via a digital quantum simulation using Gaussian operations on a set of quantum harmonic oscillator modes."
],
[
"Introduction",
"The holographic principle asserts that spacetime may be like a hologram: information contained within the volume of spacetime can be encoded on the boundary of that spacetime [1], [2], [3].",
"This heuristic principle has been used with much success in the study of black hole thermodynamics and quantum gravity, and has been precisely formulated as holographic dualities, in particular, the Anti-de-Sitter space/conformal field theory (AdS/CFT) correspondence [4].",
"A holographic duality (e.g., the AdS/CFT correspondence) is a duality or equivalence between a theory with gravity and a theory without gravity that are seen to live in the bulk and at the boundary of a spacetime manifold respectively.",
"The extra dimension (that generates the bulk) corresponds to energy scale of the boundary theory, and the gravitational dynamics in the bulk generalize the renormalization group flow equations of the boundary.",
"A characteristic prediction of the holographic principle is the existence of correlated noise in the measured positions of massive bodies, though experimental evidence so far is negative [5].",
"Recently, Qi et al.",
"proposed a concrete implementation of such a bulk/boundary correspondence.",
"Their proposal, called the Exact Holographic Mapping (EHM) [6], [7], maps the (boundary) theory of a Dirac fermion descretized on a lattice to a bulk theory with the same number of degrees of freedom (DOFs) as the boundary, but using wavelet and scale field mode operators which carry a spatial position index as well as a scale index.",
"They used Haar wavelets, which is the $\\mathcal {K}=1$ family of Daubechies wavelets [8].",
"This EHM of the fermionic theory demonstrates some key features of holographic duality such as an emergent geometry in bulk—as inferred from the scaling of correlations between bulk wavelet DOFs—that depends on the boundary theory.",
"In this way, the authors were able to demonstrate the emergence of a bulk AdS geometry from a CFT on the boundary, the emergence of a flat metric from a massive boundary quantum field theory (QFT), and some features of black hole physics in the bulk for the case of a thermal CFT at the boundary.",
"In this work, we show that the EHM extends to an entire class of wavelet transformations.",
"We do this by analyzing the Daubechies $\\mathcal {K}$ wavelet families for $\\mathcal {K}\\ge 3$ .",
"Here the $\\mathcal {K}$ index refers to the number of vanishing moments of the constituent scale and wavelet functions.",
"A larger index encodes a function more accurately at the cost of increased computational effort.",
"We show how this feature translates to changing the scaling of correlations in the bulk description of a boundary QFT such that larger $\\mathcal {K}$ provides faster renormalization of the theory, at the expense of a linear increase in the interaction neighborhood for each renormalization step.",
"Furthermore, using $\\mathcal {K}\\ge 3$ ensures that the derivative couplings can be captured exactly up to the cutoff scale since the wavelet and scale functions then have continuous first derivatives.",
"We conclude the introduction by remarking that the wavelet approach bears interesting connections with implementations[9], [6] of the holographic principle in the tensor network formalism.",
"As noted in Ref.",
"[6], the Haar wavelet transform can be described as a tree tensor network—a set of tensors (multi-linear transformations) that are contracted according to a tree graph.",
"In this description, a tensor at a given renormalization layer (radius) takes as input two nearest neighbor fields and outputs a short range wavelet field and a longer range (coarse grained) scale field used as input to the next stage of renormalization.",
"In this work, we generalize to Daubechies $\\mathcal {K}$ wavelet families, which go beyond tree tensor networks and are closer in spirit to tensor networks with loops such as the Multi-Scale Renomalization Ansatz (MERA) [10], [11] (of which, a tree tensor network is a special case corresponding to fixing some of the MERA tensors to the identity).",
"In fact, in Ref.",
"[12], the authors establish a precise way to describe various wavelet transforms as MERA tensor networks.",
"The plan of the paper is as follows.",
"We begin with a brief introduction to the basics of the wavelets functional basis in Sec.",
"and why it is a natural one for holographic encoding.",
"In Sec.",
"we describe the bulk and boundary Hamiltonians for free scalar bosonic QFT.",
"Sec.",
"presents the main results given the details of the bulk/boundary correspondence and the shape of the emergent bulk geometry.",
"Explicit circuit based constructions of the bulk and boundary states are provided in Sec.",
".",
"The requisite components are local ground or thermal state preparation of an array of bosons, linear optical gates, and single mode squeezing which could be done in a variety of engineered atomic, optical, or solid state platforms.",
"Finally, we conclude with a summary and comment on open problems."
],
[
"The wavelet basis",
"Wavelets constitute an orthonomal basis for the Hilbert space $L^2(\\mathbb {R})$ , square integrable functions on the line, and we briefly review some of their properties here.",
"For a comprehensive survey see Ref. [13].",
"Generically, wavelets are defined in terms of a mother wavelet function $w(x)$ and a father scaling function $s(x)$ by taking linear combinations of shifts and rescalings thereof.",
"For the remainder we focus on one family known as Daubechies $\\mathcal {K}$ -wavelets where the role of the integer $\\mathcal {K}$ will be described below.",
"First we introduce two unitary operators on $L^2(\\mathbb {R})$ : $\\mathcal {T}$ for discrete translation and $\\mathcal {D}$ for scaling defined by the action on a function $f\\in L^2(\\mathbb {R})$ : $\\mathcal {D}f(x)=\\sqrt{2}f(2x);\\quad \\mathcal {T}f(x)=f(x-1).$ The father scaling function $s(x)$ is a solution to the linear renormalisation group equation $s(x)=\\mathcal {D}\\left[\\sum _{n=0}^{2\\mathcal {K}-1}h_n\\mathcal {T}^n s(x)\\right],$ which performs block averaging followed by rescaling.",
"The $2\\mathcal {K}$ real coefficients $\\lbrace h_n\\rbrace $ are computed analytically for $\\mathcal {K}< 4$ and are solved for numerically otherwise.",
"For convenience, we choose language such that $\\emph {scale}$ refers to a number $r\\in \\mathbb {N}$ which describes features at a resolution $2^{-r}$ in some natural units, and higher(lower) scale means larger(smaller) $r$ .",
"Given the solution to $s(x)$ , higher scaling functions are defined by applying $n$ unit translations followed by $k$ scaling transformations on the father: $s^k_n(x) =\\mathcal {D}^k \\mathcal {T}^n s(x).$ The scaling functions are normalised so that $\\int dx\\ s^k_n(x)=1.$ The mother wavelet $w(x)$ and the father $s(x)$ have the property that they are neither localised in position or momentum.",
"The wavelets take the following form: $w(x)=\\sum _{n=0}^{2\\mathcal {K}-1}g_n\\mathcal {D}\\mathcal {T}^n s(x)=\\sum _{n=0}^{2\\mathcal {K}-1}g_ns^1_n(x),$ where the set of coefficients $\\lbrace g_n\\rbrace $ are obtained from $\\lbrace h_n\\rbrace $ by reversing the order and alternating signs: $g_n=(-1)^nh_{2\\mathcal {K}-1-n}$ (see Appendix ).",
"Scale $k$ wavelets are obtained by translating and scaling the mother: $w^k_n(x)=\\mathcal {D}^k \\mathcal {T}^n w(x).$ The index $\\mathcal {K}$ specifies the number of vanishing moments of the wavelets, i.e.",
"$\\int \\ dx\\ w(x)x^p=0\\quad p=0,..,\\mathcal {K}.$ The vanishing of the zeroth moment guarantees that the wavelet basis is square integrable [13].",
"Choosing larger $\\mathcal {K}$ means more features can be captured at a given scale, however at the expense of additional computational cost since more translations are needed during block averaging.",
"Daubechies wavelets are optimal in the sense that they have the smallest size support for a given number of vanishing moments [13].",
"The basis functions $s^k_n(x)$ and $w^k_n(x)$ have support on $[2^{-k}n,2^{-k}(n+2\\mathcal {K}-1)]$ and satisfy the following orthonormality relations: $\\begin{split}&\\int dx\\ s^k_n(x)s^k_m(x)=\\delta _{m,n},\\\\&\\int dx\\ s^k_n(x)w^{k+l}_m(x)=0\\quad (l\\ge 0),\\\\&\\int dx\\ w^k_n(x)w^l_m(x)=\\delta _{m,n}\\delta _{k,l}.\\end{split}$ By the last relation, the wavelets constitute normalised wave functions.",
"The scaling functions at scale $2^{-k}$ are complete in that $\\sum _{n=-\\infty }^{\\infty } \\frac{1}{\\sqrt{2^k}}s^k_n(x)=1.$ A final important property of the Daubechies $\\mathcal {K}$ -wavelets is that they are $\\mathcal {K}-2$ times differentiable.",
"A plot of three of the the scale and wavelet functions families is shown in Fig.",
"REF .",
"Figure: Plots of scale functions s(x)s(x) and wavelet functions w(x)w(x) for Daubechies families 𝒦=3,4,5\\mathcal {K}=3,4,5.The connection between holography and wavelets arises from the scale dependent representation of fields.",
"In a wavelet family basis, linear superpositions of scale functions $\\lbrace s^r_m(x)\\rbrace _{m=-\\infty }^{\\infty }$ (with square summable coefficients) span a subspace $\\mathcal {H}_r$ of $L^2(\\mathbb {R})$ .",
"This is a proper subspace of the higher scale space $\\mathcal {H}_r\\subset \\mathcal {H}_{r+j}\\ (j>0)$ .",
"Linear combinations of the scale $r$ wavelet functions $\\lbrace w^r_m(x)\\rbrace _{m=-\\infty }^{\\infty }$ span the orthocomplement, $\\mathcal {W}_r$ , of $\\mathcal {H}_r$ in $\\mathcal {H}_{r+1}$ : $\\mathcal {H}_{r+1}=\\mathcal {H}_{r}\\oplus \\mathcal {W}_{r}$ .",
"We can use a set of scaling functions $\\lbrace s^r_m(x)\\rbrace _{m=-\\infty }^{\\infty }$ to represent features at scale $r$ and a set of wavelets $\\lbrace w^r_m(x)\\rbrace _{m=-\\infty }^{\\infty }$ to represent features at scale $r+1$ that cannot be represented at scale $r$ .",
"The whole space has the following decomposition satisfied for any finite $r$ : $L^2(\\mathbb {R})=\\mathcal {H}_{r}\\bigoplus _{l=r}^{\\infty }\\mathcal {W}_{l},$ meaning that for a fixed scale $r$ the set $\\lbrace s^r_m(x)\\rbrace _{m=-\\infty }^{\\infty }\\bigcup \\lbrace w^l_m(x)\\rbrace _{m=-\\infty ,l=r}^{\\infty ,\\infty },$ span a basis for $L^2(\\mathbb {R})$ [13].",
"In particular, if one introduces a scale cutoff $n$ , then $\\mathcal {H}_0\\oplus _{l=0}^{n-1} \\mathcal {W}_l\\simeq \\mathcal {H}_n.$ What this means is that the Hilbert space of functions spanned by scale fields at scale $n$ is isomorphic to the Hilbert space spanned by wavelet functions at all lower scales together with scale functions at the coarsest scale.",
"We will identify the Hilbert space $\\mathcal {H}_n$ as characterizing the Hilbert space of quantum fields with a UV cutoff momentum $p\\simeq 2^n/L$ , where $L$ is the length of the system in units of distance at the coarsest scale.",
"We will call the degrees of freedom (DOFs) in this Hilbert space to be boundary DOFs.",
"Similarly, the DOFs at lower scale in $\\mathcal {H}_0\\oplus _{l=0}^{n-1} \\mathcal {W}_l$ are bulk DOFs.",
"Since the Hilbert spaces are isomorphic, there exists an unitary transformation from boundary states to bulk states.",
"Figure: Bulk/boundary corrrepondence for a time slice of a 1+1 dimensional QFT with periodic boundaries.",
"The QFT has a UV cutoff at scale nn, and the boundary state |G〉 bd \\vert G \\rangle _{\\rm bd} is the ground state of a boundary Hamiltonian H ^ bd \\hat{H}_{\\rm bd} with DOFs {Φ ^ [s]n (θ)}\\lbrace \\hat{\\Phi }^{[s]n}(\\theta )\\rbrace (red dots) .",
"The corresponding bulk state |G〉 bk \\vert G \\rangle _{\\rm bk} is the ground state of the Hamiltonian H ^ bk \\hat{H}_{\\rm bk} with DOFs {Φ ^ [s]0 (m)}∪{Φ ^ [w]r (m)}\\lbrace \\hat{\\Phi }^{[s]0}(m)\\rbrace \\cup \\lbrace \\hat{\\Phi }^{[w]r}(m)\\rbrace (green and blue dots).",
"The number LL of DOFs at the coarsest scale r=0r=0 in the bulk is determined by the wavelet family used.",
"For the field Φ ^ [w]r (m)\\hat{\\Phi }^{[w]r}(m) at scale rr and position mm, one can associate the spatial AdS 3 _{3} coördinates (ρ=L2 r 2π,θ=m2π L2 r )(\\rho =\\frac{L2^{r}}{2\\pi },\\theta =\\frac{m2\\pi }{L2^{r}}).",
"The shaded segments within the blue region enclose the bulk DOFs that contribute to the two point correlation bd 〈G|Φ ^ [s]n (m)Φ ^ [s]n (m ' )|G〉 bd _{\\rm bd}\\langle G \\vert \\hat{\\Phi }^{[\\mathbf {\\mathrm {s}}]n}(m)\\hat{\\Phi }^{[\\mathbf {\\mathrm {s}}]n}(m^{\\prime })\\vert G \\rangle _{\\rm bd}.",
"The boundary fields overlap with the same number (2𝒦-12\\mathcal {K}-1) of bulk fields at each scale, here it is 5 for the 𝒦=3\\mathcal {K}=3 wavelet family."
],
[
"1+1 D Scalar Bosonic QFT",
"To illustrate the connection between holography and wavelets we study scalar bosonic free field theory in one spatial dimension.",
"The Hamiltonian for this QFT is: $\\hat{H}=\\int dx\\ \\frac{1}{2}\\left(\\hat{\\Pi }^2(x,t)+\\mathbf {\\mathrm {\\nabla }}\\hat{\\Phi }^2(x,t)+m_0^2\\hat{\\Phi }^2(x,t)\\right),$ and the canonical momentum is $\\hat{\\Pi }(x,t)=\\frac{\\partial \\hat{\\Phi }(x,t)}{\\partial t},$ which together with the field $\\hat{\\Phi }(x,t)$ is normalised to satisfy the equal time commutation relation $[\\hat{\\Phi }(x,t),\\hat{\\Pi }(y,t)]=i\\delta (x-y)$ $(\\hbar \\equiv 1)$ .",
"Here the phase velocity of waves in this theory is set so that the speed of light is 1 and the bare mass is $m_0$ .",
"We will focus on a system of size $V$ with periodic boundaries such that $\\hat{\\Phi }(x+V,t)=\\hat{\\Phi }(x,t)$ .",
"This choice obviates the need to define special boundary wavelets and allows us to make a connection to geometry in the bulk even for systems with finite numbers of DOFs.",
"The Hamiltonian $\\hat{H}$ is diagonalized with the normal mode operators $\\hat{a}_{k_n}$ and $\\hat{a}^{\\dagger }_{k_n}$ satisfying $[\\hat{a}_{k_n},\\hat{a}^{\\dagger }_{k_m}]=\\delta _{n,m}$ , where $k_n=\\frac{2\\pi n}{V}$ with $n\\in \\mathbb {Z}\\cap [-V/2,V/2]$ .",
"The fields operators are then (henceforth we suppress the time dependence) $\\begin{array}{lll}\\hat{\\Phi }(x)&=&\\frac{1}{\\sqrt{V}}\\sum _n \\frac{1}{\\sqrt{2\\omega (k_n)}}(\\hat{a}_{k_n}e^{ik_n x}+\\hat{a}^{\\dagger }_{k_n}e^{-ik_n x}),\\\\\\hat{\\Pi }(x)&=&-\\frac{i}{\\sqrt{V}}\\sum _n \\frac{\\sqrt{\\omega (k_n)}}{2}(\\hat{a}_{k_n}e^{ik_n x}-\\hat{a}^{\\dagger }_{k_n}e^{-ik_n x}),\\\\\\end{array}$ where the dispersion relation is $\\omega (k_n)=\\sqrt{m_0^2+k_n^2}$ .",
"The ground state field-field and momentum/momentum correlations are obtained by Fourier transform (see e.g.",
"[14]) $\\begin{array}{lll}\\langle \\hat{\\Phi }(x)\\hat{\\Phi }(y)\\rangle & =& \\frac{1}{2V}\\sum _n \\frac{\\cos (k_n (x-y))}{\\omega (k_n)},\\\\\\langle \\hat{\\Pi }(x)\\hat{\\Pi }(y)\\rangle & =& \\frac{1}{2V}\\sum _n \\omega (k_n) \\cos (k_n (x-y)).\\\\\\end{array}$ In the limit $V\\rightarrow \\infty $ the continuum correlations in the massless $(m_0=0)$ phase are: $\\begin{array}{lll}\\langle \\hat{\\Phi }(x)\\hat{\\Phi }(y)\\rangle & =& -\\frac{\\ln ((x-y)^2)}{4\\pi },\\\\\\langle \\hat{\\Pi }(x)\\hat{\\Pi }(y)\\rangle & =& -\\frac{1}{2\\pi (x-y)^2}.\\\\\\end{array}$ In the massive $(m_0>0)$ phase: $\\begin{array}{lll}\\langle \\hat{\\Phi }(x)\\hat{\\Phi }(y)\\rangle & =&\\frac{1}{4\\pi }\\int _{-\\infty }^{\\infty }dk \\frac{\\cos (k (x-y))}{\\sqrt{k^2+m_0^2}}\\\\&=& \\frac{1}{2\\pi }K_0(m_0 |x-y|),\\\\\\langle \\hat{\\Pi }(x)\\hat{\\Pi }(y)\\rangle & =&\\frac{1}{4\\pi }\\int _{-\\infty }^{\\infty }dk \\cos (k (x-y))\\sqrt{k^2+m_0^2}\\\\&=& -\\frac{m_0}{2\\pi (x-y)}K_1(m_0 |x-y|),\\\\\\end{array}$ where $K_0$ and $K_1$ are modified Bessel functions of the second kind.",
"There are two distinct limiting behaviours of correlations in the massive phase.",
"For $|x-y|\\gg m_0^{-1}$ : $\\begin{array}{lll}\\langle \\hat{\\Phi }(x)\\hat{\\Phi }(y)\\rangle &\\rightarrow &-\\frac{e^{-m_0 |x-y|}}{\\sqrt{8\\pi m_0 |x-y|}},\\\\\\langle \\hat{\\Pi }(x)\\hat{\\Pi }(y)\\rangle & \\rightarrow &\\sqrt{\\frac{m_0}{8\\pi |x-y|^3}}e^{-m_0 |x-y|},\\end{array}$ whereas for $|x-y|\\ll m_0^{-1}$ : $\\begin{array}{lll}\\langle \\hat{\\Phi }(x)\\hat{\\Phi }(y)\\rangle &\\rightarrow &-\\frac{1}{2\\pi }(\\ln (\\frac{m_0 |x-y|}{2})+\\gamma ),\\\\\\langle \\hat{\\Pi }(x)\\hat{\\Pi }(y)\\rangle & \\rightarrow &-\\frac{1}{2\\pi (x-y)^2},\\end{array}$ where $\\gamma $ is the Euler gamma constant."
],
[
"Decomposition in a wavelet basis",
"The field and its conjugate can be expressed in a discrete wavelet family basis by projections onto the scaling and wavelet functions (here $r\\ge 0$ ): $\\begin{split}\\hat{\\Phi }^{[\\mathbf {\\mathrm {s}}]0}(n)&=\\int dx\\ \\hat{\\Phi }(x)s^{0}_{n}(x),~~\\hat{\\Phi }^{[w]r}(n)=\\int dx\\ \\hat{\\Phi }(x)w^r_{n}(x),\\\\\\hat{\\Pi }^{[\\mathbf {\\mathrm {s}}]0}(n)&=\\int dx\\ \\hat{\\Pi }(x)s^{0}_{n}(x),~~\\hat{\\Pi }^{[w]r}(n)=\\int dx\\ \\hat{\\Pi }(x)w^r_{n}(x).\\end{split}$ As a consequence of the orthonormality relations in Eq.",
"REF , the fields obey the following equal time commutation relations (assuming here that $0\\le r,s$ ): $\\begin{split}\\ [\\hat{\\Phi }^{[s]r}(m),\\hat{\\Pi }^{[s]r}(m^{\\prime })]&= i \\delta _{m,m^{\\prime }}\\\\\\ [\\hat{\\Phi }^{[\\mathbf {\\mathrm {w}}]r}(m),\\hat{\\Pi }^{[\\mathbf {\\mathrm {w}}]r^{\\prime }}(m^{\\prime })]&=i\\delta _{r,r^{\\prime }}\\delta _{m,m^{\\prime }}.\\\\\\end{split}$ The boundary Hamiltonian on a system with a UV cutoff scale $n$ with $V=L2^n$ modes is [15], [16]: $\\begin{array}{lll}\\hat{H}_{\\rm bd}&=&\\frac{1}{2}\\big (\\sum _{m=0}^{L2^n-1} :\\hat{\\Pi }^{[\\mathbf {\\mathrm {s}}]n}(m)\\hat{\\Pi }^{[\\mathbf {\\mathrm {s}}]n}(m):\\\\&&+m_0^2\\sum _{m=0}^{L2^n-1} :\\hat{\\Phi }^{[\\mathbf {\\mathrm {s}}]n}(m)\\hat{\\Phi }^{[\\mathbf {\\mathrm {s}}]n}(m):\\\\&&+\\sum _{m,m^{\\prime }=0}^{L2^n-1}:\\hat{\\Phi }^{[\\mathbf {\\mathrm {s}}]n}(m)D^{[ss]0}_{m,m^{\\prime }}\\hat{\\Phi }^{[\\mathbf {\\mathrm {s}}]n}(m^{\\prime }):\\big ),\\end{array}$ where $:\\hat{O}:$ indicates normal ordering of the operator $\\hat{O}$ is taken.",
"The bulk Hamiltonian is a sum of three terms involving coupling of scale/scale DOFs, scale/wavelet DOFs, and wavelet/wavelet DOFs [15], [16]: $\\hat{H}_{\\rm bk}=\\hat{H}_{\\rm ss}+\\hat{H}_{\\rm ww}+\\hat{H}_{\\rm sw},$ where $\\begin{split}\\hat{H}_{\\rm ss}&=\\frac{1}{2}\\big (\\sum _{m=0}^{L-1} :\\hat{\\Pi }^{[\\mathbf {\\mathrm {s}}]0}(m)\\hat{\\Pi }^{[\\mathbf {\\mathrm {s}}]0}(m):\\\\&+m_0^2\\sum _{m=0}^{L-1} :\\hat{\\Phi }^{[\\mathbf {\\mathrm {s}}]0}(m)\\hat{\\Phi }^{[\\mathbf {\\mathrm {s}}]0}(m):\\\\&+\\sum _{m,m^{\\prime }=0}^{L-1}:\\hat{\\Phi }^{[\\mathbf {\\mathrm {s}}]0}(m)D^{[ss]-n}_{m,m^{\\prime }}\\hat{\\Phi }^{[\\mathbf {\\mathrm {s}}]0}(m^{\\prime }):\\big ),\\\\\\hat{H}_{\\rm ww}&=\\frac{1}{2}\\big (\\sum _{m=0}^{L2^r-1}\\sum _{r=0}^{n-1}:\\hat{\\Pi }^{[w]r}(m)\\hat{\\Pi }^{[w]r}(m):\\\\&+m_0^2\\sum _{m=0}^{L2^r-1}\\sum _{r=0}^{n-1} :\\hat{\\Phi }^{[w]r}(m)\\hat{\\Phi }^{[w]r}(m):\\\\&+\\sum _{m=0}^{L2^r-1}\\sum _{m^{\\prime }=0}^{L2^{r\\prime }-1}\\sum _{r,r^{\\prime }=0}^{n-1}:\\hat{\\Phi }^{[w]r}(m)D^{[ww]r-n,r^{\\prime }-n}_{m,m^{\\prime }}\\hat{\\Phi }^{[w]r^{\\prime }}(m^{\\prime }):\\big ),\\\\\\hat{H}_{\\rm sw}&=\\frac{1}{2}\\sum _{m^{\\prime }=0}^{L-1}\\sum _{m=0}^{L2^r-1}\\sum _{r=0}^{n-1}:\\hat{\\Phi }^{[w]r}(m)D^{[sw]r-n,{-n}}_{m,m^{\\prime }}\\hat{\\Phi }^{[\\mathbf {\\mathrm {s}}]0}(m^{\\prime }).\\end{split}$ The coupling coefficients are $\\begin{split}D^{[ss]k}_{m,m^{\\prime }}&=\\int dx\\ \\partial _x s^{k}_{m}(x)\\cdot \\partial _x s^{k}_{m^{\\prime }}(x),\\\\D^{[ww]r,r^{\\prime }}_{m,m^{\\prime }}&=\\int dx\\ \\partial _x w^r_{m}(x)\\cdot \\partial _x w^{r^{\\prime }}_{m^{\\prime }}(x),\\\\D^{[sw]r,k}_{m,m^{\\prime }}&=2 \\int dx\\ \\partial _x w^r_{m}(x)\\cdot \\partial _x s^{k}_{m^{\\prime }}(x).\\\\\\end{split}$ These can be computed systematically as described in Appendix REF .",
"For wavelet family $\\mathcal {K}$ , the smallest size admissible with periodic boundaries is $L=2(2\\mathcal {K}-1)$ in order to ensure the scale and wavelet functions at the coarsest scale are orthonormal.",
"As described in Appendix , the boundary Hamiltonian can be diagonalized as $\\hat{H}_{\\rm bd}=\\sum _{j=0}^{V-1} d_j \\left(\\hat{\\tilde{a}}^{\\dagger }_j\\hat{\\tilde{a}}_j+\\frac{1}{2}\\right),$ with normal mode operators $\\hat{\\tilde{a}}_j=\\frac{1}{\\sqrt{V}}\\sum _{m=0}^{V-1}e^{i 2 \\pi jm/V}\\hat{a}^{[s]n}(m)$ and eigenenergies $d_j=\\Big [(-D^{[ss]0}_{0,0} +m_0^2)+2\\sum _{m=0}^{2\\mathcal {K}-2}D^{[ss]0}_{0,m}\\cos (k_j m)\\Big ]^{1/2}.$ In the massless case there is a boundary zero mode for $\\hat{H}_{\\rm bd}$ defined by the mode operator: $\\hat{\\tilde{a}}^{[s]n}(0)$ .",
"The ground state is the unique state that satisfies $\\hat{\\tilde{a}}^{[s]n}(j)\\vert G \\rangle _{\\rm bd}=0, \\forall j$ .",
"The bulk Hamiltonian can similarly be diagonalized as $\\hat{H}_{\\rm bk}=\\sum _{j=0}^{V-1} d_j \\left(\\hat{\\tilde{b}}^{\\dagger }_j\\hat{\\tilde{b}}_j+\\frac{1}{2}\\right),$ where the eigenenergies are the same as for the boundary and the normal mode operators are sums of operators that act on scale and wavelet DOFs.",
"The bulk normal modes are $\\hat{\\tilde{b}}_j=\\sum _{k=0}^{V-1} [M_{\\rm bk}]_{j,k}\\hat{b}_k,$ where the vector of bulk annihilation operators is $\\begin{split}\\hat{\\mathbf {b}}&=(\\hat{a}^{[s]0}(0),\\ldots , \\hat{a}^{[s]0}(2L-1),\\hat{a}^{[w]0}(0),\\ldots , \\hat{a}^{[w]0}(2L-1),\\\\&~~~~~~\\hat{a}^{[w]1}(0),\\ldots ,\\hat{a}^{[w]1}(2\\times 2L-1),\\ldots \\hat{a}^{[w]n-1}(0),\\ldots ,\\\\&~~~~~~\\hat{a}^{[w]n-1}(2^{n-1}\\times 2L-1))^T,\\nonumber \\end{split}$ and $M_{\\rm bk}$ is an orthogonal wavelet transform matrix defined in Sec.",
".",
"In the massless phase, the zero mode in the bulk is a wavelet transformation of the zero mode on the boundary, which in turn is a uniform superposition of localized modes.",
"The wavelet transform of any uniform vector has support only on the coarsest scale DOFs, and using that fact together with translational invariance of the zero mode, we have the expression for the bulk zero mode: $\\hat{\\tilde{b}}_0=\\frac{1}{\\sqrt{2L}} \\sum _{m=0}^{2L-1}\\hat{a}^{[s]0}(m)$ ."
],
[
"Bulk/boundary correspondence in the ground state",
"We want to compare the properties of the ground state $\\vert G \\rangle _{\\rm bk}$ of $\\hat{H}_{\\rm bk}$ to the ground state $\\vert G \\rangle _{\\rm bd}$ of $\\hat{H}_{\\rm bd}$ .",
"The ground state $\\rho $ of a quadratic bosonic Hamiltonian on $V$ modes is completely described by the covariance matrix: $\\Gamma _{j,k} = \\Re [ \\mathop {\\mathrm {tr}}[\\rho (\\hat{\\mathbf {\\mathrm {r}}}_j-\\langle \\hat{\\mathbf {\\mathrm {r}}}_j\\rangle ) (\\hat{\\mathbf {\\mathrm {r}}}_k-\\langle \\hat{\\mathbf {\\mathrm {r}}}_k\\rangle )]].$ Here $\\langle \\hat{\\mathbf {\\mathrm {r}}}_j\\rangle $ is the expectation value of $j$ -th component of a $2V$ dimensional vector of field operators and their conjugate momenta.",
"For the scalar bosonic theory, the means are zero so $\\Gamma _{j,k} = \\Re [ \\mathop {\\mathrm {tr}}[\\rho \\hat{\\mathbf {\\mathrm {r}}}_j \\hat{\\mathbf {\\mathrm {r}}}_k]].$ For the boundary Hamiltonian $\\hat{H}_{\\rm bd}$ , the ground state covariance matrix is $\\Gamma _{\\rm bd}=\\frac{1}{2} \\left(\\begin{array}{cc}K^{-1/2}_{\\rm bd} & 0 \\\\0 & K_{\\rm bd}^{1/2}\\end{array}\\right),$ expressed in the basis given by components of the vector $\\hat{\\mathbf {r}}_{\\rm bd}=(\\hat{\\Phi }^{[s]n}(0),\\ldots , \\hat{\\Phi }^{[s]n}(V-1),\\hat{\\Pi }^{[s]n}(0),\\ldots , \\hat{\\Pi }^{[s]n}(V-1))^T.$ Here the boundary couplings are $[K_{\\rm bd}]_{a,b}=(m_0^2-D^{[ss]0}_{0,0})\\delta _{a,b}+D^{[ss]0}_{0,a\\oplus _V(-b)}+D^{[ss]0}_{0,b\\oplus _V(-a)}.$ Similarly, for the bulk Hamiltonian, $\\hat{H}_{\\rm bk}$ , the ground state covariance matrix is [15] $\\Gamma _{\\rm bk}=\\frac{1}{2} \\left(\\begin{array}{cc}K^{-1/2}_{\\rm bk} & 0 \\\\0 & K_{\\rm bk}^{1/2}\\end{array}\\right),$ expressed in the basis given by components of the vector $\\begin{array}{lll}\\hat{\\mathbf {r}}_{\\rm bk} &=& (\\hat{\\Phi }^{[s]0}(0),\\ldots , \\hat{\\Phi }^{[s]0}(L-1),\\hat{\\Phi }^{[w]0}(0)\\ldots , \\hat{\\Phi }^{[w]0}(L-1),\\\\&&\\ldots , \\hat{\\Phi }^{[w]n-1}(0),\\ldots , \\hat{\\Phi }^{[w]n-1}(2^{n-1}L-1),\\\\&&(\\hat{\\Pi }^{[s]0}(0),\\ldots , \\hat{\\Pi }^{[s]0}(L-1),\\hat{\\Pi }^{[w]0}(0)\\ldots , \\hat{\\Pi }^{[w]0}(L-1),\\\\&&\\ldots , \\hat{\\Pi }^{[w]n-1}(0),\\ldots , \\hat{\\Pi }^{[w]n-1}(2^{n-1}L-1))^T.\\end{array}$ The bulk couplings are: $K_{\\rm bk}=\\left[\\begin{array}{cccc}K_{\\rm ss} & K_{\\rm sw}(0) & \\cdots & K_{\\rm sw}({n-1}) \\\\\\ K^T_{\\rm sw}(0) & K_{\\rm ww}(0,0) & \\ldots & K_{\\rm ww}(0,{n-1}) \\\\\\quad \\vdots & & \\ddots & \\\\\\ K^T_{\\rm sw}({n-1}) & \\dots & \\cdots & K_{\\rm ww}({n-1},{n-1})\\end{array}\\right].$ The scale-scale mode couplings are encoded in $K_{\\rm ss}$ , the scale-wavelet couplings in $K_{\\rm sw}$ and the wavelet-wavelet couplings in $K_{\\rm ww}$ .",
"These matrices are: $\\begin{split}&[K_{\\rm ss}]_{a,b}=(m_0^2-D^{[ss]-n}_{0,0})\\delta _{a,b}+D^{[ss]-n}_{0,(a-b)\\bmod {L}}+D^{[ss]-n}_{0,(b-a)\\bmod {L}}\\\\&\\quad \\quad (0\\le a,b<L),\\\\&[K_{\\rm sw}(l)]_{a,b}=D^{[sw]-n,l-n}_{a,b}\\quad \\quad (0\\le a<L, 0\\le b < L2^l, 0\\le l<n),\\\\&[K_{\\rm ww}(l,j)]_{a,b}=m_0^2\\delta _{a,b}\\delta _{j,l}+D^{[ww]l-n,j-n}_{a,b}\\\\&\\quad \\quad (0\\le a<L2^l, 0\\le b < L2^j, 0\\le j\\le l<n).\\\\\\end{split}$ Note that any local operator on the bulk can be written as a superposition of field operators on the boundary: $\\hat{A}^{[\\mathbf {\\mathrm {s}}]n}(m)=\\sum _j c_{n,j,m} \\hat{A}^{[s]0}(j)+\\sum _{j=0}^{L2^r-1} \\sum _{r=0}^{n-1} d_{n,r,j,m} \\hat{A}^{[w]r}(j),$ where for $0\\le r<n$ , $\\begin{array}{lll}c_{n,j,m}&=&\\int dx s^0_j(x)s^n_m(x),\\\\d_{n,r,j,m}&=&\\int dx w^r_j(x)s^n_m(x).\\end{array}$ Then correlation functions on the boundary can be expressed in terms of $O(n)$ correlations in the bulk: $\\begin{array}{lll}&_{\\rm bd}\\langle G \\vert \\hat{A}^{[\\mathbf {\\mathrm {s}}]n}(m)\\hat{B}^{[\\mathbf {\\mathrm {s}}]n}(m^{\\prime })\\vert G \\rangle _{\\rm bd}&\\\\&=\\sum _{j,j^{\\prime }=0}^{L-1}c_{n,j,m}c_{n,j^{\\prime },m^{\\prime }} {_{\\rm bk}}\\langle G \\vert \\hat{A}^{[\\mathbf {\\mathrm {s}}]0}(m)\\hat{B}^{[\\mathbf {\\mathrm {s}}]0}(m^{\\prime })\\vert G \\rangle _{\\rm bk}&\\\\&+\\sum _{j=0}^{L-1}\\sum _{j^{\\prime }=0}^{L2^{r\\prime }-1}\\sum _{r^{\\prime }=0}^{n-1}c_{n,j,m}d_{n,r^{\\prime },j^{\\prime },m^{\\prime }} {_{\\rm bk}}\\langle G \\vert \\hat{A}^{[\\mathbf {\\mathrm {s}}]0}(j)\\hat{B}^{[w]r^{\\prime }}(j^{\\prime })\\vert G \\rangle _{\\rm bk}&\\\\&+\\sum _{j=0}^{L2^r-1}\\sum _{j^{\\prime }=0}^{L-1}\\sum _{r=0}^{n-1}d_{n,r,j,m}c_{n,j^{\\prime },m^{\\prime }} {_{\\rm bk}}\\langle G \\vert \\hat{A}^{[w]r}(j)\\hat{B}^{[\\mathbf {\\mathrm {s}}]0}(j^{\\prime })\\vert G \\rangle _{\\rm bk}&\\\\&+\\sum _{j=0}^{L2^r-1}\\sum _{j^{\\prime }=0}^{L2^{r\\prime }-1}\\sum _{r,r^{\\prime }=0}^{n-1}d_{n,r,j,m}d_{n,r^{\\prime },j^{\\prime },m^{\\prime }} \\\\&{_{\\rm bk}}\\langle G \\vert \\hat{A}^{[w]r}(j)\\hat{B}^{[w]r^{\\prime }}(j^{\\prime })\\vert G \\rangle _{\\rm bk}.&\\\\\\end{array}$ Figure: Same scale bulk correlations as a function of position separation for a massless field theory.",
"Here the bulk scale is r=8r=8 and the boundary scale is n=15n=15.",
"Bulk field-field (A ^=Φ ^)\\hat{A}=\\hat{\\Phi }) and momentum-momentum (A ^=Π ^\\hat{A}=\\hat{\\Pi }) correlations for the 𝒦=3,5\\mathcal {K}=3,5 wavelet families are calculated using the ground state of the boundary Hamiltonian (Eq. )",
"(dots).",
"Solid lines are the derived relations for the bulk correlations from Eqs.",
", .We also want to compute the scaling of two point correlations in the bulk.",
"This can be done by calculating correlations on the wavelet DOFs only: $C^{\\hat{A},\\hat{B}}((r,j),(r^{\\prime },j^{\\prime }))={_{\\rm bk}}\\langle G \\vert \\hat{A}^{[w]r}(j)\\hat{B}^{[w]r^{\\prime }}(j^{\\prime })\\vert G \\rangle _{\\rm bk},$ where $\\hat{A}$ and $\\hat{B}$ are local wavelet mode operators.",
"The bulk correlations can be calculated as follows.",
"For $0\\le r<n$ , define $f_{n,r,j,m}=\\int dx s^n_m(x)w^r_j(x);\\\\$ then $\\begin{array}{lll}C^{\\hat{A},\\hat{B}}((r,j),(r^{\\prime },j^{\\prime }))&=&\\displaystyle {\\sum _{m,m^{\\prime }=0}^{L2^{r}-1}} f_{n,r,j,m}f_{n,r^{\\prime },j^{\\prime },m^{\\prime }}\\\\&& {_{\\rm bd}\\langle G \\vert }\\hat{A}^{[\\mathbf {\\mathrm {s}}]n}(m)\\hat{B}^{[\\mathbf {\\mathrm {s}}]n}(m^{\\prime })\\vert G \\rangle _{\\rm bd}.\\end{array}$ For $n\\gg r$ , the overlap integral effectively samples the wavelet at the location of the scale field, i.e.",
"$f_{n,r,j,m}\\approx \\int dx w^r_j(\\bar{x}) s^n_m(x)= 2^{-n/2}w^r_j(\\bar{x})$ , where $\\bar{x}=2^{-n}m$ is the approximate location of the peak of the scale field $s^n_m$ .",
"Hence, for $n\\gg r$ , the overlap coefficients can be expressed as follows: $f_{n,r,j,m}\\approx \\left\\lbrace \\begin{array}{cc}w_0^{r-n}(m-j2^{n-r}+1) & m\\in 2^{n-r}[j,j+2\\mathcal {K}-1] \\bigcap \\mathbb {Z}\\\\0 & {\\rm otherwise}.\\end{array}\\right.$ The correlator can be written for $0\\le r,r^{\\prime }<n$ as: $\\begin{array}{lll}C^{\\hat{A},\\hat{B}}((r,j),(r^{\\prime },j^{\\prime }))&=&2^{-n}2^{(r+r^{\\prime })/2}\\displaystyle {\\sum _{m=j\\times 2^{n-r}}^{(j+2\\mathcal {K}-1)\\times 2^{n-r}}}\\sum _{m^{\\prime }=j^{\\prime }\\times 2^{n-r^{\\prime }}}^{(j^{\\prime }+2\\mathcal {K}-1)\\times 2^{n-r^{\\prime }}}\\\\&&w_0^{0}((m-j\\times 2^{n-r}+1)\\times 2^{r-n}) \\\\&&w_0^{0}((m^{\\prime }-j^{\\prime }\\times 2^{n-r^{\\prime }}+1)\\times 2^{r^{\\prime }-n})\\\\&&{_{\\rm bd}\\langle G \\vert }\\hat{A}^{[\\mathbf {\\mathrm {s}}]n}(m)\\hat{B}^{[\\mathbf {\\mathrm {s}}]n}(m^{\\prime })\\vert G \\rangle _{\\rm bd}.\\\\\\end{array}$ Here we have used the property: $w_0^{r}(x)=2^{r/2}w_0^{0}(x\\times 2^{r})$ .",
"To find the boundary correlations we need to find the ground state of the boundary Hamiltonian $\\hat{H}_{\\rm bd}$ (see Appendix ).",
"When the number of modes is very large, the correlations in the ground state approach the continuum values: Eqs.",
"REF ,REF ."
],
[
"Massless case",
"The bulk correlations can be computed from the boundary correlations.",
"As derived in Appendix REF , the same scale bulk correlations for separation $j>2\\mathcal {K}-1$ and deep in the bulk are $C^{\\hat{\\Phi },\\hat{\\Phi }}((r,0),(r,j))\\approx -\\frac{2^{n-r} \\times D_{\\mathcal {K}}}{4\\pi \\mathcal {K}j^{2\\mathcal {K}}},$ where $D_{\\mathcal {K}}=\\langle x^{\\mathcal {K}}\\rangle _w^2{2\\mathcal {K} \\atopwithdelims ()\\mathcal {K}},$ and $\\langle x^{\\mathcal {K}}\\rangle _w$ is the $\\mathcal {K}$ th moment of the wavelet $w_0^0(x)$ .",
"Similarly, the momentum-momentum correlations are $C^{\\hat{\\Pi },\\hat{\\Pi }}((r,0),(r,j))\\approx \\frac{2^{r-n}\\times (2\\mathcal {K}+1)D_{\\mathcal {K}}}{2\\pi j^{2\\mathcal {K}+2}},$ For example, using the wavelet moments calculated in Appendix REF : $D_3=\\frac{225}{128}, D_4=21.53, D_5=446.04$ .",
"This fits the numerically calculated correlations in Fig.",
"REF quite well.",
"Additionally we find for self correlations $C^{\\hat{\\Phi },\\hat{\\Phi }}((r,0),(r,0))= 2^{n-r-a},$ and $C^{\\hat{\\Pi },\\hat{\\Pi }}((r,0),(r,0))= 2^{r-n+b}.$ The values $a,b$ appearing in the self correlations can be calculated from the boundary correlations (see Appendix REF ).",
"For $\\mathcal {K}\\ge 3$ , $a\\approx 3.18$ independent of $\\mathcal {K}$ , whereas $b\\approx 0.75/\\mathcal {K}^2+0.16/ \\mathcal {K}+1.24$ .",
"The mutual information between two bulk DOFs is given by $I((r,j),(r^{\\prime },j^{\\prime }))=S(\\rho _{(r,j)})+S(\\rho _{(r^{\\prime },j^{\\prime })})-S(\\rho _{(r,j),(r^{\\prime },j^{\\prime })}),$ where the von Neumann entropy $S(\\rho _A)$ of a subsystem $A$ is calculated as follows.",
"First obtain the reduced covariance matrix $\\Gamma _A$ by deleting the columns and rows of modes not contained in $A$ from the full covariance matrix $\\Gamma $ .",
"Next compute the symplectic spectrum, which are the eigenvalues of the matrix $i\\Gamma _A\\Omega $ where $\\Omega $ is the symplectic form $\\Omega =\\left(\\begin{array}{cc}0 & {\\bf 1}_{n_A} \\\\-{\\bf 1}_{n_A} & 0\\end{array}\\right),$ written the basis $\\lbrace \\hat{\\Phi }_1,\\ldots , \\hat{\\Phi }_{n_A},\\hat{\\Pi }_1,\\ldots , \\hat{\\Pi }_{n_A}\\rbrace $ of the $n_A$ modes of system $A$ .",
"These eigenvalues come in positive and negative pairs $\\lbrace \\pm \\sigma _i\\rbrace $ .",
"Taking the positive values, the entropy in bits is $S(\\rho _A)=\\sum _{\\lbrace \\sigma _i\\rbrace }[(\\sigma _i+\\frac{1}{2})\\log _2 (\\sigma _i+\\frac{1}{2})-(\\sigma _i-\\frac{1}{2})\\log _2 (\\sigma _i-\\frac{1}{2})].$ Figure: Cross scale bulk mutual information as a function of scale separation for a massless field theory.",
"Here r ' <r=14r^{\\prime }<r=14 and the boundary scale is n=15n=15.",
"Mutual information is computed from boundary correlations using Eq.",
"(dots), and the function I 0 e -d g ((r,0),(r ' ,0))/ξ r I_0e^{-d_g((r,0),(r^{\\prime },0))/\\xi _r} is plotted (solid line) with the fit ξ r =1.79ξ θ \\xi _r=1.79 \\xi _{\\theta }.From the derived values for the same scale correlations, we find the positive symplectic eigenvalues for the subsystem pair of sites $A=\\lbrace (r,0),(r,j)\\rbrace $ $\\sigma _{\\pm }=\\frac{2^{-a/2}\\sqrt{j^{2\\mathcal {K}}\\pm 2^a \\frac{D_{\\mathcal {K}}}{4\\pi \\mathcal {K} }}\\sqrt{2^b j^{2\\mathcal {K}+2}\\mp \\frac{(2\\mathcal {K}+1)D_{\\mathcal {K}}}{2\\pi }}}{j^{2\\mathcal {K}+1}}.$ while for the single mode subsystem $A=\\lbrace (r,0)\\rbrace $ , the positive symplectic eigenvalue is $\\sigma =2^{(b-a)/2}.$ To leading order in $1/j$ , for $j>2\\mathcal {K}-1$ , the mutual information between bulk sites at the same scale $r$ is $I((r,0),(r,j))=\\frac{\\left(\\frac{D_{\\mathcal {K}}}{4\\pi \\mathcal {K}}\\right)^2 F(\\mathcal {K})}{j^{4\\mathcal {K}}},$ where $\\begin{array}{lll}F(\\mathcal {K})&=&2^{2a-2}S_0+\\frac{(2^{2-2a}-2^{-a-b})^{-1}}{\\log (2)}-\\frac{2^{2a-3}\\log (2^{b-a}-\\frac{1}{4})}{\\log 2}.\\\\\\end{array}$ the $\\mathcal {K}$ dependence arising since $b$ is a function of $\\mathcal {K}$ .",
"Here the single site entropy $S_0$ is $\\begin{array}{lll}S_0\\equiv S(\\rho _{(r,j)})&=&(2^{(b-a)/2}+\\frac{1}{2})\\log _2(2^{(b-a)/2}+\\frac{1}{2})\\\\&&-(2^{(b-a)/2}-\\frac{1}{2})\\log _2(2^{(b-a)/2}-\\frac{1}{2}).\\end{array}$ By translational invariance, deep in the bulk, $S_0$ is the same at any position."
],
[
"Computing the radius of curvature",
"Following Ref.",
"[6], we make the ansatz that the mutual information falls off exponentially with the distance between two bulk modes in the AdS$_3$ metric, i.e.",
"$I((r,0),(r,j))=S_0 e^{-d_g((r,0),(r,j)/\\xi _{\\theta }}$ with geodesic distance $ d_g((r,0),(r,j))= 2R\\ln \\left[\\frac{j}{R}\\right]$ , and $\\xi _{\\theta }$ a correlation length.",
"The multiplicative constant $S_0$ is used since for zero separation, the mutual information is just the single site entropy.",
"Taking logarithms of the mutual information, $\\ln S_0 -\\frac{2R\\ln j}{\\xi _{\\theta }}+\\frac{2R\\ln R}{\\xi _{\\theta }}=\\ln \\left[\\left(\\frac{D_{\\mathcal {K}}}{4\\pi \\mathcal {K}}\\right)^2F(\\mathcal {K})\\right]-4\\mathcal {K}\\ln j.$ Hence the correlation length is $\\xi _{\\theta }=\\frac{R}{2\\mathcal {K}},$ and the the radius of curvature is $R=\\left( \\frac{D_{\\mathcal {K}}}{4\\pi \\mathcal {K}}\\right)^{1/2\\mathcal {K}}\\times \\left(\\frac{F(\\mathcal {K})}{S_0} \\right)^{1/4\\mathcal {K}}.$ Calculating the wavelet moments, for $\\mathcal {K}\\ge 3$ we find $\\left(\\frac{D_{\\mathcal {K}}}{4\\pi \\mathcal {K}}\\right)^{1/2\\mathcal {K}}\\approx 0.319\\mathcal {K}-0.357.$ Figure: Mutual information between same scale bulk DOFs as a function of position separation for a massless field theory.",
"Here the bulk scale is r=8r=8 and the boundary scale is n=15n=15.",
"The dots are computed values from the boundary correlations for the 𝒦=3,4\\mathcal {K}=3,4 wavelet families and solid lines are the prediction assuming exponential decay in geodesic distance I=S 0 e -d g ((r,0),(r,j))/ξ θ I=S_0 e^{-d_g((r,0),(r,j))/\\xi _{\\theta }} with the radius of curvature RR from Eq.",
"and correlation length ξ θ \\xi _{\\theta } from Eq. .",
"For 𝒦=3\\mathcal {K}=3: R=1.10R=1.10, ξ θ =0.18\\xi _{\\theta }=0.18, S 0 =0.22S_0=0.22.",
"For 𝒦=4\\mathcal {K}=4: R=1.49R=1.49, ξ θ =0.19\\xi _{\\theta }=0.19, S 0 =0.18S_0=0.18.As described in Sec.",
"REF , the values appearing in the self correlators are $a= 3.18$ and $b= 0.75/\\mathcal {K}^2+0.16/\\mathcal {K}+1.24$ , yielding $\\left(\\frac{F(\\mathcal {K})}{S_0}\\right)^{1/4\\mathcal {K}}\\approx \\frac{2.41}{\\mathcal {K}}+1 .$ Hence for $\\mathcal {K}\\ge 3$ the radius of curvature is approximately $R\\approx (0.32\\mathcal {K}-0.88/\\mathcal {K}+0.43).$ In Figs.",
"REF ,REF we plot mutual information for the wavelet families $\\mathcal {K}=3,4$ which shows a good match with the above calculated value for the radius of curvature.",
"The above scaling of $R$ indicates that the geometry becomes flatter with larger $\\mathcal {K}$ while the same scale correlation length approaches a constant $\\xi _{\\theta }\\rightarrow 0.16$ .",
"This is consistent with the feature that correlations fall off faster in the bulk when using wavelet transformations with larger $\\mathcal {K}$ , since for each stage of renormalization of the boundary state, more short range entanglement is removed.",
"The wavelet packet transform for $\\mathcal {K}$ wavelets can be implemented using a circuit of nearest neighbour mode couplings repeated $\\mathcal {K}$ times [17].",
"This can explains why the radius of curvature is linear in $\\mathcal {K}$ since one may view each renormalization step for large $\\mathcal {K}$ wavelets as a linear in $\\mathcal {K}$ sequence of number of nearest neighbour circuits implementing $\\mathcal {K}=2$ (e.g.",
"Haar) wavelet transformations."
],
[
"Temporal Correlations",
"To probe correlations in the time direction of the emergent bulk geometry we consider correlators of the form $\\begin{split}&C^{\\hat{A},\\hat{B}}((r,j,\\tau ),(r^{\\prime },j^{\\prime },\\tau ^{\\prime }))= {_{\\rm bk}\\langle G \\vert }T[\\hat{A}^{[w]r}(m,\\tau )\\hat{B}^{[w]r^{\\prime }}(m^{\\prime },\\tau ^{\\prime })]\\vert G \\rangle _{\\rm bd}\\\\&=\\displaystyle {\\sum _{m,m^{\\prime }=0}^{L2^{r}-1}} f_{n,r,j,m}f_{n,r^{\\prime },j^{\\prime },m^{\\prime }}{_{\\rm bd}\\langle G \\vert }T[\\hat{A}^{[\\mathbf {\\mathrm {s}}]n}(m,\\tau )\\hat{B}^{[\\mathbf {\\mathrm {s}}]n}(m^{\\prime },\\tau ^{\\prime })]\\vert G \\rangle _{\\rm bd},\\end{split}$ where $T$ is the time ordering operator.",
"For simplicity we work in imaginary time, $\\tau \\rightarrow i\\tau $ , and consider the Green's function at the same point in space.",
"For $\\tau >2^{n-r}(2\\mathcal {K}-1)$ , we find (see Appendix REF ), $C^{\\hat{a},\\hat{a}^{\\dagger }}((r,j,\\tau ),(r,j,0))\\approx \\frac{2^{(n-r)(2\\mathcal {K}+1)}0.32\\times D_{\\mathcal {K}}}{\\tau ^{(2\\mathcal {K}+1)}}.$ As before, we make the ansatz $C^{\\hat{a},\\hat{a}^{\\dagger }}((r,j,\\tau ),(r,j,0))=C_0e^{-d_g((r,j,\\tau ),(r,j,0))/\\xi _{\\tau }}$ where $\\xi _{\\tau }$ is a temporal correlation length and the geodesic distance is approximately $d_g((r,j,\\tau ),(r,0,0))=2R\\ln \\left[\\frac{\\tau 2^{r-n}}{R}\\right]$ (see Appendix ) .",
"This approximation is valid when $2^n L \\gg \\tau \\gg 2^{n-r} R$ .",
"Taking logarithms we find $\\begin{array}{lll}\\ln (C_0)-\\frac{2R}{\\xi _{\\tau }}(\\ln \\tau -(n-r)\\ln 2 -\\ln R)&=&(n-r)(2\\mathcal {K}+1)\\ln 2 \\\\&+&\\ln G_{\\mathcal {K}}-(2\\mathcal {K}+1)\\ln \\tau .\\end{array}$ From the $\\tau $ dependent term we find the correlation length $\\xi _{\\tau }=\\frac{2R}{2\\mathcal {K}+1}.$ Note this is larger than same scale spatial correlation length Eq.",
"REF ."
],
[
"Central Charge",
"The bulk/boundary correspondence also allows for computing properties which depend on long range entanglement properties, such as the central charge $c$ , which for the bosonic CFT is $c=1$ .",
"The boundary central charge can be obtained from the purity of a subsystem $A$ on the boundary consisting of an interval of $\\ell $ modes [18]: $\\mathop {\\mathrm {tr}}[\\rho ^2_A]=C \\left[\\frac{V}{\\epsilon }\\sin \\left(\\frac{\\pi \\ell }{V}\\right)\\right]^{-c/4},$ where $V$ is the total number of modes on the boundary (with periodic boundary conditions), $C$ is a constant and $\\epsilon $ is an ultraviolet cutoff size.",
"For Gaussian states, the purity of a subsystem consisting of $\\ell $ modes is $\\mathop {\\mathrm {tr}}[\\rho ^2_A]=\\frac{1}{2^{\\ell } \\sqrt{{\\rm det}\\Gamma _A}},$ where $\\Gamma _A$ is the covariance matrix for modes in region $A$ .",
"This implies that for $\\ell \\ll V$ , $\\ln [2^{2\\ell }{\\rm det}\\Gamma _A]=\\frac{c}{2}\\ln (\\ell )+{\\rm const.",
"}.$ If we consider two regions, $A_1$ and $A_2$ of lengths $\\ell _1$ and $\\ell _2>\\ell _1$ , then $c=\\frac{2}{\\ln (\\ell _2/\\ell _1)}\\left(\\ln \\left[\\frac{{\\rm det}\\Gamma _{A_2}}{{\\rm det}\\Gamma _{A_1}}\\right]+2(\\ell _2-\\ell _1)\\ln 2\\right).$ Since the elements of the covariance matrix consist of boundary correlations this data can be obtained from bulk correlations using Eq.",
"REF .",
"Computing the central charge as per Eq.",
"REF for the massless boundary CFT at scale $n=7$ and $L=10$ , with a total number of modes $V=L\\times 2^n=1280$ , we find for region sizes $\\ell _2=2\\ell _1=6$ , that $c=0.997$ ."
],
[
"Massive Case",
"For $m_02^{n-r}\\gg 1$ , the same scale field/field correlation deep in the bulk and for separations $j\\gg 2\\mathcal {K}-1$ is (see Appendix REF ) $C^{\\hat{\\Phi },\\hat{\\Phi }}((r,0),(r,j))\\approx -\\frac{2^{n-r}e^{-j \\tilde{m}}}{\\sqrt{j 8\\pi \\tilde{m}}}\\langle e^{-\\tilde{m} x}\\rangle _w \\langle e^{\\tilde{m}x}\\rangle _w.\\\\$ Similarly, the momentum/momentum field correlator is $C^{\\hat{\\Pi },\\hat{\\Pi }}((r,0),(r,j))\\approx 2^{r-n}e^{-j \\tilde{m}}\\sqrt{\\frac{\\tilde{m}}{8\\pi j^3}}\\langle e^{-\\tilde{m} x}\\rangle _w \\langle e^{\\tilde{m} x}\\rangle _w.$ For $\\mathcal {K}=3$ wavelets and $\\tilde{m}\\gg 1$ the product of averages is $\\langle e^{-\\tilde{m} x}\\rangle _w \\langle e^{\\tilde{m} x}\\rangle _w\\approx e^{4.71 \\times \\tilde{m}-20.38}$ .",
"We see that the same scale bulk correlations fall off exponentially with a renormalized mass $\\tilde{m}=m_0 2^{n-r}.$ This is consistent with the view that the boundary DOFs are renormalized over $n-r$ dyadic steps to the wavelet DOFs at the scale $r$ ."
],
[
"Ground states",
"The boundary ground state is obtained by a unitary transformation on the $V$ mode vacuum state: $\\vert G \\rangle _{\\rm bd}=\\hat{U}_{\\rm bd}\\vert {\\rm vac} \\rangle ^{\\otimes V}.$ This mapping is described by a symplectic transformation on the initially decoupled position and momentum mode operators: $\\hat{\\mathbf {r}}_{\\rm bd}\\rightarrow Y_{\\rm bd}\\hat{\\mathbf {r}}_{\\rm bd}$ .",
"The initial vacuum correlation functions are described by a covariance matrix proportional to the identity and the symplectric transformation acts on the correlation matrix as $\\Gamma _{\\rm vac}=\\frac{1}{2}\\mathbf {1}_{2V}\\rightarrow \\Gamma _{\\rm bd}=\\frac{1}{2}Y_{\\rm bd}Y_{\\rm bd}^T=\\frac{1}{2} (K_{\\rm bd}^{-1/2}\\oplus K_{\\rm bd}^{1/2}).$ Similarly for the bulk ground state, the transformation is $\\vert G \\rangle _{\\rm bk}=\\hat{U}_{\\rm bk}\\vert {\\rm vac} \\rangle ^{\\otimes V},$ which corresponds to the symplectic transformation $\\hat{\\mathbf {r}}_{\\rm bk}\\rightarrow Y_{\\rm bk}\\hat{\\mathbf {r}}_{\\rm bk}$ , acting on the covariance matrix as $\\Gamma _{\\rm vac}=\\frac{1}{2}\\mathbf {1}_{2V}\\rightarrow \\Gamma _{\\rm bk}=\\frac{1}{2}Y_{\\rm bk}Y_{\\rm bk}^T=\\frac{1}{2} (K_{\\rm bk}^{-1/2}\\oplus K_{\\rm bk}^{1/2}).$ There is a canonical Bloch-Messiah decomposition for the unitaries $\\hat{U}_{\\rm bd}$ and $\\hat{U}_{\\rm bk}$ that can be written as one round of beam splitters and phase shifters, followed by parallel single mode squeezing, followed by a second round of beam splitters and phase shifters [19].",
"The decomposition is efficient, costing $O(V^2)$ elementary operations.",
"For the construction of the boundary state: $\\hat{U}_{\\rm bd}=\\hat{R}_{\\rm bd} \\hat{D}\\hat{R}^{\\dagger }_{\\rm bd},$ where the the single mode squeezing operations are $\\hat{D}=\\prod _{j=0}^{V-1} e^{\\alpha _j (\\hat{a}_j^{2}-\\hat{a}_j^{\\dagger 2})/2},$ with squeezing parameters $\\alpha _j=-\\frac{1}{4}\\log (d_j).$ Here the $d_j$ are eigenenergies given in Eq.",
"REF , and positive values of $\\alpha $ perform momentum squeezing: $\\hat{D} \\hat{q}_j \\hat{D}^{\\dagger }=e^{\\alpha _j}\\hat{q}_j$ , $\\hat{D} \\hat{p}_j \\hat{D}^{\\dagger }=e^{-\\alpha _j}\\hat{p}_j$ .",
"The unitary $\\hat{R}_{\\rm bd}$ performs the linear transformation: $\\hat{R}_{\\rm bd}\\hat{a}_j \\hat{R}^{\\dagger }_{\\rm bd}=\\sum _{k=0}^{V-1} [M_{\\rm bd}]_{j,k} \\hat{a}_k$ , where where $M_{\\rm bd}$ is an orthogonal matrix that diagonalizes $K_{\\rm bd}$ .",
"Specifically, we can take (assuming $V$ even) $[M_{\\rm bd}]_{j,k}=\\left\\lbrace \\begin{array}{c}\\frac{1}{\\sqrt{V}}\\quad j=0 \\\\ \\sqrt{\\frac{2}{V}}\\cos (\\frac{2\\pi j k}{V})\\quad 1\\le j\\le V/2-1 \\\\ \\frac{(-1)^k}{\\sqrt{V}}\\quad j=V/2 \\\\ \\sqrt{\\frac{2}{V}}\\sin (\\frac{2\\pi j k}{V})\\quad V/2< j\\le V-1 \\end{array}\\right..$ Similarly, for the bulk state construction: $\\hat{U}_{\\rm bk}=\\hat{R}_{\\rm bk} \\hat{D} \\hat{R}^{\\dagger }_{\\rm bk}.$ The unitary $\\hat{R}_{\\rm bk}$ performs the linear transformation: $\\hat{R}_{\\rm bk}\\hat{a}_j \\hat{R}^{\\dagger }_{\\rm bk}=\\sum _{k=0}^{V-1}[M_{\\rm bk}]_{j,k} \\hat{a}_k$ , where $M_{\\rm bk}$ is the orthogonal wavelet transformation that diagonalizes $K_{\\rm bk}$ .",
"These transformed operators define the bulk normal modes $\\hat{\\tilde{b}}_j$ introduced in Eq.",
"REF .",
"In the massless case, there is a free mode with energy $d_0=0$ which, since it is completely delocalized, requires an infinite amount of squeezing.",
"However, in the wavelet basis this mode is mapped to the IR fixed point at the coarsest scale DOFs and is decoupled from the wavelet DOFs.",
"Hence one can ignore this mode if only interested in bulk wavelet correlations.",
"The maximum amount of squeezing is then dictated by the lowest, positive definite energies $d_1=d_{V-1}$ .",
"Owing to the relation $\\sum _{m=0}^{2\\mathcal {K}-2} m^2 D^{[ss]0}_{0,m}=-1$ (see Appendix REF ), in the massless case, $d_1=\\frac{2\\pi }{V}$ and the maximum squeezing needed for a $V$ mode system is $\\alpha _{\\rm max}=\\frac{1}{4}\\log (V/2\\pi )$ .",
"Figure: (a) Orthogonal wavelet transformation matrix M bk M_{\\rm bk} used to construct the bulk state of a system consisting of V=160V=160 modes with L=10L=10 and boundary scale of n=3n=3.",
"The bulk DOFs at scales r=0,1,2,3r=0,1,2,3 are evident from the recursive structure of the matrix, with the coarsest scale DOFs in the upper left hand block.",
"If matrix elements of magnitude less than 10 -4 10^{-4} are set equal to zero then M bk M_{\\rm bk} is 87%87\\% sparse.",
"(b) Squeezing parameters α j \\alpha _j from Eq.",
"for constructing ground states in the massless case (blue) and massive m 0 =1m_0=1 case (yellow).",
"For this sized system, the maximum momentum squeezing on a mode in the massless case is 7.037.03 dB (using # dB =20αlog 10 (e)\\# {\\rm dB}=20 \\alpha \\log _{10}(e))."
],
[
"Thermal states",
"The same protocol can be used to prepare thermal bulk and boundary states.",
"A thermal state of the scalar bosonic QFT at temperature $1/\\beta $ is described by a Gaussian state both on the bulk and on the boundary.",
"The covariance matrices assume a simple form (see e.g.",
"[20]) on the boundary $\\Gamma _{\\rm bd}(\\beta )=\\frac{1}{2} \\left(\\begin{array}{cc}K^{-1/2}_{\\rm bd}\\coth (\\beta K^{1/2}_{\\rm bd}) & 0 \\\\0 & K^{1/2}_{\\rm bd}\\coth (\\beta K^{1/2}_{\\rm bd})\\end{array}\\right),$ and on the bulk $\\Gamma _{\\rm bk}(\\beta )=\\frac{1}{2} \\left(\\begin{array}{cc}K^{-1/2}_{\\rm bk}\\coth (\\beta K^{1/2}_{\\rm bk}) & 0 \\\\0 & K^{1/2}_{\\rm bk}\\coth (\\beta K^{1/2}_{\\rm bk})\\end{array}\\right).$ In the limit $\\beta \\rightarrow \\infty $ these approach the ground state covariance matrices.",
"The method to construct thermal states is very similar to that for ground states.",
"The only difference is that rather than beginning in the vacuum state one should start in the separable thermal state with correlation matrix $\\Gamma (\\beta )=\\frac{1}{2}[\\oplus _{j=0}^{V-1} \\coth (\\beta d_j)]\\oplus [\\oplus _{j=0}^{V-1} \\coth (\\beta d_j)].$ Such a state can be prepared by initializing the system in a product state such that each mode $j$ is prepared in a locally thermal state with temperature $T_j=(\\beta d_j)^{-1}$ ."
],
[
"Implemetations",
"The multimode Gaussian entangled states that are needed to demonstrate the bulk/boundary correspondence considered here can be prepared in a variety of engineered quantum systems.",
"In continuous variable optical networks, large-scale Gaussian states encoded in either frequency modes [21], [22] or temporal modes [23] have been demonstrated, with the latter experiment realizing a 10000 mode Gaussian cluster state.",
"Squeezing levels of $\\sim 5$ dB are achievable with current technology [23], and proof-of-principle experiments show squeezing of up to $\\sim $ 10 dB [24][23].",
"Another available technology is circuit-QED setups using coupled arrays of microwave cavities [25].",
"Single-mode [26], [27] squeezing has already been demonstrated in these systems, and the SQUID-based controlling technology allows for very strong nonlinearities enabling high squeezing ($\\sim $ 13 dB) [28].",
"A third candidate technology is cold trapped ions.",
"In Ref.",
"[29] it was shown how to perform universal bosonic simulators using an array of trapped ions where each ion is trapped in its own local potential.",
"Single model squeezing and phase shifting operations can be done by dynamically changing local trapping potentials and beam splitter operators are enabled by making use of the Coulomb repulsion between neighboring ions brought together using time dependent pairwise potentials.",
"Additionally, thermal motional state engineering of a single ion trapped in a harmonic well has been achieved by changing the detuning of the cooling laser [30] which would allow for preparation of thermal state QFT simulators.",
"Large sized simulations can run into difficulties addressing individual modes with local gate operations.",
"Fortunately, as shown in Ref.",
"[31], the entire simulation described here consisting of state preparation, evolution by the linear optical unitaries, and measurement of quadrature moments on each mode can be done without needing mode addressability and using translationally invariant single and nearest neighbour pairwise interactions between modes arranged on a line with open boundaries.",
"The overhead for using translationally invariant operations is only linear in the number of modes.",
"Figure: Schematic of the method to construct the bulk or boundary state starting from an initial state which is a product state of quantum harmonic oscillator modes perpared in local thermal or ground (vacuum) states.",
"The unitary operator R ^ bk \\hat{R}_{\\rm bk} or R ^ bd \\hat{R}_{\\rm bd} performs a sequence of passive linear transformations on mode operators for the bulk or boundary state construction.",
"The unitary D ^\\hat{D} is the same for both state constructions and involves a set of parallel single mode squeezing operations.",
"If preparing bulk or boundary ground states, then the initial state is chosen to be the vacuum and the first round of linear operators R † R^{\\dagger } is not needed."
],
[
"Conclusions",
"In summary, wavelets are a natural basis in which to study holographic duality in QFT.",
"By way of the exact holographic mapping we have shown that ground states of $1+1$ dimensional scalar bosonic QFT can be represented in a wavelet basis where each DOF carries space-time and scale coördinates.",
"Correlations between the wavelet fields fall off exponentially with geodesic length where the metric is AdS$_3$ in the case of a massless boundary CFT and flat in the massive case.",
"The boundary and bulk states arise as ground states of two body boundary or bulk Hamiltonians but they can also be constructed via a quantum circuit acting on a lattice of bosonic modes.",
"The gate operations are Gaussian and the necessary amount of single mode squeezing grows logarithmically with system size meaning a proof of principle quantum simulation could be realized with engineered bosonic lattices.",
"While properties of the bulk corresponding to boundary thermal states was not studied in this work such states are just as easily constructed by a quantum circuit.",
"There are several areas worthy of further investigation.",
"First, we have only analysed the EHM for a single CFT with central charge $c=1$ , which is the same central charge for the free Dirac fermion studied in [6], [7].",
"In the pioneering work of Brown and Henneaux it was shown that gravitational theories in AdS$_3$ space of radius $R$ are dual to $1+1$ dimensional CFTs with central charge $c=3R/2G^{(3)}_N$ where $G^{(3)}_N$ is the Newton's gravitational constant in three spacetime dimensions.",
"While we make no claim that the bulk description described here is a theory of gravity in $2+1$ dimensions, it would be worthwhile checking if there is a dependence of radius of curvature on central charge of the boundary CFT.",
"Second, can the EHM provide useful insight for boundary gauge theories or interacting QFTs?",
"It is possible to incorporate gauge fields with a wavelet decomposition as described in Refs.",
"[16], [32] despite the fact that the basis functions are not strictly local and in Ref.",
"[15] it was shown how interacting scalar bosonic QFT can be encoded in a wavelet basis and efficiently prepared and evolved on a quantum computer.",
"Since the wavelet basis describes the state at multiple scales, this could be a useful means to infer properties of renormalization flow.",
"Finally, can the wavelet description provide a better basis for tensor network descriptions of many body states or field theories?",
"Recently, the wavelet transformation was used to construct the first analytic MERA [12] for a critical system, meaning components of the constituent tensors were analytically derived from properties of the boundary theory (and not obtained e.g.",
"from a numerical optimization).",
"On the other hand, it is likely that a wavelet basis independent description of the bulk geometry could be derived by viewing any wavelet family as constructed from a $\\emph {universal}$ nearest neighbor circuit like the binary MERA circuit.",
"Indeed, entirely new classes of wavelets can be constructed using MERA like quantum circuits [17].",
"Further investigations may reveal other interesting connections between the wavelet and tensor network descriptions (e.g., see Ref.",
"cMERA) of quantum field theories.",
"We thank Barry Sanders and Glen Evenbly for helpful comments and feedback, and acknowledge many beneficial discussions with Dean Southwood.",
"We acknowledge support from the ARC via the Centre of Excellence in Engineered Quantum Systems (EQuS), project number CE110001013 and from DP160102426."
],
[
"Boundary correlations",
"In this section we obtain the correlations for the scalar bosonic QFT at scale $n$ acting on $V=L\\times 2^n$ modes.",
"The $V\\times V$ matrix encoding the couplings in the boundary Hamiltonian can be written $K_{\\rm bd}=(-D^{[ss]0}_{0,0} +m_0^2) {\\bf 1}_V+\\sum _{m=0}^{2\\mathcal {K}-2}D^{[ss]0}_{0,m} (X_V^m+X_V^{\\dagger m}),$ where $X_V=\\sum _{j=0}^{V-1} \\vert j\\oplus _V 1 \\rangle \\langle j \\vert $ is the unitary increment operator.",
"The matrix can be diagonalized by a Fourier transform via the operator $F_V=\\frac{1}{\\sqrt{V}}\\sum _{j,k=0}^{V-1} e^{i2\\pi jk/V}\\vert j \\rangle \\langle k \\vert $ : $K_{\\rm bd} =F_V^{\\dagger } [\\oplus _{j=0}^{V-1}d^2_j] F_V,$ where the eigenvalues are the squares of $d_j=\\Big [(-D^{[ss]0}_{0,0} +m_0^2)+2\\sum _{m=0}^{2\\mathcal {K}-2}D^{[ss]0}_{0,m}\\cos (k_j m)\\Big ]^{1/2},$ where we defined $k_j=\\frac{2\\pi j}{V}$ for $j\\in \\mathbb {Z}$ (see Fig.",
"REF ).",
"Note in the massless case, the matrix is singular since $d_0=-D^{[ss]0}_{0,0}+2\\sum _m D^{[ss]0}_{0,m}=0$ .",
"The boundary Hamiltonian can then be written in terms of normal modes $\\hat{H}_{\\rm bd}=\\sum _{j=0}^{V-1} d_j \\left(\\hat{\\tilde{a}}^{[s]n\\dagger }(j)\\hat{\\tilde{a}}^{[s]n}(j)+\\frac{1}{2}\\right),$ with normal mode operators $\\hat{\\tilde{a}}^{[s]n}(j)=\\frac{1}{\\sqrt{V}}\\sum _{m=0}^{V-1}e^{i 2 \\pi jm/V}\\hat{a}^{[s]n}(m)$ .",
"In the massless case the zero mode carries the annihiltion operator $\\hat{\\tilde{a}}^{[s]n}(0)$ .",
"We define the ground state to be the unique state that satisfies $\\hat{\\tilde{a}}^{[s]n}(j)\\vert G \\rangle _{\\rm bd}=0, \\forall j$ .",
"The correlations in the ground state are $\\begin{array}{lll}_{\\rm bd}\\langle G \\vert \\hat{\\Phi }^{[\\mathbf {\\mathrm {s}}]n}(m)\\hat{\\Phi }^{[\\mathbf {\\mathrm {s}}]n}(m^{\\prime })\\vert G \\rangle _{\\rm bd}&=&\\frac{1}{2}[K^{-1/2}_{\\rm bd}]_{m,m^{\\prime }}\\\\&=&\\frac{1}{2V}\\sum _{j=0}^{V-1} \\frac{\\cos (k_j (m-m^{\\prime }) )}{d_j},\\end{array}$ $\\begin{array}{lll}_{\\rm bd}\\langle G \\vert \\hat{\\Pi }^{[\\mathbf {\\mathrm {s}}]n}(m)\\hat{\\Pi }^{[\\mathbf {\\mathrm {s}}]n}(m^{\\prime })\\vert G \\rangle _{\\rm bd}&=&\\frac{1}{2}[K^{1/2}_{\\rm bd}]_{m,m^{\\prime }}\\\\&=&\\frac{1}{2V}\\sum _{j=0}^{V-1} d_j\\cos (k_j (m-m^{\\prime }) ).\\end{array}$ Numerically, we find that these correlations are well described by the continuum values in Eqs.",
"REF ,REF up to an additive constant that depends on mass and scale.",
"Similarly, the correlations in a thermal state $\\hat{\\rho }_{\\rm bd}(\\beta )$ at temperature $T=\\beta ^{-1}$ are $\\mathop {\\mathrm {tr}}[\\hat{\\Phi }^{[\\mathbf {\\mathrm {s}}]n}(m)\\hat{\\Phi }^{[\\mathbf {\\mathrm {s}}]n}(m^{\\prime })\\hat{\\rho }_{\\rm bd}(\\beta )]=\\frac{1}{2V}\\sum _{j=0}^{V-1} \\frac{\\coth (\\beta d_j)}{d_j}\\cos (k_j (m-m^{\\prime }) ),$ $\\mathop {\\mathrm {tr}}[\\hat{\\Pi }^{[\\mathbf {\\mathrm {s}}]n}(m)\\hat{\\Pi }^{[\\mathbf {\\mathrm {s}}]n}(m^{\\prime })\\hat{\\rho }_{\\rm bd}(\\beta )]=\\frac{1}{2V}\\sum _{j=0}^{V-1} d_j \\coth (\\beta d_j)\\cos (k_j (m-m^{\\prime }) ).$ For large $V$ the correlations on the boundary in the massless case for $|m-m^{\\prime }|>2\\mathcal {K}-1$ are: $\\begin{array}{lll}{_{\\rm bd}\\langle G \\vert }\\hat{\\Phi }^{[\\mathbf {\\mathrm {s}}]n}(m)\\hat{\\Phi }^{[\\mathbf {\\mathrm {s}}]n}(m^{\\prime })\\vert G \\rangle _{\\rm bd}&=& -\\frac{\\ln ((m-m^{\\prime })^2)}{4\\pi }+\\kappa ,\\\\{_{\\rm bd}\\langle G \\vert }\\hat{\\Pi }^{[\\mathbf {\\mathrm {s}}]n}(m)\\hat{\\Pi }^{[\\mathbf {\\mathrm {s}}]n}(m^{\\prime })\\vert G \\rangle _{\\rm bd}&=&-\\frac{1}{2\\pi (m-m^{\\prime })^2}.\\end{array}$ where $\\kappa $ is a constant that depends on scale $n$ and $\\mathcal {K}$ .",
"We will show below that this constant does not impact the relevant properties of the bulk.",
"In the massive phase, $\\begin{split}_{\\rm bd}\\langle G \\vert \\hat{\\Phi }^{[\\mathbf {\\mathrm {s}}]n}(m)\\hat{\\Phi }^{[\\mathbf {\\mathrm {s}}]n}(m^{\\prime })\\vert G \\rangle _{\\rm bd}&\\approx \\frac{1}{2\\pi }K_0(m_0 |x-y|),\\\\_{\\rm bd}\\langle G \\vert \\hat{\\Pi }^{[\\mathbf {\\mathrm {s}}]n}(m)\\hat{\\Pi }^{[\\mathbf {\\mathrm {s}}]n}(m^{\\prime })\\vert G \\rangle _{\\rm bd}&\\approx -\\frac{m_0}{2\\pi (x-y)}K_1(m_0 |x-y|),\\end{split}$ Figure: Plot of the eigenvalues d j d_j as a fucntion of jj (recall the momentum k j =2πj Vk_j=\\frac{2\\pi j}{V}), for a system of size V=320V=320 and mass m 0 m_0 showing the dispersion relation obtained using Daubechies 𝒦=3,5\\mathcal {K}=3,5 wavelets.",
"Note that larger 𝒦\\mathcal {K} gives a better approximation to the linear dispersion obtained from the continuum model in the massless phase.Temporal correlations are given by the boundary Green's function (time $\\tau >0$ ): $\\begin{array}{lll}&&{_{\\rm bd}\\langle G \\vert }\\hat{a}^{[s]n}(m,\\tau )\\hat{a}^{[s]n\\dagger }(m^{\\prime },0)\\vert G \\rangle _{\\rm bd}=\\frac{1}{V}\\sum _{r,s=0}^{V-1} e^{i 2 \\pi (m^{\\prime }r-ms)/V}\\\\&&\\quad \\times {_{\\rm bd}\\langle G \\vert }\\hat{\\tilde{a}}^{[s]n}(s,\\tau )\\hat{\\tilde{a}}^{[s]n\\dagger }(r,0)\\vert G \\rangle _{\\rm bd}\\\\&&\\quad =\\frac{1}{V}\\sum _{r,s=0}^{V-1} e^{i (2 \\pi (m^{\\prime }r-ms)/V+d_s\\tau )}{_{\\rm bd}\\langle G \\vert }\\hat{\\tilde{a}}^{[s]n}(s,0)\\hat{\\tilde{a}}^{[s]n\\dagger }(r,0)\\vert G \\rangle _{\\rm bd}\\\\&&\\quad =\\frac{1}{V}\\sum _{r,s=0}^{V-1} e^{i (2 \\pi (m^{\\prime }r-ms)/V+d_s\\tau )}\\delta _{s,r}\\\\&&\\quad =\\frac{1}{V}\\sum _{r=0}^{V-1} e^{i (2 \\pi (m^{\\prime }-m)r/V+d_r\\tau )}.\\end{array}$ At this point we make a Wick rotation to imaginary time $\\tau \\rightarrow i\\tau $ .",
"In the massless case, and in the continuum limit where $d_j=2\\pi j/V$ , and with $\\tau \\gg 1$ then ${_{\\rm bd}\\langle G \\vert }\\hat{a}^{[s]n}(m,\\tau )\\hat{a}^{[s]n\\dagger }(m^{\\prime },0)\\vert G \\rangle _{\\rm bd}\\rightarrow \\frac{\\tau }{(m-m^{\\prime })^2+\\tau ^2}.$ In the wavelet basis, we find numerically that the correlations are ${_{\\rm bd}\\langle G \\vert }\\hat{a}^{[s]n}(m,\\tau )\\hat{a}^{[s]n\\dagger }(m^{\\prime },0)\\vert G \\rangle _{\\rm bd}\\approx \\frac{0.32\\tau }{(m-m^{\\prime })^2+\\tau ^2}.$"
],
[
"Same scale correlations",
"Using the bulk/boundary correspondence embodied in Eq.",
"REF , we can compute the bulk correlations from the boundary correlations.",
"For the massless case, the same scale field/field correlation is (using REF ) $\\begin{array}{lll}C^{\\hat{\\Phi },\\hat{\\Phi }}((r,0),(r,j))&=&2^{r-n}\\displaystyle {\\sum _{m=0}^{(2\\mathcal {K}-1)\\times 2^{n-r}}}\\sum _{m^{\\prime }=j\\times 2^{n-r}}^{(j+2\\mathcal {K}-1)\\times 2^{n-r}}\\\\&&w_0^{0}((m+1)\\times 2^{r-n})\\\\&&w_0^{0}((m^{\\prime }-j\\times 2^{n-r}+1)\\times 2^{r-n})\\\\&&\\Big ((1-\\delta _{m,m^{\\prime }})(-\\frac{2^{-n}\\ln ((m-m^{\\prime })^2)}{4\\pi }+\\kappa )\\\\&&+\\delta _{m,m^{\\prime }}C\\Big ).\\\\\\end{array}$ Note, we have introduced a constant $C$ to make the correlations finite at zero separation on the boundary, the final result will be independent of $C$ .",
"This equation assumes $n-r\\gg 0$ so let's write $m2^{r-n}\\rightarrow x$ , and treat $x$ as a continuous variable so that $\\delta m= 2^{n-r}dx$ and sums are replaced by integrals: $\\sum _{m=0}^{(2\\mathcal {K}-1)\\times 2^{n-r}}\\rightarrow 2^{n-r}\\int _0^{2\\mathcal {K}-1} dx$ .",
"Then we find $\\begin{array}{lll}C^{\\hat{\\Phi },\\hat{\\Phi }}((r,0),(r,j))&=&2^{n-r}\\int _0^{2\\mathcal {K}-1} dx \\int _{j}^{j+2\\mathcal {K}-1} dx^{\\prime } w_0^{0}(x) \\\\&&w_0^{0}(x^{\\prime }-j)\\Big ((1-\\delta (x-x^{\\prime }))\\\\&&\\times (-\\frac{\\ln ((2^{2(n-r)}(x-x^{\\prime })^2)}{4\\pi }+\\kappa )\\\\&&+\\delta (x-x^{\\prime })C\\Big ).\\\\\\end{array}$ Now assume $j>2\\mathcal {K}-1$ , i.e.",
"focus on correlations longer range than the size of the wavelet modes, so that the integrals satisfy $\\int _0^{2\\mathcal {K}-1} dx w_0^{0}(x) w_0^{0}(x-j)=0$ , and define $x^{\\prime \\prime }=x^{\\prime }-j$ , then $\\begin{array}{lll}C^{\\hat{\\Phi },\\hat{\\Phi }}((r,0),(r,j))&=&2^{n-r}\\int dx \\int dx^{\\prime \\prime } w_0^{0}(x) w_0^{0}(x^{\\prime \\prime })\\\\&&\\times \\Big [(\\kappa _n-\\frac{2(n-r)\\ln 2}{4\\pi })-\\frac{\\ln ((x-x^{\\prime \\prime }-j)^2)}{4\\pi }\\Big ]\\\\&=&-\\frac{2^{n-r}}{4\\pi }\\int dx \\int dx^{\\prime \\prime } w_0^{0}(x) w_0^{0}(x^{\\prime \\prime })\\\\&&\\times \\ln ((j-(x-x^{\\prime \\prime }))^2)\\\\&=&-\\frac{2^{n-r}}{2\\pi }\\int dx \\int dx^{\\prime \\prime } w_0^{0}(x) w_0^{0}(x^{\\prime \\prime })\\\\&&\\times (\\ln (1-(\\frac{x-x^{\\prime \\prime }}{j}))+\\ln (j))\\\\&=&\\frac{2^{n-r}}{2\\pi }\\int dx \\int dx^{\\prime \\prime } w_0^{0}(x) w_0^{0}(x^{\\prime \\prime })\\\\&&\\times \\sum _{k=1}^{\\infty } \\frac{(x-x^{\\prime \\prime })^k}{k j^{k}}\\\\&=&\\frac{2^{n-r}}{2\\pi }\\sum _{k=1}^{\\infty } \\frac{1}{k j^{k}}\\sum _{t=0}^k {k \\atopwithdelims ()t}(-1)^{t}\\langle x^t\\rangle _w \\langle x^{k-t}\\rangle _w.\\end{array}$ Here we have defined the wavelet moments $\\langle x^a\\rangle _w=\\int x^a w^0_0(x)dx .$ A method to calculate these moments is provided in Appendix REF .",
"A similar calculation gives the momenta-momenta correlations: $\\begin{array}{lll}C^{\\hat{\\Pi },\\hat{\\Pi }}((r,0),(r,j))&=&-2^{n-r}\\int dx \\int dx^{\\prime \\prime } w_0^{0}(x) w_0^{0}(x^{\\prime \\prime })\\\\&&\\times \\frac{1}{2\\pi 2^{2(n-r)} (j-(x-x^{\\prime \\prime }))^2}\\\\&=&-\\frac{2^{r-n}}{2\\pi j^2}\\int dx \\int dx^{\\prime \\prime } w_0^{0}(x) w_0^{0}(x^{\\prime \\prime })\\\\&&\\times \\frac{1}{(1-(x-x^{\\prime \\prime })/j)^2}\\\\&=&-\\frac{2^{r-n}}{2\\pi j^2}\\int dx \\int dx^{\\prime \\prime } w_0^{0}(x) w_0^{0}(x^{\\prime \\prime })\\\\&&\\times \\frac{1}{(1-(x-x^{\\prime \\prime })/j)^2}\\\\&=&-\\frac{2^{r-n}}{2\\pi j^2}\\int dx \\int dx^{\\prime \\prime } w_0^{0}(x) w_0^{0}(x^{\\prime \\prime })\\\\&&\\sum _{k=0}^{\\infty } \\frac{k+1}{j^k} (x-x^{\\prime \\prime })^k\\\\&=&-\\frac{2^{r-n}}{2\\pi j^2}\\sum _{k=0}^{\\infty } \\frac{k+1}{j^k} \\\\&&\\sum _{t=0}^k {k \\atopwithdelims ()t}(-1)^{t}\\langle x^t\\rangle _w \\langle x^{k-t}\\rangle _w.\\end{array}$ Keeping only the dominant term, for $j>2\\mathcal {K}-1$ , $C^{\\hat{\\Phi },\\hat{\\Phi }}((r,0),(r,j))\\approx -\\frac{2^{n-r} \\times D_{\\mathcal {K}}}{4\\pi \\mathcal {K}j^{2\\mathcal {K}}},$ where $D_{\\mathcal {K}}=\\langle x^{\\mathcal {K}}\\rangle _w^2{2\\mathcal {K} \\atopwithdelims ()\\mathcal {K}}.$ Numerically we find the self correlation $C^{\\hat{\\Phi },\\hat{\\Phi }}((r,0),(r,0))= 2^{n-r-a}.$ Similarly, for the momentum-momentum correlations, for $j>2\\mathcal {K}-1$ , $C^{\\hat{\\Pi },\\hat{\\Pi }}((r,0),(r,j))\\approx \\frac{2^{r-n}\\times (2\\mathcal {K}+1)D_{\\mathcal {K}}}{2\\pi j^{2\\mathcal {K}+2}}.$ The self correlations can be written $C^{\\hat{\\Pi },\\hat{\\Pi }}((r,0),(r,0))= 2^{r-n+b}.$ The values $a,b$ appearing in the self correlations can be calculated using Eq.",
"REF together with Eqs.",
"REF ,REF .",
"By computing self correlations for $ 3\\le \\mathcal {K}\\le 30$ we find that $a\\approx 3.18$ independent of $\\mathcal {K}$ , whereas $b$ is fit by the function $b\\approx 0.75/\\mathcal {K}^2+0.16 \\mathcal {K}+1.24$ .",
"In order for the reduced covariance matrix of a single bulk site to describe a valid quantum state, it should satisfy the positivity condition $\\Gamma _1+\\frac{i}{2}\\Omega \\ge 0,$ where $\\Omega $ is the symplectic form from Eq.",
"REF (with $n_A=1$ ) and $\\Gamma _1=\\left(\\begin{array}{cc} C^{\\hat{\\Phi },\\hat{\\Phi }}((r,0),(r,0)) & 0 \\\\0 & C^{\\hat{\\Pi },\\hat{\\Pi }}((r,0),(r,0)) \\end{array}\\right).$ It is readily verified that the computed functional form for the self correlations does satisfy positivity."
],
[
"Temporal correlations",
"For simplicity, we consider temporal correlations at the same spatial point $(r,0)$ .",
"Following the same argument in Sec.",
"REF , Then following the same argument as that leading to Eq.",
"REF we find deep in the bulk $\\begin{array}{lll}C^{\\hat{a},\\hat{a}^{\\dagger }}((r,0,\\tau ),(r,0,0))&=&2^{r-n}\\displaystyle {\\sum _{m=0}^{(2\\mathcal {K}-1)\\times 2^{n-r}}}\\sum _{m^{\\prime }=0}^{(2\\mathcal {K}-1)\\times 2^{n-r}}\\\\&&w_0^{0}((m+1)\\times 2^{r-n}) \\\\&&w_0^{0}((m^{\\prime }+1)\\times 2^{r-n})\\\\&&{_{\\rm bd}\\langle G \\vert }T[\\hat{a}^{[\\mathbf {\\mathrm {s}}]n}(m,\\tau )\\hat{a}^{\\dagger [\\mathbf {\\mathrm {s}}]n}(m^{\\prime },0)]\\vert G \\rangle _{\\rm bd}.\\\\\\end{array}$ Using the boundary Green's function from Eq.",
"REF , and assuming we are deep in the bulk so that we can approximate the sum by an integral as in Sec.",
"REF , and defining $\\tau 2^{r-n}=\\tilde{\\tau }$ then $\\begin{array}{lll}C^{\\hat{a},\\hat{a}^{\\dagger }}((r,0,\\tau ),(r,0,0))&=&0.32 \\tilde{\\tau }\\int _0^{2\\mathcal {K}-1} dx \\int _{0}^{2\\mathcal {K}-1} dx^{\\prime } w_0^{0}(x) \\\\&&w_0^{0}(x^{\\prime })((x-x^{\\prime })^2+\\tilde{\\tau }^2)^{-1}\\\\&=&0.32\\tilde{\\tau }^{-1}\\int _0^{2\\mathcal {K}-1} dx \\int _{0}^{2\\mathcal {K}-1} dx^{\\prime }w_0^{0}(x)\\\\&&w_0^{0}(x^{\\prime })\\sum _{s=0}^{\\infty }(-1)^s(\\frac{x-x^{\\prime }}{\\tilde{\\tau }})^{2s}\\\\&=&0.32\\tilde{\\tau }^{-1}\\sum _{s=0}^{\\infty }\\sum _{k=0}^{2s} (-1)^{s+k} \\tilde{\\tau }^{-2s}\\\\&&{2s \\atopwithdelims ()k}\\langle x^k\\rangle _w \\langle x^{2s-k}\\rangle _w.\\end{array}$ In the second line we have assumed that $\\tau >2^{n-r}(2\\mathcal {K}-1)$ in order to perform the power series expansion.",
"Keeping only the dominant term, $C^{\\hat{a},\\hat{a}^{\\dagger }}((r,0,\\tau ),(r,0,0))\\approx \\frac{0.32\\times D_{\\mathcal {K}}}{(2^{r-n}\\tau )^{(2\\mathcal {K}+1)}}.$ In the massive phase, the boundary ground state satisfies $\\begin{split}_{\\rm bd}\\langle G \\vert \\hat{\\Phi }^{[\\mathbf {\\mathrm {s}}]n}(m)\\hat{\\Phi }^{[\\mathbf {\\mathrm {s}}]n}(m^{\\prime })\\vert G \\rangle _{\\rm bd}&=\\frac{1}{2\\pi }K_0(m_0 |x-y|),\\\\_{\\rm bd}\\langle G \\vert \\hat{\\Pi }^{[\\mathbf {\\mathrm {s}}]n}(m)\\hat{\\Pi }^{[\\mathbf {\\mathrm {s}}]n}(m^{\\prime })\\vert G \\rangle _{\\rm bd}&= -\\frac{m_0}{2\\pi (x-y)}K_1(m_0 |x-y|),\\end{split}$ for $m=(0,1,\\ldots L2^n/2-1)$ .",
"As in Sec.",
"REF we can compute the same scale bulk correlations $C^{\\hat{\\Phi },\\hat{\\Phi }}((r,0),(r,j))$ .",
"Following the same argument that led to Eq.",
"REF , and again assuming that $j>2\\mathcal {K}-1$ , we find for the same scale correlations $\\begin{array}{lll}C^{\\hat{\\Phi },\\hat{\\Phi }}((r,0),(r,j))&=&\\frac{2^{n-r}}{2\\pi }\\int dx \\int dx^{\\prime \\prime } w_0^{0}(x) w_0^{0}(x^{\\prime \\prime })\\\\&&\\times K_0(m_0 2^{n-r} (j-(x-x^{\\prime \\prime })).\\end{array}$ For $m_02^{n-r}\\gg 1$ , this simplifies to $\\begin{array}{lll}C^{\\hat{\\Phi },\\hat{\\Phi }}((r,0),(r,j))&=&-\\frac{2^{n-r}e^{-j m_0 2^{n-r}}}{\\sqrt{8\\pi 2^{n-r} m_0}}\\int dx \\int dx^{\\prime \\prime } w_0^{0}(x) w_0^{0}(x^{\\prime \\prime })\\\\&&\\times \\frac{e^{-m_0 2^{n-r} (x^{\\prime \\prime }-x)}}{\\sqrt{j-(x-x^{\\prime \\prime })}}\\\\&\\approx &-\\frac{2^{n-r}e^{-j m_0 2^{n-r}}}{\\sqrt{j 8\\pi 2^{n-r} m_0}}\\int dx \\int dx^{\\prime \\prime } w_0^{0}(x) w_0^{0}(x^{\\prime \\prime })\\\\&&\\times e^{-m_0 2^{n-r} (x^{\\prime \\prime }-x)}\\\\&=&-\\frac{2^{n-r}e^{-j \\tilde{m}}}{\\sqrt{j 8\\pi \\tilde{m}}}\\langle e^{-\\tilde{m} x}\\rangle _w \\langle e^{\\tilde{m}x}\\rangle _w.\\\\\\end{array}$ In the second line we have assumed that $j\\gg 2\\mathcal {K}-1$ in order to simplify the denominator.",
"The error introduced in doing so is small when $m_02^{n-r}\\gg 1$ .",
"The same scale bulk correlations fall off exponentially with a renormalized mass $\\tilde{m}=m_0 2^{n-r}.$ The momentum-momentum correlations on the other hand, in the same limit $\\tilde{m}\\gg 1$ , are: $\\begin{array}{lll}C^{\\hat{\\Pi },\\hat{\\Pi }}((r,0),(r,j))&=&\\frac{2^{n-r}e^{-j m_0 2^{n-r}}\\sqrt{m_0}}{\\sqrt{8\\pi 2^{3(n-r)}}}\\int dx \\int dx^{\\prime \\prime } w_0^{0}(x) w_0^{0}(x^{\\prime \\prime })\\\\&&\\times \\frac{e^{-m_0 2^{n-r} (x^{\\prime \\prime }-x)}}{\\sqrt{(j-(x-x^{\\prime \\prime }))^3}}\\\\&\\approx &2^{r-n}e^{-j \\tilde{m}}\\sqrt{\\frac{\\tilde{m}}{8\\pi j^3}}\\langle e^{-\\tilde{m} x}\\rangle _w \\langle e^{\\tilde{m} x}\\rangle _w.\\end{array}$ For $\\mathcal {K}=3$ wavelets and $\\tilde{m}\\gg 1$ the product of averages is $\\langle e^{-\\tilde{m} x}\\rangle _w \\langle e^{\\tilde{m} x}\\rangle _w\\approx e^{4.71 \\times \\tilde{m}-20.38}$ ."
],
[
"Distances in AdS$_{3}$ space",
"The metric of Euclidean AdS$_{3}$ space (with time coördinate $t\\rightarrow i\\tau $ ) is $ds^2=\\left(\\frac{\\rho ^2}{R^2}+1\\right)d\\tau ^2+\\frac{1}{\\frac{\\rho ^2}{R^2}+1}d\\rho ^2+\\rho ^2d\\theta ^2,$ where $\\rho \\in \\mathbb {R}^+$ is a radial coördinate, $\\theta \\in [0,2\\pi )$ is the angular coördinate, $\\tau \\in \\mathbb {R}$ , and $R$ is the radius of curvature.",
"On a time slice, the geodesic distance between two points at the same radius $\\rho $ is: $\\begin{array}{lll}d_g((\\rho ,\\theta _1),(\\rho ,\\theta _2))&=&R\\cosh ^{-1}\\left(1+\\frac{2\\rho ^2}{R^2}\\sin ^2(\\frac{\\theta _1-\\theta _2}{2})\\right)\\\\&=&R\\ln \\Big [1+\\frac{2\\rho ^2}{R^2}\\sin ^2(\\frac{\\theta _1-\\theta _2}{2})\\\\&&+\\sqrt{(1+\\frac{2\\rho ^2}{R^2}\\sin ^2(\\frac{\\theta _1-\\theta _2}{2}))^2-1}\\Big ]\\\\&\\approx &R\\ln [\\frac{4\\rho ^2}{R^2}\\sin ^2(\\frac{\\theta _1-\\theta _2}{2})]\\\\&=&2R\\ln [\\frac{2\\rho }{R}\\sin (\\frac{\\theta _1-\\theta _2}{2})]\\\\&\\approx &2R\\ln [\\frac{\\rho (\\theta _1-\\theta _2)}{R}]\\end{array}$ where in third line we have assumed that we are deep in the bulk and with sufficiently small radius of curvature so that $\\rho \\gg R$ and in the fifth line we have assumed we are considering angular separations $|\\theta _1-\\theta _2|\\ll \\pi $ .",
"In our bulk description, fields localized at scale $r$ and position $m$ have AdS$_{3}$ coördinates $(\\rho =\\frac{L2^{r}}{2\\pi },\\theta =\\frac{m2\\pi }{L2^{r}})$ , so the geodesic distance between these coördinates is $d_g((r,m),(r,m^{\\prime }))\\approx 2R\\ln \\left[\\frac{|m-m^{\\prime }|}{R}\\right],$ where $R$ has the same dimensions as position $m$ (we make them dimensionless).",
"To find the geodesic distance between points at the same angle but different radii, first find the distance to the origin from a point at $(\\rho ,\\theta )$ which is one half the distance computing using Eq.",
"REF with antipodal points $\\theta _1-\\theta _2=\\pi $ .",
"This gives $d_g((r,0),(0,0))\\approx R\\left(\\ln [\\frac{L}{2\\pi R}]+r\\ln 2\\right)$ .",
"Then the geodesic distance between points is $d_g((r,0),(r^{\\prime },0))\\approx R|r^{\\prime }-r| \\ln 2 .$ where recall scale $r$ is dimensionless.",
"Finally, the geodesic distance can be computed between two events with the same spatial coördinates but separated in imaginary time by $\\tau $ .",
"The time coördinate in Euclidean AdS$_{3}$ must be rescaled according to $\\tau \\rightarrow \\tau /\\sqrt{R^2+V/((2\\pi ))^2}$ where $V$ is the size of the boundary, in order that the metric reduces to $ds^2=d\\tau ^2+\\rho ^2d\\theta ^2$ at the boundary $(\\rho =V/2\\pi )$ .",
"When this is done then for $\\rho \\gg R$ , $\\tau \\ll V$ , and $\\rho \\tau \\gg RV$ , $d_g((\\rho ,\\theta ,\\tau ),(\\rho ,\\theta ,0)\\approx 2R\\ln \\left(\\frac{2\\pi \\rho \\tau }{RV}\\right).$ In terms of the bulk geometry coördinates with boundary size $V=L2^n$ then $d_g((r,j,\\tau ),(r,j,0))\\approx 2R\\ln \\left(\\frac{\\tau 2^{n-r}}{R}\\right).$"
],
[
"Properties of Daubechies wavelets",
"The family of Daubechies wavelets are constructed from a set of scale function coefficients $\\lbrace h_j\\rbrace _{j=0}^{2\\mathcal {K}-1}$ , and wavelet coefficients $\\lbrace g_j=(-1)^j h_{2\\mathcal {K}-1-j}\\rbrace $ .",
"The coefficients for many families are available in the literature, e.g.",
"[8], and in several software packages.",
"We provide them for three families in Table REF , and plots of the scale and wavelet functions for $\\mathcal {K}=3,4,5$ are shown in Fig.",
"REF ."
],
[
"Derivative overlaps",
"The procedure to compute the derivative overlaps in Eq.",
"REF is given in Ref.",
"[16] and for completeness we include it here and evaluate them for several wavelet families.",
"Table: Values of the scale function coefficients for the wavelet families 𝒦=3,4,5\\mathcal {K}=3,4,5.The overlap between derivatives of scale/scale functions is given by the coefficients $D^{[ss]k}_{m,m^{\\prime }}$ which satisfy $D^{[ss]k}_{m,m^{\\prime }}=2^{2k}D^{[ss]0}_{0,m^{\\prime }-m}$ so it suffices to find the coeffiecients $D^{[ss]0}_{0,m}$ .",
"The cofficients are symmetric $D^{[ss]0}_{0,m}=D^{[ss]0}_{m,0}$ and because of the compact support of the scale functions, are non zero only for $|m|\\le 2\\mathcal {K}-2$ .",
"Using a resolution of the identity $\\sum _n s^0_n(x)=1$ , the coefficients can be written $D^{[ss]0}_{0,m}=\\sum _n D_{n,0,m}=\\sum _n D_{0,-n,m-n},$ where $D_{n,j,k}=\\int dx s^0_n(x)\\partial _x s^{0}_{j}(x)\\cdot \\partial _x s^{0}_{k}(x)$ .",
"These coefficients are symmetric in the last two indices $D_{n,j,k}=D_{n,k,j}$ and satisfy the following set of homogeneous equations $\\begin{array}{lll}\\left\\lbrace D_{0,r,s}=4\\sqrt{2}\\sum _{n=0}^{2\\mathcal {K}-1}\\sum _{j,k=-(2\\mathcal {K}-2)}^{2\\mathcal {K}-2}h_{n}h_{k+n-2r}h_{j+n-2s}D_{0,k,j}\\right\\rbrace \\\\\\cup \\left\\lbrace \\sum _{k=-(2\\mathcal {K}-2)}^{2\\mathcal {K}-2}D_{0,j,k}=0\\right\\rbrace ;\\ r,s\\in [-(2\\mathcal {K}-2),2\\mathcal {K}-2]\\cap \\mathbb {Z},\\end{array}$ and the set of inhomogeneous equations $\\left\\lbrace \\sum _{j=-(2\\mathcal {K}-2)}^{2\\mathcal {K}-2}jD_{0,j,k}=\\Gamma _{0,k}\\right\\rbrace ;\\ k\\in [-(2\\mathcal {K}-2),2\\mathcal {K}-2]\\cap \\mathbb {Z},$ where $\\Gamma _{n,m}=\\int dx s^0_n(x)\\partial _x s^0_m(x)=-\\Gamma _{m,n}=\\Gamma _{0,m-n}$ .",
"The coefficients $\\Gamma _{n,m}$ themselves satisfy the following set of homogeneous equations $\\begin{split}&\\left\\lbrace \\Gamma _{0,j}=2\\sum _{m=0}^{2\\mathcal {K}-1}\\sum _{n=-(2\\mathcal {K}-2)}^{2\\mathcal {K}-2}h_{m}h_{n+m-2j}\\Gamma _{0,n}\\right\\rbrace ;\\\\&j\\in [-(2\\mathcal {K}-2),2\\mathcal {K}-2]\\cap \\mathbb {Z},\\end{split}$ and one inhomogeneous equation $\\sum _{n=-(2\\mathcal {K}-2)}^{2\\mathcal {K}-2} n\\Gamma _{0,n}=1.$ First solving for the $\\Gamma _{0,m}$ one can then solve for the $D_{0,r,s}$ and finally for $D^{[ss]}_{0,m}$ .",
"Note, the coefficients satisfy $D^{[ss]}_{0,m}=D^{[ss]}_{0,-m}$ , and also $\\sum _{m=0}^{2\\mathcal {K}-2} m^2 D^{[ss]0}_{0,m}=-1$ .",
"The other overlaps of derivatives of scale/wavelet functions and wavelet/wavelet functions are given by [15] $\\begin{array}{lll}D^{[sw]l,0}_{a,b}&=&2^{2(l+1)}(\\langle a \\vert [H(l)]^{l+1} D(l) G^T(l)\\vert b \\rangle \\\\&&+\\langle a+L \\vert [H(l)]^{l+1} D(l) G^T(l)\\vert b \\rangle \\\\&&+\\langle a \\vert [H(l)]^{l+1} D(l) G^T(l)\\vert b+2^l L \\rangle ),\\\\D^{[ww]l,j}_{a,b}&=&2^{2(l+1)}(\\langle a \\vert G(l,j)[H(l,j)]^{l-j} D(l,j) G^T(l,j)\\vert b \\rangle \\\\&&+\\langle a+2^j L \\vert G(l,j)[H(l,j)]^{l-j} D(l,j) G^T(l,j)\\vert b \\rangle \\\\&&+\\langle a \\vert G(l,j)[H(l,j)]^{l-j} D(l,j) G^T(l,j)\\vert b+2^l L \\rangle ),\\end{array}$ where the scale dependent matrices are $\\begin{array}{lll}H(l)&=&\\sum _{m,n=0}^{2^{(l+2)}(L+2\\mathcal {K}-2)-(2\\mathcal {K}-1)}h_{n-2m}\\vert m \\rangle \\langle n \\vert \\\\H(l,j)&=&\\sum _{m,n=0}^{2^{(l-j+1)}2 (2^j L-2\\mathcal {K}-2))-(2\\mathcal {K}-1)}h_{n-2m}\\vert m \\rangle \\langle n \\vert \\\\D(l)&=&\\sum _{m,n=0}^{2^{(l+2)}(L+2\\mathcal {K}-2)-(2\\mathcal {K}-1)}D^{[ss]0}_{m,n}\\vert m \\rangle \\langle n \\vert \\\\D(l,j)&=&\\sum _{m,n=0}^{2^{(l-j+1)}2 (2^j L-2\\mathcal {K}-2)-(2\\mathcal {K}-1)}D^{[ss]0}_{m,n}\\vert m \\rangle \\langle n \\vert \\\\G(l)&=&\\sum _{m,n=0}^{2^{(l+2)}(L+2\\mathcal {K}-2)-(2\\mathcal {K}-1)}g_{n-2m}\\vert m \\rangle \\langle n \\vert \\\\G(l,j)&=&\\sum _{m,n=0}^{2^{(l-j+1)}2 (2^j L-2\\mathcal {K}-2)-(2\\mathcal {K}-1)}g_{n-2m}\\vert m \\rangle \\langle n \\vert .",
"\\\\\\end{array}$ Note at different scales these coefficients satisfy $D^{[ww]r-n,r^{\\prime }-n}_{m,m^{\\prime }}=2^{-2n}D^{[ww]r,r^{\\prime }}_{m,m^{\\prime }}$ , and $D^{[sw]r-n,-n}_{m,m^{\\prime }}=2^{-2n}D^{[sw]r,0}_{m,m^{\\prime }}$ ."
],
[
"Wavelet moments",
"The wavelet moments $\\langle x^a\\rangle _w=\\int x^a w^0_0(x) dx,$ satisfy the relation [16] $\\langle x^a\\rangle _w=\\frac{1}{\\sqrt{2}}\\frac{1}{2^a}\\sum _{j=0}^{2\\mathcal {K}-1} g_j \\sum _{b=0}^a {a \\atopwithdelims ()b}j^{a-b}\\langle x^b\\rangle _s,$ where $g_j=(-1)^j h_{2\\mathcal {K}-1-j}$ , Note we have defined $0^0=1$ .",
"and $h_k$ are the aforementioned scale functions coefficients.",
"The scale function moments $\\langle x^b\\rangle _s=\\int x^b s^0_0(x)dx,$ satisfy a recursion formula $\\langle x^b\\rangle _s=\\frac{1}{2^b-1}\\frac{1}{\\sqrt{2}}\\sum _{c=0}^{b-1} {b \\atopwithdelims ()c}\\sum _{k=1}^{2\\mathcal {K}-1} h_k k^{b-c} \\langle x^c\\rangle _s,$ with $\\langle x^0\\rangle _s=1$ .",
"The $\\mathcal {K}$ wavelets have vanishing moments up to $\\mathcal {K}-1$ .",
"For $\\mathcal {K}=3$ , the lowest few wavelet moments are $\\begin{array}{lll}\\langle x^a\\rangle _w&=&0,\\quad a=0,1,2\\\\\\langle x^3\\rangle _w&=&-\\frac{3\\sqrt{5/2}}{16}\\\\\\langle x^4\\rangle _w&=&\\frac{3}{32}\\Big (\\sqrt{10}(\\sqrt{5+2\\sqrt{10}}-10)-\\sqrt{5+2\\sqrt{10}}\\Big )\\\\\\langle x^5\\rangle _w&=&\\frac{15}{64}\\Big (-3-26\\sqrt{10}-5\\sqrt{5+2\\sqrt{10}}+5\\sqrt{50+20\\sqrt{10}}\\Big )\\\\\\langle x^6\\rangle _w&=&\\frac{15}{896}\\Big (-523\\sqrt{5+2\\sqrt{10}}+541\\sqrt{50+20\\sqrt{10}}\\\\&&-70(9+28\\sqrt{10})\\Big ).\\\\\\end{array}$ Table: Derivative overlap coefficients.",
"Note: D 0,m [ss] =D 0,-m [ss] D^{[ss]}_{0,m}=D^{[ss]}_{0,-m}."
]
] | 1606.05068 |
[
[
"Neutrino-Pair Exchange Long-Range Force Between Aggregate Matter"
],
[
"Abstract We study the long-range force arising between two neutral---of electric charge---aggregates of matter due to a neutrino-pair exchange, in the limit of zero neutrino mass.",
"The conceptual basis for the construction of the effective potential comes from the coherent scattering amplitude at low values of t. This amplitude is obtained using the methodology of an unsubtracted dispersion relation in t at threshold for s, where (s, t) are the Lorentz invariant scattering variables.",
"The ultraviolet behavior is irrelevant for the long-range force.",
"In turn, the absorptive part in the t-dependence is given by the corresponding unitarity relation.",
"We show that the potential describing this force decreases as $r^{-5}$ at large separation distance r. This interaction is described in terms of its own charge, which we call the weak flavor charge of the interacting systems, that depends on the flavor of the neutrino as $Q_W^e = 2Z-N$, $Q_W^\\mu = Q_W^\\tau = -N$.",
"The flavor dependence of the potential factorizes in the product of the weak charges of the interacting systems, so that the resulting force is always repulsive.",
"Furthermore, this charge is proportional to the number of constituent particles, which differs from the global mass, so this interaction could be disentangled from gravitation through deviations from the Equivalence Principle."
],
[
"[display] -1cm 1.",
"[ ] Section 0.",
"A. Segarra Figure: NO_CAPTIONFigure: NO_CAPTIONFigure: NO_CAPTION Master Thesis Neutrino-Pair Exchange Long-Range Force Between Aggregate Matter Author: Alejandro Segarra Supervisor: José Bernabéu July 2015 tocsection Abstract"
],
[
"Abstract",
"We study the long-range force arising between two neutral—of electric charge—aggregates of matter due to a neutrino-pair exchange, in the limit of zero neutrino mass.",
"The conceptual basis for the construction of the effective potential comes from the coherent scattering amplitude at low values of $t$ .",
"This amplitude is obtained using the methodology of an unsubtracted dispersion relation in $t$ at threshold for $s$ , where $(s,\\, t)$ are the Lorentz invariant scattering variables.",
"The ultraviolet behavior is irrelevant for the long-range force.",
"In turn, the absorptive part in the $t$ -dependence is given by the corresponding unitarity relation.",
"We show that the potential describing this force decreases as $r^{-5}$ at large separation distance $r$ .",
"This interaction is described in terms of its own charge, which we call the weak flavor charge of the interacting systems, that depends on the flavor of the neutrino as $Q_W^e = 2Z-N$ , $Q_W^\\mu = Q_W^\\tau = -N$ .",
"The flavor dependence of the potential factorizes in the product of the weak charges of the interacting systems, so that the resulting force is always repulsive.",
"Furthermore, this charge is proportional to the number of constituent particles, which differs from the global mass, so this interaction could be disentangled from gravitation through deviations from the Equivalence Principle."
],
[
"Introduction",
"It's been 85 years since Wolfgang Pauli postulated the existence of the neutrino in order to explain the continuous spectrum in $\\beta $ -decays, and 59 years since Reines and Cowan discovered it.",
"In those years, we've learnt many properties about this particle, such as the fact that it only interacts through weak interactions—all of its charges but weak isospin are zero.",
"In fact, in the framework of the Standard Model [1], there are only left-handed neutrinos, so Standard Model neutrinos are massless—we can't generate a neutrino mass through a Yukawa-type coupling with a Higgs doublet.",
"Other interesting phenomena related to this particle are neutrino oscillations [2], which have been well established experimentally since 1998.",
"This process is understood as the fact that there is a mismatch between mass eigenstates and flavor eigenstates, so that flavors get mixed along free propagation.",
"Indeed, the observation of neutrino oscillations is a direct measurement of the mass difference between the three states, proving that neutrinos are massive particles, which is a first signal of Physics beyond the Standard Model.",
"Therefore, the study of the origin of neutrino mass is one of the directions in which we can expect finding new Physics, even though its small value ($m_\\nu \\lesssim 1$ eV [3]) makes it hard to observe experimentally.",
"As well as determining the absolute mass of the neutrino, there's still a more fundamental question about their nature unanswered: since neutrinos can be neutral of all charges, their finite mass could be explained through a Dirac mass term (implying that neutrinos and antineutrinos are different particles, described by $4-$ component Dirac spinors) or through a Majorana one (implying that neutrinos are self-conjugate of all charges, described by 2 independent degrees of freedom).",
"In any case, the fact that their masses are very low stands, and we discuss here another property of neutrinos as mediators of a new force.",
"As is well known, the processes represented in Quantum Field Theory by the exchange of a massless particle give raise to long-range interactions.",
"An easy example is the scattering of two particles mediated by a photon, which—at tree level—describes Coulomb scattering.",
"Our objective in this work is the application of these ideas to a process mediated by neutrinos.",
"According to the Electroweak Lagrangian, the lowest-order process is a neutrino-pair exchange, which—since neutrinos are nearly massless—describes an interaction of long range.",
"With this idea in mind, we review in Section the relation between the Feynman amplitude in Born approximation and an effective potential, which is a Fourier Transform.",
"The amplitude at low $t$ , associated to the long-range behavior, is obtained by means of an unsubtracted dispersion relation.",
"Its ultraviolet dependence is of no relevance.",
"In order to simplify the calculation of the potential, in Section we exploit the untitarity of the $S$ matrix, writing the absorptive part of the $1-$ loop scattering amplitude with the amplitude of the tree-level scattering process.",
"In Section , we study the low-energy limit of the Electroweak Lagrangian in terms of a contact interaction, establishing the framework for the calculation of the scattering amplitude including both neutral current and charged current vertices.",
"We compute in detail this amplitude in Section , where it's natural to introduce the concept of a weak flavor charge of matter.",
"In terms of this amplitude, obtaining the interaction potential is straightforward, and we find in Section that it leads to a repulsive force which decreases as $r^{-6}$ .",
"We conclude this work analyzing the possibility of an experimental measurement of this interaction, which is relevant between nanometers and microns, where there are also residual electromagnetic interactions—such as Van der Waals or Casimir-Polder forces— and gravitation.",
"The measurement of this weak interaction is very compelling, since it could give information about properties of the neutrino such as its absolute mass, which is still unknown, or it could even help us to answer the most fundamental question regarding neutrinos, whether they are Dirac or Majorana particles.",
"These points are considered in Sections and ."
],
[
"From a Quantum Field Theory to an Effective Potential",
"We are interested in calculating the interaction potential resulting from a neutrino-pair exchange between aggregates of matter, which is an interaction described in the framework of a Quantum Field Theory.",
"Therefore, we will begin this work relating the concepts of interaction potential and Feynman amplitude."
],
[
"The Coulomb potential",
"It is known that the interaction between two electrically charged particles, say $A$ and $B$ , is described by the Coulomb potential, $V_C(r) = \\frac{e^2}{4\\pi }\\, \\frac{Q_A Q_B}{r}\\,,$ where $e$ is the charge of the proton, $Q_J$ the charge of the particle $J$ in units of $e$ and $r$ the distance between the two particles.",
"Throughout this work, we'll use the Natural System of Units and the Heaviside electric system—all conventions are stated in Appendix .",
"We are interested in calculating this potential using the Quantum Electrodynamics (QED), which is described by the interaction Lagrangian ${L}_\\text{QED} = -eQ\\, \\bar{\\psi }\\gamma ^\\mu \\psi \\, A_\\mu \\,.$ In this framework, the $AB\\rightarrow AB$ elastic scattering is described—at leading order—by the Feynman graph from Fig.REF .",
"Using the QED Feynman rules [1], the amplitude of the process is ${\\cal M} = e^2 Q_A Q_B \\left[ \\bar{u}(p_3) \\gamma ^\\mu u(p_1) \\right]\\, \\frac{1}{q^2} \\,\\left[ \\bar{u}(p_4) \\gamma _\\mu u(p_2) \\right]\\,.$ Since we are looking for a long-range coherent interaction, we can simplify ${\\cal M} \\approx e^2 Q_A Q_B \\left[ \\bar{u}(p_3) \\gamma ^0 u(p_1) \\right]\\, \\frac{1}{q^2} \\,\\left[ \\bar{u}(p_4) \\gamma _0 u(p_2) \\right]\\,$ taking into account the fact that $\\gamma ^0$ is related to the electric charge, which is coherent, while $\\mathbf {\\gamma }$ is related to the electromagnetic current, which is not a coherent quantity.",
"Using $\\left( \\gamma ^0 \\right)^2 = 1$ and dropping external-line factors, we get $M(q^2) = e^2 Q_A Q_B \\, \\frac{1}{q^2}\\,,$ where we defined $M(q^2)$ as ${\\cal M}(q^2) \\equiv \\bar{u}^{(A)} (p_3) \\bar{u}^{(B)} (p_4) M(q^2)u^{(B)} (p_2) u^{(A)} (p_1)$ .",
"Figure: Lowest-order Feynman diagrams for AB→ABAB\\rightarrow AB elasticscattering.",
"(a) QED interaction, mediated by a photon, whereAA and BB are particles of electric charge Q A Q_A andQ B Q_B.",
"(b) Yukawa interaction, mediated by a scalar φ\\phi of mass μ\\mu .Since this is a scattering process, we can work in the Breit reference frame (defined by $q^0 = 0$ ), which describes the non-relativistic limit (low energy transfer), where $M(q^2) = - e^2 Q_A Q_B \\, \\frac{1}{{\\mathbf {q}\\,}^2}\\,.$ We can compute the 3-dimensional Fourier Transform of this quantity (see Appendix REF ), and we find ${\\cal F}\\left\\lbrace M\\right\\rbrace (r) \\equiv \\int \\frac{\\mathrm {d}^3 q}{(2\\pi )^3}\\, e^{i\\mathbf {q}\\, \\mathbf {r}}\\, M(q^2) = - \\frac{e^2}{4\\pi }\\, \\frac{Q_A Q_B}{r} = - V_C(r)\\,.$ This expression shows the relation between the Quantum Field Theory Feynman amplitude and the interaction potential used in a potential description of the system dynamics.",
"Before considering a more general case, let's look at another simple one: the Yukawa interaction."
],
[
"The Yukawa interaction",
"Another well-known potential is Yukawa's, which describes an effective central strong nuclear force acting between nucleons, $V_Y(r) = - \\frac{g^2}{4\\pi } \\frac{e^{-\\mu r}}{r}\\,.$ From a Quantum Field Theory point of view, this interaction is described by the Lagrangian ${L}_Y = -g \\phi \\bar{\\psi }\\psi \\,,$ where $\\phi $ is a scalar field and $\\psi $ is a fermionic field.",
"Such a scalar can be physically associated to the $\\sigma $ meson for the interacting $\\pi $ -$\\pi $ mediation.",
"The $AB \\rightarrow AB$ scattering amplitude described by this Lagrangian is the one represented in Fig.REF , so it is ${\\cal M} = -g^2 \\left[ \\bar{u}(p_3) u(p_1) \\right] \\frac{1}{q^2-\\mu ^2} \\left[ \\bar{u}(p_4) u(p_2) \\right]\\,,$ where $\\mu $ is mass of the scalar, and $M(q^2) = \\frac{-g^2}{q^2-\\mu ^2}\\,.$ Again, we can work in the Breit reference frame, so that $M(q^2) = \\frac{g^2}{\\mathbf {q\\,}^2 + \\mu ^2}\\,.$ The potential must be related to the Fourier Transform of this $M(q^2)$ , which is also calculated in Appendix REF , ${\\cal F}\\left\\lbrace M\\right\\rbrace (r) = \\frac{g^2}{4\\pi } \\frac{e^{-\\mu r}}{r} = - V_Y(r)\\,,$ which is the same relation between $M(q^2)$ and $V(r)$ that we obtained in the Coulomb case."
],
[
"A more general case: particle-pair exchange",
"As we have just seen, the interaction potential between particles $A$ and $B$ is the Fourier Transform $V (r) = - \\int \\frac{\\mathrm {d}^3 q}{(2\\pi )^3} \\, e^{i \\mathbf {q}\\, \\mathbf {r}} \\, M(q^2)\\,,$ where $M$ is the lowest order Feynman amplitude for the process $AB\\rightarrow AB$ , with both $A$ and $B$ on-shell, but without external-leg factors, as is discussed in [4]Beware a minus sign between their convention for the Feynman amplitude and ours..",
"In the case of a pair exchange, this process will be the one represented in Fig.REF .",
"Figure: Feynman diagram for AB→ABAB\\rightarrow AB elastic scatteringmediated by a,ba, b exchange.In order to compute integral (REF ), we rewrite the amplitude as a dispersion relation following the steps mentioned in [5].",
"We can extend $t$ to the complex plane and expand the amplitude using Cauchy's Formula [6], $f(z) = \\frac{1}{2\\pi i} \\int _C \\mathrm {d}z^\\prime \\, \\frac{f (z^\\prime )}{z^\\prime - z}\\,,$ which is valid whenever $f(z)$ is analytic inside $C$ .",
"Figure: Integration path (in the complex plane of the ttMandelstam variable) used in the dispersion relationdecomposition of the Feynman amplitude of the process, asdiscussed in the text.The physical region of the $t$ variable of elastic scattering processes has $t<0$ , so we want the $\\mathbb {R}^-$ axis inside $C$ .",
"Also, the $t-$ channel amplitude will have a branching point at $t = (m_a + m_b)^2 \\equiv t_0 \\ge 0$ , so we can use Cauchy's Formula with the integration path shown in Fig.REF .",
"In fact, the physical region is $-s\\le t \\le t_0$ , but we are only interested in the long-range interaction, which is associated to low values of $\\left| t \\right|$ .",
"Since $\\left| t \\right| \\sim s \\sim (M_A+M_B)^2$ describes interactions of much shorter range than the nuclear size whenever $A$ and $B$ are aggregates of matter, we can take $s\\rightarrow \\infty $ without affecting the long-range amplitude, as we have done in considering the path in Fig.REF .",
"If the amplitude vanishes along the $C_\\infty $ circumference, as $\\left| t \\right| \\rightarrow \\infty $ , the only contribution is the one coming from the integral on both sides of the cut along the real $t$ axis, $\\nonumber M(t) &= \\frac{1}{2\\pi i}\\lim _{\\epsilon \\rightarrow 0}\\int ^{t_0}_{\\infty } \\mathrm {d}t^\\prime \\, \\frac{M(t^\\prime - i\\epsilon )}{t^\\prime - t} +\\frac{1}{2\\pi i} \\lim _{\\epsilon \\rightarrow 0}\\int _{t_0}^\\infty \\mathrm {d}t^\\prime \\, \\frac{M(t^\\prime + i\\epsilon )}{t^\\prime - t} = \\\\& = \\frac{1}{2\\pi i}\\lim _{\\epsilon \\rightarrow 0}\\int _{t_0}^\\infty \\mathrm {d}t^\\prime \\, \\frac{M(t^\\prime + i\\epsilon ) - M(t-i\\epsilon )}{t^\\prime - t}\\,.",
"$ If not vanishing at $C_\\infty $ , we'd have to either rewrite the dispersion relation for the subtracted amplitude or include the contribution of $C_\\infty $ .",
"We continue with the formulation without subtractions, because the contribution along $C_\\infty $ is of short range.",
"We then understand Eq.",
"(REF ) for the long-range amplitude.",
"In order to compute the analytically extended amplitude both above and below the unitarity cut, we can relate them using Schwarz Reflexion Principle [6], $M(t -i\\epsilon ) = M^*(t+i\\epsilon )\\,.$ Using this relation, we can easily write $M(t) = \\frac{1}{\\pi }\\int _{t_0}^\\infty \\, \\mathrm {d}t^\\prime \\, \\frac{\\text{Im}\\left\\lbrace M(t^\\prime )\\right\\rbrace }{t^\\prime - t}\\,,$ which is the so-called $t-$ channel dispersion relation of the Feynman amplitude.",
"Putting this expression into (REF ) and rewriting $(t^\\prime - t)^{-1}$ as (REF ) states, we get $\\nonumber V(r) &= \\frac{-1}{4\\pi ^2} \\int \\frac{\\mathrm {d}^3 q}{(2\\pi )^3} \\, e^{i \\mathbf {q}\\, \\mathbf {r}} \\, \\int _{t_0}^\\infty \\, \\mathrm {d}t^\\prime \\, \\text{Im}\\left\\lbrace M(t^\\prime )\\right\\rbrace \\int \\mathrm {d}^3r^\\prime \\, e^{-i \\mathbf {q}\\, {\\mathbf {r}\\,}^\\prime }\\, \\frac{e^{-\\sqrt{t^\\prime } r^\\prime }}{r^\\prime } = \\\\\\nonumber &= \\frac{-1}{4\\pi ^2}\\, \\int _{t_0}^\\infty \\, \\mathrm {d}t^\\prime \\, \\text{Im}\\left\\lbrace M(t^\\prime )\\right\\rbrace \\int \\mathrm {d}^3r^\\prime \\, \\frac{e^{-\\sqrt{t^\\prime }r^\\prime }}{r^\\prime } \\, \\delta ^{(3)}(\\mathbf {r} - {\\mathbf {r}\\,}^\\prime ) = \\\\&= \\frac{-1}{4\\pi ^2}\\, \\int _{t_0}^\\infty \\, \\mathrm {d}t^\\prime \\, \\text{Im}\\left\\lbrace M(t^\\prime )\\right\\rbrace \\, \\frac{ e^{-\\sqrt{t^\\prime }r}}{r} \\,.$ Therefore, the non-relativistic potential $\\boxed{\\hspace{7.11317pt} V(r) = \\frac{-1}{4\\pi ^2 r}\\, \\int _{t_0}^\\infty \\, \\mathrm {d}t^\\prime \\, \\text{Im}\\left\\lbrace M(t^\\prime )\\right\\rbrace \\, e^{-\\sqrt{t^\\prime }r} \\hspace{7.11317pt} }$ is determined by the absorptive part of the Feynman amplitude.",
"Since we are not interested in the whole $M(t)$ , but only in the Im$\\lbrace M(t)\\rbrace $ , we can make a profit from the unitarity of the $S$ matrix to simplify our calculations."
],
[
"Unitarity Relation. Absorptive Part",
"Physical processes are determined by matrix elements of the scattering matrix $S$ .",
"The $S$ matrix relates the orthonormal basis of initial states with the final states' one, so it has to be a unitary operator, $S^\\dagger S = 1.$ We define the reduced scattering matrix $T$ as $S \\equiv 1+i\\,T$ , which describes processes where there really is an interaction—initial and final states are not the same ones.",
"In terms of this operator, the unitarity relation (REF ) becomes $1 = S^\\dagger S = (1-i\\,T^\\dagger )(1+i\\,T) = 1-i\\,T^\\dagger +i\\,T+T^\\dagger T,$ so $-i(T-T^\\dagger ) = T^\\dagger T\\,.$ In order to describe a physical process, we have to consider the matrix element $\\mathinner {\\langle {f}|}S-1\\mathinner {|{i}\\rangle } = i\\, \\mathinner {\\langle {f}|} T \\mathinner {|{i}\\rangle } \\equiv i\\, (2\\pi )^4\\delta ^{(4)}(p_f-p_i) {\\cal M}(i\\rightarrow f)$ , where $\\mathinner {|{i}\\rangle }$ is the initial state and $\\mathinner {|{f}\\rangle }$ is the final one.",
"Therefore, we need to sandwich the previous relation between those states—we begin computing the left-hand side (LHS), $\\nonumber \\mathinner {\\langle {f}|} \\text{LHS} \\mathinner {|{i}\\rangle } &= -i\\,\\mathinner {\\langle {f}|} T-T^\\dagger \\mathinner {|{i}\\rangle } =\\\\\\nonumber &= -i \\left[ \\mathinner {\\langle {f}|} T \\mathinner {|{i}\\rangle } - \\mathinner {\\langle {i}|} T \\mathinner {|{f}\\rangle }^* \\right] = \\\\&= -i\\, \\times 2i \\,\\text{Im} \\left\\lbrace \\mathinner {\\langle {f}|} T \\mathinner {|{i}\\rangle } \\right\\rbrace \\,,$ where we assumed that time reversal is a good symmetry to write $T(i\\rightarrow f) - T(f\\rightarrow i)^* = 2\\,\\text{Im}\\left\\lbrace T(i\\rightarrow f) \\right\\rbrace \\,.$ On the other hand, $\\nonumber \\mathinner {\\langle {f}|} \\text{RHS} \\mathinner {|{i}\\rangle } &= \\mathinner {\\langle {f}|} T^\\dagger T \\mathinner {|{i}\\rangle } =\\\\\\nonumber &= \\mathinner {\\langle {f}|} T^\\dagger \\left[ \\sum _n \\int \\prod _{j=1}^n \\frac{\\mathrm {d}^3 q_j}{(2\\pi )^3 2E_{q_j}} \\mathinner {|{q_n}\\rangle }\\mathinner {\\langle { q_n}|} \\right] T \\mathinner {|{i}\\rangle } =\\\\&= \\sum _n \\int \\prod _{j=1}^n \\frac{\\mathrm {d}^3 q_j}{(2\\pi )^3 2E_{q_j}} \\mathinner {\\langle {f}|} T^\\dagger \\mathinner {|{ q_n}\\rangle }\\mathinner {\\langle { q_n}|} T \\mathinner {|{i}\\rangle }\\,,$ where in the second line we have inserted an identity—a sum over all possible states, with $\\mathinner {|{q_n}\\rangle }$ representing a state of $n$ particles with 4-momenta $q_1, q_2... \\,q_n$ .",
"Now we can write the unitarity relation $\\mathinner {\\langle {f}|} \\text{LHS} \\mathinner {|{i}\\rangle } = \\mathinner {\\langle {f}|} \\text{RHS} \\mathinner {|{i}\\rangle }$ as $\\boxed{ \\hspace{7.11317pt}\\begin{aligned} \\mbox{}\\\\ \\mbox{} \\end{aligned} \\text{Im} \\left\\lbrace \\mathinner {\\langle {f}|} T \\mathinner {|{i}\\rangle } \\right\\rbrace = \\frac{1}{2} \\sum _n \\int \\mathrm {d}Q_n \\mathinner {\\langle {q_n}|} T \\mathinner {|{f}\\rangle }^*\\mathinner {\\langle { q_n}|} T \\mathinner {|{i}\\rangle } \\hspace{7.11317pt} }\\,.$ Figure: Feynman diagrams for the neutrino-pair mediated (a)AB→ABAB\\rightarrow AB scattering and (b) AA ¯→BB ¯A\\bar{A} \\rightarrow B\\bar{B}scattering.",
"The labels in the figures denote the fieldswhich describe the particles in the process.Let's apply this relation to our process.",
"We are interested in calculating the absorptive part of the $AB \\rightarrow AB$ amplitude mediated by a neutrino-pair, so we need to do a $t$ -channel unitarity cut of the diagram in Fig.REF .",
"Therefore, we should write Eq.",
"(REF ) for the crossed process $A\\bar{A} \\rightarrow B\\bar{B}$ , Fig.REF , with a $\\nu \\nu $ intermediate stateSince the intermediate state is a fermionic one, there should be a spin sum.",
"However, only left-handed neutrinos exist, so in this case it is not necessary., $\\text{Im} \\left\\lbrace \\mathinner {\\langle {B\\bar{B}}|} T \\mathinner {|{A\\bar{A}}\\rangle } \\right\\rbrace = \\frac{1}{2} \\int \\frac{\\mathrm {d}^3 k_1}{(2\\pi )^3 2E_{k_1}}\\,\\frac{\\mathrm {d}^3 k_1}{(2\\pi )^3 2E_{k_1}}\\, \\mathinner {\\langle {\\nu (k_1)\\bar{\\nu }(k_2)}|} T \\mathinner {|{B\\bar{B}}\\rangle }^*\\mathinner {\\langle {\\nu (k_1) \\bar{\\nu }(k_2)}|} T \\mathinner {|{A\\bar{A}}\\rangle }\\,.$ Dropping the $(2\\pi )^4 \\delta ^{(4)}(p_f-p_i)$ global factor from both sides, this equation becomes $ \\begin{aligned}&\\text{Im} \\left\\lbrace {\\cal M}(A\\bar{A} \\rightarrow B\\bar{B}) \\right\\rbrace =\\\\&\\hspace{35.56593pt}=\\frac{1}{2} \\int \\frac{\\mathrm {d}^3 k_1}{(2\\pi )^3 2E_{k_1}}\\,\\frac{\\mathrm {d}^3 k_2}{(2\\pi )^3 2E_{k_2}}\\, (2\\pi )^4 \\delta ^{(4)}(k_1+k_2-p_i) {\\cal M}(B\\bar{B}\\rightarrow \\nu \\bar{\\nu })^*{\\cal M}(A\\bar{A}\\rightarrow \\nu \\bar{\\nu })\\,.\\end{aligned} $ Finally, we can write this expression in an explicitly Lorentz invariant manner, $\\boxed{ \\begin{aligned} \\text{Im} &\\left\\lbrace {\\cal M}(A\\bar{A} \\rightarrow B\\bar{B}) \\right\\rbrace =\\\\&\\hspace{21.33955pt} = \\frac{1}{2} \\int \\frac{\\mathrm {d}^4 k_1}{(2\\pi )^3}\\, \\delta (k_1^2)\\,\\frac{\\mathrm {d}^4 k_2}{(2\\pi )^3}\\, \\delta (k_2^2)\\, (2\\pi )^4 \\delta ^{(4)}(k_1+k_2-p_i) {\\cal M}(B\\bar{B}\\rightarrow \\nu \\bar{\\nu })^*{\\cal M}(A\\bar{A}\\rightarrow \\nu \\bar{\\nu }) \\end{aligned}}$"
],
[
"Low-Energy Contact Interaction",
"The weak interactions of fermions, charged and neutral currents, are described by the Lagrangian densities [1], [7] ${L}_\\text{CC} &= -\\frac{e}{2\\sqrt{2} \\sin \\theta _W} \\left\\lbrace W^\\dagger _\\mu \\left[ \\bar{u}_i \\gamma ^\\mu \\left( 1-\\gamma _5 \\right) V_{ij}\\, d_j + \\bar{\\nu }_i \\gamma ^\\mu \\left( 1-\\gamma _5 \\right) e_i\\right]+ \\text{h.c.} \\begin{aligned} \\mbox{}\\\\ \\mbox{} \\end{aligned} \\right\\rbrace \\,,\\\\\\nonumber &\\hspace{284.52756pt} (i,j = 1^\\text{st},\\, 2^\\text{nd},\\, 3^\\text{rd} \\text{ gen.}) \\\\{L}_\\text{NC} &= -e A_\\mu \\, Q_j \\bar{\\psi }_j \\, \\gamma ^\\mu \\, \\psi _j - \\frac{e}{4 \\sin \\theta _W \\cos \\theta _W}\\, Z_ \\mu \\, \\bar{\\psi }_j \\, \\gamma ^\\mu \\, \\left( g_{V_j} - g_{A_j} \\gamma _5 \\right) \\psi _j\\\\\\nonumber &\\equiv {L}_\\text{QED} + {L}_\\text{Z} \\,, \\hspace{213.39566pt} (\\psi _j= u, d, \\nu _e, e...)$ where $\\theta _W$ is the weak mixing angle.",
"For any elementary particle, the weak neutral couplings are given by $g_V = 2T_3 - 4Q\\sin ^2\\theta _W, \\hspace{85.35826pt} g_A = 2T_3\\,,$ where $T_3$ is the third component of weak isospin and $Q$ is the electric charge.",
"The electroweak charges of the SM fermions are written in Table REF .",
"Table: Electroweak charges of the Standard Model fermions.",
"The index i=1,2,3i=1,2,3 labels the three generations, so that u 1 =u,u 2 =c,u 3 =t...u_1 = u,\\, u_2 = c,\\, u_3 = t...We are interested in calculating the potential associated to a process at low energy, where the limit $|q^2|\\ll M_W^2, M_Z^2$ is valid, so now we'll focus in obtaining the low-energy effective interactions from the above Lagrangians."
],
[
"Effective charged\ncurrent couplings",
"We are describing neutrino scattering against an aggregate of matter, so only the $\\nu _e$ -$e$ charged current contributes to the scattering.",
"Therefore, the only two terms of the interaction Lagrangian which are interesting to our process are ${L}_\\text{CC} = M_W^2 W_\\mu ^\\dagger W^\\mu + W_\\mu ^\\dagger \\, \\bar{\\nu }_e\\, \\Gamma ^\\mu \\, e + W_\\mu \\, \\bar{e} \\, \\Gamma ^\\mu \\, \\nu _e\\,,$ where $\\Gamma ^\\mu \\equiv -\\frac{e}{2\\sqrt{2}\\sin \\theta _W}\\gamma ^\\mu (1-\\gamma _5) \\vspace{11.38092pt}$ and we also wrote the kinetic term of the $W_\\mu $ field.",
"In order to calculate the effective Lagrangian, we integrate the $W_\\mu $ degrees of freedom out of the Lagrangian using its equations of motion, $0 = \\frac{\\partial {L}_\\text{CC}}{\\partial W_\\mu ^\\dagger } = M_W^2 W^\\mu + \\bar{\\nu }_e\\, \\Gamma ^\\mu \\, e\\,,$ so $W_\\mu = -\\frac{1}{M_W^2}\\,\\bar{\\nu }_e\\, \\Gamma _\\mu \\, e =\\frac{e}{2\\sqrt{2} M_W^2 \\sin \\theta _W}\\,\\bar{\\nu }_e\\, \\gamma _\\mu \\,(1-\\gamma _5)\\, e \\,.$ Putting this relation into Eq.",
"(REF ) one easily gets ${L}^\\text{eff}_\\text{CC} = - \\frac{G_F}{\\sqrt{2}} \\left[ \\bar{\\nu }_e\\, \\gamma ^\\mu \\,(1-\\gamma _5)\\, e \\right] \\left[ \\bar{e} \\, \\gamma _\\mu \\,(1-\\gamma _5)\\, \\nu _e \\right]\\,,$ where the Fermi constant is given by $\\frac{G_F}{\\sqrt{2}} = \\frac{e^2}{8M_W^2 \\sin ^2\\theta _W}$ .",
"It is convenient to write this Lagrangian as flavour diagonal—as shown in Fig.REF —, so that we can add both CC and NC Lagrangians.",
"In order to do so, we use the Fierz identity (REF ) and the relation $\\gamma _\\mu \\gamma _\\nu \\gamma ^\\mu = -2 \\gamma _\\nu $ to write ${L}^\\text{eff}_\\text{CC} = -\\frac{G_F}{\\sqrt{2}} \\left[ \\bar{\\nu }_e\\, \\gamma ^\\mu \\,(1-\\gamma _5)\\, \\nu _e \\right] \\left[ \\bar{e} \\, \\gamma _\\mu \\,(1-\\gamma _5)\\, e \\right]\\,.$"
],
[
"Effective neutral current couplings",
"In this case, the interesting Lagrangian to our process is ${L}_\\text{NC} = \\frac{1}{2} M_Z^2 Z_\\mu Z^\\mu - \\frac{e}{4\\sin \\theta _W \\cos \\theta _W} \\,Z_ \\mu \\, \\bar{\\psi }_j \\, \\gamma ^\\mu \\, \\left( g_{V_j} - g_{A_j} \\gamma _5 \\right) \\psi _j\\,,$ where $j = u, d, e, \\nu _e, \\nu _\\mu , \\nu _\\tau $ .",
"As in the previous section, we integrate out the $Z$ degrees of freedom using its equations of motion, $0 = \\frac{\\partial {L}_\\text{NC}}{\\partial Z_\\mu } = M_Z^2 Z^\\mu - \\frac{e}{4\\sin \\theta _W \\cos \\theta _W} \\,\\bar{\\psi }_j \\, \\gamma ^\\mu \\, \\left( g_{V_j} - g_{A_j} \\gamma _5 \\right) \\psi _j\\,,$ so $Z_\\mu = \\frac{e}{4\\sin \\theta _W \\cos \\theta _W M_Z^2} \\,\\bar{\\psi }_j \\, \\gamma _\\mu \\, \\left( g_{V_j} - g_{A_j} \\gamma _5 \\right) \\psi _j\\,.$ Putting this relation in Eq.",
"(REF ) we get ${L}_\\text{NC}^\\text{eff} = -\\frac{G_F}{2\\sqrt{2}}\\left[ \\bar{\\nu }\\, \\gamma ^\\mu \\,(1-\\gamma _5)\\, \\nu \\right] \\left[ \\bar{\\psi }_j \\, \\gamma _\\mu \\,\\left( g_{V_j} - g_{A_j} \\gamma _5 \\right) \\psi _j \\right]\\,,$ as is represented in Fig.REF ."
],
[
"Low-energy effective Lagrangian for matter particles",
"Let us now consider some aggregate of matter $A$ .",
"In the scattering process $A \\nu \\rightarrow A\\nu $ at low energy, the neutrino can interact with the three “elementary” particles which matter is formed with—electrons, protons and neutrons.",
"We can consider that nucleons are point-like Dirac particles because the scattering happens at low energy—i.e.",
"the neutrino is like a large scale probe, so it cannot resolve the structure of nucleons.",
"The vector current is conserved, so both the electric charge $Q$ and the weak vector charge $g_V$ of the nucleon are the sum of its valence quarks' charges, $\\nonumber \\hspace{56.9055pt} Q_p &=1\\,,\\\\\\nonumber Q_n &=0\\,,$ $\\nonumber g_V^p &= 1-4\\sin ^2\\theta _W\\,, \\hspace{28.45274pt} \\\\g_V^n &= -1\\,.",
"$ On the other hand, the axial current is not conserved, so this argument does not apply to the weak axial charge of the nucleon.",
"In fact, Eq.",
"(REF ) shows that the axial coupling is independent of the electric charge—it only depends on the weak isospin coupled to the $W_\\mu ^3$ boson.",
"Therefore, it can be expected due to weak isospinIn fact, for the first generation of quarks, weak and strong isospin coincide.",
"symmetry that the weak neutral axial coupling at low momentum transfer, $q^2\\rightarrow 0$ , is the same as the coupling to the $W_\\mu ^\\pm $ mediated charge current responsible of the $n\\rightarrow p$ process, $g_A = 1.2723\\pm 0.0023$ [3].",
"Taking all of this into account, the Lagrangian describing the $A \\nu \\rightarrow A \\nu $ interaction has three terms, related to the processes $ \\begin{array}{ c l}\\hspace{113.81102pt} \\nu _e + e \\longrightarrow \\nu _e + e\\,, &\\\\\\hspace{113.81102pt}\\nu _i + e \\longrightarrow \\nu _i + e\\,, &\\hspace{85.35826pt} (i = \\mu , \\tau )\\\\\\hspace{113.81102pt}\\nu _j + N \\longrightarrow \\nu _j + N\\,.",
"&\\hspace{85.35826pt} (j = e, \\mu , \\tau )\\end{array} $ The first one is mediated by both charged and neutral currents.",
"Therefore, we have to add the Lagrangians (REF ) and (REF ), so we get $\\nonumber {L}_1 &= -\\frac{G_F}{\\sqrt{2}} \\left[ \\bar{\\nu }_e\\,\\gamma ^\\mu \\,(1-\\gamma _5)\\, \\nu _e \\right]\\left[ \\bar{e} \\, \\gamma _\\mu \\,(1-\\gamma _5)\\,e \\right] - \\\\\\nonumber &\\hspace{56.9055pt}- \\frac{G_F}{2\\sqrt{2}}\\left[ \\bar{\\nu }_e\\, \\gamma ^\\mu \\,(1-\\gamma _5)\\, \\nu _e \\right] \\left[ \\bar{e} \\, \\gamma _\\mu \\,\\left( g_V^e - g_A^e \\gamma _5 \\right) e \\right] = \\\\&= -\\frac{G_F}{2\\sqrt{2}}\\left[ \\bar{\\nu }_e \\, \\gamma ^\\mu \\,(1-\\gamma _5)\\, \\nu _e \\right] \\left[ \\bar{e} \\, \\gamma _\\mu \\,\\left( \\tilde{g}_V^e - \\tilde{g}_A^e \\gamma _5 \\right) e \\right]\\,, $ where we have defined $\\nonumber \\tilde{g}_V^e &= 2 + g_V^e = 1 + 4 \\sin ^2\\theta _W\\,, \\\\\\tilde{g}_A^e &= 2 + g_A^e = 1\\,.",
"$ The second and third ones are only mediated by neutral currents, so they are described by the Lagrangian (REF ), ${L}_2 &= -\\frac{G_F}{2\\sqrt{2}}\\left[ \\bar{\\nu }_i \\, \\gamma ^\\mu \\,(1-\\gamma _5)\\, \\nu _i \\right] \\left[ \\bar{e} \\, \\gamma _\\mu \\,\\left( g_V^e - g_A^e \\gamma _5 \\right) e \\right]\\,, \\hspace{85.35826pt} (i = \\mu , \\tau )\\\\{L}_3 &= -\\frac{G_F}{2\\sqrt{2}}\\left[ \\bar{\\nu }_j \\, \\gamma ^\\mu \\,(1-\\gamma _5)\\, \\nu _j \\right] \\left[ \\bar{N} \\, \\gamma _\\mu \\,\\left( g_V^N - g_A^N \\gamma _5 \\right) N \\right]\\,.",
"\\hspace{66.86414pt} \\begin{aligned} &(j = e, \\mu , \\tau )\\\\ &(N = p, n) \\end{aligned}$ Figure: Fundamental vertex of the effective low-energyLagrangian ().",
"The couplings g V ,g A g_V,\\, g_Adepend on both the neutrino flavor and which is the chargedfermion, as discussed in the text.With all this information, we finally have our whole interaction Lagrangian ${L}= {L}_1 + {L}_2 + {L}_3$ , which is $ \\boxed{\\begin{aligned}{L}= -\\frac{G_F}{2\\sqrt{2}} &\\left\\lbrace \\left[ \\bar{\\nu }_e \\, \\gamma ^\\mu \\,(1-\\gamma _5)\\, \\nu _e \\right] \\left[ \\bar{e} \\, \\gamma _\\mu \\,\\left( \\tilde{g}_V^e - \\tilde{g}_A^e \\gamma _5 \\right) e \\right] + \\begin{aligned} \\mbox{}\\\\ \\mbox{} \\end{aligned} \\right.",
"\\\\&+ \\left[ \\bar{\\nu }_i \\, \\gamma ^\\mu \\,(1-\\gamma _5)\\, \\nu _i \\right] \\left[ \\bar{e} \\, \\gamma _\\mu \\,\\left( g_V^e - g_A^e \\gamma _5 \\right) e \\right] + \\hspace{99.58464pt} (i = \\mu , \\tau ) \\\\&\\left.+ \\left[ \\bar{\\nu }_j \\, \\gamma ^\\mu \\,(1-\\gamma _5)\\, \\nu _j \\right] \\left[ \\bar{N} \\, \\gamma _\\mu \\,\\left( g_V^N - g_A^N \\gamma _5 \\right) N \\right] \\begin{aligned} \\mbox{}\\\\ \\mbox{} \\end{aligned} \\right\\rbrace \\hspace{78.24507pt} \\begin{aligned} &(j = e, \\mu , \\tau )\\\\ &(N = p, n) \\end{aligned}\\end{aligned}}$ All fundamental vertices of this Lagrangian have the same structure, as is represented in Fig.REF .",
"As a check, the couplings we obtained here are (indeed) the same ones stated in [7]."
],
[
"Neutrino-Matter Scattering",
"Once the interaction Lagrangian is written, we can focus on calculating the scattering amplitude between an aggregate of matter and a neutrino, ${\\cal M}(A\\nu \\rightarrow A\\nu )$ .",
"For simplicity, $A$ can be understood as a molecule, composed of $Z_A$ protons and electrons and $N_A$ neutrons—we'd better study electrically neutral systems, with the same number of protons and electrons, because any net-charge electric interaction would be much stronger than the weak interaction we're looking for.",
"As shown in Fig.REF , the process $A \\nu \\rightarrow A\\nu $ is described by an elementary vertex of the interaction Lagrangian (REF ).",
"The different terms of this Lagrangian show explicitly that the coupling neutrino-matter must depend on the flavor of the neutrino, so we will consider them separately."
],
[
"Electron neutrino",
"In the $A (p_1) \\, \\nu _e (k_1) \\rightarrow A(p_2) \\, \\nu _e(k_2) $ case, the amplitude is determined by $i\\,T_{\\nu _e} = \\mathinner {\\langle {A \\nu _e}|} i \\int \\mathrm {d}^4 x \\left[ {L}_1(x) +{L}_2(x) +{L}_3(x) \\right] \\mathinner {|{A \\nu _e}\\rangle } \\equiv i\\,T_{\\nu _e}^{(1)} + i\\,T_{\\nu _e}^{(2)} +i\\, T_{\\nu _e}^{(3)}\\,,$ which we can calculate separately.",
"The contribution of the first term is $\\nonumber T_{\\nu _e}^{(1)} &= \\mathinner {\\langle {A \\nu _e}|} \\int \\mathrm {d}^4 x \\, \\frac{-G_F}{2\\sqrt{2}}\\left[ \\bar{\\nu }_e \\, \\gamma ^\\mu \\,(1-\\gamma _5)\\, \\nu _e \\right] \\left[ \\bar{e} \\, \\gamma _\\mu \\,\\left( \\tilde{g}_V^e - \\tilde{g}_A^e \\gamma _5 \\right) e \\right] \\mathinner {|{A \\nu _e}\\rangle } = \\\\\\nonumber &= -\\frac{G_F}{2\\sqrt{2}}\\int \\mathrm {d}^4 x \\, \\mathinner {\\langle {\\nu _e}|} \\bar{\\nu }_e \\, \\gamma ^\\mu \\,(1-\\gamma _5)\\, \\nu _e \\mathinner {|{\\nu _e}\\rangle } \\, \\mathinner {\\langle {A}|} \\bar{e} \\, \\gamma _\\mu \\,\\left( \\tilde{g}_V^e - \\tilde{g}_A^e \\gamma _5 \\right) e \\mathinner {|{A}\\rangle } \\equiv \\\\&\\equiv -\\frac{G_F}{2\\sqrt{2}}\\int \\mathrm {d}^4 x \\, j^\\mu _{\\nu _e}(x) J_\\mu ^{(1)}(x) \\,.",
"$ Figure: Lowest order Feynman diagram for the Aν→AνA \\nu \\rightarrow A\\nu scattering in the low-energy effective weak theory.Since neutrinos are elementary particles, the leptonic current is the usual $\\nonumber j^\\mu (x) &= \\mathinner {\\langle {\\nu _e (k_2) }|} \\bar{\\nu }_e (x) \\, \\gamma ^\\mu \\,(1-\\gamma _5)\\, \\nu _e (x) \\mathinner {|{\\nu _e (k_1)}\\rangle } = \\\\&=\\left[ \\bar{u} (k_2)\\, \\gamma ^\\mu \\,(1-\\gamma _5) \\, u(k_1) \\right]\\, e^{-i(k_1-k_2)x}\\,.$ On the other hand, the molecular current matrix element will be $J_\\mu ^{(1)} (x) = e^{-i(p_1-p_2)x}\\, J_\\mu ^{(1)} = e^{-i(p_1-p_2)x}\\, \\mathinner {\\langle {A (p_2)}|} \\bar{e} (0) \\gamma _\\mu \\left(\\tilde{g}_V^e - \\tilde{g}_A^e \\gamma _5 \\right) e (0) \\mathinner {|{ A (p_1)}\\rangle } \\,.$ Since we're looking for a low-energy coherent interaction, it's interesting to analyze separately the different terms in $J_\\mu $ : $\\gamma ^0$ is a scalar quantity, related to the matrix element of $e^\\dagger e$ , which is the number operator, so its contribution is coherent.",
"$\\gamma ^0 \\gamma _5$ is a pseudo-scalar quantity, so its matrix element is related to $\\mathbf {\\sigma }\\mathbf {q} / M$ , where $\\mathbf {\\sigma }$ is the spin of $A$ , $M$ its mass and $\\mathbf {q} = \\mathbf {p}_1 - \\mathbf {p}_2$ .",
"Since this contribution depends on $\\mathbf {\\sigma }$ , it's not coherent.",
"Also, any contribution of the form $\\mathbf {q} / M$ gives a relativistic correction to the potential, so this is another reason why we can ignore this term.",
"$\\mathbf {\\gamma }$ is a polar vector, so its matrix element must be proportional to $\\mathbf {q}/M$ .",
"Again, this is a relativistic correction we won't consider.",
"$\\mathbf {\\gamma }\\gamma _5$ is an axial vector, directly related to the spin of the particle, so this contribution is not coherent.",
"Therefore, the coherent contribution to the molecular current is given by $\\nonumber J_0^{(1)} (x) &= \\mathinner {\\langle {A (p_2)}|} \\bar{e} (x)\\, \\tilde{g}_V^e \\gamma ^0 e (x) \\mathinner {|{ A (p_1)}\\rangle } = \\tilde{g}_V^e \\mathinner {\\langle {A (p_2)}|} e^\\dagger (x) \\, e (x) \\mathinner {|{ A (p_1)}\\rangle } =\\\\\\nonumber &= \\tilde{g}_V^e \\mathinner {\\langle {A (p_2)}|} \\left[ \\int \\mathrm {d}^4y \\mathinner {|{y}\\rangle }\\mathinner {\\langle {y}|} \\right] e^\\dagger (x) \\, e (x) \\left[ \\int \\mathrm {d}^4z \\mathinner {|{z}\\rangle }\\mathinner {\\langle {z}|} \\right] \\mathinner {|{ A (p_1)}\\rangle } =\\\\\\nonumber &=\\tilde{g}_V^e \\int \\mathrm {d}^4y\\, \\mathrm {d}^4 z\\, e^{ip_2y}\\, e^{-ip_1z}\\, \\mathinner {\\langle {A (y)}|} e^\\dagger (x) \\, e (x) \\mathinner {|{ A (z)}\\rangle } = \\\\\\nonumber &=\\tilde{g}_V^e \\int \\mathrm {d}^4y\\, \\mathrm {d}^4 z\\, e^{ip_2y}\\, e^{-ip_1z}\\, \\delta ^{(4)}(x-y) \\, \\delta ^{(4)}(x-z) Z_A = \\\\&= Z_A\\, \\tilde{g}_V^e\\, e^{-i(p_1-p_2)x}\\,,$ where we have inserted two Closure Relations, $I = \\int \\mathrm {d}^4x \\mathinner {|{x}\\rangle }\\mathinner {\\langle {x}|}\\,,$ and we have taken into account the fact that $e^\\dagger (x) e(x) $ is the electron number operator at $x$ .",
"Even though $J_0$ is the only relevant component of $J_\\mu $ , it is convenient to keep a relativistic framework—later we'll consider the non-relativistic limit.",
"Therefore, the $T$ matrix element (REF ) is $T_{\\nu _e}^{(1)} = (2\\pi )^4 \\delta ^{(4)}(q + k_1-k_2) \\times \\frac{-G_F}{2\\sqrt{2}} \\, J^{(1)}_\\mu \\left[ \\bar{u} (k_2)\\, \\gamma ^\\mu \\,(1-\\gamma _5) \\, u(k_1) \\right] \\,,$ where $q \\equiv p_1-p_2$ and $J_0^{(1)} = Z_A \\tilde{g}_V^e$ —this last equality, and the following giving $J_0$ values, must be understood as the coherent contribution to $J_0$ given by the number operator of the particle constituents.",
"Analogously, $T_{\\nu _e}^{(2)} &=0\\,,\\\\T_{\\nu _e}^{(3)} &= (2\\pi )^4 \\delta ^{(4)}(q + k_1-k_2) \\times \\frac{-G_F}{2\\sqrt{2}} \\, J^{(3)}_\\mu \\left[ \\bar{u} (k_2)\\, \\gamma ^\\mu \\,(1-\\gamma _5) \\, u(k_1) \\right] \\,.$ where $J_0^{(3)} = Z_A g_V^p + N_A g_V^n$ .",
"Adding all contributions and dropping the $(2\\pi )^4 \\delta (p_i-p_f)$ factor, we get ${\\cal M}(A\\nu _e \\rightarrow A\\nu _e) = -\\frac{G_F}{2\\sqrt{2}} \\, J_{A, \\mu }^e \\left[ \\bar{u} (k_2)\\, \\gamma ^\\mu \\,(1-\\gamma _5) \\, u(k_1) \\right]\\,,$ where $J_{A, \\mu }^e$ is the molecular current in the scattering with an electron neutrino.",
"Using the weak charges from (REF ) and (REF ), $\\nonumber \\hspace{56.9055pt} g_V^p &=1 - 4 \\sin ^2\\theta _W\\,,\\\\\\nonumber g_V^n &= -1\\,,\\\\\\nonumber $ $\\nonumber g_V^e &= -1+4\\sin ^2\\theta _W = - g_V^p\\,, \\hspace{28.45274pt} \\\\\\nonumber \\tilde{g}_V^e &= 2 + g_V^e = 2- g_V^p\\,,\\\\\\nonumber $ we find $J_{A, 0}^e = 2Z_A-N_A$ .",
"At this level, we remind the reader that the first term comes from charged current interaction with electrons, while the second one comes from neutral currents with neutrons—neutral currents with protons and electrons cancel out."
],
[
"Muon and tau neutrino",
"The muon and tau flavors have the same contribution to the Effective Lagrangian, so the scattering amplitudes for the processes $A \\nu _\\mu \\rightarrow A \\nu _\\mu $ and $A \\nu _\\tau \\rightarrow A \\nu _\\tau $ must be the same.",
"Therefore, we can consider both of them simultaneously and calculate $i\\,T_{\\nu _j} = \\mathinner {\\langle {A \\nu _j}|} i \\int \\mathrm {d}^4 x \\left[ {L}_1(x) +{L}_2(x) +{L}_3(x) \\right] \\mathinner {|{A \\nu _j}\\rangle } \\equiv i\\,T_{\\nu _j}^{(2)} +i\\, T_{\\nu _j}^{(3)}\\,,$ where $j=\\mu , \\tau $ and ${L}_1$ does not contribute because it only has electron neutrinos.",
"Following the same steps than in the previous section, we get $T_{\\nu _j}^{(2)} &= (2\\pi )^4 \\delta ^{(4)}(q + k_1-k_2) \\times \\frac{-G_F}{2\\sqrt{2}} \\, J_{A, \\mu }^{(2)} \\left[ \\bar{u} (k_2)\\, \\gamma ^\\mu \\,(1-\\gamma _5) \\, u(k_1) \\right] \\,,\\\\T_{\\nu _j}^{(3)} &= (2\\pi )^4 \\delta ^{(4)}(q + k_1-k_2) \\times \\frac{-G_F}{2\\sqrt{2}} \\, J_{A, \\mu }^{(3)} \\left[ \\bar{u} (k_2)\\, \\gamma ^\\mu \\,(1-\\gamma _5) \\, u(k_1) \\right] \\,,$ where $J_{A, 0}^{(2)} = Z_A g_V^e$ and $J_{A, 0}^{(3)} = Z_A g_V^p + N_A g_V^n $ .",
"Finally, dropping the $(2\\pi )^4 \\delta (p_i-p_f)$ factor, we can write ${\\cal M}(A\\nu _j \\rightarrow A\\nu _j) = -\\frac{G_F}{2\\sqrt{2}} \\,J_{A, \\mu }^j \\left[ \\bar{u} (k_2)\\, \\gamma ^\\mu \\,(1-\\gamma _5) \\, u(k_1) \\right]\\,, \\hspace{56.9055pt} \\left( j=\\mu , \\tau \\right)\\,,\\\\$ where $J_{A, 0}^j = -N_A$ .",
"As before, the neutral current interactions for electrons and protons cancel each other in neutral (of electric charge) matter."
],
[
"The weak flavor charge of aggregate matter",
"Up to this point, we have calculated the amplitudes of the processes described by all fundamental vertices of our Lagrangian, so it's convenient to sum up our results and analyze them.",
"In order to do that, it's useful to compare with well-known theories.",
"Let's consider QED.",
"In this theory, a process described by the fundamental vertex would involve two fermions and a photon.",
"If we take the photon on-shell and drop external-leg fermion factors, the $iM$ would be [line width = 1.5pt, scale = 1.7] [fermion] (-0.87, 0.5) – (0,0); t (-0.87, 0.5)[above left]$\\psi $ ; [fermionbar] (0.87, 0.5) – (0,0); t (0.87, 0.5)[above right]$\\psi $ ; [vector] (0,0) – (0, -1); t (0, -1)[above right]$\\gamma (k)$ ; t (2, -0.25)$ = -i e Q \\gamma ^\\mu \\epsilon _\\mu (k)\\,,$ ; $\\mbox{}$ where $Q$ is the electric charge of the fermion field.",
"Due to vector current conservation, this vertex does also apply to non-fundamental particles, which have an electric charge equal to the sum of its constituents' charges—and the amplitude would be this charge times the coupling $e$ .",
"This same behavior appears in weak interactions.",
"The only flavor-diagonal interaction is the one mediated by neutral currents, with the fundamental vertex [line width = 1.5pt, scale = 1.7] [fermion] (-0.87, 0.5) – (0,0); t (-0.87, 0.5)[above left]$\\psi $ ; [fermionbar] (0.87, 0.5) – (0,0); t (0.87, 0.5)[above right]$\\psi $ ; [vector] (0,0) – (0, -1); t (0, -1)[right]$Z (k)$ ; t (1, -0.425)[right]$ = -i \\frac{e}{4\\sin \\theta _W \\cos \\theta _W} \\left\\lbrace \\left( 2T_3-4Q\\sin ^2\\theta _W \\right) \\gamma ^\\mu - 2T_3 \\gamma ^\\mu \\gamma _5 \\right\\rbrace \\epsilon _\\mu (k)\\,.$ ; $\\mbox{}$ As is thoroughly discussed in [8], the vector current conservation allows us to talk about a weak charge of the fermion field $\\psi $ , which is $Q_W = 2T_3-4Q\\sin ^2\\theta _W$ , such that the weak charge of a composed particle is the sum of its constituents', as happens with electric charge.",
"However, we can't talk about the axial coupling as a charge, since the axial current is not conserved.",
"Once the concept of a weak charge has been introduced, we see that the vector part of this amplitude has the same structure as the QED one, a coupling times a charge times a mediating-particle external-leg factor.",
"According to this idea, we can expect our amplitudes to have this structure too.",
"Indeed, both Eqs.",
"(REF ) and (REF ) can be written (in the non-relativistic limit) as [line width = 1.5pt, scale = 1.7] (-1,-1) rectangle (1.2,1.2); [fermion](150:1.5) – (150:0.3cm); t (150:1.7)[left] $A$ ; [shift=(35:4pt)] [fermion](150:1.5) – (150:0.3cm); [shift=(35:-4pt)] [fermion](150:1.5) – (150:0.3cm); (150:0.3) – (150:0); [fermionbar](30:1.5) – (30:0.3cm); t (30:1.7)[right] $A$ ; [shift=(-35:-4pt)] [fermionbar](30:1.5) – (30:0.3cm); [shift=(-35:4pt)] [fermionbar](30:1.5) – (30:0.3cm); (30:0.3) – (30:0); [fill=black] (0,0) circle (.3cm); [fill=white] (0,0) circle (.29cm); (0,0) circle (.3cm); in -.9,-.8,...,.3 [line width=1 pt] (,-.3) – (+.6,.3); [shift=(0, -0.3)] [fermion] (-150:1.5) – (0,0); t (-150:1.7)[left]$\\nu _i (k_1)$ ; [fermionbar](-30:1.5) – (0,0); t (-30:1.7)[right]$\\nu _i (k_2)$ ; t (3, -0.3)$ = i \\frac{G_F}{2\\sqrt{2}}\\, J_{A, 0}^i\\, \\left[ \\bar{u} (k_2) \\gamma ^0 (1-\\gamma _5) u(k_1) \\right]\\,,$ ; $\\mbox{}$ where $Q_{W,A}^i \\equiv J_{A, 0}^i$ is the weak charge of the aggregate of matter $A$ .",
"It depends on the flavor of the neutrino, so we can speak of three weak flavor charges of aggregate matter, which are given by $\\nonumber Q_{W,A}^e &= 2Z_A-N_A\\,,\\\\Q_{W,A}^\\mu = Q_{W,A}^\\tau &= - N_A\\,.$ Eqs.",
"(REF ) state the fact that, whereas aggregate matter is neutral of electric charge, it is not neutral of weak charges!",
"It's interesting to analyze the value of those charges for “normal” matter.",
"In order to do that, we'll look at stable nuclei.",
"According to the semi-empirical mass formula [9], the $(Z,N)$ values of stable nuclei are related by $Z \\approx \\frac{A}{2+0.0157 A^{2/3}}\\,,$ where $A \\equiv Z+N$ , as is represented in Fig.REF .",
"Using those pairs of values, the weak charges of each element (neutral atom) are represented in Fig.REF , where we see that the electron neutrino weak charge is always positive, while the muon and tau neutrino charges are always negative.",
"The weak charge of aggregate matter is obtained from Fig.REF by multiplying by the number of the constituent atoms."
],
[
"Long-Range Weak Interaction Potential",
"After analyzing the $A\\nu \\rightarrow A\\nu $ scattering amplitude, we can focus on calculating the interaction potential.",
"We'll begin using Eq.",
"(REF ) to determine the absorptive part of the $AB\\rightarrow AB$ amplitude.",
"After that, we'll use Eq.",
"(REF ) to obtain the potential."
],
[
"Absorptive part of $AB \\rightarrow AB$ at low {{formula:b87f7bb6-8f21-48d8-8621-8c22d308ed36}}",
"In order to get the absorptive part of the scattering amplitude, we need to compute the crossed quantity Im$\\lbrace {\\cal M}(A \\bar{A} \\rightarrow B \\bar{B}) \\rbrace $ , as written in Eq.",
"(REF ).",
"Therefore, we need to cross the amplitude we calculated in the previous section, from ${\\cal M}(A\\nu _i \\rightarrow A\\nu _i) = -\\frac{G_F}{2\\sqrt{2}} \\,J_{A, \\mu }^i \\left[ \\bar{u} (k_2)\\, \\gamma ^\\mu \\,(1-\\gamma _5) \\, u(k_1) \\right]$ to ${\\cal M}(A \\bar{A}\\rightarrow \\nu _i \\bar{\\nu }_i) = -\\frac{G_F}{2\\sqrt{2}} \\, \\tilde{J}_{A, \\mu }^i \\left[ \\bar{u} (k_2)\\, \\gamma ^\\mu \\,(1-\\gamma _5) \\, v(k_1) \\right]\\,,$ where $\\tilde{J}_\\mu $ is the crossed molecular current, which still satisfies $\\tilde{J}^i_0 = Q^i_W$ in the non-relativistic limit.",
"According to Eq.",
"(REF ), the absorptive part of ${\\cal M}(A\\bar{A}\\rightarrow B \\bar{B})$ is determined by the quantity ${\\cal M}(A \\bar{A} \\rightarrow \\nu _i \\bar{\\nu }_i) {\\cal M}^*(B \\bar{B} \\rightarrow \\nu _i\\bar{\\nu }_i)$ , which we can now evaluate.",
"From now on we'll work in a simplified case, assuming that neutrinos are masslessThe non-vanishing mass of neutrinos will affect the behavior of the potential at the longest range—its implications will be announced in Section .",
"Prospects., so $\\nonumber {\\cal M}(A \\bar{A} \\rightarrow \\nu _i \\bar{\\nu }_i)&{\\cal M}^*(B \\bar{B} \\rightarrow \\nu _i \\bar{\\nu }_i) =\\\\\\nonumber &= \\frac{G_F^2}{8}\\, \\tilde{J}_{A, \\mu }^i \\, \\tilde{J}_{B, \\nu }^i \\left[ \\bar{u} (k_2)\\, \\gamma ^\\mu \\,(1-\\gamma _5) \\, v(k_1) \\right] \\left[ \\bar{v}(k_1)\\, \\gamma ^\\nu \\,(1-\\gamma _5) \\, u(k_2) \\right] = \\\\\\nonumber &= \\frac{G_F^2}{8}\\, \\tilde{Z}^i_{\\mu \\nu } \\text{ Tr}\\left[ {k_1} \\gamma ^\\nu (1-\\gamma _5) {k_2} \\gamma ^\\mu (1-\\gamma _5) \\right] = \\\\\\nonumber &= \\frac{G_F^2}{4}\\, \\tilde{Z}^i_{\\mu \\nu } \\text{ Tr}\\left[ {k_1} \\gamma ^\\nu {k_2} \\gamma ^\\mu (1-\\gamma _5)\\right] = \\\\&= G_F^2\\, \\tilde{Z}^i_{\\mu \\nu } \\, \\left[ k_1^\\mu k_2^\\nu + k_1^\\nu k_2^\\mu - g^{\\mu \\nu }(k_1 k_2) + a^{\\mu \\nu } \\right]\\,,$ where we defined $\\tilde{Z}^i_{\\mu \\nu } \\equiv \\tilde{J}_{A, \\mu }^i \\, \\tilde{J}_{B,\\nu }^i$ and $a^{\\mu \\nu }$ is some antisymmetric tensor which we will no longer consider because it vanishes in the non-relativistic limit, where the only relevant component is $\\mu = \\nu = 0$ .",
"Considering the contributions of the three neutrino flavors, the absorptive part is $\\nonumber \\text{Im} \\left\\lbrace {\\cal M}\\right.&\\left.\\hspace{-3.0pt}(A\\bar{A} \\rightarrow B \\bar{B}) \\right\\rbrace = \\\\\\nonumber &= \\frac{G_F^2}{8\\pi ^2}\\,\\left( \\sum _f \\tilde{Z}^f_{\\mu \\nu } \\right)\\, \\int \\mathrm {d}^4 k_1\\, \\delta (k_1^2)\\, \\delta (k_2^2)\\,\\left[ k_1^\\mu k_2^\\nu + k_1^\\nu k_2^\\mu - \\frac{1}{2} sg^{\\mu \\nu } \\right] = \\\\\\nonumber &= \\frac{G_F^2}{8\\pi ^2}\\,\\left( \\sum _f \\tilde{Z}^f_{\\mu \\nu } \\right)\\, \\int \\mathrm {d}^4 k_1\\, \\delta (k_1^2)\\, \\delta (k_2^2)\\,\\left[ -2k_1^\\mu k_1^\\nu + \\left( k_1^\\mu q^\\nu + k_1^\\nu q^\\mu \\right) - \\frac{1}{2} s g^{\\mu \\nu } \\right] = \\\\\\nonumber &= \\frac{G_F^2}{8\\pi ^2}\\,\\left( \\sum _f \\tilde{Z}^f_{\\mu \\nu } \\right)\\, \\frac{\\pi }{2}\\,\\left[ -\\frac{2}{3}\\left( q^\\mu q^\\nu - \\frac{1}{4} t g^{\\mu \\nu } \\right) + \\frac{1}{2} \\left( q^\\mu q^\\nu + q^\\nu q^\\mu \\right) - \\frac{1}{2} s g^{\\mu \\nu } \\right] = \\\\& = \\frac{G_F^2}{24\\pi }\\,\\left( \\sum _f \\tilde{Z}^f_{\\mu \\nu } \\right)\\, \\left[\\,q^\\mu q^\\nu - s g^{\\mu \\nu } \\,\\right]\\,, $ where we used $k_2 = q - k_1$ in the second line and all integrals needed in the third line are stated in [10]—we demonstrate them in Appendix REF .",
"As seen, the tensor structure of Eq.",
"(REF ) is transverse, a requirement which any quantity built from conserved currents must satisfy.",
"We can cross this result back to the $t-$ channel applying $s \\rightarrow t$ , so that $\\text{Im} \\left\\lbrace {\\cal M}(AB \\rightarrow AB) \\right\\rbrace = \\frac{G_F^2}{24\\pi }\\,\\left( \\sum _f Z^i_{\\mu \\nu } \\right)\\, \\left[ q^\\mu q^\\nu - t g^{\\mu \\nu } \\right]\\,.$ Now it's easy to evaluate the non-relativistic limit.",
"As discussed before, the only relevant component of the molecular current for coherent interactions is the scalar contribution to $J_0$ , so we can take $\\text{Im} \\left\\lbrace M(AB \\rightarrow AB) \\right\\rbrace = \\frac{G_F^2}{24\\pi }\\,\\left( \\sum _f Q_{W,A}^f \\, Q_{W,B}^f \\right)\\, \\left[ \\left(q^0\\right)^2 - t \\right]\\,.$ Besides, we are looking for a long-range interaction, so $q^0 \\approx 0$ and $\\text{Im} \\left\\lbrace M(AB \\rightarrow AB) \\right\\rbrace = - \\frac{G_F^2}{24\\pi }\\,\\left( \\sum _f Q_{W,A}^f \\, Q_{W,B}^f \\right)\\, t \\,,$ where $\\sum _f Q_{W,A}^f \\, Q_{W,B}^f = (2Z_A-N_A)(2Z_B-N_B) +2N_AN_B\\,.$ Figure: Weak coupling ∑Q W f Q W f \\sum Q_W^f Q_W^f, which is written inEq.",
"(), for the elements of the valleyof stability (Fig.",
"), each one interactingwith itself.",
"The gravitational couplingM 2 M^2/m p 2 m_p^2 ≈(Z+N) 2 \\approx (Z+N)^2, neglectingbinding energies, is also represented.As Eq.",
"(REF ) shows, all the flavor dependence of the absorptive part—and, therefore, of the potential—in the limit of massless neutrinos is factorized in the weak charges.",
"As we saw in Fig.REF , all the stable elements have the same sign for the weak charges, $Q_W^e > 0$ and $Q_W^{\\mu ,\\tau }<0$ .",
"This implies that, for any pair of elements—and therefore for any pair of molecules—this coupling has a positive sign, so the resulting force will have the same character—whether repulsive or attractive—for any pair of aggregates of matter.",
"We'll find out which of those two cases is the right one in the next section.",
"Let's see the behavior of this quantity for some cases.",
"As we did in the previous section, we'll consider only stable nuclei.",
"Since this is an interaction, we have to choose sets of two elements—we'll consider the interaction of each element with itself.",
"In Fig.REF we show the quantity $\\sum Q^f_{W,A}Q^f_{W,A}$ for the element $A$ as a function of the atomic number $Z_A$ for the stable nuclei.",
"In the same Figure we compare the weak coupling with the gravitational one.",
"It's seen that both of them increase with the number of particles of the systems interacting, but they scale differently, even when the binding energy is neglected.",
"That means that our coherent weak interaction could introduce a deviation from the Equivalence Principle, as was announced in [2]."
],
[
"Neutrino-pair exchange potential",
"After obtaining this result, the only remaining step in the calculation of the interaction potential is computing the integral (REF ), with the branching point at $t_0 = 0$ (for massless neutrinos).",
"Using the Im$\\lbrace M\\rbrace $ obtained in Eq.",
"(REF ), $V(r) = \\frac{G_F^2}{24 \\pi }\\,\\left( \\sum _f Q_{W,A}^f \\, Q_{W,B}^f \\right) \\frac{1}{4\\pi ^2 r}\\, \\int _0^\\infty \\, \\mathrm {d}t\\, t \\, e^{-\\sqrt{t} \\,r}\\,,$ so we have to evaluate this integral.",
"A primitive $P(t)$ is calculated in Appendix REF , so we obtain the integral (say $I$ ) using Barrow's Rule, $I = P(t\\rightarrow \\infty ) - P(t\\rightarrow 0)$ .",
"If we assume $r \\ne 0$ when calculating the limits—which is valid, since we are looking for a long-range interaction—this contribution to the potential is $I = \\frac{12}{r^4} \\,.$ With this result, we can finally write $\\boxed{ \\hspace{7.11317pt}\\begin{aligned} \\mbox{}\\\\ \\mbox{} \\end{aligned} V(r) = \\frac{G_F^2}{8\\pi ^3}\\,\\left( \\sum _f Q_{W,A}^f \\, Q_{W,B}^f \\right) \\frac{1}{r^5} \\hspace{7.11317pt}}\\,,$ which has an associated force given by $\\mathbf {F}(r) = - \\mathbf {\\nabla } V(r) = \\frac{5 G_F^2}{8 \\pi ^3}\\,\\left( \\sum _f Q_{W,A}^f \\, Q_{W,B}^f \\right) \\frac{\\mathbf {\\hat{r}}}{r^6}\\,,$ where $\\hat{r}$ is the radial unit vector.",
"We have obtained a long-range interaction which is repulsive for ordinary matter, since the weak coupling is always positive—as we showed in Fig.REF .",
"This is a difference with the gravitational force that we can use to distinguish them—and we can also look for the deviations from the Equivalence Principle we discussed in the previous section.",
"The other interactions that appear in electrically neutral systems are the residual electromagnetic Van der Waals ($\\sim r^{-7}$ ) or Casimir-Polder ($\\sim r^{-8}$ ) forces, which have a lower range (larger inverse power law) than our weak force, even though they're stronger.",
"It would be interesting to find systems with low electromagnetic momenta, so that this interactions became weaker, as the ones described in [11]."
],
[
"Conclusions",
"We began this work reviewing the relation between the description of an interaction process in the framework of a Quantum Field Theory and in terms of an interaction potential.",
"As shown in Section , the Feynman amplitude of an elastic scattering process and the effective potential describing this interaction are related, in Born approximation, by a Fourier Transform.",
"This is the result one would expect, taking into account that $M(q^2)$ describes the interaction process in momentum space, while $V(r)$ describes the interaction in position space.",
"Another interesting detail that stems from the fact that we are calculating a long-range interaction is that we needn't calculate the whole amplitude of the process in order to determine the potential.",
"This is due to the fact that $M(q^2)$ at $|q^2|\\rightarrow \\infty $ gives short-range contributions, so the potential is determined by Im$\\lbrace M(q^2) \\rbrace $ through an unsubtracted dispersion relation.",
"Therefore, we could use the unitarity relation from Section to avoid the calculation of $M(q^2)$ (a $1-$ loop quantity) and compute a tree-level process instead.",
"Besides, the fact that we were interested in the low-energy limit allowed us to work in the framework of an effective theory where Charged Currents and Neutral Currents could be written in the form of a contact interaction, so we worked with only one interaction vertex.",
"The results from Sections - made it clear that the effective potential was determined by the amplitude of the process $A \\nu \\rightarrow A\\nu $ , which we calculated in Section .",
"Although this was a quite straightforward calculation, it gave rise to a very interesting concept—the weak flavor charge of aggregate matter.",
"Indeed, the coupling of bulk matter to a neutrino is proportional to $G_F$ with a charge that depends on the flavor of the neutrino, $Q_W^e = 2Z-N\\,, \\hspace{85.35826pt} Q_W^\\mu = Q_W^\\tau = -N\\,.$ These are the weak flavor charges for electrically neutral matter—the case we are interested in, since any non-zero electric charge would produce an electromagnetic interaction much stronger than our weak interaction.",
"This amplitude was the last ingredient needed to compute the effective potential, which gives raise to the repulsive force $\\mathbf {F} = \\frac{5 G_F^2}{8 \\pi ^3} \\left( \\sum _f Q_{W,A}^f Q_{W,B}^f \\right) \\frac{\\mathbf {\\hat{r}}}{r^6}\\,,$ where all flavor dependence is in the weak charges.",
"This is the coherent contribution to the force, which we obtained from the vector charge $J^0$ , proportional to $\\gamma ^0$ .",
"The first correction to this result would come from the spin dependent contribution to $\\mathbf {J}$ , that comes from $\\mathbf {\\gamma }\\gamma _5$ .",
"The other two contributions to the current, proportional to $\\gamma ^0\\gamma _5$ and $\\mathbf {\\gamma }$ , give relativistic corrections $\\sim \\frac{1}{M}$ .",
"In the long-range regime we are looking at, there are two other important interactions: residual electromagnetic interactions and gravitation.",
"For ordinary molecules, Van der Waals forces are much stronger than our weak interaction at short distances, so it would be interesting to look for a system where the first electromagnetic moments are zero.",
"In the case of gravitation, there are two traits of this weak interaction that can help to distinguish between them in an experiment: this force is repulsive—while gravitation is attractive—and its charge is proportional to the number of particles but not to their mass, so it would produce a signal that deviates from the Equivalence Principle.",
"In any case, joining the previous ideas with the recent development of atomic traps [12]—not ionic traps—can be the key to observe this interaction in an experiment."
],
[
"Prospects",
"The long-range potential obtained in this work, Eq.",
"(REF ), is valid and of interest for distances between nanometers and microns.",
"The short-distance limit comes from the requirement of having neutral (of electric charge) systems of aggregate matter, while the long-distance limit is imposed by a non-vanishing value of the absolute mass of the neutrino—indeed, the range of this interaction for neutrinos of $m\\sim 0.1$ eV is of the order of $R \\sim \\frac{\\hbar c}{mc^2} = \\frac{197 \\text{ MeV fm}}{0.1 \\text{ eV}} \\sim 10^9 \\text{ fm} = 1\\, \\mu \\text{m}\\,.$ In this region, the effective potential will become of Yukawa type instead of the inverse power law.",
"We can get a first idea on the dependence of the potential with $m$ changing slightly this work's result.",
"If we had integrated Eq.",
"(REF ) from a branching point at $t_0 =~4m^2$ , the potential would have been $V(r) = \\frac{G_F^2}{8 \\pi ^3}\\,\\left( \\sum _f Q_{W,A}^f \\, Q_{W,B}^f \\right) \\left( \\frac{1}{r^5} + \\frac{2m}{r^4} + \\frac{2m^2}{r^3} + \\frac{4m^3}{3r^2}\\right) e^{-2mr}\\,,$ which depends on $m$ not only in the Yukawa exponential, but also in the preceding inverse power termsOf course, the computation of the potential at finite $m$ is not so trivial—the mass has to be included in the absorptive part of the amplitude—, but it serves for illustrating the kind of changes that will occur..",
"The neutrino mass dependence of the effective potential in the long-range behavior opens novel directions in the study of the most interesting pending questions on neutrino properties: absolute neutrino mass (from the range), flavor dependence and mixing (from the weak charges in the interaction) and, hopefully, with two neutrino exchange, the exploration of the most crucial open problem in neutrino physics: whether neutrinos are Dirac or Majorana particles.",
"The study of these problems will be the subject of my immediate future research work.",
"On the one hand, it's necessary to calculate the form of this interaction with a finite mass for the neutrino.",
"In fact, two calculations are needed: for Dirac neutrinos and for Majorana neutrinos.",
"On the other hand, the collaboration with the experimental groups involved in neutral traps will be initiated this summer during my stay at CERN, in order to find out whether the low electromagnetic interacting systems mentioned in our Conclusions could be implemented.",
"toc tocpartAppendices Appendices"
],
[
"Units",
"This work is written using the Natural System of Units, where $\\hbar = c = k_B = 1$ .",
"We describe electromagnetic quantities with the Heavyside system, $\\epsilon _0 = \\mu _0 = 1$ , so that the fine-structure constant is given by $\\alpha = e^2/4\\pi \\approx 1/137$ ."
],
[
"Relativity end Tensors",
"We define the Minkowsi metric tensor with signature $(+, -, -, -)$ , as $g_{\\mu \\nu } = g^{\\mu \\nu } = \\left( \\begin{array}{c c c c}1&0&0&0\\\\0&-1&0&0\\\\0&0&-1&0\\\\0&0&0&-1\\\\\\end{array} \\right)\\,,$ so that any 4-vector can be written as $x^\\mu = \\left( x^0, \\mathbf {x} \\right)$ .",
"We also use the Einstein Summation Convention, so scalar products can be written as $x\\cdot p = x^\\mu p_\\mu = g_{\\mu \\nu } x^\\mu p^\\nu = x^0 p^0 - \\mathbf {x}\\cdot \\mathbf {p}$ .",
"We can also write $x_\\mu = g_{\\mu \\nu } x^\\nu = \\left( x^0, -\\mathbf {x} \\right)$ , and the derivative operator is $\\partial _\\mu \\equiv \\frac{\\partial }{\\partial x^\\mu } =\\left( \\partial _t, \\mathbf {\\nabla } \\right)$ .",
"We use the totally antisymmetric tensor with the convention $\\epsilon ^{0 1 2 3} = +1 = - \\epsilon _{0123}$ ."
],
[
"Fourier Transforms",
"We define Fourier Transforms so that all $2\\pi $ factors are included in the momentum integration, $$ $4-$ dimensional FT: $\\nonumber f (x) &= \\int \\frac{\\mathrm {d}^4 k}{(2\\pi )^4}\\, e^{-ikx}\\, \\tilde{f}(k)\\,,\\\\\\nonumber \\tilde{f}(k) & = \\int \\mathrm {d}^4 x \\, e^{ikx}\\, f(x)\\,,$ $3-$ dimensional FT: $\\nonumber f (\\mathbf {x}) &= \\int \\frac{\\mathrm {d}^3 k}{(2\\pi )^3}\\, e^{i\\mathbf {k} \\mathbf {x}}\\, \\tilde{f}(\\mathbf {k})\\,,\\\\\\tilde{f}(\\mathbf {k}) & = \\int \\mathrm {d}^3 x \\, e^{-i\\mathbf {k} \\mathbf {x}}\\, f(\\mathbf {x})\\,,$ Other $2\\pi $ factors come from the following expression for the Dirac delta, $\\int \\mathrm {d}^4 x \\, e^{ikx} = (2\\pi )^4 \\delta ^{(4)}(k) \\,.$"
],
[
"Diracology",
"As is well known, Dirac gamma matrices are required to satisfy the relations $\\lbrace \\gamma ^\\mu , \\gamma ^\\nu \\rbrace = 2g^{\\mu \\nu }\\,, \\hspace{85.35826pt}[\\gamma ^\\mu , \\gamma ^\\nu ] = -2i\\sigma ^{\\mu \\nu }.$ Also, $(\\gamma ^0)^2 = - (\\gamma ^i)^2 = 1, \\hspace{85.35826pt} \\gamma _\\mu ^\\dagger = \\gamma ^0 \\gamma _\\mu \\gamma ^0.$ We define the fifth gamma matrix as $\\gamma _5 \\equiv i \\gamma ^0 \\gamma ^1 \\gamma ^2 \\gamma ^3 = -\\frac{i}{4!}",
"\\epsilon _{\\alpha \\beta \\gamma \\delta }\\, \\gamma ^\\alpha \\gamma ^\\beta \\gamma ^\\gamma \\gamma ^\\delta $ , which satisfies $(\\gamma _5)^2 = -1, \\; \\gamma _5^\\dagger =\\gamma _5$ .",
"Therefore, the quirality projectors can be written as $P_R = \\frac{1+\\gamma _5}{2}, \\hspace{85.35826pt} P_L = \\frac{1-\\gamma _5}{2}.$ Some useful contractions are $&\\gamma ^\\mu \\gamma _\\mu = 4\\,,\\hspace{193.47882pt}\\\\&\\gamma ^\\mu \\gamma ^\\nu \\gamma _\\mu = -2 \\gamma ^\\nu ,\\\\&\\gamma ^\\mu \\gamma ^\\alpha \\gamma ^\\beta \\gamma _\\mu = 4 g^{\\alpha \\beta },\\\\&\\gamma ^\\mu \\gamma ^\\nu \\gamma ^\\alpha \\gamma ^\\beta \\gamma _\\mu = -2\\gamma ^\\beta \\gamma ^\\alpha \\gamma ^\\nu .$ Some useful trace identities are $&\\text{Tr}\\left[ \\gamma ^\\mu \\gamma ^\\nu \\right] = 4 g^{\\mu \\nu },\\\\&\\text{Tr}\\left[ \\gamma ^\\mu \\gamma ^\\nu \\gamma _5 \\right] = 0\\,,\\\\&\\text{Tr}\\left[ \\gamma ^\\mu \\gamma ^\\nu \\gamma ^\\alpha \\gamma ^\\beta \\right] = 4 \\left( g^{\\mu \\nu } g^{\\alpha \\beta } - g^{\\mu \\alpha }g^{\\nu \\beta }+g^{\\mu \\beta }g^{\\nu \\alpha } \\right), \\\\&\\text{Tr}\\left[ \\gamma ^\\mu \\gamma ^\\nu \\gamma ^\\alpha \\gamma ^\\beta \\gamma _5 \\right] = -4i \\, \\epsilon ^{\\mu \\nu \\alpha \\beta },\\\\&\\text{Tr}\\left[ \\gamma ^{\\mu _1}\\gamma ^{\\mu _2}...\\gamma ^{\\mu _{2k+1}} \\right] = 0\\,.$ For any $4-$ vector $a^\\mu $ , we define ${a} \\equiv \\gamma _\\mu a^\\mu $ ."
],
[
"Fierz Identity",
"Let us consider the Dirac-scalar quantity $\\left[ \\bar{u}_1 A P_L u_2 \\right] \\left[ \\bar{u}_3 P_R B u_4 \\right]\\,,$ where $A$ and $B$ are arbitrary matrices in Dirac space, $P_{L,R}$ are the quirality projectors from Eq.",
"(REF ) and the four $u_i$ are Dirac spinorsIf any of these spinors were a $v$ spinor, nothing in this section would change—the Identity would still hold..",
"The set of matrices $\\Gamma _i = \\lbrace 1, \\gamma _5, \\gamma ^\\mu P_L, \\gamma ^\\mu P_R,\\sigma ^{\\mu \\nu } \\rbrace $ are a basis of Dirac space, so we can expand $u_2 \\bar{u}_3 = \\sum _i \\alpha _i \\Gamma _i = \\alpha _1 1 + \\alpha _5 \\gamma _5 + \\alpha _L^\\mu \\gamma _\\mu P_L + \\alpha _R^\\mu \\gamma _\\mu P_R + \\alpha _S^{\\mu \\nu } \\sigma _{\\mu \\nu }\\,.$ Since we have this expansion between quiarilty projectors, we can simplify $P_L u_2 \\bar{u}_3 P_R = P_L \\left( \\sum _i \\alpha _i \\Gamma _i \\right) P_R = \\alpha _R^\\mu \\gamma _\\mu P_R\\,,$ where we have used the relations $P_{L,R}^2 = P_{L,R}\\,,\\hspace{28.45274pt} P_{L,R} P_{R,L} = 0\\,,\\hspace{28.45274pt} P_{L,R}\\gamma _\\mu = \\gamma _\\mu P_{R,L}\\,,\\hspace{28.45274pt} P_{L,R}\\gamma _5 = \\gamma _5 P_{L,R}$ to show that all other terms are zero.",
"We need to calculate the $\\alpha _R^\\mu $ coefficient, so we evaluate the quantity $\\text{Tr}[\\gamma ^\\nu P_L u_2 \\bar{u}_3] = \\text{Tr}\\left[\\gamma ^\\nu P_L \\left( \\sum _i \\alpha _i \\Gamma _i \\right) \\right] = \\alpha _R^\\mu \\text{Tr}[\\gamma ^\\nu P_L \\gamma _\\mu P_R] = 2 \\alpha _R^\\mu \\,,$ which means that we can get the coefficient by computing $\\alpha _R^\\mu = \\frac{1}{2}\\text{Tr}[\\gamma ^\\mu P_L u_2 \\bar{u}_3] = \\frac{1}{4}\\bar{u}_3 \\gamma ^\\mu (1-\\gamma _5) u_2 = \\frac{1}{2} \\bar{u}_3 \\gamma ^\\mu P_L u_2\\,.$ Putting this relation into Eq.",
"(REF ) one gets $P_L u_2 \\bar{u}_3 P_R = \\frac{1}{2} [\\bar{u}_3 \\gamma ^\\mu P_L u_2] \\gamma _\\mu P_R\\,,$ so that Eq.",
"(REF ) can be rewritten as $\\left[ \\bar{u}_1 A P_L u_2 \\right] \\left[ \\bar{u}_3 P_R B u_4 \\right] = \\frac{1}{2} [\\bar{u}_1 A \\gamma _\\mu P_R B u_4] [\\bar{u}_3 \\gamma ^\\mu P_L u_2]\\,,$ which is the identity we wanted to prove.",
"Notice that, unlike spinors, fermionic fields anticommute, so their version of the Fierz Identity is $\\left[ \\bar{\\psi }_1 A P_L \\psi _2 \\right] \\left[ \\bar{\\psi }_3 P_R B \\psi _4 \\right] = -\\frac{1}{2} [\\bar{\\psi }_1 A \\gamma _\\mu P_R B \\psi _4] [\\bar{\\psi }_3 \\gamma ^\\mu P_L \\psi _2]\\,,$ with an extra minus sign."
],
[
"Yukawa/Coulomb propagator",
"Let's compute the Fourier Transform of the Yukawa propagator, $I \\equiv \\int \\frac{\\mathrm {d}^3 q}{(2\\pi )^3}\\, e^{i\\mathbf {q}\\, \\mathbf {r}}\\, \\frac{1}{q^2 + \\mu ^2}\\,,$ where $q \\equiv |\\mathbf {q}\\,|$ , which will also give the Fourier Transform of the Coulomb propagator taking the limit $\\mu \\rightarrow 0$ .",
"In spherical coordinates, $\\mathrm {d}^3q = q^2\\mathrm {d}q\\, \\mathrm {d}\\cos \\theta \\, \\mathrm {d}\\phi $ , we get $I = \\frac{1}{(2\\pi )^2} \\, \\int _0^\\infty \\mathrm {d}q\\,\\frac{q^2}{q^2+\\mu ^2} \\int _{-1}^{1}\\mathrm {d}\\cos \\theta e^{iqr\\cos \\theta }\\,.$ Integrating over $\\cos \\theta $ we get $\\nonumber I&= \\frac{1}{(2\\pi )^2} \\int _0^\\infty \\mathrm {d}q\\,\\frac{q^2}{q^2+\\mu ^2}\\, \\frac{2\\sin qr}{qr} = \\frac{1}{(2\\pi )^2} \\text{ Im}\\left\\lbrace \\int _{-\\infty }^\\infty \\mathrm {d}q\\,\\frac{q^2}{q^2+\\mu ^2}\\, \\frac{e^{iqr}}{qr}\\right\\rbrace =\\\\&= \\frac{1}{(2\\pi )^2r} \\text{ Im}\\left\\lbrace \\int _{-\\infty }^\\infty \\mathrm {d}y\\,\\frac{y}{y^2+\\mu ^2r^2}\\, e^{iy}\\right\\rbrace \\equiv \\frac{1}{(2\\pi )^2r} \\text{ Im}\\left\\lbrace \\int _{-\\infty }^\\infty \\mathrm {d}y\\, f(y) \\right\\rbrace \\,.$ Figure: Integration path (in the complex plane of the yyvariable) used in the Residue Theorem for the integral inexpression ().The integration of $f(y)$ along the circumference arc in Fig.REF is zero, since it goes as $e^{-|y|}/|y|$ when $y\\rightarrow i\\infty $ .",
"Therefore, using the Residue Theorem [6], $\\int _{-\\infty }^{\\infty } \\mathrm {d}y \\, f(y) = \\int _{C_\\infty } \\mathrm {d}y f(y) - 2\\pi i\\text{ Res}\\left[ f(y), y= i \\mu r \\right] = -2\\pi i \\text{ Res}\\left[ f(y), y= i \\mu r \\right]\\,.$ The pole of $f(y)$ in $y = i\\mu r$ is simple, so we can compute the residue as $\\text{ Res}\\left[ f(y), y= i \\mu r \\right] = \\lim _{y\\rightarrow i\\mu r} (y-i\\mu r)f(y) = \\lim _{y\\rightarrow i\\mu r} \\frac{y}{(y + i\\mu r)}\\, e^{iy} = \\frac{1}{2}\\, e^{-\\mu r}\\,.$ Using this result, we can trivially write $I = \\frac{1}{4\\pi }\\, \\frac{e^{-\\mu r}}{r}\\,.$"
],
[
"A spherical wave",
"Let's compute the integral $I \\equiv \\frac{1}{4\\pi }\\int \\mathrm {d}^3r\\, e^{-i \\mathbf {q}\\, \\mathbf {r}}\\, \\frac{e^{-\\sqrt{t^\\prime } r}}{r}\\,,$ where the $4\\pi $ factor has been introduced for convenience.",
"In spherical coordinates $\\mathrm {d}^3 r = r^2 \\mathrm {d}r \\, \\mathrm {d}\\Omega = r^2 \\mathrm {d}r \\,\\mathrm {d}\\cos \\theta \\, \\mathrm {d}\\phi $ , $I = \\frac{1}{2}\\int _0^\\infty \\mathrm {d}r \\, r\\,e^{-\\sqrt{t^\\prime } r} \\, \\int _{-1}^1 \\mathrm {d}\\cos \\theta \\,e^{-i q r \\cos \\theta }\\,,$ where the $\\int \\mathrm {d}\\phi $ has been trivially computed and $q\\equiv \\left| \\mathbf {q}\\, \\right|$ .",
"The remaining angular integral is also easy to calculate, so we can write $I =\\frac{1}{q} \\int _0^\\infty \\mathrm {d}r \\,e^{-\\sqrt{t^\\prime } r} \\, \\sin (qr)\\,.$ This integral can be computed integrating by parts: $\\nonumber I &= \\frac{1}{q} \\int _0^\\infty \\mathrm {d}r \\,\\left\\lbrace \\frac{\\mathrm {d}}{\\mathrm {d}r}\\left[ \\frac{-1}{\\sqrt{t^\\prime }} e^{-\\sqrt{t^\\prime } r} \\, \\sin (qr)\\right] - \\left[ \\frac{-1}{\\sqrt{t^\\prime }} e^{-\\sqrt{t^\\prime } r} \\, \\frac{\\mathrm {d}}{\\mathrm {d}r} \\sin (qr)\\right] \\right\\rbrace =\\\\\\nonumber &= 0 + \\frac{1}{\\sqrt{t^\\prime }} \\int _0^\\infty \\mathrm {d}r \\,e^{-\\sqrt{t^\\prime } r} \\, \\cos (qr) = \\\\\\nonumber &= \\frac{1}{\\sqrt{t^\\prime }} \\int _0^\\infty \\mathrm {d}r \\,\\left\\lbrace \\frac{\\mathrm {d}}{\\mathrm {d}r}\\left[ \\frac{-1}{\\sqrt{t^\\prime }} e^{-\\sqrt{t^\\prime } r} \\, \\cos (qr)\\right] - \\left[ \\frac{-1}{\\sqrt{t^\\prime }} e^{-\\sqrt{t^\\prime } r} \\, \\frac{\\mathrm {d}}{\\mathrm {d}r} \\cos (qr)\\right] \\right\\rbrace = \\\\&= \\frac{1}{t^\\prime } \\left\\lbrace 1 - q \\int _0^\\infty \\mathrm {d}r \\,e^{-\\sqrt{t^\\prime } r} \\, \\sin (qr) \\right\\rbrace $ Taking into account the fact that $q^2 \\equiv \\mathbf {q\\,}^2 = -t$ , this last relation can be written as $I = \\frac{1}{t^\\prime } (1+t I) \\hspace{28.45274pt} \\longrightarrow \\hspace{28.45274pt} (t^\\prime - t)I=1\\,.$ Therefore, we have proved the relation $\\frac{1}{t^\\prime - t} = \\frac{1}{4\\pi } \\int \\mathrm {d}^3r\\, e^{-i \\mathbf {q}\\, \\mathbf {r}}\\, \\frac{e^{-\\sqrt{t^\\prime } r}}{r}\\,.$"
],
[
"Integrals for the absorptive part",
"In this appendix we are going to calculate the integrals $&I \\equiv \\int \\mathrm {d}^4 k\\, \\delta (k^2) \\delta (\\bar{k}^2) = \\frac{\\pi }{2}\\,, \\\\&I^\\mu \\equiv \\int \\mathrm {d}^4 k\\, \\delta (k^2) \\delta (\\bar{k}^2)\\,k^\\mu = \\frac{\\pi }{4}\\,q^\\mu \\,, \\\\&I^{\\mu \\nu } \\equiv \\int \\mathrm {d}^4 k\\, \\delta (k^2)\\delta (\\bar{k}^2)\\,k^\\mu k^\\nu =\\frac{\\pi }{6}\\left( q^\\mu q^\\nu -\\frac{1}{4}\\, s\\, g^{\\mu \\nu } \\right)\\,,$ where $\\bar{k} = q-k$ and $q^2 = s$ .",
"Let's consider the first one.",
"Using $k^2 = E^2 - {\\mathbf {k}\\,}^2$ in the first delta function, we compute the $E \\equiv k^0$ integral, $I = \\int \\frac{\\mathrm {d}^3 k}{2E}\\, \\delta (\\bar{k}^2) = \\int \\frac{\\mathrm {d}^3 k}{2E}\\, \\delta \\left[q^2 - 2(kq) \\right]\\,.$ We can evaluate the integral in the CM reference frame, where $q^\\mu = ( \\sqrt{s}, \\mathbf {0} \\,)$ , so that $I = \\frac{1}{2}\\int \\mathrm {d}\\Omega \\, \\mathrm {d}E\\, E\\, \\delta \\left(s - 2E\\sqrt{s} \\right) = \\frac{1}{2} \\int \\mathrm {d}\\omega \\left.",
"\\frac{E}{2\\sqrt{s}} \\right|_{E = \\sqrt{s}/2} = \\frac{1}{8}\\int \\mathrm {d}\\Omega = \\frac{\\pi }{2}\\,,$ as we wanted to prove.",
"Due to Lorentz covariance, the $I^\\mu $ integral must be of the form $I^\\mu \\equiv \\int \\mathrm {d}^4 k\\, \\delta (k^2) \\delta (\\bar{k}^2)\\,k^\\mu = A \\, q^\\mu \\,.$ Multiplying this relation by $q_\\mu $ , we get $A\\, q^2 = \\int \\mathrm {d}^4 k\\, \\delta (k^2) \\delta (\\bar{k}^2)\\,(k\\bar{k}) = \\frac{1}{2}\\,q^2\\, I =\\frac{\\pi }{4}\\,q^2\\,,$ so $I^\\mu = \\frac{\\pi }{4}\\, q^\\mu \\,.$ Finally, we can also use Lorentz covariance to write $I^{\\mu \\nu } \\equiv \\int \\mathrm {d}^4 k\\, \\delta (k^2) \\delta (\\bar{k}^2)\\,k^\\mu k^\\nu = A g^{\\mu \\nu }+ B q^\\mu q^\\nu \\,.$ We need to multiply by $g_{\\mu \\nu }$ and $q_\\mu q_\\nu $ to determine $A$ and $B$ , $&g_{\\mu \\nu }I^{\\mu \\nu } = 4 A + q^2 B = \\int \\mathrm {d}^4 k\\, \\delta (k^2) \\delta (\\bar{k}^2)\\,k^2 = 0\\,,\\\\&\\frac{q_\\mu q_\\nu }{q^2} I^{\\mu \\nu } = A + q^2 B= \\frac{1}{q^2} \\int \\mathrm {d}^4 k\\, \\delta (k^2)\\delta (\\bar{k}^2)\\,(k\\bar{k})^2 = \\frac{1}{4}\\,^2\\, I = \\frac{\\pi }{8} q^2\\,,$ so we just need to solve the algebraic system of equations $\\nonumber 4 A + q^2 B &= 0\\,,\\\\A + q^2 B &= \\frac{\\pi }{8}\\, q^2\\,,$ which gives $A = -\\frac{\\pi }{24}\\, q^2\\,, \\hspace{113.81102pt} B = \\frac{\\pi }{6}\\,.$ Therefore, $I^{\\mu \\nu } = \\frac{\\pi }{6} \\left( q^\\mu q^\\nu - \\frac{1}{4}\\, q^2\\, g^{\\mu \\nu } \\right)\\,.$"
],
[
"Integral for the interaction potential",
"We are interested in calculating a primitive $P(t)$ of $\\int \\mathrm {d}t\\, t \\, e^{-\\sqrt{t} \\,r}\\,,$ which can be obtained quite straightforwardly using $q \\equiv \\sqrt{t}$ , $\\nonumber P(t) &= \\int \\mathrm {d}t\\, t \\, e^{-\\sqrt{t} \\,r} = 2 \\int \\,\\mathrm {d}q\\, q^3 \\, e^{- q r} = \\\\\\nonumber &= -2\\, \\frac{\\mathrm {d}^3}{\\mathrm {d}r^3} \\int \\mathrm {d}q\\, e^{-q r} = \\\\\\nonumber &= 2 \\, \\frac{\\mathrm {d}^3}{\\mathrm {d}r^3} \\left[ \\frac{e^{-qr}}{r} \\right] = \\\\\\nonumber &= 2 \\, \\frac{\\mathrm {d}^2}{\\mathrm {d}r^2} \\left[ \\left( - \\frac{1}{r^2} - \\frac{q}{r} \\right) e^{-qr} \\right] = \\\\\\nonumber &= 2 \\, \\frac{\\mathrm {d}}{\\mathrm {d}r} \\left[ \\left( \\frac{2}{r^3} + \\frac{2q}{r^2} + \\frac{q^2}{r} \\right) e^{-qr} \\right] = \\\\\\nonumber &= -\\left( \\frac{12}{r^4} + \\frac{12q}{r^3} + \\frac{6q^2}{r^2} + \\frac{2q^3}{r}\\right) e^{-qr} = \\\\&= -\\left( \\frac{12}{r^4} + \\frac{12\\sqrt{t}}{r^3} + \\frac{6t}{r^2} + \\frac{2\\sqrt{t^3}}{r}\\right) e^{-\\sqrt{t}\\,r}\\,.$"
]
] | 1606.05087 |
[
[
"Reversible part of a quantum dynamical system"
],
[
"Abstract In this work a quantum dynamical system $(\\mathfrak M,\\Phi, \\varphi)$ is constituted by a von Neumann algebra $\\mathfrak M$, by a unital Schwartz map $\\Phi:\\mathfrak{M\\rightarrow M}$ and by a $\\Phi$-invariant normal faithful state $\\varphi$ on $\\mathfrak M$.",
"The ergodic properties of a quantum dynamical system, depends on its reversible part $(\\mathfrak{D}_\\infty,\\Phi_\\infty, \\varphi_\\infty)$.",
"It is constituted by a von Neumann sub-algebra $\\mathfrak{D}_\\infty$ of $\\mathfrak M$ by an automorphism $\\Phi_\\infty$ and a normal state $\\varphi_\\infty$, the restrictions of $\\Phi$ and $\\varphi$ on $\\mathfrak{D}_\\infty$ respectively.",
"Moreover, if $\\mathfrak{D}_\\infty$ is a trivial algebra the quantum dynamical system is ergodic.",
"Furthermore we will give some properties of the reversible part of quantum dynamical system, in particular, we will study its relationships with the canonical decomposition of Nagy-Fojas of linear contraction related to the quantum dynamical system."
],
[
"Preliminares and notations",
"We consider a pair $(\\mathfrak {M},\\Phi ) $ constituted by a von Neumann algebra $\\mathfrak {M}$ and a unital Schwartz map $\\Phi :\\mathfrak {M\\rightarrow M}$ i.e.",
"a $\\sigma $ -continuous map with $\\Phi (1)=1$ which satisfies the inequality: $0 \\le \\Phi (a^*)\\Phi (a) \\le \\Phi (a^*a) \\qquad a\\in \\mathfrak {M}$ In this work the pair $(\\mathfrak {M},\\Phi ) $ will be called (discrete) quantum process and $\\Phi $ the dynamics of the quantum process.",
"A normal state $\\varphi $ on $\\mathfrak {M}$ is a stationary state for the quantum process $(\\mathfrak {M},\\Phi )$ if $\\varphi (\\Phi (a))=\\varphi (a)$ for all $a\\in \\mathfrak {M}$ , while is of asymptotic equilibrium if $\\Phi ^n(a) \\rightarrow \\varphi (a) 1$ as $n \\rightarrow \\infty $ in $\\sigma $ -topology i.e.",
"$ \\lim _{n\\rightarrow +\\infty } \\omega (\\Phi ^n(a))=\\omega (1) \\varphi (a) \\quad a\\in \\mathfrak {M} \\quad \\omega \\in \\mathfrak {M}_{*} $ We denote with $\\mathcal {B}(\\mathcal {H})$ the C*-algebra of bounded linear operator on Hilbert space $\\mathcal {H}$ and with $s$ and $\\sigma $ respectively the ultrastrong operator topology and the ultraweakly operator topology on von Neumann algebra $\\mathfrak {M}$ while with $\\mathfrak {M}_{*}$ its predual.",
"Furthermore, a normal map or a normal state are $\\sigma $ -continuous maps (see ref.",
"[6]).",
"We define the multiplicative domain $\\mathfrak {D}_{\\Phi }$ of a Schwartz map (see definition 2.1.4 and proposition 2.1.6 of [24]) as follows: $\\mathfrak {D}_{\\Phi }= \\lbrace a\\in \\mathfrak {M}: \\Phi (a^*a)=\\Phi (a^*)\\Phi (a) \\ \\ and \\ \\ \\Phi (aa^*)=\\Phi (a)\\Phi (a^*) \\rbrace $ We recall that an element $a\\in \\mathfrak {D}_{\\Phi }$ if and only if $\\Phi (ax)=\\Phi (a)\\Phi (x)$ and $\\Phi (xa)=\\Phi (x)\\Phi (a)$ for all $x\\in \\mathfrak {M}$ .",
"It follows that $\\mathfrak {D}_{\\Phi }$ is a von Neumann algebra, since it is a unital *-algebra closed in the $\\sigma $ -topology.",
"A consequence of the Schwartz's inequality is the following remark: Remark 1 If $\\Phi :\\mathfrak {M}\\rightarrow \\mathfrak {M}$ is a unital Schwartz map which admits an inverse $\\Phi ^{-1}:\\mathfrak {M}\\rightarrow \\mathfrak {M}$ (i.e.",
"a unital Schwartz map such that $\\Phi (\\Phi ^{-1}(a))=\\Phi ^{-1}(\\Phi (a))=a $ for all $a\\in \\mathfrak {M}$ ), then $\\Phi $ is an automorphism.",
"If $\\mathfrak {D}_{\\infty }^+$ is the following von Neumann algebras: $\\mathfrak {D}_{\\infty }^+=\\bigcap _{n\\in \\mathbb {N}}\\mathfrak {D}_{\\Phi ^n}$ then we have that $\\Phi (\\mathfrak {D}_{\\infty }^+)\\subset \\mathfrak {D}_{\\infty }^+$ and $\\Phi $ restricted to $\\mathfrak {D}_{\\infty }^+$ is a *-homomorphism, but it is not surjective map.",
"Moreover we have: $\\mathfrak {D}_{\\infty }^+= \\left\\lbrace a\\in \\mathfrak {D}_\\Phi : \\Phi ^n(a)\\in \\mathfrak {D}_\\Phi \\ \\textsl {for all} \\ n\\in \\mathbb {N} \\right\\rbrace $ We define the multiplicative core of $\\Phi $ (see ref.",
"[23]): $ \\mathcal {C}_\\Phi = \\bigcap _{n\\in \\mathbb {N}} \\Phi ^n (\\mathfrak {D}_{\\infty }^+)\\subset \\mathfrak {D}_{\\infty }^+ $ We have $\\Phi (\\mathcal {C}_\\Phi )\\subset \\mathcal {C}_\\Phi $ .",
"Indeed $\\Phi ^{n+1} (\\mathfrak {D}_{\\infty }^+)\\subset \\Phi ^n (\\mathfrak {D}_{\\infty }^+)$ for all $n\\ge 0$ and $ \\Phi (\\bigcap _{n\\in \\mathbb {N}}\\Phi ^{n}(\\mathfrak {D}_{\\infty }^+)) \\subset \\bigcap _{n\\in \\mathbb {N}}\\Phi (\\Phi ^{n}(\\mathfrak {D}_{\\infty }^+))=\\bigcap _{n\\in \\mathbb {N}}\\Phi ^{n+1}(\\mathfrak {D}_{\\infty }^+)=\\bigcap _{n\\in \\mathbb {N}} \\Phi ^{n}(\\mathfrak {D}_{\\infty }^+) $ It is clear that the restriction of $\\Phi $ to multiplicative core $\\mathcal {C}_\\Phi $ is a *-homomorphism and if $\\Phi $ is an injective map on $ \\mathfrak {D}_{\\infty }^+$ , then we have $\\Phi ( \\mathcal {C}_\\Phi )= \\mathcal {C}_\\Phi $ , so the restriction of $\\Phi $ to the multiplicative core is *-automorphism.",
"Since $\\Phi $ is a normal map and its restriction to $ \\mathfrak {D}_{\\infty }^+$ is a *-homomorphism, we have that the set $\\Phi ^n(\\mathfrak {D}_{\\infty }^+)$ is a von Neumann algebra (see e.g.",
"[6]), therefore $\\mathcal {C}_\\phi $ is a von Neumann algebra.",
"Let $\\varphi $ be a stationary state for the quantum processes $(\\mathfrak {M},\\Phi ) $ and $(\\mathcal {H}_{\\varphi },\\pi _{\\varphi },\\Omega _{\\varphi })$ its GNS representation.",
"It is well know (see e.g.",
"[20] ) that there is a unique linear contraction $U_{\\Phi ,\\varphi }$ of $\\mathfrak {B}(\\mathcal {H}_{\\varphi })$ such that, for any $a\\in \\mathfrak {A}$ , we have $U_{\\Phi ,\\varphi }\\pi _{\\varphi }(a)\\Omega _{\\varphi }=\\pi _{\\varphi }(\\Phi (a))\\Omega _{\\varphi }$ Furthermore if $\\varphi $ is a faithful state then there is a unital Schwartz map $\\Phi _{\\bullet }:\\pi _{\\varphi }(\\mathfrak {M})\\rightarrow \\pi _{\\varphi }(\\mathfrak {M})$ such that $\\Phi _{\\bullet }(A)\\Omega _{\\varphi }=U_{\\Phi ,\\varphi }A\\Omega _{\\varphi } \\qquad A\\in \\pi _{\\varphi }(\\mathfrak {M})$ It is simple to prove the following statements on multiplicative domains of Schwartz maps: Proposition 1 Let $(\\mathfrak {M},\\Phi ) $ be quantum processes and $\\varphi $ its faithful stationary state, we have: a] For each $d\\in \\mathfrak {D}_{\\Phi }$ results $U_{\\Phi ,\\varphi }\\pi _{\\varphi }(d)=\\pi _{\\varphi }(\\Phi (d))U_{\\Phi ,\\varphi }$ b] If $U_{\\Phi ,\\varphi }\\pi _{\\varphi }(a)=\\pi _{\\varphi }(\\Phi (a))U_{\\Phi ,\\varphi }$ then $\\Phi (ax)=\\Phi (a)\\Phi (x)$ for all $x\\in \\mathfrak {M}$ c] $U_{\\Phi ,\\varphi }^*U_{\\Phi ,\\varphi }\\in \\pi _{\\varphi }(\\mathfrak {D}_{\\Phi })^{\\prime }$ while $U_{\\Phi ,\\varphi }U_{\\Phi ,\\varphi }^*\\in \\pi _{\\varphi }(\\Phi (\\mathfrak {D}_{\\Phi }))^{\\prime }$ d] $d\\in \\mathfrak {D}_{\\Phi }$ if, and only if $||U_{\\Phi ,\\varphi }\\pi _{\\varphi }(d)\\Omega _{\\varphi }||=||\\pi _{\\varphi }(d)\\Omega _{\\varphi }||$ and $||U_{\\Phi ,\\varphi }\\pi _{\\varphi }(d^*)\\Omega _{\\varphi }||=||\\pi _{\\varphi }(d^*)\\Omega _{\\varphi }||$ It is straightforward We observe that if $\\Phi $ is a *-homomorphism, then the contraction $U_{\\Phi ,\\varphi }$ is an isometry on $\\mathcal {H}_{\\varphi }$ .",
"Another trivial consequence of Schwartz's inequality and of the existence of a faithful stationary state for quantum process $(\\mathfrak {M},\\Phi ) $ , are the following relations: $\\cdots \\mathfrak {D}_{\\Phi ^n}\\subset \\mathfrak {D}_{\\Phi ^{n-1}}\\subset \\cdots \\mathfrak {D}_{\\Phi ^2}\\subset \\mathfrak {D}_{\\Phi }\\subset \\mathfrak {M}$ for all natural numbers $n\\in \\mathbb {N}$ .",
"Furthermore, since $\\Phi $ is a injective map on $\\mathfrak {D}_{\\infty }^+$ , its restricted to $\\mathcal {C}_\\Phi $ is a *-automorphism.",
"In fact, if $a\\in \\mathfrak {D}_{\\infty }^+$ with $\\Phi (a)=0$ , then we obtain $\\varphi (\\Phi (a^*) \\Phi (a))=\\varphi (\\Phi (a^* a))=\\varphi (a^*a)=0$ Let $(\\mathfrak {M},\\Phi )$ be a quantum processes and $\\varphi $ its stationary state, we recall that the dynamics $\\Phi $ admits a $\\varphi $ -adjoint if there is a normal unital Schwartz map $\\Phi ^{\\sharp }:\\mathfrak {M}\\rightarrow \\mathfrak {M}$ such that $\\varphi (b\\Phi (a))=\\varphi (\\Phi ^{\\natural }(b)a) \\qquad a,b\\in \\mathfrak {M}$ We have the following conditions for the existence of a $\\varphi $ -adjointness of dynamics of quantum process (see proposition 3.3 in [20]): Proposition 2 Let $(\\mathfrak {M},\\Phi )$ be a quantum process and $\\varphi $ its faithful stationary state.",
"If $(\\Delta _{\\varphi },J_{\\varphi })$ denote the modular operators associated with pair $(\\pi _{\\varphi }(\\mathfrak {M}),\\Omega _{\\varphi })$ , then the following conditions are equivalent: 1 - $\\Phi $ commutes with the modular automorphism group $\\left\\lbrace \\sigma _{t}^{\\varphi } \\right\\rbrace _{t\\in \\mathbb {R}} $ i.e.",
"$\\sigma _{t}^{\\varphi }\\circ \\Phi _{\\bullet }=\\Phi _{\\bullet }\\circ \\sigma _{t}^{\\varphi } \\qquad t\\in \\mathbb {R}$ 2 - $U_{\\Phi ,\\varphi }$ commutes with modular operators: $U_{\\Phi ,\\varphi }\\Delta _{\\varphi }^{it}=\\Delta _{\\varphi }^{it}U_{\\Phi ,\\varphi } \\qquad t\\in \\mathbb {R}$ and $U_{\\Phi ,\\varphi }J_{\\varphi }=J_{\\varphi }U_{\\Phi ,\\varphi }$ 3 - $\\Phi $ admits $\\varphi $ -adjoint $\\Phi ^\\sharp $ .",
"A triple $(\\mathfrak {M},\\Phi , \\varphi )$ constituted by quantum processes $(\\mathfrak {M},\\Phi )$ , by its normal faithful stationary state $\\varphi $ and with dynamics $\\Phi $ which admits a $\\varphi $ -adjoint $\\Phi ^\\sharp $ , will be called a quantum dynamical system.",
"Decomposition theorem We consider a von Neumann algebra $\\mathfrak {M}$ and its faithful normal state $\\varphi $ and set with $(\\mathcal {H}_{\\varphi },\\pi _{\\varphi },\\Omega _{\\varphi })$ the GNS representation of $\\varphi $ and with $\\left\\lbrace \\sigma _{t}^{\\varphi }\\right\\rbrace _{t\\in \\mathbb {R}}$ its modular automorphism group.",
"Let $\\mathfrak {R}$ be a von Neumann subalgebra of $\\mathfrak {M}$ , we recall (see ref.",
"[14]) that the $\\varphi $ -orthogonal of $\\mathfrak {R}$ is the set: $\\mathfrak {R}^{\\perp _{\\varphi }}=\\lbrace a\\in \\mathfrak {M}:\\varphi (a^*x)=0 \\ \\ \\ \\text{for all}\\ \\ x\\in \\mathfrak {R} \\rbrace $ Furthermore, it is simple to prove that $\\mathfrak {R}^{\\perp _{\\varphi }}$ is a closed linear space in the $\\sigma $ -topology with $\\mathfrak {R}^{\\perp _{\\varphi }}\\cap \\mathfrak {R}$ ={0}.",
"We observe that $\\mathfrak {R}^{\\perp _{\\varphi }}\\subset \\ker \\varphi $ and if $\\mathfrak {R}=\\mathbb {C}I$ then $\\mathfrak {R}^{\\perp _{\\varphi }}= \\ker \\varphi $ , where $\\ker \\varphi =\\lbrace a\\in \\mathfrak {M}: \\ \\varphi (a)=0 \\rbrace $ .",
"Moreover if $ y \\in \\mathfrak {R} $ and $d_\\bot \\in \\mathfrak {R}^{\\perp _{\\varphi }}$ then $y d_\\bot \\in \\mathfrak {R}^{\\perp _{\\varphi }}$ since $ \\varphi ( (y d_\\bot )^* x)=\\varphi (d_\\bot ^* y ^* x)=0 \\qquad x\\in \\mathfrak {R} $ Theorem 1 The von Neumann algebra $\\mathfrak {R}$ is invariant under modular automorphism group $\\sigma _{t}^{\\varphi }$ if and only if both these conditions are fulfilled: a - the set $\\mathfrak {R}^{\\perp _{\\varphi }}$ is closed under the involution operation; b - for any $a\\in \\mathfrak {M}$ there is a unique $a_\\Vert \\in \\mathfrak {R}$ and $a_\\bot \\in \\mathfrak {R}^{\\perp _{\\varphi }}$ such that $a= a_\\Vert + a_\\bot $ .",
"In other words we have the following algebraic decomposition $\\mathfrak {M}= \\mathfrak {R} \\oplus \\mathfrak {R}^{\\perp _{\\varphi }}$ From Takesaki [25] we have $\\sigma _{t}^{\\varphi } (\\pi _\\varphi (\\mathfrak {R})) \\subset \\pi _\\varphi (\\mathfrak {R})$ for all $t\\in \\mathbb {R}$ if, and only if there exist a normal conditional expectation $\\mathcal {E}:\\mathfrak {M} \\rightarrow \\mathfrak {R} $ such that $\\varphi \\circ \\mathcal {E} = \\varphi $ .",
"Let $\\mathfrak {R}$ be invariant under modular automorphism group $\\sigma _{t}^{\\varphi }$ , it is simple to prove that $\\mathfrak {R}^{\\perp _{\\varphi }}=\\left\\lbrace a\\in \\mathfrak {M} : \\mathcal {E} (a)=0 \\right\\rbrace $ hence $\\mathfrak {R}^{\\perp _{\\varphi }}$ is closed under the involution operation.",
"For any $a\\in \\mathfrak {M}$ we set $a_\\bot =a- \\mathcal {E} (a)$ and $\\varphi (a_\\bot ^* x)= \\varphi ((a^* - \\mathcal {E} (a^*))x)=\\varphi (a^* x) - \\varphi ( \\mathcal {E} (a^*)x)= \\varphi (a^* x) - \\varphi ( \\mathcal {E} (a^*x))=0$ for all $x\\in \\mathfrak {R}$ hence $a_\\bot \\in \\mathfrak {R}^{\\perp _{\\varphi }}$ .",
"So for any $a\\in \\mathfrak {M}$ there exist a unique $a_\\Vert \\in \\mathfrak {R}$ and $a_\\bot \\in \\mathfrak {R}^{\\perp _{\\varphi }}$ such that $a= a_\\Vert + a_\\bot $ where we have set $a_\\Vert =\\mathcal {E}(a)$ .",
"The uniqueness follows because if $a=0$ then $a_\\Vert =a_\\bot =0$ .",
"Indeed we have $ \\varphi (a^* a) = \\varphi (a_\\Vert ^* a_\\Vert ) + \\varphi (a_\\bot ^*a_\\bot )=0$ since $a_\\Vert ^* a_\\bot , $ and $ a_\\bot ^*a_\\Vert $ belong to $\\mathfrak {R}^{\\perp _{\\varphi }}$ and $\\varphi $ is a faithful state.",
"For the vice-versa, if the set $\\mathfrak {R}^{\\perp _{\\varphi }}$ is closed under the involution operation and $ \\mathfrak {M}= \\mathfrak {R} \\oplus \\mathfrak {R}^{\\perp _{\\varphi }}$ then for any $a\\in \\mathfrak {M}$ there is a unique $a_\\Vert \\in \\mathfrak {R}$ and $a_\\bot \\in \\mathfrak {R}^{\\perp _{\\varphi }}$ such that $a= a_\\Vert + a_\\bot $ .",
"The map $ a \\in \\mathfrak {M} \\rightarrow a_\\Vert \\in \\mathfrak {R} $ is a projection of norm one ( i.e.",
"it is satisfies $(1)_\\Vert =1$ and $ ( (a)_\\Vert )_\\Vert =a_\\Vert $ for all $a\\in \\mathfrak {M} $ ), for Tomiyama [26] it is a normal conditional expectation (see [15] for a modern review) and $\\varphi (a)=\\varphi (a_\\Vert )$ for all $a \\in \\mathfrak {M}$ .",
"We observe that if $\\mathfrak {R}^{\\perp _{\\varphi }}$ is a *-algebra (without unit) then $\\mathfrak {R}^{\\perp _{\\varphi }}= \\left\\lbrace 0 \\right\\rbrace $ since $\\varphi (a_\\bot ^* a_\\bot )=0$ for all $a_\\bot \\in \\mathfrak {R}^{\\perp _{\\varphi }}$ and $\\varphi $ is a faithful state.",
"Moreover, if $p$ is a orthogonal projector of $\\mathfrak {M}$ then $p\\notin \\mathfrak {R}^{\\perp _{\\varphi }}$ .",
"We have the following remark: If $a\\in \\mathfrak {M}$ with $a=a_\\Vert +a_\\bot $ where $a_\\Vert \\in \\mathfrak {R}$ and $a_\\bot \\in \\mathfrak {R}^{\\perp _{\\varphi }}$ , then $||\\pi _{\\varphi }(a)\\Omega _{\\varphi }||^2=||\\pi _{\\varphi }(a_\\Vert )\\Omega _{\\varphi }||^2+||\\pi _{\\varphi }(a_\\bot )\\Omega _{\\varphi }||^2$ Proposition 3 Let $\\mathfrak {R}$ be a von Neumann algebra invariant under modular automorphism group $\\sigma _{t}^{\\varphi }$ .",
"If $\\mathcal {H}_o$ and $\\mathcal {K}_o$ are the closure of the linear space $\\pi _{\\varphi }(\\mathfrak {R})\\Omega _{\\varphi }$ and of $\\pi _{\\varphi }(\\mathfrak {R}^{\\perp _{\\varphi }})\\Omega _{\\varphi }$ respectively, then $\\mathcal {H}_{\\varphi }=\\mathcal {H}_o \\oplus \\mathcal {K}_o$ Moreover the orthogonal projection $P_o$ on Hilbert space $\\mathcal {H}_o$ belongs to $\\pi _{\\varphi }(\\mathfrak {R})^{\\prime }$ .",
"We have that $ \\mathcal {K}_o\\subset \\mathcal {H}_o^\\bot $ since for any $r_\\bot \\in \\mathfrak {R}^{\\perp _\\varphi }$ and $\\psi _o\\in \\mathcal {H}_o$ we obtain: $\\left\\langle \\pi _\\varphi (r_\\bot )\\Omega _\\varphi , \\psi _o \\right\\rangle = \\underset{\\alpha \\rightarrow \\infty }{\\lim }\\left\\langle \\pi _\\varphi (r_\\bot )\\Omega _\\varphi , \\pi _\\varphi (r_\\alpha )\\Omega _\\varphi \\right\\rangle = \\underset{\\alpha \\rightarrow \\infty }{\\lim } \\varphi (r_\\bot ^*r_\\alpha )=0$ where $\\psi _o =\\underset{\\alpha \\rightarrow \\infty }{\\lim } \\pi _\\varphi (r_\\alpha )\\Omega _\\varphi $ with $\\lbrace r_\\alpha \\rbrace _\\alpha $ net belongs to $\\mathfrak {R}$ .",
"Let $\\psi \\in \\mathcal {H}_\\varphi $ we can write $\\psi =\\underset{\\alpha \\rightarrow \\infty }{\\lim }\\pi _{\\varphi }(m_\\alpha )\\Omega _\\varphi =\\underset{\\alpha \\rightarrow \\infty }{\\lim }(\\pi _{\\varphi }(r_\\alpha )\\Omega _\\varphi +\\pi _{\\varphi }((r_{\\alpha \\bot })\\Omega _\\varphi )$ where $m_\\alpha = r_\\alpha + r_{\\alpha \\bot }$ for each $\\alpha $ .",
"The net $ \\lbrace \\pi _{\\varphi }(r_\\alpha )\\Omega _{\\varphi } \\rbrace $ has limit, since by the relation (REF ) for each $\\epsilon \\ge 0$ there is a index $\\nu $ such that for $\\alpha \\ge \\nu $ and $\\beta \\ge \\nu $ we have the Cauchy relation: $|| \\pi _\\varphi (r_\\alpha )\\Omega _\\varphi -\\pi _\\varphi (r_\\beta )\\Omega _\\varphi || \\le || \\pi _\\varphi (m_\\alpha )\\Omega _\\varphi -\\pi _\\varphi (m_\\beta )\\Omega _\\varphi ||\\le \\epsilon $ It follows that there are $\\psi _\\Vert \\in \\mathcal {H}_o$ and $\\psi _\\bot \\in \\mathcal {K}_o$ such that $\\psi =\\underset{\\alpha \\rightarrow \\infty }{\\lim }\\pi _{\\varphi }(r_\\alpha )\\Omega _\\varphi + \\underset{\\alpha \\rightarrow \\infty }{\\lim }\\pi _{\\varphi }((r_{\\alpha \\bot })\\Omega _\\varphi =\\psi _\\Vert + \\psi _\\bot \\in \\mathcal {H}_o \\oplus \\mathcal {K}_o$ It is simple to prove that $\\pi _\\varphi (\\mathfrak {R}) \\mathcal {H}_o\\subset \\mathcal {H}_o $ therefore $P_o\\in \\pi _\\varphi (\\mathfrak {R})^{\\prime }$ .",
"We have the following proposition: Proposition 4 Let $( \\mathfrak {M},\\Phi )$ be a quantum process and $\\varphi $ a normal faithful state on $\\mathfrak {M}$ .",
"For any natural number $n \\in \\mathbb {N}$ we obtain: $\\mathfrak {M} = \\mathfrak {D}_{\\Phi ^n} \\oplus \\mathfrak {D}_{\\Phi ^n}^{\\perp _\\varphi }$ and $\\mathfrak {M} = \\mathfrak {D}_\\infty ^+ \\oplus \\mathfrak {D}_\\infty ^{+ \\perp _\\varphi }$ Furthermore, if $\\varphi $ is a stationary state for $\\Phi $ , then $\\mathfrak {M} = \\mathcal {C}_\\Phi \\oplus \\mathcal {C}_\\Phi ^{\\perp _\\varphi }$ and the restriction of $\\Phi $ to $\\mathcal {C}_\\Phi $ is a *-automorphism with $\\Phi (\\mathcal {C}_\\Phi ^{\\perp _\\varphi })\\subset \\mathcal {C}_\\Phi ^{\\perp _\\varphi }$ .",
"for any $d\\in \\mathfrak {D}_{\\Phi ^n}$ and natural number $n$ we have: $\\Phi _{\\bullet }^ n(\\sigma ^{\\varphi }_{t}(\\pi _{\\varphi }(d)^*)\\sigma ^{\\varphi }_{t}(\\pi _{\\varphi }(d))) =\\Phi _{\\bullet }^n(\\sigma ^{\\varphi }_{t}(\\pi _{\\varphi }(d)^*))\\Phi _{\\bullet }^n(\\sigma ^{\\varphi }_{t}(\\pi _{\\varphi }(d))).$ since $\\Phi $ commutes with our modular automorphism group $\\sigma ^{\\varphi }_{t}$ .",
"It follows that $\\sigma ^{\\varphi }_{t}(\\pi _\\varphi (\\mathfrak {D}_{\\Phi ^n}) )$ is included in $\\pi _\\varphi (\\mathfrak {D}_{\\Phi ^n})$ for all $n \\in \\mathbb {N}$ and $t \\in \\mathbb {R}$ .",
"Let $b\\in \\mathcal {C}_\\Phi $ , we have that $\\sigma ^{\\varphi }_{t}(\\pi _{\\varphi }(b))\\in \\pi _\\varphi (\\mathcal {C}_\\Phi )$ for all real number $t$ .",
"In fact for each natural number $n$ there exist a $x_n\\in \\mathfrak {D}_\\infty ^{+}$ such that $b=\\Phi ^n(x_n)$ .",
"We can write that $\\sigma ^{\\varphi }_{t}(\\pi _\\varphi (b)) = \\sigma ^{\\varphi }_{t}(\\pi _\\varphi (\\Phi ^n(x_n))=\\Phi ^n_\\bullet (\\sigma ^{\\varphi }_{t}(\\pi _\\varphi (x_n)) $ and by above relation $\\sigma ^{\\varphi }_{t}(\\pi _\\varphi (x_n))\\in \\pi _\\varphi (\\mathfrak {D}_\\infty ^+)$ for all natural number $n$ .",
"It follows that $\\sigma ^{\\varphi }_{t}(\\pi _\\varphi (b))\\in \\pi _\\varphi (\\Phi ^n(\\mathfrak {D}_\\infty ^+)$ for all natural number $n$ .",
"Let $y\\in C_\\Phi ^{\\perp _\\varphi }$ , since $\\Phi (\\mathcal {C}_\\Phi )= \\mathcal {C}_\\Phi $ we have for any $c\\in \\mathcal {C}_\\Phi $ that $ \\varphi (\\Phi (y) c )=\\varphi (\\Phi (y) \\Phi (c_o))=\\varphi (y c_o)=0$ where $c=\\Phi (c_o)$ with $c_o\\in \\mathcal {C}_\\Phi \\subset \\mathfrak {D}_\\infty ^+$ .",
"We consider a quantum dynamical system $( \\mathfrak {M},\\Phi ,\\varphi )$ with $\\varphi $ -adjoint $\\Phi ^\\sharp $ .",
"We set with $\\mathfrak {D}_{\\infty }$ (or with $\\mathfrak {D}_{\\infty }(\\Phi )$ when we have to highlight the map $\\Phi $ ), the following von Neumann Algebra $\\mathfrak {D}_{\\infty } = \\bigcap \\limits _{k\\in \\mathbb {Z}} \\mathfrak {D}_{\\Phi _{k}}$ where for each $k$ integer we denote $\\Phi _{k}=\\left\\lbrace \\begin{array}[c]{cc}\\Phi ^{k} & k\\ge 0\\\\\\ \\ \\Phi ^{\\sharp \\left|k\\right|} & k<0\\end{array}\\right.$ while with $\\mathfrak {D}_{\\Phi _{k}}$ we have set the von Neumann algebra of the multiplicative domains of the dynamics $\\Phi _{k}$ .",
"Following [22], for each $a,b\\in \\mathfrak {M}$ and integers $k$ we define: $S_{k}(a,b)=\\Phi _{k}(a^*b)-\\Phi _{k}(a^*)\\Phi _{k}(b)\\in \\mathfrak {M}$ and we have these simple relations: a - $S_{k}(a,a)\\ge 0$ for all $a\\in \\mathfrak {M}$ and integers $k$ ; b - $S_{k}(a,b)^*=S_{k}(b,a)$ for all $a,b\\in \\mathfrak {M}$ and integers $k$ ; c - $d\\in \\mathfrak {D}_{\\infty }$ if, and only if $S_{k}(d,d)=S_{k}(d^*,d^*)=0$ for all integers $k$ ; d - $d\\in \\mathfrak {D}_{\\infty }$ if, and only if $\\varphi (S_{k}(d,d))=\\varphi (S_{k}(d^*,d^*))=0$ for all integers $k$ ; e - The map $a,b\\in \\mathfrak {M}\\rightarrow \\varphi (S_{k}(a,b))$ for all integers $k$ , is a sesquilinear form, hence $|\\varphi (S_{k}(a,b))|^2\\le \\varphi (S_{k}(a,a)) \\varphi (S_{k}(b,b)) \\qquad \\ a,b\\in \\mathfrak {M}$ We observe that $\\Phi (\\mathfrak {D}_{\\infty })\\subset \\mathfrak {D}_{\\infty }$ and $\\Phi ^{\\sharp }(\\mathfrak {D}_{\\infty })\\subset \\mathfrak {D}_{\\infty }$ .",
"Indeed for each element $d\\in \\mathfrak {D}_{\\infty }$ and integer $k$ we have $\\varphi (S_{k}(\\Phi (d),\\Phi (d))=\\varphi (S_{k+1}(d,d)=0$ and $\\varphi (S_{k}(\\Phi ^{\\sharp }(d),\\Phi ^{\\sharp }(d))=\\varphi (S_{k-1}(d,d)=0$ Furthermore $d^*\\in \\mathfrak {D}_{\\infty }$ thus we obtain also $\\varphi (S_{k}(\\Phi (d)^*,\\Phi (d)^*)=\\varphi (S_{k}(\\Phi ^{\\sharp }(d)^*,\\Phi ^{\\sharp }(d)^*)=0$ It follows that restriction of the map $\\Phi $ at von Neumann algebra $\\mathfrak {D}_{\\infty }$ it is a *-automorphism where $\\Phi (\\Phi ^{\\sharp }(d))=\\Phi ^{\\sharp }(\\Phi (d))=d$ for all $d\\in \\mathfrak {D}_{\\infty }$ .",
"We summarize the results obtained in following statement: Proposition 5 Let $( \\mathfrak {M},\\Phi ,\\varphi )$ be a quantum dynamical system.",
"The map $\\Phi _{\\infty }:\\mathfrak {D}_{\\infty }\\rightarrow \\mathfrak {D}_{\\infty }$ where $\\Phi _{\\infty }(d)=\\Phi (d)$ for all $d\\in \\mathfrak {D}_{\\infty }$ , is a *-automorphism of von Neumann algebra.",
"Furthermore if there is a von Neumann subalgebra $\\mathfrak {B}$ of $\\mathfrak {M}$ such that the restriction of $\\Phi $ to $\\mathfrak {B}$ is a *-automorphism, then we obtain $\\mathfrak {B}\\subset \\mathfrak {D}_{\\infty }$ .",
"We have a (maximal) reversible quantum dynamical systems $(\\mathfrak {D}_{\\infty },\\Phi _{\\infty },\\varphi _{\\infty })$ where the normal state $\\varphi _{\\infty }$ and the $\\varphi _{\\infty }$ -adjoint $\\Phi _{\\infty }^{\\sharp }$ , are respectively the restriction of $\\varphi $ and $\\Phi ^{\\sharp }$ to the von Neumann algebra $\\mathfrak {D}_{\\infty }$ .",
"We prove that if the restriction of $\\Phi $ to $\\mathfrak {B}$ is an automorphism, then $\\mathfrak {B} \\subset \\mathfrak {D}_\\infty $ .",
"In fact we have that $\\mathfrak {B} \\subset \\mathfrak {D}_{\\Phi ^n}$ for all natural number $n$ and if $\\Psi :\\mathfrak {B} \\rightarrow \\mathfrak {B}$ is the map such that $\\Psi (\\Phi (b))=\\Phi (\\Psi (b))=b$ for all $b \\in \\mathfrak {B}$ , then $\\Psi (b)=\\Phi ^\\sharp (b)$ , since $\\varphi (a \\Psi (b) )= \\varphi (\\Phi (a \\Psi (b))) =\\varphi (\\Phi (a) \\Phi (\\Psi (b))=\\varphi (\\Phi (a) b))=\\varphi ( a \\Phi ^\\sharp (b) )$ for all $a \\in \\mathfrak {M}$ .",
"It follows that $\\mathfrak {B}$ is also $\\Phi ^\\sharp $ -invariant, hence $\\mathfrak {B} \\subset \\mathfrak {D}_{ \\Phi ^{n \\sharp }}$ for all natural number $n$ .",
"It is clear that $\\mathfrak {D}_\\infty $ is $\\Phi _k$ -invariant for all integers $k$ and is invariant under automorphism group $\\sigma _t^\\varphi $ and by previous decomposition theorem we can say that (see [4] theorem 6): Proposition 6 If $( \\mathfrak {M},\\Phi ,\\varphi )$ is a quantum dynamical system, then there is a conditional expectation $\\mathcal {E}_\\infty :\\mathfrak {M} \\rightarrow \\mathfrak {D}_\\infty $ such that a - $\\varphi \\circ \\mathcal {E}_\\infty =\\varphi $ ; b - $\\mathfrak {D}_\\infty ^{\\perp _\\varphi }=\\ker \\mathcal {E}_\\infty $ ; c - $\\mathfrak {M} = \\mathfrak {D}_\\infty \\oplus \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ ; d - $\\Phi _k (\\mathfrak {D}_\\infty ^{\\perp _\\varphi }) \\subset \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ for all integers $k$ ; e - $\\mathcal {E}_\\infty (\\Phi _k (a))=\\Phi _k (\\mathcal {E}_\\infty (a))$ for all $a\\in \\mathfrak {M}$ and integer $k$ ; f - $\\mathcal {H}_\\varphi =\\mathcal {H}_\\infty \\oplus \\mathcal {K}_\\infty $ where $\\mathcal {H}_\\infty $ and $\\mathcal {K}_\\infty $ denotes the linear closure of $\\pi _\\varphi (\\mathfrak {D}_\\infty )\\Omega _\\varphi $ and of $\\pi _{\\varphi }(\\mathfrak {D}_\\infty ^{\\perp _{\\varphi }})\\Omega _{\\varphi }$ respectively.",
"The statements $(a)$ , $(b)$ and $(c)$ are simple consequence of theorem REF .",
"For the statement $(d)$ , if $ d_\\bot \\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ then for any integer $k$ and $x \\in \\mathfrak {D}_\\infty $ , we have: $\\varphi (\\Phi _k(d_\\bot )^*x)=\\varphi (d_\\bot ^* \\Phi _{-k}(x))=0$ since $\\Phi _{-k}(x))\\in \\mathfrak {D}_\\infty $ .",
"For the statement $(e)$ , for any $a,b \\in \\mathfrak {M}$ we obtain $\\varphi (b \\mathcal {E}_\\infty (\\Phi _k(a))) &=& \\varphi ((b_\\Vert + b_\\bot )\\mathcal {E}_\\infty (\\Phi _k(a)))=\\varphi (b_\\Vert \\mathcal {E}_\\infty (\\Phi _k(a))=\\varphi (\\mathcal {E}_\\infty (b\\Vert \\Phi _k(a)))=\\\\&=& \\varphi ( b\\Vert \\Phi _k(a)))= \\varphi (\\Phi _{-k}(b_\\Vert ) a)=\\varphi (\\mathcal {E}_\\infty (\\Phi _{-k}(b_\\Vert )a))=\\\\&=& \\varphi (\\mathcal {E}_\\infty (\\Phi _{-k} (b_\\Vert )a))=\\varphi (\\Phi _{-k}(b_\\Vert ) \\mathcal {E}_\\infty (a))= \\varphi ( b_\\Vert \\Phi _k(\\mathcal {E}_\\infty (a))=\\\\&=& \\varphi ( (b_\\Vert +b_\\bot ) \\Phi _k(\\mathcal {E}_\\infty (a))= \\varphi ( b \\Phi _k(\\mathcal {E}_\\infty (a))$ where we have write $b=b_\\Vert + b_\\bot $ with $b_\\Vert =\\mathcal {E}_\\infty (b)$ .",
"The quantum dynamical system $( \\mathfrak {D}_\\infty ,\\Phi _\\infty ,\\varphi _\\infty )$ is called the reversible part of the quantum dynamical system $( \\mathfrak {M},\\Phi ,\\varphi )$ .",
"Furthermore a quantum dynamical system is called completely irreversible if $\\mathfrak {D}_\\infty =\\mathbb {C} 1$ .",
"In this case for all $a\\in \\mathfrak {M}$ we obtain $a= \\varphi (a) 1 + a_\\bot $ and we can write $ \\mathfrak {M} = \\mathbb {C} 1 \\oplus \\ker \\varphi $ Let $( \\mathfrak {M},\\Phi ,\\varphi )$ be a completely irreversible quantum dynamical system, if the von Neumann algebra $\\mathfrak {M}$ is not trivial then there is least a not trivial projector $P\\in \\mathfrak {M}$ , such that $\\varphi (P) - \\varphi (P)^2 >0$ In fact, we can write $P=\\varphi (P) 1 + P_\\bot $ where $P_\\bot \\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ and $\\varphi (P_\\bot ^2)= \\varphi (P) - \\varphi (P)^2$ because $P_\\bot ^2 + 2 \\varphi (P) P_\\bot + \\varphi (P)^2 1= \\varphi (P) 1 + P_\\bot $ - Therefore if $\\varphi (P)=\\varphi (P)^2$ , then $P_\\bot = 0 $ .",
"In section 4 we will find the conditions when $\\mathfrak {D}_\\infty = \\mathbb {C} 1$ (see also [8] section 2 for the case $\\mathfrak {D}_\\infty ^+= \\mathbb {C} 1$ ).",
"We observe that if $\\mathcal {A}(\\mathcal {P})$ is the von Neumann algebra generated by the set of all orthogonal projections $p\\in \\mathfrak {M}$ such that $\\Phi _k(p)=\\Phi _k(p)^2$ for all integers $k$ , then $\\mathfrak {D}_\\infty =\\mathcal {A}(\\mathcal {P})$ (see [7], corollary 2).",
"In the decoherence theory the set $\\mathfrak {D}_\\infty $ is called algebra of effective observables of our quantum dynamical system (see e.g.",
"[3]) and we underline that the previous theorem is a particular case of a more general theorem that is found in [16].",
"We observe that for all natural number $n$ we obtain $\\Phi ^{\\sharp n}(\\Phi ^n(d))=d \\qquad d\\in \\mathfrak {D}_\\infty ^+$ and $\\Phi ^n(\\mathfrak {D}_\\infty ^+)\\subset \\mathfrak {D}_{\\Phi ^{\\sharp n}}$ We can say more: Remark 2 The algebra of effective observables is independent by the stationary state $\\varphi $ , since $\\mathfrak {D}_\\infty =\\mathcal {C}_\\Phi $ In fact we have that $\\mathfrak {D}_\\infty \\subset \\underset{n\\in \\mathbb {N}}{\\bigcap }\\Phi ^n(\\mathfrak {D}_\\infty ^+)$ since $\\mathfrak {D}_\\infty \\subset \\mathfrak {D}_\\infty ^+$ and $\\mathcal {C}_\\Phi \\subset \\mathfrak {D}_\\infty $ for theorem REF .",
"The next subsections are of the simple consequences of the previous propositions.",
"Ergodicity properties In this subsection we prove that the ergodic properties of a quantum dynamical system depends on its reversible part, determined from the algebra the effective observables $\\mathfrak {D}_\\infty $ .",
"We consider a quantum dynamical system $( \\mathfrak {M},\\Phi ,\\varphi )$ with $\\varphi $ -adjoint $\\Phi ^{\\sharp }$ .",
"We recall that the quantum dynamical system is ergodic if per any $a,b\\in \\mathfrak {M}$ we have: $\\underset{N\\rightarrow \\infty }{\\lim }\\dfrac{1}{N+1}\\sum \\limits _{k=0}^{N} \\left[ \\varphi (a\\Phi ^k(b))-\\varphi (a)\\varphi (b) \\right]=0$ while it is weakly mixing if $\\underset{N\\rightarrow \\infty }{\\lim }\\dfrac{1}{N+1}\\sum \\limits _{k=0}^{N} \\left| \\varphi (a\\Phi ^k(b))-\\varphi (a)\\varphi (b) \\right|=0$ We will use again the following notations $a_\\Vert =\\mathcal {E}_{\\infty }(a)$ while $a_\\bot =a-a_\\Vert $ for all $a\\in \\mathfrak {M}$ , where $\\mathcal {E}_{\\infty }:\\mathfrak {M}\\rightarrow \\mathfrak {D}_\\infty $ is the conditional expectation of decomposition theorem REF .",
"We have the following proposition: Proposition 7 The quantum dynamical system $( \\mathfrak {M},\\Phi ,\\varphi )$ is ergodic [weakly mixing] if, and only if the reversible quantum dynamical system $( \\mathfrak {D}_\\infty ,\\Phi _\\infty ,\\varphi _\\infty )$ is ergodic [weakly mixing].",
"For any $a,b\\in \\mathfrak {M}$ we have $\\varphi (a\\Phi ^k(b))-\\varphi (a)\\varphi (b)=\\varphi (a\\Phi ^k(b_\\Vert ))+ \\varphi (a\\Phi ^k(b_\\bot ))-\\varphi (a_\\Vert )\\varphi (b_\\Vert )$ Moreover $\\underset{N\\rightarrow \\infty }{\\lim }\\dfrac{1}{N+1}\\sum \\limits _{k=0}^{N} \\varphi (a\\Phi ^k(b_\\bot ))=0$ , because by relation (REF ) for every $a\\in \\mathfrak {M}$ , we have $\\underset{k\\rightarrow \\infty }{\\lim }\\varphi (a\\Phi ^k(b_\\bot ))=0$ , hence $\\underset{N\\rightarrow \\infty }{\\lim }\\dfrac{1}{N+1}\\sum \\limits _{k=0}^{N} \\left[ \\varphi (a\\Phi ^k(b)) - \\varphi (a)\\varphi (b) \\right]=\\underset{N\\rightarrow \\infty }{\\lim }\\dfrac{1}{N+1}\\sum \\limits _{k=0}^{N} \\left[ \\varphi (a\\Phi ^k(b_\\Vert ))-\\varphi (a_\\Vert )\\varphi (b_\\Vert ) \\right]$ with $\\varphi (a\\Phi ^k(b_\\Vert ))=\\varphi (a_\\Vert \\Phi ^k(b_\\Vert ))+\\varphi (a_\\bot \\Phi ^k(b_\\Vert ))$ and $\\varphi (a_\\bot \\Phi ^k(b_\\Vert ))=0$ since the element $a_\\bot \\Phi ^k(b_\\Vert )\\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ .",
"It follows that $\\underset{N\\rightarrow \\infty }{\\lim }\\dfrac{1}{N+1}\\sum \\limits _{k=0}^{N} & \\left[ \\varphi (a\\Phi ^k(b)) - \\varphi (a)\\varphi (b) \\right]=\\underset{N\\rightarrow \\infty }{\\lim }\\dfrac{1}{N+1}\\sum \\limits _{k=0}^{N} \\left[ \\varphi _\\infty (a_\\Vert \\Phi _\\infty ^k(b_\\Vert ))-\\varphi _\\infty (a_\\Vert )\\varphi _\\infty (b_\\Vert ) \\right]$ For the weakly mixing properties we have $&\\underset{N\\rightarrow \\infty }{\\lim }\\dfrac{1}{N+1}\\sum \\limits _{k=0}^{N} \\left| \\varphi (a\\Phi ^k(b))-\\varphi (a)\\varphi (b) \\right|= \\\\&=\\underset{N\\rightarrow \\infty }{\\lim }\\dfrac{1}{N+1}\\sum \\limits _{k=0}^{N} \\left| \\varphi (a_\\Vert \\Phi ^k(b_\\Vert )) + \\varphi (a_\\bot \\Phi ^k(b_\\Vert ))+ \\varphi (a\\Phi ^k(b_\\bot ))-\\varphi (a)\\varphi (b) \\right|=\\\\&=\\underset{N\\rightarrow \\infty }{\\lim }\\dfrac{1}{N+1}\\sum \\limits _{k=0}^{N} \\left| \\varphi _\\infty (a_\\Vert \\Phi _\\infty ^k(b_\\Vert )) -\\varphi _\\infty (a_\\Vert )\\varphi _\\infty (b_\\Vert ) + \\varphi (a\\Phi ^k(b_\\bot )) \\right|$ Moreover $\\underset{N\\rightarrow \\infty }{\\lim }\\dfrac{1}{N+1}\\sum \\limits _{k=0}^{N} | \\varphi (a\\Phi ^k(b_\\bot )) |=0, \\qquad a,b\\in \\mathfrak {M}$ If our quantum dynamical system $( \\mathfrak {M},\\Phi ,\\varphi )$ is weakly ergodic then $\\underset{N\\rightarrow \\infty }{\\lim }\\dfrac{1}{N+1}\\sum \\limits _{k=0}^{N} \\left| \\varphi _\\infty (a_\\Vert \\Phi _\\infty ^k(b_\\Vert )) -\\varphi _\\infty (a_\\Vert )\\varphi _\\infty (b_\\Vert ) \\right|=0$ since $\\left| \\ | \\varphi _\\infty (a_\\Vert \\Phi _\\infty ^k(b_\\Vert )) -\\varphi _\\infty (a_\\Vert )\\varphi _\\infty (b_\\Vert ) |- | \\varphi (a\\Phi ^k(b_\\bot )) | \\ \\right| \\le \\left| \\varphi (a\\Phi ^k(b))-\\varphi (a)\\varphi (b) \\right|,$ while if the reversible dynamical system $( \\mathfrak {D}_\\infty ,\\Phi _\\infty ,\\varphi _\\infty )$ is weakly mixing, then our quantum dynamical system is weakly mixing since $\\left| \\varphi (a\\Phi ^k(b))-\\varphi (a)\\varphi (b) \\right| \\le \\left| \\varphi _\\infty (a_\\Vert \\Phi _\\infty ^k(b_\\Vert )) -\\varphi _\\infty (a_\\Vert )\\varphi _\\infty (b_\\Vert ) \\right| + \\left| \\varphi (a\\Phi ^k(b_\\bot )) \\right|$ Particular *-Banach algebra Let $(\\mathfrak {M}, \\Phi , \\varphi )$ be a quantum dynamical system and $\\mathcal {E}_\\infty :\\mathfrak {M} \\rightarrow \\mathfrak {D}_\\infty $ the map of proposition REF .",
"We can define in set $\\mathfrak {M}$ another frame of *-Banach algebra changing the product between elements of $\\mathfrak {M}$ .",
"It is defined by $a\\times b =a_\\Vert \\ b_\\Vert + a_\\Vert \\ b_\\bot + a_\\bot \\ b_\\Vert \\qquad \\qquad a,b\\in \\mathfrak {M}$ where we have denoted with $a_\\Vert =\\mathcal {E}_{\\infty }(a)$ and with $a_\\bot =a-a_\\Vert $ for all $a\\in \\mathfrak {M}$ .",
"We observe again that $a_\\Vert \\ b_\\bot , a_\\bot \\ b_\\Vert \\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ since $\\mathcal {E}_{\\infty }(a_\\Vert \\ b_\\bot )=a_\\Vert \\mathcal {E}_{\\infty }(\\ b_\\bot )=0$ and $ \\mathcal {E}_{\\infty }(a_\\bot \\ b_\\Vert )=\\mathcal {E}_{\\infty }(a_\\bot ) \\ b_\\Vert =0$ .",
"Moreover we have $ a_\\bot \\times b_\\bot = 0 $ The $(\\mathfrak {M},+,\\times )$ is a Banach *-algebra with unit, since for any $a,b\\in \\mathfrak {M}$ we have: $||a \\times b||\\le ||a|| \\ ||b||$ We set with $\\mathfrak {M}^\\flat $ this Banach *-algebra.",
"We note that $\\mathfrak {M}^\\flat $ it is not a C*-algebra.",
"In fact for any $d_\\bot \\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }, \\ \\ d_\\bot \\ne 0$ we have that its spectrum in $\\mathfrak {M}^\\flat $ is $\\sigma (d_\\bot )\\subset \\lbrace 0\\rbrace $ while $ || d_\\bot ||\\ne 0$ .",
"We observe that for any $a,b\\in \\mathfrak {M}$ we have: $\\Phi (a \\times b)=\\Phi (a) \\times \\Phi (b)$ It follows that $\\Phi :\\mathfrak {M}^\\flat \\rightarrow \\mathfrak {M}^\\flat $ is a *-homomorphism of Banach Algebra.",
"For $\\varphi $ -adjoint $\\Phi ^\\sharp $ we have: $\\varphi (a \\times \\Phi (b))=\\varphi (a_\\Vert \\ \\Phi (b_\\Vert ))=\\varphi (\\Phi ^\\sharp (a_\\Vert ) \\ b_\\Vert )= \\varphi (\\Phi ^\\sharp (a) \\times b )$ with $\\Phi ^\\sharp :\\mathfrak {M}^\\flat \\rightarrow \\mathfrak {M}^\\flat $ *-homomorphism of Banach Algebra.",
"Moreover $\\varphi (a^*\\times a)=\\varphi (a_\\Vert ^* a_\\Vert )$ hence if $\\varphi (a^*\\times a)=0$ then $ a_\\Vert =0 $ , so $\\varphi $ it is not a faithful state on $\\mathfrak {M}^\\flat $ .",
"It is easily to prove that for any $a,b\\in \\mathfrak {M}^\\flat $ we obtain $\\varphi (a^* \\times b^* \\times b \\times a) = \\varphi (a^*_\\Vert \\ b^*_\\Vert \\ b_\\Vert \\ a_\\Vert )$ it follows that $\\varphi (a^* \\times b^* \\times b \\times a)\\le ||b|| \\ \\varphi (a^* \\times a)$ and we can build the GNS representation $( \\mathcal {H}_\\varphi ^\\flat ,\\pi _\\varphi ^\\flat , \\Omega _\\varphi ^\\flat )$ of the state $\\varphi $ on Banach * algebra $\\mathfrak {M}^\\flat $ that has the following properties [9]: The representation $\\pi ^\\flat _\\varphi :\\mathfrak {M}^\\flat \\rightarrow \\mathfrak {B}(\\mathcal {H}^\\flat _\\varphi )$ is a continuous map i.e.",
"$||\\pi ^\\flat _\\varphi (a)||\\le ||a||$ for all $a\\in \\mathfrak {M}^\\flat $ while $\\Omega ^\\flat _\\varphi $ is a cyclic vector for *-algebra $\\pi ^\\flat _\\varphi (\\mathfrak {M}^\\flat )$ and $\\varphi (a)=\\langle \\Omega ^\\flat _\\varphi , \\pi ^\\flat _\\varphi (a)\\Omega ^\\flat _\\varphi \\rangle _\\flat \\qquad a\\in \\mathfrak {M}^\\flat $ Furthermore we have a unitary operator $U^\\flat _\\varphi :\\mathcal {H}_\\varphi ^\\flat \\rightarrow \\mathcal {H}_\\varphi ^\\flat $ such that $\\pi _\\varphi ^\\flat (\\Phi (a)=U^\\flat _\\varphi \\pi _\\varphi ^\\flat (a) U^{\\flat *}_\\varphi \\qquad a\\in \\mathfrak {M}^\\flat $ since $\\Phi $ and $\\Phi ^\\sharp $ are *-homomorphism of Banach algebra and $U^\\flat _\\varphi \\pi _\\varphi ^\\flat (a) \\pi ^\\flat _\\varphi (b)\\Omega ^\\flat _\\varphi =\\pi _\\varphi ^\\flat (\\Phi (a \\times b)) \\Omega ^\\flat _\\varphi =\\pi _\\varphi ^\\flat (\\Phi (a )) \\pi _\\varphi ^\\flat (\\Phi (b ))\\Omega ^\\flat _\\varphi = \\pi _\\varphi ^\\flat (\\Phi (a)) U^\\flat _\\varphi \\pi ^\\flat _\\varphi (b)\\Omega ^\\flat _\\varphi $ The linear map $Z :\\mathcal {H}_\\varphi ^\\flat \\rightarrow \\mathcal {H}_\\varphi $ as defined $Z \\pi ^\\flat _\\varphi (a) \\Omega _\\varphi ^\\flat =\\pi _\\varphi ( \\mathcal {E}_\\infty (a) )\\Omega _\\varphi $ for all $a \\in \\mathfrak {M} $ it is an isometry with adjoint $Z^* \\pi _\\varphi (a)\\Omega _\\varphi =\\pi _\\varphi ^\\flat (\\mathcal {E}_\\infty (a)) \\Omega _\\varphi ^\\flat $ for all $a \\in \\mathfrak {M}^\\flat $ .",
"Furthermore we have $Z U_\\varphi ^{\\flat n} = Z U_{\\Phi , \\varphi }^n $ for all natural number $n$ .",
"Abelian algebra of effective observables We will prove that for any quantum dynamical system $(\\mathfrak {M}, \\Phi , \\varphi )$ there is an abelian algebra $\\mathcal {A} \\subset \\mathfrak {D}_\\infty $ that contains the center $Z(\\mathfrak {D}_\\infty )$ of $\\mathfrak {D}_\\infty $ and with $\\Phi (\\mathcal {A})\\subset \\mathcal {A}$ .",
"The question of the existence of an Abelian subalgebra which remains invariant under the action of a given quantum Markov semigroup are widely debated in [2] and [21].",
"We consider a discrete quantum process $(\\mathfrak {M}, \\Phi )$ with $\\Phi $ a *-automorphism.",
"We set with $\\mathfrak {P}(\\mathfrak {M})$ the pure states of $\\mathfrak {M}$ .",
"It is well know that if $\\omega (a)=0$ for all $\\omega \\in \\mathfrak {P}(\\mathfrak {M})$ then $a=0$ (see e.g.",
"[5]).",
"For any $\\omega \\in \\mathfrak {P}(\\mathfrak {M})$ with $\\mathfrak {D}_\\omega $ we set the multiplicative domain of the ucp-map $a\\in \\mathfrak {M}\\rightarrow \\omega (a)I\\in \\mathfrak {M}$ , then $\\mathfrak {D}_\\omega = \\lbrace a\\in \\mathfrak {M}: \\omega (a^*a)=\\omega (a^*) \\omega (a) \\ \\text{and} \\ \\omega (aa^*)=\\omega (a) \\omega (a^*) \\rbrace $ it is a von Neumann subalgebra of $\\mathfrak {M}$ .",
"Proposition 8 The von Neumann algebra $\\mathcal {A}=\\bigcap \\lbrace \\mathfrak {D}_\\omega : \\omega \\in \\mathfrak {P}(\\mathfrak {M}) \\rbrace $ is an abelian algebra with $\\Phi (\\mathcal {A})\\subset \\mathcal {A}$ .",
"Furthermore for any stationary state $\\varphi $ of our quantum process $(\\mathfrak {M}, \\Phi )$ , there is a $\\varphi $ -invariant conditional expectation $\\mathcal {E}_\\varphi :\\mathfrak {M}\\rightarrow \\mathcal {A}$ such that $\\mathcal {E}_\\varphi \\circ \\Phi =\\Phi $ If $a,b\\in \\mathcal {A}$ , for any pure state $\\omega $ of $\\mathfrak {M}$ we have $\\omega (ab)=\\omega (a) \\omega (b)=\\omega (ba)$ , then $\\omega (ab-ba)=0$ and it follows that $ab-ba=0$ .",
"The von Neumann algebra $\\mathcal {A}$ is $\\Phi $ -invariant $\\Phi (\\mathcal {A})\\subset \\mathcal {A}$ .",
"In fact $\\omega \\circ \\Phi \\in \\mathfrak {P}(\\mathfrak {M})$ for all $\\omega \\in \\mathfrak {P}(\\mathfrak {M})$ since $\\Phi $ is a *-automorphism.",
"Then for any $a\\in \\mathcal {A}$ we have $\\omega (\\Phi (a^*) \\Phi (a))=\\omega (\\Phi (a^*a))=\\omega (\\Phi (a^*))\\omega (\\Phi (a))$ it follows that $\\Phi (a)\\in \\mathcal {A}$ .",
"Let $\\lbrace \\sigma _\\varphi ^t \\rbrace _{t\\in \\mathbb {R}}$ be a modular group associate to GNS representation $( \\mathcal {H}_\\varphi ,\\pi _\\varphi , \\Omega _\\varphi )$ of $\\varphi $ .",
"Since the state $\\varphi $ is normal and faithful we have $\\pi _\\varphi (\\mathcal {A})^{\\prime \\prime }=\\pi _\\varphi (\\mathcal {A})$ and $\\sigma _\\varphi ^t(\\pi _\\varphi (\\mathcal {A}))\\subset \\pi _\\varphi (\\mathcal {A})$ for all $t\\in \\mathbb {R}$ .",
"In fact for any $a\\in \\mathcal {A}$ we have $\\omega (\\sigma _\\varphi ^t(a^*)\\sigma _\\varphi ^t(a))=\\omega (\\sigma _\\varphi ^t(a^*a))=\\omega (\\sigma _\\varphi ^t(a^*))\\omega (\\sigma _\\varphi ^t(a)) \\ \\qquad \\omega \\in \\mathfrak {P}(\\mathfrak {M})$ since $\\sigma _\\varphi ^t$ is a *-automorphism so $\\omega \\circ \\sigma _\\varphi ^t\\in \\mathfrak {P}(\\mathfrak {M})$ for all real number $t$ .",
"From Takesaki theorem [25] we have that there is a conditional expectation $\\mathcal {E}_\\varphi :\\mathfrak {M}\\rightarrow \\mathcal {A}$ such that $\\pi _\\varphi (\\mathcal {E}_\\varphi (m))=\\nabla ^* \\pi _\\varphi (m) \\nabla \\ \\qquad m\\in \\mathfrak {M}$ where $\\nabla :\\overline{\\pi _\\varphi (\\mathcal {A})\\Omega _\\varphi }\\longrightarrow \\mathcal {H}_\\varphi $ is the embedding map (see also [1]).",
"We recall that any pure state is multiplicative on the center $Z(\\mathfrak {M})=\\mathfrak {M} \\bigcap \\mathfrak {M}^{\\prime }$ of $\\mathfrak {M}$ (see [18]) so we have that $Z(\\mathfrak {M})\\subset \\mathfrak {D}_\\omega $ for all pure states $\\omega $ and in abelian case $\\mathcal {A}=\\mathfrak {M}$ .",
"Let $( \\mathfrak {M},\\Phi ,\\varphi )$ be a quantum dynamical system, with dynamics $\\Phi $ that admits $\\varphi $ -adjoint $\\Phi ^{\\sharp }$ .",
"By the decomposition theorem we have a *-automorphism $\\Phi _\\infty :\\mathfrak {D}_\\infty \\rightarrow \\mathfrak {D}_\\infty $ with $\\mathfrak {D}_\\infty $ von Neumann algebra, then by the previous proposition, we can say that there exist an abelian algebra $\\mathcal {A}\\subset \\mathfrak {D}_\\infty $ with $\\Phi (\\mathcal {A})\\subset \\mathcal {A}$ getting the following commutative diagram $\\begin{array}[c]{ccccc}& \\mathfrak {M} & \\overset{\\Phi }{\\longrightarrow } & \\mathfrak {M} & \\\\i_\\infty & \\uparrow & & \\downarrow &\\mathcal {E}_\\infty \\\\& \\mathfrak {D}_\\infty & \\overset{\\Phi _\\infty }{\\longrightarrow } & \\mathfrak {D}_\\infty & \\\\i_o& \\uparrow & & \\downarrow & \\mathcal {E}_\\varphi \\\\& \\mathcal {A} & \\overset{\\Phi _o}{\\longrightarrow } & \\mathcal {A} &\\end{array}$ where $i_\\infty $ and $i_o$ are the embeddig of $\\mathfrak {D}_\\infty $ and $\\mathcal {A}$ respectively, while $\\Phi _\\infty $ and $\\Phi _o$ are the restriction of $\\Phi $ to $\\mathfrak {D}_\\infty $ and $\\mathcal {A}$ respectively.",
"We observe that if the von Neumann algebra $\\mathfrak {M}$ is abelian then $\\mathcal {A}=\\mathfrak {D}_\\infty $ .",
"Dilation properties We recall that a reversible quantum dynamical system $(\\widehat{\\mathfrak {M}},\\widehat{\\Phi },\\widehat{\\varphi })$ , is said to be a dilation of the quantum dynamical system $(\\mathfrak {M},\\Phi ,\\varphi )$ , if it satisfies the following conditions: There is *-monomorphism $i:( \\mathfrak {M}, \\varphi )\\rightarrow (\\widehat{\\mathfrak {M}}, \\widehat{\\varphi })$ and a completely positive map $\\mathcal {E}:\\widehat{\\mathfrak {M}} \\rightarrow \\mathfrak {M}$ such that for each $a$ belong to $\\mathfrak {M}$ and natural number $n$ $\\mathcal {E}(\\widehat{\\Phi }^{n}(i(a)))=\\Phi ^{n}((a))$ We observe that for each $a$ belong to $\\mathfrak {M}$ and $X$ in $\\widehat{\\mathfrak {M}}$ we have: $\\mathcal {E}(i(a)X)=a \\mathcal {E} (X).$ Indeed for each $b\\in \\mathfrak {M}$ we obtain: $\\varphi (b \\mathcal {E}(i(a) X)=\\varphi (i(b) i(a) X)=\\varphi (i(b a)X)=\\varphi (b a \\mathcal {E}(X))$ So, the ucp-map $\\widehat{\\mathcal {E}} = i\\circ \\mathcal {E}$ is a conditional expectation from $\\widehat{\\mathfrak {M}}$ onto $i(\\mathfrak {M})$ which leave invariant a faithful normal state.",
"The existence of such map which characterize the range of existence of a reversible dilation of a dynamical system, be derived from a theorem of Takesaki of [25].",
"We have a proposition that establish a link between the algebra of effective observable and reversible dilation.",
"Proposition 9 If $(\\widehat{\\mathfrak {M}},\\widehat{\\Phi },\\widehat{\\varphi })$ is a dilation of quantum dynamical system $(\\mathfrak {M},\\Phi ,\\varphi )$ then $\\widehat{\\Phi }(i(a)=i(\\Phi (a)) \\quad \\textsl {if, and only if } \\quad a\\in \\mathfrak {D}_\\Phi $ We have $i(\\Phi (a)^*) \\ i(\\Phi (a))=\\widehat{\\Phi }(i(a)^*) \\widehat{\\Phi }(i(a))$ it follows that $\\Phi (a^*) \\ \\Phi (a)= \\mathcal {E} (i(\\Phi (a)^*\\Phi (a)))= \\mathcal {E}(\\widehat{\\Phi }(i(a^*a))=\\Phi (a^* a).$ For vice-versa, if $y=i(\\Phi (a))- \\widehat{\\Phi }(i(a))$ then we have $y^*y= i(\\Phi (a^*a))-\\widehat{\\Phi }(i(a^*) i(\\Phi (a))- i(\\Phi (a^*))\\widehat{\\Phi }(i(a))+ \\widehat{\\Phi }(i(a^*a))$ since $a\\in \\mathfrak {D}_\\Phi $ .",
"It follows that $\\mathcal {E}(y^*y)=0$ with $\\mathcal {E}$ faithful map, then $y=0$ .",
"Let $\\mathfrak {M}=\\mathfrak {D}_\\infty \\oplus \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ be decomposition of theorem REF of our quantum dynamical system $(\\mathfrak {M},\\Phi ,\\varphi )$ and $\\mathcal {E}_\\infty : \\mathfrak {M} \\rightarrow \\mathfrak {D}_\\infty $ the conditional expectation defined in proposition REF , we can say: Remark 3 For each $a\\in \\mathfrak {M}$ and integer $k$ we have: $\\widehat{\\Phi }^k(i(\\mathcal {E}_\\infty (a)))=i(\\Phi _k (\\mathcal {E}_\\infty (a))$ We observe that $X\\in \\ \\textit {i}(\\mathfrak {D}_\\infty )^{\\perp _{\\widehat{\\varphi }}} \\quad \\textsl {if , and only if } \\quad \\mathcal {E}(X)\\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ since $i(\\mathfrak {D}_\\infty )^{\\perp _{\\widehat{\\varphi }}} = \\lbrace X\\in \\widehat{\\mathfrak {M}}: \\widehat{\\varphi }(X^* i(d))=0 \\quad \\forall d\\in \\mathfrak {D}_\\infty \\rbrace $ and $ \\widehat{\\varphi }(X^* i(d))=\\varphi (\\mathcal {E}(X^*)d)$ for all $d\\in \\mathfrak {D}_\\infty $ .",
"We can write the following algebraic decomposition of linear spaces: $\\widehat{\\mathfrak {M}}=\\textit {i}(\\mathfrak {D}_\\infty ) \\oplus \\textit {i}(\\mathfrak {D}_\\infty )^{\\perp _{\\widehat{\\varphi }}}$ and the ucp-map $ \\widehat{\\mathcal {E}}_\\infty =i \\circ \\mathcal {E}_\\infty \\circ \\mathcal {E}$ is a conditional expectation from $\\widehat{\\mathfrak {M}}$ onto $i(\\mathfrak {D}_\\infty )$ .",
"Decomposition theorem and linear contractions We would study the relations between canonical decomposition of Nagy-Fojas of linear contraction $U_{\\Phi ,\\varphi }$ [19] and decomposition (c) of proposition REF of dynamical system $(\\mathfrak {M},\\Phi , \\varphi )$ .",
"We going to recall the main statements of these topics.",
"A contraction $T$ on the Hilbert space $\\mathcal {H}$ is called completely non-unitary if for no non zero reducing subspace $\\mathcal {K}$ for $T$ is $T_{\\mid \\mathcal {K}}$ a unitary operator, where $T_{\\mid \\mathcal {K}}$ is the restriction of contraction $T$ on the Hilbert space $\\mathcal {K}$ .",
"We set with $D_T=\\sqrt{I-T^*T}$ the defect operator of the contraction $T$ and it is well know that $TD_T=D_{T^*} T$ Moreover $ ||T\\psi |||=||\\psi || $ if, and only if $ D_T\\psi =0$ .",
"We consider the following Hilbert subspace of $\\mathcal {H}$ : $\\mathcal {H}_0= \\lbrace \\psi \\in \\mathcal {H}: ||T^n\\psi ||=||\\psi ||=||T^{* n}\\psi || \\ for \\ all \\ \\ n\\in \\mathbb {N} \\rbrace $ It is trivial show that $T^n\\mathcal {H}_0=\\mathcal {H}_0$ and $T^{* n}\\mathcal {H}_0=\\mathcal {H}_0$ for all natural number $n$ .",
"We have the following canonical decomposition (see [19]): Theorem 2 (Sz-Nagy and Fojas) To every contraction $T$ on $\\mathcal {H}$ there corresponds a uniquely determined decomposition of $\\mathcal {H}$ into a orthogonal sum of two subspace reducing $T$ we say $\\mathcal {H=H}_{0}\\mathcal {\\oplus H}_{1}$ , such that $T_0=T_{\\mid \\mathcal {H}_{0}}$ is unitary and $T_1=T_{\\mid \\mathcal {H}_{1}}$ is c.n.u., where $\\mathcal {H}_{0}=\\bigcap \\limits _{k\\in \\mathbb {Z}}\\ker \\left( D_{T_{k}}\\right) \\qquad and \\qquad \\mathcal {H}_{1}=\\mathcal {H}_{0}^{\\perp }$ with $T_{k}=\\left\\lbrace \\begin{array}[c]{cc}T^k \\ & k\\ge 0 \\\\T^{* -k } & k<0\\end{array}\\right.$ It is well know [19] that the linear operator $T_{-}= so-\\underset{n\\rightarrow +\\infty }{\\lim }T^{* n}T^n $ and $T_{+}=so-\\underset{n\\rightarrow +\\infty }{\\lim }T^n T^{* n} $ , there are in sense of strong operator ($so$ ) convergence.",
"After this brief detour on linear contractions we return to quantum dynamical systems $( \\mathfrak {M},\\Phi ,\\varphi )$ .",
"We set $V_{-}=so-\\underset{n\\rightarrow +\\infty }{\\lim }U_{\\Phi ,\\varphi }^{* n}U_{\\Phi ,\\varphi }^n $ and $V_{+}=so-\\underset{n\\rightarrow +\\infty }{\\lim }U_{\\Phi ,\\varphi }^n U_{\\Phi ,\\varphi }^{* n} $ , where $U_{\\Phi ,\\varphi }$ is the contraction defined in (REF ).",
"It follows that for each $a,b\\in \\mathfrak {M}$ we obtain: $\\underset{n\\rightarrow \\pm \\infty }{\\lim }\\varphi (S_{n}(a,b))=\\langle \\pi _{\\varphi }(a)\\Omega _\\varphi ,(I-V_\\pm )\\pi _{\\varphi }(b) \\Omega _\\varphi \\rangle $ where $S_{n}(a,b)$ is given by (REF ).",
"We recall that by proposition REF that for every integers $k$ we obtain $\\mathcal {H}_\\varphi =\\mathcal {H}_\\infty \\oplus \\mathcal {K}_\\infty $ with $U_k\\mathcal {H}_\\infty = \\mathcal {H}_\\infty $ and $U_k \\mathcal {K}_\\infty \\subset \\mathcal {K}_\\infty $ , where $U_{k}=\\left\\lbrace \\begin{array}[c]{cc}U_{\\Phi ,\\varphi }^k & k\\ge 0 \\\\\\end{array}U_{\\Phi ,\\varphi }^{* -k } & k<0\\right.$ A simple consequences of proposition REF is the following remark: For any integers $k$ we obtain $ a\\in \\mathfrak {D}_{\\Phi _k} \\quad \\textsl {if, and only if} \\quad \\pi _\\varphi (a)\\Omega _\\varphi \\in \\ker ( D_{U_k}) \\ \\textsl {and} \\ \\pi _\\varphi (a^*)\\Omega _\\varphi \\in \\ker (D_{U_k})$ Therefore $\\mathcal {H}_\\infty \\subset \\mathcal {H}_0 $ because $ \\pi _\\varphi (\\mathcal {D}_\\infty )\\Omega _\\varphi \\subset \\bigcap _{k\\in \\mathbb {Z}} \\pi _\\varphi (\\mathfrak {D}_{\\Phi _k}) \\Omega _\\varphi \\subset \\bigcap \\limits _{k\\in \\mathbb {Z}}\\ker ( D_{U_k})$ We observe that for each $a,b\\in \\mathfrak {M}$ and natural number $k$ we have (see [11] theorem 3.1): $\\underset{n\\rightarrow +\\infty }{\\lim }\\varphi (S_{k}(\\Phi ^n(a),b))=0$ Indeed for each natural numbers $k$ and $n$ , we obtain $\\varphi (S_{k}(\\Phi ^n(a),\\Phi ^n(b))=\\varphi (S_{k+n}(a,b))-\\varphi (S_{n}(a,b))$ and by the relation (REF ) result $ \\underset{n\\rightarrow +\\infty }{\\lim }(\\varphi (S_{k+n}(a,b))-\\varphi (S_{n}(a,b)))=0 $ .",
"Furthermore, for each natural number $k$ and $a,b\\in \\mathfrak {M}$ we have $| \\varphi (S_{k}(\\Phi ^n(a),b))|^2\\le \\varphi (S_{k}(\\Phi ^n(a),\\Phi ^n(a)) \\varphi (S_{k}(b,b))$ it follows that $\\underset{n\\rightarrow +\\infty }{\\lim }\\varphi (S_{k}(\\Phi ^n(a),b)=0$ .",
"We have a well-known statement (see [12], [16] and [23] ): Proposition 10 For all $a\\in \\mathfrak {M}$ any $\\sigma $ -limit point of the set $\\lbrace \\Phi ^k(a) \\rbrace _{k\\in \\mathbb {N}}$ belongs to the von Neumann algebra $\\mathfrak {D}_{\\infty }$ .",
"Moreover, for each $d_\\bot $ in $\\mathfrak {D}_{\\infty }^{\\perp _{\\varphi }}$ we have: $\\underset{k\\rightarrow +\\infty }{\\lim }\\Phi ^k(d_\\bot )=0 \\qquad and \\qquad \\underset{k\\rightarrow +\\infty }{\\lim }\\Phi ^{\\sharp k}(d_\\bot )=0$ where the limits are in $\\sigma $ -topology.",
"If $y$ is a $\\sigma $ -limit point of $\\lbrace \\Phi ^n(a) \\rbrace _{n\\in \\mathbb {N}}$ then there exists a net $\\lbrace \\Phi ^{n_{j}}(a) \\rbrace _{j\\in \\mathbb {N}}$ such that $y=\\underset{j\\rightarrow +\\infty }{\\lim }\\Phi ^{n_{j}}(a)$ in $\\sigma $ -topology.",
"Furthermore for each $b\\in \\mathfrak {M}$ we obtain $S_{k}(y,b)=\\sigma -\\underset{j\\rightarrow +\\infty }{\\lim } [ \\ \\Phi ^k(\\Phi ^{n_j}(a)b)-\\Phi ^k(\\Phi ^{n_j}(a))\\Phi ^k(b) \\ ] =\\sigma -\\underset{j\\rightarrow +\\infty }{\\lim }S_{k}(\\Phi ^{n_{j}}(a),b)$ from (REF ) we obtain $\\underset{j\\rightarrow +\\infty }{\\lim }\\varphi (S_{k}(\\Phi ^{n_{j}}(a),b))=0$ hence $\\varphi (S_{k}(y,b))=0$ .",
"It follows that $\\varphi (S_{k}(y,y))=0$ and $S_{k}(y,y)=0$ .",
"We observe that the adjoint is $\\sigma $ -continuous, then we obtain $y^*=\\underset{j\\rightarrow +\\infty }{\\lim }\\Phi ^{n_{j}}(a^*)$ , and repeating the previous steps we obtain $S_{k}(y^*,y^*)=0$ , hence $y\\in \\mathfrak {D}_{\\infty }$ .",
"For last statement we observe that for each natural number $k$ result $||\\Phi ^k(d_\\bot )||\\le ||d_\\bot ||$ and since the unit ball of the von Neumann algebra $\\mathfrak {M}$ is $\\sigma $ - compact we have that there is a subnet such that $\\Phi ^{k_\\alpha }(d_\\bot )\\rightarrow y\\in \\mathfrak {D}_{\\infty }^{\\perp _{\\varphi }}$ in $\\sigma $ -topology.",
"From previous lemma we have that $y\\in \\mathfrak {D}_{\\infty }\\cap \\mathfrak {D}_{\\infty }^{\\perp _{\\varphi }}$ it follows that $y=0$ .",
"then it can only be $\\underset{k\\rightarrow +\\infty }{\\lim }\\Phi ^k(d_\\bot )=0$ in $\\sigma $ -topology.",
"We observed that the Hilbert space $\\mathcal {H}_\\infty $ , the linear closure of $\\pi _\\varphi (\\mathfrak {D}_\\infty )\\Omega _\\varphi $ is contained in $\\mathcal {H}_0$ .",
"The next step is to understand when we have the equality of these two Hilbert spaces.",
"Let $( \\mathfrak {M},\\Phi ,\\varphi )$ be the previous quantum dynamical system, we define, for each integer $k$ , the unital Schwartz map $\\tau _k:\\mathfrak {M} \\rightarrow \\mathfrak {M}$ as $\\tau _k=\\Phi _{-k}\\circ \\Phi _k \\qquad \\qquad k\\in \\mathbb {Z}$ We have for any integer $k$ that 1 - $\\varphi \\circ \\tau _k = \\varphi $ 2 - $\\tau _k=\\tau _k^\\sharp $ , where $\\tau _k^\\sharp $ is the $\\varphi $ -adjoint of $\\tau _k$ .",
"We obtain, for any integer $k$ the dynamical system $\\left\\lbrace \\mathfrak {M}, \\tau _k, \\varphi \\right\\rbrace $ with $\\mathfrak {D}_\\infty (\\tau _k)= \\bigcap _{j\\ge 0} \\mathfrak {D}(\\tau _k^j)$ where with $ \\mathfrak {D}(\\tau _k^j) $ we have denote the multiplicative domains of map $\\tau _k ^j$ .",
"From decomposition theorem REF , for any integer $k$ we have: $\\mathfrak {M}= \\mathfrak {D}_\\infty (\\tau _k) \\oplus \\mathfrak {D}_\\infty (\\tau _k)^ {\\perp _\\varphi }$ and by the proposition REF $\\mathcal {H}_\\varphi = \\mathcal {H}_{(k)} \\oplus \\mathcal {K}_{(k)}$ where $\\mathcal {H}_{(k)} $ and $\\mathcal {K}_{(k)}$ are the closure of the linear space $\\pi _\\varphi (\\mathfrak {D}_\\infty (\\tau _k) )\\Omega _\\varphi $ and of $\\pi _{\\varphi }(\\mathfrak {D}_\\infty (\\tau _k) )^{\\perp _{\\varphi }})\\Omega _{\\varphi }$ respectively.",
"We have the following proposition: Proposition 11 If $ \\overline{\\pi _\\varphi (\\mathfrak {D}_{\\Phi _k})\\Omega _\\varphi }$ denotes the closure of linear space $\\pi _\\varphi (\\mathfrak {D}_{\\Phi _k})\\Omega _\\varphi $ then we have $\\mathcal {H}_0 = \\bigcap _{k\\in \\mathbb {Z}} \\overline{\\pi _\\varphi (\\mathfrak {D}_{\\Phi _k})\\Omega _\\varphi }$ where the $\\mathcal {H}_0$ is the Hilbert space of Nagy decomposition of theorem REF .",
"Furthermore for any $a\\in \\mathfrak {M}$ and $\\xi _0 \\in \\mathcal {H}_0$ and integr $k$ we have $U_ {\\Phi ,\\varphi }^k \\pi _\\varphi (a) \\xi _0 =\\pi _\\varphi (\\Phi ^k (a)) U_{\\Phi , \\varphi }^k \\xi _0$ We have that $\\mathfrak {D}(\\tau _k)\\subset \\mathfrak {D}_{\\Phi _k}$ for all integers $k$ .",
"In fact if $a\\in \\mathfrak {D}(\\tau _k)$ then $\\varphi (\\Phi _k(a^* a) ) & = & \\varphi (a^* a ) = \\varphi (\\tau _k(a^* a))= \\varphi (\\tau _k(a^*) \\tau _k (a)) =\\varphi (\\Phi _{-k}(\\Phi _k(a)^*) \\Phi _{-k}(\\Phi _k(a))) \\le \\\\&\\le & \\varphi (\\Phi _{-k}(\\Phi _k(a)^* \\Phi _k(a)))=\\varphi (\\Phi _k(a^*)\\Phi _k(a))\\le \\varphi (\\Phi _k(a^*a))$ It follows that $\\varphi (S_k(a,a))=0$ for all integers $k$ and in the same way proves that $\\varphi (S_k(a^*,a^*))=0$ for all integers $k$ .",
"We have proved that $\\mathfrak {D}_\\infty (\\tau _k) =\\bigcap _{j\\in \\ \\mathbb {N}} \\mathfrak {D}_{\\tau _k^j} \\subset \\mathfrak {D}_{\\tau _k} \\subset \\mathfrak {D}_{\\Phi _k}$ If $\\xi _0\\in \\mathcal {H}_0$ then for any $k$ integer and natural number $n$ we have $ (U_{\\Phi , \\varphi ,}^{* k} U_{\\Phi ,\\varphi }^{k})^n \\xi _0=\\xi _0$ and for any $r_\\bot \\in \\mathfrak {D}_\\infty (\\tau _k)^{\\perp _\\varphi }$ we can write that $\\left\\langle \\pi _\\varphi (r_\\bot )\\Omega _\\varphi , \\xi _0 \\right\\rangle = \\left\\langle (U_{\\Phi , \\varphi ,}^{* k} U_{\\Phi ,\\varphi }^{k} )^n \\ \\pi _\\varphi (r_\\bot )\\Omega _\\varphi , \\xi _0 \\right\\rangle =\\left\\langle \\pi _\\varphi (\\tau _k^n(r_\\bot )\\Omega _\\varphi , \\xi _0 \\right\\rangle $ and $\\lim _{n \\rightarrow + \\infty } \\left\\langle \\pi _\\varphi (\\tau _k^n(r_\\bot )\\Omega _\\varphi , \\xi _0 \\right\\rangle =0 \\qquad k\\in \\mathbb {Z}$ since $\\tau _k^n(r_\\bot ) \\longrightarrow 0 $ as $ n \\rightarrow \\infty $ in $\\sigma $ -topology.",
"It follows that $\\mathcal {H}_0 \\subset [ \\pi _\\varphi (\\mathfrak {D}_\\infty (\\tau _k)^{\\perp _\\varphi })\\Omega _\\varphi ] ^ \\perp = [ \\mathcal {K}_k ] ^\\perp $ .",
"Therefore for any integers $k$ we obtain: $\\mathcal {H}_0 \\subset \\mathcal {H}_{(k)} \\subset \\overline{\\pi _\\varphi (\\mathfrak {D}_{\\Phi _k})\\Omega _\\varphi } \\qquad \\Longrightarrow \\qquad \\mathcal {H}_0 \\subset \\bigcap _{k\\in \\mathbb {Z}} \\overline{\\pi _\\varphi (\\mathfrak {D}_{\\Phi _k})\\Omega _\\varphi }$ Let $\\xi _0\\in \\bigcap _{k\\in \\mathbb {Z}} \\overline{\\pi _\\varphi (\\mathfrak {D}_{\\Phi _k})\\Omega _\\varphi } $ , for any integers $k$ we have a net $d_{\\alpha ,k}\\in \\mathfrak {D}_{\\Phi _k}$ such that $\\pi _\\varphi (d_{\\alpha ,k})\\Omega _\\varphi \\rightarrow \\xi _0$ as $\\alpha \\rightarrow \\infty $ and for $k\\ge 0$ we obtain $U^{* k}_{\\Phi ,\\varphi }U^{ k}_{\\Phi ,\\varphi } \\xi _0 =U^{* k}_{\\Phi ,\\varphi }U^{ k}_{\\Phi ,\\varphi } \\lim _\\alpha \\pi _\\varphi (d_{\\alpha ,k})\\Omega _\\varphi =\\lim _\\alpha U^{* k}_{\\Phi ,\\varphi }U^{ k}_{\\Phi ,\\varphi } \\pi _\\varphi (d_{\\alpha ,k})\\Omega _\\varphi =\\lim _\\alpha \\pi _\\varphi (d_{\\alpha ,k})\\Omega _\\varphi =\\xi _0 $ in the same way for $k\\ge 0$ we have $U^{ k}_{\\Phi ,\\varphi } U^{* k}_{\\Phi ,\\varphi } \\xi _0=\\xi _0$ .",
"It follows that $ \\bigcap _{k\\in \\mathbb {Z}} \\overline{\\pi _\\varphi (\\mathfrak {D}_{\\Phi _k})\\Omega _\\varphi }\\subset \\mathcal {H}_0 $ The relation (REF ) is a straightforward.",
"We observe that for any $a\\in \\mathfrak {M}$ and $d_\\bot \\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ we have $\\underset{n\\rightarrow \\infty }{\\lim }\\varphi (a^*\\Phi _n(d_\\bot )a)=0$ since for any $d_\\bot \\in \\mathfrak {D}_{\\infty }^{\\perp _{\\varphi }}$ we obtain $\\Phi _n(d_\\bot )\\rightarrow 0$ as $n\\rightarrow \\infty $ in $\\sigma $ -topology.",
"From polarization identity we can say that $\\underset{n\\rightarrow \\infty }{\\lim }\\varphi (a\\Phi _n(d_\\bot )b)=0, \\qquad \\ a,b\\in \\mathfrak {M}, \\ d_\\bot \\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ and since $U_\\varphi $ is a contraction it follows that for any $\\xi \\in \\mathcal {H}_\\varphi $ and $\\psi \\in \\mathcal {K}_\\infty $ we have $\\underset{n\\rightarrow \\infty }{\\lim }\\langle \\xi , U_{\\Phi ,\\varphi }^n \\psi \\rangle =0 \\qquad \\text{and} \\qquad \\underset{n\\rightarrow \\infty }{\\lim }\\langle \\xi , U_{\\Phi ,\\varphi }^{*n} \\psi \\rangle =0$ We give a simple statement on the Hilbert spaces $\\mathcal {H}_\\infty $ and $\\mathcal {H}_0$ : Proposition 12 If $\\Phi ^n(d_\\bot )\\rightarrow 0$ $ [ \\Phi ^{\\sharp n}(d_\\bot )\\rightarrow 0 ] $ as $n\\rightarrow \\infty $ in $s$ -topology for all $d_\\bot \\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ ; then $\\mathcal {H}_\\infty =\\mathcal {H}_0$ and $V_+ = P_\\infty $ $ [ V_- = P_\\infty ]$ We observe that for any $\\psi \\in \\mathcal {K}_\\infty $ result $||U_{\\Phi ,\\varphi }^n \\psi ||\\rightarrow 0$ as $n\\rightarrow \\infty $ , because for any $k\\in \\mathbb {N}$ there is $d_k^\\bot \\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ such that $|| \\psi - \\pi _\\varphi (d_k^\\bot )\\Omega _\\varphi || < 1/k$ and $U_{\\Phi ,\\varphi }^n$ is a linear contraction so for all natural number $n$ we obtain: $||U_{\\Phi ,\\varphi }^n \\psi || < \\frac{1}{k}+\\varphi (\\Phi ^n(d_k^\\bot )^* \\Phi ^n(d_k^\\bot ))$ If $\\xi _0\\in \\mathcal {H}_0$ , we can write $\\xi _0=\\xi _\\Vert + \\xi _\\bot $ with $\\xi _\\Vert \\in \\mathcal {H}_\\infty $ and $\\xi _\\bot \\in \\mathcal {K}_\\infty $ .",
"Then $\\xi _\\bot =\\xi _0 - \\xi _\\Vert \\in \\mathcal {H}_0$ therefore $|| \\xi _\\Vert || + || \\xi _\\bot || = || U_{\\Phi , \\varphi }^n \\xi _0 ||= ||U_{\\Phi , \\varphi }^n \\xi _\\Vert + U_{\\Phi , \\varphi }^n \\xi _\\bot ||= ||\\xi _\\Vert || + ||U_{\\Phi , \\varphi }^n \\xi _\\bot || $ for all natural numbers $n$ it follows that $\\xi _\\bot =0$ .",
"Moreover for any $\\xi \\in \\mathcal {H}_\\varphi $ we have $U_{\\Phi ,\\varphi }^{n *} U_{\\Phi ,\\varphi }^n \\xi = \\xi _0 + U_{\\Phi ,\\varphi }^{n *} U_{\\Phi ,\\varphi }^n\\xi _1$ with $\\xi _i\\in \\mathcal {H}_i$ for $i=1,2$ and $V_+ \\xi =\\xi _0$ since $||U_{\\Phi ,\\varphi }^n \\xi _1||\\rightarrow 0$ as $n\\rightarrow \\infty $ .",
"We conclude this section with a simple observation: We recall that a dynamical system $\\lbrace \\mathfrak {M}, \\Phi , \\omega \\rbrace $ is mixing if $\\lim _{n\\rightarrow \\infty } \\varphi (a\\Phi ^n(b))=\\varphi (a) \\varphi (b) \\ , \\qquad a,b\\in \\mathfrak {M}$ by the relation (REF ) we obtain that $\\lbrace \\mathfrak {M}, \\Phi , \\omega \\rbrace $ is mixing if, and only if its reversible part $(\\mathfrak {D}_{\\infty },\\Phi _{\\infty },\\varphi _{\\infty })$ is mixing.",
"Furthermore, let $\\lbrace \\mathfrak {M}, \\Phi , \\omega \\rbrace $ be a mixing Abelian dynamical system, then there is a measurable dynamics space $(X, \\mathcal {A}, \\mu , T)$ such that $\\mathfrak {D}_{\\infty }$ is isomorphic to the von Neumann algebra $L^\\infty (X, \\mathcal {A}, \\mu )$ of the measurable bounded function on $X$ .",
"If the set $X$ is a metric space and $ \\varphi _{\\infty }$ is the unique staionary state of $\\mathfrak {D}_{\\infty }$ for the dynamics $\\Phi _{\\infty }$ , then by the corollary 4.3 of [10] we have $\\mathfrak {D}_{\\infty }=\\mathbb {C} 1$ .",
"Decomposition theorem and Cesaro mean In this section we will study the link between the decomposition theorem REF and some ergodic results which we recall briefly.",
"It is well known the following proposition (see e.g.",
"[13] par.",
"9.1 and [17] proposition 2.3).",
"Proposition 13 Let $\\lbrace \\mathfrak {M}, \\tau , \\omega \\rbrace $ be a quantum dynamical system.",
"We consider the Cesaro mean $s_n=\\frac{1}{n+1}\\sum _{k=0}^n{\\tau ^k},$ Then, there is an $\\omega $ -conditional expectation $\\mathcal {E}$ of $\\mathfrak {M}$ onto fixed point $\\mathcal {F}(\\tau )=\\left\\lbrace a\\in \\mathfrak {M} : \\tau (a)=a \\right\\rbrace $ such that $\\lim _{n\\rightarrow 0} ||\\phi \\circ s_n -\\phi \\circ \\mathcal {E} ||=0 \\qquad \\phi \\in \\mathfrak {M}_*$ A simple consequence of the previous proposition is the following remark: Remark 4 $ \\left\\lbrace \\mathfrak {M}, \\tau , \\omega \\right\\rbrace $ is ergodic if, and only if $ \\mathcal {F}(\\tau )=\\mathbb {C}1$ Let $( \\mathfrak {M},\\Phi ,\\varphi )$ be the previous quantum dynamical system, and $\\tau _k:\\mathfrak {M} \\rightarrow \\mathfrak {M}$ the Schwartz map defined in (REF ), we have a simple statement: Proposition 14 For each integer $k$ we obtain: $\\mathcal {F}(\\tau _k)=\\mathfrak {D}_{\\Phi _k}$ Without loss of generality we assume $k=1$ then $\\tau _1=\\Phi ^\\sharp \\circ \\Phi $ .",
"If $x\\in \\mathcal {F}(\\tau _1)$ we can write $\\varphi (\\Phi (x^*) \\Phi (x))=\\varphi (x^*\\tau _1(x))=\\varphi (x^*x)=\\varphi (\\Phi (x^*x))$ then $x\\in \\mathfrak {D}_\\Phi $ .",
"The converse is proved similarly.",
"Now let us ask when the algebra of effectives observables $\\mathfrak {D}_\\infty $ is trivial (see also [8] proposition 15) .",
"Proposition 15 If $\\mathcal {D}_\\infty =\\mathbb {C}1$ then the normal state $\\varphi $ is of asymptotic equilibrium and the quantum dynamical system $( \\mathfrak {M},\\Phi ,\\varphi )$ is ergodic.",
"By decomposition theorem $\\mathfrak {M}=\\mathbb {C}1 \\oplus \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ and for each $a\\in \\mathfrak {M}$ we have $a=\\varphi (a)1+a_\\perp $ with $a_\\perp \\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ .",
"It follows that $\\Phi ^n(a)= \\varphi (a) 1 + \\Phi ^n(a_\\perp )$ and $\\Phi ^n(a_\\perp )\\rightarrow 0$ in $\\sigma $ -top.",
"We have a simple consequence of the previous propositions: Corollary 1 If the quantum dynamical system $\\lbrace \\mathfrak {M}, \\tau _k, \\varphi \\rbrace $ is ergodic for some integer $k$ , then $\\mathfrak {D}_{\\infty }=\\mathbb {C}1$ .",
"If we have ergodicity then $\\mathcal {F}(\\tau _k)=\\mathfrak {D}_{\\Phi _k}=\\mathbb {C}1$ .",
"Summarizing $\\tau _1 \\ ergodic \\quad \\Longrightarrow \\quad \\Phi \\ completely \\ irreversible \\quad \\Longrightarrow \\quad \\Phi \\ ergodic$ We observe that if $( \\mathfrak {M},\\Phi ,\\varphi )$ is a quantum dynamical system with $\\Phi $ homomorphism, we have that $\\tau _1=\\Phi ^\\sharp \\circ \\Phi =id$ .",
"Hence the dynamical system $\\lbrace \\mathfrak {M}, \\tau _1, \\varphi \\rbrace $ is not ergodic (if $\\varphi $ is not multiplicative functional), while $( \\mathfrak {M},\\Phi ,\\varphi )$ can be.",
"For each integer $k$ we consider $S_{n,k}=\\frac{1}{n+1}\\sum \\limits _{j=0}^{n} \\tau ^j_k$ .",
"By previous proposition REF there is a positive map $\\mathcal {E}_k:\\mathfrak {M} \\rightarrow \\mathfrak {M}$ such that $||\\phi \\circ S_{n,k} -\\phi \\circ \\mathcal {E}_k|| \\rightarrow 0 \\qquad \\phi \\in \\mathfrak {M}_*$ and $\\mathcal {E}_k$ is the conditional expectation related of von Neumann algebra $\\mathcal {D}_{\\Phi _k}$ of theorem REF .",
"Therefore $\\mathcal {E}_k :\\mathfrak {M} \\rightarrow \\mathcal {D}_{\\Phi ^k}$ and $\\varphi \\circ \\mathcal {E}_k= \\varphi $ for all integers number $k$ .",
"Furthermore we have: $ \\mathcal {E}_h \\circ \\mathcal {E}_k= \\mathcal {E}_k \\qquad k\\ge h \\ge 0 $ because by the relation REF we have $\\mathcal {D}_{\\Phi ^k}\\subset \\mathcal {D}_{\\Phi ^h}$ for all $k\\ge h$ .",
"For each $a\\in \\mathfrak {M}$ we have $||\\mathcal {E}_k (a)||\\le ||a||$ for all integers $k$ and apply $\\sigma $ -compactness property for the bounded net $ \\left\\lbrace \\mathcal {E}_{k} (a) \\right\\rbrace _{k\\in \\mathbb {N}}$ of von Neumann algebra $\\mathfrak {M}$ , we obtain that there is at lest one $\\sigma $ -limit point $\\mathcal {E}_+ (a)$ , therefore there exist a net $ \\left\\lbrace \\mathcal {E}_{n_\\alpha } (a) \\right\\rbrace _\\alpha $ such that $\\mathcal {E}_+(a)=\\sigma -\\lim _\\alpha \\mathcal {E}_{n_\\alpha }(a)$ .",
"We obtain that $\\mathcal {E}_+(a)\\in \\mathcal {D}_{\\Phi ^k}$ for all natural number $k$ because for any $a\\in \\mathfrak {M}$ we have $\\mathcal {E}_h (\\mathcal {E}_{n_\\alpha }(a))= \\mathcal {E}_{n_\\alpha }(a)$ when $n_\\alpha \\ge h$ and since $\\mathcal {E}_h$ are normal maps follows that $\\mathcal {E}_h ( \\mathcal {E}_+(a) ) = \\mathcal {E}_+(a)$ for all natural number $h$ .",
"Furthermore, for any $x\\in \\mathfrak {D}^+_\\infty $ we have $\\varphi (xa)=\\lim _{\\alpha \\rightarrow \\infty } \\varphi (\\mathcal {E}_{n_\\alpha }(x a))= \\lim _{\\alpha \\rightarrow \\infty } \\varphi (x \\mathcal {E}_{n_\\alpha }(a)) = \\varphi (x \\mathcal {E}_+(a))$ it follows that we have a unique $\\sigma $ -limit point $\\mathcal {E}_+ (a)$ for the net $ \\left\\lbrace \\mathcal {E}_{n} (a) \\right\\rbrace _{n\\in \\mathbb {N}}$ .",
"Therefore we obtain a map $\\mathcal {E}_+ :\\mathfrak {M} \\rightarrow \\mathfrak {D}_\\infty ^+$ .",
"Moreover $\\mathcal {E}_{n_\\alpha }( \\mathcal {E}_+(a)) = \\mathcal {E}_+(a)$ for all $\\alpha $ , then $ \\mathcal {E}_+ ^2 = \\mathcal {E}_+ $ and for Tomiyama [26] the positive map $\\mathcal {E}_+$ is a conditional expectation such that $\\varphi \\circ \\mathcal {E}_+=\\varphi $ , precisely it is the conditional expectation of relation REF .",
"We can say something more: Proposition 16 Let $\\lbrace \\mathfrak {M}, \\Lambda _k, \\varphi \\rbrace _{k\\in \\mathbb {N}}$ be a family of quantum dynamical systems.",
"We consider the contraction $V_k:\\mathcal {H}_\\varphi \\rightarrow \\mathcal {H}_\\varphi $ defined in (REF ) related to Schwartz map $\\Lambda _k $ : $ V_k \\pi _\\varphi (a)\\Omega _\\varphi =\\pi _\\varphi (\\Lambda _k (a))\\Omega _\\varphi \\qquad a\\in \\mathfrak {M}$ If $|| \\left[ V_k^* - V_h ^* \\right]\\xi ||\\rightarrow 0 $ as $h,k\\rightarrow \\infty $ for all $\\xi \\in \\mathcal {H}_\\varphi $ , then there is a unital positive map $\\Lambda :\\mathfrak {M} \\rightarrow \\mathfrak {M}$ such that $|| \\phi \\circ \\Lambda _k - \\phi \\circ \\Lambda ||\\rightarrow 0$ as $k\\rightarrow \\infty $ for any $\\phi \\in \\mathfrak {M}_*$ with $ \\varphi (\\Lambda (a^*) \\Lambda (a))\\le \\varphi (a^*a) \\qquad a\\in \\mathfrak {M}$ and $\\varphi \\circ \\Lambda = \\varphi $ .",
"A simple consequence of proposition 1.1 of [17] For each natural number $n$ , we consider the following Schwartz map: $ Z_n = \\frac{1}{2n+1}\\sum \\limits _{k=-n}^{n} \\ \\tau _k$ it is obvious that $\\varphi $ is a stationary state for $Z_n$ with $\\varphi (x Z_n(y))=\\varphi (Z_n (x)y)$ for all $x,y\\in \\mathfrak {M}$ .",
"Moreover for each $a\\in \\mathfrak {M}$ we have: $\\pi _\\varphi (Z_n(a)) \\Omega _\\varphi & = & \\frac{1}{2n+1}\\sum \\limits _{k=-n}^{n} \\ \\pi _\\varphi (\\tau _k(a))\\Omega _\\varphi =\\\\& = &\\frac{1}{2n+1}\\sum \\limits _{k=0}^{n} \\ U_{\\Phi ,\\varphi }^{* k} U_{\\Phi ,\\varphi }^n \\pi _\\varphi (a))\\Omega _\\varphi +\\frac{1}{2n+1}\\sum \\limits _{k=1}^{n} \\ U_{\\Phi ,\\varphi }^k U_{\\Phi ,\\varphi }^{* k} \\pi _\\varphi (a))\\Omega _\\varphi $ and since $U_{\\Phi ,\\varphi }^{* n} U_{\\Phi ,\\varphi }^n \\rightarrow V_+$ and $U_{\\Phi ,\\varphi }^{n} U_{\\Phi ,\\varphi }^{* n}\\rightarrow V_-$ in strong operator topology, we obtain $ \\pi _\\varphi (Z_n(a)) \\Omega _\\varphi \\rightarrow \\frac{1}{2}(V_+ + V_-) \\pi _\\varphi (a)\\Omega _\\varphi $ It follows that from previous proposition that there is a $\\varphi $ invariant Schwartz map $Z:\\mathfrak {M} \\rightarrow \\mathfrak {M}$ such that $||\\phi \\circ Z_n-\\phi \\circ Z|| \\rightarrow 0 \\qquad \\phi \\in \\mathfrak {M}_*$ and $ \\pi _\\varphi (Z(a)) \\Omega _\\varphi = \\frac{1}{2}(V_+ + V_-) \\pi _\\varphi (a)\\Omega _\\varphi $ We consider the decomposition $\\mathfrak {M} = \\mathfrak {D}_\\infty \\oplus \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ for each $a=a_\\Vert + a_\\bot \\in \\mathfrak {M}$ result $Z(a_\\Vert +a_\\bot ) = a_\\Vert +Z(a_\\bot )$ with $Z(a_\\bot )\\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ .",
"We observe that if $\\Phi ^n(d_\\bot )\\rightarrow 0$ and $ \\Phi ^{\\sharp n}(d_\\bot )\\rightarrow 0 $ as $n\\rightarrow \\infty $ in $s$ -topology for all $d_\\bot \\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ (see proposition REF ); then $Z(d_\\bot )=0$ for all $d_\\bot \\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ and we have a $\\varphi $ invariant Schwartz map $Z:\\mathfrak {M} \\rightarrow \\mathfrak {D}_\\infty $ such that $ Z(xa)=xZ(a), \\qquad x\\in \\mathfrak {M}, \\ a\\in \\mathfrak {D}_\\infty $ It follows that $Z$ is the conditional expectation $\\mathcal {E}_\\infty $ of proposition REF ."
],
[
"Decomposition theorem",
"We consider a von Neumann algebra $\\mathfrak {M}$ and its faithful normal state $\\varphi $ and set with $(\\mathcal {H}_{\\varphi },\\pi _{\\varphi },\\Omega _{\\varphi })$ the GNS representation of $\\varphi $ and with $\\left\\lbrace \\sigma _{t}^{\\varphi }\\right\\rbrace _{t\\in \\mathbb {R}}$ its modular automorphism group.",
"Let $\\mathfrak {R}$ be a von Neumann subalgebra of $\\mathfrak {M}$ , we recall (see ref.",
"[14]) that the $\\varphi $ -orthogonal of $\\mathfrak {R}$ is the set: $\\mathfrak {R}^{\\perp _{\\varphi }}=\\lbrace a\\in \\mathfrak {M}:\\varphi (a^*x)=0 \\ \\ \\ \\text{for all}\\ \\ x\\in \\mathfrak {R} \\rbrace $ Furthermore, it is simple to prove that $\\mathfrak {R}^{\\perp _{\\varphi }}$ is a closed linear space in the $\\sigma $ -topology with $\\mathfrak {R}^{\\perp _{\\varphi }}\\cap \\mathfrak {R}$ ={0}.",
"We observe that $\\mathfrak {R}^{\\perp _{\\varphi }}\\subset \\ker \\varphi $ and if $\\mathfrak {R}=\\mathbb {C}I$ then $\\mathfrak {R}^{\\perp _{\\varphi }}= \\ker \\varphi $ , where $\\ker \\varphi =\\lbrace a\\in \\mathfrak {M}: \\ \\varphi (a)=0 \\rbrace $ .",
"Moreover if $ y \\in \\mathfrak {R} $ and $d_\\bot \\in \\mathfrak {R}^{\\perp _{\\varphi }}$ then $y d_\\bot \\in \\mathfrak {R}^{\\perp _{\\varphi }}$ since $ \\varphi ( (y d_\\bot )^* x)=\\varphi (d_\\bot ^* y ^* x)=0 \\qquad x\\in \\mathfrak {R} $ Theorem 1 The von Neumann algebra $\\mathfrak {R}$ is invariant under modular automorphism group $\\sigma _{t}^{\\varphi }$ if and only if both these conditions are fulfilled: a - the set $\\mathfrak {R}^{\\perp _{\\varphi }}$ is closed under the involution operation; b - for any $a\\in \\mathfrak {M}$ there is a unique $a_\\Vert \\in \\mathfrak {R}$ and $a_\\bot \\in \\mathfrak {R}^{\\perp _{\\varphi }}$ such that $a= a_\\Vert + a_\\bot $ .",
"In other words we have the following algebraic decomposition $\\mathfrak {M}= \\mathfrak {R} \\oplus \\mathfrak {R}^{\\perp _{\\varphi }}$ From Takesaki [25] we have $\\sigma _{t}^{\\varphi } (\\pi _\\varphi (\\mathfrak {R})) \\subset \\pi _\\varphi (\\mathfrak {R})$ for all $t\\in \\mathbb {R}$ if, and only if there exist a normal conditional expectation $\\mathcal {E}:\\mathfrak {M} \\rightarrow \\mathfrak {R} $ such that $\\varphi \\circ \\mathcal {E} = \\varphi $ .",
"Let $\\mathfrak {R}$ be invariant under modular automorphism group $\\sigma _{t}^{\\varphi }$ , it is simple to prove that $\\mathfrak {R}^{\\perp _{\\varphi }}=\\left\\lbrace a\\in \\mathfrak {M} : \\mathcal {E} (a)=0 \\right\\rbrace $ hence $\\mathfrak {R}^{\\perp _{\\varphi }}$ is closed under the involution operation.",
"For any $a\\in \\mathfrak {M}$ we set $a_\\bot =a- \\mathcal {E} (a)$ and $\\varphi (a_\\bot ^* x)= \\varphi ((a^* - \\mathcal {E} (a^*))x)=\\varphi (a^* x) - \\varphi ( \\mathcal {E} (a^*)x)= \\varphi (a^* x) - \\varphi ( \\mathcal {E} (a^*x))=0$ for all $x\\in \\mathfrak {R}$ hence $a_\\bot \\in \\mathfrak {R}^{\\perp _{\\varphi }}$ .",
"So for any $a\\in \\mathfrak {M}$ there exist a unique $a_\\Vert \\in \\mathfrak {R}$ and $a_\\bot \\in \\mathfrak {R}^{\\perp _{\\varphi }}$ such that $a= a_\\Vert + a_\\bot $ where we have set $a_\\Vert =\\mathcal {E}(a)$ .",
"The uniqueness follows because if $a=0$ then $a_\\Vert =a_\\bot =0$ .",
"Indeed we have $ \\varphi (a^* a) = \\varphi (a_\\Vert ^* a_\\Vert ) + \\varphi (a_\\bot ^*a_\\bot )=0$ since $a_\\Vert ^* a_\\bot , $ and $ a_\\bot ^*a_\\Vert $ belong to $\\mathfrak {R}^{\\perp _{\\varphi }}$ and $\\varphi $ is a faithful state.",
"For the vice-versa, if the set $\\mathfrak {R}^{\\perp _{\\varphi }}$ is closed under the involution operation and $ \\mathfrak {M}= \\mathfrak {R} \\oplus \\mathfrak {R}^{\\perp _{\\varphi }}$ then for any $a\\in \\mathfrak {M}$ there is a unique $a_\\Vert \\in \\mathfrak {R}$ and $a_\\bot \\in \\mathfrak {R}^{\\perp _{\\varphi }}$ such that $a= a_\\Vert + a_\\bot $ .",
"The map $ a \\in \\mathfrak {M} \\rightarrow a_\\Vert \\in \\mathfrak {R} $ is a projection of norm one ( i.e.",
"it is satisfies $(1)_\\Vert =1$ and $ ( (a)_\\Vert )_\\Vert =a_\\Vert $ for all $a\\in \\mathfrak {M} $ ), for Tomiyama [26] it is a normal conditional expectation (see [15] for a modern review) and $\\varphi (a)=\\varphi (a_\\Vert )$ for all $a \\in \\mathfrak {M}$ .",
"We observe that if $\\mathfrak {R}^{\\perp _{\\varphi }}$ is a *-algebra (without unit) then $\\mathfrak {R}^{\\perp _{\\varphi }}= \\left\\lbrace 0 \\right\\rbrace $ since $\\varphi (a_\\bot ^* a_\\bot )=0$ for all $a_\\bot \\in \\mathfrak {R}^{\\perp _{\\varphi }}$ and $\\varphi $ is a faithful state.",
"Moreover, if $p$ is a orthogonal projector of $\\mathfrak {M}$ then $p\\notin \\mathfrak {R}^{\\perp _{\\varphi }}$ .",
"We have the following remark: If $a\\in \\mathfrak {M}$ with $a=a_\\Vert +a_\\bot $ where $a_\\Vert \\in \\mathfrak {R}$ and $a_\\bot \\in \\mathfrak {R}^{\\perp _{\\varphi }}$ , then $||\\pi _{\\varphi }(a)\\Omega _{\\varphi }||^2=||\\pi _{\\varphi }(a_\\Vert )\\Omega _{\\varphi }||^2+||\\pi _{\\varphi }(a_\\bot )\\Omega _{\\varphi }||^2$ Proposition 3 Let $\\mathfrak {R}$ be a von Neumann algebra invariant under modular automorphism group $\\sigma _{t}^{\\varphi }$ .",
"If $\\mathcal {H}_o$ and $\\mathcal {K}_o$ are the closure of the linear space $\\pi _{\\varphi }(\\mathfrak {R})\\Omega _{\\varphi }$ and of $\\pi _{\\varphi }(\\mathfrak {R}^{\\perp _{\\varphi }})\\Omega _{\\varphi }$ respectively, then $\\mathcal {H}_{\\varphi }=\\mathcal {H}_o \\oplus \\mathcal {K}_o$ Moreover the orthogonal projection $P_o$ on Hilbert space $\\mathcal {H}_o$ belongs to $\\pi _{\\varphi }(\\mathfrak {R})^{\\prime }$ .",
"We have that $ \\mathcal {K}_o\\subset \\mathcal {H}_o^\\bot $ since for any $r_\\bot \\in \\mathfrak {R}^{\\perp _\\varphi }$ and $\\psi _o\\in \\mathcal {H}_o$ we obtain: $\\left\\langle \\pi _\\varphi (r_\\bot )\\Omega _\\varphi , \\psi _o \\right\\rangle = \\underset{\\alpha \\rightarrow \\infty }{\\lim }\\left\\langle \\pi _\\varphi (r_\\bot )\\Omega _\\varphi , \\pi _\\varphi (r_\\alpha )\\Omega _\\varphi \\right\\rangle = \\underset{\\alpha \\rightarrow \\infty }{\\lim } \\varphi (r_\\bot ^*r_\\alpha )=0$ where $\\psi _o =\\underset{\\alpha \\rightarrow \\infty }{\\lim } \\pi _\\varphi (r_\\alpha )\\Omega _\\varphi $ with $\\lbrace r_\\alpha \\rbrace _\\alpha $ net belongs to $\\mathfrak {R}$ .",
"Let $\\psi \\in \\mathcal {H}_\\varphi $ we can write $\\psi =\\underset{\\alpha \\rightarrow \\infty }{\\lim }\\pi _{\\varphi }(m_\\alpha )\\Omega _\\varphi =\\underset{\\alpha \\rightarrow \\infty }{\\lim }(\\pi _{\\varphi }(r_\\alpha )\\Omega _\\varphi +\\pi _{\\varphi }((r_{\\alpha \\bot })\\Omega _\\varphi )$ where $m_\\alpha = r_\\alpha + r_{\\alpha \\bot }$ for each $\\alpha $ .",
"The net $ \\lbrace \\pi _{\\varphi }(r_\\alpha )\\Omega _{\\varphi } \\rbrace $ has limit, since by the relation (REF ) for each $\\epsilon \\ge 0$ there is a index $\\nu $ such that for $\\alpha \\ge \\nu $ and $\\beta \\ge \\nu $ we have the Cauchy relation: $|| \\pi _\\varphi (r_\\alpha )\\Omega _\\varphi -\\pi _\\varphi (r_\\beta )\\Omega _\\varphi || \\le || \\pi _\\varphi (m_\\alpha )\\Omega _\\varphi -\\pi _\\varphi (m_\\beta )\\Omega _\\varphi ||\\le \\epsilon $ It follows that there are $\\psi _\\Vert \\in \\mathcal {H}_o$ and $\\psi _\\bot \\in \\mathcal {K}_o$ such that $\\psi =\\underset{\\alpha \\rightarrow \\infty }{\\lim }\\pi _{\\varphi }(r_\\alpha )\\Omega _\\varphi + \\underset{\\alpha \\rightarrow \\infty }{\\lim }\\pi _{\\varphi }((r_{\\alpha \\bot })\\Omega _\\varphi =\\psi _\\Vert + \\psi _\\bot \\in \\mathcal {H}_o \\oplus \\mathcal {K}_o$ It is simple to prove that $\\pi _\\varphi (\\mathfrak {R}) \\mathcal {H}_o\\subset \\mathcal {H}_o $ therefore $P_o\\in \\pi _\\varphi (\\mathfrak {R})^{\\prime }$ .",
"We have the following proposition: Proposition 4 Let $( \\mathfrak {M},\\Phi )$ be a quantum process and $\\varphi $ a normal faithful state on $\\mathfrak {M}$ .",
"For any natural number $n \\in \\mathbb {N}$ we obtain: $\\mathfrak {M} = \\mathfrak {D}_{\\Phi ^n} \\oplus \\mathfrak {D}_{\\Phi ^n}^{\\perp _\\varphi }$ and $\\mathfrak {M} = \\mathfrak {D}_\\infty ^+ \\oplus \\mathfrak {D}_\\infty ^{+ \\perp _\\varphi }$ Furthermore, if $\\varphi $ is a stationary state for $\\Phi $ , then $\\mathfrak {M} = \\mathcal {C}_\\Phi \\oplus \\mathcal {C}_\\Phi ^{\\perp _\\varphi }$ and the restriction of $\\Phi $ to $\\mathcal {C}_\\Phi $ is a *-automorphism with $\\Phi (\\mathcal {C}_\\Phi ^{\\perp _\\varphi })\\subset \\mathcal {C}_\\Phi ^{\\perp _\\varphi }$ .",
"for any $d\\in \\mathfrak {D}_{\\Phi ^n}$ and natural number $n$ we have: $\\Phi _{\\bullet }^ n(\\sigma ^{\\varphi }_{t}(\\pi _{\\varphi }(d)^*)\\sigma ^{\\varphi }_{t}(\\pi _{\\varphi }(d))) =\\Phi _{\\bullet }^n(\\sigma ^{\\varphi }_{t}(\\pi _{\\varphi }(d)^*))\\Phi _{\\bullet }^n(\\sigma ^{\\varphi }_{t}(\\pi _{\\varphi }(d))).$ since $\\Phi $ commutes with our modular automorphism group $\\sigma ^{\\varphi }_{t}$ .",
"It follows that $\\sigma ^{\\varphi }_{t}(\\pi _\\varphi (\\mathfrak {D}_{\\Phi ^n}) )$ is included in $\\pi _\\varphi (\\mathfrak {D}_{\\Phi ^n})$ for all $n \\in \\mathbb {N}$ and $t \\in \\mathbb {R}$ .",
"Let $b\\in \\mathcal {C}_\\Phi $ , we have that $\\sigma ^{\\varphi }_{t}(\\pi _{\\varphi }(b))\\in \\pi _\\varphi (\\mathcal {C}_\\Phi )$ for all real number $t$ .",
"In fact for each natural number $n$ there exist a $x_n\\in \\mathfrak {D}_\\infty ^{+}$ such that $b=\\Phi ^n(x_n)$ .",
"We can write that $\\sigma ^{\\varphi }_{t}(\\pi _\\varphi (b)) = \\sigma ^{\\varphi }_{t}(\\pi _\\varphi (\\Phi ^n(x_n))=\\Phi ^n_\\bullet (\\sigma ^{\\varphi }_{t}(\\pi _\\varphi (x_n)) $ and by above relation $\\sigma ^{\\varphi }_{t}(\\pi _\\varphi (x_n))\\in \\pi _\\varphi (\\mathfrak {D}_\\infty ^+)$ for all natural number $n$ .",
"It follows that $\\sigma ^{\\varphi }_{t}(\\pi _\\varphi (b))\\in \\pi _\\varphi (\\Phi ^n(\\mathfrak {D}_\\infty ^+)$ for all natural number $n$ .",
"Let $y\\in C_\\Phi ^{\\perp _\\varphi }$ , since $\\Phi (\\mathcal {C}_\\Phi )= \\mathcal {C}_\\Phi $ we have for any $c\\in \\mathcal {C}_\\Phi $ that $ \\varphi (\\Phi (y) c )=\\varphi (\\Phi (y) \\Phi (c_o))=\\varphi (y c_o)=0$ where $c=\\Phi (c_o)$ with $c_o\\in \\mathcal {C}_\\Phi \\subset \\mathfrak {D}_\\infty ^+$ .",
"We consider a quantum dynamical system $( \\mathfrak {M},\\Phi ,\\varphi )$ with $\\varphi $ -adjoint $\\Phi ^\\sharp $ .",
"We set with $\\mathfrak {D}_{\\infty }$ (or with $\\mathfrak {D}_{\\infty }(\\Phi )$ when we have to highlight the map $\\Phi $ ), the following von Neumann Algebra $\\mathfrak {D}_{\\infty } = \\bigcap \\limits _{k\\in \\mathbb {Z}} \\mathfrak {D}_{\\Phi _{k}}$ where for each $k$ integer we denote $\\Phi _{k}=\\left\\lbrace \\begin{array}[c]{cc}\\Phi ^{k} & k\\ge 0\\\\\\ \\ \\Phi ^{\\sharp \\left|k\\right|} & k<0\\end{array}\\right.$ while with $\\mathfrak {D}_{\\Phi _{k}}$ we have set the von Neumann algebra of the multiplicative domains of the dynamics $\\Phi _{k}$ .",
"Following [22], for each $a,b\\in \\mathfrak {M}$ and integers $k$ we define: $S_{k}(a,b)=\\Phi _{k}(a^*b)-\\Phi _{k}(a^*)\\Phi _{k}(b)\\in \\mathfrak {M}$ and we have these simple relations: a - $S_{k}(a,a)\\ge 0$ for all $a\\in \\mathfrak {M}$ and integers $k$ ; b - $S_{k}(a,b)^*=S_{k}(b,a)$ for all $a,b\\in \\mathfrak {M}$ and integers $k$ ; c - $d\\in \\mathfrak {D}_{\\infty }$ if, and only if $S_{k}(d,d)=S_{k}(d^*,d^*)=0$ for all integers $k$ ; d - $d\\in \\mathfrak {D}_{\\infty }$ if, and only if $\\varphi (S_{k}(d,d))=\\varphi (S_{k}(d^*,d^*))=0$ for all integers $k$ ; e - The map $a,b\\in \\mathfrak {M}\\rightarrow \\varphi (S_{k}(a,b))$ for all integers $k$ , is a sesquilinear form, hence $|\\varphi (S_{k}(a,b))|^2\\le \\varphi (S_{k}(a,a)) \\varphi (S_{k}(b,b)) \\qquad \\ a,b\\in \\mathfrak {M}$ We observe that $\\Phi (\\mathfrak {D}_{\\infty })\\subset \\mathfrak {D}_{\\infty }$ and $\\Phi ^{\\sharp }(\\mathfrak {D}_{\\infty })\\subset \\mathfrak {D}_{\\infty }$ .",
"Indeed for each element $d\\in \\mathfrak {D}_{\\infty }$ and integer $k$ we have $\\varphi (S_{k}(\\Phi (d),\\Phi (d))=\\varphi (S_{k+1}(d,d)=0$ and $\\varphi (S_{k}(\\Phi ^{\\sharp }(d),\\Phi ^{\\sharp }(d))=\\varphi (S_{k-1}(d,d)=0$ Furthermore $d^*\\in \\mathfrak {D}_{\\infty }$ thus we obtain also $\\varphi (S_{k}(\\Phi (d)^*,\\Phi (d)^*)=\\varphi (S_{k}(\\Phi ^{\\sharp }(d)^*,\\Phi ^{\\sharp }(d)^*)=0$ It follows that restriction of the map $\\Phi $ at von Neumann algebra $\\mathfrak {D}_{\\infty }$ it is a *-automorphism where $\\Phi (\\Phi ^{\\sharp }(d))=\\Phi ^{\\sharp }(\\Phi (d))=d$ for all $d\\in \\mathfrak {D}_{\\infty }$ .",
"We summarize the results obtained in following statement: Proposition 5 Let $( \\mathfrak {M},\\Phi ,\\varphi )$ be a quantum dynamical system.",
"The map $\\Phi _{\\infty }:\\mathfrak {D}_{\\infty }\\rightarrow \\mathfrak {D}_{\\infty }$ where $\\Phi _{\\infty }(d)=\\Phi (d)$ for all $d\\in \\mathfrak {D}_{\\infty }$ , is a *-automorphism of von Neumann algebra.",
"Furthermore if there is a von Neumann subalgebra $\\mathfrak {B}$ of $\\mathfrak {M}$ such that the restriction of $\\Phi $ to $\\mathfrak {B}$ is a *-automorphism, then we obtain $\\mathfrak {B}\\subset \\mathfrak {D}_{\\infty }$ .",
"We have a (maximal) reversible quantum dynamical systems $(\\mathfrak {D}_{\\infty },\\Phi _{\\infty },\\varphi _{\\infty })$ where the normal state $\\varphi _{\\infty }$ and the $\\varphi _{\\infty }$ -adjoint $\\Phi _{\\infty }^{\\sharp }$ , are respectively the restriction of $\\varphi $ and $\\Phi ^{\\sharp }$ to the von Neumann algebra $\\mathfrak {D}_{\\infty }$ .",
"We prove that if the restriction of $\\Phi $ to $\\mathfrak {B}$ is an automorphism, then $\\mathfrak {B} \\subset \\mathfrak {D}_\\infty $ .",
"In fact we have that $\\mathfrak {B} \\subset \\mathfrak {D}_{\\Phi ^n}$ for all natural number $n$ and if $\\Psi :\\mathfrak {B} \\rightarrow \\mathfrak {B}$ is the map such that $\\Psi (\\Phi (b))=\\Phi (\\Psi (b))=b$ for all $b \\in \\mathfrak {B}$ , then $\\Psi (b)=\\Phi ^\\sharp (b)$ , since $\\varphi (a \\Psi (b) )= \\varphi (\\Phi (a \\Psi (b))) =\\varphi (\\Phi (a) \\Phi (\\Psi (b))=\\varphi (\\Phi (a) b))=\\varphi ( a \\Phi ^\\sharp (b) )$ for all $a \\in \\mathfrak {M}$ .",
"It follows that $\\mathfrak {B}$ is also $\\Phi ^\\sharp $ -invariant, hence $\\mathfrak {B} \\subset \\mathfrak {D}_{ \\Phi ^{n \\sharp }}$ for all natural number $n$ .",
"It is clear that $\\mathfrak {D}_\\infty $ is $\\Phi _k$ -invariant for all integers $k$ and is invariant under automorphism group $\\sigma _t^\\varphi $ and by previous decomposition theorem we can say that (see [4] theorem 6): Proposition 6 If $( \\mathfrak {M},\\Phi ,\\varphi )$ is a quantum dynamical system, then there is a conditional expectation $\\mathcal {E}_\\infty :\\mathfrak {M} \\rightarrow \\mathfrak {D}_\\infty $ such that a - $\\varphi \\circ \\mathcal {E}_\\infty =\\varphi $ ; b - $\\mathfrak {D}_\\infty ^{\\perp _\\varphi }=\\ker \\mathcal {E}_\\infty $ ; c - $\\mathfrak {M} = \\mathfrak {D}_\\infty \\oplus \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ ; d - $\\Phi _k (\\mathfrak {D}_\\infty ^{\\perp _\\varphi }) \\subset \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ for all integers $k$ ; e - $\\mathcal {E}_\\infty (\\Phi _k (a))=\\Phi _k (\\mathcal {E}_\\infty (a))$ for all $a\\in \\mathfrak {M}$ and integer $k$ ; f - $\\mathcal {H}_\\varphi =\\mathcal {H}_\\infty \\oplus \\mathcal {K}_\\infty $ where $\\mathcal {H}_\\infty $ and $\\mathcal {K}_\\infty $ denotes the linear closure of $\\pi _\\varphi (\\mathfrak {D}_\\infty )\\Omega _\\varphi $ and of $\\pi _{\\varphi }(\\mathfrak {D}_\\infty ^{\\perp _{\\varphi }})\\Omega _{\\varphi }$ respectively.",
"The statements $(a)$ , $(b)$ and $(c)$ are simple consequence of theorem REF .",
"For the statement $(d)$ , if $ d_\\bot \\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ then for any integer $k$ and $x \\in \\mathfrak {D}_\\infty $ , we have: $\\varphi (\\Phi _k(d_\\bot )^*x)=\\varphi (d_\\bot ^* \\Phi _{-k}(x))=0$ since $\\Phi _{-k}(x))\\in \\mathfrak {D}_\\infty $ .",
"For the statement $(e)$ , for any $a,b \\in \\mathfrak {M}$ we obtain $\\varphi (b \\mathcal {E}_\\infty (\\Phi _k(a))) &=& \\varphi ((b_\\Vert + b_\\bot )\\mathcal {E}_\\infty (\\Phi _k(a)))=\\varphi (b_\\Vert \\mathcal {E}_\\infty (\\Phi _k(a))=\\varphi (\\mathcal {E}_\\infty (b\\Vert \\Phi _k(a)))=\\\\&=& \\varphi ( b\\Vert \\Phi _k(a)))= \\varphi (\\Phi _{-k}(b_\\Vert ) a)=\\varphi (\\mathcal {E}_\\infty (\\Phi _{-k}(b_\\Vert )a))=\\\\&=& \\varphi (\\mathcal {E}_\\infty (\\Phi _{-k} (b_\\Vert )a))=\\varphi (\\Phi _{-k}(b_\\Vert ) \\mathcal {E}_\\infty (a))= \\varphi ( b_\\Vert \\Phi _k(\\mathcal {E}_\\infty (a))=\\\\&=& \\varphi ( (b_\\Vert +b_\\bot ) \\Phi _k(\\mathcal {E}_\\infty (a))= \\varphi ( b \\Phi _k(\\mathcal {E}_\\infty (a))$ where we have write $b=b_\\Vert + b_\\bot $ with $b_\\Vert =\\mathcal {E}_\\infty (b)$ .",
"The quantum dynamical system $( \\mathfrak {D}_\\infty ,\\Phi _\\infty ,\\varphi _\\infty )$ is called the reversible part of the quantum dynamical system $( \\mathfrak {M},\\Phi ,\\varphi )$ .",
"Furthermore a quantum dynamical system is called completely irreversible if $\\mathfrak {D}_\\infty =\\mathbb {C} 1$ .",
"In this case for all $a\\in \\mathfrak {M}$ we obtain $a= \\varphi (a) 1 + a_\\bot $ and we can write $ \\mathfrak {M} = \\mathbb {C} 1 \\oplus \\ker \\varphi $ Let $( \\mathfrak {M},\\Phi ,\\varphi )$ be a completely irreversible quantum dynamical system, if the von Neumann algebra $\\mathfrak {M}$ is not trivial then there is least a not trivial projector $P\\in \\mathfrak {M}$ , such that $\\varphi (P) - \\varphi (P)^2 >0$ In fact, we can write $P=\\varphi (P) 1 + P_\\bot $ where $P_\\bot \\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ and $\\varphi (P_\\bot ^2)= \\varphi (P) - \\varphi (P)^2$ because $P_\\bot ^2 + 2 \\varphi (P) P_\\bot + \\varphi (P)^2 1= \\varphi (P) 1 + P_\\bot $ - Therefore if $\\varphi (P)=\\varphi (P)^2$ , then $P_\\bot = 0 $ .",
"In section 4 we will find the conditions when $\\mathfrak {D}_\\infty = \\mathbb {C} 1$ (see also [8] section 2 for the case $\\mathfrak {D}_\\infty ^+= \\mathbb {C} 1$ ).",
"We observe that if $\\mathcal {A}(\\mathcal {P})$ is the von Neumann algebra generated by the set of all orthogonal projections $p\\in \\mathfrak {M}$ such that $\\Phi _k(p)=\\Phi _k(p)^2$ for all integers $k$ , then $\\mathfrak {D}_\\infty =\\mathcal {A}(\\mathcal {P})$ (see [7], corollary 2).",
"In the decoherence theory the set $\\mathfrak {D}_\\infty $ is called algebra of effective observables of our quantum dynamical system (see e.g.",
"[3]) and we underline that the previous theorem is a particular case of a more general theorem that is found in [16].",
"We observe that for all natural number $n$ we obtain $\\Phi ^{\\sharp n}(\\Phi ^n(d))=d \\qquad d\\in \\mathfrak {D}_\\infty ^+$ and $\\Phi ^n(\\mathfrak {D}_\\infty ^+)\\subset \\mathfrak {D}_{\\Phi ^{\\sharp n}}$ We can say more: Remark 2 The algebra of effective observables is independent by the stationary state $\\varphi $ , since $\\mathfrak {D}_\\infty =\\mathcal {C}_\\Phi $ In fact we have that $\\mathfrak {D}_\\infty \\subset \\underset{n\\in \\mathbb {N}}{\\bigcap }\\Phi ^n(\\mathfrak {D}_\\infty ^+)$ since $\\mathfrak {D}_\\infty \\subset \\mathfrak {D}_\\infty ^+$ and $\\mathcal {C}_\\Phi \\subset \\mathfrak {D}_\\infty $ for theorem REF .",
"The next subsections are of the simple consequences of the previous propositions."
],
[
"Ergodicity properties",
"In this subsection we prove that the ergodic properties of a quantum dynamical system depends on its reversible part, determined from the algebra the effective observables $\\mathfrak {D}_\\infty $ .",
"We consider a quantum dynamical system $( \\mathfrak {M},\\Phi ,\\varphi )$ with $\\varphi $ -adjoint $\\Phi ^{\\sharp }$ .",
"We recall that the quantum dynamical system is ergodic if per any $a,b\\in \\mathfrak {M}$ we have: $\\underset{N\\rightarrow \\infty }{\\lim }\\dfrac{1}{N+1}\\sum \\limits _{k=0}^{N} \\left[ \\varphi (a\\Phi ^k(b))-\\varphi (a)\\varphi (b) \\right]=0$ while it is weakly mixing if $\\underset{N\\rightarrow \\infty }{\\lim }\\dfrac{1}{N+1}\\sum \\limits _{k=0}^{N} \\left| \\varphi (a\\Phi ^k(b))-\\varphi (a)\\varphi (b) \\right|=0$ We will use again the following notations $a_\\Vert =\\mathcal {E}_{\\infty }(a)$ while $a_\\bot =a-a_\\Vert $ for all $a\\in \\mathfrak {M}$ , where $\\mathcal {E}_{\\infty }:\\mathfrak {M}\\rightarrow \\mathfrak {D}_\\infty $ is the conditional expectation of decomposition theorem REF .",
"We have the following proposition: Proposition 7 The quantum dynamical system $( \\mathfrak {M},\\Phi ,\\varphi )$ is ergodic [weakly mixing] if, and only if the reversible quantum dynamical system $( \\mathfrak {D}_\\infty ,\\Phi _\\infty ,\\varphi _\\infty )$ is ergodic [weakly mixing].",
"For any $a,b\\in \\mathfrak {M}$ we have $\\varphi (a\\Phi ^k(b))-\\varphi (a)\\varphi (b)=\\varphi (a\\Phi ^k(b_\\Vert ))+ \\varphi (a\\Phi ^k(b_\\bot ))-\\varphi (a_\\Vert )\\varphi (b_\\Vert )$ Moreover $\\underset{N\\rightarrow \\infty }{\\lim }\\dfrac{1}{N+1}\\sum \\limits _{k=0}^{N} \\varphi (a\\Phi ^k(b_\\bot ))=0$ , because by relation (REF ) for every $a\\in \\mathfrak {M}$ , we have $\\underset{k\\rightarrow \\infty }{\\lim }\\varphi (a\\Phi ^k(b_\\bot ))=0$ , hence $\\underset{N\\rightarrow \\infty }{\\lim }\\dfrac{1}{N+1}\\sum \\limits _{k=0}^{N} \\left[ \\varphi (a\\Phi ^k(b)) - \\varphi (a)\\varphi (b) \\right]=\\underset{N\\rightarrow \\infty }{\\lim }\\dfrac{1}{N+1}\\sum \\limits _{k=0}^{N} \\left[ \\varphi (a\\Phi ^k(b_\\Vert ))-\\varphi (a_\\Vert )\\varphi (b_\\Vert ) \\right]$ with $\\varphi (a\\Phi ^k(b_\\Vert ))=\\varphi (a_\\Vert \\Phi ^k(b_\\Vert ))+\\varphi (a_\\bot \\Phi ^k(b_\\Vert ))$ and $\\varphi (a_\\bot \\Phi ^k(b_\\Vert ))=0$ since the element $a_\\bot \\Phi ^k(b_\\Vert )\\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ .",
"It follows that $\\underset{N\\rightarrow \\infty }{\\lim }\\dfrac{1}{N+1}\\sum \\limits _{k=0}^{N} & \\left[ \\varphi (a\\Phi ^k(b)) - \\varphi (a)\\varphi (b) \\right]=\\underset{N\\rightarrow \\infty }{\\lim }\\dfrac{1}{N+1}\\sum \\limits _{k=0}^{N} \\left[ \\varphi _\\infty (a_\\Vert \\Phi _\\infty ^k(b_\\Vert ))-\\varphi _\\infty (a_\\Vert )\\varphi _\\infty (b_\\Vert ) \\right]$ For the weakly mixing properties we have $&\\underset{N\\rightarrow \\infty }{\\lim }\\dfrac{1}{N+1}\\sum \\limits _{k=0}^{N} \\left| \\varphi (a\\Phi ^k(b))-\\varphi (a)\\varphi (b) \\right|= \\\\&=\\underset{N\\rightarrow \\infty }{\\lim }\\dfrac{1}{N+1}\\sum \\limits _{k=0}^{N} \\left| \\varphi (a_\\Vert \\Phi ^k(b_\\Vert )) + \\varphi (a_\\bot \\Phi ^k(b_\\Vert ))+ \\varphi (a\\Phi ^k(b_\\bot ))-\\varphi (a)\\varphi (b) \\right|=\\\\&=\\underset{N\\rightarrow \\infty }{\\lim }\\dfrac{1}{N+1}\\sum \\limits _{k=0}^{N} \\left| \\varphi _\\infty (a_\\Vert \\Phi _\\infty ^k(b_\\Vert )) -\\varphi _\\infty (a_\\Vert )\\varphi _\\infty (b_\\Vert ) + \\varphi (a\\Phi ^k(b_\\bot )) \\right|$ Moreover $\\underset{N\\rightarrow \\infty }{\\lim }\\dfrac{1}{N+1}\\sum \\limits _{k=0}^{N} | \\varphi (a\\Phi ^k(b_\\bot )) |=0, \\qquad a,b\\in \\mathfrak {M}$ If our quantum dynamical system $( \\mathfrak {M},\\Phi ,\\varphi )$ is weakly ergodic then $\\underset{N\\rightarrow \\infty }{\\lim }\\dfrac{1}{N+1}\\sum \\limits _{k=0}^{N} \\left| \\varphi _\\infty (a_\\Vert \\Phi _\\infty ^k(b_\\Vert )) -\\varphi _\\infty (a_\\Vert )\\varphi _\\infty (b_\\Vert ) \\right|=0$ since $\\left| \\ | \\varphi _\\infty (a_\\Vert \\Phi _\\infty ^k(b_\\Vert )) -\\varphi _\\infty (a_\\Vert )\\varphi _\\infty (b_\\Vert ) |- | \\varphi (a\\Phi ^k(b_\\bot )) | \\ \\right| \\le \\left| \\varphi (a\\Phi ^k(b))-\\varphi (a)\\varphi (b) \\right|,$ while if the reversible dynamical system $( \\mathfrak {D}_\\infty ,\\Phi _\\infty ,\\varphi _\\infty )$ is weakly mixing, then our quantum dynamical system is weakly mixing since $\\left| \\varphi (a\\Phi ^k(b))-\\varphi (a)\\varphi (b) \\right| \\le \\left| \\varphi _\\infty (a_\\Vert \\Phi _\\infty ^k(b_\\Vert )) -\\varphi _\\infty (a_\\Vert )\\varphi _\\infty (b_\\Vert ) \\right| + \\left| \\varphi (a\\Phi ^k(b_\\bot )) \\right|$"
],
[
" Particular *-Banach algebra",
"Let $(\\mathfrak {M}, \\Phi , \\varphi )$ be a quantum dynamical system and $\\mathcal {E}_\\infty :\\mathfrak {M} \\rightarrow \\mathfrak {D}_\\infty $ the map of proposition REF .",
"We can define in set $\\mathfrak {M}$ another frame of *-Banach algebra changing the product between elements of $\\mathfrak {M}$ .",
"It is defined by $a\\times b =a_\\Vert \\ b_\\Vert + a_\\Vert \\ b_\\bot + a_\\bot \\ b_\\Vert \\qquad \\qquad a,b\\in \\mathfrak {M}$ where we have denoted with $a_\\Vert =\\mathcal {E}_{\\infty }(a)$ and with $a_\\bot =a-a_\\Vert $ for all $a\\in \\mathfrak {M}$ .",
"We observe again that $a_\\Vert \\ b_\\bot , a_\\bot \\ b_\\Vert \\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ since $\\mathcal {E}_{\\infty }(a_\\Vert \\ b_\\bot )=a_\\Vert \\mathcal {E}_{\\infty }(\\ b_\\bot )=0$ and $ \\mathcal {E}_{\\infty }(a_\\bot \\ b_\\Vert )=\\mathcal {E}_{\\infty }(a_\\bot ) \\ b_\\Vert =0$ .",
"Moreover we have $ a_\\bot \\times b_\\bot = 0 $ The $(\\mathfrak {M},+,\\times )$ is a Banach *-algebra with unit, since for any $a,b\\in \\mathfrak {M}$ we have: $||a \\times b||\\le ||a|| \\ ||b||$ We set with $\\mathfrak {M}^\\flat $ this Banach *-algebra.",
"We note that $\\mathfrak {M}^\\flat $ it is not a C*-algebra.",
"In fact for any $d_\\bot \\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }, \\ \\ d_\\bot \\ne 0$ we have that its spectrum in $\\mathfrak {M}^\\flat $ is $\\sigma (d_\\bot )\\subset \\lbrace 0\\rbrace $ while $ || d_\\bot ||\\ne 0$ .",
"We observe that for any $a,b\\in \\mathfrak {M}$ we have: $\\Phi (a \\times b)=\\Phi (a) \\times \\Phi (b)$ It follows that $\\Phi :\\mathfrak {M}^\\flat \\rightarrow \\mathfrak {M}^\\flat $ is a *-homomorphism of Banach Algebra.",
"For $\\varphi $ -adjoint $\\Phi ^\\sharp $ we have: $\\varphi (a \\times \\Phi (b))=\\varphi (a_\\Vert \\ \\Phi (b_\\Vert ))=\\varphi (\\Phi ^\\sharp (a_\\Vert ) \\ b_\\Vert )= \\varphi (\\Phi ^\\sharp (a) \\times b )$ with $\\Phi ^\\sharp :\\mathfrak {M}^\\flat \\rightarrow \\mathfrak {M}^\\flat $ *-homomorphism of Banach Algebra.",
"Moreover $\\varphi (a^*\\times a)=\\varphi (a_\\Vert ^* a_\\Vert )$ hence if $\\varphi (a^*\\times a)=0$ then $ a_\\Vert =0 $ , so $\\varphi $ it is not a faithful state on $\\mathfrak {M}^\\flat $ .",
"It is easily to prove that for any $a,b\\in \\mathfrak {M}^\\flat $ we obtain $\\varphi (a^* \\times b^* \\times b \\times a) = \\varphi (a^*_\\Vert \\ b^*_\\Vert \\ b_\\Vert \\ a_\\Vert )$ it follows that $\\varphi (a^* \\times b^* \\times b \\times a)\\le ||b|| \\ \\varphi (a^* \\times a)$ and we can build the GNS representation $( \\mathcal {H}_\\varphi ^\\flat ,\\pi _\\varphi ^\\flat , \\Omega _\\varphi ^\\flat )$ of the state $\\varphi $ on Banach * algebra $\\mathfrak {M}^\\flat $ that has the following properties [9]: The representation $\\pi ^\\flat _\\varphi :\\mathfrak {M}^\\flat \\rightarrow \\mathfrak {B}(\\mathcal {H}^\\flat _\\varphi )$ is a continuous map i.e.",
"$||\\pi ^\\flat _\\varphi (a)||\\le ||a||$ for all $a\\in \\mathfrak {M}^\\flat $ while $\\Omega ^\\flat _\\varphi $ is a cyclic vector for *-algebra $\\pi ^\\flat _\\varphi (\\mathfrak {M}^\\flat )$ and $\\varphi (a)=\\langle \\Omega ^\\flat _\\varphi , \\pi ^\\flat _\\varphi (a)\\Omega ^\\flat _\\varphi \\rangle _\\flat \\qquad a\\in \\mathfrak {M}^\\flat $ Furthermore we have a unitary operator $U^\\flat _\\varphi :\\mathcal {H}_\\varphi ^\\flat \\rightarrow \\mathcal {H}_\\varphi ^\\flat $ such that $\\pi _\\varphi ^\\flat (\\Phi (a)=U^\\flat _\\varphi \\pi _\\varphi ^\\flat (a) U^{\\flat *}_\\varphi \\qquad a\\in \\mathfrak {M}^\\flat $ since $\\Phi $ and $\\Phi ^\\sharp $ are *-homomorphism of Banach algebra and $U^\\flat _\\varphi \\pi _\\varphi ^\\flat (a) \\pi ^\\flat _\\varphi (b)\\Omega ^\\flat _\\varphi =\\pi _\\varphi ^\\flat (\\Phi (a \\times b)) \\Omega ^\\flat _\\varphi =\\pi _\\varphi ^\\flat (\\Phi (a )) \\pi _\\varphi ^\\flat (\\Phi (b ))\\Omega ^\\flat _\\varphi = \\pi _\\varphi ^\\flat (\\Phi (a)) U^\\flat _\\varphi \\pi ^\\flat _\\varphi (b)\\Omega ^\\flat _\\varphi $ The linear map $Z :\\mathcal {H}_\\varphi ^\\flat \\rightarrow \\mathcal {H}_\\varphi $ as defined $Z \\pi ^\\flat _\\varphi (a) \\Omega _\\varphi ^\\flat =\\pi _\\varphi ( \\mathcal {E}_\\infty (a) )\\Omega _\\varphi $ for all $a \\in \\mathfrak {M} $ it is an isometry with adjoint $Z^* \\pi _\\varphi (a)\\Omega _\\varphi =\\pi _\\varphi ^\\flat (\\mathcal {E}_\\infty (a)) \\Omega _\\varphi ^\\flat $ for all $a \\in \\mathfrak {M}^\\flat $ .",
"Furthermore we have $Z U_\\varphi ^{\\flat n} = Z U_{\\Phi , \\varphi }^n $ for all natural number $n$ ."
],
[
"Abelian algebra of effective observables",
"We will prove that for any quantum dynamical system $(\\mathfrak {M}, \\Phi , \\varphi )$ there is an abelian algebra $\\mathcal {A} \\subset \\mathfrak {D}_\\infty $ that contains the center $Z(\\mathfrak {D}_\\infty )$ of $\\mathfrak {D}_\\infty $ and with $\\Phi (\\mathcal {A})\\subset \\mathcal {A}$ .",
"The question of the existence of an Abelian subalgebra which remains invariant under the action of a given quantum Markov semigroup are widely debated in [2] and [21].",
"We consider a discrete quantum process $(\\mathfrak {M}, \\Phi )$ with $\\Phi $ a *-automorphism.",
"We set with $\\mathfrak {P}(\\mathfrak {M})$ the pure states of $\\mathfrak {M}$ .",
"It is well know that if $\\omega (a)=0$ for all $\\omega \\in \\mathfrak {P}(\\mathfrak {M})$ then $a=0$ (see e.g.",
"[5]).",
"For any $\\omega \\in \\mathfrak {P}(\\mathfrak {M})$ with $\\mathfrak {D}_\\omega $ we set the multiplicative domain of the ucp-map $a\\in \\mathfrak {M}\\rightarrow \\omega (a)I\\in \\mathfrak {M}$ , then $\\mathfrak {D}_\\omega = \\lbrace a\\in \\mathfrak {M}: \\omega (a^*a)=\\omega (a^*) \\omega (a) \\ \\text{and} \\ \\omega (aa^*)=\\omega (a) \\omega (a^*) \\rbrace $ it is a von Neumann subalgebra of $\\mathfrak {M}$ .",
"Proposition 8 The von Neumann algebra $\\mathcal {A}=\\bigcap \\lbrace \\mathfrak {D}_\\omega : \\omega \\in \\mathfrak {P}(\\mathfrak {M}) \\rbrace $ is an abelian algebra with $\\Phi (\\mathcal {A})\\subset \\mathcal {A}$ .",
"Furthermore for any stationary state $\\varphi $ of our quantum process $(\\mathfrak {M}, \\Phi )$ , there is a $\\varphi $ -invariant conditional expectation $\\mathcal {E}_\\varphi :\\mathfrak {M}\\rightarrow \\mathcal {A}$ such that $\\mathcal {E}_\\varphi \\circ \\Phi =\\Phi $ If $a,b\\in \\mathcal {A}$ , for any pure state $\\omega $ of $\\mathfrak {M}$ we have $\\omega (ab)=\\omega (a) \\omega (b)=\\omega (ba)$ , then $\\omega (ab-ba)=0$ and it follows that $ab-ba=0$ .",
"The von Neumann algebra $\\mathcal {A}$ is $\\Phi $ -invariant $\\Phi (\\mathcal {A})\\subset \\mathcal {A}$ .",
"In fact $\\omega \\circ \\Phi \\in \\mathfrak {P}(\\mathfrak {M})$ for all $\\omega \\in \\mathfrak {P}(\\mathfrak {M})$ since $\\Phi $ is a *-automorphism.",
"Then for any $a\\in \\mathcal {A}$ we have $\\omega (\\Phi (a^*) \\Phi (a))=\\omega (\\Phi (a^*a))=\\omega (\\Phi (a^*))\\omega (\\Phi (a))$ it follows that $\\Phi (a)\\in \\mathcal {A}$ .",
"Let $\\lbrace \\sigma _\\varphi ^t \\rbrace _{t\\in \\mathbb {R}}$ be a modular group associate to GNS representation $( \\mathcal {H}_\\varphi ,\\pi _\\varphi , \\Omega _\\varphi )$ of $\\varphi $ .",
"Since the state $\\varphi $ is normal and faithful we have $\\pi _\\varphi (\\mathcal {A})^{\\prime \\prime }=\\pi _\\varphi (\\mathcal {A})$ and $\\sigma _\\varphi ^t(\\pi _\\varphi (\\mathcal {A}))\\subset \\pi _\\varphi (\\mathcal {A})$ for all $t\\in \\mathbb {R}$ .",
"In fact for any $a\\in \\mathcal {A}$ we have $\\omega (\\sigma _\\varphi ^t(a^*)\\sigma _\\varphi ^t(a))=\\omega (\\sigma _\\varphi ^t(a^*a))=\\omega (\\sigma _\\varphi ^t(a^*))\\omega (\\sigma _\\varphi ^t(a)) \\ \\qquad \\omega \\in \\mathfrak {P}(\\mathfrak {M})$ since $\\sigma _\\varphi ^t$ is a *-automorphism so $\\omega \\circ \\sigma _\\varphi ^t\\in \\mathfrak {P}(\\mathfrak {M})$ for all real number $t$ .",
"From Takesaki theorem [25] we have that there is a conditional expectation $\\mathcal {E}_\\varphi :\\mathfrak {M}\\rightarrow \\mathcal {A}$ such that $\\pi _\\varphi (\\mathcal {E}_\\varphi (m))=\\nabla ^* \\pi _\\varphi (m) \\nabla \\ \\qquad m\\in \\mathfrak {M}$ where $\\nabla :\\overline{\\pi _\\varphi (\\mathcal {A})\\Omega _\\varphi }\\longrightarrow \\mathcal {H}_\\varphi $ is the embedding map (see also [1]).",
"We recall that any pure state is multiplicative on the center $Z(\\mathfrak {M})=\\mathfrak {M} \\bigcap \\mathfrak {M}^{\\prime }$ of $\\mathfrak {M}$ (see [18]) so we have that $Z(\\mathfrak {M})\\subset \\mathfrak {D}_\\omega $ for all pure states $\\omega $ and in abelian case $\\mathcal {A}=\\mathfrak {M}$ .",
"Let $( \\mathfrak {M},\\Phi ,\\varphi )$ be a quantum dynamical system, with dynamics $\\Phi $ that admits $\\varphi $ -adjoint $\\Phi ^{\\sharp }$ .",
"By the decomposition theorem we have a *-automorphism $\\Phi _\\infty :\\mathfrak {D}_\\infty \\rightarrow \\mathfrak {D}_\\infty $ with $\\mathfrak {D}_\\infty $ von Neumann algebra, then by the previous proposition, we can say that there exist an abelian algebra $\\mathcal {A}\\subset \\mathfrak {D}_\\infty $ with $\\Phi (\\mathcal {A})\\subset \\mathcal {A}$ getting the following commutative diagram $\\begin{array}[c]{ccccc}& \\mathfrak {M} & \\overset{\\Phi }{\\longrightarrow } & \\mathfrak {M} & \\\\i_\\infty & \\uparrow & & \\downarrow &\\mathcal {E}_\\infty \\\\& \\mathfrak {D}_\\infty & \\overset{\\Phi _\\infty }{\\longrightarrow } & \\mathfrak {D}_\\infty & \\\\i_o& \\uparrow & & \\downarrow & \\mathcal {E}_\\varphi \\\\& \\mathcal {A} & \\overset{\\Phi _o}{\\longrightarrow } & \\mathcal {A} &\\end{array}$ where $i_\\infty $ and $i_o$ are the embeddig of $\\mathfrak {D}_\\infty $ and $\\mathcal {A}$ respectively, while $\\Phi _\\infty $ and $\\Phi _o$ are the restriction of $\\Phi $ to $\\mathfrak {D}_\\infty $ and $\\mathcal {A}$ respectively.",
"We observe that if the von Neumann algebra $\\mathfrak {M}$ is abelian then $\\mathcal {A}=\\mathfrak {D}_\\infty $ ."
],
[
"Dilation properties",
"We recall that a reversible quantum dynamical system $(\\widehat{\\mathfrak {M}},\\widehat{\\Phi },\\widehat{\\varphi })$ , is said to be a dilation of the quantum dynamical system $(\\mathfrak {M},\\Phi ,\\varphi )$ , if it satisfies the following conditions: There is *-monomorphism $i:( \\mathfrak {M}, \\varphi )\\rightarrow (\\widehat{\\mathfrak {M}}, \\widehat{\\varphi })$ and a completely positive map $\\mathcal {E}:\\widehat{\\mathfrak {M}} \\rightarrow \\mathfrak {M}$ such that for each $a$ belong to $\\mathfrak {M}$ and natural number $n$ $\\mathcal {E}(\\widehat{\\Phi }^{n}(i(a)))=\\Phi ^{n}((a))$ We observe that for each $a$ belong to $\\mathfrak {M}$ and $X$ in $\\widehat{\\mathfrak {M}}$ we have: $\\mathcal {E}(i(a)X)=a \\mathcal {E} (X).$ Indeed for each $b\\in \\mathfrak {M}$ we obtain: $\\varphi (b \\mathcal {E}(i(a) X)=\\varphi (i(b) i(a) X)=\\varphi (i(b a)X)=\\varphi (b a \\mathcal {E}(X))$ So, the ucp-map $\\widehat{\\mathcal {E}} = i\\circ \\mathcal {E}$ is a conditional expectation from $\\widehat{\\mathfrak {M}}$ onto $i(\\mathfrak {M})$ which leave invariant a faithful normal state.",
"The existence of such map which characterize the range of existence of a reversible dilation of a dynamical system, be derived from a theorem of Takesaki of [25].",
"We have a proposition that establish a link between the algebra of effective observable and reversible dilation.",
"Proposition 9 If $(\\widehat{\\mathfrak {M}},\\widehat{\\Phi },\\widehat{\\varphi })$ is a dilation of quantum dynamical system $(\\mathfrak {M},\\Phi ,\\varphi )$ then $\\widehat{\\Phi }(i(a)=i(\\Phi (a)) \\quad \\textsl {if, and only if } \\quad a\\in \\mathfrak {D}_\\Phi $ We have $i(\\Phi (a)^*) \\ i(\\Phi (a))=\\widehat{\\Phi }(i(a)^*) \\widehat{\\Phi }(i(a))$ it follows that $\\Phi (a^*) \\ \\Phi (a)= \\mathcal {E} (i(\\Phi (a)^*\\Phi (a)))= \\mathcal {E}(\\widehat{\\Phi }(i(a^*a))=\\Phi (a^* a).$ For vice-versa, if $y=i(\\Phi (a))- \\widehat{\\Phi }(i(a))$ then we have $y^*y= i(\\Phi (a^*a))-\\widehat{\\Phi }(i(a^*) i(\\Phi (a))- i(\\Phi (a^*))\\widehat{\\Phi }(i(a))+ \\widehat{\\Phi }(i(a^*a))$ since $a\\in \\mathfrak {D}_\\Phi $ .",
"It follows that $\\mathcal {E}(y^*y)=0$ with $\\mathcal {E}$ faithful map, then $y=0$ .",
"Let $\\mathfrak {M}=\\mathfrak {D}_\\infty \\oplus \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ be decomposition of theorem REF of our quantum dynamical system $(\\mathfrak {M},\\Phi ,\\varphi )$ and $\\mathcal {E}_\\infty : \\mathfrak {M} \\rightarrow \\mathfrak {D}_\\infty $ the conditional expectation defined in proposition REF , we can say: Remark 3 For each $a\\in \\mathfrak {M}$ and integer $k$ we have: $\\widehat{\\Phi }^k(i(\\mathcal {E}_\\infty (a)))=i(\\Phi _k (\\mathcal {E}_\\infty (a))$ We observe that $X\\in \\ \\textit {i}(\\mathfrak {D}_\\infty )^{\\perp _{\\widehat{\\varphi }}} \\quad \\textsl {if , and only if } \\quad \\mathcal {E}(X)\\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ since $i(\\mathfrak {D}_\\infty )^{\\perp _{\\widehat{\\varphi }}} = \\lbrace X\\in \\widehat{\\mathfrak {M}}: \\widehat{\\varphi }(X^* i(d))=0 \\quad \\forall d\\in \\mathfrak {D}_\\infty \\rbrace $ and $ \\widehat{\\varphi }(X^* i(d))=\\varphi (\\mathcal {E}(X^*)d)$ for all $d\\in \\mathfrak {D}_\\infty $ .",
"We can write the following algebraic decomposition of linear spaces: $\\widehat{\\mathfrak {M}}=\\textit {i}(\\mathfrak {D}_\\infty ) \\oplus \\textit {i}(\\mathfrak {D}_\\infty )^{\\perp _{\\widehat{\\varphi }}}$ and the ucp-map $ \\widehat{\\mathcal {E}}_\\infty =i \\circ \\mathcal {E}_\\infty \\circ \\mathcal {E}$ is a conditional expectation from $\\widehat{\\mathfrak {M}}$ onto $i(\\mathfrak {D}_\\infty )$ ."
],
[
" Decomposition theorem and linear contractions",
"We would study the relations between canonical decomposition of Nagy-Fojas of linear contraction $U_{\\Phi ,\\varphi }$ [19] and decomposition (c) of proposition REF of dynamical system $(\\mathfrak {M},\\Phi , \\varphi )$ .",
"We going to recall the main statements of these topics.",
"A contraction $T$ on the Hilbert space $\\mathcal {H}$ is called completely non-unitary if for no non zero reducing subspace $\\mathcal {K}$ for $T$ is $T_{\\mid \\mathcal {K}}$ a unitary operator, where $T_{\\mid \\mathcal {K}}$ is the restriction of contraction $T$ on the Hilbert space $\\mathcal {K}$ .",
"We set with $D_T=\\sqrt{I-T^*T}$ the defect operator of the contraction $T$ and it is well know that $TD_T=D_{T^*} T$ Moreover $ ||T\\psi |||=||\\psi || $ if, and only if $ D_T\\psi =0$ .",
"We consider the following Hilbert subspace of $\\mathcal {H}$ : $\\mathcal {H}_0= \\lbrace \\psi \\in \\mathcal {H}: ||T^n\\psi ||=||\\psi ||=||T^{* n}\\psi || \\ for \\ all \\ \\ n\\in \\mathbb {N} \\rbrace $ It is trivial show that $T^n\\mathcal {H}_0=\\mathcal {H}_0$ and $T^{* n}\\mathcal {H}_0=\\mathcal {H}_0$ for all natural number $n$ .",
"We have the following canonical decomposition (see [19]): Theorem 2 (Sz-Nagy and Fojas) To every contraction $T$ on $\\mathcal {H}$ there corresponds a uniquely determined decomposition of $\\mathcal {H}$ into a orthogonal sum of two subspace reducing $T$ we say $\\mathcal {H=H}_{0}\\mathcal {\\oplus H}_{1}$ , such that $T_0=T_{\\mid \\mathcal {H}_{0}}$ is unitary and $T_1=T_{\\mid \\mathcal {H}_{1}}$ is c.n.u., where $\\mathcal {H}_{0}=\\bigcap \\limits _{k\\in \\mathbb {Z}}\\ker \\left( D_{T_{k}}\\right) \\qquad and \\qquad \\mathcal {H}_{1}=\\mathcal {H}_{0}^{\\perp }$ with $T_{k}=\\left\\lbrace \\begin{array}[c]{cc}T^k \\ & k\\ge 0 \\\\T^{* -k } & k<0\\end{array}\\right.$ It is well know [19] that the linear operator $T_{-}= so-\\underset{n\\rightarrow +\\infty }{\\lim }T^{* n}T^n $ and $T_{+}=so-\\underset{n\\rightarrow +\\infty }{\\lim }T^n T^{* n} $ , there are in sense of strong operator ($so$ ) convergence.",
"After this brief detour on linear contractions we return to quantum dynamical systems $( \\mathfrak {M},\\Phi ,\\varphi )$ .",
"We set $V_{-}=so-\\underset{n\\rightarrow +\\infty }{\\lim }U_{\\Phi ,\\varphi }^{* n}U_{\\Phi ,\\varphi }^n $ and $V_{+}=so-\\underset{n\\rightarrow +\\infty }{\\lim }U_{\\Phi ,\\varphi }^n U_{\\Phi ,\\varphi }^{* n} $ , where $U_{\\Phi ,\\varphi }$ is the contraction defined in (REF ).",
"It follows that for each $a,b\\in \\mathfrak {M}$ we obtain: $\\underset{n\\rightarrow \\pm \\infty }{\\lim }\\varphi (S_{n}(a,b))=\\langle \\pi _{\\varphi }(a)\\Omega _\\varphi ,(I-V_\\pm )\\pi _{\\varphi }(b) \\Omega _\\varphi \\rangle $ where $S_{n}(a,b)$ is given by (REF ).",
"We recall that by proposition REF that for every integers $k$ we obtain $\\mathcal {H}_\\varphi =\\mathcal {H}_\\infty \\oplus \\mathcal {K}_\\infty $ with $U_k\\mathcal {H}_\\infty = \\mathcal {H}_\\infty $ and $U_k \\mathcal {K}_\\infty \\subset \\mathcal {K}_\\infty $ , where $U_{k}=\\left\\lbrace \\begin{array}[c]{cc}U_{\\Phi ,\\varphi }^k & k\\ge 0 \\\\\\end{array}U_{\\Phi ,\\varphi }^{* -k } & k<0\\right.$ A simple consequences of proposition REF is the following remark: For any integers $k$ we obtain $ a\\in \\mathfrak {D}_{\\Phi _k} \\quad \\textsl {if, and only if} \\quad \\pi _\\varphi (a)\\Omega _\\varphi \\in \\ker ( D_{U_k}) \\ \\textsl {and} \\ \\pi _\\varphi (a^*)\\Omega _\\varphi \\in \\ker (D_{U_k})$ Therefore $\\mathcal {H}_\\infty \\subset \\mathcal {H}_0 $ because $ \\pi _\\varphi (\\mathcal {D}_\\infty )\\Omega _\\varphi \\subset \\bigcap _{k\\in \\mathbb {Z}} \\pi _\\varphi (\\mathfrak {D}_{\\Phi _k}) \\Omega _\\varphi \\subset \\bigcap \\limits _{k\\in \\mathbb {Z}}\\ker ( D_{U_k})$ We observe that for each $a,b\\in \\mathfrak {M}$ and natural number $k$ we have (see [11] theorem 3.1): $\\underset{n\\rightarrow +\\infty }{\\lim }\\varphi (S_{k}(\\Phi ^n(a),b))=0$ Indeed for each natural numbers $k$ and $n$ , we obtain $\\varphi (S_{k}(\\Phi ^n(a),\\Phi ^n(b))=\\varphi (S_{k+n}(a,b))-\\varphi (S_{n}(a,b))$ and by the relation (REF ) result $ \\underset{n\\rightarrow +\\infty }{\\lim }(\\varphi (S_{k+n}(a,b))-\\varphi (S_{n}(a,b)))=0 $ .",
"Furthermore, for each natural number $k$ and $a,b\\in \\mathfrak {M}$ we have $| \\varphi (S_{k}(\\Phi ^n(a),b))|^2\\le \\varphi (S_{k}(\\Phi ^n(a),\\Phi ^n(a)) \\varphi (S_{k}(b,b))$ it follows that $\\underset{n\\rightarrow +\\infty }{\\lim }\\varphi (S_{k}(\\Phi ^n(a),b)=0$ .",
"We have a well-known statement (see [12], [16] and [23] ): Proposition 10 For all $a\\in \\mathfrak {M}$ any $\\sigma $ -limit point of the set $\\lbrace \\Phi ^k(a) \\rbrace _{k\\in \\mathbb {N}}$ belongs to the von Neumann algebra $\\mathfrak {D}_{\\infty }$ .",
"Moreover, for each $d_\\bot $ in $\\mathfrak {D}_{\\infty }^{\\perp _{\\varphi }}$ we have: $\\underset{k\\rightarrow +\\infty }{\\lim }\\Phi ^k(d_\\bot )=0 \\qquad and \\qquad \\underset{k\\rightarrow +\\infty }{\\lim }\\Phi ^{\\sharp k}(d_\\bot )=0$ where the limits are in $\\sigma $ -topology.",
"If $y$ is a $\\sigma $ -limit point of $\\lbrace \\Phi ^n(a) \\rbrace _{n\\in \\mathbb {N}}$ then there exists a net $\\lbrace \\Phi ^{n_{j}}(a) \\rbrace _{j\\in \\mathbb {N}}$ such that $y=\\underset{j\\rightarrow +\\infty }{\\lim }\\Phi ^{n_{j}}(a)$ in $\\sigma $ -topology.",
"Furthermore for each $b\\in \\mathfrak {M}$ we obtain $S_{k}(y,b)=\\sigma -\\underset{j\\rightarrow +\\infty }{\\lim } [ \\ \\Phi ^k(\\Phi ^{n_j}(a)b)-\\Phi ^k(\\Phi ^{n_j}(a))\\Phi ^k(b) \\ ] =\\sigma -\\underset{j\\rightarrow +\\infty }{\\lim }S_{k}(\\Phi ^{n_{j}}(a),b)$ from (REF ) we obtain $\\underset{j\\rightarrow +\\infty }{\\lim }\\varphi (S_{k}(\\Phi ^{n_{j}}(a),b))=0$ hence $\\varphi (S_{k}(y,b))=0$ .",
"It follows that $\\varphi (S_{k}(y,y))=0$ and $S_{k}(y,y)=0$ .",
"We observe that the adjoint is $\\sigma $ -continuous, then we obtain $y^*=\\underset{j\\rightarrow +\\infty }{\\lim }\\Phi ^{n_{j}}(a^*)$ , and repeating the previous steps we obtain $S_{k}(y^*,y^*)=0$ , hence $y\\in \\mathfrak {D}_{\\infty }$ .",
"For last statement we observe that for each natural number $k$ result $||\\Phi ^k(d_\\bot )||\\le ||d_\\bot ||$ and since the unit ball of the von Neumann algebra $\\mathfrak {M}$ is $\\sigma $ - compact we have that there is a subnet such that $\\Phi ^{k_\\alpha }(d_\\bot )\\rightarrow y\\in \\mathfrak {D}_{\\infty }^{\\perp _{\\varphi }}$ in $\\sigma $ -topology.",
"From previous lemma we have that $y\\in \\mathfrak {D}_{\\infty }\\cap \\mathfrak {D}_{\\infty }^{\\perp _{\\varphi }}$ it follows that $y=0$ .",
"then it can only be $\\underset{k\\rightarrow +\\infty }{\\lim }\\Phi ^k(d_\\bot )=0$ in $\\sigma $ -topology.",
"We observed that the Hilbert space $\\mathcal {H}_\\infty $ , the linear closure of $\\pi _\\varphi (\\mathfrak {D}_\\infty )\\Omega _\\varphi $ is contained in $\\mathcal {H}_0$ .",
"The next step is to understand when we have the equality of these two Hilbert spaces.",
"Let $( \\mathfrak {M},\\Phi ,\\varphi )$ be the previous quantum dynamical system, we define, for each integer $k$ , the unital Schwartz map $\\tau _k:\\mathfrak {M} \\rightarrow \\mathfrak {M}$ as $\\tau _k=\\Phi _{-k}\\circ \\Phi _k \\qquad \\qquad k\\in \\mathbb {Z}$ We have for any integer $k$ that 1 - $\\varphi \\circ \\tau _k = \\varphi $ 2 - $\\tau _k=\\tau _k^\\sharp $ , where $\\tau _k^\\sharp $ is the $\\varphi $ -adjoint of $\\tau _k$ .",
"We obtain, for any integer $k$ the dynamical system $\\left\\lbrace \\mathfrak {M}, \\tau _k, \\varphi \\right\\rbrace $ with $\\mathfrak {D}_\\infty (\\tau _k)= \\bigcap _{j\\ge 0} \\mathfrak {D}(\\tau _k^j)$ where with $ \\mathfrak {D}(\\tau _k^j) $ we have denote the multiplicative domains of map $\\tau _k ^j$ .",
"From decomposition theorem REF , for any integer $k$ we have: $\\mathfrak {M}= \\mathfrak {D}_\\infty (\\tau _k) \\oplus \\mathfrak {D}_\\infty (\\tau _k)^ {\\perp _\\varphi }$ and by the proposition REF $\\mathcal {H}_\\varphi = \\mathcal {H}_{(k)} \\oplus \\mathcal {K}_{(k)}$ where $\\mathcal {H}_{(k)} $ and $\\mathcal {K}_{(k)}$ are the closure of the linear space $\\pi _\\varphi (\\mathfrak {D}_\\infty (\\tau _k) )\\Omega _\\varphi $ and of $\\pi _{\\varphi }(\\mathfrak {D}_\\infty (\\tau _k) )^{\\perp _{\\varphi }})\\Omega _{\\varphi }$ respectively.",
"We have the following proposition: Proposition 11 If $ \\overline{\\pi _\\varphi (\\mathfrak {D}_{\\Phi _k})\\Omega _\\varphi }$ denotes the closure of linear space $\\pi _\\varphi (\\mathfrak {D}_{\\Phi _k})\\Omega _\\varphi $ then we have $\\mathcal {H}_0 = \\bigcap _{k\\in \\mathbb {Z}} \\overline{\\pi _\\varphi (\\mathfrak {D}_{\\Phi _k})\\Omega _\\varphi }$ where the $\\mathcal {H}_0$ is the Hilbert space of Nagy decomposition of theorem REF .",
"Furthermore for any $a\\in \\mathfrak {M}$ and $\\xi _0 \\in \\mathcal {H}_0$ and integr $k$ we have $U_ {\\Phi ,\\varphi }^k \\pi _\\varphi (a) \\xi _0 =\\pi _\\varphi (\\Phi ^k (a)) U_{\\Phi , \\varphi }^k \\xi _0$ We have that $\\mathfrak {D}(\\tau _k)\\subset \\mathfrak {D}_{\\Phi _k}$ for all integers $k$ .",
"In fact if $a\\in \\mathfrak {D}(\\tau _k)$ then $\\varphi (\\Phi _k(a^* a) ) & = & \\varphi (a^* a ) = \\varphi (\\tau _k(a^* a))= \\varphi (\\tau _k(a^*) \\tau _k (a)) =\\varphi (\\Phi _{-k}(\\Phi _k(a)^*) \\Phi _{-k}(\\Phi _k(a))) \\le \\\\&\\le & \\varphi (\\Phi _{-k}(\\Phi _k(a)^* \\Phi _k(a)))=\\varphi (\\Phi _k(a^*)\\Phi _k(a))\\le \\varphi (\\Phi _k(a^*a))$ It follows that $\\varphi (S_k(a,a))=0$ for all integers $k$ and in the same way proves that $\\varphi (S_k(a^*,a^*))=0$ for all integers $k$ .",
"We have proved that $\\mathfrak {D}_\\infty (\\tau _k) =\\bigcap _{j\\in \\ \\mathbb {N}} \\mathfrak {D}_{\\tau _k^j} \\subset \\mathfrak {D}_{\\tau _k} \\subset \\mathfrak {D}_{\\Phi _k}$ If $\\xi _0\\in \\mathcal {H}_0$ then for any $k$ integer and natural number $n$ we have $ (U_{\\Phi , \\varphi ,}^{* k} U_{\\Phi ,\\varphi }^{k})^n \\xi _0=\\xi _0$ and for any $r_\\bot \\in \\mathfrak {D}_\\infty (\\tau _k)^{\\perp _\\varphi }$ we can write that $\\left\\langle \\pi _\\varphi (r_\\bot )\\Omega _\\varphi , \\xi _0 \\right\\rangle = \\left\\langle (U_{\\Phi , \\varphi ,}^{* k} U_{\\Phi ,\\varphi }^{k} )^n \\ \\pi _\\varphi (r_\\bot )\\Omega _\\varphi , \\xi _0 \\right\\rangle =\\left\\langle \\pi _\\varphi (\\tau _k^n(r_\\bot )\\Omega _\\varphi , \\xi _0 \\right\\rangle $ and $\\lim _{n \\rightarrow + \\infty } \\left\\langle \\pi _\\varphi (\\tau _k^n(r_\\bot )\\Omega _\\varphi , \\xi _0 \\right\\rangle =0 \\qquad k\\in \\mathbb {Z}$ since $\\tau _k^n(r_\\bot ) \\longrightarrow 0 $ as $ n \\rightarrow \\infty $ in $\\sigma $ -topology.",
"It follows that $\\mathcal {H}_0 \\subset [ \\pi _\\varphi (\\mathfrak {D}_\\infty (\\tau _k)^{\\perp _\\varphi })\\Omega _\\varphi ] ^ \\perp = [ \\mathcal {K}_k ] ^\\perp $ .",
"Therefore for any integers $k$ we obtain: $\\mathcal {H}_0 \\subset \\mathcal {H}_{(k)} \\subset \\overline{\\pi _\\varphi (\\mathfrak {D}_{\\Phi _k})\\Omega _\\varphi } \\qquad \\Longrightarrow \\qquad \\mathcal {H}_0 \\subset \\bigcap _{k\\in \\mathbb {Z}} \\overline{\\pi _\\varphi (\\mathfrak {D}_{\\Phi _k})\\Omega _\\varphi }$ Let $\\xi _0\\in \\bigcap _{k\\in \\mathbb {Z}} \\overline{\\pi _\\varphi (\\mathfrak {D}_{\\Phi _k})\\Omega _\\varphi } $ , for any integers $k$ we have a net $d_{\\alpha ,k}\\in \\mathfrak {D}_{\\Phi _k}$ such that $\\pi _\\varphi (d_{\\alpha ,k})\\Omega _\\varphi \\rightarrow \\xi _0$ as $\\alpha \\rightarrow \\infty $ and for $k\\ge 0$ we obtain $U^{* k}_{\\Phi ,\\varphi }U^{ k}_{\\Phi ,\\varphi } \\xi _0 =U^{* k}_{\\Phi ,\\varphi }U^{ k}_{\\Phi ,\\varphi } \\lim _\\alpha \\pi _\\varphi (d_{\\alpha ,k})\\Omega _\\varphi =\\lim _\\alpha U^{* k}_{\\Phi ,\\varphi }U^{ k}_{\\Phi ,\\varphi } \\pi _\\varphi (d_{\\alpha ,k})\\Omega _\\varphi =\\lim _\\alpha \\pi _\\varphi (d_{\\alpha ,k})\\Omega _\\varphi =\\xi _0 $ in the same way for $k\\ge 0$ we have $U^{ k}_{\\Phi ,\\varphi } U^{* k}_{\\Phi ,\\varphi } \\xi _0=\\xi _0$ .",
"It follows that $ \\bigcap _{k\\in \\mathbb {Z}} \\overline{\\pi _\\varphi (\\mathfrak {D}_{\\Phi _k})\\Omega _\\varphi }\\subset \\mathcal {H}_0 $ The relation (REF ) is a straightforward.",
"We observe that for any $a\\in \\mathfrak {M}$ and $d_\\bot \\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ we have $\\underset{n\\rightarrow \\infty }{\\lim }\\varphi (a^*\\Phi _n(d_\\bot )a)=0$ since for any $d_\\bot \\in \\mathfrak {D}_{\\infty }^{\\perp _{\\varphi }}$ we obtain $\\Phi _n(d_\\bot )\\rightarrow 0$ as $n\\rightarrow \\infty $ in $\\sigma $ -topology.",
"From polarization identity we can say that $\\underset{n\\rightarrow \\infty }{\\lim }\\varphi (a\\Phi _n(d_\\bot )b)=0, \\qquad \\ a,b\\in \\mathfrak {M}, \\ d_\\bot \\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ and since $U_\\varphi $ is a contraction it follows that for any $\\xi \\in \\mathcal {H}_\\varphi $ and $\\psi \\in \\mathcal {K}_\\infty $ we have $\\underset{n\\rightarrow \\infty }{\\lim }\\langle \\xi , U_{\\Phi ,\\varphi }^n \\psi \\rangle =0 \\qquad \\text{and} \\qquad \\underset{n\\rightarrow \\infty }{\\lim }\\langle \\xi , U_{\\Phi ,\\varphi }^{*n} \\psi \\rangle =0$ We give a simple statement on the Hilbert spaces $\\mathcal {H}_\\infty $ and $\\mathcal {H}_0$ : Proposition 12 If $\\Phi ^n(d_\\bot )\\rightarrow 0$ $ [ \\Phi ^{\\sharp n}(d_\\bot )\\rightarrow 0 ] $ as $n\\rightarrow \\infty $ in $s$ -topology for all $d_\\bot \\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ ; then $\\mathcal {H}_\\infty =\\mathcal {H}_0$ and $V_+ = P_\\infty $ $ [ V_- = P_\\infty ]$ We observe that for any $\\psi \\in \\mathcal {K}_\\infty $ result $||U_{\\Phi ,\\varphi }^n \\psi ||\\rightarrow 0$ as $n\\rightarrow \\infty $ , because for any $k\\in \\mathbb {N}$ there is $d_k^\\bot \\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ such that $|| \\psi - \\pi _\\varphi (d_k^\\bot )\\Omega _\\varphi || < 1/k$ and $U_{\\Phi ,\\varphi }^n$ is a linear contraction so for all natural number $n$ we obtain: $||U_{\\Phi ,\\varphi }^n \\psi || < \\frac{1}{k}+\\varphi (\\Phi ^n(d_k^\\bot )^* \\Phi ^n(d_k^\\bot ))$ If $\\xi _0\\in \\mathcal {H}_0$ , we can write $\\xi _0=\\xi _\\Vert + \\xi _\\bot $ with $\\xi _\\Vert \\in \\mathcal {H}_\\infty $ and $\\xi _\\bot \\in \\mathcal {K}_\\infty $ .",
"Then $\\xi _\\bot =\\xi _0 - \\xi _\\Vert \\in \\mathcal {H}_0$ therefore $|| \\xi _\\Vert || + || \\xi _\\bot || = || U_{\\Phi , \\varphi }^n \\xi _0 ||= ||U_{\\Phi , \\varphi }^n \\xi _\\Vert + U_{\\Phi , \\varphi }^n \\xi _\\bot ||= ||\\xi _\\Vert || + ||U_{\\Phi , \\varphi }^n \\xi _\\bot || $ for all natural numbers $n$ it follows that $\\xi _\\bot =0$ .",
"Moreover for any $\\xi \\in \\mathcal {H}_\\varphi $ we have $U_{\\Phi ,\\varphi }^{n *} U_{\\Phi ,\\varphi }^n \\xi = \\xi _0 + U_{\\Phi ,\\varphi }^{n *} U_{\\Phi ,\\varphi }^n\\xi _1$ with $\\xi _i\\in \\mathcal {H}_i$ for $i=1,2$ and $V_+ \\xi =\\xi _0$ since $||U_{\\Phi ,\\varphi }^n \\xi _1||\\rightarrow 0$ as $n\\rightarrow \\infty $ .",
"We conclude this section with a simple observation: We recall that a dynamical system $\\lbrace \\mathfrak {M}, \\Phi , \\omega \\rbrace $ is mixing if $\\lim _{n\\rightarrow \\infty } \\varphi (a\\Phi ^n(b))=\\varphi (a) \\varphi (b) \\ , \\qquad a,b\\in \\mathfrak {M}$ by the relation (REF ) we obtain that $\\lbrace \\mathfrak {M}, \\Phi , \\omega \\rbrace $ is mixing if, and only if its reversible part $(\\mathfrak {D}_{\\infty },\\Phi _{\\infty },\\varphi _{\\infty })$ is mixing.",
"Furthermore, let $\\lbrace \\mathfrak {M}, \\Phi , \\omega \\rbrace $ be a mixing Abelian dynamical system, then there is a measurable dynamics space $(X, \\mathcal {A}, \\mu , T)$ such that $\\mathfrak {D}_{\\infty }$ is isomorphic to the von Neumann algebra $L^\\infty (X, \\mathcal {A}, \\mu )$ of the measurable bounded function on $X$ .",
"If the set $X$ is a metric space and $ \\varphi _{\\infty }$ is the unique staionary state of $\\mathfrak {D}_{\\infty }$ for the dynamics $\\Phi _{\\infty }$ , then by the corollary 4.3 of [10] we have $\\mathfrak {D}_{\\infty }=\\mathbb {C} 1$ ."
],
[
" Decomposition theorem and Cesaro mean ",
"In this section we will study the link between the decomposition theorem REF and some ergodic results which we recall briefly.",
"It is well known the following proposition (see e.g.",
"[13] par.",
"9.1 and [17] proposition 2.3).",
"Proposition 13 Let $\\lbrace \\mathfrak {M}, \\tau , \\omega \\rbrace $ be a quantum dynamical system.",
"We consider the Cesaro mean $s_n=\\frac{1}{n+1}\\sum _{k=0}^n{\\tau ^k},$ Then, there is an $\\omega $ -conditional expectation $\\mathcal {E}$ of $\\mathfrak {M}$ onto fixed point $\\mathcal {F}(\\tau )=\\left\\lbrace a\\in \\mathfrak {M} : \\tau (a)=a \\right\\rbrace $ such that $\\lim _{n\\rightarrow 0} ||\\phi \\circ s_n -\\phi \\circ \\mathcal {E} ||=0 \\qquad \\phi \\in \\mathfrak {M}_*$ A simple consequence of the previous proposition is the following remark: Remark 4 $ \\left\\lbrace \\mathfrak {M}, \\tau , \\omega \\right\\rbrace $ is ergodic if, and only if $ \\mathcal {F}(\\tau )=\\mathbb {C}1$ Let $( \\mathfrak {M},\\Phi ,\\varphi )$ be the previous quantum dynamical system, and $\\tau _k:\\mathfrak {M} \\rightarrow \\mathfrak {M}$ the Schwartz map defined in (REF ), we have a simple statement: Proposition 14 For each integer $k$ we obtain: $\\mathcal {F}(\\tau _k)=\\mathfrak {D}_{\\Phi _k}$ Without loss of generality we assume $k=1$ then $\\tau _1=\\Phi ^\\sharp \\circ \\Phi $ .",
"If $x\\in \\mathcal {F}(\\tau _1)$ we can write $\\varphi (\\Phi (x^*) \\Phi (x))=\\varphi (x^*\\tau _1(x))=\\varphi (x^*x)=\\varphi (\\Phi (x^*x))$ then $x\\in \\mathfrak {D}_\\Phi $ .",
"The converse is proved similarly.",
"Now let us ask when the algebra of effectives observables $\\mathfrak {D}_\\infty $ is trivial (see also [8] proposition 15) .",
"Proposition 15 If $\\mathcal {D}_\\infty =\\mathbb {C}1$ then the normal state $\\varphi $ is of asymptotic equilibrium and the quantum dynamical system $( \\mathfrak {M},\\Phi ,\\varphi )$ is ergodic.",
"By decomposition theorem $\\mathfrak {M}=\\mathbb {C}1 \\oplus \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ and for each $a\\in \\mathfrak {M}$ we have $a=\\varphi (a)1+a_\\perp $ with $a_\\perp \\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ .",
"It follows that $\\Phi ^n(a)= \\varphi (a) 1 + \\Phi ^n(a_\\perp )$ and $\\Phi ^n(a_\\perp )\\rightarrow 0$ in $\\sigma $ -top.",
"We have a simple consequence of the previous propositions: Corollary 1 If the quantum dynamical system $\\lbrace \\mathfrak {M}, \\tau _k, \\varphi \\rbrace $ is ergodic for some integer $k$ , then $\\mathfrak {D}_{\\infty }=\\mathbb {C}1$ .",
"If we have ergodicity then $\\mathcal {F}(\\tau _k)=\\mathfrak {D}_{\\Phi _k}=\\mathbb {C}1$ .",
"Summarizing $\\tau _1 \\ ergodic \\quad \\Longrightarrow \\quad \\Phi \\ completely \\ irreversible \\quad \\Longrightarrow \\quad \\Phi \\ ergodic$ We observe that if $( \\mathfrak {M},\\Phi ,\\varphi )$ is a quantum dynamical system with $\\Phi $ homomorphism, we have that $\\tau _1=\\Phi ^\\sharp \\circ \\Phi =id$ .",
"Hence the dynamical system $\\lbrace \\mathfrak {M}, \\tau _1, \\varphi \\rbrace $ is not ergodic (if $\\varphi $ is not multiplicative functional), while $( \\mathfrak {M},\\Phi ,\\varphi )$ can be.",
"For each integer $k$ we consider $S_{n,k}=\\frac{1}{n+1}\\sum \\limits _{j=0}^{n} \\tau ^j_k$ .",
"By previous proposition REF there is a positive map $\\mathcal {E}_k:\\mathfrak {M} \\rightarrow \\mathfrak {M}$ such that $||\\phi \\circ S_{n,k} -\\phi \\circ \\mathcal {E}_k|| \\rightarrow 0 \\qquad \\phi \\in \\mathfrak {M}_*$ and $\\mathcal {E}_k$ is the conditional expectation related of von Neumann algebra $\\mathcal {D}_{\\Phi _k}$ of theorem REF .",
"Therefore $\\mathcal {E}_k :\\mathfrak {M} \\rightarrow \\mathcal {D}_{\\Phi ^k}$ and $\\varphi \\circ \\mathcal {E}_k= \\varphi $ for all integers number $k$ .",
"Furthermore we have: $ \\mathcal {E}_h \\circ \\mathcal {E}_k= \\mathcal {E}_k \\qquad k\\ge h \\ge 0 $ because by the relation REF we have $\\mathcal {D}_{\\Phi ^k}\\subset \\mathcal {D}_{\\Phi ^h}$ for all $k\\ge h$ .",
"For each $a\\in \\mathfrak {M}$ we have $||\\mathcal {E}_k (a)||\\le ||a||$ for all integers $k$ and apply $\\sigma $ -compactness property for the bounded net $ \\left\\lbrace \\mathcal {E}_{k} (a) \\right\\rbrace _{k\\in \\mathbb {N}}$ of von Neumann algebra $\\mathfrak {M}$ , we obtain that there is at lest one $\\sigma $ -limit point $\\mathcal {E}_+ (a)$ , therefore there exist a net $ \\left\\lbrace \\mathcal {E}_{n_\\alpha } (a) \\right\\rbrace _\\alpha $ such that $\\mathcal {E}_+(a)=\\sigma -\\lim _\\alpha \\mathcal {E}_{n_\\alpha }(a)$ .",
"We obtain that $\\mathcal {E}_+(a)\\in \\mathcal {D}_{\\Phi ^k}$ for all natural number $k$ because for any $a\\in \\mathfrak {M}$ we have $\\mathcal {E}_h (\\mathcal {E}_{n_\\alpha }(a))= \\mathcal {E}_{n_\\alpha }(a)$ when $n_\\alpha \\ge h$ and since $\\mathcal {E}_h$ are normal maps follows that $\\mathcal {E}_h ( \\mathcal {E}_+(a) ) = \\mathcal {E}_+(a)$ for all natural number $h$ .",
"Furthermore, for any $x\\in \\mathfrak {D}^+_\\infty $ we have $\\varphi (xa)=\\lim _{\\alpha \\rightarrow \\infty } \\varphi (\\mathcal {E}_{n_\\alpha }(x a))= \\lim _{\\alpha \\rightarrow \\infty } \\varphi (x \\mathcal {E}_{n_\\alpha }(a)) = \\varphi (x \\mathcal {E}_+(a))$ it follows that we have a unique $\\sigma $ -limit point $\\mathcal {E}_+ (a)$ for the net $ \\left\\lbrace \\mathcal {E}_{n} (a) \\right\\rbrace _{n\\in \\mathbb {N}}$ .",
"Therefore we obtain a map $\\mathcal {E}_+ :\\mathfrak {M} \\rightarrow \\mathfrak {D}_\\infty ^+$ .",
"Moreover $\\mathcal {E}_{n_\\alpha }( \\mathcal {E}_+(a)) = \\mathcal {E}_+(a)$ for all $\\alpha $ , then $ \\mathcal {E}_+ ^2 = \\mathcal {E}_+ $ and for Tomiyama [26] the positive map $\\mathcal {E}_+$ is a conditional expectation such that $\\varphi \\circ \\mathcal {E}_+=\\varphi $ , precisely it is the conditional expectation of relation REF .",
"We can say something more: Proposition 16 Let $\\lbrace \\mathfrak {M}, \\Lambda _k, \\varphi \\rbrace _{k\\in \\mathbb {N}}$ be a family of quantum dynamical systems.",
"We consider the contraction $V_k:\\mathcal {H}_\\varphi \\rightarrow \\mathcal {H}_\\varphi $ defined in (REF ) related to Schwartz map $\\Lambda _k $ : $ V_k \\pi _\\varphi (a)\\Omega _\\varphi =\\pi _\\varphi (\\Lambda _k (a))\\Omega _\\varphi \\qquad a\\in \\mathfrak {M}$ If $|| \\left[ V_k^* - V_h ^* \\right]\\xi ||\\rightarrow 0 $ as $h,k\\rightarrow \\infty $ for all $\\xi \\in \\mathcal {H}_\\varphi $ , then there is a unital positive map $\\Lambda :\\mathfrak {M} \\rightarrow \\mathfrak {M}$ such that $|| \\phi \\circ \\Lambda _k - \\phi \\circ \\Lambda ||\\rightarrow 0$ as $k\\rightarrow \\infty $ for any $\\phi \\in \\mathfrak {M}_*$ with $ \\varphi (\\Lambda (a^*) \\Lambda (a))\\le \\varphi (a^*a) \\qquad a\\in \\mathfrak {M}$ and $\\varphi \\circ \\Lambda = \\varphi $ .",
"A simple consequence of proposition 1.1 of [17] For each natural number $n$ , we consider the following Schwartz map: $ Z_n = \\frac{1}{2n+1}\\sum \\limits _{k=-n}^{n} \\ \\tau _k$ it is obvious that $\\varphi $ is a stationary state for $Z_n$ with $\\varphi (x Z_n(y))=\\varphi (Z_n (x)y)$ for all $x,y\\in \\mathfrak {M}$ .",
"Moreover for each $a\\in \\mathfrak {M}$ we have: $\\pi _\\varphi (Z_n(a)) \\Omega _\\varphi & = & \\frac{1}{2n+1}\\sum \\limits _{k=-n}^{n} \\ \\pi _\\varphi (\\tau _k(a))\\Omega _\\varphi =\\\\& = &\\frac{1}{2n+1}\\sum \\limits _{k=0}^{n} \\ U_{\\Phi ,\\varphi }^{* k} U_{\\Phi ,\\varphi }^n \\pi _\\varphi (a))\\Omega _\\varphi +\\frac{1}{2n+1}\\sum \\limits _{k=1}^{n} \\ U_{\\Phi ,\\varphi }^k U_{\\Phi ,\\varphi }^{* k} \\pi _\\varphi (a))\\Omega _\\varphi $ and since $U_{\\Phi ,\\varphi }^{* n} U_{\\Phi ,\\varphi }^n \\rightarrow V_+$ and $U_{\\Phi ,\\varphi }^{n} U_{\\Phi ,\\varphi }^{* n}\\rightarrow V_-$ in strong operator topology, we obtain $ \\pi _\\varphi (Z_n(a)) \\Omega _\\varphi \\rightarrow \\frac{1}{2}(V_+ + V_-) \\pi _\\varphi (a)\\Omega _\\varphi $ It follows that from previous proposition that there is a $\\varphi $ invariant Schwartz map $Z:\\mathfrak {M} \\rightarrow \\mathfrak {M}$ such that $||\\phi \\circ Z_n-\\phi \\circ Z|| \\rightarrow 0 \\qquad \\phi \\in \\mathfrak {M}_*$ and $ \\pi _\\varphi (Z(a)) \\Omega _\\varphi = \\frac{1}{2}(V_+ + V_-) \\pi _\\varphi (a)\\Omega _\\varphi $ We consider the decomposition $\\mathfrak {M} = \\mathfrak {D}_\\infty \\oplus \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ for each $a=a_\\Vert + a_\\bot \\in \\mathfrak {M}$ result $Z(a_\\Vert +a_\\bot ) = a_\\Vert +Z(a_\\bot )$ with $Z(a_\\bot )\\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ .",
"We observe that if $\\Phi ^n(d_\\bot )\\rightarrow 0$ and $ \\Phi ^{\\sharp n}(d_\\bot )\\rightarrow 0 $ as $n\\rightarrow \\infty $ in $s$ -topology for all $d_\\bot \\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ (see proposition REF ); then $Z(d_\\bot )=0$ for all $d_\\bot \\in \\mathfrak {D}_\\infty ^{\\perp _\\varphi }$ and we have a $\\varphi $ invariant Schwartz map $Z:\\mathfrak {M} \\rightarrow \\mathfrak {D}_\\infty $ such that $ Z(xa)=xZ(a), \\qquad x\\in \\mathfrak {M}, \\ a\\in \\mathfrak {D}_\\infty $ It follows that $Z$ is the conditional expectation $\\mathcal {E}_\\infty $ of proposition REF ."
]
] | 1606.04910 |
[
[
"For Noble Gases, Energy is Positive for the Gas Phase, Negative for the\n Liquid Phase"
],
[
"Abstract We found from experimental data that for noble gases and H$_2$, the energy is positive for the gas phase, and negative for the liquid, possibly except the small vicinity of the critical point, about $(1- T/T_c) \\le 0.005$.",
"The line $E=E_c$, in the supercritical region is found to lie close to the Widom line, where $E_c$ is the critical energy."
],
[
"Introduction",
"What distinguishes the gas phase from the liquid phase?",
"For noble gases, we declare that it's the sign of the energy $E$ : $E > 0 \\,\\, \\text{for gases,} \\, \\, E < 0 \\, \\, \\text{for liquids}.$ It's because a gas is an unbound state, so it cannot be $E<0$ ; otherwise, it would condense.",
"Similarly, if $E>0$ , a liquid could not be a bound state.",
"Of course, this argument is mean-field theoretic—at best.",
"Actually it's too crude, and we cannot justify it logically.",
"It is however correct, as we will see from experimental data.",
"To be precise, it may be violated in the very small vicinity of the critical point; according to the used data, it does not hold only for $t < 0.006 $ along the saturation curve (gas-liquid coexistence curve) for 4 noble gases, where $t:= 1 - T/T_\\mathrm {c}$ is the reduced temperature, $T$ the temperature, and $T_\\mathrm {c}$ the critical temperature.",
"For the extreme case of $\\mathrm {H}_2$ ,For $T < T_\\mathrm {c}$ , the internal degrees of freedom of $\\mathrm {H}_2$ can be ignored, as is explained in section REF .",
"the violation is only for $t < 0.001$ .",
"See figure REF below.",
"We report that rule (REF ) is corroborated experimentally, but notice that it is more of theoretical value.",
"For example, it can raise interest for dynamical system theory.",
"We cannot determine from the experimental data if rule (REF ) is exact for real fluids, but if it really is, the critical energy $E_\\mathrm {c}$ is 0, or equivalently $K = |U|$ at the critical point, where $K$ is the kinetic energy and $U$ the potential energy.",
"This strongly indicates a symmetry.",
"Even if it is not exact, we conjecture that there is a symmetry, and it is weakly broken.",
"More will be discussed in section .",
"Rule (REF ) somehow seems to have been unnoticed despite its simplicity and decisive power.There is a sole exception, ref elsner, but the arguments of this paper do not make sense.",
"In section 2, they “prove” that the zero of the internal energy in thermodynamics is not arbitrary.",
"This is of course absurd.",
"What's really proven is that it's not possible to assign arbitrary zeros to each of subsystems of one entire system.",
"In spite of this assertion, they do not define the zero of the energy, nor does it mention molecule's internal excitations.",
"Then in section 4, they assume that the sign of the energy cannot change within one phase, and concludes that $E > 0$ for a gas, $E =0$ at the critical point, etc.",
"In the article in 2012 titled “What separates a liquid from a gas?”, [2] it is not mentioned.",
"It is not found in recent textbooks of statistical physics [3], [4], [5], [6], [7] nor in liquid theory textbooks.",
"[8], [9] Looser explanations like “$K \\gg |U|$ for gases, $K \\ll |U|$ for solids, and $K \\approx |U|$ for liquids” are on the other hand common.",
"Ref.",
"stishov studied these relations a bit more further for van der Waals fluid, and heuristically obtained the estimate that $|U| / (k_\\mathrm {B}T) \\approx 0.9$ at the critical point, and concluded that far from the critical point the relations $k_\\mathrm {B}T \\gtrless |U|$ are good estimates for the liquid and gas phases, but not to the precision we give.",
"This letter is organized as follows.",
"In section , the experimental data we used and the theory are explained.",
"In section , the result is stated.",
"In section , we try to extend rule (REF ) to the supercritical region, and we draw a qualitative conclusion that the line $E=E_\\mathrm {c}$ lies close to the “Widom line”,[11] the line of the maxima of $C_p$ , the constant-pressure heat capacity.",
"Section gives discussion and outlook.",
"Section is the conclusion.",
"As “experimental” data, we rely on NIST Chemistry WebBook data on fluids (hereafter “WebBook”).",
"[12] In fact they are not true experimental data, but the output of the program “REFPROP” which computes model equations.",
"Their parameters are fit to the results of experiments done in various conditions, ranging from low to high temperature and pressure, and near and far from the critical point.",
"In addition, models differ from substance to substance.",
"Thus an accurate error estimate is not available.",
"It is only stated that “These equations are the most accurate equations available worldwide.” [13] The lower bound of the temperature at which they provide data is $T_\\mathrm {tp},$ the triple point temperature, and for He, the $\\lambda $ -point temperature.",
"They provide data along the saturation curve, in addition to isotherm, isobar, etc.",
"They provide data on 75 fluids.",
"All noble gases except Rn are included."
],
[
"Definition of $E=0$",
"To define the zero of the energy, for noble gases we safely ignore internal states, i.e.",
"thermal excitation of electrons.",
"(The first excitation energy of, for example, He is about 20eV, and that of Xe is 8.3eV.)",
"In dilute limit, all fluids become an ideal gas.",
"Therefore we naturally define that $E = 3/2 Nk_\\mathrm {B}T$ in dilute limit.",
"Here, $N$ is the number of atoms.",
"WebBook provides the data on various thermodynamic properties, and we in particular need those on the internal energy.",
"The zero of the internal energy is arbitrary, and in WebBook it depends on the kind of fluid.For most fluids it says that the origin of $E$ is taken at $T=273.15$ K for “saturated liquid”, but for not few fluids it is in the supercritical region, and this explanation is dubious.",
"To interpret the energy of WebBook, we determine the zero of $E$ from the value at $T=T_\\mathrm {c}, p = 0$ Mpa for each fluid, where $p$ is the pressure.",
"(WebBook indeed provides data down to 0MPa, probably extrapolated.)",
"This choice of $T$ is arbitrary, and does not matter.",
"At these points, $ | C_v / (3/2Nk_\\mathrm {B}) - 1| < 10^{-3}$ for all available substances, where $C_v$ is the constant-volume heat capacity.",
"So they can be reliably thought as dilute limit."
],
[
"Inclusion of $\\mathrm {H}_2$",
"We also examine the behavior of $\\mathrm {H}_2$ because for $T \\le T_\\mathrm {c}$ internal excitations are almost “frozen” and can be ignored.",
"(According to WebBook, $|C_v/(3/2Nk_\\mathrm {B}) - 1| = 3 \\times 10^{-5}$ at $T=T_\\mathrm {c}, P=0$ MPa.)",
"It's because hydrogen is an exceptional molecule by having the large moments of inertia.",
"(This is not true even for D$_2$ , deuterium, for which $C_v(T_\\mathrm {c}) / (3/2Nk_\\mathrm {B}) = 1.12$ at 0MPa.)",
"From WebBook it's not clear if it is true equilibrium hydrogen, or “normal hydrogen”, i.e.",
"the 3:1 mixture of orthohydrogen and parahydrogen.",
"If it is normal hydrogen, an orthohydrogen molecule should be considered as stable, not an excited state of parahydrogen.",
"So still the zero of the energy is determined as $E=3/2Nk_\\mathrm {B}T$ at $p = 0$ MPa, where $N$ is the number of molecules."
],
[
"Other Comments",
"Helium is to some extent a quantum fluid on the saturation curve, since $\\lambda ^3 \\rho = 0.57 \\sim 1$ at the critical point, where $\\lambda $ is the thermal de Broglie wavelength and $\\rho $ is the number density.",
"But it's common to both classical and quantum mechanics that boundness is determined by the sign of the energy, so it is not necessary to modify rule (REF ) for this case.In quantum mechanics, bound states with positive energy is possible for systems.",
"See for example ref.",
"Ballentine, sec.",
"10.4.",
"But it is only for potentials which satisfy special conditions, and we ignore such cases.",
"Rule (REF ) should apply not only to pure substances but also to mixtures, as long as there is the natural definition of the origin of the energy, namely $E = 3/2Nk_\\mathrm {B}T$ in dilute limit.",
"Normal hydrogen falls into this category."
],
[
"Result",
"In figure REF we show the WebBook data of the energy of gas and liquid on the saturation curve, for 5 noble gases and $\\mathrm {H}_2$ .",
"The energy in the plot is so scaled that the energy of liquid at the triple point is $-1.$ Figure: (Color online) The energy of 5 noble gases and H 2 _2 on the saturation curve.",
"The upper curve is of the gas phase, and the lower of the liquid.",
"Horizontal dashed line is for E=0.E=0.",
"The curves are so scaled that E=-1E = -1 for the liquid phase at the triple point.",
"Inset shows the E c E_\\mathrm {c} and T c T_\\mathrm {c} for these 6 fluids, and the line is for E=-0.4Nk B TE = -0.4Nk_\\mathrm {B}T.As it can be seen, rule (REF ) is satisfied except the neighborhood of the critical point.",
"For He, the violation of rule (REF ) happens for $t < 0.02,$ for Ar, Kr, and for Xe $t < 0.006,$ and for Ne $t<0.005$ .",
"In the extreme case of $\\mathrm {H}_2$ , it is only for $t < 0.001.$ Considering the inherent uncertainty of WebBook data and experimental difficulty, this agreement is remarkable and cannot be accidental.",
"We conclude that rule (REF ) is: “Correct, possibly except very narrow regions near the critical point.” However, we cannot determine quantitatively the region where rule (REF ) does not hold, because of the lack of the error estimate in WebBook.",
"Possibly except an area close to the critical point, we are sure that rule (REF ) is correct not only on the saturation curve, but in a very wide range of $p$ when $T_\\mathrm {tp}< T < T_\\mathrm {c}$ .",
"It's because heat capacity is positive, and on isotherm $\\partial E / \\partial p < 0$ if the pressure is not too high (but if the pressure is that high probably the system crystallizes).",
"We are not sure for the gases in the region $T<T_\\mathrm {tp}$ for which WebBook doesn't provide data.",
"We also note that $E$ is always $ > E_\\mathrm {c}$ for the gas phase, and $< E_\\mathrm {c}$ for the liquid phase on the saturation curve according to WebBook."
],
[
"Critical energy",
"To assess $E_\\mathrm {c}$ , we also plot $E_\\mathrm {c}$ and $T_\\mathrm {c}$ of the same 6 fluids in the inset of figure REF .",
"(Remember an error bar is not available.)",
"We also draw the line $E=-0.4Nk_\\mathrm {B}T,$ which is simply “fit by eye.” The agreement of this line with the experimental data looks good, so we're tempted to say that $E_\\mathrm {c}$ is indeed $\\approx -0.4Nk_\\mathrm {B}T_\\mathrm {c}\\ne 0$ , but we avoid to draw any conclusion.",
"We can not tell if $E_\\mathrm {c}$ is exactly $=0$ , but the question if the line $E=E_\\mathrm {c}$ is still meaningful in the supercritical region is natural, possibly representing a crossover, dividing liquid-like and gas-like behavior.",
"In fact lines of such crossover are already proposed, dubbed the “Frenkel line”[15] and the “Widom line”.",
"[11] Actually we feel that the arguments on the Frenkel line are more convincing than those on the Widom line, but we here compare the $E=E_\\mathrm {c}$ line with the Widom line because of the data availability.",
"The Widom line is defined as the line of sharp maximum of $C_p,$ the constant pressure heat capacity, in the supercritical region, starting from the critical point.",
"More precisely, the $C_p$ divergence at the critical point does not form a round peak, but on each isothermal and isobaric line near the critical point, a sharp $C_p$ maximum exists.",
"By connecting those maxima, a “ridge” is formed, and it is the Widom line.",
"It is also characterized as the collection of maxima of various thermodynamic response functions.",
"Even though the validity of the Widom line notion is questioned,[16] there is no problem as long as we consider an area close enough to the critical point."
],
[
"Result",
"We plot in figure REF the lines $E=E_\\mathrm {c}$ , $C_p$ maximum, and also the maximum of $C_v$ , the constant volume heat capacity, for Ne and Xe.",
"Our qualitative conclusion is that the line $E=E_\\mathrm {c}$ runs near the Widom line, in the region of low enough temperature where the Widom line can be recognized without ambiguity.",
"When the system moves far away from the critical point, $C_v$ maximum disappears, and the Widom line may not be well-defined there.",
"In that region, the line $E=E_\\mathrm {c}$ departs from the $C_p$ maximum line.",
"We gave the plots of Ne and Xe, but our result applies to $\\mathrm {H}_2$ , Ar and Kr, too.",
"We cannot assert anything on He: Data close enough to the critical point are not provided by WebBook; for the region with data, the line $E=E_\\mathrm {c}$ and $C_p$ maximum do not agree well, and $C_v$ maximum cannot be observed."
],
[
"Discussion and outlook",
"Rule (REF ) which we judge almost correct, raises many questions.",
"First of all, is it exact?",
"Computer simulations should prove it; rather, a disproof will be easier than a proof—experimental verification will be difficult, because of finite-size effect and the presence of gravity.",
"[17], [18] If correct, it must be so for any interactions which have the critical point and the natural definition of $E=0$ , independent of dimensionality.",
"(Even though the physics of noble gases is usually thought to be well described by Lennard-Jones potential, the contribution of the three-body forces has to be taken into account to reproduce the third virial coefficient of real noble gases.",
"[19]) If rule (REF ) is not exact, why is its breakdown limited to the very small region near the critical point?",
"The equation $E_\\mathrm {c}=0$ can still be used as the mean-field, zeroth-order value, but how can corrections be calculated?",
"Are there any system for which exactly $E_\\mathrm {c}=0$ ?",
"If rule (REF ) is exact, $\\partial E / \\partial N \\rightarrow 0$ in thermodynamic limit at the critical point.",
"Intuitively $E$ being $=0$ is the edge of boundness, and is also the point where a dimensionful constant vanishes, so it seems to be related to the scale invariance of the critical point.",
"However the condition of $E=0$ is not sufficient, since the line of states $E=0$ does exist in the supercritical region too.",
"Rule (REF ) also means $K = |U|$ at the critical point.",
"This strongly indicates a symmetry, directly connecting $K$ and $U$ , aside from the scale invariance.",
"Even if not exact, we can say there must be an approximate symmetry.",
"What symmetry is it precisely?",
"How is it related to the scale invariance?",
"As we cautioned, rule (REF ) is very rough.",
"For example, it completely ignores the formation of atomic clusters.",
"It also treats the energy from the viewpoint of mechanics, but the energy of a fluid is a thermodynamic quantity, the (canonical) ensemble average, which is not conserved.",
"Definition of boundness is very involved, if ever possible, for many-body systems.",
"At the same time, treatment in microcanonical, or dynamical system theory may be possible.",
"Yet, its incisive simpleness allows a clear understanding, or new definitions of gas and liquid.",
"For example, consider the solution of solute A and solvent B without internal degrees of freedom.",
"Then it can be said that A is gaseous inside the solution, and B is liquid.",
"Let us write the Hamiltonian $H$ as: $H = K_A + K_B + U_{AA} + U_{BB} + U_{AB},$ where $K_A$ is the kinematic energy of A particles, $U_{AB}$ is the potential between A and B particles, and so on.",
"Now integrate out B's variables.",
"Then we obtain the effective Hamiltonian $H_{\\mathrm {eff}}$ which looks like: $H_{\\mathrm {eff}} = K_{A\\mathrm {eff}} + \\sum _i U_i,$ where $U_i$ is the $i$ -body effective potential, which is $:=0$ in dilute limit.",
"What rule (REF ) tells is that $\\langle K_{A\\mathrm {eff}} \\rangle + \\sum _{i \\ge 2} \\langle U_i \\rangle > 0,$ where the bracket is the thermal average, because A is gaseous.",
"In addition since the whole system is liquid, $\\sum _{i \\ge 0} \\langle U_i \\rangle < -\\langle K_{A\\mathrm {eff}} \\rangle < 0.$ So, $-\\langle U_0 \\rangle $ is similar to to the work function of metals, although in the current case the temperature is finite.",
"If A and B demixes so that the A-rich phase and the B-rich phase coexist, then A is gaseous in B-rich phase and liquid in A-rich one, and so on.",
"It seems almost obvious, in reality a mere heuristic though, that the critical point of binary fluid consolution belongs to the same universality class as gas-liquid's one.",
"A still easier example is the theory of dilute solutions, found in every textbook of thermodynamics.",
"When the author was a student, he felt the appearance of the gas constant $R$ was sudden and absurd.",
"“Interaction is strong, the ideal gas has nothing to do here, no?” It is the consequence of thermal average,Thermodynamics of dilute solution can be derived within pure thermodynamics, without the need of statistical mechanics.",
"See for example ref.",
"fermi-thermodynamics, chapter 7. but we have an alternative view.",
"In fact, solute is a gas trapped in the solvent, and in dilute limit, it becomes an ideal gas, because the interaction between solute molecules can be neglected.",
"The mean free path of the bare solute molecule does not matter.",
"Rule (REF ) imposes a limit on the spinodal curve, too.",
"The spinodal curve is difficult to define theoretically.",
"In textbooks, it is often explained mean-field theoretically as “the” inflection point of (the metastable branch of) the free energy.",
"(See for example ref.",
"chaikin-lubensky, section 8.7.3, or ref.",
"strobl, section 3.4.3.)",
"More careful definition is as the occurrence of negative compressibility for all wavelengths,[22] but it still suffers from the fact that it may not be well defined due to metastability.",
"We know however that the supercooling of gas and superheating of liquid cannot exceed the line $E=0.$ It is only a necessary condition, but the energy of the system is always defined.",
"At the very least it explains the existence of spinodal curves in gas-liquid transition.",
"We do not know how to extend rule (REF ) for molecular fluids.",
"Molecules have internal degrees of freedom, namely ro-vibrational modes.",
"Intermolecular interactions depend on the internal states, or in other words, they mix and it is not possible to define the quantity $U$ separately from internal states.",
"In rule (REF ) translational degrees of freedom are concerned, so to promote it to molecular fluids, we have to extract and separate them from internal degrees of freedom.",
"It must be possible, since critical points exist also for molecular fluids, but we are clueless how to do it.",
"Rule (REF ) also hints at something on the notion of cluster and percolation in lattice and off-lattice systems, which is easiest to describe from the standpoint of Monte Carlo simulation.",
"(For an introduction see for example ref.",
"landau-binder, section 5.1) “Cluster algorithms” in general update all variables in a group, called cluster, but we call it “updating cluster” (UC).",
"There is another cluster, which percolates at the critical point, which we call PC.",
"PC is used to locate the critical point in “invaded cluster algorithm”[24].",
"In Ising model, PC is the set of parallel spins which are connected.",
"It is also generalized for example to Widom-Rowlinson model,[25], [26] but not for general fluids.",
"UC is a subset of PC, and it has to satisfy detailed-balance.",
"It is usually chosen to make the algorithm most efficient, but it is not necessary.",
"Because percolation is deeply connected to criticality, the current situation where PC is lacking for general systems is unsatisfactory.",
"Our questions are, how to define PC for general systems, and does UC have a physical meaning beyond a mere computational utility?",
"Is it possible to define an analogue of the kinetic energy for lattice systems?",
"By answering them, it may be possible to obtain more insight on the opaque relations between the lattice-gas models and fluids.",
"In physics, models, even toy models, have served to make various advances, and we inevitably pose this question: Is there any one-particle, central force system, classical or quantum, which has a phase transition at $T=T_\\mathrm {c},$ and for $T \\gtrless T_\\mathrm {c}, E \\gtrless 0$ ?",
"For classical cases, natural order parameters are $\\langle 1/r \\rangle $ and $\\langle U \\rangle .$ Figure: (Color online) The energy difference of evaporation ΔE(t)\\Delta E(t) of 75 fluids along the saturation curve, normalized to 1 at the triple point T=T tp T = T_\\mathrm {tp}.",
"The solid line means t 0.44 t^{0.44} with the same normalization.",
"The data of He, H 2 _2, and D 2 _2 are represented by specially thick dots, which substantially differ from others.Rule (REF ) also suggests that the energy may be an order parameter.",
"What we have discovered recently [27] is that the energy difference $\\Delta E(t)$ of evaporation along the saturation curve is universal, by being well approximated by the power law $\\propto t^a$ , where $a \\approx 0.44$ , including molecular fluids.",
"Figure REF shows $\\Delta E$ for 75 fluids of which data is provided by WebBook, This is surprising and uncanny, because ro-vibrational modes are diverse among substances.In the figure, $\\Delta E$ is normalized to 1 at the triple point.",
"This normalization was inspired by the work by Torquato et al.",
"[28] which reported similar universality for the latent heat $T\\Delta S$ .",
"They represented $T\\Delta S$ by 6-term polynomials with non-universal, fluid dependent coefficients.",
"We also found that $T\\Delta S \\propto t^{0.38}$ .",
"They are not critical phenomena; these two power laws apply to the entire saturation curve except an area near the critical point, but instead down to the triple point.",
"The critical exponents of them and of $p\\Delta V = -\\Delta F$ are $\\beta $ .$\\Delta V(t)$ and $\\Delta F(t)$ are differences of volume and Helmholtz energy along the saturation curve.",
"The critical exponents of $\\Delta V$ and hence of $p\\Delta V$ are strictly $=\\beta $ .",
"If asymptotically $p_\\text{c} -p \\sim t$ , which is almost certain, by the Clausius-Clapeyron relation $\\Delta S, T\\Delta S \\sim t^\\beta $ .",
"Then by noting $\\Delta G = 0$ , the difference of Gibbs energy along the saturation curve, it is likely that $\\Delta E \\sim t^\\beta .$ Here $\\Delta E$ and $T\\Delta S$ are per particle; contrary to our motivation, we couldn't find anything conclusive on the spatial energy density.",
"We pointed out that $E$ is a quantity that can be defined purely in mechanics, without thermodynamics.",
"But not only $\\Delta E,$ but also $\\Delta 1/V,$ the density difference, is an order parameter along the saturation curve, as known very well, and $V$ is a pure mechanical quantity, too.",
"Some mysterious truth seems to be still hidden."
],
[
"Conclusion",
"We found from experimental data that for noble gases and $\\mathrm {H}_2$ , the energy is positive for the gas phase, and negative for the liquid phase.",
"According to the used data, this rule dose not hold for $1 - T/T_\\mathrm {c}< 0.02$ for He, for other 4 noble gases, $1 - T/T_\\mathrm {c}< 0.006$ , and for $\\mathrm {H}_2$ , $1 - T/T_\\mathrm {c}< 0.001$ , along the saturation curve.",
"The used data does not provide any error estimate.",
"In the supercritical region, we draw the qualitative conclusion that the line $E=E_c$ runs close to the Widom line in the region not so much far from the critical point."
]
] | 1606.05148 |
[
[
"Analysis of pattern forming instabilities in an ensemble of two-level\n atoms optically excited by counter-propagating fields"
],
[
"Abstract We explore various models for the pattern forming instability in a laser-driven cloud of cold two-level atoms with a plane feedback mirror.",
"Focus is on the combined treatment of nonlinear propagation in a diffractively thick medium and the boundary condition given by feedback.",
"The combined presence of purely transverse transmission gratings and reflection gratings on wavelength scale is addressed.",
"Different truncation levels of the Fourier expansion of the dielectric susceptibility in terms of these gratings are discussed and compared to literature.",
"A formalism to calculate the exact solution for the homogenous state in presence of absorption is presented.",
"The relationship between the counterpropagating beam instability and the feedback instability is discussed.",
"Feedback reduces the threshold by a factor of two under optimal conditions.",
"Envelope curves which bound all possible threshold curves for varying mirror distances are calculated.",
"The results are comparing well to experimental results regarding the observed length scales and threshold conditions.",
"It is clarified where the assumption of a diffractively thin medium is justified."
],
[
"Introduction",
"Optical pattern formation in driven nonlinear media has been studied extensively in late 1980s and 1990s.",
"After observations in four-wave mixing experiments with sodium vapors in counter-propagating (CP) beam configuration reported by Grynberg et.",
"al.",
"[1], transverse patterns have been observed in liquid crystals [2], [3], [4], [5], thin organic films [6], photorefractives [7], [8] and alkali vapors [9], [10], [11] in the single feedback mirror (SFM) configuration proposed in [12], [13] (see Fig.",
"REF a for a scheme).",
"Recent years have seen a resurgence of interest in study of transverse self-organization with cold atomic gases in both CP [14], [15] and SFM configurations [16], [17] with potential application in condensed matter simulation [18], [19], [20].",
"Significance of the experiments of Refs.",
"[16], [17] (depicted in Fig.",
"REF ) is in using optomechanical [21], [22] and two-level nonlinearities, respectively.",
"For long pulses ($>10\\, \\mu $ s), with blue detuning, optomechanical density modulation was shown to be dominant in optimum conditions [16].",
"For shorter pulses ($<2\\,\\mu $ s), pattern formation (see Fig.",
"REF b) was found to be consistent with the standard two-level electronic nonlinearity [17].",
"Indeed it was already recognized that a two-level instability, though not necessary, was an efficient seed for the optomechanical patterns [16].",
"Here we consider the theory of this two-level instability.",
"A particularly simple model of the SFM configuration, for a diffractively-thin slice of Kerr medum, was analyzed by Firth [12].",
"However, as was highlighted in Ref.",
"[23] and later in [16], [17], the full analysis of pattern properties for small mirror distances demands a \"thick-medium\" approach, i.e.",
"inclusion of diffraction within the non-linear medium.",
"The requisite theory is closely related to that used to analyze pattern formation in a mirrorless thick-medium (slab) with two counterpropagating input fields.",
"Such CP systems have been analyzed for Kerr media by by Firth et al [24] and Geddes et al [25].",
"For a two-level CP system, Muradyan et al [26], in an extended abstract for NLGW 2005, describe pattern formation in cold atoms with counter-propagating fields, including both electronic and optomechanical mechanisms, the latter in a molasses model.",
"These works [24], [25], [26] provide the main theoretical background to the present paper, though mention should be made of early analysis [27] aimed at modeling oscillatory spatial instabilities observed in a (hot) sodium vapor SFM experiment [28].",
"The present paper concentrates at the modelling of the simultaneous presence of transmission (purely transverse gratings resulting from the interference of the pump with copropagating sidebands) and reflection gratings (wavelength scaled gratings which result from the interference of counterpropagating beams) in presence of the feedback mirror, whereas earlier treatments only utilized pure transmission gratings [12], [13], [11].",
"For the analysis of photorefractive experiments two-beam coupling via pure reflection gratings were considered [23].",
"As in the Muradyan model (MM) [26], we use a time-independent susceptibility approach to the two-level nonlinearity.",
"This precludes consideration of growth rates or oscillatory instabilities [27], but leads to reasonably tractable and transparent models which allow the parameter dependences of pattern thresholds to be investigated.",
"We include absorption, so as to allow for arbitrary atom-field detunings.",
"We also consider the inclusion of reflection-grating effects at different orders (MM include such effects, but only at lowest order).",
"This analysis is then applied to the calculation of thresholds for transverse instability in various thick-medium models.",
"These include the Kerr limit, used for the thick-medium calculations presented in Fig.",
"3B of [16].",
"In [17] preliminary two-level results were presented for two cases: quasi-Kerr (i.e.",
"large detuning, neglecting absorption, but not saturation of the refractive nonlinearity) for the pattern size vs mirror distance; and absorptive thin-slice for the threshold vs atomic detuning.",
"The theory behind all these models, as well as additional results for these and related models, will be presented.",
"As well as elaborating previous preliminary results and presenting generalizations of previous Kerr threshold formulae, we mention two useful and general results which emerge.",
"First, the SFM threshold curve of intensity vs diffraction parameter, for a Kerr-like medium with its feedback mirror directly at its output, coincides with the threshold for the CP instability in a medium of twice the length.",
"This might seem obvious from a 'mirror-image' picture, but there's a twist.",
"The CP thresholds are actually described by two separate but intertwined curves, e.g.",
"[25], but only one of these corresponds to a SFM configuration, for symmetry reasons.",
"Secondly, each member of the family of such threshold curves generated by varying the mirror distance is tangent to an envelope curve, which can be analytically calculated in many cases.",
"This gives useful insight into the mirror-distance dependence of pattern scales, but also enables a quantitative examination of the thin-slice limit, in which diffraction within the medium is neglected.",
"It turns out that the thin-medium approximation works only at zero order, i.e.",
"the threshold at large mirror distance is linear, not quadratic or higher, in $1/D$ , where $L$ is the medium thickness and $d=DL$ the mirror distance.",
"Figure: (Color online) (a) Experimental SFMscheme : a linearly-polarized light pulse is sent into an atomic cloud; the transmitted beam is retro-reflected by a mirror with an adjustable distance DLDL beyond the end of the cloud.",
"(b) Typical single-shot light distributions observed in the transverse instability regime, in the near (left) and far (right) field.",
"Parameters: cloud of 87 ^{87}Rb atoms at T=200μT=200\\,\\mu K driven at a detuning of δ=+6.5Γ\\delta =+6.5 \\Gamma to the F=2→F ' =3F=2\\rightarrow F^{\\prime }=3 transition of the D 2 _2-line with an intensity of 0.47 W/cm 2 ^2, optical density in line center OD=210=210, effective sample size (FWHM of cloud) L=8.5L=8.5 mm."
],
[
"System and Model",
"As in [26], we consider the counter-propagating fields $A$ (forward field) and $B$ (backward field, see Fig.",
"REF a) to be coupled by a nonlinear susceptibility $\\chi _{NL}= - \\frac{6\\pi }{k_0^3} n_a \\frac{2\\delta /\\Gamma -i}{1+4\\delta ^2/\\Gamma ^2}\\frac{1}{1+I/I_{s\\delta }}$ Here $n_a$ is the atomic density (considered constant here).",
"$I$ is the intensity, which will be a standing wave: $I/I_{s\\delta } = |A e^{ikz} + B e^{-ikz}|^2$ .",
"We can conveniently rewrite (REF ) as $\\chi _{NL}= \\chi _l \\frac{1}{1+I/I_{s\\delta }}$ where $\\chi _l$ is the linear susceptibility (and is complex, though absorption is neglected in the MM model, making the system Kerr-like).",
"The next step is to expand the nonlinear factor in a Fourier series: $\\frac{1}{1+I/I_{s\\delta }}= \\sigma _0 +\\sigma _+ e^{2ikz} +\\sigma _-e^{-2ikz} + h.o.t.$ The higher-order terms do not lead to any phase matched couplings, and so can reasonably be neglected whatever the intensity.",
"The coefficients $\\sigma _{\\pm }$ evidently describe a $2k$ longitudinal modulation of the susceptibility, i.e.",
"a reflection grating, which will scatter the forward field into the backward one and vice versa.",
"The field equations (M3) of [26] can then be written as $ \\left\\lbrace \\begin{array}{l}\\frac{\\partial A}{\\partial z} - \\frac{ i}{2k}\\nabla ^2_\\perp A =i \\frac{k}{2}\\chi _l(\\sigma _0 A +\\sigma _+ B) ,\\\\\\\\ \\frac{\\partial B}{\\partial z} + \\frac{ i}{2k}\\nabla ^2_\\perp B = -i \\frac{k}{2}\\chi _l(\\sigma _- A +\\sigma _0 B) \\\\\\end{array} \\right.$ To calculate $\\sigma _{0,\\pm }$ , we write the exact expansion of the saturation term (REF ) as $\\frac{1}{1+I/I_{s\\delta }}= \\frac{1}{1+a^2+b^2} (1+r(e_{+} + e_{+}^{*}))^{-1}$ where $a=|A|$ , $b=|B|$ , $e_{+} = e^{2ikz} e^{i(\\theta _A-\\theta _B)}$ , with $\\theta _{A,B} = arg(A,B)$ .",
"We have introduced a coupling parameter $r=hab/(1+a^2+b^2)$ , where the \"grating parameter\" $h$ [24] has been introduced to allow consistent consideration of the cases of no reflection grating ($h=0$ ), and of a full grating ($h=1$ ).",
"In the former case $\\sigma _{\\pm }=0$ , which would correspond to the standing-wave modulation of the susceptibility being washed out by drift or diffusion.",
"Partial wash-out could be accommodated by intermediate values of $h$ , but would need some associated physical justification.",
"The MM model includes the full grating, so corresponds to $h=1$ .",
"The series expansion of $(1+r(e_{+} + e_{+}^{*}))^{-1}$ is always convergent, because $r< 1/2$ .",
"Even terms contribute to $\\sigma _0$ , odd terms to $\\sigma _{\\pm }$ .",
"Using the binomial theorem, we find $ \\left\\lbrace \\begin{array}{l}(1+a^2+b^2)\\sigma _0 = 1 +2r^2 +6r^4 + 20r^6 + ...\\\\\\\\ (1+a^2+b^2)\\sigma _+ = - e^{i(\\theta _A-\\theta _B)}(r + 3r^3 + 10r^5 + ...) \\\\\\end{array} \\right.$ with $\\sigma _- = \\sigma _+^{*}$ .",
"Inserting these expressions into (REF ) gives $ \\left\\lbrace \\begin{array}{l}\\frac{\\partial A}{\\partial z} - \\frac{ i}{2k}\\nabla ^2_\\perp A =i \\frac{k}{2}\\chi _lA(\\frac{(1+2r^2 + ...) -(br/a)(1+3r^2 + ...)}{1+a^2+b^2}) ,\\\\\\\\ \\frac{\\partial B}{\\partial z} + \\frac{ i}{2k}\\nabla ^2_\\perp B = -i \\frac{k}{2}\\chi _l B(\\frac{(1+2r^2 + ...) -(ar/b)(1+3r^2 + ...)}{1+a^2+b^2}) \\\\\\end{array} \\right.$ Note that both sums are positive definite, so higher-order terms reduce the saturation (first term), but increase the strength of the cross-coupling (second term).",
"To lowest order (i.e.",
"cubic nonlinearity), the brackets become $(1-a^2-(1+h)b^2)$ and $(1-(1+h)a^2-b^2)$ for the $A$ and $B$ equations respectively, showing the expected factor of two enhancement of the cross-coupling due to the grating when $h=1$ .",
"In the absence of the grating $r=0$ , and the bracketed expressions reduce to $(1+s)^{-1}$ in both cases, where $s=a^2+b^2$ is the usual saturation parameter.",
"The MM model effectively truncates the series expansions in (REF ) at the first term, eventually leading to their equation (M8) for the \"transverse eigenvalues\" [26], which include saturation denominators $\\sim (1+s)^{-1}$ .",
"However, because $r^2 \\sim s^2$ , the terms neglected in the MM model are of the same order as the terms $\\sim s^5$ which saturate the cubic nonlinearity.",
"For $s= 0.4$ (the value in Fig.",
"1 of [26]), $r^2$ is only about 0.02, so its neglect is not especially serious in that case.",
"The series in (REF ) can be summed.",
"In fact several papers, going back to the 1970s, have obtained analytic solutions to the system (REF ), or closely equivalent systems (in the plane-wave limit).",
"For our purposes, the papers of van Wonderen et al [29], [30], who were addressing optical bistability in a Fabry-Perot cavity, are most directly relevant, and underpin the analytic zero-order (no diffraction) solution obtained in the next section.",
"Summing the series and combining both terms leads to a set of field evolution equations: $ \\left\\lbrace \\begin{array}{l}\\frac{\\partial A}{\\partial z} - \\frac{ i}{2k}\\nabla ^2_\\perp A =i \\frac{k}{2}\\chi _lA(1-\\frac{1-2a^2h/(1+a^2+b^2)}{(1-4r^2)^{\\frac{1}{2}}})/2a^2h ,\\\\\\\\ \\frac{\\partial B}{\\partial z} + \\frac{ i}{2k}\\nabla ^2_\\perp B = -i \\frac{k}{2}\\chi _l B(1-\\frac{1-2b^2h/(1+a^2+b^2)}{(1-4r^2)^{\\frac{1}{2}}})/2b^2h \\\\\\end{array} \\right.$ In the limit of no grating, $h,r \\rightarrow 0$ , both brackets reduce to the expected saturation denominator.",
"For finite $h$ , there is explicit nonreciprocity, since the susceptibilities for A and B are different, because of the susceptibility grating.",
"However, the amplitudes $A$ and $B$ are slowly varying in $z$ , allowing the propagation in the medium to be approximated by comparitively few longitudinal spatial steps.",
"In all the cases discussed above, we can write the propagation equations in the form $ \\left\\lbrace \\begin{array}{l}\\frac{\\partial A}{\\partial z} - \\frac{ i}{2k}\\nabla ^2_\\perp A =-\\frac{\\alpha _l}{2}(1+i\\Delta ) F(a^2,b^2)A ,\\\\\\\\ \\frac{\\partial B}{\\partial z} + \\frac{ i}{2k}\\nabla ^2_\\perp B = \\frac{\\alpha _l}{2}(1+i\\Delta ) F(b^2,a^2)B \\\\\\end{array} \\right.$ where $\\alpha _l$ is the linear absorption coefficient, $\\Delta (= 2\\delta /\\Gamma )$ is the scaled detuning, and the function $F$ describes the nonlinearity of the atomic susceptibility, as modelled by e.g.",
"(REF ) or (REF ), by the cubic ($\\chi ^{(3)}$ ) approximation, or some other model.",
"By definition, $F(0,0) =1$ , but $F(a^2,b^2) \\ne F(b^2,a^2) $ in general, because of non-reciprocity due to standing-wave effects.",
"The cubic model ($F(a^2,b^2) = 1-a^2 -(1+h)b^2$ ) is the simplest example, explicitly non-reciprocal if $h \\ne 0$ .",
"Fig.",
"REF illustrates the intensity dependence of the susceptibility and cross-coupling for $h=1$ , for the cubic, MM and full models.",
"The cubic (i.e.",
"$\\chi ^{(3)}$ ) model evidently has a very limited range of validity, whereas the MM and full models are broadly similar over a broad range, though quantitatively distinct.",
"Figure: (color online) Susceptibility and cross-coupling functions against a 2 a^2 for the equal-intensity case a=ba=b, with h=1h=1, i.e.",
"full grating.",
"Green: cubic approximation, F=1-3a 2 F =1-3a^2.",
"Red: MM model: () with series truncated at 1.",
"Blue: full model, no truncation.",
"Susceptibility curves start from 1.01.0 at a=0a=0, cross-coupling from zero."
],
[
"Zero-order equations and solutions",
"To find the pattern-formation thresholds, we first drop diffraction, and solve the plane-wave, zero-order problem in which $A,B$ depend on $z$ alone.",
"For convenience we set $|A(z)|^2 = p(z)$ and $|B(z)|^2 = q(z)$ , and scale $z$ to the medium length $L$ .",
"From (REF ) it follows that the plane-wave intensities $p(z), q(z)$ obey the real equations: $ \\left\\lbrace \\begin{array}{l}\\frac{dp}{d z} =-\\alpha _l L F(p,q)p ,\\\\\\\\ \\frac{dq}{d z} =\\alpha _l L F(q,p)q \\\\\\\\\\end{array} \\right.$ leading to the expected exponential absorption of the intensities in the linear limit.",
"We define the input intensity $p(0)=p_0$ and transmitted intensity $p(1)=p_1$ , and similarly $q(0)=q_0$ , $q(1)=q_1$ .",
"In the SFM configuration $q_1 = R p_1$ , where $R$ is the mirror reflection coefficient, whereas in the CP problem we usually have $q_1 = p_0$ .",
"We now solve (REF ) for three different models: no grating $(h=0)$ ; the MM model ($h=1$ , but truncated summations); and the full-grating $h=1$ model based on (REF ).",
"For $h=0$ , $F=1/(1+s)=1/(1+p+q)$ is symmetric in its arguments, and it follows that the product of the counter-propagating intensities (and indeed of the fields, $AB$ ) is independent of $z$ , simplifying the analysis.",
"We set $p(z)q(z) = K$ , where $K$ is constant, and thus $K = p_1q_1 = Rp_1^2 $ for a feedback mirror of reflectivity $R$ .",
"It follows that the backward intensity $q(z)$ is given by $K/p(z)$ , enabling the first member of (REF ) to be written in terms of $p(z)$ alone.",
"It can then be integrated analytically, giving $ln(p/p_0)+p-K/p - p_0 + K/p_0 +\\alpha _lLz = 0 ,$ and hence, for the transmitted power $p_1$ (using the explicit value of $K$ ): $ \\\\ln(p_1/p_0)+(1-R)p_1=p_0-Rp_1^2/p_0 -\\alpha _lL .\\\\\\\\$ For $h=1$ , $F(p,q)=(1+p)/(1+s)^2$ in the MM model.",
"We can again find a propagation constant, in this case given by $K=pq/(1+s)$ , again leading to a an integrable first-order equation in $p(z)$ alone: $\\frac{p(1+p)}{(p-K)^2} \\frac{dp}{d z} =-\\alpha _l L.$ This leads, for the transmitted power $p_1$ , to $ \\\\H(p_1,K)-H(p_0,K) +\\alpha _lL =0.\\\\\\\\$ where $H(p,K)= p +(2K+1)ln(p-K) - \\frac{K(K+1)}{p-K}$ .",
"Finally, it turns out that the all-grating system given by (REF ) also possesses a propagation constant, given by $K = W(z) - s(z)$ , where $ W(z) = (1+2s+\\xi ^2)^{\\frac{1}{2}}$ , and $\\xi (z) = p(z) - q(z)$ .",
"Essentially the same conservation law was noted by Van Wonderen et al in the context of optical bistability in a Fabry-Perot resonator [29], for which the propagation equations are identical to the present case, though the boundary conditions are different.",
"In terms of $W, s,\\xi $ the all-grating function $F_{all} (p,q)$ becomes $F_{all} = (1+(\\xi -1)/W)/(s+\\xi )$ , with its transpose $F_{all} (q,p)$ obtained by $\\xi \\rightarrow -\\xi $ .",
"Recasting equations (REF ), it turns out that the propagation equations for $s$ and $\\xi $ take a fairly simple form: $ \\left\\lbrace \\begin{array}{l}\\frac{ds}{d z} = - \\alpha _l \\xi /W ,\\\\\\\\ \\frac{d\\xi }{d z} = - \\alpha _l (1-1/W) \\\\\\\\\\end{array} \\right.$ from which one easily deduces $dW/dz =ds/dz$ , and thus the constancy of $K = W(z) -s(z)$ .",
"One can then obtain an integrable differential equation in just one variable.",
"For example, by using the definition of $W$ and of $K$ to express $W$ in terms of $K$ and $\\xi $ , the second of equations (REF ) is easily integrated to yield: $ \\\\\\xi + ln(\\xi + (\\xi ^2+2-2K)^{\\frac{1}{2}}) +\\alpha _l L z = const.\\\\\\\\$ For the important case $R=1$ , we have $s_1 = 2p_1$ , $\\xi _1 = 0$ , hence $W_1 = (1+4p_1)^\\frac{1}{2}$ and thus $K = (1+4p_1)^\\frac{1}{2} -4p_1$ .",
"Using this data in (REF ) yields an implicit expression for $\\xi _0$ in terms of $K$ (and thus $p_1$ ): $ \\\\\\xi _0 + ln(\\xi _0 + (\\xi _0^2+2-2K)^{\\frac{1}{2}}) -{\\frac{1}{2}} ln(2-2K) = \\alpha _l L .\\\\\\\\$ Given $\\xi _0$ , it is straightforward to calculate $W_0$ and $s_0$ , and thus the input intensity $p_0$ and the backward output intensity $q_0$ , all in terms of the given transmitted intensity $p_1$ , thus completing the solution of the plane-wave problem for the all-gratings model.",
"A problem with the MM model arises as the tuning $\\Delta $ approaches resonance.",
"It turns out that the transmission determined from (REF ) shows \"bistability\", i.e.",
"the output $p_1$ is not a single-valued function of the input $p_0$ , if the optical density is high enough.",
"This is surprising and counterintuitive, and turns out to be a flaw in the model: including more terms in the series expansion (REF ) eventually makes $p_1$ single-valued.",
"In particular the all-gratings formula (REF ) and its $R=1$ sub-case (REF ) give single-valued transmission characteristics."
],
[
"Transverse perturbations",
"We now assume that a solution has been found for the plane wave case $A=A_0(z)$ , $B=B_0(z)$ , obeying appropriate longitudinal boundary conditions.",
"This solution may be numerical, or a solution to some special-case or approximate version of (REF ).",
"We now turn our attention to the stability of such a plane wave solution against transverse perturbations.",
"We suppose that the solution of (REF ), subject to the appropriate boundary conditions, is known, and consider transverse perturbations of the form $A=A_0(1+f)$ , $B=B_0(1+g)$ , where $\\nabla ^2_\\perp (f,g)=-Q^2(f,g)$ , i.e.",
"the transverse perturbation has wave vector $Q$ , corresponding to a diffraction angle $Q/k$ in the far field.",
"Assuming $|f|,|g| << 1$ , we obtain the linearised propagation equations: $ \\left\\lbrace \\begin{array}{l}\\frac{df}{dz} + \\frac{ iQ^2}{2k}f = -\\alpha _l(1+i\\Delta )(F_{11}f^{\\prime }+F_{12}g^{\\prime }) ,\\\\\\frac{dg}{dz} - \\frac{ iQ^2}{2k}g = \\alpha _l (1+i\\Delta )(F_{21}f^{\\prime }+F_{22}g^{\\prime }) \\\\\\end{array} \\right.$ Here $f = f^{\\prime } +i f^{\\prime \\prime }$ , $g=g^{\\prime } + ig^{\\prime \\prime }$ , and , and the real quantities $F_{ij}$ are defined as $F_{11}=p\\frac{\\partial F(p,q)}{\\partial p}$ , $F_{12}=q\\frac{\\partial F(p,q)}{\\partial q}$ , $F_{21}=p\\frac{\\partial F(q,p)}{\\partial p}$ , $F_{22}=q\\frac{\\partial F(q,p)}{\\partial q}$ .",
"We assume that the fields are time-independent, adequate to calculate the threshold of a zero-frequency pattern-forming (Turing) instability at wavevector $Q$ .",
"To find Hopf instabilities, or to properly account for dynamical behavior of the field-atom system, we would have to start from the Maxwell-Bloch equations, rather than our susceptibility model.",
"It is worth mentioning that van Wonderen and Suttorp, in a later paper on dispersive optical bistability [30], perform a perturbation analysis of the full Maxwell-Bloch equations with all grating orders included.",
"The resulting model is very involved, and beyond our present scope.",
"Meantime, we are content to address the Turing pattern formation problem."
],
[
"Quasi-Kerr case",
"Let's begin with the case of large detuning, where the absorption is negligible.",
"The linear absorption coefficient can be written as $\\alpha _l = \\alpha _0/(1+\\Delta ^2)$ , where $\\alpha _0$ is the on-resonance absorption.",
"Formally, as a quasi-Kerr model, we suppose that $|\\Delta |$ is large enough that $\\alpha _l L$ can be neglected, but with $\\alpha _l \\Delta L$ finite, so that the nonlinearity is purely refractive.",
"For example, recent experiments [16], [17] employed optical densities $\\alpha _0 L$ of order 100, so neglect of absorption is reasonable for $|\\Delta | \\sim 20$ , which is at the high end of the experimental range.",
"With this assumption, the forward and backward intensities $p,q$ can be considered constant, and so are the $F_{ij}$ .",
"For feedback mirror boundary conditions, we have $q = R p$ , where $R$ is the mirror reflectivity, while for the CP problem $q=p$ if the system is symmetrically pumped.",
"Following [24], we set $\\theta =Q^2L/2k$ , and recast equations (REF ) in this quasi-Kerr limit as $ \\left\\lbrace \\begin{array}{l}\\frac{df}{dz} = -i\\theta f -i\\alpha _l L\\Delta (F_{11} f^{\\prime }+ F_{12}g^{\\prime }) ,\\\\\\frac{dg}{dz} = i\\theta g + i\\alpha _l L\\Delta (F_{21}f^{\\prime }+F_{22}g^{\\prime }) \\\\\\end{array} \\right.$ It is convenient to define a $2 \\times 2$ matrix $\\hat{F}$ formed from the $F_{ij}$ .",
"The nonlinearly-driven terms in equations (REF ) are pure imaginary, as for Kerr media, because of our assumption on $\\Delta $ .",
"Importantly, we have not imposed any restrictions on the magnitude of the intensities.",
"If the linear absorption is small, the saturated absorption is even smaller, so our approximation becomes better, not worse, for high intensity.",
"Nor is there any restriction on the form of $F$ , so that we can examine and compare different models of nonlinearity and of standing-wave response.",
"In order to align with previous work on the CP Kerr case, we first consider the symmetric equal intensity case ($p=q$ ), for which $F_{11}=F_{22}=F_{sym}$ and $F_{12}=F_{21}= GF_{sym}$ .",
"Both $F_{sym}$ and $G$ are in general functions of $s=2p$ , but are independent of $z$ .",
"Thus for any given input(s) (REF ) are formally equivalent to a corresponding Kerr problem, with renormalized intensity and grating factor, and can be solved by the same methods.",
"To develop the Kerr analogy further, we can write $& & \\hat{F}_{sym}= F_{sym} \\left(\\begin{array}{cccc}1 & G \\\\G & 1\\end{array}\\right).",
"\\nonumber $ The eigenvalues of $ \\hat{F}_{sym}$ are simply given by $F_{sym}(1 \\pm G)$ , with corresponding eigenvectors proportional to [$1, \\pm 1$ ].",
"We now define $\\psi _{1,2}^2 = \\theta (\\theta +\\kappa \\phi _{1,2})$ , where the effective Kerr coefficient $\\kappa = \\alpha _l L \\Delta $ .",
"($\\phi _1, \\phi _2$ ) are the eigenvalues of $\\hat{F}$ , chosen such that ($\\phi _1, \\phi _2) \\rightarrow F_{sym}(1-G,1+G)$ as $q \\rightarrow p$ .",
"This ensures that $\\psi _{1,2}$ coincide exactly with the quantities $\\psi _{1,2}$ used in [24], [25] in analyzing the Kerr CP case.",
"It follows that the analysis and results established in these papers for the symmetrically-pumped CP Kerr problem extend to the present quasi-Kerr case, in which both the strength of the nonlinearity and of the grating-coupling $G$ can be intensity dependent (see Appendix for details).",
"Hence the quasi-Kerr CP threshold condition is given by the expression familiar from, e.g., [25]: $2+2cos\\psi _1 cos\\psi _2 +\\left(\\frac{\\psi _1}{\\psi _2} +\\frac{\\psi _2}{\\psi _1} \\right)sin\\psi _1 sin\\psi _2 = 0.$ While this expression is indeed familiar for a Kerr medium, our discussion shows that it applies much more generally, i.e.",
"to any medium (including saturating media) which can be described by a nonlinearity function of the form $F(p,q)$ , subject to absorption being negligible.",
"Muradyan et al [26] implied this result in the context of the MM model extended to include some optomechancal effects, but did not explicitly demonstrate it.",
"We now present the explicit forms of the matrix $\\hat{F}$ for various models of interest here.",
"For the Kerr case, we have $\\hat{F}_{kerr}= -\\left(\\begin{array}{cccc}p & (1+h)q \\\\(1+h)p & q\\end{array}\\right).$ For $p=q$ this leads to $F_{sym} = -p$ and $G=1+h$ as expected.",
"For the MM model, we obtain $& & \\hat{F}_{MM}= - \\frac{1}{(1+s)^3}\\\\& & \\left(\\begin{array}{cccc}p(1+s)-2hpq & (1+h)q(1+s) -2hq^2 \\\\(1+h)p(1+s)-2hp^2 & q(1+s) -2hpq\\end{array}\\right).",
"\\nonumber $ For $p=q=s/2$ and $h=1$ the above expression for $\\hat{F}_{MM}$ leads to $F_{sym} = - \\frac{p}{(1+s)^3}$ , while we find an intensity-dependent grating factor $G=2+s$ .",
"This differs from the results of [26], wherein the given formulae imply $G=2$ .",
"The general (all grating terms) function $F$ given in (REF ) also leads to explicit expressions for the matrix $\\hat{F}_{all}$ .",
"In the absence of grating terms, i.e.",
"for $h=0$ , it simplifies to $& & \\hat{F}_{h=0}= - \\frac{1}{(1+s)^2} \\left(\\begin{array}{cccc}p & q \\\\p & q\\end{array}\\right) \\nonumber $ which leads to $F_{sym} = - \\frac{p}{(1+s)^2}$ , while $G=1$ as expected, implying a zero eigenvalue for $\\hat{F}_{h=0}$ , and hence $\\psi _1 = \\theta $ .",
"The MM model gives identical results for $h=0$ .",
"With all grating terms included, i.e.",
"for $h=1$ , we obtain $& & \\hat{F}_{all}= \\left(\\begin{array}{cccc}(1+s)/W^3 - F & -2q/W^3 \\\\-2p/W^3 &(1+s)/W^3 -F^T\\end{array}\\right)$ where $F^T(p,q)= F(q,p)$ .",
"For equal intensities $W=\\sqrt{1+2s }$ and $\\xi =0$ .",
"Some calculation then shows that $G$ is approximately $2+2s$ for small $s$ .",
"The behavior for larger $s$ is dominated by the fact that $F_{11}$ changes sign at $s= 1+ \\sqrt{2}$ , as does $G$ .",
"Turning now to the SFM problem, we note that in the system and models discussed in [16], the origin of the mirror distance coordinate $d$ was at the centre of the cloud.",
"In the present work it is more natural to set the origin at the cloud exit, and to use a dimensionless coordinate $D$ , i.e.",
"the mirror is at distance $D L$ beyond the medium.",
"Evidently $d = L(D+\\frac{1}{2})$ .",
"$D$ can be negative if the feedback optics involves a telescope.",
"The boundary conditions at the output then become $ g(1) = e^{-2\\psi _D} f(1)$ , where $\\psi _D= D \\theta $ .",
"Using the analysis presented in the Appendix this, along with $f(0)=0$ , leads to the SFM threshold condition for perfect mirror reflection ($R=1$ ) $c_1 c_2 +\\left(\\frac{\\psi _2}{\\psi _1}c_D^2+\\frac{\\psi _1}{\\psi _2} s_D^2 \\right)s_1 s_2 = c_D s_D\\left( \\beta _1 s_1 c_2 -\\beta _2 s_2 c_1\\right).$ Here $c_i=cos\\psi _i$ ; $s_i=sin\\psi _i$ : $c_D=cos\\psi _D$ ; $s_D=sin\\psi _D$ , and $\\beta _n = \\left(\\frac{\\psi _n}{\\theta }-\\frac{\\theta }{\\psi _n}\\right)$ .",
"As a first example, we consider $D=0$ , i.e.",
"the mirror is directly at the output.",
"Since $s_D =0$ for $D=0$ , (REF ) simplifies to $c_1 c_2 +\\left(\\frac{\\psi _2}{\\psi _1} \\right)s_1 s_2 = 0.$ Now, it is known (e.g.",
"[25]) that (REF ) can be written as the product of two factors, $H_1 H_2 =0$ .",
"An interesting and important feature of (REF ) is that it is identical to the condition $H_2 =0$ , but with $\\psi _i \\rightarrow 2\\psi _i$ .",
"Since this corresponds to doubling the length of the medium, we conclude that the transverse instability threshold conditions for a medium with a lossless feedback mirror at its output corresponds exactly to a threshold condition for a medium of twice the length with balanced counterpropagating inputs.",
"This is consistent with the intuitive idea that the SFM system is somehow the \"mirror image\" of a CP system.",
"There is a twist, however.",
"$H_1 =0$ does not define a threshold for the SFM system at $D=0$ , due to the fact that (REF ) is not symmetric under $1 \\rightarrow 2$ .",
"Geddes et al [25] show, using the parity symmetry of the symmetrically pumped CP system, that $H_1 =0$ and $H_2 =0$ correspond to perturbation eigenmodes which are respectively odd and even, i.e.",
"$f=-g$ and $f=g$ respectively at the centre of the medium.",
"Only the latter corresponds to the SFM boundary condition, and hence only the even-mode instabilities of the CP system correspond to SFM instabilities.",
"This breaking of parity (and hence $1 \\leftrightarrow 2$ ) symmetry explains why we had to be careful in defining $\\psi _{1,2}$ , so as to align them with the Kerr definitions.",
"Figure: Threshold intensity (in units of α l LΔp/2\\alpha _l L \\Delta p/2) vs diffraction parameter θ=Q 2 L/2k\\theta =Q^2L/2k, calculated from () for a Kerr medium described by F ^ kerr \\hat{F}_{kerr}, with h=0h=0 (green) and h=1h=1 (orange).",
"Positive and negative intensity values, respectively, correspond to self-focusing and self-defocusing Kerr media.",
"There is a feedback mirror placed directly at the end of the medium (D=0D=0, R=1R=1).Fig.",
"REF shows threshold curves for a Kerr medium with a lossless feedback mirror at its output, calculated from (REF ) using $D=0$ and $G=1+h=2$ .",
"The SFM threshold curves are identical to one of the two intertwined curves found in the CP problem, see, e.g., Fig.",
"2 of [25], allowing for the factor of two in length $L$ needed to align the SFM and SP problems.",
"The intensity unit used in Fig.",
"REF , $\\alpha _l L \\Delta p/2$ , is the single-beam, single-pass self-phase-shift of the forward field, as used in [25], so the twofold reduction in thresholds compared with Fig.",
"2 of [25] is a real advantage of the SFM configuration over the CP one.",
"Turning now to the saturable two-level case, Fig.",
"REF shows threshold curves for a two-level medium with a lossless feedback mirror at its output, calculated from (REF ) using the response function $\\hat{F}_{All}$ .",
"We have taken advantage of the invariance of the $\\psi _i$ , and hence of (REF ), under simultaneous sign changes of $\\theta $ and $\\Delta $ to combine red and blue detuning cases in a single graph, with negative $\\theta $ corresponding to red detuning.",
"The effect of saturation is clearly seen in the presence of upper, as well as lower, thresholds.",
"The cases shown ($|\\alpha _l L \\Delta |= 8$ ) are fairly close to the minimum quasi-Kerr phase shift to allow a transverse instability, so that the unstable domains are closed curves forming distinct bands of unstable transverse wave vectors $Q$ .",
"These bands are located in fairly close correspondence to local threshold minima in the Kerr case (Fig.",
"REF ), with red and blue detuning corresponding to self-defocusing and self-focusing Kerr cases respectively.",
"Hence one effect of the $f=g$ mode constraint is that the unstable bands for red detuning (left panel) are complementary to (and generally have larger $Q$ than) those for blue detuning (right panel).",
"The scaling relation between the $D=0$ SFM and CP systems applies also to two-level media.",
"Fig.",
"REF illustrates the scaling property by doubling the quasi-Kerr coefficient $\\alpha _l L \\Delta $ , equivalent to doubling $L$ , compared to Fig.",
"REF .",
"The CP thresholds appear as closed loops, rather than intertwined open curves as in a Kerr-medium (cf Fig.",
"REF ), because saturation implies existence of upper, as well as lower instability thresholds.",
"For blue detuning, the loops do intersect at low $\\theta $ , reminiscent of the Kerr case, though with upper intersections also.",
"The main point to notice, however, is that the even modes (orange) are identical to the corresponding SFM loops in Fig.",
"REF , while the odd modes (green, dashed) are absent from Fig.",
"REF .",
"(Note the change of scale of $\\theta $ , necessary to align the two cases, since $\\theta $ is $ \\sim L$ .)",
"As well as vividly illustrating the scaling relation between the $D=0$ SFM and CP systems, the fact that the (lowest) SFM threshold (blue) is much lower than the CP one (orange, green), illustrates a major practical advantage of the SFM over the CP configuration in terms of achieving instability.",
"Indeed there is no CP instability for the parameters of Fig.",
"REF .",
"Additionally, of course, the SFM configuration needs only one laser (or half the power compared to splitting a single laser beam to make the two inputs required in the CP configuration).",
"The SFM thresholds in Fig.",
"REF are rather Kerr-like (though single), with the upper threshold at high enough $s$ to make the curves appear open.",
"The small blue loop is actually an island of SFM stability, which grows as $\\alpha _l L \\Delta $ is increased.",
"It will occur for CP also, but at still-larger $\\alpha _l L \\Delta $ and beyond, because of the scaling property.",
"Figure: Threshold saturation intensity ss vs diffraction parameter θ=Q 2 L/2k\\theta =Q^2L/2k, calculated from () for a two-level medium described by F ^ All \\hat{F}_{All}, with h=1h=1.",
"There is a feedback mirror placed directly at the end of the medium (D=0D=0 case).",
"The quasi-Kerr coefficient |α l LΔ||\\alpha _l L \\Delta | is 8, with negative and positive θ\\theta corresponding to red and blue detuning respectively at diffraction parameter |θ||\\theta |.Figure: Comparison, for the same two-level medium, between counterpropagation (CP) and single feedback mirror (SFM) pattern thresholds.",
"Parameters as for Fig.",
"except |α l LΔ|=16|\\alpha _l L \\Delta | =16.",
"Therefore the closed loops (solid) marking the even-mode CP instability are identical to those for the SFM instability in Fig.",
", consistent with the scaling relationship discussed in the text.",
"Note the change of θ\\theta scale from Fig.",
".",
"The odd-mode CP threshold loops (dashed) are absent in Fig.",
", as expected.",
"The SFM thresholds for this case (open curves) are significantly lower than the CP thresholds, while the upper thresholds are beyond the plot range for ss.",
"(The small loop at (θ,s)∼(4,1)(\\theta , s) \\sim (4,1) is an island of SFM stability.)Fig.",
"REF shows the effect of finite mirror distance $D$ , in this case negative, for blue/red detuning.",
"Because the finite-$D$ formula (REF ) is also invariant under simultaneous sign changes of $\\theta $ and $\\Delta $ , we can again use negative $\\theta $ to display both blue and red detuning thresholds on the same graph.",
"The thresholds are somewhat lowered in comparison with $D=0$ , and the unstable bands are shifted as well as broadened.",
"Indeed the red detuning now has an unstable band at small $Q$ , corresponding to a small-angle scattering cone in the far field.",
"This sensitivity to mirror distance can be interpreted as a phase-matching effect: the external phase shift $\\psi _D$ provides an extra flexibility in comparison with the CP problem, enabling instability in cases where the internal nonlinear and diffractive phases are ill-matched.",
"Figure: Threshold saturation intensity ss vs diffraction parameter θ=Q 2 L/2k\\theta =Q^2L/2k, calculated from () for a two-level medium described by F ^ All \\hat{F}_{All}, with h=1h=1.",
"There is a feedback mirror at negative effective distance (D=-1.3D=-1.3) from the end of the medium.",
"The quasi-Kerr coefficient |α l LΔ|=8|\\alpha _l L \\Delta | = 8, with negative and positive θ\\theta corresponding to red and blue detuning respectively at diffraction parameter |θ||\\theta |.Figure: Saturation intensity ss vs diffraction parameter θ=Q 2 L/2k\\theta =Q^2L/2k.",
"Blue curves: Envelope curves calculated from () for a two-level medium described by F ^ All \\hat{F}_{All}, with h=1h=1.",
"The quasi-Kerr coefficient |α l LΔ|=16|\\alpha _l L \\Delta | =16, corresponding to blue detuning.",
"Orange curve: Threshold curve with a feedback mirror at negative effective distance (D=-0.5D=-0.5, i.e.",
"at the centre of the medium), which touches the envelope curves.Figure: Saturation intensity ss vs diffraction parameter θ=Q 2 L/2k\\theta =Q^2L/2k.",
"Blue curves (dashed): Envelope curves calculated from () for a two-level medium described by F ^ All \\hat{F}_{All}, with h=1h=1.",
"Quasi-Kerr coefficient |α l LΔ|=7.1|\\alpha _l L \\Delta | =7.1.",
"Orange curves: Threshold curves with a feedback mirror at negative effective distance (D=-1.3D=-1.3) from the end of the medium, which touches the envelope curves.The presence of mirror distance as an additional parameter in the SFM formula (REF ) as compared to the CP formula (REF ) makes it harder to see what is going on.",
"Especially at small $|D|$ , the transverse wavelength with lowest threshold varies strongly with mirror distance.",
"The threshold intensity, however, varies much less, and we now show that the threshold curve (intensity vs diffraction parameter $\\theta $ ) is bounded below by an envelope curve.",
"Indeed there are a set of upper and lower envelope curves, which can be calculated analytically from (REF ).",
"In deriving (REF ), we naturally assumed that the feedback phase functions $c_D, s_D$ are real.",
"If the intensity is just below a threshold minimum, however, we can still find a solution with complex $\\psi _D$ , which corresponds physically to introducing some gain into the feedback loop.",
"Just above the minimum, there are generally two adjacent values of $\\theta $ which solve (REF ).",
"Since the threshold curves oscillate, with maxima as well as minima, we note that two real roots just below a maximum become no roots just above, again with complex roots corresponding to feedback gain.",
"We can quantify this scenario by observing that (REF ) can be turned into a quadratic equation in $tan \\psi _D$ .",
"Vanishing discriminant for this equation corresponds to the transition between complex and real $\\psi _D$ .",
"This results in the following equation: $4(c_1 c_2 +\\frac{\\psi _1}{\\psi _2} s_1 s_2)(c_1 c_2 +\\frac{\\psi _2}{\\psi _1} s_1 s_2) = ( \\beta _1 s_1 c_2 -\\beta _2 s_2 c_1)^2.$ As a first example, Fig.",
"REF illustrates the envelope curves for the all-grating quasi-Kerr model, together with a part of the threshold curve for $D= -0.5$ , which indeed touches both curves.",
"The system parameters here are similar to those in Fig.",
"REF , and thus enable a more detailed view of the shape of the SFM threshold curve, as well as the very low $s$ values at which thrshold can be reached.",
"Contact with the envelope is not necessarily at the extrema of the threshold curve, but for all $D$ values, and all cases, considered the threshold curves are bounded by, and tangent to, the envelope curves given by (REF ).",
"In particular, the absolute minimum of the envelope, approximately at $\\Theta =1.5$ , $s=0.08$ in Fig.",
"REF , defines the minimum attainable threshold for $|\\alpha _l L \\Delta | = 16$ with other medium parameters fixed.",
"Whereas the threshold curves either asymptote to, or are distinct from the axis $\\theta =0$ , the envelope curve in Fig.",
"REF seems to approach the axis at finite $s$ .",
"This is confirmed and eloborated in Fig.",
"REF , where the trick of plotting also for $\\theta < 0$ nicely exhibits the finiteness of the intercept of the envelope, as well as continuity across the $\\theta =0$ axis between red and blue detuning.",
"Our interpretation is that the envelope intercept corresponds to $L \\rightarrow 0$ , i.e.",
"the \"thin medium\" limit, which we will discuss further shortly.",
"First, however, we note that the thresholds for $D=-1.3$ are neatly tangential to the envelope curves, including the small envelope loop at $\\theta \\sim 5$ .",
"Since there is no corresponding loop for red detuning, we can conclude that only a single band of patterns can be found, at any $D$ , for red detuning.",
"Another important feature of Fig.",
"REF is that the finite slope of the envelope at the axis means that one or other of the detunings has its absolute minimum threshold at finite $\\theta $ , whereas for the other the threshold decreases as $D$ is increased, with minimum threshold being found in the thin-medium limit.",
"Figure REF further illustrates how the envelope curves capture the essential behavior of the threshold curves, this time for a Kerr medium with no grating term ($h=0$ ).",
"Here two distances ($D=-1.5,-3.0$ ) are shown, and we begin to see how the faster oscillations of the threshold for larger mirror distances allow a better exploration of the envelope, and thus potentially lower thresholds.",
"For the self-focusing case, where the envelope has a minimum at finite $\\theta $ , we can see, for $D=3$ , the transition of the lowest threshold from the lowest-Q to the second-lowest-Q band.",
"Assuming that the dominant pattern is determined by the lowest threshold, we would expect a sudden drop in the observed pattern perod as $D$ is increased.",
"This phenomenon is indeed observed (see Fig.",
"REF below for an example).",
"Conversely, for self-defocusing the lowest threshold always decreases as $D$ is increased, so that the patterns with lowest threshold are found at large mirror distances, and have large spatial scales, with pattern wavelength scaling like $\\sqrt{d/k}$ , as is well known from thin-medium theory [12].",
"In contrast, CP thresholds for $G=1$ defocusing Kerr media decrease with increasing $Q$ , see, e.g.",
"[25].",
"The same is true, of course, for the SFM with $D=0$ , as shown in Fig.",
"REF .",
"This finite-$D$ advantage can be attributed to the ability of the feedback phase to compensate for both the diffractive and nonlinear phase shifts in the medium, which have the same sign for defocusing, and thus cannot cancel each other as they can for self-focusing.",
"This no-grating Kerr case is also interesting in that the envelope curves cross, and hence the threshold curves must thread through the intersection (Fig.",
"REF ).",
"It follows that the threshold is actually independent of mirror distance at these crossings.",
"Note that the threshold will normally be lower at a different diffraction parameter (as occurs in Fig.",
"REF ), and observing the phenomenon would require isolating the specific wavenumber by Fourier filtering in the feedback loop [31].",
"The finite limit for small diffraction, $\\theta \\rightarrow 0$ , of the envelope is ($\\pm 0.5$ ) in Fig.",
"REF , and corresponds exactly to the thin-slice value [12], but the finite slope at $\\theta =0$ means that the pattern-forming modes are not, in fact, threshold-degenerate when the medium thickness is taken into account.",
"Fig.",
"REF shows this in more detail for a moderately large mirror distance ($D=10$ ).",
"For self-focusing the envelope curve falls for increasing diffraction parameter in the range displayed.",
"The minimum is reached at $\\theta =\\pi /2$ .",
"For negative detuning the lowest wavenumber is selected.",
"In both cases, therefore, the multi-fractal patterns predicted in the thin-slice limit [32] and dependent on mode-degeneracy are not expected to occur in practice, unless other mechanisms or devices are able to restore degeneracy.",
"Figure: Threshold intensity (in units of α l LΔp/2\\alpha _l L \\Delta p/2) vs diffraction parameter θ=Q 2 L/2k\\theta =Q^2L/2k.",
"Blue curves: Envelope curves calculated from () for a Kerr medium with h=0h=0, i.e.",
"G=1G=1.",
"Positive and negative intensity values, respectively, correspond to self-focusing and self-defocusing Kerr media.",
"Also threshold curves with a feedback mirror at negative effective distance from the end of the medium.",
"Orange curves: D=-1.5D=-1.5.",
"Green curves: D=-3.0D=-3.0.",
"In both cases the threshold curves touch the envelope curves, and are confined by them.Figure: Threshold intensity (in units of α l LΔp/2\\alpha _l L \\Delta p/2) vs diffraction parameter θ=Q 2 L/2k\\theta =Q^2L/2k.",
"Blue curves: envelope curves calculated from () for a Kerr medium described by F ^ kerr \\hat{F}_{kerr}, with h=0h=0, i.e.",
"G=1G=1.",
"Positive and negative intensity values, respectively, correspond to self-focusing and self-defocusing Kerr media.",
"Orange curves: D=10D=10.",
"The feedback mirror is quite far from the medium, which is thus a quite-thin slice.",
"Note that the mode thresholds are not degenerate, as they are in simple thin-slice SFM models .Further envelope properties are illustrated by the envelope curves for a Kerr medium with grating (Fig.",
"REF ), this time plotted along with threshold curves for positive mirror distances.",
"In this case higher-order modes are visible, but the corresponding envelope curves again confine the corresponding threshold curves.",
"Here the envelopes of the lowest order modes do not actually cross, though there are still values of $\\theta $ for which the threshold is almost distance independent.",
"Again the small-diffraction limit corresponds to the standard thin-slice threshold, but this limit is approached with finite slope.",
"Figure: Threshold intensity (in units of α l LΔp/2\\alpha _l L \\Delta p/2) vs diffraction parameter θ=Q 2 L/2k\\theta =Q^2L/2k for a Kerr medium with h=1h=1.",
"Negative intensities correspond to negative Kerr, i.e.",
"self-defocusing.",
"Blue curves: Envelope curves calculated from ().",
"Also threshold curves with a feedback mirror at positive effective distance.",
"Orange curves: D=0.5D=0.5.",
"Green curves: D=1.0D= 1.0.",
"In both cases the threshold curves touch the envelope curves, and are confined by them.The above figures demonstrate how the threshold extrema move vs $\\theta $ as mirror distance $D$ is varied.",
"An interesting and relevant way to examine this is to plot pattern scale ( $ \\sim 1/\\sqrt{\\theta }$ ) vs $D$ for fixed intensity.",
"This is demonstrated in Fig.",
"REF , where the parameters are chosen to match those of [17], and the intensity $s=0.085$ is just above the minimum threshold, so that the unstable regions appear as long narrow islands.",
"The \"fan\" shape of the island group is due to the Talbot effect: the threshold values satisfying (REF ) are evidently periodic in $\\psi _D = D \\theta $ , which means that at fixed $\\theta $ (size) and intensity, threshold values are periodic in $D$ .",
"This is particularly clear at the bottom of the fan in Fig.",
"REF , where the tips of the islands are equally-spaced in $D$ .",
"The Talbot periodicity is inversely proportional to $\\theta $ , which is why the islands fan out as the pattern scale increases (i.e.",
"as $\\theta $ decreases).",
"Figure: Pattern period (arb.",
"units) vs mirror distance DD at fixed intensity s=0.085s=0.085.",
"Threshold curves calculated from () for a two-level medium described by F ^ All \\hat{F}_{All}, with h=1h=1.",
"The quasi-Kerr coefficient α l LΔ=13.94\\alpha _l L \\Delta =13.94, corresponding to blue detuning.",
"For optical density 210 , this corresponds to detuning Δ=2δ/Γ=15\\Delta = 2\\delta /\\Gamma = 15.Such \"Talbot fans\" are readily observed experimentally.",
"The fan reported in [17] is shown in Fig.",
"REF , where the experimental data fit well to threshold data from (REF ) using our two-level all-grating model based on $\\hat{F}_{all}$ .",
"Fig.",
"REF b plots the pattern period against mirror distance.",
"Around $D\\approx 0$ the lengthscale with the smallest wavenumber (largest period) is selected.",
"At higher $|D|$ , two lengthscales are found in the pattern.",
"Both are in good agreement with the prediction from the theory.",
"The inset shows excellent agreement between the measured and calculated $D$ -periodicities.",
"In the earlier optomechanical patterns paper [16], there is a more limited fan, to which threshold data from (REF ) are fitted using a Kerr model (h=0, because the slow time scale allows atomic motion to wash out the longitudinal grating).",
"Fig.",
"REF a plots the power diffracted into the first and second unstable wavenumber obtained by integrating the measured far field intensity distributions over an annulus with the respective radius.",
"We did not measure thresholds, but to a first approximation one can argue that the diffracted power increases with increasing distance to threshold and hence the measured data can be interpreted as indicators of inverted threshold curves.",
"We compare them with the threshold curves obtained from the all grating quasi-Kerr model as the detuning is reasonably large and absorption not very important.",
"As indicated in the discussion of Fig.",
"REF a, around $D\\approx 0$ , only the lowest wavenumber (i.e.",
"the one from the first Talbot balloon) is excited.",
"For a mirror within the medium ($D=-1 \\dots 0$ ), the diffracted power is low and the predicted thresholds are high.",
"For increasing $|D|$ threshold are predicted to fall dramatically and indeed well developed patterns, indicated by high diffracted power, are observed.",
"For further increasing $|D|$ the theory predicts that the second Talbot balloon at higher wavenumber has the lowest threshold.",
"Indeed excitation of this length scale is observed but it does not take over completely in the experimental data.",
"Figure: (Color online) a) Diffracted power (experiment, left axis) and predicted threshold saturation intensity (theory, right axis) vs scaled mirror distance DD.",
"The cloud thickness is L=9L=9 mm.",
"b) Pattern period Λ\\Lambda vs mirror distance.",
"In physical units, the x-axis corresponds to -60 mm to +40 mm measured from the center of the cloud.",
"Parameters: blue detuning, Δ=15\\Delta = 15, see .",
"The diffracted power is normalized to its maximal value.",
"Red solid dots: experimental data for first Talbot balloon (lowest wavenumber), black circles: experimental data for second Talbot balloon (next highest wavenumber excited, in a) enhanced by factor of 5).",
"The red and black curves are the corresponding theoretical predictions and are calculated from () using the all-grating two-level model.Inset: The measured DD period as a function of the pattern size (stars), together with the Talbot effect prediction (line).For a further investigation of the Talbot fan phenomenon we analyze a somewhat different experimental SFM situation in which optical pumping between Zeeman substates, rather than two-level electronic excitation, is the main nonlinearity [9], [33], [34], [35].",
"Experimental parameters are an effective medium length of $L=3.2$ mm, beam intensity $I=18$ mW/cm$^2$ and detuning $\\Delta = -14$ .",
"The homogenous solution is not saturated in this case [36], so it is reasonable to compare the data to the length scales and threshold curves obtained from a self-focusing thick medium Kerr theory.",
"Experimental measurements of diffracted power and pattern lengthscale vs mirror distance are shown in Fig.",
"REF .",
"It is apparent that the behavior is very similar to the one observed for the electronic 2-level case in Fig.",
"REF , but there is one crucial difference.",
"For large enough $|D|$ ($D> 0.7$ , $D<-2.5$ ) the length scale from the first Talbot balloon is completely suppressed and the length scale of the second balloon takes over completely.",
"This is in good, although not quantitative, agreement with the thick medium model as discussed earlier in connection with Figure REF , though the transition is predicted to occur at somewhat larger $|D|$ .",
"Nevertheless, it is an important confirmation of the importance of the diffraction within the medium influencing length scale selection.",
"In view of the fact that the atomic clouds have an approximately Gaussian density distribution and the theory assumes a rectangular distribution, quantitative deviations between theory and experiment are not surprising.",
"Figure: (Color online) a) Predicted threshold, b) experimentally observed diffracted power (normalized to its maximal value) and c) pattern period vs mirror distance DD.",
"In unscaled parameters, the x-axis corresponds to -12.8 mm to +10.2 mm measured from cloud center.",
"Parameters: effective medium length is L=3.2L=3.2 mm, beam intensity I=18I=18 mW/cm 2 ^2 and detuning Δ=-14\\Delta = -14.",
"Red solid dots: experimental data for first Talbot balloon (lowest wavenumber), blue circles: experimental data for second Talbot balloon (next highest wavenumber excited).",
"The red and blue curves are the corresponding theoretical predictions and are calculated for a self-focusing Kerr medium with h=1h=1 described by F ^ Kerr \\hat{F}_{Kerr}.",
"The insets show far field patterns obtained at the mirror positions indicated illustrating the length scale competition.Figures REF and REF indicate that a change of mirror distance can drag the pattern period along qualitatively as in a diffractively thin medium but only up to a point.",
"Then the system jumps back to a smaller length scale it seems to prefer, which can be changed again to some extent by changing mirror distance.",
"The origin of this behavior lies in the interaction between the threshold curves and the envelope as discussed before.",
"For increasing $|D|$ the threshold curves move to lower $Q$ and have more wiggles in a certain range of $\\theta $ on the envelope curve, which means they can explore more effectively the potentially lowest threshold condition.",
"Another way to illustrate this point is visualized in Fig.",
"REF .",
"The red solid curve in Fig.",
"REF a denotes the length scale of the minimum threshold mode vs mirror distance.",
"For $D=-3 \\ldots 1$ it mirrors the first Talbot balloon, until it jumps to the second and follows it for $D=-6 \\ldots -4$ and $D=1.5 \\ldots 4$ .",
"Afterwards it jumps again and wiggles around a horizontal, which is very close to the value for the CP instability at twice the medium length or the SFM instability at $D=0$ (Fig.",
"REF ).",
"The changes of lengthscale imply that the minimum of the envelope curve is at finite $\\theta $ and the system is trying to stay close to this value as far as compatible with the specific boundary conditions, i.e.",
"diffractive phase shift $\\theta $ at the feedback distance $D$ .",
"These considerations are maybe even more apparent for the thresholds (Fig.",
"REF b) where the SFM and CP threshold curves are nearly indistinguishable at large $|D|$ .",
"Figure: (Color online) a) Pattern length scale (characterized by diffraction parameter θ\\theta ) and b) threshold intensity vs mirror distance DD for a self-focusing Kerr medium with h=1h=1 described by F ^ Kerr \\hat{F}_{Kerr}.",
"Red solid curve: minimum threshold, blue dashed curve: lowest wavenumber (first Talbot) balloon, blue dotted curve: second lowest wavenumber (second Talbot) balloon, black solid curve: minimum threshold condition for CP instability, Fig.",
"."
],
[
"Saturable absorption and approach to atomic resonance",
"As mentioned, the quasi-Kerr treatment of the atomic susceptibility is valid only for large atomic detunings.",
"Nonlinear effects typically strengthen as detuning is decreased and atomic resonance is approached, but resonant absorption kills the feedback.",
"It is therefore important to extend our models to address the absorptive response at finite detunings, which implies using z-dependent forward and backward intensities in the transverse perturbation problem.",
"From the structure of (REF ), it is evident that the presence of the diffraction parameter $\\theta $ mixes the real and imaginary parts of the perturbations ($f,g$ ) and thus adds significant mathematical complication.",
"As a first approach to inclusion of absorption, therefore, it is worthwhile to analyze the case in which $\\theta $ is set equal to zero.",
"Physically, this corresponds to neglecting diffraction within the medium, often referred to as the \"thin medium\" approximation.",
"As well as linking to work in which the medium is regarded as a thin slice, this approach also enables consideration of multi-slice models [24], where the medium is approximated by a sequence of slices with free-space diffraction in-between.",
"Split-step numerical algorithms typically adopt such an approach, so there is also computational interest in this approximation.",
"In our earlier discussion of the envelope functions in the quasi-Kerr approximation, we saw that the thin-slice limit $\\theta =0$ is typically approached with finite slope, but nevertheless with a threshold of the same order as those found for optimum mirror distances.",
"We can therefore expect that the thin-medium approximation will offer a worthwhile qualitative picture of the effect of linear and nonlinear absorption on thresholds and tuning ranges as atomic resonance is approached, and indeed we will find behaviors in rather good agreement with the cold-atom patterns reported in [17].",
"Dropping diffraction and assuming threshold conditions, the system (REF ) becomes: $ \\left\\lbrace \\begin{array}{l}\\frac{df}{dz} = -\\alpha _lL(1+i\\Delta )(F_{11}f^{\\prime }+F_{12}g^{\\prime }) ,\\\\\\frac{dg}{dz} = \\alpha _lL(1+i\\Delta )(F_{21}f^{\\prime }+F_{22}g^{\\prime }) \\\\\\end{array} \\right.$ In the presence of absorption, the elements of $\\hat{F}$ are z-dependent, for example obeying the zero-order solutions derived above for various models.",
"Note that the imaginary parts of both $f$ and $g$ are slaved to the real parts.",
"In particular, if $f(0)=f_0=0$ , as for the input to a SFM system, then $f(1)=f_1=(1+i\\Delta ) f_1^{\\prime }$ .",
"The usual mirror feedback conditions for a transverse perturbation then imply $dq_1= R(cos\\psi _D+\\Delta sin\\psi _D)dp_1$ , where $dp_1$ and $dq_1$ are the intensity changes associated with $f_1$ and $g_1$ .",
"Instead of integrating the system (REF ) we adopt a different approach.",
"Since we have neglected diffraction in the medium, the perturbed system obeys the same equation as the homogeneous solution, but with perturbed boundary conditions.",
"Specifically, for a given output $p_1$ , and corresponding feedback $q_1 = R p_1$ , we can analytically and/or numerically calculate the corresponding input $p_0$ .",
"Using the same algorithm, we can formally calculate the change in $p_0$ due to a small change $dp_1$ in $p_1$ with no change in $q_1$ , and conversely.",
"We can thus find the ratio of $dp_1$ to $dq_1$ which leaves $p_0$ unchanged to first order - which is the input boundary condition.",
"Only for specific values of $p_1$ will this ratio be equivalent to the feedback phase relation $dq_1= R(cos\\psi _D+\\Delta sin\\psi _D)dp_1$ identified above.",
"Finding such $p_1$ values, and the corresponding input values $p_0$ , gives pattern thresholds for the assumed values of $R$ , $\\Delta $ , $\\theta $ and $D$ .",
"We then eliminate $D$ and $\\theta $ by requiring that that the perturbation gain is maximised, which implies $ cos\\psi _D+\\Delta sin\\psi _D = \\sqrt{(}1+\\Delta ^2)$ .",
"With these choices we find that the maximal pattern-forming region is a closed domain in the remaining parameter space ($p_0, \\Delta $ ) for both the no-grating and all-grating two-level models.",
"Figure REF compares the threshold domains for these two thin-slice, all-tuning, absorptive models with experimental data [17] on the detuning behavior of the diffracted power observed under pattern formation conditions in a cold Rb cloud with single feedback mirror.The agreement for the all-grating model is rather satisfactory, bearing in mind that the theory only calculates threshold conditions, while the experiment detects diffracted power only if the perturbation gain is large enough to build a strong pattern from noise within the microsecond or so duration of the pump pulse.",
"Moreover, we note that the no-grating threshold domain is smaller than that in which transverse structure is observed.",
"This provides firm evidence that reflection gratings are present in the cold-atom cloud, in agreement with expectations based on the inability of transport mechanisms to wash out susceptibility gratings at such low temperatures when such short input pulses are used.",
"Figure: (Color online) (a) Two-level instability domain (δ>0\\delta > 0) reported in .Diffracted power P d P_d is measured as a function of δ>0\\delta > 0 (note that Δ=2δ/Γ\\Delta =2\\delta /\\Gamma ) and input intensity II, and the data plottedas isolines.",
"Note the logarithmic horizontal scale.",
"The dotted loops indicate maximal instability domains calculated in the thin-medium approximation as described in the text: (black) domain calculated from (), i.e.",
"with all reflection gratings included (h=1h=1); (red) domain calculated from (), i.e.",
"with no reflection gratings (h=0h=0).",
"Both dotted traces are rescaled to absolute values of intensity and detuning."
],
[
"Conclusion",
"In this paper we have undertaken a largely analytic investigation of thresholds and lengthscales for pattern formation in a saturable two-level medium, optically-excited close to resonance from one side, and with a feedback mirror to reflect and phase-shift the light fields after they have traversed the medium.",
"In that scenario, we have established a number of results, in encouraging agreement with recent experimental results in several cases.",
"Perhaps our main result is that thresholds for the feedback mirror (SFM) configuration are in exact correspondence with one set of threshold curves for symmetrically-excited counterpropagation (CP) in a medium of twice the length, when the mirror plane in the SFM system is at the output of the medium.",
"One important consequence of this is that SFM thresholds are significantly lower than CP thresholds in the same sample (e.g.",
"cloud of cold atoms).",
"Since large cold-atom clouds are difficult to produce, this can make the difference between observing well-developed patterns and failing to reach threshold at all.",
"While this scaling result is derived for a saturable nonlinearity with absorption neglected, it is a consequence of parity symmetry in the CP system, and should hold in relation to any CP system with parity symmetry.",
"Assuming that there is a stable zero-order solution of the CP system equations which exhibits parity symmetry, any perturbation eigenmode of the system must be either symmetric or anti-symmetric at the central symmetry plane.",
"For an even mode, and also for the zero-order solution, one can replace the CP system with a perfect mirror at the symmetry plane without essential change to the equations or the solutions, and the CP/SFM scaling follows.",
"Hence in a wide class of nonlinear optical systems, the SFM system offers an approximately fourfold advantage in power over a CP configuration using the same medium (twofold reduction in pumping power, and approximately twofold reduction in threshold power).",
"There is a further advantage of the SFM system, in that the mirror location $D$ can be varied continuously over a wide range around and beyond the medium length.",
"As well as allowing the observed pattern scale to be quasi-continuously varied, which is at the very least useful for diagnostics, it is also found that the minimum threshold usually occurs for $D \\ne 0$ , essentially because a non-zero feedback phase allows optimum matching between forward and backward perturbation growth rates.",
"We have considered, and compared to experiment, the \"Talbot fan\" characteristics which characterize the evolution of pattern scales as $D$ is varied, and explained observed sudden changes of scale in terms of mode competition in the neighborhood of the minimum possible (in $D$ ) threshold.",
"The additional degree of freedom offered by finite $D$ also implies an additional complexity in the analysis.",
"We have shown, however, that thresholds are constrained by envelope curves to which the threshold curves are tangent, and along which they evolve as $D$ is varied.",
"Hence important properties of the SFM system such as the minimum possible threshold, and the domains within which pattern formation is possible (or impossible) can be found, often analytically.",
"Again, the envelope propery is likely to be general, even though we have derived it only in the quasi-Kerr limit, because it follows from the structure of the feedback boundary condition.",
"Importantly, the envelope functions enable a quantitative investigation of the limit $D/L \\rightarrow \\infty $ , which correspond to diffraction in the medium being negligible compared to that in the feedback loop, i.e the thin-slice limit.",
"We find that threshold values tend to precisely the thin-medium values, but with finite slope.",
"As a consequence we have demonstrated that the degeneracy of the unstable modes predicted in thin-medium theory does not survive inclusion of finite medium length, even at lowest order.",
"Diffusive damping removing the degeneracy was introduced in the first treatments [12], [13] to model carrier diffusion in semiconductors or elasto-viscous coupling in liquid crystals, which will make these media deviate from purely local Kerr media.",
"In hot atom experiments [9], [10], [11] the thermal motion of the atoms, which can be modelled under appropriate conditions [10], [11] as diffusive motion, will in tendency provide a stronger wash-out for transverse gratings at larger wavenumber and thus remove the degeneracy.",
"In cold atoms this effect is not very strong and the finite medium thickness appears to be the main mechanism responsible for the emergence of a defined length scale [16], [17].",
"In the specific context of the two-level nonlinearity we have analyzed different models to take account of wavelength scale (reflection) gratings in the steady-state susceptibility applicable to counterpropagation problems.",
"We have found that models in which only the lowest-order (2k) gratings are considered predict a zero-order bistability as resonance is approached.",
"This bistability disappears when all orders (m$\\times $ 2k) of gratings are included, and is therefore probably spurious.",
"We have been able to develop models which include all grating orders, in particular in the quasi-Kerr and thin-medium limits, and have demonstrated reasonable agreement with experiment using these all-grating models.",
"In summary, we have developed a firm and systematic foundation for the analysis of the effects of in-medium diffraction, and of reflection gratings, in SFM pattern formation.",
"Though we have focused here on the saturable two-level electronic nonlinearity, our approach and techniques have applicability across a wide class of nonlinearities.",
"While our present analysis deals only with thresholds and steady-state instabilities, these are an important, and even essential, preliminary to more extensive numerical simulations, necessarily involving many additional parameters and many spatial and temporal scales.",
"We already showed [16] that a simple thick-medium Kerr model gives useful insight into optomechanical SFM patterns, and in this work we have shown that a similar analysis helps understand important features of polarization-mediated SFM patterns in cold atoms.",
"Patterns in cold-atom clouds with laser irradiation and mirror feedback are proving to a be a very rich field, with diverse implications, and a secure basis for the interpretation of experimental results and the development of appropriate theoretical models is therefore very important.",
"The Strathclyde group is grateful for support by the Leverhulme Trust and an university studentship for IK by the University of Strathclyde.",
"The Sophia Antipolis group is supported by CNRS, UNS, and Région PACA.",
"The collaboration between the two groups was supported by Strathclyde Global Exchange Fund and CNRS.",
"WJF also acknowledges sharing of unpublished work by M. Saffman.",
"We are grateful to A. Arnold and P. Griffin for experimental support, to G.R.W.",
"Robb, G.-L. Oppo and R. Kaiser for fruitful discussions."
],
[
"Appendix",
"In this Appendix we present a matrix approach to the analytic solution of (REF ) leading to the threshold formulae (REF ,REF ) for the CP and SFM problems respectively in the quasi-Kerr case.",
"Our methods and results are broadly similar to those of [24], [25], but because of slight notational differences, and our extension to more general nonlinearities and the SFM problem, it is perhaps worthwhile to present the details of the analysis.",
"We analyze the system in terms of a real 4-component vector $U = [f^{\\prime },g^{\\prime },f^{\\prime \\prime },-g^{\\prime \\prime }]^{tr}$ , which obeys $\\frac{dU}{dz} = M U.$ where $M$ is a real $4 \\times 4$ matrix with constant coefficients: $M=\\left(\\begin{array}{cccc}0 & 0 & \\theta & 0 \\\\0 & 0 & 0 & \\theta \\\\-\\theta -\\kappa F_{11} & -\\kappa F_{12} & 0 & 0 \\\\-\\kappa F_{21} & -\\theta -\\kappa F_{22}&0 & 0\\end{array}\\right) \\nonumber $ Here $\\kappa = \\alpha _{l}L\\Delta $ is an effective Kerr coefficient.",
"The formal solution to (REF ) is $U(1) = exp(M) U(0)$ or $U(0) = exp(- M) U(1)$ .",
"For both CP and SFM cases, $f(0) =0$ is assumed, giving two conditions on the solution.",
"The boundary conditions at $z=1$ provide the necessary two additional equations.",
"For the CP case, with input fields at both ends, this condition is simply $g(1)=0$ .",
"For the feedback mirror case, however, the condition is that $f=g$ on the mirror, and hence $g(1) = exp(-2i\\psi _D) f(1)$ , where $\\psi _D =D \\theta /L$ governs the phase shift of the perturbation field in propagating an effective distance $DL$ to the mirror.",
"The relative mirror distance $D$ can be negative if the feedback optics involves a telescope.",
"For both types of boundary condition the solution to (REF ) leads to a pair of homogeneous linear equations for ($g^{\\prime }(0),g^{\\prime \\prime }(0)$ ) which have a non-trivial solution only if the determinant of the coefficients vanishes.",
"This condition determines the pattern formation threshold as a function of $Q^2$ and system parameters.",
"Hence, given $expM$ , the quasi-Kerr limit is fully solvable for all the two-level models we have discussed, for both the CP and SFM cases.",
"The problem thus hinges on exponentiation of the matrix $M$ .",
"It has has a similar form to that analysed in the Appendix to [24], and can be analytically exponentiated in a similar fashion.",
"Squaring $M$ , we obtain a block-diagonal matrix, its diagonal submatrices both being $-C$ , where the $2 \\times 2$ matrix $C$ is given by $C=\\theta (\\theta +\\kappa \\hat{F}$ ).",
"The eigenvalues of $C$ are given by the parameters $\\psi _{i}^2 = \\theta (\\theta +\\kappa \\phi _{i})$ introduced in the main text, where the $\\phi _{i}$ are the eigenvalues of $\\hat{F}$ .",
"It follows that any unitary transformation that diagonalizes $\\hat{F}$ also diagonalizes $C$ , which provides one route to calculation of $expM$ .",
"As mentioned above, for equal intensities the eigenvectors of $\\hat{F}$ are proportional to $(1, \\pm 1)$ , which enables an intensity-independent transformation on $(f,g)$ leading to explicit expressions for $expM$ (and $exp(Mz)$ ) in terms of the $\\psi _i$ , equivalent to those obtained in [25].",
"Because the SFM boundary conditions are more involved than the CP ones, and also to enable consideration of mirror reflectivity $R \\ne 1$ , we choose to use the ($f,g$ ) basis described by $U$ .",
"Because $M^2$ is block diagonal, we write $exp(M)= 1+ \\frac{M^2}{2!}",
"+ \\frac{M^4}{4!}",
"+ .... + M (1+ \\frac{M^2}{3!}",
"+ \\frac{M^4}{5!}",
"+ ....)\\nonumber $ The power series in $M^2$ can be expressed as block-diagonal cosine and sinc functions of $\\sqrt{C}$ , a 2x2 matrix obeying $(\\sqrt{C})^2 = C$ .",
"As in [24], we can then write an explicit expression for $expM$ as a $2 \\times 2$ block matrix: $exp(M)=\\left(\\begin{array}{cccc}cos\\sqrt{C} & \\theta sinc\\sqrt{C} \\\\-( C / \\theta ) sinc\\sqrt{C} & cos\\sqrt{C}\\end{array}\\right)$ .",
"Because the cosine and sinc are even functions, this expression for $exp(M)$ is unique in terms of $C$ , even though $\\sqrt{C}$ is not uniquely defined.",
"Suppose that the $2 \\times 2$ matrix $E$ diagonalizes $\\hat{F}$ , i.e.",
"$E\\hat{F}E^{-1} = diag(\\phi _1,\\phi _2$ ).",
"Then E also diagonalizes $C$ , as $ diag(\\psi _1^2,\\psi _2^2)$ , and hence any matrix function of $C$ , such as those occurring in $exp(M)$ .",
"Defining $E_2$ as a diagonal $2 \\times 2$ block matrix with $E$ as its diagonal blocks, some manipulation readily leads to $E_2 U(1)=\\left(\\begin{array}{cccc}c_1 & 0 & \\theta s_1/\\psi _1 & 0 \\\\0 & c_2 & 0 & \\theta s_2/\\psi _2 \\\\-\\psi _1 s_1/\\theta & 0 & c_1 & 0 \\\\0 & -\\psi _2 s_2/\\theta &0 & c_2\\end{array}\\right) E_2 U(0)$ where $c_i = cos \\psi _i$ and $s_i = sin \\psi _i$ .",
"A similar equation holds for $U(z)$ at any position $0< z<1$ within the medium, with the arguments of the sines and cosines replaced by $\\psi _i z$ , so the evolution of the perturbations within the medium can also be calculated.",
"This analytic solution can be applied to any quasi-Kerr \"slab\" system, for any boundary conditions, whether CP or SFM, including the unequal intensity case $p \\ne q$ (e.g.",
"$R \\ne 1$ for SFM).",
"It can also be used to calculate probe gain, for example, i.e.",
"for non-zero input perturbations.",
"Here we will only consider equal intensities, for which, as mentioned in the main text, the eigenvectors of $\\hat{F}$ are simply given by $(1,\\pm 1)$ , leading to a simple explicit expression for $E$ : $& & E= \\frac{1}{\\sqrt{2}} \\left(\\begin{array}{cccc}1 & -1 \\\\1 & 1\\end{array}\\right).",
"\\nonumber $ For the case of counterpropagating inputs with the usual boundary conditions $f(0) = g(1) =0$ , (REF ) leads, after some algebra, to the usual CP threshold formula (REF ).",
"Since $E$ is a simple constant matrix independent of any system parameters, one can conveniently consider $EU$ as a change of variables in (REF ), which is effectively the approach of Geddes et al [25].",
"For the $R=1$ feedback mirror, the right side of (REF ) is the same as for the CP problem ($f(0)=0$ ), but the left side needs to express the feedback-phase relationship between $f(1)$ and $g(1)$ .",
"Using the appropriate boundary conditions leads to the threshold expression (REF ) in the main text.",
"For unequal intensities, the CP threshold expression was presented in [24].",
"It leads to an interesting phenomenon whereby the crossings of the two threshold curves $H_1=0$ and $H_2=0$ become anti-crossings, with oscillatory solutions along a line of Hopf bifurcation joining the static threshold curves.",
"Because the SFM problem is not parity-symmetric, no such scenario exists in the $R \\ne 1$ feedback mirror situation, and only quantitative effects on the threshold are expected."
]
] | 1606.04885 |
[
[
"On the Calculation of the Incomplete MGF with Applications to Wireless\n Communications"
],
[
"Abstract The incomplete moment generating function (IMGF) has paramount relevance in communication theory, since it appears in a plethora of scenarios when analyzing the performance of communication systems.",
"We here present a general method for calculating the IMGF of any arbitrary fading distribution.",
"Then, we provide exact closed-form expressions for the IMGF of the very general {\\kappa}-{\\mu} shadowed fading model, which includes the popular {\\kappa}-{\\mu}, {\\eta}-{\\mu}, Rician shadowed and other classical models as particular cases.",
"We illustrate the practical applicability of this result by analyzing several scenarios of interest in wireless communications: (1) Physical layer security in the presence of an eavesdropper, (2) Outage probability analysis with interference and background noise, (3) Channel capacity with side information at the transmitter and the receiver, and (4) Average bit-error rate with adaptive modulation, when the fading on the desired link can be modeled by any of the aforementioned distributions."
],
[
"Introduction",
"The moment generating function (MGF) has played a pivotal role in communication theory for decades, as a tool for evaluating the performance of communication systems in very different scenarios [1], [2], [3], [4].",
"The MGF of the signal-to-noise ratio (SNR) $\\gamma $ , defined as the Laplace transform of the Probability Density Function (PDF) of $\\gamma $ , is well-known for most popular fading distributions [5], [6], [7] and hence enables for a simple characterization of the performance metrics of interest in closed-form.",
"A more general function extensively used in communication theory is the so-called incomplete MGF (IMGF), also referred to as truncated MGF, or interval MGF.",
"This function has an additional degree of freedom by allowing the lower (or equivalently, upper) limit of the integral in the Laplace transform, say $\\zeta $ , to be greater than zero (or equivalently, lower than $\\infty $ ), and also appears when characterizing the performance in a number of scenarios of interest: order statistics [8], [9], [10], [11], symbol and bit error rate calculation with multiantenna reception [12], [13], capacity analysis in fading channels [2], outage probability analysis in cellular systems [14], [15], adaptive scheduling techniques [16], cognitive relay networks [17] or physical layer security [18].",
"Despite its usefulness, to the best of our knowledge the expressions of the IMGF are largely unknown for fading distributions other than the classical ones: Rayleigh, Rice, Nakagami-$m$ and Hoyt.",
"In fact, a general and systematic way to find analytical expressions for the IMGF does not yet exist, thus requiring the use of state-of-the-art numerical techniques for its evaluation [2].",
"Thence, there is a twofold motivation for this paper from a purely communication-theoretic perspective: first, we present a general theory for deriving the IMGF of the SNR for an arbitrary distribution.",
"Specifically, we show that the IMGF of any fading distribution is given in terms of an inverse Laplace transform of a shifted version of the MGF scaled by the Laplace domain variable.",
"This implies that the IMGF should have a similar functional form as the cumulative distribution function (CDF).",
"Secondly, we exemplify the usefulness of this result by deriving a closed-form expression for the IMGF of the $\\kappa $ -$\\mu $ shadowed distribution [7], [19], which includes popular fading distributions such as $\\kappa $ -$\\mu $, $\\eta $ -$\\mu $ [20], [21] or Rician-shadowed [22] as particular cases, and for all of which the IMGF had not been previously reported.",
"In order to illustrate its practical applicability, we introduce several scenarios of interest in wireless communications: As a main application, we focus on a physical-layer security set-up on which two legitimate peers (Alice and Bob) wish to communicate in the presence of an external eavesdropper (Eve).",
"The characterization of the maximum rate at which a secure communication can be attained, i.e.",
"the secrecy capacity $C_S$ , is a classical problem [23], [24] in communication theory.",
"Remarkably, the research on physical-layer security of communication systems operating in the presence of fading has been boosted in the last years ever since the original works in [25], [26].",
"The fact that Eve and Bob observe independent fading realizations adds an additional layer of security to communication compared to the conventional set-up for the Gaussian wiretap channel [24].",
"Hence, a secure communication is feasible even when the average SNR at Bob is lower than the average SNR at Eve.",
"In the literature, there is a great interest on understanding how the consideration of more sophisticated fading models than Rayleigh may impact the secrecy performance attending to different metrics: the outage probability of secrecy capacity (OPSC), the probability of strictly-positive secrecy capacity (SPSC) or the average secrecy capacity (ASC).",
"Specifically, fading models such as Nakagami-$m$ [27], Rician [28], Weibull [29], two-wave with diffuse power [30], Nakagami-$q$ [31] or $\\kappa $ -$\\mu $ [32] have been considered.",
"However, the statistical characterization of the OPSC in a tractable form is often unfeasible due to the involved mathematical derivations.",
"Thus, approximations are usually required for evaluating the OPSC [30], [32], being only possible the calculation of the SPSC in closed-form [28], [29], [32].",
"Very recent works have proposed novel approaches to deriving the secrecy performance metrics in a general way: in [33], a duality between the OPSC and the outage probability analysis in the presence of interference and background noise was presented, which greatly simplifies the analysis of the OPSC calculation for an arbitrary choice of fading distributions for the desired and eavesdropper links.",
"In [34], a unified MGF approach to the analysis of physical-layer security in wireless systems was introduced, allowing for a numerical evaluation of the secrecy performance metrics for arbitrary fading distributions.",
"As we will later see, the practical application of our results allows that the OPSC can be evaluated directly by specializing the IMGF of the SNR at Bob at some specific values.",
"Thus, the OPSC (and hence the SPSC as a special case) can be evaluated in closed-form provided that such IMGF is given in closed-form.",
"This is exemplified for the very general case of the $\\kappa $ -$\\mu $ shadowed distribution [7].",
"We also illustrate how the IMGF can be used to obtain other performance metrics in relevant scenarios in communication theory which, despite looking rather dissimilar at a first glance, they all require for the computation of the IMGF.",
"The first one of these additional scenarios is the outage probability analysis of wireless communications systems affected by interference and background noise.",
"As previously stated, this problem was recently shown to be mathematically equivalent to the OPSC in [33]; thus, the outage probability in this scenario will also be expressed in terms of the IMGF under the same conditions assumed when computing the OPSC.",
"The second additional scenario is related to the analysis of the channel capacity when side information is available at both the transmitter and the receiver sides [35].",
"According to the framework introduced in [2], the capacity in this scenario can be expressed in terms of the IMGF.",
"Finally, we also analyze a classical performance metric in communication theory, which is the average bit-error rate (ABER) with adaptive modulation [36].",
"The ABER in this scenario under arbitrary fading is also given in terms of the IMGF of the fading distribution.",
"The remainder of this paper is structured as follows: in Section , the main mathematical contributions of this paper are presented, within the most relevant is a general way to deriving the IMGF of any arbitrary fading distribution.",
"Closed-form expressions for the IMGF of the $\\kappa $ -$\\mu $ shadowed distribution are also given, as well as for all the special cases included therein ($\\kappa $ -$\\mu $ , $\\eta $ -$\\mu $ , Rician shadowed, Rician, Nakagami-$m$ , Nakagami-$q$ , Rayleigh and one-sided Gaussian).",
"Then, in Section these mathematical results are used to present an IMGF-based approach to the physical-layer security analysis in wireless systems.",
"Section is devoted to illustrate how additional scenarios of interest in wireless communications can also be analyzed by using the IMGF.",
"Numerical results are given in Section , whereas the main conclusions are outlined in Section ."
],
[
"Mathematical Results",
"Definition 1 (Lower IMGF) Let $X$ be a non-negative random variable and $\\zeta $ a non-negative real number, the lower IMGF of $X$ is defined as $ {\\mathcal {M}} _{X}^{l}(s,\\zeta )\\triangleq \\int _{0}^{\\zeta } {e^{sx} f_X \\left( x \\right)dx}.$ Definition 2 (Upper IMGF) Let $X$ be a non-negative random variable and $\\zeta $ a non-negative real number, the upper IMGF of $X$ is defined as $ {\\mathcal {M}} _{X}^{u}(s,\\zeta )\\triangleq \\int _{\\zeta }^\\infty {e^{sx} f_X \\left( x \\right)dx}.$ Obviously, the MGF of $X$ is obtained from these IMGFs as $ {\\mathcal {M}} _{X}(s)= {\\mathcal {M}} _{X}^{l}(s,\\infty )= {\\mathcal {M}} _{X}^{u}(s,0)$ .",
"Both incomplete IMGFs are easily related through the following equation ${ {\\mathcal {M}} }_X^u \\left( {s,\\zeta } \\right) &= { {\\mathcal {M}} }_X \\left( s \\right) - { {\\mathcal {M}} }_X^l \\left( {s,\\zeta } \\right),$ In the following Lemma, we present a general expression for the IMGF of an arbitrarily distributed non-negative random variable.",
"Lemma 1 Let $X$ be a non-negative random variable with MGF $ {\\mathcal {M}} _{\\gamma }(s)$ .",
"Its lower IMGFs can be computed by the inverse Laplace transform of the scaled-shifted MGF, i.e.",
"${ {\\mathcal {M}} }_X^l \\left( {s,\\zeta } \\right) &= {{\\mathcal {L} }}^{ - 1} \\left\\lbrace {\\frac{1}{p}{ {\\mathcal {M}} }_X \\left( {s - p} \\right);p,z=\\zeta } \\right\\rbrace ,$ where ${{\\mathcal {L} }}\\left\\lbrace {h\\left( z \\right);z,p} \\right\\rbrace \\triangleq \\int _0^\\infty {e^{ - pz} h\\left( z \\right)dz}$ represents the Laplace transform from the $z$ -domain to the $p$ -domain, and ${{\\mathcal {L} }}^{ - 1} \\left\\lbrace {H\\left( z \\right);p,z} \\right\\rbrace $ the inverse Laplace transform from the $p$ -domain to the $z$ -domain.",
"Let us consider the upper IMGF in (REF ) as a function of the upper integration limit, i.e.",
"$\\Lambda (z)=\\int _0^{z} {e^{sx} f_X \\left( x \\right)dx}.$ The Laplace transform of $\\Lambda (z)$ in the $p$ -domain can be expressed as ${\\begin{array}{c}{{\\mathcal {L} }}\\left\\lbrace \\Lambda \\left( z \\right);z,p\\right\\rbrace = {{\\mathcal {L} }}\\left\\lbrace {\\int _0^z {e^{sx} f_X \\left( x \\right)dx;z,p} } \\right\\rbrace = \\frac{1}{p}{{\\mathcal {L} }}\\left\\lbrace {e^{sz} f_X \\left( z \\right);z,p} \\right\\rbrace \\hfill \\\\= \\frac{1}{p}{{\\mathcal {L} }}\\left\\lbrace {f_X \\left( x \\right);z,p - s} \\right\\rbrace = \\frac{1}{p}{ {\\mathcal {M}} }_X \\left( {s - p} \\right), \\hfill \\\\\\end{array}}$ where both the definite integral property and the modulation property of the Laplace transform have been applied in order to complete the proof.",
"This Lemma provides a general way of computing the IMGF of any fading distribution in terms of the conventional MGF.",
"Interestingly, (REF ) involves an inverse Laplace transform of a shifted version of the MGF, scaled by the Laplace domain variable $p$ .",
"Thus, it is expectable that the IMGF has a functional form similar to the CDF, since the CDF arises as a particular case of the lower IMGF when evaluated in ${s=0}$ , i.e.",
"$F_X(\\zeta )= {\\mathcal {M}} _{X}^{l}(0,\\zeta ).$ This result also suggests that the IMGF of any distribution for which either the CDF or the MGF are not available in closed-form, will not be likely to have a closed-form expression.",
"Otherwise, we would be finding an expression for such CDF or MGF as a special case.",
"This is the case of some relevant fading distributions in the literature like Durgin and Rappaport's Two-Wave with Diffuse Power (TWDP) fading model [37], or the Beckmann distribution [38], [39].",
"The CDF for these distributions is only available in integral form, but their MGFs have a closed-form expression [5], [4].",
"Another valid example is the Lognormal distribution, for which the CDF has a closed-form expression in terms of the Gaussian $Q$ -function, but its MGF has been largely unknown [40].",
"As consequences of Lemma REF , the IMGF of any non-negative RV can be obtained in a very general form by an inverse Laplace transform operation over the MGF.",
"As we will now show, this Laplace transform can be computed in closed-form for a number of fading distributions of interest.",
"In the following corollaries (and summarized in Table REF ), we derive closed-form expressions for the IMGF of the $\\kappa $ -$\\mu $ shadowed distribution and the special cases included therein, namely the Rician shadowed, $\\kappa $ -$\\mu $ and $\\eta $ -$\\mu $ distributions.",
"All these expressions are new in the literature to the best of our knowledge.",
"Corollary 1 Let $\\gamma $ be a $\\kappa $ -$\\mu $ shadowed random variable with $E\\left\\lbrace \\gamma \\right\\rbrace =\\bar{\\gamma }$ , and non-negative real shape parameters $\\kappa $ , $\\mu $ and $m$ , i.e, ${\\gamma \\sim \\mathcal {S}_{\\kappa \\mu m}(\\gamma ;\\kappa ,\\mu ,m)}$The symbol $\\sim $ reads as statistically distributed as.",
"[7].",
"Then, its lower IMGF is given in the first entry of Table REF , where $\\Phi _2(\\cdot )$ is the bivariate confluent hypergeometric function defined in [41].",
"For the sake of clarity let us write the MGF of the $\\kappa $ -$\\mu $ shadowed distribution as follows ${\\begin{array}{c}{ {\\mathcal {M}} }_{\\gamma } \\left( s \\right) = ( - 1)^\\mu A(s - a)^{m - \\mu } (s - b)^{ - m} \\hfill \\\\\\left\\lbrace {\\begin{array}{c}A = \\frac{{\\mu ^\\mu m^m \\left( {1 + \\kappa } \\right)^\\mu }}{{\\bar{\\gamma }^\\mu \\left( {\\mu \\kappa + m} \\right)^m }} \\hfill \\\\a = \\frac{{\\mu \\left( {1 + \\kappa } \\right)}}{{\\bar{\\gamma }}} \\hfill \\\\b = \\frac{{\\mu \\left( {1 + \\kappa } \\right)}}{{\\bar{\\gamma }}}\\frac{m}{{\\mu \\kappa + m}} \\hfill \\\\\\end{array}} \\right.",
"\\hfill \\\\\\end{array}}.$ In order to apply Lemma REF , the scaled-shifted MGF of $\\gamma $ is first expressed as follows ${\\begin{array}{c}\\frac{1}{p}{ {\\mathcal {M}} }_{\\gamma } \\left( {s - p} \\right) = \\frac{{A( - 1)^\\mu }}{p}(s - p - a)^{m - \\mu } (s - p - b)^{ - m} \\hfill \\\\= \\frac{A}{p}(p - s + a)^{m - \\mu } (p - s + b)^{ - m} \\hfill \\\\= \\frac{A}{{p^{1 + \\mu } }}\\left( {1 - \\left( {\\frac{{s - a}}{p}} \\right)} \\right)^{m - \\mu } \\left( {1 - \\left( {\\frac{{s - b}}{p}} \\right)} \\right)^{ - m} \\hfill \\\\= \\frac{A}{{p^{1 + \\mu } }}\\left( {1 - \\left( {\\frac{{s - a}}{p}} \\right)} \\right)^{ - (\\mu - m)} \\left( {1 - \\left( {\\frac{{s - b}}{p}} \\right)} \\right)^{ - m}.",
"\\hfill \\\\\\end{array}}$ To perform the inverse Laplace transform the following pair can be considered [42] ${\\begin{array}{c}{{\\mathcal {L} }}\\left\\lbrace {x^{\\gamma - 1} \\Phi _2^{\\left( n \\right)} \\left( {\\beta _1 ,...,\\beta _n ;\\gamma ;\\lambda _1 x,...,\\lambda _n x} \\right);x,p} \\right\\rbrace = \\hfill \\\\= \\frac{{\\Gamma \\left( \\gamma \\right)}}{{p^\\gamma }}\\left( {1 - \\frac{{\\lambda _1 }}{p}} \\right)^{ - \\beta _1 } ...\\left( {1 - \\frac{{\\lambda _n }}{p}} \\right)^{ - \\beta _n } ,\\quad \\hfill \\\\\\operatorname{Re} \\left\\lbrace \\gamma \\right\\rbrace > 0,\\operatorname{Re} \\left\\lbrace p \\right\\rbrace > \\max \\left\\lbrace {0,\\operatorname{Re} \\left\\lbrace {\\lambda _1 } \\right\\rbrace ,...,\\operatorname{Re} \\left\\lbrace {\\lambda _n } \\right\\rbrace } \\right\\rbrace .",
"\\hfill \\\\\\end{array}}$ Joining (REF ), (REF ) and Lemma REF the proof is completed.",
"It must be noted that the IMGF here obtained has the same functional form as the CDF given in [7]; thus, evaluating the IMGF has the same complexity as evaluating the CDF.",
"Table: Lower IMGFs for the κ\\kappa -μ\\mu shadowed fading model and particular cases included therein.",
"Upper IMGFs can be obtained using the MGFs given in , , and restated in the table, and then using ().Corollary 2 Let $\\gamma $ be a Rician shadowed random variable with $E\\left\\lbrace \\gamma \\right\\rbrace =\\bar{\\gamma }$ , and non-negative real shape parameters $K$ and $m$ , i.e, ${\\gamma \\sim \\mathcal {S}_{Km}(\\gamma ;K,m)}$ [22].",
"Then, its lower IMGF is given in the second entry of Table REF .",
"Specializing the results for the $\\kappa $ -$\\mu $ shadowed distribution for $\\mu =1$ , the IMGF for the Rician-shadowed case is obtained with $K=\\kappa $ .",
"Again, the IMGF of the Rician Shadowed distribution has the same functional form as its CDF [43].",
"We must note that for the case of $m\\in \\mathbb {Z}$ , we have that $\\Phi _2(1-m,m;2;\\cdot ,\\cdot )$ function reduces to a finite sum of Laguerre polynomials [43]; similarly, for $m$ being a positive half integer, then $\\Phi _2(1-m,m;2;\\cdot ,\\cdot )$ function reduces to a finite sum of Kummer hypergeometric functions, modified Bessel functions and Marcum-$Q$ functions [43].",
"Corollary 3 Let $\\gamma $ be a $\\kappa $ -$\\mu $ random variable with ${E\\left\\lbrace \\gamma \\right\\rbrace =\\bar{\\gamma }}$ and non-negative real shape parameters $\\kappa $ and $\\mu $ , i.e, ${\\gamma \\sim \\mathcal {S}_{\\kappa \\mu }(\\gamma ;\\kappa ,\\mu )}$ [20].",
"Then, its lower IMGF is given in the third entry of Table REF , where $Q_{\\mu }$ is the generalized Marcum $Q$ function of $\\mu $ -th order [5].",
"Using the MGF of a $\\kappa $ -$\\mu $ distributed RV $\\gamma $ in [6], and according to Lemma REF , the lower IMGF can be expressed as $&{ {\\mathcal {M}} }_{\\gamma }^l \\left( {s,z} \\right) = {\\mathcal {L} }^{-1}\\left\\lbrace \\tfrac{1}{p} {\\mathcal {M}} _{\\gamma }(s-p);p,z\\right\\rbrace \\nonumber \\\\&={{\\mathcal {L} }^{ - 1}}\\left\\lbrace {\\tfrac{1}{p}\\tfrac{{{\\mu ^\\mu }{{\\left( {1 + \\kappa }\\right)}^\\mu }}}{{{{\\left( {\\mu \\left( {1 + \\kappa } \\right) - \\bar{\\gamma }s + \\bar{\\gamma }p} \\right)}^\\mu }}}\\exp \\left( {\\tfrac{{\\mu \\kappa \\bar{\\gamma }s - \\mu \\kappa \\bar{\\gamma }p}}{{\\mu \\left( {1 + \\kappa } \\right) -\\bar{\\gamma }s + \\bar{\\gamma }p}}} \\right);p,z} \\right\\rbrace ,$ After some algebra, the terms in (REF ) can be conveniently rearranged so the inverse Laplace transform has the following form: $\\begin{array}{l}{ {\\mathcal {M}} }_{\\gamma }^l \\left( {s,z} \\right) = \\exp \\left( { - \\mu \\kappa } \\right){\\left( {\\frac{{\\mu \\left( {1 + \\kappa } \\right)}}{{\\bar{\\gamma }}}} \\right)^\\mu }\\exp \\left( {\\left( {s -\\frac{{\\mu \\left( {1 + \\kappa } \\right)}}{{\\bar{\\gamma }}}} \\right)z}\\right) \\times \\\\{{\\mathcal {L} }^{ - 1}}\\left\\lbrace {\\frac{1}{{{p^{\\mu + 1}}}}\\frac{1}{{1 - \\left({\\frac{{\\mu \\left( {1 + \\kappa } \\right)}}{{\\bar{\\gamma }}} - s}\\right)/p}}\\exp \\left( {\\frac{1}{p}{{\\frac{{{\\mu ^2}\\kappa \\left( {1 + \\kappa }\\right)}}{{\\bar{\\gamma }}}}}} \\right);p,z} \\right\\rbrace ,\\end{array}$ This inverse Laplace transform can be calculated in terms of the bivariate confluent hypergeometric function $\\Phi _3(\\cdot )$ defined in [41] by using the transform pair in [44].",
"This yields $\\begin{array}{l} { {\\mathcal {M}} }_{\\gamma }^l \\left( {s,z} \\right) = \\frac{\\mu ^{\\mu }(1+\\kappa )^{\\mu }z^{\\mu }}{\\Gamma (\\mu +1)\\bar{\\gamma }^{\\mu }\\exp {(\\kappa \\mu )}}\\exp \\left( {\\left( {s -\\frac{{\\mu \\left( {1 + \\kappa } \\right)}}{{\\bar{\\gamma }}}} \\right)z}\\right) \\\\ \\times {\\Phi _3}\\left({1,\\mu + 1;\\left( {\\frac{{\\mu \\left( {1 + \\kappa } \\right)}}{{\\bar{\\gamma }}} - s} \\right)z,\\frac{{{\\mu ^2}\\kappa \\left( {1 + \\kappa } \\right)}}{{\\bar{\\gamma }}}z} \\right),\\end{array}$ Finally, we make use of the existing connection between the $\\Phi _3(1,\\mu +1;a;b)$ function and the generalized Marcum $Q$ -function, which holds for $\\mu \\in \\mathbb {R}$ .",
"Adapting [45] to the specific case here addressed, we have $\\frac{\\Phi _3\\left(1,\\mu +1;a,b\\right)}{\\Gamma (\\mu +1)}=\\exp \\left(a+\\tfrac{b}{a}\\right)a^{-\\mu }\\left[1-Q_{\\mu }\\left(\\sqrt{2\\tfrac{b}{a}},\\sqrt{2a}\\right)\\right].$ Combining this equation with (REF ) yields the final expression given in the third entry of Table REF.",
"This completes the proof.",
"Note that the IMGF is given in terms of the well-known Marcum $Q$ -function, just like the CDF of the $\\kappa $ -$\\mu $ distribution originally derived by Yacoub [20].",
"Corollary 4 Let $\\gamma $ be a $\\eta $ -$\\mu $ random variable with ${E\\left\\lbrace \\gamma \\right\\rbrace =\\bar{\\gamma }}$ and non-negative real shape parameters $\\eta $ and $\\mu $ , i.e, ${\\gamma \\sim \\mathcal {S}_{\\eta \\mu }(\\gamma ;\\eta ,\\mu )}$ [20].",
"Then, its lower IMGF is given in the fourth entry of Table REF .",
"Leveraging the recent connection between the $\\kappa $ -$\\mu $ shadowed distribution and the $\\eta $ -$\\mu $ distribution [21], the IMGF of the $\\eta $ -$\\mu $ power envelope in format 1 is obtainedAccording to the original definition in [20], this format implies that $\\eta \\in [0,\\infty )$ by setting the parameters of the $\\kappa $ -$\\underline{\\mu }$ shadowed distribution to $\\underline{\\mu }= 2\\mu $ , $\\kappa =\\frac{1-\\eta }{2\\eta }$ and $m=\\mu $ .",
"This expression is coincident with the one obtained in [45], and also has the same complexity as the original CDF for the $\\eta $ -$\\mu $ distribution derived in [46] ."
],
[
"Single-antenna scenario",
"Aided by the previous mathematical results, we will now show how the IMGF can be used to directly analyze the performance in wireless scenarios.",
"Specifically, we aim to determine the physical layer security of a wireless link between two legitimate peers (Alice and Bob) in the presence of an external eavesdropper (Eve) [25], [26].",
"We first consider that Bob, Eve and Alice are equipped with single-antenna devices; the inclusion of multiple antennas at Bob or Eve will be later discussed in this section.",
"Depending on the characteristics of the propagation environment, random fluctuations affecting the desired and wiretap links need to be modeled with a specific distribution.",
"The performance in this scenario can be characterized by the secrecy capacity $C_S$ , defined as $C_S\\triangleq C_b-C_e = \\log _2\\left({\\frac{1+\\gamma _b}{1+\\gamma _e}}\\right)>0,$ where $\\gamma _b$ and $\\gamma _e$ are the instantaneous SNRs at Bob and Eve, respectively, and $C_b$ and $C_e$ denote the capacities of the communication links between Alice and Bob, and between Alice and Eve, respectively.",
"In many cases, Eve's channel state information (CSI) is not available at Alice and hence information-theoretic security cannot be guaranteed.",
"This may be the case on which Eve is a passive eavesdropper, and hence Alice has no way to have access to Eve's CSI.",
"Conversely, perfect knowledge of Bob's CSI by Alice can be assumed.",
"Thus, Alice selects a constant secrecy rate $R_S$ for transmission; in this situation, the outage probability of the secrecy capacity (OPSC) gives the probability that communication at a certain secrecy rate ${R_S > 0}$ cannot be securely attained.",
"This metric is computed as follows $\\mathcal {P}_{R_S}\\triangleq \\Pr \\left\\lbrace {C_S \\leqslant R_S } \\right\\rbrace &=\\Pr \\left\\lbrace {\\gamma _b \\leqslant \\left(2^{R_S}-1\\right)\\left(\\frac{2^{R_S}}{2^{R_S}-1}\\gamma _e+1\\right) } \\right\\rbrace \\\\&= 1 - \\Pr \\left\\lbrace {C_S > R_S } \\right\\rbrace .$ The probability of strictly positive secrecy capacity can be obtained as a particular case of (REF ) by setting ${R_S=0}$ .",
"Finally, another secrecy performance metric of interest is the $\\epsilon $ -outage secrecy capacity $C_{\\epsilon }$ .",
"This is defined as the largest secrecy rate $R_S$ for which the OPSC satisfies ($\\mathcal {P}_{R_S}{\\le \\epsilon }$ ), with $0\\le \\epsilon \\le 1$ .",
"For a certain $\\epsilon $ , this metric is computed as $C_{\\epsilon }\\triangleq \\underset{R_S:\\mathcal {P}_{R_S}\\le \\epsilon }{\\sup } \\left\\lbrace R_S\\right\\rbrace ,$ We will first assume that the fading experienced by the eavesdropper can be modeled by the $\\kappa $ -$\\mu $ shadowed distribution, i.e.",
"${\\gamma _e\\sim \\mathcal {S}_{\\kappa \\mu m}(\\gamma _e;\\kappa ,\\mu ,m)}$ .",
"This distribution [7] is well suited to model both line-of sight (LOS) and non-LOS (NLOS) scenarios, and also includes most popular fading distributions in the literature as special cases.",
"We will also consider that the fading severity parameters $\\mu $ and $m$ take integer valuesThis can be justified as follows: the parameter $\\mu $ in the $\\kappa $ -$\\mu $ distribution introduced by Yacoub [20] was defined as the number of clusters of multipath waves propagating in a certain environment; thus, according to this definition the consideration of integer $\\mu $ is related to the physical model for the $\\kappa $ -$\\mu $ distribution.",
"Equivalently, the restriction of $m$ to take integer values does not have a major impact unless the LOS component is affected by heavy shadowing (i.e.",
"very low values of $m$ ).",
"In practice, this restriction has a negligible effect, and specially when the $\\kappa $ -$\\mu $ shadowed distribution is used to approximate the $\\kappa $ -$\\mu $ distribution in a more tractable form [47]., which allows for a simpler mathematical tractability.",
"With this only restriction, and for any arbitrary fading distribution at the legitimate link between Alice and Bob, the OPSC can be computed using the following Lemma: Lemma 2 Let us consider the communication between two legitimate peers A and B in the presence of an external eavesdropper E. Let $\\gamma _b$ and $\\gamma _e$ be the instantaneous SNRs at B and E, respectively, and $\\bar{\\gamma }_b$ and $\\bar{\\gamma }_e$ the average SNRs at B and E, respectively.",
"If ${\\gamma _e\\sim \\mathcal {S}_{\\kappa \\mu m}(\\gamma _e;\\kappa _e,\\mu _e,m_e)}$ with $\\lbrace \\mu ,m\\rbrace \\in \\mathbb {Z}^+$ , then for any finite rate $R_S\\ge 0$ the OPSC can be expressed in terms of the IMGF of $\\gamma _b$ as $\\mathcal {P}_{R_S}&=\\sum _{i=0}^{M}C_i e^{\\alpha \\beta _i}\\sum _{r=0}^{m_i-1}\\sum _{k=0}^{r} \\tfrac{(-\\alpha \\beta _i)^{r}}{k!(r-k)!}",
"\\tfrac{\\partial ^k {\\mathcal {M}} _{\\gamma _b}^{u}\\left( s,z\\right)}{\\partial s^k}\\bigg {|}_{\\stackrel{s=-\\beta _i}{z=\\alpha }}+ {\\mathcal {M}} _{\\gamma _b}^{l}(0,2^{R_S}-1)\\nonumber \\\\&=\\sum _{i=0}^{M}C_i e^{\\alpha \\beta _i}\\sum _{r=0}^{m_i-1}\\sum _{k=0}^{r} \\tfrac{(-\\alpha \\beta _i)^{r}}{k!(r-k)!}",
"\\tfrac{\\partial ^k {\\mathcal {M}} _{\\gamma _b}^{u}\\left( s,z\\right)}{\\partial s^k}\\bigg {|}_{\\stackrel{s=-\\beta _i}{z=\\alpha }}+F_{\\gamma _b}\\left(\\alpha \\right).$ where $F_{\\gamma _b}(\\cdot )$ is the CDF of $\\gamma _b$ , $\\alpha =2^{R_S}-1$ , $\\beta _i=1/(2^{R_S}\\Omega _i)$ , and the parameters $M$ , $m_i$ , $\\Omega _i$ and $C_i$ are related to $\\kappa $ , $\\mu $ , $m$ and $\\bar{\\gamma }_e$ as described in Table REF in Appendix REF .",
"See Appendix .",
"Expression (REF ) yields the OPSC in any scenario on which the eavesdropper's fading channel can be modeled with the $\\kappa $ -$\\mu $ shadowed distribution, for any arbitrary choice of the fading distribution for the legitimate channel.",
"Thence, we can use the results in Table REF combined with (REF ) to derive analytical expressions for the secrecy performance in those scenarios on which the fading at the desired link can be modeled by the general $\\kappa $ -$\\mu $ shadowed distribution, or any of the particular cases included therein.",
"Note that there is no need to restrict the parameters $\\mu $ and $m$ of the legitimate channel to take integer values, as the IMGF derived in Table REF holds for any $\\lbrace \\mu ,m\\rbrace \\in \\mathbb {R}$ .",
"In some scenarios, Eve may only have access to signals arriving from NLOS paths [48], [49].",
"Under this premise, we can assume that the eavesdropper link can be modeled by the Rayleigh distribution.",
"This leads $\\gamma _e$ to be exponentially distributed with average SNR $\\bar{\\gamma }_e$ , i.e.",
"$\\gamma _e\\sim \\text{Exp}{(\\bar{\\gamma }_e)}$ .",
"Thus, the following corollary arises as a special case of Lemma 2.",
"Corollary 5 Let us consider the communication between two legitimate peers A and B in the presence of an external eavesdropper E. Let $\\gamma _b$ and $\\gamma _e$ be the instantaneous SNRs at B and E, respectively, and $\\bar{\\gamma }_b$ and $\\bar{\\gamma }_e$ the average SNRs at B and E, respectively.",
"If ${\\gamma _e\\sim \\text{Exp}{(\\bar{\\gamma }_e)}}$ , then for any finite rate $R_S\\ge 0$ the OPSC can be expressed in terms of the IMGF of $\\gamma _b$ as $\\mathcal {P}_{R_S}=& {\\mathcal {M}} _{\\gamma _b}^{l}(0,2^{R_S}-1)+e^{\\frac{2^{R_S}-1}{2^{R_S}\\bar{\\gamma }_e}} {\\mathcal {M}} _{\\gamma _b}^{u}\\left(-\\frac{1}{2^{R_S}\\bar{\\gamma }_e},2^{R_S}-1\\right)\\nonumber \\\\=&F_{\\gamma _b}\\left(2^{R_S}-1\\right)+e^{\\frac{2^{R_S}-1}{2^{R_S}\\bar{\\gamma }_e}} {\\mathcal {M}} _{\\gamma _b}^{u}\\left(-\\frac{1}{2^{R_S}\\bar{\\gamma }_e},2^{R_S}-1\\right),$ where $F_{\\gamma _b}(\\cdot )$ is the CDF of $\\gamma _b$ .",
"Following the same derivation in Appendix and setting $\\kappa =0$ and $\\mu =1$ yields the desired result.",
"Note that these results provide a systematic way to derive the OPSC for any arbitrary fading distribution in the legitimate link, provided that the IMGF of the SNR at Bob is known.",
"We must also note the OPSC for $\\bar{\\gamma }_b\\gg \\bar{\\gamma }_e$ does not depend on the distribution of $\\gamma _e$ , but only on the distribution of $\\gamma _b$ and the average SNR at Eve $\\bar{\\gamma }_e$ [31].",
"Thus, the consideration of Rayleigh fading for the eavesdropper link as performed in Corollary 5 has a negligible effect in practice, while simplifying the analysis.",
"If now assuming Rayleigh fading also for the legitimate channel, the OPSC expression in (REF ) reduces to the one originally calculated in [25].",
"This can be checked by setting $\\kappa =0$ and $\\mu =1$ in the IMGF of the $\\kappa $ -$\\mu $ distribution in the third entry of Table REF .",
"Using the equivalence $Q_1(0,\\sqrt{2x})=e^{-x}$ given in [5], and after some manipulations we obtain $\\mathcal {P}_{R_S}=1-e^{-\\frac{2^{R_S}-1}{\\bar{\\gamma }_b}}\\frac{\\bar{\\gamma }_b}{\\bar{\\gamma }_b+2^{R_S}\\bar{\\gamma }_e}.$"
],
[
"Extension to the multi-antenna scenario",
"In the previous analysis, we have explicitly assumed that all the agents in the system are equipped with single-antenna devices.",
"We here show that the extension for the case on which Eve and Bob are multi-antenna devices can be straightforwardly carried out.",
"First, note that in the derivation carried out in Appendix in order to prove Lemma 2, there is neither any restriction related to the number of antennas used by Bob, nor related to the combining strategy carried out.",
"Hence, Lemma 2 can be applied as is for a multi-antenna configuration at Bob.",
"In this case, the only requirement to derive the OPSC is to determine the IMGF of the SNR after combining.",
"For instance, if maximal ratio combining (MRC) is used by Bob we have that $\\gamma _b=\\sum _{i=1}^{N_B}{\\gamma _b}_i$ and $ {\\mathcal {M}} _{\\gamma _b}^{u}\\left(s,z\\right)=\\prod _{i=1}^{N_B} {\\mathcal {M}} _{{\\gamma _b}_i}^{u}\\left(s_i,z_i\\right),$ where $N_B$ is the number of receive antennas at Bob, ${{\\gamma _b}_i}$ denote the per-branch instantaneous SNRs, and independence between receive branches has been considered.",
"When assuming multiple antennas at Eve with i.i.d.",
"branches under $\\kappa $ -$\\mu $ shadowed fading, the extension is also straightforward when considering that Eve performs MRC reception in order to maximize the receive SNR; since the sum of $N_E$ i.i.d.",
"$\\kappa $ -$\\mu $ shadowed random variables is also $\\kappa $ -$\\mu $ shadowed-distributed with $\\mu _{eq}=N_E\\cdot {\\mu }$ , $m_{eq}=N_E\\cdot {m}$ and $\\gamma _e=N_E\\cdot {\\gamma _e}$ , the OPSC in the multiantenna scenario can be expressed as in Lemma 2."
],
[
"Outage probability analysis with interference and background noise",
"The performance characterization of wireless communication systems in the presence of interference is a very important problem in communication theory, ever since the advent of digital cellular systems, whose performance is known to be limited by the interference received from nearby cells.",
"Let us denote as $\\gamma _d$ the instantaneous SNR at the intended receiver, and let us denote as $\\gamma _i$ the aggregate instantaneous interference-to-noise ratio corresponding to the set of interfering signals affecting the receiver.",
"The outage probability in this scenario, defined as the probability that the signal-to-noise plus interference ratio is below a given threshold $\\gamma _{\\text{th}}$ , can be calculated as $\\text{OP}_{\\text{NI}}=\\Pr \\left\\lbrace \\gamma _d \\leqslant \\gamma _{\\text{th}}\\left(\\gamma _i+1\\right) \\right\\rbrace ,$ As pointed out in [33], expressions (REF ) and (REF ) are formally equivalent, up to some scaling of the random variables and setting $\\gamma _{\\text{th}}=2^{R_S}-1$ .",
"This means that both $\\text{OP}_{\\text{NI}}$ and $\\mathcal {P}_{R_S}$ will have the same functional form for the same choice of distributions.",
"The derivation of the outage probability of communication systems in the presence of interference and background noise in a $\\kappa $ -$\\mu $ shadowed/$\\kappa $ -$\\mu $ shadowed scenario, which is an open problem in the literature, is directly obtained by using the IMGF given in the first entry of Table I, under the same conditions assumed in Section .",
"Recent results in the literature arise as special cases [50]."
],
[
"Channel capacity with side information at the transmitter and the receiver",
"Let us now consider the scenario on which a transmitter, subject to an average transmit power constraint, communicates with a receiver through a fading channel.",
"Assuming perfect channel knowledge at both the transmitter and receiver sides, the transmitter can optimally adapt its power and rate.",
"The Shannon capacity in this scenario is known to be given by the following expression [35], $C= \\int _{0}^{\\infty }\\log \\left(\\frac{\\gamma }{\\gamma _0}\\right)f_{\\gamma }(\\gamma )d\\gamma ,$ where a normalized bandwidth $B=1$ was assumed for simplicity.",
"In (REF ), $\\gamma $ is the instantaneous SNR and $\\gamma _0$ is a cut-off SNR determined by the average power constraint.",
"An alternative expression for this capacity was proposed in [2] in terms of the $E_i$ -transform, which makes use of the exponential integral function $E_i(\\cdot )$ as integration kernel, yielding $C= \\frac{1}{\\log (2)} \\int _{0}^{\\infty }E_i(-x)\\exp (x)\\Psi (x,\\gamma _0)dx,$ where the ancillary function $\\Psi (x,\\gamma _0)$ is defined as $\\Psi (x,\\gamma _0)\\triangleq M_{\\gamma }^{u}\\left(\\frac{x}{\\gamma _0},x\\right) + \\frac{1}{\\gamma _0} \\frac{\\partial {\\mathcal {M}} _{\\gamma }^{u}\\left( s,z\\right)}{\\partial s}\\bigg {|}_{\\stackrel{s=\\frac{x}{\\gamma _0}}{z=x}}.$ Thus, equation (REF ) provides an alternative way of computing the capacity in this scenario for an arbitrary distribution of the SNR, in terms of the IMGF and its first derivative.",
"Using the expressions for the IMGF derived in Table I, capacity results are obtained for the $\\kappa $ -$\\mu $ shadowed fading channels, and all the special cases included therein.",
"These results are also new in the literature."
],
[
"Average bit-error rate with adaptive modulation",
"Adaptive modulation makes use of channel knowledge at the transmitter side in order to optimally design system parameters such as constellation size, transmit power, coding rates and schemes, and many others [36].",
"One extended alternative is the design of the constellation size and power in order to maximize the average throughput, for a certain instantaneous bit-error rate (BER) constraint.",
"In this scenario, the average BER $\\bar{P}_b$ of adaptive modulation with $M$ -QAM is well approximated using [36] and [36], as $\\bar{P}_b \\approx \\frac{{\\sum \\limits _{j = 1}^{N - 1} {k_j \\int _{\\gamma _{j - 1} }^{\\gamma _j } {0.2\\exp \\left( { - 1.5\\frac{{\\gamma }}{{2^{k_j } - 1}}} \\right)f_{\\gamma }\\left( \\gamma \\right)d\\gamma } } }}{{\\sum \\limits _{j = 1}^{N - 1} {k_j \\int _{\\gamma _{j - 1} }^{\\gamma _j } {f_{\\gamma }\\left( \\gamma \\right)d\\gamma } } }},$ where $\\gamma $ represents the instantaneous SNR, $\\bar{\\gamma }$ is the average SNR, $N$ is the number of fading regions, $\\lbrace \\gamma _j\\rbrace $ are the SNR switching thresholds and $k_j$ is the number of bits per complex symbol employed when $\\gamma _{j-1}\\le \\gamma <\\gamma _j$ .",
"Note that the denominator in (REF ) represents the exact average spectral efficiency and, for convenience, $\\gamma _{N-1}\\triangleq \\infty $ .",
"Using the IMGF of $\\gamma $ , the following closed-form expression is obtained as $\\bar{P}_b &\\approx \\frac{{0.2\\sum \\limits _{j = 1}^{N - 1} {k_j \\left\\lbrace { {\\mathcal {M}} _{\\gamma }^{l}(- \\frac{{1.5}}{{2^{k_j } - 1}},\\gamma _{j - 1}) - {\\mathcal {M}} _{\\gamma }^{l}(- \\frac{{1.5}}{{2^{k_j } - 1}},\\gamma _{j})} \\right\\rbrace } }}{{\\sum \\limits _{j = 1}^{N - 1} {k_j \\left\\lbrace { {\\mathcal {M}} _{\\gamma }^{l}(0,\\gamma _{j - 1}) - {\\mathcal {M}} _{\\gamma }^{l}(0,\\gamma _{j})} \\right\\rbrace } }}\\nonumber \\\\&\\approx \\frac{{0.2\\sum \\limits _{j = 1}^{N - 1} {k_j \\left\\lbrace { {\\mathcal {M}} _{\\gamma }^{l}(- \\frac{{1.5}}{{2^{k_j } - 1}},\\gamma _{j - 1}) - {\\mathcal {M}} _{\\gamma }^{l}(- \\frac{{1.5}}{{2^{k_j } - 1}},\\gamma _{j})} \\right\\rbrace } }}{{\\sum \\limits _{j = 1}^{N - 1} {k_j \\left\\lbrace {F_{\\gamma }(\\gamma _{j - 1}) - F_{\\gamma }(\\gamma _{j})} \\right\\rbrace } }}.$ Therefore, the average BER in this scenario for any fading distribution can be easily obtained by evaluating a finite number of terms involving the IMGF.",
"More specifically, closed-form results new in the literature can be obtained for the case of the $\\kappa $ -$\\mu $ shadowed fading distribution and special cases."
],
[
"Numerical Results",
"In this section we provide numerical results for some of the practical scenarios previously analyzed.",
"Specifically, we focus on the outage probability of the secrecy capacity studied in Section under different fading scenarios.",
"We assume that Bob and Eve are only equipped with one antenna, and for the eavesdropper's channel we set $\\kappa =0$ and $\\mu =1$ .",
"The effect of system parameters on the $\\epsilon $ -outage secrecy capacity is also investigated.",
"All the results shown here have been analytically obtained by the direct evaluation of the expressions developed in this paper: Additionally, Monte Carlo simulations have been performed to validate the derived expressions, and are also presented in all figures, showing an excellent agreement with the analytical results.",
"Details on how to compute the confluent bivariate function $\\Phi _2$ are given in [7].",
"In Figs.",
"REF -REF , the OPSC is represented considering different fading models as a function of the average SNR at Bob $\\bar{\\gamma }_b$ , for different sets of values of the fading parameters.",
"We assume in these figures that the normalized rate threshold value used to declare an outage is $R_S = 0.1$ , and an average SNR at Eve $\\bar{\\gamma }_e$ = 15 dB.",
"Figs.",
"REF and REF , show results for the $\\kappa $ -$\\mu $ shadowed fading considering, respectively, small ($\\kappa =1.5$ ) and large ($\\kappa =10$ ) LOS components in the received wave clusters for different values of the $\\mu $ parameter and also considering light ($m=12$ ) or heavy ($m=0.5$ ) shadowing for the LOS components.",
"As expected, as the fading parameter $\\mu $ increases, the diversity gain increases too, resulting in a higher slope of the curves in the high SNR regime, and with diminishing returns as $\\mu $ increases.",
"Note that $\\mu $ represents the number of received wave clusters when it takes an integer value.",
"It can also be observed that the performance is always better when the LOS components are lightly shadowed, and this improvement is much more noticeable for large LOS components.",
"Figure: Outage probability of secrecy capacity under κ\\kappa -μ\\mu shadowed fading as a function of γ ¯ b \\bar{\\gamma }_b, for different values of mm and μ\\mu .",
"Parameter values: κ=1.5\\kappa =1.5, γ ¯ e =15\\bar{\\gamma }_e=15 dB and R S =0.1R_S=0.1.Figure: Outage probability of secrecy capacity under κ\\kappa -μ\\mu shadowed fading as a function of γ ¯ b \\bar{\\gamma }_b, for different values of mm and μ\\mu .",
"Parameter values: κ=10\\kappa =10, γ ¯ e =15\\bar{\\gamma }_e=15 dB and R S =0.1R_S=0.1.The impact of shadowed LOS components on performance can be observed in Fig.",
"REF , where the outage probability of the secrecy capacity under Rician shadowed fading is presented for different values of the $m$ and $K$ parameters.",
"It can be observed that it is more beneficial for the performance to have small LOS components ($K=1.5$ ) if they are affected by heavy shadowing.",
"Conversely, if the shadowing is mild, large LOS components always yield a lower outage probability.",
"This appreciation can be confirmed by observing the results in Fig.",
"REF , which depicts the outage probability under $\\kappa $ -$\\mu $ fading, i.e., when the LOS components do not experience any shadowing.",
"Figure: Outage probability of secrecy capacity under Rician shadowed fading as a function of γ ¯ b \\bar{\\gamma }_b, for different values of KK and mm.",
"Parameter values: γ ¯ e =15\\bar{\\gamma }_e=15 dB and R S =0.1R_S=0.1.Figure: Outage probability of secrecy capacity under κ\\kappa -μ\\mu fading as a function of γ ¯ b \\bar{\\gamma }_b, for different values of κ\\kappa and μ\\mu .",
"Parameter values: γ ¯ e =15\\bar{\\gamma }_e=15 dB and R S =0.1R_S=0.1.Fig.",
"REF presents results for the $\\eta $ -$\\mu $ fading, i.e.. in a NLOS scenario on which the in-phase and quadrature components of the scattered waves are not necessarily equally distributed.",
"We consider format 1 of this distribution, for which $\\eta $ represents the scattered-wave power ratio between the in-phase and quadrature components of each cluster of multipath, and the number of multipath clusters, when $\\mu $ is a semi-integer, is represented by $2\\mu $ .",
"It can be observed that, when the in-phase and quadrature components are highly imbalanced ($\\eta =0.04$ ), the performance is poorer.",
"On the other hand, increasing the number of multipath clusters have always a beneficial impact on performance, as the instantaneous received signal power is smoothed.",
"Figure: Outage probability of secrecy capacity under η\\eta -μ\\mu fading as a function of γ ¯ b \\bar{\\gamma }_b, for different values of η\\eta and μ\\mu .",
"Parameter values: γ ¯ e =15\\bar{\\gamma }_e=15 dB and R S =0.1R_S=0.1.Fig.",
"REF shows the normalized $\\epsilon $ -outage secrecy capacity $C_{\\epsilon }$ under $\\kappa $ -$\\mu $ shadowed fading as a function of $\\bar{\\gamma }_b$ , where the capacity normalization has been computed with respect to the capacity of an AWGN channel with SNR equal to $\\bar{\\gamma }_b$ .",
"We have assumed $m=2$ and an average SNR at Eve of $\\bar{\\gamma }_e=-10$ dB.",
"The corresponding results for $\\eta $ -$\\mu $ fading are presented in Fig.",
"REF , also for $\\bar{\\gamma }_e=-10$ dB.",
"In both figures it can be observed that when the outage probability is set to a high value ($\\epsilon =0.8$ ), better channel conditions (i.e.",
"$\\mu =6$ , $\\kappa =10$ for $\\kappa $ -$\\mu $ shadowed fading and $\\mu =4$ , $\\eta =0.9$ for $\\eta $ -$\\mu $ fading) yield a lower $\\epsilon $ -outage capacity.",
"Conversely, better channel conditions results in a higher capacity for lower values of $\\epsilon $ .",
"Further insight can be obtained from Fig.",
"REF , where it is shown the normalized $\\epsilon $ -outage secrecy capacity $C_{\\epsilon }$ under $\\kappa $ -$\\mu $ fading as a function of the outage probability $\\epsilon $ and for different average SNRs at Eve, assuming an average SNR of 10 dB at the desired link.",
"We observe that higher outage probability $\\epsilon $ leads to higher $C_{\\epsilon }$ , having an important influence the average SNR of the eavesdropper's channel.",
"It can also be observed that, as the channel conditions improve, the normalized $\\epsilon $ -outage secrecy capacity tends to one for all values of the outage probability for low values of $\\bar{\\gamma }_e$ .",
"Figure: Normalized ϵ\\epsilon -outage secrecy capacity C ϵ C_{\\epsilon } under κ\\kappa -μ\\mu shadowed fading as a function of γ ¯ b \\bar{\\gamma }_b.",
"Parameter values; m=2m=2, γ ¯ e =-10\\bar{\\gamma }_e=-10 dB.Figure: Normalized ϵ\\epsilon -outage secrecy capacity C ϵ C_{\\epsilon } under η\\eta -μ\\mu fading as a function of γ ¯ b \\bar{\\gamma }_b.",
"Parameter value γ ¯ e =-10\\bar{\\gamma }_e=-10 dB.Figure: Normalized ϵ\\epsilon -outage secrecy capacity C ϵ C_{\\epsilon } under κ\\kappa -μ\\mu fading as a function of ϵ\\epsilon .",
"Parameter value γ b =10\\gamma _b=10 dB.Note that wireless communication over fading channels does not require necessarily the average SNR of the channel between Alice and Bob to be greater than the average SNR of the channel between Alice and Eve, since there is certain probability that the instantaneous SNR of the main channel being higher than the instantaneous SNR of the eavesdropper's channel ($\\gamma _b>\\gamma _e$ ) even when $\\bar{\\gamma }_b<\\bar{\\gamma }_e$ .",
"In fact, there is a trade off between the outage probability of the secrecy capacity ${ \\Pr \\left\\lbrace C_S\\le R_S\\right\\rbrace }$ and the $\\epsilon $ -outage secrecy capacity $C_{\\epsilon }$ , where a higher $C_{\\epsilon }$ corresponds to a higher outage probability $\\epsilon $ , and viceversa.",
"For that reason, results in Figs.",
"REF , REF and REF show that the normalized outage secrecy capacity may have non-zero values even when $\\bar{\\gamma }_b \\le \\bar{\\gamma }_e$ (for high values of $\\epsilon $ )."
],
[
"Conclusions",
"A fundamental connection between the incomplete MGF of a positive random variable and its complete MGF has been presented.",
"The main takeaway is that the IMGF is expected to have a similar functional form as the CDF, and hence its evaluation should not require any additional complexity.",
"Using this novel connection, closed-form expressions for the IMGF of the $\\kappa $ -$\\mu $ shadowed distribution (and all the special cases included therein) have been derived for the first time in the literature.",
"This has enabled us to introduce a new framework for the analysis of the physical layer security in scenarios on which the desired link is affected by any arbitrary fading distribution, and the eavesdropper's link undergoes $\\kappa $ -$\\mu $ shadowed fading.",
"We hope that the results in this paper may facilitate the performance evaluation for more practical setups related to physical layer security, such as those using artificial noise transmission or the collaboration of a friendly jammer, and, in general for all the situations and scenarios on which the IMGF makes appearances."
],
[
"Acknowledgment",
"This work has been funded by the Consejería de Economía, Innovación, Ciencia y Empleo of the Junta de Andalucía, the Spanish Government and the European Fund for Regional Development FEDER (projects P2011-TIC-7109, P2011-TIC-8238, TEC2013-42711-R, TEC2013-44442-P and TEC2014-57901-R)."
],
[
"Proof of Lemma 2",
"From the definition of OPSC in (REF ), the probability of achieving a successful secure communication is given by $& \\Pr \\left\\lbrace C_S>R_S\\right\\rbrace = \\Pr \\left\\lbrace \\gamma _e<\\frac{1}{2^{R_S}}(1+\\gamma _b)-1\\right\\rbrace .$ Note that $\\gamma _e$ only takes non-negative values; hence, for this condition to occur in (REF ), then the inequality ${\\gamma _b \\ge 2^{R_S}-1}$ must be satisfied, i.e.",
"$\\Pr \\left\\lbrace \\gamma _e<\\frac{1}{2^{R_S}}(1+\\gamma _b)-1 | \\gamma _b < 2^{R_S}-1\\right\\rbrace = 0.$ Therefore we can write $\\Pr \\left\\lbrace C_S>R_S\\right\\rbrace &= \\int _{2^{R_S}-1}^\\infty f_{\\gamma _b}(x) \\left(\\int _0^{\\frac{1}{2^{R_S}}(1+x)-1} f_{\\gamma _e}(y) dy\\right) dx \\nonumber \\\\&=\\int _{2^{R_S}-1}^\\infty f_{\\gamma _b}(x) F_{\\gamma _e}\\left(\\frac{1}{2^{R_S}}(1+x)-1\\right) dx,$ where $F_{X}(\\cdot )$ and $f_{X}(\\cdot )$ are the CDF and PDF of the random variable $X$ , respectively.",
"Let us first assume that the fading experienced by the eavesdropper's is modeled by th $\\kappa $ -$\\mu $ shadowed distribution [7].",
"For integer values of $\\mu $ and $m$ , the CDF of the $\\kappa $ -$\\mu $ shadowed distribution can be expressed as a mixture of gamma distributions as described in Appendix REF , as follows: $F_{\\gamma _e}(\\gamma _e)=1-\\sum _{i=0}^{M}C_i e^{-\\frac{\\gamma }{\\Omega _i}}\\sum _{r=0}^{m_i-1}\\frac{1}{r!",
"}\\left(\\frac{\\gamma }{\\Omega _i}\\right)^{r},$ Plugging (REF ) in (REF ) yields $&\\Pr \\left\\lbrace C_S>R_S\\right\\rbrace = 1- F_{\\gamma _b}(2^{R_S}-1)-\\sum _{i=0}^{M}C_i \\exp {\\left(-\\frac{1-2^{R_S}}{2^{R_S}\\bar{\\Omega }_i}\\right)} \\nonumber \\\\& \\times \\underbrace{\\int _{2^{R_S}-1}^\\infty f_{\\gamma _b}(x) e^{ -\\frac{ x}{2^{R_S} \\bar{\\Omega }_i}} \\sum _{r=0}^{m_i-1} \\frac{1}{r!",
"}\\left(\\frac{1+x-2^{R_S}}{2^{R_S}\\Omega _i}\\right)^r dx}_{\\mathcal {I}},$ where the parameters $M$ , $m_i$ , $C_i$ and $\\Omega _i$ are defined in Table REF in the Appendix REF in terms of the parameters $\\kappa $ , $\\mu $ , $m$ and $\\gamma _e$ of the eavesdropper's fading distribution.",
"Using the binomial expansion, and defining $\\alpha =2^{R_S}-1$ , $\\beta _i=1/(2^{R_S}\\Omega _i)$ , the integral term $\\mathcal {I}$ can be reexpressed as $\\mathcal {I} &=\\sum _{r=0}^{m_i-1}\\sum _{k=0}^{r} \\frac{(-\\alpha )^{r-k}}{k!(r-k)!}",
"\\beta _i^{r} \\int _{\\alpha }^\\infty x^k f_{\\gamma _b}(x) e^{ -\\beta _i x} dx,\\nonumber \\\\&=\\sum _{i=0}^{m_i-1}\\sum _{k=0}^{r} \\frac{\\alpha ^{r-k}}{k!(r-k)!}",
"\\beta _i^{r} \\frac{\\partial ^k {\\mathcal {M}} _{\\gamma _b}^{u}\\left( s,z\\right)}{\\partial s^k}\\bigg {|}_{\\stackrel{s=-\\beta _i}{z=\\alpha }}$ where the general derivative property in the transform domain was used, in order to identify the $k^{th}$ derivative of the IMGF.",
"Finally, using (REF ) in (REF ) and (REF ) yields $\\mathcal {P}_{R_S}= \\sum _{i=0}^{M}C_i e^{\\alpha \\beta _i}\\sum _{r=0}^{m_i-1}\\sum _{k=0}^{r} \\tfrac{(-\\alpha \\beta _i)^{r}}{k!(r-k)!}",
"\\frac{\\partial ^k {\\mathcal {M}} _{\\gamma _b}^{u}\\left( s,z\\right)}{\\partial s^k}\\bigg {|}_{\\stackrel{s=-\\beta _i}{z=\\alpha }}+F_{\\gamma _b}\\left(\\alpha \\right).$ This completes the proof."
],
[
"CDF of the $\\kappa $ -{{formula:75fe90f6-8db4-4c8c-8771-1308b69d9820}} shadowed distribution for integer fading parameters",
"The CDF of the $\\kappa $ -$\\mu $ shadowed fading model was originally given in [7] as $F_{\\gamma }(\\gamma )=&\\frac{{\\mu ^{\\mu -1} m^m \\left( {1 + \\kappa } \\right)^\\mu }}{{\\Gamma (\\mu )\\bar{\\gamma }^\\mu \\left( {\\mu \\kappa + m} \\right)^m }}z^\\mu \\times \\nonumber \\\\& \\Phi _2 \\left( {\\mu - m,m;1 + \\mu ; { - \\tfrac{{\\mu \\left( {1 + \\kappa } \\right)}}{{\\bar{\\gamma }}}} \\gamma , { - \\tfrac{{\\mu \\left( {1 + \\kappa } \\right)}}{{\\bar{\\gamma }}}\\tfrac{m}{{\\mu \\kappa + m}}} \\gamma } \\right).$ If the fading parameters $\\mu $ and $m$ take integer values, this CDF can be expressed as a finite mixture of squared Nakagami distributions, i.e.",
"as a finite sum of exponentials and powers [47].",
"Manipulating the expressions in [47], we can compactly express the CDF as $F_{\\gamma }(\\gamma )=1-\\sum _{i=0}^{M}C_i e^{-\\frac{\\gamma }{\\Omega _i}}\\sum _{r=0}^{m_i-1}\\frac{1}{r!",
"}\\left(\\frac{\\gamma }{\\Omega _i}\\right)^{r},$ where the parameters $m_i$ , $M$ and $\\Omega _i$ are expressed in Table REF in terms of the parameters of the $\\kappa $ -$\\mu $ shadowed distribution, namely $\\kappa $ , $\\mu $ , $m$ and $\\bar{\\gamma }$ .",
"Table: Parameter values for the κ\\kappa -μ\\mu shadowed distribution with integer μ\\mu and mm,"
]
] | 1606.05127 |
[
[
"Estimating mutual information in high dimensions via classification\n error"
],
[
"Abstract Multivariate pattern analyses approaches in neuroimaging are fundamentally concerned with investigating the quantity and type of information processed by various regions of the human brain; typically, estimates of classification accuracy are used to quantify information.",
"While a extensive and powerful library of methods can be applied to train and assess classifiers, it is not always clear how to use the resulting measures of classification performance to draw scientific conclusions: e.g.",
"for the purpose of evaluating redundancy between brain regions.",
"An additional confound for interpreting classification performance is the dependence of the error rate on the number and choice of distinct classes obtained for the classification task.",
"In contrast, mutual information is a quantity defined independently of the experimental design, and has ideal properties for comparative analyses.",
"Unfortunately, estimating the mutual information based on observations becomes statistically infeasible in high dimensions without some kind of assumption or prior.",
"In this paper, we construct a novel classification-based estimator of mutual information based on high-dimensional asymptotics.",
"We show that in a particular limiting regime, the mutual information is an invertible function of the expected $k$-class Bayes error.",
"While the theory is based on a large-sample, high-dimensional limit, we demonstrate through simulations that our proposed estimator has superior performance to the alternatives in problems of moderate dimensionality."
],
[
"Introduction",
"A fundamental challenge of computational neuroscience is to understand how information about the external world is processed and represented in the brain.",
"Each individual neuron aggregates the incoming information into a single sequence of spikes–an output which is too simplistic by itself to capture the full complexity of sensory input.",
"Only by combining the signals from massive ensembles of neurons is it possible to reconstruct our complex representation of the world.",
"Nevertheless, neurons form hierarchies of specialization within neural circuits, which are further organized in various specialized regions of the brain.",
"At the lowest level of the hierarchy–individual neurons, it is possible to infer and interpret the functional relationship between a neuron and stimulus features of interest using single-cell recording technologies.",
"Due to the inherent stochasticity of the neural output, it is natural to view the neuron as a noisy channel, and use mutual information to quantify how much of the stimulus information is encoded by the neuron.",
"Moving up the hierarchy to the the macroscale level of organization in the brain requires both different experimental methodologies and new approaches for summarizing and inferring measures of information in the brain.",
"Shannon's mutual information $I(X; Y)$ is fundamentally a measure of dependence between random variables $X$ and $Y$ , and is defined as $I(X;Y) = \\int p(x, y) \\log \\frac{p(x, y)}{p(x)p(y)}dxdy.$ Various properties of $I(X; Y)$ make it ideal for quantifying the information between a random stimulus $X$ and the signaling behavior of an ensembles of neurons, $Y$ [1].",
"A leading metaphor is that of a noisy communications channel; the mutual information describes the rate at which $Y$ can communicate bits from $X$ .",
"This framework is well-suited for summarizing the properties of a single neuron coding external stimulus information; indeed, experiments studying the properties of a single or a small number of neurons often make use of the concept of mutual information in summarizing or interpreting their results [2].",
"See discussions in [3].",
"However, estimating mutual information for multiple channels requires large and over-parameterized generative models.",
"Machine learning algorithms showed a way forward: a seminal work by Haxby [4] proposed to quantify the information in multiple channels by measuring how well the stimulus can be identified from the brain responses, in what is known as “multivariate pattern analysis” (MVPA).",
"To demonstrate that a particular brain region responds to a certain type of sensory information, one employs supervised learning to build a classifier that classifies the stimulus class from the brain activation in that region.",
"Classifiers that achieve above-chance classification accuracy indicate that information from the stimulus is represented in the brain region.",
"In principle, one could just as well test the statistical hypothesis that the Fisher information or mutual information between the stimulus and the activation patterns is nonzero.",
"But in practice, the machine learning approach enjoys several advantages: First, it is invariant to the parametric representation of the stimulus space, and is opportunistic in the parameterization of the response space.",
"This is an important quality for naturalistic stimulus-spaces, such as faces or natural images.",
"Second, it scales better with the dimensionality of both the stimulus space and the responses space, because a slimmer discriminative model can be used rather than a fully generative model.",
"Nevertheless, classification error is problematic for quantifying the strength of the relation between stimulus and outputs due to its arbitrary scale and strong dependence on experimental choices.",
"Classification accuracy depends on the particular choice of stimuli exemplars employed in the study and the number of partitions ($k$ ) used to define the classes for the classification task.",
"The difficulty of the classification task depends on the number of classes defined: high classification accuracy can be achieved relatively easily by using a coarse partition of stimuli exemplars into classes.",
"Often $k$ is an arbitrary design constraint, and researchers try to extrapolate the error for alternative number of classes [5].",
"In a meta-analysis on visual decoding, Coutanche et al (2016) [6] quantified the strength of a classification study using the formula $\\text{decoding strength} = \\frac{\\text{accuracy} - \\text{chance}}{\\text{chance}}.$ Such an approach may compensate for the differences in accuracy due purely to choice of number of classes defined; however, no theory is provided to justify the formula.",
"In contrast, mutual information has ideal properties for quantitatively comparing information between different studies, or between different brain regions, subjects, feature-spaces, or modalities.",
"Not only is the mutual information defined independently of the arbitrary definition of stimulus classes (albeit still dependent on an implied distribution over stimuli), it is even meaningful to discuss the difference between the mutual information measured for one system and the mutual information for a second system.",
"Hence, a popular approach which combines the strengths of the machine learning approach and the advantages of the information theoretic approach is to obtain a lower bound on the mutual information by using the confusion matrix of a classifier.",
"Treves [7] first proposed using the empirical mutual information of the classification matrix in order to obtain a lower bound of the mutual information $I(X; Y)$ ; this confusion-matrix-based lower bound has subsequently enjoyed widespread use in the MVPA literature [2].",
"Even earlier that this, the idea of linking classification performance to mutual information can be found in the beginnings of information theory.",
"Fano's inequality provides a lower bound on mutual information in relation to the optimal prediction error, or Bayes error.",
"In practice, the bound obtained may be a vast underestimate [8]."
],
[
"Our contributions",
"In this paper, we propose a new way to link classification performance to the implied mutual information.",
"To create this link we need to overcome the arbitrary choice of exemplars, and the arbitrary number of classes k. Towards this end, we define a notion of $k$ -class average Bayes error which is uniquely defined for any given stimulus distribution and stochastic mapping from stimulus to response.",
"The $k$ -class average Bayes error is the expectation of the Bayes error (the classification error of the optimal classifier) when $k$ stimuli exemplars are drawn i.i.d.",
"from the stimulus distribution, and treated as distinct classes.",
"Hence the average Bayes error can in principle be estimated if the appropriate randomization is employed for designing the experiment.",
"Specifically, we establish a relationship between the mutual information $I(X; Y)$ and the average $k$ -class Bayes error, $e_{ABE, k}$ .",
"In short, we will identify a function $\\pi _k$ (which depends on $k$ ), $e_{ABE, k} \\approx \\pi _k(\\sqrt{2 I(X; Y)})$ and that this approximation becomes accurate under a limit where $I(X;Y)$ is small relative to the dimensionality of $X$ , and under the condition that the components of $X$ are approximately independent.",
"The function $\\pi _k$ is given by $\\pi _k(c) = 1 - \\int _{\\mathbb {R}} \\phi (z - c) \\Phi (z)^{k-1} dz.$ This formula is not new to the information theory literature: it appears as the error rate of an orthogonal constellation [9].",
"What is surprising is that the same formula can be used to approximate the error rate in much more general class of classification problemsAn intuitive explanation for this fact is that points from any high-dimensional distribution lie in an orthogonal configuration with high probability.–this is precisely the universality result which provides the basis for our proposed estimator.",
"Figure REF displays the plot of $\\pi _k$ for several values of $k$ .",
"For all values of $k$ , $\\pi _k(\\mu )$ is monotonically decreasing in $\\mu $ , and tends to zero as $\\mu \\rightarrow \\infty $ , which is what we expect since if $I(X; Y)$ is large, then the average Bayes error should be small.",
"Another intuitive fact is that $ \\pi _k(0) = 1 -\\frac{1}{k}, $ since after all, an uninformative response cannot lead to above-chance classification accuracy.",
"Figure: Left: The function π k (μ)\\pi _k(\\mu ) for k={2,10}k = \\lbrace 2, 10\\rbrace .Right: I ^ HD \\hat{I}_{HD} with I ^ Fano \\hat{I}_{Fano} as functions of e ^ gen \\hat{e}_{gen}, for k=3k = 3.While I ^ Fano \\hat{I}_{Fano} is bounded from above by log(k)\\log (k) (dotted line), I ^ HD \\hat{I}_{HD} is unbounded.The estimator we propose is $\\hat{I}_{HD} = \\frac{1}{2}(\\pi _{k}^{-1}(\\hat{e}_{gen, \\alpha }))^2,$ obtained by inverting the relation (REF ), then substituting an estimate of generalization error $\\hat{e}_{gen, \\alpha }$ for the $e_{ABE, k}$ .",
"As such, our estimator can be directly compared to the $\\hat{I}_{Fano}$ , since both are functions of $\\hat{e}_{gen,\\alpha }$ (Figure 1.)",
"As the estimate of generalization error goes to zero, $\\hat{I}_{Fano}$ approaches $\\log (k)$ while $\\hat{I}_{HD}$ goes to infinity.",
"This difference in behavior is due to the fact that in contrast to Fano's inequality, the asymptotic relationship (REF ) is independent of the number of classes $k$ .",
"In the paper we argue for the advantages of our method in comparison to alternative discriminative estimators under the assumption that the discriminative model approximates the Bayes rule.",
"While this is an unrealistic assumption, it simplifies the theoretical discussion, and allows us to clearly discuss the principles behind our method.",
"Alternatively, we can take the view that any observed classification error is a lower bound on the Bayes prediction, therefore interpreting our result as establishing a usually tighter lower bound on $I(X,Y)$ .",
"The organization of the paper is as follows.",
"We outline our framework in Section 2.1.",
"In Section 2.2 we present our key result, which links the asymptotic average Bayes error to the mutual information, under an asymptotic setting intended to capture the notion of high dimensionalityNamely, one where the number of classes is fixed, and where the information $I(X; Y)$ remains fixed, while the dimensionality of the input $X$ and output $Y$ both grow to infinity.",
"We make a number of additional regularity conditions to rule out scenarios where $(X, Y)$ is really less “high-dimensional” than it appears, since most of the variation is captured a low-dimensional manifold..",
"In Section 2.3 we apply this result to derive our proposed estimator, $\\hat{I}_{HD}$ (where HD stands for “high-dimensional.”) Section 3 presents simulation results, and Section 4 concludes.",
"All proofs are given in the supplement."
],
[
"Setting",
"Let us assume that the variables $X, Y$ have a joint distribution $F$ , and that one can define a conditional distribution of $Y$ given $X$ , $Y|X \\sim F_X,$ and let $G$ denote the marginal distribution of $X$ .",
"We assume that data is collected using stratified sampling.",
"For $j = 1,\\hdots , k$ , sample i.i.d.",
"exemplars $X^{(1)},\\hdots , X^{(k)} \\sim G$ .",
"For $i =1,\\hdots , n$ , draw $Z^i$ iid from the uniform distribution on $1,\\hdots , k$ , then draw $Y^i$ from the conditional distribution $F_{X^{(Z_i)}}$ .",
"Stratified sampling is commonly seen in controlled experiments, where an experimenter chooses an input $X$ to feed into a black box, which outputs $Y$ .",
"An example from fMRI studies is an experimental design where the subject is presented a stimulus $X$ , and the experimenter measures the subject's response via the brain activation $Y$ .",
"Note the asymmetry in our definition of stratified sampling: our convention is to take $X$ to be the variable preceding $Y$ in causal order.",
"Such causal directionality constrains the stratified sampling to have repeated $X$ rather than repeated $Y$ values, but has no consequence for the mutual information $I(X; Y)$ , which is a symmetric function.",
"When stratified sampling is employed, one can define an exemplar-based classification task.",
"One defines the class function $Z$ by $Z: \\lbrace X^{(1)}, \\hdots , X^{(k)}\\rbrace \\rightarrow \\lbrace 1,\\hdots , k\\rbrace ,$ $Z(X^{(i)}) = i\\text{ for }i = 1, \\hdots , k.$ One defines the generalization error by $e_{gen}(f) = \\frac{1}{k} \\sum _{i=1}^k\\Pr [f(Y) \\ne Z|X = X^{(i)}].$ In an exemplar-based classification, there is no need to specify an arbitrary partition on the input space (as is the case in category-based classification), but note that the $k$ classes are randomly defined.",
"One consequence is that the Bayes error $e_{Bayes}$ is a random variable: when the sampling produces $k$ similar exemplars, $e_{Bayes}$ will be higher, and when the sampling produces well-separated exemplars $e_{Bayes}$ may be lower.",
"Therefore, it is useful to consider the average Bayes error, $e_{ABE, k} = \\textbf {E}_{X^{(1)},\\hdots , X^{(k)}}[e_{Bayes}],$ where the expectation is taken over the joint distribution of $X^{(1)},\\hdots , X^{(k)} \\stackrel{iid}{\\sim } G$ .",
"We use the terminology classifier to refer to any algorithm which takes data as input, and produces a classification rule $f$ as output.",
"Mathematically speaking, the classifier is a functional which maps a set of observations to a classification rule, $ \\mathcal {F}:\\lbrace (x^{1},y^{1}),\\hdots , (x^{m}, y^{m})\\rbrace \\mapsto f(\\cdot ).",
"$ The data $(x^1,y^1),\\hdots , (x^m, y^m)$ used to obtain the classification rule is called training data.",
"When the goal is to obtain inference about the generalization error $e_{gen}$ of the classification rule $f$ , it becomes necessary to split the data into two independent sets: one set to train the classifier, and one to evaluate the performance.",
"The reason that such a splitting is necessary is because using the same data to test and train a classifier introduces significant bias into the empirical classification error [10].",
"The classification rule is obtained via $ f = \\mathcal {F}(S_{train}), $ where $S_{train}$ is the training set, and the performance of the classifier is evaluated by predicting the classes of the test set.",
"The results of this test are summarized by a $k\\times k$ confusion matrix $M$ with $ M_{ij} = \\sum _{\\ell =r_1 +1}^r I(f(y^{(i), r}) = j).",
"$ The $i, j$ th entry of $M$ counts how many times a output in the $i$ th class was classified to the $j$ th class.",
"The test error is the proportion of off-diagonal terms of $M$ , $ e_{test} = \\frac{1}{kr} \\sum _{i \\ne j} M_{ij}, $ and is an unbiased estimator of $e_{gen}$ .",
"However, in small sampling regimes the quantity $e_{test}$ may be too variable to use as an estimator of $e_{gen}$ .",
"We recommend the use of Bayesian smoothing, defining an $\\alpha $ -smoothed estimate $\\hat{e}_{gen, \\alpha }$ by $ \\hat{e}_{gen,\\alpha } = (1 - \\alpha ) e_{test} + \\alpha \\frac{k-1}{k}, $ which takes a weighted average of the unbiased estimate $e_{test}$ , and the natural prior of chance classification.",
"We define a discriminative estimator to be a function which maps the misclassification matrix to a positive number, $ \\hat{I}:\\mathbb {N}^{k \\times k} \\rightarrow \\mathbb {R}.",
"$ We are aware of the following examples of discriminative estimators: (1) estimators $\\hat{I}_{Fano}$ derived from using Fano's inequality, and (2) the empirical information of the confusion matrix, $\\hat{I}_{CM}$ , as introduced by Treves [7].",
"We discuss these estimators in Section 3."
],
[
"Universality result",
"We obtain the universality result in two steps.",
"First, we link the average Bayes error to the moments of some statistics $Z_i$ .",
"Secondly, we use taylor approximation in order to express $I(X; Y)$ in terms of the moments of $Z_i$ .",
"Connecting these two pieces yields the formula (REF ).",
"Let us start by rewriting the average Bayes error: $e_{ABE, k} = \\Pr [p(Y|X_1) \\le \\max _{j \\ne 1} p(Y|X_j)| X = X_1].$ Defining the statistic $Z_i = \\log p(Y|X_i) - \\log p(Y|X_1)$ , where $Y\\sim p(y|X_1)$ , we obtain $ e_{ABE} = \\Pr [\\max _{j > 1} Z_i > 0].",
"$ The key assumption we need is that $Z_2,\\hdots , Z_k$ are asymptotically multivariate normal.",
"If so, the following lemma allows us to obtain a formula for the misclassification rate.",
"Lemma 1.",
"Suppose $(Z_1, Z_2, \\hdots , Z_k)$ are jointly multivariate normal, with $\\textbf {E}[Z_1 - Z_i]= \\alpha $ , $\\text{Var}(Z_1) = \\beta \\ge 0$ , $\\text{Cov}(Z_1, Z_i) = \\gamma $ , $\\text{Var}(Z_i)= \\delta $ , and $\\text{Cov}(Z_i, Z_j) = \\epsilon $ for all $i, j = 2, \\hdots ,k$ , such that $\\beta + \\epsilon - 2\\gamma > 0$ .",
"Then, letting $\\mu = \\frac{\\textbf {E}[Z_1 - Z_i]}{\\sqrt{\\frac{1}{2}\\text{Var}(Z_i - Z_j)}} = \\frac{\\alpha }{\\sqrt{\\delta - \\epsilon }},$ $\\nu ^2 = \\frac{\\text{Cov}(Z_1 -Z_i, Z_1 - Z_j)}{\\frac{1}{2}\\text{Var}(Z_i - Z_j)} = \\frac{\\beta + \\epsilon - 2\\gamma }{\\delta - \\epsilon },$ we have $\\Pr [Z_1 < \\max _{i=2}^k Z_i] &= \\Pr [W < M_{k-1}]\\\\&= 1 - \\int \\frac{1}{\\sqrt{2\\pi \\nu ^2}} e^{-\\frac{(w-\\mu )^2}{2\\nu ^2}} \\Phi (w)^{k-1} dw,$ where $W \\sim N(\\mu , \\nu ^2)$ and $M_{k-1}$ is the maximum of $k-1$ independent standard normal variates, which are independent of $W$ .",
"To see why the assumption that $Z_2,\\hdots , Z_k$ are multivariate normal might be justified, suppose that $X$ and $Y$ have the same dimensionality $d$ , and that joint density factorizes as $p(x^{(j)}, y) = \\prod _{i=1}^d p_i(x^{(j)}_i, y_i)$ where $x_i^{(j)}, y_i$ are the $i$ th scalar components of the vectors $x^{(j)}$ and $y$ .",
"Then, $Z_i = \\sum _{m=1}^d \\log p_m(y_m | x^{(i)}_m) - \\log p_m(y_m | x^{(m)}_1)$ where $x_{i, j}$ is the $i$ th component of $x_j$ .",
"The $d$ terms $\\log p_m(y_m | x_{m, i}) - \\log p_m(y_m | x_{m, 1})$ are independent across the indices $m$ , but dependent between the $i = 1,\\hdots , k$ .",
"Therefore, the multivariate central limit theorem can be applied to conclude that the vector $(Z_2,\\hdots , Z_k)$ can be scaled to converge to a multivariate normal distribution.",
"While the componentwise independence condition is not a realistic assumption, the key property of multivariate normality of $(Z_2,\\hdots , Z_k)$ holds under more general conditions, and appears reasonable in practice.",
"It remains to link the moments of $Z_i$ to $I(X;Y)$ .",
"This is accomplished by approximating the logarithmic term by the Taylor expansion $\\log \\frac{p(x, y)}{p(x) p(y)} \\approx \\frac{p(x, y) - p(x) p(y)}{p(x) p(y)} - \\left(\\frac{p(x, y) - p(x) p(y)}{p(x) p(y)}\\right)^2 + \\hdots .$ A number of assumptions are needed to ensure that needed approximations are sufficiently accurate; and additionally, in order to apply the central limit theorem, we need to consider a limiting sequence of problems with increasing dimensionality.",
"We now state the theorem.",
"Theorem 1.",
"Let $p^{[d]}(x, y)$ be a sequence of joint densities for $d = 1,2,\\hdots $ .",
"Further assume that A1.",
"$\\lim _{d \\rightarrow \\infty } I(X^{[d]}; Y^{[d]}) = \\iota < \\infty .$ A2.",
"There exists a sequence of scaling constants $a_{ij}^{[d]}$ and $b_{ij}^{[d]}$ such that the random vector $(a_{ij}\\ell _{ij}^{[d]} +b_{ij}^{[d]})_{i, j = 1,\\hdots , k}$ converges in distribution to a multivariate normal distribution, where $\\ell _{ij} = \\log p(y^{(i)}|x^{(i)})$ for independent $y^{(i)} \\sim p(y|x^{(i)})$ .",
"A3.",
"Define $u^{[d]}(x, y) = \\log p^{[d]}(x, y) - \\log p^{[d]}(x) - \\log p^{[d]}(y).$ There exists a sequence of scaling constants $a^{[d]}$ , $b^{[d]}$ such that $a^{[d]}u^{[d]}(X^{(1)}, Y^{(2)}) + b^{[d]}$ converges in distribution to a univariate normal distribution.",
"A4.",
"For all $i \\ne k$ , $\\lim _{d \\rightarrow \\infty }\\text{Cov}[u^{[d]}(X^{(i)}, Y^{(j)}), u^{[d]}(X^{(k)}, Y^{(j)})] = 0.$ Then for $e_{ABE, k}$ as defined above, we have $\\lim _{d \\rightarrow \\infty } e_{ABE, k} = \\pi _k(\\sqrt{2 \\iota })$ where $\\pi _k(c) = 1 - \\int _{\\mathbb {R}} \\phi (z - c) \\Phi (z)^{k-1} dz$ where $\\phi $ and $\\Phi $ are the standard normal density function and cumulative distribution function, respectively.",
"Assumptions A1-A4 are satisfied in a variety of natural models.",
"One example is a multivariate Gaussian sequence model where $X \\sim N(0,\\Sigma _d)$ and $ Y = X + E $ with $ E \\sim N(0, \\Sigma _e), $ where $\\Sigma _d$ and $\\Sigma _e$ are $d \\times d$ covariance matrices, and where $X$ and $E$ are independent.",
"Then, if $d \\Sigma _d$ and $\\Sigma _e$ have limiting spectra $H$ and $G$ respectively, the joint densities $p(x, y)$ for $d = 1,\\hdots , $ satisfy assumptions A1 - A4.",
"Another example is the multivariate logistic model, which we describe in Section 3.",
"We further discuss the rationale behind A1-A4 in the supplement, along with the detailed proof."
],
[
"High-dimensional estimator",
"As stated in the introduction, we propose the estimator $\\hat{I}_{HD}(M) = \\frac{1}{2}(\\pi _{k}^{-1}(\\hat{e}_{gen, \\alpha }))^2.$ For sufficiently high-dimensional problems, $\\hat{I}_{HD}$ can accurately recover $I(X; Y) > \\log k$ , supposing also that the classifier $\\mathcal {F}$ consistently estimates the Bayes rule.",
"The number of observations needed depends on the convergence rate of $\\mathcal {F}$ and also the complexity of estimating $e_{gen, \\alpha }$ .",
"Therefore, without making assumptions on $\\mathcal {F}$ , the sample complexity is at least exponential in $I(X; Y)$ .",
"This is because when $I(X; Y)$ is large relative to $\\log (k)$ , the Bayes error $e_{ABE, k}$ is exponentially small.",
"Hence $O(1/e_{ABE, k})$ observations in the test set are needed to recover $e_{ABE, k}$ to sufficient precision.",
"While the sample complexity exponential in $I(X; Y)$ is by no means ideal, by comparison, the nonparametric estimation approaches have a complexity exponential in the dimensionality.",
"Hence, $\\hat{I}_{HD}$ is favored over nonparametric approaches in settings with high dimensionality and low signal-to-noise ratio."
],
[
"Simulation",
"We compare the discriminative estimators $\\hat{I}_{CM}$ , $\\hat{I}_{Fano}$ , $\\hat{I}_{HD}$ with a nonparametric estimator $\\hat{I}_0$ in the following simulation, and the correctly specified parametric estimator $\\hat{I}_{MLE}$ .",
"We generate data according to a multiple-response logistic regression model, where $ X \\sim N(0,I_p) $ , and $Y$ is a binary vector with conditional distribution $Y_i|X = x \\sim \\text{Bernoulli}(x^T B_i)$ where $B$ is a $p \\times q$ matrix.",
"One application of this model might be modeling neural spike count data $Y$ arising in response to environmental stimuli $X$ [12].",
"We choose the naive Bayes for the classifier $\\mathcal {F}$ : it is consistent for estimating the Bayes rule.",
"Figure: Sampling distributions of I ^\\hat{I} for data generated from the multiple-response logistic model.",
"p=q=10p = q = 10; k=20k= 20; B=sI 10 B = sI_{10}, where s∈[0,200]s \\in [0, \\sqrt{200}]; and r=1000r = 1000.The estimator $\\hat{I}_{Fano}$ is based on Fano's inequality, which reads $H(Z|Y) \\le H(e_{Bayes}) + e_{Bayes} \\log ||\\mathcal {Z}| - 1|$ where $H(e)$ is the entropy of a Bernoulli random variable with probability $e$ .",
"Replacing $H(Z|Y)$ with $H(X|Y)$ and replacing $e_{Bayes}$ with $\\hat{e}_{gen, \\alpha }$ , we get the estimator $\\hat{I}_{Fano}(M) = log(K) - \\hat{e}_{gen, \\alpha } log(K-1) + \\hat{e}_{gen, \\alpha } log(p) + (1-\\hat{e}_{gen, \\alpha }) log(1-\\hat{e}_{gen, \\alpha }).$ Meanwhile, the confusion matrix estimator computes $\\hat{I}_{CM}(M) = \\frac{1}{k^2} \\sum _{i=1}^k \\sum _{j=1}^k \\log \\frac{M_{ij}}{r/k},$ which is the empirical mutual information of the discrete joint distribution $(Z, f(Y))$ .",
"It is known that $\\hat{I}_{CM}$ , $\\hat{I}_0$ tend to underestimate the mutual information.",
"Quiroga et al.",
"[2] discussed two sources of `information loss' which lead to $\\hat{I}_{CM}$ underestimating the mutual information: the discretization of the classes, and the error in approximating the Bayes rule.",
"Meanwhile, Gastpar et al.",
"[11] showed that $\\hat{I}_0$ is biased downwards due to undersampling of the exemplars: to counteract this bias, they introduce the anthropic correction estimator $\\hat{I}_\\alpha $However, without a principled approach to choose the parameter $\\alpha \\in (0,1]$ , $\\hat{I}_\\alpha $ could still vastly underestimate or overestimate the mutual information..",
"In addition to the sources of information loss discussed by Quiroga et al., an additional reason why $\\hat{I}_{CM}$ and $\\hat{I}_{Fano}$ underestimate the mutual information is that they are upper bounded by $\\log (k)$ , where $k$ is the number of classes.",
"As $I(X; Y)$ exceeds $\\log (k)$ , the estimate $\\hat{I}$ can no longer approximate $I(X; Y)$ , even up to a constant factor.",
"In contrast, $\\hat{I}_{HD}$ is unbounded and may either underestimate or overestimate the mutual information in general, but performs well when the high-dimensionality assumption is met.",
"In Figure 2 we show the sampling distributions of the five estimators as $I(X; Y)$ is varied in the interval $[0, 4]$ .",
"The estimator $\\hat{I}_{MLE}$ is a plug-in estimator using $\\hat{B}$ , the coefficient matrix estimated via multinomial regression of $Y$ on $X$ ; it recovers the true mutual information within $\\pm 2\\%$ with a probability of 90%.",
"We see that $\\hat{I}_{CM}$ , $\\hat{I}_{Fano}$ , and $\\hat{I}_0$ indeed begin to asymptote as they approach $\\log (k) = 2.995$ .",
"In contrast, $\\hat{I}_{HD}$ remains a good approximation of $I(X; Y)$ within the range, although it begins to overestimate at the right endpoint.",
"The reason why $\\hat{I}_{HD}$ loses accuracy as the true information $I(X; Y)$ increases is that the multivariate normality approximation used to derive the estimator becomes less accurate when the conditional distribution $p(y|x)$ becomes highly concentrated."
],
[
"Discussion",
"Discriminative estimators of mutual information have the potential to estimate mutual information in high-dimensional data without resorting to fully parametric assumptions.",
"However, a number of practical considerations also limit their usage.",
"First, one has to find a good classifier $\\mathcal {F}$ for the data: techniques for model selection can be used to choose $\\mathcal {F}$ from a large library of methods.",
"However, there is no way to guarantee how well the chosen classifier approximates the optimal classification rule.",
"Secondly, one has to estimate the generalization error from test data: the complexity of estimating $e_{gen}$ could become the bottleneck when $e_{gen}$ is close to 0.",
"Thirdly, for previous estimators $\\hat{I}_{Fano}$ and $\\hat{I}_{CM}$ , the ability of the estimator to distinguish high values of $I(X; Y)$ is limited by the number of classes $k$ .",
"Our estimator $\\hat{I}_{HD}$ is subject to the first two limitations, along with any conceivable discriminative estimator, but overcomes the third limitation under the assumption of stratified sampling and high dimensionality.",
"It can be seen that additional assumptions are indeed needed to overcome the third limitation, the $\\log (k)$ upper bound.",
"Consider the following worst-case example: let $X$ and $Y$ have joint density $p(x, y) = \\frac{1}{k}I(\\lfloor kx \\rfloor = \\lfloor ky \\rfloor ) $ on the unit square.",
"Under partition-based classification, if we set $Z(x) = \\lfloor kx \\rfloor + 1$ , then no errors are made under the Bayes rule.",
"We therefore have a joint distribution which maximizes any reasonable discriminative estimator but has finite information $I(X; Y) = \\log (k)$ .",
"The consequence of this is that under partition-based classification, we cannot hope to distinguish distributions with $I(X; Y) > \\log (k)$ .",
"The situation is more promising if we specialize to stratified sampling: in the same example, a Bayes of zero is no longer likely due to the possibility of exemplars being sampled from the same bin (`collisions')–we obtain an approximation to the average Bayes error through a Poisson sampling model: $e_{ABE, k} \\approx \\frac{1}{e}\\sum _{j=1}^\\infty \\frac{1}{j(j!",
")}= 0.484$ .",
"By specializing further to the high-dimensional regime, we obtain even tighter control on the relation between Bayes error and mutual information.",
"Our estimator therefore provides more accurate estimation at the cost of more additional assumptions, but just how restrictive are these assumptions?",
"The assumption of stratified sampling is usually not met in the most common applications of classification where the classes are defined a priori.",
"For instance, if the classes consist of three different species of iris, it does not seem appropriate to model the three species as i.i.d.",
"draws from some distribution on a space of infinitely many potential iris species.",
"Yet, when the classes have been pre-defined in an arbitrary manner, the mutual information between a latent class-defining variable $X$ and $Y$ may be only weakly related to the classification accuracy.",
"We rely on the stratified sampling assumption to obtain the necessary control on how the classes in the classification task are defined.",
"Fortunately, in many applications where one is interested in estimating $I(X; Y)$ , a stratified sampling design can be practically implemented.",
"The assumption of high dimensionality is not easy to check: having a high-dimension response $Y$ does not suffice, since even then $Y$ could still lie close to a low-dimensional manifold.",
"In such cases, $\\hat{I}_{HD}$ could either overestimate or underestimate the mutual information.",
"In situations where $(X, Y)$ lie on a manifold, one could effectively estimate mutual information by would be to combining dimensionality reduction with nonparametric information estimation [13].",
"We suggest the following diagnostic to determine if our method is appropriate: subsample within the classes collected and check that $\\hat{I}_{HD}$ does not systematically increase or decrease with the number of classes $k$ .",
"The assumption of approximating the Bayes rule is impractical to check, as any nonparametric estimate of the Bayes error requires exponentially many observations.",
"Hence, while the present paper studies the `best-case' scenario where the model is well-specified, it is even more important to understand the robustness of our method in the more realistic case where the model is misspecified.",
"We leave this question to future work.",
"Even given a classifier which consistently estimates the Bayes error, the estimator $\\hat{I}_{HD}$ can still be improved.",
"One can employ more sophisticated methods to estimate $e_{ABE, k}$ : for example, extrapolating from learning curves [14].",
"Furthermore, depending on the risk function, one may debias or shrink the estimate $\\hat{I}_{HD}$ to achieve a more favorable bias-variance tradeoff.",
"All of the necessary assumptions are met in our simulation experiment, hence our proposed estimator is seen to dramatically outperform existing estimators.",
"It remains to assess the utility of our estimation procedure in a real-world example, where both the high-dimensional assumption and the model specification assumption are likely to be violated.",
"In a forthcoming work, we apply our framework to evaluate visual encoding models in human fMRI data."
],
[
"Acknowledgments",
"We thank John Duchi, Youngsuk Park, Qingyun Sun, Jonathan Taylor, Trevor Hastie, Robert Tibshirani for useful discussion.",
"CZ is supported by an NSF graduate research fellowship.",
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[
"Appendix",
"Lemma 1.",
"Suppose $(Z_1, Z_2, \\hdots , Z_k)$ are jointly multivariate normal, with $\\textbf {E}[Z_1 - Z_i]= \\alpha $ , $\\text{Var}(Z_1) = \\beta $ , $\\text{Cov}(Z_1, Z_i) = \\gamma $ , $\\text{Var}(Z_i)= \\delta $ , and $\\text{Cov}(Z_i, Z_j) = \\epsilon $ for all $i, j = 2, \\hdots ,k$ , such that $\\beta + \\epsilon - 2\\gamma > 0$ .",
"Then, letting $\\mu = \\frac{\\textbf {E}[Z_1 - Z_i]}{\\sqrt{\\frac{1}{2}\\text{Var}(Z_i - Z_j)}} = \\frac{\\alpha }{\\sqrt{\\delta - \\epsilon }},$ $\\nu ^2 = \\frac{\\text{Cov}(Z_1 -Z_i, Z_1 - Z_j)}{\\frac{1}{2}\\text{Var}(Z_i - Z_j)} = \\frac{\\beta + \\epsilon - 2\\gamma }{\\delta - \\epsilon },$ we have $\\Pr [Z_1 < \\max _{i=2}^k Z_i] &= \\Pr [W < M_{k-1}]\\\\&= 1 - \\int \\frac{1}{\\sqrt{2\\pi \\nu ^2}} e^{-\\frac{(w-\\mu )^2}{2\\nu ^2}} \\Phi (w)^{k-1} dw,$ where $W \\sim N(\\mu , \\nu ^2)$ and $M_{k-1}$ is the maximum of $k-1$ independent standard normal variates, which are independent of $W$ .",
"Proof.",
"We can construct independent normal variates $G_1$ , $G_2,\\hdots , G_k$ such that $G_1 \\sim N(0, \\beta + \\epsilon - 2 \\gamma )$ $G_i \\sim N(0, \\delta - \\epsilon )\\text{ for }i > 1$ such that $Z_1 - Z_i = \\alpha + G_1 + G_i \\text{ for }i > 1.$ Hence $\\Pr [Z_1 < \\max _{i=2}^k Z_i] &= \\Pr [\\min _{i > 1} Z_1 - Z_i < 0].\\\\&= \\Pr [\\min _{i=2}^{k} G_1 + G_i + \\alpha < 0]\\\\&= \\Pr [\\min _{i=2}^{k} G_i < -\\alpha - G_1]\\\\&= \\Pr [\\min _{i=2}^{k} \\frac{G_i}{\\sqrt{\\delta - \\epsilon }} < -\\frac{\\alpha - G_1}{\\sqrt{\\delta - \\epsilon }}].$ Since $\\frac{G_i}{\\sqrt{\\delta - \\epsilon }}$ are iid standard normal variates, and since $-\\frac{\\alpha - G_1}{\\sqrt{\\delta - \\epsilon }} \\sim N(\\mu ,\\nu ^2)$ for $\\mu $ and $\\nu ^2$ given in the statement of the Lemma, the proof is completed via a straightforward computation.",
"$\\Box $ Theorem 1.",
"Let $p^{[d]}(x, y)$ be a sequence of joint densities for $d = 1,2,\\hdots $ as given above.",
"Further assume that A1.",
"$\\lim _{d \\rightarrow \\infty } I(X^{[d]}; Y^{[d]}) = \\iota < \\infty .$ A2.",
"There exists a sequence of scaling constants $a_{ij}^{[d]}$ and $b_{ij}^{[d]}$ such that the random vector $(a_{ij}\\ell _{ij}^{[d]} +b_{ij}^{[d]})_{i, j = 1,\\hdots , k}$ converges in distribution to a multivariate normal distribution.",
"A3.",
"There exists a sequence of scaling constants $a^{[d]}$ , $b^{[d]}$ such that $a^{[d]}u(X^{(1)}, Y^{(2)}) + b^{[d]}$ converges in distribution to a univariate normal distribution.",
"A4.",
"For all $i \\ne k$ , $\\lim _{d \\rightarrow \\infty }\\text{Cov}[u(X^{(i)}, Y^{(j)}), u(X^{(k)}, Y^{(j)})] = 0.$ Then for $e_{ABE, k}$ as defined above, we have $\\lim _{d \\rightarrow \\infty } e_{ABE, k} = \\pi _k(\\sqrt{2 \\iota })$ where $\\pi _k(c) = 1 - \\int _{\\mathbb {R}} \\phi (z - c) \\Phi (z)^{k-1} dz$ where $\\phi $ and $\\Phi $ are the standard normal density function and cumulative distribution function, respectively.",
"Proof.",
"For $i = 2,\\hdots , k$ , define $Z_i = \\log p(Y^{(1)}|X^{(i)}) - \\log p(Y^{(1)}|X^{(1)}).$ Then, we claim that $\\vec{Z} = (Z_2,\\hdots , Z_k)$ converges in distribution to $\\vec{Z} \\sim N\\left(-2\\iota ,\\begin{bmatrix}4\\iota & 2\\iota & \\cdots & 2\\iota \\\\2\\iota & 4\\iota & \\cdots & 2\\iota \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\2\\iota & 2\\iota & \\cdots & 4\\iota \\end{bmatrix}\\right).$ Combining the claim with the lemma (stated below this proof) yields the desired result.",
"To prove the claim, it suffices to derive the limiting moments $\\textbf {E}[Z_i] \\rightarrow -2\\iota ,$ $\\text{Var}[Z_i] \\rightarrow 4\\iota ,$ $\\text{Cov}[Z_i, Z_j] \\rightarrow 2\\iota ,$ for $i \\ne j$ , since then assumption A2 implies the existence of a multivariate normal limiting distribution with the given moments.",
"Before deriving the limiting moments, note the following identities.",
"Let $X^{\\prime } = X^{(2)}$ and $Y = Y^{(1)}$ .",
"$\\textbf {E}[e^{u(X^{\\prime }, Y)}] = \\int p(x) p(y) e^{u(x, y)} dx dy = \\int p(x, y) dx dy = 1.$ Therefore, from assumption A3 and the formula for gaussian exponential moments, we have $\\lim _{d \\rightarrow \\infty } \\textbf {E}[u(X^{\\prime }, Y)]-\\frac{1}{2}\\text{Var}[u(X^{\\prime }, Y)] = 0.$ Let $\\sigma ^2 = \\lim _{d \\rightarrow \\infty } \\text{Var}[u(X^{\\prime }, Y)]$ .",
"Meanwhile, by applying assumption A2, $\\lim _{d \\rightarrow \\infty } I(X; Y) &= \\lim _{d \\rightarrow \\infty } \\int p(x, y) u(x, y) dx dy= \\lim _{d \\rightarrow \\infty } \\int p(x) p(y) e^{u(x, y)} u(x, y) dx dy\\\\&= \\lim _{d \\rightarrow \\infty } \\textbf {E}[e^{u(X, Y^{\\prime })}u(X, Y^{\\prime })]\\\\&= \\int _{\\mathbb {R}} e^z z \\frac{1}{\\sqrt{2\\pi \\sigma ^2}}e^{-\\frac{(z + \\sigma ^2/2)^2}{2\\sigma ^2}} \\text{ (applying A2)}\\\\&= \\int _{\\mathbb {R}} z \\frac{1}{\\sqrt{2\\pi \\sigma ^2}}e^{-\\frac{(z - \\sigma ^2/2)^2}{2\\sigma ^2}}\\\\&= \\frac{1}{2}\\sigma ^2.$ Therefore, $\\sigma ^2 = 2\\iota ,$ and $\\lim _{d \\rightarrow \\infty } \\textbf {E}[u(X^{\\prime }, Y)] = -\\iota .$ Once again by applying A2, we get $\\lim _{d \\rightarrow \\infty } \\text{Var}[u(X, Y)]&= \\lim _{d \\rightarrow \\infty } \\int (u(x, y) - \\iota )^2 p(x, y) dx dy\\\\&= \\lim _{d \\rightarrow \\infty } \\int (u(x, y) - \\iota )^2 e^{u(x, y)} p(x) p(y) dx dy\\\\&= \\lim _{d \\rightarrow \\infty } \\textbf {E}[(u(X^{\\prime }, Y) - \\iota )^2 e^{u(X^{\\prime }, Y)}]\\\\&= \\int (z - \\iota )^2 e^z \\frac{1}{\\sqrt{4\\pi \\iota }} e^{-\\frac{(z+\\iota )^2}{4\\iota }} dz \\text{ (applying A2)}\\\\&= \\int (z - \\iota )^2 \\frac{1}{\\sqrt{4\\pi \\iota }} e^{-\\frac{(z-\\iota )^2}{4\\iota }} dz\\\\&= 2\\iota .$ We now proceed to derive the limiting moments.",
"We have $\\lim _{d \\rightarrow \\infty } \\textbf {E}[Z]&= \\lim _{d \\rightarrow \\infty } \\textbf {E}[ \\log p(Y|X^{\\prime }) - \\log p(Y|X)]\\\\&= \\lim _{d \\rightarrow \\infty } \\textbf {E}[ u(X^{\\prime }, Y) - u(X, Y) ] = -2\\iota .$ Also, $\\lim _{d \\rightarrow \\infty } \\text{Var}[Z]&= \\lim _{d \\rightarrow \\infty } \\text{Var}[ u(X^{\\prime }, Y) - u(X, Y) ]\\\\&= \\lim _{d \\rightarrow \\infty } \\text{Var}[ u(X^{\\prime }, Y)] +\\text{Var}[ u(X, Y) ]\\text{ (using assumption A4) }\\\\&= 4\\iota ,$ and similarly $\\lim _{d \\rightarrow \\infty } \\text{Cov}[Z_i, Z_j]&= \\lim _{d \\rightarrow \\infty } \\text{Var}[ u(X, Y)]\\text{ (using assumption A4) }\\\\&= 2\\iota .$ This concludes the proof.",
"$\\Box $ ."
],
[
"Assumptions of theorem 1",
"Assumptions A1-A4 are satisfied in a variety of natural models.",
"One example is a multivariate Gaussian model where $X \\sim N(0, \\Sigma _d)$ $E \\sim N(0, \\Sigma _e)$ $Y = X + E$ where $\\Sigma _d$ and $\\Sigma _e$ are $d \\times d$ covariance matrices, and where $X$ and $E$ are independent.",
"Then, if $d \\Sigma _d$ and $\\Sigma _e$ have limiting spectra $H$ and $G$ respectively, the joint densities $p(x, y)$ for $d = 1,\\hdots , $ satisfy assumptions A1 - A4.",
"We can also construct a family of densities satisfying A1 - A4, which we call an exponential family sequence model since each joint distribution in the sequence is a member of an exponential family.",
"A given exponential family sequence model is specified by choice of a base carrier function $b(x, y)$ and base sufficient statistic $t(x, y)$ , with the property that carrier function factorizes as $b(x, y) = b_x(x) b_y(y)$ for marginal densities $b_x$ and $b_y$ .",
"Note that the dimensions of $x$ and $y$ in the base carrier function are arbitrary; let $p$ denote the dimension of $x$ and $q$ the dimension of $y$ for the base carrier function.",
"Next, one specifies a sequence of scalar parameters $\\kappa _1, \\kappa _2,\\hdots $ such that $\\lim _{d \\rightarrow \\infty } d \\kappa _d = c < \\infty .$ for some constant $c$ .",
"For the $d$ th element of the sequence, $X^{[d]}$ is a $pd$ -dimensional vector, which can be partitioned into blocks $X^{[d]} = (X_1^{[d]},\\hdots , X_d^{[d]})$ where each $X_i^{[d]}$ is $p$ -dimensional.",
"Similarly, $Y^{[d]}$ is partitioned into $Y_i^{[d]}$ for $i = 1,\\hdots , d$ .",
"The density of $(X^{[d]}, Y^{[d]})$ is given by $p^{[d]}(x^{[d]}, y^{[d]}) = Z_d^{-1} \\left(\\prod _{i=1}^d b(x_i^{[d]}, y_i^{[d]}) \\right)\\exp \\left[\\kappa _d \\sum _{i=1}^d t(x_i^{[d]}, y_i^{[d]}) \\right],$ where $Z_d$ is a normalizing constant.",
"Hence $p^{[d]}$ can be recognized as the member of an exponential family with carrier measure $\\left(\\prod _{i=1}^d b(x_i^{[d]}, y_i^{[d]}) \\right)$ and sufficient statistic $\\sum _{i=1}^d t(x_i^{[d]}, y_i^{[d]}).$ One example of such an exponential family sequence model is a multivariate Gaussian model with limiting spectra $H = \\delta _1$ and $G = \\delta _1$ , but scaled so that the marginal variance of the components of $X$ and $Y$ are equal to one.",
"This corresponds to a exponential family sequence model with $b_x(x) = b_y(x) = \\frac{1}{\\sqrt{2\\pi }} e^{-x^2/2}$ and $t(x, y) = xy.$ Another example is a multivariate logistic regression model, given by $X \\sim N(0, I)$ $Y_i \\sim \\text{Bernoulli}(e^{\\beta X_i}/(1 + e^{\\beta X_i}))$ This corresponds to an exponential family sequence model with $b_x(x) = \\frac{1}{\\sqrt{2\\pi }} e^{-x^2/2}$ $b_y(y) = \\frac{1}{2}\\text{ for }y = \\lbrace 0, 1\\rbrace ,$ and $t(x, y) = x\\delta _1(y) - x\\delta _0(y).$ The multivariate logistic regression model (and multivariate Poisson regression model) are especially suitable for modeling neural spike count data; we simulate data from such a multivariate logistic regression model in section X."
],
[
"Additional simulation results",
"Multiple-response logistic regression model $X \\sim N(0, I_p)$ $Y \\in \\lbrace 0,1\\rbrace ^q$ $Y_i|X = x \\sim \\text{Bernoulli}(x^T B_i)$ where $B$ is a $p \\times q$ matrix.",
"Multiple-response logistic regression model $X \\sim N(0, I_p)$ $Y \\in \\lbrace 0,1\\rbrace ^q$ $Y_i|X = x \\sim \\text{Bernoulli}(x^T B_i)$ where $B$ is a $p \\times q$ matrix.",
"Methods.",
"Nonparametric: $\\hat{I}_0$ naive estimator, $\\hat{I}_\\alpha $ anthropic correction.",
"ML-based: $\\hat{I}_{CM}$ confusion matrix, $\\hat{I}_F$ Fano, $\\hat{I}_{HD}$ high-dimensional method.",
"Sampling distribution of $\\hat{I}$ for $\\lbrace p = 3$ , $B= \\frac{4}{\\sqrt{3}} I_3$ , $K = 20$ , $r = 40\\rbrace $ .",
"True parameter $I(X; Y) = 0.800$ (dotted line.)",
"Figure: NO_CAPTION Naïve estimator performs best!",
"$\\hat{I}_{HD}$ not effective.",
"Sampling distribution of $\\hat{I}$ for $\\lbrace p = 50$ , $B = \\frac{4}{\\sqrt{50}} I_{50}$ , $K = 20$ , $r = 8000\\rbrace $ .",
"True parameter $I(X; Y) = 1.794$ (dashed line.)",
"Figure: NO_CAPTION Non-parametric methods extremely biased.",
"Estimation path of $\\hat{I}_{HD}$ and $\\hat{I}_\\alpha $ as $n$ ranges from 10 to 8000.",
"$\\lbrace p = 10$ , $B = \\frac{4}{\\sqrt{10}} I_{10}$ , $K = 20\\rbrace $ .",
"True parameter $I(X; Y) = 1.322$ (dashed line.)",
"Figure: NO_CAPTION Estimated $\\hat{I}$ vs true $I$ .",
"Figure: NO_CAPTION Sampling distribution of $\\hat{I}_{HD}$ for $\\lbrace p = 10$ , $B = \\frac{4}{\\sqrt{10}} I_{10}$ , $N = 80000\\rbrace $ , and $K = \\lbrace 5, 10, 15, 20, \\hdots , 80\\rbrace $ , $r = N/k$ .",
"True parameter $I(X; Y) = 1.322$ (dashed line.)",
"Figure: NO_CAPTION Decreasing variance as $K$ increases.",
"Bias at large and small $K$ .",
"$p = 20$ and $q = 40$ , entries of $B$ are iid $N(0, 0.025)$ .",
"$K=20$ , $r = 8000$ , true $I(X; Y) = 1.86$ (dashed line.)",
"Sampling distribution of $\\hat{I}$ .",
"Figure: NO_CAPTION"
]
] | 1606.05229 |
[
[
"C II radiative cooling of the Galactic diffuse interstellar medium:\n Insight about the star formation in Damped Lyman-alpha systems"
],
[
"Abstract The far-infrared [C II] 158 micrometer fine structure transition is considered to be a dominant coolant in the interstellar medium.",
"For this reason, under the assumption of a thermal steady state, it may be used to infer the heating rate and, in turn, the star formation rate in local, as well as in high redshift systems.",
"In this work, radio and ultraviolet observations of the Galactic interstellar medium are used to understand whether C II is indeed a good tracer of the star formation rate.",
"For a sample of high Galactic latitude sightlines, direct measurements of the temperature indicate the presence of C II in both the cold and the warm phases of the diffuse interstellar gas.",
"The cold gas fraction (~ 10 - 50% of the total neutral gas column density) is not negligible even at high Galactic latitude.",
"It is shown that, to correctly estimate the star formation rate, C II cooling in both the phases should hence be considered.",
"The simple assumption, that the [C II] line originates only from either the cold or the warm phase, significantly underpredicts or overpredicts the star formation rate, respectively.",
"These results are particularly important in the context of the Damped Lyman-alpha systems for which a similar method is often used to estimate the star formation rate.",
"The derived star formation rates in such cases may not be reliable if the temperature of the gas under consideration is not constrained independently."
],
[
"Introduction",
"In the standard model of the Galactic diffuse interstellar medium (ISM), a balance of the heating and cooling processes leads to a thermal steady state [15], [16], [40], [66], [67].",
"Thus, in a multiphase medium, different phases coexist at different temperature but in an approximate thermal pressure equilibrium.",
"Broadly, the diffuse multiphase medium consists of the cold neutral medium (CNM), the warm neutral medium (WNM), the warm ionized medium (WIM) and the hot ionized medium (HIM).",
"The local physical conditions of the ISM are determined by a host of factors, including the local radiation field and cosmic ray energy density, the dust grain abundance, composition and size distribution, material and mechanical energy transfer from both impulsive disturbances such as the ejection of the outer mantle in the late stages of stellar evolution and supernova explosions, as well as more steady sources such as stellar winds.",
"Here we focus on understanding the [C ii] 158 $\\mu $ m fine structure cooling in the diffuse ISM.",
"Based on the assumption of thermal steady state, observation of C ii$^*$ absorption can be used to estimate the cooling/heating rate and, in turn, to infer the star formation rate (SFR).",
"In detail however, for a given estimated cooling rate, the inferred ultraviolet (UV) and cosmic ray flux (and thus, the SFR) change significantly depending on the assumed physical conditions of the gas.",
"Whether the gas is in the cold or warm phase, for example, is one of the important factors in this regard.",
"Thus, to estimate the SFR from C ii$^*$ absorption, direct measurement of the temperature of the diffuse ISM for the same lines of sight is also necessary.",
"In this work we present temperature measurements from Galactic H i 21 cm observations toward a sample of high latitude extra-galactic radio sources.",
"For these lines of sight, both H i 21 cm single dish emission spectra and UV spectroscopic data covering C ii$^*$ absorption are available.",
"Background motivation of this work is presented in §.",
"The details of observation and analysis techniques used here are described in §.",
"The results and relevant discussions are presented in § and §, respectively.",
"Finally, we summarize the conclusions in §.",
"Major sources of heating in the diffuse ISM are (i) photoelectric heating due to ejection of electrons from the dust grains by the far ultraviolet (FUV) radiation field, (ii) heating due to ionization by cosmic rays and soft X-rays, (iii) photoionization of species like C i, Si i, Fe i etc.",
"(for which the ionization potential is less than 13.6 eV) by the FUV radiation field and (iv) collisional ionization of H and He by impact with H and $e^-$ (e.g., [16], [25], [4], [66]; [61]).",
"Since both photoelectric heating and heating due to ionization by cosmic rays and X-rays are related to the interstellar radiation field, the total heating rate is a function of the SFR ($\\dot{\\psi _*})$ .",
"Dominant cooling mechanisms in the ISM, on the other hand, are (i) cooling by the fine structure lines of C i, C ii, O i, Si i, Si ii, S i, Fe i and Fe ii, (ii) metastable transitions of C ii, O i, Si ii, S ii etc.",
"(iii) collisional excitation of Lyman-$\\alpha $ and (iv) radiative recombination of $e^-$ onto dust grains and polycyclic aromatic hydrocarbons (PAHs) [4], [66], [61].",
"Various cooling mechanisms become important at different temperature and thus the total cooling rate depends on the physical conditions like temperature and density of the gas.",
"In thermal steady state, the total cooling rate is equal to the total heating rate, and the physical conditions in different phases can be deduced by considering the thermal and ionization equilibrium."
],
[
"Cooling by the [C",
"The [Cii] 158 $\\mu $ m transition is a dominant contributor to the cooling in the ISM because of (1) the high abundance of carbon (second most abundant metal in gas phase), (2) high abundance of its singly ionized stage, (3) relatively low optical depth of the transition, and (4) the easy excitation of the $^2P_{3/2}$ fine structure state ($h\\nu /k = 91$ K) by collisions under typical conditions in the diffuse ISM.",
"Please see [18] and references therein for an extensive review.",
"The C ii radiative cooling rate can be determined directly from the [C ii] line intensity of the $^2P_{3/2}$ to $^2P_{1/2}$ 157.7 $\\mu $ m transition in the far-infrared (FIR).",
"Alternately, the measured column density of C ii$^*$ per H i atom is also believed to be a direct measure of the cooling rate of the gas ([45]; [61]; [37]).",
"C ii$^*$ column density can be measured from the C ii$^*$ absorption lines at 1037.018 Å and 1335.708 Å in the FUV originating in the same $^2P_{3/2}$ state."
],
[
"C ",
"For thermal steady state condition, one can infer the heating rate from the measured Cii cooling rate.",
"The heating rate, in turn, depends on the fluxes of UV photons and cosmic rays, and thus on the SFR.",
"Based on this reasoning, observation of C ii or C ii$^*$ is often considered as a tracer of star formation in the local as well as in the high redshift Universe [6], [47], [57], [31], [44], [52], [53].",
"A similar argument has been used for high redshift damped Lyman-$\\alpha $ systems (DLAs) to deduce $\\dot{\\psi _*}$ [61], [63].",
"One of the main sources of uncertainty, while estimating the SFR for DLAs, is the physical condition, more importantly the temperature, of the gas.",
"For example, in absence of any direct measurement of temperature, the SFR calculations for the DLAs by [61] were based on the assumption that most of the gas giving rise to the C ii$^*$ absorption is in the CNM phase.",
"Otherwise, if it is assumed that all the gas is in the WNM phase, then the inferred SFR per unit area in DLAs is significantly higher than that of the Milky way.",
"On the other hand, [37] studied the C ii$^*$ absorption along high Galactic latitude extra-galactic sources, and, in conjunction with observations of the diffuse H$\\alpha $ emission along these same line of sight, concluded that most of the C ii$^*$ absorption occurs in the WNM or the WIM.",
"This conclusion, that most of the C ii$^*$ along high Galactic latitudes comes from the WNM/WIM, rests in large part on the assumption that there is negligible amount of gas in the CNM phase along these sight lines.",
"However, there exist several high latitude lines of sight with a high CNM fraction [48], [50], [51].",
"One way to critically re-examine this uncertainty regarding the inferred SFR, would be to directly measure the temperature of the diffuse H i in our Galaxy for lines of sight with C ii$^*$ absorption.",
"This will constrain the cold gas fraction for lines of sight with H i column density similar to that of DLAs.",
"With this, and the known C ii$^*$ column density, it will also be possible, following the same reasoning used for the DLAs, to estimate the SFR, and compare it with the Galactic SFR derived using other methods.",
"From the sample of [37] and [60] with UV spectroscopic observations, 15 sources were selected in the declination range accessible to the Giant Metrewave Radio Telescope [58] and the Karl G. Jansky Very Large Array [42].",
"Figure REF shows the position of the background sources in the Galactic coordinate system.",
"These are all high Galactic latitude sources ($|b|>20^\\circ $ ), with 1.4 GHz flux density greater than 100 mJy and a substantial flux in the compact components.",
"Table REF lists the names of the background sources, Galactic coordinates, interstellar reddening, and the column density of H i, H$_2$ and C ii$^*$ for these lines of sight.",
"High spectral resolution H i emission spectra along these lines of sight are already available from the Leiden/Argentine/Bonn (LAB) Galactic H i survey [21], [1], [3], [28].",
"The H i column densities from the LAB survey for these lines of sight are $(1.4 - 3.7)\\times 10^{20} {\\rm ~cm}^{-2}$ .",
"So, for extra-galactic sources, similar lines of sight will have $\\sim (3 - 7)\\times 10^{20} {\\rm ~cm}^{-2}$ H i column density - very much like the typical DLA lines of sight.",
"Figure REF shows the relation between total hydrogen column density and reddening for the lines of sight in our sample.",
"The Galactic reddening E(B-V) is derived from the infrared dust maps with recent recalibration [56], [55].",
"The total hydrogen column density N(H) = N(H i) + 2N(H$_2$ ) is based on LAB N(H i), and N(H$_2$ ) from [60].",
"The solid line corresponds to the expected value of N(H i)/E(B-V) = $8.3\\times 10^{21}{\\rm ~cm}^{-2} {\\rm mag}^{-1}$ [38].",
"This gas column density to reddening ratio is $\\sim 40$ % higher than the corresponding value derived, e.g., by [5], from optical/UV observations.",
"However, for this high latitude sample, it is more appropriate to instead compare with the value derived from the radio/IR observations for similar lines of sight by [38].",
"From Figure REF , reddening for these lines of sight seems to have no unusual deviation from the typical dust to gas ratio."
],
[
"Radio observations and the data reduction",
"The GMRT observations were carried out in cycle 8 (2005) for five of the sources using a total 1.0 MHz bandwidth with 128 spectral channels (i.e.",
"a velocity resolution of $\\sim $ 1.6 km s$^{-1}$ ).",
"The VLA B-configuration observations (project code 12A-428) for ten sources were carried out in 2012 using 256 channels over 1.0 MHz bandwidth (i.e.",
"$\\sim $ 0.8 km s$^{-1}$ per channel).",
"Depending on the target continuum flux density, on-source time was from 20 minutes to 6 hours.",
"Short scans on calibrator sources were used for flux calibration, phase calibration and also to determine the bandpass shape.",
"Unfortunately, a significant amount of data are affected by interference, and has to be excluded.",
"Standard data analysis including flagging bad data, calibration, and imaging was done using the Astronomical Image Processing System (AIPS; produced and maintained by the National Radio Astronomy Observatory).",
"The continuum emission, estimated by averaging data from line-free channels, was subtracted from the multi-channel visibility data.",
"The residual data were then used to make the image cubes, and any small residual continuum was subtracted in the image plane by fitting a linear baseline to the line-free regions.",
"The absorption spectra toward the compact component were then extracted from the high resolution image cubes where the smooth H i emission was resolved out.",
"Finally, the absorption spectra were converted from flux density to optical depth ($\\tau $ ) using the 1.4 GHz flux density value at the corresponding location of the continuum image.",
"The H i emission and absorption spectra for all 15 lines of sight are shown in Figure REF .",
"For each line of sight, the top and the bottom panels show the LAB H i emission spectra and the GMRT or VLA H i absorption spectra, respectively.",
"Name of the background continuum source and the telescope names are also mentioned at the top.",
"For 10 out of these 15 lines of sight, H i 21 cm absorption is clearly detected.",
"For the detections, (multi-)Gaussian model of the absorption spectra are also overplotted in Figure REF ."
],
[
"Temperature estimation",
"In the radio regime, the classical method to determine the temperature of the gas consists of observing the H i 21 cm line in absorption towards a bright radio continuum source, and 21 cm emission spectrum along a nearby line of sight.",
"H i emission and absorption spectra allow one to measure the spin temperature [35], which is often used as a proxy for the kinetic temperature (T$_{\\rm k}$ ) of the gas.",
"For the CNM, T$_{\\rm s}$ is expected to be tightly coupled to T$_{\\rm k}$ via collisions [14].",
"It is also possible to estimate the temperature from the observed linewidth of the H i emission and absorption components.",
"This method is more useful for absorption spectra where the opacity is additive for a multi-Gaussian component fit.",
"For emission spectra, the relative position of different components along the line of sight being a-priori unknown, multi-component decomposition is more complicated, and often there is no straightforward and unique interpretation.",
"Due to possible non-thermal broadening, the observed linewidth provides only an upper limit to T$_{\\rm k}$ .",
"For the 10 cases with a detection in the present sample, Gaussian components are fitted to the absorption spectra.",
"Table REF presents the integrated H i column densities (from the LAB survey), the integrated optical depth values (or upper limits) from this study, and also shows the best fit parameter values (peak optical depth, centre and width of the components) for all the spectra.",
"The width of each component then provides an upper limit T$_{\\rm k, max}$ .",
"We also used the emission and the absorption spectra to compute spin temperature spectra (at a resolution of $\\sim 1.0$ and 1.6 km s$^{-1}$ for VLA and GMRT sample, respectively).",
"The local minimum of spin temperature, T$_{\\rm s, min}$ from these spectra over the velocity range of any absorption component is taken as an estimator of T$_{\\rm s}$ for the corresponding “cold” component.",
"For velocity intervals with only H i emission (and corresponding C ii$^*$ absorption; see below), but no detections of H i absorption, the same method is used to estimate the lower limit of T$_{\\rm s}$ from the $3\\sigma $ upper limit of the optical depth.",
"Please note that the H i emission may have some contribution due to blending with components unrelated to the absorption.",
"Hence, in general, using T$_{\\rm s, min}$ as an estimator of T$_{\\rm s}$ may result in an overestimation of temperature.",
"Thus, low T$_{\\rm s}$ values will be a conservative indicator of cold gas.",
"Table: Details of the absorption spectra and the Gaussian fit parametersFigure: Estimated H i spin temperature (T s _{\\rm s}) of the cold component, and the ortho-para temperature (T 01 _{\\rm 01}) of molecular hydrogen vs. N(H i) for different velocity components.",
"T s _{\\rm s} measurements, from this work, are shown as filled circles with errorbars (and with arrow for lower limits).",
"T 01 _{\\rm 01} from for corresponding components are shown as open squares.Figure: Fraction of cold gas for the lines of sight estimated from H i absorption.",
"For non-detection of absorption component, an average value of T s =200_{\\rm s} = 200 K is adopted to compute the upper limit of the cold gas column density over a velocity range of V 90 V_{90} of the corresponding LAB spectra.Figure REF presents the summary of T$_{\\rm s}$ measurements for this sample.",
"Considering H i 21 cm emission and absorption spectra, along with the C ii$^*$ absorption spectra, for these 15 lines of sight there are 21 components with distinct velocity range, with 12 cases of detection of H i absorption, and 9 cases of non-detection.",
"The values (and the lower limits) of T$_{\\rm s}$ , derived as outlined above, are plotted (filled circles with errorbars, with arrow for lower limits) against N(H i) for these 21 components.",
"Eight out of 10 components with detected H i absorption have T$_{\\rm s}$ below 200 K, and only one component is above 300 K with T$_{\\rm s}\\approx 490 \\pm 110$ .",
"The upper limits of T$_{\\rm s}$ are not very tight due to low T$_{\\rm B}$ and/or high RMS $\\tau $ .",
"Overall, however, the presence of cold ISM with temperature $\\lesssim 200$ K for these lines of sight is very clear from the T$_{\\rm s}$ measurements.",
"As expected for the turbulent ISM, for all the absorption components, T$_{\\rm k, max}$ is always higher than T$_{\\rm s}$ due to non-thermal broadening.",
"For 16 out of these 21 components, [60] presents measurements of column density, and the “ortho-para temperature” (T$_{\\rm 01}$ ) of molecular hydrogen covering the same velocity range as the diffuse H i. T$_{\\rm 01}$ is coupled to the kinetic temperature or the formation temperature of H$_2$ depending on whether the observed ortho-to-para ratio is achieved mainly by proton and hydrogen exchange collision or by reactions in the dust grain surface [9], [59].",
"For the diffuse ISM with conditions similar to these lines of sight, T$_{\\rm 01}$ is expected to trace the kinetic temperature [48].",
"Measured T$_{\\rm 01}$ for these 16 components, shown in Figure REF as open squares, also consistently indicate the presence of gas with temperature around 200 K for all of these lines of sight."
],
[
"Cold and warm gas fraction",
"Next, we use this estimated T$_{\\rm s}$ of the detected CNM absorption components for each line of sight to compute the column density in the cold phase, and fraction of CNM (using the total H i column density from the LAB spectrum).",
"Note that the blending effect mentioned above may cause a similar overestimation of CNM column density as well.",
"Also, the total H i column density from the LAB emission spectrum is derived assuming optically thin condition ($\\tau <<1$ ).",
"For lines of sight with large optical depth, this may result in an underestimation of the column density [7].",
"For these low optical depth lines of sight, however, this assumption is reasonable, and the correction to the total H i column density due to optical depth is negligible.",
"For non-detection, the upper limit of the CNM column density is computed assuming an average T$_{\\rm s} = 200$ K and a velocity width same as the $V_{90}$ of the corresponding LAB emission spectra.",
"The derived CNM and WNM column densities are given in Table REF , and the CNM fraction for the sample is shown in Figure REF .",
"Although in one case the cold gas fraction is as high as $\\sim 50\\%$ , most of these lines of sight contains only $\\sim 10-30\\%$ CNM.",
"In §, we further probe any plausible correlation between the C ii$^*$ column density and the CNM/WNM/total H i column density to understand if C ii cooling happens preferably in the cold or warm phase."
],
[
"The UV data analysis",
"UV spectroscopy for these lines of sight with the Far Ultraviolet Spectroscopic Explorer (FUSE) and the Space Telescope Imaging Spectrograph (STIS) on board the Hubble Space Telescope (HST) is reported earlier by [37] and [60].",
"However, for nine of these sightlines, archival data from the HST Cosmic Origin Spectrograph (COS) covering the C ii$^*$ 1335.708 Å transition is now also available.",
"Hence we used the COS data for these nine lines of sight, and adopted the column density values from literature for the remaining six lines of sight.",
"For the COS data, after rebinning to a common wavelength grid, and coadding individual exposures, we fit a local continuum to each spectrum at the location of the Galactic C ii 1334Å and C ii$^*$ 1335Å absorption, and then measure both the restframe equivalent width as well as the column density, line-centre and Doppler parameter b for the C ii$^*$ component via a Voigt profile fit.",
"The observed spectra as well as the best fit models are shown in Figure REF , and the total C ii$^*$ column densities from these fits are presented in Table REF .",
"In some cases, certain fit parameters had to be fixed due to the complicated velocity structures and/or blending.",
"Note that, for FUSE and STIS spectra of these sources, [37] employ two methods (equivalent width based on the curve of growth and apparent optical depth) very different from ours (Voigt profile fit), and therefore some discrepancy in the values may be expected."
],
[
"H ",
"The H i spin temperature measurements shows the presence of $\\sim 10-30\\%$ cold gas along these lines of sight at the same velocities as the C ii$^*$ absorption components.",
"This, however, does not mean that C ii$^*$ coexists with only cold gas.",
"One way to investigate if C ii cooling takes place preferably in the cold or warm phase is to check how the derived total C ii$^*$ column density depends on the cold and/or warm H i column density for these lines of sight.",
"If the C ii$^*$ absorption arises dominantly in cold or warm phase, then one will expect a tight correlation between N(C ii$^*$ ) and N(H i)$_{\\rm CNM}$ or N(H i)$_{\\rm WNM}$ , respectively.",
"Alternatively, if the C ii$^*$ abundance is similar in the cold and warm phases, there will be stronger correlation between N(C ii$^*$ ) and the total N(H i).",
"The three panels in Figure REF show the total C ii$^*$ column density along the lines of sight of this sample with respect to the cold, warm and the total H i column density (left, middle and right panel, respectively).",
"We do not see any obvious strong correlation in these plots.",
"This is expected when a significant fraction of C ii$^*$ is in the WIM phase, as suggested by, e.g.",
"[37].",
"A careful and quantitative statistical analysis, however, shows that the correlation is relatively stronger between N(C ii$^*$ ) and the total N(H i) compared to the other two.",
"The Kendall $\\tau $ coefficient is $0.56$ for the correlation with the total N(H i) with a two-sided p-value of $0.0087$ .",
"The $\\tau $ coefficients for N(H i)$_{\\rm CNM}$ and N(H i)$_{\\rm WNM}$ are $0.11$ (p $= 0.80$ ) and $0.40$ (p $= 0.21$ ), respectively.",
"We also computed the Spearman's rank correlation coefficients.",
"The correlation coefficient in this case is $\\rho = 0.74$ for the total N(H i), but $0.12$ and $0.51$ for the cold and warm gas column density, respectively.",
"Clearly, the correlation is relatively tighter for the total N(H i).",
"The above values of $\\tau $ , p and $\\rho $ are computed using lines of sight excluding the non-detections.",
"Including the non-detections (i.e.",
"treating the $3\\sigma $ limits as “measured values”), and/or excluding NGC 5236 (for which C ii$^*$ column density measurement was problematic) in the analysis, do not change the results significantly.",
"Even then, N(C ii$^*$ ) shows a relatively tighter correlation with the total N(H i).",
"This correlation, though not visually obvious, indicates that the C ii$^*$ abundance is not significantly different in the cold and warm phases.",
"As a simple consistency check, we considered the correlation between N(C ii$^*$ ) and N(H i) for individual velocity components.",
"The result is shown in Figure REF .",
"Here also the data are consistent with a significant correlation between the C ii$^*$ column density and the total H i column density (Kendall $\\tau = 0.64$ with p $=5.383\\times 10^{-5}$ using generalized Kendall's $\\tau $ test to include the non-detectionsThis is done using the Nondetects And Data Analysis (`NADA') package [36] in the R statistical software environment [46] available from the Comprehensive R Archive Network (CRAN) site.",
"), and with no obvious separation of CNM/WNM components.",
"This again suggests a similar abundance of C ii$^*$ in the cold and warm phases.",
"The thin blue line in Figure REF shows a constant value of N(C ii$^*$ )/N(H i) = $4.7\\times 10^{-7}$ corresponding to the median abundance of the sample, and the thick magenta line is the best fit Akritas-Thiel-Sen regression line calculated in NADA by consistently including the non-detections also.",
"For the range of column densities of our interest, as shown in Figure REF , these two do not differ much.",
"As an extension to this, we have also tried a multivariate linear regression analysis to separate out CNM, WNM and WIM contribution to the total observed C ii$^*$ column density.",
"We have used the measured CNM and WNM column densities from this study, and the H$\\alpha $ intensities from the Wisconsin H-Alpha Mapper (WHAM) Survey [20] as a proxy for WIM column densities.",
"This analysis suggests a relation $\\frac{N(C II^*)}{10^{13}~cm^{-2}} &=& (1.43 \\pm 0.84)\\times \\frac{N(CNM)}{10^{19}~cm^{-2}}\\nonumber \\\\& + & (0.49 \\pm 0.12)\\times \\frac{N(WNM)}{10^{19}~cm^{-2}}\\nonumber \\\\& + & (0.02 \\pm 0.49)\\times \\frac{I(H\\alpha )}{1~Rayleigh}$ indicating a very weak dependence on N(WIM)in contrast to the [37] assertion that most of the C ii$^*$ is in the WIM phase.",
"The coefficient for N(CNM) also has a large uncertainty (effectively consistent with zero at $< 2\\sigma $ level), indicating strongest ($> 4\\sigma $ ) dependence of N(C ii$^*$ ) on N(WNM).",
"However, if we drop the dependence on I(H$\\alpha $ ), the regression analysis results in $\\frac{N(C II^*)}{10^{13}~cm^{-2}} &=& (0.59 \\pm 0.12)\\times \\frac{N(CNM)}{10^{19}~cm^{-2}}\\nonumber \\\\& + & (0.10 \\pm 0.31)\\times \\frac{N(WNM)}{10^{19}~cm^{-2}}$ implying most of the C ii$^*$ to be existing in the CNM.",
"In this case, the coefficient for N(WNM) has a large uncertainty and a low statistical significance.",
"The inconsistent and contradictory result of this analysis may be due to the small number of lines of sight in our study, and it should be carried out for a larger sample to properly separate out C ii$^*$ column densities in different ISM phases.",
"As the multivariate regression analysis remains inconclusive in this case, based on the observed correlations mentioned earlier, the abundance of C ii$^*$ is assumed to be same in the cold and warm neutral medium for all further analysis."
],
[
"The estimated star formation rate",
"We have also derived, by a method similar to that used for the DLAs, the SFR per unit area from the observed quantities for our lines of sight.",
"An accurate estimation of the interstellar radiation field, and therefore the SFR from the observed C ii$^*$ and H i column density, involves computation of thermal and ionization equilibrium conditions in the presence of all of the relevant heating and cooling mechanisms.",
"This is possible only via detailed numerical simulations, which is beyond the scope of the present work.",
"Rather we used some simplifying assumptions, and results from existing simulations [66], [67] to estimate an approximate value of the SFR.",
"So, the following result should be considered as an order of magnitude consistency check.",
"Table: Summery of the resultsFollowing [45], the cooling rate due to the [Cii] 158 $\\mu $ m fine structure transition for a given C ii$^*$ column density N(C ii$^*$ ) and a H i column density N(H i) is given by $l_{c}&=&{\\rm N(C~II^*)}h\\nu _{ul}A_{ul}/{\\rm N(H~I)} \\\\&=&2.89 \\times 10^{-20} {\\rm N(C~II^*)}/{\\rm N(H~I)}~~ {\\rm ergs~s}^{-1}({\\rm ~H~atom})^{-1} \\nonumber $ where $h\\nu _{ul}$ and $A_{ul}$ are the upper level energy and coefficient for spontaneous decay of the $^2P_{3/2}$ to $^2P_{1/2}$ transition.",
"On the other hand, the photoelectric heating rate is a function of the FUV field [4], [66], [67], and given by $\\Gamma _{\\rm d} = 1.3 \\times 10^{-24} \\epsilon G_0 ~~ {\\rm ergs~s}^{-1}({\\rm ~H~atom})^{-1}$ where $\\epsilon $ is the heating efficiency.",
"The FUV field strength $G_0$ , normalized to the local interstellar value [19], is proportional to the SFR per unit area.",
"For an SFR of $\\log _{10}\\dot{\\psi _*} = -2.4$ M$_{\\odot }$ yr$^{-1}$ kpc$^{-2}$ , $G_0 = 1.7$ [12].",
"To estimate $\\dot{\\psi _*}$ using the total N(H i) from the LAB survey, and the CNM/WNM column density from this work, we now consider three situations where these lines of sight have (i) only cold gas (“CNM model”), (ii) only warm gas (“WNM model”), and (iii) a mix of CNM and WNM (“two-phase model”) with the same abundance ratio of C ii$^*$ to H i column density.",
"The photoelectric heating rate from dust grains is approximately equal to the [C ii] fine structure cooling rate in the CNM.",
"On the other hand, the [C ii] fine structure cooling rate in the WNM is about an order of magnitude lower than the photoelectric heating rate.",
"On average, the photoelectric heating accounts for $\\approx 60 - 65\\%$ of the total heating rate.",
"The heating efficiencies are $\\epsilon \\sim 0.05$ and $\\sim 0.005$ for the cold and the warm phase, respectively [66].",
"Putting these numbers together, and using the observed N(C ii$^*$ ) and N(H i) in the cold and warm phases, the SFR per unit area is estimated for our lines of sight (see Table REF ).",
"As shown in Figure REF , $\\dot{\\psi _*}$ is about $0.1 - 2 \\times 10^{-3}$ M$_{\\odot }$ yr$^{-1}$ kpc$^{-2}$ for the “CNM model”, and about $1 - 24\\times 10^{-2}$ M$_{\\odot }$ yr$^{-1}$ kpc$^{-2}$ for the “WNM model”.",
"The median values are $0.37 \\times 10^{-3}$ and $4.1 \\times 10^{-2}$ M$_{\\odot }$ yr$^{-1}$ kpc$^{-2}$ for the CNM and WNM model, respectively.",
"For the “two-phase model”, the estimated range is about $2 - 35 \\times 10^{-3}$ M$_{\\odot }$ yr$^{-1}$ kpc$^{-2}$ , with a median value of $14.6 \\times 10^{-3}$ M$_{\\odot }$ yr$^{-1}$ kpc$^{-2}$ .",
"Clearly, the “two-phase model” provides a relatively better match with the Milky Way value of $\\dot{\\psi _*} \\sim 4 \\times 10^{-3}$ M$_{\\odot }$ yr$^{-1}$ kpc$^{-2}$ derived from other independent observations [32], [33].",
"The assumption, that all the C ii$^*$ absorption arises in the WNM or CNM, respectively overpredicts or underpredicts $\\dot{\\psi _*}$ by more than an order of magnitude.",
"There are studies [11], [22], based mostly on nearby galaxy samples, to calibrate C ii surface brightness and/or luminosity against other independent measurements of SFR.",
"For a comparison, we have also used such correlation to derive SFR for our sample.",
"This is done by first converting C ii$^*$ column density to a cooling rate (using equation 3), and then converting it to $\\dot{\\psi _*}$ using equation 2 from [22].",
"The estimated range of $\\dot{\\psi _*}$ , shown in Figure REF , is $\\sim 0.2 - 10.5 \\times 10^{-4}$ M$_{\\odot }$ yr$^{-1}$ kpc$^{-2}$ , with a median (mean) value of $0.84$ ($1.56$ ) $\\times 10^{-4}$ M$_{\\odot }$ yr$^{-1}$ kpc$^{-2}$ .",
"Similar to that of the “CNM model”, this underpredicts $\\dot{\\psi _*}$ by more than an order of magnitude.",
"This discrepancy is not surprising because the high Galactic latitude lines of sight of our sample may have different physical conditions (including cold/warm phase fraction) than the ones used to derive the relation.",
"Similarly, without considering these properties in detail, such relations will not be readily useful in estimating the SFR for the high redshift DLAs as well."
],
[
"Discussion",
"It has been earlier established by [37] that the WIM is one of the major ISM components along high Galactic latitude lines of sight with C ii$^*$ absorption.",
"The current study establishes from direct H i observations that the lines of sight certainly pass through a non-negligible fraction of CNM, coexisting in the same velocity range as that of the C ii$^*$ absorption.",
"Computing the SFR, assuming the ISM to be in the warm phase (without taking the CNM fraction into consideration), results in a higher value of $\\dot{\\psi _*}$ .",
"However, even after including the CNM fraction in the calculation, the median value of $\\dot{\\psi _*}$ is more than a factor of three higher than the average SFR of the Milky Way.",
"This mismatch is not surprising because, in our simplified order of magnitude calculation of the SFR, we have neglected the WIM component which is likely to contain a significant fraction of the observed C ii$^*$ column density [37].",
"Also, the derived values of $\\dot{\\psi _*}$ vary by a factor of about 20 for different lines of sight.",
"Earlier studies have suggested a correlation between [C ii] luminosity and the SFR both at galaxy scale [6], [10], as well as at smaller scales [41], [31], [44].",
"But, it is also well-known that the scatter in this relation is fairly large [39], [33], [11], [22].",
"For a sample of galaxies, [10] found that the data are consistent with C ii primarily being associated with cold ISM; [43] found only $\\sim 4\\%$ of the C ii to be from ionized gas in the Galactic plane.",
"[44], on the other hand, concluded that multiple ISM components significantly contribute to the [C ii] luminosity of the Milky Way disk (but also see [17] who concluded that, in the Galactic plane, C ii has the same filling factor as that of the CNM).",
"[44] have suggested that the [C ii] luminosity – SFR scaling relation is different for different ISM phases.",
"This may be due to varying energetics, or different timescale of SFR that C ii is sensitive to in these various phases [31].",
"When averaged over galaxy scale, this gives rise to the observed extra-galactic scaling.",
"Considering all these aspects, the large scatter in the derived $\\dot{\\psi _*}$ for our sample is not surprising.",
"Also, instead of averaging over galactic scale, or restricting to the CNM dominated disk of the Galaxy, here we are rather considering high Galactic latitude lines of sight with larger WNM and WIM fraction.",
"So, in principle, a different N(C ii) - SFR relation for this sample is quite possible.",
"There are two more possible reasons for the large scatter in estimated values of $\\dot{\\psi _*}$ for this sample.",
"The Galactic value of $\\dot{\\psi _*} \\sim 3.3- 4 \\times 10^{-3}$ M$_{\\odot }$ yr$^{-1}$ kpc$^{-2}$ [32], [8] is only an average value over the entire extent of the disk.",
"In reality, there is spatial variation of the SFR [33].",
"The other possibility is related to the basic assumption of the thermal steady state itself.",
"Recently, theoretical as well as observational studies have raised doubts on the validity of the steady state model itself [29], [23], [24], [2], [51], [54], [34].",
"There are evidences that much of the “warm gas” is in the so-called unstable phase with temperature lower than the WNM temperature of the classical model.",
"More detailed studies are necessary to understand how this may affect the [C ii] luminosity – SFR scaling relation.",
"What are the possible connections of these results with the SFR for the DLAs?",
"It is indeed true that, for the DLAs, the Cii cooling rate is not entirely balanced by the background radiation only, and requires contribution from a local radiation field which, in turn, is related to the star formation in these systems [62], [65], [13].",
"Interestingly, the C ii cooling rate itself shows a bimodal distribution.",
"This is proposed to be related to the mode of heating - in situ star formation for the `low cool' population (cooling rate $< 10^{-27}$ ergs s$^{-1}$ H$^{-1}$ ), and the star formation in the central Lyman Break Galaxy for the `high cool' population [65].",
"However, to estimate the SFR, the cold fraction of the gas along the line of sight still remains an important parameter (particularly for the low cool systems).",
"As shown here, for the sample of these DLA-like lines of sight, the inferred SFR is not a good estimator of the Galactic average value of the true $\\dot{\\psi _*}$ without a proper characterization of the phases of the ISM in consideration.",
"Hence, a more detailed analysis of the energetics of the individual systems [13], as well as direct and unambiguous measurements of the temperature [26], [27], [49], [30] is necessary to get a handle on the issue (please also see [64] for a comprehensive discussion)."
],
[
"Conclusions",
"As the [C ii] fine structure transition is one of the main cooling mechanisms in the ISM, the observed C ii column density is often used, for local and high redshift systems (including DLAs), to infer the SFR.",
"Here, we have used radio and UV data to study the Galactic ISM along a sample of high Galactic latitude “DLA-like” sightlines with C ii$^*$ absorption.",
"The H i 21 cm absorption and emission spectra were used to directly constrain the temperature of the gas along these lines of sight.",
"A good fraction ($\\sim 10 - 50\\%$ ) of the neutral gas, even for these high latitude lines of sight, is in the cold phase.",
"The correlation of the C ii$^*$ column density is tighter with the total H i column density (compared to that with only the cold or the warm gas column density).",
"This suggests that C ii is coexisting with both the CNM and the WNM.",
"The derived SFR values, with the assumption that the [C ii] fine structure cooling is important only in either the cold or the warm phase, do not match with the known value of the Galactic SFR.",
"We conclude that C ii may not be a reliable tracer of star formation without adequate constraints on the temperature of the gas where the cooling is happening.",
"More such multiwavelength studies of the Milky Way and other nearby systems, probing a wide range of ISM conditions, will be useful to better understand the connection between C ii and star formation for the DLAs.",
"This work is dedicated to the memory of A. M. Wolfe, who deceased on 2014 February 17, at the age of 74.",
"We thank the anonymous reviewer for useful comments and suggestions that helped us improving the quality of this manuscript significantly.",
"We are grateful to Dr. Eric Feigelson for his valuable inputs to improve the statistical analysis.",
"We are also grateful to Jayaram N. Chengalur, Raghunathan Srianand, Eric Greisen, Nissim Kanekar and Rajaram Nityananda for useful discussions, and to Andreas Brunthaler for his comments on an earlier version of this paper.",
"We thank the Center for Astrophysics and Space Sciences, University of California, San Diego (CASS/UCSD) for prompt response and for permitting us to list late Prof. A. M. Wolfe as a coauthor.",
"NR acknowledges support from MPIfR and the Alexander von Humboldt Foundation during his stay at MPIfR where part of this research was carried out.",
"This research has made use of NASA's Astrophysics Data System.",
"We thank the staff of the GMRT and the VLA who have made these observations possible.",
"The GMRT is run by the National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research.",
"The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.",
"The Wisconsin H-Alpha Mapper is funded by the National Science Foundation.",
"Some of the data used in this paper were obtained from the Leiden/Argentina/Bonn Galactic H i survey.",
"The results reported here are also based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the data archive at the Space Telescope Science Institute.",
"STScI is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract NAS 5-26555."
]
] | 1606.04966 |
[
[
"Spinodal instabilities of baryon-rich quark matter in heavy ion\n collisions"
],
[
"Abstract Using the test-particle method to solve the transport equation derived from the Nambu-Jona-Lasino (NJL) model, we study how phase separation occurs in an expanding quark matter like that in a heavy ion collision.",
"To test our method, we first investigate the growth rates of unstable modes of quark matter in a static cubic box and find them to agree with the analytical results that were previously obtained using the linear response theory.",
"In this case, we also find the higher-order scaled density moments to increase with time and saturate at values significantly larger than one, which corresponds to a uniform density distribution, after the phase separation.",
"The skewness of the quark number event-by-event distribution in a small sub-volume of the system is also found to increase, but this feature disappears if the sub-volume is large.",
"For the expanding quark matter, two cases are considered with one using a blast-wave model for the initial conditions and the other using initial conditions from a mulple-phase transport (AMPT) model.",
"In both cases, we find the expansion of the quark matter is slowed down by the presence of a first-order phase transition.",
"Also, density clumps appear in the system and the momentum distribution of partons becomes anisotropic , which can be characterized by large scaled density moments and non-vanishing anisotropic elliptic and quadrupolar flows, respectively.",
"The large density fluctuations further lead to an enhancement in the dilepton yield.",
"In the case with the AMPT initial conditions, the presence of a first-order phase transition also results in a narrower rapidity distribution of partons after their freeze out.",
"These effects of density fluctuations can be regarded as possible signals for a first-order phase transition that occurs in the baryon-rich quark matter formed in relativistic heavy ion collisions."
],
[
"Introduction",
"Studying the properties of baryon-rich quark-gluon plasma (QGP) is the main focus of the beam energy scan (BES) experiments [1], [2], [3] at the Relativistic Heavy Ion Collider (RHIC) as well as at the future Facility for Antiproton and Ion Research (FAIR).",
"These experiments are expected to shed light on whether the phase transition from the baryon-rich QGP to the hadronic matter is a first-order one and the location of the critical end point in the QCD phase diagram if the phase transition is first-order.",
"To help understand what could happen in a baryon-rich QGP, we have recently used the Polyakov-Nambu-Jona-Lasinia (PNJL) model to study its spinodal instability [4].",
"We have found via the linear response theory that the spinodal boundary in the temperature and density plane of the QCD phase diagram shrinks with increasing wave number of the unstable mode and is also reduced in the absence of the Polyakov loop.",
"In the small wave number or long wavelength limit, the spinodal boundary coincides with that determined from the isothermal spinodal instability in the thermodynamic approach.",
"We have further found that the quark vector interaction suppresses unstable modes of all wave numbers.",
"For the wave number dependence of the growth rate of unstable modes, it initially increases with the wave number but decreases when the wave number is large.",
"For the collisional effect from quark scattering, we have included it via the linearized Boltzmann equation and found it to decrease the growth rate of unstable modes of all wave numbers.",
"In the present study, we continue the above work by investigating how unstable modes would grow if one goes beyond the linear response or small amplitude limit.",
"This is carried out by using the transport equation derived from the NJL model to study the time evolution of density fluctuations in a confined as well as in an expanding quark matter.",
"Specifically, we study the time evolution of higher-order density moments in the quark matter, the distribution of quark number in a sub-volume of the quark matter, the quark momentum anisotropy, and dilepton production rate from quark-antiiquark annihilation.",
"As shown below, these observables could serve as signatures for a first-order phase transition of the baryon-rich quark matter produced in heavy ion collisions.",
"The paper is organized as follows: In the next section, we give a brief review on the NJL model and the transport equations based on its Lagrangian.",
"The transport equations are solved by the test-particle method in Section III to study both the short and long time behavior of the spinodal instability of a quark matter in a periodic box.",
"The same method is applied in Section IV to an expanding quark matter to study how density fluctuations are affected by the expansion of the system as in heavy ion collisions.",
"Finally, a summary is given in Section V. In the Appendix, we describe in detail the effect due to the finite grids used in the numerical calculations on the growth rate of unstable modes."
],
[
"The NJL Lagragian And The Transport Model",
"The NJL Lagrangian containing only the scalar interaction for three quark flavors has the form [5]: $&&\\mathcal {L}^S_{NJL}=\\bar{q}(i\\lnot {\\partial }-m_0)q+\\frac{G_S}{2}\\sum _{a=0}^{8}\\bigg [(\\bar{q}\\lambda ^aq)^2+(\\bar{q}i\\gamma _5\\lambda ^aq)^2\\bigg ]\\nonumber \\\\&&\\quad -K\\bigg [{\\rm det}_f\\bigg (\\bar{q}(1+\\gamma _5)q\\bigg )+{\\rm det}_f\\bigg (\\bar{q}(1-\\gamma _5)q\\bigg )\\bigg ],$ where $q=(u,d,s)^T$ , $m_0={\\rm diag}(m_{0u}, m_{0d}, m_{0s})$ and $\\lambda ^a$ are the Gell-Mann matrices for $a=1,2 \\cdots 8$ , with $\\lambda ^0$ being the identity matrix multiplied by $\\sqrt{2/3}$ .",
"The Lagrangian preserves $U(1)\\times SU(N_f)_L\\times SU(N_f)_R$ symmetry but breaks the axial symmetry, which is broken in QCD by the axial anomaly, by the Kobayashi-Masakawa-t'Hooft (KMT) interaction given by the last term in Eq.",
"(REF ) [6].",
"The ${\\rm det}_f$ in this term denotes the determinant in the flavor space [7], that is ${\\rm det}_f (\\bar{q}\\Gamma q)=\\sum _{i,j,k}\\varepsilon _{ijk}(\\bar{u}\\Gamma q_i)(\\bar{d}\\Gamma q_j)(\\bar{s}\\Gamma q_k),$ where $\\Gamma $ denotes either a Dirac gamma or the identity matrix.",
"The determinantal term is responsible for obtaining the correct splitting in the masses of $\\eta $ and $\\eta ^\\prime $ mesons.",
"Because the NJL model is not renormalizable, a regularization scheme is required to remove infinities in the momentum integrations.",
"In this study, we assume that all interactions are among quarks of 3-momenta with magnitudes below the cutoff momentum $\\Lambda $ .",
"Taking $\\Lambda =0.6023~\\mathrm {GeV}$ , the values of the scalar coupling $G_S$ and the KMT interaction $K$ can be determined from fitting the pion mass, the kaon mass, and the pion decay constant, and their values are $G_S\\Lambda ^2=3.67$ , and $K\\Lambda ^5=12.36$ if the current quark masses are taken to be $m_{0u}=m_{0d}=3.6$ and $m_{0s}=87$ MeV [8].",
"A flavor-singlet vector interaction can be added to the NJL Lagrangian as follows: $\\mathcal {L}^V_{NJL}=-G_V(\\bar{q}\\gamma ^\\mu q)^2,$ where the coupling strength $G_V$ is assumed to be independent of the temperature $T$ and the net quark chemical potential $\\mu $ .",
"The value of $G_V$ affects the order of quark matter phase transition.",
"If $G_V$ is large, the first-order phase transition induced by the attractive scalar interaction could disappear [4].",
"In the present study, we treat it as a parameter to change the equation of state of quark matter.",
"For describing the quark matter produced in a heavy ion collision, we use the Boltzmann (or transport) equations that can be derived from the NJL Lagrangian in terms of the non-equilibrium Green's functions for quarks and antiqaurks [9], and they are: $&&\\partial _{X^0}f_a(X,\\mathbf {p})+\\frac{p^{i\\pm }}{E_{\\mathbf {p}^\\pm }}\\partial _{X^i}f_a(X,\\mathbf {p})\\nonumber \\\\&&-\\partial _{X^i}V_a^S(X)\\frac{M_a}{E_{\\mathbf {p}^\\pm }}\\partial _{p_i}f_a(X,\\mathbf {p})\\mp \\partial _{X^i}V^V_0(X)\\partial _{p_i}f_a(X,\\mathbf {p})\\nonumber \\\\&&\\mp \\partial _{X^i}V^V_j(X)\\frac{p^{j\\pm }}{E_{\\mathbf {p}^\\pm }}\\partial _{p_i}f_a(X,\\mathbf {p})=\\mathcal {C}[f_a],$ where $f_a(X,{\\bf p})$ is the phase-space distribution function of quarks or antiquarks of flavor $a$ .",
"In the above, $\\mathbf {p}^\\pm \\equiv \\mathbf {p}\\pm \\mathbf {V}^V$ is the kinetic momentum with the subscript $+$ referring to quarks and $-$ referring to antiquarks, $V^V_\\mu =-2G_V\\sum _a\\langle \\bar{q}\\gamma _\\mu q\\rangle _a$ is the vector potential, and $M_a=m_{0a}-V^S_a$ is the effective quark mass with $V^S_a=2G_S\\langle \\bar{q}q\\rangle _a+2K\\langle \\bar{q}q\\rangle _b\\langle \\bar{q}q\\rangle _c$ being the scalar potential with $a\\ne b\\ne c$ .",
"The right hand side of Eq.",
"(REF ), $\\mathcal {C}[f_a]&\\equiv & \\sum _{bcd}\\frac{1}{1+\\delta _{ab}}\\int \\frac{d^3\\mathbf {p}_b}{(2\\pi )^32E_b}\\frac{d^3\\mathbf {p}_c}{(2\\pi )^32E_c}\\frac{d^3\\mathbf {p}_d}{(2\\pi )^32E_d}\\frac{(2\\pi )^4}{2E_a}\\delta ^4(p_a+p_b-p_c-p_d)\\nonumber \\\\&&\\times |\\mathcal {M}_{ab}|^2\\left[f_cf_d(1-f_a)(1-f_b)-f_af_b(1-f_c)(1-f_d)\\right],$ is the collisional term that describe the scatterings among quarks and antiquarks, with the subscripts $a$ , $b$ , $c$ , and $d$ now denoting not only the flavor but also the spin and color, and baryon charge (quark or anti-quark) of a parton.",
"The above equation can be solved using the test particle method [10] by expressing the distribution function in terms of the density of test particles, whose equations of motions are determined by the left hand side of Eq.",
"(REF ), and they are $\\dot{\\mathbf {x}}&=&\\frac{\\mathbf {p}^\\pm }{E_{\\mathbf {p}^\\pm }},\\\\\\dot{\\mathbf {p}}&=&\\nabla V^S(\\mathbf {x})\\frac{M}{E_{\\mathbf {p}^\\pm }}\\pm \\nabla V^V_0(\\mathbf {x})\\pm \\nabla V^V_j(\\mathbf {x})\\frac{p^{j\\pm }}{E_{\\mathbf {p}^\\pm }}.$ The second equation in the above can also be written as $\\dot{\\mathbf {p}}^\\pm =\\nabla V^S(\\mathbf {x})\\frac{M}{E_{\\mathbf {p}}}\\mp \\dot{\\mathbf {}x\\times \\mathbf {B}\\pm \\mathbf {E},}where \\mathbf {B}=\\nabla \\times \\mathbf {V}^V is the strong magnetic field and \\mathbf {E}=\\partial _t\\mathbf {V}^V+\\nabla V^S is the strong electric field.$ Besides the mean fields, test particles are also affected by collisions, which can be treated geometrically by generalizing the method of Ref.",
"[11] to use the particle scattering cross section $\\sigma $ in the quark matter frame to check whether the impact parameter between two colliding particles is smaller than $\\sqrt{\\sigma /\\pi }$ and if the two colliding particles pass through each other at the next time step during the evolution of the system.",
"For two particles of masses $m_A$ and $m_B$ , momenta ${\\bf p}_A$ and ${\\bf p}_B$ , and energies $E_A$ and $E_B$ , this cross section is related to the cross section in their center-of-mass frame $\\sigma _{\\rm CM}(\\sqrt{s})$ with $s=(p_A+p_B)^2$ being the square of their invariant mass, which is the one used in Ref.",
"[11], by $\\sigma =\\sigma _\\mathrm {CM}(\\sqrt{s})\\frac{\\sqrt{(s-(m_A+m_B)^2)(s-(m_A-m_B)^2)}}{2E_AE_B|\\mathbf {v}_A-\\mathbf {v}_B|}.$ In the above, ${\\bf v}_A={\\bf p}_A/E_A$ and ${\\bf v}_B={\\bf p}_B/E_B$ are the velocities of the two particles.",
"The 3-momenta of the two particles after the scattering are taken to be isotropic in their center-of-mass frame.",
"Because of the high quark baryon chemical potential considered in the present study, the Pauli blocking effect on scatterings is also included by checking the available phase space for the final states [11].",
"We have checked that the above treatment of parton scattering reproduces the expected scattering rate evaluated via direct numerical integrations."
],
[
"QUARK MATTER IN A BOX",
"This section serves as a bridge between the studies of the spinodal instabilities in the small and large amplitude limits.",
"Although the case of small amplitude has already been discussed in Ref.",
"[4], we can develop an intuitive picture for how an initial sinusoidal fluctuation in a baryon-rich quark matter grows during the early stage of its time evolution from solving the Boltzmann equation in the test particle method as discussed in the previous section.",
"For the large amplitude case, which also includes the growth of instabilities during the late stage, solving the Boltzamnn equation allows us to follow the whole phase separation process to see how dense clusters develop inside a box of initially uniform quark matter and finally lead to the formation of large scale structures.",
"It also provides the possibility to find the appropriate observables to characterize these structures."
],
[
"Small amplitude density fluctuations",
"We consider a quark matter that is confined in a cubic box with periodic boundary conditions.",
"The system is prepared by distributing many test particles inside the box according to the density of the system with their momenta given by the Fermi-Dirac distribution at certain temperature.",
"We then study the growth of density fluctuations from an initial distribution with density and temperature corresponding to that inside the spinodal region.",
"Results obtained from solving the Boltzmann equation by following the classical motions of these test particles are compared with those obtained from the linear response theory in Ref. [4].",
"Specifically, we introduce an initial density fluctuation that has a sinusoidal oscillation in the z direction, $\\rho _\\mathrm {ini}=\\rho _0 (1+0.1\\sin (2\\pi z/L))$ , where $\\rho _0$ is the average initial density and $L$ is the length of the box with $L=10,~20,~30,~40,~50~\\mathrm {fm}$ corresponding to wave numbers $k=0.63,~0.31,~0.21,~0.16,~0.13~\\mathrm {fm}^{-1}$ , respectively.",
"As an example, Fig.",
"REF shows how the amplitude of the sinusoidal wave grows with time in the case of $L=20~\\mathrm {fm}$ , the average density $\\rho _0=0.7~\\mathrm {fm}^{-3}$ , and an initial temperature $T=45$ MeV.",
"Since the amplitude of density fluctuation at early times is expected to grow exponentially, it can be approximated by a hyperbolic cosine function of time, i.e., $\\delta \\rho (t)=\\delta \\rho _0\\mathrm {cosh}(\\Gamma _k t),$ where $\\Gamma _k$ is the growth rate and can be extracted directly from the numerical results, and they are shown in Fig.",
"REF by solid circles.",
"They are seen to agree very well with those obtained from an analytical calculation based on the linearized Boltzmann equation [4] after including the finite grid size effect as described in the Appendix, shown by the solid and dashed lines for the cases with and without the collision term in the Boltzmann equation, respectively."
],
[
"Large amplitude density fluctuations",
"To study how density fluctuations emerge and grow, we compare results from two calculations based on the same initial conditions but with and without the spinodal instability in the equation of state.",
"This is achieved by introducing a vector interaction in the NJL model, which is known to move the state of a quark matter from inside the spinodal region to the outside if its strength is sufficiently large [4].",
"For example, for a quark matter of temperature $T_0=$ 20 MeV and net quark density $\\rho _0=0.5~\\mathrm {fm}^{-3}$ , the spinodal region disappears if the vector coupling $G_V$ has the same value as the scalar coupling $G_S$ , although the state of the quark matter is well inside the spinodal instability region for $G_V=0$ .",
"Figure REF shows the time evolution of the density distribution in a box of size $20\\times 20\\times 20~\\mathrm {fm}^3$ for the two cases of $G_V=0$ (left column) and $G_V=G_S$ (right column), with the darker color denoting the high density regions and the lighter color denoting the low density regions.",
"Although the system is initially uniform in space, some dense spots are present due to statistical fluctuations as a result of finite number of test particles used in the calculation.",
"In the case of $G_V=G_S$ without a first-order phase transition or spinodal instability, the density distribution in the box remains unchanged with time as shown in the right column.",
"This changes dramatically, however, for the case of $G_V=0$ .",
"Due to the spinodal instability, the initial dense spots act like \"seeds\", which create several small low pressure regions that attract nearby partons and lead to the formation of many clusters at $t=20$ fm$/c$ .",
"These clusters further grow in size by connecting with each other and form stable large structures at $t=40$ fm$/c$ , when the system clearly separates into two phases of matter with one of high density and the other of low density.",
"Figure: Demonstration of the phase separation in the phase diagram.A clearer picture can be obtained by taking a cross sectional view on the $z=0$ plane as shown by the density distribution contours in Fig.",
"REF .",
"The two phases are now distinguishable with the dilute phase having a density of about $0.25~\\mathrm {fm}^{-3}$ and the dense phase having a density of about $1.0~\\mathrm {fm}^{-3}$ .",
"According to the phase diagram in Fig.",
"REF , the initial location of the system is indicated by the circle inside the spinodal region.",
"During the phase separation, the location of most part of the system moves towards the left boundary of the spinodal instability region that has a density of about $0.2~\\mathrm {fm}^{-3}$ , while that of the small part of the system moves towards the right boundary of the spinodal instability region that has a density of about $0.9~\\mathrm {fm}^{-3}$ , consistent with the picture shown by the density evolution.",
"Figure: Time evolution of the density-density correlation function in a quark matter of temperature T=20T=20 MeV and average net quark density n q =0.5 fm -3 n_q=0.5~\\mathrm {fm}^{-3} inside the spinodal region.As the large scale structure forms, we expect the density-density correlation $\\overline{\\rho (r)\\rho (0)}$ to get stronger and the correlation length to become larger.",
"This is indeed the case as shown in Fig.",
"REF , where it is seen that both the amplitude of the correlation function and the correlation length increases with time.",
"Figure: Time evolution of the scaled density moments in a quark matter of temperature T=20T=20 MeV and average net quark density n q =0.5 fm -3 n_q=0.5~\\mathrm {fm}^{-3} inside the spinodal region.The density fluctuations can also be quantified by the scaled density moments $\\langle \\rho ^N\\rangle /\\langle \\rho \\rangle ^N$ [12], where $\\langle \\rho ^N\\rangle \\equiv \\frac{\\int d^3\\mathbf {r} \\rho (\\mathbf {r})^{N+1}}{\\int d^3\\mathbf {r} \\rho (\\mathbf {r})}.$ This quantity is scale invariant since its value remains unchanged under a scale transfomation $\\mathbf {r}\\rightarrow \\lambda \\mathbf {r}$ , where $\\lambda $ can be any positive number.",
"The scaled density moments are all equal to one for a uniform density distribution but become greater than one as the density fluctuations grow.",
"In Fig.",
"REF , we show by dotted, dashed, and solid lines the scaled density moments for $N=2$ , 4 and 6, respectively.",
"Our results show that the scaled moments increase during the phase separation and reach their saturated values at about $t=40$ fm$/c$ , when the phase separation almost ends.",
"Also, moments with larger $N$ increase faster and saturate at larger values.",
"The final saturation values can be estimated as follows.",
"For a system of an initial density $\\rho _0$ that separates into two phases of density $\\rho _1$ and $\\rho _2$ with volumes $V_1$ and $V_2$ , respectively, the scaled density moments are then $\\frac{\\langle \\rho ^N\\rangle }{\\langle \\rho \\rangle ^N}=\\frac{\\rho _1^{N+1}V_1+\\rho _2^{N+1}V_2}{\\left(\\rho _1^2 V_1+\\rho _2^2 V_2\\right)^N/\\left(\\rho _1V_1+\\rho _2V_2\\right)^{N-1}}.$ Using the condition of particle number conservation $\\rho _1V_1+\\rho _2V_2=\\rho _0(V_1+V_2),$ the scaled density moments after the phase separation is thus $\\frac{\\langle \\rho ^N\\rangle }{\\langle \\rho \\rangle ^N}=\\frac{[\\rho _1^{N+1}(\\rho _2-\\rho _0)+\\rho _2^{N+1}(\\rho _0-\\rho _1)][\\rho _0(\\rho _2-\\rho _1)]^{N-1}}{[\\rho _1^{2}(\\rho _2-\\rho _0)+\\rho _2^{2}(\\rho _0-\\rho _1)]^N}.$ For our case of $\\rho _0=0.5~\\mathrm {fm}^{-3}$ , $\\rho _1\\approx 0.25~\\mathrm {fm}^{-3}$ , and $\\rho _2\\approx 1.0~\\mathrm {fm}^{-3}$ , we have $\\langle \\rho ^2\\rangle /\\langle \\rho \\rangle ^2\\approx 1.22$ ,$\\langle \\rho ^4\\rangle /\\langle \\rho \\rangle ^4\\approx 2.11$ , and $\\langle \\rho ^6\\rangle /\\langle \\rho \\rangle ^6\\approx 3.75$ , which are close to the final saturation values shown in Fig.",
"REF .",
"Other quantities of interest are the skewness and kurtosis of the particle multiplicity distribution, which were proposed as possible signals for the critical phenomena [13] and have been studied in the beam energy scan experiments at RHIC[1], [2].",
"They are defined as follows: $\\mathrm {skewness}&\\equiv &\\frac{\\langle \\delta N_q^3\\rangle }{\\langle \\delta N_q^2\\rangle ^{3/2}}, \\nonumber \\\\\\mathrm {kurtosis}&\\equiv &\\frac{\\langle \\delta N_q^4\\rangle }{\\langle \\delta N_q^2\\rangle ^2}-3.$ Both quantities characterize how far an event-by-event multiplicity distribution deviates from a normal distribution.",
"A positive skewness means a long tail on the right side of the distribution, i.e., most events have the net quark number below the mean value, while some events have an extreme high net quark number.",
"A positive kurtosis implies a sharper peak than the peak in a normal distribution, while a negative kurtosis corresponds to a flatter one.",
"Theoretical calculations based on the grand canonical picture predict that both quantities diverge with the correlation length when a system approaches its critical point [13], with the kurtosis diverging faster than the skewness.",
"Therefore, they have thus been suggested as the signals for the existence of a critical end point in the QCD phase diagram.",
"Figure: Time evolution of the event-by-event distribution of the number of quarks in a sub-volume of size 0.6 fm 3 ^3 (upper window) and 30 fm 3 ^3 (lower window) for a quark matter of temperature T=20T=20 MeV and average net quark density n q =0.5 fm -3 n_q=0.5~\\mathrm {fm}^{-3} inside the spinodal region.",
"The total number of events is 1000.To be consistent with the grand canonical picture, we consider quarks in a sub-volume of the box in our study, such as its central cell, and treat the remaining part as the reservoir.",
"When the system is initially inside the spinodal instability region, quarks in the reservoir can sometimes move into the sub-volume, but in most of the times quarks would leave from the sub-volume to the reservoir.",
"The number of quarks inside this sub-volume thus varies drastically from event to event, leading to large values for the skewness and kurtosis in its event-by-event distribution.",
"In Figs.",
"REF , we show the event-by-event distribution of the number of quarks in the central cell from 1000 events at $t=0$ , 20, and 40 fm$/c$ by the solid, dashed and dotted lines, respectively, for the two cases of sub-volume of size $0.6~\\mathrm {fm}^3$ (upper window) and $30~\\mathrm {fm}^3$ (lower window).",
"The upper window of Fig.",
"REF clearly shows that the distribution for the small sub-volume becomes asymmetric as time increases, starting with an initial skewness of 0.11 and increasing to 0.60 at 20 fm$/c$ and 0.75 at 40 fm/$c$ .",
"This feature is absent in the lower window of Fig.",
"REF for the larger sub-volume, where the distribution remains essentially symmetric with increasing time, with the skewness changing slowly from -0.001 (t=0) to 0.086 (t=20 fm$/c$ ) and 0.132 (t=40 fm$/c$ ), and there is no apparent increase or decrease in the kurtosis."
],
[
"Blast wave initial conditions",
"To study how large density fluctuations due to the spinodal instability as a result of a first-order phase transition obtained from the box calculation in the previous section are affected by the expansion of the system as in a heavy ion collision, we carry out a dynamical calculation using the transport model that includes parton scatterings besides the mean-field potentials described in Section II.",
"For the initial parton distributions, their positions are taken to follow that of a spherical Wood-Saxon form: $\\rho (r)=\\frac{\\rho _0}{1+\\exp ((r-R)/a)}$ with a radius $R=5~\\mathrm {fm}$ and a surface thickness parameter $a=0.5~\\mathrm {fm}$ , similar to that expected from a central Au+Au collisions.",
"The momenta of these patons are again taken to be that of a Fermi-Dirac distribution at certain temperature.",
"Calculations are then carried out with two different equations of state with and without a first-order phase transition, which can be realized by adjusting the coupling strength for the vector interaction.",
"Figure: Phase trajectory of the central cell of an expanding quark matter for the two cases with (solid line) and without (dashed line) a first-order phase transition using the blast wave initial conditions.",
"The spinodal region is shown by the gray color.To see how the expanding system goes into the spinodal region in the QCD phase diagram, we first study the time evolution of the temperature and net quark density in the central volume of 42.875 fm$^3$ , which has an initial density $\\rho _0=1.5~\\mathrm {fm}^3$ and temperature $T=70$ MeV, and trace its phase trajectory as shown in Fig.",
"REF for the two cases with (solid line) and without (dashed line) a phase transition.",
"Although the quark matter described by the transport model may not always be in perfect thermal equilibrium, we approximate its temperature by that of an equilibrated one that has the same energy density and net quark density in the NJL model.",
"As expected, the quark matter with a first-order phase transition (solid curve) enters the spinodal instability region, which is shown by the gray color, at about $6.5~\\mathrm {fm}/c$ and leaves the region at about $17.4~\\mathrm {fm}/c$ after spending about $10~\\mathrm {fm}/c$ inside this region.",
"How the central density decreases with time is shown by the solid line in Fig.",
"REF , which is seen to decrease slower than in the case without a first-order phase transition shown by the dashed line obtained with $G_V=G_S$ Figure: Time evolution of the density of the central cell of an expanding quark matter for the two cases with (solid line) and without (dashed line) a first-order phase transition.Figure: Density distributions of an expanding quark matter on the z=0z=0 plane at t=20t=20 fm/cc for the case with a first-order phase transition (left window) and at t=10t=10 fm/cc for the case without a first-order phase transition (right window).The density fluctuations can be seen from the density distribution on a plane such as the one at $z=0$ shown in Fig.",
"REF .",
"The left window shows the density distribution at $t=20$ fm/$c$ for the case with a first-order phase transition, while the right window shows that at $t=10$ fm/$c$ for the case without a first-order phase transition, when the density of the central cell is about $0.2$ fm$^{-3}$ in both cases.",
"Although density clumps appear in both cases, those in the one with a first-order phase transition are significantly larger.",
"As in the case of quark matter in a box, we can quantify the density fluctuations by the scaled density moments [14].",
"They are shown in Fig.",
"REF by the black and red lines for the cases with and without a first-order phase transition, respectively.",
"The dotted, dashed, and solid lines are for $N=2$ , 4, and 6, respectively.",
"In both cases, the scaled density moments first increase and then decrease with time.",
"In the case without a first-order phase transition, this is caused by the fast increase of the surface of the quark matter and the quick deviation from its initial smooth Wood-Saxon density distribution.",
"To the contrary, the scaled density moments in the case with a first-order phase transition becomes much larger with time and only decreases slightly afterwards, reflecting the effect due to density clumps that distribute randomly inside the expanding quark matter.",
"Therefore, the saturated scaled density moments, which are larger for larger $N$ , can be regarded as signals for a first-order phase transition in a baryon-rich quark matter [12].",
"Figure: (Color Online).",
"Scaled density moments as functions of time for the cases with (black lines) and without (red lines) a first-order phase transition.Figure: Final anisotropic flow coefficients v 2 v_2 (upper window) and v 4 v_4 (lower window) distributions for 100 events of an expanding quark matter with the same blast wave initial conditions.Since density fluctuations can lead to spatial anisotropy even in central heavy ion collisions, it has been suggested that they may affect the anisotropic flows in the transverse plane [15], [16].",
"The latter are defined by the coefficients $v_n$ in the expansion of the transverse momentum distribution $f(p_T,\\phi )$ as a Fourier series in the azimuthal angle $\\phi $ , $f(p_T,\\phi )=\\frac{N(p_T)}{2\\pi }\\lbrace 1+2\\sum _{n=1}^\\infty v_n(p_T)\\cos [n(\\phi -\\psi _n)]\\rbrace ,$ where $\\psi _n$ is the event plane angle [17].",
"To calculate the anisotropic flow coefficients, we use the two particle cumulant method [18], [19], namely, $v_n\\lbrace 2\\rbrace =\\sqrt{\\langle \\cos (n\\Delta \\phi )\\rangle }$ by averaging over all particle pairs in an event.",
"We have calculated $v_2\\lbrace 2\\rbrace $ and $v_4\\lbrace 2\\rbrace $ for 100 events of an expanding quark matter with the same blast wave initial conditions, and their final event distributions are shown, respectively, in the upper and lower windows of Fig.",
"REF with the solid and dashed lines for the cases with and without first order phase transition, respectively.",
"Both distributions peak at a larger value for the case with a first-order phase transition, particularly for $v_4$ , thus providing a plausible signal for the first-oder phase transition.",
"However, the values of the fluctuation induced $v_2$ and $v_4$ are much smaller than those in non-central heavy ion collisions.",
"Figure: Dilepton yield as a function of the invariant mass s\\sqrt{s} for the cases with (solid line) and without (dashed line) a first-order phase transition in an expanding quark matter with the blast wave initial conditions.We have also studied the effect of density fluctuations on dilepton production from a quark matter.",
"Since the dilepton production rate is proportional to the square of parton density, more dileptons are produced when the density fluctuation is large.",
"Also, a longer partonic phase as a result of a first-order phase transition would increase the depletion yield as well.",
"As usually done in studying dilepton production in heavy ion collisions [20], we use the perturbative approach to calculate the dilepton yield from the quark-antiquark scattering by neglecting its effect on the dynamics of the expanding quark matter.",
"Using the dilepton production cross section, $\\sigma _{q\\bar{q}\\rightarrow e^+e^-}&=&\\frac{4\\pi \\alpha ^2}{3s}\\sqrt{\\frac{1-4m_e^2/s}{1-4m_q^2/s}}\\nonumber \\\\&&\\times \\bigg (1+2\\frac{m_e^2+m_q^2}{s}+4\\frac{m_e^2m_q^2}{s^2}\\bigg ),$ where $s=(p_{e^-}+p_{e^+})^2$ is the square of the dilepton invariant mass, we have calculated the dilepton invariant mass spectrum from the expanding quark matter, and they are shown in Fig.",
"REF by the solid and dashed lines for the cases with and without first-order phase transition, respectively.",
"As expected, more dileptions are produced from the quark matter with a first-order phase transition.",
"We note the dilepton invariant mass spectrum peaks at $\\sqrt{s}\\approx 0.5$ GeV with the peak value being about $3.5\\times 10^{-4}$ GeV$^{-1}$ , which is comparable with the result obtained from a hadronic transport model [21].",
"This enhancement in dilepton production may thus be detectable in experiments.",
"We also note that most dileptons are produced from quark-antiquark annihilation as very few pions are present in the system due to the low phase transition temperature $T_c$ in the SU(3) NJL model."
],
[
"AMPT initial conditions",
"In this subsection, we use a more realistic initial parton distribution for heavy ion collisions.",
"Specifically, the initial partons are obtained from a multiphase transport (AMPT) model with string melting [22] that uses the heavy ion jet interaction generator (HIJING) [23], [24], [25] as the input.",
"This model includes not only the mini-jet partons from initial hard collisions but also hadrons produced from excited strings, which are projectile and target nucleons that have suffered interactions, by converting them to partons according to the flavor and spin structures of their valence quarks.",
"In particular, a meson is converted to a quark and an anti-quark, while a baryon is first converted to a quark and a diquark, and the diquark is then decomposed into two quarks.",
"The quark masses are taken to be $m_u=5.6$ , $m_d=9.9$ , and $m_s=199~\\mathrm {MeV}/c^2$ as in the PYTHIA program [26].",
"The above two-body decomposition is isotropic in the rest frame of the parent hadron or diquark.",
"These partons are produced after a formation time of $t_f=E_H/m^2_{T,H}$ , with $E_H$ and $m_{T,H}$ denoting, respectively, the energy and transverse mass of the parent hadron.",
"We obtain these partons as the initial conditions for our study of an expanding quark matter by running the AMPT program with vanishing parton scattering cross sections in Zhang's parton cascade (ZPC)[27] and with the hadronic afterburner based on a relativistic transport (ART) [28], [29] turned off.",
"Using the partons from Au+Au collisions at zero impact parameter and a center-of-mass energy $\\sqrt{s_{NN}}=2.5$ GeV as the initial distribution, we have found that some parts of the system go through the spinodal region when the SU(3) NJL model with $G_V=0$ is used in the Boltzmann equation and in constructing the phase diagram.",
"Figure: Phase trajectories of the central part of an expanding quark matter for the cases with (solid line) and without (dashed line) a first-order phase transition using the initial parton distribution from the AMPT model.",
"The spinodal region is shown by the gray color.As shown by the solid line in Fig.",
"REF , the trajectory of the central part of the system goes into the spinodal instability region at about $4.4~\\mathrm {fm}/c$ after expansion, and moves out of this region at about $5~\\mathrm {fm}/c$ .",
"Although $0.6~\\mathrm {fm}/c$ is too short for the spinodal instability to develop in the central part of the quark matter, its other parts may stay longer in the spinodal instability region due to both the spatial distribution of initial partons and the correlations between their rapidities and longitudinal ($z$ ) coordinates.",
"Figure: Rapidity and longitudinal coordinate correlations of initial partons from the AMPT model for central Au+Au collisions at s NN =2.5 GeV \\sqrt{s_{NN}}=2.5~\\mathrm {GeV}.Figure: Time evolution of the density distributions in central Au+Au collisions at s NN =2.5 GeV \\sqrt{s_{NN}}=2.5~\\mathrm {GeV} using initial conditions from the AMPT for the cases of free streaming (upper row) and including quark scattering as well as mean fields from the NJL model with G V =G S G_V=G_S (middle row) and G V =0G_V=0 (lower row).Figure: Density distribution of an expanding quark matter on the y=0y=0 plane at t=10t=10 fm/c with (upper window) and without (lower window) a first-order phase transition using the AMPT initial conditions.Figure: Final rapidity distribution of quarks for the cases with (solid curve) and without (dashed curve) a first-order phase transition from an expanding quark matter using the AMPT initial conditions.Figure REF shows the rapidity and longitudinal coordinate correlation of initial partons from a typical AMPT event for central Au+Au collisions at $\\sqrt{s_{NN}}=2.5~\\mathrm {GeV}$ .",
"This correlation can be quantified as follows: $r_{yz}\\equiv \\frac{\\sum _i (y_i-\\bar{y})(z_i-\\bar{z})}{\\sqrt{\\sum _i (y_i-\\bar{y})^2\\sum _i(z_i-\\bar{z})^2}}=0.355.$ This positive correlation indicates that partons initially on the right side of the quark matter are more likely to have momenta pointing to the right or forward direction, while partons initially in the left of the quark matter are more likely to have momenta pointing to the left or backward dirction.",
"This correlation helps the initially disc-shaped quark matter to expand, leading to a fast decrease of the density in the center of the quark matter as shown in the upper row of Fig.",
"REF .",
"Here, the quark matter is initially largely confined in a thin disk of thickness less than 0.5 fm.",
"When it is allowed to free streaming without any interactions, there appear two high density clumps that fly apart in the opposite directions.",
"This feature becomes less prominent after the inclusion of quark scattering and mean-field potentials but without a phase transition in the quark matter, i.e., taking $G_V=G_S$ , as shown in the middle row of Fig.",
"REF .",
"With a first-order phase transition in the quark matter by setting $G_V=0$ , the lower row of Fig.",
"REF shows that the initial central disk evolves into three disks of dense matter with one in the middle due to the strong attractions that keep some partons from moving away, besides the two forward and backward moving disks.",
"As the quark matter expands, these disks transform into rings and finally turn into disjointed clumps.",
"Furthermore, the density distribution of the quark matter in the reaction plane ($y=0$ ) shown in Fig.",
"REF indicates that the quark matter with a first-order phase transition expands twice as slow as that without a first-order phase transition.",
"Because of the non-trivial spatial distribution even in the case of free-streaming quark matter, the scaled density moments are no longer useful quantities to characterize the density fluctuations of an expanding quark matter due to its spinodal instability or a first-order phase transition.",
"On the other hand, the different density variations along the beam ($z$ ) axis shown in Fig.",
"REF are expected to affect the parton rapidity distribution.",
"This is because partons in the middle disc, which is present only in the case with a first-order phase transition, have a small rapidity and due to the attractive quark interactions, they attract partons from the other two discs and slow down their expansion in the longitudinal direction, thus restricting their rapidities to a narrow region around the midrapidity.",
"As shown by the solid line in Fig.",
"REF , the parton rapidity distribution in the case with a first-order phase transition is indeed much narrower than that in the case without a first-order phase transition, shown by the dashed line.",
"This effect can be regarded as a possible signal of a first-order phase transition and is worth studying in experiments.",
"Figure: Dilepton yield as a function of invariant mass s\\sqrt{s} for the cases with (solid curve) and without (dashed curve) a first-order phase transition from an expanding quark matter using the AMPT initial conditions.We have also studied the dilepton invariant mass spectrum from an expanding quark matter with initial conditions from the AMPT model.",
"This is shown in Fig.",
"REF by the solid and dashed lines for the cases with and without a first-order phase transition, respectively.",
"As in the previous section using the blast-wave initial conditions, the presence of a first-order phase transition enhances the dilepton yield as a result of density fluctuations and a longer partonic phase.",
"However, the dilepton yield is lower than that obtained from the calculation with the blast wave initial condition by two orders of magnitude because there are very few antiquarks in the partonic matter produced in heavy ion collisions at such a low energy and also because we have not included the bremsstrahlung contribution to dilepton production from the quark-quark scattering."
],
[
"conclusions",
"The spinodal instability is a thermodynamic feature of a first-order phase transition in a many-body system.",
"It occurs when its pressure in some parts decreases with increasing density.",
"This can amplify the density fluctuations and lead to a phase separation in the system.",
"We have studied this phenomenon by solving the Boltzmann equations using the test particle method.",
"The calculations are based on the NJL model, which has been shown to give good a description of the vacuum properties of the hadrons and also predicts the existence of a first-order phase transition in baryon-rich quark matter.",
"We have obtained some intuitive pictures on the phase separation in a quark matter that is either in a static box or undergoes expansion.",
"For the case of a static box, we have found that the growth rates extracted from the early growth of a sinusoidal density fluctuation agree with the analytical results obtained from the linearized Boltzmann equation.",
"We have also calculated the higher-order density moments of the quark matter and found them to increase and saturate at large values after phase separation, making them possible signals for the first-order phase transition.",
"The skewness of the quark number event-by-event distribution in a small sub-volume of the quark matter is also found to increase, but this feature disappears if the sub-volume is large.",
"As for the expanding quark matter, two cases have been studied.",
"One is based on the blast-wave initial conditions, while the other using the AMPT initial conditions, which are disc-like as a result of the strong correlations between the parton rapidity and longitudinal coordinate.",
"In both cases, we have found that the expansion of the quark matter is slowed down by the presence of a first-order phase transition.",
"Density clumps are found to appear and lead to an anisotropy in the momentum space, which can be characterized by the scaled density moments and the anisotropic flows $v_2$ and $v_4$ , respectively.",
"An enhancement in the dilepton yield is also observed.",
"The expansion of the quark matter with the AMPT initial conditions is more complex.",
"Normally, the initial disc-like quark matter splits into two discs, moving along the beam axis in opposite directions.",
"If the expanding quark matter undergoes a first order-phase transition, a third disc appears in the middle and pulls the other two discs towards it, resulting in a narrower rapidity distribution.",
"In the future, we plan to develop a more consistent transport model, in which all cross sections are calculated self-consistently from the NJL model, so that the temperature and density dependence of the collisional effect can be taken into account.",
"The dilepton production through the $qq\\rightarrow qqe^+e^-$ process will also be included, since it could be the main contribution to the dilepton yield from a quark matter of high baryon chemical potential.",
"We also plan to extend the transport model using the PNJL model[30], which is more realistic and agrees better with the lattice results for a quark matter with low baryon chemical potential.",
"We hope that our study will help to understand the phase transition in the baryon-rich matter by comparing theoretical predictions with available and future experimental data."
],
[
"Acknowledgements",
"This work was supported by the US Department of Energy under Contract No.",
"DE-SC0015266 and the Welch Foundation under Grant No.",
"A-1358.",
"*"
],
[
"Finite grid size effects",
"Counting partons in a grid of finite size in evaluating the mean fields effectively allows the partons in the grid interact with each other, thus modifying the contact interactions in the NJL model to finite-range ones.",
"To study this effect, we need to calculate the probability for two partons in the same grid to have a separation $\\Delta \\mathbf {x}$ .",
"Given a parton located at $x\\in [0,a]$ in a 1-dimensional grid $[0,a]$ , the probability to find another parton located at $x+\\Delta x$ in the same grid is $P(\\Delta x)&=&\\frac{1}{a}\\int dx \\theta (x)\\theta (a-x)\\theta (x+\\Delta x)\\theta (a-x-\\Delta x)\\nonumber \\\\&=&\\mathrm {tri}\\left(\\frac{\\Delta x}{a}\\right),$ where $\\mathrm {tri}(x)\\stackrel{\\Delta }{=}\\mathrm {max}(0,1-|x|).$ The above expression can be straightforwardly generalized to the 3-dimensional case to give $P(\\Delta \\mathbf {x})=\\prod _i\\mathrm {tri}\\left(\\frac{\\Delta x^i}{a^i}\\right),$ where $\\lbrace a^1,a^2,a^3\\rbrace $ are the grid lengths.",
"The interaction between two partons at $\\mathbf {x}$ and $\\mathbf {y}$ is then replaced by $G_S\\delta ^3(\\mathbf {x}-\\mathbf {y})&\\rightarrow &\\frac{G_S}{\\prod _i a^i}\\prod _i\\mathrm {tri}\\left(\\frac{x^i-y^i}{a_i}\\right),\\nonumber \\\\K\\delta ^3(\\mathbf {x}-\\mathbf {y})&\\rightarrow &\\frac{K}{\\prod _i a^i}\\prod _i\\mathrm {tri}\\left(\\frac{x^i-y^i}{a_i}\\right).$ Transforming Eq.",
"(REF ) from $\\mathbf {x}$ -space to $\\mathbf {k}$ -space gives $G_S\\rightarrow \\tilde{G}_S&=&G_S\\prod _i \\frac{2\\cos (a^i k_i)-2}{a^i k_i},\\nonumber \\\\K\\rightarrow \\tilde{K}&=&K\\prod _i \\frac{2\\cos (a^i k_i)-2}{a^i k_i}.$ Note that in the limit that $a^i k_i\\rightarrow 0$ for all the $i$ , $\\tilde{G}_S\\rightarrow G_S$ and $\\tilde{K}\\rightarrow K$ , which means the modification does not affect the long wavelength modes.",
"Replacing $G_S$ and $K$ in Eq.",
"(35) in Ref.",
"[4] with $\\tilde{G}_S$ and $\\tilde{K}$ , respectively, and solving the resulting equation, we obtain the modified dispersion relation, and they are shown in Fig.",
"REF for a grid size $a^i=2/3~\\mathrm {fm}$ by the solid and dashed lines for the cases with and without the collision term, respectively.",
"As expected, the growth rate $\\Gamma _k$ is not much affected in the small $k$ region but is significantly suppressed in the large $k$ region.",
"The finite grid size effect is thus similar to the quantum effect shown in Ref. [4].",
"Using a finite grid size essentially allows partons to interact at finite separation, resulting in an effective finite-range interaction."
]
] | 1606.05012 |
[
[
"CLEAR: Covariant LEAst-square Re-fitting with applications to image\n restoration"
],
[
"Abstract In this paper, we propose a new framework to remove parts of the systematic errors affecting popular restoration algorithms, with a special focus for image processing tasks.",
"Generalizing ideas that emerged for $\\ell_1$ regularization, we develop an approach re-fitting the results of standard methods towards the input data.",
"Total variation regularizations and non-local means are special cases of interest.",
"We identify important covariant information that should be preserved by the re-fitting method, and emphasize the importance of preserving the Jacobian (w.r.t.",
"the observed signal) of the original estimator.",
"Then, we provide an approach that has a \"twicing\" flavor and allows re-fitting the restored signal by adding back a local affine transformation of the residual term.",
"We illustrate the benefits of our method on numerical simulations for image restoration tasks."
],
[
"=1 [pages=1-last]arxiv"
]
] | 1606.05158 |
[
[
"Superconformal SU(1,1|n) mechanics"
],
[
"Abstract Recent years have seen an upsurge of interest in dynamical realizations of the superconformal group SU(1,1|2) in mechanics.",
"Remarking that SU(1,1|2) is a particular member of a chain of supergroups SU(1,1|n) parametrized by an integer n, here we begin a systematic study of SU(1,1|n) multi-particle mechanics.",
"A representation of the superconformal algebra su(1,1|n) is constructed on the phase space spanned by m copies of the (1,2n,2n-1) supermultiplet.",
"We show that the dynamics is governed by two prepotentials V and F, and the Witten-Dijkgraaf-Verlinde-Verlinde equation for F shows up as a consequence of a more general fourth-order equation.",
"All solutions to the latter in terms of root systems reveal decoupled models only.",
"An extension of the dynamical content of the (1,2n,2n-1) supermultiplet by angular variables in a way similar to the SU(1,1|2) case is problematic."
],
[
"ITP–UH–14/16 Superconformal SU($1,1|n$ ) mechanics $\\textrm {\\LARGE Anton Galajinsky\\ }^{a} \\quad \\textrm {\\Large and} \\quad \\textrm {\\LARGE Olaf Lechtenfeld\\ }^{b}$ ${}^{a}$ Laboratory of Mathematical Physics, Tomsk Polytechnic University, 634050 Tomsk, Lenin Ave. 30, Russian Federation Email: galajin@tpu.ru ${}^{b}$ Institut für Theoretische Physik und Riemann Center for Geometry and Physics, Leibniz Universität Hannover, Appelstrasse 2, 30167 Hannover, Germany Email: lechtenf@itp.uni-hannover.de Recent years have seen an upsurge of interest in dynamical realizations of the superconformal group SU$(1,1|2)$ in mechanics.",
"Remarking that SU$(1,1|2)$ is a particular member of a chain of supergroups SU$(1,1|n)$ parametrized by an integer $n$ , here we begin a systematic study of SU$(1,1|n)$ multi-particle mechanics.",
"A representation of the superconformal algebra $su(1,1|n)$ is constructed on the phase space spanned by $m$ copies of the $(1,2n,2n{-}1)$ supermultiplet.",
"We show that the dynamics is governed by two prepotentials $V$ and $F$ , and the Witten-Dijkgraaf-Verlinde-Verlinde equation for $F$ shows up as a consequence of a more general fourth-order equation.",
"All solutions to the latter in terms of root systems reveal decoupled models only.",
"An extension of the dynamical content of the $(1,2n,2n{-}1)$ supermultiplet by angular variables in a way similar to the SU$(1,1|2)$ case is problematic.",
"PACS: 11.30.Pb; 12.60.Jv Keywords: superconformal mechanics, SU$(1,1|n)$ superconformal algebra 1.",
"Introduction The recent increase of interest in dynamical realizations of the superconformal group SU$(1,1|2)$ [1]–[14] and its $D(2,1|\\alpha )$ extension [15]–[23] was motivated by the proposal in [24], [25] that a study of superconformal mechanics may have applications to the quantum mechanics of black holes.",
"In particular, according to [25] the large-$m$ limit of the $m$ -particle SU$(1,1|2)$ superconformal Calogero model may provide a microscopic description of the extreme Reissner–Nordström black hole in the near-horizon limit.",
"The explicit construction of the $m$ -particle SU$(1,1|2)$ superconformal Calogero model reduces to solving a variant of the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equation [1], [4].",
"Although plenty of interesting solutions to the WDVV equation were found in terms of root systems and their deformations [1], [6], [7], [26], [27], [28], [29], the construction of interacting models seems unfeasible beyond $m=3$ .",
"Since, in the context of [25], it is the structure of the superconformal group which matters, any multi-particle SU$(1,1|2)$ mechanics appears to be a good candidate.",
"Yet, no attempt has been made to link the large-$m$ limit of any known superconformal many-body quantum mechanics to the extreme Reissner–Nordström black hole in the near-horizon limit.",
"The studies in [1]–[23] proved useful for understanding the structure of interactions of various SU$(1,1|2)$ and $D(2,1|\\alpha )$ supermultiplets.",
"Supersymmetric couplings in $d{=}1$ are of interest on their own right because of novel features which are absent in higher dimensions.",
"The superconformal group SU$(1,1|2)$ is a particular member of a chain of supergroups SU$(1,1|n)$ parametrized by an integer $n$ .",
"The corresponding superconformal algebra $su(1,1|n)$ involves $n^2{+}3$ bosonic and $4n$ fermionic generators.",
"In particular, its bosonic sector includes $so(2,1)$ and $su(n)$ subalgebras.",
"The natural question arises whether the interesting features revealed for the SU$(1,1|2)$ case persist for higher values of $n$ .",
"Interacting many-body SU$(1,1|n)$ mechanics may also have applications to the quantum mechanics of higher-dimensional black holes.",
"Specifically, the bosonic subgroup SO$(2,1)\\times SU(n)$ of SU$(1,1|n)$ coincides with the near-horizon symmetry group of the Myers–Perry black hole with all rotation parameters set equal (see, e.g., the discussion in [30], [31]).",
"In this work, we initiate a systematic study of SU$(1,1|n)$ many-body mechanics.",
"There are two competing approaches to analyzing superconformal mechanics, namely the direct construction of an $su(1,1|n)$ representation within the Hamiltonian framework, and the superfield approach combined with the method of nonlinear realizations.",
"In [32] for example, the second approach has been used to describe a single supermultiplet of type $(1,2n,2n{-}1)$ .",
"Although the superfield formulation is more powerful, the Hamiltonian approach yields on-shell components and allows one to comprehend the basic dynamical features and the structure of interactions in a simpler and more transparent way.",
"In some instances it also offers notable technical simplifications in building interacting models [14].",
"In this work we thus adhere to the Hamiltonian formalism.",
"The paper is organized as follows.",
"In Section 2 we fix our notation and represent the structure relations of the superconformal algebra $su(1,1|n)$ in a form analogous to the previously studied case of $su(1,1|2)$ .",
"Section 3 is devoted to the construction of an $su(1,1|n)$ representation on the phase space spanned by $m$ copies of the $(1,2n,2n-1)$ supermultiplet.",
"It is shown that similarly to the SU$(1,1|2)$ case the dynamics is governed by two prepotentials $V$ and $F$ .",
"However, the WDVV equation appears as a consequence of a more general fourth-order equation for $F$ .",
"The latter is absent in SU$(1,1|2)$ mechanics because of a specific Fierz identity which exists for SU$(2)$ spinors only.",
"In Section 4 we consider prepotentials $F$ constructed from root systems.",
"It is demonstrated that the fourth-order structure equation characterizing SU$(1,1|n)$ mechanics forces all root vectors to be mutually orthogonal.",
"This implies decoupled dynamics.",
"In Section 5 we try to generalize also the analysis of [14] from SU$(1,1|2)$ to SU$(1,1|n)$ .",
"More specifically, we attempt to extend an arbitrary phase-space representation of $su(n)$ to an $su(1,1|n)$ representation, with a negative result.",
"The concluding Section 6 contains a summary and an outlook.",
"Throughout the paper summation over repeated indices is understood.",
"2.",
"Superconformal algebra $su(1,1|n)$ The superconformal algebra $su(1,1|n)$ involves $n^2+3$ bosonic and $4n$ fermionic generators.",
"Its even part is the direct sum $so(2,1)\\oplus su(n)\\oplus u(1)$ .",
"The generators of $so(2,1)$ , which we designate as $H$ , $D$ , $K$ , correspond to the time translation, dilatation and special conformal transformation, respectively.",
"The $R$ –symmetry subalgebra $su(n)\\oplus u(1)$ is generated by $J_a$ , with $a=1,\\dots ,n^2-1$ , and $L$ .",
"The odd part of the superalgebra includes the supersymmetry generators $Q_\\alpha $ , $\\bar{Q}^{\\alpha }$ , where $\\alpha =1,\\dots ,n$ , and their superconformal partners $S_\\alpha $ , $\\bar{S}^{\\alpha }$ .",
"It is assumed that the fermions are hermitian conjugates of each other ${(Q_\\alpha )}^{\\dagger }=\\bar{Q}^{\\alpha }, \\qquad {(S_\\alpha )}^{\\dagger }=\\bar{S}^{\\alpha }.$ $Q_\\alpha $ and $S_\\alpha $ transform as $su(n)$ spinors.",
"The structure relations of $su(1,1|n)$ read $&\\lbrace H,D \\rbrace =H, && \\lbrace H,K \\rbrace =2D,\\nonumber \\\\[2pt]&\\lbrace D,K\\rbrace =K, && \\lbrace J_a,J_b \\rbrace =f_{abc} J_c,\\nonumber \\\\[2pt]&\\lbrace Q_\\alpha , \\bar{Q}^\\beta \\rbrace =-2 i H {\\delta _\\alpha }^\\beta , &&\\lbrace Q_\\alpha , \\bar{S}^\\beta \\rbrace =2{{(\\lambda _a)}_\\alpha }^\\beta J_a+\\left(2iD-{\\textstyle \\frac{n{-}2}{n}} L\\right){\\delta _\\alpha }^\\beta ,\\nonumber \\\\[2pt]&\\lbrace S_\\alpha , \\bar{S}^\\beta \\rbrace =-2i K {\\delta _\\alpha }^\\beta , &&\\lbrace \\bar{Q}^\\alpha , S_\\beta \\rbrace =-2{{(\\lambda _a)}_\\beta }^\\alpha J_a+\\left(2iD+{\\textstyle \\frac{n{-}2}{n}} L\\right) {\\delta _\\beta }^\\alpha ,\\nonumber \\\\[2pt]& \\lbrace D,Q_\\alpha \\rbrace = -{\\textstyle \\frac{1}{2}} Q_\\alpha , && \\lbrace D,S_\\alpha \\rbrace ={\\textstyle \\frac{1}{2}} S_\\alpha ,\\nonumber \\\\[2pt]&\\lbrace K,Q_\\alpha \\rbrace =S_\\alpha , && \\lbrace H,S_\\alpha \\rbrace =-Q_\\alpha ,\\nonumber \\\\[2pt]&\\lbrace J_a,Q_\\alpha \\rbrace ={\\textstyle \\frac{i}{2}} {{(\\lambda _a)}_\\alpha }^\\beta Q_\\beta , && \\lbrace J_a,S_\\alpha \\rbrace ={\\textstyle \\frac{i}{2}} {{(\\lambda _a)}_\\alpha }^\\beta S_\\beta ,\\nonumber \\\\[2pt]& \\lbrace D,\\bar{Q}^\\alpha \\rbrace =-{\\textstyle \\frac{1}{2}} \\bar{Q}^\\alpha , && \\lbrace D,\\bar{S}^\\alpha \\rbrace ={\\textstyle \\frac{1}{2}} \\bar{S}^\\alpha ,\\nonumber \\\\[2pt]& \\lbrace K,\\bar{Q}^\\alpha \\rbrace =\\bar{S}^\\alpha , && \\lbrace H,\\bar{S}^\\alpha \\rbrace =-\\bar{Q}^\\alpha ,\\nonumber \\\\[2pt]&\\lbrace J_a,\\bar{Q}^\\alpha \\rbrace =-{\\textstyle \\frac{i}{2}} \\bar{Q}^\\beta {{(\\lambda _a)}_\\beta }^\\alpha , && \\lbrace J_a,\\bar{S}^\\alpha \\rbrace =-{\\textstyle \\frac{i}{2}}\\bar{S}^\\beta {{(\\lambda _a)}_\\beta }^\\alpha ,\\nonumber \\\\[2pt]&\\lbrace L,Q_\\alpha \\rbrace =i Q_\\alpha , && \\lbrace L,S_\\alpha \\rbrace =i S_\\alpha ,\\nonumber \\\\[2pt]&\\lbrace L,\\bar{Q}^\\alpha \\rbrace =-i \\bar{Q}^\\alpha , && \\lbrace L,\\bar{S}^\\alpha \\rbrace =-i \\bar{S}^\\alpha ,$ where $f_{abc}$ are the totally antisymmetric structure constants of $su(n)$ and $\\lambda _a$ are the hermitian and traceless $n\\times n$ –matrices which obey the (anti)commutation relations $[\\lambda _a,\\lambda _b]=2 i f_{abc} \\lambda _c, \\qquad \\lbrace \\lambda _a,\\lambda _b \\rbrace ={\\textstyle \\frac{4}{3}} \\delta _{ab}+2 d_{abc} \\lambda _c,$ with the totally symmetric coefficients $d_{abc}$ .",
"In what follows the Fierz identity $\\frac{1}{2} {{(\\lambda _a)}_\\alpha }^\\beta {{(\\lambda _a)}_\\gamma }^\\sigma =-\\frac{1}{n} {\\delta _\\alpha }^\\beta {\\delta _\\gamma }^\\sigma + {\\delta _\\gamma }^\\beta {\\delta _\\alpha }^\\sigma $ proves to be helpful.",
"3.",
"Realization of $su(1,1|n)$ in many-body mechanics In order to realize the $su(1,1|n)$ superconformal algebra in many-body mechanics, let us consider a phase space parametrized by $m$ bosonic canonical pairs $(x^i, p^i)$ , and $m$ self–conjugate fermions ${(\\psi ^i_\\alpha )}^{\\dagger }=\\bar{\\psi }^{ i \\alpha }$ , $i=1,\\dots ,m$ , $\\alpha =1,\\dots ,n$ , which obey the conventional Poisson brackets $\\lbrace x^i,p^j \\rbrace =\\delta ^{ij} , \\qquad \\lbrace \\psi ^i_\\alpha , \\bar{\\psi }^{j \\beta } \\rbrace =-i{\\delta _\\alpha }^\\beta \\delta ^{ij}.$ It is assumed that each fermion belongs to the fundamental representation of SU$(n)$ .",
"Guided by the previous studies of the $su(1,1|2)$ –case [1], [4], [6], let us introduce two prepotentials $V(x^1,\\dots ,x^n)$ , $F(x^1,\\dots ,x^n)$ and consider the following functions: $&H={\\textstyle \\frac{1}{2}} \\left(p^i p^i+\\partial ^i V \\partial ^i V \\right)+\\partial ^i \\partial ^j V (\\bar{\\psi }^i \\psi ^j)+{\\textstyle \\frac{1}{2}} \\partial ^i \\partial ^j \\partial ^k \\partial ^l F (\\bar{\\psi }^i \\psi ^j) (\\bar{\\psi }^k \\psi ^l), && D=tH-{\\textstyle \\frac{1}{2}} x^i p^i,\\nonumber \\\\[2pt]&K=t^2 H-t x^i p^i +{\\textstyle \\frac{1}{2}} x^i x^i, && J_a={\\textstyle \\frac{1}{2}} (\\bar{\\psi }^i \\lambda _a \\psi ^i),\\nonumber \\\\[2pt]&Q_\\alpha =(p^i+i \\partial ^i V) \\psi ^i_\\alpha +i \\partial ^i \\partial ^j \\partial ^k F \\psi ^i_\\alpha (\\bar{\\psi }^j \\psi ^k) , && S_\\alpha =x^i \\psi ^i_\\alpha -t Q_\\alpha ,\\nonumber \\\\[2pt]&\\bar{Q}^\\alpha =(p^i-i \\partial ^i V) \\bar{\\psi }^{i \\alpha }-i \\partial ^i \\partial ^j \\partial ^k F \\bar{\\psi }^{i \\alpha } (\\bar{\\psi }^j \\psi ^k), &&\\bar{S}^\\alpha =x^i \\bar{\\psi }^{i \\alpha } -t \\bar{Q}^\\alpha ,\\nonumber \\\\[2pt]&L=\\bar{\\psi }^i \\psi ^i,$ where $\\bar{\\psi }^i \\psi ^j=\\bar{\\psi }^{i \\alpha } \\psi ^j_\\alpha $ , $\\bar{\\psi }^i \\lambda _a \\psi ^i=\\bar{\\psi }^{i \\alpha } {{(\\lambda _a)}_\\alpha }^\\beta \\psi _\\beta ^i$ .",
"It is straightforward to verify that these functions do obey the structure relations (REF ) under the Poisson bracket (REF ) provided the restrictions on the prepotentials $&&(\\partial ^i \\partial ^j \\partial ^k F)( \\partial ^k \\partial ^l \\partial ^m F)\\ =\\ (\\partial ^m \\partial ^j \\partial ^k F)( \\partial ^k \\partial ^l \\partial ^i F), \\qquad x^i (\\partial ^i \\partial ^j \\partial ^k F)\\ =\\ -\\delta ^{jk},\\\\[4pt]&&x^i \\partial ^i V=C, \\qquad \\partial ^i \\partial ^j V\\ =\\ (\\partial ^i \\partial ^j \\partial ^k F) \\partial ^k V, \\qquad \\partial ^i \\partial ^j \\partial ^k \\partial ^l F\\ =\\ (\\partial ^i \\partial ^j \\partial ^p F)(\\partial ^p \\partial ^k \\partial ^l F).\\nonumber $ hold, with $C$ being an arbitrary constant.",
"Note that all the constraints in (REF ) coincide with those characterizing the $su(1,1|2)$ case, but for the rightmost equation entering the second line which is new.",
"It arises when computing the bracket $\\lbrace Q_\\alpha , \\bar{Q}^\\beta \\rbrace $ which explicitly involves the term $\\bigl (\\partial ^i \\partial ^j \\partial ^k \\partial ^l F-(\\partial ^i \\partial ^j \\partial ^p F)(\\partial ^p \\partial ^k \\partial ^l F)\\bigr )\\,\\psi ^i_\\alpha \\bar{\\psi }^{j \\beta } (\\bar{\\psi }^k \\psi ^l).$ For $n=2$ the spinor index $\\alpha $ takes only two values, and the spinors in the previous formula can be reordered so as to yield the piece proportional to $(\\bar{\\psi }^i \\psi ^j)(\\bar{\\psi }^k \\psi ^l) {\\delta _\\alpha }^\\beta $ , thus providing a contribution to the Hamiltonian which is quartic in fermions.",
"For $n>2$ such reordering is no longer possible, and one has to impose the extra condition $\\partial ^i \\partial ^j \\partial ^k \\partial ^l F\\ =\\ (\\partial ^i \\partial ^j \\partial ^p F)(\\partial ^p \\partial ^k \\partial ^l F),$ which yields the main difference from the $su(1,1|2)$ case.",
"Note that, by antisymmetrization of the indices $i$ and $l$ , (REF ) actually implies the WDVV equation visible in the first line in (REF ).",
"Hence, the additional requirement as compared to the $n=2$ case is the totally symmetric projection of (REF ), $\\partial ^i \\partial ^j \\partial ^k \\partial ^l F \\ =\\ (\\partial ^{(i} \\partial ^j \\partial ^p F)(\\partial ^p \\partial ^k \\partial ^{l)} F) ,$ where the symmetrisation (with weight $\\frac{1}{4!",
"}$ ) excludes the summation index $p$ .",
"Further differentiation of this relation, together with the WDVV equation, yields a hierarchy of equations, $\\partial ^{i_1}\\partial ^{i_2}\\cdots \\partial ^{i_{r+3}} F \\ =\\ r!",
"(\\partial ^{i_1}\\partial ^{i_2}\\partial ^{k_1} F)(\\partial ^{k_1}\\partial ^{i_3}\\partial ^{k_2} F)(\\partial ^{k_2}\\partial ^{i_4}\\partial ^{k_3} F) \\cdots (\\partial ^{k_r}\\partial ^{i_{r+2}}\\partial ^{i_{r+3}} F)$ together with $x^i(\\partial ^i\\partial ^{i_2}\\cdots \\partial ^{i_{r+3}} F)=-r\\,\\partial ^{i_2}\\cdots \\partial ^{i_{r+3}} F$ , for $r=1,2,\\ldots .$ When computing the brackets $\\lbrace Q_\\alpha , \\bar{S}^\\beta \\rbrace $ and $\\lbrace \\bar{Q}^\\alpha , S_\\beta \\rbrace $ , one has to use the Fierz identity (REF ).",
"In particular, the constant $C$ , which enters the homogeneity condition $x^i \\partial ^i V=C$ , appears in the algebra as the central charge, $&&\\lbrace Q_\\alpha , \\bar{S}^\\beta \\rbrace \\ =\\ \\phantom{-}2{{(\\lambda _a)}_\\alpha }^\\beta J_a+\\left(2iD-{\\textstyle \\frac{n{-}2}{n}} L+C\\right){\\delta _\\alpha }^\\beta ,\\nonumber \\\\[4pt]&&\\lbrace \\bar{Q}^\\alpha , S_\\beta \\rbrace \\ =\\ -2{{(\\lambda _a)}_\\beta }^\\alpha J_a+\\left(2iD+{\\textstyle \\frac{n{-}2}{n}} L-C\\right) {\\delta _\\beta }^\\alpha .$ If desirable, $C$ can be removed by redefining $L$ .",
"In the latter case the bosonic limit of $L$ yields a constant rather than zero.",
"4.",
"Prepotentials $F$ related to root systems The leftmost equation in the first line in (REF ) is a variant of the WDVV equation.",
"With regard to the SU($1,1|2)$ mechanics it has been extensively studied in [4], [6], [7], [27], [28].",
"In particular, each solution of the WDVV equation satisfying (REF ) will qualify to describe some SU$(1,1|n)$ superconformal mechanics.",
"The known WDVV solutions are based on so-called $\\vee $ -systems [29], which are certain deformations of Coxeter root systems.",
"For these, the prepotential $F$ takes the form $F \\ =\\ -{\\textstyle \\frac{1}{4}} \\sum _\\alpha h_\\alpha \\, (\\alpha \\cdot x)^2 \\ln (\\alpha \\cdot x)^2 ,$ where $\\lbrace \\alpha \\rbrace $ is a set of positive $m$ -dimensional root vectors, subject to the usual constraints for reflection groups or their $\\vee $ -system deformations, and $h_\\alpha $ are real weights to be determined.",
"Inserting (REF ) into (REF ), we obtain the condition $\\sum _\\alpha h_\\alpha \\, \\frac{\\alpha ^i\\,\\alpha ^j\\,\\alpha ^k\\,\\alpha ^l}{(\\alpha \\cdot x)^2} \\ +\\ \\sum _{\\alpha ,\\beta } h_\\alpha h_\\beta \\, \\frac{\\alpha ^i\\,\\alpha ^j\\,(\\alpha \\cdot \\beta )\\,\\beta ^k\\,\\beta ^l}{(\\alpha \\cdot x)(\\beta \\cdot x)} \\ =\\ 0 .$ The diagonal terms in this double sum fix the weights, $(\\alpha \\cdot \\alpha )\\,h_\\alpha = 1.$ The projection antisymmetric in $i$ and $l$ ensures the WDVV equation; it is assumed to be fulfilled for our root systems.",
"The symmetric projection gives further algebraic conditions: the vanishing of the double residues of the poles $(\\alpha \\cdot x)^{-1}(\\beta \\cdot x)^{-1}$ for any pair $(\\alpha ,\\beta )$ yields $(\\alpha \\cdot \\beta )\\,(\\alpha ^i \\alpha ^j \\beta ^k \\beta ^l + \\beta ^i \\beta ^j \\alpha ^k \\alpha ^l) \\ =\\ 0 .$ Contracting this with $\\alpha ^i \\beta ^j \\alpha ^k \\beta ^l$ produces $(\\alpha \\cdot \\alpha ) (\\beta \\cdot \\beta ) (\\alpha \\cdot \\beta )^2 \\ =\\ 0\\qquad \\Longrightarrow \\qquad \\alpha \\cdot \\beta = 0$ for any pair of distinct roots $(\\alpha ,\\beta )$ .",
"This admits only the direct sum of mutually orthogonal one-dimensional (i.e.",
"rank-one) systems.",
"By a rigid rotation of coordinates $x^i$ , one can always bring it into the form $\\lbrace \\alpha \\rbrace \\ =\\ \\lbrace (1,0,0,\\ldots ,0), (0,1,0,\\ldots ,0), \\ldots , (0,0,0,\\ldots ,1) \\rbrace .$ So we have arrived at a no-go theorem for interacting SU$(1,1|n)$ mechanics based on the on-shell supermultiplet of type $(1,2n,2n{-}1)$ .",
"5.",
"Angular variables For SU$(1,1|2)$ mechanics one can extend the dynamical content of the simplest $(1,4,3)$ supermultiplet by introducing angular variables providing some realization of $su(2)$ in a purely group-theoretic way [9], [14].",
"It suffices to consider a phase space parametrized by the canonical pairs $(\\theta ^A, p_{\\theta A})$ , $A=1,\\dots ,n$ , which obey the conventional Poisson brackets $\\lbrace \\theta ^A,p_{\\theta B} \\rbrace ={\\delta ^A}_B$ and realize on such a phase space the functions $J_a=J_a (\\theta ,p_\\theta )$ , $a=1,2,3$ , which obey the structure relations of the $su(2)$ $R$ -symmetry subalgebra $\\lbrace J_a,J_b \\rbrace =\\epsilon _{abc} J_c.$ Then, the supersymmetry charges involve the angular variables only via the currents $J_a$ , $&&Q_\\alpha \\ =\\ p \\,\\psi _\\alpha +\\frac{2i}{x} {(\\sigma _a \\psi )}_\\alpha J_a -\\frac{i}{x} \\psi _\\alpha (\\bar{\\psi }\\psi ),$ where ${{(\\sigma _a)}_\\alpha }^\\beta $ are the Pauli matrices [14].",
"Let us try to generalize this construction to the case of the superconformal algebra $su(1,1|n)$ .",
"Introducing functions $J_ a$ of the angular variables subject to the $su(n)$ structure relations $\\lbrace J_a,J_b \\rbrace =f_{abc} J_c$ , and employing the matrices ${{(\\lambda _a)}_\\alpha }^\\beta $ from (REF ), it is straightforward to verify that the obvious candidate supersymmetry charge, $&&Q_\\alpha \\ =\\ p \\,\\psi _\\alpha +\\frac{2i}{x} {(\\lambda _a \\psi )}_\\alpha J_a -\\frac{i}{x} \\psi _\\alpha (\\bar{\\psi }\\psi ),$ is indeed nilpotent, i.e.",
"$\\lbrace Q_\\alpha ,Q_\\beta \\rbrace =0.$ However, in view of the properties of the $\\lambda $ –matrices in (REF ), the bracket of $Q_\\alpha $ with $\\bar{Q}^\\beta $ yields not just the Hamiltonian: $\\lbrace Q_\\alpha , \\bar{Q}^\\beta \\rbrace \\ =\\ -2 i\\,H {\\delta _\\alpha }^\\beta -\\frac{4 i}{x^2} {{(\\lambda _a)}_\\alpha }^\\beta d_{abc} J_b J_c,$ where $d_{abc}$ are the symmetric structure coefficients appearing in (REF ).",
"This means that the algebra does not close.",
"One might try to modify the troublesome second term in $Q_\\alpha $ .",
"However, such a term seems indispensable for providing the structure relations (REF ).",
"We thus conclude that an extension of the $(1,2n,2n{-}1)$ supermultiplet by angular variables in a way similar to the $su(1,1|2)$ case is problematic.",
"Perhaps a more sophisticated construction involving extra auxiliary variables will help to circumvent the problem.",
"6.",
"Discussion To summarize, in this work we made the first step towards a systematic description of SU$(1,1|n)$ multi-particle superconformal mechanics.",
"Our consideration was primarily focused on the possibilities offered by the Hamiltonian formalism.",
"The structure relations of the superconformal algebra $su(1,1|n)$ were established in a form analogous to the previously studied $su(1,1|2)$ case.",
"A representation of $su(1,1|n)$ on the phase space spanned by $m$ copies of the $(1,2n,2n{-}1)$ supermultiplet was constructed.",
"It was shown that the dynamics is governed by two prepotentials $V$ and $F$ , and that the WDVV equation for $F$ arises as a consequence of a more restrictive fourth-order equation.",
"Solutions to the latter in terms of root systems allow decoupled models only.",
"An attempt to extend the dynamical content of the $(1,2n,2n{-}1)$ supermultiplet by adding angular variables in a way similar to the $su(1,1|2)$ case compromised the closure of the $su(1,1|n)$ superconformal algebra.",
"Hence, our results indicate that the construction of interacting SU$(1,1|n)$ models with $n>2$ appears to be a more difficult task than in the SU$(1,1|2)$ case.",
"The Hamiltonian formulation adopted in this work automatically yields on-shell models.",
"It is tempting to investigate SU$(1,1|n)$ mechanics off-shell within the superfield approach combined with the method of nonlinear realizations, along the lines proposed in [32].",
"The key problem within the superfield method will be to guess the superfield constraints which will result in interacting dynamics.",
"A possible link of SU$(1,1|n)$ mechanics to the near-horizon Myers–Perry black hole with equal rotation parameters is worth studying as well.",
"Finally, it might be rewarding to investigate the integrability of (REF ) on its own, which is more special than the WDVV equation.",
"Acknowledgements We thank S. Krivonos for bringing [32] to our attention.",
"A.G. is grateful to the Institute for Theoretical Physics at Hannover University for the hospitality extended to him at different stages of this research.",
"The work was supported by the DFG grant Le-838/12-2, the MSE program Nauka under the project 3.825.2014/K, the RFBR grant 15-52-05022, and the Action MP1405 QSPACE from the European Cooperation in Science and Technology (COST)."
]
] | 1606.05230 |
[
[
"Absolutely Continuous Spectrum for Parabolic Flows/Maps"
],
[
"Abstract We provide an abstract framework for the study of certain spectral properties of parabolic systems; specifically, we determine under which general conditions to expect the presence of absolutely continuous spectral measures.",
"We use these general conditions to derive results for spectral properties of time-changes of unipotent flows on homogeneous spaces of semisimple groups regarding absolutely continuous spectrum as well as maximal spectral type; the time-changes of the horocycle flow are special cases of this general category of flows.",
"In addition we use the general conditions to derive spectral results for twisted horocycle flows and to rederive certain spectral results for skew products over translations and Furstenberg transformations."
],
[
"Motivation",
"Spectral theory of dynamical systems has long been studied [13]; of particular interest, is the notion of when to expect the presence of absolutely continuous spectral measures.",
"Since absolutely continuous spectrum implies mixing, this property can be thought of as an indicator of how chaotic, or how far from orderly, a system is.",
"In the hyperbolic setting, systems are characterized as having a correlation decay that is exponential.",
"As a result, techniques derived from the existence of a spectral gap as well as probabilistic tools are available for the study of spectral properties, and therefore, it is in the hyperbolic setting where the existence of absolutely continuous spectrum predominantly occurs.",
"Interestingly, certain parabolic systems also share this property despite having at most polynomial decay of correlations.",
"This slower decay of correlations precludes the use of the tools available in the spectral study of hyperbolic systems, and consequently, spectral theory of smooth parabolic flows and smooth perturbations of well known parabolic flows has been much less studied.",
"This work is devoted to creating an abstract framework for the study of certain spectral properties of parabolic systems.",
"Specifically, we attempt to answer the question: under what general conditions can we expect the existence of absolutely continuous spectral measures?"
],
[
"Statement of results",
"In Theorem REF in we present general conditions under which we expect a skew-adjoint operator to have absolutely continuous spectrum.",
"The proof of this theorem is a general application of the method in [5], in which the authors show that the Fourier Transform of the spectral measures of smooth coboundaries are square integrable.",
"The method in [5] was inspired by Marcus's proof of mixing of horocycle flows [14] which requires a specific form of tangent dynamics from which one can exploit shear of nearby trajectories.",
"In addition, we rely on the bootstrap technique from [5] to estimate the decay of correlations of specific smooth coboundaries.",
"Our choice of coboundaries depends upon a growth condition involving the commutator of the skew-adjoint operator and a certain auxiliary operator.",
"We use this general, functional analytic result to derive the following results for certain parabolic dynamical systems.",
"Theorem REF states that time-changes of unipotent flows on homogeneous spaces of semisimple groups have absolutely continuous spectrum.",
"In the compact case, we also show that the maximal spectral type is Lebesgue, following the method in [5].",
"The time-changes of the horocycle flow are special cases of this general category of flows, and spectral properties of the time-changes of horocycle flows were shown in [5], [18], and [19].",
"In addition, we use the general conditions to prove Theorem REF and Corollary REF regarding spectral results for twisted horocycle flows, combining the horocycle time-change with a circle rotation.",
"Lastly, we rederive certain spectral results for skew products over translations and Furstenberg transformations, originally shown in [19].",
"Our results are slightly weaker as we prove results for functions of class $C^{2}$ while the author in [19] considers functions of class $C^{1}$ with an added Dini Condition.",
"Remark 1 It is standard to consider spectral decomposition in the setting of self-adjoint operators.",
"When we consider flows that are represented as strongly continuous one-parameter unitary groups, the generating operators (vector fields) are essentially skew-adjoint.",
"Since multiplication by $i$ gives an essentially self-adjoint operator we make no distinction from the standard setting, and thus, directly apply the theory for self-adjoint operators."
],
[
"Preliminary assumptions",
"Suppose that a closed operator $X$ on a Hilbert Space $\\mathcal {H}$ , defined on a dense subspace $D$ such that $X(D) \\subset D$ , generates a strongly continuous, one parameter group $\\lbrace e^{sX}\\rbrace $ .",
"Suppose also that $e^{tU}(D)\\subset D$ is a strongly continuous, unitary group with infinitesimal generator $U$ , and that the commutator $H(t)=e^{-tU}\\left[X,e^{tU} \\right]$ is defined on $D$ .",
"For $u \\in D$ , let $\\frac{H(t)}{t^{\\beta }}u\\overset{\\mathcal {H}}{\\underset{t \\rightarrow \\infty }{\\longrightarrow }} Hu,$ such that $H(D) \\subset D$ and $\\overline{Ran(H)}=\\overline{\\left\\lbrace Hu:u \\in D \\right\\rbrace }=\\mathcal {H}$ .",
"For $B_{1}$ , $B_{2}$ bounded operators on $\\mathcal {H}$ such that $B_{2}: D \\rightarrow D$ , let $\\left\\Vert \\; \\langle e^{tU}f,f \\rangle _{\\mathcal {H}}\\; \\right\\Vert _{L^{2}(\\mathbb {R})} \\; \\le \\; \\left\\Vert \\frac{1}{\\sigma }\\int _{0}^{\\sigma } \\langle e^{sX}e^{tU}f, B_{1}e^{sX}B_{2}f \\rangle _{\\mathcal {H}} ds\\: \\right\\Vert _{L^{2}(\\mathbb {R})}.$ Theorem 1 If for $\\beta > \\frac{1}{2}$ , $H(t)$ and $H$ satisfy: (i) $\\frac{H(t)}{t^{\\beta }}H^{-1}$ is defined on $Ran(H)$ , extends by continuity to a bounded operator $\\frac{\\tilde{H}(t)H^{-1}}{t^{\\beta }}$ on $\\mathcal {H}$ with uniformly bounded $\\left\\Vert \\cdot \\right\\Vert _{op}$ norm in $t$ , and satisfies $\\limsup _{t \\rightarrow \\infty } \\left\\Vert I-\\frac{\\tilde{H}(t)H^{-1}}{t^{\\beta }} \\right\\Vert _{op} < 1.$ (ii) $ \\left[X,\\frac{H(t)}{t^{\\beta }}H^{-1} \\right]$ is defined on $Ran(H)$ and extends by continuity to a bounded operator on $\\mathcal {H}$ with uniformly bounded $\\left\\Vert \\cdot \\right\\Vert _{op}$ norm in $t$ (iii) $\\big [H(t),H \\big ]H^{-1}$ is defined on $Ran(H)$ and extends by continuity to a bounded operator on $\\mathcal {H}$ with uniformly bounded $\\left\\Vert \\cdot \\right\\Vert _{op}$ norm in $t$ then for all $f \\in \\overline{Ran(H)}=\\mathcal {H}$ , the associated spectral measures, $\\mu _{f}$ , of $U$ are absolutely continuous.",
"Remark 2 Often in ergodic theory, $\\mathcal {H}$ is a subspace of a larger Hilbert Space; for example, $\\mathcal {H}=L^{2}_{0}(M)$ the space of zero-average functions in $L^{2}(M)$ .",
"While in this setting the Theorem doesn't give a result for purely absolutely continuous spectrum it implies the existence of an absolutely continuous component.",
"Remark 3 The utilization of the operator $H(t)$ is suggested by the method in [5] which was based on Marcus's shear mechanism in [14].",
"The authors in [20] and [17] have used a similar term to derive criteria for strong mixing.",
"Let $f \\in Ran(H)$ and let $\\hat{\\mu }_{f}(t)=\\int _{\\mathbb {R}} e^{it \\xi } d\\mu _{f}(\\xi )$ be the Fourier Transform of the spectral measure $\\mu _{f}$ .",
"$\\left\\Vert \\hat{\\mu }_{f}(t) \\right\\Vert _{L^{2}(\\mathbb {R})} =\\left\\Vert \\; \\langle e^{tU}f,f \\rangle _{\\mathcal {H}} \\: \\right\\Vert _{L^{2}(\\mathbb {R})}\\; \\le \\;\\left\\Vert \\frac{1}{\\sigma }\\int _{0}^{\\sigma } \\langle e^{sX}e^{tU}f, B_{1}e^{sX}B_{2}f \\rangle _{\\mathcal {H}} \\: ds \\: \\right\\Vert _{L^{2}(\\mathbb {R})}.$ For $s \\in [0, \\sigma ]$ , we integrate by parts: $\\hspace{56.9055pt} B_{1}e^{sX}B_{2}f \\hspace{56.9055pt} \\frac{d}{ds}\\left(B_{1}e^{sX}B_{2}f \\right) = B_{1}e^{sX}X(B_{2}f)$ $e^{sX}e^{tU}f \\hspace{28.45274pt} \\hspace{56.9055pt} \\int _{0}^{\\sigma } e^{sX}e^{tU}fds.$ $\\begin{split}\\frac{1}{\\sigma }\\int _{0}^{\\sigma } \\langle e^{sX}e^{tU}f, B_{1}e^{sX}B_{2}f \\rangle _{\\mathcal {H}} \\:ds=& \\: \\frac{1}{\\sigma } \\langle \\int _{0}^{\\sigma } e^{sX}e^{tU}fds, B_{1}e^{\\sigma X}B_{2}f \\rangle _{\\mathcal {H}}\\\\-& \\frac{1}{\\sigma }\\int _{0}^{\\sigma } \\langle \\int _{0}^{S}e^{sX}e^{tU}fds, B_{1}e^{sX}X(B_{2}f) \\rangle _{\\mathcal {H}} dS\\end{split}$ From our assumptions, both $B_{1}e^{sX}B_{2}f$ and $ B_{1}e^{sX}X(B_{2}f)$ are bounded in $\\mathcal {H}$ .",
"Thus, in order to show that $\\hat{\\mu }_{f}(t) = O(\\frac{1}{t^{\\beta }})$ , we need a bound (in $t$ ) for $\\left\\Vert \\int _{0}^{\\sigma }e^{sX}e^{tU}f\\:ds \\right\\Vert _{\\mathcal {H}}.$ Suppose that conditions $(i)$ , $(ii)$ , and $(iii)$ hold, and let $f$ be a coboundary of the form $f=Hg$ , for $g \\in Dom(H)$ : $\\begin{split}\\int _{0}^{\\sigma }e^{sX}e^{tU}f\\:ds =& \\int _{0}^{\\sigma }e^{sX}e^{tU}Hg\\:ds \\\\=&\\underbrace{\\int _{0}^{\\sigma }e^{sX}e^{tU}\\left(H-\\frac{H(t)}{t^{\\beta }} \\right)g\\:ds}_\\text{I.",
"}+\\underbrace{\\int _{0}^{\\sigma }e^{sX}e^{tU}\\frac{H(t)}{t^{\\beta }}g\\: ds}_\\text{II.",
"}.\\end{split}$ I.",
"Let $\\tilde{H}(s,t)=e^{sX}e^{tU}\\left(I-\\frac{\\tilde{H}(t)H^{-1}}{t^{\\beta }}\\right)e^{-tU}e^{-sX},$ where $\\frac{\\tilde{H}(t)H^{-1}}{t^{\\beta }}$ is the bounded extension of $\\frac{H(t)H^{-1}}{t^{\\beta }}$ .",
"Since $\\lbrace e^{sX}\\rbrace $ is strongly continuous, for $f \\in \\mathcal {D}$ , $s\\!\\!-\\!\\!\\lim _{s \\rightarrow 0} e^{sX}f = If.$ Thus, $\\sup _{s \\in [0,\\sigma ]} \\Vert e^{sX} \\Vert _{op} \\le 1 + k(\\sigma )$ where $\\lim _{\\sigma \\rightarrow 0} k(\\sigma ) = 0.$ Since $\\limsup \\limits _{t \\rightarrow \\infty } \\left\\Vert I-\\frac{\\tilde{H(t)}H^{-1}}{t^{\\beta }} \\right\\Vert _{op} < 1$ , then for large $t$ and small enough $\\sigma $ , $\\Vert \\tilde{H}(s,t) \\Vert _{\\mathcal {H}} < C_{1} < 1.$ We compute, for $f=e^{sX}(e^{tU}(Hu))$ , $\\begin{split}\\lim _{\\Delta s \\rightarrow 0} & \\frac{\\tilde{H}(s+ \\Delta s,t)f -\\tilde{H}(s,t)f}{\\Delta s}\\\\=&\\lim _{\\Delta s \\rightarrow 0} \\left( \\frac{e^{(s+\\Delta s)X}e^{tU}\\left(\\frac{\\tilde{H(t)}H^{-1}}{t^{\\beta }}\\right)e^{-tU}e^{-(s+ \\Delta s)X}f}{\\Delta s} \\right.\\\\-& \\left.",
"\\frac{e^{sX}e^{tU}\\left(\\frac{\\tilde{H(t)}H^{-1}}{t^{\\beta }}\\right)e^{-tU}e^{-sX}f}{\\Delta s} \\right)\\\\=&\\lim _{\\Delta s \\rightarrow 0} \\left( \\frac{e^{(s+\\Delta s)X}e^{tU}\\left(\\frac{\\tilde{H(t)}H^{-1}}{t^{\\beta }}\\right)e^{-tU}e^{-(s+ \\Delta s)X}f}{\\Delta s} \\right.\\\\\\pm & \\frac{e^{(s+\\Delta s)X}e^{tU}\\left(\\frac{\\tilde{H(t)}H^{-1}}{t^{\\beta }}\\right)e^{-tU}e^{-sX}f}{\\Delta s}\\\\-& \\left.",
"\\frac{e^{sX}e^{tU}\\left(\\frac{\\tilde{H(t)}H^{-1}}{t^{\\beta }}\\right)e^{-tU}e^{-sX}f}{\\Delta s} \\right)\\\\=& \\lim _{\\Delta s \\rightarrow 0} \\left( \\frac{(e^{(s+\\Delta s)X})e^{tU}\\left(\\frac{\\tilde{H(t)}H^{-1}}{t^{\\beta }}\\right)e^{-tU}(e^{-sX}-e^{-(s + \\Delta )X})f}{\\Delta s} \\right.",
"\\end{split}$ $\\begin{split}+& \\left.",
"\\frac{(e^{(s+\\Delta s)X}-e^{sX})e^{tU}\\left(\\frac{\\tilde{H(t)}H^{-1}}{t^{\\beta }}\\right)e^{-tU}e^{-sX}f}{\\Delta s} \\right)\\\\=&\\lim _{\\Delta s \\rightarrow 0} \\left( \\frac{e^{sX}e^{\\Delta s X}e^{tU}\\left(\\frac{\\tilde{H(t)}H^{-1}}{t^{\\beta }}\\right)e^{-tU}e^{-sX}(I-e^{-\\Delta sX})f}{\\Delta s}\\right.",
"\\\\+& \\left.",
"\\frac{e^{sX}(e^{\\Delta s X}-I)e^{tU}\\left(\\frac{\\tilde{H(t)}H^{-1}}{t^{\\beta }}\\right)e^{-tU}e^{-sX}f}{\\Delta s} \\right)\\\\\\pm &\\lim _{\\Delta s \\rightarrow 0} e^{sX}e^{\\Delta s X} e^{tU} \\left( \\frac{\\tilde{H(t)}H^{-1}}{t^{\\beta }} \\right) e^{-tU}e^{-sX}Xf \\\\=&\\lim _{\\Delta s \\rightarrow 0} e^{sX}e^{\\Delta s X}e^{tU}\\left(\\frac{\\tilde{H(t)}H^{-1}}{t^{\\beta }}\\right)e^{-tU}e^{-sX}\\left(\\frac{I-e^{-\\Delta sX}}{\\Delta s}-X\\right)f\\\\+ & \\lim _{\\Delta s \\rightarrow 0} e^{sX}e^{\\Delta s X}e^{tU}\\left(\\frac{\\tilde{H(t)}H^{-1}}{t^{\\beta }}\\right)e^{-tU}e^{-sX}Xf\\\\+ &\\lim _{\\Delta s \\rightarrow 0} e^{sX} \\left( \\frac{e^{\\Delta s X}-I}{\\Delta s} \\right) e^{tU} \\left(\\frac{\\tilde{H(t)}H^{-1}}{t^{\\beta }}\\right)e^{-tU}e^{-sX}f.\\end{split}$ The first limit, $\\lim _{\\Delta s \\rightarrow 0} e^{sX}e^{\\Delta s X}e^{tU}\\left(\\frac{\\tilde{H(t)}H^{-1}}{t^{\\beta }}\\right)e^{-tU}e^{-sX}\\left(\\frac{I-e^{-\\Delta sX}}{\\Delta s}-X\\right)f = 0,$ follows from the preliminary assumption, $\\lim _{\\Delta s \\rightarrow 0} \\left( \\frac{e^{\\Delta s}-I}{\\Delta s}\\right) = X.$ and $\\lim _{\\Delta s \\rightarrow 0} e^{\\Delta s X} = I.$ The second and third limits follow from above, $\\begin{split}\\lim _{\\Delta s \\rightarrow 0} e^{sX}e^{\\Delta s X}e^{tU}\\left(\\frac{\\tilde{H(t)}H^{-1}}{t^{\\beta }}\\right)e^{-tU}e^{-sX}Xf\\\\=& \\: e^{sX}e^{tU}\\left(\\frac{\\tilde{H(t)}H^{-1}}{t^{\\beta }}\\right)e^{-tU}e^{-sX}Xf\\end{split}$ and $\\begin{split}\\lim _{\\Delta s \\rightarrow 0} e^{sX} \\left( \\frac{e^{\\Delta s X}-I}{\\Delta s} \\right) e^{tU} \\left(\\frac{\\tilde{H(t)}H^{-1}}{t^{\\beta }}\\right)e^{-tU}e^{-sX}f\\\\= \\: e^{sX} X & e^{tU} \\left(\\frac{\\tilde{H(t)}H^{-1}}{t^{\\beta }}\\right)e^{-tU}e^{-sX}f.\\end{split}$ Since $f=e^{sX}(e^{tU}(Hu))$ , $\\begin{split}e^{sX}Xe^{tU}\\tilde{H(t)}H^{-1}e^{-tU}e^{-sX}f\\\\=& \\: e^{sX}Xe^{tU}\\tilde{H(t)}H^{-1}e^{-tU}e^{-sX}e^{sX}(e^{tU}(Hu))\\\\=& \\: e^{sX}Xe^{tU}H(t)u \\in \\mathcal {H}\\end{split}$ and $\\begin{split}e^{sX}e^{tU}\\tilde{H(t)}H^{-1}e^{-tU}Xe^{-sX}f\\\\=& \\: e^{sX}e^{tU}\\tilde{H(t)}H^{-1}e^{-tU}Xe^{-sX}e^{sX}e^{tU}(Hu)\\\\=& \\: e^{sX}e^{tU}\\tilde{H(t)}H^{-1}e^{-tU}Xe^{tU}(Hu) \\in \\mathcal {H}\\end{split}$ since $e^{tU}(Hu) \\subset D$ and $\\tilde{H(t)}H^{-1}$ is a bounded operator on $\\mathcal {H}$ .",
"Since $Ran(U)$ is dense in $\\mathcal {H}$ and $e^{tU}$ and $e^{sX}$ are bounded, invertible operators, $e^{sX}\\left(e^{tU}(Ran(H))\\right)$ is dense in $\\mathcal {H}$ .",
"Thus, $\\frac{\\partial \\tilde{H(s,t)}}{\\partial s} =\\frac{1}{t^{\\beta }}e^{sX}\\left[X,e^{tU}\\tilde{H(t)}H^{-1}e^{-tU} \\right]e^{-sX}$ is defined on $\\mathcal {H}$ .",
"Now we can rewrite $\\int _{0}^{\\sigma }e^{sX}e^{tU}\\left(H-\\frac{H(t)}{t^{\\beta }} \\right)gds = \\int _{0}^{\\sigma }e^{sX}e^{tU}\\left(I-\\frac{H(t)H^{-1}}{t^{\\beta }} \\right)Hg \\: ds$ and again consider the extension $\\int _{0}^{\\sigma }e^{sX}e^{tU}\\left(I-\\frac{\\tilde{H(t)}H^{-1}}{t^{\\beta }} \\right)Hgds = \\int _{0}^{\\sigma }\\tilde{H}(s,t)e^{sX}e^{tU}f \\: ds.$ Integration by parts gives $\\begin{split}\\int _{0}^{\\sigma }\\tilde{H}(s,t)e^{sX}e^{tU}fds\\\\=&\\tilde{H}(\\sigma ,t)\\int _{0}^{\\sigma }e^{sX}e^{tU}fds-\\int _{0}^{\\sigma }\\frac{\\partial \\tilde{H}(S,t)}{\\partial S}\\left[\\int _{0}^{S}e^{sX}e^{tU}fds \\right]dS.\\end{split}$ So now we must ensure that $\\frac{\\partial \\tilde{H}(s,t)}{\\partial s} = \\frac{1}{t^{\\beta }}e^{sX}\\left[X,e^{tU}\\tilde{H(t)}H^{-1}e^{-tU} \\right]e^{-sX}$ is uniformly bounded in $\\Vert \\cdot \\Vert _{op}$ .",
"Let $h \\in \\mathcal {H}$ .",
"$\\begin{split}\\frac{1}{t^{\\beta }}e^{sX}\\left(\\left[X,e^{tU}\\tilde{H(t)}H^{-1}e^{-tU} \\right] \\right)e^{-sX}h\\\\=& \\: \\frac{1}{t^{\\beta }}e^{sX}\\left(\\left[X,e^{tU}\\right]\\tilde{H(t)}H^{-1}e^{-tU}\\right.\\\\+& \\: e^{tU}\\left[X,\\tilde{H(t)}H^{-1} \\right]e^{-tU}\\\\+& \\left.",
"e^{tU}\\tilde{H(t)}H^{-1}\\left[X,e^{-tU}\\right]\\right) e^{-sX}h.\\end{split}$ Using the identity $e^{-tU}\\left[X,e^{tU}\\right]=-\\left[X,e^{-tU}\\right]e^{tU}$ we can simplify and combine terms: $\\frac{1}{t^{\\beta }}e^{sX}e^{tU}\\left(H(t)\\tilde{H(t)}H^{-1}+\\left[X,\\tilde{H(t)}H^{-1}\\right]-\\tilde{H(t)}H^{-1}H(t) \\right)e^{-tU}e^{-sX}h$ $= e^{sX}e^{tU}\\left(\\left[X,\\frac{\\tilde{H(t)}H^{-1}}{t^{\\beta }}\\right]-\\frac{\\tilde{H(t)}H^{-1}}{t^{\\beta }}\\left[\\tilde{H(t)},H \\right]H^{-1} \\right)e^{-tU}e^{-sX}h$ where $\\left[X,\\frac{\\tilde{H(t)}H^{-1}}{t^{\\beta }} \\right]$ and $\\left[\\tilde{H(t)},H \\right]H^{-1}$ are the bounded extensions of $\\left[X,\\frac{H(t)H^{-1}}{t^{\\beta }} \\right]$ and $\\left[H(t),H \\right]H^{-1}$ .",
"Conditions $(i)$ , $(ii)$ , and $(iii)$ imply that $\\begin{split}\\left\\Vert \\frac{\\partial \\tilde{H}(s,t)}{\\partial s} h \\right\\Vert _{\\mathcal {H}}\\\\\\le & C\\left(\\left\\Vert \\frac{\\tilde{H(t)}}{t^{\\beta }} H^{-1} \\right\\Vert _{op} \\cdot \\: \\left\\Vert [\\tilde{H(t)}, H]H^{-1} \\right\\Vert _{op}+ \\left\\Vert [X,\\frac{\\tilde{H(t)}}{t^{\\beta }}H^{-1}] \\right\\Vert _{op}\\right) \\cdot \\left\\Vert h \\right\\Vert _{\\mathcal {H}}\\\\\\le & C_{2}\\cdot \\Vert h \\Vert _{\\mathcal {H}}\\end{split}$ for some constants $C$ and $C_{2}$ , and thus, $\\left\\Vert \\frac{\\partial \\tilde{H}(s,t)}{\\partial s} \\right\\Vert _{op} \\le C_{2}.$ II.",
"For $g \\in Dom(H)$ , $\\frac{1}{t^{\\beta }}\\int _{0}^{\\sigma }e^{sX}e^{tU}H(t)gds= \\frac{1}{t^{\\beta }}\\int _{0}^{\\sigma }e^{sX}e^{tU}Xgds - \\frac{1}{t^{\\beta }}\\int _{0}^{\\sigma }\\frac{d}{ds}e^{sX}e^{tU}gds$ which implies that $\\left\\Vert \\frac{1}{t^{\\beta }}\\int _{0}^{\\sigma }e^{sX}e^{tU}H(t)gds \\right\\Vert _{\\mathcal {H}} \\le \\frac{C_{3}}{t^{\\beta }} \\left\\Vert Xg \\right\\Vert _{\\mathcal {H}} + \\frac{C_{4}}{t^{\\beta }}\\left\\Vert g \\right\\Vert _{\\mathcal {H}}.$ Finally, from I. and II., $\\begin{split}\\sup _{s \\in [0,\\sigma ]} \\Vert \\int _{0}^{s}e^{sX}e^{tU}fds \\Vert _{\\mathcal {H}}\\le & \\sup _{s\\in [0,\\sigma ]} \\left(\\left\\Vert \\tilde{H}(s,t) \\right\\Vert _{op} \\cdot \\left\\Vert \\int _{0}^{s} e^{sX}e^{tU}fds \\right\\Vert _{\\mathcal {H}} \\right)\\\\+ & \\: \\sigma \\cdot \\sup _{s\\in [0,\\sigma ]} \\left(\\left\\Vert \\frac{\\partial \\tilde{H}(s,t)}{\\partial s} \\right\\Vert _{op} \\cdot \\left\\Vert \\int _{0}^{s} e^{sX}e^{tU}fds \\right\\Vert _{\\mathcal {H}} \\right)\\\\+ & \\frac{C_{3} \\left\\Vert Xg \\right\\Vert _{\\mathcal {H}} + C_{4}\\left\\Vert g \\right\\Vert _{\\mathcal {H}}}{t^{\\beta }}.\\end{split}$ So for $\\sigma >0$ , chosen such that $0 < C_{1}+ \\sigma C_{2} < 1$ , for all $t$ sufficiently large, $\\sup _{s\\in [0,\\sigma ]} \\left\\Vert \\int _{0}^{s}e^{sX}e^{tU}fds \\right\\Vert _{\\mathcal {H}} \\le \\frac{C_{3}\\left\\Vert Xg \\right\\Vert _{\\mathcal {H}} + C_{4} \\left\\Vert g \\right\\Vert _{\\mathcal {H}}}{t^{\\beta }}\\frac{1}{(1-C_{1} - \\sigma C_{2})} = O(\\frac{1}{t^{\\beta }}).$ Thus, since $\\hat{\\mu }_{f}(t) \\in L^{2}(\\mathbb {R})$ , $\\mu _{f}$ is absolutely continuous.",
"As this holds for $f \\in Ran(H)$ , $\\mu _{f}$ is absolutely continuous for any $f \\in \\mathcal {H}$ .",
"Remark 4 The proof for the discrete case is the same aside from the replacement of the continuous parameter $t$ and norm $\\Vert \\cdot \\Vert _{L^{2}(\\mathbb {R})}$ by the discrete parameter $n$ and norm $\\Vert \\cdot \\Vert _{\\ell ^{2}(\\mathbb {Z})}$ .",
"The conclusion becomes $\\left\\Vert \\; \\langle e^{nU}f,f \\rangle _{\\mathcal {H}}\\; \\right\\Vert _{\\ell ^{2}(\\mathbb {Z})}=O(\\frac{1}{n^{\\beta }})$ for $\\beta > \\frac{1}{2}$ , and thus, $\\mu _{f}(n) \\in \\ell ^{2}(\\mathbb {Z})$ ."
],
[
"Time-changes of unipotent flows on\nhomogeneous spaces of semisimple groups",
"As a direct consequence of Theorem REF , we derive a result for a specific category of generating operators.",
"Let $G$ be a semisimple Lie group and let the manifold $M = \\Gamma \\setminus G$ for some lattice $\\Gamma $ in $G$ such that $M$ has finite area.",
"By the Jacobson-Morozov Theorem, any nilpotent element $U$ of the semisimple Lie algebra of $G$ is contained in a subalgebra isomorphic to $\\mathfrak {sl}_{2}$ .",
"This means that this subalgebra contains an element $X$ , such that $[U,X]=U$ .",
"Let $e^{tU}$ be a unitary operator of the Hilbert space $L^{2}(M,vol)$ .",
"Thus, if the unipotent flow generated by $U$ , $f \\circ \\phi _{t}^{U} = e^{tU}f$ , $f \\in L^{2}(M,vol)$ , is ergodic, then from Lemma 5.1 in [15], it has purely absolutely continuous spectrum on $L^{2}_{0}(M,vol)=\\left\\lbrace f \\in L^{2}(M) \\: | \\int _{M} f \\: vol=0 \\right\\rbrace .$ Let $\\tau : M \\times \\mathbb {R} \\rightarrow \\mathbb {R}$ such that $\\tau \\in C^{\\infty }(M, \\mathbb {R})$ and $\\tau (x,t+t^{\\prime })=\\tau (x,t)+\\tau (\\phi _{t}^{U}(x),t^{\\prime }).$ Let $\\alpha : M \\rightarrow \\mathbb {R}^{+}$ , be the infinitesimal generator of $\\tau $ , such that $\\alpha \\in C^{\\infty }(M)$ and $\\int _{M} \\alpha vol = \\int _{M} vol_{\\alpha }=1$ where $vol$ is the $\\phi _{t}^{U}$ -invariant volume form and $vol_{\\alpha }$ is the $\\phi _{t}^{U_{\\alpha }}$ -invariant volume form.",
"Now we consider a time-changed flow, $\\lbrace \\phi ^{U_{\\alpha }}_{t}\\rbrace $ , generated by $U_{\\alpha }=:U / \\alpha .$ Let $e^{tU_{\\alpha }}$ be a unitary operator on the Hilbert space $L^{2}(M, vol_{\\alpha })$ .",
"The following formulas hold on $D = C^{\\infty }(M)$ .",
"$[X,U_{\\alpha }]=G(\\alpha )U_{\\alpha }=(\\frac{X\\alpha }{\\alpha }-1)U_{\\alpha }=H$ $e^{-tU_{\\alpha }}\\left[X,e^{tU_{\\alpha }} \\right]=G(\\alpha , t)U_{\\alpha }=\\boxed{\\left(\\int _{0}^{t} (\\frac{X\\alpha }{\\alpha }-1) \\circ \\phi _{\\tau }^{U_{\\alpha }}(x) d\\tau \\right)U_{\\alpha }=H(t).",
"}$ The ergodicity of $\\phi _{t}^{U_{\\alpha }}$ gives us the following limit a.e., $\\begin{split}\\lim _{t \\rightarrow \\infty } \\frac{G(\\alpha ,t)}{t}=& \\lim _{t \\rightarrow \\infty } \\frac{1}{t}\\int _{0}^{t} \\left(\\frac{X\\alpha }{\\alpha }-1 \\right) \\circ \\phi _{\\tau }^{U_{\\alpha }}(x) d\\tau \\\\= & \\lim _{t \\rightarrow \\infty }\\left( \\frac{1}{t}\\int _{0}^{t} \\frac{X\\alpha }{\\alpha } \\circ \\phi _{\\tau }^{U_{\\alpha }}(x) d\\tau +\\frac{1}{t}\\int _{0}^{t} -1 \\circ \\phi _{\\tau }^{U_{\\alpha }}(x) d\\tau \\right)\\\\=& \\lim _{t \\rightarrow \\infty }\\left( \\frac{1}{t}\\int _{0}^{t} \\frac{X\\alpha }{\\alpha } \\circ \\phi _{\\tau }^{U_{\\alpha }}(x) d\\tau -1 \\right)\\\\=&\\int _{M} \\frac{X\\alpha }{\\alpha }\\: dvol_{\\alpha } -1=-1.\\end{split}$ From the Dominated Convergence Theorem, with dominating function $2 \\Vert G(\\alpha ) \\Vert _{\\infty }$ , we have convergence in $L^{2}(M)$ .",
"Thus, for $u \\in C^{\\infty }(M)$ , $\\lim _{t \\rightarrow \\infty } \\frac{G(\\alpha ,t)}{t}U_{\\alpha } u = \\boxed{ -U_{\\alpha }u=Hu.",
"}$ Lastly, $\\begin{split}\\int _{M}e^{tU_{\\alpha }}f \\: \\cdot \\overline{f} \\: dvol_{\\alpha }= \\int _{M}e^{tU_{\\alpha }}f \\: \\cdot \\overline{\\alpha f} \\: dvol\\\\= &\\int _{M} e^{sX}e^{tU_{\\alpha }}f \\: \\cdot \\overline{e^{sX}\\alpha f} \\: dvol\\\\= & \\int _{M} e^{sX}e^{tU_{\\alpha }}f \\: \\cdot \\overline{\\frac{1}{\\alpha }e^{sX}\\alpha f} \\: dvol_{\\alpha }.\\end{split}$ So if we integrate both sides of $\\langle e^{tU_{\\alpha }}f,f \\rangle _{L^{2}(M, vol_{\\alpha })} = \\langle e^{sX}e^{tU_{\\alpha }}f, \\frac{1}{\\alpha }e^{sX}\\alpha f \\rangle _{L^{2}(M, vol_{\\alpha })}$ with respect to $s$ , we obtain the following equality $\\begin{split}\\Vert \\; \\langle e^{tU_{\\alpha }}f,f \\rangle _{L^{2}(M, vol_{\\alpha })}\\; \\Vert _{L^{2}(\\mathbb {R}, dt)}\\; \\\\= & \\; \\Vert \\frac{1}{\\sigma }\\int _{0}^{\\sigma } \\langle e^{sX}e^{tU_{\\alpha }}f, \\frac{1}{\\alpha }e^{sX}\\alpha f \\rangle _{L^{2}(M, vol_{\\alpha })}\\: \\: ds \\: \\: \\Vert _{L^{2}(\\mathbb {R}, dt)}.\\end{split}$ Thus, the preliminary assumptions for Theorem REF are satisfied with $B_{1}=\\frac{1}{\\alpha }I$ and $B_{2}=\\alpha I$ .",
"Theorem 2 a.",
"Any smooth time-change of an ergodic flow on $M$ generated by a nilpotent element of a semisimple Lie algebra has absolutely continuous spectrum on $L^{2}_{0}(M,vol_{\\alpha })$ if $\\Vert \\frac{X\\alpha }{\\alpha } \\Vert _{\\infty } < 1 $ .",
"b.",
"Any smooth time-change of a uniquely ergodic flow on $M$ generated by a nilpotent element of a semisimple Lie algebra has absolutely continuous spectrum on $L^{2}_{0}(M,vol_{\\alpha })$ .",
"Remark 5 The condition in part a. is equivalent to the condition employed by Kushnirenko [12] (Theorem 2) to prove mixing for the time-changes of the horocycle flow.",
"As shown by Marcus, this condition is unnecessarily restrictive.",
"In the compact setting, the authors in [5] prove spectral results using the implicit unique ergodicity instead of requiring such a condition.",
"The author in [18] proves similar spectral results by imposing this Kushnirenko condition; the author later substitutes this condition by a utilization of unique ergodicity [19] in the compact case.",
"In the noncompact setting it remains open as to whether or not spectral results can be derived without imposing a Kushnirenko-type condition.",
"We show that the conditions of Theorem REF hold.",
"a.",
"$(i)$ Let $f=U_{\\alpha }g$ for $g \\in C^{\\infty }(M)$ .",
"$\\begin{split}\\left\\Vert \\frac{H(t)}{t}H^{-1}f \\right\\Vert _{L^{2}(M, vol_{\\alpha })}=&\\left\\Vert \\frac{G(\\alpha ,t)}{t}U_{\\alpha }(-U_{\\alpha }^{-1}f) \\right\\Vert _{L^{2}(M, vol_{\\alpha })}\\\\=&\\left\\Vert \\frac{G(\\alpha ,t)}{t}f \\right\\Vert _{L^{2}(M, vol_{\\alpha })}\\\\\\le & \\: 2 \\left\\Vert G(\\alpha ) \\right\\Vert _{\\infty } \\cdot \\Vert f \\Vert _{L^{2}(M, vol_{\\alpha })}\\\\\\le & \\: 2 \\Vert f \\Vert _{L^{2}(M, vol_{\\alpha })}\\end{split}$ Since the above holds for $f \\in Ran(U_{\\alpha })$ , $\\frac{H(t)}{t}H^{-1}$ extends to a bounded operator on $\\overline{Ran(U_{\\alpha })}=L^{2}_{0}(M,vol_{\\alpha })$ with uniformly bounded norm in $t$ , $\\left\\Vert \\frac{H(t)}{t}H^{-1} \\right\\Vert _{op}\\le 2.$ Also, $\\begin{split}\\left\\Vert (I - \\frac{H(t)}{t}H^{-1})f \\right\\Vert _{L^{2}(M,vol_{\\alpha })}=&\\left\\Vert \\left(1+\\frac{G(\\alpha ,t)}{t} \\right)f \\right\\Vert _{L^{2}(M,vol_{\\alpha })}\\\\=&\\left\\Vert \\left(1+ \\left(\\frac{1}{t}\\int _{0}^{t} (\\frac{X\\alpha }{\\alpha }-1) \\circ \\phi _{\\tau }^{U_{\\alpha }}(x) d\\tau \\right) \\right)f \\right\\Vert _{L^{2}(M,vol_{\\alpha })}\\\\=&\\left\\Vert \\left(\\frac{1}{t}\\int _{0}^{t} \\frac{X\\alpha }{\\alpha } \\circ \\phi _{\\tau }^{U_{\\alpha }}(x) d\\tau \\right)f \\right\\Vert _{L^{2}(M,vol_{\\alpha })}\\\\\\le & \\left\\Vert \\left(\\frac{1}{t}\\int _{0}^{t} \\frac{X\\alpha }{\\alpha } \\circ \\phi _{\\tau }^{U_{\\alpha }}(x) d\\tau \\right) \\right\\Vert _{\\infty } \\cdot \\Vert f \\Vert _{L^{2}(M,vol_{\\alpha })}\\\\\\le & \\left\\Vert \\frac{X\\alpha }{\\alpha } \\right\\Vert _{\\infty }\\cdot \\left\\Vert f \\right\\Vert _{L^{2}(M,vol_{\\alpha })}\\\\\\le & \\: \\Vert f \\Vert _{L^{2}(M,vol_{\\alpha })}.\\end{split}$ Since the above holds on $Ran(U_{\\alpha })$ the following is true on $\\overline{Ran(U_{\\alpha })}=L^{2}_{0}(M,vol_{\\alpha }),$ $\\limsup _{t \\rightarrow \\infty } \\left\\Vert I - \\frac{H(t)}{t}H^{-1} I \\right\\Vert _{op}=\\limsup _{t \\rightarrow \\infty } \\left\\Vert I + \\frac{G(\\alpha ,t)}{t} I \\right\\Vert _{op} < 1.$ $\\textit {(ii)}$ In the following calculation we use that $D\\phi _{t}^{U_{\\alpha }}(X)=G(\\alpha ,t)U_{\\alpha }\\circ \\phi _{t}^{U_{\\alpha }}+ X \\circ \\phi _{t}^{U_{\\alpha }}$ where $D\\phi _{t}^{U_{\\alpha }}$ denotes the differential of the diffeomorphism $\\phi _{t}^{U_{\\alpha }}$ .",
"$\\begin{split}\\left[X, \\frac{H(t)}{t}H^{-1} \\right]=& X\\left(\\frac{1}{t}\\int _{0}^{t}(\\frac{X\\alpha }{\\alpha }-1)\\circ \\phi _{\\tau }^{U_\\alpha } d\\tau \\right)-\\left(\\frac{1}{t}\\int _{0}^{t}(\\frac{X\\alpha }{\\alpha }-1)\\circ \\phi _{\\tau }^{U_\\alpha } d\\tau \\right)X\\\\=&\\frac{1}{t}\\int _{0}^{t}\\left(D\\phi _{\\tau }^{U_{\\alpha }}(X) \\circ \\phi _{-\\tau }^{U_\\alpha } \\right)\\left(\\frac{X\\alpha }{\\alpha } \\right) \\circ \\phi _{\\tau }^{U_\\alpha } d\\tau \\\\=&\\frac{1}{t}\\int _{0}^{t}\\left(G(\\alpha ,\\tau )U_{\\alpha }+X \\right)\\left(\\frac{X\\alpha }{\\alpha } \\right) \\circ \\phi _{\\tau }^{U_\\alpha } d\\tau \\\\=&\\frac{1}{t}\\int _{0}^{t}G(\\alpha ,\\tau )U_{\\alpha }\\left(\\frac{X\\alpha }{\\alpha } \\right)\\circ \\phi _{\\tau }^{U_\\alpha }(x)d\\tau + \\frac{1}{t}\\int _{0}^{t}X \\left(\\frac{X\\alpha }{\\alpha } \\right)\\circ \\phi _{\\tau }^{U_\\alpha }(x)d\\tau \\\\=&\\frac{1}{t}\\int _{0}^{t}G(\\alpha ,\\tau )\\frac{d}{d\\tau }\\left(\\frac{X\\alpha }{\\alpha } \\right)\\circ \\phi _{\\tau }^{U_\\alpha }(x)d\\tau + \\frac{1}{t}\\int _{0}^{t}X \\left(\\frac{X\\alpha }{\\alpha } \\right)\\circ \\phi _{\\tau }^{U_\\alpha }(x)d\\tau \\end{split}$ We integrate $\\frac{1}{t}\\int _{0}^{t}G(\\alpha ,\\tau )\\frac{d}{d\\tau }\\left(\\frac{X\\alpha }{\\alpha } \\right)\\circ \\phi _{\\tau }^{U_{\\alpha }}(x)d\\tau $ by parts, $\\begin{split}\\frac{1}{t}\\int _{0}^{t}G(\\alpha ,\\tau )\\frac{d}{d\\tau }\\left(\\frac{X\\alpha }{\\alpha } \\right)\\circ \\phi _{\\tau }^{U_{\\alpha }}(x)d\\tau \\\\= \\: \\frac{G(\\alpha ,t)}{t} \\left(\\frac{X\\alpha }{\\alpha } \\right)&\\circ \\phi _{t}^{U_\\alpha } - \\frac{1}{t}\\int _{0}^{t} (\\frac{X\\alpha }{\\alpha }-1)(\\frac{X\\alpha }{\\alpha }) \\circ \\phi _{\\tau }^{U_{\\alpha }}(x)d\\tau ,\\end{split}$ and obtain the bound, $\\begin{split}&\\left\\Vert \\frac{1}{t}\\int _{0}^{t}(G(\\alpha ,\\tau )U_{\\alpha }+X)\\left(\\frac{X\\alpha }{\\alpha } \\right) \\circ \\phi _{\\tau }^{U_{\\alpha }} d\\tau \\right\\Vert _{\\infty }\\\\\\le & \\left\\Vert \\frac{G(\\alpha ,t)}{t}\\left(\\frac{X\\alpha }{\\alpha }\\right)\\right\\Vert _{\\infty }\\\\+& \\left\\Vert \\frac{1}{t}\\int _{0}^{t} (\\frac{X\\alpha }{\\alpha }-1)\\left(\\frac{X\\alpha }{\\alpha } \\right) \\circ \\phi _{\\tau }^{U_{\\alpha }}(x)d\\tau \\right\\Vert _{\\infty }\\\\+& \\left\\Vert \\frac{1}{t}\\int _{0}^{t}X\\left(\\frac{X\\alpha }{\\alpha }\\right)\\circ \\phi _{\\tau }^{U_{\\alpha }}(x)d\\tau \\right\\Vert _{\\infty }\\\\\\le & \\: 2 \\: \\cdot \\left\\Vert \\frac{X\\alpha }{\\alpha }-1 \\right\\Vert _{\\infty } \\cdot \\left\\Vert \\left(\\frac{X\\alpha }{\\alpha } \\right) \\right\\Vert _{\\infty }\\\\+ & \\left\\Vert X\\left(\\frac{X\\alpha }{\\alpha } \\right) \\right\\Vert _{\\infty }\\\\\\le & \\: 2(2) + C_{\\alpha }^{\\prime \\prime }\\end{split}$ where $C_{\\alpha }^{\\prime \\prime }$ depends on the second derivative of $\\alpha $ .",
"Since $\\left[X, \\frac{H(t)}{t}H^{-1} \\right]$ is the multiplication operator given by $\\left(\\frac{1}{t}\\int _{0}^{t}(G(\\alpha ,\\tau )U_{\\alpha }+X)\\left(\\frac{X\\alpha }{\\alpha }\\right) \\circ \\phi _{\\tau }^{U_{\\alpha }} d\\tau \\right)\\cdot I,$ we obtain the following bound, $\\begin{split}&\\left\\Vert \\left[X, \\frac{H(t)}{t}H^{-1} \\right] f \\right\\Vert _{L^{2}(M,vol_{\\alpha })}\\\\\\le & \\left\\Vert \\frac{1}{t}\\int _{0}^{t}(G(\\alpha ,\\tau )U_{\\alpha }+X)\\left(\\frac{X\\alpha }{\\alpha } \\right) \\circ \\phi _{\\tau }^{U_{\\alpha }} d\\tau \\right\\Vert _{\\infty } \\cdot \\Vert f\\Vert _{L^{2}(M,vol_{\\alpha })}\\\\\\le & (4 + C_{\\alpha }^{\\prime \\prime }) \\cdot \\Vert f\\Vert _{L^{2}(M,vol_{\\alpha })}.\\end{split}$ Thus, $\\left[X, \\frac{H(t)}{t}H^{-1} \\right]$ extends to a bounded operator on $\\overline{Ran(U_{\\alpha })}$ with operator norm uniformly bounded in $t$ : $\\left\\Vert \\left[X, \\frac{H(t)}{t}H^{-1} \\right] \\right\\Vert _{op} \\le 4 + C_{\\alpha }^{\\prime \\prime }.$ $\\textit {(iii)}$ $\\begin{split}\\left\\Vert \\left[H(t),H \\right]H^{-1}f \\right\\Vert _{L^{2}(M, vol_{\\alpha })}=&\\left\\Vert \\left[G(\\alpha ,t)U_{\\alpha },-U_{\\alpha } \\right](-U_{\\alpha }^{-1}f) \\right\\Vert _{L^{2}(M, vol_{\\alpha })}\\\\=&\\left\\Vert \\left[U_{\\alpha }(\\int _{0}^{t} (\\frac{X\\alpha }{\\alpha }-1) \\circ \\phi _{\\tau }^{U_{\\alpha }} d\\tau ) \\right] \\cdot f \\right\\Vert _{L^{2}(M, vol_{\\alpha })}\\\\=& \\left\\Vert \\left[\\int _{0}^{t} \\frac{d}{d\\tau }(\\frac{X\\alpha }{\\alpha }-1) \\circ \\phi _{\\tau }^{U_{\\alpha }} d\\tau ) \\right] \\cdot f \\right\\Vert _{L^{2}(M, vol_{\\alpha })}\\\\\\le & \\: \\Vert G(\\alpha )\\circ \\phi _{t}^{U_{\\alpha }} -G(\\alpha ) \\Vert _{\\infty } \\cdot \\Vert f \\Vert _{L^{2}(M, vol_{\\alpha })}\\\\\\le & \\: 2 \\Vert G(\\alpha ) \\Vert _{\\infty } \\cdot \\Vert f \\Vert _{L^{2}(M, vol_{\\alpha })}\\\\ \\le & \\: 2(2) \\cdot \\Vert f \\Vert _{L^{2}(M, vol_{\\alpha })}\\end{split}$ The above holds on coboundaries of the form $f=U_{\\alpha }g$ , so on $\\overline{Ran(U_\\alpha )}=L^{2}_{0}(M, vol_{\\alpha })$ , $\\left\\Vert [H(t),H]H^{-1} \\right\\Vert _{op} \\le 4.$ Since conditions $(i)$ , $(ii)$ , and $(iii)$ of Theorem REF .",
"are satisfied on $Ran(U_{\\alpha })$ , the time-changed flow, $\\lbrace \\phi ^{U_{\\alpha }}_{t}\\rbrace $ , has purely absolutely continuous spectrum on $\\overline{Ran(U_{\\alpha })}$ .",
"Since $\\lbrace \\phi ^{U_{\\alpha }}_{t}\\rbrace $ is ergodic, $\\overline{Ran(U_{\\alpha })}=L^{2}_{0}(M,vol_{\\alpha })$ .",
"This concludes the proof of part a. b.",
"Now we assume that the flow $\\lbrace \\phi _{t}^{U} \\rbrace $ , and hence $\\lbrace \\phi _{t}^{U_{\\alpha }} \\rbrace $ , are uniquely ergodic.",
"$\\begin{split}&\\left\\Vert \\left(I - \\frac{H(t)}{t}H^{-1} \\right)f \\: \\right\\Vert _{L^{2}(M,vol_{\\alpha })}\\\\[2mm]=&\\left\\Vert \\left(1+\\frac{G(\\alpha ,t)}{t} \\right)f \\: \\right\\Vert _{L^{2}(M,vol_{\\alpha })}\\\\[2mm]=&\\left\\Vert \\left(1+ \\left(\\frac{1}{t}\\int _{0}^{t} (\\frac{X\\alpha }{\\alpha }-1) \\circ \\phi _{\\tau }^{U_{\\alpha }}(x) d\\tau \\right) \\right)f \\: \\right\\Vert _{L^{2}(M,vol_{\\alpha })}\\\\[2mm]=&\\left\\Vert \\left(\\frac{1}{t}\\int _{0}^{t} \\frac{X\\alpha }{\\alpha } \\circ \\phi _{\\tau }^{U_{\\alpha }}(x) d\\tau \\right) f \\: \\right\\Vert _{L^{2}(M,vol_{\\alpha })}\\\\[2mm]\\le & \\left\\Vert \\left(\\frac{1}{t}\\int _{0}^{t} \\frac{X\\alpha }{\\alpha } \\circ \\phi _{\\tau }^{U_{\\alpha }}(x) d\\tau \\right) \\right\\Vert _{\\infty } \\cdot \\Vert f \\Vert _{L^{2}(M,vol_{\\alpha })}.\\end{split}$ If $\\lbrace \\phi _{t}^{U_{\\alpha }} \\rbrace $ is uniquely ergodic, then the following converges uniformly, $\\lim _{t \\rightarrow \\infty } \\frac{1}{t}\\int _{0}^{t} \\frac{X\\alpha }{\\alpha }\\circ \\phi _{\\tau }^{U_{\\alpha }}d\\tau = \\int _{M}\\frac{X\\alpha }{\\alpha } \\: dvol_{\\alpha } = 0,$ and thus, $\\begin{split}\\limsup _{t \\rightarrow \\infty } \\left\\Vert I + \\frac{H(t)}{t}H^{-1} \\right\\Vert _{op}\\le & \\limsup _{t \\rightarrow \\infty } \\left\\Vert \\left(\\frac{1}{t}\\int _{0}^{t} \\frac{X\\alpha }{\\alpha } \\circ \\phi _{\\tau }^{U_{\\alpha }}(x) d\\tau \\right) \\right\\Vert _{\\infty }\\\\[2mm]=&\\left\\Vert \\int _{M}\\frac{X\\alpha }{\\alpha } \\: dvol_{\\alpha } \\right\\Vert _{\\infty }=0.\\end{split}$ Hence, $\\limsup _{t \\rightarrow \\infty } \\left\\Vert I + \\frac{H(t)}{t}H^{-1} \\right\\Vert _{op}<1$ is satisfied on $\\overline{Ran(U_{\\alpha })}=L^{2}_{0}(M,vol_{\\alpha })$ without imposing any further conditions on $\\frac{X\\alpha }{\\alpha }$ .",
"The remainder of the proof is the same as in a except that $\\left\\Vert \\frac{X\\alpha }{\\alpha } \\right\\Vert _{\\infty } \\le C_{\\alpha }^{\\prime } < \\infty $ where $C_{\\alpha }^{\\prime }$ is finite but not necessarily equal to 1.",
"Theorem 3 (Maximal Spectral Type) The maximal spectral type of the uniqely ergodic flow $\\lbrace \\phi ^{U_{\\alpha }}_{t}\\rbrace $ is Lebesgue on the subspace $\\overline{Ran(U_{\\alpha })}$ .",
"We follow the method in [5].",
"Lemma 1 [5] Suppose that the maximal spectral type of $\\lbrace \\phi ^{U_{\\alpha }}_{t} \\rbrace $ is not Lebesgue.",
"Then there exists a smooth non-zero function $\\omega \\in L^{2}(\\mathbb {R},dt)$ such that for all functions $g \\in C^{\\infty }(M)$ the following holds: $\\int _{\\mathbb {R}} \\omega (t) \\int ^{\\sigma }_{0} e^{sX}e^{tU_{\\alpha }}U_{\\alpha }g\\: ds\\: dt=0$ Since the maximal spectral type is not Lebesgue, then there exists a compact set $A \\subset \\mathbb {R}$ such that $A$ has positive Lebesgue measure but measure 0 with respect to the maximal spectral type.",
"So we let $\\omega \\in L^{2}(\\mathbb {R})$ be the complex conjugate of the Fourier transform of the characteristic function $\\chi _{A}$ of the set $A \\subset \\mathbb {R}$ .",
"For $f,h \\in Ran(U_{\\alpha })$ , let $\\mu _{f,h}$ denote the joint spectral measure (which we know is absolutely continuous with respect to Lebesgue since $f, h \\in Ran(U_{\\alpha })$ .",
"Thus, $\\int _{\\mathbb {R}} \\omega (t)\\langle e^{tU_{\\alpha }}f, h \\rangle _{L^{2}(M,vol)}dt=\\int _{\\mathbb {R}} \\chi _{A}(\\xi )d\\mu _{f,h}(\\xi )=0.$ In particular, when $f=U_{\\alpha }g$ we have $\\begin{split}\\int _{0}^{\\sigma } \\int _{\\mathbb {R}} \\omega (t) \\langle e^{sX}e^{tU_{\\alpha }}U_{\\alpha }g,h\\rangle _{L^{2}(M,vol)}\\: dt\\: ds\\\\= \\langle \\int _{\\mathbb {R}} \\omega (t) & \\int _{0}^{\\sigma } e^{sX}e^{tU_{\\alpha }}U_{\\alpha }g\\:ds\\:dt,h\\rangle _{L^{2}(M,vol)}=0.\\end{split}$ Recall that satisfying conditions $(i)$ and $(ii)$ and $(iii)$ in Theorem REF results in the bound $\\begin{split}\\sup _{s \\in [0, \\sigma ]} \\left\\Vert \\int _{0}^{\\sigma } e^{sX}e^{tU_{\\alpha }}U_{\\alpha }gds \\right\\Vert _{L^{2}(\\mathbb {R},dt)}\\\\\\le & \\frac{C_{ \\sigma }(\\alpha )}{t^{\\beta }}\\max \\left\\lbrace \\Vert g \\Vert _{L^{2}(M)}, \\Vert Xg \\Vert _{L^{2}(M)}, \\Vert U_{\\alpha }g \\Vert _{L^{2}(M)} \\right\\rbrace \\end{split}$ where $\\beta =1$ and $C_{\\sigma }(\\alpha )$ is a constant that depends on the time-change function $\\alpha $ and parameter $\\sigma > 0$ .",
"Because $\\int _{0}^{\\sigma } e^{sX}e^{tU_{\\alpha }}U_{\\alpha }g\\:ds$ is bounded on $M$ , it follows that $\\int _{\\mathbb {R}} \\omega (t) \\int _{0}^{\\sigma } e^{sX}e^{tU_{\\alpha }}U_{\\alpha }g\\:ds\\:dt$ vanishes.",
"Lemma 2 [5] Let $\\omega \\in L^{2}(\\mathbb {R},dt)$ .",
"If for some $x \\in M$ and for all $g \\in C^{\\infty }(M)$ , $\\int _{\\mathbb {R}} \\omega (t) \\int _{0}^{\\sigma } e^{sX}e^{tU_{\\alpha }}U_{\\alpha }g(x) \\: ds \\: dt=0,$ then $\\omega $ vanishes identically.",
"Fix $x \\in M$ and $\\sigma > 0$ .",
"Since $U$ is contained in a subalgebra isomorphic to $\\mathfrak {sl}_{2}$ , there exists an element $V$ such that $[U,V]=2X$ and $[X,V]=-V$ .",
"For any $T >0$ , $\\rho >0$ , and $\\frac{1}{2}>\\gamma > 0$ , let $E^{T}_{\\rho ,\\sigma }$ be the flow-box for the the flow $\\lbrace \\phi ^{U_{\\alpha }}_{t} \\rbrace $ defined as follows: $E^{T}_{\\rho , \\sigma }=(\\phi ^{U_{\\alpha }}_{t} \\circ \\phi ^{X}_{s} \\circ \\phi ^{V}_{r})(x)$ , for all $(r,s,t) \\in (-\\gamma , \\gamma ) \\times (-\\rho , \\rho ) \\times (-\\sigma , \\sigma )$ .",
"For any $\\chi \\in C^{\\infty }_{0}(-1,1)$ and any $\\psi \\in C^{\\infty }_{0}(-T,T)$ , let $\\tilde{g}(r,s,t):=\\chi (\\frac{r}{\\rho }) \\chi (\\frac{s}{\\sigma }) \\psi (t).$ Let $g \\in C^{\\infty }(M)$ such that $g=0$ on $ M\\setminus Im(E^{T}_{\\rho , \\sigma })$ and $g \\circ E^{T}_{\\rho , \\sigma } = \\left\\lbrace \\begin{array}{lr}0 & on \\; M\\setminus Im(E^{T}_{\\rho , \\sigma })\\\\\\tilde{g}(r,s,t) & on \\; Im(E^{T}_{\\rho , \\sigma })\\end{array}\\right.$ Let $T_{\\rho , \\sigma } >0$ be defined as: $T_{\\rho , \\sigma }:= \\min \\left\\lbrace |t|>T: \\cup _{s \\in [-\\sigma ,\\sigma ]} (\\phi ^{U_{\\alpha }}_{t} \\circ \\phi ^{X}_{s})(x) \\cap Im(E^{T}_{\\rho , \\sigma }) \\ne \\hbox{;} \\right\\rbrace .$ From unique ergodicity, $lim_{\\rho \\rightarrow 0^{+}} T_{\\rho , \\sigma } = +\\infty .$ The composition of the flow box with $U_{\\alpha }g$ and $Xg$ follow from the commutation relations: $(U_{\\alpha }g) \\circ E^{T}_{\\rho , \\sigma }: = \\chi \\left(\\frac{r}{\\rho } \\right) \\chi \\left(\\frac{s}{\\sigma } \\right) \\frac{d\\psi (t)}{dt}(t)$ and $ \\begin{split}(Xg) \\circ E^{T}_{\\rho , \\sigma } = \\frac{1}{\\sigma }\\chi \\left(\\frac{r}{\\rho } \\right) \\frac{d\\chi }{ds}\\left(\\frac{s}{\\sigma } \\right)\\psi (t)\\\\- \\left( \\int _{0}^{t} (\\frac{X\\alpha }{\\alpha }-1) \\circ \\phi _{\\tau }^{U_{\\alpha }}\\circ \\phi _{s}^{X} \\circ \\phi ^{V}_{r}(x) d\\tau \\right) & \\chi \\left(\\frac{r}{\\rho }\\right) \\chi \\left(\\frac{s}{\\sigma }\\right)\\frac{d\\psi }{dt}(t).\\end{split}$ From the assumptions of Lemma REF and by integrating REF , we have $ \\begin{split}\\chi (0)\\: \\left(\\int _{0}^{\\sigma } \\chi \\left(\\frac{s}{\\sigma }\\right)\\:ds \\right) \\left(\\int _{-T}^{T}\\omega (t) \\frac{d\\psi (t)}{dt}\\:dt \\right)\\\\+ \\int _{\\mathbb {R}\\setminus [-T_{\\rho , \\sigma }, T_{\\rho , \\sigma }]} & \\omega (t) \\int _{0}^{\\sigma } e^{sX}e^{tU_{\\alpha }}(U_{\\alpha }g) \\: ds \\: dt = 0.\\end{split}$ The bound $C_{\\sigma }(\\alpha )$ of $\\int _{0}^{\\sigma } e^{sX}e^{tU_{\\alpha }}U_{\\alpha }gds$ derived for the spectral results, combined with REF and REF , give us the following $L^{2}$ bound, $\\Vert \\int _{0}^{\\sigma } e^{sX}e^{tU_{\\alpha }}U_{\\alpha }gds \\Vert _{L^{2}(\\mathbb {R}, dt)} \\le \\frac{C_{\\sigma }(\\alpha )}{t}\\max \\lbrace \\Vert g \\Vert _{\\infty }, \\Vert Xg \\Vert _{\\infty }, \\Vert U_{\\alpha }g \\Vert _{\\infty } \\rbrace $ $\\le \\frac{C_{\\sigma }(\\alpha )}{t}\\max \\left\\lbrace 1,T \\right\\rbrace \\times \\max \\hspace{0.02864pt} ^{2} \\left\\lbrace \\Vert \\chi \\Vert _{L^{\\infty }(\\mathbb {R})}, \\Vert \\chi ^{\\prime } \\Vert _{L^{\\infty }(\\mathbb {R})}, \\Vert \\psi \\Vert _{L^{\\infty }(\\mathbb {R})}, \\Vert \\psi ^{\\prime } \\Vert _{L^{\\infty }(\\mathbb {R})} \\right\\rbrace .$ Since the above bound is uniform with respect to $\\rho $ , we can conclude that the following limit holds, $ \\lim _{\\rho \\rightarrow 0^{+}} \\int _{\\mathbb {R}\\setminus [-T_{\\rho , \\sigma },T_{\\rho , \\sigma }]}\\omega (t) \\int _{0}^{\\sigma } e^{sX}e^{tU_{\\alpha }}(U_{\\alpha }g)dsdt = 0.$ Combining equation REF with the limit result in REF implies that $\\int _{\\mathbb {R}}\\omega (t) \\frac{d\\psi (t)}{dt}\\:dt=0$ and thus, $\\omega \\equiv 0$ ."
],
[
"Time changes of the horocycle flow - compact and finite area",
"On $M=\\Gamma \\setminus PSL(2, \\mathbb {R})$ , where $M$ is either compact or of finite area, we consider the basis $ \\left\\lbrace U=\\begin{pmatrix}0 & 1 \\\\ 0 & 0\\end{pmatrix}, \\:V=\\begin{pmatrix}0 & 0 \\\\ 1 & 0\\end{pmatrix}, \\:X=\\begin{pmatrix}\\frac{1}{2} & 0 \\\\ 0 & -\\frac{1}{2}\\end{pmatrix} \\right\\rbrace $ of the Lie algebra $\\mathfrak {sl}_{2}(\\mathbb {R})$ , where $U$ and $V$ are the generators of the positive and negative horocycle flows, $\\lbrace h_{t}^{U}\\rbrace $ and $\\lbrace h_{t}^{V}\\rbrace $ respectively, and $X$ is the generator of the geodesic flow, $\\lbrace \\phi _{s}^{X}\\rbrace $ .",
"From [1], we know that $iU,iV,iX$ are essentially self-adjoint on $C^{\\infty }(M)$ , and thus, $U,V,X$ are essentially skew-adjoint on $C^{\\infty }(M)$ .",
"It follows that time-changes of the horocycle flow are special cases of Theorem REF (when $M$ is of finite volume, $\\lbrace h_{t}^{U_{\\alpha }} \\rbrace $ is ergodic, and when $M$ is compact, $\\lbrace h_{t}^{U_{\\alpha }} \\rbrace $ is uniquely ergodic).",
"The following Corollary was already proved in [5], [18], [19] under slightly weaker regularity assumptions; in this paper we have not attempted to optimize the regularity.",
"Corollary 1 a.",
"Any smooth time-change $\\lbrace h_{t}^{U_{\\alpha }}\\rbrace $ of the horocycle flow on $M$ (finite volume) has absolutely continuous spectrum on $L^{2}_{0}(M,vol_{\\alpha })$ if $\\Vert \\frac{X\\alpha }{\\alpha }\\Vert _{\\infty }<1$ .",
"b.",
"Any smooth time-change $\\lbrace h_{t}^{U_{\\alpha }}\\rbrace $ of the horocycle flow on $M$ (compact) has Lebesgue spectrum on $L^{2}_{0}(M,vol_{\\alpha })$ ."
],
[
"Twisted horocycle flows",
"Much work has been done on the spectral analysis of skew products on tori, for example, [3], [9], [10], [11].",
"We would like to consider a skew product for which the base dynamics are ergodic (in fact uniquely ergodic), but not an action on $S^{1}$ .",
"For such an example, we will examine the conditions under which the spectral properties persist or do not persist after we combine the horocycle time-change with a circle rotation.",
"Our new space is $\\hat{M}=(\\Gamma \\setminus PSL(2, \\mathbb {R})) \\times S^{1}$ for $\\Gamma $ a cocompact lattice.",
"We define the following operators: $\\hat{X} =(X,0)$ where $X$ is the generator of the geodesic flow.",
"$\\hat{V}=(V,0)$ where $V$ is the generator of the negative horocycle flow.",
"$\\hat{\\frac{d}{d\\theta }}=(0, \\frac{d}{d\\theta })$ where $\\frac{d}{d\\theta }$ is a rotation on $S^{1}$ .",
"$W=(U,0 )+ (0,\\alpha \\frac{d}{d\\theta })$ where $U$ is the generator of the positive horocycle flow and $\\alpha = \\alpha (x)$ , $x \\in \\Gamma \\setminus PSL(2, \\mathbb {R})$ , is the time change function as in 3.1.",
"Proposition 1 The flow $\\lbrace \\phi _{t}^{W}\\rbrace $ is uniquely ergodic.",
"Consider the time-change $\\lbrace \\phi _{t}^{W_{\\alpha }}\\rbrace = \\frac{1}{\\alpha }W = \\hat{U}_{\\alpha } \\times \\frac{\\hat{d}}{d\\theta }$ .",
"Since $\\lbrace h_{t}^{U_{\\alpha }}\\rbrace $ is mixing [14], then it is weakly mixing, and thus $\\lbrace \\phi _{t}^{W_{\\alpha }} \\rbrace $ is ergodic [4].",
"This implies the ergodicity of $\\lbrace \\phi _{t}^{W}\\rbrace $ .",
"Since $\\lbrace \\phi _{t}^{W}\\rbrace $ is ergodic and $\\lbrace h_{t}^{U}\\rbrace $ is uniquely ergodic [7], then from [6] (applied to flows), $\\lbrace \\phi _{t}^{W}\\rbrace $ is uniquely ergodic.",
"We are interested in the spectrum of the flow $\\lbrace \\phi _{t}^{W} \\rbrace $ , so we compute the commutator with $\\hat{X}$ .",
"$e^{-tW}\\left[\\hat{X}, e^{tW} \\right] = \\boxed{tW + \\left(\\int _{0}^{t}(\\hat{X}\\alpha -\\alpha )\\circ \\phi _{\\tau }^{W}(x)\\: d\\tau \\right)\\hat{\\frac{d}{d\\theta }} = H(t)}$ For $u \\in C^{\\infty }(\\hat{M})$ , $\\lim _{t \\rightarrow \\infty } \\frac{H(t)}{t} u=\\boxed{\\left(W - \\frac{\\hat{d}}{d\\theta }\\right)\\:u = Hu}$ Since, $\\left\\Vert \\; \\langle e^{tW}f,f\\rangle \\; \\right\\Vert _{L^{2}(\\mathbb {R}, dt)}\\; = \\: \\left\\Vert \\frac{1}{\\sigma }\\int _{0}^{\\sigma } \\langle e^{s\\hat{X}}e^{tW}f, e^{s\\hat{X}}f\\rangle \\: ds \\: \\right\\Vert _{L^{2}(\\mathbb {R}, dt)},$ the preliminary assumptions are satisfied with $B_{1}=B_{2}=I$ .",
"However, if we proceed with verifying the conditions of Theorem REF for functions in the range of $H$ , we are unable to extend pointwise bounds in $L^{2}(\\hat{M})$ to uniform bounds in the operator norm.",
"Instead we modify our operators by introducing an operator $P$ , defined in such a way that it not only acts as a projection operator but also preserves regularity.",
"Let $\\chi \\in C_{0}^{\\infty }(\\mathbb {R}\\setminus \\lbrace 0\\rbrace )$ such that the support of $\\chi $ is a compact subset of the spectrum of $H$ away from 0.",
"For $f,g \\in L^{2}(\\hat{M})$ , $\\begin{split}\\langle Pf,g \\rangle _{L^{2}(\\hat{M})} = & \\int _{\\mathbb {R}} \\chi (x) d\\mu _{f,g}(x)\\\\= & \\int _{\\mathbb {R}} \\hat{\\chi }(t) \\hat{\\mu _{f,g}} (t) \\: dt\\\\= & \\int _{\\mathbb {R}} \\hat{\\chi }(t) \\langle e^{tH}f,g \\rangle _{L^{2}(\\hat{M})} \\: dt\\end{split}$ since $H$ is a vector field, and thus, $Pf = \\int _{\\mathbb {R}} \\hat{\\chi }(t)e^{tH}f \\: dt.$ The decay of $e^{tH}f=f \\circ \\phi _{t}^{H}$ is at most polynomial in $t$ , however, since $\\chi \\in C_{0}^{\\infty }(\\mathbb {R}\\setminus \\lbrace 0\\rbrace )$ , $\\hat{\\chi } \\in \\mathcal {S}(\\mathbb {R})$ , and thus, must decay faster than any power of $\\frac{1}{t}.$ In this way, we guarantee that $P: C^{\\infty }(\\hat{M}) \\rightarrow C^{\\infty }(\\hat{M}),$ and we take $D = P(C^{\\infty }(\\hat{M})).$ Now we introduce our modified operators.",
"Let $\\hat{X}_{p}= P\\hat{X}P.$ Since $P$ commutes with $e^{tH}$ , $P$ commutes with $H$ .",
"Thus, $P$ commutes with $W$ and $e^{tW}$ .",
"Therefore, $e^{-tW}\\left[\\hat{X}_{P}, e^{tW} \\right]=Pe^{-tW}\\left[\\hat{X},e^{tW} \\right]P= PH(t)P=H_{P}(t).$ For $u \\in C^{\\infty }(\\hat{M})$ , $\\lim _{t \\rightarrow \\infty } \\frac{H_{P}(t)}{t} u=PHPu=HP^{2}u= H_{P}u.$ Note that now $H_{P}$ is a bounded, invertible operator.",
"Let $C_{H_{P}}=\\left\\Vert H_{P} \\right\\Vert _{op}$ $C_{H_{P}}^{-1}=\\Vert H_{P}^{-1} \\Vert _{op}$ $C_{P}^{k} = \\Vert P^{k} \\Vert _{op}$ $C_{H}^{P} = \\Vert HP \\Vert _{op}$ $C_{\\alpha }^{\\prime }=\\Vert \\hat{X}\\alpha - \\alpha \\Vert _{\\infty }$ Theorem 4 The flow $\\lbrace \\phi _{t}^{W}\\rbrace $ has absolutely continuous spectrum on $\\overline{Ran(H)}$ .",
"We will verify the conditions of Theorem REF on each subspace $\\mathbb {E}_{n} = \\left\\lbrace \\hat{\\frac{d}{d \\theta }}u=inu: u \\in L^{2}(\\hat{M}) \\right\\rbrace .$ $(i)$ $\\begin{split}\\frac{H_{P}(t)}{t} =& PWP + Pin\\left(\\frac{1}{t}\\int _{0}^{t}(\\hat{X}\\alpha - \\alpha )\\circ \\phi _{\\tau }^{W}(x) \\: d\\tau \\right)P\\\\=& PHP + PinP + Pin \\left(\\frac{1}{t}\\int _{0}^{t}(\\hat{X}\\alpha - \\alpha )\\circ \\phi _{\\tau }^{W}(x) \\: d\\tau \\right)P\\\\=& H_{P} + Pin\\left( \\frac{L(t)}{t}+1 \\right)P\\end{split}$ for $L(t) = \\int _{0}^{t}(\\hat{X}\\alpha - \\alpha )\\circ \\phi _{\\tau }^{W}(x) \\: d\\tau .$ Let $f=H_{P}g$ , $\\begin{split}\\left\\Vert \\frac{H_{P}(t)}{t}H_{P}^{-1}f \\right\\Vert _{L^{2}(\\hat{M})} = & \\left\\Vert \\frac{H_{P}(t)}{t}g \\right\\Vert _{L^{2}(\\hat{M})}\\\\= & \\left\\Vert H_{P}g + inP\\left(\\frac{L(t)}{t}+1 \\right)P g \\right\\Vert _{L^{2}(\\hat{M})}\\\\\\le & \\left\\Vert H_{P}g \\right\\Vert _{L^{2}(\\hat{M})} + \\left\\Vert inP\\left(\\frac{L(t)}{t}+1 \\right)P g \\right\\Vert _{L^{2}(\\hat{M})} \\\\\\le & C_{H_{P}}\\big \\Vert g \\big \\Vert _{L^{2}(\\hat{M})} + \\: nC_{P}^{2}\\left\\Vert \\hat{X}\\alpha -\\alpha \\right\\Vert _{\\infty } \\left\\Vert g \\right\\Vert _{L^{2}(\\hat{M})}\\\\+ & \\: n C_{P}^{2} \\left\\Vert g \\right\\Vert _{L^{2}(\\hat{M})}\\\\= & \\left(C_{H_{P}} + nC_{P}^{2}(C_{\\alpha }^{\\prime }+1) \\right) \\left\\Vert g \\right\\Vert _{L^{2}(\\hat{M})}.\\end{split}$ So, $\\left\\Vert \\frac{H_{P}(t)}{t}H_{P}^{-1} \\right\\Vert _{op} \\le C_{H_{P}} + nC_{P}^{2}(C_{\\alpha }^{\\prime }+1).$ Also, $\\begin{split}\\left\\Vert \\left(I - \\frac{H_{p}(t)}{t}H_{p}^{-1}\\right)f \\right\\Vert _{L^{2}(\\hat{M})} = & \\left\\Vert \\left(I - I - inP\\left(\\frac{L(t)}{t}+1 \\right)PH_{p}^{-1} \\right)f \\right\\Vert _{L^{2}(\\hat{M})}\\\\= & \\left\\Vert inP\\left( \\frac{L(t)}{t}+ 1 \\right)Pg \\right\\Vert _{L^{2}(\\hat{M})}\\\\\\le & \\: n \\: C_{P}^{2} \\left\\Vert \\frac{L(t)}{t} + 1 \\right\\Vert _{\\infty }\\cdot \\Vert g \\Vert _{L^{2}(\\hat{M})}.\\end{split}$ Since $\\lbrace \\phi _{t}^{W}\\rbrace $ is uniquely ergodic, the following converges uniformly, $\\lim _{t \\rightarrow \\infty } \\frac{L(t)}{t} + 1 = \\lim _{t \\rightarrow \\infty }\\frac{1}{t}\\int _{0}^{t}(\\hat{X}\\alpha - \\alpha ) \\circ \\phi _{\\tau }^{W}(x) \\:d\\tau + 1 = 0.$ So, $\\limsup _{t \\rightarrow \\infty } \\left\\Vert I - \\frac{H_{p}(t)}{t}H_{p}^{-1} \\right\\Vert _{op} \\le \\limsup _{t \\rightarrow \\infty } n(C_{P})^{2} \\left\\Vert \\left( \\frac{L(t)}{t} +1 \\right) \\right\\Vert _{\\infty }=0,$ and hence, $\\limsup _{t \\rightarrow \\infty } \\left\\Vert I - \\frac{H_{p}(t)}{t}H_{p}^{-1} \\right\\Vert _{op}<1.$ $(ii)$ $\\begin{split}\\left[\\hat{X}_{P}, \\frac{H_{p}(t)}{t}(H_{P})^{-1} \\right]= & \\underbrace{P \\left[\\hat{X},P \\right]P\\frac{H(t)}{t}PH_{P}^{-1}}_{a}\\\\+ & \\underbrace{P^{2} \\left[\\hat{X},P \\right]\\frac{H(t)}{t}PH_{P}^{-1}}_{b}\\\\+ & \\underbrace{P^{3}\\left[\\hat{X},\\frac{H(t)}{t}\\right]PH_{P}^{-1}}_{c}\\\\+ & \\underbrace{P^{3}\\frac{H(t)}{t}\\left[\\hat{X},P \\right]H_{P}^{-1}}_{d}\\\\+ & \\underbrace{P^{3}\\frac{H(t)}{t}P\\left[\\hat{X},H_{P}^{-1} \\right]}_{e}.\\end{split}$ Before we bound terms a-e, we show bounds for the terms $\\left[\\hat{X},P \\right]$ and $\\left[\\hat{X}, \\frac{H(t)}{t} \\right]P$ .",
"$\\begin{split}\\left[\\hat{X},P \\right]f = & \\int _{\\mathbb {R}} \\hat{\\chi }(t) \\left[\\hat{X},e^{tH} \\right]f \\: dt\\\\= & \\int _{\\mathbb {R}} \\hat{\\chi }(t)e^{tW}e^{-tW}\\left[\\hat{X},e^{tH+tin-tin} \\right]f \\: dt\\\\= & \\int _{\\mathbb {R}} \\hat{\\chi }(t)e^{tW}e^{-tW}\\left[\\hat{X},e^{tW} \\right]e^{-itn}f \\: dt\\\\= & \\int _{\\mathbb {R}} \\hat{\\chi }(t)e^{tW}H(t)e^{-itn}f \\: dt \\\\= & \\int _{\\mathbb {R}} \\hat{\\chi }(t)e^{tW}tWe^{-itn}f \\: dt + \\int _{\\mathbb {R}} \\hat{\\chi }(t)e^{tW}(in L(t))e^{-itn}f \\: dt\\\\= & \\int _{\\mathbb {R}} \\hat{\\chi }(t)e^{-itn}t\\frac{d}{dt}(e^{tW}f) \\: dt + \\int _{\\mathbb {R}} \\hat{\\chi }(t)e^{tW}(in L(t))e^{-itn}f \\: dt.\\end{split}$ The first term we integrate by parts: $\\hspace{28.45274pt} \\hat{\\chi }(t)e^{-itn}t \\hspace{28.45274pt} \\hat{\\chi }^{^{\\prime }}(t)e^{-itn}t + \\hat{\\chi }(t)e^{-itn}-in\\hat{\\chi }(t)e^{-itn}t$ $\\hspace{-71.13188pt} e^{tW}f \\hspace{28.45274pt} \\frac{d}{dt}(e^{tW}f)$ $\\begin{split}\\int _{\\mathbb {R}} \\hat{\\chi }(t)e^{-itn}t\\frac{d}{dt}(e^{tW}f) \\: dt= & \\: \\hat{\\chi }(t)e^{-itn}te^{tW}f\\biggr \\vert _{\\infty }^{-\\infty }\\\\+ & \\int _{\\mathbb {R}} \\left( \\hat{\\chi }^{^{\\prime }}(t)e^{-itn}t + \\hat{\\chi }(t)e^{-itn}-in\\hat{\\chi }(t)e^{-itn}t \\right)e^{tW}f \\: dt\\\\= & \\int _{\\mathbb {R}} \\left( \\hat{\\chi }^{^{\\prime }}(t)e^{-itn}t + \\hat{\\chi }(t)e^{-itn}-in\\hat{\\chi }(t)e^{-itn}t \\right)e^{tW}f \\: dt.\\end{split}$ So, $\\begin{split}& \\left\\Vert \\int _{\\mathbb {R}} \\left( \\hat{\\chi }^{^{\\prime }}(t)e^{-itn}t + \\hat{\\chi }(t)e^{-itn}-in\\hat{\\chi }(t)e^{-itn}t \\right)e^{tW}f \\: dt \\right\\Vert _{L^{2}(\\hat{M})}\\\\\\le & \\left(\\int _{\\mathbb {R}} \\left| \\hat{\\chi }^{^{\\prime }}(t)t \\right| \\: dt + \\int _{\\mathbb {R}} \\left|\\hat{\\chi }(t) \\right| \\: dt + \\int _{\\mathbb {R}} \\left|n\\hat{\\chi }(t) \\right| \\: dt \\right) \\left\\Vert f \\right\\Vert _{L^{2}(\\hat{M})}\\\\\\le & C_{1} \\left\\Vert f \\right\\Vert _{L^{2}(\\hat{M})} .\\end{split}$ The boundedness of the second term follows immediately, $\\begin{split}\\left\\Vert \\int _{\\mathbb {R}} \\hat{\\chi }(t)e^{tW}(in L(t))e^{-itn}f \\: dt \\right\\Vert _{L^{2}(\\hat{M})}\\le & \\int _{\\mathbb {R}} \\left| \\hat{\\chi }(t)(in L(t)) \\right| \\: dt \\Vert f \\Vert _{L^{2}(\\hat{M})}\\\\\\le & C_{2} \\Vert f \\Vert _{L^{2}(\\hat{M})}.\\end{split}$ Thus, $\\left\\Vert \\left[\\hat{X},P \\right] f \\right\\Vert _{L^{2}(\\hat{M})} \\le (C_{1}+C_{2}) \\Vert f \\Vert _{L^{2}(\\hat{M})} = C \\Vert f \\Vert _{L^{2}(\\hat{M})},$ and hence, $\\left\\Vert \\left[\\hat{X},P \\right] \\right\\Vert _{op}\\le C.$ Also, $\\begin{split}\\left[\\hat{X},\\frac{H(t)}{t} \\right]P = & \\left[\\hat{X},W \\right]P + \\left[\\hat{X},\\frac{1}{t}\\int _{0}^{t}(\\hat{X}\\alpha - \\alpha ) \\circ \\phi _{\\tau }^{W}(x) \\: d\\tau \\: in \\right]P\\\\= & \\: (\\hat{U} + \\hat{X}\\alpha \\: in)P + [\\hat{X},\\frac{1}{t}\\int _{0}^{t}(\\hat{X}\\alpha - \\alpha ) \\circ \\phi _{\\tau }^{W}(x) \\: d\\tau \\: in]P\\\\= & \\left(\\hat{U} + (\\alpha -1)in -(\\alpha -1)in + \\hat{X}\\alpha \\: in \\right)P\\\\+ & \\left[\\hat{X},\\frac{1}{t}\\int _{0}^{t}(\\hat{X}\\alpha - \\alpha ) \\circ \\phi _{\\tau }^{W}(x) \\: d\\tau \\: in \\right]P\\\\= & \\: HP + \\left(\\hat{X}\\alpha - \\alpha +1 \\right)\\: inP + \\left[\\hat{X},\\frac{1}{t}\\int _{0}^{t}(\\hat{X}\\alpha - \\alpha ) \\circ \\phi _{\\tau }^{W}(x) \\: d\\tau \\: in \\right]P.\\end{split}$ Now we bound the following term, $\\begin{split}& \\left[\\hat{X},\\frac{1}{t}\\int _{0}^{t}(\\hat{X}\\alpha - \\alpha ) \\circ \\phi _{\\tau }^{W}(x) \\: d\\tau \\: in \\right] \\\\= & \\: \\hat{X} \\left(\\frac{1}{t}\\int _{0}^{t}(\\hat{X}\\alpha - \\alpha )\\circ \\phi _{\\tau }^{W}(x) \\:d\\tau \\right)in\\\\- & \\left(\\frac{1}{t}\\int _{0}^{t}(\\hat{X}\\alpha - \\alpha )\\phi _{\\tau }^{W}(x) \\:d\\tau \\right)\\hat{X}in\\\\= & \\: \\frac{1}{t}\\int _{0}^{t}\\left(D\\phi _{\\tau }^{W}(\\hat{X})\\circ \\phi _{-\\tau }^{W}(x) \\right)(\\hat{X}\\alpha - \\alpha )\\circ \\phi _{\\tau }^{W}(x) \\:d\\tau \\: in.\\end{split}$ Since $\\hat{X}\\alpha - \\alpha $ is a function on $M$ , $\\begin{split}\\frac{1}{t}\\int _{0}^{t}\\left(D\\phi _{\\tau }^{W}(\\hat{X})\\circ \\phi _{-\\tau }^{W}(x) \\right)&(\\hat{X}\\alpha - \\alpha )\\circ \\phi _{\\tau }^{W}(x) \\:d\\tau \\: in\\\\= & \\: \\frac{1}{t}\\int _{0}^{t} \\left(D\\phi _{\\tau }^{U}(X)\\circ \\phi _{-\\tau }^{U}(x) \\right) (X\\alpha - \\alpha )\\circ \\phi _{\\tau }^{U}(x) \\:d\\tau \\: in\\end{split}$ $\\begin{split}= & \\: \\frac{1}{t}\\int _{0}^{t}(X+ \\tau U)(X\\alpha -\\alpha )\\circ \\phi _{\\tau }^{U}(x) \\:d\\tau \\: in.\\end{split}$ We consider the $L^{\\infty }(M)$ norm and let $\\left\\Vert X\\alpha \\right\\Vert _{\\infty } = C_{\\alpha ,M} < \\infty $ $\\left\\Vert X\\alpha - \\alpha \\right\\Vert _{\\infty } = C_{\\alpha ,M}^{^{\\prime }} < \\infty $ and $\\left\\Vert X(X \\alpha ) \\right\\Vert _{\\infty }= C_{\\alpha ,M}^{\\prime \\prime } < \\infty .$ Hence, $\\begin{split}& \\left\\Vert \\frac{1}{t}\\int _{0}^{t}(X+ \\tau U)(X\\alpha - \\alpha )\\circ \\phi _{\\tau }^{U}(x) \\:d\\tau \\: in \\right\\Vert _{\\infty }\\\\\\le & \\left\\Vert \\frac{1}{t}\\int _{0}^{t}X(X\\alpha - \\alpha )\\circ \\phi _{\\tau }^{U}(x) \\:d\\tau \\: in \\Vert _{\\infty } + \\Vert \\frac{1}{t}\\int _{0}^{t}\\tau U(X\\alpha - \\alpha )\\circ \\phi _{\\tau }^{U}(x) \\:d\\tau \\: in \\right\\Vert _{\\infty }\\\\\\le & \\: n \\Vert X(X\\alpha - \\alpha ) \\Vert _{\\infty } \\: + \\: n \\Vert (X\\alpha - \\alpha ) \\circ \\phi _{t}^{U}-(X\\alpha - \\alpha ) \\Vert _{\\infty } \\\\\\le & \\: n \\left\\Vert X(X\\alpha - \\alpha ) \\Vert _{\\infty } + 2n \\Vert X\\alpha - \\alpha \\right\\Vert _{\\infty } \\\\\\le & \\: n \\Vert X(X\\alpha ) \\Vert _{\\infty } + \\: n \\Vert X \\alpha \\Vert _{\\infty } + 2n \\Vert X\\alpha - \\alpha \\Vert _{\\infty } \\\\\\le & \\: nC_{\\alpha ,M}^{\\prime \\prime } \\: + n C_{\\alpha ,M} + 2nC_{\\alpha ,M}^{^{\\prime }} .\\end{split}$ Thus, $\\left[\\hat{X}, \\frac{H(t)}{t} \\right]P$ extends to a bounded operator on $\\overline{Ran(H_{p})}$ with operator norm uniformly bounded in $t$ : $\\left\\Vert \\left[\\hat{X}, \\frac{H(t)}{t} \\right]P \\right\\Vert _{op} \\le \\left(C_{H}^{P}+n(C_{\\alpha }^{^{\\prime }}+1)C_{P}^{1} \\right) + n\\left(C_{\\alpha ,M}^{\\prime \\prime } \\: + C_{\\alpha ,M} + 2C_{\\alpha ,M}^{^{\\prime }} \\right)C_{P}^{1}.$ $a$ : $\\begin{split}\\left\\Vert P \\left[\\hat{X},P \\right]P\\frac{H(t)}{t}PH_{P}^{-1} \\right\\Vert _{op}\\le & \\: C^{1}_{P}\\cdot C \\cdot \\left\\Vert \\frac{H_{P}(t)}{t}H_{P}^{-1} \\right\\Vert _{op}\\\\\\le & \\: C^{1}_{P}\\cdot C \\cdot \\left(C_{H_{P}} + nC_{P}^{2}(C_{\\alpha }^{\\prime }+1) \\right).\\end{split}$ $b:$ $\\begin{split}\\left\\Vert P^{2} \\left[\\hat{X},P \\right]\\frac{H(t)}{t}PH_{P}^{-1} \\right\\Vert _{op}= & \\left\\Vert P^{2} \\left[\\hat{X},P \\right](H + \\left(\\frac{L(t)}{t} + 1 \\right) in)PH_{P}^{-1} \\right\\Vert _{op}\\\\\\le & \\: C_{P}^{2} \\cdot C \\cdot \\left( \\left\\Vert HP \\right\\Vert _{op} +\\: n\\left( \\left\\Vert \\hat{X}\\alpha - \\alpha \\right\\Vert _{\\infty } + 1 \\right) C_{P}^{1} \\right)\\\\ \\cdot & \\Vert H_{P}^{-1} \\Vert _{op}\\\\= & \\: C_{P}^{2} \\cdot C \\cdot \\left(C_{H}^{P} + n(C_{\\alpha }^{\\prime }+1)C_{P}^{1} \\right)C_{H_{P}}^{-1}\\end{split}$ $c:$ $\\left\\Vert P^{3} \\left[\\hat{X},\\frac{H(t)}{t} \\right]PH_{P}^{-1} \\right\\Vert _{op}$ $\\le C_{P}^{3}\\left((C_{H}^{P}+n \\left(C_{\\alpha }^{^{\\prime }}+1)C_{P}^{1} \\right) + n(C_{\\alpha ,M}^{\\prime \\prime } \\: + C_{\\alpha ,M} + 2C_{\\alpha ,M}^{^{\\prime }})C_{P}^{1} \\right) C_{H_{P}}^{-1}.$ $d:$ $\\left\\Vert P^{3}\\frac{H(t)}{t}\\left[\\hat{X},P \\right]H_{P}^{-1} \\right\\Vert _{op} \\le C_{P}^{2}\\cdot \\left(C_{H}^{P} + nC_{P}^{1}(C_{\\alpha }^{\\prime }+1) \\right)\\cdot C \\cdot C_{H_{P}}^{-1}$ $e:$ $\\begin{split}\\left[\\hat{X},H_{P}^{-1} \\right]= & H_{P}^{-1}\\left[H_{P},\\hat{X} \\right]H_{P}^{-1}\\\\= & H_{P}^{-1}\\left[P,\\hat{X} \\right]PHH_{P}^{-1} + H_{P}^{-1}P\\left[P,\\hat{X} \\right]HH_{P}^{-1} + H_{P}^{-1}P^{2}\\left[\\hat{X},H \\right]H_{P}^{-1}.\\end{split}$ $\\begin{split}\\left\\Vert \\left[\\hat{X},H_{P}^{-1} \\right] \\right\\Vert _{op}\\le & \\: C_{H_{P}}^{-1}\\cdot C \\cdot C_{H}^{P}C_{H_{P}}^{-1} + C_{H_{P}}^{-1}C_{P}^{1}\\cdot C \\cdot C_{H}^{P}C_{P}^{-1}C_{H_{P}}^{-1}\\\\+ & \\: C_{H_{P}}^{-1}(C_{H_{P}}+C_{P}^{2}n(C_{\\alpha }^{^{\\prime }}+1))C_{H_{P}}^{-1}.\\end{split}$ $\\begin{split}& \\left\\Vert P^{3}\\frac{H(t)}{t}P\\left[\\hat{X},H_{P}^{-1}\\right] \\right\\Vert _{op}\\le C_{P}^{2}\\left(C_{H_{P}}+nC_{P}^{2}(C_{\\alpha }^{\\prime }+1) \\right)\\\\\\cdot & \\left( C_{H_{P}}^{-1}\\cdot C \\cdot C_{H}^{P}C_{H_{P}}^{-1} + C_{H_{P}}^{-1}C_{P}^{1}\\cdot C \\cdot C_{H}^{P}C_{P}^{-1}C_{H_{P}}^{-1} + C_{H_{P}}^{-1}(C_{H_{P}}+C_{P}^{2}n(C_{\\alpha }^{^{\\prime }}+1)C_{H_{P}}^{-1} \\right).\\end{split}$ $(iii)$ .",
"$\\begin{split}\\left[ H_{P}(t),H_{P} \\right] H_{P}^{-1} = & \\left[tH_{P} + P(L(t)+t)Pin, H_{P} \\right]H_{P}^{-1}\\\\= & \\left[PL(t)Pin,H_{P} \\right]H_{P}^{-1}\\\\= & P^{3}HL(t)PinH_{P}^{-1}=P^{3}WL(t)PinH_{P}^{-1}-P^{3}inL(t)PinH_{P}^{-1}\\\\= & P^{3}\\left((\\hat{X}\\alpha -\\alpha )\\circ \\phi _{t}^{W} - (\\hat{X}\\alpha - \\alpha )\\right)PinH_{P}^{-1}-P^{3}inL(t)PinH_{P}^{-1}.\\end{split}$ $\\begin{split}\\left\\Vert \\left[H_{p}(t),H_{P} \\right]H_{P}^{-1} \\right\\Vert _{op}\\\\\\le & \\: 2n \\: C_{P}^{4}C_{\\alpha }^{\\prime }C_{H_{P}}^{-1}+n^{2}C_{P}^{4}C_{\\alpha }^{^{\\prime }}C_{H_{P}}^{-1}.\\end{split}$ We have shown that the conditions of Theorem REF are satisfied on $Ran(P)$ .",
"We would like to extend this to $Ran(H)$ .",
"Recall that $H_{P}$ depends upon a choice of $\\chi \\in C_{0}^{\\infty }(\\mathbb {R}\\setminus \\lbrace 0\\rbrace )$ .",
"For $f \\in Dom(H)$ , we can express the following in terms of integrals involving the spectral projector as $Hf = \\int _{\\mathbb {R}} x \\: dE(x)f,$ $H_{P}f = \\int _{\\mathbb {R}}x\\chi (x) \\: dE(x)f,$ and since $f \\in Dom(H)$ , $\\int _{\\mathbb {R}} x^{2} \\: dE(x)f< + \\infty .$ Let $\\chi $ be such that $\\chi (x) = 1$ for $x \\in (-K,-\\epsilon )\\cup (\\epsilon , K)=I_{\\epsilon ,K}$ and $supp(\\chi )$ vanishes outside of $I_{\\epsilon ,K}$ .",
"Since on $I_{\\epsilon ,K}$ , $H_{P}f=Hf,$ we consider $H_{P}f - Hf$ on $\\mathbb {R}\\setminus I_{\\epsilon ,K}$ , i.e., $\\int _{\\mathbb {R}\\setminus I_{\\epsilon ,K}}x(\\chi (x)-1)\\:dE(x)f.$ For $|x|\\le \\epsilon $ , $\\begin{split}\\lim _{\\epsilon \\rightarrow 0} \\left\\Vert \\int _{|x|\\le \\epsilon } x\\left(\\chi (x)-1 \\right)\\:dE(x)f \\: \\right\\Vert ^{2}_{L^{2}(\\mathbb {R})} = & \\lim _{\\epsilon \\rightarrow 0} \\int _{|x|\\le \\epsilon } \\left|x(\\chi (x)-1) \\right|^{2}\\:dE(x)f\\\\\\le & \\lim _{\\epsilon \\rightarrow 0} 4\\epsilon ^{2} \\int _{|x|\\le \\epsilon }\\:dE(x)f\\\\\\le & \\lim _{\\epsilon \\rightarrow 0} 4\\epsilon ^{2} \\Vert f \\Vert _{L^{2}(\\hat{M})}\\\\= & 0.\\end{split}$ For $|x| \\ge K$ , $\\begin{split}\\lim _{K \\rightarrow \\infty } \\left\\Vert \\int _{|x| \\ge K} x(\\chi (x)-1)\\:dE(x)f \\right\\Vert ^{2}_{L^{2}(\\mathbb {R})} = & \\lim _{K \\rightarrow \\infty } \\int _{|x| \\ge K} \\left| x(\\chi (x)-1) \\right|^{2}\\:dE(x)f\\\\\\le & \\lim _{K \\rightarrow \\infty } \\int _{|x| \\ge K} 4x^{2}\\:dE(x)f =0\\end{split}$ since $\\int _{\\mathbb {R}} x^{2} \\:dE(x)f< + \\infty .$ Thus, $\\inf _{\\chi \\in C^{\\infty }_{0}(\\mathbb {R} \\setminus \\lbrace 0\\rbrace )} \\left\\Vert H_{P}f-Hf \\right\\Vert _{L^{2}(\\hat{M})} =0.$ So, for any $Hf$ , there exists a sequence $\\lbrace H_{P_{n}}\\rbrace $ such that $H_{P_{n}} \\rightarrow Hf,$ and thus, $\\overline{Ran(H)} = \\overline{\\left\\lbrace \\bigcup _{P}Ran(H_{P}) \\right\\rbrace }.$ Consequently, for every $f \\in \\overline{Ran(H)}$ , $\\mu _{f}$ is absolutely continuous.",
"The characteristics of $\\overline{Ran(H)}$ are linked to the properties of the cocycle $a(x,t)=\\int _{0}^{t}(\\alpha -1) \\circ h^{U}_{s}(x) \\: ds$ since $\\phi _{t}^{H}(x,\\theta )=\\left(h^{U}_{t}(x), \\theta + \\int _{0}^{t}(\\alpha -1) \\circ h^{U}_{s}(x) \\: ds \\right).$ For $\\pi $ a projection from $\\mathbb {R}$ to $S^{1}$ , let $\\hat{a} = \\pi (a).$ Let $k_{0}=\\min \\left\\lbrace k: k\\hat{a}(x,t) = g\\circ h_{t}^{U}(x)-g(x) \\right\\rbrace ,$ for $k \\in \\mathbb {Z}^{+}$ and $g: M \\rightarrow S^{1}$ .",
"From [3] we have the following cases.",
"If $k_{0}=\\infty $ then $\\lbrace \\phi _{t}^{W} \\rbrace $ has purely absolutely continuous spectrum on $L^{2}_{0}(\\hat{M})$ ; this is the case when $\\lbrace \\phi _{t}^{H} \\rbrace $ is ergodic, and thus, $\\overline{Ran(H)}=L^{2}_{0}(\\hat{M}).$ When, $k_{0}<\\infty $ , we have a nontrivial pure point component of the spectrum.",
"To show this, we consider the subspaces given by $E_{nk_{0}}=\\left\\lbrace f(x)e^{ink_{0}\\theta } \\right\\rbrace \\subset L^{2}(\\hat{M})$ for $f \\in L^{2}(M)$ and $n \\in \\mathbb {Z}$ .",
"The operator $H_{t}:f(x) \\rightarrow f \\circ h_{t}^{U}(x)\\:e^{ink_{0}\\hat{a}(x,t)}$ is unitarily equivalent to the restriction of the unitary group $\\lbrace \\phi _{t}^{H}\\rbrace $ to $E_{nk_{0}}$ .",
"Let $S_{t}:f(x) \\rightarrow f \\circ h_{t}^{U}(x)$ and $V_{t}:f(x) \\rightarrow f(x)\\:e^{-ing(x)}.$ We compute $H_{t} \\circ V_{t}$ : $\\begin{split}\\left(H_{t} \\circ V_{t} \\right)(f)(x)= & f \\circ h_{t}^{U}(x)\\:e^{-ing \\circ h_{t}^{U}(x) }\\:e^{ink_{0}\\hat{a}(x,t)}\\\\= & f \\circ h_{t}^{U}(x)\\:e^{-ing \\circ h_{t}^{U}(x)}\\:e^{ing \\circ h_{t}^{U}(x)-ing(x)}\\\\= & f \\circ h_{t}^{U}(x)\\:e^{-ing(x)} \\\\= & \\left(V_{t} \\circ S_{t} \\right)(f)(x).\\end{split}$ Hence, on $E_{nk_{0}}$ , $H_{t}$ is unitarily equivalent to $S_{t}$ .",
"Since $e^{ink_{0}\\theta }$ is an invariant function for $S_{t}$ , $H_{t}$ has an eigenfunction in $E_{nk_{0}}$ for every $n$ .",
"Thus, the spectrum on $E_{nk_{0}}$ has an absolutely continuous component as well as an infinite dimensional pure point component.",
"This leads to the following Corollary.",
"Corollary 2 The spectrum of $\\lbrace \\phi _{t}^{W} \\rbrace $ on $C^\\perp $ , the orthocomplement of the constants, has a pure point component (possibly trivial) and an absolutely continuous component, but no singularly continuous component."
],
[
"Applications to maps",
"The author in [19] uses the Mourre Estimate [2], [16] to prove the following spectral results.",
"Here we rederive similar results by showing that the conditions of Theorem REF are satisfied; our results are slightly weaker as we prove results for functions of class $C^{2}$ while the author in [19] considers functions of class $C^{1}$ with an added Dini Condition.",
"We use the notation and description from [19]."
],
[
"Skew products over translations",
"Let $X$ be a compact metric abelian Lie group with normalized Haar measure $\\mu $ .",
"Let $\\lbrace F_{t}\\rbrace $ be a uniquely ergodic [6] translation flow (we assume that $F_{1}$ is ergodic), $F_{t} = y_{t}x$ with vector field $Y$ .",
"The associated operators $\\lbrace V_{t}\\rbrace $ are given by $V_{t}\\psi =\\psi \\circ F_{t}$ with generator $P=-i{L}_{Y}$ .",
"Let $G$ be a compact metric abelian group.",
"Let $\\phi :$ $X \\rightarrow G$ such that $\\phi $ can be written as $\\phi =\\xi \\eta $ where $\\xi $ is a group homomorphism and $\\eta $ satisfies $\\sup _{t > 0} \\left\\Vert \\frac{{L}_{Y}(\\chi \\circ \\eta ) \\circ F_{t} - {L}_{Y}(\\chi \\circ \\eta )}{t}) \\right\\Vert _{\\infty } < \\infty $ and $\\chi \\circ \\eta = e^{i\\tilde{\\eta }_{\\chi }}$ for $\\tilde{\\eta }_{\\chi } \\in Dom(P)$ a real-valued function determined by $\\chi $ and $\\eta $ .",
"The skew product, $T:X \\times G \\rightarrow X \\times G$ , is defined by $T(x,z)=(y_{1}x,\\phi (x) z)$ with corresponding unitary operator $W \\Psi = \\Psi \\circ T$ .",
"Let $\\hat{G}$ be the character group of $G$ .",
"The decomposition $L^{2}(X \\times G)=\\bigoplus _{\\chi \\in \\hat{G}}L_{\\chi }$ , $L_{\\chi } = \\lbrace \\varphi \\otimes \\chi : \\varphi \\in L^{2}(X)\\rbrace $ and the restriction of $W$ to the subspaces $L_{\\chi }$ allow us to study the spectrum of convenient, unitarily equivalent operators to $W|_{L_{\\chi }}$ , namely, $U_{\\chi } \\psi = (\\chi \\circ \\psi ) V_{1} \\psi $ for $\\chi \\circ \\xi \\lnot \\equiv 1$ .",
"(Here $U_{\\chi }$ takes the place of $e^{U}$ as given in the conditions.)",
"We will choose to take the commutator with $P$ ; it follows from [1] that $P$ is essentially self-adjoint on $D = C^{\\infty }(X)$ .",
"$\\begin{split}\\left[P, U_{\\chi }\\right]= &\\left[P,(\\chi \\circ \\phi )V_{1}\\right]\\\\= & \\left[P, (\\chi \\circ \\phi )I \\right]V_{1}\\\\= & -i{L}_{Y}\\left(\\chi \\circ \\phi \\right)V_{1}\\\\= & -i\\left[{L}_{Y}(\\chi \\circ \\xi )(\\chi \\circ \\eta ) + (\\chi \\circ \\xi ){L}_{Y}(\\chi \\circ \\eta ) \\right]V_{1}\\\\= & -i\\left[\\xi _{0} + \\frac{{L}_{Y}(\\chi \\circ \\eta )}{(\\chi \\circ \\eta )} \\right](\\chi \\circ \\phi )V_{1}\\\\= & -i\\left[\\xi _{0} + \\frac{{L}_{Y}(\\chi \\circ \\eta )}{(\\chi \\circ \\eta )} \\right]U_{\\chi }\\end{split}$ where $\\xi _{0}=\\frac{d}{dt}(\\chi \\circ \\xi )(y_{t})|_{t=0} \\in i\\mathbb {R}\\setminus \\lbrace 0\\rbrace $ .",
"So, $[P, U_{\\chi }] =\\left(- i\\xi _{0}-\\frac{i{L}_{Y}(\\chi \\circ \\eta )}{(\\chi \\circ \\eta )} \\right)U_{\\chi } = GU_{\\chi }.$ Thus, $U_{\\chi }^{-n}[P, U_{\\chi }^{n}] = \\sum _{k=1}^{n}U_{\\chi }^{-k}GU_{\\chi }^{k}$ $=\\boxed{\\sum _{k=1}^{n}\\left(- i\\xi _{0}-\\frac{i{L}_{Y}(\\chi \\circ \\eta )}{(\\chi \\circ \\eta )}\\right)\\circ F_{-k}=\\sum _{k=1}^{n}G\\circ F_{-k}= H(n).",
"}$ Note that $\\begin{split}\\frac{-i{L}_{Y}(\\chi \\circ \\eta )}{(\\chi \\circ \\eta )} = & \\frac{-i{L}_{Y}(e^{i\\tilde{\\eta }_{\\chi }})}{e^{i\\tilde{\\eta }_{\\chi }}}\\\\= & \\frac{-ie^{i\\tilde{\\eta }_{\\chi }} \\cdot {L}_{Y}(\\tilde{\\eta }_{\\chi })}{e^{i\\tilde{\\eta }_{\\chi }}}\\\\= & -i{L}_{Y}(\\tilde{\\eta }_{\\chi })\\\\= & \\: P\\tilde{\\eta }_{\\chi }.\\end{split}$ From unique ergodicity we get the following convergence $\\lim _{n \\rightarrow \\infty } \\frac{H(n)}{n}u =\\left(-i\\xi _{0} + \\int _{X} P \\tilde{\\eta }_{\\chi } \\: d\\mu \\right)u =\\boxed{-i\\xi _{0}u=Hu}$ uniformly in $n$ for $u \\in L_{\\chi }$ .",
"Since $\\left\\Vert \\; \\langle U_{\\chi }^{n}f,f \\rangle _{L_{\\chi }}\\; \\right\\Vert _{\\ell ^{2}(\\mathbb {Z})} \\; =\\; \\left\\Vert \\frac{1}{\\sigma }\\int _{0}^{\\sigma } \\langle e^{sP}U_{\\chi }^{n}f, e^{sP}f \\rangle _{L_{\\chi }} ds\\: \\right\\Vert _{\\ell ^{2}(\\mathbb {Z})},$ the preliminary assumptions for Theorem REF are satisfied for $B_{1}=B_{2}=I$ .",
"Now we proceed by showing that the conditions of Theorem REF hold.",
"$(i)$ It is unnecessary to consider coboundaries since both $H=-i\\xi _{0}I$ and $H^{-1}=\\frac{i}{\\xi _{0}}I$ are constants.",
"Instead we take any $f \\in L_{\\chi }$ .",
"$\\begin{split}\\frac{H(n)}{n}H^{-1}f = & \\left(\\frac{1}{n}\\sum _{k=1}^{n}\\left(- i\\xi _{0}-\\frac{i{L}_{Y}(\\chi \\circ \\eta )}{(\\chi \\circ \\eta )} \\right)\\circ F_{-k} \\right) \\cdot \\frac{i}{\\xi _{0}}f\\\\= & f + \\left(\\frac{1}{n}\\sum _{k=1}^{n} P\\tilde{\\eta }_{\\chi } \\circ F_{-k} \\right) \\cdot \\frac{i}{\\xi _{0}}f .\\end{split}$ So, $\\left\\Vert \\frac{H(n)}{n}H^{-1}f \\right\\Vert _{L_{\\chi }} \\le \\left(1 + \\frac{\\Vert P\\tilde{\\eta }_{\\chi } \\Vert _{L_{\\chi }}}{|\\xi _{0}|} \\right) \\cdot \\Vert f \\Vert _{L_{\\chi }}.$ Since $\\tilde{\\eta }_{\\chi } \\in Dom(P)$ , $\\Vert P\\tilde{\\eta }_{\\chi } \\Vert _{L_{\\chi }} \\le C_{1}.$ Thus, $\\frac{H(n)}{n}H^{-1}$ is a bounded operator with uniformly bounded norm in $n$ , $\\left\\Vert \\frac{H(n)H^{-1}}{n} \\right\\Vert _{op} \\le 1 + \\frac{C_{1}}{|\\xi _{0}|}.$ Also, $\\begin{split}\\left\\Vert \\left(I - \\frac{H(n)}{n}H^{-1} \\right) f \\right\\Vert _{L_{\\chi }}= &\\left\\Vert \\left(1- \\left(1+ \\left(\\frac{1}{n}\\sum _{k=1}^{n} P\\tilde{\\eta }_{\\chi } \\circ F_{-k}\\right) \\cdot \\frac{i}{\\xi _{0}}\\right)\\right)f \\right\\Vert _{L_{\\chi }}\\\\= &\\left\\Vert \\left(\\frac{1}{n}\\sum _{k=1}^{n} P\\tilde{\\eta }_{\\chi } \\circ F_{-k} \\cdot \\frac{i}{\\xi _{0}} \\right)f \\right\\Vert _{L_{\\chi }}\\\\\\le & \\left\\Vert \\left(\\frac{1}{n}\\sum _{k=1}^{n} P\\tilde{\\eta }_{\\chi } \\circ F_{-k} \\right) \\cdot \\frac{i}{\\xi _{0}} \\right\\Vert _{\\infty } \\cdot \\Vert f \\Vert _{L_{\\chi }}.\\end{split}$ As a result of unique ergodicity, the following converges uniformly, $\\lim _{n \\rightarrow \\infty } \\left(\\frac{1}{n}\\sum _{k=1}^{n} P\\tilde{\\eta }_{\\chi } \\circ F_{-k} \\right) \\cdot \\frac{i}{\\xi _{0}}= \\frac{i}{\\xi _{0}}\\int _{X} P \\tilde{\\eta }_{\\chi } \\: d\\mu =0,$ and thus, $\\begin{split}\\limsup _{n \\rightarrow \\infty } \\left\\Vert I - \\frac{H(n)}{n}H^{-1} \\right\\Vert _{op}\\le & \\limsup _{n \\rightarrow \\infty } \\left\\Vert \\left(\\frac{1}{n}\\sum _{k=1}^{n} P\\tilde{\\eta }_{\\chi } \\circ F_{-k} \\right) \\cdot \\frac{i}{\\xi _{0}} \\right\\Vert _{\\infty }\\\\= & \\left\\Vert \\frac{i}{\\xi _{0}} \\int _{X} P \\tilde{\\eta }_{\\chi } \\: d\\mu \\right\\Vert _{\\infty }\\\\= & \\: 0.\\end{split}$ Hence, $\\limsup _{n \\rightarrow \\infty } \\left\\Vert I + \\frac{H(n)}{n}H^{-1} \\right\\Vert _{op}<1.$ $(ii)$ $\\begin{split}\\left[P,\\frac{H(n)}{n}H^{-1}\\right]= & \\left[P, I + (\\frac{1}{n}\\sum _{k=1}^{n} P\\tilde{\\eta }_{\\chi } \\circ F_{-k}) \\cdot \\frac{i}{\\xi _{0}}I \\right]\\\\= & \\left[P, \\left(\\frac{1}{n}\\sum _{k=1}^{n} P\\tilde{\\eta }_{\\chi } \\circ F_{-k} \\right) \\cdot \\frac{i}{\\xi _{0}}I \\right]\\\\= & \\left(\\frac{1}{n}\\sum _{k=1}^{n} P(P\\tilde{\\eta }_{\\chi }) \\circ F_{-k} \\right) \\cdot \\frac{i}{\\xi _{0}}I.\\end{split}$ Since $\\sup _{t > 0} \\left\\Vert \\frac{{L}_{Y}(\\chi \\circ \\eta ) \\circ F_{t} - {L}_{Y}(\\chi \\circ \\eta )}{t}) \\right\\Vert _{\\infty } < \\infty $ , $P(P\\tilde{\\eta }_{\\chi })$ is bounded in $L_{\\chi }$ and $\\left\\Vert \\left[P,\\frac{H(n)}{n}H^{-1} \\right] f \\right\\Vert _{L_{\\chi }} \\le \\frac{\\Vert P(P\\tilde{\\eta }_{\\chi }) \\Vert _{L_{\\chi }}}{|\\xi _{0}|} \\cdot \\Vert f \\Vert _{L_{\\chi }} \\le \\frac{C_{2}}{|\\xi _{0}|} \\Vert f \\Vert _{L_{\\chi }}.$ Thus, $\\left[P,\\frac{H(n)}{n}H^{-1} \\right]$ extends to a bounded operator on $L_{\\chi }$ with uniformly bounded norm in $n$ , $\\left\\Vert \\left[P,\\frac{H(n)}{n}H^{-1} \\right] \\right\\Vert _{op} \\le \\frac{C_{2}}{|\\xi _{0}|}.$ $(iii)$ Since the operator $H$ is just multiplication by the constant $-i\\xi _{0}$ , $\\left[H(n),H\\right]H^{-1}=0$ .",
"Thus, condition $(iii)$ is immediately satisfied.",
"Since conditions $(i)$ , $(ii)$ , and $(iii)$ of Theorem REF are satisfied on each $L_{\\chi }$ , we have shown that the operator $U_{\\chi }$ has purely absolutely continuous spectrum on $L_{\\chi }$ .",
"Thus, $W$ has purely absolutely continuous spectrum when restricted to the subspace $\\bigoplus _{\\chi \\in \\hat{G},\\chi \\circ \\xi \\lnot \\equiv 1 }L_{\\chi }$ .",
"In addition, from the purity law in [8] extended to translations, the maximal spectral type is either purely Lebesgue, purely singularly continuous, or purely discrete with respect to $\\mu $ (the Haar measure).",
"Since we know that the spectrum is absolutely continuous from above, we have proved the following theorem.",
"Theorem 5 The operator $U_{\\chi }$ has Lebesgue spectrum on $L_{\\chi }$ .",
"Thus, $W$ has countable Lebesgue spectrum when restricted to the subspace $\\bigoplus _{\\chi \\in \\hat{G},\\chi \\circ \\xi \\lnot \\equiv 1 }L_{\\chi }$ .",
"Similar results with less restrictive assumptions on $\\eta $ have been derived in [11] and [19]."
],
[
"Furstenberg transformations",
"Let $\\mu _{n}$ be the normalized Haar measure on $\\mathbb {T}^{n} \\simeq \\mathbb {R}^{n}/\\mathbb {Z}^{n}$ and ${H}_{n}=L^{2}(\\mathbb {T}^{n}, \\mu _{n})$ .",
"Let $T_{d}:\\mathbb {T}^{d} \\rightarrow \\mathbb {T}^{d}$ , $d \\ge 2$ , be the uniquely ergodic map [6] $T_{d}(x_{1}, x_{2}, ..., x_{d})$ $=(x_{1}+y, x_{2}+b_{2,1}x_{1}+h_{1}(x_{1}), ..., x_{d}+b_{d,1}x_{1}+ \\cdot \\cdot \\cdot + b_{d,d-1}x_{d-1}+h_{d-1}(x_{1},x_{2},...,x_{d-1}))$ $\\mathbb {\\unknown.",
"}\\mod {\\mathbb {}{Z}^{d}for y \\in \\mathbb {R}\\setminus \\mathbb {Q}, b_{j,k} \\in \\mathbb {Z}, b_{l,l-1} \\ne 0, and l \\in \\lbrace 2,...,d\\rbrace .",
"(For n=2, we get the skew product in 4.1).",
"Let each h_{j}:\\mathbb {T}^{j} \\rightarrow \\mathbb {R} satisfy a uniform Lipschitz condition in x_{j} and be in C^{2}(\\mathbb {T}^{j}).",
"What follows is very similar to the case of the skew products over translations.",
"We begin by considering the operatorW_{d}: {H}_{d} \\rightarrow {H}_{d}.The space {H}_{d} can be decomposed into\\begin{center} {H}_{d} = {H}_{1} \\bigoplus _{j \\in \\lbrace 2,...d \\rbrace ,\\: k \\in \\mathbb {Z} \\setminus \\lbrace 0 \\rbrace } {H}_{j,k} \\end{center}for {H}_{j,k}=\\overline{Span\\left\\lbrace \\eta \\bigotimes \\chi _{k}|\\eta \\in {H}_{j-1}\\right\\rbrace } and \\chi _{k}(x_{j}) = e^{2 \\pi i k x_{j}} \\in \\hat{\\mathbb {T}}.",
"}The restriction of $ d$, $ Wd|Hj,k$ is unitarily equivalent to the operator$$U_{j,k} \\eta = e^{2\\pi i k \\phi _{j}}W_{j-1}\\eta $$for$$\\eta \\in {H}_{j-1}$$and$$\\phi _{j}(x_{1},x_{2},...,x_{j-1})=b_{j,1}x_{1}+ \\cdot \\cdot \\cdot + b_{j,j-1}x_{j-1} + h_{j-1}(x_{1},x_{2},...,x_{j-1}).$$We will choose to take the commutator with $ Pj-1 = -i j-1$, the essentially self-adjoint \\cite {amreinnelson} generator of the translation group $ {Vt,j-1}t R$ in $ Hj-1$.",
"The following formulas hold on $ D = C(Tj-1)$.$$\\begin{split}\\left[P_{j-1},U_{j,k} \\right]= &\\left[P_{j-1}, e^{2 \\pi i k \\phi _{j}}I \\right]W_{j-1}\\\\= & -i\\partial _{j-1}(e^{2 \\pi i k \\phi _{j}}) W_{j-1}\\\\= & (2 \\pi k b_{j,j-1} + 2 \\pi k \\partial _{j-1} h_{j-1})e^{2 \\pi i k \\phi _{j}}W_{j-1}.\\end{split}$$So,$$\\left[P_{j-1}, U_{j,k}\\right] =(2 \\pi kb_{j,j-1}+ 2 \\pi k \\partial _{j-1}h_{j-1})U_{j,k}= G U_{j,k}.$$Thus,$$ U_{j,k}^{-n}\\left[P_{j,k}, U_{j,k}^{n} \\right] = \\boxed{\\sum ^{n}_{l=1} U^{-l}_{j,k}GU^{l}_{j,k} =\\sum ^{n}_{l=1} G \\circ T_{j-1}^{-l}=H(n).",
"}$$$ From unique ergodicity we get the following convergence $\\lim _{n \\rightarrow \\infty } \\frac{H(n)}{n}u =2 \\pi k b_{j,j-1} + 2\\pi k \\int _{\\mathbb {T}^{j-1}} \\partial _{j-1}h_{j-1} \\: d\\mu =\\boxed{2 \\pi k b_{j,j-1}u=Hu}$ uniformly in $n$ for $u \\in D={H}_{j-1}$ .",
"Since $\\left\\Vert \\; \\langle U_{j,k}^{n}f,f \\rangle _{{H}_{j-1}}\\; \\right\\Vert _{\\ell ^{2}(\\mathbb {Z})} \\; = \\; \\left\\Vert \\frac{1}{\\sigma }\\int _{0}^{\\sigma } \\langle e^{sP_{j-1}}U_{j,k}^{n}f, e^{sP_{j-1}}f \\rangle _{{H}_{j-1}} ds\\: \\right\\Vert _{\\ell ^{2}(\\mathbb {Z})},$ the preliminary assumptions for Theorem REF are satisfied for $B_{1}=B_{2}=I$ .",
"Now we proceed by showing that the conditions of Theorem REF hold.",
"$(i)$ It is unnecessary to consider coboundaries since both $H=2 \\pi k b_{j,j-1}I$ and $H^{-1}=\\frac{1}{2 \\pi k b_{j,j-1}}I$ are constants.",
"Instead we take any $f \\in {H}_{j-1}$ .",
"$\\begin{split}\\frac{H(n)}{n}H^{-1}f = &\\left(\\frac{1}{n}\\left(\\sum ^{n}_{l=1} \\left(2 \\pi kb_{j,j-1}+ 2 \\pi k \\partial _{j-1}h_{j-1} \\right) \\circ T_{j-1}^{-l} \\right) \\right) \\cdot \\frac{1}{2 \\pi kb_{j,j-1}}f\\\\= & f + \\left(\\frac{1}{n}\\sum ^{n}_{l=1} \\left(2 \\pi k \\partial _{j-1}h_{j-1} \\right) \\circ T_{j-1}^{-l} \\right)\\frac{1}{2 \\pi kb_{j,j-1}}f.\\end{split}$ Hence, $\\left\\Vert \\frac{H(n)}{n}H^{-1}f \\right\\Vert _{{H}_{j-1}} \\le \\left(1+\\frac{\\Vert 2 \\pi k \\partial _{j-1}h_{j-1} \\Vert _{{H}_{j-1}}}{|2 \\pi kb_{j,j-1}|} \\right) \\Vert f \\Vert _{{H}_{j-1}}.$ Since $h_{j-1}$ satisfies a uniform Lipschitz condition in $x_{j-1}$ , $\\left\\Vert \\partial _{j-1}h_{j-1} \\right\\Vert _{{H}_{j-1}}\\le C_{1}.$ So $\\frac{H(n)}{n}H^{-1}$ extends to a bounded operator on ${H}_{j,k}$ with uniformly bounded norm in $n$ , $\\left\\Vert \\frac{H(n)}{n}H^{-1} \\right\\Vert _{op} \\le 1+\\frac{| 2 \\pi k| \\left\\Vert \\partial _{j-1}h_{j-1} \\right\\Vert _{{H}_{j-1}}}{|2 \\pi kb_{j,j-1}|} \\le \\frac{C_{1}}{|b_{j,j-1}|}.$ Also, $\\begin{split}& \\left\\Vert \\left(I - \\frac{H(n)}{n}H^{-1} \\right) f \\right\\Vert _{{H}_{j-1}}\\\\= & \\left\\Vert \\left(1- \\left(1+\\left(\\frac{1}{n}\\sum ^{n}_{l=1} (2 \\pi k \\partial _{j-1}h_{j-1}\\right) \\circ T_{j-1}^{-l}\\right)\\frac{1}{2 \\pi kb_{j,j-1}}\\right)f \\right\\Vert _{{H}_{j-1}}\\\\= & \\left\\Vert \\left(\\frac{1}{n}\\sum ^{n}_{l=1} \\left(2 \\pi k \\partial _{j-1}h_{j-1} \\right) \\circ T_{j-1}^{-l}\\right)\\frac{1}{2 \\pi kb_{j,j-1}}f \\right\\Vert _{{H}_{j-1}}\\\\\\le & \\left\\Vert \\left(\\frac{1}{n}\\sum ^{n}_{l=1} \\left(2 \\pi k \\partial _{j-1}h_{j-1} \\right) \\circ T_{j-1}^{-l} \\right)\\frac{1}{2 \\pi kb_{j,j-1}} \\right\\Vert _{\\infty } \\cdot \\Vert f \\Vert _{{H}_{j-1}}.\\end{split}$ As a result of unique ergodicity, the following converges uniformly, $\\lim _{n \\rightarrow \\infty } \\left(\\frac{1}{n}\\sum ^{n}_{l=1} (2 \\pi k \\partial _{j-1}h_{j-1}) \\circ T_{j-1}^{-l} \\right)\\frac{1}{2 \\pi kb_{j,j-1}} =\\frac{1}{b_{j,j-1}} \\int _{\\mathbb {T}^{j-1}} \\partial _{j-1}h_{j-1} \\: d\\mu =0,$ and thus, $\\begin{split}\\limsup _{n \\rightarrow \\infty }\\left\\Vert I - \\frac{H(n)}{n}H^{-1} \\right\\Vert _{op}\\le & \\limsup _{n \\rightarrow \\infty } \\left\\Vert \\left(\\frac{1}{n}\\sum ^{n}_{l=1} (2 \\pi k \\partial _{j-1}h_{j-1}) \\circ T_{j-1}^{-l} \\right)\\frac{1}{2 \\pi kb_{j,j-1}} \\right\\Vert _{\\infty }\\\\= &\\left\\Vert \\frac{1}{b_{j,j-1}} \\int _{\\mathbb {T}^{j-1}} \\partial _{j-1}h_{j-1} \\: d\\mu \\: \\right\\Vert _{\\infty }=0\\end{split}$ Hence, $\\limsup _{n \\rightarrow \\infty } \\left\\Vert I + \\frac{H(n)}{n}H^{-1} \\right\\Vert _{op}<1.$ $(ii)$ $\\begin{split}\\left[P_{j-1},\\frac{H(n)}{n}H^{-1} \\right]= & \\left[P_{j-1}, I + \\left(\\frac{1}{n}\\sum _{k=1}^{n} (2 \\pi k \\partial _{j-1}h_{j-1}) \\circ T_{j-1}^{-l} \\right) \\cdot \\frac{1}{2 \\pi kb_{j,j-1}}I \\right]\\\\= & \\left[P_{j-1}, \\left(\\frac{1}{n}\\sum _{k=1}^{n} (2 \\pi k \\partial _{j-1}h_{j-1}) \\circ T_{j-1}^{-l} \\right) \\cdot \\frac{1}{2 \\pi kb_{j,j-1}}I \\right]\\\\= & \\left(\\frac{1}{n}\\sum _{k=1}^{n} (2 \\pi k \\partial _{j-1}(\\partial _{j-1}h_{j-1})) \\circ T_{j-1}^{-l} \\right) \\cdot \\frac{1}{2 \\pi kb_{j,j-1}}I.\\end{split}$ Since $h_{j-1} \\in C^{2}(\\mathbb {T}^{j-1})$ , $\\partial _{j-1}(\\partial _{j-1} h_{j-1})$ is bounded in ${H}_{j-1}$ , $\\begin{split}\\Vert [P,\\frac{H(n)}{n}H^{-1}] f \\Vert _{{H}_{j-1}}\\le & \\frac{|2\\pi k|\\Vert \\partial _{j-1}(\\partial _{j-1} h_{j-1})\\Vert _{{H}_{j-1}}}{|2 \\pi kb_{j,j-1}|} \\cdot \\Vert f \\Vert _{{H}_{j-1}}\\\\\\le & \\frac{C_{2}}{|b_{j,j-1}|} \\Vert f \\Vert _{{H}_{j-1}}.\\end{split}$ Thus, $[P,\\frac{H(n)}{n}H^{-1}]$ extends to a bounded operator on ${H}_{j-1}$ with uniformly bounded norm in $n$ , $\\left\\Vert [P,\\frac{H(n)}{n}H^{-1}] \\right\\Vert _{op} \\le \\frac{C_{2}}{|b_{j,j-1}|}.$ $(iii)$ Since the operator $H$ is just multiplication by $2 \\pi k b_{j,j-1}$ , $\\left[H(n),H \\right]H^{-1}=0.$ Thus, condition $(iii)$ is immediately satisfied.",
"Since conditions $(i)$ , $(ii)$ , and $(iii)$ of Theorem REF are satisfied, the operator $U_{j,k}$ has purely absolutely continuous spectrum on each ${H}_{j,k}$ .",
"Thus, $W_{d}$ has purely absolutely continuous spectrum on the orthocomplement of ${H}_{1}$ .",
"Applying the the purity law in [8], we derive, with slightly stronger regularity assumptions, a similar result to the ones found in [11] and [19].",
"Theorem 6 $W_{d}$ has countable Lebesgue spectrum on the orthocomplement of ${H}_{1}$ ."
],
[
"Acknowledgments",
"The author is extremely grateful to her advisor, Professor Giovanni Forni, for his patience, guidance and support.",
"The author would also like to thank the anonymous referee for various useful comments and suggestions.",
"Much of this work was completed at the University of Maryland, College Park, and was partially supported by Giovanni Forni's NSF Grant DMS 1201534."
]
] | 1606.04962 |
[
[
"Critical behavior of quantum magnets with long-range interactions in the\n thermodynamic limit"
],
[
"Abstract Quasiparticle properties of quantum magnets with long-range interactions are investigated by high-order linked-cluster expansions in the thermodynamic limit.",
"It is established that perturbative continuous unitary transformations on white graphs are a promising and flexible approach to treat long-range interactions in quantum many-body systems.",
"We exemplify this scheme for the one-dimensional transverse-field Ising chain with long-range interactions.",
"For this model the elementary Quasiparticle gap is determined allowing to access the quantum-critical regime including critical exponents and multiplicative logarithmic corrections for the ferro- and antiferromagnetic case."
],
[
"White-graph expansion of the long-range TFIM",
"We investigated the critical behavior of the one-dimensional TFIM with algebraically decaying long-range interactions $\\operatorname{\\mathcal {H}}=-\\frac{1}{2}\\sum _{j}\\sigma _j^z -\\lambda \\sum _{i\\ne j} \\frac{1}{|i-j|^\\alpha }\\sigma _i^x\\sigma _{j}^x\\quad .",
"$ using perturbative continuous unitary transformations about the high-field limit.",
"To this end we perform a Matsubara-Matsuda transformation [28] and replace the Pauli matrices $\\sigma _i^\\kappa $ , $\\kappa \\in \\lbrace x,z\\rbrace $ with hardcore-boson annihilation (creation) operators $b_i^{(\\dagger )}$ $\\sigma _i^x = b_i^\\dagger +b_i, && \\sigma _i^z=1-2\\hat{n}_i,\\quad \\text{with}~\\hat{n}_i=b_i^\\dagger b_i^{\\phantom{\\dagger }}\\quad .$ The ground state of polarized spins in the limit $\\lambda \\rightarrow 0$ becomes the vacuum state in the bosonic Quasiparticle picture while spin-flip excitations correspond to hardcore bosons located on the lattice sites.",
"In this formulation we end up with Eq.",
"(5) in the main body of the manuscript $\\operatorname{\\mathcal {H}}=\\sum _{j} \\hat{n}_j -\\lambda \\sum _{i\\ne j} g_\\alpha (i-j)\\left( b^\\dagger _i b^\\dagger _j + b^\\dagger _i b^{\\phantom{\\dagger }}_j + {\\rm H.c.}\\right) \\, , $ which is of the form (2) with $N_{\\rm max}=2$ and $g_\\alpha (i-j)\\equiv |i-j|^{-\\alpha }$.",
"In pCUTs, Hamiltonian (2) is mapped up to high orders in perturbation to an effective Hamiltonian $\\mathcal {H}_\\text{eff}$ with $[\\mathcal {H}_{\\rm eff},\\mathcal {Q}]=0$ .",
"The block-diagonal $\\mathcal {H}_\\text{eff}$ conserves therefore the number of Quasiparticles which correspond to dressed spin-flip excitations in our case.",
"Here we focus on the one-qp sector where the effective Hamiltonian is given as a hopping Hamiltonian of the form $\\operatorname{\\mathcal {H}}^{\\rm 1qp} =\\sum _{i}\\sum _\\delta a_\\delta \\left( b^\\dagger _i b^{\\phantom{\\dagger }}_{i+\\delta } + {\\rm H.c.}\\right) \\, , $ with $a_\\delta $ denoting the hopping amplitude of distance $\\delta $ between two sites on the chain.",
"In pCUTs, these hopping amplitudes are derived up to high orders in perturbation.",
"Using the Fourier transformation $b_{j}^\\dagger = \\frac{1}{\\sqrt{N_{\\mathrm {s}}}}\\sum \\limits _q \\mathrm {e}^{\\mathrm {i}q j}b_{q}^\\dagger ,\\quad b_{j}^{\\phantom{\\dagger }} &= \\frac{1}{\\sqrt{N_{\\mathrm {s}}}}\\sum \\limits _q \\mathrm {e}^{-\\mathrm {i}q j}b_{q}^{\\phantom{\\dagger }}$ with the number of lattice sites $N_{\\mathrm {s}}$ , the one-qp Hamiltonian (REF ) is readily diagonalized $\\operatorname{\\mathcal {H}}^{\\rm 1qp} = \\sum \\limits _{q} \\omega _q \\,b_{q}^\\dagger b_{q}^{\\phantom{\\dagger }}\\quad .$ Here $\\omega _q=a_0+2\\sum _{\\delta >0} a_\\delta \\cos (q\\,\\delta )$ is the one-qp dispersion.",
"The minimum of the dispersion corresponds to the one-qp gap $\\Delta \\equiv {\\rm min}_q\\,\\omega _q$ .",
"For the long-range TFIM the one-qp gap $\\Delta $ is located at momentum $q_{\\Delta }=0$ for a ferromagnetic and $q_\\Delta =\\pi $ for an antiferromagnetic Ising interaction, respectively.",
"We have calculated this Quasiparticle gap $\\Delta $ as a series in the perturbation parameter $\\lambda $ $\\Delta (\\lambda ) = 1 + p_1 \\lambda + p_2 \\lambda ^2 +\\dots ~+ p_{k} \\lambda ^{k} $ up to order $k=8$ .",
"All prefactors $p_r$ depend on $q_{\\Delta }$ and can be analytically expressed as $p_r=\\sum _{\\gamma } t_{r,\\gamma }$ where the sum runs over all graphs $\\gamma $ contributing to the given order $r$ (c.f.",
"Fig.",
"(1) in the main body of the text for an overview of all graphs up to order 3).",
"In order 8 there are 358 graphs in total.",
"The parameter $t_{r,\\gamma }$ is the unique contribution of graph $\\gamma $ to the coefficient $p_r$ in which the aforementioned infinite sums appear due to the embedding process.",
"In practice, we introduce a different coupling $\\lambda _{j}$ for each link $l_j$ of a given graph $\\gamma $ .",
"The pCUT calculation in order $r$ then yields hopping amplitudes between sites $\\nu $ and $\\nu +\\delta $ of the form $\\sum _{\\lbrace r_j\\rbrace } A_{\\nu ,\\nu +\\delta ,\\gamma }\\left({\\lbrace r_j\\rbrace }\\right)\\,\\lambda _{1}^{r_1}\\cdots \\lambda ^{r_{\\rm max}}_{{\\rm max}}$ where $\\sum _{j}r_j=r$ holds for each summand and the coefficients $A_{\\nu ,\\nu +\\delta ,\\gamma }\\left({\\lbrace r_j\\rbrace }\\right)$ are exact fractions.",
"In the next step one has to embed the graph links $l_j$ into the infinite chain which implies $\\lambda _{j}^{r_j} \\longrightarrow -\\lambda ^{r_j} \\left(\\frac{1}{|\\delta _{l_j}|^{\\alpha }}\\right)^{r_j}$ and summing over all possible embeddings of graph $\\gamma $ .",
"Fourier transformation of all hopping processes yields the parameter $t_{r,\\gamma }$ which can be written for general momentum $q$ as $t_{r,\\gamma } = a_{0,\\gamma }^{(r)} + 2 \\sum _{\\begin{array}{c}\\delta \\in \\gamma \\\\\\delta >0\\end{array}} a_{\\delta ,\\gamma }^{(r)} \\cos (q\\,\\delta )\\quad ,$ where $a_{\\delta ,\\gamma }^{(r)} = \\xi _\\gamma &\\sum _{\\nu <N_\\gamma }\\sum _{\\lbrace r_j\\rbrace } A_{\\nu ,\\nu +\\delta ,\\gamma }\\left({\\lbrace r_j\\rbrace }\\right)\\,\\\\ & ~\\sum _{s_{N_\\gamma }}\\dots \\sum _{s_2}\\sum _{s_1}\\, f_{\\nu ,\\nu +\\delta ,\\gamma }^{{\\lbrace r_j\\rbrace }}(\\lbrace s_j\\rbrace )\\quad .$ Here $\\nu =0..(N_\\gamma -1)$ where $N_\\gamma $ is the total number of the graph's lattice sites, $\\xi _\\gamma $ is a factor compensating the overcounting in the summation due to the graph symmetry, and $A_{\\nu ,\\nu +\\delta ,\\gamma }\\left({\\lbrace r_j\\rbrace }\\right)$ is the pCUT graph-dependent hopping amplitude from graph-site $\\nu $ to $\\nu +\\delta $ .",
"The lattice-site indices on the infinite chain are denoted by $s_\\nu $ .",
"For the local hopping $a_{0,\\gamma }^{(r)}$ the graph's ground-state energy is subtracted from the one-qp energy.",
"The factor $f_{\\nu ,\\nu +\\delta ,\\gamma }^{{\\lbrace r_j\\rbrace }}(\\lbrace s_j\\rbrace )$ is a graph-dependent product of fractions arising from the long-range interactions $f_{\\nu ,\\nu +\\delta ,\\gamma }^{{\\lbrace r_j\\rbrace }}(\\lbrace s_j\\rbrace )=\\lambda ^r \\prod _{\\lbrace r_j\\rbrace }\\frac{1}{\\left|s_{\\nu _j}-s_{\\nu ^{\\prime }_j}\\right|^{r_j\\alpha }}$ where the sum over all $r_m$ equals the order $r$ and $s_{\\nu _j}-s_{\\nu ^{\\prime }_j}=\\delta _{l_j}\\ne 0$.",
"Figure: Embedding of a graph with three sites s ν s_\\nu into the one-dimensional lattice in the thermodynamic limit.",
"One after the other, each of these sites have to be set to any of the (still unoccupied) lattice sites to get the contribution of all the realizations of the graph in the actual lattice.As a simple example, which arises from the pCUT calculation in order $r=3$ , let us consider graph (ii) in Fig.",
"(1) in the main body of the manuscript denoted from now on by $\\gamma _{\\rm (ii)}$ .",
"This chain graph has three sites $s_0$ , $s_1$ , and $s_2$ and two links $l_1$ between the first two sites and $l_2$ between the last two sites.",
"Here we focus on a specific nearest-neighbor hopping between site $s_0$ to $s_1$ with a certain set of $\\lbrace r_j\\rbrace $ in order to illustrate the embedding procedure and we want to calculate in the following contribution of this process to the parameter $t_{3,\\gamma _{\\rm (ii)}}$ .",
"The corresponding contribution for that hopping on graph $\\gamma _{\\rm (ii)}$ is given as $-\\frac{1}{4}\\lambda _{1}\\lambda _{2}^2\\quad .$ The embedding process, illustrated in Fig.",
"REF , means a summation over all possible realizations of that graph on the actual lattice.",
"For a long-range interaction there are clearly infinitely many possibilities.",
"In our example we get after embedding the following contribution to the parameter $t_{3,\\gamma _{\\rm (ii)}}$ $\\frac{1}{4}\\lambda ^3 \\sum _{ \\begin{array}{c} \\delta _{l_2} =-\\infty \\\\ \\delta _{l_2}\\ne -\\delta _{l_1} \\\\ \\delta _{l_2} \\ne 0 \\end{array}}^\\infty \\sum _{ \\begin{array}{c} \\delta _{l_1}=-\\infty \\\\ \\delta _{l_1} \\ne 0 \\end{array}}^\\infty \\frac{1}{|\\delta _{l_1}|^\\alpha }\\frac{1}{|\\delta _{l_2}|^{2\\alpha }}\\cos (q\\delta _{l_1})$ where the factor $\\xi _{\\gamma _{\\rm (ii)}}=1/2$ comes from the graph's symmetry and accounts for a double counting of each realization of the graph on the lattice.",
"This factor is canceled with the factor 2 in Eq.",
"(REF ).",
"The conditions $\\delta _{l_2}\\ne -\\delta _{l_1}$ and $\\delta _{l_j}\\ne 0$ in the sums ensure that the possibility of two graph sites being located on the same lattice site is excluded.",
"For a quantitative evaluation of this expression the infinite sums still need to be calculated.",
"This task proves to be difficult for a general value of $q$ .",
"Here we are only interested in the two specific momenta $q=0$ and $q=\\pi $ .",
"In both cases expression (REF ) can be evaluated analytically to a product of two Riemann zeta functions.",
"For the ferromagnetic case $q=0$ one obtains $\\left(2\\lambda ^3\\zeta (\\alpha )\\zeta (2\\alpha )-1\\lambda ^3\\zeta (3\\alpha )\\right)$ and for the antiferromagnetic case one finds $\\lambda ^3 \\left( 2 \\left(2^{1-2 \\alpha }-1\\right) \\zeta (\\alpha ) \\zeta (2 \\alpha ) +2^{-3 \\alpha } \\left(8^\\alpha -2\\right) \\zeta (3 \\alpha ) \\right) \\, .$"
],
[
"Extrapolation of data sequences",
"The nested infinite sums appearing at perturbative orders $r>2$ cannot be evaluated analytically.",
"Therefore we have calculated the various contributions by cutting the sums at finite limits $\\mathcal {N}$ .",
"In this situation one has to find proper schemes to extrapolate the data sequences for different $\\mathcal {N}$ to $\\mathcal {N}\\rightarrow \\infty $ .",
"In practice we have applied the Wynn algorithm and performed proper scalings in $1/\\mathcal {N}$ to the coefficients $p_r$ of the one-qp gap.",
"We haven chosen to extrapolate the $p_r$ to minimize the number of extrapolations which have to be done in order to obtain $\\Delta $ .",
"We found that the behavior of the ferromagnetic data sequences is fundamentally different from the antiferromagnetic ones.",
"The ferromagnetic sequences converge monotonically for large enough $\\mathcal {N}$ while in the antiferromagnetic case one observes an alternating behavior about the exact value at $\\mathcal {N}\\rightarrow \\infty $ .",
"As a consequence, the antiferromagnetic coefficients $p_r$ converge faster with $\\mathcal {N}$ than the ferromagnetic parameters and the scaling behavior of both cases is different."
],
[
"Wynn algorithm",
"The sums are evaluated for fixed values of $\\alpha $ as partial sums up to the upper boundary $\\mathcal {N}$ .",
"In the antiferromagnetic case the partial sums are alternating.",
"Therefore we consider only every second data point to get a monotonically converging series of data points (see also next section).",
"These data points are extrapolated using Wynn's epsilon method [29].",
"Several extrapolations using a subset of the full series of points from $S_1$ up to $S_{\\mathcal {N}}$ are made for each $p_r$ .",
"These are shown as red crosses in the figures.",
"Afterwards the Wynn results are averaged using the best converged data points which is marked by a vertical black line in the figures (see e.g., Fig.",
"REF ).",
"Wynn's epsilon method is an acceleration method for series which are converging slowly, as is the case especially for small values of $\\alpha $ .",
"Setting the start values of the algorithm to $\\epsilon _0(S_n)=S_n$ and $\\epsilon _{-1}(S_n)=0$ the iteration reads $\\epsilon _{k+1}(S_n)=\\epsilon _{k-1}(S_{n+1})+\\frac{1}{\\epsilon _k(S_{n+1})-\\epsilon _k(S_n)}\\quad .$"
],
[
"Scaling",
"As discussed above, each coefficient $p_r$ of the gap is a sum of various nested infinite sums.",
"Truncating the infinite sums at a finite limit $\\mathcal {N}$ , one might wonder how the coefficients $p_r$ scale to the infinite-sum limit for different $\\alpha $ .",
"Here we argue that each term of infinite sums scales similarly to the scaling of a product of Riemann zeta functions, which can be derived analytically and is therefore used as the proper scaling of the numerical data sequences."
],
[
"ferromagnetic case",
"If one sets $q=0$ in the coefficients $p_r$ relevant for ferromagnetic Ising interactions, then all infinite sums become monotonic (see for example Eq.",
"(REF )).",
"We therefore start by considering a single harmonic sum of the form $\\sum _{\\delta =1}^{\\mathcal {N}} \\frac{1}{\\delta ^\\alpha }$ which converges to the Riemann zeta function $\\zeta (\\alpha )$ for $\\mathcal {N}\\rightarrow \\infty $ .",
"We are interested in the leading asymptotics for large $\\mathcal {N}$ of the full sum, i.e.",
"we consider the difference $\\sum _{\\delta =\\mathcal {N}+1}^{\\infty } \\frac{1}{\\delta ^\\alpha }= \\zeta (\\alpha )-\\sum _{\\delta =1}^{\\mathcal {N}} \\frac{1}{\\delta ^\\alpha }\\quad .$ We therefore replace the sum by an integral and find for large $\\mathcal {N}$ and $\\alpha >1$ $\\int _{\\mathcal {N}+1}^\\infty {\\rm d}\\,\\delta \\,\\frac{1}{\\delta ^\\alpha } = \\frac{(\\mathcal {N}+1)^{-\\alpha +1}}{-\\alpha +1}\\propto \\frac{\\mathcal {N}^{-\\alpha +1}}{-\\alpha +1}\\quad .$ In the coefficients $p_r$ there are sums of terms with a different number of infinite sums.",
"If these sums are independent, then one can factorize them and obtains generically a product of harmonic sums of the form $\\left(\\sum _{\\delta _1=1}^{\\mathcal {N}} \\frac{1}{\\delta _1^\\alpha }\\right)\\left(\\sum _{\\delta _2=1}^{\\mathcal {N}} \\frac{1}{\\delta _2^\\alpha }\\right)\\cdots \\left(\\sum _{\\delta _m=1}^{\\mathcal {N}} \\frac{1}{\\delta _m^\\alpha }\\right)\\, .$ Each terms scales for large $\\mathcal {N}$ as $\\zeta (\\alpha )+\\frac{\\mathcal {N}^{-\\alpha +1}}{-\\alpha +1}$ so that the leading scaling of the product is $\\zeta (\\alpha )^m+m\\zeta (\\alpha ) \\frac{\\mathcal {N}^{-\\alpha +1}}{-\\alpha +1} + \\ldots \\quad .$ So all products scale with the same exponent $(1-\\alpha )$ independent of $m$ which we also confirmed numerically.",
"In the following we used this scaling for the coefficients $p_r$ of the gap.",
"Here we assume that the nested conditions in the sum, which usually spoil the possibility to factorize the sums, do not alter the scaling behavior.",
"First, one can rewrite a nested product of sums often as a sum of unnested sums.",
"Second, the term with the largest number of sums arises always from the longest chain graph contributing in a given order and the contribution of this chain graph contains always the factorized product of independent sums."
],
[
"antiferromagnetic case",
"If one sets $q=\\pi $ in the coefficients $p_r$ relevant for antiferromagnetic Ising interactions, then all infinite sums become alternating (see for example Eq.",
"(REF )).",
"We therefore start by considering a single sum of the form $\\sum _{\\delta =1}^{\\mathcal {N}}\\, (-1)^\\delta \\, \\frac{1}{\\delta ^\\alpha }$ and we denote the limiting value of the sum as $\\epsilon (\\alpha )$ for $\\mathcal {N}\\rightarrow \\infty $ .",
"We are again interested in the leading asymptotics for large $\\mathcal {N}$ of the full sum, i.e.",
"we consider the difference $\\sum _{\\delta =\\mathcal {N}+1}^{\\infty } (-1)^\\delta \\, \\frac{1}{\\delta ^\\alpha } = \\epsilon (\\alpha )-\\sum _{\\delta =1}^{\\mathcal {N}} (-1)^\\delta \\, \\frac{1}{\\delta ^\\alpha }\\quad .$ We then separate odd and even orders corresponding to negative and positive contributions and we assume $N$ to be even $\\sum _{\\delta =\\mathcal {N}+1}^{\\infty }\\, (-1)^\\delta \\, \\frac{1}{\\delta ^\\alpha }=\\sum _{\\delta =\\frac{\\mathcal {N}}{2}+1}^{\\infty } \\left( \\frac{1}{(2\\delta )^\\alpha } - \\frac{1}{(2\\delta -1)^\\alpha }\\right) \\quad .$ This sum is again monotonic as above for the ferromagnetic case.",
"The involved $\\delta $ are large, since $\\mathcal {N}$ is supposed to be large.",
"We therefore perform the Taylor expansion $1/(2\\delta -1)^\\alpha \\approx 1/(2\\delta )^\\alpha (1+\\alpha /2\\delta +\\ldots )$ for the second term so that the sum is taken over $\\alpha /(2\\delta )^{\\alpha +1}$ .",
"In the next step we replace the sum again by an integral and find the following scaling behavior $\\int _{\\frac{\\mathcal {N}}{2}+1}^\\infty {\\rm d}\\,\\delta \\frac{\\alpha }{(2\\delta )^{\\alpha +1}} = -\\alpha \\frac{\\left(\\frac{\\mathcal {N}}{2}+1\\right)^{-\\alpha }}{2^{\\alpha +1}} \\propto -\\frac{\\alpha }{2}\\mathcal {N}^{-\\alpha } \\quad .$ As for the ferromagnetic case, this can be generalized for products of independent sums to $\\epsilon (\\alpha )^m-m\\frac{\\alpha }{2}\\epsilon (\\alpha )\\,\\,\\mathcal {N}^{-\\alpha }+\\ldots \\quad ,$ where $\\epsilon (\\alpha )$ denotes the exact value for $\\mathcal {N}\\rightarrow \\infty $ .",
"So all products scale with the same exponent $-\\alpha $ independent of $m$ which we also confirmed numerically.",
"We used this scaling for the coefficients $p_r$ of the gap in the antiferromagnetic case."
],
[
"Wynn extrapolation and scaling analysis",
"This section contains an exemplary overview of the extrapolations and scalings of the prefactors $p_r$ (c.f.",
"(REF )) for both, a ferromagnetic and an antiferromagnetic Ising interaction.",
"Representative data for $\\alpha =3/2$ and $\\alpha =5/2$ are shown in Figs.",
"REF to REF for the highest orders 6, 7, and 8.",
"The contributions from all relevant graphs that are given as nested sums are evaluated up to an upper boundary $\\mathcal {N}$ which is only limited by computation time.",
"These partial sums $S_n$ are shown as green circles in the figures.",
"They are plotted against $n=\\frac{1}{\\mathcal {N}^{\\alpha -1}}$ ($n=\\frac{1}{\\mathcal {N}^\\alpha }$ ) for a ferromagnetic (antiferromagnetic) Ising interaction.",
"As derived in the previous section the series of points then should display a linear behavior for large $\\mathcal {N}$ .",
"The last two points (corresponding to the largest $\\mathcal {N}$ ) are used to define a linear curve which gives an estimation for the value of the prefactor for $\\mathcal {N}\\rightarrow \\infty $ .",
"The curve is shown as a solid green line.",
"For the calculation of the Wynn extrapolants a subset of partial sums $(S_1,\\dots ,S_{\\mathcal {N}})$ is used and shown as red crosses in the figures.",
"The antiferromagnetic series display an alternating behavior due to the location of the gap at $q=\\pi $ (see Eq.",
"(REF )).",
"Only every second value is used to obtain a monotonically converging series.",
"While they give the general tendency, they deviate from the scaled result considerably when looking at small values of $\\alpha $ in the ferromagnetic case.",
"However, we found that the differences between the two extrapolation/scaling schemes do influence the final results for the critical values and exponents only marginally.",
"For a better comparison of Wynn extrapolation and scaling value the Wynn results are averaged from a minimum $\\mathcal {N}$ when they seem to have converged.",
"This minimum $\\mathcal {N}$ is illustrated by a vertical solid black line in the figures REF to REF .",
"The standard deviation of these points is illustrated by a gray area.",
"It can be clearly seen that the prefactors for the antiferromagnetic interaction converge much faster than their ferromagnetic counterpart.",
"Also, as a result, they are in much better agreement with the Wynn extrapolations.",
"Figure: Wynn extrapolation & fit for the highest-order prefactors in the ferromagnetic case for α=1.5\\alpha =1.5.",
"The black vertical line marks the point after which Wynn extrapolation points are used for calculating the average (dashed black line).",
"The gray area around the mean refers to the standard deviation of those Wynn points.Figure: Wynn extrapolation & fit for the highest-order prefactors in the ferromagnetic case for α=2.5\\alpha =2.5.",
"The black vertical line marks the point after which Wynn extrapolation points are used for calculating the average (dashed black line).",
"The gray area around the mean refers to the standard deviation of those Wynn points.Figure: Wynn extrapolation & fit for the highest-order prefactors in the antiferromagnetic case for α=1.5\\alpha =1.5.",
"The black vertical line marks the point after which Wynn extrapolation points are used for calculating the average (dashed black line).",
"The gray area around the mean refers to the standard deviation of those Wynn points.Figure: Wynn extrapolation & fit for the highest-order prefactors in the antiferromagnetic case for α=2\\alpha =2.",
"The black vertical line marks the point after which Wynn extrapolation points are used for calculating the average (dashed black line).",
"The gray area around the mean refers to the standard deviation of those Wynn points."
],
[
"Extrapolation of high-order series",
"Once the energy gap is given as a power series (c.f.",
"Eq.",
"(REF )), we perform standard dLog-Padé extrapolations.",
"We refer to the literature for general review of this topic, as for example given in Ref. Guttmann1989.",
"Here we give specific information which is relevant for the particular extrapolation we performed in the main body of the manuscript, which is essentially the information given in Ref. Coester2016.",
"Our series are all of the form $F(\\lambda )=\\sum _{n\\ge 0}^k a_n \\lambda ^n=a_0+a_1\\lambda +a_2\\lambda ^2+\\dots a_k\\lambda ^k,$ with $\\lambda \\in \\mathbb {R}$ and $a_i \\in \\mathbb {R}$ .",
"If one has power-law behavior near a critical value $\\lambda _{\\rm c}$ , the true physical function $\\tilde{F}(\\lambda )$ close to $\\lambda _{\\rm c}$ is given by $\\tilde{F}(\\lambda )\\approx \\left(1-\\frac{\\lambda }{\\lambda _{\\rm c}}\\right)^{-\\theta } A(\\lambda ),$ where $\\theta $ is the associated critical exponent.",
"If $A(\\lambda )$ is analytic at $\\lambda =\\lambda _{\\rm c}$ , we can write $\\tilde{F}(\\lambda )\\approx \\left(1-\\frac{\\lambda }{\\lambda _{\\rm c}}\\right)^{-\\theta }A|_{\\lambda =\\lambda _{\\rm c}}\\left(1+\\mathcal {O}\\left(1-\\frac{\\lambda }{\\lambda _{\\rm c}}\\right)\\right).$ Near the critical value $\\lambda _{\\rm c}$ , the logarithmic derivative is then given by $\\tilde{D}(\\lambda )&:=\\frac{\\text{d}}{\\text{d}\\lambda }\\ln {\\tilde{F}(\\lambda )}\\\\&\\approx \\frac{\\theta }{\\lambda _{\\rm c}-\\lambda }\\left\\lbrace 1+ \\mathcal {O}(\\lambda -\\lambda _{\\rm c})\\right\\rbrace \\nonumber .$ In the case of power-law behavior, the logarithmic derivative $\\tilde{D}(\\lambda )$ is therefore expected to exhibit a single pole at $\\lambda \\equiv \\lambda _{\\rm c}$ .",
"The latter is the reason why so-called Dlog-Padé extrapolation is often used to extract critical points and critical exponents from high-order series expansions.",
"Dlog-Padé extrapolants of $F(\\lambda )$ are defined by $dP[L/M]_F(\\lambda )=\\exp \\left(\\int _{0}^\\lambda P[L/M]_{D}\\,\\,\\text{d}\\lambda ^{\\prime }\\right)$ and represent physically grounded extrapolants in the case of a second-order phase transition.",
"Here $P[L/M]_{D}$ denotes a standard Padé extrapolation of the logarithmic derivative $P[L/M]_{D}:=\\frac{P_L(\\lambda )}{Q_M(\\lambda )}=\\frac{p_0+p_1\\lambda +\\dots + p_L \\lambda ^L}{q_0+q_1\\lambda +\\dots q_M \\lambda ^M}\\quad ,$ with $p_i\\in \\mathbb {R}$ and $q_i \\in \\mathbb {R}$ and $q_0=1$ .",
"Additionally, $L$ and $M$ have to be chosen so that $L+M-1\\le k$ .",
"Physical poles of $P[L/M]_{D}(\\lambda )$ then indicate critical values $\\lambda _{\\rm c}$ while the corresponding critical exponent of the pole $\\lambda _{\\rm c}$ can be deduced by $\\theta \\equiv \\left.\\frac{P_L(\\lambda )}{\\frac{\\text{d}}{\\text{d}\\lambda } Q_M(\\lambda )}\\right|_{\\lambda =\\lambda _{\\rm c}} .$ If the exact value (or a quantitative estimate from other approaches) of $\\lambda _{\\rm c}$ is known, one can obtain better estimates of the critical exponent by defining $\\theta ^*(\\lambda )&\\equiv (\\lambda _{\\rm c}-\\lambda )D(\\lambda )\\\\&\\approx \\theta +\\mathcal {O}(\\lambda -\\lambda _{\\rm c}),$ where $D(\\lambda )$ is given by Eq.",
"(REF ).",
"Then $P[L/M]_{\\theta ^*}\\big |_{\\lambda =\\lambda _{\\rm c}}=\\theta $ yields a (biased) estimate of the critical exponent.",
"In the ferromagnetic case at the upper critical $\\alpha =5/3$ , the long-range TFIM displays multiplicative corrections close to the quantum critical point so that one expects the following critical behavior $\\bar{F}(\\lambda )\\approx \\left(1-\\frac{\\lambda }{\\lambda _{\\rm c}}\\right)^{-\\theta } \\left(\\ln \\left( 1-\\frac{\\lambda }{\\lambda _{\\rm c}}\\right)\\right)^{p} \\bar{A}(\\lambda ),$ where $\\lambda _{\\rm c}$ ($\\theta $ ) is the associated critical point (exponent) as before while $p$ yields the exponent of multiplicative logarithmic corrections.",
"Clearly, the extraction of $p$ from a high-order series expansion is very demanding.",
"The only reasonable approach is to bias the extrapolation by fixing $\\theta $ .",
"In our case the critical exponent $\\theta $ is given by the well-known mean-field value $1/2$ .",
"Assuming again that the function $\\bar{A}(\\lambda )$ is analytic close to $\\lambda _{\\rm c}$ , Eq.",
"(REF ) transforms into $\\bar{F}(\\lambda )&\\approx & \\left(1-\\frac{\\lambda }{\\lambda _{\\rm c}}\\right)^{-\\theta } \\left(\\ln \\left( 1-\\frac{\\lambda }{\\lambda _{\\rm c}}\\right)\\right)^{p}\\bar{A}|_{\\lambda =\\lambda _{\\rm c}}\\nonumber \\\\&& \\cdot \\left(1+\\mathcal {O}\\left(1-\\frac{\\lambda }{\\lambda _{\\rm c}}\\right)\\right).$ and the logarithmic derivative Eq.",
"(REF ) becomes $\\bar{D}(\\lambda )&\\approx \\frac{\\theta }{\\lambda _{\\rm c}-\\lambda } + \\frac{-p}{\\ln \\left(1-\\lambda /\\lambda _{\\rm c}\\right)\\left(\\lambda _{\\rm c}-\\lambda \\right)} + \\mathcal {O}\\left(\\lambda -\\lambda _{\\rm c}\\right)\\nonumber .$ One can then estimate the multiplicative logarithmic correction $p$ by defining $p^{*}(\\lambda )&\\equiv -\\ln \\left( 1-\\lambda /\\lambda _{\\rm c}\\right) \\left[ \\left( \\lambda _{\\rm c}-\\lambda \\right) D(\\lambda )-\\theta \\right]\\\\&\\approx p +\\mathcal {O}(\\lambda -\\lambda _{\\rm c}),$ and by performing Padé extrapolants of this function $P[L/M]_{p*}\\big |_{\\lambda =\\lambda _{\\rm c}}=p \\quad .$"
]
] | 1606.05111 |
[
[
"Machine Learning Across Cultures: Modeling the Adoption of Financial\n Services for the Poor"
],
[
"Abstract Recently, mobile operators in many developing economies have launched \"Mobile Money\" platforms that deliver basic financial services over the mobile phone network.",
"While many believe that these services can improve the lives of the poor, a consistent difficulty has been identifying individuals most likely to benefit from access to the new technology.",
"Here, we combine terabyte-scale data from three different mobile phone operators from Ghana, Pakistan, and Zambia, to better understand the behavioral determinants of mobile money adoption.",
"Our supervised learning models provide insight into the best predictors of adoption in three very distinct cultures.",
"We find that models fit on one population fail to generalize to another, and in general are highly context-dependent.",
"These findings highlight the need for a nuanced approach to understanding the role and potential of financial services for the poor."
],
[
"Introduction",
"Billions of people around the world live without access to formal financial institutions.",
"Over the last several years, mobile phone operators across the globe have launched “Mobile Money\" platforms, which make it possible for many of the world's poor to conduct basic financial transactions from inexpensive feature phones.",
"However, outside of Kenya and a few other countries, worldwide adoption of Mobile Money has been extremely anemic.",
"The vast majority of deployments have struggled to promote sustained product adoption, and an industry report from 2014 estimates that 66% of registered customers were inactive .",
"An open and important question thus revolves around understanding what drives customers to adopt and use Mobile Money, and whether patterns observed in one country will generalize to another.",
"Our work builds on several distinct strands in the academic literature.",
"The first strand that is concerned with understanding the determinants of mobile money adoption has historically been the domain of development researchers, and includes both macroeconomic studies , , as well as qualitative, ethnographic work , , .",
"A second strand of literature seeks to derive general insights from mobile phone transactions logs.",
"This encompasses a wide array of applications, including predicting the socioeconomic status , gender , and age of individual mobile phone users.",
"Most relevant to this study, a third area of prior work mines digital transactions logs to model the social and behavioral determinants of product adoption , , with a few recent papers written in the context of mobile phone data , , .",
"Relative to these studies, our study moves this literature forward by (a) innovating in the method used to generate features, thereby providing a systematic and comprehensive approach to feature engineering; (b) leveraging data from three different contexts to calibrate the external validity and generalizability of our results; and (c) carefully articulating the experiments in a way that will enable other researchers to replicate and extend these methods."
],
[
"Data and Context",
"The data that we have used in this project consists of anonymized Call Detail Records (CDR) and Mobile Money Transaction Records (MMTR) of all the subscribers from three different operators in Ghana, Pakistan, and Zambia.",
"All three countries rank in the bottom third of the Human Development Index and Financial Inclusion Index as shown in Table REF .",
"The CDR and MMTR contain basic metadata on every event that occurs on the mobile phone network, including phone calls, text messages, and any form of Mobile Money activity.",
"In total, the original data contains billions of transactions conducted by tens of millions of unique individuals.",
"Each dataset spans several months of activity, which we divide into a “training” period and an “evaluation” period.",
"CDR from a 10-day training period was used to engineer features and fit a predictive model, where the target variables (based on Mobile Money activity) were measured in a subsequent 3-month evaluation period.",
"Based on activity during the evaluation period, each subscribers is categorized as a “Voice Only User” (no MM activity), a “Registered Mobile Money User” (one or more MMTR's), or an “Active Mobile Money User” (at least one MMTR in each month).",
"Table: Discussion and Conclusions"
]
] | 1606.05105 |
[
[
"On some determinant and matrix inequalities with a geometrical flavour"
],
[
"Abstract In this paper we study some determinant inequalities and matrix inequalities which have a geometrical flavour.",
"We first examine some inequalities which place work of Macbeath [13] in a more general setting and also relate to recent work of Gressman [8].",
"In particular, we establish optimisers for these determinant inequalities.",
"We then use these inequalities to establish our main theorem which gives a geometric inequality of matrix type which improves and extends some inequalities of Christ in [5]."
],
[
"Notation and Preliminaries",
"Let $\\mathbb {R}^{n}$ be the $n$ -dimensional Euclidean space, $n \\ge 1$ .",
"$|\\cdot |$ denotes the Lebesgue measure on $\\mathbb {R}^{n}$ and the absolute value on $\\mathbb {R}$ .",
"Denote $\\mathfrak {M}^{n \\times n}(\\mathbb {R})$ by a set of all $n \\times n$ real matrices.",
"Let $B (0, r)$ be the ball centred at 0 with radius $r$ .",
"For $A\\subset \\mathbb {R}^{n} $ of finite Lebesgue measure, we define the symmetric rearrangement of $A$ as $A^{\\ast }:=\\lbrace x: |x|<r \\rbrace \\equiv B (0, r)$ , with $|A^{\\ast }| =|A|$ .",
"That is, $v_{n} r^{n}=|A|$ , where $v_{n}$ is the volume of unit ball in $\\mathbb {R}^{n}$ .",
"We then define the symmetric decreasing rearrangement of a nonnegative measurable function $f$ as $f^{\\ast } (x):= \\int _{0}^{\\infty } \\chi _{\\lbrace f>t\\rbrace ^{\\ast }} (x) dt, $ where $ \\chi _{\\lbrace f>t\\rbrace }$ is the characteristic function of the level set $ \\lbrace x: f(x)>t \\rbrace $ , and define the Steiner symmetrisation of $f$ with respect to the $j$ -th coordinate as $\\mathcal {R}_{j}f(x_{1}, \\dots ,x_{n})=f^{\\ast j}(x_{1}, \\dots ,x_{n}) := \\int _{0}^{\\infty } \\chi _{\\lbrace f(x_{1}, \\dots ,x_{j-1}, \\cdot , x_{j+1}, \\dots , x_{n})>t\\rbrace ^{\\ast }}(x_{j}) dt.$ Let $u\\in \\mathbb {R}^{n}$ be a unit vector, $u^{\\perp }$ be its orthogonal complement.",
"Then for any $x\\in \\mathbb {R}^{n}$ , it can be uniquely written as $x=tu+y$ where $y\\in u^{\\perp }$ .",
"We define the Steiner symmetrisation of $A$ with respect to the direction $u$ as $\\mathcal {S}_{u}(A):= \\lbrace tu+y: A\\cap (\\mathbb {R}u+y)\\ne \\phi , |t| \\le \\frac{|A\\cap (\\mathbb {R}u+y)|}{2}\\rbrace .$ Obviously, $\\mathcal {R}_{j}\\chi _{A}$ is the Steiner symmetrisation of $A$ with respect to the direction $e_{j}$ , $1\\le j \\le n$ .",
"For simplicity, we denote $\\mathcal {S}_{e_{n}} \\mathcal {S}_{e_{n-1}} \\dots \\mathcal {S}_{e_{1}} (E)$ by $\\mathcal {S}E$ , where $\\lbrace e_{1}, \\dots , e_{n}\\rbrace $ is the standard orthonormal basis in $\\mathbb {R}^{n}$ .",
"One easily sees that for any measurable set $E \\subset \\mathbb {R}^{n}$ $\\sup \\limits _{x \\in E^{\\ast }} \\ |x| \\le \\displaystyle {\\sup _{x \\in E }} \\ |x|, \\qquad \\mathrm {(1.1)}$ and from this it is not hard to see that $\\sup \\limits _{x,y \\in E^{\\ast }} \\ |x-y| \\le \\displaystyle {\\sup _{x,y \\in E }} \\ |x-y|.",
"\\qquad \\mathrm {(1.2)}$ One way to obtain this is as follows.",
"$\\sup \\limits _{x,y \\in E} \\ |x-y|= \\sup \\limits _{z \\in E-E} \\ |z| \\ge \\sup \\limits _{z \\in (E-E)^{\\ast }} \\ |z|.",
"\\qquad \\mathrm {(1.3)}$ For any $A,B \\in \\mathbb {R}^{n}$ of finite Lebesgue measure, it follows from the Brunn-Minkowski inequality that $A^{\\ast }+B^{\\ast } \\subset (A+B)^{\\ast }.",
"\\qquad \\mathrm {(1.4)}$ Applying (1.4) in (1.3) implies $\\sup \\limits _{x,y \\in E} \\ |x-y|\\ge \\sup \\limits _{z\\in (E-E)^{\\ast }} \\ |z|\\ge \\sup \\limits _{x\\in E^{\\ast }, y\\in E^{\\ast }} \\ |x-y|,$ which completes (1.2).",
"Let $E$ be a measurable set of finite volume in $\\mathbb {R}^{n}$ .",
"By the definition of the symmetric rearrangement, $E^{\\ast }=B(0,r)$ , with $v_{n} r^{n} = |E|$ .",
"Clearly, $\\sup \\limits _{x\\in E^{\\ast }} \\ |x|=r, \\ \\sup \\limits _{x,y \\in E^{\\ast }} \\ |x-y| =2r.$ By (1.1) and (1.2) we have the following sharp inequality $|E| \\le v_{n} \\sup \\limits _{x\\in E} \\ |x|^{n}, \\qquad \\mathrm {(1.5)}$ $|E| \\le \\frac{v_{n}}{2^{n}} \\sup \\limits _{x,y \\in E} \\ |x-y|^{n}.",
"\\qquad \\mathrm {(1.6)}$ Moreover, optimisers of both (1.5) and (1.6) are balls in $\\mathbb {R}^{n}$ .",
"Inequality (1.6) is an isodiametric inequality, that is, amongst all sets with given diameter the ball has maximal volume."
],
[
"Macbeath's inequalities",
"We now go on to study the analogues of (1.5) and (1.6) where we replace the distance norm by a volume or determinant, so the question becomes that of studying inequalities of the form $|E| \\le A_{n} \\sup \\limits _{\\begin{array}{c}y_{j} \\in E \\\\ j=1, \\dots , n\\end{array}} \\det (0, y_{1}, \\dots , y_{n}), \\qquad \\mathrm {(1.7)}$ and $|E| \\le B_{n} \\sup \\limits _{\\begin{array}{c}y_{j} \\in E \\\\ j=1, \\dots , n+1\\end{array}} \\det (y_{1}, \\dots , y_{n+1}), \\qquad \\mathrm {(1.8)}$ which are supposed to hold for any measurable set $E$ in $\\mathbb {R}^{n}$ .",
"Here $\\det (y_{1}, \\dots , y_{n+1}):=n!",
"\\mathrm {vol} (\\mathrm {co}\\lbrace y_{1}, \\dots , y_{n+1}\\rbrace ).$ So $\\det (y_{1}, \\dots , y_{n+1}) \\ge 0$ .",
"The precise value of $\\det (y_{1}, \\dots , y_{n+1})$ is the absolute value of the determinant of the matrix $(y_{1}-y_{n+1}, \\dots , y_{n}-y_{n+1})_{n\\times n}$ .",
"In the special case when $n=1$ , they become of the type (1.5) and (1.6) automatically.",
"Note that both (1.7) and (1.8) are $\\mathrm {GL}_{n}(\\mathbb {R})$ invariant, and (1.8) is translation invariant while (1.7) is not.",
"Actually, it is enough to study convex measurable sets in $\\mathbb {R}^{n}$ , since $\\sup \\limits _{\\begin{array}{c}y_{j} \\in E \\\\ j=1, \\dots , n\\end{array}} \\det (0, y_{1}, \\dots , y_{n})=\\sup \\limits _{\\begin{array}{c}y_{j} \\in \\mathrm {co}(E) \\\\ j=1, \\dots , n\\end{array}} \\det (0, y_{1}, \\dots , y_{n}),$ and $\\sup \\limits _{\\begin{array}{c}y_{j} \\in E \\\\ j=1, \\dots , n+1\\end{array}} \\det (y_{1}, \\dots , y_{n+1})=\\sup \\limits _{\\begin{array}{c}y_{j} \\in \\mathrm {co}(E) \\\\ j=1, \\dots , n+1\\end{array}} \\det (y_{1}, \\dots , y_{n+1}).$ We are interested in the best constants $A_n$ , $B_n$ and their optimsers.",
"It is not hard to deduce that the best constant $A_n$ and $B_n$ are related by $B_n \\le A_n \\le (n+1)B_n.",
"\\qquad \\mathrm {(1.9)}$ Indeed, the translation invariance of (1.8) allows us to assume that $0 \\in E$ .",
"Then $B_n \\le A_n$ follows immediately.",
"On the other hand, by the basic determinant property we have $\\det (y_{1}, \\dots , y_{n+1})\\le \\sum \\limits _{j=1}^{n+1} \\det (0, y_{1}, \\dots , y_{j-1}, y_{j+1}, \\dots , y_{n}),$ which implies that $\\sup \\limits _{\\begin{array}{c}y_{j} \\in E \\\\ j=1, \\dots , n+1\\end{array}} \\det (y_{1}, \\dots , y_{n+1})\\le (n+1) \\sup \\limits _{\\begin{array}{c}y_{j} \\in E \\\\ j=1, \\dots , n\\end{array}} \\det (0, y_{1}, \\dots , y_{n}).$ That completes $A_n \\le (n+1)B_n$ .",
"So in the special case when $n=1$ , we have $A_1=2$ , $B_1= 1$ that follows from (1.5) and (1.6).",
"Geometrically, the right side of (1.8) relates to the maximal volume of $n$ -simplex whose vertices are in $E$ .",
"The relationship between the maximal volume of the $n$ -simplex whose vertices are in $E$ and the measure of $E$ has been studied before (see [10], [13]).",
"It is well known that by compactness given a compact convex set $E\\subset \\mathbb {R}^{n}$ , there exists a simplex $T \\subset E$ of maximal volume.",
"Let $F$ be a facet of $T$ , $v$ the opposite vertex, and $H$ the hyperplane through $v$ parallel to $F$ .",
"Then $H$ supports $E$ , since otherwise one would obtain a contradiction to the maximality of the volume of $T$ .",
"Since $F$ is an arbitrary facet of $T$ , $T$ is contained in the simplex $-n(T-c)+c$ , where $c$ is the centroid of $T$ .",
"See [10] for details.",
"So $T \\subset E \\subset -n(T-c)+c$ , and thus $ |E| \\le n^{n}|T|.",
"\\qquad \\mathrm {(1.10)}$ which implies that $B_n \\le n^{n}, \\ A_n \\le (n+1)n^{n}.$ In 1950, Macbeath [13] already gave the sharp version of (1.10) and (1.8) as follows.",
"Given a compact convex set $E \\subset \\mathbb {R}^{n}$ , denote $\\mathfrak {B}_{m}$ the set of convex polytopes with at most $m$ vertices in $E$ , and denote $\\mathfrak {C}_{m}$ the set of convex polytopes with at most $m$ vertices in $E^{\\ast }$ .",
"Then $\\sup \\limits _{T^{\\prime } \\in \\mathfrak {C}_{m}} |T^{\\prime }| \\le \\sup \\limits _{T\\in \\mathfrak {B}_{m}} |T|.",
"\\qquad \\mathrm {(1.11)}$ So when $m=n+1$ , (1.11) gives $\\sup \\limits _{\\begin{array}{c}y_{j} \\in E^{\\ast } \\\\ j=1, \\dots , n+1\\end{array}} \\det (y_{1}, \\dots , y_{n+1})\\le \\sup \\limits _{\\begin{array}{c}y_{j} \\in E \\\\ j=1, \\dots , n+1\\end{array}} \\det (y_{1}, \\dots , y_{n+1}).$ Moreover the problem is clearly affine invariant, thus the extremising sets turn out to be balls and ellipsoids for (1.8).",
"Because the maximal simplex with vertices on a ball is the regular simplex with all sides equal, we can obtain the corresponding best constant $B_n$ .",
"However, we do not believe that the sharp value of $A_n$ in (1.7) has been given previously."
],
[
"Our Results",
"In this paper we shall give an alternative method to derive (1.7) and (1.8) with sharp constants $A_n$ , $B_n$ .",
"In Section 2, we will study some rearrangement inequalities which together with some work in [4] establish this.",
"A key ingredient will be Lemma 4.7 of [4], stating that for any $E_{j}\\subset \\mathbb {R}$ of finite Lebesgue measure, and $a_{j}\\in \\mathbb {R}$ , $j=1, \\dots , l$ , $\\sup \\limits _{x_{j}\\in E_{j}^{\\ast }} \\ |\\sum _{j=1}^{l} a_{j}x_{j}| \\le \\sup \\limits _{x_{j}\\in E_{j}} \\ |\\sum _{j=1}^{l} a_{j}x_{j}|.",
"\\qquad \\mathrm {(1.12)}$ See Lemma 2.2 for the proof.",
"More generally, returning to the inequalities (1.1), (1.2), we see there are functional versions.",
"One can consider a bilinear functional rearrangement version of (1.2).",
"For all nonnegative measurable functions $f, g $ defined on $\\mathbb {R}^{n}$ , $\\displaystyle {\\sup _{x,y}} \\ f^{\\ast }(x) g^{\\ast }(y) |x-y| \\le \\displaystyle {\\sup _{x,y}} \\ f(x) g(y) |x-y| \\qquad \\mathrm {(1.13)}$ holds.",
"Likewise, by the same argument as in its proof we also have $\\displaystyle {\\sup _{x}} \\ f^{\\ast }(x) |x| \\le \\displaystyle {\\sup _{x}} \\ f(x) |x|.",
"\\qquad \\mathrm {(1.14)}$ For the proof, see Lemma 4.2 in [4].",
"In Section 2, generalizing them we arrive at the following multilinear functional rearrangement inequalities, $\\displaystyle {\\sup _{y_{j}}} \\prod _{j=1}^{n} f_{j}^{\\ast }(y_{j}) \\det (0, y_{1}, \\dots , y_{n})\\le \\displaystyle {\\sup _{y_{j}}} \\prod _{j=1}^{n} f_{j}(y_{j}) \\det (0, y_{1}, \\dots , y_{n}), \\qquad \\mathrm {(1.15)}$ and $\\displaystyle {\\sup _{y_{j}}} \\prod _{j=1}^{n+1} f_{j}^{\\ast }(y_{j}) \\det (y_{1}, \\dots , y_{n+1})\\le \\displaystyle {\\sup _{y_{j}}} \\prod _{j=1}^{n+1} f_{j}(y_{j}) \\det (y_{1}, \\dots , y_{n+1}), \\qquad \\mathrm {(1.16)}$ which hold for any nonnegative measurable functions vanishing at infinity $f_{j} $ defined on $\\mathbb {R}^{n}$ , in the sense that all its positive level sets have finite measure, $| \\lbrace x: |f(x)| >t \\rbrace | < \\infty $ , for all $t >0$ .",
"As a matter of fact, we establish much more general inequalities in Theorem 2.5 below.",
"Then we get (1.7), (1.8) with the sharp constants by specialising to $f_{j}=\\chi _{E}$ in (1.15)-(1.16), which also includes Macbeath's work (1.11) when $m=n+1$ .",
"There is another class of inequalities concerning analogues of (1.5), (1.6) where we replace the underlying Euclidean space $\\mathbb {R}^{n}$ by the space of $n \\times n$ real matrices, and the Euclidean norm by $| \\det (A) |$ .",
"For example, Christ first studied this type of inequality in [5].",
"Here “$\\det $ \" becomes ordinary determinant of a matrix.",
"Sublemma 14.1.",
"[5] For any $n \\ge 1$ there exists $C \\in \\mathbb {R}^{+}$ with the following property.",
"Let $E \\subset \\mathfrak {M}^{n \\times n}(\\mathbb {R})$ be a compact convex set satisfying $|E| < \\infty $ and $E=-E$ .",
"Then there exists $A\\in E$ satisfying $ | \\det (A) | \\ge C |E|^{\\frac{1}{n}}, \\qquad \\mathrm {(1.17)}$ where $|\\cdot |$ denotes the Lebesgue measure on Euclidean space $\\mathbb {R}^{n^{2}}$ and the absolute value on $\\mathbb {R}$ .",
"Lemma 13.2.",
"[5] For any $n \\ge 1$ there exists $c, C \\in \\mathbb {R}^{+}$ and $k \\in \\mathbb {N}$ with the following property.",
"Let $E$ be a measurable set in $\\mathfrak {M}^{n \\times n}(\\mathbb {R})$ satisfying $|E| < \\infty $ .",
"Then there exist $T_{1}, \\dots , T_{k} \\in E$ and coefficients $s_{j} \\in \\mathbb {Z}$ satisfying $|s_{j}| \\le c$ , $\\sum \\limits _{j=1}^{k} s_{j}=0$ , such that $| \\det (\\sum \\limits _{j=1}^{k} s_{j}T_{j}) | \\ge C |E|^{\\frac{1}{n}}.",
"\\qquad \\mathrm {(1.18)}$ Remarks 1.",
"1.",
"Let $\\widetilde{E}=E-A := \\lbrace T-A: T \\in E\\rbrace $ with $A \\in \\mathfrak {M}^{n \\times n}(\\mathbb {R})$ , then by Lemma 13.2 there exist $T_{1}, \\dots ,T_{k} \\in E$ and $s_{j} \\in \\mathbb {Z}$ satisfying $|s_{j}| \\le c$ , $\\sum \\limits _{j=1}^{k} s_{j}=0$ , such that $| \\det (\\sum \\limits _{j=1}^{k} s_{j}(T_{j}-A)) | = | \\det (\\sum \\limits _{j=1}^{k} s_{j}T_{j}) | \\ge C |E|^{\\frac{1}{n}}=C |\\widetilde{E}|^{\\frac{1}{n}}, \\qquad \\mathrm {(1.19)}$ which shows (1.18) has a translation invariance property that (1.17) lacks.",
"2.",
"Based on the translation variance property, we have an equivalent form of Lemma 13.2: there exists $c, C \\in \\mathbb {R}^{+}$ such that for any $E \\subset \\mathfrak {M}^{n \\times n}(\\mathbb {R})$ we can always select $T_{1}, \\dots ,T_{k} \\in E$ and coefficients $s_{j} \\in \\mathbb {Z}$ satisfying $|s_{j}| \\le c$ , such that $| \\det (\\sum \\limits _{j=1}^{k} s_{j}T_{j}) | \\ge C |E|^{\\frac{1}{n}}$ .",
"The equivalence is as follows.",
"Supposing $A \\in E$ , denote $\\widetilde{E}=E-A$ .",
"Then if there exist $\\overline{T}_{1}=T_{1}-A, \\dots , \\overline{T}_{k}=T_{k}-A \\in \\widetilde{E}$ , where $T_{j}\\in E$ , $1\\le j\\le k$ , and there exist $s_{j} \\in \\mathbb {Z}$ satisfying $|s_{j}| \\le c$ , such that $| \\det (\\sum \\limits _{j=1}^{k} s_{j}\\overline{T}_{j}) | \\ge C |\\widetilde{E}|^{\\frac{1}{n}}.$ That is, $| \\det ( s_{1}T_{1}+ \\dots + s_{k}T_{k} - (s_{1}+ \\dots + s_{k}) A)| \\ge C |\\widetilde{E}|^{\\frac{1}{n}}=C |E|^{\\frac{1}{n}},$ which satisfies the conditions of Lemma 13.2.",
"More specifically, when proving Lemma 13.2 Christ [5] gave that under the same hypothesis of Lemma 13.2, there exist $A_{j} \\in E$ , and $s_{j} \\in \\lbrace 0, 1\\rbrace $ , $j= 1, \\dots , n$ , such that $|\\det (\\sum \\limits _{j=1}^{n} s_{j}A_{j}) | \\ge C |E|^{\\frac{1}{n}} $ which implies that for any measurable $E \\subset \\mathfrak {M}^{n \\times n}$ , $\\sup \\limits _{\\begin{array}{c}A_{1}, \\dots , A_{n} \\in E \\\\ s_{1}, \\dots , s_{n} \\in \\lbrace 0, 1\\rbrace \\end{array}} |\\det ( s_{1}A_{1}+ \\dots + s_{n}A_{n}) |\\gtrsim _{n} |E|^{\\frac{1}{n}}.",
"\\qquad \\mathrm {(1.20)}$ In this paper we will improve (1.17)-(1.18) as follows, mainly relying on the rearrangement inequality (1.12).",
"Main Theorem.",
"There exists a finite constant $\\mathcal {C}_{n}$ such that for any measurable sets $E_j \\subset \\mathfrak {M}^{n \\times n}(\\mathbb {R})$ of finite measure, $j=1, \\dots , n$ , $ \\prod \\limits _{j=1}^{n}|E_j|^{\\frac{1}{n^{2}}}\\le \\mathcal {C}_{n} \\sup \\limits _{\\begin{array}{c}A_{j} \\in E_{j} \\\\ j=1, \\dots , n\\end{array}} | \\det (A_{1}+ \\dots + A_{n} ) |.",
"\\qquad \\mathrm {(1.21)}$ The main theorem implies (1.17) holds for all compact convex sets in $\\mathfrak {M}^{n \\times n}(\\mathbb {R})$ and extends Lemma 13.2 as described below.",
"In particular, we see from the main Theorem that all the $s_j$ in (1.20) can be taken to be 1.",
"Corollary A.",
"There exists a finite constant $\\mathcal {A}_{n}$ such that for any measurable set $E \\subset \\mathfrak {M}^{n \\times n}(\\mathbb {R})$ of finite measure, for any non-zero scalar $\\lambda _{j} \\in \\mathbb {R}$ , $j=1, \\dots , n$ , $(\\prod _{j=1}^{n} |\\lambda _{j}|) |E|^{\\frac{1}{n}}\\le \\mathcal {A}_{n} \\displaystyle {\\sup _{A_{j} \\in E}} \\ | \\det (\\lambda _{1} A_{1}+ \\dots + \\lambda _{n} A_{n} ) |.",
"\\qquad \\mathrm {(1.22)}$ Corollary B.",
"There exists a finite constant $\\mathcal {B}_{n}$ such that for any measurable compact convex set $E \\subset \\mathfrak {M}^{n \\times n}(\\mathbb {R})$ of finite measure, $|E|^{\\frac{1}{n}} \\le \\mathcal {B}_{n} \\displaystyle {\\sup _{A \\in E}} \\ | \\det (A) |.",
"\\qquad \\mathrm {(1.23)}$ See Section 3 for the proof of Corollary B.",
"Remarks 2.",
"1.",
"One can easily check that $ \\sup \\limits _{A \\in \\mathrm {co} \\lbrace 0, E \\rbrace } \\ | \\det (A)|=\\sup \\limits _{A \\in E} \\ | \\det (A)|.$ This is because $ |\\det ( \\lambda A)|= \\lambda ^{n} |\\det ( A)|$ for any $\\lambda \\in [0,1]$ , so we can always assume that $0 \\in E$ .",
"Given a measurable $E \\subset \\mathfrak {M}^{n \\times n}(\\mathbb {R})$ , by scaling let $\\widetilde{E}= r E$ , $0 \\ne r \\in \\mathbb {R}$ , then $(| \\widetilde{E}|)^{\\frac{1}{n}}= (r^{n^{2}} |E|)^{\\frac{1}{n}}=r^{n}|E|^{\\frac{1}{n}},$ and $\\displaystyle {\\sup _{A \\in \\widetilde{E}}} \\ | \\det (A)|= r^{n} \\displaystyle {\\sup _{A \\in E}} \\ | \\det (A)|.$ However, (1.23) is not translation invariant.",
"2.",
"We use a counterexample to show that (1.23) fails without the convex condition.",
"Take $n=2$ as an example, and let $E= \\lbrace (a,b,c,d): 0\\le ad \\le 1, 0 \\le bc \\le 1, \\ $ and$ \\ 1/N \\le a \\le N, 1/N \\le b \\le N\\rbrace $ .",
"Then we have $\\displaystyle {\\sup _{A \\in E}} \\ | \\det (A)|= \\displaystyle {\\sup _{A \\in E}} \\ | \\det \\left(\\begin{array}{cc}a & c \\\\b & d \\\\\\end{array}\\right) |\\le 2.$ and $|E|= (2 \\ln N)^{2}$ .",
"Let $N \\rightarrow \\infty $ , then we get the contradiction to (1.23).",
"Remarks 3.",
"1.",
"An open problem is what the best constants $\\mathcal {A}_{n}$ , $\\mathcal {B}_{n}$ , $\\mathcal {C}_{n}$ are.",
"We prove in this paper that balls or ellipsoids are not their optimisers.",
"2.",
"Note that inequalities of matrix type introduced in this part do not enjoy an obvious affine invariance.",
"Nevertheless, there is an important action of $\\mathrm {SL}_{n}(\\mathbb {R})$ on $\\mathfrak {M}^{n \\times n}(\\mathbb {R})$ by premultiplication.",
"That is, if $T\\in \\mathrm {GL}_{n}(\\mathbb {R})$ , $A\\in \\mathfrak {M}^{n \\times n}(\\mathbb {R})$ and $E \\subset \\mathfrak {M}^{n \\times n}(\\mathbb {R})$ , then $\\det (TA)=\\det (T) \\det (A)$ and $|TE|=|\\det (T)|^{n} |E|.$ So both matrix inequalities in this paper are invariant under premultiplication by a matrix of unimodular determinant.",
"We do not use the invariance of the entire problem under the action of left-multiplication by members of $\\mathrm {SL}_{n}(\\mathbb {R})$ but instead the facts which underly this invariance, i.e.",
"that this action preserves determinants of individual matrices and preserves volumes of sets.",
"It enters as a “catalyst\" in order to obtain a measure theoretic consequence and its presence vanishes without trace."
],
[
"Determinant inequalities",
"In this section we study the determinant inequalities discussed in the introduction.",
"First we recall an estimate by Gressman [8] as follows.",
"Lemma 2.1 [8] There exists a finite constant $C_{n}$ such that for any $y \\in \\mathbb {R}^{n}$ , for any measurable sets $E_{1}, \\dots , E_{n}$ in $\\mathbb {R}^{n}$ and for any $\\delta >0$ $| \\lbrace (y_{1}, \\dots , y_{n})\\in E_{1} \\times \\cdots \\times E_{n}:\\det (y, y_{1}, \\dots , y_{n}) < \\delta \\rbrace | \\le C_{n} \\delta \\displaystyle {\\prod _{j=1}^{n}} |E_{j}|^{1-\\frac{1}{n}}.",
"\\qquad \\mathrm {(2.1)}$ As an immediate consequence of (2.1), we obtain the following inequality (2.2).",
"With the same constant $C_n$ , we have for any $y \\in \\mathbb {R}^{n}$ , for any measurable sets $E_{j} \\subset \\mathbb {R}^{n}$ , $1 \\le j \\le n$ , $ \\displaystyle {\\prod _{j=1}^{n}} \\ |E_{j}|^{\\frac{1}{n}} \\le C_{n}\\sup \\limits _{y_{1} \\in E_{1}, \\dots , y_{n} \\in E_{n} } \\det (y, y_{1}, \\dots , y_{n}).",
"\\qquad \\mathrm {(2.2)}$ One way to see this is as follows.",
"Let $y \\in \\mathbb {R}^{n}$ and suppose $\\sup \\limits _{y_{1} \\in E_{1}, \\dots , y_{n} \\in E_{n} } \\det (y, y_{1}, \\dots , y_{n})=s < \\infty .$ It follows from Lemma 2.1 that for all measurable sets $E_{j} \\subset \\mathbb {R}^{n}$ , $1 \\le j \\le n$ , $| \\lbrace (y_{1}, \\dots , y_{n}) \\in E_{1} \\times \\cdots \\times E_{n}: \\det (y, y_{1}, \\dots , y_{n}) \\le s ) \\rbrace |\\le C_{n} s \\displaystyle {\\prod _{j=1}^{n}} |E_{j}|^{1-\\frac{1}{n}}.$ Note that $s = \\sup \\limits _{y_{1} \\in E_{1}, \\dots , y_{n} \\in E_{n}} \\det (y, y_{1}, \\dots , y_{n})$ , so $| \\lbrace (y_{1}, \\dots , y_{n}) \\in E_{1} \\times \\cdots \\times E_{n}: \\det (y, y_{1}, \\dots , y_{n}) \\le s ) \\rbrace |= \\displaystyle {\\prod _{j=1}^{n}} \\ |E_{j}|.$ Therefore, $\\displaystyle {\\prod _{j=1}^{n}} \\ |E_{j}| \\le C_{n} s \\displaystyle {\\prod _{j=1}^{n}} |E_{j}|^{1-\\frac{1}{n}}.$ That is, $\\displaystyle {\\prod _{j=1}^{n}} |E_{j}|^{\\frac{1}{n}} \\le C_{n} s=C_{n} \\sup \\limits _{y_{1} \\in E_{1}, \\dots , y_{n} \\in E_{n} } \\det (y, y_{1}, \\dots , y_{n}),$ which completes (2.2).",
"This motivates a multilinear perspective.",
"Later on, we will prove the sharp version of (2.1)-(2.2).",
"More generally, functional versions of (2.2) have been studied in [4].",
"As shown in Theorem 3.1 of [4], for any nonnegative measurable functions $ f_{j} \\in L^{p_{j}}(\\mathbb {R}^{n}) $ , $ \\displaystyle {\\prod _{j=1}^{n+1}} \\Vert f_{j}\\Vert _{p_{j}} \\le C_{n, p_{j}} \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ f_{j}(y_{j}) \\det (y_{1}, \\dots , y_{n+1})^{\\gamma } \\qquad \\mathrm {(2.3)}$ holds, if and only if $p_{j}$ satisfy $ \\frac{1}{p_{j}} < \\frac{\\gamma }{n}$ for all $1 \\le j \\le n+1$ and $\\gamma = \\displaystyle { \\sum _{j=1}^{n+1} } \\ \\frac{1}{p_{j}}$ .",
"And Lemma 3.2 in [4] gives an endpoint case of the multilinear inequality (2.3).",
"That is, for any nonnegative measurable functions $ f_{j} \\in L^{p_{j}}(\\mathbb {R}^{n}) $ $ \\displaystyle {\\prod _{j=1}^{n}} \\Vert f_{j}\\Vert _{L^{n, \\infty } (\\mathbb {R}^{n})} \\Vert f_{n+1}\\Vert _{L^{\\infty }}\\le C_{n} \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{n+1}} \\ f_{j}(y_{j}) \\ \\det (y_{1}, \\dots , y_{n+1}).",
"\\qquad \\mathrm {(2.4)}$ It is not hard to see (2.4) implies for any $y \\in \\mathbb {R}^{n}$ $ \\displaystyle {\\prod _{j=1}^{n}} \\Vert f_{j}\\Vert _{L^{n, \\infty } (\\mathbb {R}^{n})}\\le C_{n} \\displaystyle {\\sup _{y_{j}}} \\ \\displaystyle {\\prod _{j=1}^{n}} \\ f_{j}(y_{j}) \\ \\det (y, y_{1}, \\dots , y_{n}), \\qquad \\mathrm {(2.5)}$ which also concludes (2.2) by specialising to $f_{j}=\\chi _{E_{j}}$ .",
"For the proof of (2.3)- (2.5) and more general multilinear cases, we refer to [4].",
"Before studying the sharp versions of inequalities (2.2), we recall some useful tools in [4] which were already stated in the introduction.",
"Lemma 2.2 [4] Let $E_{j}$ be measurable sets in $\\mathbb {R}$ and $a_{j}\\in \\mathbb {R}$ , $j=1, \\dots , l$ .",
"Then $\\sup \\limits _{x_{j}\\in E_{j}^{\\ast }} |\\sum _{j=1}^{l} a_{j}x_{j}| \\le \\sup \\limits _{x_{j}\\in E_{j}} |\\sum _{j=1}^{l} a_{j}x_{j}|.",
"\\qquad \\mathrm {(2.6)}$ From the Brunn-Minkowski inequality $|E+F| \\ge |E| + |F|$ where $E, F \\subset \\mathbb {R}$ , it follows that $|E_1+ \\dots + E_l| \\ge |E_1| + \\dots + |E_l|.$ Because $E_{j}^{\\ast }= (-|E_j|/2, |E_j|/2)$ , $1 \\le j \\le l$ , then $E_{1}^{\\ast }+ \\dots + E_{l}^{\\ast } = (-\\sum \\limits _{j=1}^{l} \\frac{|E_j|}{2}, \\sum \\limits _{j=1}^{l} \\frac{|A_j|}{2}).$ Thus we have $|(E_{1}+ \\dots + E_{l})^{\\ast }| = |E_{1}+ \\dots + E_{l}| \\ge |E_1| + \\dots + |E_l| =| E_{1}^{\\ast }+ \\dots + E_{l}^{\\ast }|,$ which implies $(E_{1}+ \\dots + E_{l})^{\\ast } \\supset E_{1}^{\\ast }+ \\dots + E_{l}^{\\ast }.",
"\\qquad \\mathrm {(2.7)}$ Clearly, for any non-zero $a \\in \\mathbb {R}$ and any measurable subset $E$ in $ \\mathbb {R}$ $(aE)^{\\ast }=a E^{*}.",
"\\qquad \\mathrm {(2.8)}$ Combining with (2.7)-(2.8) we have $(a_1 E_{1}+ \\dots + a_l E_{l})^{\\ast } \\supset a_1 E_{1}^{\\ast }+ \\dots + a_l E_{l}^{\\ast }.",
"\\qquad \\mathrm {(2.9)}$ Apply (1.1) and (2.9), $\\sup \\limits _{x_{j}\\in E_{j}} |\\sum _{j=1}^{l} a_{j}x_{j}|= \\sup \\limits _{\\bar{x} \\in \\sum \\limits _{j=1}^{l} a_j E_{j}} |\\bar{x}|\\ge \\sup \\limits _{\\bar{x} \\in (\\sum \\limits _{j=1}^{l} a_j E_{j})^{\\ast }} |\\bar{x}|\\ge \\sup \\limits _{\\bar{x} \\in \\sum \\limits _{j=1}^{l} a_j E_{j}^{\\ast }} |\\bar{x}|.$ Besides, $\\sup \\limits _{\\bar{x} \\in \\sum \\limits _{j=1}^{l} a_j E_{j}^{\\ast }} |\\bar{x}|= \\sup \\limits _{x_j \\in a_{j}E_{j}^{*} } | \\sum _{j=1}^{l} x_{j}|=\\sup \\limits _{x_j \\in E_{j}^{*} } | \\sum _{j=1}^{l} a_{j}x_{j}|.$ Therefore, $\\sup \\limits _{x_{j}\\in E_{j}} |\\sum _{j=1}^{l} a_{j}x_{j}| \\ge \\sup \\limits _{x \\in E_{j}^{*} } | \\sum _{j=1}^{l} a_{j}x_{j}|.$ It follows from Lemma 2.2 we have inequalities (2.10)-(2.12).",
"Let $E_1, \\dots , E_l$ be measurable sets in $\\mathbb {R}^{n}$ .",
"Let $l \\ge n$ and let $ A=\\lbrace a_{ik}\\rbrace $ be an $l \\times n$ real matrix.",
"Then for each $1 \\le t \\le n$ , $\\sup \\limits _{\\begin{array}{c}y_{j} \\in \\mathcal {S}_{e_{t}}(E_j) \\\\ j=1, \\dots , l\\end{array}} \\det (0, \\sum \\limits _{i=1}^{l} a_{i1}y_{i}, \\dots , \\sum \\limits _{i=1}^{l} a_{in}y_{i})\\le \\sup \\limits _{\\begin{array}{c}y_{j} \\in E_j \\\\ j=1, \\dots , l\\end{array}} \\det (0, \\sum \\limits _{i=1}^{l} a_{i1}y_{i}, \\dots , \\sum \\limits _{i=1}^{l} a_{in}y_{i}), \\qquad \\mathrm {(2.10)}$ where $\\lbrace e_{1}, \\dots , e_{n}\\rbrace $ is the standard basis for $\\mathbb {R}^{n}$ .",
"Let $l=n$ and $a_{ik}={\\left\\lbrace \\begin{array}{ll}1 &\\mbox{if $i=k$} \\\\0 &\\mbox{otherwise,}\\end{array}\\right.",
"}$ so (2.10) gives $\\sup \\limits _{\\begin{array}{c}y_{j} \\in \\mathcal {S}_{e_{t}}(E_j) \\\\ j=1, \\dots , n\\end{array}} \\det (0, y_{1}, \\dots , y_{n})\\le \\sup \\limits _{\\begin{array}{c}y_{j} \\in E_j \\\\ j=1, \\dots , n\\end{array}} \\det (0, y_{1}, \\dots , y_{n}).",
"\\qquad \\mathrm {(2.11)}$ If we set $l=n+1$ and $a_{ik}={\\left\\lbrace \\begin{array}{ll}1 &\\mbox{if $i=k$} \\\\-1 &\\mbox{if $i=n+1$} \\\\0 &\\mbox{otherwise,}\\end{array}\\right.",
"}$ thus $\\sup \\limits _{\\begin{array}{c}y_{j} \\in \\mathcal {S}_{e_{t}}(E_j) \\\\ j=1, \\dots , n+1\\end{array}} \\det (y_{1}, \\dots , y_{n+1})\\le \\sup \\limits _{\\begin{array}{c}y_{j} \\in E_j \\\\ j=1, \\dots , n+1\\end{array}} \\det (y_{1}, \\dots , y_{n+1}).",
"\\qquad \\mathrm {(2.12)}$ For simplicity, we just see (2.10) holds for $e_{1}$ .",
"Define the projection $\\pi $ : $\\mathbb {R}^{n} \\rightarrow \\mathbb {R}^{n-1}$ by $\\pi (x)=(x_2, \\dots , x_n), \\ \\forall \\ x=(x_1, \\dots , x_n) \\in \\mathbb {R}^{n}.$ For any $x\\in \\mathbb {R}^{n}$ , write $x= (x_1, x^{\\prime })$ where $x^{\\prime } \\in \\mathbb {R}^{n-1}$ .",
"For $y_j \\in E_j$ , $\\det (0, y_{1}, \\dots , y_{n})=| \\det \\left(\\begin{array}{cccc}y_{11} & y_{21} & \\dots & y_{n1} \\\\\\vdots & \\vdots & \\ & \\vdots \\\\y_{1n} & y_{2n} & \\dots & y_{nn}\\end{array}\\right) |= |y_{11} A_1 +y_{21} A_2 + \\dots y_{n1} A_{n}|,$ where $A_{j}$ depend only on $\\lbrace y_{1}^{\\prime }, \\dots , y_{n}^{\\prime }\\rbrace $ .",
"Hence, $\\det (0, \\sum \\limits _{i=1}^{l} a_{i1}y_{i}, \\dots , \\sum \\limits _{i=1}^{l} a_{in}y_{i})$ is the linear combination of $y_{11}, \\dots , y_{l1}$ .",
"That is, $\\det (0, \\sum \\limits _{i=1}^{l} a_{i1}y_{i}, \\dots , \\sum \\limits _{i=1}^{l} a_{in}y_{i})=|y_{11} B_1 +y_{21} B_2 + \\dots y_{l1} B_{l}|,$ where $B_{j}$ depend only on $\\lbrace y_{1}^{\\prime }, \\dots , y_{l}^{\\prime }\\rbrace $ .",
"For each $j$ , fix $y_{j}^{\\prime }:=(y_{j2}, \\dots , y_{jn}) \\in \\pi (E_j)$ , $1 \\le j \\le l$ .",
"Let $E_{j}(y_{j}^{\\prime }) =\\lbrace y_{j1}\\in \\mathbb {R}: (y_{j1}, y_{j}^{\\prime }) \\in E_{j}\\rbrace .",
"$ It follows from Lemma 2.2 that $\\sup \\limits _{y_{j1} \\in E_{j}(y_{j}^{\\prime })^{\\ast }} |\\sum _{j=1}^{l} B_{j}y_{j1}| \\le \\sup \\limits _{y_{j1} \\in E_{j}(y_{j}^{\\prime })} |\\sum _{j=1}^{l} B_{j}y_{j1}|.",
"\\qquad \\mathrm {(2.13)}$ Since $\\mathcal {S}_{e_{1}}(E_j)= \\bigcup \\limits _{y_{j}^{\\prime } \\in \\pi (E_j)} \\lbrace (y_{j1}, y_{j}^{\\prime }): y_{j1} \\in E_{j}(y_{j}^{\\prime })^{\\ast }\\rbrace , $ together with (2.13) gives $\\sup \\limits _{\\begin{array}{c}y_{j} \\in \\mathcal {S}_{e_{1}}(E_j) \\\\ j=1, \\dots , l\\end{array}} \\det (0, \\sum \\limits _{i=1}^{l} a_{i1}y_{i}, \\dots , \\sum \\limits _{i=1}^{l} a_{in}y_{i})\\le \\sup \\limits _{\\begin{array}{c}y_{j} \\in E_j \\\\ j=1, \\dots , l\\end{array}} \\det (0, \\sum \\limits _{i=1}^{l} a_{i1}y_{i}, \\dots , \\sum \\limits _{i=1}^{l} a_{in}y_{i}).$ More generally, togehter with the rotation invariance we have the following rearrangement theorem.",
"Theorem 2.3 Let $ A=\\lbrace a_{ik}\\rbrace $ be an $l \\times n$ real matrix with $l \\ge n$ .",
"Let $u$ be a unit vector in $\\mathbb {R}^{n}$ .",
"Then for any measurable sets $E_j \\subset \\mathbb {R}^{n}$ , $1 \\le j \\le l$ , $\\sup \\limits _{\\begin{array}{c}y_{j} \\in \\mathcal {S}_{u}(E_j) \\\\ j=1, \\dots , l\\end{array}} \\det (0, \\sum \\limits _{i=1}^{l} a_{i1}y_{i}, \\dots , \\sum \\limits _{i=1}^{l} a_{in}y_{i})\\le \\sup \\limits _{\\begin{array}{c}y_{j} \\in E_j \\\\ j=1, \\dots , l\\end{array}} \\det (0, \\sum \\limits _{i=1}^{l} a_{i1}y_{i}, \\dots , \\sum \\limits _{i=1}^{l} a_{in}y_{i}).$ Suppose $u=\\rho e_{t}$ , where $\\rho $ is a rotation around the origin in $\\mathbb {R}^{n}$ .",
"By definition, $\\mathcal {S}_{\\rho e_{t}}(E)&=\\lbrace m \\rho e_{t}+y: E\\cap [\\mathbb {R}(\\rho e_{t})+y ] \\ne \\phi , |m| \\le \\frac{|E\\cap [\\mathbb {R}(\\rho e_{t})+y] |}{2}\\rbrace \\\\&= \\lbrace \\rho (me_{t}+\\rho ^{-1}y): \\rho ^{-1}(E) \\cap (\\mathbb {R}e_{t}+\\rho ^{-1}y)\\ne \\phi , |m| \\le \\frac{ |\\rho [\\rho ^{-1}(E)\\cap (\\mathbb {R}e_{t}+\\rho ^{-1}y)]|}{2}\\rbrace \\\\&= \\lbrace \\rho (me_{t}+\\rho ^{-1}y): \\rho ^{-1}(E) \\cap (\\mathbb {R}e_{t}+\\rho ^{-1}y)\\ne \\phi , |m| \\le \\frac{ |\\rho ^{-1}(E)\\cap (\\mathbb {R}e_{t}+\\rho ^{-1}y)|}{2}\\rbrace .$ Note that $\\mathcal {S}_{e_{t}}(\\rho ^{-1}(E))= \\lbrace m e_{t}+\\rho ^{-1}y: \\rho ^{-1}(E) \\cap (\\mathbb {R}e_{t}+\\rho ^{-1}y)\\ne \\phi ,|m| \\le \\frac{| \\rho ^{-1}(E) \\cap (\\mathbb {R}e_{t}+\\rho ^{-1}y)|}{2} \\rbrace .$ Hence we obtain $\\mathcal {S}_{\\rho e_{t}}(E)=\\rho \\circ \\mathcal {S}_{e_{t}}(\\rho ^{-1}(E)).",
"\\qquad \\mathrm {(2.14)}$ By the invariance under rotation $\\rho $ $\\sup \\limits _{\\begin{array}{c}y_{j} \\in \\mathcal {S}_{u}(E_j) \\\\ j=1, \\dots , l\\end{array}}\\det (0, \\sum \\limits _{i=1}^{l} a_{i1}y_{i}, \\dots , \\sum \\limits _{i=1}^{l} a_{in}y_{i})&= \\sup \\limits _{\\begin{array}{c}y_{j} \\in \\rho \\circ \\mathcal {S}_{e_{t}}(\\rho ^{-1}(E_j)) \\\\ j=1, \\dots , l\\end{array}}\\det (0, \\sum \\limits _{i=1}^{l} a_{i1}y_{i}, \\dots , \\sum \\limits _{i=1}^{l} a_{in}y_{i}) \\\\&=\\sup \\limits _{\\begin{array}{c}y_{j} \\in \\mathcal {S}_{e_{t}}(\\rho ^{-1}(E_j)) \\\\ j=1, \\dots , l\\end{array}} \\det (0, \\sum \\limits _{i=1}^{l} a_{i1}y_{i}, \\dots , \\sum \\limits _{i=1}^{l} a_{in}y_{i}).$ Applying (2.10) gives $\\sup \\limits _{\\begin{array}{c}y_{j} \\in \\mathcal {S}_{e_{t}}(\\rho ^{-1}(E_j)) \\\\ j=1, \\dots , l\\end{array}} \\det (0, \\sum \\limits _{i=1}^{l} a_{i1}y_{i}, \\dots , \\sum \\limits _{i=1}^{l} a_{in}y_{i})&\\le \\sup \\limits _{\\begin{array}{c}y_{j} \\in \\rho ^{-1}(E_j) \\\\ j=1, \\dots , l\\end{array}} \\det (0, \\sum \\limits _{i=1}^{l} a_{i1}y_{i}, \\dots , \\sum \\limits _{i=1}^{l} a_{in}y_{i}) \\\\&= \\sup \\limits _{\\begin{array}{c}y_{j} \\in E_j \\\\ j=1, \\dots , l\\end{array}} \\det (0, \\sum \\limits _{i=1}^{l} a_{i1}y_{i}, \\dots , \\sum \\limits _{i=1}^{l} a_{in}y_{i}).$ Therefore, we conclude $\\sup \\limits _{\\begin{array}{c}y_{j} \\in \\mathcal {S}_{u}(E_j) \\\\ j=1, \\dots , l\\end{array}}\\det (0, \\sum \\limits _{i=1}^{l} a_{i1}y_{i}, \\dots , \\sum \\limits _{i=1}^{l} a_{in}y_{i})= \\sup \\limits _{\\begin{array}{c}y_{j} \\in E_j \\\\ j=1, \\dots , l\\end{array}} \\det (0, \\sum \\limits _{i=1}^{l} a_{i1}y_{i}, \\dots , \\sum \\limits _{i=1}^{l} a_{in}y_{i}).$ Now we can decide the sharp versions of the determinant inqualities in this section.",
"It is known that, given a compact convex set $K \\subset \\mathbb {R}^{n}$ , there exists a sequence of iterated Steiner symmetrisations of $K$ that converges in the Hausdorff metric to a ball of the same volume.",
"For example, given a basis of unit directions $u_1, \\dots , u_n$ for $\\mathbb {R}^{n}$ having mutually irrational multiple of $\\pi $ radian differences, then the sequence $\\mathcal {S}_{u_{n}} \\dots \\mathcal {S}_{u_{2}} \\mathcal {S}_{u_{1}}(K)$ iterated infinitely many times to $K$ will converge to a ball of the same volume as $K$ .",
"For the convergence of Steiner symmetrisation, refer to [1], [2], [6], [11], [15], etc.",
"One can easily verify that the suprema function on the right side of inequalities (2.10) are continuous under the Hausdorff metric, and they do not change if we replace each $E_j$ by $\\overline{\\mathrm {co}}(E_j)$ .",
"Therefore, applying the convergence of Steiner symmetrisation together with Theorem 2.3 we have shown the following lemma.",
"Lemma 2.4 Let $l \\ge n$ and let $ A=\\lbrace a_{ik}\\rbrace $ be an $l \\times n$ real matrix.",
"Then for any measurable sets $E_{j} \\subset \\mathbb {R}^{n}$ , $1 \\le j \\le l$ , $\\sup \\limits _{y_{1} \\in E_{1}^{*}, \\dots , y_{l} \\in E_{l}^{*} } \\det (0, \\sum \\limits _{i=1}^{l} a_{i1}y_{i}, \\dots , \\sum \\limits _{i=1}^{l} a_{in}y_{i})\\le \\sup \\limits _{y_{1} \\in E_{1}, \\dots , y_{l} \\in E_{l} } \\det (0, \\sum \\limits _{i=1}^{l} a_{i1}y_{i}, \\dots , \\sum \\limits _{i=1}^{l} a_{in}y_{i}).",
"$ Obviously, it follows from Lemma 2.4 that $\\sup \\limits _{y_{1} \\in E_{1}^{*}, \\dots , y_{n} \\in E_{n}^{*} } \\det (0, y_{1}, \\dots , y_{n})\\le \\sup \\limits _{y_{1} \\in E_{1}, \\dots , y_{n} \\in E_{n} } \\det (0, y_{1}, \\dots , y_{n}), \\qquad \\mathrm {(2.15)}$ and $\\sup \\limits _{y_{1} \\in E_{1}^{*}, \\dots , y_{n+1} \\in E_{n+1}^{*} } \\det (y_{1}, \\dots , y_{n+1})\\le \\sup \\limits _{y_{1} \\in E_{1}, \\dots , y_{n+1} \\in E_{n+1} } \\det (y_{1}, \\dots , y_{n+1}) \\qquad \\mathrm {(2.16)}$ hold for any measurable sets $E_{j} \\subset \\mathbb {R}^{n}$ , $1 \\le j \\le n+1$ .",
"From Lemma 2.4 we obtain the multilinear functional rearrangement inequalities.",
"Theorem 2.5 Let $f_{j}$ be nonnegative measurable functions vanishing at infinity on $\\mathbb {R}^{n}$ .",
"Let $A=\\lbrace a_{ij}\\rbrace \\in \\mathrm {GL}_{n}(\\mathbb {R})$ , then $\\displaystyle {\\sup _{y_{j}}} \\prod _{j=1}^{n} f_{j}^{\\ast }(\\sum \\limits _{i=1}^{n}a_{ij}y_{i}) \\det (0, y_{1}, \\dots , y_{n})\\le \\displaystyle {\\sup _{y_{j}}} \\prod _{j=1}^{n} f_{j}(\\sum \\limits _{i=1}^{n}a_{ij}y_{i}) \\det (0, y_{1}, \\dots , y_{n}).",
"\\qquad \\mathrm {(2.17)}$ Let $A=\\lbrace a_{ij}\\rbrace \\in \\mathrm {GL}_{(n+1)}(\\mathbb {R})$ , then $\\displaystyle {\\sup _{y_{j}}} \\prod _{j=1}^{n+1} f_{j}^{\\ast }(\\sum \\limits _{i=1}^{n+1}a_{ij}y_{i}) \\det (y_{1}, \\dots , y_{n+1})\\le \\displaystyle {\\sup _{y_{j}}} \\prod _{j=1}^{n+1} f_{j}(\\sum \\limits _{i=1}^{n+1}a_{ij}y_{i}) \\det (y_{1}, \\dots , y_{n+1}), \\qquad \\mathrm {(2.18)}$ where the $\\sup $ is the essential supremum.",
"Let $\\tilde{y}_j =\\sum \\limits _{i=1}^{n}a_{ij}y_{i}$ , $1 \\le j \\le n$ , so $\\det (0, y_{1}, \\dots , y_{n})= \\det (0, \\tilde{y}_{1}, \\dots , \\tilde{y}_{n}) |\\det (A)|^{-1}.",
"$ Then for (2.17) it suffices to prove $\\displaystyle {\\sup _{\\tilde{y}_{j}}} \\prod _{j=1}^{n} f_{j}^{\\ast }(\\tilde{y}_{j}) \\det (0, \\tilde{y}_{1}, \\dots , \\tilde{y}_{n})\\le \\displaystyle {\\sup _{\\tilde{y}_{j}}} \\prod _{j=1}^{n} f_{j}(\\tilde{y}_{j}) \\det (0, \\tilde{y}_{1}, \\dots , \\tilde{y}_{n}).",
"\\qquad \\mathrm {(2.19)}$ Similarly, for (2.18) denote $\\tilde{y}_j =\\sum \\limits _{i=1}^{n+1}a_{ij}y_{i}$ , $1 \\le j \\le n+1$ .",
"Since $\\left(\\begin{array}{ccc}y_1 & \\dots & y_{n+1}\\end{array}\\right) =\\left(\\begin{array}{ccc}\\tilde{y}_1 & \\dots & \\tilde{y}_{n+1}\\end{array}\\right) A^{-1},$ $\\det (y_{1}, \\dots , y_{n+1})$ can be written as the form $\\det (0, \\sum \\limits _{i=1}^{n+1} c_{i1} \\tilde{y}_{i}, \\sum \\limits _{i=1}^{n+1} c_{i2} \\tilde{y}_{i}, \\dots , \\sum \\limits _{i=1}^{n+1} c_{in} \\tilde{y}_{i}).$ Specifically, suppose $A^{-1}=\\lbrace b_{ij}\\rbrace _{n+1}$ , then by calculation we have $c_{ik}=b_{ik}-b_{i(n+1)}$ with $1 \\le k \\le n$ , $1 \\le i \\le n+1$ .",
"Hence (2.18) becomes $\\begin{array}{ll}& \\ \\ \\ \\ \\ \\displaystyle {\\sup _{\\tilde{y}_{j}}} \\prod _{j=1}^{n+1} f_{j}^{\\ast }(\\tilde{y}_{j})\\det (0, \\sum \\limits _{i=1}^{n+1} c_{i1} \\tilde{y}_{i}, \\sum \\limits _{i=1}^{n+1} c_{i2} \\tilde{y}_{i}, \\dots , \\sum \\limits _{i=1}^{n+1} c_{in} \\tilde{y}_{i}) \\\\& \\le \\displaystyle {\\sup _{\\tilde{y}_{j}}} \\prod _{j=1}^{n+1} f_{j}(\\tilde{y}_{j})\\det (0, \\sum \\limits _{i=1}^{n+1} c_{i1} \\tilde{y}_{i}, \\sum \\limits _{i=1}^{n+1} c_{i2} \\tilde{y}_{i}, \\dots , \\sum \\limits _{i=1}^{n+1} c_{in} \\tilde{y}_{i}).\\end{array}\\qquad \\mathrm {(2.20)}$ We claim that for any $l \\ge n$ , for any $l \\times n$ real matrix $B=\\lbrace c_{ik}\\rbrace $ $\\displaystyle {\\sup _{y_{j}}} \\prod _{j=1}^{l} f_{j}^{\\ast }(y_{j}) \\det (0, \\sum \\limits _{i=1}^{l} c_{i1} y_{i}, \\dots , \\sum \\limits _{i=1}^{l} c_{in} y_{i})\\le \\displaystyle {\\sup _{y_{j}}} \\prod _{j=1}^{l} f_{j}(y_{j}) \\det (0, \\sum \\limits _{i=1}^{l} c_{i1} y_{i}, \\dots , \\sum \\limits _{i=1}^{l} c_{in} y_{i}) $ holds.",
"Suppose $\\displaystyle {\\sup _{y_{j}}} \\prod _{j=1}^{l} f_{j}(y_{j}) \\det (0, \\sum \\limits _{i=1}^{l} c_{i1} y_{i}, \\dots , \\sum \\limits _{i=1}^{l} c_{in} y_{i})=s< \\infty .$ We assume for a contradiction that $\\displaystyle {\\sup _{y_{j}}} \\prod _{j=1}^{l} f_{j}^{\\ast }(y_{j}) \\det (0, \\sum \\limits _{i=1}^{l} c_{i1} y_{i}, \\dots , \\sum \\limits _{i=1}^{l} c_{in} y_{i})>s.$ Then there exist positive $\\varepsilon $ and a set $ G \\subset \\mathbb {R}^{n} \\times \\dots \\times \\mathbb {R}^{n}$ such that $|G|>0$ and for all $(x_1, \\dots , x_l) \\in G$ we have $ \\prod _{j=1}^{l} f_{j}^{\\ast }(x_{j}) \\det (0, \\sum \\limits _{i=1}^{l} c_{i1} x_{i}, \\dots , \\sum \\limits _{i=1}^{l} c_{in} x_{i})> s+\\varepsilon , \\qquad \\mathrm {(2.21)}$ which gives $f_{1}^{\\ast }(x_1) > (s+ \\varepsilon ) (\\prod _{j=2}^{l} f_{j}^{\\ast }(x_j)\\det (0, \\sum \\limits _{i=1}^{l} c_{i1} x_{i}, \\dots , \\sum \\limits _{i=1}^{l} c_{in} x_{i}))^{-1}.",
"\\qquad \\mathrm {(2.22)}$ Define the set $E_{1}:=\\lbrace y_1: f_{1}(y_1) > (s+\\varepsilon ) (\\prod _{j=2}^{l} f_{j}^{\\ast }(x_j) \\det (0, \\sum \\limits _{i=1}^{l} c_{i1} x_{i}, \\dots , \\sum \\limits _{i=1}^{l} c_{in} x_{i}))^{-1} \\rbrace ,$ so by the property of decreasing rearrangement together with (2.22) $ |E_1|> v_{n} |x_{1}|^{n}.",
"$ From the definition of $E_1$ $ f_{2}^{\\ast }(x_2) >( s+\\frac{\\varepsilon }{2}) \\ (\\displaystyle {\\inf _{y_{1} \\in E_1 }} f_{1}(y_1) \\prod _{j=3}^{l} f_{j}^{\\ast }(x_j)\\det (0, \\sum \\limits _{i=1}^{l} c_{i1} x_{i}, \\dots , \\sum \\limits _{i=1}^{l} c_{in} x_{i}) )^{-1}.$ We then define $E_2= \\lbrace y_2: f_{2}(y_2)>( s+\\frac{\\varepsilon }{2}) (\\displaystyle {\\inf _{y_{1} \\in E_1 }} f_{1}(y_1) \\prod _{j=3}^{l} f_{j}^{\\ast }(x_j)\\det (0, \\sum \\limits _{i=1}^{l} c_{i1} x_{i}, \\dots , \\sum \\limits _{i=1}^{l} c_{in} x_{i}) \\rbrace ,$ so $ |E_2|> v_{n} |x_{2}|^{n}.",
"$ Overall, we can take the similar arguments to define sets $E_t$ , $1< t< l$ $E_t =\\lbrace y_t: f_{t}(y_t) >(s+\\frac{\\varepsilon }{t}) (\\prod _{j=1}^{t-1} \\inf \\limits _{y_{j} \\in E_j } f_{j}(y_j) \\prod _{j=t+1}^{l} f_{j}^{\\ast }(x_j)\\det (0, \\sum \\limits _{i=1}^{l} c_{i1} x_{i}, \\dots , \\sum \\limits _{i=1}^{l} c_{in} x_{i}))^{-1} \\rbrace ,$ and $E_l=\\lbrace y_l: f_{l}(y_l) > (s+\\frac{\\varepsilon }{l}) (\\prod _{j=1}^{l-1} \\inf \\limits _{y_{j} \\in E_j } f_{j}(y_j)\\det (0, \\sum \\limits _{i=1}^{l} c_{i1} x_{i}, \\dots , \\sum \\limits _{i=1}^{l} c_{in} x_{i}) )^{-1} \\rbrace .$ It is easily seen that for each $j=1, \\dots , l$ $|E_j|> v_{n} |x_{j}|^{n}, \\qquad \\mathrm {(2.23)}$ and thus $x_j \\in E_{j}^{\\ast }$ .",
"It follows from Lemma 2.4 that $\\sup \\limits _{y_{1} \\in E_{1}^{*}, \\dots , y_{l} \\in E_{l}^{*}} \\det (0, \\sum \\limits _{i=1}^{l} c_{i1} y_{i}, \\dots , \\sum \\limits _{i=1}^{l} c_{in} y_{i}) \\le \\sup \\limits _{y_{1} \\in E_1, \\dots , y_{l} \\in E_l} \\det (0, \\sum \\limits _{i=1}^{l} c_{i1} y_{i}, \\dots , \\sum \\limits _{i=1}^{l} c_{in} y_{i}).$ That together with $x_{j} \\in E_{j}^{\\ast }$ , $j=1, \\dots , l$ , implies $ \\det (0, \\sum \\limits _{i=1}^{l} c_{i1} x_{i}, \\dots , \\sum \\limits _{i=1}^{l} c_{in} x_{i}) \\le \\sup \\limits _{y_{1} \\in E_1, \\dots , y_{n} \\in E_n} \\det (0, \\sum \\limits _{i=1}^{l} c_{i1} y_{i}, \\dots , \\sum \\limits _{i=1}^{l} c_{in} y_{i}).",
"\\qquad \\mathrm {(2.24)}$ From the definition of $E_l$ we have for any $y_j \\in E_j$ , $1 \\le j \\le l$ $& \\ \\ \\ \\ \\prod \\limits _{j=1}^{l} f_{j}(y_{j}) \\det (0, \\sum \\limits _{i=1}^{l} c_{i1} y_{i}, \\dots , \\sum \\limits _{i=1}^{l} c_{in} y_{i}) \\\\&> (s + \\frac{\\varepsilon }{l} ) (\\det (0, \\sum \\limits _{i=1}^{l} c_{i1} x_{i}, \\dots , \\sum \\limits _{i=1}^{l} c_{in} x_{i}) )^{-1}\\det (0, \\sum \\limits _{i=1}^{l} c_{i1} y_{i}, \\dots , \\sum \\limits _{i=1}^{l} c_{in} y_{i}).$ Therefore, together with (2.24) we obtain $s &\\ge \\sup \\limits _{y_{1} \\in E_1, \\dots , y_{l} \\in E_l} \\prod _{j=1}^{l} f_{j}(y_{j})\\det (0, \\sum \\limits _{i=1}^{l} c_{i1} y_{i}, \\dots , \\sum \\limits _{i=1}^{l} c_{in} y_{i}) \\\\&>(s + \\frac{\\varepsilon }{l} ) (\\det (0, \\sum \\limits _{i=1}^{l} c_{i1} x_{i}, \\dots , \\sum \\limits _{i=1}^{l} c_{in} x_{i}) )^{-1}\\sup \\limits _{y_{1} \\in E_1, \\dots , y_{l} \\in E_l}\\det (0, \\sum \\limits _{i=1}^{l} c_{i1} y_{i}, \\dots , \\sum \\limits _{i=1}^{l} c_{in} y_{i}) \\\\&>s,$ which gives a contradiction.",
"That completes the proof of claim.",
"Therefore, (2.19)-(2.20) hold.",
"Remark 2.6 We use a counterexample to show that Theorem 2.5 is false if $\\det (A)=0$ .",
"Let $f_1=\\chi _{A}$ , $f_2= \\chi _{B}$ where $A, B$ are disjoint measurable sets in $\\mathbb {R}^{2}$ with non-zero measure.",
"Obviously, $\\sup \\limits _{y_1, y_2 \\in \\mathbb {R}^{2}} f_1 (y_1 +y_2) f_2 (y_1 +y_2) \\det (0, y_{1}, y_{2})=0,$ while $\\sup \\limits _{y_1, y_2} f_{1}^{\\ast } (y_1 +y_2) f_{2}^{\\ast } (y_1 +y_2) \\det (0, y_{1}, y_{2}) \\ne 0.$ Likewise, for the same sets $A, B$ above, let $f_1=\\chi _{A}$ , $f_2=f_3= \\chi _{B}$ .",
"Then $\\sup \\limits _{y_1, y_2, y_3 \\in \\mathbb {R}^{2}} f_1 (y_1 +y_2+y_3) f_2 (y_1 +y_2+y_3)f_ 3(y_3) \\det (y_{1}, y_{2}, y_{3})=0,$ while $\\sup \\limits _{y_1, y_2, y_3 \\in \\mathbb {R}^{2}} f_{1}^{\\ast } (y_1 +y_2+y_3) f_{2}^{\\ast } (y_1 +y_2+y_3) f_{3}^{\\ast }(y_3) \\det (y_{1}, y_{2}, y_{3}) \\ne 0.$ Let $A=I$ .",
"From Theorem 2.5 it is straightforward to see that $\\displaystyle {\\sup _{y_{j}}} \\prod _{j=1}^{n} f_{j}^{\\ast }(y_{j}) \\det (0, y_{1}, \\dots , y_{n})\\le \\displaystyle {\\sup _{y_{j}}} \\prod _{j=1}^{n} f_{j}(y_{j}) \\det (0, y_{1}, \\dots , y_{n}), \\qquad \\mathrm {(2.25)}$ $\\displaystyle {\\sup _{y_{j}}} \\prod _{j=1}^{n+1} f_{j}^{\\ast }(y_{j}) \\det (y_{1}, \\dots , y_{n+1})\\le \\displaystyle {\\sup _{y_{j}}} \\prod _{j=1}^{n+1} f_{j}(y_{j}) \\det (y_{1}, \\dots , y_{n+1}).",
"\\qquad \\mathrm {(2.26)}$ Let $f_{j}= \\chi _{E_{j}}$ , and $E_j$ be measurable sets in $\\mathbb {R}^{n}$ .",
"Applying (2.25)-(2.26) we obtain the following two sharp “multilinear\" determinant inequalties suggested by the multilinear perspective of (2.2): $ \\displaystyle {\\prod _{j=1}^{n}} \\ |E_{j}|^{\\frac{1}{n}} \\le A_n\\sup \\limits _{y_{1} \\in E_{1}, \\dots , y_{n} \\in E_{n} } \\det (0, y_{1}, \\dots , y_{n}), \\qquad \\mathrm {(2.27)}$ and $ \\displaystyle {\\prod _{j=1}^{n+1}} \\ |E_{j}|^{\\frac{1}{n+1}} \\le B_n\\sup \\limits _{y_{1} \\in E_{1}, \\dots , y_{n+1} \\in E_{n+1} } \\det (y_{1}, \\dots , y_{n+1}).",
"\\qquad \\mathrm {(2.28)}$ Moreover, they are both extremised by balls centred at 0.",
"It follows from (2.25)-(2.26) that we also obtain the optimisers for (1.7) and (1.8) which is the special case when $E_{j}=E$ .",
"It should be pointed out that (2.25)-(2.26) improves multilinear rearrangement inequalities (2.29), (2.30) given in [4].",
"For each $1 \\le i \\le n$ $\\displaystyle {\\sup _{y_{j}}} \\prod _{j=1}^{n} f_{j}^{\\ast i}(y_{j}) \\det (0, y_{1}, \\dots , y_{n})\\le \\displaystyle {\\sup _{y_{j}}} \\prod _{j=1}^{n} f_{j}(y_{j}) \\det (0, y_{1}, \\dots , y_{n}), \\qquad \\mathrm {(2.29)}$ and $\\displaystyle {\\sup _{y_{j}}} \\prod _{j=1}^{n+1} f_{j}^{\\ast i}(y_{j}) \\det (y_{1}, \\dots , y_{n+1})\\le \\displaystyle {\\sup _{y_{j}}} \\prod _{j=1}^{n+1} f_{j}(y_{j}) \\det (y_{1}, \\dots , y_{n+1}), \\qquad \\mathrm {(2.30)}$ where $f_{j}^{\\ast i}$ is the Steiner symmetrisation of $f_j$ with respect to the $i$ -th coordinate.",
"Finally we give the best constant of inequality (2.1) mainly applying the Brascamp-Lieb-Luttinger rearrangement inequality.",
"In 1974, Brascamp, Lieb and Luttinger [3] proved the following inequality (2.31) which is a generalisation of Riesz's rearrangement inequality [14].",
"Let $f_{j}$ be nonnegative measurable functions on $\\mathbb {R}^{n}$ that vanish at infinity, $j=1, \\dots , m$ .",
"Let $k \\le m$ and let $B=\\lbrace b_{ij}\\rbrace $ be a $k\\times m$ matrix with $1 \\le i \\le k$ , $1 \\le j \\le m$ .",
"Define $I(f_{1}, \\dots , f_{m}):= \\int _{(\\mathbb {R}^{n})^{k}} \\prod \\limits _{j=1}^{m} f_{j}(\\sum \\limits _{i=1}^{k} b_{ij}x_{i}) dx_{1} \\dots dx_{k}.$ Then $I(f_{1}, \\dots , f_{m}) \\le I(f_{1}^{\\ast }, \\dots , f_{m}^{\\ast }).",
"\\qquad \\mathrm {(2.31)}$ Theorem 2.7 Let $f_{j}$ be nonnegative measurable functions vanishing at infinity on $\\mathbb {R}^{n}$ , Define $J(f_1, \\dots , f_{n+1})=\\int _{(\\mathbb {R}^{n})^{n}} \\prod _{j=1}^{n} f_{j}(y_{j}) f_{n+1} (\\det (0, y_{1}, \\dots , y_{n})) dy_{1} \\dots dy_{n}$ and $G(f_1, \\dots , f_{n+2})= \\int _{(\\mathbb {R}^{n})^{n+1}} \\prod _{j=1}^{n+1} f_{j}(y_{j}) f_{n+2} (\\det (y_{1}, \\dots , y_{n+1})) dy_{1} \\dots dy_{n+1}$ Then $J(f_1, \\dots , f_{n+1}) \\le J(f_{1}^{\\ast }, \\dots , f_{n+1}^{\\ast }), \\qquad \\mathrm {(2.32)}$ and $G(f_1, \\dots , f_{n+2}) \\le G(f_{1}^{\\ast }, \\dots , f_{n+2}^{\\ast }).",
"\\qquad \\mathrm {(2.33)}$ By the layer cake representation, it suffices to show that for any $E_{j}$ of finite volume in $\\mathbb {R}^{n}$ , $1 \\le j \\le n+2$ , $J(E_{1}, \\dots , E_{n+1}) \\le J(E_{1}^{\\ast }, \\dots , E_{n+1}^{\\ast }), \\ \\ G(E_{1}, \\dots , E_{n+2}) \\le G(E_{1}^{\\ast }, \\dots , E_{n+2}^{\\ast }).",
"$ For any measurable $F_{j} \\subset \\mathbb {R}$ , $1 \\le j \\le n+1$ , Brascamp-Lieb-Luttinger rearrangement inequality implies that $& \\ \\ \\ \\int _{(\\mathbb {R}^{n})^{n}} \\prod \\limits _{j=1}^{n} \\chi _{F_j}(x_j) \\chi _{F_{n+1}} (\\sum \\limits _{j=1}^{n} a_{j}x_{j} ) dx_{1} \\dots dx_{n} \\\\&\\le \\int _{(\\mathbb {R}^{n})^{n}} \\prod \\limits _{j=1}^{n} \\chi _{F_{j}^{\\ast }}(x_j) \\chi _{F_{n+1}^{*}} (\\sum \\limits _{j=1}^{n} a_{j}x_{j} ) dx_{1} \\dots dx_{n}.$ As before, since $\\det (0, y_{1}, \\dots , y_{n})$ is the linear combination of $y_{11}, \\dots , y_{n1}$ , similar to the proof of (2.10) we have $J(E_{1}, \\dots , E_{n+1}) \\le J(\\mathcal {S}_{e_1}(E_1), \\dots , \\mathcal {S}_{e_1}(E_{n+1}) ).",
"\\qquad \\mathrm {(2.34)}$ Note that $J(E_{1}, \\dots , E_{n+1})$ is invariant under $O(n)$ .",
"By the property of $\\mathcal {S}_{\\rho e_{i}}(E)=\\rho \\circ \\mathcal {S}_{e_{i}}(\\rho ^{-1}(E)),$ we obtain for any $u \\in \\mathbb {S}^{n-1}$ that is a unit vector in $\\mathbb {R}^{n}$ , $J(E_{1}, \\dots , E_{n+1}) \\le J(\\mathcal {S}_{u}(E_1), \\dots , \\mathcal {S}_{u}(E_{n+1}) ).",
"\\qquad \\mathrm {(2.35)}$ Likewise, since $\\det ( y_{1}, \\dots , y_{n+1})$ can be seen as the linear combination of $y_{11}, \\dots , y_{(n+1)1}$ , and the Brascamp-Lieb-Luttinger rearrangement inequality $& \\ \\ \\ \\int _{(\\mathbb {R}^{n})^{n}} \\prod \\limits _{j=1}^{n+1} \\chi _{F_j}(x_j) \\chi _{F_{n+2}} (\\sum \\limits _{j=1}^{n+1} a_{j}x_{j} ) dx_{1} \\dots dx_{n+1} \\\\&\\le \\int _{(\\mathbb {R}^{n})^{n}} \\prod \\limits _{j=1}^{n+1} \\chi _{F_{j}^{\\ast }}(x_j) \\chi _{F_{n+2}^{*}} (\\sum \\limits _{j=1}^{n+1} a_{j}x_{j} ) dx_{1} \\dots dx_{n+1},$ we also have $G(E_{1}, \\dots , E_{n+2}) \\le G(\\mathcal {S}_{e_1}(E_1), \\dots , \\mathcal {S}_{e_1}(E_{n+2})).",
"\\qquad \\mathrm {(2.36)}$ Hence by (2.14) together with the invariance of $G(E_{1}, \\dots , E_{n+2})$ $G(E_{1}, \\dots , E_{n+2}) \\le G(\\mathcal {S}_{u}(E_1), \\dots , \\mathcal {S}_{u}(E_{n+2})).",
"\\qquad \\mathrm {(2.37)}$ Let $H$ be the semigroup of all finite products of $\\mathcal {S}_{u}$ 's.",
"Brascamp, Lieb and Luttinger [3] proved for any bounded measurable $E \\subset \\mathbb {R}^{n}$ , there exists $\\lbrace h_m\\rbrace _{m=0}^{\\infty } \\subset G$ such that $E_{m}:=h_m (E)$ converges to $E^{\\ast }$ in symmetric difference.",
"That is, $\\lim \\limits _{m \\rightarrow \\infty } |E_m \\triangle E^{\\ast }| =0, \\qquad \\mathrm {(2.38)}$ where $\\triangle $ denotes the symmetric difference of two sets.",
"Here we sketch the sequence of sets $\\lbrace E_m\\rbrace $ .",
"Let $E_{0}=h_0 E=E$ .",
"Given $E_{m}$ , choose unit vector $u_{1}$ such that $|\\mathcal {S}_{u_1} (E_m) \\triangle E^{\\ast }| < \\inf \\limits _{u \\in \\mathbb {S}^{n-1}} |\\mathcal {S}_{u} (E_m) \\triangle E^{\\ast }| + \\frac{1}{m}.$ Hence we select $u_2$ , ..., $u_n \\in \\mathbb {S}^{n-1}$ such that $\\lbrace u_1, \\dots , u_n\\rbrace $ becomes an orthonormal basis in $\\mathbb {R}^{n}$ , and then construct $E_{m+1}=h_{m+1} (E)= \\mathcal {S}_{u_n} \\mathcal {S}_{u_{n-1}} \\dots \\mathcal {S}_{u_1} (E_m).$ The sequence of sets $\\lbrace E_m\\rbrace $ constructed above converges to $E^{\\ast }$ in symmetric difference.",
"See [3] for the detailed proof.",
"Therefore, we apply the convergence of Steiner symmetrisation together with (2.35) and (2.37) to conclude $J(E_1, \\dots , E_{n+1}) \\le J(E_{1}^{\\ast }, \\dots , E_{n+1}^{\\ast }),$ and $G(E_1, \\dots , E_{n+2}) \\le G(E_{1}^{\\ast }, \\dots , E_{n+2}^{\\ast }).",
"$ Lastly, applying the layer cake representation for $f_j$ together with Fubini's theorem gives $J(f_1, \\dots , f_{n+1})=\\int _{0}^{\\infty } \\dots \\int _{0}^{\\infty }J( \\chi _{\\lbrace f_1 >t_1\\rbrace }, \\dots , \\chi _{\\lbrace f_{n+1} >t_{n+1}\\rbrace })dt_{1} \\dots dt_{n+1}.$ Since (2.32)-(2.33) hold for characteristic functions of sets of finite Lebesgue measure, for any $t_{j}$ , $1\\le j \\le n+1$ $J( \\chi _{\\lbrace f_1 >t_1\\rbrace }, \\dots , \\chi _{\\lbrace f_{n+1} >t_{n+1}\\rbrace })\\le J( \\chi _{\\lbrace f_1 >t_1\\rbrace }^{\\ast }, \\dots , \\chi _{\\lbrace f_{n+1} >t_{n+1}\\rbrace }^{\\ast }).",
"\\qquad \\mathrm {(2.39)}$ Thus $J(f_1, \\dots , f_{n+1}) &\\le \\int _{0}^{\\infty } \\dots \\int _{0}^{\\infty }J( \\chi _{\\lbrace f_1 >t_1\\rbrace }^{\\ast }, \\dots , \\chi _{\\lbrace f_{n+1} >t_{n+1}\\rbrace }^{\\ast }) dt_{1} \\dots dt_{n+1} \\\\&= J(f_{1}^{\\ast }, \\dots , f_{n+1}^{\\ast }).$ Similarly, $G(f_1, \\dots , f_{n+2}) \\le G(f_{1}^{\\ast }, \\dots , f_{n+2}^{\\ast }).$ This completes Theorem 2.7.",
"Let $f_j= \\chi _{E_{j}}$ , $1 \\le j \\le n$ , and $f_{n+1}=\\chi _{ (| \\cdot | < \\delta )}$ .",
"Theorem 2.7 gives $& \\ \\ \\ \\ \\ |\\lbrace (y_{1}, \\dots , y_{n})\\in E_{1} \\times \\cdots \\times E_{n}:\\det ( 0, y_{1}, \\dots , y_{n}) < \\delta \\rbrace | \\\\&\\le | \\lbrace (y_{1}, \\dots , y_{n})\\in E_1^{\\ast } \\times \\cdots \\times E_n^{\\ast } : \\det (0, y_{1}, \\dots , y_{n}) < \\delta \\rbrace |.$ This implies that inequality (2.1) is extremised by balls centred at $y$ , where $y\\in \\mathbb {R}^{n}$ .",
"Let $f_{n+2}=| \\cdot |^{-1}$ , then Theorem 2.7 implies $& \\ \\ \\ \\ \\ \\ \\int _{(\\mathbb {R}^{n})^{n+1}} \\prod _{j=1}^{n+1} f_{j}(y_{j}) \\det (y_{1}, \\dots , y_{n+1})^{-1} dy_{1} \\dots dy_{n+1} \\\\&\\le \\int _{(\\mathbb {R}^{n})^{n+1}} \\prod _{j=1}^{n+1} f_{j}^{\\ast }(y_{j}) \\det (y_{1}, \\dots , y_{n+1})^{-1} dy_{1} \\dots dy_{n+1}.$"
],
[
" Matrix inequalities",
"Now we turn to see the analogues of (1.5) and (1.6) replacing the Euclidean space $\\mathbb {R}^{n}$ by the space of $n \\times n$ real matrices.",
"We remark that the proof of Theorem 3.1 mainly relies on the rearrangement inequality (2.6) and an invariance under the action of $O(n)$ by premultiplication as described in the introduction.",
"Theorem 3.1 There exists a finite constant $\\mathcal {C}_{n}$ such that for any measurable set $E_j \\subset \\mathfrak {M}^{n \\times n}$ of finite measure, $j=1, \\dots , n$ , $ \\prod \\limits _{j=1}^{n}|E_j|^{\\frac{1}{n^{2}}}\\le \\mathcal {C}_{n} \\sup \\limits _{\\begin{array}{c}A_{j} \\in E_{j} \\\\ j=1, \\dots , n\\end{array}} | \\det (A_{1}+ \\dots + A_{n} ) |, \\qquad \\mathrm {(3.1)}$ where $|\\cdot |$ denotes the Lebesgue measure on Euclidean space $\\mathbb {R}^{n^{2}}$ and the absolute value on $\\mathbb {R}$ .",
"Suppose $\\sup \\limits _{\\begin{array}{c}A_{j} \\in E_{j} \\\\ j=1, \\dots , n\\end{array}} | \\det (A_{1}+ \\dots + A_{n} ) |=s < \\infty .$ First we give some definition and notation.",
"Let $F \\subset \\mathfrak {M}^{n \\times m}$ , define $v(F)=\\lbrace \\left(\\begin{array}{cccc}a_{11} & a_{21} & \\dots & a_{(m-1)1} \\\\\\vdots & \\vdots & \\ & \\vdots \\\\a_{1n} & a_{2n} & \\dots & a_{(m-1)n}\\end{array}\\right): \\exists \\ \\left(\\begin{array}{c}a_{m1} \\\\ \\vdots \\\\ a_{mn}\\end{array} \\right)\\ \\mathrm {such \\ that} \\ \\left(\\begin{array}{ccc}a_{11} & \\dots & a_{m1} \\\\\\vdots & \\ & \\vdots \\\\a_{1n} & \\dots & a_{mn}\\end{array} \\right)\\in F \\rbrace ,$ so $ v(F) \\subset \\mathfrak {M}^{n \\times (m-1)}$ .",
"For any $n$ -by-$(m-1)$ matrix $x=\\left(\\begin{array}{cccc}a_{11} & a_{21} & \\dots & a_{(m-1)1} \\\\\\vdots & \\vdots & \\ & \\vdots \\\\a_{1n} & a_{2n} & \\dots & a_{(m-1)n}\\end{array}\\right) \\in v(F),$ we denote $F^{x}=\\lbrace \\left(\\begin{array}{c}a_{m1} \\\\ \\vdots \\\\ a_{mn}\\end{array} \\right): \\ \\left(\\begin{array}{ccc}a_{11} & \\dots & a_{m1} \\\\\\vdots & \\ & \\vdots \\\\a_{1n} & \\dots & a_{mn}\\end{array} \\right)\\in F \\rbrace \\subset \\mathfrak {M}^{n \\times 1}.$ Let $E \\subset \\mathfrak {M}^{n \\times n}$ .",
"For any rotation around the origin $T$ in $\\mathbb {R}^{n}$ , consider $\\Phi _{T} : A \\mapsto TA, \\ \\forall \\ A \\in E,$ where $T$ is a $n$ -by-$n$ matrix with $\\det (T)=1$ .",
"Note that $\\Phi _{T}$ does not change $|E|$ and $\\sup \\limits _{A \\in E} |\\det (A)|$ .",
"This is because $\\sup \\limits _{A \\in \\Phi _{T}(E)} |\\det (A)|= \\sup \\limits _{A \\in E} |\\det (TA)|= \\sup \\limits _{A \\in E} |\\det (A)|.",
"\\qquad \\mathrm {(3.2)}$ Besides, if we see the matrix $A= \\left(\\begin{array}{ccc}a_{11} & \\dots & a_{n1} \\\\\\vdots & \\ & \\vdots \\\\a_{1n} & \\dots & a_{nn}\\end{array} \\right) \\in E$ as a vector $(a_{11}, \\dots , a_{1n}, a_{21}, \\dots , a_{2n}, \\dots , a_{n1}, \\dots , a_{nn}) \\in \\mathbb {R}^{n^{2}},$ then the matrix $\\Phi _{T}(A)$ becomes $\\left(\\begin{array}{cccc}T & \\ & \\ & \\ \\\\\\ & T & \\ & \\ \\\\\\ & \\ & \\ddots & \\ \\\\\\ & \\ & \\ & T\\end{array} \\right)\\left(\\begin{array}{c}a_{11} \\\\ \\vdots \\\\ a_{nn}\\end{array} \\right).$ Thus $|\\Phi _{T}(E)|=|T|^{n}|E|=|E|.",
"\\qquad \\mathrm {(3.3)}$ From $|E|=\\int _{v(E)}|E^{x}|dx$ it follows that there always exists $\\overline{x} \\in v(E)$ such that $|v(E)| |E^{\\overline{x}}| \\gtrsim _n |E|.",
"\\qquad \\mathrm {(3.4)}$ By John Ellipsoid, for any compact convex $G\\subset \\mathbb {R}^{n}$ there exists an ellipsoid $G^{\\prime } \\subset G$ such that $|G^{\\prime }| \\gtrsim _n |G|.",
"\\qquad \\mathrm {(3.5)}$ For the John ellipsoid $G^{\\prime }$ , we choose a rotation $T \\in O(n)$ such that $T G^{\\prime }$ is an ellipsoid with principal axes parallel to the coordinate axes.",
"As well known, for every ellipsoid $T G^{\\prime }$ with principal axes parallel to the coordinate axes, there exists an axis-parallel rectangle $H \\subset T G^{\\prime } $ such that $|H| \\gtrsim _n |T G^{\\prime }|.",
"\\qquad \\mathrm {(3.6)}$ Hence if $E^{\\overline{x}}$ is convex, from (3.5)-(3.6) we may assume that there exists $T \\in O(n)$ such that $E^{\\overline{x}}$ is an axis-parallel rectangle in $\\mathbb {R}^{n}$ .",
"Take $n=2$ .",
"By (3.4) there exists $x_{10}\\in v(E_{1}) \\subset \\mathfrak {M}^{2 \\times 1}$ , $x_{20}\\in v(E_{2}) \\subset \\mathfrak {M}^{2 \\times 1}$ such that $|v(E_{1})| |E_{1}^{x_{10}}| \\gtrsim |E_1|, \\ \\ |v(E_{2})| |E_{2}^{x_{20}}| \\gtrsim |E_2|.",
"\\qquad \\mathrm {(3.7)}$ Then $\\max \\lbrace |v(E_{2})| |E_{1}^{x_{10}}|, |v(E_{1})| |E_{2}^{x_{20}}| \\rbrace \\gtrsim (|E_1||E_2|)^{1/2}.$ For simplicity, suppose $|v(E_{2})| |E_{1}^{x_{10}}| \\gtrsim (|E_1||E_2|)^{1/2}.",
"\\qquad \\mathrm {(3.8)}$ To study the suprema, we consider 2-by-2 matrix $\\overline{A}_{1}:=\\left(\\begin{array}{cc}(x_{10})_1 & (x_{10})_2 \\\\\\end{array}\\right) \\in E_{1}$ with $(x_{10})_1 =x_{10} \\in \\mathfrak {M}^{n \\times 1}$ and $(x_{10})_2 \\in E_{1}^{x_{10}}$ .",
"For any $\\overline{A}_{2}:=\\left(\\begin{array}{cc}x_1 & x_2 \\\\\\end{array}\\right) \\in E_{2}$ , for any constructed $\\overline{A}_{1}$ above $s &\\ge | \\det (\\overline{A}_{1} + \\overline{A}_{2}) | \\\\&=| \\det \\left(\\begin{array}{ccc}x_1+ (x_{10})_1 & x_2+ (x_{10})_2 \\\\\\end{array}\\right)|.$ So fix the first column, we have for any $x_1 \\in v(E_2)$ , $x_2 \\in E_{2}^{x_1}$ $s \\ge \\sup \\limits _{(x_{10})_{2} \\in E_{1}^{x_{10}}}| \\det \\left(\\begin{array}{cc}x_1+ (x_{10})_1 & x_2+ (x_{10})_2 \\\\\\end{array}\\right)|.",
"\\qquad \\mathrm {(3.9)}$ Because fix all the columns except one, the $|\\det |$ function is convex function of the remaining column.",
"Thus $s \\ge \\sup \\limits _{(x_{10})_{2} \\in \\mathrm {co}E_{1}^{x_{10}}}| \\det \\left(\\begin{array}{cc}x_1+ (x_{10})_1 & x_2+ (x_{10})_2 \\\\\\end{array}\\right)|.",
"\\qquad \\mathrm {(3.10)}$ By (3.5) we may assume $\\mathrm {co}E_{1}^{x_{10}}$ is an ellipsoid in $\\mathbb {R}^{2}$ .",
"Choose a rotation $T_{0} \\in O(2)$ such that $T_{0} \\mathrm {co}E_{1}^{x_{10}}$ is an ellipsoid with principal axes parallel to the coordinate axes.",
"From (3.6) we may assume $T_{0} \\mathrm {co}E_{1}^{x_{10}}$ is an axis-parallel rectangle.",
"Note that (3.10) is invariant under $O(2)$ as discussed in (3.2), so $\\begin{array}{ll}s &\\ge \\sup \\limits _{(x_{10})_{2} \\in \\mathrm {co}E_{1}^{x_{10}}}| \\det \\left(\\begin{array}{cc}x_1+ (x_{10})_1 & x_2+ (x_{10})_2 \\\\\\end{array}\\right)| \\\\&=\\sup \\limits _{(x_{10})_{2} \\in \\mathrm {co}E_{1}^{x_{10}}}| \\det \\left(\\begin{array}{cc}T_{0} x_1+ T_0 (x_{10})_1 & T_0 x_2+ T_0 (x_{10})_2 \\\\\\end{array}\\right)|.\\end{array}\\qquad \\mathrm {(3.11)}$ Since $T_{0} \\mathrm {co}E_{1}^{x_{10}}$ is an axis-parallel rectangle in $\\mathbb {R}^{2}$ , it can be written as $A_1 \\times A_2$ , where $A_1, A_2$ are intervals in $\\mathbb {R}$ , and then $\\mathcal {S}(T_{0} \\mathrm {co}E_{1}^{x_{10}})=\\mathcal {S}(T_{0} \\mathrm {co}E_{1}^{x_{10}}+ T_0 x_2 )=A_{1}^{\\ast } \\times A_{2}^{\\ast }, \\ \\forall \\ x_2\\in E_{2}^{x_1}.$ Similar to the proof of (2.10), applying (2.6) gives for any $x_1 \\in v(E_{2})$ $s \\ge \\sup \\limits _{(x_{10})_{2} \\in \\mathcal {S} (T_0 \\mathrm {co}E_{1}^{x_{10}})}| \\det \\left(\\begin{array}{cc}T_{0} x_1+ T_0 (x_{10})_1 & (x_{10})_{2} \\\\\\end{array}\\right)|.",
"\\qquad \\mathrm {(3.12)}$ Therefore, by (2.2) we deduce that $s \\ge C |T_{0} v(E_{2})+ T_0 (x_{10})_1 |^{1/2} |\\mathcal {S} (T_0 \\mathrm {co}E_{1}^{x_{10}})|^{1/2}= C |v(E_{2})|^{1/2} |\\mathrm {co}E_{1}^{x_{10}}|^{1/2}.$ This together with (3.8) implies $s \\ge C |v(E_{2})|^{1/2} |\\mathrm {co}E_{1}^{x_{10}}|^{1/2} \\ge C |v(E_{2})|^{1/2} |E_{1}^{x_{10}}|^{1/2} \\ge C(|E_1||E_2|)^{1/2},$ which completes (3.1) for $n=2$ .",
"Take $n=3$ .",
"By (3.4) for each $E_{j}$ there exists $x_{j0} \\in v(E_{j}) \\subset \\mathfrak {M}^{3 \\times 2}$ such that $|v(E_{j})| |E_{j}^{x_{j0}}| \\gtrsim |E_j|, 1 \\le j \\le 3.",
"\\qquad \\mathrm {(3.13)}$ Denote $F_{j}=v(E_{j}) \\subset \\mathfrak {M}^{3 \\times 2}$ , there exists fixed $x_{j1} \\in v(F_{j}) \\subset \\mathfrak {M}^{3 \\times 1}$ such that $|v(F_{j})| |F_{j}^{x_{j1}}| \\gtrsim |F_{j}| =v(E_{j}).",
"\\qquad \\mathrm {(3.14)}$ From (3.13)-(3.14), we have $x_{j0} \\in v(E_{j}) , x_{j1} \\in v(F_{j})$ $|v(F_{j})| |F_{j}^{x_{j1}}| |E_{j}^{x_{j0}}| \\gtrsim |E_j|, 1 \\le j \\le 3.",
"\\qquad \\mathrm {(3.15)}$ It is not hard to see there exists $\\lbrace i_1, i_2, i_3\\rbrace $ with $i_1 \\ne i_2 \\ne i_3$ such that $(v(F_{i_3})| |F_{i_2}^{x_{i_2 1}}| |E_{i_1}^{x_{i_1 0}}| )^{3}\\ge \\prod \\limits _{j=1}^{3} (|v(F_{j})| |F_{j}^{x_{j1}}| |E_{j}^{x_{j0}}|)\\gtrsim \\prod \\limits _{j=1}^{3} |E_j|.",
"\\qquad \\mathrm {(3.16)}$ For simplicity, suppose $|v(F_{3})| |F_{2}^{x_{21}}| |E_{1}^{x_{10}}| \\gtrsim (|E_1||E_2||E_3|)^{1/3}.",
"\\qquad \\mathrm {(3.17)}$ Now we consider 3-by-3 matrices $\\overline{A}_{1}:=\\left(\\begin{array}{ccc}(x_{10})_1 & (x_{10})_2 & (x_{10})_3 \\\\\\end{array}\\right) \\in E_{1}$ with $\\left(\\begin{array}{cc}(x_{10})_1 & (x_{10})_2 \\\\\\end{array}\\right) =x_{10} \\in \\mathfrak {M}^{3 \\times 2}$ and $(x_{10})_3 \\in E_{1}^{x_{10}}$ ; $\\overline{A}_{2}:=\\left(\\begin{array}{ccc}(x_{21})_1 & (x_{21})_2 & (x_{21})_3 \\\\\\end{array}\\right) \\in E_{2}$ with the condition $(x_{21})_1 =x_{21} \\in \\mathfrak {M}^{3 \\times 1}$ and $(x_{21})_2 \\in F_{2}^{x_{21}}$ .",
"For any $\\overline{A}_{3}:=\\left(\\begin{array}{ccc}x_1 & x_2 & x_3 \\\\\\end{array}\\right) \\in E_{3}$ , for any constructed $\\overline{A}_{1}, \\overline{A}_{2}$ above, $s &\\ge | \\det (\\overline{A}_{1} + \\overline{A}_{2}+ \\overline{A}_{3} ) | \\\\&=| \\det \\left(\\begin{array}{ccc}x_1+ (x_{10})_1+ (x_{21})_1 & x_2+ (x_{10})_2 + (x_{21})_2 & x_3+ (x_{10})_3 + (x_{21})_3 \\\\\\end{array}\\right)|.$ So fix all columns except the 3rd column, we have $s \\ge \\sup \\limits _{(x_{10})_{3} \\in E_{1}^{x_{10}}}| \\det \\left(\\begin{array}{ccc}x_1+ (x_{10})_1+ (x_{21})_1 & x_2+ (x_{10})_2 + (x_{21})_2 & x_3+ (x_{10})_3 + (x_{21})_3 \\\\\\end{array}\\right)|.$ Obviously, $s \\ge \\sup \\limits _{(x_{10})_{3} \\in \\mathrm {co}E_{1}^{x_{10}}}| \\det \\left(\\begin{array}{ccc}x_1+ (x_{10})_1+ (x_{21})_1 & x_2+ (x_{10})_2 + (x_{21})_2 & x_3+ (x_{10})_3 + (x_{21})_3 \\\\\\end{array}\\right)|.",
"$ As before, by (3.5) we assume there exists $T_{0} \\mathrm {co}E_{1}^{x_{10}}$ is an ellipsoid with principal axes parallel to the coordinate axes in $\\mathbb {R}^{3}$ .",
"From (3.6) we may assume $T_{0} \\mathrm {co}E_{1}^{x_{10}}$ is an axis-parallel rectangle.",
"Because of the invariance under $O(3)$ , $s &\\ge \\sup \\limits _{(x_{10})_{3} \\in \\mathrm {co}E_{1}^{x_{10}}}| \\det \\left(\\begin{array}{ccc}x_1+ (x_{10})_1+ (x_{21})_1 & x_2+ (x_{10})_2 + (x_{21})_2 & x_3+ (x_{10})_3 + (x_{21})_3 \\\\\\end{array}\\right)| \\\\&=\\sup \\limits _{(x_{10})_{3} \\in \\mathrm {co}E_{1}^{x_{10}}}| \\det \\left(\\begin{array}{ccc}T_{0}(x_1+ (x_{10})_1+ (x_{21})_1) & T_{0}( x_2+ (x_{10})_2 + (x_{21})_2 ) & T_{0} (x_3+ (x_{10})_3 + (x_{21})_3) \\\\\\end{array}\\right)|.$ Since $T_{0} \\mathrm {co}E_{1}^{x_{10}}$ is an axis-parallel rectangle in $\\mathbb {R}^{3}$ , it can be written as $A_1 \\times A_2 \\times A_3$ , where $A_1, A_2, A_3$ are intervals in $\\mathbb {R}$ .",
"Similar to the proof of (2.10) together with $\\mathcal {S}(T_{0} \\mathrm {co}E_{1}^{x_{10}})=\\mathcal {S}(T_{0} \\mathrm {co}E_{1}^{x_{10}}+h)=A_{1}^{\\ast } \\times A_{2}^{\\ast } \\times A_{3}^{\\ast }, \\ \\forall \\ h \\in \\mathbb {R}^{3},$ applying (2.6) gives for any $\\left(\\begin{array}{cc}x_1 & x_2 \\\\\\end{array}\\right) \\in v(E_{3})$ , $s \\ge \\sup \\limits _{(x_{10})_{3} \\in \\mathcal {S} (T_0 \\mathrm {co}E_{1}^{x_{10}})}| \\det \\left(\\begin{array}{ccc}T_{0}(x_1+ (x_{10})_1+ (x_{21})_1) & T_{0}( x_2+ (x_{10})_2 + (x_{21})_2 ) & (x_{10})_{3} \\\\\\end{array}\\right)|.",
"$ Then fix all columns except the 2nd column, $s \\ge \\sup \\limits _{(x_{21})_{2} \\in F_{2}^{x_{21}}}| \\det \\left(\\begin{array}{ccc}T_{0}(x_1+ (x_{10})_1+ (x_{21})_1) & T_{0}( x_2+ (x_{10})_2 + (x_{21})_2 ) & (x_{10})_{3} \\\\\\end{array}\\right)| $ holds for any $(x_{10})_{3} \\in \\mathcal {S} (T_0 \\mathrm {co}E_{1}^{x_{10}})$ .",
"Similarly, by the convex property of $|\\det |$ function when fixing other columns $s \\ge \\sup \\limits _{(x_{21})_{2} \\in \\mathrm {co}F_{2}^{x_{21}}}| \\det \\left(\\begin{array}{ccc}T_{0}(x_1+ (x_{10})_1+ (x_{21})_1) & T_{0}( x_2+ (x_{10})_2 + (x_{21})_2 ) & (x_{10})_{3} \\\\\\end{array}\\right)|.$ By (3.5) we may assume $T_0 \\mathrm {co}F_{2}^{x_{21}}$ is an ellipsoid in $\\mathbb {R}^{3}$ .",
"Choose a rotation $T_{1} \\in O(3)$ such that $T_1 T_0 \\mathrm {co}F_{2}^{x_{21}}$ is an ellipsoid with principal axes parallel to the coordinate axes.",
"From (3.6) we may assume $T_1 T_0 \\mathrm {co}F_{2}^{x_{21}}$ is an axis-parallel rectangle.",
"By the invariance of $O(3)$ , $s &\\ge \\sup \\limits _{(x_{21})_{2} \\in \\mathrm {co}F_{2}^{x_{21}}}| \\det \\left(\\begin{array}{ccc}T_{0}(x_1+ (x_{10})_1+ (x_{21})_1) & T_{0}( x_2+ (x_{10})_2 + (x_{21})_2 ) & (x_{10})_{3} \\\\\\end{array}\\right)| \\\\&= \\sup \\limits _{(x_{21})_{2} \\in \\mathrm {co}F_{2}^{x_{21}}}| \\det \\left(\\begin{array}{ccc}T_1 T_{0}(x_1+ (x_{10})_1+ (x_{21})_1) & T_1 T_{0}( x_2+ (x_{10})_2 + (x_{21})_2 ) & T_1 (x_{10})_{3} \\\\\\end{array}\\right)|.$ Since $T_1 T_0 \\mathrm {co}F_{2}^{x_{21}}$ is an axis-parallel rectangle, together with $\\mathcal {S}(T_1 T_{0} \\mathrm {co}F_{2}^{x_{21}})=\\mathcal {S}(T_1 T_{0} \\mathrm {co}F_{2}^{x_{21}}+h), \\ \\ \\forall \\ h \\in \\mathbb {R}^{3}$ apply inequality (2.6) again to obtain $s \\ge \\sup \\limits _{\\begin{array}{c} (x_{10})_{3} \\in \\mathcal {S} (T_0 \\mathrm {co}E_{1}^{x_{10}}) \\\\ (x_{21})_{2} \\in \\mathcal {S} (T_1 T_0 \\mathrm {co}F_{2}^{x_{21}})\\end{array}}| \\det \\left(\\begin{array}{ccc}T_1 T_{0}(x_1+ (x_{10})_1+ (x_{21})_1) & (x_{21})_2 & T_1 (x_{10})_{3} \\\\\\end{array}\\right)| $ holds for any $x_1 \\in v(F_3) \\subset \\mathfrak {M}^{3 \\times 1}$ .",
"Lastly, applying (2.2) we conclude $s &\\ge C |T_1 T_{0} v(F_3)+T_1 T_{0} (x_{10})_1+T_1 T_{0} (x_{21})_1|^{1/3} | \\mathcal {S} (T_1 T_0 \\mathrm {co}F_{2}^{x_{21}})|^{1/3} |T_1 \\mathcal {S} (T_0 \\mathrm {co}E_{1}^{x_{10}}) |^{1/3} \\\\&= C |v(F_3)|^{1/3} |\\mathrm {co} F_{2}^{x_{21}})|^{1/3} | \\mathrm {co}E_{1}^{x_{10}}) |^{1/3}.$ This together with (3.17) implies $s \\ge C |v(F_{3})|^{1/3} |\\mathrm {co}F_{2}^{x_{21}}|^{1/3} |\\mathrm {co}E_{1}^{x_{10}}|^{1/3}\\ge C |v(F_{3})|^{1/3} |F_{2}^{x_{21}}|^{1/3} |E_{1}^{x_{10}}|^{1/3}\\ge C(|E_1||E_2||E_3|)^{1/3}.$ This completes (3.1) for $n=3$ .",
"For the general $n$ , for each $E_{j}$ , denote $F_{j0}= E_{j}$ , $1 \\le j \\le n$ .",
"Given $1 \\le k \\le n-2$ , let $F_{jk}= v(F_{j(k-1)}) \\subset \\mathfrak {M}^{n \\times (n-k)}, $ then by (3.4) there exists fixed $x_{jk}\\in v(F_{jk}) \\subset \\mathfrak {M}^{n \\times (n-k-1)}$ , $0 \\le k \\le n-2$ , such that $|v(F_{jk})| |F_{jk}^{x_{jk}}| \\gtrsim |F_{jk}|=|v(F_{j(k-1)})|.",
"\\qquad \\mathrm {(3.18)}$ That is, for each $E_j$ there exist $\\lbrace x_{j0}, \\dots , x_{j(n-2)}\\rbrace $ such that for each $k=0, \\dots , n-2$ $x_{jk} \\in v(F_{jk}) \\subset \\mathfrak {M}^{n \\times (n-k-1)}, $ and $|v(F_{j(n-2)})| |F_{j(n-2)}^{x_{j(n-2)}}| |F_{j(n-3)}^{x_{j(n-3)}}| \\dots |F_{j1}^{x_{j1}}| |F_{j0}^{x_{j0}}| \\gtrsim _{n} |E_j|.",
"\\qquad \\mathrm {(3.19)}$ It is not hard to see there exist $\\lbrace i_j\\rbrace _{j=1}^{n}$ with $1 \\le i_{j} \\le n$ and $i_{j} \\ne i_{k}$ for $j \\ne k$ such that $& \\ \\ \\ \\ \\ (|v(F_{i_{n}(n-2)})| |F_{i_{n-1}(n-2)}^{x_{i_{n-1} (n-2)}}| |F_{i_{n-2} (n-3)}^{x_{i_{n-2} (n-3)}}| \\dots |F_{i_{2}1}^{x_{i_{2} 1}}| |F_{i_{1}0}^{x_{i_{1} 0}}|)^{n} \\\\& \\ge \\prod \\limits _{j=1}^{n} (|v(F_{j(n-2)})| |F_{j(n-2)}^{x_{j(n-2)}}| |F_{j(n-3)}^{x_{j(n-3)}}| \\dots |F_{j1}^{x_{j1}}| |F_{j0}^{x_{j0}}|)\\gtrsim _{n} \\prod \\limits _{j=1}^{n}|E_j|.$ For simplicity, denote $i_j=j, 1 \\le j \\le n$ .",
"That is, $|v(F_{n(n-2)})| |F_{(n-1)(n-2)}^{x_{(n-1) (n-2)}}| |F_{(n-2) (n-3)}^{x_{(n-2) (n-3)}}| \\dots |F_{2 1}^{x_{2 1}}| |F_{10}^{x_{1 0}}|\\gtrsim _{n} \\prod \\limits _{j=1}^{n}|E_j|^{1/n}.",
"\\qquad \\mathrm {(3.20)}$ To study the suprema, we consider the following $n$ -by-$n$ matrices $\\overline{A}_{1}:=\\left(\\begin{array}{ccc}(x_{10})_1 & \\dots & (x_{10})_n \\\\\\end{array}\\right) \\in E_{1}$ with $\\left(\\begin{array}{ccc}(x_{10})_1 & \\dots & (x_{10})_{(n-1)} \\\\\\end{array}\\right)=x_{10} \\in \\mathfrak {M}^{n \\times (n-1)}$ and $(x_{10})_n \\in F_{10}^{x_{10}}$ ; $\\overline{A}_{2}:=\\left(\\begin{array}{ccc}(x_{21})_1 & \\dots & (x_{21})_n \\\\\\end{array}\\right) \\in E_{2}$ with $\\left(\\begin{array}{ccc}(x_{21})_1 & \\dots & (x_{21})_{(n-2)} \\\\\\end{array}\\right)=x_{21} \\in \\mathfrak {M}^{n \\times (n-2)}$ and $(x_{21})_{n-1} \\in F_{21}^{x_{21}}$ .",
"That is, construct $\\lbrace \\overline{A}_{1}, \\dots , \\overline{A}_{n-1}\\rbrace $ such that for each $1 \\le k \\le n-1$ $\\overline{A}_{k}:=\\left(\\begin{array}{ccc}(x_{k(k-1)})_1 & \\dots & (x_{k(k-1)})_n \\\\\\end{array}\\right) \\in E_{k}, $ with the condition that $\\left(\\begin{array}{ccc}x_{k(k-1)})_1 & \\dots & (x_{k(k-1)})_{n-k} \\\\\\end{array}\\right) =x_{k(k-1)} \\in \\mathfrak {M}^{n \\times (n-k)}, \\ \\ (x_{k(k-1)})_{n-k+1} \\in F_{k(k-1)}^{x_{k(k-1)}}.$ For any $\\overline{A}_{n}:=\\left(\\begin{array}{ccc}x_1 & \\dots & x_n \\\\\\end{array}\\right) \\in E_{n}$ , for any constructed $\\overline{A}_{1}, \\dots , \\overline{A}_{n-1}$ above, $s &\\ge | \\det (\\overline{A}_{1} + \\dots + \\overline{A}_{n-1}+ \\overline{A}_{n}) | \\\\&=| \\det \\left(\\begin{array}{ccc}x_1+\\sum \\limits _{k=1}^{n-1} (x_{k(k-1)})_1 & \\dots & x_n+\\sum \\limits _{k=1}^{n-1} (x_{k(k-1)})_n \\\\\\end{array}\\right)|.$ Taking the same arguments as in the case $n=3$ , there exist $T_0, T_1 \\in O(n)$ $s \\ge \\sup \\limits _{\\begin{array}{c} (x_{10})_n \\in \\mathcal {S}(T_0 \\mathrm {co}F_{10}^{x_{10}}) \\\\ (x_{21})_{(n-1)} \\in \\mathcal {S}(T_1 T_0 \\mathrm {co}F_{21}^{x_{21}} )\\end{array}}| \\det \\left(\\begin{array}{cc}B & B^{\\prime } \\\\\\end{array}\\right)|, \\qquad \\mathrm {(3.21)}$ where $B = T_1 T_0\\left(\\begin{array}{ccc}x_1+\\sum \\limits _{k=1}^{n-1} (x_{k(k-1)})_1 & \\dots & x_{n-2}+\\sum \\limits _{k=1}^{n-1} (x_{k(k-1)})_{n-2} \\\\\\end{array}\\right) \\in \\mathfrak {M}^{n \\times (n-2)},$ $B^{\\prime } =\\left(\\begin{array}{cc}(x_{21})_{(n-1)} & T_1 (x_{10})_n \\\\\\end{array}\\right) \\in \\mathfrak {M}^{n \\times 2}.$ Applying the same arguments again to (3.21), there exist $T_2 \\in O(n)$ $s \\ge \\sup \\limits _{\\begin{array}{c} (x_{10})_n \\in \\mathcal {S}(T_0 \\mathrm {co}F_{10}^{x_{10}}) \\\\ (x_{21})_{(n-1)} \\in \\mathcal {S}(T_1 T_0 \\mathrm {co}F_{21}^{x_{21}} )\\\\ (x_{32})_{n-2} \\in \\mathcal {S}(T_2 T_1 T_0 \\mathrm {co}F_{32}^{x_{32}}) \\end{array}}| \\det \\left(\\begin{array}{cc}C & C^{\\prime } \\\\\\end{array}\\right)|, \\qquad \\mathrm {(3.22)}$ where $C = T_2 T_1 T_0\\left(\\begin{array}{ccc}x_1+\\sum \\limits _{k=1}^{n-1} (x_{k(k-1)})_1 & \\dots & x_{n-3}+\\sum \\limits _{k=1}^{n-1} (x_{k(k-1)})_{n-3} \\\\\\end{array}\\right) \\in \\mathfrak {M}^{n \\times (n-3)},$ $C^{\\prime } =\\left(\\begin{array}{ccc}(x_{32})_{(n-2)} & T_2 (x_{21})_{(n-1)} & T_2 T_1 (x_{10})_n \\\\\\end{array}\\right) \\in \\mathfrak {M}^{n \\times 3}.$ Keep repeating the same arguments above and finally we have there exists $T_0, \\dots , T_{n-2} \\in O(n)$ , such that for any $x_1 \\in v(F_{n(n-2)}) \\subset \\mathfrak {M}^{1}$ $s \\ge \\sup \\limits _{\\begin{array}{c} (x_{10})_n \\in \\mathcal {S}(T_0 \\mathrm {co}F_{10}^{x_{10}}) \\\\ (x_{21})_{(n-1)} \\in \\mathcal {S}(T_1 T_0 \\mathrm {co}F_{21}^{x_{21}} )\\\\ \\dots \\dots \\\\(x_{(n-1)(n-2)})_{2} \\in \\mathcal {S}(T_{n-2} T_{n-3} \\dots T_0 \\mathrm {co}F_{(n-1)(n-2)}^{x_{(n-1)(n-2)}} )\\end{array}}| \\det \\left(\\begin{array}{cc}D & D^{\\prime } \\\\\\end{array}\\right)|, \\qquad \\mathrm {(3.23)}$ where $D \\in \\mathfrak {M}^{n \\times 1}$ , $D^{\\prime } \\in \\mathfrak {M}^{n \\times (n-1)}$ : $D= (T_{n-2} \\dots T_0) (x_1 + \\sum \\limits _{k=1}^{n-1} (x_{k(k-1)})_1),$ $D^{\\prime }=\\left(\\begin{array}{cccccc}(x_{(n-1)(n-2)})_2 & T_{n-2}(x_{(n-2)(n-3)})_3 & (T_{n-2} T_{n-3}) (x_{(n-3)(n-4)})_4 & \\dots & (T_{n-2} \\dots T_1) (x_{10})_n \\\\\\end{array}\\right).$ It follows from (2.2) together with the invariance under $O(n)$ that $s\\ge C |v(F_{n(n-2)})|^{1/n} |\\mathrm {co}F_{(n-1)(n-2)}^{x_{n-1)(n-2)}}|^{1/n}| \\mathrm {co}F_{(n-2)(n-3)}^{x_{(n-2)(n-3)}}|^{1/n} \\dots |\\mathrm {co}F_{21}^{x_{21}}|^{1/n} |\\mathrm {co}F_{10}^{x_{10}}|^{1/n}.$ Obviously, $|\\mathrm {co}F_{k(k-1)}^{x_{k(k-1)}}| \\ge |F_{k(k-1)}^{x_{k(k-1)}}|, \\ 1 \\le k \\le n-1.$ This together with (3.20) implies $s &\\ge C(|v(F_{n(n-2)})| | \\mathrm {co}F_{(n-1)(n-2)}^{x_{(n-1) (n-2)}}| | \\mathrm {co}F_{(n-2) (n-3)}^{x_{(n-2) (n-3)}}|\\dots | \\mathrm {co}F_{2 1}^{x_{2 1}}| |F_{10}^{x_{1 0}}|)^{1/n} \\\\&\\ge C (|v(F_{n(n-2)})| |F_{(n-1)(n-2)}^{x_{(n-1) (n-2)}}| |F_{(n-2) (n-3)}^{x_{(n-2) (n-3)}}| \\dots |F_{2 1}^{x_{2 1}}| |F_{10}^{x_{1 0}}| )^{1/n}\\ge C \\prod \\limits _{j=1}^{n}|E_j|^{\\frac{1}{n^{2}}}.$ This completes Theorem 3.1.",
"Corollary 3.2 There exists a finite constant $\\mathcal {A}_{n}, \\mathcal {B}_{n}$ such that for any measurable set $E \\subset \\mathfrak {M}^{n \\times n}$ of finite measure, for any non-zero scalar $\\lambda _{j} \\in \\mathbb {R}$ , $j=1, \\dots , n$ , $(\\prod _{j=1}^{n} |\\lambda _{j}|) |E|^{\\frac{1}{n}}\\le \\mathcal {A}_{n} \\displaystyle {\\sup _{\\begin{array}{c}A_{j} \\in E \\\\ j=1, \\dots , n\\end{array}}} \\ | \\det (\\lambda _{1} A_{1}+ \\dots + \\lambda _{n} A_{n} ) |.",
"\\qquad \\mathrm {(3.24)}$ If $E$ is a compact convex set in $\\mathfrak {M}^{n \\times n}$ , then $|E|^{\\frac{1}{n}} \\le \\mathcal {B}_{n} \\displaystyle {\\sup _{A \\in E}} \\ | \\det (A) |.",
"\\qquad \\mathrm {(3.25)}$ To see (3.25), let $E_{j}=\\lambda _{j}E$ .",
"Applying Theorem 3.1 gives $\\prod _{j=1}^{n} |\\lambda _{j} E|^{\\frac{1}{n^{2}}}\\le \\mathcal {C}_{n} \\displaystyle {\\sup _{\\begin{array}{c}A_{j} \\in E \\\\ j=1, \\dots , n\\end{array}}} \\ | \\det (\\lambda _{1} A_{1}+ \\dots + \\lambda _{n} A_{n} ) |,$ which implies (3.25).",
"In particular, if $E \\subset \\mathfrak {M}^{n \\times n}$ is a compact convex set, setting $\\lambda _{j}=\\frac{1}{n}$ , $j=1, \\dots , n$ , it follows from (3.24) that $ (\\frac{1}{n})^{n} |E|^{1/n} \\le \\mathcal {A}_{n} \\displaystyle {\\sup _{\\begin{array}{c}A_{j} \\in E \\\\ j=1, \\dots , n\\end{array}}} | \\det (\\frac{1}{n}A_{1}+ \\dots +\\frac{1}{n} A_{n} ) |.$ On the other hand, since $E$ is convex, $\\displaystyle {\\sup _{A\\in E}} | \\det (A) |\\ge \\displaystyle {\\sup _{\\begin{array}{c}A_{j} \\in E \\\\ j=1, \\dots , n\\end{array} }} | \\det (\\frac{1}{n}A_{1}+ \\dots +\\frac{1}{n} A_{n} ) |.$ Thus we get (3.25).",
"Here we give a direct way to see Lemma 13.2 [5] which follows from (3.25).",
"Let $E \\subset \\mathfrak {M}^{n \\times n}$ be a measurable set.",
"The inequality (1.18) in Lemma 13.2 has translation invariance property, so we assume that $0 \\in E$ .",
"Given any matrices $A_{1}, \\dots , A_{n^{2}}$ in $E$ , from (3.25) it follows that $ |\\mathrm {co} \\lbrace 0,A_{1}, \\dots , A_{n^{2}} \\rbrace |^{\\frac{1}{n}} \\lesssim _{n} \\sup \\limits _{A \\in \\mathrm {co} \\lbrace 0,A_{1}, \\dots , A_{n^{2}} \\rbrace } \\ | \\det (A) |, \\qquad \\mathrm {(3.26)}$ By (2.2), there exist $A_{1}, \\dots , A_{n^{2}}$ such that $|E| \\lesssim _{n} | \\mathrm {co} \\lbrace 0,A_{1}, \\dots , A_{n^{2}} \\rbrace |,$ together with (3.26) we obtain that $|E|^{\\frac{1}{n}} \\lesssim _{n} \\sup \\limits _{A \\in \\mathrm {co} \\lbrace 0,A_{1}, \\dots , A_{n^{2}} \\rbrace } \\ | \\det (A) |.",
"\\qquad \\mathrm {(3.27)}$ For any convex set $F \\subset \\mathfrak {M}^{n \\times n}$ $ \\sup \\limits _{A \\in \\mathrm {co} \\lbrace 0, F \\rbrace } \\ | \\det (A)|=\\sup \\limits _{A \\in F} \\ | \\det (A)|,$ since $ |\\det ( \\lambda A)|= \\lambda ^{n} |\\det ( A)| \\le | \\det (A)|$ for any $\\lambda \\in [0,1]$ .",
"So $\\sup \\limits _{A \\in \\mathrm {co} \\lbrace 0,A_{1}, \\dots , A_{n^{2}} \\rbrace } \\ | \\det (A) |= \\sup \\limits _{A \\in \\mathrm {co} \\lbrace A_{1}, \\dots , A_{n^{2}} \\rbrace } \\ | \\det (A) |.",
"\\qquad \\mathrm {(3.28)}$ Denote $A^{(k)}$ by the $k$ -th column vector of the matrix $A$ , $1 \\le k \\le n$ .",
"Then there exist $\\widetilde{A}_{1}, \\dots , \\widetilde{A}_{n} \\in \\lbrace A_{1}, \\dots , A_{n^{2}}\\rbrace $ ($\\widetilde{A}_{i}$ , $\\widetilde{A}_{j}$ might be the same matrix), such that for any $ \\lbrace \\lambda _{1}, \\dots ,\\lambda _{n^{2}} \\rbrace $ satisfying $\\sum \\limits _{j=1}^{n^{2}} \\lambda _{j} =1$ and $0 \\le \\lambda _{j} \\le 1$ , $| \\det ( \\lambda _{1}A_{1}+ \\dots + \\lambda _{n^{2}}A_{n^{2}} ) |\\le | \\sum \\limits _{ \\begin{array}{c}i_{j} \\in \\lbrace 1, \\dots , n \\rbrace \\\\ i_{j} \\ne i_{k}, \\forall j \\ne k \\end{array}} \\det ( \\widetilde{A}_{i_{1}}^{(1)}, \\dots , \\widetilde{A}_{i_{n}}^{(n)} ) | \\qquad \\mathrm {(3.29)}$ holds, this is because $\\sum \\limits _{1 \\le l_{1}, \\dots , l_{n} \\le n^{2}} \\lambda _{l_{1}} \\dots \\lambda _{l_{n}}\\le \\sum \\limits _{1 \\le l_{1}, \\dots , l_{n-1} \\le n^{2}} \\lambda _{l_{1}} \\dots \\lambda _{l_{n-1}}\\le \\dots \\le \\sum \\limits _{1 \\le l_{1}, l_{2} \\le n^{2}} \\lambda _{l_{1}} \\lambda _{l_{2}}\\le \\sum \\limits _{1 \\le l_{1} \\le n^{2}} \\lambda _{l_{1}}= 1.$ Hence from (3.27)-(3.29) $|E|^{\\frac{1}{n}} \\lesssim _{n} | \\sum \\limits _{ \\begin{array}{c}i_{j} \\in \\lbrace 1, \\dots , n \\rbrace \\\\ i_{j} \\ne i_{k}, \\forall j \\ne k \\end{array}} \\det ( \\widetilde{A}_{i_{1}}^{(1)}, \\dots , \\widetilde{A}_{i_{n}}^{(n)} ) |.",
"\\qquad \\mathrm {(3.30)}$ As mentioned in the proof of Lemma 13.2 [5], $ \\sum \\limits _{ \\begin{array}{c}i_{j} \\in \\lbrace 1, \\dots , n \\rbrace \\\\ i_{j} \\ne i_{k}, \\forall j \\ne k \\end{array}} \\det ( \\widetilde{A}_{i_{1}}^{(1)}, \\dots , \\widetilde{A}_{i_{n}}^{(n)} ) $ is $\\mathbb {Z}$ -linear combination of $\\lbrace \\det ( \\sum \\limits _{j=1}^{n} s_{j} \\widetilde{A}_{j} ): s_{j} \\in \\lbrace 0,1\\rbrace \\rbrace $ .",
"This gives (1.20): $ |E|^{\\frac{1}{n}}\\lesssim _{n} \\sup \\limits _{\\begin{array}{c}A_{1}, \\dots , A_{n} \\in E \\\\ s_{1}, \\dots , s_{n} \\in \\lbrace 0, 1\\rbrace \\end{array}} |\\det ( s_{1}A_{1}+ \\dots + s_{n}A_{n})|.$ Obviously, (3.25) is not affine invariant.",
"The following example shows balls or ellipsoids are not the optimisers.",
"Example 3.2.",
"(i) Let $n=2$ , $E=B(0,r)$ , $A=\\left(\\begin{array}{cc}a & c \\\\b & d \\\\\\end{array}\\right) \\in E$ .",
"Then $\\sup \\limits _{A\\in E} |\\det (A)|=\\frac{r^{2}}{2}$ by calculation.",
"Consider the ellipsoid $F$ in $\\mathbb {R}^{4}$ with $|F|=|B(0,r)|$ , $F= \\lbrace \\left(\\begin{array}{cc}a & c \\\\b & d \\\\\\end{array}\\right): \\frac{a^{2}}{l_{1}^{2}}+ \\frac{b^{2}}{l_{2}^{2}}+\\frac{c^{2}}{l_{3}^{2}}+\\frac{d^{2}}{l_{4}^{2}} \\le 1 \\rbrace .$ It is easy to obtain $\\sup \\limits _{A\\in F} |\\det (A)| \\ge \\frac{l_{1}l_{4}+l_{2}l_{3}}{4} \\ge \\frac{r^{2}}{2}$ by GM-AM inequality.",
"(ii) Let $r=1$ .",
"Since $A\\mapsto |\\det (A)|$ is a continuous function on $E=B(0,1)$ under the natural topology on Euclidean space $\\mathbb {R}^{4}$ , there exists $0<\\delta <\\frac{1}{25}$ such that $|\\det (A)|\\le \\frac{1}{4}$ for all $A\\in E$ satisfying $|A-\\left(\\begin{array}{cc}1 & 0 \\\\0 & 0 \\\\\\end{array}\\right)|= (a-1)^{2}+b^{2}+c^{2}+d^{2}\\le 2\\delta .$ Then for all $A\\in E$ satisfying $\\sqrt{1-\\delta } \\le a \\le 1$ , we have $b^{2}+c^{2}+d^{2} \\le 1-a^{2} \\le 1- (1-\\delta ) = \\delta .$ Thus $|A-\\left(\\begin{array}{cc}1 & 0 \\\\0 & 0 \\\\\\end{array}\\right)|= (a-1)^{2}+b^{2}+c^{2}+d^{2}\\le (1- \\sqrt{1-\\delta })^{2} + \\delta \\le 2\\delta $ which implies that $|\\det (A)|\\le \\frac{1}{4}$ for any $A\\in E$ satisfying $\\sqrt{1-\\delta } \\le a \\le 1$ .",
"Let $P=\\left(\\begin{array}{cc}0 & 0 \\\\0 & p \\\\\\end{array}\\right)$ with $p=\\frac{1}{\\sqrt{1-\\delta }}$ and then consider $\\sup \\limits _{A\\in \\mathrm {co} \\lbrace P\\cup E\\rbrace } |\\det (A)|$ , $\\ \\ \\ \\ \\ \\sup \\limits _{A\\in \\mathrm {co} \\lbrace P\\cup E\\rbrace } |\\det (A)|&=\\sup \\limits _{A\\in E, \\lambda \\in [0,1]} |\\det (\\lambda A+(1-\\lambda )P) |\\\\&= \\sup \\limits _{A\\in E, \\lambda \\in [0,1]}|\\det \\left(\\begin{array}{cc}\\lambda a & \\lambda c \\\\\\lambda b & \\lambda d+ (1-\\lambda )p \\\\\\end{array}\\right)|\\\\&= \\sup \\limits _{A\\in E, \\lambda \\in [0,1]}|\\det \\left(\\begin{array}{cc}\\lambda a & \\lambda c \\\\\\lambda b & \\lambda d \\\\\\end{array}\\right)+\\det \\left(\\begin{array}{cc}\\lambda a & 0 \\\\\\lambda b & (1-\\lambda )p \\\\\\end{array}\\right)|.$ When $a\\notin [\\sqrt{1-\\delta }, 1]$ , $&\\ \\sup \\limits _{A\\in E, \\lambda \\in [0,1]}|\\det \\left(\\begin{array}{cc}\\lambda a & \\lambda c \\\\\\lambda b & \\lambda d \\\\\\end{array}\\right)+\\det \\left(\\begin{array}{cc}\\lambda a & 0 \\\\\\lambda b & (1-\\lambda )p \\\\\\end{array}\\right)| \\\\&\\le \\sup \\limits _{\\lambda \\in [0,1]}\\lambda ^{2} \\frac{1}{2} + \\lambda (1-\\lambda ) a p \\\\&\\le \\sup \\limits _{\\lambda \\in [0,1]}\\lambda ^{2} \\frac{1}{2} + \\lambda (1-\\lambda ) \\sqrt{1-\\delta } \\frac{1}{\\sqrt{1-\\delta }} \\\\&= \\sup \\limits _{\\lambda \\in [0,1]}\\lambda ^{2} \\frac{1}{2} + \\lambda (1-\\lambda ) \\le \\frac{1}{2}.$ When $a \\in [\\sqrt{1-\\delta }, 1]$ , $&\\ \\sup \\limits _{A\\in E, \\lambda \\in [0,1]}|\\det \\left(\\begin{array}{cc}\\lambda a & \\lambda c \\\\\\lambda b & \\lambda d \\\\\\end{array}\\right)+\\det \\left(\\begin{array}{cc}\\lambda a & 0 \\\\\\lambda b & (1-\\lambda )p \\\\\\end{array}\\right)| \\\\&\\le \\sup \\limits _{\\lambda \\in [0,1]}\\lambda ^{2} \\frac{1}{4} + \\lambda (1-\\lambda ) p \\\\&=\\sup \\limits _{\\lambda \\in [0,1]}\\lambda ^{2} \\frac{1}{4} + \\lambda (1-\\lambda ) \\frac{1}{\\sqrt{1-\\delta }}.$ It is easy to see for $0<\\delta <\\frac{1}{25}$ given above, $\\sup \\limits _{\\lambda \\in [0,1]}\\lambda ^{2} \\frac{1}{4} + \\lambda (1-\\lambda ) \\frac{1}{\\sqrt{1-\\delta }} \\le \\frac{1}{2}.$ Therefore, $\\sup \\limits _{A\\in \\mathrm {co} \\lbrace P\\cup E\\rbrace } |\\det (A)| = \\sup \\limits _{A\\in E} |\\det (A)|,$ which implies balls can not be the optimisers.",
"Remark 3.3.",
"Let $E \\subset \\mathfrak {M}^{n \\times n}$ be a compact convex set.",
"If we compare the maximal volume of simiplicies $\\sup \\limits _{A_{0}, \\dots , A_{n^{2}} \\in E} \\mathrm {vol}( \\mathrm {co} \\lbrace A_{0}, \\dots , A_{n^{2}} \\rbrace )$ contained in $E$ with the $\\displaystyle {\\sup _{A \\in E}} \\ | \\det (A) |$ , it follows from (3.25) that $\\sup \\limits _{A_{0}, \\dots , A_{n^{2}} \\in E} \\mathrm {vol}( \\mathrm {co} \\lbrace A_{0}, \\dots , A_{n^{2}} \\rbrace )\\lesssim _{n} \\displaystyle {\\sup _{A \\in E}} \\ | \\det (A) |^{n}.",
"\\qquad \\mathrm {(3.31)}$ Indeed by John ellipsoids, it is enough to consider the case when $E$ is a ellipsoid in $\\mathfrak {M}^{n \\times n}$ .",
"For any ellpsoid $E\\equiv \\lbrace x\\in \\mathbb {R}^{n^{2}}: \\displaystyle {\\sum _{i}^{n^{2}}} \\frac{ | \\langle x-x_{0}, \\omega _{i} \\rangle |^{2}}{l_{i}^{2}} \\le 1 \\rbrace ,$ where $x_{0}\\in \\mathbb {R}^{n^{2}}$ , $\\lbrace \\omega _{i}\\rbrace $ is an orthonormal basis in $\\mathbb {R}^{n^{2}}$ .",
"By the affine invariance of $\\sup \\limits _{A_{0}, \\dots , A_{n^{2}} \\in E} \\mathrm {vol}( \\mathrm {co} \\lbrace A_{0}, \\dots , A_{n^{2}} \\rbrace )$ , it is enough to see balls centred at 0.",
"Apply the Hadamard inequality, for any $A_{j} \\in B(0,r) \\subset \\mathbb {R}^{n^{2}}$ , $j=0, \\dots , n^{2}$ $\\mathrm {vol}( \\mathrm {co} \\lbrace A_{0}, \\dots , A_{n^{2}} \\rbrace )\\le |A_0-A_1 | |A_0-A_2| \\dots |A_0-A_{n^{2}}| \\lesssim _{n} r^{n^{2}} \\sim |B(0,r)| .$ Hence for any ellipsoid $E \\subset \\mathbb {R}^{n^{2}}$ , $\\sup \\limits _{A_{0}, \\dots , A_{n^{2}} \\in E} \\mathrm {vol}( \\mathrm {co} \\lbrace A_{0}, \\dots , A_{n^{2}} \\rbrace )\\lesssim _{n} |E|.$ On the other hand, by (3.25) $|E| \\lesssim _{n} \\displaystyle {\\sup _{A \\in E}} \\ | \\det (A) |^{n}.$ Therefore, we have the following relation $\\sup \\limits _{A_{0}, \\dots , A_{n^{2}} \\in E} \\mathrm {vol}( \\mathrm {co} \\lbrace A_{0}, \\dots , A_{n^{2}} \\rbrace )\\lesssim _{n} \\displaystyle {\\sup _{A \\in E}} \\ | \\det (A) |^{n}.$ Similarly, we have $\\sup \\limits _{A_{1}, \\dots , A_{n^{2}} \\in E} \\mathrm {vol}( \\mathrm {co} \\lbrace 0, A_{1}, \\dots , A_{n^{2}} \\rbrace )\\lesssim _{n} \\displaystyle {\\sup _{A \\in E}} \\ | \\det (A) |^{n}.",
"\\qquad \\mathrm {(3.32)}$ If $0 \\in E$ , it is true which mainly due to the Hadamard inequality and the $\\mathrm {GL}_{n}(\\mathbb {R})$ invariance of $\\sup \\limits _{A_{1}, \\dots , A_{n^{2}} \\in E} \\mathrm {vol}( \\mathrm {co} \\lbrace 0, A_{1}, \\dots , A_{n^{2}} \\rbrace )$ .",
"If $0 \\notin E$ , the relation above still holds because of the fact $\\sup \\limits _{A \\in E} | \\det (A) |^{n}= \\sup \\limits _{A \\in \\mathrm {co}\\lbrace 0, E\\rbrace } | \\det (A) |^{n}.$ Acknowledgments.",
"I am grateful to my supervisor Professor Carbery for his helpful suggestions and revision on this paper.",
"This work was supported by the scholarship from China Scholarship Council."
]
] | 1606.05208 |
[
[
"A Burau-Alexander 2-functor on tangles"
],
[
"Abstract We construct a weak 2-functor from the bicategory of oriented tangles to a bicategory of Lagrangian cospans.",
"This functor simultaneously extends the Burau representation of the braid groups, its generalization to tangles due to Turaev and the first-named author, and the Alexander module of 1 and 2-dimensional links."
],
[
"Introduction",
"Since its introduction in 1935, the Burau representation [5] has been one of the most studied representations of the braid groups.",
"In its reduced version, it takes the form of a homomorphism $\\rho _n\\colon B_n\\rightarrow \\mathit {GL}_{n-1}(\\Lambda )$ with $\\Lambda ={Z}[t^{\\pm 1}]$ , which preserves some non-degenerate skew-hermitian form on $\\Lambda ^{n-1}$ [16].",
"The construction of $\\rho _n$ , whether the algebraic one [5], [4] or the homological one [13], easily extend to oriented braids, i.e.",
"braids where different strands can be oriented in different directions.",
"In a slightly pedantic style, one can therefore say that the Burau representations constitute a functor $\\rho $ from the groupoid $\\mathbf {Braids}$ , with objects finite sequences of signs $\\pm 1$ and morphisms oriented braids, to the groupoid $\\mathbf {U}_\\Lambda $ , with objects $\\Lambda $ -modules equipped with a non-degenerate skew-hermitian form and morphisms unitary $\\Lambda $ -isomorphisms.",
"In this context, it is natural to ask whether this Burau functor extends to the category $\\mathbf {Tangles}$ of oriented tangles (whose formal definition can be found in subsection REF ).",
"Such an extension was constructed by Turaev and the first-named author in [6].",
"In a nutshell, they defined a category $\\mathbf {Lagr}_\\Lambda $ of Lagrangian relations in which $\\mathbf {U}_\\Lambda $ embeds via the graph functor: if $f$ is a unitary isomorphism, then its graph $\\Gamma _f$ is a Lagrangian relation.",
"Then, they constructed a functor $\\mathcal {F}\\colon \\mathbf {Tangles}\\rightarrow \\mathbf {Lagr}_\\Lambda $ such that $\\mathcal {F}(\\beta )=\\Gamma _{\\rho (\\beta )}$ for any oriented braid $\\beta $ .",
"Note that the groupoid $\\mathbf {Braids}$ is nothing but the core of $\\mathbf {Tangles}$ : both categories have the same objects, and the isomorphisms of the latter are the morphisms of the former.",
"Therefore, one might wonder if there is an extension $\\mathcal {B}$ of $\\rho $ to oriented tangles taking values in a category whose core is (equivalent to) $\\mathbf {U}_\\Lambda $ .",
"More importantly, oriented surfaces between oriented tangles turn $\\mathbf {Tangles}$ into a (weak) 2-category (see subsection REF for a discussion of this fact), and several functors, such as the one coming from Khovanov homology [12], have been shown to extend to 2-functors.",
"Hence, one can also hope to extend $\\mathcal {B}$ to a 2-functor.",
"This is what we achieve in the present paper, building on the homological definition of the Burau representation.",
"The idea is to consider cospans [3], [8] of $\\Lambda $ -modules, i.e.",
"diagrams of the form $H\\rightarrow T\\leftarrow H^{\\prime }$ .",
"More precisely, we start by defining the category $\\mathbf {L}_\\Lambda $ of Lagrangian cospans, whose core is shown to be equivalent to $\\mathbf {U}_\\Lambda $ .",
"This category should be understood as a generalization of the category of Lagrangian relations, in the sense that there is a full (non-faithful) functor $F\\colon \\mathbf {L}_\\Lambda \\rightarrow \\mathbf {Lagr}_\\Lambda $ which is the identity on objects.",
"The functor $\\mathcal {F}$ then lifts in a very natural way to a functor $\\mathcal {B}$ taking values in $\\mathbf {L}_\\Lambda $ .",
"In summary, we have the commutative diagram of functors $@R0.5cm{& \\mathbf {L}_\\Lambda [d]^F \\\\\\mathbf {Tangles} [ur]^{\\mathcal {B}} [r]_{\\mathcal {F}} & \\mathbf {Lagr}_\\Lambda \\,,}$ where $\\mathcal {B}$ extends the Burau functor $\\rho $ in the following sense: the restriction of $\\mathcal {B}$ to $\\mathbf {Braids}=\\mathit {core}(\\mathbf {Tangles})$ fits in the commutative diagram $@R0.5cm{& \\mathit {core}(\\mathbf {L}_\\Lambda ) [d]_\\simeq ^{F|} \\\\\\mathbf {Braids} [ur]^{\\mathcal {B}|} [r]_{\\rho } & \\mathbf {U}_\\Lambda \\,,}$ with the vertical arrow an equivalence of categories.",
"Furthermore, the category $\\mathbf {L}_\\Lambda $ can be modified in a natural way and endowed with a weak 2-category structure yielding a bicategory, and $\\mathcal {B}$ extends to a weak 2-functor on the bicategory of oriented tangles and surfaces.",
"Finally, when restricted to 1 and 2-endomorphisms of the empty set, i.e.",
"oriented links and closed surfaces, $\\mathcal {B}$ is nothing but the Alexander module.",
"The paper is organized as follows.",
"In Section , we recall the definition of the category $\\mathbf {Lagr}_\\Lambda $ of Lagrangian relations, we define our category $\\mathbf {L}_\\Lambda $ of Lagrangian cospans together with the full functor $F\\colon \\mathbf {L}_\\Lambda \\rightarrow \\mathbf {Lagr}_\\Lambda $ , and prove that the core of $\\mathbf {L}_\\Lambda $ is equivalent to $\\mathbf {U}_\\Lambda $ .",
"In Section , we show that $\\mathbf {L}_\\Lambda $ naturally extends to a bicategory.",
"In Section , we give the definition of the category of oriented tangles, discuss its 2-category extension, construct the functor $\\mathcal {B}$ and show that it extends to a weak 2-functor.",
"Finally, in Section , we briefly explain how other versions of the Burau representation (namely, unreduced and multivariable versions) can be extended to weak 2-functors using the same ideas."
],
[
"Acknowledgments",
"The authors wish to thank Louis-Hadrien Robert for useful discussions, and the anonymous referee for pointing out a mistake in an earlier version of this article.",
"The first author was supported by the Swiss National Science Foundation.",
"The second author was supported by the NCCR SwissMAP, funded by the Swiss National Science Foundation."
],
[
"Lagrangian categories",
"The aim of this section is to introduce the various algebraic categories that appear in our construction.",
"In a first subsection, we briefly recall the definition of the category $\\mathbf {Lagr}_\\Lambda $ of Lagrangian relations over a ring $\\Lambda $ and explain why it should be understood as a generalization of the groupoid $\\mathbf {U}_\\Lambda $ of unitary automorphisms of Hermitian $\\Lambda $ -modules, following [6].",
"In subsection REF , we recall the theory of cospans in a category with pushouts.",
"In subsection REF , we define the category $\\mathbf {L}_\\Lambda $ of Lagrangian cospans, and relate it to the category of Lagrangian relations via a full functor $F\\colon \\mathbf {L}_\\Lambda \\rightarrow \\mathbf {Lagr}_\\Lambda $ .",
"In subsection REF , we show that this functor restricts to an equivalence of categories between the core groupoid of $\\mathbf {L}_\\Lambda $ and $\\mathbf {U}_\\Lambda $ ."
],
[
"The category of Lagrangian relations",
"Fix an integral domain $\\Lambda $ endowed with a ring involution $\\lambda \\mapsto \\overline{\\lambda }$ .",
"A skew-Hermitian form on a $\\Lambda $ -module $H$ is a map $\\omega \\colon H \\times H\\rightarrow \\Lambda $ such that for all $x,y,z \\in H$ and all $\\lambda ,\\lambda ^{\\prime }\\in \\Lambda $ , i $\\omega (\\lambda x + \\lambda ^{\\prime } y,z)=\\lambda \\omega (x,z)+\\lambda ^{\\prime } \\omega (y,z)$ , ii $\\omega (x,y)=-\\overline{\\omega (y,x)}$ , iii if $\\omega (x,y)=0$ for all $y \\in H$ , then $x=0$ .",
"A Hermitian $\\Lambda $ -module $H$ is a finitely generated $\\Lambda $ -module endowed with a skew-Hermitian form $\\omega $ .",
"The same module $H$ with the opposite form $-\\omega $ will be denoted by $-H$ .",
"The annihilator of a submodule $A \\subset H$ is the submodule$\\mathrm {Ann}(A)=\\lbrace x \\in H \\ | \\ \\omega (v,x)=0 \\text{ for all } v \\in A \\rbrace \\,.$ A submodule is called Lagrangian if it is equal to its annihilator.",
"Given a submodule $A$ of a Hermitian $\\Lambda $ -module $H$ , set $\\overline{A}= \\lbrace x \\in H \\ | \\ \\lambda x \\in A \\ \\text{for a non-zero} \\ \\lambda \\in \\Lambda \\rbrace \\,.$ Observe that if $A$ is Lagrangian, then $\\overline{A}=A$ .",
"If $H$ and $H^{\\prime }$ are Hermitian $\\Lambda $ -modules, a Lagrangian relation from $H$ to $H^{\\prime }$ is a Lagrangian submodule of $(-H)\\oplus H^{\\prime }$ .",
"For instance, given a Hermitian $\\Lambda $ -module $H$ , the diagonal relation $\\Delta _H= \\lbrace h \\oplus h \\in H \\oplus H \\ | \\ h \\in H \\rbrace $ is a Lagrangian relation from $H$ to $H$ .",
"Given two Lagrangian relations $N_1$ from $H$ to $H^{\\prime }$ and $N_2$ from $H^{\\prime }$ to $H^{\\prime \\prime }$ , their composition is defined as $N_2 \\circ N_1 := \\overline{N_2N_1}\\subset (-H) \\oplus H^{\\prime \\prime }$ , where $N_2N_1 = \\lbrace x \\oplus z \\ | \\ x \\oplus y \\in N_1 \\ \\text{and} \\ y \\oplus z \\in N_2 \\text{ for some~$y\\in H^{\\prime }$}\\rbrace \\,.$ The proof of the next theorem can be found in [6].",
"Theorem 2.1 Hermitian $\\Lambda $ -modules, as objects, and Lagrangian relations, as morphisms, form a category.",
"$\\Box $ Following [6], we shall denote this category by $\\mathbf {Lagr}_\\Lambda $ and call it the category of Lagrangian relations over $\\Lambda $.",
"We shall say that a $\\Lambda $ -linear map between two Hermitian $\\Lambda $ -modules is unitary if it preserves the corresponding skew-Hermitian forms.",
"Let us now briefly recall why Lagrangian relations can be understood as a generalization of unitary $\\Lambda $ -isomorphisms and unitary $Q$ -isomorphisms, where $Q=Q(\\Lambda )$ is the field of fractions of $\\Lambda $ .",
"Let $\\mathbf {U}_\\Lambda $ be the category of Hermitian $\\Lambda $ -modules and unitary $\\Lambda $ -isomorphisms.",
"Also, let $\\mathbf {U}_\\Lambda ^0$ be the category of Hermitian $\\Lambda $ -modules, where the morphisms between $H$ and $H^{\\prime }$ are the unitary $Q$ -isomorphisms between $H\\otimes Q$ and $H^{\\prime } \\otimes Q$ .",
"The graph of a $\\Lambda $ -linear map $f \\colon H \\rightarrow H^{\\prime }$ is the submodule $\\Gamma _f=\\lbrace x \\oplus f(x) \\rbrace $ of $H \\oplus H^{\\prime }$ .",
"Similarly the restricted graph of a $Q$ -linear map $\\varphi \\colon H \\otimes Q \\rightarrow H^{\\prime } \\otimes Q$ is $\\Gamma _\\varphi ^0=\\Gamma _\\varphi \\cap (H \\oplus H^{\\prime }).$ The proof of the following theorem can be found in [6].",
"Theorem 2.2 The maps $f \\mapsto f\\otimes \\mathit {id}_Q$ , $f \\mapsto \\Gamma _f$ and $\\varphi \\mapsto \\Gamma ^0_\\varphi $ define faithful functors which are the identity on objects, and fit in the commutative diagram ${\\mathbf {U}_\\Lambda [r]^{- \\otimes Q} @/_1pc/[rr]_\\Gamma & \\mathbf {U}_\\Lambda ^0 [r]^{\\Gamma ^0}& \\mathbf {Lagr}_\\Lambda \\,.",
"}$ $\\Box $ We shall call such functors embeddings of categories.",
"Cospans in a category with pushouts Among the arguments that will be used in this article, some are well-known and of purely categorical nature.",
"This subsection contains a quick review of these results (see [3], [15] for further detail).",
"Let us fix a category $\\mathbf {C}$ .",
"Throughout this subsection, all objects, morphisms, diagrams, and the like will be in this fixed category $\\mathbf {C}$ .",
"Recall that a span is a diagram of the form $T_1\\stackrel{i_1}{\\longleftarrow }H\\stackrel{i_2}{\\longrightarrow }T_2$ .",
"A pushout of such a span is an object $P$ together with morphisms $T_1\\stackrel{j_1}{\\longrightarrow }P\\stackrel{j_2}{\\longleftarrow }T_2$ such that $j_1 i_1=j_2 i_2$ , which satisfies the following universal property: for any $T_1\\stackrel{k_1}{\\longrightarrow }Q\\stackrel{k_2}{\\longleftarrow }T_2$ such that $k_1 i_1=k_2 i_2$ , there exists a unique morphism $u\\colon P\\rightarrow Q$ with $u j_1=k_1$ and $u j_2=k_2$ .",
"This is illustrated in the following commutative diagram: $@R0.5cm{& Q & & \\\\& P @{.>}[u]^{u} & & \\\\T_1 [ru]^{j_1} @/^1pc/[uur]^{k_1} & & T_2.",
"[ul]_{j_2}@/_1pc/[uul]_{k_2} \\\\& H[lu]_{i_1}[ru]^{i_2} & }$ If a span admits a pushout, then the latter is unique up to canonical isomorphism.",
"However, not all spans admit pushouts in general.",
"From now on, we shall assume that $\\mathbf {C}$ is a category with pushouts, i.e.",
"that any span admits a pushout.",
"Moreover, we fix for each span a pushout.",
"Let $H,H^{\\prime }$ be two objects.",
"A cospan from $H$ to $H^{\\prime }$ is a diagram $H\\stackrel{i}{\\longrightarrow }T\\stackrel{i^{\\prime }}{\\longleftarrow }H^{\\prime }$ .",
"Two cospans $H\\stackrel{i_1}{\\longrightarrow }T_1\\stackrel{i_1^{\\prime }}{\\longleftarrow }H^{\\prime }$ and $H\\stackrel{i_2}{\\longrightarrow }T_2\\stackrel{i_2^{\\prime }}{\\longleftarrow }H^{\\prime }$ are isomorphic if there is an isomorphism $f\\colon T_1\\rightarrow T_2$ such that $f i_1=i _2$ and $f i_1^{\\prime } = i_2^{\\prime }$ .",
"The composition of two cospans $H\\stackrel{i_1}{\\longrightarrow }T_1\\stackrel{i_1^{\\prime }}{\\longleftarrow }H^{\\prime }$ and $H^{\\prime }\\stackrel{i^{\\prime }_2}{\\longrightarrow }T_2\\stackrel{i_2^{\\prime \\prime }}{\\longleftarrow }H^{\\prime \\prime }$ is the cospan from $H$ to $H^{\\prime \\prime }$ given by the (fixed) pushout diagram $@R0.5cm{& & T_2 \\circ T_1 & & & \\\\& T_1 [ru]^{j_1} & & T_2 [ul]_{j_2} \\\\H [ru]^{i_1} & & H^{\\prime } [ru]^{i^{\\prime }_2}[lu]_{i_1^{\\prime }} & &H^{\\prime \\prime }\\,.",
"[lu]_{i_2^{\\prime \\prime }}&}$ Finally, the identity cospan of an object $H$ is defined as the cospan $I_H:=(H\\stackrel{\\mathit {id}}{\\longrightarrow }H\\stackrel{\\mathit {id}}{\\longleftarrow }H)$ .",
"Remark 2.3 Given any morphism $H^{\\prime }\\stackrel{i^{\\prime }}{\\longrightarrow }H^{\\prime \\prime }$ , one easily checks that $@R0.5cm{& H^{\\prime \\prime } & \\\\H^{\\prime }[ru]^{i^{\\prime }} & & \\quad H^{\\prime \\prime } [lu]_{\\mathit {id}}\\\\& H^{\\prime }[lu]_{\\mathit {id}}[ru]^{i^{\\prime }} & } \\ \\ \\ @R0.5cm{& H^{\\prime \\prime } & \\\\H^{\\prime \\prime }[ru]^{\\mathit {id}} & & H^{\\prime } [lu]_{i^{\\prime }}\\\\& H^{\\prime }[lu]_{i^{\\prime }}[ru]^{\\mathit {id}} & }$ are pushout diagrams.",
"Therefore, if one makes this choice of pushout for spans of the form $H^{\\prime }\\stackrel{\\mathit {id}}{\\longleftarrow }H^{\\prime }\\stackrel{i^{\\prime }}{\\longrightarrow }H^{\\prime \\prime }$ and $H^{\\prime \\prime }\\stackrel{i^{\\prime }}{\\longleftarrow }H^{\\prime }\\stackrel{\\mathit {id}}{\\longrightarrow }H^{\\prime }$ , then the composition of $H\\stackrel{i}{\\longrightarrow }H^{\\prime }\\stackrel{\\mathit {id}}{\\longleftarrow }H^{\\prime }$ and $H^{\\prime }\\stackrel{i^{\\prime }}{\\longrightarrow }T\\stackrel{i^{\\prime \\prime }}{\\longleftarrow }H^{\\prime \\prime }$ is given by $H\\stackrel{i^{\\prime }i}{\\longrightarrow }T\\stackrel{i^{\\prime \\prime }}{\\longleftarrow }H^{\\prime \\prime }$ , and the composition of $H\\stackrel{i}{\\longrightarrow }T\\stackrel{i^{\\prime }}{\\longleftarrow }H^{\\prime }$ and $H^{\\prime }\\stackrel{\\mathit {id}}{\\longrightarrow }H^{\\prime }\\stackrel{i^{\\prime \\prime }}{\\longleftarrow }H^{\\prime \\prime }$ is given by $H\\stackrel{i}{\\longrightarrow }T\\stackrel{i^{\\prime }i^{\\prime \\prime }}{\\longleftarrow }H^{\\prime \\prime }$ .",
"For this reason, cospans should be understood as generalizing morphisms in the category $\\mathbf {C}$ .",
"Note that the composition of cospans depends on the choice of a pushout for each span; therefore, it cannot be associative for all such choices.",
"For the same reason, the composition does not admit $I_H$ as a two-sided unit in general.",
"However, for any fixed choice of pushouts, the corresponding composition does satisfy these properties up to canonical isomorphisms of cospans.",
"We refer the reader to [15] for a proof of this standard fact in the dual context of spans.",
"There are two possible strategies at this point.",
"The first one, which we will use in the remaining part of Section , is to consider the category given by the objects of $\\mathbf {C}$ , as objects, and the isomorphism classes of cospans in $\\mathbf {C}$ , as morphisms.",
"The second one, which we will use in the next sections, is to follow the “main principle of category theory \" as stated in [10], that is: not to identify isomorphic cospans, but to view these canonical isomorphisms as part of the (higher) structure.",
"This naturally leads to the concept of a bicategory, that will be reviewed in subsection REF and used in subsections REF and REF .",
"The category $\\mathbf {L}_\\Lambda $ of Lagrangian cospans We now take $\\mathbf {C}$ to be the category of $\\Lambda $ -modules, with $\\Lambda $ any integral domain.",
"After observing that this is a category with pushouts, we impose further conditions on our cospans and work with isomorphism classes thereof.",
"We begin with the following standard result, whose easy proof is left to the reader.",
"Lemma 2.4 The square $@R0.5cm{& P & \\\\T_1 [ru]^{j_1} & & T_2 [ul]_{j_2} \\\\& H [ru]_{i_2}[lu]^{i_1} &}$ is a pushout diagram in the category of $\\Lambda $ -modules if and only if the sequence $@R0.5cm{H[r]^{\\!\\!\\!\\!\\!\\!\\!",
"(-i_1,i_2)} & T_1\\oplus T_2[r]^{\\;\\;\\;\\;\\;{j_1\\atopwithdelims ()j_2}} & P [r]& 0\\,}$ is exact.$\\Box $ In particular, a pushout is given by the cokernel of the map $(-i_1,i_2) \\colon H \\rightarrow T_1 \\oplus T_2$ sending $x$ to $(-i_1(x)) \\oplus i_2(x)$ , so this is a category with pushouts.",
"By abuse of notation, we shall sometimes simply denote by $T$ (the isomorphism class of) a cospan of the form $H\\rightarrow T\\leftarrow H^{\\prime }$ .",
"For a cospan $H\\stackrel{i}{\\longrightarrow }T\\stackrel{i^{\\prime }}{\\longleftarrow }H^{\\prime }$ , consider the submodule $N_T:=\\mathit {Ker}{-i\\atopwithdelims ()\\phantom{-}i^{\\prime }}$ of $H \\oplus H^{\\prime }$ , where ${-i\\atopwithdelims ()\\phantom{-}i^{\\prime }}\\colon H \\oplus H^{\\prime } \\rightarrow T$ maps $(x,y)$ to $i^{\\prime }(y)-i(x)$ .",
"Note that if $T_1$ and $T_2$ are isomorphic cospans, then $N_{T_1}$ and $N_{T_2}$ are equal.",
"Lemma 2.5 For any two composable cospans $T_1$ and $T_2$ , we have $N_{T_2 \\circ T_1}=N_{T_2} N_{T_1}$ .",
"Consider two cospans $H\\stackrel{i_1}{\\longrightarrow }T_1\\stackrel{i_1^{\\prime }}{\\longleftarrow }H^{\\prime }$ and $H^{\\prime }\\stackrel{i^{\\prime }_2}{\\longrightarrow }T_2\\stackrel{i_2^{\\prime \\prime }}{\\longleftarrow }H^{\\prime \\prime }$ .",
"By definition, $N_{T_2 \\circ T_1}$ is the kernel of the map $H\\oplus H^{\\prime \\prime }\\rightarrow T_2\\circ T_1$ given by $(x,z)\\mapsto j_2(i_2^{\\prime \\prime }(z))-j_1(i_1(x))$ .",
"Since $T_2 \\circ T_1$ is represented by the cokernel of the map $(-i_1^{\\prime },i^{\\prime }_2)\\colon H^{\\prime } \\rightarrow T_1 \\oplus T_2$ , $N_{T_2 \\circ T_1}$ consists of the elements $x \\oplus z\\in H\\oplus H^{\\prime \\prime }$ for which $(-i_1(x))\\oplus i_2^{\\prime \\prime }(z)$ lies in the image of $(-i_1^{\\prime },i^{\\prime }_2)$ .",
"Therefore, $N_{T_2 \\circ T_1}$ is equal to $\\lbrace x \\oplus z \\in H \\oplus H^{\\prime \\prime } | \\ i_1(x)=i_1^{\\prime }(y) \\text{ and } i^{\\prime }_2(y)=i_2^{\\prime \\prime }(z) \\text{ for some } y \\in H^{\\prime } \\rbrace \\,.$ In other words, $N_{T_2 \\circ T_1}$ is equal to $\\mathit {Ker}{-i_2^{\\prime }\\atopwithdelims ()\\phantom{-}i_2^{\\prime \\prime }}\\mathit {Ker}{-i_1\\atopwithdelims ()\\phantom{-}i_1^{\\prime }}=N_{T_2}N_{T_1}$ .",
"Recall from the subsection REF that if $A$ is a submodule of a Hermitian $\\Lambda $ -module $H$ , then $ \\overline{A}$ consists of all $x \\in H$ such that $\\lambda x\\in A$ for a non-zero $\\lambda \\in \\Lambda $ .",
"We shall say that a cospan $H\\stackrel{i}{\\longrightarrow }T\\stackrel{i^{\\prime }}{\\longleftarrow }H^{\\prime }$ is Lagrangian if $\\overline{N_T}$ is a Lagrangian submodule of $(-H)\\oplus H^{\\prime }$ .",
"For instance, the identity cospan $I_H$ is a Lagrangian cospan, since $\\overline{N_H}=N_H$ is equal to the diagonal relation $\\Delta _H$ .",
"Proposition 2.6 Hermitian $\\Lambda $ -modules, as objects, and isomorphism classes of Lagrangian cospans, as morphisms, form a category $\\mathbf {L}_\\Lambda $ .",
"Moreover, the map $T\\mapsto \\overline{N_T}$ gives rise to a full functor $F\\colon \\mathbf {L}_\\Lambda \\rightarrow \\mathbf {Lagr}_\\Lambda $ .",
"As explained in subsection REF , it is a standard fact that the composition of isomorphism classes of cospans by pushouts is well-defined (i.e.",
"does not depend on the choice of the pushouts), is associative, with the identity cospan acting trivially.",
"Therefore, we only need to check that the composition of two Lagrangian cospans $H\\rightarrow T_1\\leftarrow H^{\\prime }$ and $H^{\\prime }\\rightarrow T_2\\leftarrow H^{\\prime \\prime }$ is also Lagrangian, and that $F$ is a full functor.",
"By Lemma REF , we have $\\overline{N_{T_2\\circ T_1}}=\\overline{N_{T_2}N_{T_1}}=N_{T_2}\\circ N_{T_1}\\,.$ Since $N_{T_2}$ are $N_{T_1}$ are Lagrangian, $N_{T_2}=\\overline{N_{T_2}}, N_{T_1}=\\overline{N_{T_1}}$ and $N_{T_2}\\circ N_{T_1}$ is also Lagrangian by Theorem REF .",
"Therefore, the cospan $H\\rightarrow T_2 \\circ T_1\\leftarrow H^{\\prime \\prime }$ is Lagrangian and $F$ is a functor.",
"Finally, given a Lagrangian relation $N$ from $H$ to $H^{\\prime }$ , consider the cospan $H\\stackrel{i}{\\longrightarrow }T\\stackrel{i^{\\prime }}{\\longleftarrow }H^{\\prime }$ where $T=(H\\oplus H^{\\prime })/N$ and $i$ (resp.",
"$i^{\\prime }$ ) is the inclusion of $H$ (resp.",
"$H^{\\prime }$ ) into $H\\oplus H^{\\prime }$ composed with the canonical projection.",
"By construction, it is a Lagrangian cospan with $\\overline{N_T}=\\overline{N}=N$ , so the functor $F$ is full.",
"Let us conclude this subsection by noting that the functor $F\\colon \\mathbf {L}_\\Lambda \\rightarrow \\mathbf {Lagr}_\\Lambda $ is not faithful.",
"Indeed, given any cospan $H\\stackrel{i}{\\longrightarrow }T\\stackrel{i^{\\prime }}{\\longleftarrow }H^{\\prime }$ and any $\\Lambda $ -module $\\widetilde{T}$ , consider the cospan given by $H\\stackrel{(i,0)}{\\longrightarrow }T\\oplus \\widetilde{T}\\stackrel{(i^{\\prime },0)}{\\longleftarrow }H^{\\prime }$ .",
"One immediately checks the equality $N_{T\\oplus \\widetilde{T}}=N_T$ .",
"Therefore, if the first cospan is Lagrangian and $\\widetilde{T}$ is non-trivial, then these two cospans represent different morphisms in $\\mathbf {L}_\\Lambda $ mapped by $F$ to the same morphism in $\\mathbf {Lagr}_\\Lambda $ .",
"The core of the category $\\mathbf {L}_\\Lambda $ Recall that the core of a category $\\mathcal {C}$ is the maximal sub-groupoid of $\\mathcal {C}$ .",
"In other words, $\\mathit {core}(\\mathcal {C})$ is the subcategory of $\\mathcal {C}$ consisting of all objects of $\\mathcal {C}$ and with morphisms all the isomorphisms of $\\mathcal {C}$ .",
"We shall say that a cospan $H\\stackrel{i}{\\longrightarrow }T\\stackrel{i^{\\prime }}{\\longleftarrow }H^{\\prime }$ is invertible if $i$ and $i^{\\prime }$ are $\\Lambda $ -isomorphisms.",
"Proposition 2.7 The core of $\\mathbf {L}_\\Lambda $ consists of Hermitian $\\Lambda $ -modules, as objects, and isomorphism classes of invertible Lagrangian cospans, as morphisms.",
"Furthermore, the map assigning to such a cospan $H\\stackrel{i}{\\longrightarrow }T\\stackrel{i^{\\prime }}{\\longleftarrow }H^{\\prime }$ the $\\Lambda $ -isomorphism $i^{\\prime -1}i\\colon H\\rightarrow H^{\\prime }$ gives rise to an equivalence of categories $\\mathit {core}(\\mathbf {L}_\\Lambda ) \\stackrel{\\simeq }{\\longrightarrow }\\mathbf {U}_\\Lambda $ which fits in the commutative diagram $@R0.5cm{\\mathit {core}(\\mathbf {L}_\\Lambda )[r][d]_\\simeq &\\mathbf {L}_\\Lambda [d]^F\\\\\\mathbf {U}_\\Lambda [r]^{\\Gamma }&\\mathbf {Lagr}_\\Lambda \\,.",
"}$ Given a cospan $H\\stackrel{i}{\\longrightarrow }T\\stackrel{i^{\\prime }}{\\longleftarrow }H^{\\prime }$ with $i$ an isomorphism, one easily checks that the diagram $@R0.5cm{& & H & & \\\\& T [ru]^{i^{-1}} & & T [ul]_{i^{-1}} & \\\\H [ru]^{i} & & H^{\\prime } [ru]^{i^{\\prime }}[lu]_{i^{\\prime }} & &H [lu]_{i}}$ satisfies the universal property for the pushout defining the composition of $H\\stackrel{i}{\\longrightarrow }T\\stackrel{i^{\\prime }}{\\longleftarrow }H^{\\prime }$ with $H^{\\prime }\\stackrel{i^{\\prime }}{\\longrightarrow }T\\stackrel{i}{\\longleftarrow }H$ .",
"Hence, if both $i$ and $i^{\\prime }$ are isomorphisms, then these cospans are inverse of one another, and therefore isomorphisms in $\\mathbf {L}_\\Lambda $ , i.e.",
"morphisms in $\\mathit {core}(\\mathbf {L}_\\Lambda )$ .",
"Conversely, let $H\\stackrel{i_1}{\\longrightarrow }T_1\\stackrel{i_1^{\\prime }}{\\longleftarrow }H^{\\prime }$ be a morphism in $\\mathit {core}(\\mathbf {L}_\\Lambda )$ , and let $H^{\\prime }\\stackrel{i_2^{\\prime }}{\\longrightarrow }T_2\\stackrel{i_2}{\\longleftarrow }H$ be its inverse.",
"Working with the cokernel representatives of $T_2\\circ T_1$ and $T_1\\circ T_2$ , this means that there exist $\\Lambda $ -isomorphisms $C:=\\mathit {Coker}(-i_1^{\\prime },i^{\\prime }_2)\\stackrel{\\varphi }{\\rightarrow } H$ and $C^{\\prime }:=\\mathit {Coker}(-i_2,i_1)\\stackrel{\\varphi ^{\\prime }}{\\rightarrow } H^{\\prime }$ such that the following diagrams commute: $@R0.5cm{& & H & & \\\\& & C [u]^{\\varphi }_{\\simeq } & & \\\\& T_1 [ru]^{j_1} & & T_2 [ul]_{j_2} &\\\\H [ru]^{i_1} @/^2pc/[rruuu]^{\\mathit {id}_H} & & H^{\\prime } [ru]^{i^{\\prime }_2}[lu]_{i_1^{\\prime }} & &H [lu]_{i_2}@/_2pc/[lluuu]_{\\mathit {id}_H}}\\ @R0.5cm{& & H^{\\prime } & & \\\\& & C^{\\prime } [u]^{\\varphi ^{\\prime }}_{\\simeq } & & \\\\& T_2 [ru]^{j^{\\prime }_2} & & T_1 [ul]_{j^{\\prime }_1} &\\\\H^{\\prime } [ru]^{i^{\\prime }_2} @/^2pc/[rruuu]^{\\mathit {id}_{H^{\\prime }}} & & H [ru]^{i_1}[lu]_{i_2} & &H^{\\prime } [lu]_{i^{\\prime }_1}@/_2pc/[lluuu]_{\\mathit {id}_{H^{\\prime }}}\\,.",
"}$ This implies that the following diagram has exact rows, and is commutative: ${0 [r] & H[r]^{\\!\\!\\!\\!\\!\\!\\!",
"(-1,1)}[d]_{\\varphi ^{\\prime }j_1^{\\prime } i_1}& H\\oplus H [r]^{\\;\\;\\;\\;\\;{1\\atopwithdelims ()1}}[d]^{i_1\\oplus i_2}&H[d]^{\\varphi ^{-1}}_\\simeq [r]& 0\\phantom{\\,.",
"}\\\\0 [r] & H^{\\prime }[r]_{\\!\\!\\!\\!\\!\\!\\!",
"(-i_1^{\\prime },i^{\\prime }_2)} & T_1\\oplus T_2[r]_{\\;\\;\\;\\;\\;{j_1\\atopwithdelims ()j_2}} & C [r]& 0\\,.",
"}$ Using the universal property of the pushouts $T_2\\circ T_1$ and $T_1\\circ T_2$ , one can check that the maps $\\varphi ^{\\prime } j_1^{\\prime }i_1=\\varphi ^{\\prime } j_2^{\\prime }i_2\\colon H\\rightarrow H^{\\prime }$ and $\\varphi j_1i_1^{\\prime }=\\varphi j_2i^{\\prime }_2\\colon H^{\\prime }\\rightarrow H$ are inverse of each other, and therefore isomorphisms.",
"By the five-lemma applied to the diagram above, $i_1$ and $i_2$ are also isomorphisms.",
"Exchanging the roles of $T_1$ and $T_2$ leads to the same conclusion for $i_1^{\\prime }$ and $i^{\\prime }_2$ , so both these cospans are invertible.",
"Now, let $G\\colon \\mathit {core}(\\mathbf {L}_\\Lambda ) \\rightarrow \\mathbf {U}_\\Lambda $ be defined by assigning to the invertible cospan $H\\stackrel{i}{\\longrightarrow }T\\stackrel{i^{\\prime }}{\\longleftarrow }H^{\\prime }$ the $\\Lambda $ -isomorphism $i^{\\prime -1}i\\colon H\\rightarrow H^{\\prime }$ .",
"First note that isomorphic cospans are mapped to the same isomorphism.",
"Next, observe that for any two invertible cospans $H\\stackrel{i_1}{\\longrightarrow }T_1\\stackrel{i_1^{\\prime }}{\\longleftarrow }H^{\\prime }$ and $H^{\\prime }\\stackrel{i^{\\prime }_2}{\\longrightarrow }T_2\\stackrel{i_2}{\\longleftarrow }H^{\\prime \\prime }$ , we have $G(T_2 \\circ T_1)=(j_2i_2)^{-1}(j_1i_1)=i_2^{-1}j_2^{-1}j_1i_1=i_2^{-1}i^{\\prime }_2i_1^{\\prime -1}i_1 = G(T_2) \\circ G(T_1)\\,.$ (Here, we used the fact that since $j_2i_2$ and $i_2$ are isomorphisms, so is $j_2$ .)",
"We now check that if a cospan $H\\stackrel{i}{\\longrightarrow }T\\stackrel{i^{\\prime }}{\\longleftarrow }H^{\\prime }$ is Lagrangian, then $i^{\\prime -1}i$ is unitary.",
"Indeed, $F(T)=\\overline{N_T}=\\overline{\\mathit {Ker}{\\textstyle {-i\\atopwithdelims ()\\phantom{-}i^{\\prime }}}}=\\mathit {Ker}{\\textstyle {-i\\atopwithdelims ()\\phantom{-}i^{\\prime }}}=\\mathit {Ker}{\\textstyle {-i^{\\prime -1}i\\atopwithdelims ()\\mathit {id}}}=\\Gamma _{i^{\\prime -1}i}$ is a Lagrangian subspace of $(-H)\\oplus H^{\\prime }$ .",
"Therefore, for any $x\\in H$ and $y\\in H^{\\prime }$ , we have $0=(-\\omega \\oplus \\omega ^{\\prime })(x \\oplus i^{\\prime -1}(i(x)), y \\oplus i^{\\prime -1}(i(y)))=-\\omega (x,y)+\\omega ^{\\prime }(i^{\\prime -1}(i(x)),i^{\\prime -1}(i(y)))\\,,$ so $i^{\\prime -1}i$ is indeed unitary.",
"The equality $F(T)=\\Gamma _{i^{\\prime -1}i}$ displayed above also shows the commutativity of the diagram in the statement.",
"It only remains to check that $G$ is a fully-faithful functor.",
"Given any unitary isomorphism $f\\colon H\\rightarrow H^{\\prime }$ , the cospan $H\\stackrel{f}{\\longrightarrow }H^{\\prime }\\stackrel{\\mathit {id}}{\\longleftarrow }H^{\\prime }$ is invertible, Lagrangian, and is mapped to $f$ by $G$ .",
"Finally, if $H\\stackrel{i_1}{\\longrightarrow }T_1\\stackrel{i_1^{\\prime }}{\\longleftarrow }H^{\\prime }$ and $H\\stackrel{i_2}{\\longrightarrow }T_2\\stackrel{i_2^{\\prime }}{\\longleftarrow }H^{\\prime }$ are invertible cospans with $i_1^{\\prime -1}i_1=i_2^{\\prime -1}i_2$ , then the map $i_2i_1^{-1}=i^{\\prime }_2i_1^{\\prime -1}\\colon T_1\\rightarrow T_2$ defines an isomorphism between these two cospans.",
"Since $\\Lambda $ is an integral domain, we can consider its quotient field $Q=Q(\\Lambda )$ .",
"We shall call a cospan $H\\stackrel{i}{\\longrightarrow }T\\stackrel{i^{\\prime }}{\\longleftarrow }H^{\\prime }$ rationally invertible if $i_Q:= i \\otimes Q$ and $i_Q^{\\prime }:=i^{\\prime } \\otimes Q$ are $Q$ -isomorphisms.",
"The next proposition can be checked by the same arguments as Proposition REF .",
"We therefore leave the proof to the reader.",
"Proposition 2.8 Hermitian $\\Lambda $ -modules, as objects, and isomorphism classes of rationally invertible Lagrangian cospans, as morphisms, form a category $\\mathit {core}(\\mathbf {L}_\\Lambda )^0$ .",
"Furthermore, the map assigning to such a cospan $H\\stackrel{i}{\\longrightarrow }T\\stackrel{i^{\\prime }}{\\longleftarrow }H^{\\prime }$ the $Q$ -isomorphism $i_Q^{\\prime -1}i_Q$ gives rise to a full functor $\\mathit {core}(\\mathbf {L}_\\Lambda )^0\\rightarrow \\mathbf {U}^0_\\Lambda $ which fits in the following commutative diagram: $@R0.5cm{\\mathit {core}(\\mathbf {L}_\\Lambda )^0[r][d]&\\mathbf {L}_\\Lambda [d]^F\\\\\\mathbf {U}^0_\\Lambda [r]^{\\Gamma ^0}&\\mathbf {Lagr}_\\Lambda \\,.",
"}$ $\\Box $ Summarizing this section, we have six categories which all have Hermitian $\\Lambda $ -modules as objects.",
"They fit in the following commutative diagram $@R0.5cm{\\mathit {core}(\\mathbf {L}_\\Lambda ) [r] [d]_\\simeq & \\mathit {core}(\\mathbf {L}_\\Lambda )^0 [d][r] & \\mathbf {L}_\\Lambda [d]^F\\\\\\mathbf {U}_\\Lambda [r]^{-\\otimes Q} @/_2pc/[rr]_\\Gamma & \\mathbf {U}_\\Lambda ^0 [r]^{\\Gamma ^0}& \\mathbf {Lagr}_\\Lambda \\,,}$ where the horizontal arrows are embeddings of categories, the left-most vertical arrow is an equivalence of categories, and the two remaining ones are full functors.",
"A bicategory of Lagrangian cospans The aim of this section is to endow $\\mathbf {L}_\\Lambda $ with the structure of a bicategory.",
"We begin by recalling in subsection REF the notions of bicategory and weak 2-functor, before defining the bicategory of Lagrangian cospans in subsection REF .",
"2-categories and 2-functors Following the original work of Bénabou [3], it is a traditional practice to use the term “2-category” for what Kapranov and Voevodsky call a “strict 2-category” [10].",
"As it turns out, the categories that appear in our work are not of this type, but have a richer structure: that of some type of weak 2-category known as a bicategory.",
"We now recall the definition of this structure, following [3].",
"A bicategory $\\mathcal {C}$ consists of the following data: A set $\\mathrm {Ob}\\mathcal {C}$ whose elements are called objects.",
"For each pair of objects $(X,Y)$ , a category $\\mathcal {C}(X,Y)$ whose objects are called 1-morphisms and denoted by $f\\colon X\\rightarrow Y$ or by $X\\stackrel{f}{\\rightarrow }Y$ , whose morphisms are called 2-morphisms and denoted by $\\alpha \\colon f\\Rightarrow g$ , or by $@C+2pc{X^{f}_{g}{\\;\\;\\;\\alpha }&Y}$ , and whose composition is called vertical composition and denoted by $\\left(@C+2pc{X^{f}_{g}{\\;\\;\\;\\alpha }&Y},@C+2pc{X^{g}_{h}{\\;\\;\\;\\beta }&Y}\\right)\\mapsto @C+2pc{X^{f}_{h}{\\;\\;\\;\\;\\;\\;\\beta \\star \\alpha }&Y}\\,.$ We shall denote the identity morphism for $f$ by $\\mathit {Id}_f\\colon f\\Rightarrow f$ .",
"For each object $X$ , an identity 1-morphism $I_X\\colon X\\rightarrow X$ .",
"For each triple of objects $(X,Y,Z)$ , a functor $\\mathcal {C}(X,Y)\\times \\mathcal {C}(Y,Z)\\rightarrow \\mathcal {C}(X,Z)$ denoted by $\\left(@C+2pc{X^{f}_{g}{\\;\\;\\;\\alpha }&Y},@C+2pc{Y^{k}_{\\ell }{\\;\\;\\;\\beta }&Z}\\right)\\mapsto @C+2pc{X^{k\\circ f}_{\\ell \\circ g}{\\;\\;\\;\\;\\;\\;\\beta \\bullet \\alpha }&Z\\,,}$ and called the horizontal composition functor.",
"Note that the functoriality of this composition boils down to the identity $\\mathit {Id}_f \\bullet \\mathit {Id}_g = \\mathit {Id}_{g\\circ f}$ for any composable 1-morphisms $f$ and $g$ , and to the interchange law $(\\delta \\bullet \\beta ) \\star (\\gamma \\bullet \\alpha )=(\\delta \\star \\gamma ) \\bullet (\\beta \\star \\alpha )\\,,$ for each composable 2-morphisms $\\alpha ,\\beta ,\\gamma $ and $\\delta $ .",
"This last condition is best understood by saying that the following 2-morphism is well-defined, i.e.",
"independent of the order of the compositions: $@C+2pc{X^{}{\\;\\;\\;\\alpha } [r] _{}{\\;\\;\\;\\beta } & Y ^{}{\\;\\;\\;\\gamma } [r] _{}{\\;\\;\\;\\delta } & Z\\,.",
"}$ If this horizontal composition is associative (both on the 1-morphisms and 2-morphisms) and admits $I_X$ as a two-sided unit, then we are in the presence of a (strict) 2-category.",
"As mentioned above, a bicategory has a richer structure: the horizontal composition is associative and unital only up to natural isomorphisms, which are part of the structure.",
"To be more precise, a bicategory also contains the following data: For any triple of composable 1-morphisms $f,g,h$ , an invertible 2-morphism $a=a_{fgh}\\colon (h\\circ g)\\circ f\\Rightarrow h\\circ (g\\circ f)$ which is natural in $f,g$ and $h$ , and called the associativity isomorphism.",
"For any 1-morphism $X\\stackrel{f}{\\rightarrow }Y$ , two invertible 2-morphisms $\\ell =\\ell _f\\colon I_Y\\circ f\\Rightarrow f$ and $r=r_f\\colon f\\circ I_X \\Rightarrow f$ which are natural in $f$ .",
"These natural isomorphisms must satisfy the following two coherence axioms.",
"Given four composable 1-morphisms $e,f,g,h$ , there are two natural ways to pass from $((h\\circ g)\\circ f)\\circ e$ to $h\\circ (g\\circ (f\\circ e))$ using the associativity isomorphisms, one in two steps, the other one in three.",
"The associativity coherence axiom requires that these two compositions coincide.",
"Finally, given any 1-morphisms $X\\stackrel{f}{\\rightarrow }Y\\stackrel{g}{\\rightarrow }Z$ , the identity coherence axiom requires the composition $(g\\circ I_Y)\\circ f\\stackrel{a}{\\Longrightarrow } g\\circ (I_Y\\circ f)\\stackrel{\\mathit {Id}_g\\bullet \\ell _f}{\\Longrightarrow }g\\circ f$ to coincide with $r_g\\bullet \\mathit {Id}_f$ .",
"Let us now recall the definition of a weak 2-functor, also known as a pseudofunctor [9] and originally called a homomorphism of bicategories by Bénabou [3].",
"Given two bicategories $\\mathcal {C}$ and $\\mathcal {D}$ , a weak 2-functor $\\mathcal {F}\\colon \\mathcal {C}\\rightarrow \\mathcal {D}$ consists of the following data: A map $\\mathcal {F}\\colon \\mathrm {Ob}\\mathcal {C}\\rightarrow \\mathrm {Ob}\\mathcal {D}$ .",
"For each pair of objects $(X,Y)$ in $\\mathcal {C}$ , a functor $\\mathcal {C}(X,Y)\\rightarrow \\mathcal {D}(\\mathcal {F}(X),\\mathcal {F}(Y))$ denoted by $@C+2pc{X^{f}_{g}{\\;\\;\\;\\alpha }&Y}\\mapsto @C+2pc{\\mathcal {F}(X)^{\\mathcal {F}(f)}_{\\mathcal {F}(g)}{\\;\\;\\;\\;\\;\\;\\mathcal {F}(\\alpha )}&\\mathcal {F}(Y)}\\,.$ Note that this functoriality is equivalent to the identities $\\mathcal {F}(\\beta \\star \\alpha )=\\mathcal {F}(\\beta )\\star \\mathcal {F}(\\alpha )\\quad \\text{and}\\quad \\mathcal {F}(\\mathit {Id}_f)=\\mathit {Id}_{\\mathcal {F}(f)}\\,.$ If we also have the identities $\\mathcal {F}(I_X)=I_{\\mathcal {F}(X)}$ , $\\mathcal {F}(g\\circ f)=\\mathcal {F}(g)\\circ \\mathcal {F}(f)$ and $\\mathcal {F}(\\beta \\bullet \\alpha )=\\mathcal {F}(\\beta )\\bullet \\mathcal {F}(\\alpha )$ , then we are in the presence of a (strict) 2-functor.",
"Our functor has a finer structure: once again, these identities hold up to isomorphisms of functors, which are part of the data as follows.",
"For each object $X$ of $\\mathcal {C}$ , an invertible 2-morphism $\\varphi _X\\colon I_{\\mathcal {F}(X)}\\Rightarrow \\mathcal {F}(I_X)$ in $\\mathcal {D}$ .",
"For each $X\\stackrel{f}{\\rightarrow }Y\\stackrel{g}{\\rightarrow }Z$ , an invertible 2-morphism $\\varphi =\\varphi _{fg}\\colon \\mathcal {F}(g)\\circ \\mathcal {F}(f)\\Rightarrow \\mathcal {F}(g\\circ f)$ in $\\mathcal {D}$ such that for any $@C+2pc{X^{f}_{f^{\\prime }}{\\;\\;\\;\\alpha }&Y^{g}_{g^{\\prime }}{\\;\\;\\;\\beta }&Y\\,,}$ we have the following equality of 2-morphisms in $\\mathcal {D}$ : $\\varphi _{f^{\\prime }g^{\\prime }}\\star (\\mathcal {F}(\\beta )\\bullet \\mathcal {F}(\\alpha ))=\\mathcal {F}(\\beta \\bullet \\alpha )\\star \\varphi _{fg}\\,\\colon \\,\\mathcal {F}(g)\\circ \\mathcal {F}(f)\\Rightarrow \\mathcal {F}(g^{\\prime }\\circ f^{\\prime })\\,.$ Finally, these isomorphisms of functors are required to satisfy two coherence axioms.",
"For any composable 1-morphisms $f,g,h$ of $\\mathcal {C}$ , the composition $(\\mathcal {F}(h)\\circ \\mathcal {F}(g))\\circ \\mathcal {F}(f)\\stackrel{a}{\\Rightarrow }\\mathcal {F}(h)\\circ (\\mathcal {F}(g)\\circ \\mathcal {F}(f))\\stackrel{\\mathit {Id}\\bullet \\varphi }{\\Longrightarrow }\\mathcal {F}(h)\\circ \\mathcal {F}(g\\circ f)\\stackrel{\\varphi }{\\Rightarrow }\\mathcal {F}(h\\circ (g\\circ f))$ is equal to the composition $(\\mathcal {F}(h)\\circ \\mathcal {F}(g))\\circ \\mathcal {F}(f)\\stackrel{\\varphi \\bullet \\mathit {Id}}{\\Longrightarrow }\\mathcal {F}(h\\circ g)\\circ \\mathcal {F}(f)\\stackrel{\\varphi }{\\Rightarrow }\\mathcal {F}((h\\circ g)\\circ f)\\stackrel{\\mathcal {F}(a)}{\\Longrightarrow }\\mathcal {F}(h\\circ (g\\circ f))\\,.$ For any $X\\stackrel{f}{\\rightarrow }Y$ , the composition $I_{\\mathcal {F}(Y)}\\circ \\mathcal {F}(f)\\stackrel{\\varphi _Y\\bullet \\mathit {Id}}{\\Longrightarrow }\\mathcal {F}(I_Y)\\circ \\mathcal {F}(f)\\stackrel{\\varphi }{\\Rightarrow }\\mathcal {F}(I_Y\\circ f)\\stackrel{\\mathcal {F}(\\ell )}{\\Longrightarrow }\\mathcal {F}(f)$ coincides with $\\ell _{\\mathcal {F}(f)}$ , and similarly for $r$ .",
"A bicategory of cospans Our goal is now to define a bicategory of Lagrangian cospans.",
"To do so, we will first work in the more general setting of a category with pushouts.",
"We wish to emphasize that our resulting bicategory of cospans differs from the usual definition considered in the literature, where the 2-morphisms are usually taken to be morphisms of cospans (see e.g. [3]).",
"On the other hand, a notion dual to the 2-morphisms that we consider was already studied by Morton [14] in another context (see also [15]).",
"Throughout this section, $\\mathbf {C}$ is a category with pushouts in which we fix a pushout for each span.",
"The objects of our bicategory are the objects of $\\mathbf {C}$ and the 1-morphisms are the cospans in $\\mathbf {C}$ , where the horizontal composition is given by our choice of a fixed pushout.",
"It remains to define the 2-morphisms, the vertical composition, the associativity and identity isomorphisms, and what is left of the horizontal composition.",
"A 2-cospan in $\\mathbf {C}$ from $H\\stackrel{i_1}{\\longrightarrow }T_1\\stackrel{i_1^{\\prime }}{\\longleftarrow }H^{\\prime }$ to $H\\stackrel{i_2}{\\longrightarrow }T_2\\stackrel{i_2^{\\prime }}{\\longleftarrow }H^{\\prime }$ consists of a cospan $T_1 \\stackrel{\\alpha _1}{\\longrightarrow }A\\stackrel{\\alpha _2}{\\longleftarrow }T_2$ in $\\mathbf {C}$ for which the two following squares commute $@R0.5cm{& T_1[d]^{\\alpha _1} & \\\\H [ur]^{i_1} [dr]_{i_2} & A & \\;H^{\\prime }\\,.",
"[ul]_{i^{\\prime }_1} [dl]^{i^{\\prime }_2} \\\\& T_2 [u]_{\\alpha _2} &}$ Two such 2-cospans $T_1 \\stackrel{\\alpha _1}{\\longrightarrow }A\\stackrel{\\alpha _2}{\\longleftarrow }T_2$ and $T_1 \\stackrel{\\alpha _1^{\\prime }}{\\longrightarrow }A^{\\prime }\\stackrel{\\alpha _2^{\\prime }}{\\longleftarrow }T_2$ are said to be isomorphic if there is a $\\mathbf {C}$ -isomorphism $f \\colon A \\rightarrow A^{\\prime }$ such that $f\\alpha _1=\\alpha _1^{\\prime }$ and $f\\alpha _2=\\alpha _2^{\\prime }$ .",
"Abusing notation, we shall often denote the isomorphism class of such a 2-cospan by $A\\colon T_1\\Rightarrow T_2$ .",
"These will be the 2-morphisms in our 2-category.",
"Let us now proceed with the definition of the vertical composition of the 2-morphisms $A\\colon T_1\\Rightarrow T_2$ and $B\\colon T_2\\Rightarrow T_3$ .",
"It is best explained by the diagram $@R0.5cm{& T_1[d]_{\\alpha _1} & \\\\& A & \\\\H @/^1pc/[uur]^{i_1} [r]^{i_2} & T_2 [u]^{\\alpha _2} & H^{\\prime }[l]_{i_2^{\\prime }}@/_1pc/[uul]_{i^{\\prime }_1}\\\\& \\star & \\\\H @/_1pc/[ddr]_{i_3} [r]^{i_2} & T_2 [d]_{\\beta _2} & H^{\\prime }[l]_{i_2^{\\prime }} @/^1pc/[ddl]^{i^{\\prime }_3} \\\\& B & \\\\& T_3[u]^{\\beta _3} & \\\\} \\ \\ @R0.5cm{\\\\ \\\\ \\\\ \\\\ = \\\\ \\\\ \\\\}\\ \\ @R0.5cm{\\\\ \\\\& T_1[d]^{v_A\\alpha _1} & \\\\H @/^1pc/[ur]^{i_1} @/_1pc/[dr]_{i_3} & B \\star A & H^{\\prime },@/_1pc/[ul]_{i_1^{\\prime }} @/^1pc/[dl]^{i_3^{\\prime }} \\\\& T_3 [u]_{v_B\\beta _3} & \\\\\\\\\\\\}$ where $B \\star A$ and $v_A, v_B$ are given by the pushout diagram $@R0.5cm{& & B \\star A & & & \\\\& A [ru]^{v_A} & & B [ul]_{v_B} \\\\T_1[ru]^{\\alpha _1} & & T_2 [ru]^{\\beta _2}[lu]_{\\alpha _2} & &T_3.",
"[lu]_{\\beta _3}&}$ One can easily check that this indeed defines a 2-cospan.",
"Remark 3.1 In the special case where $\\alpha _2=\\mathit {id}_{T_2}$ (resp.",
"$\\beta _2=\\mathit {id}_{T_2}$ ), this vertical composition $T_1\\stackrel{v_A\\alpha _1}{\\longrightarrow }B\\star A\\stackrel{v_B\\beta _3}{\\longleftarrow }T_3$ is isomorphic to $T_1\\stackrel{\\beta _2\\alpha _1}{\\longrightarrow }B\\stackrel{\\beta _3}{\\longleftarrow }T_3$ (resp.",
"$T_1\\stackrel{\\alpha _1}{\\longrightarrow }A\\stackrel{\\alpha _2\\beta _3}{\\longleftarrow }T_3$ ).",
"This is a direct consequence of Remark REF .",
"On the level 2-morphisms, the horizontal composition of $A\\colon T_1\\Rightarrow T_2$ and $B\\colon T_3\\Rightarrow T_4$ is described by the diagram $@R0.5cm{ & T_3 [d]^{\\beta _3} & \\\\H^{\\prime } [ur]^{i_3^{\\prime }} [dr]_{i_4^{\\prime }} & B & H^{\\prime \\prime }[ul]_{i^{\\prime \\prime }_3} [dl]^{i^{\\prime \\prime }_4} \\\\& T_4 [u]_{\\beta _4} }\\ @R0.5cm{ \\\\ \\bullet \\\\ }\\ @R0.5cm{& T_1[d]^{\\alpha _1} & \\\\H [ur]^{i_1} [dr]_{i_2} & A & H^{\\prime }[ul]_{i^{\\prime }_1} [dl]^{i^{\\prime }_2} \\\\& T_2 [u]_{\\alpha _2} &}\\ @R0.5cm{ \\\\ = \\\\ }\\ @R0.5cm{& T_3\\circ T_1[d]^{h_{31}} & \\\\H @/^1pc/[ur]^{j_1i_1} @/_1pc/[dr]_{j_2i_2} & B \\bullet A & \\;H^{\\prime \\prime }\\,,@/_1pc/[ul]_{j_3i_3^{\\prime \\prime }} @/^1pc/[dl]^{j_4i_4^{\\prime \\prime }} \\\\& T_4\\circ T_2 [u]_{h_{42}} &}$ where $j_1,\\dots ,j_4$ are the maps that arise in the compositions $T_3\\circ T_1$ and $T_4\\circ T_2$ (see the diagrams below), $B \\bullet A$ is given by the pushout $@R0.5cm{& & B \\bullet A & & & \\\\& A [ru]^{h_A} & & B [ul]_{h_B} \\\\H[ru]^{\\alpha _1i_1} & & H^{\\prime } [ru]^{\\beta _3i_3^{\\prime }}[lu]_{\\alpha _2i_2^{\\prime }} & &H^{\\prime \\prime }\\,,[lu]_{\\beta _4i_4^{\\prime \\prime }}&}$ and the maps $h_{31}$ and $h_{42}$ are obtained as follows.",
"Since $h_A \\alpha _1 i_1^{\\prime }=h_A \\alpha _2 i_2^{\\prime }=h_B \\beta _3 i_3^{\\prime }$ and $h_A\\alpha _2 i_2^{\\prime }=h_B\\beta _3i_3^{\\prime }=h_B\\beta _4i_4^{\\prime }$ , the pushout diagrams $@R0.4cm{& & B \\bullet A & & & \\\\& & T_3\\circ T_1 [u]^{h_{31}} & & & \\\\& A @/^1pc/[uur]^{h_A} & & B @/_1pc/[uul]_{h_B} & & \\\\& T_1 [ruu]^{j_1} [u]^{\\alpha _1} & & T_3 [uul]_{j_3} [u]_{\\beta _3} \\\\& & H^{\\prime } [lu]_{i_1^{\\prime }}[ru]^{i_3^{\\prime }} & &}@R0.4cm{& & B \\bullet A & & & \\\\& & T_4\\circ T_2 [u]^{h_{42}} & & & \\\\& A @/^1pc/[uur]^{h_A} & & B @/_1pc/[uul]_{h_B} & & \\\\& T_2 [ruu]^{j_2} [u]^{\\alpha _2} & & T_4 [uul]_{j_4} [u]_{\\beta _4} \\\\& & H^{\\prime } [lu]_{i_2^{\\prime }}[ru]^{i_4^{\\prime }} & &}$ provide maps $h_{31}$ and $h_{42}$ which turn $T_3 \\circ T_1 \\stackrel{h_{31}}{\\longrightarrow }B \\bullet A\\stackrel{h_{42}}{\\longleftarrow }T_4 \\circ T_2$ into a 2-cospan, as one easily checks.",
"The proof of the following theorem can be found in [15] in the dual context of spans.",
"It applies without change to the present setting.",
"Theorem 3.2 Let $\\mathbf {C}$ be a category with pushouts in which a choice of pushout is fixed for each span.",
"Objects in $\\mathbf {C}$ , as objects, cospans in $\\mathbf {C}$ , as morphisms, and isomorphism classes of 2-cospans in $\\mathbf {C}$ , as 2-morphisms, form a bicategory.",
"Note that strictly speaking, this bicategory depends on the choice of pushouts.",
"However, another choice would give a bicategory isomorphic in an obvious sense (see e.g. [3]).",
"The special case where $\\mathbf {C}$ is the category of $\\Lambda $ -modules and the morphisms are Lagrangian cospans yields the following corollary.",
"Corollary 3.3 Fix a pushout for each span of $\\Lambda $ -modules.",
"Hermitian $\\Lambda $ -modules, as objects, Lagrangian cospans, as morphisms, and isomorphism classes of 2-cospans, as 2-morphisms, form a bicategory.$\\Box $ We shall call it “the” bicategory of Lagrangian cospans.",
"The Burau-Alexander 2-functor The aim of this section is to define a weak 2-functor $\\mathcal {B}$ from the bicategory of oriented tangles to the bicategory of Lagrangian cospans where $\\Lambda $ is the ring ${Z}[t^{\\pm 1}]$ of Laurent polynomials in one variable with integer coefficients.",
"We proceed in two steps: in Subsection REF , we recall the definition of the category of oriented tangles, and construct a functor $\\mathcal {B} \\colon \\textbf {Tangles} \\rightarrow \\textbf {L}_\\Lambda $ .",
"In Subsections REF and REF , we study the bicategory of tangles, and convert $\\mathcal {B}$ into a weak 2-functor with values in the bicategory of Lagrangian cospans.",
"The functor $\\mathcal {B}$ on objects and 1-morphisms We start by recalling the definition of the category of oriented tangles.",
"Let $D^2$ be the closed unit disk in ${R}^2$ .",
"Given a non-negative integer $n$ , let $x_j$ be the point $((2j-n-1)/n,0)$ in $D^2$ , for $j=1,\\dots ,n$ .",
"Let $\\varepsilon $ and $\\varepsilon ^{\\prime }$ be sequences of $\\pm 1$ 's of respective length $n$ and $n^{\\prime }$ .",
"An $(\\varepsilon ,\\varepsilon ^{\\prime })$ -tangle is a pair consisting of the cylinder $D^2\\times [0,1]$ and an oriented smooth 1-submanifold $\\tau $ whose oriented boundary is $\\sum _{j=1}^{n^{\\prime }}\\varepsilon ^{\\prime }_j(x^{\\prime }_j,1)-\\sum _{j=1}^{n}\\varepsilon _i(x_j,0)$ .",
"Note that a $(\\emptyset ,\\emptyset )$ -tangle is nothing but an oriented link.",
"Two $(\\varepsilon ,\\varepsilon ^{\\prime })$ -tangles $\\tau _1$ and $\\tau _2$ are isotopic if there exists an isotopy $h_t$ of $D^2\\times [0,1]$ , keeping $D^2\\times \\lbrace 0,1\\rbrace $ fixed, such that $h_1\\vert _{\\tau _1}\\colon \\tau _1\\simeq \\tau _2$ is an orientation-preserving homeomorphism.",
"We shall denote by $I_\\varepsilon $ the isotopy class of the trivial $(\\varepsilon ,\\varepsilon )$ -tangle $(D^2,\\lbrace x_1,\\dots ,x_n\\rbrace )\\times [0,1]$ .",
"Given an $(\\varepsilon ,\\varepsilon ^{\\prime })$ -tangle $\\tau _1$ and an $(\\varepsilon ^{\\prime },\\varepsilon ^{\\prime \\prime })$ -tangle $\\tau _2$ , their composition is the $(\\varepsilon ,\\varepsilon ^{\\prime \\prime })$ -tangle $\\tau _2\\circ \\tau _1$ obtained by gluing the two cylinders along the disk corresponding to $\\varepsilon ^{\\prime }$ , smoothing it if needed, and shrinking the length of the resulting cylinder by a factor 2 (see Figure REF ).",
"Clearly, the composition induces a composition on the isotopy classes of tangles, which is associative and admits $I_\\varepsilon $ as a 2-sided unit.",
"Therefore, the sequences of $\\pm 1$ 's, as objects, and the isotopy classes of tangles, as morphisms, form a category denoted by $\\mathbf {Tangles}$ and called the category of oriented tangles.",
"Figure: An (ε,ε ' )(\\varepsilon ,\\varepsilon ^{\\prime })-tangle τ 1 \\tau _1 with ε=(+1,-1)\\varepsilon =(+1,-1) and ε ' =(-1,+1,-1,+1)\\varepsilon ^{\\prime }=(-1,+1,-1,+1), an (ε ' ,ε '' )(\\varepsilon ^{\\prime },\\varepsilon ^{\\prime \\prime })-tangle τ 2 \\tau _2with ε '' =(-1,+1)\\varepsilon ^{\\prime \\prime }=(-1,+1), and their composition, the (ε,ε '' )(\\varepsilon ,\\varepsilon ^{\\prime \\prime })-tangle τ 2 ∘τ 1 \\tau _2\\circ \\tau _1.Recall that a tangle $\\tau \\subset D^2 \\times [0,1]$ is called an oriented braid if every component of $\\tau $ is strictly increasing or strictly decreasing with respect to the projection onto $[0,1]$ .",
"The finite sequences of $\\pm 1$ 's as objects, and the isotopy classes of oriented braids, as morphisms, form a subcategory $\\mathbf {Braids}$ of $\\mathbf {Tangles}$ , which is nothing but its core.",
"Finally a tangle $\\tau \\subset D^2\\times [0,1]$ is called an oriented string link if every component of $\\tau $ joins $D^2 \\times \\lbrace 0 \\rbrace $ and $D^2 \\times \\lbrace 1 \\rbrace $ .",
"Isotopy classes of oriented string links are the morphisms of a category $\\mathbf {Strings}$ which satisfies $\\mathbf {Braids} \\subset \\mathbf {Strings} \\subset \\mathbf {Tangles}\\,,$ where all the inclusions denote embeddings of categories.",
"We are now ready to define our Burau-Alexander functor $\\mathcal {B}\\colon \\mathbf {Tangles}\\rightarrow \\mathbf {L}_\\Lambda $ .",
"We start by defining it on objects, following the construction of [6].",
"Denote by $\\mathcal {N}(\\lbrace x_1,\\dots ,x_n\\rbrace )$ an open tubular neighborhood of $\\lbrace x_1,\\dots , x_n \\rbrace $ in $D^2\\subset {R}^2$ , and let $S^2$ be the 2-sphere obtained by the one-point compactification of ${R}^2$ .",
"Given a sequence $\\varepsilon =(\\varepsilon _1,\\dots ,\\varepsilon _n)$ of $\\pm 1$ , set $\\ell _\\varepsilon =\\sum _{i=1}^n \\varepsilon _i$ and endow the compact surface $D_\\varepsilon = \\left\\lbrace \\begin{array}{lr}D^2\\setminus \\mathcal {N}(\\lbrace x_1,\\dots ,x_n\\rbrace ) & \\text{if } \\ell _\\varepsilon \\ne 0\\\\S^2\\setminus \\mathcal {N}(\\lbrace x_1,\\dots ,x_n\\rbrace ) & \\text{if } \\ell _\\varepsilon =0\\end{array}\\right.$ with an orientation (pictured counterclockwise), a base point $z$ , and the generating family $\\lbrace e_1,\\dots , e_n \\rbrace $ of $\\pi _1(D_\\varepsilon ,z)$ , where $e_i$ is a simple loop turning once around $x_i$ counterclockwise if $\\varepsilon _i=+1$ , clockwise if $\\varepsilon _i=-1$ .",
"The same space with the opposite orientation will be denoted by $-D_\\varepsilon $ .",
"The natural epimomorphism $H_1(D_\\varepsilon )\\rightarrow \\mathbb {Z}$ , given by $e_j \\mapsto 1$ induces an infinite cyclic covering $\\widehat{D}_\\varepsilon \\rightarrow D_\\varepsilon $ whose homology is endowed with a structure of module over $\\Lambda ={Z}[t^{\\pm 1}]$ .",
"If $\\ell _\\varepsilon \\ne 0$ , then $D_\\varepsilon $ obviously retracts by deformation on the wedge of $n$ circles representing the generators $e_1,\\dots ,e_n$ of $\\pi _1(D_\\varepsilon ,z)$ , and one can check that $H_1(\\widehat{D}_\\varepsilon )$ is a free $\\Lambda $ -module of rank $n-1$ .",
"(It is free of rank $n-2$ if $\\ell _\\varepsilon $ vanishes.)",
"If $\\langle \\ , \\ \\rangle \\colon H_1(\\widehat{D}_\\varepsilon )\\times H_1(\\widehat{D}_\\varepsilon )\\rightarrow {Z}$ denotes the skew-symmetric intersection form obtained by lifting the orientation of $D_\\varepsilon $ to $\\widehat{D}_\\varepsilon $ , then the formula $\\omega _\\varepsilon (x,y)=\\sum _{k \\in \\mathbb {Z}} \\langle t^kx,y \\rangle t^{-k}$ defines a skew-Hermitian $\\Lambda $ -valued pairing on $H_1(\\widehat{D}_\\varepsilon )$ which is non-degenerate by [6].",
"(This is the only reason for considering $S^2$ instead of $D^2$ when $\\ell _\\varepsilon $ vanishes.)",
"Therefore, following the terminology of subsection REF , $\\mathcal {B}(\\varepsilon ):=(H_1(\\widehat{D}_\\varepsilon ),\\omega _\\varepsilon )$ is a free Hermitian $\\Lambda $ -module for any object $\\varepsilon $ of the category of oriented tangles.",
"Note that this coincides with the definition of the Lagrangian functor $\\mathcal {F}\\colon \\mathbf {Tangles}\\rightarrow \\mathbf {Lagr}_\\Lambda $ of [6] at the level of objects.",
"Let us now turn to morphisms.",
"First note that the existence of an $(\\varepsilon ,\\varepsilon ^{\\prime })$ -tangle $\\tau \\subset D^2 \\times [0,1]$ implies that $\\ell _\\varepsilon =\\ell _{\\varepsilon ^{\\prime }}$ .",
"Denote by $\\mathcal {N}(\\tau )$ an open tubular neighborhood of $\\tau $ in $D^2 \\times [0,1]$ .",
"We shall orient the exterior $X_\\tau = \\left\\lbrace \\begin{array}{lr}(D^2 \\times [0,1]) \\setminus \\mathcal {N}(\\tau ) & \\text{if } \\ell _\\varepsilon \\ne 0\\\\(S^2 \\times [0,1]) \\setminus \\mathcal {N}(\\tau ) & \\text{if } \\ell _\\varepsilon =0\\end{array}\\right.$ of $\\tau $ so that the induced orientation on $\\partial X_\\tau $ extends the orientation on the space $(-D_\\varepsilon ) \\sqcup D_{\\varepsilon ^{\\prime }}$ .",
"Clearly, the abelian group $H_1(X_\\tau )$ is generated by the oriented meridians of the connected components of $\\tau $ .",
"The homomorphism $H_1(X_\\tau )\\rightarrow \\mathbb {Z}$ mapping these meridians to 1 extends the previously defined homomorphisms $H_1(D_\\varepsilon ) \\rightarrow \\mathbb {Z}$ and $H_1(D_{\\varepsilon ^{\\prime }})\\rightarrow \\mathbb {Z}$ .",
"It determines an infinite cyclic covering $\\widehat{X}_\\tau \\rightarrow X_\\tau $ whose homology is endowed with a structure of module over $\\Lambda $ .",
"Let $i_\\tau \\colon H_1(\\widehat{D}_\\varepsilon ) \\rightarrow H_1(\\widehat{X}_\\tau )$ and $i_\\tau ^{\\prime }\\colon H_1(\\widehat{D}_{\\varepsilon ^{\\prime }}) \\rightarrow H_1(\\widehat{X}_\\tau )$ be the homomorphisms induced by the inclusions of $\\widehat{D}_\\varepsilon $ and $\\widehat{D}_{\\varepsilon ^{\\prime }}$ into $\\widehat{X}_\\tau $ .",
"Since $\\mathcal {F}(\\tau )=\\overline{\\mathit {Ker}{-i_\\tau \\atopwithdelims ()\\phantom{-}i_\\tau ^{\\prime }}}$ is a Lagrangian submodule of $(-H_1(\\widehat{D}_\\varepsilon ))\\oplus H_1(\\widehat{D}_\\varepsilon ^{\\prime })$ [6], it follows that $H_1(\\widehat{D}_\\varepsilon ) \\stackrel{i_\\tau }{\\longrightarrow }H_1(\\widehat{X}_\\tau ) \\stackrel{i_\\tau ^{\\prime }}{\\longleftarrow }H_1(\\widehat{D}_{\\varepsilon ^{\\prime }})$ is a Lagrangian cospan for any 1-morphism $\\tau $ in the category of oriented tangles.",
"Note that the equality above is nothing but the definition of the Lagrangian functor $\\mathcal {F}$ of [6] at the level of morphisms.",
"Theorem 4.1 For any sequence $\\varepsilon $ of $\\pm 1$ 's, set $\\mathcal {B}(\\varepsilon )=(H_1(\\widehat{D}_\\varepsilon ),\\omega _\\varepsilon )$ and for any isotopy class $\\tau $ of tangles, let $\\mathcal {B}(\\tau )$ denote the isomorphism class of the Lagrangian cospan $H_1(\\widehat{D}_\\varepsilon ) \\stackrel{i_\\tau }{\\longrightarrow }H_1(\\widehat{X}_\\tau ) \\stackrel{i_\\tau ^{\\prime }}{\\longleftarrow }H_1(\\widehat{D}_{\\varepsilon ^{\\prime }})$ .",
"This defines a functor $\\mathcal {B}\\colon \\mathbf {Tangles}\\rightarrow \\mathbf {L}_\\Lambda $ which fits in the commutative diagram $@R0.5cm{\\mathbf {Braids} [d] [r] @/_3pc/[dd]_\\rho & \\mathbf {String} [d] [r] & \\mathbf {Tangles} [d]^{\\mathcal {B}} @/^2.5pc/[dd]^{\\mathcal {F}}\\ \\\\\\mathit {core}(\\mathbf {L}_\\Lambda ) [d]_\\simeq [r] & \\mathit {core}(\\mathbf {L}_\\Lambda )^0 [d][r] & \\mathbf {L}_\\Lambda [d]^F \\\\\\mathbf {U}_\\Lambda [r]^{- \\otimes Q} @/_1.5pc/[rr]_\\Gamma & \\mathbf {U}^0_\\Lambda [r]^{\\Gamma ^0} & \\mathbf {Lagr}_\\Lambda ,}$ where the left-most vertical arrow is the Burau functor, the horizontal arrows are the embeddings of categories described in Subsections REF and REF , and $F$ is the full functor defined in Subsection REF (recall diagram (REF )).",
"Furthermore, if $\\tau $ is an oriented link, then $\\mathcal {B}(\\tau )$ is nothing but its Alexander module.",
"For any object $\\varepsilon $ , the cospan associated to the identity tangle $I_\\varepsilon $ is canonically isomorphic to the identity cospan $I_{\\mathcal {B}(\\varepsilon )}$ .",
"Let us now check that given $\\tau _1 \\in T(\\varepsilon ,\\varepsilon ^{\\prime })$ and $\\tau _2 \\in T(\\varepsilon ^{\\prime },\\varepsilon ^{\\prime \\prime })$ , we have the equality $\\mathcal {B}(\\tau _2\\circ \\tau _1)=\\mathcal {B}(\\tau _2)\\circ \\mathcal {B}(\\tau _1)$ .",
"Let $H_1(\\widehat{D}_\\varepsilon ) \\stackrel{i_1}{\\rightarrow }H_1(\\widehat{X}_{\\tau _1}) \\stackrel{i_1^{\\prime }}{\\leftarrow }H_1(\\widehat{D}_{\\varepsilon ^{\\prime }})$ and $H_1(\\widehat{D}_{\\varepsilon ^{\\prime }}) \\stackrel{i_2^{\\prime }}{\\rightarrow }H_1(\\widehat{X}_{\\tau _2}) \\stackrel{i_2^{\\prime \\prime }}{\\leftarrow }H_1(\\widehat{D}_{\\varepsilon ^{\\prime \\prime }})$ be the Lagrangian cospans arising from $\\tau _1$ and $\\tau _2$ .",
"We must show that $H_1(\\widehat{D}_\\varepsilon ) \\stackrel{k_1i_1}{\\longrightarrow } H_1(\\widehat{X}_{\\tau _3 \\circ \\tau _1}) \\stackrel{k_2i_2^{\\prime \\prime }}{\\longleftarrow }H_1(\\widehat{D}_{\\varepsilon ^{\\prime \\prime }})$ is isomorphic to the composition $H_1(\\widehat{D}_\\varepsilon ) \\stackrel{j_1i_1}{\\longrightarrow } H_1(\\widehat{X}_{\\tau _3})\\circ H_1(\\widehat{X}_{\\tau _1}) \\stackrel{j_2i_2^{\\prime \\prime }}{\\longleftarrow }H_1(\\widehat{D}_{\\varepsilon ^{\\prime \\prime }})$ , where $k_1,k_2$ are the inclusion induced maps and $j_1,j_2$ are maps resulting from any representative of the pushout $H_1(\\widehat{X}_{\\tau _3})\\circ H_1(\\widehat{X}_{\\tau _1})$ .",
"Observe that $\\widehat{X}_{\\tau _2\\circ \\tau _1}$ decomposes as the union of $\\widehat{X}_{\\tau _1}$ and $\\widehat{X}_{\\tau _2}$ glued along $\\widehat{D}_{\\varepsilon ^{\\prime }}$ .",
"Therefore, the associated Mayer-Vietoris exact sequence $@R0.5cm{H_1(\\widehat{D}_{\\varepsilon ^{\\prime }})[r]^{\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!",
"(-i_1^{\\prime },i^{\\prime }_2)}&H_1(\\widehat{X}_{\\tau _1})\\oplus H_1(\\widehat{X}_{\\tau _2})[r]^{\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;{k_1\\atopwithdelims ()k_2}}&H_1(\\widehat{X}_{\\tau _2\\circ \\tau _1})[r]& 0}$ together with Lemma REF imply that $H_1(\\widehat{X}_{\\tau _1}) \\stackrel{k_1}{\\longrightarrow } H_1(\\widehat{X}_{\\tau _2 \\circ \\tau _1}) \\stackrel{k_2}{\\longleftarrow }H_1(\\widehat{X}_{\\tau _2})$ is a representative of the pushout $H_1(\\widehat{X}_{\\tau _1}) \\circ H_1(\\widehat{X}_{\\tau _2})$ .",
"The claim follows.",
"Then, observe that the Lagrangian functor $\\mathcal {F}$ is by definition the composition of the functors $\\mathcal {B}\\colon \\mathbf {Tangles}\\rightarrow \\mathbf {L}_\\Lambda $ and $F\\colon \\mathbf {L}_\\Lambda \\rightarrow \\mathbf {Lagr}_\\Lambda $ .",
"Also, if $\\tau $ is an oriented string link, then $\\mathcal {B}(\\tau )$ is a rationally invertible cospan by [13], and thus belongs to $\\mathit {core}(\\mathbf {L}_\\Lambda )^0$ by definition.",
"If $\\tau $ is an oriented braid on the other hand, then $\\mathcal {B}(\\tau )$ is obviously an invertible cospan, and therefore belongs to $\\mathit {core}(\\mathbf {L}_\\Lambda )$ by Proposition REF .",
"Finally, if $\\tau $ is a $(\\emptyset ,\\emptyset )$ -tangle, that is, an oriented link $L$ , then the associated Lagrangian cospan is given by $0\\rightarrow H_1(\\widehat{X}_L)\\leftarrow 0$ , with $X_L$ the complement of $L$ in the 3-ball.",
"A straightforward Mayer-Vietoris argument shows that considering $L$ in the 3-ball or in the 3-sphere does not change the Alexander module, and the proof is completed.",
"The bicategory of tangles The aim is now to convert $\\mathcal {B}$ to a weak 2-functor.",
"To do so, we first need to understand how tangles form a (possibly weak) 2-category.",
"Once this is done, we will switch from the category $\\textbf {L}_\\Lambda $ to the bicategory of Lagrangian cospans and define the weak 2-functor in subsection REF .",
"One might think that tangles produce a 2-category in a straightforward way [7]: simply define the objects and 1-morphisms as in $\\mathbf {Tangles}$ , and the 2-morphisms as isotopy classes of oriented surfaces in $D^2\\times [0,1]\\times [0,1]$ .",
"However, the corresponding vertical composition is not well-defined: indeed, one needs to paste two surfaces along isotopic tangles, and since the space of tangles isotopic to a fixed one is not necessarily simply-connected, different choices of isotopies can lead to different surfaces.",
"There are a couple of ways to circumvent this difficulty.",
"One of them is to restrict the space of tangles whose isotopy classes form the 1-morphisms, so that the corresponding space of isotopic tangles has trivial fundamental group.",
"Such a construction was given by Kharlamov and Turaev in [11] (see also [1]): they considered the class of so-called generic tangles, and proved that the space of generic tangles isotopic to a fixed one (through generic tangles) is simply-connected, thus obtaining a strict 2-category.",
"However, it is more natural in our setting to take the following alternative approach: define 1-morphisms as oriented tangles, and consider isotopies bewteen tangles as part of the “higher structure”.",
"Figure: A cobordism Σ⊂D 2 ×[0,1]×[0,1]\\Sigma \\subset D^2\\times [0,1]\\times [0,1] between two (ε,ε ' )(\\varepsilon ,\\varepsilon ^{\\prime })-tangles τ 1 \\tau _1 and τ 2 \\tau _2, with ε=(+1,+1,-1)\\varepsilon =(+1,+1,-1)and ε ' =(+1)\\varepsilon ^{\\prime }=(+1).Let us be more precise.",
"The objects of this bicategory are sequences $\\varepsilon $ of $\\pm 1$ 's, while the 1-morphisms from $\\varepsilon $ to $\\varepsilon ^{\\prime }$ are the $(\\varepsilon ,\\varepsilon ^{\\prime })$ -tangles in $D^2\\times [0,1]$ that are trivial near the top and bottom of the cylinder.",
"(This is to ensure that the composition of two tangles remains a smooth 1-submanifold.)",
"Given two $(\\varepsilon ,\\varepsilon ^{\\prime })$ -tangles $\\tau _1$ and $\\tau _2$ , a $(\\tau _1,\\tau _2)$ -cobordism is a pair consisting of the 4-ball $D^2\\times [0,1]\\times [0,1]$ together with a proper oriented smooth 2-submanifold $\\Sigma $ whose oriented boundary is given by $\\partial \\Sigma =(\\tau _2 \\times \\lbrace 0\\rbrace )\\cup (\\varepsilon ^{\\prime }\\times \\lbrace 1\\rbrace \\times [0,1])\\cup ((-\\tau _1)\\times \\lbrace 1\\rbrace )\\cup ((-\\varepsilon )\\times \\lbrace 0\\rbrace \\times [0,1])\\,,$ as illustrated in Figure REF .",
"Note that a $(\\emptyset ,\\emptyset )$ -cobordism is nothing but a closed oriented surface embedded in the 4-ball.",
"Two $(\\tau _1,\\tau _2)$ -cobordisms $\\Sigma $ and $\\Sigma ^{\\prime }$ are isotopic if there exists an isotopy $h_t$ of $D^2\\times [0,1] \\times [0,1]$ , keeping $\\partial (D^2\\times [0,1]\\times [0,1])$ fixed, such that $h_1\\vert _{\\Sigma }\\colon \\Sigma \\simeq \\Sigma ^{\\prime }$ is an orientation-preserving homeomorphism and $h_t(\\Sigma )$ is a $(\\tau _1,\\tau _2)$ -cobordism for all $t$ .",
"We shall denote by $\\Sigma \\colon \\tau _1\\Rightarrow \\tau _2$ the isotopy class of a $(\\tau _1,\\tau _2)$ -cobordism $\\Sigma $ , and by $\\mathit {Id}_\\tau $ the isotopy class of the trivial $(\\tau ,\\tau )$ -cobordism $(D^2\\times [0,1],\\tau )\\times [0,1]$ .",
"Fix a $(\\tau _1,\\tau _2)$ -cobordism $\\Sigma $ and a $(\\tau _2,\\tau _3)$ -cobordism $\\Sigma ^{\\prime }$ .",
"Their vertical composition is the $(\\tau _1,\\tau _3)$ -cobordism $\\Sigma _2\\star \\Sigma _1$ obtained by gluing the two 4-balls along the cylinders containing $\\tau _2$ , and shrinking the height of the resulting 4-ball $D^2\\times [0,1]\\times [0,2]$ by a factor 2 (see Figure REF ).",
"Finally, fix $(\\varepsilon ,\\varepsilon ^{\\prime })$ -tangles $\\tau _1,\\tau _2$ and $(\\varepsilon ^{\\prime },\\varepsilon ^{\\prime \\prime })$ -tangles $\\tau _3,\\tau _4$ .",
"Given a $(\\tau _1,\\tau _2)$ -cobordism $\\Sigma _1$ and a $(\\tau _3,\\tau _4)$ -cobordism $\\Sigma _2$ , their horizontal composition is the $(\\tau _3\\circ \\tau _1,\\tau _4\\circ \\tau _2)$ -cobordism $\\Sigma _2\\bullet \\Sigma _1$ obtained by gluing the two 4-balls along the cylinder $D^2 \\times [0,1]$ corresponding to $\\varepsilon ^{\\prime }$ , and shrinking the length of the resulting 4-ball by a factor 2 (Figure REF ).",
"Figure: The vertical composition of a (τ 1 ,τ 2 )(\\tau _1,\\tau _2)-cobordism Σ 1 \\Sigma _1 and a (τ 2 ,τ 3 )(\\tau _2,\\tau _3)-cobordism Σ 2 \\Sigma _2, the (τ 1 ,τ 3 )(\\tau _1,\\tau _3)-cobordism Σ 2 ☆Σ 1 \\Sigma _2\\star \\Sigma _1.Figure: The horizontal composition of a (τ 1 ,τ 2 )(\\tau _1,\\tau _2)-cobordism Σ 1 \\Sigma _1 and a (τ 3 ,τ 4 )(\\tau _3,\\tau _4)-cobordism Σ 2 \\Sigma _2,the (τ 3 ∘τ 1 ,τ 4 ∘τ 2 )(\\tau _3\\circ \\tau _1,\\tau _4\\circ \\tau _2)-cobordism Σ 2 •Σ 1 \\Sigma _2\\bullet \\Sigma _1.The bicategory of oriented tangles can now be defined as follows: the objects are the finite sequences of $\\pm 1$ 's, the 1-morphisms are given by the tangles, and the 2-morphisms are given by isotopy classes of cobordisms as described above.",
"Finally, the associativity and identity isomorphisms $a\\colon (\\tau _3\\circ \\tau _2)\\circ \\tau _1\\Rightarrow \\tau _3\\circ (\\tau _2\\circ \\tau _1),\\quad \\ell _{\\tau }\\colon I_{\\varepsilon ^{\\prime }}\\circ \\tau \\Rightarrow \\tau ,\\quad r_\\tau \\colon \\tau \\circ I_{\\varepsilon }\\Rightarrow \\tau $ are given by the trace of the obvious isotopies.",
"It is a routine check to verify that all the axioms of a bicategory are satisfied.",
"The weak 2-functor We are now ready to define our weak 2-functor from the bicategory of oriented tangles to the bicategory of Lagrangian cospans.",
"Recall from subsection REF that we must associate a Hermitian $\\Lambda $ -module $\\mathcal {B}(\\varepsilon )$ to each object $\\varepsilon $ , a cospan $\\mathcal {B}(\\tau )$ to each tangle $\\tau $ and an isomorphism class of 2-cospans to each cobordism $\\Sigma $ .",
"Additionally, for each $\\varepsilon $ , we must define an invertible 2-morphism $\\varphi _\\varepsilon \\colon I_{\\mathcal {B}(\\varepsilon )} \\Rightarrow \\mathcal {B}(I_\\varepsilon )$ and for each pair $\\tau _1, \\tau _2$ of composable tangles, an invertible 2-morphism $\\varphi _{\\tau _1\\tau _2} \\colon \\mathcal {B}(\\tau _2) \\circ \\mathcal {B}(\\tau _1) \\Rightarrow \\mathcal {B}(\\tau _2\\circ \\tau _1)$ .",
"Let us associate to each object $\\varepsilon $ the Hermitian $\\Lambda $ -module $\\mathcal {B}(\\varepsilon )=(H_1(\\widehat{D}_\\varepsilon ),\\omega _\\varepsilon )$ and to each $(\\varepsilon ,\\varepsilon ^{\\prime })$ -tangle $\\tau $ the Lagrangian cospan $\\mathcal {B}(\\tau )$ given by $H_1(\\widehat{D}_\\varepsilon ) \\stackrel{i_\\tau }{\\longrightarrow }H_1(\\widehat{X}_\\tau ) \\stackrel{i_\\tau ^{\\prime }}{\\longleftarrow }H_1(\\widehat{D}_{\\varepsilon ^{\\prime }})$ .",
"(Note that we slightly abuse notations here, as $\\mathcal {B}(\\tau )$ now no longer stands for the isomorphism class of this cospan, but for the cospan itself.)",
"As for 2-morphisms, we proceed as follows.",
"Fix two $(\\varepsilon ,\\varepsilon ^{\\prime })$ -tangles $\\tau _1,\\tau _2$ .",
"Given a $(\\tau _1,\\tau _2)$ -cobordism $\\Sigma \\subset D^2 \\times [0,1] \\times [0,1]$ , denote by $\\mathcal {N}(\\Sigma )$ an open tubular neighborhood of $\\Sigma $ in $D^2 \\times [0,1] \\times [0,1]$ .",
"We shall orient the exterior $W_\\Sigma = \\left\\lbrace \\begin{array}{lr}(D^2 \\times [0,1] \\times [0,1]) \\setminus \\mathcal {N}(\\Sigma ) & \\text{if } \\ell _\\varepsilon \\ne 0\\\\(S^2 \\times [0,1] \\times [0,1]) \\setminus \\mathcal {N}(\\Sigma ) & \\text{if } \\ell _\\varepsilon =0\\end{array}\\right.$ of $\\Sigma $ so that the induced orientation on $\\partial W_\\Sigma $ extends the orientation on the space $(-X_{\\tau _1}) \\sqcup X_{\\tau _2}$ .",
"Clearly, $H_1(W_\\Sigma )$ is generated by the (oriented) meridians of the connected components of $\\Sigma $ .",
"The homomorphism $H_1(W_\\Sigma )\\rightarrow \\mathbb {Z}$ obtained by mapping these meridians to 1 extends the previously defined homomorphisms $H_1(X_{\\tau _1}) \\rightarrow \\mathbb {Z}$ and $H_1(X_{\\tau _2})\\rightarrow \\mathbb {Z}$ .",
"It determines an infinite cyclic covering $\\widehat{W}_\\Sigma \\rightarrow W_\\Sigma $ whose homology is endowed with a structure of module over $\\Lambda $ .",
"Denote by $H_1(\\widehat{D}_\\varepsilon ) \\stackrel{i_1}{\\longrightarrow }H_1(\\widehat{X}_{\\tau _1}) \\stackrel{i_1^{\\prime }}{\\longleftarrow }H_1(\\widehat{D}_{\\varepsilon ^{\\prime }})$ and $H_1(\\widehat{D}_\\varepsilon ) \\stackrel{i_2}{\\longrightarrow }H_1(\\widehat{X}_{\\tau }) \\stackrel{i_2^{\\prime }}{\\longleftarrow }H_1(\\widehat{D}_{\\varepsilon ^{\\prime }})$ the Lagrangian cospans arising from $\\tau _1$ and $\\tau _2$ , and let $\\alpha _{1}\\colon H_1(\\widehat{X}_{\\tau _1}) \\rightarrow H_1(\\widehat{W}_\\Sigma )$ and $\\alpha _{2}\\colon H_1(\\widehat{X}_{\\tau _2}) \\rightarrow H_1(\\widehat{W}_\\Sigma )$ be the homomorphisms induced by the inclusions of $\\widehat{X}_{\\tau _1}$ and $\\widehat{X}_{\\tau _2}$ into $\\widehat{W}_\\Sigma $ .",
"Combining all these inclusion induced maps, the following diagram commutes $@R0.5cm{& H_1(\\widehat{X}_{\\tau _1}) [d]^{\\alpha _1} & \\\\H_1(\\widehat{D}_{\\varepsilon }) [ur]^{i_1} [dr]_{i_2} & H_1(\\widehat{W}_\\Sigma ) & H_1(\\widehat{D}_{\\varepsilon ^{\\prime }})[ul]_{i_1^{\\prime }}[dl]^{i_2^{\\prime }}\\,.",
"\\\\& H_1(\\widehat{X}_{\\tau _2}) [u]_{\\alpha _2} &}$ Hence, $H_1(\\widehat{X}_{\\tau _1}) \\stackrel{\\alpha _1}{\\longrightarrow }H_1(\\widehat{W}_\\Sigma ) \\stackrel{\\alpha _2}{\\longleftarrow }H_1(\\widehat{X}_{\\tau _2})$ is a 2-cospan, whose isomorphism class we denote by $\\mathcal {B}(\\Sigma )\\colon \\mathcal {B}(\\tau _1)\\Rightarrow \\mathcal {B}(\\tau _2)$ .",
"Given any object $\\varepsilon $ , let $\\alpha _\\varepsilon \\colon H_1(\\widehat{D}_{\\varepsilon })\\rightarrow H_1(\\widehat{X}_{I_\\varepsilon })$ denote the isomorphism of $\\Lambda $ -modules induced by the inclusion of $D_\\varepsilon $ in $D_\\varepsilon \\times [0,1]=X_{I_\\varepsilon }$ .",
"This isomorphism fits in the commutative diagram $@R0.5cm{& H_1(\\widehat{D}_{\\varepsilon }) [d]^{\\alpha _\\varepsilon } & \\\\H_1(\\widehat{D}_{\\varepsilon }) [ur]^{\\mathit {id}} [dr]_{\\alpha _\\varepsilon } & H_1(\\widehat{X}_{I_\\varepsilon }) & H_1(\\widehat{D}_{\\varepsilon })[ul]_\\mathit {id}[dl]^{\\alpha _\\varepsilon } \\,.",
"\\\\& H_1(\\widehat{X}_{I_\\varepsilon }) [u]_\\mathit {id}&}$ By Remark REF , the 2-morphism $\\varphi _\\varepsilon \\colon I_{\\mathcal {B}(\\varepsilon )}\\Rightarrow \\mathcal {B}(I_\\varepsilon )$ defined by this diagram is invertible, as required in the definition of a weak 2-functor.",
"Given an $(\\varepsilon ,\\varepsilon ^{\\prime })$ -tangle $\\tau _1$ and an $(\\varepsilon ^{\\prime },\\varepsilon ^{\\prime \\prime })$ -tangle $\\tau _2$ , the first part of the proof of Theorem REF actually shows that there is a canonical isomorphism $\\alpha _{\\tau _1\\tau _2}\\colon H_1(\\widehat{X}_{\\tau _2})\\circ H_1(\\widehat{X}_{\\tau _1})\\rightarrow H_1(\\widehat{X}_{\\tau _2\\circ \\tau _1})$ which fits in the commutative diagram $@R0.5cm{& & H_1(\\widehat{X}_{\\tau _2})\\circ H_1(\\widehat{X}_{\\tau _1}) [d]^{\\alpha _{\\tau _1\\tau _2}} & & \\\\H_1(\\widehat{D}_{\\varepsilon }) @/^1pc/[urr]^{j_1i_1} @/_1pc/[drr]_{k_1i_1} [r]^{i_1} & H_1(\\widehat{X}_{\\tau _1}) [ur]_{j_1} [dr]^{k_1} &H_1(\\widehat{X}_{\\tau _2\\circ \\tau _1}) &H_1(\\widehat{X}_{\\tau _2}) [ul]^{j_2} [dl]_{k_2} & H_1(\\widehat{D}_{\\varepsilon ^{\\prime \\prime }})@/_1pc/[ull]_{j_2i_2^{\\prime \\prime }}@/^1pc/[dll]^{k_2i_2^{\\prime \\prime }}[l]_{i_2^{\\prime \\prime }}\\,, \\\\& & H_1(\\widehat{X}_{\\tau _2\\circ \\tau _1}) [u]_\\mathit {id}& &}$ where we follow the notations of the aforementioned proof.",
"Hence, this defines a canonical 2-morphism $\\varphi _{\\tau _1\\tau _2}\\colon \\mathcal {B}(\\tau _1)\\circ \\mathcal {B}(\\tau _2)\\Rightarrow \\mathcal {B}(\\tau _2\\circ \\tau _1)$ , which is invertible by Remark REF .",
"Theorem 4.2 $\\mathcal {B}$ together with the isomorphisms $\\varphi _\\varepsilon $ and $\\varphi _{\\tau _1\\tau _2}$ gives rise to a weak 2-functor from the bicategory of oriented tangles to the bicategory of Lagrangian cospans, whose restriction to oriented surfaces is given by the Alexander module.",
"First note that isotopic cobordisms define isomorphic 2-cospans, so $\\mathcal {B}$ is well-defined at the level of 2-morphisms.",
"Also, for any tangle $\\tau $ , $\\mathcal {B}$ clearly maps the trivial concordance $\\mathit {Id}_\\tau $ to a 2-cospan canonically isomorphic to the identity 2-cospan $\\mathit {Id}_{\\mathcal {B}(\\tau )}$ .",
"Let us now verify that $\\mathcal {B}$ preserves the vertical composition.",
"Fix a $(\\tau _1,\\tau _2)$ -cobordism $A$ and a $(\\tau _2,\\tau _3)$ -cobordism $B$ .",
"Let $H_1(\\widehat{X}_{\\tau _1}) \\stackrel{\\alpha _1}{\\rightarrow }H_1(\\widehat{W}_A) \\stackrel{\\alpha _2}{\\leftarrow }H_1(\\widehat{X}_{\\tau _2})$ and $H_1(\\widehat{X}_{\\tau _2}) \\stackrel{\\beta _2}{\\rightarrow }H_1(\\widehat{W}_B) \\stackrel{\\beta _3}{\\leftarrow }H_1(\\widehat{X}_{\\tau _3})$ be the 2-cospans arising from $A$ and $B$ .",
"We need to show that $H_1(\\widehat{X}_{\\tau _1}) \\stackrel{k_A\\alpha _1}{\\longrightarrow } H_1(\\widehat{W}_{B \\star A}) \\stackrel{k_B\\beta _3}{\\longleftarrow }H_1(\\widehat{X}_{\\tau _3})$ is isomorphic to the vertical composition $H_1(\\widehat{X}_{\\tau _1}) \\stackrel{v_A\\alpha _1}{\\longrightarrow } H_1(\\widehat{W}_B) \\star H_1(\\widehat{W}_A) \\stackrel{v_B\\beta _3}{\\longleftarrow } H_1(\\widehat{X}_{\\tau _3})$ , where $k_A,k_B$ are the inclusion induced maps and $v_A,v_B$ are maps resulting from any representative of the pushout $H_1(\\widehat{W}_B) \\star H_1(\\widehat{W}_A)$ .",
"Observe that $\\widehat{W}_{B \\star A}$ decomposes as the union of $\\widehat{W}_B$ and $\\widehat{W}_A$ glued along $\\widehat{X}_{\\tau _2}$ .",
"Therefore, the associated Mayer-Vietoris exact sequence $@R0.5cm{H_1(\\widehat{X}_{\\tau _2})[r]^{\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!",
"(-\\alpha _2,\\beta _2)}&H_1(\\widehat{W}_A)\\oplus H_1(\\widehat{W}_B)[r]^{\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;{k_A \\atopwithdelims ()k_B}}&H_1(\\widehat{W}_{B \\star A})[r]& 0}$ together with Lemma REF imply that $H_1(\\widehat{W}_A) \\stackrel{k_A}{\\longrightarrow } H_1(\\widehat{W}_{B \\star A}) \\stackrel{k_B}{\\longleftarrow }H_1(\\widehat{W}_B)$ is a representative for the pushout $H_1(\\widehat{W}_A) \\star H_1(\\widehat{W}_B)$ .",
"Consequently, these two cospans are canonically isomorphic and the claim follows.",
"Given tangles and cobordisms as illustrated below $@C+2pc{\\varepsilon ^{\\tau _1}_{\\tau _2}{\\;\\;\\;A}&\\varepsilon ^{\\prime }^{\\tau _3}_{\\tau _4}{\\;\\;\\;B}&\\varepsilon ^{\\prime \\prime }\\,,}$ our next goal is to prove the equality $\\varphi _{\\tau _2\\tau _4}\\star (\\mathcal {B}(B)\\bullet \\mathcal {B}(A))=\\mathcal {B}(B\\bullet A)\\star \\varphi _{\\tau _1\\tau _3}$ up to isomorphism of 2-cospans.",
"Since the 2-morphism $\\varphi _{\\tau _1\\tau _3}$ is represented by the 2-cospan $H_1(\\widehat{X}_{\\tau _3}) \\circ H_1(\\widehat{X}_{\\tau _1})\\stackrel{\\alpha _{\\tau _1\\tau _3}}{\\longrightarrow } H_1(\\widehat{X}_{\\tau _3\\circ \\tau _1}) \\stackrel{\\mathit {id}}{\\longleftarrow }H_1(\\widehat{X}_{\\tau _3\\circ \\tau _1})$ , Remark REF implies that the right hand side of equation (REF ) is represented by the 2-cospan $H_1(\\widehat{X}_{\\tau _3}) \\circ H_1(\\widehat{X}_{\\tau _1}) \\stackrel{k_{31}\\alpha _{\\tau _1\\tau _3}}{\\longrightarrow } H_1(\\widehat{W}_{B \\bullet A}) \\stackrel{k_{42}}{\\longleftarrow }H_1(\\widehat{X}_{\\tau _4\\circ \\tau _2})\\,,$ where $k_{31}$ and $k_{42}$ are induced by the inclusion maps.",
"A similar argument shows that the left hand side of equation (REF ) is represented by the 2-cospan $H_1(\\widehat{X}_{\\tau _3}) \\circ H_1(\\widehat{X}_{\\tau _1}) \\stackrel{h_{31}}{\\longrightarrow } H_1(\\widehat{W}_B) \\bullet H_1(\\widehat{W}_A) \\stackrel{h_{42}\\alpha ^{-1}_{\\tau _2\\tau _4}}{\\longleftarrow } H_1(\\widehat{X}_{\\tau _4\\circ \\tau _2})\\,,$ where this time, the maps $h_{31}$ and $h_{42}$ are the ones which arise from the definition of horizontal composition.",
"It now remains to find an isomorphism $f$ of $\\Lambda $ -modules which fits in the following commutative diagram: $@R0.5cm{& H_1(\\widehat{X}_{\\tau _3}) \\circ H_1(\\widehat{X}_{\\tau _1}) [dl]_{h_{31}} [dr]^{k_{31}\\alpha _{\\tau _1\\tau _3}} & \\\\H_1(\\widehat{W}_B) \\bullet H_1(\\widehat{W}_A) [rr]^f & & \\;H_1(\\widehat{W}_{B\\bullet A})\\,.",
"\\\\& H_1(\\widehat{X}_{\\tau _4\\circ \\tau _2}) [lu]^{h_{42}\\alpha _{\\tau _2\\tau _4}^{-1}}[ur]_{k_{42}} &}$ In order to construct $f$ , first observe that the following diagram commutes $@R0.5cm{H_1(\\widehat{X}_{\\tau _2})[d]_{\\alpha _2} & H_1(\\widehat{D}_{\\varepsilon ^{\\prime }}) [d]^{\\cong }[r]^{i_3^{\\prime }}[l]_{i_2^{\\prime }}& H_1(\\widehat{X}_{\\tau _3})[d]^{\\beta _3} \\\\H_1(\\widehat{W}_A) & H_1(\\widehat{D}_{\\varepsilon ^{\\prime }} \\times [0,1])[l][r] & H_1(\\widehat{W}_B),}$ where all the maps are induced by inclusions.",
"Hence, identifying $H_1(\\widehat{D}_{\\varepsilon ^{\\prime }}\\times [0,1])$ with $H_1(\\widehat{D}_{\\varepsilon ^{\\prime }})$ , the first map in the Mayer-Vietoris exact sequence $@R0.5cm{H_1(\\widehat{D}_{\\varepsilon ^{\\prime }})[r]^{\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!",
"}&H_1(\\widehat{W}_A)\\oplus H_1(\\widehat{W}_B)[r]^{\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;}&H_1(\\widehat{W}_{B \\bullet A})[r]& 0}$ is given by $(-\\alpha _2i_2^{\\prime },\\beta _3i_3^{\\prime })$ .",
"It now follows from Lemma REF that the cospan of inclusion induced maps $H_1(\\widehat{W}_A) \\stackrel{k_A}{\\rightarrow } H_1(\\widehat{W}_{B \\bullet A}) \\stackrel{k_B}{\\leftarrow } H_1(\\widehat{W}_B)$ is a representative of the pushout $H_1(\\widehat{W}_A) \\stackrel{h_A}{\\rightarrow }H_1(\\widehat{W}_B) \\bullet H_1(\\widehat{W}_A)\\stackrel{h_B}{\\leftarrow } H_1(\\widehat{W}_B)$ .",
"Invoking the corresponding universal property, this produces a $\\Lambda $ -module isomorphism $f \\colon H_1(\\widehat{W}_B) \\bullet H_1(\\widehat{W}_A) \\rightarrow H_1(\\widehat{W}_{B\\bullet A})$ with $fh_A=k_A$ and $fh_B=k_B$ .",
"Using successively the definition of $\\alpha _{\\tau _1\\tau _3}$ (see diagram (REF ) for the relevant notations), the commutativity of inclusion induced maps, and the equalities above, one gets $f^{-1}k_{31} \\alpha _{\\tau _1\\tau _3}j_3=f^{-1}k_{31}k_3=f^{-1}k_B\\beta _3=h_B\\beta _3\\,.$ The equality $f^{-1}k_{31} \\alpha _{\\tau _1\\tau _3}j_1=h_A\\alpha _1$ is proved similarly.",
"Hence, the universal property of diagram (REF ) implies that $h_{31}=f^{-1}k_{31} \\alpha _{\\tau _1\\tau _3}$ .",
"The equality $h_{42}=f^{-1}k_{42} \\alpha _{\\tau _2\\tau _4}$ can be dealt with in the same way, and equation (REF ) is proved.",
"Given an $(\\varepsilon ,\\varepsilon ^{\\prime })$ -tangle $\\tau $ , we must now show that the 2-morphism $\\mathcal {B}(r_\\tau ) \\star \\varphi _{I_{\\varepsilon }\\tau } \\star (I_{\\mathcal {B}(\\tau )} \\bullet \\varphi _\\varepsilon )$ coincides with $r_{\\mathcal {B}(\\tau )} \\colon \\mathcal {B}(\\tau ) \\circ I_{\\mathcal {B}(\\varepsilon )} \\Rightarrow \\mathcal {B}(\\tau )$ .",
"First observe that by Remark REF , one can choose representatives of the pushouts so that for any cospan $H \\rightarrow T \\leftarrow H^{\\prime }$ , one has $T \\circ I_H=T$ .",
"In particular, we only need to prove that, for this choice of pushouts, $\\mathcal {B}(r_\\tau ) \\star \\varphi _{I_{\\varepsilon }\\tau } \\star (I_{\\mathcal {B}(\\tau )} \\bullet \\varphi _\\varepsilon ) =I_{\\mathcal {B}(\\tau )}.$ As a first step, using the definition of the horizontal composition and Remark REF , we deduce that $I_{\\mathcal {B}(\\tau )} \\bullet \\varphi _\\varepsilon $ is represented by the 2-cospan $H_1(\\widehat{X}_\\tau ) \\stackrel{\\mathit {id}}{\\longrightarrow } H_1(\\widehat{X}_\\tau ) \\stackrel{h}{\\longleftarrow } H_1(\\widehat{X}_\\tau ) \\circ H_1(\\widehat{X}_{I_\\varepsilon })\\,,$ where $h$ is the unique morphism which fits in the following commutative diagram (recall diagram (REF )): $@R0.5cm{& & H_1(\\widehat{X}_\\tau ) & & & \\\\& & H_1(\\widehat{X}_\\tau ) \\circ H_1(\\widehat{X}_{I_\\varepsilon }) [u]^{h} & & & \\\\& H_1(\\widehat{D}_\\varepsilon ) @/^1pc/[uur]^{i} & & H_1(\\widehat{X}_\\tau ) @/_1pc/[uul]_{\\mathit {id}} & & \\\\& H_1(\\widehat{X}_{I_\\varepsilon }) [ruu]^{j_1} [u]^{\\alpha _\\varepsilon ^{-1}} & & H_1(\\widehat{X}_\\tau ).",
"[uul]_{j_2} [u]_{\\mathit {id}} \\\\& & H_1(\\widehat{D}_\\varepsilon ) [lu]_{\\alpha _\\varepsilon }[ru]^{i} & &}$ A short computation using Remark REF then shows that the left hand side of equation (REF ) is represented by the 2-cospan $H_1(\\widehat{X}_\\tau ) \\stackrel{\\alpha _{I_\\varepsilon \\tau } h^{-1}}{\\longrightarrow } H_1(\\widehat{X}_{\\tau \\circ I_\\varepsilon }) \\stackrel{r^{-1}}{\\longleftarrow } H_1(\\widehat{X}_\\tau )\\,,$ where $r\\colon H_1(\\widehat{X}_{\\tau \\circ I_\\varepsilon })\\rightarrow H_1(\\widehat{X}_\\tau )$ is the isomorphism induced by the obvious isotopy from $\\tau \\circ I_\\varepsilon $ to $\\tau $ .",
"We now claim that $r$ induces a 2-cospan isomorphism from $I_{\\mathcal {B}(\\tau )}$ to this cospan.",
"To prove this claim, we only need to show the equality $r\\alpha _{I_\\varepsilon \\tau } h^{-1}=\\mathit {id}_{H_1(\\widehat{X}_{\\tau })}$ , i.e.",
"to check that $r\\alpha _{I_\\varepsilon \\tau }$ satisfies the defining property of $h$ displayed above.",
"Since $\\alpha _{I_\\varepsilon \\tau }j_1$ and $\\alpha _{I_\\varepsilon \\tau }j_2$ are the inclusion induced homomorphisms (recall diagram (REF )), this follows from the functoriality of homology.",
"The proof of the equality $\\mathcal {B}(\\ell _\\tau ) \\star \\varphi _{\\tau I_{\\varepsilon ^{\\prime }}} \\star (\\varphi _{\\varepsilon ^{\\prime }} \\bullet I_{\\mathcal {B}(\\tau )})=\\ell _{\\mathcal {B}(\\tau )}$ is dealt with in the same way.",
"Finally, the axiom involving the associativity isomorphisms is left to the reader: although the proof is tedious, it involves no other ideas than the ones presented up to now.",
"Therefore we have proved that $\\mathcal {B}$ is a weak 2-functor and we turn to the last statement of the theorem.",
"If $\\Sigma $ is a $(\\emptyset ,\\emptyset )$ -cobordism, that is, a closed oriented surface in the 4-ball, then the associated 2-cospan is given by $@R0.5cm{& 0 [d] & \\\\0 [ur] [dr] & H_1(\\widehat{W}_\\Sigma ) & 0 [ul] [dl]\\,, \\\\& 0 [u] &}$ with $W_\\Sigma $ the complement of $\\Sigma $ in the 4-ball.",
"This is nothing but the Alexander module of $\\Sigma $ .",
"Unreduced and multivariable versions Recall that the representation originally defined by Burau takes the form of a homomorphism $\\overline{\\rho }_n\\colon B_n\\rightarrow \\mathit {GL}_n(\\Lambda )$ , which is the direct sum of a trivial 1-dimensional representation with $\\rho _n\\colon B_n\\rightarrow \\mathit {GL}_{n-1}(\\Lambda )$ , the reduced Burau representation.",
"Also, these representations admit multivariable extensions, the so-called Gassner representations of the pure braid groups.",
"It is therefore natural to ask whether these variations of the Burau representation can also be extended to weak 2-functors.",
"This is indeed the case, and is the subject of this slightly informal last section.",
"More precisely, we start in subsection REF by explaining how $\\overline{\\rho }$ can be extended to a functor $\\overline{\\mathcal {B}}$ on tangles.",
"This functor is no longer Lagrangian ($\\overline{\\rho }$ is not unitary) but it is monoidal and behaves well with respect to traces.",
"In subsection REF , we indicate how to extend it to a weak 2-functor.",
"Finally, in subsection REF , we briefly explain how all of these constructions can be extended to multivariable versions, defined on the category of colored tangles.",
"Extending the unreduced Burau representation to a monoidal functor Given an integral domain $\\Lambda $ , let $\\mathbf {C}_\\Lambda $ denote the category with finitely generated $\\Lambda $ -modules as objects, and isomorphism classes of cospans as morphisms, composed by pushouts.",
"Also, let $\\mathbf {GL}_\\Lambda $ denote the groupoid with the same objects as $\\mathbf {C}_\\Lambda $ and $\\Lambda $ -isomorphisms as morphisms.",
"As in Section , one can check that the map assigning to an invertible cospan $H\\stackrel{i}{\\longrightarrow }T\\stackrel{i^{\\prime }}{\\longleftarrow }H^{\\prime }$ the $\\Lambda $ -isomorphism $i^{\\prime -1}i\\colon H\\rightarrow H^{\\prime }$ defines an equivalence of categories $\\mathit {core}(\\mathbf {C}_\\Lambda ) \\stackrel{\\simeq }{\\longrightarrow }\\mathbf {GL}_\\Lambda $ .",
"Note that the direct sum endows these categories with a monoidal structure, with the trivial $\\Lambda $ -module $H=0$ being the identity object.",
"Given an endomorphism of $\\mathbf {C}_\\Lambda $ , i.e.",
"a cospan of the form $H\\stackrel{i}{\\longrightarrow }T\\stackrel{i^{\\prime }}{\\longleftarrow }H$ , define the trace of $T$ as the coequalizer $@R0.5cm{H @/^/[rr]^{i}@/_/[rr]_{i^{\\prime }}& & T [r]^{\\!\\!\\!j}&\\mathrm {tr}(T)\\,.",
"}$ Viewing $\\mathrm {tr}(T)$ as the isomorphism class of the cospan $0\\rightarrow \\mathrm {tr}(T)\\leftarrow 0$ , the trace actually defines a map $\\mathrm {tr}\\colon \\mathit {End}(H) \\rightarrow \\mathit {End}(0)$ .",
"It is an amusing exercise to check that it satisfies the following properties, as it should (see e.g. [17]).",
"i If $T_1$ is a cospan from $H$ to $H^{\\prime }$ and $T_2$ from $H^{\\prime }$ to $H$ , then $\\mathrm {tr}(T_1\\circ T_2)=\\mathrm {tr}(T_2 \\circ T_1)$ .",
"ii If $T_1$ and $T_2$ are two endomorphisms, then $\\mathrm {tr}(T_1\\oplus T_2)=\\mathrm {tr}(T_1)\\circ \\mathrm {tr}(T_2)$ .",
"iii If $T$ is an endomorphism of 0, then $\\mathrm {tr}(T)=T$ .",
"These additional structures are also present in the category of tangles.",
"Indeed, the juxtaposition endows $\\mathbf {Tangles}$ with a monoidal structure, with the empty set $\\varepsilon =\\emptyset $ being the identity object.",
"Furthermore, the closure of a tangle defines a natural trace function $\\mathit {End}(\\varepsilon )\\rightarrow \\mathit {End}(\\emptyset )$ .",
"In this context, the unreduced Burau representation can be understood as a monoidal functor $\\overline{\\rho }\\colon \\mathbf {Braids}\\rightarrow \\mathbf {GL}_\\Lambda $ , where $\\Lambda ={Z}[t^{\\pm 1}]$ .",
"We now sketch the construction of a monoidal functor $\\overline{\\mathcal {B}}\\colon \\mathbf {Tangles}\\rightarrow \\mathbf {C}_\\Lambda $ extending $\\overline{\\rho }$ , and behaving well with respect to traces.",
"We shall follow the notation of Section , apart from the fact that all exteriors will be considered in the unit disc $D^2$ , and not the sphere $S^2$ even when $\\ell _\\varepsilon $ vanishes.",
"Let $x_0$ be the point $(-1,0)$ in $D^2$ .",
"For any sequence $\\varepsilon $ of $\\pm 1$ 's, set $\\overline{\\mathcal {B}}(\\varepsilon )=H_1(\\widehat{D}_\\varepsilon ,\\widehat{x_0})$ and for any isotopy class $\\tau $ of tangles, let $\\overline{\\mathcal {B}}(\\tau )$ denote the isomorphism class of the cospan $H_1(\\widehat{D}_\\varepsilon ,\\widehat{x_0}) \\stackrel{i_\\tau }{\\longrightarrow }H_1(\\widehat{X}_\\tau ,\\widehat{x_0\\times I})\\stackrel{i_\\tau ^{\\prime }}{\\longleftarrow }H_1(\\widehat{D}_{\\varepsilon ^{\\prime }},\\widehat{x_0})$ , where $\\widehat{Y}$ stands for the inverse image of a subspace $Y\\subset X_\\tau $ by the infinite cyclic covering map $\\widehat{X_\\tau }\\rightarrow X_\\tau $ .",
"Following almost verbatim the proof of Theorem REF , one checks that this defines a functor $\\overline{\\mathcal {B}}\\colon \\mathbf {Tangles}\\rightarrow \\mathbf {C}_\\Lambda $ which fits in the commutative diagram $@R0.5cm{& \\mathbf {Braids} [d] [r] @/_1pc/[ld]_{\\overline{\\rho }} & \\mathbf {Tangles} [d]^{\\overline{\\mathcal {B}}} \\\\\\mathbf {GL}_\\Lambda &\\mathit {core}(\\mathbf {C}_\\Lambda ) [l]_\\simeq [r] & \\mathbf {C}_\\Lambda \\,.",
"}$ Furthermore, an additional application of Mayer-Vietoris shows that this functor is monoidal.",
"(The basepoint $x_0$ is chosen so that the juxtaposition of tangles can be realized in a natural way by gluing discs along intervals, with $x_0$ a common endpoint of these intervals.)",
"Finally, if $\\tau $ is an $(\\varepsilon ,\\varepsilon )$ -tangle, then $\\mathrm {tr}(\\overline{\\mathcal {B}}(\\tau ))$ is nothing but the relative Alexander module of the oriented link in $D^2\\times I$ (or equivalently, in $S^3$ ) obtained by the closure of $\\tau $ .",
"$\\overline{\\mathcal {B}}$ as a monoidal weak 2-functor One can modify $\\mathbf {C}_\\Lambda $ to obtain a bicategory in the exact same way as we did for $\\mathbf {L}_\\Lambda $ , with 2-morphisms given by isomorphism classes of 2-cospans (recall subsection REF ).",
"Furthermore, the direct sum endows this bicategory with a monoidal structure.",
"Also, the juxtaposition endows the bicategory of tangles with a monoidal structure.",
"Here again, some care is needed, as different conventions such as the ones in [11] and [1] will lead to different monoidal bicategories.",
"We will not go into these details, but only mention that our construction is robust enough to be valid in these different settings.",
"Let us sketch how the functor $\\overline{\\mathcal {B}}$ can be extended to a weak 2-functor, following the notation of subsection REF .",
"Given a $(\\tau _1,\\tau _2)$ -cobordism $\\Sigma $ , let us denote by $\\overline{\\mathcal {B}}(\\Sigma )\\colon \\overline{\\mathcal {B}}(\\tau _1)\\Rightarrow \\overline{\\mathcal {B}}(\\tau _2)$ the isomorphism class of the 2-cospan $H_1(\\widehat{X}_{\\tau _1},\\widehat{x_0\\times I}) \\stackrel{\\alpha _1}{\\longrightarrow }H_1(\\widehat{W}_\\Sigma ,\\widehat{x_0\\times I\\times I}) \\stackrel{\\alpha _2}{\\longleftarrow }H_1(\\widehat{X}_{\\tau _2},\\widehat{x_0\\times I})\\,.$ One can check that this defines a weak 2-functor, that is monoidal in a sense that, once again, we shall not discuss in detail here.",
"Multivariable versions Let $\\mu $ be a positive integer.",
"Recall that a $\\mu $ -colored tangle consists in an oriented tangle $\\tau $ together with a surjective map assigning to each component of $\\tau $ an integer in $\\lbrace 1,\\dots ,\\mu \\rbrace $ .",
"As explained in [6], $\\mu $ -colored tangles naturally form a category $\\mathbf {Tangles}_\\mu $ , with the $\\mu =1$ case being nothing but $\\mathbf {Tangles}$ .",
"Obviously, assigning a color to the cobordisms and proceeding as in subsection REF , one obtains a bicategory of $\\mu $ -colored tangles.",
"All the results of the present paper extend to this multivariable setting in a straightforward way, that we now very briefly summarize.",
"The coloring of points, tangles and cobordisms induces homomorphisms from the homology of the corresponding exterior onto ${Z}^\\mu $ , thus defining free abelian covers whose homology is a module over the ring of multivariable Laurent polynomials ${Z}[{Z}^\\mu ]={Z}[t_1^{\\pm 1},\\dots ,t_\\mu ^{\\pm 1}]=:\\Lambda _\\mu $ .",
"This allows one to construct a weak 2-functor from the bicategory of $\\mu $ -colored tangles to the bicategory of Lagrangian cospans over $\\Lambda _\\mu $ , which extends the colored Gassner representation of $\\mu $ -colored braids, and whose restriction to $\\mu $ -colored links and surfaces is nothing but the multivariable Alexander module.",
"The results of subsections REF and REF can be extended in the same way."
],
[
"A bicategory of Lagrangian cospans",
"The aim of this section is to endow $\\mathbf {L}_\\Lambda $ with the structure of a bicategory.",
"We begin by recalling in subsection REF the notions of bicategory and weak 2-functor, before defining the bicategory of Lagrangian cospans in subsection REF ."
],
[
"2-categories and 2-functors",
"Following the original work of Bénabou [3], it is a traditional practice to use the term “2-category” for what Kapranov and Voevodsky call a “strict 2-category” [10].",
"As it turns out, the categories that appear in our work are not of this type, but have a richer structure: that of some type of weak 2-category known as a bicategory.",
"We now recall the definition of this structure, following [3].",
"A bicategory $\\mathcal {C}$ consists of the following data: A set $\\mathrm {Ob}\\mathcal {C}$ whose elements are called objects.",
"For each pair of objects $(X,Y)$ , a category $\\mathcal {C}(X,Y)$ whose objects are called 1-morphisms and denoted by $f\\colon X\\rightarrow Y$ or by $X\\stackrel{f}{\\rightarrow }Y$ , whose morphisms are called 2-morphisms and denoted by $\\alpha \\colon f\\Rightarrow g$ , or by $@C+2pc{X^{f}_{g}{\\;\\;\\;\\alpha }&Y}$ , and whose composition is called vertical composition and denoted by $\\left(@C+2pc{X^{f}_{g}{\\;\\;\\;\\alpha }&Y},@C+2pc{X^{g}_{h}{\\;\\;\\;\\beta }&Y}\\right)\\mapsto @C+2pc{X^{f}_{h}{\\;\\;\\;\\;\\;\\;\\beta \\star \\alpha }&Y}\\,.$ We shall denote the identity morphism for $f$ by $\\mathit {Id}_f\\colon f\\Rightarrow f$ .",
"For each object $X$ , an identity 1-morphism $I_X\\colon X\\rightarrow X$ .",
"For each triple of objects $(X,Y,Z)$ , a functor $\\mathcal {C}(X,Y)\\times \\mathcal {C}(Y,Z)\\rightarrow \\mathcal {C}(X,Z)$ denoted by $\\left(@C+2pc{X^{f}_{g}{\\;\\;\\;\\alpha }&Y},@C+2pc{Y^{k}_{\\ell }{\\;\\;\\;\\beta }&Z}\\right)\\mapsto @C+2pc{X^{k\\circ f}_{\\ell \\circ g}{\\;\\;\\;\\;\\;\\;\\beta \\bullet \\alpha }&Z\\,,}$ and called the horizontal composition functor.",
"Note that the functoriality of this composition boils down to the identity $\\mathit {Id}_f \\bullet \\mathit {Id}_g = \\mathit {Id}_{g\\circ f}$ for any composable 1-morphisms $f$ and $g$ , and to the interchange law $(\\delta \\bullet \\beta ) \\star (\\gamma \\bullet \\alpha )=(\\delta \\star \\gamma ) \\bullet (\\beta \\star \\alpha )\\,,$ for each composable 2-morphisms $\\alpha ,\\beta ,\\gamma $ and $\\delta $ .",
"This last condition is best understood by saying that the following 2-morphism is well-defined, i.e.",
"independent of the order of the compositions: $@C+2pc{X^{}{\\;\\;\\;\\alpha } [r] _{}{\\;\\;\\;\\beta } & Y ^{}{\\;\\;\\;\\gamma } [r] _{}{\\;\\;\\;\\delta } & Z\\,.",
"}$ If this horizontal composition is associative (both on the 1-morphisms and 2-morphisms) and admits $I_X$ as a two-sided unit, then we are in the presence of a (strict) 2-category.",
"As mentioned above, a bicategory has a richer structure: the horizontal composition is associative and unital only up to natural isomorphisms, which are part of the structure.",
"To be more precise, a bicategory also contains the following data: For any triple of composable 1-morphisms $f,g,h$ , an invertible 2-morphism $a=a_{fgh}\\colon (h\\circ g)\\circ f\\Rightarrow h\\circ (g\\circ f)$ which is natural in $f,g$ and $h$ , and called the associativity isomorphism.",
"For any 1-morphism $X\\stackrel{f}{\\rightarrow }Y$ , two invertible 2-morphisms $\\ell =\\ell _f\\colon I_Y\\circ f\\Rightarrow f$ and $r=r_f\\colon f\\circ I_X \\Rightarrow f$ which are natural in $f$ .",
"These natural isomorphisms must satisfy the following two coherence axioms.",
"Given four composable 1-morphisms $e,f,g,h$ , there are two natural ways to pass from $((h\\circ g)\\circ f)\\circ e$ to $h\\circ (g\\circ (f\\circ e))$ using the associativity isomorphisms, one in two steps, the other one in three.",
"The associativity coherence axiom requires that these two compositions coincide.",
"Finally, given any 1-morphisms $X\\stackrel{f}{\\rightarrow }Y\\stackrel{g}{\\rightarrow }Z$ , the identity coherence axiom requires the composition $(g\\circ I_Y)\\circ f\\stackrel{a}{\\Longrightarrow } g\\circ (I_Y\\circ f)\\stackrel{\\mathit {Id}_g\\bullet \\ell _f}{\\Longrightarrow }g\\circ f$ to coincide with $r_g\\bullet \\mathit {Id}_f$ .",
"Let us now recall the definition of a weak 2-functor, also known as a pseudofunctor [9] and originally called a homomorphism of bicategories by Bénabou [3].",
"Given two bicategories $\\mathcal {C}$ and $\\mathcal {D}$ , a weak 2-functor $\\mathcal {F}\\colon \\mathcal {C}\\rightarrow \\mathcal {D}$ consists of the following data: A map $\\mathcal {F}\\colon \\mathrm {Ob}\\mathcal {C}\\rightarrow \\mathrm {Ob}\\mathcal {D}$ .",
"For each pair of objects $(X,Y)$ in $\\mathcal {C}$ , a functor $\\mathcal {C}(X,Y)\\rightarrow \\mathcal {D}(\\mathcal {F}(X),\\mathcal {F}(Y))$ denoted by $@C+2pc{X^{f}_{g}{\\;\\;\\;\\alpha }&Y}\\mapsto @C+2pc{\\mathcal {F}(X)^{\\mathcal {F}(f)}_{\\mathcal {F}(g)}{\\;\\;\\;\\;\\;\\;\\mathcal {F}(\\alpha )}&\\mathcal {F}(Y)}\\,.$ Note that this functoriality is equivalent to the identities $\\mathcal {F}(\\beta \\star \\alpha )=\\mathcal {F}(\\beta )\\star \\mathcal {F}(\\alpha )\\quad \\text{and}\\quad \\mathcal {F}(\\mathit {Id}_f)=\\mathit {Id}_{\\mathcal {F}(f)}\\,.$ If we also have the identities $\\mathcal {F}(I_X)=I_{\\mathcal {F}(X)}$ , $\\mathcal {F}(g\\circ f)=\\mathcal {F}(g)\\circ \\mathcal {F}(f)$ and $\\mathcal {F}(\\beta \\bullet \\alpha )=\\mathcal {F}(\\beta )\\bullet \\mathcal {F}(\\alpha )$ , then we are in the presence of a (strict) 2-functor.",
"Our functor has a finer structure: once again, these identities hold up to isomorphisms of functors, which are part of the data as follows.",
"For each object $X$ of $\\mathcal {C}$ , an invertible 2-morphism $\\varphi _X\\colon I_{\\mathcal {F}(X)}\\Rightarrow \\mathcal {F}(I_X)$ in $\\mathcal {D}$ .",
"For each $X\\stackrel{f}{\\rightarrow }Y\\stackrel{g}{\\rightarrow }Z$ , an invertible 2-morphism $\\varphi =\\varphi _{fg}\\colon \\mathcal {F}(g)\\circ \\mathcal {F}(f)\\Rightarrow \\mathcal {F}(g\\circ f)$ in $\\mathcal {D}$ such that for any $@C+2pc{X^{f}_{f^{\\prime }}{\\;\\;\\;\\alpha }&Y^{g}_{g^{\\prime }}{\\;\\;\\;\\beta }&Y\\,,}$ we have the following equality of 2-morphisms in $\\mathcal {D}$ : $\\varphi _{f^{\\prime }g^{\\prime }}\\star (\\mathcal {F}(\\beta )\\bullet \\mathcal {F}(\\alpha ))=\\mathcal {F}(\\beta \\bullet \\alpha )\\star \\varphi _{fg}\\,\\colon \\,\\mathcal {F}(g)\\circ \\mathcal {F}(f)\\Rightarrow \\mathcal {F}(g^{\\prime }\\circ f^{\\prime })\\,.$ Finally, these isomorphisms of functors are required to satisfy two coherence axioms.",
"For any composable 1-morphisms $f,g,h$ of $\\mathcal {C}$ , the composition $(\\mathcal {F}(h)\\circ \\mathcal {F}(g))\\circ \\mathcal {F}(f)\\stackrel{a}{\\Rightarrow }\\mathcal {F}(h)\\circ (\\mathcal {F}(g)\\circ \\mathcal {F}(f))\\stackrel{\\mathit {Id}\\bullet \\varphi }{\\Longrightarrow }\\mathcal {F}(h)\\circ \\mathcal {F}(g\\circ f)\\stackrel{\\varphi }{\\Rightarrow }\\mathcal {F}(h\\circ (g\\circ f))$ is equal to the composition $(\\mathcal {F}(h)\\circ \\mathcal {F}(g))\\circ \\mathcal {F}(f)\\stackrel{\\varphi \\bullet \\mathit {Id}}{\\Longrightarrow }\\mathcal {F}(h\\circ g)\\circ \\mathcal {F}(f)\\stackrel{\\varphi }{\\Rightarrow }\\mathcal {F}((h\\circ g)\\circ f)\\stackrel{\\mathcal {F}(a)}{\\Longrightarrow }\\mathcal {F}(h\\circ (g\\circ f))\\,.$ For any $X\\stackrel{f}{\\rightarrow }Y$ , the composition $I_{\\mathcal {F}(Y)}\\circ \\mathcal {F}(f)\\stackrel{\\varphi _Y\\bullet \\mathit {Id}}{\\Longrightarrow }\\mathcal {F}(I_Y)\\circ \\mathcal {F}(f)\\stackrel{\\varphi }{\\Rightarrow }\\mathcal {F}(I_Y\\circ f)\\stackrel{\\mathcal {F}(\\ell )}{\\Longrightarrow }\\mathcal {F}(f)$ coincides with $\\ell _{\\mathcal {F}(f)}$ , and similarly for $r$ ."
],
[
"A bicategory of cospans",
"Our goal is now to define a bicategory of Lagrangian cospans.",
"To do so, we will first work in the more general setting of a category with pushouts.",
"We wish to emphasize that our resulting bicategory of cospans differs from the usual definition considered in the literature, where the 2-morphisms are usually taken to be morphisms of cospans (see e.g. [3]).",
"On the other hand, a notion dual to the 2-morphisms that we consider was already studied by Morton [14] in another context (see also [15]).",
"Throughout this section, $\\mathbf {C}$ is a category with pushouts in which we fix a pushout for each span.",
"The objects of our bicategory are the objects of $\\mathbf {C}$ and the 1-morphisms are the cospans in $\\mathbf {C}$ , where the horizontal composition is given by our choice of a fixed pushout.",
"It remains to define the 2-morphisms, the vertical composition, the associativity and identity isomorphisms, and what is left of the horizontal composition.",
"A 2-cospan in $\\mathbf {C}$ from $H\\stackrel{i_1}{\\longrightarrow }T_1\\stackrel{i_1^{\\prime }}{\\longleftarrow }H^{\\prime }$ to $H\\stackrel{i_2}{\\longrightarrow }T_2\\stackrel{i_2^{\\prime }}{\\longleftarrow }H^{\\prime }$ consists of a cospan $T_1 \\stackrel{\\alpha _1}{\\longrightarrow }A\\stackrel{\\alpha _2}{\\longleftarrow }T_2$ in $\\mathbf {C}$ for which the two following squares commute $@R0.5cm{& T_1[d]^{\\alpha _1} & \\\\H [ur]^{i_1} [dr]_{i_2} & A & \\;H^{\\prime }\\,.",
"[ul]_{i^{\\prime }_1} [dl]^{i^{\\prime }_2} \\\\& T_2 [u]_{\\alpha _2} &}$ Two such 2-cospans $T_1 \\stackrel{\\alpha _1}{\\longrightarrow }A\\stackrel{\\alpha _2}{\\longleftarrow }T_2$ and $T_1 \\stackrel{\\alpha _1^{\\prime }}{\\longrightarrow }A^{\\prime }\\stackrel{\\alpha _2^{\\prime }}{\\longleftarrow }T_2$ are said to be isomorphic if there is a $\\mathbf {C}$ -isomorphism $f \\colon A \\rightarrow A^{\\prime }$ such that $f\\alpha _1=\\alpha _1^{\\prime }$ and $f\\alpha _2=\\alpha _2^{\\prime }$ .",
"Abusing notation, we shall often denote the isomorphism class of such a 2-cospan by $A\\colon T_1\\Rightarrow T_2$ .",
"These will be the 2-morphisms in our 2-category.",
"Let us now proceed with the definition of the vertical composition of the 2-morphisms $A\\colon T_1\\Rightarrow T_2$ and $B\\colon T_2\\Rightarrow T_3$ .",
"It is best explained by the diagram $@R0.5cm{& T_1[d]_{\\alpha _1} & \\\\& A & \\\\H @/^1pc/[uur]^{i_1} [r]^{i_2} & T_2 [u]^{\\alpha _2} & H^{\\prime }[l]_{i_2^{\\prime }}@/_1pc/[uul]_{i^{\\prime }_1}\\\\& \\star & \\\\H @/_1pc/[ddr]_{i_3} [r]^{i_2} & T_2 [d]_{\\beta _2} & H^{\\prime }[l]_{i_2^{\\prime }} @/^1pc/[ddl]^{i^{\\prime }_3} \\\\& B & \\\\& T_3[u]^{\\beta _3} & \\\\} \\ \\ @R0.5cm{\\\\ \\\\ \\\\ \\\\ = \\\\ \\\\ \\\\}\\ \\ @R0.5cm{\\\\ \\\\& T_1[d]^{v_A\\alpha _1} & \\\\H @/^1pc/[ur]^{i_1} @/_1pc/[dr]_{i_3} & B \\star A & H^{\\prime },@/_1pc/[ul]_{i_1^{\\prime }} @/^1pc/[dl]^{i_3^{\\prime }} \\\\& T_3 [u]_{v_B\\beta _3} & \\\\\\\\\\\\}$ where $B \\star A$ and $v_A, v_B$ are given by the pushout diagram $@R0.5cm{& & B \\star A & & & \\\\& A [ru]^{v_A} & & B [ul]_{v_B} \\\\T_1[ru]^{\\alpha _1} & & T_2 [ru]^{\\beta _2}[lu]_{\\alpha _2} & &T_3.",
"[lu]_{\\beta _3}&}$ One can easily check that this indeed defines a 2-cospan.",
"Remark 3.1 In the special case where $\\alpha _2=\\mathit {id}_{T_2}$ (resp.",
"$\\beta _2=\\mathit {id}_{T_2}$ ), this vertical composition $T_1\\stackrel{v_A\\alpha _1}{\\longrightarrow }B\\star A\\stackrel{v_B\\beta _3}{\\longleftarrow }T_3$ is isomorphic to $T_1\\stackrel{\\beta _2\\alpha _1}{\\longrightarrow }B\\stackrel{\\beta _3}{\\longleftarrow }T_3$ (resp.",
"$T_1\\stackrel{\\alpha _1}{\\longrightarrow }A\\stackrel{\\alpha _2\\beta _3}{\\longleftarrow }T_3$ ).",
"This is a direct consequence of Remark REF .",
"On the level 2-morphisms, the horizontal composition of $A\\colon T_1\\Rightarrow T_2$ and $B\\colon T_3\\Rightarrow T_4$ is described by the diagram $@R0.5cm{ & T_3 [d]^{\\beta _3} & \\\\H^{\\prime } [ur]^{i_3^{\\prime }} [dr]_{i_4^{\\prime }} & B & H^{\\prime \\prime }[ul]_{i^{\\prime \\prime }_3} [dl]^{i^{\\prime \\prime }_4} \\\\& T_4 [u]_{\\beta _4} }\\ @R0.5cm{ \\\\ \\bullet \\\\ }\\ @R0.5cm{& T_1[d]^{\\alpha _1} & \\\\H [ur]^{i_1} [dr]_{i_2} & A & H^{\\prime }[ul]_{i^{\\prime }_1} [dl]^{i^{\\prime }_2} \\\\& T_2 [u]_{\\alpha _2} &}\\ @R0.5cm{ \\\\ = \\\\ }\\ @R0.5cm{& T_3\\circ T_1[d]^{h_{31}} & \\\\H @/^1pc/[ur]^{j_1i_1} @/_1pc/[dr]_{j_2i_2} & B \\bullet A & \\;H^{\\prime \\prime }\\,,@/_1pc/[ul]_{j_3i_3^{\\prime \\prime }} @/^1pc/[dl]^{j_4i_4^{\\prime \\prime }} \\\\& T_4\\circ T_2 [u]_{h_{42}} &}$ where $j_1,\\dots ,j_4$ are the maps that arise in the compositions $T_3\\circ T_1$ and $T_4\\circ T_2$ (see the diagrams below), $B \\bullet A$ is given by the pushout $@R0.5cm{& & B \\bullet A & & & \\\\& A [ru]^{h_A} & & B [ul]_{h_B} \\\\H[ru]^{\\alpha _1i_1} & & H^{\\prime } [ru]^{\\beta _3i_3^{\\prime }}[lu]_{\\alpha _2i_2^{\\prime }} & &H^{\\prime \\prime }\\,,[lu]_{\\beta _4i_4^{\\prime \\prime }}&}$ and the maps $h_{31}$ and $h_{42}$ are obtained as follows.",
"Since $h_A \\alpha _1 i_1^{\\prime }=h_A \\alpha _2 i_2^{\\prime }=h_B \\beta _3 i_3^{\\prime }$ and $h_A\\alpha _2 i_2^{\\prime }=h_B\\beta _3i_3^{\\prime }=h_B\\beta _4i_4^{\\prime }$ , the pushout diagrams $@R0.4cm{& & B \\bullet A & & & \\\\& & T_3\\circ T_1 [u]^{h_{31}} & & & \\\\& A @/^1pc/[uur]^{h_A} & & B @/_1pc/[uul]_{h_B} & & \\\\& T_1 [ruu]^{j_1} [u]^{\\alpha _1} & & T_3 [uul]_{j_3} [u]_{\\beta _3} \\\\& & H^{\\prime } [lu]_{i_1^{\\prime }}[ru]^{i_3^{\\prime }} & &}@R0.4cm{& & B \\bullet A & & & \\\\& & T_4\\circ T_2 [u]^{h_{42}} & & & \\\\& A @/^1pc/[uur]^{h_A} & & B @/_1pc/[uul]_{h_B} & & \\\\& T_2 [ruu]^{j_2} [u]^{\\alpha _2} & & T_4 [uul]_{j_4} [u]_{\\beta _4} \\\\& & H^{\\prime } [lu]_{i_2^{\\prime }}[ru]^{i_4^{\\prime }} & &}$ provide maps $h_{31}$ and $h_{42}$ which turn $T_3 \\circ T_1 \\stackrel{h_{31}}{\\longrightarrow }B \\bullet A\\stackrel{h_{42}}{\\longleftarrow }T_4 \\circ T_2$ into a 2-cospan, as one easily checks.",
"The proof of the following theorem can be found in [15] in the dual context of spans.",
"It applies without change to the present setting.",
"Theorem 3.2 Let $\\mathbf {C}$ be a category with pushouts in which a choice of pushout is fixed for each span.",
"Objects in $\\mathbf {C}$ , as objects, cospans in $\\mathbf {C}$ , as morphisms, and isomorphism classes of 2-cospans in $\\mathbf {C}$ , as 2-morphisms, form a bicategory.",
"Note that strictly speaking, this bicategory depends on the choice of pushouts.",
"However, another choice would give a bicategory isomorphic in an obvious sense (see e.g. [3]).",
"The special case where $\\mathbf {C}$ is the category of $\\Lambda $ -modules and the morphisms are Lagrangian cospans yields the following corollary.",
"Corollary 3.3 Fix a pushout for each span of $\\Lambda $ -modules.",
"Hermitian $\\Lambda $ -modules, as objects, Lagrangian cospans, as morphisms, and isomorphism classes of 2-cospans, as 2-morphisms, form a bicategory.$\\Box $ We shall call it “the” bicategory of Lagrangian cospans."
],
[
"The Burau-Alexander 2-functor",
"The aim of this section is to define a weak 2-functor $\\mathcal {B}$ from the bicategory of oriented tangles to the bicategory of Lagrangian cospans where $\\Lambda $ is the ring ${Z}[t^{\\pm 1}]$ of Laurent polynomials in one variable with integer coefficients.",
"We proceed in two steps: in Subsection REF , we recall the definition of the category of oriented tangles, and construct a functor $\\mathcal {B} \\colon \\textbf {Tangles} \\rightarrow \\textbf {L}_\\Lambda $ .",
"In Subsections REF and REF , we study the bicategory of tangles, and convert $\\mathcal {B}$ into a weak 2-functor with values in the bicategory of Lagrangian cospans."
],
[
"The functor $\\mathcal {B}$ on objects and 1-morphisms",
"We start by recalling the definition of the category of oriented tangles.",
"Let $D^2$ be the closed unit disk in ${R}^2$ .",
"Given a non-negative integer $n$ , let $x_j$ be the point $((2j-n-1)/n,0)$ in $D^2$ , for $j=1,\\dots ,n$ .",
"Let $\\varepsilon $ and $\\varepsilon ^{\\prime }$ be sequences of $\\pm 1$ 's of respective length $n$ and $n^{\\prime }$ .",
"An $(\\varepsilon ,\\varepsilon ^{\\prime })$ -tangle is a pair consisting of the cylinder $D^2\\times [0,1]$ and an oriented smooth 1-submanifold $\\tau $ whose oriented boundary is $\\sum _{j=1}^{n^{\\prime }}\\varepsilon ^{\\prime }_j(x^{\\prime }_j,1)-\\sum _{j=1}^{n}\\varepsilon _i(x_j,0)$ .",
"Note that a $(\\emptyset ,\\emptyset )$ -tangle is nothing but an oriented link.",
"Two $(\\varepsilon ,\\varepsilon ^{\\prime })$ -tangles $\\tau _1$ and $\\tau _2$ are isotopic if there exists an isotopy $h_t$ of $D^2\\times [0,1]$ , keeping $D^2\\times \\lbrace 0,1\\rbrace $ fixed, such that $h_1\\vert _{\\tau _1}\\colon \\tau _1\\simeq \\tau _2$ is an orientation-preserving homeomorphism.",
"We shall denote by $I_\\varepsilon $ the isotopy class of the trivial $(\\varepsilon ,\\varepsilon )$ -tangle $(D^2,\\lbrace x_1,\\dots ,x_n\\rbrace )\\times [0,1]$ .",
"Given an $(\\varepsilon ,\\varepsilon ^{\\prime })$ -tangle $\\tau _1$ and an $(\\varepsilon ^{\\prime },\\varepsilon ^{\\prime \\prime })$ -tangle $\\tau _2$ , their composition is the $(\\varepsilon ,\\varepsilon ^{\\prime \\prime })$ -tangle $\\tau _2\\circ \\tau _1$ obtained by gluing the two cylinders along the disk corresponding to $\\varepsilon ^{\\prime }$ , smoothing it if needed, and shrinking the length of the resulting cylinder by a factor 2 (see Figure REF ).",
"Clearly, the composition induces a composition on the isotopy classes of tangles, which is associative and admits $I_\\varepsilon $ as a 2-sided unit.",
"Therefore, the sequences of $\\pm 1$ 's, as objects, and the isotopy classes of tangles, as morphisms, form a category denoted by $\\mathbf {Tangles}$ and called the category of oriented tangles.",
"Figure: An (ε,ε ' )(\\varepsilon ,\\varepsilon ^{\\prime })-tangle τ 1 \\tau _1 with ε=(+1,-1)\\varepsilon =(+1,-1) and ε ' =(-1,+1,-1,+1)\\varepsilon ^{\\prime }=(-1,+1,-1,+1), an (ε ' ,ε '' )(\\varepsilon ^{\\prime },\\varepsilon ^{\\prime \\prime })-tangle τ 2 \\tau _2with ε '' =(-1,+1)\\varepsilon ^{\\prime \\prime }=(-1,+1), and their composition, the (ε,ε '' )(\\varepsilon ,\\varepsilon ^{\\prime \\prime })-tangle τ 2 ∘τ 1 \\tau _2\\circ \\tau _1.Recall that a tangle $\\tau \\subset D^2 \\times [0,1]$ is called an oriented braid if every component of $\\tau $ is strictly increasing or strictly decreasing with respect to the projection onto $[0,1]$ .",
"The finite sequences of $\\pm 1$ 's as objects, and the isotopy classes of oriented braids, as morphisms, form a subcategory $\\mathbf {Braids}$ of $\\mathbf {Tangles}$ , which is nothing but its core.",
"Finally a tangle $\\tau \\subset D^2\\times [0,1]$ is called an oriented string link if every component of $\\tau $ joins $D^2 \\times \\lbrace 0 \\rbrace $ and $D^2 \\times \\lbrace 1 \\rbrace $ .",
"Isotopy classes of oriented string links are the morphisms of a category $\\mathbf {Strings}$ which satisfies $\\mathbf {Braids} \\subset \\mathbf {Strings} \\subset \\mathbf {Tangles}\\,,$ where all the inclusions denote embeddings of categories.",
"We are now ready to define our Burau-Alexander functor $\\mathcal {B}\\colon \\mathbf {Tangles}\\rightarrow \\mathbf {L}_\\Lambda $ .",
"We start by defining it on objects, following the construction of [6].",
"Denote by $\\mathcal {N}(\\lbrace x_1,\\dots ,x_n\\rbrace )$ an open tubular neighborhood of $\\lbrace x_1,\\dots , x_n \\rbrace $ in $D^2\\subset {R}^2$ , and let $S^2$ be the 2-sphere obtained by the one-point compactification of ${R}^2$ .",
"Given a sequence $\\varepsilon =(\\varepsilon _1,\\dots ,\\varepsilon _n)$ of $\\pm 1$ , set $\\ell _\\varepsilon =\\sum _{i=1}^n \\varepsilon _i$ and endow the compact surface $D_\\varepsilon = \\left\\lbrace \\begin{array}{lr}D^2\\setminus \\mathcal {N}(\\lbrace x_1,\\dots ,x_n\\rbrace ) & \\text{if } \\ell _\\varepsilon \\ne 0\\\\S^2\\setminus \\mathcal {N}(\\lbrace x_1,\\dots ,x_n\\rbrace ) & \\text{if } \\ell _\\varepsilon =0\\end{array}\\right.$ with an orientation (pictured counterclockwise), a base point $z$ , and the generating family $\\lbrace e_1,\\dots , e_n \\rbrace $ of $\\pi _1(D_\\varepsilon ,z)$ , where $e_i$ is a simple loop turning once around $x_i$ counterclockwise if $\\varepsilon _i=+1$ , clockwise if $\\varepsilon _i=-1$ .",
"The same space with the opposite orientation will be denoted by $-D_\\varepsilon $ .",
"The natural epimomorphism $H_1(D_\\varepsilon )\\rightarrow \\mathbb {Z}$ , given by $e_j \\mapsto 1$ induces an infinite cyclic covering $\\widehat{D}_\\varepsilon \\rightarrow D_\\varepsilon $ whose homology is endowed with a structure of module over $\\Lambda ={Z}[t^{\\pm 1}]$ .",
"If $\\ell _\\varepsilon \\ne 0$ , then $D_\\varepsilon $ obviously retracts by deformation on the wedge of $n$ circles representing the generators $e_1,\\dots ,e_n$ of $\\pi _1(D_\\varepsilon ,z)$ , and one can check that $H_1(\\widehat{D}_\\varepsilon )$ is a free $\\Lambda $ -module of rank $n-1$ .",
"(It is free of rank $n-2$ if $\\ell _\\varepsilon $ vanishes.)",
"If $\\langle \\ , \\ \\rangle \\colon H_1(\\widehat{D}_\\varepsilon )\\times H_1(\\widehat{D}_\\varepsilon )\\rightarrow {Z}$ denotes the skew-symmetric intersection form obtained by lifting the orientation of $D_\\varepsilon $ to $\\widehat{D}_\\varepsilon $ , then the formula $\\omega _\\varepsilon (x,y)=\\sum _{k \\in \\mathbb {Z}} \\langle t^kx,y \\rangle t^{-k}$ defines a skew-Hermitian $\\Lambda $ -valued pairing on $H_1(\\widehat{D}_\\varepsilon )$ which is non-degenerate by [6].",
"(This is the only reason for considering $S^2$ instead of $D^2$ when $\\ell _\\varepsilon $ vanishes.)",
"Therefore, following the terminology of subsection REF , $\\mathcal {B}(\\varepsilon ):=(H_1(\\widehat{D}_\\varepsilon ),\\omega _\\varepsilon )$ is a free Hermitian $\\Lambda $ -module for any object $\\varepsilon $ of the category of oriented tangles.",
"Note that this coincides with the definition of the Lagrangian functor $\\mathcal {F}\\colon \\mathbf {Tangles}\\rightarrow \\mathbf {Lagr}_\\Lambda $ of [6] at the level of objects.",
"Let us now turn to morphisms.",
"First note that the existence of an $(\\varepsilon ,\\varepsilon ^{\\prime })$ -tangle $\\tau \\subset D^2 \\times [0,1]$ implies that $\\ell _\\varepsilon =\\ell _{\\varepsilon ^{\\prime }}$ .",
"Denote by $\\mathcal {N}(\\tau )$ an open tubular neighborhood of $\\tau $ in $D^2 \\times [0,1]$ .",
"We shall orient the exterior $X_\\tau = \\left\\lbrace \\begin{array}{lr}(D^2 \\times [0,1]) \\setminus \\mathcal {N}(\\tau ) & \\text{if } \\ell _\\varepsilon \\ne 0\\\\(S^2 \\times [0,1]) \\setminus \\mathcal {N}(\\tau ) & \\text{if } \\ell _\\varepsilon =0\\end{array}\\right.$ of $\\tau $ so that the induced orientation on $\\partial X_\\tau $ extends the orientation on the space $(-D_\\varepsilon ) \\sqcup D_{\\varepsilon ^{\\prime }}$ .",
"Clearly, the abelian group $H_1(X_\\tau )$ is generated by the oriented meridians of the connected components of $\\tau $ .",
"The homomorphism $H_1(X_\\tau )\\rightarrow \\mathbb {Z}$ mapping these meridians to 1 extends the previously defined homomorphisms $H_1(D_\\varepsilon ) \\rightarrow \\mathbb {Z}$ and $H_1(D_{\\varepsilon ^{\\prime }})\\rightarrow \\mathbb {Z}$ .",
"It determines an infinite cyclic covering $\\widehat{X}_\\tau \\rightarrow X_\\tau $ whose homology is endowed with a structure of module over $\\Lambda $ .",
"Let $i_\\tau \\colon H_1(\\widehat{D}_\\varepsilon ) \\rightarrow H_1(\\widehat{X}_\\tau )$ and $i_\\tau ^{\\prime }\\colon H_1(\\widehat{D}_{\\varepsilon ^{\\prime }}) \\rightarrow H_1(\\widehat{X}_\\tau )$ be the homomorphisms induced by the inclusions of $\\widehat{D}_\\varepsilon $ and $\\widehat{D}_{\\varepsilon ^{\\prime }}$ into $\\widehat{X}_\\tau $ .",
"Since $\\mathcal {F}(\\tau )=\\overline{\\mathit {Ker}{-i_\\tau \\atopwithdelims ()\\phantom{-}i_\\tau ^{\\prime }}}$ is a Lagrangian submodule of $(-H_1(\\widehat{D}_\\varepsilon ))\\oplus H_1(\\widehat{D}_\\varepsilon ^{\\prime })$ [6], it follows that $H_1(\\widehat{D}_\\varepsilon ) \\stackrel{i_\\tau }{\\longrightarrow }H_1(\\widehat{X}_\\tau ) \\stackrel{i_\\tau ^{\\prime }}{\\longleftarrow }H_1(\\widehat{D}_{\\varepsilon ^{\\prime }})$ is a Lagrangian cospan for any 1-morphism $\\tau $ in the category of oriented tangles.",
"Note that the equality above is nothing but the definition of the Lagrangian functor $\\mathcal {F}$ of [6] at the level of morphisms.",
"Theorem 4.1 For any sequence $\\varepsilon $ of $\\pm 1$ 's, set $\\mathcal {B}(\\varepsilon )=(H_1(\\widehat{D}_\\varepsilon ),\\omega _\\varepsilon )$ and for any isotopy class $\\tau $ of tangles, let $\\mathcal {B}(\\tau )$ denote the isomorphism class of the Lagrangian cospan $H_1(\\widehat{D}_\\varepsilon ) \\stackrel{i_\\tau }{\\longrightarrow }H_1(\\widehat{X}_\\tau ) \\stackrel{i_\\tau ^{\\prime }}{\\longleftarrow }H_1(\\widehat{D}_{\\varepsilon ^{\\prime }})$ .",
"This defines a functor $\\mathcal {B}\\colon \\mathbf {Tangles}\\rightarrow \\mathbf {L}_\\Lambda $ which fits in the commutative diagram $@R0.5cm{\\mathbf {Braids} [d] [r] @/_3pc/[dd]_\\rho & \\mathbf {String} [d] [r] & \\mathbf {Tangles} [d]^{\\mathcal {B}} @/^2.5pc/[dd]^{\\mathcal {F}}\\ \\\\\\mathit {core}(\\mathbf {L}_\\Lambda ) [d]_\\simeq [r] & \\mathit {core}(\\mathbf {L}_\\Lambda )^0 [d][r] & \\mathbf {L}_\\Lambda [d]^F \\\\\\mathbf {U}_\\Lambda [r]^{- \\otimes Q} @/_1.5pc/[rr]_\\Gamma & \\mathbf {U}^0_\\Lambda [r]^{\\Gamma ^0} & \\mathbf {Lagr}_\\Lambda ,}$ where the left-most vertical arrow is the Burau functor, the horizontal arrows are the embeddings of categories described in Subsections REF and REF , and $F$ is the full functor defined in Subsection REF (recall diagram (REF )).",
"Furthermore, if $\\tau $ is an oriented link, then $\\mathcal {B}(\\tau )$ is nothing but its Alexander module.",
"For any object $\\varepsilon $ , the cospan associated to the identity tangle $I_\\varepsilon $ is canonically isomorphic to the identity cospan $I_{\\mathcal {B}(\\varepsilon )}$ .",
"Let us now check that given $\\tau _1 \\in T(\\varepsilon ,\\varepsilon ^{\\prime })$ and $\\tau _2 \\in T(\\varepsilon ^{\\prime },\\varepsilon ^{\\prime \\prime })$ , we have the equality $\\mathcal {B}(\\tau _2\\circ \\tau _1)=\\mathcal {B}(\\tau _2)\\circ \\mathcal {B}(\\tau _1)$ .",
"Let $H_1(\\widehat{D}_\\varepsilon ) \\stackrel{i_1}{\\rightarrow }H_1(\\widehat{X}_{\\tau _1}) \\stackrel{i_1^{\\prime }}{\\leftarrow }H_1(\\widehat{D}_{\\varepsilon ^{\\prime }})$ and $H_1(\\widehat{D}_{\\varepsilon ^{\\prime }}) \\stackrel{i_2^{\\prime }}{\\rightarrow }H_1(\\widehat{X}_{\\tau _2}) \\stackrel{i_2^{\\prime \\prime }}{\\leftarrow }H_1(\\widehat{D}_{\\varepsilon ^{\\prime \\prime }})$ be the Lagrangian cospans arising from $\\tau _1$ and $\\tau _2$ .",
"We must show that $H_1(\\widehat{D}_\\varepsilon ) \\stackrel{k_1i_1}{\\longrightarrow } H_1(\\widehat{X}_{\\tau _3 \\circ \\tau _1}) \\stackrel{k_2i_2^{\\prime \\prime }}{\\longleftarrow }H_1(\\widehat{D}_{\\varepsilon ^{\\prime \\prime }})$ is isomorphic to the composition $H_1(\\widehat{D}_\\varepsilon ) \\stackrel{j_1i_1}{\\longrightarrow } H_1(\\widehat{X}_{\\tau _3})\\circ H_1(\\widehat{X}_{\\tau _1}) \\stackrel{j_2i_2^{\\prime \\prime }}{\\longleftarrow }H_1(\\widehat{D}_{\\varepsilon ^{\\prime \\prime }})$ , where $k_1,k_2$ are the inclusion induced maps and $j_1,j_2$ are maps resulting from any representative of the pushout $H_1(\\widehat{X}_{\\tau _3})\\circ H_1(\\widehat{X}_{\\tau _1})$ .",
"Observe that $\\widehat{X}_{\\tau _2\\circ \\tau _1}$ decomposes as the union of $\\widehat{X}_{\\tau _1}$ and $\\widehat{X}_{\\tau _2}$ glued along $\\widehat{D}_{\\varepsilon ^{\\prime }}$ .",
"Therefore, the associated Mayer-Vietoris exact sequence $@R0.5cm{H_1(\\widehat{D}_{\\varepsilon ^{\\prime }})[r]^{\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!",
"(-i_1^{\\prime },i^{\\prime }_2)}&H_1(\\widehat{X}_{\\tau _1})\\oplus H_1(\\widehat{X}_{\\tau _2})[r]^{\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;{k_1\\atopwithdelims ()k_2}}&H_1(\\widehat{X}_{\\tau _2\\circ \\tau _1})[r]& 0}$ together with Lemma REF imply that $H_1(\\widehat{X}_{\\tau _1}) \\stackrel{k_1}{\\longrightarrow } H_1(\\widehat{X}_{\\tau _2 \\circ \\tau _1}) \\stackrel{k_2}{\\longleftarrow }H_1(\\widehat{X}_{\\tau _2})$ is a representative of the pushout $H_1(\\widehat{X}_{\\tau _1}) \\circ H_1(\\widehat{X}_{\\tau _2})$ .",
"The claim follows.",
"Then, observe that the Lagrangian functor $\\mathcal {F}$ is by definition the composition of the functors $\\mathcal {B}\\colon \\mathbf {Tangles}\\rightarrow \\mathbf {L}_\\Lambda $ and $F\\colon \\mathbf {L}_\\Lambda \\rightarrow \\mathbf {Lagr}_\\Lambda $ .",
"Also, if $\\tau $ is an oriented string link, then $\\mathcal {B}(\\tau )$ is a rationally invertible cospan by [13], and thus belongs to $\\mathit {core}(\\mathbf {L}_\\Lambda )^0$ by definition.",
"If $\\tau $ is an oriented braid on the other hand, then $\\mathcal {B}(\\tau )$ is obviously an invertible cospan, and therefore belongs to $\\mathit {core}(\\mathbf {L}_\\Lambda )$ by Proposition REF .",
"Finally, if $\\tau $ is a $(\\emptyset ,\\emptyset )$ -tangle, that is, an oriented link $L$ , then the associated Lagrangian cospan is given by $0\\rightarrow H_1(\\widehat{X}_L)\\leftarrow 0$ , with $X_L$ the complement of $L$ in the 3-ball.",
"A straightforward Mayer-Vietoris argument shows that considering $L$ in the 3-ball or in the 3-sphere does not change the Alexander module, and the proof is completed."
],
[
"The bicategory of tangles",
"The aim is now to convert $\\mathcal {B}$ to a weak 2-functor.",
"To do so, we first need to understand how tangles form a (possibly weak) 2-category.",
"Once this is done, we will switch from the category $\\textbf {L}_\\Lambda $ to the bicategory of Lagrangian cospans and define the weak 2-functor in subsection REF .",
"One might think that tangles produce a 2-category in a straightforward way [7]: simply define the objects and 1-morphisms as in $\\mathbf {Tangles}$ , and the 2-morphisms as isotopy classes of oriented surfaces in $D^2\\times [0,1]\\times [0,1]$ .",
"However, the corresponding vertical composition is not well-defined: indeed, one needs to paste two surfaces along isotopic tangles, and since the space of tangles isotopic to a fixed one is not necessarily simply-connected, different choices of isotopies can lead to different surfaces.",
"There are a couple of ways to circumvent this difficulty.",
"One of them is to restrict the space of tangles whose isotopy classes form the 1-morphisms, so that the corresponding space of isotopic tangles has trivial fundamental group.",
"Such a construction was given by Kharlamov and Turaev in [11] (see also [1]): they considered the class of so-called generic tangles, and proved that the space of generic tangles isotopic to a fixed one (through generic tangles) is simply-connected, thus obtaining a strict 2-category.",
"However, it is more natural in our setting to take the following alternative approach: define 1-morphisms as oriented tangles, and consider isotopies bewteen tangles as part of the “higher structure”.",
"Figure: A cobordism Σ⊂D 2 ×[0,1]×[0,1]\\Sigma \\subset D^2\\times [0,1]\\times [0,1] between two (ε,ε ' )(\\varepsilon ,\\varepsilon ^{\\prime })-tangles τ 1 \\tau _1 and τ 2 \\tau _2, with ε=(+1,+1,-1)\\varepsilon =(+1,+1,-1)and ε ' =(+1)\\varepsilon ^{\\prime }=(+1).Let us be more precise.",
"The objects of this bicategory are sequences $\\varepsilon $ of $\\pm 1$ 's, while the 1-morphisms from $\\varepsilon $ to $\\varepsilon ^{\\prime }$ are the $(\\varepsilon ,\\varepsilon ^{\\prime })$ -tangles in $D^2\\times [0,1]$ that are trivial near the top and bottom of the cylinder.",
"(This is to ensure that the composition of two tangles remains a smooth 1-submanifold.)",
"Given two $(\\varepsilon ,\\varepsilon ^{\\prime })$ -tangles $\\tau _1$ and $\\tau _2$ , a $(\\tau _1,\\tau _2)$ -cobordism is a pair consisting of the 4-ball $D^2\\times [0,1]\\times [0,1]$ together with a proper oriented smooth 2-submanifold $\\Sigma $ whose oriented boundary is given by $\\partial \\Sigma =(\\tau _2 \\times \\lbrace 0\\rbrace )\\cup (\\varepsilon ^{\\prime }\\times \\lbrace 1\\rbrace \\times [0,1])\\cup ((-\\tau _1)\\times \\lbrace 1\\rbrace )\\cup ((-\\varepsilon )\\times \\lbrace 0\\rbrace \\times [0,1])\\,,$ as illustrated in Figure REF .",
"Note that a $(\\emptyset ,\\emptyset )$ -cobordism is nothing but a closed oriented surface embedded in the 4-ball.",
"Two $(\\tau _1,\\tau _2)$ -cobordisms $\\Sigma $ and $\\Sigma ^{\\prime }$ are isotopic if there exists an isotopy $h_t$ of $D^2\\times [0,1] \\times [0,1]$ , keeping $\\partial (D^2\\times [0,1]\\times [0,1])$ fixed, such that $h_1\\vert _{\\Sigma }\\colon \\Sigma \\simeq \\Sigma ^{\\prime }$ is an orientation-preserving homeomorphism and $h_t(\\Sigma )$ is a $(\\tau _1,\\tau _2)$ -cobordism for all $t$ .",
"We shall denote by $\\Sigma \\colon \\tau _1\\Rightarrow \\tau _2$ the isotopy class of a $(\\tau _1,\\tau _2)$ -cobordism $\\Sigma $ , and by $\\mathit {Id}_\\tau $ the isotopy class of the trivial $(\\tau ,\\tau )$ -cobordism $(D^2\\times [0,1],\\tau )\\times [0,1]$ .",
"Fix a $(\\tau _1,\\tau _2)$ -cobordism $\\Sigma $ and a $(\\tau _2,\\tau _3)$ -cobordism $\\Sigma ^{\\prime }$ .",
"Their vertical composition is the $(\\tau _1,\\tau _3)$ -cobordism $\\Sigma _2\\star \\Sigma _1$ obtained by gluing the two 4-balls along the cylinders containing $\\tau _2$ , and shrinking the height of the resulting 4-ball $D^2\\times [0,1]\\times [0,2]$ by a factor 2 (see Figure REF ).",
"Finally, fix $(\\varepsilon ,\\varepsilon ^{\\prime })$ -tangles $\\tau _1,\\tau _2$ and $(\\varepsilon ^{\\prime },\\varepsilon ^{\\prime \\prime })$ -tangles $\\tau _3,\\tau _4$ .",
"Given a $(\\tau _1,\\tau _2)$ -cobordism $\\Sigma _1$ and a $(\\tau _3,\\tau _4)$ -cobordism $\\Sigma _2$ , their horizontal composition is the $(\\tau _3\\circ \\tau _1,\\tau _4\\circ \\tau _2)$ -cobordism $\\Sigma _2\\bullet \\Sigma _1$ obtained by gluing the two 4-balls along the cylinder $D^2 \\times [0,1]$ corresponding to $\\varepsilon ^{\\prime }$ , and shrinking the length of the resulting 4-ball by a factor 2 (Figure REF ).",
"Figure: The vertical composition of a (τ 1 ,τ 2 )(\\tau _1,\\tau _2)-cobordism Σ 1 \\Sigma _1 and a (τ 2 ,τ 3 )(\\tau _2,\\tau _3)-cobordism Σ 2 \\Sigma _2, the (τ 1 ,τ 3 )(\\tau _1,\\tau _3)-cobordism Σ 2 ☆Σ 1 \\Sigma _2\\star \\Sigma _1.Figure: The horizontal composition of a (τ 1 ,τ 2 )(\\tau _1,\\tau _2)-cobordism Σ 1 \\Sigma _1 and a (τ 3 ,τ 4 )(\\tau _3,\\tau _4)-cobordism Σ 2 \\Sigma _2,the (τ 3 ∘τ 1 ,τ 4 ∘τ 2 )(\\tau _3\\circ \\tau _1,\\tau _4\\circ \\tau _2)-cobordism Σ 2 •Σ 1 \\Sigma _2\\bullet \\Sigma _1.The bicategory of oriented tangles can now be defined as follows: the objects are the finite sequences of $\\pm 1$ 's, the 1-morphisms are given by the tangles, and the 2-morphisms are given by isotopy classes of cobordisms as described above.",
"Finally, the associativity and identity isomorphisms $a\\colon (\\tau _3\\circ \\tau _2)\\circ \\tau _1\\Rightarrow \\tau _3\\circ (\\tau _2\\circ \\tau _1),\\quad \\ell _{\\tau }\\colon I_{\\varepsilon ^{\\prime }}\\circ \\tau \\Rightarrow \\tau ,\\quad r_\\tau \\colon \\tau \\circ I_{\\varepsilon }\\Rightarrow \\tau $ are given by the trace of the obvious isotopies.",
"It is a routine check to verify that all the axioms of a bicategory are satisfied."
],
[
"The weak 2-functor",
"We are now ready to define our weak 2-functor from the bicategory of oriented tangles to the bicategory of Lagrangian cospans.",
"Recall from subsection REF that we must associate a Hermitian $\\Lambda $ -module $\\mathcal {B}(\\varepsilon )$ to each object $\\varepsilon $ , a cospan $\\mathcal {B}(\\tau )$ to each tangle $\\tau $ and an isomorphism class of 2-cospans to each cobordism $\\Sigma $ .",
"Additionally, for each $\\varepsilon $ , we must define an invertible 2-morphism $\\varphi _\\varepsilon \\colon I_{\\mathcal {B}(\\varepsilon )} \\Rightarrow \\mathcal {B}(I_\\varepsilon )$ and for each pair $\\tau _1, \\tau _2$ of composable tangles, an invertible 2-morphism $\\varphi _{\\tau _1\\tau _2} \\colon \\mathcal {B}(\\tau _2) \\circ \\mathcal {B}(\\tau _1) \\Rightarrow \\mathcal {B}(\\tau _2\\circ \\tau _1)$ .",
"Let us associate to each object $\\varepsilon $ the Hermitian $\\Lambda $ -module $\\mathcal {B}(\\varepsilon )=(H_1(\\widehat{D}_\\varepsilon ),\\omega _\\varepsilon )$ and to each $(\\varepsilon ,\\varepsilon ^{\\prime })$ -tangle $\\tau $ the Lagrangian cospan $\\mathcal {B}(\\tau )$ given by $H_1(\\widehat{D}_\\varepsilon ) \\stackrel{i_\\tau }{\\longrightarrow }H_1(\\widehat{X}_\\tau ) \\stackrel{i_\\tau ^{\\prime }}{\\longleftarrow }H_1(\\widehat{D}_{\\varepsilon ^{\\prime }})$ .",
"(Note that we slightly abuse notations here, as $\\mathcal {B}(\\tau )$ now no longer stands for the isomorphism class of this cospan, but for the cospan itself.)",
"As for 2-morphisms, we proceed as follows.",
"Fix two $(\\varepsilon ,\\varepsilon ^{\\prime })$ -tangles $\\tau _1,\\tau _2$ .",
"Given a $(\\tau _1,\\tau _2)$ -cobordism $\\Sigma \\subset D^2 \\times [0,1] \\times [0,1]$ , denote by $\\mathcal {N}(\\Sigma )$ an open tubular neighborhood of $\\Sigma $ in $D^2 \\times [0,1] \\times [0,1]$ .",
"We shall orient the exterior $W_\\Sigma = \\left\\lbrace \\begin{array}{lr}(D^2 \\times [0,1] \\times [0,1]) \\setminus \\mathcal {N}(\\Sigma ) & \\text{if } \\ell _\\varepsilon \\ne 0\\\\(S^2 \\times [0,1] \\times [0,1]) \\setminus \\mathcal {N}(\\Sigma ) & \\text{if } \\ell _\\varepsilon =0\\end{array}\\right.$ of $\\Sigma $ so that the induced orientation on $\\partial W_\\Sigma $ extends the orientation on the space $(-X_{\\tau _1}) \\sqcup X_{\\tau _2}$ .",
"Clearly, $H_1(W_\\Sigma )$ is generated by the (oriented) meridians of the connected components of $\\Sigma $ .",
"The homomorphism $H_1(W_\\Sigma )\\rightarrow \\mathbb {Z}$ obtained by mapping these meridians to 1 extends the previously defined homomorphisms $H_1(X_{\\tau _1}) \\rightarrow \\mathbb {Z}$ and $H_1(X_{\\tau _2})\\rightarrow \\mathbb {Z}$ .",
"It determines an infinite cyclic covering $\\widehat{W}_\\Sigma \\rightarrow W_\\Sigma $ whose homology is endowed with a structure of module over $\\Lambda $ .",
"Denote by $H_1(\\widehat{D}_\\varepsilon ) \\stackrel{i_1}{\\longrightarrow }H_1(\\widehat{X}_{\\tau _1}) \\stackrel{i_1^{\\prime }}{\\longleftarrow }H_1(\\widehat{D}_{\\varepsilon ^{\\prime }})$ and $H_1(\\widehat{D}_\\varepsilon ) \\stackrel{i_2}{\\longrightarrow }H_1(\\widehat{X}_{\\tau }) \\stackrel{i_2^{\\prime }}{\\longleftarrow }H_1(\\widehat{D}_{\\varepsilon ^{\\prime }})$ the Lagrangian cospans arising from $\\tau _1$ and $\\tau _2$ , and let $\\alpha _{1}\\colon H_1(\\widehat{X}_{\\tau _1}) \\rightarrow H_1(\\widehat{W}_\\Sigma )$ and $\\alpha _{2}\\colon H_1(\\widehat{X}_{\\tau _2}) \\rightarrow H_1(\\widehat{W}_\\Sigma )$ be the homomorphisms induced by the inclusions of $\\widehat{X}_{\\tau _1}$ and $\\widehat{X}_{\\tau _2}$ into $\\widehat{W}_\\Sigma $ .",
"Combining all these inclusion induced maps, the following diagram commutes $@R0.5cm{& H_1(\\widehat{X}_{\\tau _1}) [d]^{\\alpha _1} & \\\\H_1(\\widehat{D}_{\\varepsilon }) [ur]^{i_1} [dr]_{i_2} & H_1(\\widehat{W}_\\Sigma ) & H_1(\\widehat{D}_{\\varepsilon ^{\\prime }})[ul]_{i_1^{\\prime }}[dl]^{i_2^{\\prime }}\\,.",
"\\\\& H_1(\\widehat{X}_{\\tau _2}) [u]_{\\alpha _2} &}$ Hence, $H_1(\\widehat{X}_{\\tau _1}) \\stackrel{\\alpha _1}{\\longrightarrow }H_1(\\widehat{W}_\\Sigma ) \\stackrel{\\alpha _2}{\\longleftarrow }H_1(\\widehat{X}_{\\tau _2})$ is a 2-cospan, whose isomorphism class we denote by $\\mathcal {B}(\\Sigma )\\colon \\mathcal {B}(\\tau _1)\\Rightarrow \\mathcal {B}(\\tau _2)$ .",
"Given any object $\\varepsilon $ , let $\\alpha _\\varepsilon \\colon H_1(\\widehat{D}_{\\varepsilon })\\rightarrow H_1(\\widehat{X}_{I_\\varepsilon })$ denote the isomorphism of $\\Lambda $ -modules induced by the inclusion of $D_\\varepsilon $ in $D_\\varepsilon \\times [0,1]=X_{I_\\varepsilon }$ .",
"This isomorphism fits in the commutative diagram $@R0.5cm{& H_1(\\widehat{D}_{\\varepsilon }) [d]^{\\alpha _\\varepsilon } & \\\\H_1(\\widehat{D}_{\\varepsilon }) [ur]^{\\mathit {id}} [dr]_{\\alpha _\\varepsilon } & H_1(\\widehat{X}_{I_\\varepsilon }) & H_1(\\widehat{D}_{\\varepsilon })[ul]_\\mathit {id}[dl]^{\\alpha _\\varepsilon } \\,.",
"\\\\& H_1(\\widehat{X}_{I_\\varepsilon }) [u]_\\mathit {id}&}$ By Remark REF , the 2-morphism $\\varphi _\\varepsilon \\colon I_{\\mathcal {B}(\\varepsilon )}\\Rightarrow \\mathcal {B}(I_\\varepsilon )$ defined by this diagram is invertible, as required in the definition of a weak 2-functor.",
"Given an $(\\varepsilon ,\\varepsilon ^{\\prime })$ -tangle $\\tau _1$ and an $(\\varepsilon ^{\\prime },\\varepsilon ^{\\prime \\prime })$ -tangle $\\tau _2$ , the first part of the proof of Theorem REF actually shows that there is a canonical isomorphism $\\alpha _{\\tau _1\\tau _2}\\colon H_1(\\widehat{X}_{\\tau _2})\\circ H_1(\\widehat{X}_{\\tau _1})\\rightarrow H_1(\\widehat{X}_{\\tau _2\\circ \\tau _1})$ which fits in the commutative diagram $@R0.5cm{& & H_1(\\widehat{X}_{\\tau _2})\\circ H_1(\\widehat{X}_{\\tau _1}) [d]^{\\alpha _{\\tau _1\\tau _2}} & & \\\\H_1(\\widehat{D}_{\\varepsilon }) @/^1pc/[urr]^{j_1i_1} @/_1pc/[drr]_{k_1i_1} [r]^{i_1} & H_1(\\widehat{X}_{\\tau _1}) [ur]_{j_1} [dr]^{k_1} &H_1(\\widehat{X}_{\\tau _2\\circ \\tau _1}) &H_1(\\widehat{X}_{\\tau _2}) [ul]^{j_2} [dl]_{k_2} & H_1(\\widehat{D}_{\\varepsilon ^{\\prime \\prime }})@/_1pc/[ull]_{j_2i_2^{\\prime \\prime }}@/^1pc/[dll]^{k_2i_2^{\\prime \\prime }}[l]_{i_2^{\\prime \\prime }}\\,, \\\\& & H_1(\\widehat{X}_{\\tau _2\\circ \\tau _1}) [u]_\\mathit {id}& &}$ where we follow the notations of the aforementioned proof.",
"Hence, this defines a canonical 2-morphism $\\varphi _{\\tau _1\\tau _2}\\colon \\mathcal {B}(\\tau _1)\\circ \\mathcal {B}(\\tau _2)\\Rightarrow \\mathcal {B}(\\tau _2\\circ \\tau _1)$ , which is invertible by Remark REF .",
"Theorem 4.2 $\\mathcal {B}$ together with the isomorphisms $\\varphi _\\varepsilon $ and $\\varphi _{\\tau _1\\tau _2}$ gives rise to a weak 2-functor from the bicategory of oriented tangles to the bicategory of Lagrangian cospans, whose restriction to oriented surfaces is given by the Alexander module.",
"First note that isotopic cobordisms define isomorphic 2-cospans, so $\\mathcal {B}$ is well-defined at the level of 2-morphisms.",
"Also, for any tangle $\\tau $ , $\\mathcal {B}$ clearly maps the trivial concordance $\\mathit {Id}_\\tau $ to a 2-cospan canonically isomorphic to the identity 2-cospan $\\mathit {Id}_{\\mathcal {B}(\\tau )}$ .",
"Let us now verify that $\\mathcal {B}$ preserves the vertical composition.",
"Fix a $(\\tau _1,\\tau _2)$ -cobordism $A$ and a $(\\tau _2,\\tau _3)$ -cobordism $B$ .",
"Let $H_1(\\widehat{X}_{\\tau _1}) \\stackrel{\\alpha _1}{\\rightarrow }H_1(\\widehat{W}_A) \\stackrel{\\alpha _2}{\\leftarrow }H_1(\\widehat{X}_{\\tau _2})$ and $H_1(\\widehat{X}_{\\tau _2}) \\stackrel{\\beta _2}{\\rightarrow }H_1(\\widehat{W}_B) \\stackrel{\\beta _3}{\\leftarrow }H_1(\\widehat{X}_{\\tau _3})$ be the 2-cospans arising from $A$ and $B$ .",
"We need to show that $H_1(\\widehat{X}_{\\tau _1}) \\stackrel{k_A\\alpha _1}{\\longrightarrow } H_1(\\widehat{W}_{B \\star A}) \\stackrel{k_B\\beta _3}{\\longleftarrow }H_1(\\widehat{X}_{\\tau _3})$ is isomorphic to the vertical composition $H_1(\\widehat{X}_{\\tau _1}) \\stackrel{v_A\\alpha _1}{\\longrightarrow } H_1(\\widehat{W}_B) \\star H_1(\\widehat{W}_A) \\stackrel{v_B\\beta _3}{\\longleftarrow } H_1(\\widehat{X}_{\\tau _3})$ , where $k_A,k_B$ are the inclusion induced maps and $v_A,v_B$ are maps resulting from any representative of the pushout $H_1(\\widehat{W}_B) \\star H_1(\\widehat{W}_A)$ .",
"Observe that $\\widehat{W}_{B \\star A}$ decomposes as the union of $\\widehat{W}_B$ and $\\widehat{W}_A$ glued along $\\widehat{X}_{\\tau _2}$ .",
"Therefore, the associated Mayer-Vietoris exact sequence $@R0.5cm{H_1(\\widehat{X}_{\\tau _2})[r]^{\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!",
"(-\\alpha _2,\\beta _2)}&H_1(\\widehat{W}_A)\\oplus H_1(\\widehat{W}_B)[r]^{\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;{k_A \\atopwithdelims ()k_B}}&H_1(\\widehat{W}_{B \\star A})[r]& 0}$ together with Lemma REF imply that $H_1(\\widehat{W}_A) \\stackrel{k_A}{\\longrightarrow } H_1(\\widehat{W}_{B \\star A}) \\stackrel{k_B}{\\longleftarrow }H_1(\\widehat{W}_B)$ is a representative for the pushout $H_1(\\widehat{W}_A) \\star H_1(\\widehat{W}_B)$ .",
"Consequently, these two cospans are canonically isomorphic and the claim follows.",
"Given tangles and cobordisms as illustrated below $@C+2pc{\\varepsilon ^{\\tau _1}_{\\tau _2}{\\;\\;\\;A}&\\varepsilon ^{\\prime }^{\\tau _3}_{\\tau _4}{\\;\\;\\;B}&\\varepsilon ^{\\prime \\prime }\\,,}$ our next goal is to prove the equality $\\varphi _{\\tau _2\\tau _4}\\star (\\mathcal {B}(B)\\bullet \\mathcal {B}(A))=\\mathcal {B}(B\\bullet A)\\star \\varphi _{\\tau _1\\tau _3}$ up to isomorphism of 2-cospans.",
"Since the 2-morphism $\\varphi _{\\tau _1\\tau _3}$ is represented by the 2-cospan $H_1(\\widehat{X}_{\\tau _3}) \\circ H_1(\\widehat{X}_{\\tau _1})\\stackrel{\\alpha _{\\tau _1\\tau _3}}{\\longrightarrow } H_1(\\widehat{X}_{\\tau _3\\circ \\tau _1}) \\stackrel{\\mathit {id}}{\\longleftarrow }H_1(\\widehat{X}_{\\tau _3\\circ \\tau _1})$ , Remark REF implies that the right hand side of equation (REF ) is represented by the 2-cospan $H_1(\\widehat{X}_{\\tau _3}) \\circ H_1(\\widehat{X}_{\\tau _1}) \\stackrel{k_{31}\\alpha _{\\tau _1\\tau _3}}{\\longrightarrow } H_1(\\widehat{W}_{B \\bullet A}) \\stackrel{k_{42}}{\\longleftarrow }H_1(\\widehat{X}_{\\tau _4\\circ \\tau _2})\\,,$ where $k_{31}$ and $k_{42}$ are induced by the inclusion maps.",
"A similar argument shows that the left hand side of equation (REF ) is represented by the 2-cospan $H_1(\\widehat{X}_{\\tau _3}) \\circ H_1(\\widehat{X}_{\\tau _1}) \\stackrel{h_{31}}{\\longrightarrow } H_1(\\widehat{W}_B) \\bullet H_1(\\widehat{W}_A) \\stackrel{h_{42}\\alpha ^{-1}_{\\tau _2\\tau _4}}{\\longleftarrow } H_1(\\widehat{X}_{\\tau _4\\circ \\tau _2})\\,,$ where this time, the maps $h_{31}$ and $h_{42}$ are the ones which arise from the definition of horizontal composition.",
"It now remains to find an isomorphism $f$ of $\\Lambda $ -modules which fits in the following commutative diagram: $@R0.5cm{& H_1(\\widehat{X}_{\\tau _3}) \\circ H_1(\\widehat{X}_{\\tau _1}) [dl]_{h_{31}} [dr]^{k_{31}\\alpha _{\\tau _1\\tau _3}} & \\\\H_1(\\widehat{W}_B) \\bullet H_1(\\widehat{W}_A) [rr]^f & & \\;H_1(\\widehat{W}_{B\\bullet A})\\,.",
"\\\\& H_1(\\widehat{X}_{\\tau _4\\circ \\tau _2}) [lu]^{h_{42}\\alpha _{\\tau _2\\tau _4}^{-1}}[ur]_{k_{42}} &}$ In order to construct $f$ , first observe that the following diagram commutes $@R0.5cm{H_1(\\widehat{X}_{\\tau _2})[d]_{\\alpha _2} & H_1(\\widehat{D}_{\\varepsilon ^{\\prime }}) [d]^{\\cong }[r]^{i_3^{\\prime }}[l]_{i_2^{\\prime }}& H_1(\\widehat{X}_{\\tau _3})[d]^{\\beta _3} \\\\H_1(\\widehat{W}_A) & H_1(\\widehat{D}_{\\varepsilon ^{\\prime }} \\times [0,1])[l][r] & H_1(\\widehat{W}_B),}$ where all the maps are induced by inclusions.",
"Hence, identifying $H_1(\\widehat{D}_{\\varepsilon ^{\\prime }}\\times [0,1])$ with $H_1(\\widehat{D}_{\\varepsilon ^{\\prime }})$ , the first map in the Mayer-Vietoris exact sequence $@R0.5cm{H_1(\\widehat{D}_{\\varepsilon ^{\\prime }})[r]^{\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!",
"}&H_1(\\widehat{W}_A)\\oplus H_1(\\widehat{W}_B)[r]^{\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;}&H_1(\\widehat{W}_{B \\bullet A})[r]& 0}$ is given by $(-\\alpha _2i_2^{\\prime },\\beta _3i_3^{\\prime })$ .",
"It now follows from Lemma REF that the cospan of inclusion induced maps $H_1(\\widehat{W}_A) \\stackrel{k_A}{\\rightarrow } H_1(\\widehat{W}_{B \\bullet A}) \\stackrel{k_B}{\\leftarrow } H_1(\\widehat{W}_B)$ is a representative of the pushout $H_1(\\widehat{W}_A) \\stackrel{h_A}{\\rightarrow }H_1(\\widehat{W}_B) \\bullet H_1(\\widehat{W}_A)\\stackrel{h_B}{\\leftarrow } H_1(\\widehat{W}_B)$ .",
"Invoking the corresponding universal property, this produces a $\\Lambda $ -module isomorphism $f \\colon H_1(\\widehat{W}_B) \\bullet H_1(\\widehat{W}_A) \\rightarrow H_1(\\widehat{W}_{B\\bullet A})$ with $fh_A=k_A$ and $fh_B=k_B$ .",
"Using successively the definition of $\\alpha _{\\tau _1\\tau _3}$ (see diagram (REF ) for the relevant notations), the commutativity of inclusion induced maps, and the equalities above, one gets $f^{-1}k_{31} \\alpha _{\\tau _1\\tau _3}j_3=f^{-1}k_{31}k_3=f^{-1}k_B\\beta _3=h_B\\beta _3\\,.$ The equality $f^{-1}k_{31} \\alpha _{\\tau _1\\tau _3}j_1=h_A\\alpha _1$ is proved similarly.",
"Hence, the universal property of diagram (REF ) implies that $h_{31}=f^{-1}k_{31} \\alpha _{\\tau _1\\tau _3}$ .",
"The equality $h_{42}=f^{-1}k_{42} \\alpha _{\\tau _2\\tau _4}$ can be dealt with in the same way, and equation (REF ) is proved.",
"Given an $(\\varepsilon ,\\varepsilon ^{\\prime })$ -tangle $\\tau $ , we must now show that the 2-morphism $\\mathcal {B}(r_\\tau ) \\star \\varphi _{I_{\\varepsilon }\\tau } \\star (I_{\\mathcal {B}(\\tau )} \\bullet \\varphi _\\varepsilon )$ coincides with $r_{\\mathcal {B}(\\tau )} \\colon \\mathcal {B}(\\tau ) \\circ I_{\\mathcal {B}(\\varepsilon )} \\Rightarrow \\mathcal {B}(\\tau )$ .",
"First observe that by Remark REF , one can choose representatives of the pushouts so that for any cospan $H \\rightarrow T \\leftarrow H^{\\prime }$ , one has $T \\circ I_H=T$ .",
"In particular, we only need to prove that, for this choice of pushouts, $\\mathcal {B}(r_\\tau ) \\star \\varphi _{I_{\\varepsilon }\\tau } \\star (I_{\\mathcal {B}(\\tau )} \\bullet \\varphi _\\varepsilon ) =I_{\\mathcal {B}(\\tau )}.$ As a first step, using the definition of the horizontal composition and Remark REF , we deduce that $I_{\\mathcal {B}(\\tau )} \\bullet \\varphi _\\varepsilon $ is represented by the 2-cospan $H_1(\\widehat{X}_\\tau ) \\stackrel{\\mathit {id}}{\\longrightarrow } H_1(\\widehat{X}_\\tau ) \\stackrel{h}{\\longleftarrow } H_1(\\widehat{X}_\\tau ) \\circ H_1(\\widehat{X}_{I_\\varepsilon })\\,,$ where $h$ is the unique morphism which fits in the following commutative diagram (recall diagram (REF )): $@R0.5cm{& & H_1(\\widehat{X}_\\tau ) & & & \\\\& & H_1(\\widehat{X}_\\tau ) \\circ H_1(\\widehat{X}_{I_\\varepsilon }) [u]^{h} & & & \\\\& H_1(\\widehat{D}_\\varepsilon ) @/^1pc/[uur]^{i} & & H_1(\\widehat{X}_\\tau ) @/_1pc/[uul]_{\\mathit {id}} & & \\\\& H_1(\\widehat{X}_{I_\\varepsilon }) [ruu]^{j_1} [u]^{\\alpha _\\varepsilon ^{-1}} & & H_1(\\widehat{X}_\\tau ).",
"[uul]_{j_2} [u]_{\\mathit {id}} \\\\& & H_1(\\widehat{D}_\\varepsilon ) [lu]_{\\alpha _\\varepsilon }[ru]^{i} & &}$ A short computation using Remark REF then shows that the left hand side of equation (REF ) is represented by the 2-cospan $H_1(\\widehat{X}_\\tau ) \\stackrel{\\alpha _{I_\\varepsilon \\tau } h^{-1}}{\\longrightarrow } H_1(\\widehat{X}_{\\tau \\circ I_\\varepsilon }) \\stackrel{r^{-1}}{\\longleftarrow } H_1(\\widehat{X}_\\tau )\\,,$ where $r\\colon H_1(\\widehat{X}_{\\tau \\circ I_\\varepsilon })\\rightarrow H_1(\\widehat{X}_\\tau )$ is the isomorphism induced by the obvious isotopy from $\\tau \\circ I_\\varepsilon $ to $\\tau $ .",
"We now claim that $r$ induces a 2-cospan isomorphism from $I_{\\mathcal {B}(\\tau )}$ to this cospan.",
"To prove this claim, we only need to show the equality $r\\alpha _{I_\\varepsilon \\tau } h^{-1}=\\mathit {id}_{H_1(\\widehat{X}_{\\tau })}$ , i.e.",
"to check that $r\\alpha _{I_\\varepsilon \\tau }$ satisfies the defining property of $h$ displayed above.",
"Since $\\alpha _{I_\\varepsilon \\tau }j_1$ and $\\alpha _{I_\\varepsilon \\tau }j_2$ are the inclusion induced homomorphisms (recall diagram (REF )), this follows from the functoriality of homology.",
"The proof of the equality $\\mathcal {B}(\\ell _\\tau ) \\star \\varphi _{\\tau I_{\\varepsilon ^{\\prime }}} \\star (\\varphi _{\\varepsilon ^{\\prime }} \\bullet I_{\\mathcal {B}(\\tau )})=\\ell _{\\mathcal {B}(\\tau )}$ is dealt with in the same way.",
"Finally, the axiom involving the associativity isomorphisms is left to the reader: although the proof is tedious, it involves no other ideas than the ones presented up to now.",
"Therefore we have proved that $\\mathcal {B}$ is a weak 2-functor and we turn to the last statement of the theorem.",
"If $\\Sigma $ is a $(\\emptyset ,\\emptyset )$ -cobordism, that is, a closed oriented surface in the 4-ball, then the associated 2-cospan is given by $@R0.5cm{& 0 [d] & \\\\0 [ur] [dr] & H_1(\\widehat{W}_\\Sigma ) & 0 [ul] [dl]\\,, \\\\& 0 [u] &}$ with $W_\\Sigma $ the complement of $\\Sigma $ in the 4-ball.",
"This is nothing but the Alexander module of $\\Sigma $ ."
],
[
"Unreduced and multivariable versions",
"Recall that the representation originally defined by Burau takes the form of a homomorphism $\\overline{\\rho }_n\\colon B_n\\rightarrow \\mathit {GL}_n(\\Lambda )$ , which is the direct sum of a trivial 1-dimensional representation with $\\rho _n\\colon B_n\\rightarrow \\mathit {GL}_{n-1}(\\Lambda )$ , the reduced Burau representation.",
"Also, these representations admit multivariable extensions, the so-called Gassner representations of the pure braid groups.",
"It is therefore natural to ask whether these variations of the Burau representation can also be extended to weak 2-functors.",
"This is indeed the case, and is the subject of this slightly informal last section.",
"More precisely, we start in subsection REF by explaining how $\\overline{\\rho }$ can be extended to a functor $\\overline{\\mathcal {B}}$ on tangles.",
"This functor is no longer Lagrangian ($\\overline{\\rho }$ is not unitary) but it is monoidal and behaves well with respect to traces.",
"In subsection REF , we indicate how to extend it to a weak 2-functor.",
"Finally, in subsection REF , we briefly explain how all of these constructions can be extended to multivariable versions, defined on the category of colored tangles."
],
[
"Extending the unreduced Burau representation to a monoidal functor",
"Given an integral domain $\\Lambda $ , let $\\mathbf {C}_\\Lambda $ denote the category with finitely generated $\\Lambda $ -modules as objects, and isomorphism classes of cospans as morphisms, composed by pushouts.",
"Also, let $\\mathbf {GL}_\\Lambda $ denote the groupoid with the same objects as $\\mathbf {C}_\\Lambda $ and $\\Lambda $ -isomorphisms as morphisms.",
"As in Section , one can check that the map assigning to an invertible cospan $H\\stackrel{i}{\\longrightarrow }T\\stackrel{i^{\\prime }}{\\longleftarrow }H^{\\prime }$ the $\\Lambda $ -isomorphism $i^{\\prime -1}i\\colon H\\rightarrow H^{\\prime }$ defines an equivalence of categories $\\mathit {core}(\\mathbf {C}_\\Lambda ) \\stackrel{\\simeq }{\\longrightarrow }\\mathbf {GL}_\\Lambda $ .",
"Note that the direct sum endows these categories with a monoidal structure, with the trivial $\\Lambda $ -module $H=0$ being the identity object.",
"Given an endomorphism of $\\mathbf {C}_\\Lambda $ , i.e.",
"a cospan of the form $H\\stackrel{i}{\\longrightarrow }T\\stackrel{i^{\\prime }}{\\longleftarrow }H$ , define the trace of $T$ as the coequalizer $@R0.5cm{H @/^/[rr]^{i}@/_/[rr]_{i^{\\prime }}& & T [r]^{\\!\\!\\!j}&\\mathrm {tr}(T)\\,.",
"}$ Viewing $\\mathrm {tr}(T)$ as the isomorphism class of the cospan $0\\rightarrow \\mathrm {tr}(T)\\leftarrow 0$ , the trace actually defines a map $\\mathrm {tr}\\colon \\mathit {End}(H) \\rightarrow \\mathit {End}(0)$ .",
"It is an amusing exercise to check that it satisfies the following properties, as it should (see e.g. [17]).",
"i If $T_1$ is a cospan from $H$ to $H^{\\prime }$ and $T_2$ from $H^{\\prime }$ to $H$ , then $\\mathrm {tr}(T_1\\circ T_2)=\\mathrm {tr}(T_2 \\circ T_1)$ .",
"ii If $T_1$ and $T_2$ are two endomorphisms, then $\\mathrm {tr}(T_1\\oplus T_2)=\\mathrm {tr}(T_1)\\circ \\mathrm {tr}(T_2)$ .",
"iii If $T$ is an endomorphism of 0, then $\\mathrm {tr}(T)=T$ .",
"These additional structures are also present in the category of tangles.",
"Indeed, the juxtaposition endows $\\mathbf {Tangles}$ with a monoidal structure, with the empty set $\\varepsilon =\\emptyset $ being the identity object.",
"Furthermore, the closure of a tangle defines a natural trace function $\\mathit {End}(\\varepsilon )\\rightarrow \\mathit {End}(\\emptyset )$ .",
"In this context, the unreduced Burau representation can be understood as a monoidal functor $\\overline{\\rho }\\colon \\mathbf {Braids}\\rightarrow \\mathbf {GL}_\\Lambda $ , where $\\Lambda ={Z}[t^{\\pm 1}]$ .",
"We now sketch the construction of a monoidal functor $\\overline{\\mathcal {B}}\\colon \\mathbf {Tangles}\\rightarrow \\mathbf {C}_\\Lambda $ extending $\\overline{\\rho }$ , and behaving well with respect to traces.",
"We shall follow the notation of Section , apart from the fact that all exteriors will be considered in the unit disc $D^2$ , and not the sphere $S^2$ even when $\\ell _\\varepsilon $ vanishes.",
"Let $x_0$ be the point $(-1,0)$ in $D^2$ .",
"For any sequence $\\varepsilon $ of $\\pm 1$ 's, set $\\overline{\\mathcal {B}}(\\varepsilon )=H_1(\\widehat{D}_\\varepsilon ,\\widehat{x_0})$ and for any isotopy class $\\tau $ of tangles, let $\\overline{\\mathcal {B}}(\\tau )$ denote the isomorphism class of the cospan $H_1(\\widehat{D}_\\varepsilon ,\\widehat{x_0}) \\stackrel{i_\\tau }{\\longrightarrow }H_1(\\widehat{X}_\\tau ,\\widehat{x_0\\times I})\\stackrel{i_\\tau ^{\\prime }}{\\longleftarrow }H_1(\\widehat{D}_{\\varepsilon ^{\\prime }},\\widehat{x_0})$ , where $\\widehat{Y}$ stands for the inverse image of a subspace $Y\\subset X_\\tau $ by the infinite cyclic covering map $\\widehat{X_\\tau }\\rightarrow X_\\tau $ .",
"Following almost verbatim the proof of Theorem REF , one checks that this defines a functor $\\overline{\\mathcal {B}}\\colon \\mathbf {Tangles}\\rightarrow \\mathbf {C}_\\Lambda $ which fits in the commutative diagram $@R0.5cm{& \\mathbf {Braids} [d] [r] @/_1pc/[ld]_{\\overline{\\rho }} & \\mathbf {Tangles} [d]^{\\overline{\\mathcal {B}}} \\\\\\mathbf {GL}_\\Lambda &\\mathit {core}(\\mathbf {C}_\\Lambda ) [l]_\\simeq [r] & \\mathbf {C}_\\Lambda \\,.",
"}$ Furthermore, an additional application of Mayer-Vietoris shows that this functor is monoidal.",
"(The basepoint $x_0$ is chosen so that the juxtaposition of tangles can be realized in a natural way by gluing discs along intervals, with $x_0$ a common endpoint of these intervals.)",
"Finally, if $\\tau $ is an $(\\varepsilon ,\\varepsilon )$ -tangle, then $\\mathrm {tr}(\\overline{\\mathcal {B}}(\\tau ))$ is nothing but the relative Alexander module of the oriented link in $D^2\\times I$ (or equivalently, in $S^3$ ) obtained by the closure of $\\tau $ .",
"$\\overline{\\mathcal {B}}$ as a monoidal weak 2-functor One can modify $\\mathbf {C}_\\Lambda $ to obtain a bicategory in the exact same way as we did for $\\mathbf {L}_\\Lambda $ , with 2-morphisms given by isomorphism classes of 2-cospans (recall subsection REF ).",
"Furthermore, the direct sum endows this bicategory with a monoidal structure.",
"Also, the juxtaposition endows the bicategory of tangles with a monoidal structure.",
"Here again, some care is needed, as different conventions such as the ones in [11] and [1] will lead to different monoidal bicategories.",
"We will not go into these details, but only mention that our construction is robust enough to be valid in these different settings.",
"Let us sketch how the functor $\\overline{\\mathcal {B}}$ can be extended to a weak 2-functor, following the notation of subsection REF .",
"Given a $(\\tau _1,\\tau _2)$ -cobordism $\\Sigma $ , let us denote by $\\overline{\\mathcal {B}}(\\Sigma )\\colon \\overline{\\mathcal {B}}(\\tau _1)\\Rightarrow \\overline{\\mathcal {B}}(\\tau _2)$ the isomorphism class of the 2-cospan $H_1(\\widehat{X}_{\\tau _1},\\widehat{x_0\\times I}) \\stackrel{\\alpha _1}{\\longrightarrow }H_1(\\widehat{W}_\\Sigma ,\\widehat{x_0\\times I\\times I}) \\stackrel{\\alpha _2}{\\longleftarrow }H_1(\\widehat{X}_{\\tau _2},\\widehat{x_0\\times I})\\,.$ One can check that this defines a weak 2-functor, that is monoidal in a sense that, once again, we shall not discuss in detail here.",
"Multivariable versions Let $\\mu $ be a positive integer.",
"Recall that a $\\mu $ -colored tangle consists in an oriented tangle $\\tau $ together with a surjective map assigning to each component of $\\tau $ an integer in $\\lbrace 1,\\dots ,\\mu \\rbrace $ .",
"As explained in [6], $\\mu $ -colored tangles naturally form a category $\\mathbf {Tangles}_\\mu $ , with the $\\mu =1$ case being nothing but $\\mathbf {Tangles}$ .",
"Obviously, assigning a color to the cobordisms and proceeding as in subsection REF , one obtains a bicategory of $\\mu $ -colored tangles.",
"All the results of the present paper extend to this multivariable setting in a straightforward way, that we now very briefly summarize.",
"The coloring of points, tangles and cobordisms induces homomorphisms from the homology of the corresponding exterior onto ${Z}^\\mu $ , thus defining free abelian covers whose homology is a module over the ring of multivariable Laurent polynomials ${Z}[{Z}^\\mu ]={Z}[t_1^{\\pm 1},\\dots ,t_\\mu ^{\\pm 1}]=:\\Lambda _\\mu $ .",
"This allows one to construct a weak 2-functor from the bicategory of $\\mu $ -colored tangles to the bicategory of Lagrangian cospans over $\\Lambda _\\mu $ , which extends the colored Gassner representation of $\\mu $ -colored braids, and whose restriction to $\\mu $ -colored links and surfaces is nothing but the multivariable Alexander module.",
"The results of subsections REF and REF can be extended in the same way."
]
] | 1606.05166 |
[
[
"Clustering high dimensional mixed data to uncover sub-phenotypes:joint\n analysis of phenotypic and genotypic data"
],
[
"Abstract The LIPGENE-SU.VI.MAX study, like many others, recorded high dimensional continuous phenotypic data and categorical genotypic data.",
"LIPGENE-SU.VI.MAX focuses on the need to account for both phenotypic and genetic factors when studying the metabolic syndrome (MetS), a complex disorder that can lead to higher risk of type 2 diabetes and cardiovascular disease.",
"Interest lies in clustering the LIPGENE-SU.VI.MAX participants into homogeneous groups or sub-phenotypes, by jointly considering their phenotypic and genotypic data, and in determining which variables are discriminatory.",
"A novel latent variable model which elegantly accommodates high dimensional, mixed data is developed to cluster LIPGENE-SU.VI.MAX participants using a Bayesian finite mixture model.",
"A computationally efficient variable selection algorithm is incorporated, estimation is via a Gibbs sampling algorithm and an approximate BIC-MCMC criterion is developed to select the optimal model.",
"Two clusters or sub-phenotypes (`healthy' and `at risk') are uncovered.",
"A small subset of variables is deemed discriminatory which notably includes phenotypic and genotypic variables, highlighting the need to jointly consider both factors.",
"Further, seven years after the LIPGENE-SU.VI.MAX data were collected, participants underwent further analysis to diagnose presence or absence of the MetS.",
"The two uncovered sub-phenotypes strongly correspond to the seven year follow up disease classification, highlighting the role of phenotypic and genotypic factors in the MetS, and emphasising the potential utility of the clustering approach in early screening.",
"Additionally, the ability of the proposed approach to define the uncertainty in sub-phenotype membership at the participant level is synonymous with the concepts of precision medicine and nutrition."
],
[
"Keywords",
"clustering, mixed data, phenotypic data, SNP data, metabolic syndrome."
],
[
"Introduction",
"Many large cohort based studies collect high dimensional continuous phenotypic and categorical genotypic data.",
"The pan European LIPGENE-SU.VI.MAX (SUpplementation en VItamines et Minéraux AntioXydants) study (www.ucd.ie/lipgene) is one such study which focuses on the need to account for both phenotypic and genetic factors when studying the metabolic syndrome (MetS).",
"The MetS is a complex disorder that can lead to increased risk of developing type 2 diabetes and cardiovascular disease.",
"The MetS is the term used to describe a clustering of several risk factors for cardiovascular disease, namely obesity, abnormal blood lipids, insulin resistance and high blood pressure.",
"Obesity is on the rise globally and is considered to be a principle factor in the development of insulin resistance and the metabolic syndrome.",
"The World Health Organisation estimates that the global prevalence of diabetes will almost double from 171 million people in 2000 to 300 million people by the year 2030.",
"Given the strain this will place on health and health systems all over the world there is a need to gain greater understanding of the MetS, thereby reducing its adverse health effects.",
"In particular, the influence of both phenotypic and genetic factors (and their interaction) on the MetS has recently come to the fore, and is the focus of the LIPGENE-SU.VI.MAX project.",
"Valuable introductions and contributions to the LIPGENE-SU.VI.MAX project include [79] and [26].",
"Under the LIPGENE-SU.VI.MAX study, high dimensional data of mixed type were collected on a group of participants.",
"Continuous phenotypic variables (e.g.",
"anthropometric and biochemical variables such as waist circumference and plasma fatty acid levels) as well as categorical (binary and nominal) genotypic single nucleotide-polymorphism (SNP) variables were recorded.",
"Here, interest lies in clustering the participants into homogeneous groups or sub-phenotypes, based on jointly modelling their phenotypic and genotypic data, to uncover groups with similar phenotypic-genotypic profiles.",
"In the LIPGENE-SU.VI.MAX study, a large number of phenotypic and genotypic variables were recorded; determining which variables discriminate between the resulting sub-phenotypes is therefore of interest.",
"Moreover, given the ethos of the LIPGENE-SU.VI.MAX study, whether the set of discriminatory variables includes both phenotypic and genotypic variables is of key interest.",
"The developed methodology has wide applicability beyond the LIPGENE-SU.VI.MAX study, in any setting seeking to uncover subgroups in a cohort on which high dimensional data of mixed type have been recorded.",
"Joint modelling approaches for data of mixed type are gaining attention in a range of statistical and applied areas (see [22], [25], [105], [20], [18], among others, for example).",
"In particular [21] provides a comprehensive overview of recent methodological and applied advances in the mixed data modelling area.",
"Latent factor models in particular have been successfully employed to jointly model mixed data; [90], [44] and [74] use factor analytic models to analyse mixed data but not in a clustering context.",
"In a similar vein to the approach taken here, [50] consider a joint analysis of SNP and gene expression data in studies of complex diseases such as asthma, but again not in the clustering context.",
"The MetS has had recent exposure in the statistical and computational literature – [64], [102] and [43] employ computational approaches to learn about the disease, but mainly from a genetic point of view.",
"Latent variable based clustering models have been successfully utilised to analyse high dimensional data.",
"For example, [37] propose a mixture of factor analysers model with a cluster specific parsimonious covariance matrix.",
"A suite of similar models with varying levels of parsimony is developed in [68] and the mixture of factor analysers model is fitted in a Bayesian framework in [27].",
"Mixtures of structural equation models are developed in [109] and [110].",
"More recent developments in this area include those in [7], [8] and [104], among others.",
"However, while these models can efficiently model high dimensional data, none of them can cluster observed mixed data while also correctly handling each variable type.",
"Clustering data of mixed type is a challenging statistical problem.",
"Early attempts to address the problem include the use of mixture models and location mixture models [23], [24], [75], [52], [53] as well as non-model based approaches [51], [1]; [103] clusters mixed categorical data using a latent class analysis approach.",
"More recently [15], [13], [72], [14], [39] attempt to cluster mixed categorical data using latent variable models and [12] cluster multivariate ordinal data using a stochastic binary search algorithm.",
"However none of these can analyse the specific combination of continuous and categorical variables without transforming the original variables in some way, or can feasibly accommodate high dimensional data.",
"An alternative model-based approach to clustering mixed continuous and categorical data, clustMD, is introduced in [70].",
"While this approach can explicitly model the inherent nature of continuous and categorical variables directly, it is again computationally infeasible to use for high dimensional data.",
"In particular, clustMD cannot accommodate large numbers of nominal variables.",
"Copula models for clustering mixed data [63], [57] while showing distinct promise, also have limitations in high dimensional settings.",
"The recent mixture of factor analysers for mixed data (MFA-MD) [71] is a hybrid of latent variable models for different data types and provides the machinery for clustering mixed categorical data.",
"Here, the MFA-MD model is extended to facilitate clustering of high dimensional, mixed continuous and categorical data.",
"Specifically, the joint model is composed of a factor analysis model for continuous data, an item response theory model for binary/ordinal data and a multinomial probit type model is used for nominal data.",
"The clustering machinery is provided by a finite mixture model.",
"The MFA-MD model is ideal for high dimensional data settings as its factor analytic roots provide a parsimonious covariance structure.",
"However large numbers of variables, as are present in the LIPGENE-SU.VI.MAX data, hamper the substantive interpretability of the resulting clusters and place a heavy computational burden on model fitting.",
"Existing approaches to variable selection in a clustering context include reversible jump Markov chain Monte Carlo methods [99], approximate Bayes factors are used in [92] and [65] to compare nested sets of variables and [106] use penalised model based clustering in the context of microarray data.",
"Such methods would be computationally expensive given the latent variable aspect of the MFA-MD model, and given the large number of variables in LIPGENE-SU.VI.MAX data.",
"Therefore, here an efficient novel online variable selection algorithm is incorporated when fitting the extended MFA-MD model, improving substantive interpretability and computational costs.",
"Inspired by [5], variable selection is based on a within cluster variance to overall variance criterion, efficiently leading to an interpretative clustering solution.",
"Model fitting is performed in the Bayesian paradigm and is achieved via a Gibbs sampling algorithm.",
"As in any clustering setting, uncovering the number of underlying clusters is a key, and often difficult, question.",
"In the context of the extended MFA-MD model, the dimension of the latent factor aspect of the model also requires selection.",
"Typical likelihood based model selection criteria such as the Bayesian Information Criterion (BIC) [95], [56] have been demonstrated to perform well in many general clustering settings [30], [41], and marginal likelihood evaluation [32] or the use of over fitting mixture models have gained warranted attention [101], [62] in the Bayesian literature.",
"The likelihood function of the MFA-MD model is intractable however, rendering such approaches unusable.",
"Therefore, here a novel approximation of the likelihood function is incorporated with the BIC-MCMC criterion [34], to efficiently select the optimal model (i.e.",
"the optimal number of clusters and the optimal number of latent factors) in the context of the extended MFA-MD model.",
"The extended MFA-MD model, with variable selection, is used to cluster the LIPGENE-SU.VI.MAX participants within a Bayesian framework.",
"A range of models with varying numbers of clusters and latent factor dimensions are fitted.",
"The BIC-MCMC criterion suggests two clusters or sub-phenotypes of participants, and a set of just 25 of the original 738 variables are deemed discriminatory.",
"Examination of the cluster specific parameters reveals a `healthy' sub-phenotype and an `at risk' sub-phenotype.",
"Notably the set of discriminatory variables contains both phenotypic and genotypic variables, highlighting the need to jointly consider both data types.",
"Some of the discriminatory variables are intuitive and have been highlighted previously in the literature, but some of the discriminating SNPs in particular are novel discoveries.",
"Seven years after the LIPGENE-SU.VI.MAX data analysed here were collected, each of the participants underwent further analysis to diagnose the presence or absence of the MetS, based on a criterion which considers continuous phenotypic data only.",
"The two clusters uncovered here from the initial LIPGENE-SU.VI.MAX data strongly correspond to the seven year follow up disease classification, highlighting the role of phenotypic and genetic factors in the MetS and, perhaps most importantly, the potential utility of the clustering approach in early screening.",
"The model-based nature of the MFA-MD approach to clustering provides a global view of the group structure in the set of LIPGENE-SU.VI.MAX participants.",
"However, it additionally provides detailed insight to sub-phenotype membership at the participant level, through quantification of the probability of sub-phenotype membership for each participant.",
"This ability to define the uncertainty of cluster membership is an important development for the application of the metabotyping concept in precision medicine and nutrition [76].",
"The remainder of the paper is organised into the following sections.",
"Section provides background to the LIPGENE-SU.VI.MAX study, and specific details on the data collected.",
"Section contains the three modelling contributions of the paper: (i) details of the extended MFA-MD model for high dimensional, mixed continuous and categorical data (ii) an outline of the variable selection and inference procedure and (iii) the development of the approximate BIC-MCMC model selection tool.",
"Simulation studies, diverted to the Supplementary Material for clarity, provide evidence to support the modelling and selection approaches taken.",
"Section discusses the results of fitting the extended MFA-MD model for high dimensional data to the LIPGENE-SU.VI.MAX data, and considers model fit.",
"The paper concludes in Section with a discussion and some future research directions."
],
[
"The LIPGENE-SU.VI.MAX study",
"LIPGENE-SU.VI.MAX is a European Union Sixth Framework Integrated Programme entitled `Diet, genomics and the metabolic syndrome: an integrated nutrition, agro-food, social and economic analysis' conducted by 25 research centres across Europe.",
"The primary focus of LIPGENE-SU.VI.MAX is the interaction of nutrients and genotype in the metabolic syndrome (MetS).",
"The MetS is the term used to describe a clustering of several risk factors for cardiovascular disease, namely obesity, abnormal blood lipids (such as high blood cholesterol and raised triglyceride levels), insulin resistance and high blood pressure (hypertension).",
"One quarter of the world's adult population have the metabolic syndrome and increasing numbers of children and adolescents have it as the worldwide obesity epidemic accelerates.",
"Table REF details the MetS diagnosis criterion used here which relates to insulin resistance, dyslipidaemia, cholesterol, blood pressure and abdominal obesity.",
"Many closely related definitions of the MetS are also in use [3], [4], [49].",
"Table: A person with 3 or more of the abnormalities listed below is diagnosed as having the MetS.Under LIPGENE-SU.VI.MAX, data from a prospective population-based study were available [26], [46].",
"Twenty-six continuous phenotypic measurements in addition to 801 categorical SNP variables were recorded for each of 1754 participants.",
"Examples of the continuous phenotypic measurements include fasting glucose concentration, waist circumference and plasma fatty acid levels.",
"An example of a categorical genotypic variable is the nominal SNP rs512535 of the APOB gene which has three genotypes, $AA$ , $GG$ or $AG$ in the data.",
"The 801 genotypic variables were selected using a candidate gene approach based on pathways adversely affected in the metabolic syndrome, and their relevant genes, as previously described in [19], [80].",
"Biological variables were based on characteristics of the metabolic syndrome [3], [4], [49] and plasma fatty acid profiles were determined as biomarkers of habitual dietary intake as previously described [79].",
"Some data cleaning was conducted prior to analysis.",
"Without loss of generality, the nominal SNP variables were coded with the convention 0 = dominant homozygous, 1 = recessive homozygous and 2 = heterozygous.",
"Any SNP variable with more than 100 missing values was removed, as were SNPs for which all 3 genotypes were not observed in the data (most of which only had one observed genotype and therefore are non-discriminatory in a clustering setting).",
"A total of 990 participants were then removed as they still had at least one missing value across the remaining SNPs.",
"Some of the remaining SNPs had a small number ($<$ 10% of the number of participants) of counts of the recessive homozygous genotype.",
"In such cases, for reasons of computational efficiency and stability, the recessive homozygous and the heterozygous categories were merged, thus resulting in some SNPs becoming binary variables.",
"The merged category can be thought of as a `compound genotype'.",
"For example, the SNP rs17777371 of the ADD1 gene became a binary SNP with genotypes $GG$ and $CG/CC$ in the data.",
"While losing some information, merging of at least one sparsely observed genotype with another will not largely impact the findings in terms of uncovering clusters, or highlighting variables which discriminate between clusters.",
"A total of 371 SNPs were collapsed to binary variables, leaving 341 nominal SNP variables.",
"Finally the SNP data were combined with the continuous phenotypic data and participants that had any missing values for the continuous variables were removed.",
"This left a final complete data set of 505 participants and 738 variables (26 continuous variables, 371 binary SNPs and 341 nominal SNPs); this data set is analysed here.",
"No genotypes were removed solely as a result of missing data from other variables, and the continuous variables were standardised before any analysis was performed.",
"The full list of 738 variables analysed here is given in the Supplementary Material.",
"As stated LIPGENE-SU.VI.MAX was a prospective study.",
"Seven years after the data analysed here were collected, new continuous phenotypic data were recorded on the LIPGENE-SU.VI.MAX participants in order to diagnose the presence or absence of the MetS, according to the criterion detailed in Table REF .",
"The correspondence between the clusters uncovered from the initial phenotypic and genotypic data and the seven year follow-up disease diagnosis is examined in Section ."
],
[
"Modelling and inference",
"A model-based approach is taken to cluster the LIPGENE-SU.VI.MAX participants, based on their initial mixed continuous, binary and nominal data.",
"The mixture of factor analysers model for mixed ordinal and nominal data, MFA-MD, is introduced in [71].",
"Here the MFA-MD model is extended to also allow for continuous data, a variable selection procedure is proposed which facilitates feasible handling of high dimensional data, details of Bayesian inference are provided, and an approximate BIC-MCMC criterion for model selection is developed."
],
[
"Modelling the continuous phenotypic variables",
"A factor analysis model [96] is used to model the multivariate continuous phenotypic data.",
"Specifically, the observed $J$ continuous phenotypic measurements, $\\underline{z}_i$ , on participant $i$ are modelled as $\\underline{z}_i = \\underline{\\mu } + \\Lambda \\underline{\\theta }_i + \\underline{\\epsilon }_i$ where $\\underline{\\mu }$ is a mean vector, $\\Lambda $ is a loadings matrix and $\\underline{\\theta }_i$ is a participant specific latent trait.",
"The error vector $\\underline{\\epsilon }_i$ follows a zero mean multivariate Gaussian distribution with diagonal covariance matrix $\\Psi $ .",
"The dimension of the latent trait $\\underline{\\theta }_i$ is $Q$ where $Q \\ll J$ .",
"The factor analysis model offers parsimony as the marginal covariance $\\Sigma = \\Lambda \\Lambda ^T + \\Psi $ requires estimation of only $J(Q+1)$ parameters."
],
[
"Modelling the binary SNP variables",
"As described in Section some SNPs are treated as binary variables and are modelled using item response theory (IRT) models.",
"Suppose that SNP rs17777371 is the $j^{th}$ variable (for $j = 1, \\ldots , J$ ).",
"IRT models assume that, for participant $i$ , a latent Gaussian variable $z_{ij}$ corresponds to each observed binary response $y_{ij}$ .",
"A Gaussian link function is assumed, though other link functions, such as the logit, are detailed in the IRT literature [61], [28].",
"If $z_{ij} < 0$ then the binary response will be $y_{ij} = 0$ while if $z_{ij} > 0$ then $y_{ij} = 1$ .",
"Relating this to SNP rs17777371, say, if $z_{ij} < 0$ then the observed genotype for participant $i$ will be $GG$ while if $z_{ij} > 0$ then observed genotype will be $CG/CC$ .",
"In a standard IRT model, a factor analytic structure is then used to model the underlying latent variable $z_{ij}$ .",
"It is assumed that the value of $z_{ij}$ depends on a $Q$ dimensional, participant specific, latent trait $\\underline{\\theta }_i$ (often termed the ability parameter) and on some variable specific parameters.",
"Specifically, the underlying latent variable $z_{ij}$ for respondent $i$ and variable $j$ is assumed to be distributed as $z_{ij}| \\underline{\\theta }_i \\sim N(\\mu _j + \\underline{\\lambda }_j^T\\underline{\\theta }_i, 1).$ The parameters $\\underline{\\lambda }_j$ and $\\mu _j$ are usually termed the item discrimination parameters and the negative item difficulty parameter respectively.",
"As in [2], a probit link function is used so the conditional variance of $z_{ij}$ is 1.",
"Under this model, the conditional probability that $y_{ij}=1$ is $P(y_{ij}=1 |\\mu _j, \\underline{\\lambda }_j, \\underline{\\theta }_i) = \\Phi \\left(\\mu _j + \\underline{\\lambda }_j^T\\underline{\\theta }_i\\right)$ where $\\Phi $ denotes the standard Gaussian cumulative distribution function."
],
[
"Modelling nominal SNP variables",
"Modeling nominal data is challenging, due to the fact that the set of possible responses is not ordered.",
"In the LIPGENE-SU.VI.MAX data, the possible responses for nominal SNPs is a set of three genotypes.",
"For example, the nominal SNP rs512535 of the APOB gene has three levels or genotypes, $AA$ , $GG$ or $AG$ , in the data.",
"These response levels are coded as 0, 1 and 2 respectively, but no order is implied.",
"As detailed in Section REF , the IRT model for binary SNP variables posits a one dimensional latent variable for each observed binary SNP.",
"In the factor analytic model for nominal SNP variables, a two-dimensional latent vector is required for each observed nominal SNP.",
"That is, the latent vector for participant $i$ corresponding to nominal SNP $j$ is denoted $\\underline{z}_{ij} = (z_{ij}^1, z_{ij}^{2})^T.$ The observed nominal response is then assumed to be a manifestation of the values of the elements of $\\underline{z}_{ij}$ relative to each other and to a cut-off point, assumed to be 0.",
"That is, $y_{ij} = \\left\\lbrace \\begin{array}{ll}0 & \\mbox{if $ \\displaystyle \\max \\lbrace z_{ij}^1, z_{ij}^{2}\\rbrace < 0$} \\vspace{2.84544pt}\\\\1 & \\mbox{if $z_{ij}^{1} = \\displaystyle \\max \\lbrace z_{ij}^1, z_{ij}^{2}\\rbrace $ and $z_{ij}^{1} > 0$} \\vspace{2.84544pt}\\\\2 & \\mbox{if $z_{ij}^{2} = \\displaystyle \\max \\lbrace z_{ij}^1, z_{ij}^{2}\\rbrace $ and $z_{ij}^{2} > 0$}.\\\\\\end{array} \\right.",
"$ Similar to the IRT model, the latent vector $\\underline{z}_{ij}$ is modelled via a factor analytic model.",
"The mean of the conditional distribution of $\\underline{z}_{ij}$ depends on a respondent specific, $Q$ -dimensional, latent trait, $\\underline{\\theta }_i$ , and item specific parameters i.e.",
"$\\underline{z}_{ij}| \\underline{\\theta }_i \\sim \\mbox{MVN}_{2}(\\underline{\\mu }_j + \\Lambda _j \\underline{\\theta }_i, \\mathbf {I})$ where $\\mathbf {I}$ denotes the identity matrix.",
"The loadings matrix $\\Lambda _{j}$ is a $2 \\times Q$ matrix, analogous to the item discrimination parameter in the IRT model of Section REF ; likewise, the mean $\\underline{\\mu }_{j}$ is analogous to the item difficulty parameter in the IRT model.",
"It should be noted that binary data could also be regarded as nominal.",
"The model proposed here is equivalent to the model proposed in Section REF when the number of possible levels is two."
],
[
"A factor analysis model for mixed mixed continuous and categorical data",
"The factor analysis model for continuous phenotypic variables, the IRT model for binary SNPs and the factor analysis model for nominal SNPs all have a common structure.",
"These models are combined to produce a unifying model for mixed continuous, binary and nominal data.",
"For each participant $i$ there are $A = 26$ observed continuous phenotypic variables, $B = 371$ latent continuous variables corresponding to the binary SNP variables and $C = 341$ latent continuous vectors corresponding to the nominal SNPs.",
"These are collected together in a single $D$ dimensional vector $\\underline{z}_{i}$ where $D = A + B + 2C$ .",
"That is, underlying participant $i$ 's set of $J = 738 (=A+B+C)$ continuous, binary and nominal variables lies $\\underline{z}_i & = &\\left(z_{i1}, \\ldots , z_{iA}, z_{i(A+1)} \\ldots , z_{i(A+B)}, z_{i(A+B+1)}^1, z_{i(A+B+1)}^2\\ldots , z_{iJ}^1, z_{iJ}^{2} \\right).$ The first $A$ entries of this vector are the observed continuous measurements.",
"The remaining entries are latent data underlying the categorical responses.",
"This vector is then modelled using a factor analytic structure i.e.",
"$\\underline{z}_{i}| \\underline{\\theta }_i \\sim \\mbox{MVN}_{D}(\\underline{\\mu } + \\Lambda \\underline{\\theta }_i, \\Psi ).$ The $D \\times Q$ dimensional matrix $\\Lambda $ is termed the loadings matrix and $\\underline{\\mu }$ is the mean vector.",
"The entries of the diagonal covariance matrix $\\Psi $ are 1 along the diagonal, with the exception of the first $A$ entries which correspond to the continuous variables.",
"This model provides a parsimonious factor analysis model for the high dimensional latent vector $\\underline{z}_{i}$ which underlies the observed mixed data.",
"Marginally the latent vector is distributed as $\\underline{z}_{i} \\sim \\mbox{MVN}_{D}(\\underline{\\mu }, \\Lambda \\Lambda ^{T} + \\Psi )$ resulting in a parsimonious covariance structure for $\\underline{z}_{i}$ ."
],
[
"A mixture of factor analyzers model for mixed continuous and categorical data",
"To facilitate clustering, the hybrid model defined in Section REF is placed within a mixture modeling framework resulting in the extended mixture of factor analyzers model for mixed data (MFA-MD).",
"In the MFA-MD model, clustering occurs at the latent variable level.",
"That is, under the MFA-MD model the distribution of the observed and latent data $\\underline{z}_i$ is modeled as a mixture of $G$ Gaussian densities i.e.",
"$f(\\underline{z}_i) = \\sum _{g=1}^{G}\\pi _g \\mbox{MVN}_{D}\\left( \\underline{\\mu }_g, \\:\\: \\Lambda _g \\Lambda _g^T +\\Psi \\right).$ The probability of belonging to cluster $g$ is denoted by $\\pi _g$ ($\\sum _{g=1}^{G}\\pi _g = 1$ , $\\pi _g > 0$ $\\forall $ $g$ ).",
"The mean and loadings are cluster specific, while $\\Psi $ is equal across clusters for additional parsimony.",
"Constraining the loadings matrices to be equal across clusters, similar in ethos to the mixture of common factor analysers [7], [8], would offer further parsimony but result in a subtly yet importantly different model.",
"As is standard in a model-based approach to clustering [66], [29], [67], [16], [30], a latent indicator variable, $\\underline{\\ell }_i = (\\ell _{i1}, \\ldots , \\ell _{iG})$ is introduced for each participant $i$ .",
"This binary vector indicates the cluster to which participant $i$ belongs i.e.",
"$l_{ig} = 1$ if $i$ belongs to cluster $g$ ; all other entries in the vector are 0.",
"Under the MFA-MD model for mixed continuous and categorical data, the augmented likelihood function for the $N = 505$ participants is $\\nonumber & &\\prod _{i=1}^{N} \\prod _{g=1}^{G} \\left\\lbrace \\pi _g \\left[ \\prod _{j=1}^{A} N(z_{ij}|\\tilde{\\underline{\\lambda }}_{gj}^{T} \\tilde{\\underline{\\theta }}_{i}, \\psi _{jj})\\right] \\right.\\\\ \\nonumber && \\left.",
"\\times \\left[ \\prod _{j=A+1}^{B} \\prod _{k=0}^{1 }N^{T}(z_{ij}|\\tilde{\\underline{\\lambda }}_{gj}^{T} \\tilde{\\underline{\\theta }}_{i}, 1)^{\\mathbb {I}\\lbrace y_{ij} = k\\rbrace } \\right] \\right.",
"\\\\&& \\left.",
"\\times \\left[\\prod _{j=A+B+1}^{J} \\prod _{k=1}^{2} \\prod _{s=0}^{2} N^T(z_{ij}^{k} |\\tilde{\\underline{\\lambda }}_{gj}^{k^{T}} \\tilde{\\underline{\\theta }}_{i}, 1)^{\\mathbb {I}(y_{ij}=s)}\\right] \\right\\rbrace ^{\\ell _{ig}}$ where $\\tilde{\\underline{\\theta }}_i = (1, \\theta _{i1}, \\ldots , \\theta _{iq})^T$ and $\\tilde{\\Lambda }_g$ is the matrix resulting from the combination of $\\underline{\\mu }_g$ and $\\Lambda _g$ so that the first column of $\\tilde{\\Lambda }_g$ is $\\underline{\\mu }_g$ .",
"In the binary part of the model, the Gaussian is truncated between $-\\infty $ and 0 if $y_{ij} = 0$ , and between 0 and $\\infty $ otherwise.",
"In the nominal part of the model, The Gaussian is also truncated, depending on the observed $y_{ij}$ i.e.",
"If $y_{ij}=0$ then $ \\displaystyle \\max \\lbrace z_{ij}^1, z_{ij}^{2}\\rbrace < 0$ .",
"If $y_{ij}=1$ then $z_{ij}^{1} = \\displaystyle \\max \\lbrace z_{ij}^1, z_{ij}^{2}\\rbrace $ and $z_{ij}^{1} > 0$ , $z_{ij}^2$ is restricted so that $z_{ij}^2 < z_{ij}^{1}$ .",
"If $y_{ij}=2$ then $z_{ij}^{2} = \\displaystyle \\max \\lbrace z_{ij}^1, z_{ij}^{2}\\rbrace $ and $z_{ij}^{2} > 0$ , $z_{ij}^1$ is restricted so that $z_{ij}^1 < z_{ij}^{2}$ .",
"The MFA-MD model proposed here is related to the mixture of factor analyzers model [37], [67] which is appropriate when the observed data are all continuous in nature.",
"A Bayesian treatment of such a model is detailed in [27]; [68] detail a suite of parsimonious mixture of factor analyzer models."
],
[
"Variable selection",
"The LIPGENE-SU.VI.MAX data contain a large number of variables, particularly categorical variables.",
"A variable selection algorithm that removes variables which have no clustering information would ease the computational burden of the model fitting process and also provide substantive interpretation advantages by only retaining variables which discriminate between clusters.",
"A simple but effective online variable selection procedure is incorporated here.",
"For an informative or discriminatory variable, the within cluster variance will be lower than the overall variance for that variable in the data.",
"Variables for which the within cluster and overall variances are similar do not discriminate between clusters and are not interesting in a clustering context.",
"Specifically, for each variable $j$ , a variance ratio $VR_{j}$ is computed where ${VR}_j & = & \\displaystyle \\sum _{g=1}^{G}\\sum _{\\begin{array}{c}i=1\\\\ \\forall i \\in g\\end{array}}^{n_g}(z_{ij} - \\bar{z}_{gj})^2 / \\sum _{i=1}^N(z_{ij} - \\bar{z}_j)^2.$ The variance ratio is computed in an online manner in that at an iteration of the model fitting algorithm $n_g$ denotes the number of participants currently classified as members of cluster $g$ .",
"In turn, the empirical cluster mean for cluster $g$ and variable $j$ is denoted by $\\bar{z}_{gj}$ , while the overall mean for variable $j$ is denoted by $\\bar{z}_j$ .",
"Small values for $VR_j$ indicate that variable $j$ discriminates between clusters while larger values indicate that variable $j$ takes similar values across all clusters and therefore contains no clustering information.",
"A user specified threshold $\\varepsilon $ is set such that if $VR_j > \\varepsilon $ , variable $j$ is dropped from the model and otherwise it is retained.",
"Selection of $\\varepsilon $ is application specific and its choice within the LIPGENE-SU.VI.MAX analysis is discussed in Section .",
"The choice of $\\epsilon $ can be thought of as the choice of how many variables the model will highlight as discriminatory; $\\epsilon $ doesn't have an `optimal' value as is typical of many tuning parameters.",
"Decreasing $\\epsilon $ is equivalent to indicating that a more aggressive variable selection is desirable.",
"This variable selection method is shown to perform well in simulation studies, provided in the Supplementary Material."
],
[
"Bayesian inference",
"The Bayesian paradigm is a natural framework for the estimation of latent variable models.",
"Fitting the proposed MFA-MD model in a Bayesian framework requires specification of prior distributions for all parameters.",
"Conjugate prior distributions are employed here.",
"Specifically, $ \\tilde{\\underline{\\lambda }}_{gd} \\sim \\mbox{MVN}_{(Q+1)}(\\underline{\\mu }_{\\lambda }, \\Sigma _{\\lambda }), \\hspace{5.69046pt} \\underline{\\pi } \\sim \\mbox{Dir}(\\underline{\\alpha })$ and $\\psi _{jj} \\sim \\mathcal {G}^{-1}(\\beta _1, \\beta _2)$ .",
"For participant $i$ , it is assumed the latent trait $\\underline{\\theta }_i$ follows a standard multivariate Gaussian distribution while the latent indicator variable $\\underline{\\ell }_i$ follows a Multinomial$(1, \\underline{\\pi })$ distribution.",
"Further, conditional on membership of cluster $g$ , the latent variable $\\underline{z}_{i}| l_{ig} = 1 \\sim \\mbox{MVN}_{D}(\\underline{\\mu }_{g}, \\Lambda _{g} \\Lambda _{g}^{T} + \\mathbf {\\Psi })$ .",
"Combining these latent variable distributions and prior distributions with the augmented likelihood function specified in (REF ) results in the joint posterior distribution, from which samples of the model parameters and latent variables are drawn using a Gibbs sampling MCMC scheme.",
"Full conditional distributions for the latent variables and model parameters are detailed below; derivations and definitions of the distributional parameters are given in the Supplementary Material.",
"Allocation vectors: $\\underline{\\ell }_i|\\ldots \\sim \\mbox{Multinomial}(1,\\underline{p})$ .",
"Mixing proportions: $\\underline{\\pi } | \\ldots \\sim \\mbox{Dirichlet}(\\underline{\\delta }_\\pi )$ .",
"Latent traits: $\\underline{\\theta }_i| \\ldots \\sim \\mbox{MVN}_Q\\left( \\underline{\\mu }_\\theta , \\Sigma _\\theta \\right)$ .",
"Item parameters: $\\tilde{\\underline{\\lambda }}_{gd} | \\ldots \\sim \\mbox{MVN}_{(q+1)} \\left(\\underline{\\zeta }_\\lambda , \\Omega _\\lambda \\right)$ .",
"Error variance parameters: $\\psi _{jj} \\sim \\mathcal {G}^{-1}(b_{1j}, b_{2j})$ .",
"The full conditional distribution for the latent data $\\mathbf {z}$ follows a truncated Gaussian distribution.",
"The point of truncation depends on the form of the corresponding variable, the observed response, and the previously sampled values of $\\mathbf {z}$ in the MCMC chain.",
"The distributions are truncated to satisfy the conditions detailed in Section REF .",
"The latent variable $z_{ij}$ is therefore updated as detailed below.",
"Note that $z_{ij}$ is not sampled for $j = 1, \\ldots , A$ as in the case of the continuous variables $y_{ij} = z_{ij}$ .",
"If variable $j$ is binary and $y_{ij}=0$ then $z_{ij}| \\ldots \\sim N^T\\left(\\tilde{\\underline{\\lambda }}_{gj}^T\\tilde{\\underline{\\theta }}_i, 1 \\right)$ where the distribution is truncated on the interval $(-\\infty , 0)$ .",
"The truncation interval is $(0, \\infty )$ if $y_{ij}=1$ .",
"If item $j$ is nominal then $z_{ij}^k | \\ldots \\sim N^T \\left( \\tilde{\\underline{\\lambda }}_{gj}^{k^T}\\tilde{\\underline{\\theta }}_i, 1\\right)$ where $\\tilde{\\underline{\\lambda }}_{gj}^k$ is the row of $\\tilde{\\Lambda }_g$ corresponding to $z_{ij}^k$ and the truncation intervals are defined as follows: if $y_{ij}=0$ then $z_{ij}^k \\in (-\\infty ,0)$ for $k = 1,2$ .",
"if $y_{ij}=k$ for $k=1,2$ then: $z_{ij}^{k} \\in (\\tau , \\infty )$ where $\\tau = \\max \\left(0, \\displaystyle \\max _{l \\ne k}\\lbrace z_{ij}^l\\rbrace \\right)$ .",
"for $l \\ne k$ then $z_{ij}^{l} \\in \\left( - \\infty , z_{ij}^{k}\\right)$ .",
"Note that in the case of $y_{ij} = k \\ne 0$ , the value $z_{ij}^{l}$ in the evaluation of $\\tau $ in step 1 is the previously sampled value in the MCMC chain.",
"The value of $z_{ij}^{k}$ in step 2 is the value sampled in step 1.",
"The variable selection method presented in Section REF is incorporated into the outlined Gibbs sampler, and thus the proposed MFA-MD model is fitted in three stages: Burn in phase: In the first phase of the model fitting procedure all variables are included and the Gibbs sampling algorithm is run until convergence.",
"Variable selection phase: After the burn in phase the algorithm moves into the variable selection phase.",
"Given the current clustering, the variance ratio $VR_j$ is computed for each variable $j$ .",
"All variables for which $VR_j$ is greater than $\\varepsilon $ are dropped from the model.",
"The algorithm is allowed to burn in again before another variable selection step is performed, with a user specified frequency.",
"The variable selection phase ends when no variables are removed from the model at a number of successive variable selection steps.",
"Posterior sampling phase: During this phase the Gibbs sampling algorithm proceeds, given the discriminating variables.",
"Given its factor analytic roots, the MFA-MD model is not identifiable.",
"Here, the loadings matrices are unconstrained and a Procrustean rotation is employed to solve the problem of their rotational invariance, following ideas in [47], [45] and as detailed in [71].",
"Further, the well known clustering label switching problem is addressed using a loss function approach as in [97]."
],
[
"Model selection via an approximate BIC-MCMC criterion",
"As with any clustering problem, the number $G$ of clusters is unknown.",
"Moreover, in the case of the MFA-MD model, the dimension of the latent trait $Q$ is also unknown.",
"Under a model based approach to clustering, such as that taken here, the use of principled, statistical model selection tools to choose both $G$ and $Q$ are available.",
"Formal likelihood based criteria such as the Bayesian Information Criterion (BIC) [95], [56] have been demonstrated to perform well in many general clustering settings (e.g.",
"[30], [41]), and also in clustering settings involving latent factor models (e.g.",
"[68]) and variable selection (e.g.",
"[92], [48]).",
"There is also a rich Bayesian literature regarding model evidence; model selection tools based on the marginal likelihood [31], [33], [69] are a natural approach to general model selection within the Bayesian paradigm, with reversible jump MCMC methods [93] and Markov birth-death methods [98] popular in the context of clustering.",
"More recently, overfitting approaches to model selection within clustering using Bayesian finite mixtures have gained warranted attention [101], [62].",
"In the context of choosing $Q$ in latent factor models, [60] provide a comprehensive overview of Bayesian model assessment.",
"Such approaches naturally require evaluation of the joint likelihood of the observed continuous and categorical data $Y$ which for the MFA-MD model is intractable as it requires integrating a multidimensional truncated Gaussian distribution, where truncation limits differ and are dependent across the dimensions.",
"These approaches also require the variables in the data to be the same when comparing models.",
"Thus, in order to select the optimal MFA-MD model an approximation of the observed data likelihood is constructed which involves both variables retained and removed during the variable selection steps.",
"Recall that for participant $i$ their observed data consists of $A$ continuous phenotypic variables, $B$ binary SNP variables and $C$ nominal SNP variables collected in $\\underline{y}_i = (y_{i1}, \\ldots , y_{iJ})$ where $J = A + B + C.$ Denoting the $\\ddot{A}$ continuous, $\\ddot{B}$ binary and $\\ddot{C}$ nominal variables with clustering information collectively as $\\underline{\\ddot{y}}_{i}$ and the $\\dot{A}$ continuous, $\\dot{B}$ binary and $\\dot{C}$ nominal variables with no clustering information collectively by $\\underline{\\dot{y}}_{i}$ , the contribution to the likelihood function for participant $i$ is approximated as $\\nonumber \\mathcal {\\tilde{L}}_i & = & f(\\underline{\\ddot{y}}_{i}) f(\\underline{\\dot{y}}_{i})\\\\\\nonumber & = & \\left[ \\sum _{g=1}^{G} \\pi _g \\left\\lbrace \\mbox{MVN}_{\\ddot{A}}(\\mu _g, \\Lambda _{g} \\Lambda _{g}^{T} + \\Psi ) \\prod _{j=1}^{\\ddot{B}+\\ddot{C}} P(\\ddot{y}_{ij}| i \\in g) \\right\\rbrace \\right] \\\\& & \\times \\left[ \\mbox{MVN}_{\\dot{A}}(\\mu , \\Lambda \\Lambda ^{T} + \\Psi ) \\prod _{j=1}^{\\dot{B}+\\dot{C}} P(\\dot{y}_{ij}) \\right].$ That is, independence is first assumed between the set of discriminating and the set of non-clustering variables.",
"Further, for the discriminating variables, conditional independence between the set of $\\ddot{A}$ continuous and the set of $\\ddot{B}+\\ddot{C}$ categorical variables is assumed, and within the set of $\\ddot{B}+\\ddot{C}$ categorical variables.",
"Additionally, independence between the set of continuous and the set of categorical variables without clustering information is also assumed, and within the set of non-clustering categorical variables.",
"The multivariate Gaussian densities for the continuous variables in (REF ) are straight forward to evaluate; a single Bayesian factor analysis model is fitted to the $\\dot{A}$ removed variables.",
"For the categorical variables in (REF ), simple empirical probabilities are calculated from the observed data.",
"For the $\\ddot{B}+\\ddot{C}$ categorical discriminating variables, these probabilities are the observed response probabilities, within each cluster.",
"For the $\\dot{B}+\\dot{C}$ non-clustering categorical variables, the probabilities are the observed response probabilities among the $N$ participants.",
"Thus a tractable approximation to the intractable likelihood is available, and can be used to compare models with varying values of $G$ and $Q$ , and with varying sets of discriminating variables.",
"This approximated observed likelihood function is incorporated in the BIC-MCMC [34] criterion to perform model selection with the MFA-MD model.",
"Analogous to the traditional BIC, the BIC-MCMC is derived from the largest observed log likelihood value generated across the MCMC draws, penalised for lack of parsimony.",
"In the context of the MFA-MD model the approximate BIC-MCMC is defined as: $\\mbox{BIC-MCMC } = 2 \\times \\log \\mathcal {\\tilde{L}} - \\nu \\times \\log (N)$ where $\\mathcal {\\tilde{L}} = \\prod _{i=1}^{N} \\mathcal {\\tilde{L}}_{i}$ denotes the largest observed approximate likelihood value across the MCMC draws and $\\nu $ denotes the number of parameters in (REF ).",
"Thus for $G = 1, \\ldots , G_{\\max }$ and $Q = 1, \\ldots , Q_{\\max }$ , the approximate observed likelihood function $\\mathcal {\\tilde{L}}$ is evaluated at each MCMC iteration, and the largest value used to compute the associated BIC-MCMC.",
"The model with largest BIC-MCMC is chosen as the optimal model.",
"The BIC-MCMC has been shown to perform well in the context of mixture models generally [34]; its performance in combination with the likelihood approximation within the MFA-MD context is also shown to perform well in the simulation studies provided in the Supplementary Material."
],
[
"Results",
"In order to cluster the set of LIPGENE-SU.VI.MAX participants, a number of MFA-MD models with $G=1, \\ldots , G_{\\max } = 4$ and $Q=1, \\ldots , Q_{\\max } = 10$ were fitted to the initial mixed phenotypic and genotypic data.",
"The maximum value considered for $G$ was motivated by expert opinion on the expected structure of the set of participants; the maximum value of $Q$ considered was motivated by the observed performance of the $G = 1$ model (see Figure REF ), and by run time considerations.",
"The Jeffreys prior, Dirichlet$(0.5, \\ldots , 0.5)$ , was specified for the mixing proportions $\\underline{\\pi }$ .",
"An inverse gamma prior, with shape and scale parameters of 7, was specified for the $A$ diagonal elements of $\\Psi $ corresponding to continuous variables.",
"The mode of this relatively uninformative prior is just less than 1.",
"A zero mean multivariate Gaussian prior was specified for $\\tilde{\\underline{\\lambda }}_{gd}$ with $\\Sigma _\\lambda = 5\\mathbf {I}$ , which again is relatively uninformative.",
"Prior sensitivity was assessed by trialling different values of the hyperparameters.",
"The results were relatively insensitive to changes in the hyperparameters for $\\underline{\\pi }$ and $\\tilde{\\underline{\\lambda }}_{gd}$ but somewhat sensitive to the hyper parameters for $\\psi _{jj}$ .",
"Sensitivity to these inverse gamma hyperparameter values is a known problem for Bayesian inference of models of this type [36].",
"For each of the forty models fitted, the burn in phase was run for 20,000 iterations and in the variable selection phase the variance ratio criterion was computed every 1000 iterations.",
"This period between variable selection steps allowed the MCMC algorithm to `burn in' again after variables have been removed.",
"In the LIPGENE-SU.VI.MAX setting, the variable selection threshold $\\varepsilon $ was fixed at $0.95$ for continuous phenotypic variables and at $0.99$ for categorical SNP variables.",
"These thresholds are very conservative so that only the most uninformative variables were removed.",
"The thresholds could be lowered to facilitate a more aggressive variable selection procedure.",
"The model fitting algorithm remained in the variable selection phase until no variables were removed from the model for four successive variable selection steps.",
"When this occurred the algorithm moved into the posterior sampling phase which was then run for 100,000 iterations, thinned every 100th iteration.",
"Convergence of the Markov chains was assessed using trace and auto-correlation plots.",
"Computation times for these models are variable as the speed will depend on how many variables are removed and on both the dimension of the latent trait and the number of clusters fitted to the data.",
"The $G=1$ , $Q=1$ model took approximately 11 hours while the $G=4$ , $Q=10$ model took approximately 25 hours.",
"It should be noted that no variable selection can be applied if only one cluster is fitted to the data.",
"The optimal model described below took less than 5 hours to fit as only a small number of variables were deemed discriminatory.",
"These timings were measured by fitting the model using one processor of a quad core (2.83GHz) desktop PC with 4GB of RAM.",
"The optimal MFA-MD model was selected using the approximate BIC-MCMC criterion developed in Section REF .",
"Figure REF illustrates the approximate BIC-MCMC for each of the forty models fitted; the optimal model is indicated to have $G = 2$ clusters and $Q = 8$ latent factors.",
"During fitting of the optimal $G = 2, Q=8$ MFA-MD model, a large number of variables is dropped at the beginning of the variable selection phase but as the phase proceeds the model converges on a relatively small number of discriminatory variables.",
"A plot showing the evolution of the number of variables retained during the variable selection phase is given in the Supplementary Material.",
"Only 25 of the original 738 variables are retained under the $G = 2, Q = 8$ model.",
"Of those retained, 12 are continuous phenotypic variables, 2 are binary SNP variables and 11 are nominal SNP variables.",
"Notably, in the nearest competing model $G = 2, Q = 9$ , a total of 22 variables were retained, 16 of which were the same as those retained in the optimal $G = 2, Q = 8$ model; this pattern was observed in general within models with the same number of groups.",
"Alternative variable selection criteria to (REF ) are possible: the set of 40 models were also fitted using a weighted version of $VR_j$ where each squared difference in the numerator in (REF ) is multiplied by the posterior probability that observation $i$ belongs to cluster $g$ and a `fuzzy' clustering version of the cluster specific means $\\bar{z}_{gj}$ is also used.",
"This fuzzy version of $VR_j$ allows observations that are not assigned to a cluster with a high degree of certainty to contribute to the within cluster variances of multiple clusters.",
"In the case of the LIPGENE-SU.VI.MAX study, it was found that this fuzzy version of $VR_j$ had no effect on the optimal models: the same variables were chosen and the same clustering solutions were found, thus giving the same interpretation.",
"Of particular note, in the context of the LIPGENE-SU.VI.MAX study, is that both phenotypic and genotypic variables are deemed to be informative.",
"The reduction from 738 to 25 variables aids the substantive interpretation of the model output significantly and ensures model fitting efficiency.",
"Examination of the cluster specific parameters under the optimal model provides insight to the clustering structure in the set of LIPGENE-SU.VI.MAX participants; posterior inferences from the optimal MFA-MD model are discussed in what follows.",
"Figure: The approximate BIC-MCMC for each of the MFA-MD models fitted to the set of LIPGENE-SU.VI.MAX participants.",
"The dashed grey line indicates the largest approximate BIC-MCMC value achieved; the optimal model has two clusters and eight latent dimensions."
],
[
"Examining the cluster specific parameters for the set of discriminatory variables",
"The reduced cardinality of the set of variables facilitates interpretation of the substantive differences between the resulting clusters or `sub-phenotypes'.",
"Figure: Box plots of the MCMC samples of mean parameters in each cluster, for the discriminating continuous phenotypic variables.",
"All variables were standardised prior to analysis.",
"The original units for each variable are detailed in the Supplementary Material.Table: THYN1 (rs570113)The means of the retained continuous phenotypic variables for each cluster are illustrated in Figure REF .",
"Examination of these posterior parameter estimates provides particular insight to the structure of the two clusters.",
"Cluster 1 appears to be a `healthy' sub-phenotype in that the phenotypic variable means are lower in general in cluster 1 than in cluster 2.",
"It is well known that lower values of such phenotypic variables are typically associated with better health.",
"For example, the mean levels of triglycerides, waist circumference, body mass index (BMI) and systolic and diastolic blood pressure variables (SBP and DBP respectively) are notably lower in cluster 1 than cluster 2.",
"The exception is Apo A-1, the major structural protein of the high density lipoprotein (HDL) particle, low levels of which are a recognised risk factor for cardiovascular disease [107], [40].",
"Apo A-1 levels are usually low when HDL cholesterol levels are reduced, thus it is intuitive that higher Apo A-1 levels are reported in the healthy cluster.",
"Table REF details the empirical posterior probability of each genotype across the thirteen retained SNPs, conditional on cluster membership.",
"Clear differences in the distributions between clusters are visible.",
"For example, in both retained binary SNPs rs17777371 of the ADD1 gene and rs1050289 of the OLR1 gene participants in both clusters are most likely to take the dominant homozygous genotype.",
"However, for both SNPs, cluster 1 is more likely to take the compound recessive homozygous/heterozygous genotype (the second level) than cluster 2.",
"In terms of retained nominal SNPs, the probability distributions between clusters for the rs4784744 SNP of the CETP gene and the rs2235800 SNP of the SLC25A14 gene also show some disparities, for example.",
"For the rs4784744 SNP of the CETP gene, participants in cluster 1 are more likely to have the dominant homozygous genotype than those in cluster 2, with those in cluster 2 more likely to have the heterozygous genotype.",
"For the rs2235800 SNP of the SLC25A14 gene, 60% of participants assigned to cluster 2 have the dominant homozygous genotype compared to 38% of those in cluster 1.",
"The probability distribution is much more evenly spread across the genotypes for participants in cluster 1 than for those in cluster 2.",
"The 13 SNP variables deemed to be discriminatory are also listed in Table REF , which provides details on characteristics of the discriminating SNPs and the biological pathways to which they are associated.",
"Most of the SNPs deemed to be discriminatory are involved in lipid metabolism, glucose homeostasis or blood pressure regulation.",
"Associations between polymorphisms of a number of genes involved in fatty acid and lipid metabolism, inflammation, appetite control and adiposity with risk of the MetS or its features have previously been identified in the LIPGENE-SU.VI.MAX cohort [88], [89], [87], [83], [84], [85], [86], [79], [81], [82]; some of these SNPs are also highlighted here, in addition to some novel discoveries.",
"Of particular interest in the current analysis is the APOB rs512535 SNP which has previously been reported to have association with MetS risk [87].",
"Apo B is the main apolipoprotein associated with low density lipoprotein and the triglyceride rich lipoproteins [17].",
"Other findings of note are rs9770068 of the INSIG1 gene which is involved in cholesterol metabolism [91] and rs4784744 of the CETP gene which is involved in mediating exchange of lipids between lipoproteins and reverse cholesterol transport [58]; rs2544377 of the LRP2 gene and the rs1050289 SNP of the OLR1 gene, both of which are involved in lipid homeostasis [59], [94]; rs2970901 of the FABP1 gene and rs185411 of the SLC27A6 gene both of which are involved in fatty acid metabolism [78], [6] and rs17777371 of the ADD1 gene which is involved in blood pressure regulation [9].",
"Examination of the posterior parameter estimates across all discriminating variables suggests that cluster 1 could be termed a `healthy' sub-phenotype and cluster 2 an `at risk' sub-phenotype.",
"Further, some of the phenotypic and SNP variables deemed to be discriminatory appear intuitive, while others are suggestive of potentially interesting relationships for further research.",
"Table: Characteristics of the set of 13 binary and nominal SNP variables deemed to be discriminatory.",
"(Source: NCBI SNP data base http://www.ncbi.nlm.nih.gov/SNP/)"
],
[
"Correspondence between sub-phenotype membership and the seven year follow-up MetS diagnosis",
"As stated, the data analysed here are an initial set of measurements under the LIPGENE-SU.VI.MAX study.",
"At a seven year follow-up, new continuous phenotypic data on each of the 505 participants were recorded.",
"Each participant was then diagnosed as having the MetS or not based on the criterion in Table REF , which considers continuous phenotypic data only.",
"It is therefore of interest to compare the cluster or sub-phenotype membership of each LIPGENE-SU.VI.MAX participant based on their initial phenotypic and genotypic data to their subsequent MetS diagnosis, seven years later.",
"The cluster or sub-phenotype membership for each participant is obtained by first computing the conditional probability that participant $i$ belongs to each cluster based on the MCMC samples, and a `hard' clustering is then obtained by assigning each participant to the cluster for which they have largest membership probability.",
"Table REF details the cross tabulation of the initial sub-phenotypes and the follow-up MetS diagnosis.",
"It can be seen that traits of the MetS are apparent in the initial data, as the cross-tabulation shows good agreement, with a Rand index of 0.73 (and an adjusted Rand index of 0.46).",
"Notably, Figure REF suggests there are five closely competing models to the optimal $G = 2, Q = 8$ model i.e.",
"the $G = 2, Q = 5, 6, 7, 9, 10$ models.",
"Comparing the resulting clusterings from these models to the follow-up MetS diagnosis results in Rand indices ranging from 0.71 to 0.74 and in adjusted Rand indices ranging from 0.42 to 0.48, suggesting that the models deemed optimal by the BIC-MCMC criterion all indeed have similar performance and perform well.",
"Table: Cross tabulation of sub-phenotype membership (based on fitting the MFA-MD model to the initial phenotypic and genotypic data) and MetS diagnosis (based on the diagnosis criterion in Table on seven year follow up phenotypic data only).",
"The Rand index is 0.73 (adjusted Rand index = 0.46).Of further interest is whether the level of correspondence between the sub-phenotypes and the follow-up MetS diagnosis is stronger than that observed between the MetS diagnoses from both time points based on the phenotypic data only.",
"One of the abnormalities required for diagnosis involves HDL cholesterol – HDL cholesterol data are not available in the initial measurements however.",
"Therefore the current diagnosis criterion in Table REF cannot be applied to the initial data.",
"Hence, participants are diagnosed as MetS cases if they satisfy two or more of the remaining four diagnostic conditions relating to waist circumference, blood pressure, TAG and glucose concentration.",
"Table REF details the cross tabulation of the `initial diagnosis' compared to the `follow-up diagnosis' based on the phenotypic data only.",
"Notably, the follow up diagnosis does not change here if it is based on 2 of the 4 available variables rather than on the criterion outlined in Table REF .",
"Table REF also suggests that the traits of the MetS are apparent in the initial data, as the MetS diagnoses from the two time points agree well, with a Rand index of 0.69 (adjusted Rand index of 0.38).",
"However, the level of agreement is lower in Table REF than that observed in Table REF , highlighting the importance of utilising both phenotypic and genotypic factors, and the potential utility of the clustering approach in early screening.",
"Table: Cross tabulation of MetS diagnoses from initial and follow up data.",
"The Rand index is 0.69 (adjusted Rand index is 0.38)Further, to explore the influence of modelling each data type in its innate form, a $k$ -means clustering algorithm with $k = 2$ was applied to all the 738 variables, treating all the SNP variable codes as continuous values.",
"Comparing the resulting clustering to the follow up MetS diagnosis gave a Rand index of 0.60 (adjusted Rand index = 0.21).",
"Applying $k$ -means clustering (again with $k = 2$ ) to the set of 25 variables selected as discriminatory under the optimal $G = 2, Q = 8$ model gave a Rand index of 0.68 (adjusted Rand index = 0.37) when compared to the follow-up MetS diagnosis.",
"As noted the MFA-MD model achieved a Rand index of 0.73 (adjusted Rand index = 0.46) highlighting the benefit of modelling the variables in their innate form.",
"Finally, the MFA-MD model outlined above was fitted to only the continuous phenotypic variables from the initial LIPGENE-SU.VI.MAX data.",
"The optimal model, according to the approximate BIC-MCMC, was the $G = 2, Q = 7$ model, which gave a Rand index of 0.50 (adjusted Rand of 0.005) with the follow-up MetS diagnosis.",
"This model under-performs when compared to analysing the phenotypic and genetic data jointly, again highlighting the importance of considering phenotypic and genotypic factors simultaneously with regard to early screening for the MetS."
],
[
"Quantifying uncertainty in sub-phenotype membership at the participant level",
"One of the main advantages of a model-based approach to clustering is the inherent assessment of the uncertainty about cluster membership [10], [41].",
"In the LIPGENE-SU.VI.MAX context, the model-based approach allows quantification of the probability of sub-phenotype membership for each participant.",
"As stated, the cluster membership for each participant is obtained by first computing the conditional probability that participant $i$ belongs to each cluster based on the MCMC samples, and a `hard' clustering is then obtained by assigning each participant to the cluster for which they have largest membership probability.",
"The uncertainty with which participant $i$ is assigned to its cluster may then be estimated by $U_i = \\min _{g=1,\\ldots , G} \\lbrace 1 - \\mathbf {P}(\\mbox{cluster } g \\ | \\ \\mbox{participant }i)\\rbrace .",
"$ If participant $i$ is strongly associated with cluster $g$ then $U_i$ will be close to zero.",
"Figure REF illustrates the clustering uncertainties under the optimal MFA-MD model.",
"Figure REF illustrates the clustering uncertainty for each LIPGENE-SU.VI.MAX participant.",
"The maximum uncertainty observed is 0.496, associated with participant number 445.",
"This participant is clustered with the `healthy' sub-phenotype, but there is high uncertainty associated with this clustering.",
"Examination of this participant's data provides insight to this high clustering uncertainty – participant 445 has much higher SBP and DBP, and much lower Apo A-1 levels than the mean levels in the `healthy' sub-phenotype.",
"Further, participant 445 differs from the modal genotypes observed in the `healthy' sub-phenotype for SNPs APOB (rs512535), FABP1 (rs2970901) and INSIG1 (rs9770068).",
"Thus while this participant is clustered with the `healthy' sub-phenotype they have large probability of being `at risk'.",
"Thus, the model-based nature of the MFA-MD approach to clustering provides a global view of the group structure in the LIPGENE-SU.VI.MAX participants, but also provides detailed insight to sub-phenotype membership at the participant level; the ability to define the uncertainty in cluster membership is an important development for the application of the metabotyping concept in precision medicine and nutrition [76].",
"Overall, the vast majority of LIPGENE-SU.VI.MAX participants have very small clustering uncertainty, as illustrated by Figure REF .",
"Figure: (a) The participant specific clustering uncertainties and (b) the histogram of the clustering uncertainties across all participants, under the optimal MFA-MD model."
],
[
"Assessing model fit",
"In order to assess how well the selected MFA-MD model fits the LIPGENE-SU.VI.MAX data, Bayesian residuals and Bayesian latent residuals are utilised [55], [28].",
"For continuous phenotypic variables the Bayesian residual for participant $i$ on variable $j$ is $\\epsilon _{ij} = \\left(z_{ij} - \\tilde{\\underline{\\lambda }}_{gj}^T\\tilde{\\theta }_i\\right)/\\psi _{jj}.$ The continuous phenotypic data are observed so this residual may be calculated explicitly by subtracting $\\tilde{\\underline{\\lambda }}_{gj}^T\\tilde{\\theta }_i$ at each MCMC iteration from $z_{ij}$ and dividing this quantity by $\\psi _{jj}$ from that iteration.",
"For a well fitting model, this residual follows a standard Gaussian distribution.",
"However, $z_{ij}$ corresponding to a categorical SNP variable is not observed but sampled during the MCMC scheme.",
"A Bayesian latent residual for these variables may be defined as $\\epsilon _{ij} = z_{ij} - \\tilde{\\underline{\\lambda }}_{gj}^T\\tilde{\\theta }_i.$ The sampled values of $z_{ij}$ , $ \\tilde{\\underline{\\lambda }}_{gj}$ and $\\tilde{\\theta }_i$ are used to calculate this residual at each MCMC iteration.",
"If the model fits well such residuals should follow a standard Gaussian distribution.",
"For the nominal SNP variables this residual will be multivariate since two latent dimensions are required to model each nominal SNP.",
"The Bayesian residuals and latent residuals follow their theoretical distribution reasonably well for the optimal $G=2$ , $Q=8$ MFA-MD model.",
"As an example, Figure REF illustrates kernel density estimates of Bayesian latent residuals corresponding to the ADD1 (rs17777371) SNP for 50 randomly selected participants.",
"The densities are estimated based on the residuals calculated at each MCMC iteration.",
"Curves that do not follow a standard Gaussian distribution correspond to participants whose genotype was unusual given the cluster to which they were assigned.",
"Kernel density estimate plots for other Bayesian residuals and Bayesian latent residuals are provided in the Supplementary Material.",
"Figure: Density estimates of the Bayesian latent residuals for the rs17777371 SNP of the ADD1 gene for 50 randomly selected participants.",
"The standard Gaussian density curve is shown by the black dashed line."
],
[
"Discussion",
"The primary focus of the pan European LIPGENE-SU.VI.MAX project is to study the interaction of nutrients and genotype in the metabolic syndrome.",
"Data collected under LIPGENE-SU.VI.MAX are high dimensional and of mixed type, and interest lies in exploring the set of LIPGENE-SU.VI.MAX participants to uncover subgroups with homogeneous phenotypic and genotypic profiles.",
"Examining the link between the resulting clusters and seven-year follow-up MetS diagnosis aids understanding of the role of both phenotypic and genotypic factors in the MetS and provides the opportunity to identify subjects at risk.",
"A clustering method that takes account of different data types and models each one appropriately is therefore necessary.",
"While factor analytic methods for data of mixed type and latent factor based clustering methods have already been well developed, the proposed MFA-MD methodology contributes a number of novel advances to the area: the MFA-MD model provides a single, unifying and elegant model for data which notably includes any combination of continuous, binary or nominal response variables.",
"the MFA-MD approach models nominal response variables in their innate form, rather than requiring a dummy variable representation as is typically necessary in other approaches to clustering nominal response variables.",
"the variable selection approach permits high dimensional data to be feasibly and efficiently handled, which is theoretically possible but practically challenging for some latent factor models.",
"the model based approach to clustering and the novel likelihood function approximation facilitates the use of an objective model selection criterion to select the optimal number of clusters and factors rather than relying on subjective heuristic tools.",
"The MFA-MD approach proposed here jointly and elegantly models continuous phenotypic, binary SNP and nominal SNP data, while providing clustering facilities.",
"The suitability of the MFA-MD model for this task is due to its basis in and the relations between a factor analysis model for continuous data, item response theory for binary data and multinomial probit models for nominal data.",
"Further, the parsimonious factor analysis covariance structure is ideal for modelling such high dimensional data.",
"Most of the large number of LIPGENE-SU.VI.MAX data set variables have little to offer in terms of clustering information; a simple and efficient variable selection algorithm is intertwined with the MFA-MD fitting process, thereby highlighting variables that contribute clustering information.",
"This greatly simplifies the task of interpreting the clusters substantively.",
"A key aspect of the proposed approach to variable selection is that variables are removed from the model online, thus dramatically reducing the computational burden of fitting the MFA-MD model to high-dimensional data.",
"Several penalisation based variable selection approaches have previously been proposed for latent factor clustering models, for example in [77], [35], [108]; these only consider continuous data in a maximum likelihood framework however.",
"The fact that non-discriminating variables are removed from the MFA-MD model rather than shrinking their associated parameters to zero (meaning all variables are still included in the modelling procedure) ensures the dramatic increase in computational efficiency of the proposed approach.",
"As with any clustering problem, of key interest is inferring the number of clusters present in the set of LIPGENE-SU.VI.MAX participants.",
"Standard information criteria approaches in a model based clustering setting involve the evaluation of the observed likelihood function and are not feasible under the MFA-MD model – it employs latent variables and evaluation of the observed likelihood function relies on intractable multidimensional integrals.",
"Here an approximation of the observed data likelihood is constructed, and employed in the BIC-MCMC criterion to select both the number of clusters and the dimension of the underlying latent factors in the MFA-MD model.",
"Simulation studies suggest the approximate model selection criterion exhibits desirable performance, as does the variable selection approach taken.",
"When applied to the initial mixed phenotypic and genotypic LIPGENE-SU.VI.MAX data, the MFA-MD model uncovers two clusters or `sub-phenotypes' of participants; exploration of the cluster specific parameters suggests one cluster is a `healthy' sub-phenotype and the other an `at risk' sub-phenotype.",
"Both phenotypic and genotypic variables are identified as discriminatory; some are novel discoveries and are indicative of further directions of research.",
"Further, when comparing the resulting clusters to the MetS diagnosis seven years later, the proposed approach out-performs both the use of the standard MetS diagnosis criterion, and the result when clustering using the continuous phenotypic data only, thus emphasising the importance of jointly considering both phenotypic and genotypic profiles when screening for MetS.",
"The proposed MFA-MD approach to clustering provides a global view of the group structure in the set of LIPGENE-SU.VI.MAX participants, but also provides detailed insight to sub-phenotype membership at the participant level, synonymous with the concepts of precision medicine and nutrition.",
"The developed methodology has wide applicability beyond the LIPGENE-SU.VI.MAX study, in any setting seeking to uncover subgroups in a cohort on which high dimensional data of mixed type have been recorded.",
"There are many potential areas of future research for the MFA-MD methodology proposed here.",
"Covariate data such as ethnicity and gender are potentially important when studying MetS, and are currently involved in some of the varying MetS diagnosis criteria [3], [4].",
"Incorporating such covariate information in the MFA-MD model could provide understanding of cause-effect relationships in the clustering context.",
"Such information could be incorporated into the MFA-MD model in a mixture of experts framework [54], [42].",
"Within the LIPGENE-SU.VI.MAX cohort a large number of participants were removed from the original data set prior to analysis due to the presence of missing data.",
"To ensure generalisability of the proposed approach it would be advantageous to address such missingness in a more elegant manner.",
"The latent variable and Bayesian origins of the developed model and methodology would allow missing data to be treated as latent variables that can be naturally imputed as part of the MCMC inferential sampling scheme.",
"Such missing data would be required to be missing at random, which was deemed not to be the case in the LIPGENE-SU.VI.MAX cohort.",
"The approximate model selection criterion developed demonstrated good performance but can be computationally expensive to compute and other approaches have potential merit.",
"Non-parametric approaches to clustering such as the Dirichlet process (or infinite) mixture model [100] provide an alternative to the finite mixture approach taken here, and do not require a model selection tool to choose $G$ .",
"However, in the case of MFA-MD the value of $Q$ still requires inference; considering an infinite factor model [11] would again avoid the need for a model selection criterion for $Q$ , and allow the latent factor dimension to vary across clusters, in a similar manner to that considered in [73].",
"Such approaches may provide computationally cheaper ways to find the optimal values of $G$ and $Q$ without requiring an expensive grid search.",
"Considering more parsimonious versions of the model [68] would increase modelling flexibility, as would extending the model to include other data types, such as count data, for example.",
"Including such further complexity in the MFA-MD methodology would serve to increase the computational cost of model fitting which, even with the efficiency inducing variable selection procedure, is still somewhat onerous.",
"A variational Bayes approach to estimation of the MFA-MD model [38] may have potential in terms of feasibly implementing the model at increased scale and complexity, and may also aid some of the intractable likelihood difficulties.",
"Further, exploring other latent variable representations, for nominal variables in particular, may be fruitful in terms of achieving parsimony and computational efficiency."
],
[
"Acknowledgements",
"The authors would like to acknowledge the members of the Working Group on Statistical Learning at University College Dublin and the members of the Working Group on Model-based Clustering at the University of Washington for numerous discussions that contributed enormously to this work.",
"The authors would also like to acknowledge the LIPGENE-SU.VI.MAX cohort as the source of these data (FOOD-CT-2003-505944), our cohort co-investigators Serge Hercberg, Denis Larion, Richard Planells, Sandrine Bertrais and Emmanuelle Kesse-Guyot; as well as the LIPGENE-SU.VI.MAX participants.",
"Finally, the authors would like to thank the reviewers and editors who added considerably to this work through their thorough and thought provoking reviews.",
"The authors acknowledge LIPGENE-SU.VI.MAX subjects and investigators, funded by European Commission FP6 (FOOD-CT-2003-505944).",
"ICG was supported by Science Foundation Ireland (SFI/09/RFP/MTH2367) and the Insight Research Centre (SFI/12/RC/2289).",
"DMcP was supported by Science Foundation Ireland (SFI/09/RFP/MTH2367).",
"LB was supported by Science Foundation Ireland (SFI/14/JPI_HDHL/B3075).",
"HMR was supported by Science Foundation Ireland (SFI/PI/11/1119)."
]
] | 1606.05107 |
[
[
"Cooperative spontaneous emission from indistinguishable atoms in\n arbitrary motional quantum states"
],
[
"Abstract We investigate superradiance and subradiance of indistinguishable atoms with quantized motional states, starting with an initial total state that factorizes over the internal and external degrees of freedom of the atoms.",
"Due to the permutational symmetry of the motional state, the cooperative spontaneous emission, governed by a recently derived master equation [F. Damanet et al., Phys.",
"Rev.",
"A 93, 022124 (2016)], depends only on two decay rates $\\gamma$ and $\\gamma_0$ and a single parameter $\\Delta_{\\mathrm{dd}}$ describing the dipole-dipole shifts.",
"We solve the dynamics exactly for $N=2$ atoms, numerically for up to 30 atoms, and obtain the large-$N$-limit by amean-field approach.",
"We find that there is a critical difference $\\gamma_0-\\gamma$ that depends on $N$ beyond which superradiance is lost.",
"We show that exact non-trivial dark states (i.e.",
"states other than the ground state with vanishing spontaneous emission) only exist for $\\gamma=\\gamma_0$, and that those states (dark when $\\gamma=\\gamma_0$) are subradiant when $\\gamma<\\gamma_0$."
],
[
"Introduction",
"Cooperative spontaneous emission of light from excited atoms, which results from their common coupling with the surrounding electromagnetic field, is a central field of research in quantum optics.",
"In a seminal paper [2], Dicke showed that spontaneous emission can be strongly enhanced when atoms are close to each other in comparison to the wavelength of the emitted radiation.",
"This phenomenon, called superradiance, and its counterpart corresponding to reduced spontaneous emission, called subradiance, were first considered for distinguishable atoms at fixed positions.",
"Depending on the geometrical arrangement of the atoms in space, deeper analyses showed later that virtual photon exchanges between atoms were likely to destroy superradiance [3], [4], [5].",
"Due to the complexity involved by the exact treatment of these dipole-dipole interactions, analytical characterizations of superradiance have been found only for particular geometries or for a small number of atoms [6], [5], [7], [8], [9], [10].",
"Yet, cooperative emission processes are not restricted to small atomic samples.",
"They can also be observed in dilute atomic systems for which the near-field contributions of the dipole-dipole interactions are insignificant.",
"Recent studies concern single-photon superradiance [11], [12], [13], [14], subradiance in cold atomic gases [15], [16], collective Lamb-shift [17], [18] and localization of light [19], [20].",
"Moreover, super- and subradiance can be explained using a quantum multipath interference approach and can be simulated from the measurement of higher-order-intensity-correlation functions on atoms separated by a distance larger than the emission wavelength [21], [22].",
"The atomic motion can have a significant influence on the spontaneous emission and scattering of light [23], [24], [25], [26], [27], [28], and vice versa (see e.g.",
"[29], [30], [31], [32], [33], [34], [35]).",
"However, its role on cooperative emission processes is not yet fully understood, especially in large laser driven atomic systems [36].",
"The interplay of atomic motion and cooperative processes has been the subject of recent experiments [37], [38], [39], [40], [41] and can lead to interesting effects such as supercooling of atoms [42] or superradiant Rayleigh scattering from a Bose-Einstein condensate (BEC) [43].",
"In hot atomic samples, the motion can be treated classically and leads to Doppler broadenings of the spectral lines.",
"In (ultra)cold atomic samples, the quantum nature of the motion and the indistinguishability must be taken into account, as they also lead to strong modifications of the dynamics.",
"In this paper, we study super- and subradiance from indistinguishable atoms, taking into account recoil, quantum fluctuations of the atomic positions, and quantum statistics.",
"To this end, we solve a recently derived master equation describing the cooperative spontaneous emission of light by $N$ two-level atoms in arbitrary quantum motional states [1].",
"Indistinguishability of atoms profoundly changes their internal dynamics as compared to that of distinguishable atoms.",
"For distinguishable atoms with classical positions, the solution of the master equation depends considerably on the geometrical arrangement of the atoms in space.",
"When describing each atom as a two-level atom, the internal state of the atoms thus evolves in the full Hilbert space of dimension $2^N$ .",
"The same general considerations can be made when the atomic positions are treated quantum mechanically since despite changed rates and level shifts the master equation [1] then retains the same global form with the same Lindblad operators.",
"Hence, each configuration must be dealt with case by case.",
"However, for indistinguishable atoms, the global state has to be invariant under exchange of the atoms.",
"For initial states that are separable between the internal degrees of freedom and the motional degrees of freedom, both internal and motional states must be invariant under permutation of atoms.",
"Furthermore, on the time scale of the spontaneous emission in the optical domain, the motional state can be considered frozen such that the permutational symmetry of the motional state prevails throughout the entire emission process.",
"This leads to permutationally invariant average Lindblad-Kossakowski matrix of emission rates and permutationally invariant dipole shifts, which limits the quantum dynamics also of the internal degrees of freedom to the permutation-invariant subspace of dimension $\\mathcal {O}(N^2)$ of the global Hilbert space [44], [45], thus greatly simplifying the problem.",
"However, the quantum fluctuations of the positions of the atoms modify the cooperative effect of collective emission, thus leading back from superradiance to individual spontaneous emission for large enough quantum uncertainty in the positions.",
"The paper is organized as follows.",
"In section II, we discuss the general form of the master equation for the atomic internal dynamics derived in [1] in the case of indistinguishable atoms.",
"In particular, we show that the master equation preserves permutation invariance of the internal state.",
"In section III, we write the master equation in the coupled spin basis as it is particularly suited for permutation-invariant states.",
"Finally, in section IV, we solve the master equation analytically for $N=2$ and numerically for up to $N=30$ atoms in order to study the impact of the quantization of the atomic motion on super- and subradiance.",
"Let us consider $N$ indistinguishable atoms in a mixture $\\rho _A$ .",
"Each wave function of the mixture has to be either symmetric (bosons) or antisymmetric (fermions) under exchange of atoms.",
"Let $P_\\pi $ denote the permutation operator of a permutation $\\pi $ defined through exchange of the atomic labels.",
"We have [see Eq.",
"(REF ) in Appendix A] $P_\\pi \\rho _A P_{\\pi ^{\\prime }}^\\dagger = (\\pm 1)^{p_\\pi + p_{\\pi ^{\\prime }}}\\rho _A\\quad \\forall \\,\\pi , \\pi ^{\\prime }$ where $p_\\pi $ is the parity of the permutation $\\pi $ (even or odd), and $(\\pm 1)^{p_\\pi }$ the phase factor picked up accordingly for bosons (upper sign) or fermions (lower sign).",
"Moreover, the Born approximation performed in [1] assumes that the initial atomic state is separable, i.e.",
"$\\rho _A(0)=\\rho _A^\\mathrm {in}(0)\\otimes \\rho _A^\\mathrm {ex}$ with $\\rho _A^\\mathrm {ex}$ the motional density operator at time $t=0$ .",
"This implies that both internal and external states are invariant under permutation of atoms [see Eq.",
"(REF ) in Appendix A], $\\begin{aligned}&P^\\mathrm {in}_\\pi \\rho ^\\mathrm {in}_A(0) P_{\\pi }^{\\mathrm {in}\\dagger }=\\rho ^\\mathrm {in}_A(0) \\quad \\forall \\, \\pi , \\\\[2pt]&P^\\mathrm {ex}_\\pi \\rho ^\\mathrm {ex}_A P_{\\pi }^{\\mathrm {ex}\\dagger }=\\rho ^\\mathrm {ex}_A \\quad \\forall \\, \\pi .\\end{aligned}$"
],
[
"Standard form",
"In the interaction picture, the master equation for the reduced density matrix $\\rho _A^\\mathrm {in}(t)$ describing the internal dynamics of the system $A$ composed of $N$ indistinguishable atoms takes the standard form [1] $\\begin{aligned}\\frac{d\\rho _A^\\mathrm {in}(t)}{dt} &= \\mathcal {L}\\left[\\rho _A^\\mathrm {in}(t) \\right] = - \\frac{i}{\\hbar }\\left[ H_\\mathrm {dd}, \\rho _A^\\mathrm {in}(t) \\right] + \\mathcal {D}\\left[\\rho _A^\\mathrm {in}(t)\\right],\\end{aligned}$ with the Liouvillian superoperator $\\mathcal {L}\\left[\\cdot \\right]$ involving the dipole-dipole Hamiltonian $H_\\mathrm {dd} = \\hbar \\Delta _{\\mathrm {dd}}\\, \\sum _{i \\ne j }^{N} \\sigma _+^{(i)}\\sigma _-^{(j)},$ with $\\Delta _{\\mathrm {dd}}$ the dipole-dipole shifts, and the dissipator $\\begin{aligned}\\mathcal {D}\\left[\\rho _A^\\mathrm {in}\\right] ={}& \\gamma \\, \\sum _{i\\ne j}^N \\left(\\sigma _-^{(j)}\\rho _A^\\mathrm {in}\\sigma _+^{(i)}-\\frac{1}{2}\\left\\lbrace \\sigma _+^{(i)}\\sigma _-^{(j)},\\rho _A^\\mathrm {in}\\right\\rbrace \\right) \\\\& + \\gamma _0\\, \\sum _{i=1}^N \\left(\\sigma _-^{(i)}\\rho _A^\\mathrm {in}\\sigma _+^{(i)}-\\frac{1}{2}\\left\\lbrace \\sigma _+^{(i)}\\sigma _-^{(i)},\\rho _A^\\mathrm {in}\\right\\rbrace \\right),\\end{aligned}$ with $\\gamma _0$ the single-atom spontaneous emission rate and $\\gamma $ the cooperative (off-diagonal) decay rates.",
"In Eqs.",
"(REF ) and (REF ), $\\sigma _{+}^{(j)} = (|e\\rangle \\langle g|)_j$ and $\\sigma _{-}^{(j)} =(|g\\rangle \\langle e|)_j$ are the ladder operators for atom $j$ with $|g\\rangle $ ($|e\\rangle $ ) the lower (upper) atomic level of energy $-\\hbar \\omega _0/2$ ($\\hbar \\omega _0/2$ ).",
"Note that in Eq.",
"(REF ), we do not consider diagonal terms ($i = j$ ) corresponding to the Lamb-shifts.",
"They can be discarded by means of a renormalization of the atomic frequency.",
"The fact that all off-diagonal ($i\\ne j$ ) decay rates are equal and all dipole-dipole shifts are equal for any pairs of atoms is merely a consequence of the indistinguishability of atoms (see Appendix B for a formal derivation).",
"The master equation (REF ) is valid for arbitrary motional quantum states and can be applied well-beyond the Lamb-Dicke regime.",
"All effects related to the quantization of the atomic motion are encoded in the values taken by the dipole-dipole shift $\\Delta _{\\mathrm {dd}}$ and the decay rate $\\gamma $ .",
"We give their exact expressions for arbitrary motional symmetric or antisymmetric states in Appendix B [see Eqs.",
"(REF ) and (REF )].",
"They depend not only on the average atomic positions (classical atomic positions) but also on their quantum fluctuations and correlations as described by the quantum motional (external) state $\\rho _A^\\mathrm {ex}$ of the atoms.",
"In particular, their values can strongly depend on the statistical nature (bosonic or fermionic) of the atoms.",
"In the next section, we give analytical expressions of $\\gamma $ for BECs in different regimes."
],
[
"Off-diagonal decay rates $\\gamma $ for Bose-Einstein condensates",
"We first consider the case of a non-interacting BEC confined in an isotropic harmonic trap at zero temperature.",
"In this case, all atoms are in the same motional state $\\phi (\\mathbf {r}) = e^{-|\\mathbf {r}|^2/4 \\ell ^2}/(\\sqrt{2\\pi }\\ell )^{3/2}$ with $\\ell = \\sqrt{\\hbar /2M\\Omega }$ the width of the spatial density, $M$ the atomic mass and $\\Omega $ the trap frequency.",
"The decay rate $\\gamma $ for this motional state follows from Eq.",
"(REF ) with $\\rho _1(\\mathbf {r})=|\\phi (\\mathbf {r})|^2$ and is given by $\\gamma = \\gamma _0\\, e^{-\\eta ^2}$ with $\\eta = k_0 \\ell $ the Lamb-Dicke parameter and $k_0$ the radiation wavenumber.",
"Since the size of a BEC typically lies in the range $10-10^3\\mu \\mathrm {m}$ [46], significant modifications of the decay rate $\\gamma $ should be observable for internal transitions in the visible and near-infrared domain.",
"We now consider the case of a BEC with strong repulsive interactions at zero temperature, for which the spatial density $\\rho _1(\\mathbf {r})$ is given in the Thomas-Fermi approximation by $\\rho _1(\\mathbf {r}) = {\\left\\lbrace \\begin{array}{ll} \\frac{M}{4\\pi \\hbar ^2 a}\\left[\\mu - V_\\mathrm {ext}(\\mathbf {r})\\right] & \\mbox{for }\\mu \\geqslant V_\\mathrm {ext}(\\mathbf {r}), \\\\[4pt]0 & \\mbox{for }\\mu < V_\\mathrm {ext}(\\mathbf {r}),\\end{array}\\right.",
"}$ where $a$ is the scattering length, $\\mu $ is the chemical potential and $V_\\mathrm {ext}(\\mathbf {r}) = M \\Omega ^2 r^2/2$ is the harmonic trap potential.",
"Inserting Eq.",
"(REF ) into Eq.",
"(REF ) yields after integration $\\gamma = 225 \\gamma _0 \\frac{\\left(3x \\cos x + (x^2 - 3) \\sin x\\right)^2}{x^{10}}$ where $x = \\eta \\@root 5 \\of {60 N a/\\ell }$ with $N$ the number of atoms in the BEC.",
"Figure REF shows Eq.",
"(REF ) as a function of $x$ .",
"In particular, when $x\\rightarrow 0$ (small recoil), $\\gamma $ tends to $\\gamma _0$ .",
"We finally study the transition from a thermal cloud to a non-interacting BEC by considering a gas of trapped bosonic atoms in thermal equilibrium at finite temperature $T$ .",
"The spatial density of the gas is given by [47] $\\rho _1(\\mathbf {r}) = \\frac{1}{N(2\\pi \\ell ^2)^{\\frac{3}{2}}}\\sum _{k = 1}^{\\infty } \\frac{z^k}{\\left( 1 - e^{-2 k \\beta \\hbar \\Omega } \\right)^{\\frac{3}{2}}}\\, e^{-\\frac{r^2}{2\\ell ^2} \\tanh \\left(\\frac{k\\beta \\hbar \\Omega }{2}\\right)},$ where $z=e^{\\beta \\mu }$ is the fugacity and $\\beta = 1/k_BT$ with $k_B$ the Boltzmann constant.",
"Inserting Eq.",
"(REF ) into Eq.",
"(REF ) yields after integration $\\gamma = \\frac{\\gamma _0}{N^2} \\Bigg [\\sum _{k =1}^{\\infty } \\frac{z^{k} e^{3 k \\beta \\hbar \\Omega }}{(1- e^{k \\beta \\hbar \\Omega })^3}\\, e^{-\\frac{\\eta ^2}{2}\\coth \\left(\\frac{k\\beta \\hbar \\Omega }{2}\\right) }\\Bigg ]^2.$ For $z\\rightarrow 0$ , Eq.",
"(REF ) tends to a thermal cloud profile $\\rho _1(\\mathbf {r})=e^{-(r/2R)^2}/(2\\pi R^2)^{3/2}$ with $R=\\sqrt{k_BT/m\\Omega ^2}$ and we get $\\gamma = \\gamma _0\\, e^{-k_0^2R^2}$ where $e^{-k_0^2R^2}$ is the Debye-Waller factor.",
"Figure REF shows Eq.",
"(REF ) as a function of $\\eta $ for $\\beta \\hbar \\Omega = 1/10$ and different values of the fugacity.",
"The curves $\\gamma (\\eta )$ switch gradually from Eq.",
"(REF ) for $z=0$ to Eq.",
"(REF ) for $z=1$ as the fugacity is increased, as a consequence of the formation of a condensed phase (see the inset of Fig.",
"REF ).",
"For fixed Lamb-Dicke parameter and $\\beta \\hbar \\Omega $ , $\\gamma $ increases monotonically with the fugacity.",
"Hence, cooperative effects will be more pronounced when all atoms are in the condensed phase (pure BEC).",
"Figure: Off-diagonal decay rate γ\\gamma as a function of the dimensionless variable x=η60Na/ℓ 5x = \\eta \\@root 5 \\of {60 N a/\\ell } for a BEC with strong repulsive interactions in the Thomas-Fermi limit.Figure: Off-diagonal decay rate γ\\gamma as a function of the Lamb-Dicke parameter η\\eta for a gas of trapped bosonic atoms at thermal equilibrium for βℏΩ=1/10\\beta \\hbar \\Omega = 1/10 and different values of the fugacity, from z=0z=0 (left) to z=1z=1 (right).",
"The inset shows spatial density profiles for the same parameters."
],
[
"Lindblad form",
"Before we discuss the Lindblad form of the master equation (REF ), let us note that the dipole-dipole Hamiltonian (REF ) can be rewritten in terms of collective spin operators only as $H_\\mathrm {dd} = \\hbar \\Delta _{\\mathrm {dd}} \\Big [ J_+ J_- - \\frac{1}{2} \\left( N \\mathbb {1} + 2 J_z \\right) \\Big ],$ where $\\mathbb {1}$ is the identity operator acting on the internal atomic states, $J_\\pm = \\sum _{j = 1}^N \\sigma _\\pm ^{(j)}$ are the collective spin ladder operators, and $J_z = \\frac{1}{2}\\sum _{j = 1}^N \\sigma _z^{(j)}$ with $\\sigma _z^{(j)} = (|e\\rangle \\langle e| - |g\\rangle \\langle g|)_j$ .",
"As for the dissipator (REF ), it also involves individual spin operators and can be rewritten as $\\mathcal {D}\\left[\\rho _A^\\mathrm {in}\\right] = \\gamma \\left( J_- \\rho _A^\\mathrm {in} J_+ - \\frac{1}{2} \\Big \\lbrace J_+ J_-, \\rho _A^\\mathrm {in} \\Big \\rbrace \\right) \\\\+ (\\gamma _0-\\gamma ) \\left( \\sum _{i=1}^N \\sigma _-^{(i)}\\rho _A^\\mathrm {in} \\sigma _+^{(i)}-\\frac{1}{4}\\Big \\lbrace N \\mathbb {1} + 2 J_z,\\rho _A^\\mathrm {in}\\Big \\rbrace \\right).$ The Lindblad form is obtained from the diagonalization of the $N \\times N$ matrix of decay rates ${\\gamma }= \\begin{pmatrix}\\gamma _0& \\gamma & \\ldots & \\gamma \\\\\\gamma & \\gamma _0 & \\ldots & \\gamma \\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\\\gamma & \\gamma & \\ldots & \\gamma _0 \\\\\\end{pmatrix},$ with $|\\gamma | \\leqslant \\gamma _0$ [1].",
"Associated with each eigenvector with non-zero eigenvalue $\\Gamma _\\ell $ is a Lindblad operator $F_\\ell $ .",
"Degenerate eigenvalues give rise to several Lindblad operators.",
"The matrix (REF ) has eigenvalues 1 = 0 + (N-1)N+, 2 = 0 - , with onefold and $(N-1)$ -fold degeneracy, respectively.",
"For the dynamics to be Markovian, the matrix ${\\gamma }$ has to be positive.",
"This implies that $-\\gamma _0/(N-1)\\leqslant \\gamma \\leqslant \\gamma _0$ and $0\\leqslant \\Delta \\gamma \\leqslant \\gamma _0\\,N/(N-1)$ .",
"An eigenvector $\\mathbf {v}_1$ with the largest eigenvalue ($\\Gamma _1$ ) is the vector with all components equal to $1/\\sqrt{N}$ .",
"The corresponding Lindblad operator is $F_1=J_-/\\sqrt{N}$ .",
"The remaining eigenvectors with degenerate eigenvalue $\\Delta \\gamma $ span the subspace $\\mathbb {C}^{N-1}$ orthogonal to $\\mathbf {v}_1$ and lead to Lindblad operators $F_\\ell $ .",
"The Lindblad form of the dissipator thus reads $\\mathcal {D}\\left[\\rho _A^\\mathrm {in}\\right] = \\frac{\\Gamma _1}{N} \\left( J_- \\rho _A^\\mathrm {in} J_+ - \\frac{1}{2} \\Big \\lbrace J_+ J_-, \\rho _A^\\mathrm {in} \\Big \\rbrace \\right) \\\\+ \\Delta \\gamma \\left( \\sum _{\\ell =2}^N F_\\ell \\rho _A^\\mathrm {in} F_\\ell ^\\dagger -\\frac{1}{2}\\Big \\lbrace F_\\ell ^\\dagger F_\\ell ,\\rho _A^\\mathrm {in}\\Big \\rbrace \\right).$ When $\\Delta \\gamma = 0$ ($\\gamma = \\gamma _0$ ), the system evolves under the sole action of the collective spin operator $J_-$ .",
"As a consequence, starting from an internal symmetric state, the dynamics is restricted to the symmetric subspace of dimension $N+1$ .",
"This is the superradiant regime [5].",
"When $\\Delta \\gamma > 0$ ($\\gamma < \\gamma _0$ ), all the additional Lindblad operators are involved in the dynamics.",
"The superoperator multiplying $\\Delta \\gamma $ in Eq.",
"(REF ) can be rewritten as $\\sum _{i=1}^N \\sigma _-^{(i)} \\,{\\cdot }\\, \\sigma _+^{(i)} - \\frac{J_- {\\cdot }\\, J_+}{N}-\\frac{1}{2} \\left\\lbrace \\frac{N \\mathbb {1}}{2} + J_z - \\frac{J_+J_-}{N}, {\\cdot } \\right\\rbrace .$ Hence, it cannot be expressed as a function of collective spin operators only.",
"However, it affects each atom identically.",
"Therefore, the Liouvillian superoperator does not distinguish between atoms and commutes with $P_\\pi $ for all permutations $\\pi $ , i.e.",
"$P_\\pi \\mathcal {L}[\\rho ] P^{\\dagger }_\\pi = \\mathcal {L}[P_\\pi \\rho P^{\\dagger }_\\pi ] \\quad \\forall \\, \\rho , \\forall \\,\\pi .$ It couples symmetric states with the broader class of permutation-invariant states.",
"These states, denoted hereafter by $\\rho _{_\\mathrm {PI}}$ , are states satisfying [48] $\\rho _{_\\mathrm {PI}}= P_\\pi \\rho _{_\\mathrm {PI}}P_\\pi ^{\\dagger } \\hspace{28.45274pt} \\forall \\, \\pi .$ They act on a subspace whose dimension grows only as $N^2$ [44], [45]."
],
[
"Master equation in the coupled spin basis",
"From now on, we will denote the internal density matrix $\\rho _A^\\mathrm {in}$ by $\\rho $ .",
"In this section, we express the master equation (REF ) in the coupled spin basis, which is particularly suited for the study of permutation-invariant states."
],
[
"Coupled spin basis",
"The Hilbert space $\\mathcal {H}$ of an ensemble of $N$ two-level systems admits the Wedderburn decomposition [49], [50], [51], [48] $\\mathcal {H} = (\\mathbb {C}^2)^{\\otimes N} \\simeq \\bigoplus _{J = J_\\mathrm {min}}^{N/2} \\mathcal {H}_J \\otimes \\mathcal {K}_J,$ with $J_\\mathrm {min} = 0$ for even $N$ and $1/2$ for odd $N$ .",
"In Eq.",
"(REF ), $\\mathcal {H}_J$ is the representation space of dimension $2J+1$ on which the irreducible representations of the group SU(2) act.",
"The number of degenerate irreducible representations with total angular momentum $J$ is equal to the dimension $d_N^J = \\frac{(2J+1)N!",
"}{(N/2 - J)!(N/2+J+1)!",
"}$ of the multiplicity space $\\mathcal {K}_J$ on which the irreducible representations of the symmetric group $S_{N}$ act.",
"The total Hilbert space $\\mathcal {H}$ is therefore spanned by the states $|J,M,k_J\\rangle \\equiv |J,M\\rangle \\otimes |k_J\\rangle $ , where $|J, M\\rangle $ are basis states of the subspaces $\\mathcal {H}_J$ ($J = J_\\mathrm {min}, \\cdots , N/2$ ; $M = -J, \\cdots , J$ ) and $|k_J\\rangle $ are basis states of the subspaces $\\mathcal {K}_J$ ($k_J= 1, \\cdots , d_N^J$ ).",
"The $2^N$ basis states $\\left\\lbrace |J,M,k_J\\rangle \\right\\rbrace $ form the coupled spin basis [45].",
"By construction, $|J,M,k_J\\rangle $ are spin-$J$ states satisfying $\\begin{aligned}& \\mathbf {J}^2 |J,M,k_J\\rangle = J(J+1) |J,M, k_J\\rangle , \\\\& J_z |J,M,k_J\\rangle = M |J,M, k_J\\rangle , \\\\& J_\\pm |J,M,k_J\\rangle = \\sqrt{(J\\mp M)(J\\pm M + 1)}|J,M\\pm 1, k_J\\rangle \\end{aligned}$ with $\\mathbf {J}^2 = J_x^2 + J_y^2 + J_z^2$ and $J_m =\\frac{1}{2}\\sum _{j = 1}^N \\sigma _m^{(j)}$ ($m = x,y,z$ ).",
"The degenerate structure of the decomposition (REF ) is depicted in the Bratteli diagram shown in Fig.",
"REF .",
"There are $d_N^J$ ways to obtain an angular momentum $J$ from the coupling of $N$ spins $1/2$ , each way being associated with a path in the Bratteli diagram.",
"The quantum number $k_J= 1,\\cdots , d_N^J$ enables one to distinguish these different paths.",
"Figure: Bratteli diagram representing the degeneracy structure d n J ×Jd_n^J \\times J of NN coupled spins 1/21/2.",
"The two colored paths leading to the same angular momentum J=0J = 0 correspond to two different values of the quantum number k J=0 k_{J=0}."
],
[
"Permutation-invariant states in the coupled spin basis",
"According to the Schur-Weyl duality [52], [53], any permutation $P_\\pi $ acts only on the multiplicity subspaces $\\mathcal {K}_J$ [see decomposition (REF )] and thus has the form $P_\\pi = \\bigoplus _{J = J_\\mathrm {min}}^{N/2} \\mathbb {1}_{\\mathcal {H}_J} \\otimes P_J(\\pi ),$ where $\\mathbb {1}_{\\mathcal {H}_J}$ is the identity operator on $\\mathcal {H}_J$ and $P_J(\\pi )$ is an irreducible representation of $S_N$ of dimension $d_N^J$ .",
"A permutation-invariant mixed state $\\rho _{_\\mathrm {PI}}$ commutes with $P_\\pi $ for any permutation $\\pi $ [see Eq.",
"(REF )] and thus admits in the coupled spin basis a block-diagonal form [48], $\\begin{aligned}\\rho _{_\\mathrm {PI}}&= \\bigoplus _{J = J_\\mathrm {min}}^{N/2} \\rho _J \\otimes \\mathbb {1}_{\\mathcal {K}_J}, \\\\\\end{aligned}$ where $\\mathbb {1}_{\\mathcal {K}_J}$ is the identity operator on $\\mathcal {K}_J$ and $\\begin{aligned}\\rho _J &= \\sum _{M,M^{\\prime } = -J}^J \\rho _{J}^{M,M^{\\prime }} \\, |J, M \\rangle \\langle J, M^{\\prime }|,\\end{aligned}$ with the density matrix elements $\\rho _{J}^{M,M^{\\prime }} \\equiv \\langle J,M,k_J| \\rho _{_\\mathrm {PI}}| J,M^{\\prime },k_J\\rangle \\quad \\forall \\, k_J.$ The block-diagonal form illustrated in Fig.",
"REF shows that $\\rho _{_\\mathrm {PI}}$ does not contain any coherences between blocks of different angular momenta $J$ .",
"For each $J$ , there are $d_N^J$ identical subblocks.",
"Since the matrix elements in these blocks do not depend on the label $k_J= 1, \\cdots , d_N^J$ , the number of real parameters needed to specify a permutation-invariant state $\\rho _{_\\mathrm {PI}}$ corresponds to the sum of the density matrix elements of all $\\rho _J$ $\\sum _{J = J_\\mathrm {min}}^{N/2} (2J+1)^2 = \\frac{1}{6}(N+1)(N+2)(N+3)=\\mathcal {O}(N^3).$ This number is much smaller than the total number ($2^{2N}-1$ ) of a general $N$ -atom density operator and highlights the convenience of this representation.",
"Figure: Block-diagonal form of the density matrix representing apermutation-invariant state in the coupled spin basis.",
"To each value of the angular momentum JJ corresponds d N J d_N^J subblocks of dimension (2J+1)×(2J+1)(2J+1)\\times (2J+1).",
"The block with J=N/2J = N/2 is unique and is spanned by symmetric states.Note that a symmetric mixed state $\\rho _S$ is just a particular case of permutation-invariant state (REF ) with $ \\rho _{J}^{M,M^{\\prime }} = 0$ for $J \\ne N/2$ .",
"The non-vanishing matrix elements of symmetric states all lie in the upper block $\\rho _{N/2}$ of dimension $N+1$ depicted in Fig.",
"REF ."
],
[
"Projection of the master equation in the coupled spin basis",
"Let us write the master equation (REF ) in terms of matrix elements of the density operator in the coupled spin basis.",
"By inserting Eq.",
"(REF ) into Eq.",
"(REF ) and upon using Eqs.",
"(REF ) and (REF ), we get $\\left[H_\\mathrm {dd}, \\rho (t)\\right]= \\hbar \\Delta _{\\mathrm {dd}} \\sum _{J = J_\\mathrm {min}}^{N/2} \\sum _{M,M^{\\prime } = -J}^{J} \\rho _{J}^{M,M^{\\prime }}(t) \\big (M^{\\prime 2}-M^2\\big )\\, |J, M\\rangle \\langle J, M^{\\prime }| \\otimes \\mathbb {1}_{\\mathcal {K}_J},$ $\\begin{aligned}\\mathcal {D}\\left[\\rho (t)\\right] = \\sum _{J = J_\\mathrm {min}}^{N/2} \\sum _{M,M^{\\prime } = -J}^{J} \\rho _{J}^{M,M^{\\prime }}(t) \\Bigg [ & \\, \\gamma \\, A_-^{J,M} A_-^{J,M^{\\prime }}\\, |J, M-1\\rangle \\langle J, M^{\\prime }-1|\\otimes \\mathbb {1}_{\\mathcal {K}_J} \\Bigg .",
"\\\\&\\, - \\frac{1}{2}\\left( \\gamma \\big (A_-^{J,M}\\big )^{2}+\\gamma \\big (A_-^{J,M^{\\prime }}\\big )^{2}+ \\Delta \\gamma (N+M+M^{\\prime }) \\right) |J, M\\rangle \\langle J, M^{\\prime }| \\otimes \\mathbb {1}_{\\mathcal {K}_J}\\\\&\\, + \\Delta \\gamma \\sum _{j=1}^N\\sigma _-^{(j)} \\Big [|J, M\\rangle \\langle J, M^{\\prime }|\\otimes \\mathbb {1}_{\\mathcal {K}_J}\\Big ] \\sigma _+^{(j)} \\,\\Bigg ].\\end{aligned}$ The last term in Eq.",
"(REF ) cannot be written solely in terms of collective spin operators but affects each atom identically.",
"It has been evaluated in [44] and reads $\\begin{aligned}\\sum _{j=1}^N\\sigma _-^{(j)} \\Big [|J, M\\rangle \\langle J, M^{\\prime }|\\otimes \\mathbb {1}_{\\mathcal {K}_J}\\Big ] \\,\\sigma _+^{(j)} = & \\, \\frac{1}{2J}A_-^{J,M}A_-^{J,M^{\\prime }}\\left( 1 + \\frac{\\alpha _N^{J+1}(2J+1)}{d_{N}^J (J+1)} \\right) \\,|J, M-1\\rangle \\langle J, M^{\\prime }-1|\\otimes \\mathbb {1}_{\\mathcal {K}_J} \\\\& \\, + \\frac{B_-^{J,M}B_-^{J,M^{\\prime }} \\alpha _N^J}{2 J d_N^{J-1}} \\,|J-1, M-1\\rangle \\langle J-1, M^{\\prime }-1|\\otimes \\mathbb {1}_{\\mathcal {K}_{J-1}} \\\\& \\, + \\frac{D_-^{J,M}D_-^{J,M^{\\prime }} \\alpha _N^{J+1}}{2 (J +1) d_N^{J+1}} \\,|J+1, M-1\\rangle \\langle J+1, M^{\\prime }-1|\\otimes \\mathbb {1}_{\\mathcal {K}_{J+1}},\\end{aligned}$ where AJ,M = (JM)(JM+1), B-J,M = -(J+M)(J+M-1) , D-J,M = (J-M+1)(J-M+2) , and $\\alpha _{N}^{J} = \\sum _{J^{\\prime } = J}^{N/2} d_{N}^{J^{\\prime }}.$ Equation (REF ) shows that dipole-dipole interactions do not couple blocks of different angular momentum $J$ , but couple non-diagonal ($M \\ne M^{\\prime }$ ) density matrix elements within a block.",
"The term (REF ) describes transitions giving rise to energy loss due to photon emissions, since it reduces the value of the quantum numbers $M$ and $M^{\\prime }$ by one unit.",
"Such transitions from a block of angular momentum $J$ occur either within a same block or in neighboring blocks of angular momentum $J \\pm 1$ .",
"The former preserve the symmetry of the state while the latter modify it.",
"By injecting Eqs.",
"(REF ) and (REF ) into the master equation (REF ) and projecting onto the states $|J, M, k_J\\rangle $ , we get a system of $\\mathcal {O}(N^3)$ [see Eq.",
"(REF )] differential equations for the density matrix elements $\\rho _{J}^{MM^{\\prime }}(t)$ that reads $\\frac{d\\rho _{J}^{M,M^{\\prime }}(t)}{dt} = - \\Gamma _{{}_J^{M,M^{\\prime }}}^{(1)} \\,\\rho _{J}^{M,M^{\\prime }}(t)+ \\Gamma _{{}_J^{M+1,M^{\\prime }+1}}^{(2)} \\, \\rho _{J}^{M+1,M^{\\prime }+1}(t)+ \\Gamma _{{}_{J+1}^{M+1,M^{\\prime }+1}}^{(3)} \\,\\rho _{J+1}^{M+1,M^{\\prime }+1}(t)+ \\Gamma _{{}_{J-1}^{M+1,M^{\\prime }+1}}^{(4)} \\,\\rho _{J-1}^{M+1,M^{\\prime }+1}(t),$ with $\\begin{aligned}& \\Gamma _{{}_J^{M,M^{\\prime }}}^{(1)} = i \\Delta _{\\mathrm {dd}}(M^{\\prime 2}-M^2) + \\frac{\\gamma }{2} \\Big [\\big (A_-^{J,M}\\big )^{2}+\\big (A_-^{J,M^{\\prime }}\\big )^{2}\\Big ] + \\frac{\\Delta \\gamma }{2}(N+M+M^{\\prime }), \\\\& \\Gamma _{{}_J^{M+1,M^{\\prime }+1}}^{(2)} = A_{+}^{J,M} A_{+}^{J,M^{\\prime }} \\left[ \\gamma \\, + \\frac{ \\Delta \\gamma }{2J}\\left( 1 + \\frac{\\alpha _N^{J+1}(2J+1)}{d_{N}^J (J+1)} \\right) \\right],\\\\& \\Gamma _{{}_{J+1}^{M+1,M^{\\prime }+1}}^{(3)} = \\Delta \\gamma \\, \\frac{B_-^{J+1,M+1}B_-^{J+1,M^{\\prime }+1} \\alpha _N^{J+1}}{2 (J+1) d_N^{J}},\\\\& \\Gamma _{{}_{J-1}^{M+1,M^{\\prime }+1}}^{(4)} = \\Delta \\gamma \\, \\frac{D_-^{J-1,M+1}D_-^{J-1,M^{\\prime }+1} \\alpha _N^{J}}{2 J d_N^{J}}.\\end{aligned}$ Equations (REF ) for the transition rates show that the populations $\\rho _{J}^{M,M}$ are decoupled from the coherences $\\rho _{J}^{M,M^{\\prime }}$ ($M \\ne M^{\\prime }$ ).",
"More specifically, coherences specified by $M,M^{\\prime }$ are only coupled to coherences with the same difference $M-M^{\\prime }$ , and populations $\\rho _{J}^{M,M}$ can only feed populations $\\rho _{J^{\\prime }}^{M^{\\prime },M^{\\prime }}$ with $M^{\\prime } \\leqslant M$ and $J^{\\prime } \\geqslant (J-M)/2$ .",
"This can be seen from Eqs.",
"(REF ) and (REF ) and Fig.",
"REF , which shows the couplings between the populations together with the corresponding rates.",
"Indeed, in Eq.",
"(REF ), the derivative of $\\rho _{J^{\\prime }}^{M^{\\prime },M^{\\prime }}$ depends only on density matrix elements with equal or larger quantum numbers $M$ , which implies that starting from a state with a given $M$ , only states with $M^{\\prime } \\leqslant M$ can be populated during the dynamics.",
"As for the quantum number $J^{\\prime }$ , it can decrease or increase through the channels with rates $\\Gamma ^{(3)}$ and $\\Gamma ^{(4)}$ (see Fig.",
"REF ).",
"However, it cannot decrease indefinitely.",
"Consider the initial state $|J,M\\rangle $ : All states $|J-Q, M-Q\\rangle $ with positive half-integer $Q$ can be populated provided that $J-Q\\geqslant J_\\mathrm {min}$ and $J-Q\\geqslant M-Q\\geqslant -(J-Q)$ .",
"The first inequality of the latter expression is always satisfied since $M \\leqslant J$ , but the second inequality imposes $Q \\leqslant (J+M)/2$ .",
"This in turn implies the minimal value $(J-M)/2$ for the quantum number $J^{\\prime } \\equiv J-Q$ .",
"Figure: Couplings between the populations ρ J M,M \\rho _{J}^{M,M} (small closed circles) lying in the different blocks ρ J \\rho _J of angular momentum J=N/2,N/2-1,N/2-2,⋯J=N/2,N/2-1,N/2-2,\\cdots (gray squares), as described by Eq. ().",
"The large closed and open circles at the bottom right of each block are the populations ρ J -J,-J \\rho _{J}^{-J,-J} corresponding to subradiant states when Δγ=0\\Delta \\gamma = 0 (see Sec. ).",
"The arrows show the different couplings between populations characterized by the rates Γ (r) \\Gamma ^{(r)} (with r=1,2,3,4r = 1, 2,3,4 and where the subscripts have been dropped for the sake of clarity).",
"The rates Γ (1) \\Gamma ^{(1)} and Γ (2) \\Gamma ^{(2)} are related to transitions within a block while the rates Γ (3) \\Gamma ^{(3)} and Γ (4) \\Gamma ^{(4)} (proportional to Δγ\\Delta \\gamma ) are related to transitions between different blocks ρ J \\rho _J.",
"This diagram shows that starting with the initial condition ρ J M,M (0)=1\\rho _{J}^{M,M}(0) = 1, only populations ρ J ' M ' ,M ' \\rho _{J^{\\prime }}^{M^{\\prime },M^{\\prime }} with M ' ⩽MM^{\\prime } \\leqslant M and J ' ⩾(J-M)/2J^{\\prime } \\geqslant (J-M)/2 can be non-zero during the radiative decay.",
"When Δγ>0\\Delta \\gamma > 0, Γ (3) \\Gamma ^{(3)} and Γ (4) \\Gamma ^{(4)} are non-zero and the state |N/2,-N/2〉|N/2,-N/2\\rangle (large closed circles) is the only stationary state for any initial conditions."
],
[
"Solutions of the master equation",
"The solutions of the master equation for indinstinguishable atoms only involve the rates $\\gamma $ , $\\Delta \\gamma = \\gamma _0 - \\gamma $ , and $\\Delta _\\mathrm {dd}$ .",
"In this section, we compute numerical solutions up to 30 atoms for different values of these rates.",
"The solutions allow us to study the modifications of super- and subradiance arising from a proper quantum treatment of the atomic motion.",
"In addition, we obtain analytical results for large $N$ by applying a mean-field approximation.",
"In order to quantify the modifications in the release of energy from the atomic system, we calculate the normalized radiated energy rate [5] $I(t) = -\\frac{d}{dt}\\langle J_z \\rangle (t).$ For permutation-invariant states (REF ), Eq.",
"(REF ) can be expressed in terms of the populations $\\rho _{J}^{M,M}$ as $I(t) = -\\sum _{J = J_\\mathrm {min}}^{N/2} d_{N}^{J} \\sum _{M = -J}^{J} M \\frac{d \\rho _{J}^{M,M}(t)}{dt}.$ By inserting Eq.",
"(REF ) into (REF ) and after algebraic manipulations, we get $I(t) = \\sum _{J = J_\\mathrm {min}}^{N/2} d_{N}^{J} \\sum _{M = -J}^{J} c_{J}^{M} \\, \\rho _{J}^{M,M}(t)$ with positive coefficients $c_{J}^{M}$ given by $c_{J}^{M} = \\big (J+M\\big )\\big (J-M+1\\big ) \\, \\gamma + \\left(M + \\frac{N}{2}\\right) \\Delta \\gamma .$"
],
[
"Superradiance",
"The superradiance phenomenon is usually observed when the atoms are initially in a symmetric internal state $|N/2, M\\rangle $ .",
"In this section, we choose for initial state the symmetric state $|N/2,N/2\\rangle \\equiv |e,e,\\cdots ,e\\rangle $ .",
"This choice allows us to study the superradiant radiative cascade starting from the highest energy level.",
"For two atoms, a simple analytical solution of the master equation can be obtained and is given in Appendix C. For the initial condition $\\rho (0) = |1,1\\rangle \\langle 1,1|\\equiv |e,e\\rangle \\langle e,e|$ , the radiated energy rate (REF ) resulting from the solution (REF ) given in the Appendix reads $\\begin{aligned}I(t) = \\frac{e^{-2(\\gamma + \\Delta \\gamma )t}}{(2 \\gamma + \\Delta \\gamma ) \\Delta \\gamma } \\bigg [ & \\, (2 \\gamma + \\Delta \\gamma )^2 \\Delta \\gamma + \\Delta \\gamma ^2 (2 \\gamma + \\Delta \\gamma ) \\\\& \\,+ (2 \\gamma + \\Delta \\gamma )^3 \\Big (e^{\\Delta \\gamma t} - 1\\Big ) \\\\& \\, + \\Delta \\gamma ^3 \\Big (e^{(2 \\gamma + \\Delta \\gamma ) t} - 1\\Big ) \\bigg ].\\end{aligned}$ In the absence of quantum fluctuations of the atomic positions and for colocated atoms [1], i.e.",
"when $\\Delta \\gamma =0$ ($\\gamma =\\gamma _0$ ), pure superradiance occurs during which all symmetric Dicke states $|1,1\\rangle ,|1,0\\rangle $ and $|1,-1\\rangle $ are gradually populated.",
"In this case, Eq.",
"(REF ) reduces to the superradiant radiated energy rate $I(t) = 2 \\gamma _0\\, e^{-2 \\gamma _0 t} (1 + 2 \\gamma _0 t).$ When $\\Delta \\gamma > 0$ , the singlet state $|0,0\\rangle $ is coupled to the symmetric Dicke states and the radiated energy rate is reduced at small times as can be seen in Fig.",
"REF .",
"When $\\gamma =0$ , $\\Delta \\gamma =\\gamma _0$ and Eq.",
"(REF ) reduces to the pure exponential decay characteristic of individual spontaneous emission $I(t) = 2 \\gamma _0\\, e^{- \\gamma _0 t}.",
"$ Figure: Radiated energy rate for two atoms in the initial state |ee〉|ee\\rangle as a function of time for γ=γ 0 \\gamma =\\gamma _0 (blue solid curve), γ=3γ 0 /4\\gamma =3\\gamma _0/4 (green dottedcurve), γ=0\\gamma =0 (orange dashed curve).",
"The blue solid curvecorresponds to pure superradiance [Eq.",
"()], the orangedashed curve to independent spontaneous emission[Eq.",
"()] and the green dotted curve to alteredsuperradiance [Eq. ()].",
"The inset shows the radiated energy rate for the initial state |0,0〉|0,0\\rangle and the same parameters."
],
[
"Numerical results for $N > 2$",
"In this section, we solve numerically the set of coupled equations (REF ) for the initial condition $\\rho (0) = \\rho _{N/2}^{N/2,N/2} = |e,e,\\cdots ,e\\rangle \\langle e,e,\\cdots ,e|$ and for different values of $\\Delta \\gamma $ .",
"We then compute the radiated energy rate (REF ).",
"Figure REF shows $I(t)$ as a function of time from 3 to 30 atoms, where each panel corresponds to a different value of $\\Delta \\gamma $ .",
"For $\\Delta \\gamma =0$ , pure superradiance occurs (first panel).",
"For $\\Delta \\gamma =\\gamma _0$ , the radiated energy rate decreases according to $I(t) = N \\gamma _0\\, e^{-\\gamma _0 t}$ , as is typical of individual spontaneous emission (last panel).",
"The middle panels show the crossover between these two regimes.",
"Figure REF is a three-dimensional plot of $I(t)$ showing the crossover for $N=30$ .",
"Figure: Radiated energy rate as a function of time for different values of Δγ=γ 0 -γ\\Delta \\gamma = \\gamma _0 - \\gamma (corresponding to the different panels) and different number of atoms (N=3,⋯,30N=3,\\cdots ,30 from bottom to top on the left of each panel).",
"The case Δγ=0\\Delta \\gamma = 0 (pure superradiance) is illustrated in the first panel while the case Δγ=γ 0 \\Delta \\gamma = \\gamma _0 corresponding to independent spontaneous emissions is illustrated in the last panel.Figure: Radiated energy rate as a function of time and Δγ\\Delta \\gamma for N=30N = 30 atoms.",
"The superradiant pulse progressively disappears as Δγ\\Delta \\gamma increases from 0 to Δγ * =0.817γ 0 \\Delta \\gamma ^*=0.817\\,\\gamma _0.",
"For Δγ=γ 0 \\Delta \\gamma = \\gamma _0, I(t)I(t) decays exponentially at a rate γ 0 \\gamma _0.",
"The white line indicates the location of the maximum of the pulse.In order to characterize the superradiant pulse in the intermediate regime, we compute its relative height $A_I$ and the time $t_I$ at which its maximum occurs.",
"These quantities are defined as $A_I = \\underset{t}{\\mathrm {max}}[I(t)] - I(0) = I(t_I)-N \\gamma _0,$ Our results, displayed in Fig.",
"REF , show that the height $A_I$ of the pulse is maximal for $\\Delta \\gamma =0$ , decreases monotonically with $\\Delta \\gamma $ and vanishes for $\\Delta \\gamma \\geqslant \\Delta \\gamma ^{*}$ , where the critical value $\\Delta \\gamma ^{*}$ depends only on the number of atoms.",
"The decrease as a function of $\\Delta \\gamma $ is more and more linear as $N$ increases.",
"We explain this behavior in the next subsection on the basis of a mean-field approximation.",
"For sufficiently large $N$ , the time $t_I$ at which the maximum occurs increases as a function of $\\Delta \\gamma $ before dropping to zero at $\\Delta \\gamma =\\Delta \\gamma ^*$ .",
"The critical value $\\Delta \\gamma ^{*}$ increases as the number of atoms increases, as shown in Fig.",
"REF , and tends to $\\gamma _0$ for $N \\rightarrow \\infty $ .",
"This means that for a fixed value of $\\Delta \\gamma $ , superradiance can always be observed for a sufficiently large number of atoms.",
"Indeed, the derivative of the radiated energy rate (REF ) reads $\\frac{dI(t)}{dt} = \\sum _{J = J_\\mathrm {min}}^{N/2} d_{N}^{J} \\sum _{M = -J}^{J} \\tilde{c}_{J}^{M} \\, \\rho _{J}^{M,M}(t)$ with $\\tilde{c}_{J}^{M} = 2 \\big (J+M\\big )\\big (J-M+1\\big ) \\big [(M-1)\\gamma - \\Delta \\gamma \\big ] \\gamma \\\\ - \\left(M + \\frac{N}{2}\\right) \\Delta \\gamma ^2.$ If the derivative of the radiated energy rate at initial time is strictly positive, a non-zero superradiant pulse height ($A_{I} > 0$ ) is always obtained.",
"For an initial fully excited state, this sufficient condition in terms of the critical value $\\Delta \\gamma ^{*}(N)$ reads $\\Delta \\gamma < \\gamma _0\\left(1 - \\frac{1}{\\sqrt{N-1}}\\right) \\equiv \\Delta \\gamma ^{*}(N).$ As shown in Fig.",
"REF , our numerical results are in excellent agreement with Eq.",
"(REF ).",
"Figure: Shown on top is the height A I A_I of the superradiant pulse rescaled by N 2 γ 0 /4N^2\\gamma _0/4 as a function of Δγ=γ 0 -γ\\Delta \\gamma = \\gamma _0 - \\gamma for N=3,⋯,30N = 3, \\cdots , 30 (from left to right).",
"The bottom shows the delay time t I (Δγ)t_I(\\Delta \\gamma ) after which the radiated intensity attains a maximum, rescaled by t I (0)t_I(0).",
"The dashed green curves correspond to the mean-field results [Eqs.",
"() and ()].Figure: Critical value Δγ * \\Delta \\gamma ^* at which the superradiant pulse height A I A_I drops to zero and remains zero for Δγ>Δγ * \\Delta \\gamma >\\Delta \\gamma ^*, plotted as a function of the number of atoms.",
"The circles show the values extracted from numerical computations.",
"The solid line shows the analytical prediction given by Eq.",
"()."
],
[
"Mean field approach",
"When the number of atoms is large, a mean-field approximation can be made [54], [55] that assumes an internal state of the form $\\rho (t) \\approx \\sigma (t) \\otimes \\cdots \\otimes \\sigma (t).$ In the mean-field approximation, all atoms lie in the same quantum state $\\sigma (t)$ .",
"The global state $\\rho (t)$ is permutation invariant at any time $t$ but not necessarily symmetric.",
"However, when $\\sigma (t)$ is a pure state, $\\rho (t)$ is symmetric and has only components in the block of maximal angular momentum $J=N/2$ .",
"When $\\Delta \\gamma =0$ , the superradiant cascade takes place only in the block $J=N/2$ and $\\sigma (t)$ is usually chosen pure [54].",
"When $\\Delta \\gamma \\ne 0$ , the ratio between the transition rates within the block $J = N/2$ and the neighboring block $J = N/2 - 1$ for the emission of the $s$ -th photon with $s\\gg 1$ is much larger than 1, i.e.",
"$\\frac{\\Gamma _{{}_{N/2}^{N/2 - s + 1,N/2 - s + 1}}^{(2)} }{\\Gamma _{{}_{N/2}^{N/2 - s+1,N/2 - s+1}}^{(3)}} \\approx s\\frac{\\gamma }{\\Delta \\gamma }\\gg 1.$ Hence, during the main part of the radiative cascade (when $s$ is large), the dynamics takes place essentially in the block $J=N/2$ , so that we also choose $\\sigma (t)$ to be a pure state.",
"By inserting Eq.",
"(REF ) into the master equation (REF ) and by tracing over $N-1$ atoms, we get the following non-linear equation for $\\sigma (t)$ $\\frac{d\\sigma (t)}{dt} = -\\frac{i}{\\hbar } \\Big [ V_H\\left[\\sigma (t) \\right] + V_D\\left[ \\sigma (t) \\right] , \\sigma (t) \\Big ] + \\mathcal {D}_\\mathrm {se}\\left[ \\sigma (t) \\right].$ In Eq.",
"(REF ), $V_H$ is the non-linear Hartree potential (proportional to the dipole-dipole shift $\\Delta _{\\mathrm {dd}}$ ) $V_H\\left[ \\sigma (t) \\right] = \\hbar \\Delta _{\\mathrm {dd}} (N-1) \\, \\big (\\langle \\sigma _+\\rangle \\sigma _- +\\langle \\sigma _-\\rangle \\sigma _+ \\big )$ where $\\langle \\, {\\cdot }\\,\\rangle =\\mathrm {Tr}[{\\cdot }\\,\\sigma (t)]$ , $V_D$ is the non-linear dissipative potential $V_D\\left[ \\sigma (t) \\right] = i \\hbar \\gamma \\frac{N-1}{2} \\, \\big (\\langle \\sigma _+\\rangle \\sigma _- - \\langle \\sigma _-\\rangle \\sigma _+ \\big ),$ and $ \\mathcal {D}_\\mathrm {se}$ is the single-atom dissipator accounting for spontaneous emission $\\mathcal {D}_\\mathrm {se}\\left[ \\sigma (t) \\right] = \\gamma _0 \\left( \\sigma _- \\sigma (t) \\sigma _+ - \\frac{1}{2} \\left\\lbrace \\sigma _+ \\sigma _-, \\sigma (t) \\right\\rbrace \\right).$ Equation (REF ) cannot be solved analytically because of the presence of the term (REF ).",
"However, as $N$ gets large, this one can be neglected in comparison to (REF ) and (REF ) provided $\\gamma \\ne 0$ and $N-1$ can be replaced by $N$ .",
"Equation (REF ) can then be related for pure states $\\sigma _\\psi (t)=|\\psi (t)\\rangle \\langle \\psi (t)|$ to a non-linear Schrödinger equation for $|\\psi (t)\\rangle $ of the form (in the interaction picture) [54], [56], [57] $\\frac{d |\\psi (t)\\rangle }{dt} = -\\frac{i}{\\hbar } \\big (V_H[\\sigma _\\psi (t)] + V_D[\\sigma _\\psi (t)]\\big ) |\\psi (t)\\rangle .$ As in [54], we parametrize the state $|\\psi (t)\\rangle $ by $|\\psi (t)\\rangle = \\sqrt{p(t)} \\, e^{i \\theta (t)} |e\\rangle + \\sqrt{1-p(t)}\\, |g\\rangle $ with $p(t)= |\\langle e | \\psi (t)\\rangle |^2$ the mean number of atoms in the excited state.",
"By inserting Eq.",
"(REF ) into (REF ) we get dp(t)dt = - N p(t) [1-p(t)], d(t)dt = - N dd [1-p(t)].",
"With the conditions $p(t_I) = 1/2$ and $\\theta (0) = \\theta _0$ , the system of equations (REF ) has the unique solution p(t) = 11+eN (t-tI), (t) = 0 + dd [p(t)(1 + e-N tI)], where $t_I$ corresponds to the time at which half the photons have been emitted and is identified with the delay time of superradiance [5].",
"The phase $\\theta (t)$ depend on both the dipole-dipole shift $\\Delta _{\\mathrm {dd}}$ and the decay rate $\\gamma $ , while the population $p(t)$ depends only on the rate $\\gamma $ .",
"In other words, the dissipative dynamics is not affected by the dipole-dipole shift.",
"The radiated energy rate in the mean-field approximation reads $I_{\\mathrm {mf}}(t) = -N \\frac{d p(t)}{dt} = \\frac{N^2 \\gamma }{4} \\cosh ^{-2}{\\left[ \\frac{N \\gamma }{2} (t-t_I)\\right]},$ and is of the same form as the pure superradiant pulse for colocated atoms [54], [5], but with $\\gamma _0$ replaced by $\\gamma $ and a priori a different delay time $t_I$ .",
"The quantum fluctuations of the atomic positions modify the value of $\\gamma $ as compared to $\\gamma _0$ , and thus the shape of the superradiant pulse, which is however always present except for $\\gamma =0$ .",
"The height $A_{I,\\mathrm {mf}}$ of the pulse (REF ), given by $A_{I,\\mathrm {mf}} = \\frac{N^2 \\gamma }{4} = \\frac{N^2 \\gamma _0}{4}\\left(1-\\frac{\\Delta \\gamma }{\\gamma _0}\\right),$ is always smaller than the height $N^2 \\gamma _0/4$ of the pure superradiant pulse since $\\gamma \\leqslant \\gamma _0$ .",
"Equation (REF ) is compared with numerical simulations in Fig.",
"REF (green dashed curve, top panel).",
"As for the delay time $t_I$ , it cannot be evaluated precisely in the mean-field approach.",
"Nevertheless, an approximation can be obtained in the limit $N\\rightarrow \\infty $ and for $\\gamma \\ne 0$ by evaluating the sum of the typical times between each photon emission [5].",
"We find $t_I \\sim \\frac{\\ln N}{N \\gamma }$ which corresponds to the result of Gross and Haroche [5] but with $\\gamma _0$ replaced by $\\gamma $ .",
"The ratio between the delay time for $\\Delta \\gamma \\ne 0$ and the one for $\\Delta \\gamma = 0$ (pure superradiance) is thus given by $\\frac{t_I(\\Delta \\gamma )}{t_I(0)} = \\frac{\\gamma _0}{\\gamma } = \\frac{1}{1-(\\Delta \\gamma /\\gamma _0)}.$ It is always larger than 1 and increases with $\\Delta \\gamma $ , meaning that the larger $\\Delta \\gamma $ is, the longer it takes before the radiated energy rate attains a maximum.",
"Equation (REF ) is compared with numerical simulations in Fig.",
"REF (green dashed curve, bottom panel)."
],
[
"Subradiance",
"Subradiant states are states for which the radiated energy rate decays slowly as compared to the one corresponding to independent spontaneous emission.",
"Dark (or decoherence-free) states are a particular class of subradiant states for which the radiated energy rate (REF ) vanishes.",
"According to Eq.",
"(REF ), their only non-zero populations $\\rho _J^{M,M}$ are those for which $J$ and $M$ are such that $c_{J}^{M}=0$ .",
"When $\\Delta \\gamma = 0$ , the condition $c_{J}^{M}=0$ is satisfied for $M = -J$ [58].",
"As a consequence, all states $|J,-J\\rangle $ (in number $\\alpha _N^{J_\\mathrm {min}}$ ; see, e.g., [59]) are dark states.",
"When $\\Delta \\gamma > 0$ , the only dark state is obtained for $J = M = N/2$ and corresponds to the ground state $|g,\\cdots ,g\\rangle $ .",
"In the following, we study the time evolution of the state $|J_0,-J_0\\rangle $ (with $J_0\\in \\lbrace J_{\\mathrm {min}},\\cdots ,N/2\\rbrace $ ) when $\\Delta \\gamma > 0$ .",
"The initial non-zero matrix element $\\rho _{J_0}^{-J_0,-J_0}$ is only coupled to the matrix elements $\\rho _{J}^{-J,-J}$ with higher angular momenta $J$ , i.e.",
"$J_0 \\leqslant J\\leqslant N/2$ , as can be seen from Fig.",
"REF .",
"The system will thus gradually populate all states $|J, -J\\rangle $ with $J>J_0$ before finally reaching the ground state $|N/2, -N/2\\rangle $ .",
"The populations $\\rho _{J}^{-J,-J}$ are obtained from Eq.",
"(REF ), which simplifies to $\\begin{aligned}\\frac{d\\rho _{J}^{-J,-J}(t)}{dt} = & -\\Delta \\gamma \\Bigg [ \\, \\left(\\frac{N}{2} - J\\right)\\rho _{J}^{-J,-J}(t) \\\\&\\, - \\frac{d_N^{J-1}}{d_{N}^{J}}\\left( \\frac{N}{2} - J+1\\right) \\rho _{J-1}^{-J+1,-J+1}(t) \\Bigg ]\\end{aligned}$ and admits the solution $\\rho _{J}^{-J,-J}(t) = \\frac{\\left(\\frac{N}{2}-J_0\\right)!\\;e^{-\\Delta \\gamma \\left( \\frac{N}{2} - J_0 \\right) t}}{d_{N}^{J}\\left(\\frac{N}{2}-J\\right)!\\left(J-J_0 \\right)!}",
"\\left(e^{\\Delta \\gamma \\, t} - 1 \\right)^{J-J_0}.$ Inserting this expression into Eq.",
"(REF ) for the radiated energy rate yields after some calculations $I(t) = \\Delta \\gamma \\left( \\frac{N}{2} - J_0 \\right) e^{-\\Delta \\gamma t}.$ Hence, $I(t)$ decreases exponentially regardless of the initial angular momentum $J_0$ , except for the case $J_0 = N/2$ (ground state) for which $I(t)=0$ at any time $t$ .",
"We also see that the states $|J_0,-J_0\\rangle $ are subradiant, since the emission rate $\\Delta \\gamma $ is always smaller than $\\gamma _0$ , the single-atom spontaneous emission rate."
],
[
"Conclusions",
"We have investigated superradiance and subradiance from indistinguishable atoms with quantized motional state based on the master equation derived in [1].",
"The indistinguishability of the atoms implies that for an initially factorized state between the external (center-of-mass) and internal degrees of freedom the motional state must be invariant under permutation of atoms.",
"As a consequence, the whole dynamics is parametrized only by three real numbers, namely the diagonal $\\gamma _0$ and off-diagonal $\\gamma \\leqslant \\gamma _0$ decay rates, and a dipole-dipole shift $\\Delta _{\\mathrm {dd}}$ that is identical for all atoms.",
"All three parameters can be “quantum-programmed” by appropriate choice of the motional state of the atoms.",
"For $\\gamma =\\gamma _0$ standard superradiance results, whereas for $\\gamma \\rightarrow 0$ individual spontaneous emission of the atoms prevails.",
"A continuous transition between these two extreme cases can be achieved.",
"A superradiant enhancement of the emitted intensity is always observed for $\\gamma >\\gamma _0/\\sqrt{N-1}$ where $N$ is the number of atoms.",
"All non-trivial dark states (i.e.",
"states other than the ground state with strictly vanishing emission of radiation) are immediately lost as soon as $\\gamma <\\gamma _0$ .",
"This implies that for harmonically trapped atoms, exact decoherence free subspaces that protect against spontaneous emission through destructive interference of individual spontaneous emission amplitudes exist only in the limit of classically localized atoms, i.e.",
"atoms in infinitely steep traps.",
"Finally, we showed that the states that are dark when $\\gamma =\\gamma _0$ are only subradiant when $\\gamma <\\gamma _0$ .",
"FD would like to thank the FRS-FNRS (Belgium) for financial support.",
"FD is a FRIA (Belgium) grant holder of the Fonds de la Recherche Scientifique-FNRS (Belgium)."
],
[
"Appendix A : Symmetry of density matrix under permutation of indistinguishable atoms",
"In this Appendix, we give general properties under permutation of atoms of the (reduced) density matrices describing the states of indistinguishable atoms.",
"Consider a set of $N$ indistinguishable atoms (bosons or fermions) with internal and external degrees of freedom.",
"We define the orthonormal basis vectors as $|{\\nu }\\rangle |{\\phi }\\rangle \\equiv |\\nu _1,\\ldots ,\\nu _N\\rangle |\\phi _1,\\ldots ,\\phi _N\\rangle $ , where $|\\nu _j\\rangle $ (resp.",
"$|\\phi _j\\rangle $ ) are the internal (resp.",
"external) orthonormal basis states of the particle $j$ .",
"The permutation operator $P_\\pi $ corresponding to the permutation $\\pi $ is defined through exchange of the particle labels in the basis states, i.e.",
"$P_\\pi |{\\nu }\\rangle |{\\phi }\\rangle =|\\nu _{\\pi _1},\\ldots ,\\nu _{\\pi _N}\\rangle |\\phi _{\\pi _1},\\ldots ,\\phi _{\\pi _N}\\rangle \\equiv |{\\nu }_\\pi \\rangle |{\\phi }_\\pi \\rangle \\,.$ We have $P_\\pi = P_\\pi ^\\mathrm {in}\\otimes P_\\pi ^\\mathrm {ex}$ , where $P_\\pi ^\\mathrm {in}$ and $P_\\pi ^\\mathrm {ex}$ are such that $P_\\pi ^\\mathrm {in} |{\\nu }\\rangle = |{\\nu }_\\pi \\rangle $ and $P_\\pi ^\\mathrm {ex} |{\\phi }\\rangle = |{\\phi }_\\pi \\rangle $ .",
"An arbitrary pure state $|\\psi \\rangle $ of the full system can be written as $|\\psi \\rangle =\\sum _{{\\nu }{\\phi }}\\alpha _{{\\nu }{\\phi }}|{\\nu }\\rangle |{\\phi }\\rangle \\,$ and must be invariant under permutations up to a global phase, i.e.",
"$P_\\pi |\\psi \\rangle =(\\pm )^{p_\\pi }|\\psi \\rangle $ , where $p_\\pi $ is the parity of the permutation (even or odd), and $(\\pm )^{p_\\pi }$ the phase factor picked up accordingly for bosons ($+$ ) or fermions ($-$ ).",
"Then we have the following: Lemma 1 An arbitrary mixed state $\\rho $ of indistinguishable bosons or fermions (density operator on the full Hilbert space) satisfies $P_\\pi \\rho P_{\\pi ^{\\prime }}^\\dagger =(\\pm )^{p_\\pi + p_{\\pi ^{\\prime }}}\\rho \\;\\; \\forall \\pi ,\\,\\pi ^{\\prime }\\,.$ The mixed state of a system of indistinguishable bosons (fermions) must be a mixture of pure states that have all the full permutation symmetry (antisymmetry), i.e.",
"$\\rho =\\sum _i p_i |\\psi ^{(i)}\\rangle \\langle \\psi ^{(i)}|$ where $p_i$ are probabilities and $P_\\pi |\\psi ^{(i)}\\rangle =(\\pm )^{p_\\pi }|\\psi ^{(i)}\\rangle $ for all $i$ .",
"Applying $P_\\pi $ from the left and $P_{\\pi ^{\\prime }}^\\dagger $ from the right immediately yields the claim.",
"Consider now the reduced density matrix corresponding to the internal degrees of freedom only.",
"Inserting the decomposition (REF ) for each state $|\\psi ^{(i)}\\rangle $ in the convex sum (REF ), we obtain $\\rho ^\\mathrm {in}\\equiv {\\rm Tr}_{\\rm ex}\\rho =\\sum _{{\\phi }}\\langle {\\phi }|\\rho |{\\phi }\\rangle = \\sum _ip_i\\sum _{{\\phi },{\\nu },{\\mu }}\\alpha _{{\\nu }{\\phi }}^{(i)}\\alpha _{{\\mu }{\\phi }}^{(i)*}|{\\nu }\\rangle \\langle {\\mu }|\\,.",
"$ Similarly, the reduced density matrix corresponding to the external degrees of freedom reads $\\rho ^\\mathrm {ex}\\equiv {\\rm Tr}_{\\rm in}\\rho =\\sum _{{\\nu }}\\langle {\\nu }|\\rho |{\\nu }\\rangle = \\sum _ip_i\\sum _{{\\nu },{\\phi },{\\psi }}\\alpha _{{\\nu }{\\phi }}^{(i)}\\alpha _{{\\nu }{\\psi }}^{(i)*}|{\\phi }\\rangle \\langle {\\psi }|\\,.$ Then we have the following Lemma 2 The arbitrary reduced density matrices $\\rho ^\\mathrm {in}$ and $\\rho ^\\mathrm {ex}$ of indistinguishable atoms (bosons or fermions) satisfy $\\begin{aligned}&P^\\mathrm {in}_\\pi \\rho ^\\mathrm {in} P_{\\pi }^{\\mathrm {in}\\dagger }=\\rho ^\\mathrm {in} \\quad \\forall \\, \\pi \\,, \\\\&P^\\mathrm {ex}_\\pi \\rho ^\\mathrm {ex} P_{\\pi }^{\\mathrm {ex}\\dagger }=\\rho ^\\mathrm {ex} \\quad \\forall \\, \\pi \\,.\\end{aligned}$ We present here the proof for $\\rho ^\\mathrm {in}$ .",
"The symmetry of the full state implies the symmetry of the coefficients $\\alpha _{{\\nu }{\\phi }}^{(i)}$ : $P_\\pi |\\psi ^{(i)}\\rangle &=&\\sum _{{\\nu }{\\phi }}\\alpha _{{\\nu }{\\phi }}^{(i)}|{\\nu }_\\pi \\rangle |{\\phi }_\\pi \\rangle \\\\&=&\\sum _{{\\nu }{\\phi }}\\alpha _{{\\nu }_{\\pi ^{-1}}{\\phi }_{\\pi ^{-1}}}^{(i)}|{\\nu }\\rangle |{\\phi }\\rangle =(\\pm )^{p_\\pi }|\\psi ^{(i)}\\rangle \\,,$ and projecting onto the basis states gives $(\\pm )^{p_\\pi }\\alpha ^{(i)}_{{\\nu }{\\phi }}=\\alpha ^{(i)}_{{\\nu }_{\\pi ^{-1}}{\\phi }_{\\pi ^{-1}}}\\,.$ As a consequence, $\\begin{aligned}P_\\pi ^\\mathrm {in}\\rho ^\\mathrm {in} P_\\pi ^{\\mathrm {in}\\dagger }&= \\sum _{i, {\\phi },{\\nu },{\\mu }}p_i\\alpha ^{(i)}_{{\\nu }{\\phi }}\\alpha ^{(i)*}_{{\\mu }{\\phi }}|{\\nu }_\\pi \\rangle \\langle {\\mu }_\\pi |\\\\&=\\sum _{i, {\\phi },{\\nu },{\\mu }}p_i\\alpha ^{(i)}_{{\\nu }_{\\pi ^{-1}}{\\phi }}\\alpha ^{(i)*}_{{\\mu }_{\\pi ^{-1}}{\\phi }}|{\\nu }\\rangle \\langle {\\mu }|\\\\&=\\sum _{i, {\\phi },{\\nu },{\\mu }}p_i\\alpha ^{(i)}_{{\\nu }_{\\pi ^{-1}}{\\phi }_{\\pi ^{-1}}}\\alpha ^{(i)*}_{{\\mu }_{\\pi ^{-1}}{\\phi }_{\\pi ^{-1}}}|{\\nu }\\rangle \\langle {\\mu }|\\nonumber =\\rho ^{\\mathrm {in}}\\,,\\end{aligned}$ where in the penultimate step permutation $\\pi $ was absorbed in the sum over all ${\\phi }$ , and the last step follows from Eqs.",
"(REF ) and (REF ).",
"Note that in general for the reduced density matrix $\\rho ^\\mathrm {in}$ the statement corresponding to Eq.",
"(REF ) does not hold, i.e.",
"$P_\\pi \\rho ^\\mathrm {in} P_{\\pi ^{\\prime }}^\\dagger \\ne \\rho ^\\mathrm {in}$ for $\\pi \\ne \\pi ^{\\prime }$ : Going through the last proof again with the second $\\pi $ replaced by $\\pi ^{\\prime }$ , one realizes that in at least one of the coefficients $\\alpha ^{(i)}_{{\\nu }_{\\pi ^{-1}}{\\phi }}$ or $\\alpha ^{(i)*}_{{\\mu }_{\\pi ^{^{\\prime }-1}}{\\phi }}$ , ${\\phi }$ cannot be replaced by ${\\phi }_{\\pi ^{-1}}$ or ${\\phi }_{\\pi ^{^{\\prime }-1}}$ if $\\pi \\ne \\pi ^{\\prime }$ , and in general $\\alpha _{{\\nu }{\\phi }}\\ne \\alpha _{{\\nu }_{\\pi ^{-1}}{\\phi }_{\\pi ^{^{\\prime }-1}}}$ even for bosons.",
"In this Appendix, we show that all off-diagonal ($i\\ne j$ ) decay rates $\\gamma _{ij}$ and all dipole-dipole shifts $\\Delta _{ij}$ are equal for any pair of indistinguishable atoms $i$ and $j$ in arbitrary permutation invariant motional states.",
"Then, we give their general expressions for arbitrary symmetric or antisymmetric motional states.",
"As shown in [1], the diagonal decay rates are equal to the single-atom spontaneous emission rate $\\gamma _0$ for any motional state while the off-diagonal decay rates and dipole-dipole shifts are given, respectively, by ij = R3 cl(r) F-1r [Cijex(k)] dr, ij = R3 cl(r) F-1r [Cijex(k)] dr, with $\\mathcal {F}^{-1}_{\\mathbf {r}} \\left[\\mathcal {C}_{ij}^\\mathrm {ex}(\\mathbf {k})\\right]$ the inverse Fourier transform of the motional correlation function [60] $\\mathcal {C}_{ij}^\\mathrm {ex}(\\mathbf {k})= \\mathrm {Tr}_{\\mathrm {ex}}\\left[ e^{i\\mathbf {k}{\\cdot } \\hat{\\mathbf {r}}_{ij}} \\rho _A^\\mathrm {ex} \\right],$ where $\\hat{\\mathbf {r}}_{ij} = \\hat{\\mathbf {r}}_{i} - \\hat{\\mathbf {r}}_j$ is the difference between the position operators of atoms $i$ and $j$ .",
"In Eqs.",
"(REF ) and (REF ), $\\gamma ^{\\mathrm {cl}}(\\mathbf {r}) $ and $\\Delta ^{\\mathrm {cl}}(\\mathbf {r})$ are the classical expressions of the decay rates and dipole-dipole shifts, respectively, for a pair of atoms connected by $\\mathbf {r}$ and a radiation of wavenumber $k_0$ [61], [62], [63], $\\gamma ^{\\mathrm {cl}}(\\mathbf {r}) =\\frac{3 \\gamma _0 }{2}\\Bigg [ p \\,\\frac{\\sin (k_0 r)}{k_0 r} + q \\left( \\frac{\\cos (k_0 r)}{(k_0 r)^2} - \\frac{\\sin (k_0 r)}{(k_0 r)^3}\\right)\\Bigg ]$ and $\\Delta ^{\\mathrm {cl}}(\\mathbf {r}) =\\frac{3 \\gamma _0 }{4}\\Bigg [ - p \\,\\frac{\\cos (k_0 r)}{k_0 r} + q \\left( \\frac{\\sin (k_0 r)}{(k_0 r)^2} + \\frac{\\cos (k_0 r)}{(k_0 r)^3}\\right) \\Bigg ].$ with $p$ and $q$ angular factors given by $p={\\left\\lbrace \\begin{array}{ll}\\sin ^2 \\alpha & \\mbox{for a $\\pi $ transition}\\\\\\tfrac{1}{2}(1+\\cos ^2 \\alpha ) & \\mbox{for a $\\sigma ^\\pm $ transition}\\end{array}\\right.",
"}$ and $q={\\left\\lbrace \\begin{array}{ll}1-3 \\cos ^2 \\alpha & \\mbox{for a $\\pi $ transition}\\\\\\tfrac{1}{2}(3 \\cos ^2 \\alpha -1) & \\mbox{for a $\\sigma ^\\pm $transition,}\\end{array}\\right.",
"}$ where $\\alpha =\\arccos (\\mathbf {e}_r{\\cdot }\\mathbf {e}_z)$ is the angle between the quantization axis and $\\mathbf {r}$ .",
"Indistinguishability of atoms implies that their motional state is invariant under permutation [see Appendix A], i.e.",
"$P^\\mathrm {ex}_\\pi \\rho _A^\\mathrm {ex} P_{\\pi }^{\\mathrm {ex}\\dagger }=\\rho _A^\\mathrm {ex} \\quad \\forall \\, \\pi \\,.$ Upon using the latter equation, the motional correlation function (REF ) is found to satisfy $\\begin{aligned}\\mathcal {C}_{ij}^\\mathrm {ex}(\\mathbf {k})&= \\mathrm {Tr}_{\\mathrm {ex}}\\left[ e^{i\\mathbf {k}{\\cdot } \\hat{\\mathbf {r}}_{ij}} P^\\mathrm {ex}_\\pi \\rho _A^\\mathrm {ex} P^{\\mathrm {ex}\\dagger }_\\pi \\right] \\\\&= \\mathrm {Tr}_{\\mathrm {ex}}\\left[P^{\\mathrm {ex}\\dagger }_\\pi e^{i\\mathbf {k}{\\cdot } \\hat{\\mathbf {r}}_{ij}} P^\\mathrm {ex}_\\pi \\rho _A^\\mathrm {ex} \\right] \\\\&= \\mathrm {Tr}_{\\mathrm {ex}}\\left[ e^{i\\mathbf {k}{\\cdot } \\hat{\\mathbf {r}}_{\\pi (i)\\pi (j)}} \\rho _A^\\mathrm {ex} \\right] = \\mathcal {C}_{\\pi (i)\\pi (j)}^\\mathrm {ex}(\\mathbf {k}).\\end{aligned}$ The equality of $\\mathcal {C}_{ij}^\\mathrm {ex}(\\mathbf {k})$ for any pair of atoms [Eq.",
"(REF )] implies the equality of the decay rates (REF ) [or the dipole-dipole shifts (REF )] for any pair of atoms.",
"Consider now an arbitrary symmetric or antisymmetric motional state of the form $\\rho _A^{\\mathrm {ex},\\pm } = \\sum _{m = 1}^M p_m \\big |\\Phi _A^{(m),\\pm }\\big \\rangle \\big \\langle \\Phi _A^{(m),\\pm }\\big |,$ where $p_m$ are the weights of the statistical mixture ($p_m \\ge 0$ and $\\sum _m p_m = 1$ ) and $\\big |\\Phi _A^{(m),\\pm }\\big \\rangle $ ($m = 1, \\cdots , M$ ) are symmetric ($+$ ) or antisymmetric ($-$ ) $N$ -atom motional pure states.",
"Any state $\\big |\\Phi _A^{(m),\\pm }\\big \\rangle $ can be written as $\\begin{aligned}\\big |\\Phi _A^{(m),\\pm }\\big \\rangle =\\sqrt{\\frac{n_{\\phi _1^{(m)}}!\\cdots n_{\\phi _N^{(m)}}!}{N!",
"}}\\,\\sum _{\\pi } (\\pm 1)^{p_\\pi } \\,\\big |\\phi _{\\pi (1)}^{(m)}\\cdots \\phi _{\\pi (N)}^{(m)}\\big \\rangle \\end{aligned}$ where $\\big |\\phi _{j}^{(m)}\\big \\rangle $ ($j = 1, \\cdots , N$ ) are normalized (but not necessarily orthogonal) single-atom motional states, $n_{\\phi _j^{(m)}}$ is the number of atoms occupying the state $\\big |\\phi _j^{(m)}\\big \\rangle $ , and the sum runs over all permutations $\\pi $ of the atoms.",
"The off-diagonal decay rates and the dipole-dipole shifts for the motional state (REF ) can be expressed in terms of exchange integrals as [1] ij = m = 1M pm , ' ij,'(m), R3R3 cl(r-r') (i)(m)(r) '(i)(m)*(r) (j)(m)(r') '(j)(m)*(r') dr dr', ij = m = 1M pm , ' ij,'(m), R3R3 cl(r-r') (i)(m)(r) '(i)(m)*(r) (j)(m)(r') '(j)(m)*(r') dr dr', with $\\phi _j^{(m)}(\\mathbf {r}) = \\big \\langle \\mathbf {r} | \\phi _j^{(m)}\\big \\rangle $ the single-atom motional states in the position representation, $\\lambda _{ij,\\pi \\pi ^{\\prime }}^{(m),\\pm } = \\frac{\\displaystyle (\\pm 1)^{p_\\pi + p_{\\pi ^{\\prime }}} \\, \\prod _{n = 1 \\atop n\\ne i,j}^N \\big \\langle \\phi _{\\pi ^{\\prime }(n)}^{(m)}\\big |\\phi _{\\pi (n)}^{(m)}\\big \\rangle }{\\displaystyle \\sum _{\\tilde{\\pi },\\tilde{\\pi }^{\\prime }} (\\pm 1)^{p_{\\tilde{\\pi }} + p_{\\tilde{\\pi }^{\\prime }}} \\prod _{n = 1}^N\\big \\langle \\phi _{\\tilde{\\pi }^{\\prime }(n)}^{(m)}\\big |\\phi _{\\tilde{\\pi }(n)}^{(m)}\\big \\rangle }.$ The cooperative decay rates and dipole-dipole shifts (REF ) and (REF ) depend on their classical expressions (REF ) and (REF ), which oscillate and decrease as a function of the interatomic distance on a length scale of the order of the wavelength of the emitted radiation.",
"In addition, they depend on the single-atom wavepackets and can vary as a function of their extensions and overlaps.",
"The indistinguishability of atoms is reflected by the summations over all permutations of the atoms, which implies the equality of all off-diagonal decay rates $\\gamma _{ij}$ and all dipole-dipole shifts $\\Delta _{ij}$ .",
"Note that when all atoms occupy the same motional state $\\rho _1$ with spatial density $\\rho _1(\\mathbf {r})=\\langle \\mathbf {r}|\\rho _1|\\mathbf {r}\\rangle $ , the global motional state $\\rho _A^\\mathrm {ex}=\\rho _1^{\\otimes N}$ is symmetric and separable and the decay rates (REF ) and dipole-dipole shifts (REF ) merely read $\\gamma _{ij} = \\iint _{\\mathbb {R}^3\\times \\mathbb {R}^3} \\gamma ^{\\mathrm {cl}}(\\mathbf {r}-\\mathbf {r}^{\\prime }) \\:\\rho _1(\\mathbf {r}) \\:\\rho _1(\\mathbf {r}^{\\prime }) \\,d\\mathbf {r}\\, d\\mathbf {r}^{\\prime },$ $\\Delta _{ij} = \\iint _{\\mathbb {R}^3\\times \\mathbb {R}^3} \\Delta ^{\\mathrm {cl}}(\\mathbf {r}-\\mathbf {r}^{\\prime }) \\:\\rho _1(\\mathbf {r}) \\:\\rho _1(\\mathbf {r}^{\\prime }) \\,d\\mathbf {r}\\, d\\mathbf {r}^{\\prime }.$ In this Appendix, we give the most general solution of the master equation (REF ) for $N =2 $ atoms.",
"In this case, $J = 0,1$ and the decomposition (REF ) of the internal Hilbert space of the atomic system reads $\\mathcal {H} = \\mathbb {C}^2 \\otimes \\mathbb {C}^2 \\simeq \\left(\\mathcal {H}_0 \\otimes \\mathcal {K}_0 \\right) \\oplus \\left(\\mathcal {H}_1 \\otimes \\mathcal {K}_1\\right),$ where the dimensions of $\\mathcal {K}_0$ and $\\mathcal {K}_1$ are $d_0 = d_1 = 1$ .",
"The value $J = 1$ defines the triplet states $\\left\\lbrace |1,1\\rangle , |1,0\\rangle , |1,-1\\rangle \\right\\rbrace $ which are all symmetric while the value $J = 0$ corresponds to the singlet state $|0,0\\rangle $ , which is antisymmetric.",
"In the standard basis $\\left\\lbrace |e,e\\rangle ,|e,g\\rangle ,|g,e\\rangle ,|g,g\\rangle \\right\\rbrace $ , they read $\\begin{array}{l} |1,1\\rangle = |e,e\\rangle , \\\\[6pt]\\displaystyle |1,0\\rangle = \\frac{|e,g\\rangle +|g,e\\rangle }{\\sqrt{2}}, \\\\[10pt]|1,-1\\rangle = |g,g\\rangle .\\end{array}\\quad |0,0\\rangle = \\frac{|e,g\\rangle -|g,e\\rangle }{\\sqrt{2}},$ The solutions of (REF ) for the density matrix elements $\\rho _{J}^{M,M^{\\prime }}(t)$ in terms of $\\gamma , \\Delta \\gamma $ and $\\Delta _{\\mathrm {dd}}$ are in this case given by $\\begin{aligned}&\\rho _{1}^{1,1}(t) = \\rho _{1}^{1,1}(0)\\, e^{-2(\\gamma +\\Delta \\gamma )t}, \\\\[5pt]&\\rho _{1}^{0,0}(t) = \\rho _{1}^{0,0}(0)\\, e^{-(2 \\gamma + \\Delta \\gamma ) t} + \\frac{2 \\gamma + \\Delta \\gamma }{\\Delta \\gamma } \\rho _{1}^{1,1}(t) \\left(e^{\\Delta \\gamma t} -1\\right), \\\\[5pt]&\\rho _{1}^{-1,-1}(t) = 1 - \\rho _{1}^{1,1}(t)- \\rho _{1}^{0,0}(t) - \\rho _{0}^{0,0}(t), \\\\[5pt]&\\rho _{0}^{0,0}(t) = \\rho _{0}^{0,0}(0)\\, e^{-\\Delta \\gamma t} + \\frac{\\Delta \\gamma }{2 \\gamma + \\Delta \\gamma } \\rho _{1}^{1,1}(t) \\left(e^{(2 \\gamma + \\Delta \\gamma ) t} -1\\right), \\\\[5pt]&\\rho _{1}^{1,0}(t) = \\rho _{1}^{1,0}(0) e^{-(4\\gamma + 3\\Delta \\gamma + 2i \\Delta _{\\mathrm {dd}})t/2}, \\\\[5pt]&\\rho _{1}^{1,-1}(t) = \\rho _{1}^{1,-1}(0) e^{-(\\gamma + \\Delta \\gamma )t},\\\\[5pt]&\\rho _{1}^{0,-1}(t) = \\rho _{1}^{0,-1}(0) e^{-(2 \\gamma + \\Delta \\gamma - 2 i \\Delta _{\\mathrm {dd}}) t/2} \\\\&\\hspace{14.22636pt}+ \\rho _{1}^{1,0}(t) \\frac{2 \\gamma + \\Delta \\gamma }{\\gamma + \\Delta \\gamma + 2 i \\Delta _{\\mathrm {dd}}} \\left(e^{( \\gamma + \\Delta \\gamma + 2i \\Delta _{\\mathrm {dd}})t} - 1 \\right).\\end{aligned}$"
]
] | 1606.05102 |
[
[
"Constraining modified theories of gravity with gravitational wave\n stochastic background"
],
[
"Abstract The direct discovery of gravitational waves has finally opened a new observational window on our Universe, suggesting that the population of coalescing binary black holes is larger than previously expected.",
"These sources produce an unresolved background of gravitational waves, potentially observables by ground-based interferometers.",
"In this paper we investigate how modified theories of gravity, modeled using the ppE formalism, affect the expected signal, and analyze the detectability of the resulting stochastic background by current and future ground-based interferometers.",
"We find the constraints that AdLIGO would be able to set on modified theories, showing that they may significantly improve the current bounds obtained from astrophysical observations of binary pulsars."
],
[
"Constraining Modified Theories of Gravity with Gravitational-Wave Stochastic Background Andrea Maselliandrea.maselli@uni-tuebingen.de Theoretical Astrophysics, Eberhard Karls University of Tuebingen, Tuebingen 72076, Germany Stefania Marassi INAF - Osservatorio Astronomico di Roma, Via di Frascati 33, I-00040 Monteporzio, Italy Valeria Ferrari Dipartimento di Fisica, Sapienza Universita di Roma & Sezione INFN Roma 1, P.A.",
"Moro 5, 00185, Roma, Italy Kostas Kokkotas Theoretical Astrophysics, Eberhard Karls University of Tuebingen, Tuebingen 72076, Germany Center for Relativistic Astrophysics and School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA Raffaella Schneider INAF - Osservatorio Astronomico di Roma, Via di Frascati 33, I-00040 Monteporzio, Italy 04.30.-w, 04.80.Cc, 04.50.Kd The direct discovery of gravitational waves has finally opened a new observational window on our Universe, suggesting that the population of coalescing binary black holes is larger than previously expected.",
"These sources produce an unresolved background of gravitational waves, potentially observable by ground-based interferometers.",
"In this Letter we investigate how modified theories of gravity, modeled using the parametrized post-Einsteinian formalism, affect the expected signal, and analyze the detectability of the resulting stochastic background by current and future ground-based interferometers.",
"We find the constraints that Advanced LIGO would be able to set on modified theories, showing that they may significantly improve the current bounds obtained from astrophysical observations of binary pulsars.",
"I. Introduction.— The LIGO and Virgo Collaborations has recently announced the first direct detection of gravitational waves (GW) [1].",
"The loudness of the GW 150914 event, with an unexpected high signal-to-noise ratio (SNR), has allowed us to associate GW 150914 to the coalescence of a binary black hole (BBH) system with (source-frame) masses $\\rm 36_{-4}^{+5}M_\\odot $ and $\\rm 29_{-4}^{+4}M_{\\odot }$ at a luminosity distance of $\\rm \\sim 400 Mpc$ .",
"This binary detection implies BBH masses and coalescence rates higher than previous theoretical predictions [2], [3], and in agreement with recent estimates, obtained with a population synthesis approach, which predicts the formation of a detectable BBH in the early Universe, and in low metallicity environments [4], [5].",
"As a consequence, also the stochastic gravitational wave background (GWB) produced by these coalescing cosmological BBH sources should be at the higher end of previous estimates [6], [7], [8], [9], [10], [11], and could be potentially detectable by advanced detectors [12].",
"In this Letter we explore for the first time the ability of terrestrial interferometers to constrain the fundamental parameters of modified theories of gravity, through the detection of the GWB generated by the coalescing BBH.",
"To this aim, we compare the fiducial GWB computed assuming general relativity (GR) [12] with the signal produced by modified theories.",
"We model deviations from GR using the parametrized post-Einsteinian (PPE) formalism [13], which has been developed to capture GR modifications in the GW data.",
"More specifically, similar to the post-Newtonian (PN) formalism, which is a low-velocity and weak-field expansion of the metric and matter variables [14], the PPE approach traces back model independent deviations from GR directly into GW templates.",
"The relevance of such corrections in the gravitational emission of compact binaries has been deeply investigated in the literature [15], [16], [17], [18], [19], [20], [21], showing that GW parameter estimation can be strongly affected by GR deviations, if they are not properly taken into account [22].",
"Astrophysical constraints on the lowest order PPE coefficients have been set using observations of relativistic binary pulsars [23].",
"We refer the reader to the seminal manuscript [13], and to the review papers [24], [25] (and references therein), for an exhaustive description of this topic.",
"Following the PPE approach, in this work we do not focus on any specific theory of gravity.",
"Conversely, we carry out a completely agnostic analysis.",
"We investigate the regions of the PPE parameter space that are more likely to contribute to the GWB, producing significant deviations from GR.",
"We consider both second and third generation detectors, analyzing their ability to detect the modified signal and to extract its physical properties.",
"Finally, we show how terrestrial GW interferometers can improve the bounds on the PPE coefficients set by binary pulsar observations [23].",
"II.",
"The stochastic background.— The normalized GWB spectral energy density is defined as $\\Omega _\\textnormal {GW}(f)= \\frac{1}{\\rho _c}\\frac{\\rm d\\rho _\\textnormal {GW}}{d\\ln f},$ where $\\rho _\\textnormal {GW}$ is the GW energy density and $\\rm \\rho _{c}=\\frac{\\rm 3H_{0}^{2}}{\\rm 8\\pi G}$ is the critical energy density required to close the Universe.",
"We assume $\\rm H_{0} = 70$ km/s/Mpc, $\\rm \\Omega _{m}=0.3$ , $\\rm \\Omega _{\\Lambda }=0.7$ .",
"The function $\\Omega _\\textnormal {GW}$ can also be written as $\\Omega _\\textnormal {GW}(f)=\\frac{f}{\\rm H_0\\rho _c}\\int \\frac{ \\frac{dE_\\textnormal {GW}}{df}[f(1+z)]{\\cal R}_{coal}(z)}{(1+z)E(z)}dz,$ where $\\frac{\\rm dE_{GW}}{df}$ is the rest-frame GW spectrum emitted by a single source, $E(z)=\\sqrt{\\rm \\Omega _{m}(1+z)^3+\\Omega _{\\Lambda }}$ for a flat Universe, and ${\\cal R}_{coal}(z)$ is the BBH observed event rate per comoving volume.",
"Following [12], we assume as the fiducial model ${\\cal R}_{coal}(z)$ proportional to the cosmic star formation rate [26], [27], weighted by the fraction of stars with metallicity $Z<0.5 Z_{\\odot }$ (see APPEndix B in [28]).",
"III.",
"The waveform model.— In the PPE approach the gravitational waveform in the frequency domain is modified both in amplitude and phase with respect to the PN waveforms, $h(f)=h_\\textnormal {GR}(f)(1+\\alpha u^\\beta )e^{i\\delta u^\\zeta }\\ ,$ where $u=(\\pi {\\cal M} f)^{1/3}$ , ${\\cal M}$ is the chirp mass of the system, and $(\\alpha ,\\beta ,\\delta ,\\zeta )$ are PPE parameters.",
"$\\beta $ and $\\zeta $ define the type of modification introduced in the theory, while $\\alpha $ and $\\delta $ control the magnitude of the deviation, and have to be constrained by data.",
"(In [15] Yunes and collaborators have shown the equivalence between the framework used by the LIGO Collaboration to test GR in [29], and the PPE formalism adopted in our paper.",
"As pointed out in [15], the waveforms employed by the LIGO Collaboration represent a subset of the PPE models, since they allow only positive GR modifications in the phase.)",
"In this Letter, we focus on the information which can be extracted from the wave amplitude, therefore on $\\alpha $ and $\\beta $ .",
"$h_\\textnormal {GR}(f)$ is the phenomenological waveform described in [30], which combines the PN approximation with numerical relativity results to describe the whole binary coalescence.",
"Moreover, we only consider the inspiral part of the signal, deferring to a forthcoming paper a detailed analysis of the impact of GR modifications on the merger and ringdown phases [31].",
"For $\\alpha =\\delta =0$ we recover the standard PN waveform which, at lowest order, has amplitude $ {\\cal A}(f)=\\sqrt{\\frac{5}{24}}\\frac{{\\cal M}^{5/6}}{\\pi ^{2/3}d}f^{-7/6}\\ , $ where $d$ is the source distance.",
"In our analysis we truncate the template at the merger frequency in GR, which can be parametrized in terms of the mass components of the binary [30].",
"The waveform (REF ) enters quadratically into the GWB through the GW single source spectrum $dE_\\textnormal {GW}/df$ , i.e., $\\Omega _\\textnormal {GW}(f)\\propto h_\\textnormal {GR}(f)^2(1+\\alpha u^\\beta )^2$ .",
"As noted in [13], Eq.",
"(REF ) does not describe the most general modified waveform, and can be thought of as a single-parameter deformation of GR.",
"Although multiple PPE coefficients may enter into the gravitational signals of a certain theory of gravity, the templates we consider parametrize the effects that are more relevant in the interferometer's bandwidth.",
"It should be mentioned that a feature of this approach is that the map between the PPE parameters and a specific theory is not unique; thus, there could be more than one model yielding the same result [15].",
"However, the detection of these coefficients would provide precious information on the theory of gravity.",
"For example, a measurement of $\\alpha \\ne 0$ for $\\beta =1$ would identify a parity violation, while for $\\beta =-8$ it would be a hint of anomalous acceleration, or violation of position invariance [24], [15].",
"In this Letter we do not choose any particular modified theory of gravity.",
"Rather, being completely agnostic on the real nature of gravity, we explore the PPE parameter space to study how the modified waveform affects the GWB produced by the BBH.",
"However, we assume the GR corrections in Eq.",
"REF as perturbative terms, and accordingly, in our analysis we consistently consider values of $\\alpha $ and $\\beta $ that satisfy the bound $\\vert \\alpha u^{\\beta }\\vert <1$ .",
"The PPE parameters have already been constrained by astrophysical observations.",
"Using the data of double binary pulsars, strong bounds have been set on the amplitude $\\alpha $ as a function of the specific considered theory, identified by the exponent $\\beta $ [23].",
"It is shown that negative values of $\\beta $ yield very tight constraints on $\\alpha $ .",
"For gravity theories with $\\beta =-2$ , which gives $-1$ PN corrections in the amplitude, these observations imply $\\vert \\alpha \\vert \\lesssim 10^{-9}$ .",
"In general, theories with $\\beta <0$ predict corrections to GR that affect the low frequency regime and therefore, are well constrained by electromagnetic observations of binary systems far from coalescence, or by future GW space interferometers [32].",
"In this Letter we will focus our analysis on modified waveforms with $\\beta >0$ .",
"In these models the gravitational waveforms exhibit corrections at higher frequencies, and therefore are ideal candidates to be tested in the near future by ground-based GW interferometers.",
"To quantify the differences between the GR and the modified background, we introduce the optimized SNR for a given integration time $\\rm T$ [6]: $\\textnormal {SNR}=\\frac{3H^2_0}{\\sqrt{50}\\pi ^2}\\sqrt{\\rm T}\\int _{0}^{\\infty }df\\left[\\frac{\\gamma ^2(f)\\Omega _\\textnormal {GW}^2(f)}{f^6S_1(f) S_2(f)}\\right]^{1/2}\\ ,$ where $S_{1}(f),S_2(f)$ are the power spectral noise densities of two detectors, and $\\gamma (f)$ is the normalized overlap reduction function.",
"We have computed the SNR for Advanced LIGO (AdLIGO) and the Einstein Telescope (ET), assuming the ZERO_DET_high_P anticipated sensitivity for both Livingston and Hanford sites [33], and the ET-B configuration for the [34].",
"For the latter, $\\gamma (f)$ is assumed to be constant, i.e., $\\gamma =-3/8$ , while for AdLIGO the overlap function is given by the numerical results described in [35].",
"IV.",
"Results.— In Fig.",
"REF we show the GWB spectra for modified theories with exponent $\\beta =(2,1)$ and different values of the parameter $\\alpha $ , compared to the fiducial GR case, (We note that the fiducial model considered in this work differs from the GWB of [12], which is computed also taking into account the merger and the ringdown phases.)",
"and assuming a mean chirp mass of ${\\cal M}=28M_\\odot $ .",
"The power-law integrated sensitivity curve for AdLIGO with an integration time of 1 year is also shown.",
"Figure: The spectral energy density Ω G W(f)\\Omega _\\textnormal {GW}(f) is plotted as afunction of frequency.",
"Ω G W(f)\\Omega _\\textnormal {GW}(f) is computed in GR and in modifiedtheories with PPE parameter β=2\\beta =2, and β=1\\beta =1, and different values ofα\\alpha .",
"The power-law integrated sensitivity curve for 1 year of integrationwith AdLIGO is also shown.The net effect of positive (negative) values of $\\alpha $ is to increase (decrease) the spectral energy density of the background, which, for certain values, is significantly different from that predicted by GR.",
"As an example, a gravity theory with $\\beta =2$ , which yields a 1 PN correction to the amplitude of the waveform, and $\\alpha \\sim 6$ , would produce a background three times larger than the fiducial.",
"A similar behaviour is shown for GWBs with $\\beta =1$ .",
"As expected by the PN character of the PPE approach, for a fixed $\\alpha $ , smaller values of the exponent $\\beta $ yields larger deviations.",
"When $\\alpha <0$ the amplitude of the GWB decreases, limiting the possibility to detect these backgrounds with advanced detectors.",
"However, they are potentially observable by a third generation of ground based interferometers.",
"The left panel of Fig.",
"REF shows the GWB for some values of $\\alpha <0$ and $\\beta =(2,1.5,1)$ , compared to the power-law integrated sensitivity curve of ET, assuming 1 year of observation.",
"Figure: (Left) Same as Fig.",
"(), for values of negativeα\\alpha and different values of β\\beta , compared to the ETsensitivity curve for one year of observation.",
"(Right)The SNR with which AdLIGO would detect a PPE background is plottedfor some of the gravity theories considered in Sec.",
"IV versus the integrationtime.In the right panel of Fig.",
"REF we show how the SNR changes as a function of the integration time, for AdLIGO and different PPE models.",
"For some of the considered configurations the SNR increases to a factor $\\gtrsim 10$ after 24 months.",
"The fiducial GR background would require $\\sim $ 30 years to reach the same value.",
"Figure: Contour linescorresponding to different SNR, for gravity theories with PPE parametersα>0\\alpha >0 and β∈[0.2,2]\\beta \\in [0.2,2] assuming 1 and 3 years of integration withAdLIGO.",
"The shaded region identifies the parameter space where the PPEparameters satisfy the bound |αu β |<1\\vert \\alpha u^{\\beta }\\vert <1,while the long-dashed curves correspond to thepulsar constraint .",
"The allowed parameterspace is the colored region on the right side of the pulsarconstraint.In Fig.",
"REF we extend our analysis, showing the contour lines for detection thresholds SNR$=(3,5,8)$ , computed for AdLIGO with 1 and 3 years of integration, for theories with $\\alpha >0$ and $\\beta \\in [0.2,2]$ .",
"The long-dashed curve identifies the region where the parameters $\\alpha $ and $\\beta $ are constrained by binary pulsar observations, as computed in [23]: the allowed region is on the right of this curve.",
"The shaded region defines the range where $\\alpha $ and $\\beta $ satisfy the condition $\\vert \\alpha u^{\\beta }\\vert <1$ .",
"After 1 year of integration, AdLIGO would be able to identify GWBs with SNR = 5, produced by modified theories with $\\beta \\gtrsim 0.9$ and values of $\\alpha $ lying on the red dashed curve.",
"Three years of observation would be needed to detect the same signals with $8\\lesssim \\textnormal {SNR}\\lesssim 10$ .",
"In Table REF we show the SNR computed for the advanced and third generation interferometers, for different integration times, for the GWB computed using the PPE waveforms with $\\beta =2$ .",
"Large SNRs are expected for ET, (We note that such SNRs may be biased since Eq.",
"(REF ) is defined in the small signal approximation, whereas ET should be able to detect these backgrounds directly.)",
"but for some values of $\\alpha $ and $\\beta $ the GWB could be potentially detectable also by AdLIGO.",
"Table: SNR of AdLIGO and ET computed for different integration times,β=2\\beta =2 and different values of α\\alpha .The analysis presented above shows that a region of the PPE parameter space does exist, where the spectral energy density $\\Omega _\\textnormal {GW}(f)$ of the GWB produced by binary black hole coalescence could be detected by AdLIGO.",
"To further clarify this point, we assess the ability of current interferometers to distinguish these GWB from the GR counterpart, and to extract physical information.",
"We follow the strategy adopted in [28], where it has recently been pointed out that second generation detectors may not be able to distinguish between a BBH GWB and a generic power-law background.",
"This would strongly affect our ability to extract information on the background shape.",
"Here, we apply a model selection procedure to determine whether the modified GWB can be discerned by one computed in GR, or assuming a power-law behavior.",
"To this aim we compare the likelihood functions between two models $\\Omega _{1,2}(f)$ : $ {\\cal L}(\\Omega _1\\vert \\Omega _2)\\propto \\textnormal {Exp}\\left[-\\frac{1}{4}(\\Omega _1-\\Omega _2\\vert \\Omega _1-\\Omega _2)\\right]\\ , $ where $ (A\\vert B)=2T\\left(\\frac{3H_0^2}{10\\pi ^2}\\right)^2\\int _{f_\\textnormal {min}}^\\infty df\\gamma ^2(f)\\frac{A(f)B(f)}{f^6 S_1(f)S_2(f)}$ .",
"Then, we compute their likelihood ratios, $ {\\cal R}_\\textnormal {PPE}=\\ln \\frac{{\\cal L}(\\Omega _\\textnormal {GR}\\vert \\Omega _\\textnormal {GR})}{{\\cal L}(\\Omega _\\textnormal {GR}\\vert \\Omega _\\textnormal {PPE})}\\ \\ ,\\ {\\cal R}_\\textnormal {PL}=\\ln \\frac{{\\cal L}(\\Omega _\\textnormal {PPE}\\vert \\Omega _\\textnormal {PPE})}{{\\cal L}(\\Omega _\\textnormal {PPE}\\vert \\Omega _\\textnormal {PL})}\\ , $ where $\\Omega _\\textnormal {PPE}=\\Omega _\\textnormal {PPE}(\\alpha ,\\beta )$ and as usual $\\Omega _\\textnormal {GR}=\\Omega _\\textnormal {PPE}(\\alpha =0)$ , while the power-law density is given by $\\Omega _\\textnormal {PL}=\\Omega _0(f/f_0)^{2/3}$ , with $f_0$ being the arbitrary reference frequency and $\\Omega _0$ the amplitude that can be computed analytically [28].",
"If the likelihood ratio approaches 1, the two GWBs cannot be distinguished, while large values of ${\\cal R}$ identify a preferred model.",
"In particular ${\\cal R}_\\textnormal {PPE}\\gg 1$ suggests that $\\Omega _\\textnormal {PPE}\\ne \\Omega _\\textnormal {GR}$ , and ${\\cal R}_\\textnormal {PL}\\gg 1$ reveals that the detected GWB differs significantly from a power-law energy spectrum.",
"To assess the full detectability of the features of the GW signal, both ratios in Eq.",
"(REF ) must be greater than 1.",
"The top panels of Fig.",
"REF show the values of $\\alpha $ and $\\beta $ for which ${\\cal R}_\\textnormal {PPE}=1$ and ${\\cal R}_\\textnormal {PL}=1$ , with 1 and 3 years of integration with AdLIGO, compared against the binary pulsar constraints (long-dashed curve).",
"Figure: (Top) Values of α\\alpha and β\\beta yielding ℛ P L=1{\\cal R}_\\textnormal {PL}=1and ℛ P PE=1{\\cal R}_\\textnormal {PPE}=1, computed for AdLIGO with 1 and 3 years ofintegration.",
"The long-dashed line and the shaded region correspond to the pulsarconstraints and the parameter space where |αu β |<1\\vert \\alpha u^{\\beta }\\vert <1,respectively.",
"The green curves represent the contour line for the GWB detectablewith SNR =3 and 5.",
"(Bottom) Likelihood ratios ℛ P PE{\\cal R}_\\textnormal {PPE} andℛ P L{\\cal R}_\\textnormal {PL}, computed for ET as a function ofthe integration time, for PPE models with α=0.1\\alpha =0.1 and β=(2,1.5)\\beta =(2,1.5).We note that for $\\beta \\gtrsim 1$ all configurations in the allowed region (shaded zone on the right of the long-dashed curve) lead to ${\\cal R}_\\textnormal {PPE}>1$ , and then can potentially be distinguished from the fiducial model, with a cumulative SNR $\\lesssim 4$ .",
"Three years of observation would improve this picture, allowing us to discern among gravity theories with ${\\cal R}_\\textnormal {PPE}>>1$ and SNR $\\gtrsim 5$ .",
"However, larger values of the PPE amplitude, outside the permitted parameter space, are needed to satisfy the condition ${\\cal R}_\\textnormal {PL}\\gg 1$ .",
"For example, a gravity theory with $\\beta =1.5$ requires $\\alpha \\gtrsim 12$ .",
"Third generation detectors, would be able to fully extract the physical information of the GWB in the allowed parameter space, and constrain the change in slope in Figs.",
"REF -REF due to the PPE correction, which is frequency dependent.",
"In the bottom panel of Fig.",
"REF we show ${\\cal R}_\\textnormal {PPE}$ and ${\\cal R}_\\textnormal {PL}$ for two PPE theories with $\\alpha =0.1$ and $\\beta =(1.5,2)$ , as a function of the integration time, for ET.",
"For both theories we find high values of both likelihood ratios, even after 6 months of observation.",
"It is interesting to note that even a null detection of the GW signal would provide information on the allowed space for the PPE parameters.",
"We propose here a simple strategy to exploit this feature.",
"As a rule of thumb we can assume that the GWB is potentially observable if the SNR is greater than a defined threshold SNR$_\\textnormal {T}$ .",
"Looking at Fig.",
"REF , after 1 year of integration and assuming SNR$_\\textnormal {T}=3$ , if no GWB is detected, we could exclude the parameter space outside the two green dot-dashed curves.",
"A SNR threshold of 5 rules out the values of $\\alpha $ and $\\beta $ outside the red short-dashed curves.",
"Two or more years of observation would provide additional restrictions on the PPE coefficients.",
"This simple strategy would constrain the PPE amplitude $\\alpha $ to values $\\mathcal {O}(10)$ , with a large impact on models with $\\beta \\gtrsim 1 $ , where current bounds are quite loose.",
"In fact, for a gravity theory with $\\beta =2$ , binary pulsar observations can only constrain $\\alpha $ to be $\\vert \\alpha \\vert \\lesssim 2000$ .",
"Our approach would improve this bound by 2 orders of magnitude.",
"IV.",
"Conclusions.— In this paper we have analyzed how GR modifications affect the GWB produced by the coalescence of BBH systems, showing that the parameter space available for modified theories may yield an enhancement of the background energy density.",
"As pointed out in [12] the fiducial GWB has an uncertainty band which depends on different assumptions on the formation and evolution of the binary progenitors.",
"Alternative theories introduce another source of degeneracy.",
"However, for a given astrophysical scenario, every GWB computed in GR has a modified PPE counterpart that, for $\\alpha >0$ , is larger in amplitude and different in slope.",
"These features imply that a GWB detection will still be able to constrain the PPE parameters in large regions of the parameter space.",
"In addition, it should be mentioned that in the future more detectors, Virgo [36], KAGRA [37] and LIGO-India [38], will become operational; the sensitivity of the network of these detectors will significantly increase with respect to that of AdLIGO alone, making accessible further regions of the PPE parameter space.",
"A detailed investigation of the regions of $\\alpha $ and $\\beta $ where the GWB is distinguishable from the GR background requires a more sophisticated statistical analysis, like the one presented in [39].",
"We plan to include this analysis in a forthcoming extended publication [31], in which we will also consider the effect of a network of detectors.",
"Finally, we remark that a comprehensive study of how the merger and the ringdown may affect the GWB in GR was carried out in [10], [7], pointing out that only ET would be able to identify the contribution of these two phases.",
"It would be interesting to reexamine these results for AdLIGO, as far as modified theories of gravity are considered.",
"Acknowledgements.— The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Program (FP/2007-2013)/ERC Grant No.",
"306476.",
"It is a pleasure to thank Nico Yunes, Emanuele Berti and Paolo Pani for having carefully read the manuscript and for their useful comments."
]
] | 1606.04996 |
[
[
"Designing a Human-Machine Hybrid Computing System for Unstructured Data\n Analytics"
],
[
"Abstract Current machine algorithms for analysis of unstructured data typically show low accuracies due to the need for human-like intelligence.",
"Conversely, though humans are much better than machine algorithms on analyzing unstructured data, they are unpredictable, slower and can be erroneous or even malicious as computing agents.",
"Therefore, a hybrid platform that can intelligently orchestrate machine and human computing resources would potentially be capable of providing significantly better benefits compared to either type of computing agent in isolation.",
"In this paper, we propose a new hybrid human-machine computing platform with integrated service level objectives (SLO) management for complex tasks that can be decomposed into a dependency graph where nodes represent subtasks.",
"Initial experimental results are highly encouraging.",
"To the best of our knowledge, ours is the first work that attempts to design such a hybrid human-machine computing platform with support for addressing the three SLO parameters of accuracy, budget and completion time."
],
[
"0pt*2.0*2.0"
],
[
"0pt*1.8*1.8 Abstract Current machine algorithms for analysis of unstructured data typically show low accuracies due to the need for human-like intelligence.",
"Conversely, though humans are much better than machine algorithms on analyzing unstructured data, they are unpredictable, slower and can be erroneous or even malicious as computing agents.",
"Therefore, a hybrid platform that can intelligently orchestrate machine and human computing resources would potentially be capable of providing significantly better benefits compared to either type of computing agent in isolation.",
"In this paper, we propose a new hybrid human-machine computing platform with integrated service level objectives (SLO) management for complex tasks that can be decomposed into a dependency graph where nodes represent subtasks.",
"Initial experimental results are highly encouraging.",
"To the best of our knowledge, ours is the first work that attempts to design such a hybrid human-machine computing platform with support for addressing the three SLO parameters of accuracy, budget and completion time.",
"keywords: Crowdsourcing, task scheduling, human augmented computing, service level objectives, microtask, data analytics."
],
[
"Introduction",
"With the proliferation of the Internet, mobile devices, social media and video cameras, traditional methods of data analytics are facing disruptions.",
"There has been a paradigm shift in data sources from the traditional format of structured data (typically arranged in rows and columns with a well-defined data model) to unstructured data where either the data does not have a pre-defined data model or is not organized in a pre-defined manner.",
"These unstructured data comes in the form of natural language text, speech, images and videos, among others.",
"Most state of the art automated algorithms available today do not meet the expectations of quality when it comes to analysis of these forms and volume of unstructured data.This results in irregularities and ambiguities that make it difficult to derive interpretations using traditional analytics techniques that have been designed to work with data stored in fielded form in databases or annotated (semantically tagged) in documents.",
"The problem is even more accentuated by the volume of data that has to be dealt with, often in gigabytes or petabytes of data per day.",
"Most studies state that today, unstructured information might account for more than $80\\%$ of all data.",
"Therefore, there is a growing requirement for human-like analytics with machine-like throughput and predictability.",
"As an illustrative example, a company whose products are sold via millions of small retail outlets might want to gain insights from in-store video data into shopper demographics, brand/product perception and purchase behavior, in order to come up with effective sales and marketing strategies, develop new products, and optimize its supply chain.",
"State of the art automated solutions for automatically spotting event of interests in audio-video streams perform poorly in the presence of poor lighting, clutter and crowd, ambient noise levels, multiple simultaneous conversations, and the use of multiple languages and dialects.",
"As another example, a marketing campaign may like to perform richer qualitative analysis (theme, sentiment and intent analysis) of social media from Twitter and Facebook.",
"Most existing automated solutions for both the above illustrative cases today can not match the quality of analysis by humans.",
"We therefore believe that for problems such as the above, the use of human intelligence via crowdsourcing can significantly analytics solutions.",
"Such an approach is particularly attractive for developing economies given the large pool of human resources and high levels of under-employment.",
"Statistics from the popular crowdsourcing platform Amazon Mechanical Turk (mTurk) reveal that the majority ($46\\%$ ) of workers on mTurk are from developing nations, and for $30\\%$ of these workers, mTurk is their main source of income [1], [8].",
"Our objective is thus to design a next generation hybrid computing platform that will utilize on-demand scalable human intelligence through crowdsourcing to complement scalable machine computation on the cloud.",
"Such a platform would theoretically be able to deliver smarter, richer and more sophisticated analytics on unstructured data.",
"Creating such a platform poses a number of technical and software engineering challenges when we try to provide service level guarantees in terms of budget (payment for on-demand human and machine resources), completion time (humans with required skills are not always available), and accuracy (automated technology is brittle, and humans can be unreliable and error-prone).",
"In this paper, we present a new approach for designing such an hybrid computing platform, analogous to an operating system with machine and human processors, over which a given task workload needs to be intelligently deployed to meet the service level objectives (SLOs).",
"Analogous to the elastic, on-demand machine computing resources enabled by the cloud, the proposed platform with its set of API libraries for i) task definition, execution and monitoring and, ii) connector to existing crowdsourcing and social media platforms would enable scalable, on-demand human processors in a transparent and seamless manner.",
"To the best of our knowledge, ours is the first work that attempts to build a computing platform that transparently uses human and machine computing agents to address the three SLO parameters of accuracy, budget and completion time deadline for solving complex, workflow based tasks."
],
[
"Related Work",
"A crowdsourcing platform acts as an intermediary/broker or marketplace between task requesters and workers for short-term microtask assignments that usually require a low degree of cognitive load and skills.",
"While micro-tasking based crowdsourcing platforms are increasingly seeing substantial use - such as Amazon Mechanical Turk (AMT) [1], CrowdFlower [2] and MobileWorks [3] - none of them focus on automated SLO management using a hybrid man-machine computation system.",
"AMT is the largest and most popular paid crowdsourcing platform.",
"But, it does not provide any guarantee on either accuracy or time and the pay per microtask is fixed during human intelligence task (HIT) creation [5].",
"Compared to AMT, CrowdFlower provides automatic quality control through the insertion of gold data - data whose answers are known a priori.",
"However, timeliness and budget adjustments are done manually by the job submitter/requester in CrowdFlower.",
"MobileWorks uses a combination of captive workers and local managers on ground to control accuracy and time constraints.",
"CrowdForge [9] uses general purpose framework for complex, interdependent tasks map-reduce like abstraction to create dynamic partitioning of microtasks among workers - workers decide a task partition and once the submit their answers, their results in turn generate new subtasks for other workers.",
"Crowdforge uses a variety of quality control methods such as voting, verification or merging items and intelligent aggregation of results using both machine algorithms as well as human workers.",
"Clowder [11] and its predecessor TurKontrol [7], [14] use a new approach of decision-theoretic control methodology for iterative workflows (workflows where multiple passes/workers iterate over previous results) wherein each controller runs a partially observable markov decision process (POMDP) [14].",
"However, due to high-dimensional and continuous state space, solving a POMDP is a notoriously hard problem, thus making these approaches computationally intensive.",
"Turkomatic [10] is another example of iterative workflow based quality control in crowdsourcing.",
"The AutoMan system provides an environment where the job requester can program a confidence level for the overall computation and a budget.",
"The AutoMan runtime system then transparently manages all details necessary for scheduling, pricing, and quality control through automatic scheduling of human tasks for each computation until it achieves the desired confidence level.",
"The runtime system monitors, reprices, and restarts human tasks as necessary with the ability to parallely schedule the same task across multiple human workers to achieve the specified confidence level while staying under budget.",
"The system periodically determines the minimum number of tasks necessary to meet the confidence SLO with remaining budget and spawns more tasks if required (same pay and time-out).",
"However, AutoMan focuses only on human based computation and ignores the use of a heterogeneous computing model using machine and human agents in parallel.",
"In CDAS [13], an analytics job is first transformed into human jobs and computer jobs, which are then processed by different modules.",
"The human jobs are handled by the crowdsourcing engine.",
"However, CDAS focusses only on accuracy/quality of the results.",
"BudgetFix [15] aims at crowdsourcing at minimal cost and with predictable accuracy for complex tasks that involve different types of interdependent microtasks structured into complex workflows.",
"BudgetFix determines the number of interdependent micro-tasks and the price to pay for each task given budget constraints.",
"It also provides quality guarantees on the accuracy of the output of each phase of a given workflow.",
"However, it does not consider deadline constraint and focusses only on the budget and accuracy constraints."
],
[
"Preliminaries",
"We first present a brief introduction to some of the common crowdsourcing terminology.",
"A crowd refers to a group of workers willing to voluntarily do small duration and simple tasks on a crowdsourcing platform.",
"This group is characterized by being heterogeneous and by the fact that its members do not know each other.",
"An individual who is a member of such a crowd is known as a crowdworker or simply a worker.",
"Microtasking is the process of breaking down a task into smaller, well defined sub-tasks known as microtasks.",
"The following characteristics of a task are usually required for it to qualify as a microtask: A microtask can be performed independent of other microtasks.",
"A microtask requires human participation or intelligence and can be done in a short period of time by a human (typically ranging from a few seconds to minutes of cognitive load).",
"A microtask is either not solvable by a machine algorithm or the quality of the machine solution is unsatisfactory for the application for which the microtask was generated or would take significantly longer time than a human.",
"Examples of microtasks include image tagging and categorization, digitization and validation of text in images, object tagging in images, sentiment analysis of a text snippet, text classification, language translation, event detection in video, keyword spotting in audio, etc.",
"to name a few.",
"Figure: Example of a paid microtaskIn this paper, we shall use the term crowdsourcing and paid crowdsourcing interchangeably for human based computing.",
"Platforms like the AMT exhibit a list of available tasks - each task being a collection of multiple microtasks (usually tens or hundreds).",
"A task typically includes instructions for the workers, reward per microtask and deadlines.",
"Most tasks have two types of deadlines - one after which the task expires; the second is the time to completion before which a worker must complete her microtask to be considered for payment.",
"Compensation per microtask is generally low, since requesters expect that work can be completed on a time scale ranging from seconds to minutes.",
"Pay per microtask ranges from $\\$0.01$ to to several dollars.",
"As an example, on AMT, which is one of the largest paid crowdsourcing platforms, most microtasks or human intelligence tasks (HITs) as they are known on AMT, are priced between $\\$0.01$ and $\\$0.05$ .",
"A typical example of a microtask from [16] is shown in Figure REF .",
"We now give a definition of workflow that we will use for the rest of the paper: Definition 1 The workflow for a given task is a dependency graph consisting of a sequence of activities, each executed by some computing entity, in order to transform raw data into useful, application specific information.",
"The activities are represented as nodes in the dependency graph, with edges depicting data/execution dependencies/order between the activities."
],
[
"Proposed Platform",
"Our goal in this paper is to solve complex workflow-based tasks of the following form on a human-machine computing platform: Given a task $S$ that can be represented as a dependency graph consisting of nodes representing subtasks that are solvable using a human-machine agent system and the SLO metrics ($A^*$ , $B^*$ , $T^*$ ) specified by the task requester, complete $S$ with confidence/accuracy of the results being at least $A^*$ , while ensuring that the total money spent is less than $B^*$ and the total time taken is less than $T^*$ .",
"Towards this end, we propose a hybrid computing platform that uses machine computation as well as crowd-intelligence to solve workflow-based complex analytics problems on unstructured data under a fundamental assumption that there exists a feasible solution for the specified SLOs and enough number of human workers are reachable through our platform.",
"The most important components of our proposed platform are outlined below.",
"Figure: Representative workflow description as task dependency graph"
],
[
"Platform Interfaces",
"The platform will provide three main interface libraries for users to interact with the platform in order to create, manage, execute and monitor tasks: Crowd Access Layer: The crowd access layer (CAL) will be used for bringing crowd workers into our platform.",
"The larger and more diverse the crowd, more the chances of scaling computation and meeting the SLO requirements of the tasks executing on our platform.",
"Hence, the platform will not only provide a dedicated crowdsourcing channel for creating a private crowd for a given task, but also have connectors to tap into existing crowdsourcing platforms like the AMT and CrowdCloud.",
"As part of the CAL, the platform will also have APIS that support a subset of the APIs of popular social media platforms (e.g., Facebook, Twitter, Instagram) so that the platform can exploit social media to recruit suitable workers for a given task.",
"Machine Abstraction Layer: The machine abstraction layer (MAL) would provide the necessary APIs that would allow platform users to plugin/register their own machine algorithms/software.",
"To make it easy to use the platform, we intend to also provide a set of popular analytics software and APIs to access them, for text and image analytics using a software as a service (SaaS) model.",
"Task Management Library: The primary goal of the task management library (TML) would be to expose a task-workflow specification interface for hybrid tasks, whereby a complex, multi-stage task that can be decomposed into subtasks and specified as a task-dependency graph with subtask nodes tagged either as machine only or human only or either and then processed using a workflow engine.",
"The SLOs for individual subtasks and the specific algorithms or crowd to use for a subtask node would also form a part of the task-workflow.",
"Figure REF shows a representative workflow definition where a task is decomposed into subtasks along with their individual SLOs and execution agent specifications.",
"Once the workflow is submitted to the system, the platform would automatically schedule and manage the execution of the subtasks specified in the dependency graph with the goal of satisfying the specified SLO values.",
"The task management library would also provide APIs that can be used for monitoring the execution progress of a submitted task."
],
[
"Task Execution Management Engine",
"The heart of the proposed platform is defined by its ability to provide service level guarantees on accuracy, time and budget while using crowd and cloud computing agents.",
"In order to provide such guarantees while using a combination of human and machine computing agents, a task execution management engine is required that can provide the following two fundamental functionalities: Intelligently partition work between human and machine computing agents.",
"Provide continuous monitoring and management of task execution to guarantee SLOs of accuracy ($A^*$ ), budget ($B^*$ ) and deadline ($T^*$ ).",
"For example, in the event of an exception (defined by time-out, unacceptable results submitted by a computing agents, etc.",
"), reschedule the work to a different agent.",
"As a first step towards building such a task execution management engine, we consider in this paper only the problem of solving data-parallel microtasks, i.e., similar but independent microtasks with different inputs.",
"Our goal is: Given a task set $S$ consisting of $n$ microtasks that are solvable using a human-machine agent system and the SLO metrics ($A^*$ , $B^*$ , $T^*$ ) specified by the task requester, complete $S$ with confidence/accuracy of the results being at least $A^*$ , while ensuring that the total money spent is within budget $B^*$ and the total time taken is less than $T^*$ .",
"A task $\\mathcal {S}$ consists of $n$ independent, homogeneous microtasks each of which can be executed in parallel using either human or machine computing agents.",
"An example would be a sentiment analysis task on a set of independent tweets - analyzing the sentiment of each tweet would then constitute a microtask and each tweet can be analyzed by either a human or a sentiment analysis algorithm.",
"The $n$ microtasks are to be executed on a payment-based task execution platform.",
"The task execution management engine uses the crowd access layer (CAL) or the machine abstraction layer (MAL) APIs to access the different types of crowd workers as well as machine algorithms for microtask execution.",
"We assume the total allocated time interval $T^*$ (for finishing the task set $S$ ) to be divided into $K$ polling intervals with the polling instances being $t_0$ , $t_1$ , $t_2$ , $\\ldots $ , $t_{K-1} = T^*$ .",
"We denote by $n_H(t)$ the number of microtasks assigned to humans at time instance $t$ .",
"$n_M(t) = n - n_H(t)$ is the number of microtasks assigned to machine agents at time instance $t$ .",
"Since, humans are unpredictable [8] and there is no ground truth to establish the correctness of an answer, we assume that every microtask that is assigned for human agent based execution on a crowdsourcing platform is replicated $w$ times, i.e., every human-assigned microtask is done in parallel by $w$ humans.",
"We denote a replicated human assignment of a microtask by $w$ -task.",
"A similar assumption for replicated assignments per machine assigned tasks can be made if required, where each of the assignments is executed using a different machine algorithm.",
"A $w$ -task is said to be in picked state if it has been assigned to an agent but has either not been completed or the $w$ -task completion deadline has not elapsed.",
"done indicates a picked $w$ -task that has been returned to the system by an agent.",
"Note that a $w$ -task in done state does not necessarily imply that the submitted answer is correct.",
"The correctness of a submitted answer is determined only after the microtask result evaluation phase in which the answers for all the done $w$ -tasks are considered for correctness.",
"A separate result evaluation module within the task execution management engine would be responsible for aggregating the results of individual $w$ -tasks and arriving at a consolidated accuracy/confidence level for the microtasks completed at any given point in time.",
"At present, we are using a simple majority voting scheme for result aggregation - the answer choice of the majority workers is taken as the correct answer.",
"For microtasks that can be scheduled to run in parallel over either machine or human computing agents, we define an internal metric called $HM$ -Ratio ($\\lambda $ ) and distribute the microtasks workload such that the number of human tasks is $\\lambda $ times that of machine tasks.",
"The rational being that while humans can produce more accurate results for the type of tasks that we are interested in, they are also slower and more expensive as computational agents.",
"Hence, the $\\lambda $ parameter can be used to manage the accuracy SLO goal while keeping the total cost under the budget $B^*$ .",
"Unfortunately, as humans are unpredictable, there is a need for continuous feedback and control of the execution.",
"Therefore, for dynamic control of the SLO, we have defined a second probe called the Microtask Completion Rate ($\\rho $ ), which reflects the rate of completion ($\\rho $ ) of microtasks by humans and machines.",
"At regular polling intervals, a risk estimate of meeting $A^*$ or $T^*$ is made based on the current value of $\\rho $ and appropriate corrective actions are taken subject to the budget constraint of $B^*$ .",
"Examples of such corrective actions are: changing $\\lambda $ to re-allocated more microtasks to machine agents, or to more workers, or to workers with higher capability, or increasing the incentive per microtask so as to attract/employ agents with better quality/speed.",
"These concepts have been further refined in [16] and an early prototype has been built and validated through simulation with actual performance data generated from anonymous crowd workers on Amazon Mechanical Turk Machine and Hewlett Packard's Autonomy IDOL."
],
[
"Results",
"We carried out multiple experiments using AMT to assign tasks to the crowd, and Hewlett-Packard's Autonomy IDOL [4] for automated analysis.",
"For evaluation purpose, we used a set of 1000 tweets, each of which was to be categorized into six intent categories and subcategories.",
"We experimented with three types of crowd workers - i) known, expert workers consisting of team members, ii) anonymous workers from the public crowd on AMT and, (iii) anonymous workers on AMT who had passed a short training before being allowed to work on our microtasks.",
"A majority voting scheme was used for result aggregation for all worker types.",
"For expert workers, we used 3 assignments per microtask.",
"The 3000 microtasks were completed by the expert workers over 21 days, with $91.8\\%$ of the 1000 tweets achieving majority consensus.",
"For experiment on AMT, we used a subset of 250 randomly selected tweets from the 100 tweets dataset.",
"Untrained workers on AMT - with both 3 and 5 assignments per tweet - categorized these 250 tweets within a day.",
"We paid each worker $0.02$ for successfully completing a microtask assignment.",
"In the absence of gold data, the majority voting scheme on the labeled data from the expert crowd was used to determine the correct answer for each tweet.",
"An analysis of the results showed that for AMT crowd with 3 workers per tweet, the accuracy was only $57.2\\%$ and with 5 workers per tweet it was $78\\%$ .",
"For the same dataset of 250 tweets, the qualified workers generated an accuracy of $80.4\\%$ but took 7 days, with 3 assignments per microtask.",
"The performance of each type of crowd on the three SLO parameters is depicted in Figure REF .",
"Figure: Worker arrival pattern on AMTThe results from the expert crowd were also used for training the multi-class classifier of Autonomy IDOL, which in turn was used to categorize the 250 tweets given to the AMT crowd.",
"The corresponding accuracy for Autonomy IDOL was only $67.2\\%$ .",
"We also studied the arrival pattern of the workers on AMT to pick up our microtasks.",
"Figure REF shows the distribution of the arrival pattern of the AMT workers.",
"It is clear from the plot that the arrival pattern of workers on AMT follows a Poisson distribution and the best fit curve had a mean arrival rate of $0.039084$ .",
"Figure: Performance comparison of different worker classes"
],
[
"Conclusion",
"In this paper, we have proposed a new human-machine hybrid computational platform that will allow the transparent and seamless use of machine algorithms and crowdsourcing channels to solve complex, workflow-based analytics tasks on unstructured data while ensuring that the specified service level objectives of accuracy, budget and timeliness are taken into account in the task execution plan.",
"Our initial experiments on performance data collected for anonymous crowd workers on Amazon Mechanical Turk, expert workers and machine algorithms for text analytics provide indications that such a platform would indeed be feasible and provide significant benefits for SLO driven analytics on unstructured data.",
"We do acknowledge that more extensive experiments and development is needed to establish the complete effectiveness of our proposed platform and further work on developing a task execution management system have shown encouraging results [16].",
"To the best of our knowledge, ours is the first work that attempts to simultaneously attempt to build a hybrid computing platform to address the three SLO parameters of accuracy, budget and deadline for data-parallel microtasks."
]
] | 1606.04929 |
[
[
"Search for giant planets in M67 III: excess of hot Jupiters in dense\n open clusters"
],
[
"Abstract Since 2008 we used high-precision radial velocity (RV) measurements obtained with different telescopes to detect signatures of massive planets around main-sequence and evolved stars of the open cluster (OC) M67.",
"We aimed to perform a long-term study on giant planet formation in open clusters and determine how this formation depends on stellar mass and chemical composition.",
"A new hot Jupiter (HJ) around the main-sequence star YBP401 is reported in this work.",
"An update of the RV measurements for the two HJ host-stars YBP1194 and YBP1514 is also discussed.",
"Our sample of 66 main-sequence and turnoff stars includes 3 HJs, which indicates a high rate of HJs in this cluster (~5.6% for single stars and ~4.5% for the full sample ).",
"This rate is much higher than what has been discovered in the field, either with RV surveys or by transits.",
"High metallicity is not a cause for the excess of HJs in M67, nor can the excess be attributed to high stellar masses.",
"When combining this rate with the non-zero eccentricity of the orbits, our results are qualitatively consistent with a HJ formation scenario dominated by strong encounters with other stars or binary companions and subsequent planet-planet scattering, as predicted by N-body simulations."
],
[
"Introduction",
"Hot Jupiters (HJs) are defined as giant planets ($M_{p} > 0.3$$M_{\\mathrm {Jup}}$ ) on short-period orbits (P < 10 days).",
"They show an occurrance rate of $\\sim $ 1.2% around Sun-like field stars [53], [24].",
"These close-in giant planets are highly unlikely to have formed in situ, and it is believed that they form beyond the snow line where solid ices are more abundant, allowing the planet cores to grow several times more massive than in the inner part of the proto-planetary disk before undergoing an inward migration.",
"Of the mechanisms that are able to trigger migration, the two supported most often are dynamical interaction with the circumstellar disk [12], [19], [51] and gravitational scattering caused by other planets [41], [18].",
"Other ideas include violent migration mechanism such as dynamical encounters with a third body (multi-body dynamical interaction).",
"In particular, recent N-body simulations have shown that a planetary system inside a crowded birth-environment can be strongly destabilized by stellar encounters and dynamical interaction, which also favours the formation of HJs [6], [22], [46].",
"Open clusters (OCs) hold great promise as laboratories in which properties of exoplanets and theories of planet formation and migration can be explored.",
"In Paper I [31] we described a radial velocity (RV) survey to detect the signature of giant planets around a sample of main-sequence (MS) and giant stars in M67.",
"The first three planets discovered were presented in [3].",
"One goal of this project is to investigate whether and how planet formation is influenced by the environment.",
"Recent planet search surveys in OCs support that the statistics in OCs is compatible with the field [21], [3], [26], [39], [38].",
"In this work we show that for M67 the frequency of HJs is even higher than in the field."
],
[
"Observations and orbital solutions",
"Of the 88 stars in the original M67 sample, 12 have been found to be binaries [31].",
"Two additional binaries have recently been discovered (Brucalassi et al.",
"2016).",
"The final sample therefore comprises 74 single stars (53 MS and turnoff stars and 21 giants) that are all high-probability members (from proper motion and radial velocity) of the cluster according to [54] and [42].",
"Table: Stellar parameters of the three M67 stars newly found to host planet candidates.The star YBP401 shows significant indications of a HJ companion and is analysed here in detail.",
"We also present an update of the RV measurements for the stars YBP1194 and YBP1514, for which two other HJs were announced in our previous work [3].",
"Basic stellar parameters ($V$ , $B-V$ , $T_{\\mathrm {eff}}$ , $\\log g$ ) with their uncertainties were taken from the literature.",
"A distance modulus of 9.63$\\pm $ 0.05 [30] and a reddening of E(B-V)=0.041$\\pm $ 0.004 [47] were assumed, stellar masses and radii were derived using the 4 Gyr theoretical isochrones from [36] and [11].",
"The parameters estimated from isochrone fitting agree within the errors with the values adopted from the literature.",
"The main characteristics of the three host stars are listed in Table REF .",
"Figure: Phased RV measurements and Keplerian best fit, best-fit residuals, andbisector variation for YBP401.",
"Black dots: HARPS measurements, reddots: SOPHIE measurements, orange dots: HARPS-N measurements.The RV measurements were carried out using the HARPS spectrograph [25] at the ESO 3.6m telescope in high-efficiency mode (with R=90 000 and a spectral range of 378-691 nm), with the SOPHIE spectrograph [2] at the OHP 1.93 m telescope in high-efficiency mode (with R=40 000 and a range of 387-694 nm), with the HRS spectrograph [48] at the Hobby Eberly Telescope (with R=60 000 and a range of 407.6-787.5 nm), and with the HARPS-N spectrograph at the TNG on La Palma of the Canary Islands (spectral range of 383-693 nm and R=115 000).",
"Additional RV data points for giant stars have been observed between 2003 and 2005 [20] with the CORALIE spectrograph at the 1.2 m Euler Swiss telescope.",
"HARPS, SOPHIE, and HARPS-N are provided with a similar automatic pipeline.",
"The spectra are extracted from the detector images and cross-correlated with a numerical G2-type mask.",
"Radial velocities are derived by fitting each resulting cross-correlation function (CCF) with a Gaussian [1], [35].",
"For the HRS, the radial velocities were computed using a series of dedicated routines based on IRAF and by cross-correlating the spectra with a G2 star template [4].",
"We used nightly observations of the RV standard star HD32923 to correct all observations for each star to the zero point of HARPS [31] and to take into account any instrument instability or systematic velocity shifts between runs.",
"An additional correction was applied to the SOPHIE data to consider the low signal-to-noise ratio (S/N) of the observations [43].",
"Figure: Phased RV measurements and Keplerian best fit, best-fit residuals, andbisector variation for YBP1194.Same symbols as in Fig., green dots: HRS measurements.The RV measurements of our target stars were studied by computing the Lomb-Scargle periodogram [45], [13] and by using a Levenberg-Marquardt analysis [52] to fit Keplerian orbits to the radial velocity data.",
"The orbital solutions were independently checked using the Yorbit program (Segransan et al.",
"2011) and a simple Markov chain Monte Carlo (MCMC) analysis (see Table REF ).",
"We investigated the presence and variability of chromospheric active regions in these stars by measuring the variations of the core of the H$\\alpha $ line with respect to the continuum, following a method similar to the one described in [33].",
"The more sensitive Ca II H and K lines were not accessible because of the low S/N ratio of our observations.",
"For each case we verified the correlation between the RVs and the bisector span of the CCF [37] or with the full width at half maximum (FWHM) of the CCF.",
"YBP401.",
"According to [54], this F9V MS star has a membership probability of 97% and a proper motion shorter than 6 mas/yr with respect to the average.",
"[50] revised the membership list of [54] and expressed doubts about the cluster membership for YBP401.",
"However, the RV value considered for YBP401 in [50] has an uncertainty of $\\sigma =130$ $\\mathrm {m\\, s^{-1}}$ and is not consistent with our measurements by more than 1$\\sigma $ .",
"Recently, [10] confirmed YBP401 as a single cluster member.",
"This target has been observed since January 2008: 19 RV points have been obtained with HARPS with a typical S/N of 15 (per pixel at 550 nm) and a mean measurement uncertainty of 15 $\\mathrm {m\\, s^{-1}}$ including calibration errors.",
"Five additional RV measurements were obtained with SOPHIE and two with HARPS-N, with measurement uncertainties of 9.0 $\\mathrm {m\\, s^{-1}}$ and 11.0 $\\mathrm {m\\, s^{-1}}$ , respectively.",
"Figure: Phased RV measurements and Keplerian best fit, best-fit residuals, andbisector variation for YBP1514.",
"Same symbols as in Fig..The final 26 RV measurements of YBP401 show a variability of $\\sim $ 35 $\\mathrm {m\\, s^{-1}}$ and an average uncertainty of $\\sim $ 14 $\\mathrm {m\\, s^{-1}}$ for the individual RV values.",
"A clear peak is present in the periodogram (see Fig.",
"REF ) at 4.08 days.",
"A Keplerian orbit was adjusted to the RV data of YBP401 (see Fig.",
"REF ), and the resulting orbital parameters for the planet candidate are reported in Tables REF and REF .",
"We note that the non-zero eccentricity is consistent with e$=$ 0 within 2$\\sigma $ and the other parameters change by less than 1$\\sigma $ when fixing e$=$ 0.",
"We included the eccentricity in the data analysis, which resulted in a better fit ($\\chi _{red}^{2}\\sim 1$ ) and in reduced RV residuals.",
"However, more precise observations are needed to constrain small non-zero eccentricities and to avoid overinterpreting the results (see discussions in Zakamska et al.",
"2011; Pont et al.",
"2011).",
"The residuals have an rms amplitude of $\\sim $ 13 $\\mathrm {m\\, s^{-1}}$ and the periodogram of the residuals does not show any clear periodicity when the main signal is removed (see Fig.",
"REF ).",
"Neither the bisector span nor the activity index present correlations with the RV variations (see Fig.",
"REF ); this excludes activity-induced variations of the shape or the spectral lines as the source of the RV measurements.",
"YBP1194 and YBP1514.",
"We have now collected 29 measurements for both YBP1194 and YBP1514, spanning seven years.",
"The average RV uncertainty is $\\sim $ 13.0 $\\mathrm {m\\, s^{-1}}$ for HARPS and SOPHIE, $\\sim $ 26.0 $\\mathrm {m\\, s^{-1}}$ for HRS and $\\sim $ 8.0 $\\mathrm {m\\, s^{-1}}$ for HARPS-N .",
"Figures REF and REF show the phase-folded data points together with the best-fit solution and the residual over the time.",
"The peak in the periodogram is more pronounced and the RV signal is better determined (see Fig.",
"REF ) than in [3].",
"When the planet signature is removed, the rms of the residuals is $\\sim $ 12.3 $\\mathrm {m\\, s^{-1}}$ for YBP1194 and $\\sim $ 14.4 $\\mathrm {m\\, s^{-1}}$ for YBP1514.",
"We note that the resulting updated orbital parameters are consistent within the errors with the previously published data (see Tables REF and REF ).",
"Table: Orbital parameters of the planetary candidates.",
"PP: period, TT: time at periastron passage,ee: eccentricity, ω\\omega : argument of periastron, KK: semi-amplitude of the RV curve,msinim\\sin {i}: planetary minimum mass,γ\\gamma : average radial velocity, σ\\sigma (O-C): dispersion of Keplerian fit residuals."
],
[
"Frequency of hot Jupiters in OCs",
"The most striking result is that with the star YBP401 we have found three HJs around 66 MS and subgiant stars in M67 (53 stars if we only consider single stars).",
"This gives a frequency of HJs of 4.5$^{+4.5}_{-2.5}$ % and 5.6$^{+5.4}_{-2.6}$ %.",
"These results also agree with the HJs frequency (5.5$^{+5.5}_{-2.5}$ %) obtained by a Monte Carlo analysis in our parallel work (Brucalassi et al.",
"2016 sub.).",
"Our values are higher than those derived from the RV surveys around FGK stars [53].",
"The comparison is even more striking when considering that the Kepler http://kepler.nasa.gov/ statistics of HJs is lower, around 0.4$\\%$ [14].",
"However, the comparison between different samples and between simulations and observations is not trivial.",
"The analysis of the Kepler and the RV surveys for instance use different selection criteria (radii vs. masses) and different intervals of orbital periods.",
"[7] showed that the discrepancy might be partially due to the different metallicity of the samples.",
"Another effect to take into account is in the definition of the comparison samples.",
"RV surveys are performed on pre-selected samples that have been corrected for the presence of binaries, while the Kepler statistics (and most of the simulations) refer to all FGK stars in the Cygnus field, without any previous selection for binaries.",
"For M67 our survey sample was heavily pre-selected with the aim to eliminate all known and suspected binaries in advance.",
"We therefore expect that when we compare our results on the whole sample (3/66 or 4.5$^{+4.5}_{-2.5}$ %) with the Kepler statistics (0.4$\\%$ ), an upper limit of the planet frequency will be provided, while the comparison of the frequency of the single-star sample (3/53 or 5.6$^{+5.4}_{-2.6}$ % ) is expected to compare well with the 1.2$\\%$ of the radial velocity surveys because they have gone through a similar selection process.",
"Finally, based on a HJ occurrance rate of 1.2% like for field stars, one or two additional HJs may exist in M67 with a non-negligible probability of 5%.",
"For several years, the lack of detected planets in OCs was in contrast with the field results, but the recent discoveries [21], [3], [26], [39], [38] have completely changed the situation.",
"These results are difficult to reconcile with the early survey of [34], who did not detect any HJs around 94 G-M stars of the Hyades cluster.",
"Since the extrapolation from non-detection to non-existence of HJs critically depends on several assumptions such as stellar noise and real measurement errors, constraints on the allocated time, number of observations per star, sampling and planet mass, we cannot state at present whether the discrepancy is real until larger surveys are performed.",
"If we were to add the results of the RV surveys in the OCs M67, Hyades, and Praesepe, we would determine a rate of 6 out of 240 HJs per surveyed stars (including some binaries), which is a high percentage when compared to 10 out of 836 HJ per surveyed stars in the field sample of [53].",
"We can conclude that, contrary to early reports, the frequency of HJs discovered in the three OCs subject of recent RV surveys is higher than amongst the field stars.",
"To explain the high frequency of HJs in M67, Hyades, and Praesepe, we may argue that the frequency of HJs depends on stellar metallicity, mass, or on dynamical history, and therefore environment.",
"The dependence of planet frequency on stellar metallicity is complex: even if established very early [17], [49], [9], [44], a real correlation seems to be present only for Jupiters around MS stars, while it does not hold for giant planets around evolved stars [32] or for low-mass planets [24].",
"Both Hyades and Praesepe are metal rich [29], [8], and this may explain the higher frequency of HJs in these clusters, but this is not the case of M67, which has a well-established solar metallicity and abundance pattern [40].",
"The hypothesis that the high frequency of HJs in M67 or in OCs in general originates from the higher mass of the host star can be also excluded: the stars hosting HJs in M67 all have masses around one solar mass, which is very similar to the masses of the HJ hosts discovered in the field.",
"A similar argument holds for the stars hosting HJs in Praesepe and in the Hyades.",
"Finally, environment is left as the most suitable option to explain the HJs excess.",
"It has been suggested that dense birth-environments such as stellar clusters can have a significant effect on the planet formation process and the resulting orbital properties of single planets or planetary systems.",
"Close stellar fly-by or binary companions can alter the structure of any planetary system and may also trigger subsequent planet-planet scattering over very long timescales [6], [22].",
"This leads to the ejection of some planets, but it also seems to favour the conditions for the formation of HJs [46].",
"As predicted by such mechanisms, M67 HJs show orbits with non-zero eccentricity, which is also true for the HJ found in the Hyades.",
"The importance of the encounters is primarly determined by the local stellar density, the binary fraction, the collisional cross-section of the planetary system, and the timescale on which the planet is exposed to external perturbations.",
"[22] produced simulations for a cluster of 700 stars and an initial half-mass radius of 0.38 parsecs, showing that a non-negligible number of stars spend long enough as a binary system and also that the majority of the stars is affected by at least one fly-by.",
"M67 has more than 1400 stars at present, it is dominated by a high fraction of binaries [5] after loosing at least three quarters of its original stellar mass, and has suffered mass segregation.",
"[46] have recently completed N-body simulations for a case similar to the one of M67, but with only a 10$\\%$ of binaries, finding that HJs can be produced in 0.4$\\%$ of cluster planetary systems when only considering initial fly-by encounters.",
"This fraction is smaller than what we find in M67, and the influence of other migration mechanisms probably needs to be considered as well to explain our results.",
"However, the same authors acknowledged that a higher fraction of binaries will strongly enhance the probability of HJ formation and therefore their frequency.",
"Given that the binary fraction of M67 stars is currently very high [31], [23] and that models show that it must also have been high at the origin [16], the high fraction of M67 HJs seems to qualitatively agree with the N-body simulations.",
"The same simulations predict that after 5 Gyr the percentage of stars hosting HJs retained by the cluster is substantially higher than the percentage of stars not hosting HJs.",
"Two factors can contribute to enhance HJ planet formation: the capability of producing HJs, and the capability of the cluster to retain stars hosting HJs.",
"The interaction takes part well within the first Gyr of the cluster lifetime, so that the stars with HJs do not require to still have a stellar companion at the age of M67.",
"[10] reported no evidence for nearby companions at the present epoch.",
"Finally, considering that about one of ten HJs produces a transit, we suggest to carefully examine the Kepler/K2 observations [15] for any transit.",
"LP acknowledges the Visiting Researcher program of the CNPq Brazilian Agency, at the Fed.",
"Univ.",
"of Rio Grande do Norte, Brazil.",
"RPS thanks ESO DGDF, the HET project, the PNPS and PNP of INSU - CNRS for allocating the observations.",
"MTR received support from PFB06 CATA (CONICYT).",
"Figure: Top: Lomb-Scargle periodogram of the RV measurements, residuals, bisector span, and FWHM for YBP401.Central: Same plots for YBP1194.",
"Bottom: same plots for YBP1514.The dashed lines correspond to 5% and 1% false-alarm probabilities, calculated accordingto Horne & Baliunas (1986) and white noise simulations.Figure: Top: RV measurements versus bisector span, residuals versus bisector span, RV measurements versus CCF FWHM andRV measurements versus Hα\\alpha activity indicator for YBP401.",
"The Hα\\alpha activity indicator is computed as the areabelow the core of Hα\\alpha line with respect to the continuum.CCF FWHM values are calculated by subtracting the respective instrumental FWHM in quadrature.Same symbols as in Fig.",
".Middle: The same plots for YBP1194.",
"Bottom: The same plots for YBP1514.Table: RV measurements, RV uncertainties, bisector span, and ratio of the H α _{\\alpha } corewith respect to the continuum for YBP401.All the RV data points are corrected to the zero point of HARPS.Table: Orbital parameters of the planetary candidates usinga simple MCMC analysis to fit Keplerian orbits to the RV data.We considered as free parameters the orbital period PP, the time of transit T c _{c}, the radialvelocity semi-amplitude KK, the centre-of-mass velocity γ\\gamma , andthe orthogonal quantities ecosω\\sqrt{e}\\cos \\omega and esinω\\sqrt{e}\\sin \\omega , whereee is the eccentricity and ω\\omega is the argument of periastron.We quote the mode of the resulting parameter distributions as the final value andthe 68.3%68.3\\% interval with equal probability density at the ±1σ\\pm 1\\sigma bound to derive the uncertainty.TT: time at periastron passage,KK: semi-amplitude of the RV curve,msinim\\sin {i}: planetary minimum mass."
]
] | 1606.05247 |
[
[
"Suzaku Observations of Moderately Obscured (Compton-thin) Active\n Galactic Nuclei Selected by Swift/BAT Hard X-ray Survey"
],
[
"Abstract We report the results obtained by a systematic, broadband (0.5--150 keV) X-ray spectral analysis of moderately obscured (Compton-thin; $22 \\leq \\log N_{\\rm H} < 24$) active galactic nuclei (AGNs) observed with Suzaku and Swift/Burst Alert Telescope (BAT).",
"Our sample consists of 45 local AGNs at $z<0.1$ with $\\log L_{\\rm 14-195\\1mmkeV} > 42$ detected in the Swift/BAT 70-month survey, whose Suzaku archival data are available as of 2015 December.",
"All spectra are uniformly fit with a baseline model composed of an absorbed cutoff power-law component, reflected emission accompanied by a narrow fluorescent iron-K$\\alpha$ line from cold matter (torus), and scattered emission.",
"Main results based on the above analysis are as follows.",
"(1) The photon index is correlated with Eddington ratio, but not with luminosity or black hole mass.",
"(2) The ratio of the iron-K$\\alpha$ line to X-ray luminosity, a torus covering fraction indicator, shows significant anti-correlation with luminosity.",
"(3) The averaged reflection strength derived from stacked spectra above 14 keV is larger in less luminous ($\\log L_{\\rm 10-50\\1mmkeV} \\leq 43.3$; $R= 1.04^{+0.17}_{-0.19}$) or highly obscured AGNs ($\\log N_{\\rm H} > 23$; $R = 1.03^{+0.15}_{-0.17}$) than in more luminous ($\\log L_{\\rm 10-50\\1mmkeV} > 43.3$; $R= 0.46^{+0.08}_{-0.09}$) or lightly obscured objects ($\\log N_{\\rm H} \\leq 23$; $R = 0.59^{+0.09}_{-0.10}$), respectively.",
"(4) The [O IV] 25.89 $\\mu$m line to X-ray luminosity ratio is significantly smaller in AGNs with lower soft X-ray scattering fractions, suggesting that the [O IV] 25.89 $\\mu$m luminosity underestimates the intrinsic power of an AGN buried in a small opening-angle torus."
],
[
"Introduction",
"High-quality broadband X-ray spectral observations are essential to unveil the structure of active galactic nuclei (AGNs).",
"The main X-ray continuum, which can be well approximated by a power law with an exponential cutoff, is thought to be Comptonized photons by a hot corona in the vicinity of the supermassive black hole (SMBH).",
"This emission interacts with the surrounding cold matter, the putative “dusty torus” invoked by the AGN unified model [3].",
"When one observes the central engine through the torus (so-called type-2 or Compton-thin obscured AGNs; $22 \\le \\log N_{\\rm H} <24$ ), the spectrum shows a low energy cutoff due to photoelectric absorption.",
"The torus also produces a reflected component, which is seen as a hump at $\\sim 30$ keV, accompanied by a narrow iron-K$\\alpha $ fluorescent line at $\\simeq $ 6.4 keV [32], [66].",
"The reflection component from the inner accretion disk with a relativistically broadened iron-K$\\alpha $ line is often reported in type-1 AGNs [95], [73], [77], although it is more difficult to robustly confirm its existence in type-2 AGNs.",
"It is because when we see an AGN through the torus with an edge-on view, the features are smeared out by the absorption and more significant broadening.",
"A scattered component by gas surrounding the torus is present, which is observed as a weak unabsorbed continuum in the soft X-ray band in obscured AGNs.",
"The column density, the reflection strengths from the torus and disk, the equivalent width (EW) of an iron-K$\\alpha $ line, and the scattering fraction all carry information on the distribution of surrounding matter.",
"Moderately obscured (Compton-thin) AGNs, defined as those with line-of-sight column densities of $22 \\le \\log N_{\\rm H} < 24$ , are the most abundant AGN population in the universe [102], [1], [80].",
"Also, they are ideal targets to study gas distribution around the nucleus.",
"Unlike the case of type-1 or unobscured AGNs ($\\log N_{\\rm H} < 22$ ), the photoelectric absorption feature enables us to accurately measure $N_{\\rm H}$ and to observe the scattered X-ray light thanks to the suppression of the direct component.",
"Because effects by Compton scattering can be neglected, it is possible to accurately estimate the intrinsic X-ray luminosity in Compton-thin AGNs, whose measurement would become unavoidably somewhat model-dependent in Compton-thick AGNs ($\\log N_{\\rm H} > 24$ ).",
"Moreover, AGNs sometimes show time variation of the intrinsic luminosity and/or absorption [84], [36].",
"In that case, they provide us with valuable information such as the locus and structure of the dusty torus.",
"Hard X-ray all-sky surveys performed with INTEGRAL IBIS/ISGRI and Swift/Burst Alert Telescope (BAT) give least biased AGN samples against the obscuration [8], [7], thanks to the high penetrating power of hard X-ray photons ($> 10$ keV).",
"The Suzaku observatory [71] was capable of simultaneously observing broadband X-ray spectra of AGNs, covering typically the 0.5–40 keV band.",
"It achieved the best sensitivity at energies above 10 keV before NuSTAR as a pointing observatory [37].",
"The combination of Suzaku data and time averaged Swift/BAT spectra covering the 14–195 keV band is very powerful for studying the broadband X-ray spectra of local AGNs selected by Swift/BAT, allowing to improve the understanding of the absorbing and reprocessing material for individual obscured AGNs [104], [114], [26], [25], [97], [31], [30], [96] This article is a summary paper reporting the data of essentially all local moderately-obscured AGNs observed with both Suzaku and Swift/BAT, except for a few objects whose data have been already intensively analyzed and published.",
"The number of the targets is 45, all originally selected from the Swift/BAT 70-month catalog [7].",
"Among them, the Suzaku broadband spectra of 19 objects are reported here for the first time.",
"Our main goal is to investigate the properties of matter around the nuclei through a uniform analysis of the broadband X-ray spectra.",
"Suzaku summary papers for low luminosity AGNs and Compton-thick AGNs are presented by [49] and Tanimoto et al.",
"(in prep.",
"), respectively.",
"This paper is organized as follows.",
"Section describes the details of our sample and the overview of the data.",
"We explain our procedure of the spectral analysis in Section .",
"The results and discussion are presented in Section .",
"Section summarizes our findings.",
"We adopt the cosmological parameters of ($H_0$ , $\\Omega _{\\rm m}$ , $\\Omega _{\\rm lambda}$ ) = (70 km s$^{-1}$ Mpc$^{-1}$ , 0.3, 0.7) when calculating a distance from a redshift.",
"Unless otherwise noted, all errors are quoted at the 1$\\sigma $ confidence level for a single parameter of interest.",
"Our sample of moderately-obscured (Compton-thin) AGNs consists of 45 Swift/BAT selected AGNs [7] at $z<0.1$ whose Suzaku archival data are available as of 2015 December.",
"The advantage of this sample is its high-quality broadband X-ray spectra (0.5–150 keV) that allow us to robustly constrain the X-ray spectral features.",
"As for the sample selection by $N_{\\rm H}$ , we firstly check previous Suzaku papers that are complied by [41] and [28].",
"For the rest of objects, we refer to the results with other satellites listed in [41] and [60].",
"We exclude those that turned out to be not moderately-obscured ($22 \\le \\log N_{\\rm H} < 24$ ) AGNs from our spectral analysis.",
"As described in Section , time variation of $N_{\\rm H}$ does not affect our sample selection.",
"The hard X-ray luminosity averaged for 70 months is limited to $\\log L_{\\rm 14-195\\hspace{2.84526pt}keV} > 42 $ , since objects with $\\log L_{\\rm 14-195\\hspace{2.84526pt}keV} < 42 $ are reported in [49].",
"Radio-loud (e.g., PKS, 3C, or 4C sources) or blazar type objects, which possibly possess jets, are not included because of possible contamination of the X-ray emission due to the presence of a jet.",
"Also, we exclude 3 bright objects with complex spectra, NGC 3227, NGC 3516, and NGC 4151, which have been intensively analyzed with different models such as relativistic reflection components from the inner disk, complex absorbers, and multiple power-law components [77], [50], [74], [10] .",
"The basic information of the sample (i.e., galaxy name, position, redshift, distance, black hole mass $M_{\\rm BH}$ ) is listed in Table REF .",
"The distances are the mean value available in the NASA/IPAC Extragalactic Database (NED), or calculated from the redshift when the distances are not available in the NED.",
"We compile the black hole masses estimated by the gas dynamics around the SMBH, reverberation mapping, empirical formula using the broad-line width and luminosity in the optical band (e.g., H$\\beta $ and $\\lambda $ 5100), or relations of $M_{\\rm BH}$ with bulge properties (e.g., velocity dispersion and $K$ -band luminosity).",
"When multiple SMBH masses are available for a single object, the mean value is taken.",
"The $M_{\\rm BH}$ distribution of our sample is represented in Figure REF .",
"The mean and standard deviation of $\\log (M_{\\rm BH}/M_{\\odot })$ , where $M_{\\odot }$ is the solar mass, is $8.1\\pm 0.1$ and $0.6\\pm 0.1$ , respectively.",
"Throughout this paper, we adopt the 2–10 keV to bolometric correction factor of 20 [106], which is applicable to AGNs with Eddington ratios of $\\lambda _{\\rm Edd}< 0.1$ .",
"As described in Section REF , almost all of our objects show $\\lambda _{\\rm Edd} < 0.1$ .",
"Adoption of the luminosity-dependent bolometric correction factor of [62] does not affect our main conclusions, except for the hard X-ray luminosity and Eddington ratio correlation (Section REF ).",
"Figure: Distribution of black hole mass."
],
[
"Data reduction",
"Suzaku carries the X-ray Imaging Spectrometers [52] and the Hard X-ray Detector [94], sensitive to soft ($< 10 $ keV) and hard X-ray photons ($> 10$ keV), respectively.",
"Three of the four XISs are frontside-illuminated camera (FI-XISs; XIS-0, XIS-2, and XIS-3), and the other is a backside-illuminated one (BI-XIS; XIS-1).",
"The HXD consists of the PIN diodes and GSO scintillators.",
"Table REF lists the IDs of the Suzaku observation data we analyze.",
"If Suzaku observed an object on several occasions, we adopt the data with the longest exposure.",
"For 2MASX J0350-5018, we utilize all observations because the exposure of a single observation is found to be too short for meaningful spectral analysis.",
"FTOOLS v6.15.1 and the Suzaku calibration database released on 2015 Jan 5 are used for the data reduction.",
"We reprocess the unfiltered XIS event data in the standard manner, as described in the ABC guide[1].",
"The XIS source events are extracted from a circular region with radii of 1'–4' depending on the flux, whereas the background is taken from an off-source region within the XIS field-of-view, where no other source is present.",
"All the FI-XISs spectra available in each observation are combined into one to increase the signal-to-noise ratio.",
"We generate the XIS response matrix and ancillary response files with xisrmfgen and xissimarfgen [44], respectively.",
"We bin the XIS spectra with minimum counts of 100 per bin.",
"For HXD/PIN, we start with the “cleaned” event files provided by the Suzaku/HXD team.",
"We make the background spectrum including the “tuned” non X-ray background model [27] and the simulated Cosmic X-ray background spectrum based on [34].",
"We basically use the HXD/PIN data in the 16–40 keV band for the spectral analysis.",
"We further limit them to an energy band where the source signals are sufficiently higher than the uncertainty in the non X-ray background model: $\\sim $ 3% for exposures of less than 40 ksec and $\\sim $ 1% for longer exposures [27].",
"[1]https://heasarc.gsfc.nasa.gov/docs/suzaku/analysis/abc/ lllllrccccc Information of Targets 0pt Galaxy Name Swift ID RA.",
"Dec. Redshift $D$ $\\log M_{\\rm BH}/M_{\\odot }$ $M_{\\rm BH}$ Ref.",
"Suzaku ID Suzaku Ref.",
"(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 2MASX J0216+5126 J0216.3+5128 34.124333 51.440194 0.0288 126.1 $...$ $...$ 705006010 $$ 2MASX J0248+2630 J0249.1+2627 42.247199 26.510890 0.057997 259.3 $...$ $...$ 704013010 $$ 2MASX J0318+6829 J0318.7+6828 49.579079 68.492062 0.090100 412.0 $...$ $...$ 702075010 1 2MASX J0350-5018 J0350.1-5019 57.599042 -50.309917 0.036492 160.6 8.8 1 701017010$^\\dagger $ 2 2MASX J0444+2813 J0444.1+2813 71.037542 28.216861 0.011268 38.6 7.4 2 703021010 $$ 2MASX J0505-2351 J0505.8-2351 76.440542 -23.853889 0.035041 154.1 7.5 1 701014010 2 2MASX J0911+4528 J0911.2+4533 137.874863 45.468331 0.026782 117.1 7.5 1 703008010 $$ 2MASX J1200+0648 J1200.8+0650 180.241393 6.806423 0.036045 158.6 8.5 1 703009010 1 Ark 347 J1204.5+2019 181.1236551 20.3162130 0.022445 97.8 8.1 1 705002010 $$ ESO 103-035 J1838.4-6524 279.584750 -65.427556 0.013286 57.5 7.5 1,3 703031010 3 ESO 263-G013 J1009.3-4250 152.450875 -42.811222 0.033537 147.3 8.0 4 702120010 1,4 ESO 297-G018 J0138.6-4001 24.654833 -40.011417 0.025227 110.1 9.7 1 701015010 2 ESO 506-G027 J1238.9-2720 189.727458 -27.307833 0.025024 109.2 8.6 1 702080010 1,5 Fairall 49 J1836.9-5924 279.242875 -59.402389 0.020021 87.1 $...$ $...$ 702118010 1,6,7 Fairall 51 J1844.5-6221 281.224917 -62.364833 0.014178 45.9 8.0 5 708046010 8 IC 4518A J1457.8-4308 224.421583 -43.132111 0.016261 70.5 7.5 3 706012010 $$ LEDA 170194 J1239.3-1611 189.72 -16.23 0.040000 161.5 8.9 1,3 703007010 $$ MCG +04-48-002 J2028.5+2543 307.146083 25.733333 0.013900 60.2 7.1 4 702081010 1,5 MCG -01-05-047 J0152.8-0329 28.204167 -3.446833 0.017197 68.5 7.6 4 704043010 $$ MCG -02-08-014 J0252.7-0822 43.097481 -8.510413 0.016752 72.7 $...$ $...$ 704045010 $$ MCG -05-23-016 J0947.6-3057 146.917319 -30.948734 0.008486 36.6 7.4 1,3 700002010 1,9,10,11 Mrk 1210 J0804.2+0507 121.0244092 5.1138450 0.013496 58.4 7.9 6 702111010 1,12 Mrk 1498 J1628.1+5145 247.016937 51.775390 0.054700 244.0 8.6 1 701016010 1,2 Mrk 18 J0902.0+6007 135.493323 60.151709 0.011088 47.9 7.5 1 705001010 $$ Mrk 348 J0048.8+3155 12.1964225 31.9569681 0.015034 65.1 8.0 1 703029010 13 Mrk 417 J1049.4+2258 162.378861 22.964555 0.032756 143.8 8.0 1 702078010 1,5 Mrk 520 J2200.9+1032 330.17458 10.54972 0.026612 108.0 8.3 7 407014010 $$ Mrk 915 J2236.7-1233 339.193768 -12.545162 0.024109 105.2 8.1 5 708029010 $$ NGC 1052 J0241.3-0816 40.2699937 -8.2557642 0.005037 19.7 8.7 8 702058010 1,10,14 NGC 1142 J0255.2-0011 43.8008169 -0.1835573 0.028847 126.3 9.2 1,3,4 701013010 1,2 NGC 2110 J0552.2-0727 88.047420 -7.456212 0.007789 35.6 8.3 1 100024010 1,9,10,15 NGC 235A J0042.9-2332 10.720042 -23.541028 0.022229 96.8 8.8 1 708026010 $$ NGC 3081 J0959.5-2248 149.873080 -22.826277 0.007976 26.5 7.7 1,4,9 703013010 1,16 NGC 3431 J1051.2-1704A 162.812667 -17.008028 0.017522 76.1 $...$ $...$ 707012010 $$ NGC 4388 J1225.8+1240 186.444780 12.662086 0.008419 20.5 8.0 1,3,4,9 800017010 1,10,17 NGC 4507 J1235.6-3954 188.9026308 -39.9092628 0.011801 51.0 8.0 1,3,4,9 702048010 1,18 NGC 4992 J1309.2+1139 197.2733500 11.6341550 0.025137 109.7 8.4 1,3,4 701080010 1,4 NGC 5252 J1338.2+0433 204.5665139 4.5425817 0.022975 83.6 8.9 1,3 707028010 $$ NGC 526A J0123.8-3504 20.9766408 -35.0655289 0.019097 83.0 8.0 1,5 705044010 $$ NGC 5506 J1413.2-0312 213.3120500 -3.2075769 0.006181 23.8 7.5 1,3,4,9 701030020 1,3,10,11 NGC 6300 J1717.1-6249 259.247792 -62.820556 0.003699 13.9 7.3 3,4 702049010 1 NGC 7172 J2201.9-3152 330.5078800 -31.8696658 0.008683 33.9 8.0 1,3,4,9 703030010 1 NGC 788 J0201.0-0648 30.2768639 -6.8155172 0.013603 58.9 8.2 1,3,4 703032010 $$ UGC 03142 J0443.9+2856 70.944958 28.971917 0.021655 94.3 8.3 10 707032010 $$ UGC 12741 J2341.8+3033 355.481083 30.581750 0.017445 76.4 $...$ $...$ 704014010 $$ (1) Galaxy name.",
"(2) Swift/BAT name in the 70-month catalog [7].",
"(3)–(5) Position in units of degree and redshift taken from the NED.",
"(6) Distance in units of Mpc.",
"(7)–(8) Black hole mass and the reference.",
"(9) Observation ID of the Suzaku data we analyze.",
"(10) Paper already reporting the Suzaku spectral analysis.",
"References for black hole masses.",
"(1) [113] (2) [107] (3) [76] (4) [51] (5) [9] (6) [118] (7) [115] (8) [23] (9) [22] (10) [109] References for papers.",
"(1) [27] (2) [26] (3) [33] (4) [17] (5) [114] (6) [100] (7) [56] (8) [93] (9) [78] (10) [72] (11) [77] (12) [69] (13) [61] (14) [13] (15) [86] (16) [25] (17) [90] (18) [12] ($$ ) The Suzaku spectra are reported for the first time in this paper.",
"We also analyze the data, whose observation IDs are 701017020 and 701017030. llcrrrrr Correlations Y X Sample $N$ $\\rho ($ X$,$ Y$)$ $P($ X$,$ Y$)$ $a$ $b$ (1) (2) (3) (4) (5) (6) (7) (8) $\\log \\lambda ^{\\rm BAT}_{\\rm Edd}$ $\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}$ All 38 $0.17$ $3.1\\times 10^{-1}$ $ ... $ $ ... $ $\\Gamma $ $\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}$ All 45 $-0.26$ $8.6\\times 10^{-2}$ $...$ $...$ $\\log N_{\\rm H}$ $\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}$ All 45 $0.08$ $5.9\\times 10^{-1}$ $ ... $ $ ... $ $\\Gamma $ $\\log \\lambda ^{\\rm BAT}_{\\rm Edd}$ All 38 $0.41$ $9.7\\times 10^{-3}$ $2.11\\pm 0.01$ $0.20\\pm 0.01$ $\\log N_{\\rm H}$ $\\log \\lambda ^{\\rm BAT}_{\\rm Edd}$ All 38 $-0.04$ $7.9\\times 10^{-1}$ $ ... $ $ ... $ $\\Gamma $ $\\log N_{\\rm H}$ All 45 $-0.03$ $8.6\\times 10^{-1}$ $ ... $ $ ... $ $\\log L_{\\rm K\\alpha }$ $\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV} $ All 45 $0.89$ $4.1\\times 10^{-16}$ $0.2\\pm 1.8$ $0.94\\pm 0.04$ $R$ $\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV} $ All 45 $-0.29$ $5.6\\times 10^{-2}$ $ ... $ $ ... $ $R$ $\\log (L_{\\rm K\\alpha }/L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}) $ All 45 $0.38$ $1.0\\times 10^{-2}$ $ ... $ $ ... $ $R$ $\\log N_{\\rm H}$ All 45 $0.04$ $7.9\\times 10^{-1}$ $ ... $ $ ... $ $\\log \\lambda L_{\\lambda \\hspace{2.84526pt}12 \\mu {\\rm m}}$ $\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV} $ All 43 $0.67$ $1.0\\times 10^{-6}$ $3.7\\pm 3.0$ $0.92\\pm 0.07$ $\\log \\lambda L_{\\lambda \\hspace{2.84526pt}12 \\mu {\\rm m}}$ $\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV} $ Nuc.",
"28 $0.53$ $3.8\\times 10^{-3}$ $3.8\\pm 4.1$ $0.91\\pm 0.10$ $\\log \\lambda F_{\\lambda \\hspace{2.84526pt}12 \\mu {\\rm m}}$ $\\log F^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV} $ All 43 $0.63$ $5.8\\times 10^{-6}$ $0.9\\pm 1.0$ $1.06\\pm 0.09$ $\\log \\lambda F_{\\lambda \\hspace{2.84526pt}12 \\mu {\\rm m}}$ $\\log F^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV} $ Nuc.",
"28 $0.60$ $6.9\\times 10^{-4}$ $1.8\\pm 1.3$ $1.15\\pm 0.13$ $\\log L_{\\rm [O\\hspace{2.84526pt}{\\footnotesize IV}] }$ $\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV} $ All 33 $0.10$ $5.8\\times 10^{-1}$ $-7.8\\pm 4.9$ $1.13\\pm 0.12$ $\\log (L_{\\rm [O\\hspace{2.84526pt}{\\footnotesize IV}] }/L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}) $ $\\log f_{\\rm scat} $ All 32 $0.35$ $4.9\\times 10^{-2}$ $-2.25\\pm 0.11$ $0.98\\pm 0.07$ (1) Y varibale.",
"(2) X variable.",
"(3) Sample used for the fitting.",
"Nucleus (Nuc.)",
"corresponds to the sample whose 12 $\\mu $ m luminosities were measured at high spatial resolution.",
"(4) Number of objects of the sample.",
"(5) Spearman's Rank coefficient for the correlation.",
"(6) Null hypothesis probability of obtaining no correlation.",
"(7)-(8) Fitting parameters of Y $= a + b$ X, which are derived for the correlations with the null hypothesis probabilities smaller than 5% except for the correlations of $R$ , and that of the [O IV] versus X-ray luminosity."
],
[
"Broadband Spectral Analysis",
"In addition to the Suzaku spectra, we also utilize the Swift/BAT spectra averaged for 70 months [7].",
"XSPEC (version 12.8.1.g) is used for the spectral analysis.",
"We perform a simultaneous fit to the FI-XISs, BI-XIS, HXD/PIN, and Swift/BAT spectra, which cover the 1–10 keV, 0.5–8 keV, 16–40 keV, and 14–150 keV bands, respectively.",
"The 1.7–1.9 keV XIS spectra are excluded to avoid the response uncertainties at the Si-K edge energy.",
"Also, we do not use energy ranges where source photons are not significantly detected at 1$\\sigma $ .",
"The cross-normalizations of the Swift/BAT and HXD/PIN spectra with respect to the FI-XISs spectrum are set to 1.0 and 1.16 (1.18) for the XIS (HXD) nominal position observation, respectively, whereas that of the BI-XIS one is left as a free parameter.",
"By using the phabs model, we always consider the Galactic absorption ($N^{\\rm Gal}_{\\rm H}$ ), whose value is estimated from the H I map of [47].",
"Solar abundances given by [2] are assumed."
],
[
"Baseline Model",
"To reproduce the broadband X-ray spectra covering the 0.5–150 keV band, we start with the following baseline model: constant*zphabs*zpowerlw*zhighect +constant*zhighect*zpowerlw+pexrav+zgauss.",
"This model consists of an absorbed cutoff power law (transmitted component), a scattered component, and a reflection component from distant, cold matter accompanied by a narrow fluorescence iron-K$\\alpha $ line.",
"We fix the cutoff energy at 300 keV, a canonical value for nearby AGNs [18].",
"Through the first constant model ($N_{\\rm XIS}$ ), we take into account possible time variation of the cutoff power-law component between the Suzaku and Swift/BAT spectra.",
"The zphabs model is used to represent photoelectric absorption.",
"The second term represents scattered emission of the primary X-ray component by gas located outside the torus.",
"This unabsorbed component is assumed to have the same shape as the transmitted component with a fractional normalization of $f_{\\rm scat}$ .",
"The pexrav code [58] reproduces reflected continuum emission, whose relative strength to the transmitted component is defined by $R = \\Omega /2\\pi $ ($\\Omega $ is the solid angle of the reflector).",
"The inclination angle to the reflector is fixed at 60$^\\circ $ .",
"To avoid unphysical fitting results, we impose an upper limit of $R = 2$ , corresponding to the extreme case where the nucleus is covered by the reflector in all directions.",
"The zgauss component represents an iron-K$\\alpha $ fluorescent line, where the line width and energy is fixed at 20 eV and 6.4 keV, respectively.",
"The width corresponds to a typical velocity dispersion of $\\sim $ 2000 km s$^{-1}$ measured in local Seyfert galaxies with Chandra/HETGS by [91].",
"If the line energy is allowed to vary, the resultant value is consistent with 6.4 keV within the 99% confidence interval except for Fairall 51, Mrk 1498, and NGC 5506.",
"For the three objects, we leave the line energy as a free parameter.",
"We assume that the reflection components did not vary in accordance with the primary emission between the Suzaku and Swift/BAT observations, considering the large size of the reflector ($\\sim $ a pc scale).",
"After fitting the spectra with the above baseline model, we systematically test if inclusion of other model components improves the fit.",
"We adopt a new model if the improvement is found to be significant at a 99% confidence level (i.e., $\\Delta \\chi ^2 < -6.64$ and $< -9.21$ , which correspond to the 99% limits of the $\\chi ^2$ distribution with degrees of freedom of 1 and 2, respectively.).",
"The additional model components we consider are as follows: (1) optically-thin thermal emission from the host galaxy (apec in XSPEC), (2) partial absorption of the cutoff power-law component (zpcfabs), (3) absorption of the reflection components (zphabs), by considering that emission from a large-scale reflector like a dusty torus [42] may be subject to absorption different from that in the line of sight, and (4) emission/absorption lines (zgauss) of He-like iron ions at $6.70$ keV, H-like iron ions at $6.97$ keV, iron-K$\\beta $ at 7.06 keV, and nickel-K$\\alpha $ at 7.48 keV.",
"The Compton shoulder of an iron-K$\\alpha $ line is also considered, which is modelled by a gaussian (zgauss) at 6.31 keV [68], [90].",
"The line width ($1 \\sigma $ ) of these lines is set to 20 eV.",
"Moreover, we systematically survey other emission or absorption lines (e.g., those of ultra fast outflow) at energies above 6.4 keV by adding a line component (zgauss) with two additional free parameters (line energy and normalization).",
"We include lines if the improvement of $\\chi ^2$ is larger than $\\Delta \\chi ^2$ = 9.21.",
"We also check the absorption lines reported by [99], who analyzed XMM-Newton data, but we do not detect any of them in our Suzaku spectra at $>$ 99% confidence level.",
"It would be because they are too weak or variable.",
"In the Appendix, Table summarizes the results of the spectral analysis.",
"We obtain good fits for all targets, which fulfill either $\\chi ^2/d.o.f < 1.2$ or null hypothesis probability larger than $1\\%$ .",
"Figure REF and Figure REF show the unfolded spectra and best-fit models in the 0.5–150 keV and 4–9 keV bands, respectively.",
"Table lists the flux, absorption-corrected luminosity, Eddington ratio, and EW of the iron-K$\\alpha $ line with respect to the total contnuum.",
"Here, we define the Eddington luminosity as $L_{\\rm Edd} = 1.26 \\times 10^{38}(M_{\\rm BH}/M_{\\odot })$ erg s$^{-1}$ .",
"The information of the detected emission/absorption lines is listed in Table ."
],
[
"Relativistic Reflection Component from the Accretion Disk",
"We further examine whether the spectra statistically require relativistically blurred reflection component from the inner optically-thick accretion disk.",
"For this purpose, we use the model constant*kdblur*reflionx, where reflionx calculates a reflected continuum and emission lines from an ionized disk [87] and kdblur reproduces relativistic effects in the vicinity of a SMBH.",
"Compared with type-1 AGNs, this component, if any, would be more difficult to detect and characterize in type-2 AGNs because of the absorption and lower flux contribution in edge-on geometry.",
"Thus, we minimize the number of the free parameters.",
"Among the parameters of the reflionx model, the photon index ($\\Gamma $ ) and normalization are liked to those of the primary cutoff power-law component in the baseline model.",
"We assume two ionization parameters of the disk ($\\xi =10$ and 100).",
"The parameters of the kdblur model are a radial emissivity index $q$ (emissivity $\\propto r^{-q}$ ), inner and outer radii ($r_{\\rm in}$ and $r_{\\rm out}$ ), and inclination angle ($\\theta _{\\rm inc}$ ).",
"We allow $r_{\\rm in}$ to vary within 1–100$r_{\\rm g}$ ($r_{\\rm g}$ is the Gravitational radius); this upper limit is imposed to avoid strong coupling with the narrow iron line from distant matter.",
"We fix $q$ , $r_{\\rm out}$ , and $\\theta _{\\rm inc}$ at 3, $400r_{\\rm g}$ , and $60^\\circ $ , respectively.",
"Because the reflionx model does not have an explicit parameter of the reflection strength ($\\Omega /2\\pi $ ), we quantify it by multiplying the constant model.",
"Its upper limit is set to $3.2\\times 10^{-3}$ and $3.2\\times 10^{-4}$ for $\\xi = 10$ and 100, respectively, which reproduces the same 10–100 keV flux as the pexrav model with $\\Omega /2\\pi $ = 1 for a photon index of 1.7.",
"The disk reflection component is subject to the same (partial) absorption models as for the primary component.",
"In summary, only the inner radius ($r_{\\rm in}$ ) and normalization (constant) are left as free parameters.",
"Adding the disk-reflection component to the best-fit models obtained in Section REF , we find that fits are significantly improved at $>$ 99% confidence level in four sources (2MASX J1200+0648, Fairall 49, NGC 526A, and NGC 6300).",
"Their unfolded spectra that give a smaller $\\chi ^2$ value between the assumptions of $\\xi =10$ or 100 are shown in Figure REF .",
"Except for NGC 6300, the disk-reflection component better improves the fit at energies below 10 keV than above 10 keV.",
"That is, a broad iron-K$\\alpha $ line feature is more essential to reproduce the spectra than the reflection hump at $\\sim $ 30 keV in the first three objects.",
"Possible presence of an ionized-disk reflection component in 2MASX J1200+648 and NGC 6300 is suggested here for the first time, while it was reported for Fairall 49 by [46], consistent with our result.",
"[73] analyzed the XMM-Newton spectra of NGC 526A and found that the relativistic reflection can well reproduce the spectra but is indistinguishable from absorption models in terms of statistics.",
"Table summarizes the resultant parameters of the disk-reflection component.",
"For convenience, the normalization factor is converted into equivalent reflection strength in units of $\\Omega /2\\pi $ .",
"The fraction of these AGNs (4 out of 45) should be regarded as a conservative lower limit, because of the small ranges of parameters we have investigated and of the limitation in the spectral quality.",
"We leave detailed discussion on the relativistic reflection components in obscured AGNs for future studies.",
"Figure: (a) distribution of absorption-corrected 10–50 keV luminosity.",
"(b) distribution of Eddington ratio.",
"(c) distribution of photon index.",
"(d) distribution of hydrogen column density.The solid and dashed histograms in the upper figures refer to the luminosities measuredwith Suzaku and Swift/BAT, respectively.",
"For clarity, the dashed histograms areslightly shifted to the right.Figure: Distribution of the time variation constant of the cutoff power-law componentbetween the Suzaku and Swift/BAT observations.Figure: (a) correlation of Eddington ratio with 10–50 keV luminosityand regression function (dashed line).",
"(b) correlation of photon index with 10–50 keV luminosityofΓ=8.6-0.16logL 10-50 keV BAT \\Gamma = 8.6 - 0.16 \\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}.",
"(c) correlation of absorption column density with 10–50 keV luminosity.",
"(d) correlation of photon index with Eddington ratio and regression function (dashed line)ofΓ=2.11+0.20logλ Edd BAT \\Gamma = 2.11 + 0.20 \\log \\lambda ^{\\rm BAT}_{\\rm Edd}.",
"(e) correlation of absorption column density with Eddington ratio.",
"(f) correlation of photon index with absorption column density.The blue and black circles represent the MLAGNs(L 10-50 keV BAT ≤43.3L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV} \\le 43.3) and HLAGNs(L 10-50 keV BAT >43.3L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV} > 43.3), respectively."
],
[
"Results and Discussion",
"In this section, we summarize X-ray properties of our sample obtained from the spectral analysis described in Section REF , and investigate correlations among them.",
"We always refer to the results with the best-fit model without the relativistic disk-reflection component (Section REF ) for all targets.",
"Inclusion of the disk-reflection components little affects the other spectral parameters for the four objects reported in Section REF .",
"Then we also compare them with the MIR properties (12 $\\mu $ m and [O IV] 25.89 $\\mu $ m, hereafter [O IV] , luminosity).",
"To statistically quantify the correlation strength between two variables $($ X$,$ Y$)$ , we calculate the Spearman's rank coefficients $\\rho ($ X$,$ Y$)$ and standard Student's t-null significance levels $P($ X$,$ Y$)$ .",
"To derive a linear regression line with a form of Y$ = a + b$ X, we adopt the ordinary least-square (OLS) bisector method [45] for luminosity-luminosity correlations, or the least chi-square method for the others, unless otherwise noted.",
"Table REF gives the results for different combinations of parameters.",
"If the correlation is found to be significant at $>$ 95% confidence level, then its best-fit regression line is plotted in the corresponding figure.",
"We examine possible time variability of $N_{\\rm H}$ by compiling previous results in the literature obtained with XMM-Newton, Chandra, or NuSTAR.",
"Also, if an object was observed with Suzaku on multiple occasions, we analyze all available data with the same spectral model and derive $N_{\\rm H}$ for each epoch.",
"These results are summarized in Table .",
"Although we have to bear in mind that the best-fit $N_{\\rm H}$ depends on the continuum model adopted, 4 objects (Mrk 1210, Mrk 348, NGC 1052, and NGC 4507) seem to show significant time variability of $N_{\\rm H}$ by a factor of $>2$ within 15 years.",
"Nevertheless, we confirm that even if we adopt the averaged value of $N_{\\rm H}$ instead of the Suzaku only result, it does not affect the sample selection $(22 \\le \\log N_{\\rm H} < 24)$ and our results on the correlations of $N_{\\rm H}$ with other X-ray properties (Sections REF and REF ) and on the stacked X-ray spectral analysis (Section REF )."
],
[
"Basic X-ray Properties",
"Figure REF shows distributions of the absorption-corrected 10–50 keV luminosity ($L_{\\rm 10-50\\hspace{2.84526pt}keV}$ ), Eddington ratio ($\\lambda _{\\rm Edd}$ ), photon index ($\\Gamma $ ), and hydrogen column density ($N_{\\rm H}$ ) of the whole sample.",
"The mean and standard deviation are summarized in Table REF .",
"In Figure REF , we plot the distribution of the time-variation constant, $N_{\\rm XIS}$ , which represents the luminosity change of the primary cutoff power-law component between the Suzaku and Swift/BAT observations.",
"The mean and standard deviation are $0.05\\pm 0.03$ and $0.21\\pm 0.04$ .",
"This suggests that a typical level of variability of the primary X-ray emission on timescales of $\\sim $ a day to several years is $\\sim $ 0.2 dex.",
"ccc Mean and Standard Deviation of Spectral Parameters 0pt X $r($ X$)$ $\\sigma ($ X$)$ (1) (2) (3) $\\log L^{\\it Suzaku}_{\\rm 10-50\\hspace{2.84526pt}keV}$ $43.3\\pm 0.08$ $0.53\\pm 0.11$ $\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}$ $43.3\\pm 0.08$ $0.51\\pm 0.11$ $\\log \\lambda ^{\\it Suzaku}_{\\rm Edd}$ $-1.9\\pm 0.1$ $0.6\\pm 0.1$ $\\log \\lambda ^{\\rm BAT}_{\\rm Edd}$ $-1.9\\pm 0.1$ $0.6\\pm 0.1$ $\\Gamma $ $1.74\\pm 0.03$ $0.19\\pm 0.04$ $\\log N_{\\rm H}$ $23.1\\pm 0.09$ $0.59\\pm 0.12$ $\\log N_{\\rm XIS}$ $0.05\\pm 0.03$ $0.21\\pm 0.04$ Columns: (1) Parameter.",
"(2) Mean of X.",
"(3) Standard deviation of X.",
"Figure REF shows correlations among $\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}$ , $\\log \\lambda ^{\\rm BAT}_{\\rm Edd}$ , $\\Gamma $ , and $\\log N_{\\rm H}$ .",
"For easy check of any luminosity dependence, we divide our sample into two groups, moderate luminosity AGNs (MLAGNs) with $\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV} \\le 43.3$ and high luminosity ones (HLAGNs) with $\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}> 43.3$ , which consists of 24 and 21 objects, respectively.",
"The criterion of $\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}$ is determined so that the source numbers of the two subsamples become the same when we make spectral stacking analysis (Section REF ).",
"As inferred from Figure REF (a), there is no significant $\\log \\lambda ^{\\rm BAT}_{\\rm Edd}$ –$\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}$ correlation.",
"However, when adopting the luminosity-dependent correction factor of [62], we find a significant correlation with $P (\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}, \\log \\lambda _{\\rm Edd})= 2.9\\times 10^{-2}$ .",
"We find that photon index increases with Eddington ratio ($P (\\log \\lambda ^{\\rm BAT}_{\\rm Edd}, \\Gamma ) = 9.7 \\times 10^{-3}$ ), but does not significantly correlates with luminosity ($P (\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}, \\Gamma ) = 8.6\\times 10^{-2}$ ).",
"The dependence of photon index on black hole mass is found to be rather weak, with $P (\\log M_{\\rm BH}, \\Gamma ) = 5.0 \\times 10^{-2}$ and $\\rho (\\log M_{\\rm BH}, \\Gamma ) = -0.32$ .",
"The positive $\\Gamma $ –$\\log \\lambda _{\\rm Edd}$ correlation for luminous AGNs was also reported previously [89], [15], [117].",
"The slope we obtain ($b = 0.20\\pm 0.01$ ) is flatter than those obtained in previous studies [89], however.",
"It may be because our sample lacks high-Eddington ratio, high-luminosity AGNs ($\\log L_{\\rm X} > 44$ ), which likely show much softer spectra.",
"Most of them are identified as type-1 AGNs, as expected from the luminosity dependence of the type-1 AGN fraction [103], [102], [38], [8], [14], [16], [79].",
"We also confirm that even if we adopt the bolometric correction factor of [62], the positive correlation remains tight with $P (\\log \\lambda ^{\\rm BAT}_{\\rm Edd},\\Gamma ) = 2.1 \\times 10^{-2}$ .",
"Hence, the Eddington ratio may be an important parameter that determines the nature of the X-ray emitting corona (optical depth end electron temperature).",
"Theoretically, this correlation can be explained if Compton cooling of the corona by seed photons becomes more efficient with increasing accretion rate, leading to a smaller Compton y-parameter, and hence a softer spectrum.",
"The hydrogen column density correlates with neither $\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}$ nor $\\log \\lambda ^{\\rm BAT}_{\\rm Edd}$ .",
"Also, there is no significant correlation between $\\log N_{\\rm H}$ and $\\Gamma $ , supporting that the underlying continuum shape is properly determined without strong coupling with the absorption unlike the case of narrow band spectral analysis with limited photon statistics.",
"Figure: Correlation between the iron-Kα\\alpha EW and absorption column density.The blue and black circles represent the MLAGNs(L 10-50 keV BAT ≤43.3L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV} \\le 43.3) and HLAGNs(L 10-50 keV BAT >43.3L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV} > 43.3), respectively."
],
[
"Reflected Emission from Distant, Cold Matter",
"The narrow iron-K$\\alpha $ fluorescence line at $\\approx $ 6.4 keV is originated from distant, cold matter around the nucleus.",
"We conventionally call it “torus”, although its shape and size are still largely unknown.",
"We significantly detect the iron-K$\\alpha $ line for all the objects of our sample.",
"Except for 5 objects, the EW with respect to the reflection continuum is found to be in a range of 0.5–3 keV, which is consistent with a theoretical prediction from reflection by optically thick, cold matter within variations of inclination (EW $\\sim $ 1–2 keV; [66]) and of iron abundance by a factor of 2 [67].",
"In the rest of objects (NGC 4388, NGC 5252, LEDA 170194, Mrk 520, and Mrk 915), the EW is very large ($>3$ keV) because the reflection continuum (i.e., $R$ ) is apparently very weak.",
"We interpret that the tori in these objects are Compton-thin, thus producing a much weaker hump at $\\sim $ 30 keV and absorption iron-K edge features than what the pexrav model predicts.",
"In fact, we confirm that a Monte-Carlo based torus model by [42] can well reproduce the spectra of these objects including the continuum and iron-K$\\alpha $ emission line, as done in [98], [48], and [49].",
"Systematic application of numerical torus models to all spectra of our sample is a subject of future works.",
"The relative intensity of the iron-K$\\alpha $ line to the underlying continuum contains critical information on the covering fraction and column density of the torus.",
"Figure REF plots the observed EW with respect to the total continuum against $N_{\\rm H}$ .",
"A systematic increase in EW with $N_{\\rm H}$ is seen in log $N_{\\rm H} > 23$ , confirming previous results [28], [14].",
"This can be explained by the attenuation of the transmitted component by absorption at the same energy, which makes the EW of the iron K$\\alpha $ line larger.",
"Thus, EW is not a good indicator to discuss the torus structure.",
"To correct for the absorption effect in the continuum flux, we adopt the $L_{\\rm K\\alpha }/L_{\\rm 10-50\\hspace{2.84526pt} keV}$ ratio instead of the EW as proposed by [81].",
"Figure REF shows the correlations of $\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}$ with $\\log L_{\\rm K\\alpha }$ and $\\log ($$L_{\\rm K\\alpha }/L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt} keV}$$)$ .",
"We first calculate the OLS bisector regression line of the $\\log L_{\\rm K\\alpha }$ –$\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}$ correlation.",
"The correlation is significant with $P(\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV},\\log L_{\\rm K\\alpha }) = 4.1\\times 10^{-16} $ .",
"The regression line we obtain gives a negative correlation between $\\log ($$L_{\\rm K\\alpha }/L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt} keV}$$)$ and $\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}$ as shown in Figure REF (b).",
"The slope ($-0.06\\pm 0.04$ ) is consistent with $-0.11\\pm 0.01$ derived by [81].",
"Figure: (a) correlation of iron-Kα\\alpha line luminosity with 10–50 keV luminosityand regression function (dashed line) oflogL Kα =0.2+0.94logL 10-50 keV BAT \\log L_{\\rm K\\alpha } = 0.2 + 0.94 \\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}.",
"(b) correlation of the iron-Kα\\alpha to10–50 keV luminosity ratio with 10–50 keV luminosityand regression function (dashed line) oflog(L Kα /L 10-50 keV BAT )=0.2-0.06logL 10-50 keV BAT \\log (L_{\\rm K\\alpha }/L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}) =0.2 - 0.06 \\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}.The blue and black circles represent the MLAGNs(L 10-50 keV BAT ≤43.3L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV} \\le 43.3) and HLAGNs(L 10-50 keV BAT >43.3L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV} > 43.3), respectively.The dashed lines represent the regression line.Strength of the reflection component from the torus, $R$ $(=\\Omega /2\\pi )$ , can be also used as an indicator of the torus covering fraction.",
"Figure REF shows correlations of $R$ with $\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}$ , $\\log ($$L_{\\rm K\\alpha }/L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt} keV}$$)$ , and $\\log N_{\\rm H}$ .",
"We overplot the mean values of $R$ with 1$\\sigma $ errors calculated in each region of the X-axis parameter from all objects (black) and from only objects whose $R$ values are significantly measured (pink) (i.e., the lower and upper limits of $R$ are larger than 0 and smaller than 2, respectively), by adopting the best-fit value listed in Table .",
"Most of the results are consistent within the errors between the two calculations and show the same trends against the X-axis parameter.",
"Although the negative correlation of $R$ with luminosity is not strong ($P(\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}, R) = 5.6\\times 10^{-2}$ ), this trend is confirmed in Section REF by analyzing the stacked hard X-ray spectra of MLAGNs ($L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}\\le 43.3$ ) and HLAGNs ($L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV} > 43.3$ ).",
"On the other hand, the correlation between $R$ and $\\log ($$L_{\\rm K\\alpha }/L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt} keV}$$)$ is found to be significant with $P(\\log ($$L_{\\rm K\\alpha }/L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt} keV}$$), R) =1.0\\times 10^{-2}$ and $\\rho (\\log ($$L_{\\rm K\\alpha }/L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt} keV}$$), R) = 0.38$ .",
"This result supports that the reflection continuum and narrow iron-K$\\alpha $ emission line originate from the same material (i.e., torus).",
"We also find the trend that $R$ is larger in more obscured AGNs, although the significance is not high in this plot (but see Section REF for an analysis of averaged hard X-ray spectra of the moderately obscured AGNs with $\\log N_{\\rm H} \\le 23$ and highly obscured ones with $\\log N_{\\rm H} > 23$ ; hereafter MOAGNs and HOAGNs, respectively).",
"This is expected if the direction-averaged column density and/or covering fraction of the torus is larger in AGNs with larger line-of-sight absorptions.",
"Figure: (a) correlation of reflection strength with 10–50 keV luminosity.",
"(b) correlation of reflection strength with the iron-Kα\\alpha to 10–50 keV luminosity ratio.",
"(c) correlation of reflection strength with absorption column density.The cyan and gray circles represent the MLAGNs(L 10-50 keV BAT ≤43.3L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV} \\le 43.3) and HLAGNs(L 10-50 keV BAT >43.3L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV} > 43.3), respectively.The open circles represent data with upper limits.The black lines represent the mean and 1σ1\\sigma error of RR in each regionfor all objects, while the pink lines(slightly shifted to the right for clarity) for only objects whoseRR values are significantly measured."
],
[
"Average Hard X-ray Spectra",
"To investigate the average reflection strength in an alternative way, we analyze the Swift/BAT and HXD/PIN stacked hard X-ray spectra for the subsamples of MLAGNs and HLAGNs or those of MOAGNs and HOAGNs.",
"Because only continuum emission is present above 14 keV, we do not make any K-correction in the summation, for simplicity, by excluding 3 distant AGNs at $z>0.05$ .",
"A cutoff power-law plus its reflected component (zpowerlw*zhighect+pexrav in the XSPEC terminology) is used to reproduce the continuum.",
"The redshift is fixed to the mean value of each subsample.",
"Even at energies above 14 keV, large absorption with $\\log N_{\\rm H} \\gtrsim 23.5$ is not negligible.",
"To take into account this effect, we multiply the partial covering model, zpcfabs, to the above continuum.",
"The covering fraction is set to the fraction of the integrated fluxes of AGNs with $\\log N_{\\rm H} > 23.5$ in each subsample and the column density is fixed at their average value.",
"Thus, the reflection strength, photon index, and normalization are left as free parameters.",
"Figure REF shows the unfolded spectra of Swift/BAT and HXD/PIN for the MLAGN ($L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV} \\le 43.3$ ) and HLAGN ($L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV} > 43.3$ ) subsamples, which contain the same number of sources (21), together with their best-fit models.",
"Here, when fitting the HXD/PIN spectra, we fix the photon index at the best-fitting value obtained from each Swift/BAT spectrum.",
"The MLAGN spectrum shows significantly stronger reflection strength ($R = 1.04^{+0.17}_{-0.19}$ ) than that of the HLAGNs ($R = 0.46^{+0.08}_{-0.09}$ ), consistent with the results suggected in Section REF .",
"The confidence contours between photon index and $R$ obtained from the Swift/BAT and HXD/PIN spectra are plotted in Figure REF (c) and (f).",
"As noticed, the two results are compatible with each other, although the constraints obtained with the HXD/PIN data are much weaker owing to the limited energy band.",
"Our findings support the luminosity-dependent unified AGN scheme.",
"We also make the same analysis to the subsamples of MOAGNs ($\\log N_{\\rm H} \\le 23$ ) and HOAGNs ($\\log N_{\\rm H} > 23$ ).",
"The unfolded spectra with best-fit models and confidence contours between photon index and $R$ are plotted in Figure REF .",
"We find a significant difference in $R$ between the MOAGNs ($R = 0.59^{+0.09}_{-0.10}$ ) and HOAGNs ($R =1.03^{+0.15}_{-0.17}$ ) from the Swift/BAT spectra, confirming the trend already reported in Section REF more robustly.",
"This is consistent with the previous work by [82], who carried out stacking INTEGRAL data.",
"As we already mentioned, this implies that on average the covering fraction and/or average column density of a torus is larger in AGNs with larger line-of-sight absorptions.",
"In summary, the average reflection strength of the MLAGNs and HOAGNs is larger than that of the HLAGNs and MOAGNs, respectively.",
"Figure: Upper: for the MLAGNs (logL 10-50 keV BAT ≤43.3\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV} \\le 43.3).Lower: for HLAGNs (logL 10-50 keV BAT >43.3\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV} > 43.3).",
"(a), (d) the stacked Swift/BAT spectra.",
"(b), (e) the stacked HXD/PIN spectra.",
"(c), (f) confidence contours between photon index and reflection strength.In the left and central figures, the upperpanel shows the spectrum with the best-fit model consisting of acutoff power-law (black dashed line) and its reflection component (bluedashed line), whereas the lower panel plots the ratioof the data to the model.",
"The reduced chi-squared statistic(χ 2 /dof\\chi ^2/dof) for eachfitting result is represented in the parenthesis.In the right figures, the constraint obtainedfrom the Swift/BAT (HXD/PIN) spectrum is represented with theblack (gray) and red (blue) lines, corresponding to Δχ 2 =2.3\\Delta \\chi ^2=2.3and 4.64.6, or the 68% and 90% confidence levels, respectively.Figure: Same as Figure but for the MOAGNs (logN H ≤23\\log N_{\\rm H} \\le 23) inthe upper figures and for the HOAGNs (logN H >23\\log N_{\\rm H} > 23) in thelower figures."
],
[
"Correlations among the X-ray and MIR properties",
"AGNs are also bright in the MIR band owing to emission of hot dust in the torus heated by the primary radiation from the central engine.",
"In fact, a good correlation between the MIR and X-ray luminosities in AGNs has been reported by several works [29], [41], [65], [4].",
"Using our sample, we examine correlations of $\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}$ with 12 $\\mu $ m MIR luminosity $\\log \\lambda L_{\\rm 12\\hspace{2.84526pt}\\mu m}$ and $\\log (\\lambda L_{\\rm 12\\hspace{2.84526pt}\\mu m}/L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV})$ as plotted in Figure REF .",
"Here we refer to the nucleus 12 $\\mu $ m luminosities compiled by [4] based on subarcsecond resolution imaging.",
"For AGNs whose luminosities are not available in [4], we calculate those using the photometric data of Wide-field Infrared Survey Explorer [116].",
"Because the spatial resolution of WISE is limited ($\\sim 6^{\\prime \\prime }.5$ in the 12 $\\mu $ m band), contamination from the host galaxy may not be ignorable in these data.",
"Table summarizes the compiled $\\lambda L_{\\rm 12\\hspace{2.84526pt}\\mu m}$ luminosities.",
"We obtain a tight $\\log \\lambda L_{\\rm 12\\hspace{2.84526pt}\\mu m}$ –$\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}$ correlation with a slope $b= 0.92\\pm 0.07$ $ (0.91\\pm 0.10) $ from the whole sample (that only with the high resolution MIR data).",
"The slope is consistent with previous results [41], [4].",
"We confirm that the flux-flux correlation is also significant, indicating that the luminosity-luminosity correlation is robust against the Malmquist bias.",
"The regression line yields a negative slope ($b = -0.08\\pm 0.07$ ) in the correlation between the MIR to 10–50 keV luminosity ratio $(\\lambda L_{\\rm 12\\hspace{2.84526pt}\\mu m}/L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV})$ and the 10–50 keV luminosity.",
"As investigated in detail by [92], $\\lambda L_{\\rm 12\\hspace{2.84526pt}\\mu m}/L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}$ is predicted to monotonically increase with increasing covering factor, or decreasing half opening-angle, of the torus in the edge-on view (i.e., type-2 AGN) case.",
"Here, the half opening-angle is defined as that between the polar axis and the upper edge of the torus.",
"It is because the solid angle of the torus illuminated by the accretion disk increases with decreasing half opening-angle and consequently the reprocessed emission in the infrared band becomes stronger.",
"Hence, the negative correlation of the MIR to X-ray luminosity ratio with X-ray luminosity is consistent with the luminosity-dependent AGN unified model [103], [59], [57], [79].",
"Figure: (a) correlation between 12 μ\\mu m and 10–50 keV luminositiesand regression function (dashed line) oflogλL 12μm =3.7+0.92logL 10-50 keV BAT \\log \\lambda L_{\\rm 12\\hspace{2.84526pt}\\mu m} = 3.7 + 0.92 \\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}.",
"(b) correlation between the 12 μ\\mu m to 10–50 keV luminosity ratio and 10–50 keV luminosityand regression function (dashed line) oflog(λL 12μm /L 10-50 keV BAT )=3.7-0.08logL 10-50 keV BAT \\log (\\lambda L_{\\rm 12\\hspace{2.84526pt}\\mu m}/L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}) = 3.7 - 0.08\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}.The blue and black circles represent the MLAGNs(L 10-50 keV BAT ≤43.3L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV} \\le 43.3) and HLAGNs(L 10-50 keV BAT >43.3L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV} > 43.3), respectively.The dashed lines represent the regression line.Figure: (a) correlation between [O IV] and 10–50 keV luminositiesand regression function (dashed line) oflogL [O IV ] =-7.8+1.13logL 10-50 keV BAT \\log L_{\\rm [O\\hspace{2.84526pt}{\\footnotesize IV}] } = -7.8 + 1.13 \\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}.The dotted and dot-dashed lines represent the regression linesobtained from the type-1 and type-2 AGN samples of , respectively.",
"(b) correlation between the [O IV] to 10–50 keV luminosity ratio and scattered fractionand regression function (dashed line) oflog(L [O IV ] /L 10-50 keV BAT )=-2.25+0.98logf scat \\log (L_{\\rm [O\\hspace{2.84526pt}{\\footnotesize IV}] }/L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}) = -2.25 + 0.98 \\log f_{\\rm scat}.The blue, black, and red circles (only in the left figure)represent the MLAGNs (L 10-50 keV BAT ≤43.3L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV} \\le 43.3), HLAGNs(L 10-50 keV BAT >43.3L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV} > 43.3), and low scattering-fraction AGNs(f scat <0.5%f_{\\rm scat} < 0.5\\%), respectively."
],
[
"X-ray and [O IV] Luminosity as AGN Power Indicator",
"Gas around the torus is excited by irradiation from the central engine and scatters a part of incident photons.",
"Hence, it provides us with the information of the intrinsic luminosity even in obscured AGNs.",
"To estimate the intrinsic AGN power, some authors proposed the usage of the [O IV] line [70], [83].",
"This is because, compared with optical emission lines, [O IV] is much less affected by dust extinction in the interstellar matter and is less contaminated by starlight from the host galaxy due to the high ionization potential ($54.9$ eV).",
"To investigate the correlation of the [O IV] line to 10–50 keV luminosity, another proxy of AGN luminosity, we compile the [O IV] luminosities ($L_{\\rm [O\\hspace{2.84526pt}{\\footnotesize IV}] }$ ) from the literature [110], [111], [43], [55], as listed in Table .",
"Figure REF (a) plots $\\log L_{\\rm [O\\hspace{2.84526pt}{\\footnotesize IV}] }$ against $\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}$ .",
"The correlation is insignificant with $P (\\log L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}, \\log L_{\\rm [O\\hspace{2.84526pt}{\\footnotesize IV}] }) = 5.8\\times 10^{-1}$ .",
"Previous studies reported that the $L_{\\rm [O\\hspace{2.84526pt}{\\footnotesize IV}] }/L_{\\rm X}$ ratio of type-2 AGNs may be higher than that of type-1 AGNs [70], [83], [55].",
"They suggested that the difference is ascribed to an underestimation of the X-ray luminosity due to obscuration in type-2 AGNs or to anisotropy of the intrinsic X-ray emission.",
"For comparison, we overplot the relations of [55] by converting the 14–195 keV luminosity into the 10–50 keV one with a power-law photon index of 1.7.",
"As shown in Figure REF (a), the $L_{\\rm [O\\hspace{2.84526pt}{\\footnotesize IV}] }/L_{\\rm X}$ ratio of our sample is more similar to that of type-1 AGNs than that of the type-2 AGNs in [55].",
"As a result, we do not see significant difference in the $L_{\\rm [O\\hspace{2.84526pt}{\\footnotesize IV}] }/L_{\\rm X}$ ratio between type-1 and Compton-thin type-2 AGNs.",
"This suggests that the anisotropy of X-ray emission is unlikely.",
"A notable finding is that most of low scattering-fraction AGNs (with best-fit $f_{\\rm scat} < 0.5\\%$ ) show systematically low values of $L_{\\rm [O\\hspace{2.84526pt}{\\footnotesize IV}] }/L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}$ ratio than the average.",
"Indeed, we find a significant correlation between the $L_{\\rm [O\\hspace{2.84526pt}{\\footnotesize IV}] }/L^{\\rm BAT}_{\\rm 10-50\\hspace{2.84526pt}keV}$ ratio and the scattered fraction (see Figure REF (b)).",
"The regression line is calculated with the OLS bisector method by ignoring objects without significant detection of the scattered component.",
"We exclude Mrk 915 because of its apparently very high scattering fraction ($\\sim 40\\%$ ), which is much higher than typical values in obscured AGNs [11] and should be attributed to leaky or ionized absorbers.",
"These results well agree with that by [105] that low scattering-fraction AGNs show low [O III] to hard X-ray luminosity ratios on average.",
"This supports the picture that a significant fraction of this population of AGNs are deeply “buried” in small opening-angle tori.",
"This also implies that the [O IV] luminosity may not be an ideal indicator of the intrinsic AGN power for the whole AGN population.",
"We have analyzed the broadband (0.5–150 keV) X-ray spectra of 45 local, moderately obscured (Compton-thin) AGNs observed with Suzaku and Swift/BAT in a uniform manner.",
"The broadband X-ray spectra are basically well reproduced with the baseline model composed of an absorbed cutoff power-law component, a scattered component, and a (unabsorbed/absorbed) reflection component with a fluorescent iron-K$\\alpha $ line.",
"Additional components such as emission/absorption lines and optically-thin thermal emission in the host galaxy are also taken into account if required.",
"The main conclusions of our work are summarized as follows.",
"We evaluate time variation of the luminosity of the primary power-law component between the Suzaku and 70-month averaged Swift/BAT observations.",
"The standard deviation is $\\sim $ 0.2 dex, which can be regarded as typical variability on timescales of $\\sim $ a day to several years.",
"We find a significant correlation of photon index with Eddington ratio, but not with luminosity or black hole mass.",
"This is consistent with previous results [89], [15], [117].",
"A narrow iron-K$\\alpha $ line is significantly detected in all objects.",
"The $L_{\\rm K\\alpha }/L_{\\rm 10-50\\hspace{2.84526pt} keV}$ ratio decreases with luminosity, supporting the luminosity-dependent AGN unified model where the covering fraction of tori decreases with luminosity.",
"The average reflection strength derived from stacked spectra above 14 keV is found to be larger in less luminous ($\\log L_{\\rm 10-50\\hspace{2.84526pt}keV}\\le 43.3$ ) or heavily obscured ($\\log N_{\\rm H} > 23$ ) AGNs than in more luminous ($\\log L_{\\rm 10-50\\hspace{2.84526pt}keV} > 43.3$ ) or lightly obscured AGNs ($\\log N_{\\rm H} \\le 23$ ), respectively.",
"We confirm strong correlation between the X-ray and MIR luminosities ($\\log \\lambda L_{\\rm \\lambda 12\\hspace{2.84526pt}\\mu m}$ –$\\log L_{\\rm 10-50\\hspace{2.84526pt}keV}$ ), which results in a negative $\\log (\\lambda L_{\\rm \\lambda 12\\hspace{2.84526pt}\\mu m}/L_{\\rm 10-50\\hspace{2.84526pt}keV})$ –$\\log L_{\\rm 10-50\\hspace{2.84526pt}keV}$ correlation.",
"This is again consistent with the luminosity-dependent unified model.",
"The average [O IV] line to hard X-ray luminosity ratio obtained from our sample is lower than previous estimates using other samples of type-2 AGNs.",
"In particular, this ratio is found to be significantly lower in low scattering-fraction AGNs.",
"This suggests that the [O IV] luminosity may significantly underestimate the intrinsic luminosity of AGNs deeply buried in small opening-angle tori.",
"crcccc MIR luminosity 0pt Target Name $\\log L_{12\\hspace{1.0pt}\\mu {\\rm m}}$ Ref.",
"$\\log L_{12\\hspace{1.0pt}\\mu {\\rm m}}$ $\\log L_{\\rm [O\\hspace{2.84526pt}{\\footnotesize IV}] }$ Ref.",
"$\\log L_{\\rm [O\\hspace{2.84526pt}{\\footnotesize IV}] }$ (1) (2) (3) (4) (5) 2MASX J0216+5126 $ ... $ $ ... $ $ ... $ $ ... $ 2MASX J0248+2630 $ 44.17\\pm 0.01 $ W $ ... $ $ ... $ 2MASX J0318+6829 $ 43.90\\pm 0.01 $ W $ ... $ $ ... $ 2MASX J0350-5018 $ 43.04\\pm 0.01 $ W $ 40.5 $ 1 2MASX J0444+2813 $ 42.58\\pm 0.01 $ W $ ... $ $ ... $ 2MASX J0505-2351 $ 43.57\\pm 0.13 $ A $ 40.7 $ 1 2MASX J0911+4528 $ 43.03\\pm 0.01 $ W $ 41.0 $ 1 2MASX J1200+0648 $ 43.56\\pm 0.01 $ W $ ... $ $ ... $ Ark 347 $ 43.27\\pm 0.11 $ A $ 41.5 $ 1 ESO 103-035 $ 43.68\\pm 0.19 $ A $ 41.1 $ 2 ESO 263-G013 $ 43.56\\pm 0.03 $ A $ ... $ $ ... $ ESO 297-G018 $ 43.03\\pm 0.07 $ A $ 40.7 $ 1 ESO 506-G027 $ 43.80\\pm 0.04 $ A $ 40.8 $ 1 Fairall 49 $ 43.93\\pm 0.20 $ A $ 41.5 $ 2 Fairall 51 $ 43.39\\pm 0.04 $ A $ 40.8 $ 2 IC 4518A $ 43.47\\pm 0.06 $ A $ 41.7 $ 3 LEDA 170194 $ 43.52\\pm 0.08 $ A $ ... $ $ ... $ MCG +04-48-002 $ 43.59\\pm 0.01 $ W $ 40.7 $ 4 MCG -01-05-047 $ 42.88\\pm 0.15 $ A $ ... $ $ ... $ MCG -02-08-014 $ 42.84\\pm 0.07 $ A $ ... $ $ ... $ MCG -05-23-016 $ 43.45\\pm 0.04 $ A $ 40.6 $ 2 Mrk 1210 $ 43.67\\pm 0.01 $ W $ 40.9 $ 1 Mrk 1498 $ 44.24\\pm 0.01 $ W $ ... $ $ ... $ Mrk 18 $ 42.84\\pm 0.01 $ W $ 39.9 $ 2 Mrk 348 $ 43.59\\pm 0.01 $ W $ 41.0 $ 2 Mrk 417 $ 43.58\\pm 0.01 $ W $ 41.1 $ 1 Mrk 520 $ ... $ $ ... $ $ 41.8 $ 1 Mrk 915 $ 43.45\\pm 0.04 $ A $ 41.7 $ 1 NGC 1052 $ 42.20\\pm 0.06 $ A $ 39.0 $ 2 NGC 1142 $ 43.08\\pm 0.20 $ A $ 41.0 $ 3 NGC 2110 $ 43.08\\pm 0.06 $ A $ 40.8 $ 2 NGC 235A $ 43.30\\pm 0.16 $ A $ 41.3 $ 4 NGC 3081 $ 42.49\\pm 0.07 $ A $ 41.0 $ 2 NGC 3431 $ 42.91\\pm 0.01 $ W $ ... $ $ ... $ NGC 4388 $ 42.38\\pm 0.07 $ A $ 41.2 $ 2 NGC 4507 $ 43.68\\pm 0.04 $ A $ 41.0 $ 2 NGC 4992 $ 43.44\\pm 0.09 $ A $ 40.2 $ 1 NGC 5252 $ 43.16\\pm 0.04 $ A $ 40.9 $ 1 NGC 526A $ 43.68\\pm 0.05 $ A $ 41.1 $ 2 NGC 5506 $ 43.16\\pm 0.03 $ A $ 41.2 $ 2 NGC 6300 $ 42.51\\pm 0.11 $ A $ 39.9 $ 3 NGC 7172 $ 42.81\\pm 0.04 $ A $ 40.8 $ 2 NGC 788 $ 43.15\\pm 0.05 $ A $ 41.0 $ 2 UGC 03142 $ 43.25\\pm 0.01 $ W $ ... $ $ ... $ UGC 12741 $ 42.63\\pm 0.01 $ W $ 40.4 $ 2 (1) Galaxy name.",
"(2) 12 $\\mu $ m luminoisty.",
"(3) References for the 12 $\\mu $ m luminosity: (A) [4], (W) the data taken from the ALLWISE Source Catalog [116].",
"(4) [O IV] luminosity.",
"(5) References for the [O IV] luminosity: (1) [111] (2) [110] (3) [55] (4) [43].",
"Part of this work was financially supported by the Grant-in-Aid for JSPS Fellows for young researchers (T.K.)",
"and for Scientific Research 26400228 (Y.U.).",
"We acknowledge financial support from the CONICYT-Chile grants “EMBIGGEN\" Anillo ACT1101 (CR), FONDECYT 1141218 (CR), and Basal-CATA PFB–06/2007 (CR).",
"This research has made use of the NASA/ IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration."
],
[
"Broadband X-ray spectra and best-fitting models",
"cccccccccccccccccccc[h] Best-fit parameters -0.5pt Target Name $N^{\\rm Gal}_{\\rm H}$ $A_{\\rm pl}$ $N_{\\rm XIS}$ $N_{\\rm H}$ $N^{\\rm ref}_{\\rm H}$ $N^{\\rm pc}_{\\rm H}$ $f_{\\rm pc}$ $\\Gamma $ $f_{\\rm scat}$ $R$ $kT_1$ $kT_2$ $\\chi ^2/dof$ (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) 2MASX J0216+5126 $ 14.2 $ $ 2.96^{+0.40}_{-0.35} $ $ 0.90^{+0.11}_{-0.07} $ $ 1.54^{+0.05}_{-0.04} $ $ ... $ $ ... $ $ ... $ $ 1.84^{+0.03}_{-0.02} $ $ 0.64^{+0.37}_{-0.38} $ $ 0.00^{+0.11}_{} $ $ ... $ $ ... $ $ 430.3/403 $ 2MASX J0248+2630 $ 10.3 $ $ 1.02^{+0.27}_{-0.22} $ $ 1.84^{+0.42}_{-0.16} $ $ 22.01^{+1.01}_{-0.74} $ $ ... $ $ ... $ $ ... $ $ 1.42^{+0.08}_{-0.04} $ $ 2.41^{+0.67}_{-0.53} $ $ 0.05^{+0.48}_{-0.05} $ $ ... $ $ ... $ $ 96.0/115 $ 2MASX J0318+6829 $ 30.8 $ $ 0.88^{+0.10}_{-0.16} $ $ 1.45^{+0.33}_{-0.12} $ $ 5.41^{+0.25}_{-0.20} $ $ ... $ $ ... $ $ ... $ $ 1.52^{+0.06}_{-0.02} $ $ 3.63^{+0.92}_{-0.56} $ $ 0.00^{+0.39}_{} $ $ ... $ $ ... $ $ 278.6/234 $ 2MASX J0350-5018 $ 1.16 $ $ 0.40^{+0.85}_{-0.23} $ $ 0.73^{+0.51}_{-0.37} $ $ 41^{+33}_{-10} $ $ ... $ $ ... $ $ ... $ $ 1.53^{+0.30}_{-0.25} $ $ 9.7^{+11.5}_{-3.7} $ $ 2.00^{}_{-0.90} $ $ ... $ $ ... $ $ 31.4/41 $ 2MASX J0444+2813 $ 17.8 $ $ 1.47^{+0.31}_{-0.24} $ $ 0.86^{+0.09}_{-0.08} $ $ 8.97^{+0.35}_{-0.34} $ $ ... $ $ ... $ $ ... $ $ 1.45\\pm 0.07 $ $ 0.83^{+0.25}_{-0.22} $ $ 0.33^{+0.31}_{-0.23} $ $ ... $ $ ... $ $ 239.1/230 $ 2MASX J0505-2351 $ 2.12 $ $ 4.47^{+0.29}_{-0.20} $ $ 0.89^{+0.04}_{-0.05} $ $ 5.65^{+0.10}_{-0.12} $ $ ... $ $ 12.0^{+5.0}_{-3.9} $ $ 0.29\\pm 0.01 $ $ 1.67^{+0.04}_{-0.01} $ $ 1.06^{+0.06}_{-0.12} $ $ 0.12^{+0.05}_{-0.12} $ $ ... $ $ ... $ $ 722.3/711 $ 2MASX J0911+4528 $ 1.23 $ $ 6.7^{+2.8}_{-2.0} $ $ 0.69^{+0.14}_{-0.10} $ $ 40.3^{+2.2}_{-3.2} $ $ ... $ $ ... $ $ ... $ $ 2.12^{+0.10}_{-0.09} $ $ 0.00^{+0.35}_{} $ $ 0.29^{+0.12}_{-0.10} $ $ ... $ $ ... $ $ 110.2/96 $ 2MASX J1200+0648 $ 1.18 $ $ 1.44^{+0.32}_{-0.21} $ $ 2.78^{+0.34}_{-0.38} $ $ 8.09^{+0.21}_{-0.18} $ $ ... $ $ ... $ $ ... $ $ 1.81^{+0.04}_{-0.03} $ $ 0.19^{+0.33}_{-0.19} $ $ 2.00^{}_{-0.31} $ $ ... $ $ ... $ $ 406.9/435 $ Ark 347 $ 2.30 $ $ 0.73^{+0.50}_{-0.32} $ $ 0.50^{+0.17}_{-0.14} $ $ 24.0^{+3.8}_{-3.3} $ $ ... $ $ ... $ $ ... $ $ 1.54\\pm 0.12 $ $ 6.1^{+4.4}_{-2.4} $ $ 1.28^{+0.72}_{-0.68} $ $ 1.26^{+0.35}_{-0.20} $ $ ... $ $ 56.7/46 $ ESO 103-035 $ 5.71 $ $ 28.6^{+3.4}_{-3.6} $ $ 1.27^{+0.07}_{-0.05} $ $ 20.45^{+0.37}_{-0.44} $ $ 4.6^{+1.3}_{-1.1} $ $ 55.9^{+8.8}_{-9.5} $ $ 0.33^{+0.03}_{-0.04} $ $ 2.07^{+0.02}_{-0.03} $ $ 0.10^{+0.03}_{-0.02} $ $ 0.87^{+0.26}_{-0.22} $ $ 1.07^{+0.27}_{-0.15} $ $ ... $ $ 1126.1/1096 $ ESO 263-G013 $ 10.2 $ $ 2.80^{+0.79}_{-0.61} $ $ 0.94^{+0.13}_{-0.11} $ $ 25.64^{+1.17}_{-0.96} $ $ ... $ $ ... $ $ ... $ $ 1.67^{+0.08}_{-0.06} $ $ 0.71^{+0.24}_{-0.25} $ $ 0.05^{+0.25}_{-0.05} $ $ 0.93^{+0.13}_{-0.17} $ $ ... $ $ 96.7/108 $ ESO 297-G018 $ 1.63 $ $ 5.23^{+1.13}_{-0.95} $ $ 0.96^{+0.10}_{-0.08} $ $ 63.8^{+3.6}_{-3.4} $ $ ... $ $ ... $ $ ... $ $ 1.70\\pm 0.05 $ $ 0.26^{+0.15}_{-0.14} $ $ 0.34^{+0.13}_{-0.12} $ $ ... $ $ ... $ $ 42.9/48 $ ESO 506-G027 $ 5.45 $ $ 7.4^{+1.7}_{-1.4} $ $ 0.54^{+0.05}_{-0.04} $ $ 83.7^{+5.1}_{-4.8} $ $ ... $ $ ... $ $ ... $ $ 1.70^{+0.06}_{-0.05} $ $ 0.34^{+0.10}_{-0.08} $ $ 0.22\\pm 0.05 $ $ ... $ $ ... $ $ 70.0/60 $ Fairall 49 $ 6.47 $ $ 8.5^{+2.1}_{-2.4} $ $ 2.06^{+0.88}_{-0.43} $ $ 1.02^{+0.03}_{-0.05} $ $ ... $ $ 3.20^{+0.85}_{-0.95} $ $ 0.22^{+0.03}_{-0.04} $ $ 2.28\\pm 0.05 $ $ 1.69^{+0.45}_{-0.37} $ $ 0.98^{+0.80}_{-0.54} $ $ ... $ $ ... $ $ 1949.8/1753 $ Fairall 51 $ 6.97 $ $ 5.98^{+1.12}_{-0.89} $ $ 2.56^{+0.26}_{-0.24} $ $ 2.67^{+0.50}_{-0.63} $ $ ... $ $ 5.30^{+0.99}_{-0.47} $ $ 0.77^{+0.10}_{-0.12} $ $ 2.03\\pm 0.04 $ $ 2.00^{+0.49}_{-0.47} $ $ 2.00^{}_{-0.23} $ $ 0.19^{+0.02}_{-0.01} $ $ ... $ $ 554.8/534 $ IC 4518A $ 8.78 $ $ 6.7^{+2.5}_{-1.9} $ $ 0.71^{+0.18}_{-0.14} $ $ 20.25^{+0.97}_{-0.98} $ $ 3.53^{+1.09}_{-0.93} $ $ ... $ $ ... $ $ 2.11^{+0.09}_{-0.08} $ $ 0.66^{+0.27}_{-0.18} $ $ 2.00^{}_{-0.22} $ $ 0.24^{+0.04}_{-0.05} $ $ 1.06^{+0.21}_{-0.12} $ $ 234.1/212 $ LEDA 170194 $ 3.00 $ $ 2.27^{+0.23}_{-0.24} $ $ 0.92^{+0.08}_{-0.06} $ $ 5.24^{+0.10}_{-0.12} $ $ ... $ $ ... $ $ ... $ $ 1.57^{+0.02}_{-0.03} $ $ 1.84^{+0.22}_{-0.18} $ $ 0.00^{+0.05}_{} $ $ ... $ $ ... $ $ 738.4/643 $ MCG +04-48-002 $ 20.7 $ $ 3.38^{+0.95}_{-0.74} $ $ 0.95^{+0.12}_{-0.10} $ $ 73.5^{+6.9}_{-8.4} $ $ 26^{+152}_{-13} $ $ ... $ $ ... $ $ 1.62^{+0.06}_{-0.05} $ $ 0.97^{+0.30}_{-0.25} $ $ 0.81^{+0.50}_{-0.35} $ $ 1.01^{+0.18}_{-0.28} $ $ ... $ $ 69.9/57 $ MCG -01-05-047 $ 2.72 $ $ 2.41^{+1.26}_{-0.83} $ $ 0.50^{+0.16}_{-0.11} $ $ 18.4^{+1.4}_{-1.2} $ $ ... $ $ ... $ $ ... $ $ 1.88\\pm 0.11 $ $ 2.65^{+1.37}_{-0.92} $ $ 1.13^{+0.72}_{-0.50} $ $ ... $ $ ... $ $ 110.0/87 $ MCG -02-08-014 $ 4.46 $ $ 3.7^{+1.2}_{-1.0} $ $ 1.04^{+0.22}_{-0.18} $ $ 11.90^{+0.63}_{-0.64} $ $ ... $ $ ... $ $ ... $ $ 2.00^{+0.08}_{-0.10} $ $ 0.00^{+0.63}_{} $ $ 1.58^{+0.42}_{-0.64} $ $ ... $ $ ... $ $ 178.9/160 $ MCG -05-23-016 $ 8.70 $ $ 36.1^{+1.4}_{-1.5} $ $ 1.29^{+0.04}_{-0.03} $ $ 1.57\\pm 0.01 $ $ ... $ $ 50.6\\pm 3.0 $ $ 0.27\\pm 0.02 $ $ 1.96^{+0.02}_{-0.01} $ $ 0.47^{+0.04}_{-0.03} $ $ 0.84\\pm 0.12 $ $ ... $ $ ... $ $ 3845.9/3479 $ Mrk 1210 $ 3.45 $ $ 3.75^{+1.01}_{-0.72} $ $ 1.04^{+0.11}_{-0.14} $ $ 43.9^{+2.1}_{-2.0} $ $ ... $ $ ... $ $ ... $ $ 1.80\\pm 0.04 $ $ 0.70^{+0.31}_{-0.24} $ $ 1.81^{+0.19}_{-0.35} $ $ 0.27\\pm 0.03 $ $ 1.11^{+0.08}_{-0.07} $ $ 193.8/158 $ Mrk 1498 $ 1.83 $ $ 2.97^{+0.68}_{-0.30} $ $ 1.64^{+0.18}_{-0.23} $ $ 14.85^{+0.54}_{-0.45} $ $ ... $ $ ... $ $ ... $ $ 1.81^{+0.06}_{-0.05} $ $ 1.59^{+0.39}_{-0.42} $ $ 2.00^{}_{-0.52} $ $ 0.14^{+0.08}_{-0.06} $ $ ... $ $ 183.4/168 $ Mrk 18 $ 4.37 $ $ 0.40^{+0.57}_{-0.29} $ $ 1.12^{+1.18}_{-0.46} $ $ 10.6^{+1.6}_{-1.3} $ $ ... $ $ ... $ $ ... $ $ 1.62^{+0.31}_{-0.26} $ $ 5.2^{+12.7}_{-3.3} $ $ 1.18^{+0.82}_{-1.18} $ $ ... $ $ ... $ $ 40.4/34 $ Mrk 348 $ 5.79 $ $ 11.94^{+0.26}_{-0.14} $ $ 1.40^{+0.03}_{-0.04} $ $ 6.01^{+0.11}_{-0.05} $ $ ... $ $ 7.64^{+0.83}_{-0.66} $ $ 0.60^{0.00}_{-0.02} $ $ 1.77^{+0.01}_{} $ $ 0.13^{+0.05}_{-0.04} $ $ 0.93^{+0.12}_{-0.13} $ $ 0.85^{+0.10}_{-0.08} $ $ ... $ $ 1699.4/1767 $ Mrk 417 $ 1.88 $ $ 0.94^{+0.40}_{-0.30} $ $ 2.39^{+0.30}_{-0.51} $ $ 44.9\\pm 3.2 $ $ ... $ $ ... $ $ ... $ $ 1.60^{+0.08}_{-0.09} $ $ 0.78^{+0.84}_{-0.59} $ $ 1.76^{+0.24}_{-0.77} $ $ 1.04^{+0.20}_{-0.23} $ $ ... $ $ 74.3/70 $ Mrk 520 $ 4.30 $ $ 1.63^{+0.16}_{-0.14} $ $ 1.25^{+0.11}_{-0.07} $ $ 1.87^{+0.04}_{-0.05} $ $ ... $ $ ... $ $ ... $ $ 1.53^{+0.02}_{-0.01} $ $ 5.27^{+0.59}_{-0.54} $ $ 0.00^{+0.13}_{} $ $ ... $ $ ... $ $ 699.6/688 $ Mrk 915 $ 5.35 $ $ 0.60^{+0.13}_{-0.11} $ $ 1.28^{+0.22}_{-0.18} $ $ 1.59\\pm 0.11 $ $ ... $ $ ... $ $ ... $ $ 1.40\\pm 0.03 $ $ 44.1^{+10.1}_{-7.8} $ $ 0.00^{+0.20}_{} $ $ ... $ $ ... $ $ 304.5/277 $ NGC 1052 $ 2.83 $ $ 2.41^{+0.32}_{-0.37} $ $ 1.21\\pm 0.14 $ $ 4.86^{+0.29}_{-0.16} $ $ ... $ $ 19.8^{+2.5}_{-2.6} $ $ 0.71^{+0.02}_{-0.01} $ $ 1.75^{+0.06}_{-0.04} $ $ 4.81^{+0.29}_{-0.80} $ $ 0.35^{+0.28}_{-0.35} $ $ 0.82\\pm 0.05 $ $ ... $ $ 416.1/425 $ NGC 1142 $ 5.81 $ $ 4.48^{+0.74}_{-0.64} $ $ 0.96\\pm 0.07 $ $ 60.9^{+2.4}_{-2.2} $ $ 7.0^{+2.5}_{-2.3} $ $ ... $ $ ... $ $ 1.66\\pm 0.04 $ $ 0.25^{+0.09}_{-0.08} $ $ 0.78^{+0.22}_{-0.17} $ $ 0.19\\pm 0.07 $ $ 0.97\\pm 0.05 $ $ 200.6/204 $ NGC 2110 $ 2.18 $ $ 18.70\\pm 0.42 $ $ 1.74\\pm 0.04 $ $ 2.32^{+0.07}_{-0.06} $ $ ... $ $ 2.80\\pm 0.09 $ $ 0.64\\pm 0.03 $ $ 1.65\\pm 0.01 $ $ 0.28\\pm 0.04 $ $ 0.38\\pm 0.06 $ $ 0.98\\pm 0.02 $ $ ... $ $ 4276.3/3823 $ NGC 235A $ 1.41 $ $ 5.0^{+1.8}_{-1.3} $ $ 0.61^{+0.10}_{-0.09} $ $ 65.2^{+7.0}_{-6.3} $ $ ... $ $ ... $ $ ... $ $ 1.78^{+0.07}_{-0.08} $ $ 0.51^{+0.23}_{-0.18} $ $ 0.34^{+0.16}_{-0.14} $ $ 0.59^{+0.07}_{-0.08} $ $ ... $ $ 37.8/28 $ NGC 3081 $ 3.88 $ $ 7.6^{+1.6}_{-1.3} $ $ 0.72\\pm 0.05 $ $ 82.5^{+4.0}_{-3.8} $ $ ... $ $ ... $ $ ... $ $ 1.73\\pm 0.05 $ $ 0.52^{+0.13}_{-0.10} $ $ 0.20^{+0.05}_{-0.06} $ $ 0.19^{+0.01}_{-0.02} $ $ 0.98\\pm 0.08 $ $ 80.1/69 $ NGC 3431 $ 4.17 $ $ 0.52^{+0.26}_{-0.12} $ $ 3.01^{+0.68}_{-1.05} $ $ 7.08^{+0.28}_{-0.29} $ $ ... $ $ ... $ $ ... $ $ 1.61^{+0.02}_{-0.07} $ $ 1.42^{+0.78}_{-0.75} $ $ 2.0^{}_{-1.3} $ $ ... $ $ ... $ $ 245.9/234 $ NGC 4388 $ 2.58 $ $ 19.2\\pm 1.1 $ $ 0.88^{+0.01}_{-0.02} $ $ 23.78^{+0.69}_{-0.70} $ $ ... $ $ 43.1\\pm 3.6 $ $ 0.56\\pm 0.03 $ $ 1.65\\pm 0.01 $ $ 0.65^{+0.04}_{-0.03} $ $ 0.08^{+0.05}_{-0.04} $ $ 0.24^{+0.01}_{-0.02} $ $ 0.97\\pm 0.02 $ $ 1669.1/1703 $ NGC 4507 $ 7.04 $ $ 19.7^{+2.5}_{-2.3} $ $ 0.57^{+0.03}_{-0.02} $ $ 26.9^{+5.2}_{-3.9} $ $ ... $ $ 79.9^{+5.1}_{-5.7} $ $ 0.91\\pm 0.02 $ $ 1.79^{+0.03}_{-0.02} $ $ 0.31^{+0.05}_{-0.04} $ $ 0.43^{+0.08}_{-0.07} $ $ 0.15\\pm 0.01 $ $ 0.81\\pm 0.02 $ $ 387.0/360 $ NGC 4992 $ 1.93 $ $ 2.17^{+0.65}_{-0.51} $ $ 1.05^{+0.15}_{-0.13} $ $ 60.1^{+3.6}_{-3.4} $ $ ... $ $ ... $ $ ... $ $ 1.57\\pm 0.06 $ $ 0.00^{+0.17}_{} $ $ 0.52^{+0.18}_{-0.13} $ $ ... $ $ ... $ $ 74.0/61 $ NGC 5252 $ 2.14 $ $ 8.88^{+0.87}_{-0.77} $ $ 0.47\\pm 0.02 $ $ 2.11^{+0.53}_{-0.52} $ $ ... $ $ 5.86^{+0.36}_{-0.33} $ $ 0.81^{+0.05}_{-0.06} $ $ 1.66^{+0.03}_{-0.02} $ $ 0.39^{+0.24}_{-0.32} $ $ 0.00^{+0.02}_{} $ $ 0.16\\pm 0.02 $ $ 0.84^{+0.06}_{-0.05} $ $ 472.6/443 $ NGC 526A $ 2.31 $ $ 3.12\\pm 0.39 $ $ 3.78^{+0.58}_{-0.43} $ $ 1.22\\pm 0.01 $ $ ... $ $ ... $ $ ... $ $ 1.68\\pm 0.01 $ $ 6.46^{+0.71}_{-0.84} $ $ 0.99^{+0.43}_{-0.33} $ $ ... $ $ ... $ $ 2467.0/2335 $ NGC 5506 $ 4.08 $ $ 26.29^{+0.83}_{-0.59} $ $ 1.73\\pm 0.03 $ $ 3.10^{+0.01}_{-0.02} $ $ ... $ $ ... $ $ ... $ $ 1.95^{0.00}_{-0.01} $ $ 1.09^{+0.04}_{-0.05} $ $ 2.00^{}_{-0.07} $ $ ... $ $ ... $ $ 3398.7/3185 $ NGC 6300 $ 7.79 $ $ 11.43^{+1.00}_{-0.93} $ $ 1.04^{+0.06}_{-0.05} $ $ 22.21^{+0.35}_{-0.34} $ $ ... $ $ ... $ $ ... $ $ 1.86\\pm 0.02 $ $ 0.56\\pm 0.08 $ $ 0.83^{+0.12}_{-0.10} $ $ 0.85^{+0.06}_{-0.07} $ $ ... $ $ 765.9/723 $ NGC 7172 $ 1.95 $ $ 14.18^{+0.53}_{-0.51} $ $ 1.44^{+0.04}_{-0.05} $ $ 8.90\\pm 0.07 $ $ ... $ $ ... $ $ ... $ $ 1.74^{+0.01}_{-0.02} $ $ 0.12\\pm 0.03 $ $ 0.34^{+0.10}_{-0.09} $ $ 0.33^{+0.14}_{-0.04} $ $ ... $ $ 2056.7/2033 $ NGC 788 $ 2.12 $ $ 5.35^{+1.16}_{-0.99} $ $ 1.19^{+0.13}_{-0.11} $ $ 73.4^{+4.0}_{-3.8} $ $ 11.9^{+3.4}_{-3.1} $ $ ... $ $ ... $ $ 1.77\\pm 0.05 $ $ 0.71^{+0.19}_{-0.14} $ $ 1.21^{+0.48}_{-0.35} $ $ 0.75^{+0.08}_{-0.11} $ $ ... $ $ 88.8/87 $ UGC 03142 $ 17.6 $ $ 1.11^{+0.34}_{-0.17} $ $ 1.45^{+0.17}_{-0.26} $ $ 1.59^{+0.37}_{-0.32} $ $ ... $ $ 9.67^{+0.94}_{-0.65} $ $ 0.79^{+0.01}_{-0.03} $ $ 1.58^{+0.07}_{-0.08} $ $ 3.92^{+0.81}_{-1.45} $ $ 2.00^{}_{-0.77} $ $ ... $ $ ... $ $ 205.8/231 $ UGC 12741 $ 5.79 $ $ 1.88^{+1.28}_{-0.80} $ $ 1.47^{+0.55}_{-0.33} $ $ 60.8^{+5.0}_{-4.7} $ $ ... $ $ ... $ $ ... $ $ 1.79^{+0.12}_{-0.11} $ $ 0.00^{+0.78}_{} $ $ 0.62^{+0.42}_{-0.24} $ $ ... $ $ ... $ $ 52.5/41 $ (1) Galaxy name.",
"(2) Galactic absorption in units of $10^{20}$ cm$^{-2}$ .",
"(3) Normalization of the cutoff power-law component at 1 keV in units of $10^{-3}$ photons keV$^{-1}$ cm$^{-2}$ s$^{-1}$ .",
"(4) Time variability of the cutoff power-law component between the Suzaku and Swift/BAT spectra.",
"(5) Intrinsic absorption in units of $10^{22}$ cm$^{-2}$ .",
"(6) Absorption of the reflection components in units of $10^{22}$ cm$^{-2}$ .",
"(7) Partial absorption of the cutoff power-law component in units of $10^{22}$ cm$^{-2}$ .",
"(8) Covering fraction of the partial absorption of the cutoff power-law component.",
"(9) Photon index of the cutoff power-law component.",
"(10) Scattered fraction in units of %.",
"(11) Relative reflection strength ($R = \\Omega /2\\pi $ ) of the pexrav model.",
"(12)–(13) Temperatures of the apec models in units of keV.",
"(14) Chi squared and degrees of freedom.",
"cccccccccccccccccccc Flux and luminosity 0pt Target Name $\\log F^{\\rm BI-XIS}_{\\rm 0.5-2}$ $\\log F^{\\rm FI-XIS}_{\\rm 2-10}$ $\\log F^{\\rm PIN\\ast }_{\\rm 10-50}$ $\\log F^{\\rm BAT}_{\\rm 10-50}$ $\\log L^{\\rm BI-XIS}_{\\rm 0.5-2}$ $\\log L^{\\rm FI-XIS}_{\\rm 2-10}$ $\\log L^{\\rm PIN\\ast }_{\\rm 10-50}$ $\\log L^{\\rm BAT}_{\\rm 10-50}$ EW $L_{{\\rm K}\\alpha }/L^{\\rm BAT}_{\\rm 10-50}$ $\\lambda ^{\\it Suzaku}_{\\rm Edd}/\\lambda ^{\\rm BAT}_{\\rm Edd}$ (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) 2MASX J0216+5126 $ -12.2 $ $ -11.1 $ $ -11.0 $ $ -11.0 $ $ 43.0 $ $ 43.2 $ $ 43.3 $ $ 43.3 $ $ 35^{+13}_{-12} $ $ 2.58^{+0.93}_{-0.86} $ $ .../ ... $ 2MASX J0248+2630 $ -13.4 $ $ -11.3 $ $ -10.6 $ $ -10.9 $ $ 43.5 $ $ 43.9 $ $ 44.3 $ $ 44.1 $ $ 61\\pm 16 $ $ 3.56\\pm 0.91 $ $ .../ ... $ 2MASX J0318+6829 $ -13.3 $ $ -11.4 $ $ -10.9 $ $ -11.1 $ $ 43.7 $ $ 44.1 $ $ 44.4 $ $ 44.2 $ $ 42^{+13}_{-12} $ $ 3.08^{+0.92}_{-0.89} $ $ .../ ... $ 2MASX J0350-5018 $ -13.1 $ $ -12.1 $ $ -11.1 $ $ -11.1 $ $ 42.3 $ $ 42.7 $ $ 43.4 $ $ 43.4 $ $ 376^{+65}_{-61} $ $ 5.97^{+1.03}_{-0.96} $ $ -2.8/-2.7 $ 2MASX J0444+2813 $ -13.6 $ $ -11.3 $ $ -10.7 $ $ -10.6 $ $ 41.7 $ $ 42.1 $ $ 42.6 $ $ 42.6 $ $ 119\\pm 13 $ $ 3.82\\pm 0.40 $ $ -2.0/-2.0 $ 2MASX J0505-2351 $ -12.8 $ $ -11.0 $ $ -10.6 $ $ -10.5 $ $ 43.4 $ $ 43.7 $ $ 43.9 $ $ 43.9 $ $ 60\\pm 6 $ $ 2.96\\pm 0.31 $ $ -0.7/-0.6 $ 2MASX J0911+4528 $ -14.2 $ $ -11.7 $ $ -11.1 $ $ -11.0 $ $ 43.2 $ $ 43.2 $ $ 43.2 $ $ 43.3 $ $ 43\\pm 15 $ $ 1.67\\pm 0.57 $ $ -1.2/-1.0 $ 2MASX J1200+0648 $ -13.5 $ $ -11.1 $ $ -10.7 $ $ -10.9 $ $ 43.4 $ $ 43.6 $ $ 43.8 $ $ 43.6 $ $ 57^{+9}_{-8} $ $ 5.67^{+0.88}_{-0.76} $ $ -1.7/-2.1 $ Ark 347 $ -12.9 $ $ -12.0 $ $ -11.0 $ $ -10.9 $ $ 42.0 $ $ 42.4 $ $ 43.0 $ $ 43.2 $ $ 149\\pm 46 $ $ 2.43\\pm 0.74 $ $ -2.5/-2.2 $ ESO 103-035 $ -13.1 $ $ -10.6 $ $ -10.1 $ $ -10.1 $ $ 43.5 $ $ 43.5 $ $ 43.6 $ $ 43.5 $ $ 52\\pm 5 $ $ 3.50\\pm 0.36 $ $ -0.8/-0.9 $ ESO 263-G013 $ -13.2 $ $ -11.4 $ $ -10.8 $ $ -10.8 $ $ 43.2 $ $ 43.4 $ $ 43.7 $ $ 43.7 $ $ 85\\pm 16 $ $ 3.35\\pm 0.64 $ $ -1.3/-1.3 $ ESO 297-G018 $ -13.5 $ $ -11.5 $ $ -10.5 $ $ -10.5 $ $ 43.2 $ $ 43.5 $ $ 43.7 $ $ 43.7 $ $ 154\\pm 23 $ $ 3.14\\pm 0.47 $ $ -3.0/-3.0 $ ESO 506-G027 $ -13.3 $ $ -11.7 $ $ -10.6 $ $ -10.4 $ $ 43.1 $ $ 43.3 $ $ 43.6 $ $ 43.8 $ $ 465^{+33}_{-32} $ $ 4.26\\pm 0.30 $ $ -2.0/-1.8 $ Fairall 49 $ -11.3 $ $ -10.6 $ $ -10.7 $ $ -10.9 $ $ 43.5 $ $ 43.4 $ $ 43.3 $ $ 43.0 $ $ 49\\pm 6 $ $ 10.3\\pm 1.2 $ $ .../ ... $ Fairall 51 $ -12.1 $ $ -10.7 $ $ -10.3 $ $ -10.6 $ $ 42.9 $ $ 42.9 $ $ 43.1 $ $ 42.8 $ $ 40\\pm 8 $ $ 5.21^{+1.03}_{-0.98} $ $ -1.8/-2.2 $ IC 4518A $ -12.8 $ $ -11.3 $ $ -10.7 $ $ -10.7 $ $ 42.8 $ $ 42.8 $ $ 43.0 $ $ 43.1 $ $ 56\\pm 17 $ $ 2.53^{+0.77}_{-0.76} $ $ -1.5/-1.3 $ LEDA 170194 $ -12.9 $ $ -11.2 $ $ -10.7 $ $ -10.7 $ $ 43.1 $ $ 43.5 $ $ 43.8 $ $ 43.8 $ $ 66^{+8}_{-7} $ $ 3.30^{+0.40}_{-0.36} $ $ -2.2/-2.2 $ MCG +04-48-002 $ -13.1 $ $ -11.7 $ $ -10.5 $ $ -10.5 $ $ 42.5 $ $ 42.8 $ $ 43.2 $ $ 43.2 $ $ 183^{+2590}_{-46} $ $ 4.0^{+57.1}_{-1.0} $ $ -1.2/-1.1 $ MCG -01-05-047 $ -12.9 $ $ -11.7 $ $ -11.0 $ $ -10.9 $ $ 42.2 $ $ 42.4 $ $ 42.7 $ $ 42.9 $ $ 221\\pm 26 $ $ 6.07\\pm 0.71 $ $ -2.0/-1.7 $ MCG -02-08-014 $ -13.7 $ $ -11.3 $ $ -10.8 $ $ -10.8 $ $ 42.7 $ $ 42.8 $ $ 43.0 $ $ 43.0 $ $ 117\\pm 15 $ $ 6.77\\pm 0.85 $ $ .../ ... $ MCG -05-23-016 $ -11.1 $ $ -10.1 $ $ -9.8 $ $ -9.9 $ $ 43.2 $ $ 43.3 $ $ 43.4 $ $ 43.3 $ $ 58\\pm 3 $ $ 4.65\\pm 0.22 $ $ -0.9/-1.0 $ Mrk 1210 $ -12.6 $ $ -11.3 $ $ -10.5 $ $ -10.5 $ $ 42.6 $ $ 42.8 $ $ 43.1 $ $ 43.1 $ $ 190\\pm 15 $ $ 5.15\\pm 0.40 $ $ -1.9/-1.9 $ Mrk 1498 $ -12.3 $ $ -11.1 $ $ -10.5 $ $ -10.6 $ $ 43.9 $ $ 44.1 $ $ 44.4 $ $ 44.3 $ $ 72\\pm 12 $ $ 4.00\\pm 0.68 $ $ -1.3/-1.5 $ Mrk 18 $ -13.4 $ $ -11.8 $ $ -11.3 $ $ -11.3 $ $ 41.4 $ $ 41.8 $ $ 42.2 $ $ 42.2 $ $ 248^{+55}_{-56} $ $ 10.0\\pm 2.2 $ $ -2.4/-2.5 $ Mrk 348 $ -12.7 $ $ -10.5 $ $ -10.0 $ $ -10.1 $ $ 43.3 $ $ 43.5 $ $ 43.7 $ $ 43.6 $ $ 45^{+4}_{-5} $ $ 2.93^{+0.25}_{-0.30} $ $ -1.3/-1.4 $ Mrk 417 $ -13.3 $ $ -11.5 $ $ -10.6 $ $ -10.8 $ $ 43.1 $ $ 43.4 $ $ 43.8 $ $ 43.6 $ $ 125\\pm 21 $ $ 4.72\\pm 0.78 $ $ -1.4/-1.8 $ Mrk 520 $ -12.2 $ $ -11.1 $ $ -10.7 $ $ -10.8 $ $ 42.8 $ $ 43.1 $ $ 43.5 $ $ 43.4 $ $ 92\\pm 9 $ $ 6.20^{+0.62}_{-0.59} $ $ -2.0/-2.1 $ Mrk 915 $ -12.1 $ $ -11.2 $ $ -10.8 $ $ -10.9 $ $ 42.5 $ $ 42.9 $ $ 43.3 $ $ 43.2 $ $ 133^{+18}_{-17} $ $ 7.39^{+1.01}_{-0.93} $ $ -2.0/-2.0 $ NGC 1052 $ -12.5 $ $ -11.3 $ $ -10.8 $ $ -10.8 $ $ 41.5 $ $ 41.7 $ $ 41.9 $ $ 41.9 $ $ 114\\pm 10 $ $ 6.63^{+0.58}_{-0.60} $ $ -3.7/-3.8 $ NGC 1142 $ -13.0 $ $ -11.4 $ $ -10.4 $ $ -10.4 $ $ 43.2 $ $ 43.6 $ $ 43.9 $ $ 43.9 $ $ 226^{+16}_{-15} $ $ 4.66^{+0.33}_{-0.31} $ $ -2.5/-2.5 $ NGC 2110 $ -11.6 $ $ -10.0 $ $ -9.6 $ $ -9.8 $ $ 43.0 $ $ 43.3 $ $ 43.6 $ $ 43.4 $ $ 34\\pm 2 $ $ 3.04\\pm 0.17 $ $ -1.7/-2.0 $ NGC 235A $ -12.8 $ $ -11.7 $ $ -10.8 $ $ -10.6 $ $ 42.9 $ $ 43.1 $ $ 43.3 $ $ 43.5 $ $ 171\\pm 69 $ $ 2.7\\pm 1.1 $ $ -2.5/-2.3 $ NGC 3081 $ -12.6 $ $ -11.5 $ $ -10.5 $ $ -10.4 $ $ 42.0 $ $ 42.3 $ $ 42.5 $ $ 42.6 $ $ 313\\pm 25 $ $ 3.90\\pm 0.31 $ $ -2.2/-2.1 $ NGC 3431 $ -13.5 $ $ -11.3 $ $ -10.8 $ $ -11.1 $ $ 42.4 $ $ 42.7 $ $ 43.1 $ $ 42.8 $ $ 102\\pm 13 $ $ 8.4\\pm 1.1 $ $ .../ ... $ NGC 4388 $ -12.3 $ $ -10.7 $ $ -9.9 $ $ -9.9 $ $ 42.3 $ $ 42.6 $ $ 42.8 $ $ 42.9 $ $ 190\\pm 5 $ $ 4.92\\pm 0.12 $ $ -2.2/-2.2 $ NGC 4507 $ -12.3 $ $ -11.2 $ $ -10.2 $ $ -10.0 $ $ 42.9 $ $ 43.1 $ $ 43.3 $ $ 43.5 $ $ 421\\pm 14 $ $ 4.30\\pm 0.15 $ $ -1.7/-1.5 $ NGC 4992 $ -14.3 $ $ -11.6 $ $ -10.6 $ $ -10.6 $ $ 42.8 $ $ 43.3 $ $ 43.6 $ $ 43.6 $ $ 168\\pm 30 $ $ 3.17\\pm 0.57 $ $ -2.0/-2.1 $ NGC 5252 $ -12.4 $ $ -11.0 $ $ -10.6 $ $ -10.2 $ $ 42.9 $ $ 43.1 $ $ 43.4 $ $ 43.7 $ $ 83^{+10}_{-9} $ $ 2.29^{+0.28}_{-0.24} $ $ -2.5/-2.2 $ NGC 526A $ -11.3 $ $ -10.4 $ $ -10.1 $ $ -10.5 $ $ 43.3 $ $ 43.6 $ $ 43.9 $ $ 43.4 $ $ 48\\pm 4 $ $ 7.85\\pm 0.67 $ $ -1.2/-1.7 $ NGC 5506 $ -11.4 $ $ -10.0 $ $ -9.7 $ $ -9.8 $ $ 42.8 $ $ 43.0 $ $ 43.1 $ $ 43.0 $ $ 44^{+4}_{-2} $ $ 3.87^{+0.32}_{-0.22} $ $ -1.3/-1.5 $ NGC 6300 $ -12.7 $ $ -10.8 $ $ -10.2 $ $ -10.2 $ $ 41.8 $ $ 42.0 $ $ 42.2 $ $ 42.1 $ $ 67\\pm 7 $ $ 3.22\\pm 0.33 $ $ -2.1/-2.2 $ NGC 7172 $ -12.8 $ $ -10.3 $ $ -9.9 $ $ -10.1 $ $ 42.8 $ $ 43.0 $ $ 43.2 $ $ 43.1 $ $ 52\\pm 4 $ $ 4.04\\pm 0.33 $ $ -1.7/-1.9 $ NGC 788 $ -12.9 $ $ -11.4 $ $ -10.4 $ $ -10.4 $ $ 42.8 $ $ 43.0 $ $ 43.3 $ $ 43.2 $ $ 223^{+24}_{-22} $ $ 5.45^{+0.58}_{-0.54} $ $ -2.0/-2.0 $ UGC 03142 $ -12.8 $ $ -11.3 $ $ -10.6 $ $ -10.7 $ $ 42.6 $ $ 42.9 $ $ 43.4 $ $ 43.3 $ $ 156\\pm 15 $ $ 6.24^{+0.60}_{-0.59} $ $ -2.1/-2.3 $ UGC 12741 $ -14.4 $ $ -11.8 $ $ -10.9 $ $ -11.0 $ $ 42.6 $ $ 42.8 $ $ 43.0 $ $ 42.9 $ $ 150\\pm 28 $ $ 4.96^{+0.92}_{-0.93} $ $ .../ ... $ (1) Galaxy name.",
"(2)–(5) Logarithmic observed flux in the 0.5–2 keV (BI-XIS), 2–10 keV (FI-XISs), 10–50 keV (PIN), and 10–50 keV (BAT) bands in units of erg cm$^{-2}$ s$^{-1}$ .",
"(6)–(9) Logarithmic absorption-corrected luminosity in the same energy bands as (2)–(5) in units of erg s$^{-1}$ , respectively.",
"(10) Equivalent width of the iron-K$\\alpha $ line in units of eV.",
"(11) Ratio of the iron-K$\\alpha $ line to 10–50 keV continuum luminosity in units of $\\times 10^{-3}$ .",
"(12) Logarithmic Eddington ratio based on the 2–10 keV luminosity measured with Suzaku and Swift/BAT.",
"* According to the XIS or HXD nominal position observation, the flux and luminosity are divided by 1.16 or 1.18 to take into account the instrumental cross-calibration factor between the FI-XISs and HXD/PIN spectra.",
"cccccccccccccccccccc Information of detected emission/absorption lines -2pt Target Name $N_{\\rm 6.4\\hspace{2.84526pt}keV}$ $N_{\\rm 6.31\\hspace{2.84526pt}keV}$ $N_{\\rm 6.70\\hspace{2.84526pt}keV}$ $N_{\\rm 6.97\\hspace{2.84526pt}keV}$ $N_{\\rm 7.06\\hspace{2.84526pt}keV}$ $N_{\\rm 7.48\\hspace{2.84526pt}keV}$ $E_{1}$ $N_{1}$ $E_{2}$ $N_{2}$ $E_{3}$ $N_{3}$ $E_{4}$ $N_{4}$ (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) 2MASX J0216+5126 $ 2.9^{+1.1}_{-1.0} $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ 2MASX J0248+2630 $ 5.7\\pm 1.4 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ 2MASX J0318+6829 $ 3.0\\pm 0.9 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ 2MASX J0350-5018 $ 5.7^{+1.0}_{-0.9} $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ 2MASX J0444+2813 $ 9.1\\pm 1.0 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ 2MASX J0505-2351 $ 9.3\\pm 1.0 $ $ ... $ $ ... $ $ ... $ $ 3.3\\pm 0.9 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ 2MASX J0911+4528 $ 2.0\\pm 0.7 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ 2MASX J1200+0648 $ 7.4^{+1.1}_{-1.0} $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ Ark 347 $ 3.2\\pm 1.0 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ ESO 103-035 $ 27.1\\pm 2.8 $ $ 6.4\\pm 2.4 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ ESO 263-G013 $ 6.4\\pm 1.2 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ ESO 297-G018 $ 11.7\\pm 1.7 $ $ ... $ $ -4.5^{+1.5}_{-1.6} $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ ESO 506-G027 $ 21.4\\pm 1.5 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ Fairall 49 $ 12.7\\pm 1.5 $ $ ... $ $ 8.7\\pm 1.5 $ $ 7.2\\pm 1.5 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ Fairall 51 $ ... $ $ ... $ $ -24.7\\pm 2.3 $ $ -14.1^{+2.5}_{-2.4} $ $ ... $ $ ... $ $ 6.31\\pm 0.03 $ $ 14.1^{+2.8}_{-2.7} $ $ 8.00\\pm 0.04 $ $ -13.2^{+2.9}_{-3.2} $ $ 8.29^{+0.06}_{-0.05} $ $ -11.1^{+3.6}_{-3.2} $ $ 8.67^{+0.04}_{-0.05} $ $ -13.4^{+3.5}_{-3.3} $ IC 4518A $ 5.5^{+1.7}_{-1.6} $ $ 4.7\\pm 1.5 $ $ -5.9^{+0.8}_{-0.9} $ $ -2.6\\pm 0.9 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ LEDA 170194 $ 6.9\\pm 0.8 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ 0.88\\pm 0.02 $ $ 5.5^{+2.1}_{-1.9} $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ MCG +04-48-002 $ 15.3^{+217.2}_{-3.9} $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ MCG -01-05-047 $ 8.2\\pm 1.0 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ MCG -02-08-014 $ 10.6\\pm 1.3 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ 7.46^{+0.09}_{-0.05} $ $ -4.8\\pm 1.2 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ MCG -05-23-016 $ 62.6\\pm 3.0 $ $ 14.5\\pm 2.9 $ $ ... $ $ ... $ $ 7.0\\pm 2.2 $ $ 8.4\\pm 2.2 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ Mrk 1210 $ 16.8\\pm 1.3 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ 1.18\\pm 0.01 $ $ 4.2\\pm 1.0 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ Mrk 1498 $ ... $ $ ... $ $ ... $ $ ... $ $ 6.6\\pm 1.9 $ $ ... $ $ 6.29\\pm 0.02 $ $ 11.3\\pm 1.9 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ Mrk 18 $ 5.3\\pm 1.2 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ Mrk 348 $ 25.3^{+2.2}_{-2.6} $ $ 10.3^{+1.9}_{-2.6} $ $ -5.9^{+1.7}_{-1.8} $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ Mrk 417 $ 7.6\\pm 1.3 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ Mrk 520 $ 10.9^{+1.1}_{-1.0} $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ Mrk 915 $ 10.0^{+1.4}_{-1.3} $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ NGC 1052 $ 10.1\\pm 0.9 $ $ ... $ $ -1.3\\pm 0.8 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ NGC 1142 $ 20.8^{+1.5}_{-1.4} $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ NGC 2110 $ 48.6\\pm 2.8 $ $ 18.1\\pm 2.7 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ NGC 235A $ 7.5\\pm 3.0 $ $ 8.1\\pm 3.1 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ NGC 3081 $ 17.8\\pm 1.4 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ NGC 3431 $ 7.6\\pm 1.0 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ NGC 4388 $ 70.3\\pm 1.7 $ $ 6.9\\pm 1.5 $ $ -8.1\\pm 1.1 $ $ -6.1\\pm 1.5 $ $ 10.5\\pm 1.4 $ $ 8.7\\pm 1.1 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ NGC 4507 $ 47.0\\pm 1.6 $ $ 5.4\\pm 1.3 $ $ ... $ $ ... $ $ 6.6\\pm 0.9 $ $ 5.1\\pm 0.8 $ $ 1.21\\pm 0.01 $ $ 5.6\\pm 0.8 $ $ 1.35\\pm 0.01 $ $ 5.8\\pm 0.6 $ $ 2.45^{+0.03}_{-0.04} $ $ 1.6\\pm 0.5 $ $ 3.72^{+0.01}_{-0.02} $ $ 1.9\\pm 0.5 $ NGC 4992 $ 8.3\\pm 1.5 $ $ 4.5\\pm 1.4 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ NGC 5252 $ 13.9^{+1.7}_{-1.5} $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ NGC 526A $ 25.2\\pm 2.2 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ NGC 5506 $ ... $ $ 42.8^{+3.3}_{-6.1} $ $ 24.1^{+3.5}_{-3.0} $ $ 11.8^{+4.2}_{-4.7} $ $ 14.7^{+4.8}_{-4.2} $ $ ... $ $ 6.45\\pm 0.01 $ $ 56.6^{+4.7}_{-3.2} $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ NGC 6300 $ 19.2\\pm 2.0 $ $ 6.9\\pm 1.9 $ $ -4.0\\pm 1.3 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ NGC 7172 $ 36.5\\pm 3.0 $ $ 8.0\\pm 2.9 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ NGC 788 $ 23.6^{+2.5}_{-2.3} $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ UGC 03142 $ 13.4\\pm 1.3 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ UGC 12741 $ 5.6\\pm 1.1 $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ $ ... $ (1) Galaxy name.",
"(2)-(7) Normalization of the emission lines at 6.40 keV, 6.31 keV, 6.68 keV, 6.93 keV, 7.06 keV, and 7.48 keV in units of $\\times 10^{-6}$ photons cm$^{-2}$ s$^{-1}$ .",
"(8)-(13) Line Energy and normalization of emission/absorption lines in units of keV and $\\times 10^{-6}$ photons cm$^{-2}$ s$^{-1}$ , respectively."
],
[
"Spectra fitted with the relativistic reflection component",
"cccc[H] Disk parameters Target Name $r_{\\rm in}$ $R_{\\rm Disk}$ $-\\Delta \\chi ^2$ (1) (2) (3) (4) 2MASX J0216+5126 $ 100^{}_{-99}/100^{}_{-99} $ $ 0.00^{+0.15}_{}/0.00^{+0.09}_{} $ $ 0.0/0.0 $ 2MASX J0248+2630 $ 1^{+99}_{}/1^{+99}_{} $ $ 0.40^{+0.57}_{-0.40}/0.04^{+0.27}_{-0.04} $ $ 0.1/0.0 $ 2MASX J0318+6829 $ 10^{+22}_{-5}/10^{+23}_{-6} $ $ 0.62^{+0.32}_{-0.46}/0.55^{+0.16}_{-0.42} $ $ 1.7/1.6 $ 2MASX J0350-5018 $ 100^{}_{-99}/100^{}_{-99} $ $ 0.0^{+1.0}_{}/0.0^{+1.0}_{} $ $ 0.0/0.0 $ 2MASX J0444+2813 $ 25^{+19}_{-8}/25^{+19}_{-9} $ $ 1.00^{}_{-0.27}/1.00^{}_{-0.26} $ $ 5.3/5.7 $ 2MASX J0505-2351 $ 33^{+67}_{-32}/35^{+65}_{-34} $ $ 0.12^{+0.05}_{-0.12}/0.00^{+0.20}_{} $ $ 0.6/0.0 $ 2MASX J0911+4528 $ 100^{}_{-61}/100^{}_{-99} $ $ 0.22^{+0.19}_{-0.17}/0.22^{+0.35}_{-0.22} $ $ 1.6/0.7 $ 2MASX J1200+0648 ${\\bf 29^{+24}_{-10}/20^{+23}_{-5}} $ ${\\bf 0.62^{+0.18}_{-0.19}/0.51^{+0.13}_{-0.10} } $ ${\\bf 19.1/13.0 } $ Ark 347 $ 1^{+99}_{}/1^{+99}_{} $ $ 0.00^{+0.54}_{}/0.0^{+1.0}_{} $ $ 0.0/0.0 $ ESO 103-035 $ 4^{+3}_{-2}/4^{+3}_{-2} $ $ 0.17^{+0.08}_{-0.03}/0.25^{+0.09}_{-0.07} $ $ 5.8/5.7 $ ESO 263-G013 $ 3^{+97}_{-2}/3^{+97}_{-2} $ $ 0.13^{+0.50}_{-0.13}/0.12^{+0.52}_{-0.12} $ $ 0.4/0.4 $ ESO 297-G018 $ 63^{+37}_{-35}/64^{+36}_{-42} $ $ 0.58^{+0.42}_{-0.40}/0.48^{+0.52}_{-0.37} $ $ 2.4/1.8 $ ESO 506-G027 $ 100^{}_{-99}/100^{}_{-99} $ $ 0.00^{+0.32}_{}/0.00^{+0.27}_{} $ $ 0.0/0.0 $ Fairall 49 $ 30^{+70}_{-17}/{\\bf 4\\pm 2} $ $ 0.11^{+0.09}_{-0.10}/{\\bf 0.30^{+0.08}_{-0.06}} $ $ 1.0/{\\bf 23.2} $ Fairall 51 $ 100^{}_{-56}/20^{+14}_{-7} $ $ 0.25^{+0.12}_{-0.09}/0.34^{+0.12}_{-0.14} $ $ 7.5/5.3 $ IC 4518A $ 16^{+4}_{-5}/16^{+17}_{-5} $ $ 0.89^{+0.11}_{-0.36}/0.99^{+0.01}_{-0.43} $ $ 6.5/5.1 $ LEDA 170194 $ 26^{+74}_{-25}/92^{+8}_{-91} $ $ 0.00^{+0.09}_{}/0.00^{+0.15}_{} $ $ 0.0/0.0 $ MCG +04-48-002 $ 1^{+99}_{}/1^{+99}_{} $ $ 0.0^{+1.0}_{}/0.27^{+0.73}_{-0.27} $ $ 0.0/0.0 $ MCG -01-05-047 $ 25^{+36}_{-9}/25^{+75}_{-23} $ $ 1.00^{}_{-0.58}/1.00^{}_{-0.47} $ $ 3.0/3.0 $ MCG -02-08-014 $ 20^{+80}_{-9}/25^{+75}_{-10} $ $ 0.53^{+0.34}_{-0.33}/0.89^{+0.11}_{-0.41} $ $ 3.1/6.5 $ MCG -05-23-016 $ 100^{}_{-36}/4^{+2}_{-3} $ $ 0.07\\pm 0.04/0.03\\pm 0.01 $ $ 3.7/0.9 $ Mrk 1210 $ 10^{+15}_{-5}/16^{+16}_{-9} $ $ 1.00^{}_{-0.39}/1.00^{}_{-0.21} $ $ 7.1/9.0 $ Mrk 1498 $ 2\\pm 1/2^{+12}_{-1} $ $ 0.96^{+0.04}_{-0.38}/1.00^{}_{-0.44} $ $ 6.5/6.3 $ Mrk 18 $ 100^{}_{-99}/100^{}_{-99} $ $ 0.00^{+0.47}_{}/0.00^{+0.58}_{} $ $ 0.0/0.0 $ Mrk 348 $ 77^{+23}_{-76}/18^{+82}_{-17} $ $ 0.00^{+0.06}_{}/0.00^{+0.06}_{} $ $ 0.1/0.1 $ Mrk 417 $ 1^{+99}_{}/3^{+97}_{-2} $ $ 0.00^{+0.61}_{}/0.00^{+0.45}_{} $ $ 0.0/0.0 $ Mrk 520 $ 46^{+54}_{-45}/30^{+70}_{-29} $ $ 0.02^{+0.27}_{-0.02}/0.00^{+0.14}_{} $ $ 0.1/0.0 $ Mrk 915 $ 100^{}_{-99}/100^{}_{-99} $ $ 0.03^{+0.53}_{-0.03}/0.04^{+0.56}_{-0.04} $ $ 0.0/0.0 $ NGC 1052 $ 100^{}_{-37}/100^{+-100}_{-100} $ $ 0.39^{+0.20}_{-0.17}/0.32^{+-0.32}_{-0.32} $ $ 4.3/2.7 $ NGC 1142 $ 70^{+30}_{-69}/28^{+72}_{-27} $ $ 0.00^{+0.35}_{}/0.00^{+0.16}_{} $ $ 0.0/0.0 $ NGC 2110 $ 25^{+75}_{-15}/1^{+6}_{} $ $ 0.05^{+0.03}_{-0.02}/0.08\\pm 0.04 $ $ 1.1/1.3 $ NGC 235A $ 11^{+31}_{-10}/1^{+99}_{} $ $ 0.86^{+0.14}_{-0.72}/0.73^{+0.27}_{-0.65} $ $ 1.5/1.3 $ NGC 3081 $ 100^{}_{-87}/100^{}_{-88} $ $ 0.57^{+0.43}_{-0.30}/0.63^{+0.37}_{-0.32} $ $ 4.1/4.2 $ NGC 3431 $ 25^{+19}_{-13}/7^{+5}_{-6} $ $ 0.66^{+0.32}_{-0.35}/1.00^{}_{-0.28} $ $ 2.5/5.6 $ NGC 4388 $ 100^{}_{-99}/100^{}_{-99} $ $ 0.00^{+0.03}_{}/0.00^{+0.03}_{} $ $ 0.0/0.0 $ NGC 4507 $ 1^{+6}_{}/100^{}_{-32} $ $ 0.59^{+0.28}_{-0.23}/0.28^{+0.18}_{-0.15} $ $ 8.8/3.9 $ NGC 4992 $ 65^{+35}_{-21}/65^{+35}_{-21} $ $ 1.00^{}_{-0.20}/1.00^{}_{-0.20} $ $ 8.4/8.2 $ NGC 5252 $ 5^{+95}_{-4}/40^{+60}_{-39} $ $ 0.00^{+0.08}_{}/0.00^{+0.06}_{} $ $ 0.0/0.0 $ NGC 526A ${\\bf 44^{+33}_{-11}}/33^{+67}_{-32} $ ${\\bf 0.26^{+0.07}_{-0.03}}/0.00^{+0.05}_{} $ ${\\bf 9.3}/0.0 $ NGC 5506 $ 100^{}_{-36}/100^{}_{-99} $ $ 0.08^{+0.06}_{-0.04}/0.00^{+0.01}_{} $ $ 5.1/0.0 $ NGC 6300 ${\\bf 2^{}_{-1}/2^{}_{-1}} $ ${\\bf 0.39\\pm 0.11/0.56\\pm 0.16} $ $ {\\bf 9.7/13.7} $ NGC 7172 $ 100^{}_{-36}/100^{}_{-83} $ $ 0.10^{+0.03}_{-0.02}/0.07\\pm 0.06 $ $ 3.5/1.1 $ NGC 788 $ 3^{+97}_{-2}/3^{+97}_{-2} $ $ 0.21^{+0.55}_{-0.21}/0.19^{+0.57}_{-0.19} $ $ 0.2/0.1 $ UGC 03142 $ 44^{+56}_{-43}/32^{+68}_{-31} $ $ 0.54^{+0.46}_{-0.54}/0.55^{+0.45}_{-0.55} $ $ 0.9/0.7 $ UGC 12741 $ 100^{}_{-49}/100^{}_{-99} $ $ 0.49^{+0.51}_{-0.42}/0.46^{+0.54}_{-0.46} $ $ 1.4/1.0 $ (1) Galaxy name.",
"(2) Inner radius in units of $r_{\\rm g}$ for the assumed ionization parameters, $\\xi = 10$ and 100.",
"(3) Equivalent reflection strength for the same ionization parameters as (2).",
"(4) Difference of the chi-squred value before and after adding the disk components for the same ionization parameters as (2).",
"AGNs for which fitting results are significantly improved by inclusion of the relativistic reflection components from a disk are represented in boldface."
],
[
"Information of hydrogen column density",
"cccccccccccc Information of hydrogen column density 0pt Target Name $N_{\\rm H}$ Obs.",
"date Observatory Ref.",
"$N_{\\rm H}$ (1) (2) (3) (4) (5) 2MASX J0216+5126 $ 1.74^{+0.06}_{-0.07} $ 2006-01-24 XMM-Newton 1 2MASX J0318+6829 $ < 14 $ 2006-01-29 XMM-Newton 1 2MASX J0505-2351 $ 9.90\\pm 0.30 $ 2009-08-06 XMM-Newton 2 2MASX J0911+4528 $ 48^{+28}_{-25} $ 2006-04-10 XMM-Newton 1 2MASX J1200+0648 $ 10.60^{+0.80}_{-1.01} $ 2006-06-26 XMM-Newton 1 Ark 347 $ 19.2^{+4.4}_{-3.5} $ 2003-01-02 XMM-Newton 3 ESO 103-035 $ 18.9^{+0.6}_{-1.1} $ 2002-03-15 XMM-Newton 4 ESO 263-G013 $ 25.7^{+1.5}_{-1.4} $ 2007-06-14 XMM-Newton 4 ESO 506-G027 $ 66.0^{+5.0}_{-4.8} $ 2006-01-24 XMM-Newton 3 Fairall 49 $ 1.08\\pm 0.02 $ 2001-03-05 XMM-Newton 5 $ 1.06\\pm 0.02 $ 2001-03-06 XMM-Newton 5 $ 1.46\\pm 0.01 $ 2013-09-04 XMM-Newton 6 $ 1.29^{+0.01}_{-0.02} $ 2013-10-15 XMM-Newton 6 Fairall 51 $ 3.43^{+0.29}_{-0.31} $ 2013-09-05 Suzaku this work $ 4.49^{+0.46}_{-0.49} $ 2013-09-07 Suzaku this work $ 2.67^{+1.09}_{-0.99} $ 2013-09-13 Suzaku this work IC 4518A $ 14.0^{+3.0}_{-1.0} $ 2006-08-07 XMM-Newton 4 LEDA 170194 $ 2.9^{+1.3}_{-0.3} $ 2005-07-25 Chandra 4 MCG +04-48-002 $ 57.4^{+9.6}_{-6.6} $ 2006-04-23 XMM-Newton 4 MCG -01-05-047 $ 26.3\\pm 1.0 $ 2009-07-24 XMM-Newton 7 MCG -05-23-016 $ 1.25^{+0.29}_{-0.18} $ 2000-11-14 Chandra 8 $ 1.94^{+0.38}_{-0.40} $ 2001-05-13 XMM-Newton 8 $ 1.80\\pm 0.23 $ 2001-12-01 XMM-Newton 8 $ 1.49\\pm 0.01 $ 2013-06-01 Suzaku this work $ 1.50\\pm 0.01 $ 2013-06-05 Suzaku this work Mrk 1210 $ 17.8^{+7.8}_{-7.9} $ 2001-05-05 XMM-Newton 9 $ 29.6^{+1.8}_{-1.7} $ 2008-02-15 Chandra 10 $ 25.5^{+3.3}_{-2.9} $ 2008-02-17 Chandra 10 $ 37.6^{+4.4}_{-4.6} $ 2008-03-06 Chandra 10 Mrk 18 $ 18.3^{+7.2}_{-5.7} $ 2006-03-23 XMM-Newton 1 Mrk 348 $ 13.4^{+0.20}_{-0.46} $ 2002-07-18 XMM-Newton 11 $ 12.9\\pm 1.2 $ 2013-01-04 XMM-Newton 11 Mrk 417 $ 54^{+25}_{-11} $ 2006-06-15 XMM-Newton 1 NGC 1052 $ 13.8^{+2.0}_{-1.8} $ 2001-08-15 XMM-Newton 12 $ 5.3\\pm 1.5 $ 2005-09-18 Chandra 12 $ 9.30^{+0.52}_{-0.51} $ 2006-01-12 XMM-Newton 12 $ 8.96^{+0.43}_{-0.42} $ 2009-01-14 XMM-Newton 12 $ 9.47^{+0.39}_{-0.38} $ 2009-08-12 XMM-Newton 12 NGC 1142 $ 47.0^{+3.5}_{-3.2} $ 2006-01-28 XMM-Newton 13 $ 73.9^{+7.9}_{-7.0} $ 2007-07-21 Suzaku this work NGC 2110 $ 4.0\\pm 1.8 $ 2001-12-19 Chandra 14 $ < 4.5 $ 2003-03-05 Chandra 14 $ 3.90\\pm 0.4 $ 2003-03-05 XMM-Newton 14 $ 2.53\\pm 0.13 $ 2012-08-31 Suzaku this work $ 4.0\\pm 0.4 $ 2012-10-05 NuSTAR 14 $ 4.0\\pm 0.7 $ 2013-02-14 NuSTAR 14 NGC 4388 $ 25.6^{+3.1}_{-2.9} $ 2001-06-08 Chandra 13 $ 24.3^{+1.1}_{-1.0} $ 2002-12-12 XMM-Newton 13 NGC 4507 $ 42.8^{+0.9}_{-0.7} $ 2001-01-04 XMM-Newton 3 $ 90\\pm 10 $ 2005-07-25 Chandra 15 $ 68.5^{+14.9}_{-9.6} $ 2006-06-27 XMM-Newton 1 $ 87^{+7}_{-8} $ 2010-06-24 XMM-Newton 16 $ 97\\pm 9 $ 2010-07-03 XMM-Newton 16 $ 76^{+10}_{-13} $ 2010-07-13 XMM-Newton 16 $ 94\\pm 11 $ 2010-07-23 XMM-Newton 16 $ 80^{+8}_{-6} $ 2010-08-03 XMM-Newton 16 $ 65\\pm 7 $ 2010-12-02 Chandra 16 NGC 5252 $ 2.32^{+0.13}_{-0.15} $ 2003-08-11 Chandra 17 NGC 526A $ 1.14\\pm 0.26 $ 2003-06-21 XMM-Newton 18 NGC 5506 $ 2.69^{+0.02}_{-0.03} $ 2004-07-11 XMM-Newton 13 $ 2.80^{+0.01}_{-0.02} $ 2004-08-07 XMM-Newton 13 $ 3.09\\pm 0.03 $ 2006-08-08 Suzaku this work $ 3.16\\pm 0.03 $ 2007-01-31 Suzaku this work NGC 6300 $ 25.4^{+4.3}_{-3.7} $ 2001-03-02 XMM-Newton 11 $ 14.1^{+1.3}_{-2.0} $ 2009-06-10 Chandra 11 $ 19.8^{+1.4}_{-2.7} $ 2009-06-14 Chandra 11 NGC 7172 $ 8.45^{+0.36}_{-0.33} $ 2002-11-18 XMM-Newton 11 $ 8.75\\pm 0.27 $ 2004-11-11 XMM-Newton 11 $ 8.34^{+0.16}_{-0.15} $ 2007-04-24 XMM-Newton 11 NGC 788 $ 44.4^{+8.7}_{-7.8} $ 2009-09-06 Chandra 11 $ 50.3^{+6.1}_{-5.7} $ 2010-01-15 XMM-Newton 11 (1) Galaxy name.",
"(2) Hydrogen column density of the neutral full-covering absorption model.",
"Errors correspond to the 90% confidence interval.",
"The confidence level of the errors compiled from [39], [40], [53], and [88], is not clear because it is not described.",
"(3) Observation date.",
"(4) Observatory.",
"(5) References for $N_{\\rm H}$ .",
"References.",
"(1) [112].",
"(2) [108].",
"(3) [75].",
"(4) [20].",
"(5) [100].",
"(6) [56].",
"(7) [101].",
"(8) [6].",
"(9) [5].",
"(10) [85].",
"(11) [40].",
"(12) [39].",
"(13) [53].",
"(14) [64].",
"(15) [88].",
"(16) [63].",
"(17) [19].",
"(18) [14]."
]
] | 1606.04941 |
[
[
"Improving Power Generation Efficiency using Deep Neural Networks"
],
[
"Abstract Recently there has been significant research on power generation, distribution and transmission efficiency especially in the case of renewable resources.",
"The main objective is reduction of energy losses and this requires improvements on data acquisition and analysis.",
"In this paper we address these concerns by using consumers' electrical smart meter readings to estimate network loading and this information can then be used for better capacity planning.",
"We compare Deep Neural Network (DNN) methods with traditional methods for load forecasting.",
"Our results indicate that DNN methods outperform most traditional methods.",
"This comes at the cost of additional computational complexity but this can be addressed with the use of cloud resources.",
"We also illustrate how these results can be used to better support dynamic pricing."
],
[
"Introduction",
"Currently, most of the energy produced worldwide uses coal or natural gas.",
"However, much of this energy is wasted.",
"In the United States of America, approximately 58% of energy produced is wasted [1].",
"Furthermore, 40% of this wasted energy is due to industrial and residential buildings.",
"By reducing energy wastage in the electric power industry, we reduce damage to the environment and reduce the dependence on fossil fuels.",
"Short-term load forecasting (STLF) (i.e., one hour to a few weeks) can assist since, by predicting load, one can do more precise planning, supply estimation and price determination.",
"This leads to decreased operating costs, increased profits and a more reliable electricity supply for the customer.",
"Over the past decades of research in STLF there have been numerous models proposed to solve this problem.",
"These models have been classified into classical approaches like moving average [5] and regression models [10], as well as machine learning based techniques, regression trees [15], support vector machines [16] and Artificial Neural Networks [13].",
"In recent years, many deep learning methods have been shown to achieve state-of-the-art performance in various areas such as speech recognition [9], computer vision [11] and natural language processing [3].",
"This promise has not been demonstrated in other areas of computer science due to a lack of thorough research.",
"Deep learning methods are representation-learning methods with multiple levels of representation obtained by composing simple but non-linear modules that each transform the representation at one level (starting with the raw input) into a representation at a higher, slightly more abstract level [12].",
"With the composition of enough such transformations, very complex functions can be learned.",
"In this paper, we compare deep learning and traditional methods when applied to our STLF problem and we also provide a comprehensive analysis of numerous deep learning models.",
"We then show how these methods can be used to assist in the pricing of electricity which can lead to less energy wastage.",
"To the best of our knowledge, there is little work in such comparisons for power usage in an electrical grid.",
"The data we use is based on one year of smart meter data collected from residential customers.",
"We apply each of the deep and traditional algorithms to the collected data while also noting the corresponding computational runtimes.",
"Due to differences in electricity usage between the week and the weekend, we then split the data into two new datasets: weekends and weekly data.",
"The algorithms are applied to these new datasets and the results are analyzed.",
"The results show that the deep architectures are superior to the traditional methods by having the lowest error rate, but they do have the longest run-time.",
"Due to space limitations we do not provide details of the traditional approaches but do provide references."
],
[
"Data Description",
"Our dataset consists of 8592 samples of 18 features that were collected from several households.",
"The dataset was broken into 3 parts for training, validation and testing of sizes 65%, 15%, 20% respectively.",
"The readings were recorded at hourly intervals throughout the year.",
"Some of the features were electrical load readings for the previous hour, the previous two hours, the previous three hours, the previous day same hour, the previous day previous hour, the previous day previous two hours, the previous 2 days same hour, the previous 2 days previous hour, the previous 2 days previous two hours, the previous week same hour, the average of the past 24 hours and the average of the past 7 days.",
"The rest of the features (which do not contain electrical load readings) are the day of the week, hour of the day, if it is a weekend, if it is a holiday, temperature and humidity.",
"These features were selected as they are typically used for STLF.",
"In addition, the total electrical load does not change significantly throughout the year since the households are located in a tropical country where the temperature remains fairly constant throughout the year.",
"Table: Baseline algorithms"
],
[
"Comparison Method",
"As a preprocessing step, the data is cleaned and scaled to zero mean and unit variance.",
"All traditional methods use cross-validation to determine appropriate values for the hyper-parameters.",
"A random grid search was used to determine the hyper-parameters for the deep learning methods.",
"Several baseline algorithms were chosen.",
"They include the Weighted Moving Average (WMA) where $y_{t+1} = \\alpha y_{i} + \\beta y_{i-167}$ with $\\alpha = 0.05$ and $\\beta = 0.95$ , Multiple Linear Regression (MLR) and quadratic regression (MQR), Regression Tree (RT) with the minimum number of branch nodes being 8, Support Vector Regression (SVR) with a linear kernel and Multilayer Perception (MLP), with the number of hidden neurons being 100.",
"For our Deep Neural Network methods we used Deep Neural Network without pretraining (DNN-W), DNN with pretraining using Stacked Autoencoders (DNN-SA) [18], Recurrent Neural Networks (RNN) [8], RNNs and Long Short Term Memory (RRN-LSTM) [7], Convolutional Neural Networks (CNN) [19] and CNNs and Long Short Term Memory (CNN-LSTM)] [17] To evaluate the goodness of fit of these algorithms we use the Mean Absolute Percentage Error (MAPE) defined as: $\\text{MAPE} = \\frac{100}{n} \\sum _{t=1}^{n} \\frac{|y_t - \\widehat{y}_t|}{y_t}$ where $n$ is the number of data points, $t$ is the particular time step, $y_t$ is the target or actual value and $\\widehat{y}_t$ is the predicted value.",
"In order to determine the cost of the prediction errors (i.e.",
"whether the prediction is above or below the actual value) the Mean Percentage Error (MPE) is used, which is defined as: $\\text{MPE} = \\frac{100}{n} \\sum _{t=1}^{n} \\frac{y_t - \\widehat{y}_t}{y_t}$"
],
[
"Numerical Results",
"We first look at the baseline methods, (with the exception of MLP) in Table REF .",
"From the table we see that MLR performs the worst, with a MAPE of 24.25%, which would indicate that the problem is not linear (see Figure REF ).",
"However, the RT algorithm outperforms the rest of the methods by a noticeable margin.",
"This shows that the problem can be split into some discrete segments which would accurately forecast the load.",
"This can be confirmed by looking at the load in Figure REF where it is clear that, depending on the time of day, there is significant overlap of the value of the load between days.",
"Thus, having a node in the RT determining the time of the day would significantly improve accuracy.",
"The run-time for these algorithms was quite short with WMA taking the longest due to the cross-validation step where we determined all possible coefficients in steps of 0.05.",
"Due to the typically long running time of DNN architectures, the algorithms were restricted to 200 and 400 epocs.",
"From Table REF , there is a clear difference when looking at the 200 epocs and the 400 epocs MAPE columns, as most of the algorithms have a lower MAPE after running for 400 epocs when compared with 200 epocs.",
"This is especially true for the $\\text{DNN-SA}_3$ which saw significant drops in the MAPE.",
"The MLP did not perform the worst in both epocs but it was always in the lower half of accuracy.",
"This indicates that the shallow network might not be finding the patterns or structure of the data as quickly as the DNN architectures.",
"However, it outperformed RT in both the 200 and 400 epocs.",
"This alludes to the fact that the hidden layer is helping to capture some of the underlying dynamics that a RT cannot.",
"Table: Daily MAPE ValuesLooking at the 200 epocs column, we see that $\\text{DNN-W}_3$ performs the best with a MAPE of 2.64%.",
"On the other hand, the most stable architecture is the DNN-SA with a MAPE consistently less than 3%.",
"This robustness is shown when the epocs are increased to 400 where the DNN-SA architecture outperforms all the other methods (both the baseline and deep methods).",
"The pretraining certainly gave these methods a boost over the other methods as it guides the learning towards basins of attraction of minima that support better generalization from the training data set [6].",
"RNNs, and to an extent LSTM, have an internal state which gives it the ability to exhibit dynamic temporal behavior.",
"However, they require a much longer time to compute which is evident in Table REF since these methods had trouble mapping those underlying dynamics of the data in such a small number of epocs.",
"CNNs do not maintain internal state, however with load forecasting data, one can expect a fair amount of auto-correlation that requires memory.",
"This could explain their somewhat low but unstable MAPE for 200 and 400 epocs.",
"Taking both tables into consideration, most of the DNN architectures vastly outperform the traditional approaches, but DNNs require significantly more time to run and thus there is a trade-off.",
"For STLF, which is a very dynamic environment, one cannot wait for a new model to complete its training stage.",
"Hence, this is another reason we limited the number of epocs to 200 and 400.",
"Table REF shows that limiting the epocs did not adversely affect many of the DNN architectures as most were able to surpass the accuracy of the traditional methods (some by a lot).",
"When selecting a model, one would have to determine if the length of time to run the model is worth the trade-off between accuracy and runtime."
],
[
"Daily Analysis",
"We know that people have different electrical usage patterns on weekdays when compared to weekends.",
"This difference can be seen in Figure REF which illustrates usage for a sample home.",
"This household uses more energy during the weekdays than on weekends.",
"There are electrical profiles that may be opposite, i.e., where the weekend electrical load is more.",
"Whatever the scenario, there are usually different profiles for weekdays and weekends.",
"Figure: Electrical ProfilesTo see how our models handle weekdays and weekends, we calculated the average MAPE for each day of the week in the test set (the 400 epoc models was used for the DNNs calculations).",
"The average for each day of the week is tabulated in Table REF .",
"From the table, it is clear that most of the DNN algorithms have their lowest MAPE during the week.",
"This is indicative that the patterns for weekdays are similar and as a result have more data.",
"By having more data, DNNs are better able to capture the underlying structure of the data and thus are able to predict the electrical load with greater accuracy.",
"Weekend predictions have a higher MAPE since DNNs require a lot of data to perform accurate predictions and for weekends this data is limited.",
"The WMA and MQR seem to have their best day on Sunday, but have a very poor MAPE for the rest of the days.",
"This indicates that the models have an internal bias towards Sunday and as a result fail to accurately predict the values for other days.",
"It is clear, again, that DNNs outperform the traditional methods."
],
[
"Mean Percentage Error",
"In this particular domain, an electricity provider will also be interested in changes of electrical load, as opposed to absolute error, in order to adjust generation accordingly, mostly because starting up additional plants takes time.",
"This is why the Mean Percentage Error (MPE) was used.",
"The MPE would tell that a model with a positive value \"under-predicts\" the load while a negative value \"over-predicts\" the actual value and they can then adjust their operations accordingly.",
"Many of the traditional methods had predicted more electrical load than the actual load, including MLP.",
"However, most of the DNNs have under-predicted the load value.",
"Looking at the best in Table REF , DNN-SAs MPE values (for 400 epocs), they are all under 1% and positive, which indicates that it under-predicts the value.",
"However, one should not use the MPE alone.",
"An example is RNNs which have a low positive MPE, however it's MAPE in both epocs is around 5%.",
"This indicates that RNN had a slightly larger sum of values that \"under-predicts\" than \"over-predicts\", but its overall accuracy is not as good as other deep architectures."
],
[
"Applications to Energy Efficiency",
"Using the results from STLF (MAPE and MPE), a company can now accurately predict upcoming load.",
"This would mean that a power generating company can now produce energy at a much more precise amount rather than producing excess energy that would be wasted.",
"Since most of these companies use fossil fuels which are non-renewable sources of energy, we would be conserving them as well as reducing levels of carbon dioxide released into the atmosphere and the toxic byproducts of fossil fuels.",
"Another benefit of accurate load forecasting is that of dynamic pricing.",
"Many residential customers pay a fixed rate per kilowatt.",
"Dynamic pricing is an approach that allows the cost of electricity to be based on how expensive this electricity is to produce at a given time.",
"The production cost is based on many factors, which in this paper, is characterized by the algorithms for STLF.",
"By having a precise forecast of electrical load, companies now have the ability to determine trends, especially at peak times.",
"An example of this would be in the summer months when many people may want to turn on their air conditioners and thus electricity now becomes expensive to produce as the company could have to start up additional power generating plants to account for this load.",
"If the algorithms predict that there would be this increase in electrical load around the summer months, this would be reflected in the higher price that consumers would need to pay.",
"As a result, most people would not want to keep their air conditioner on all the time (as per usual) but use it only when necessary.",
"Taking this example and adding on washing machines, lights and other appliances, we can see the immense decrease in energy that can be achieved on the consumer side."
],
[
"Related Work",
"The area of short-term load forecasting (STLF) has been studied for many decades but deep learning has only recently seen a surge of research into its applications.",
"Significant research has been focused on Recurrent Neural Networks (RNNs).",
"In the thesis by [14], RNNs was used to compare other methods for STLF.",
"These methods included modifications of MLP by training with algorithms like Particle Swarm Optimization, Genetic Algorithms and Artificial Immune Systems.",
"Two other notable papers that attempt to apply DNN for STLF are [2] and [4].",
"In [2], they compare Deep Feedfoward Neural Networks, RNNs and kernelized regression.",
"In the paper by [4] a RNN is used for forecasting loads and the result is compared to a Feedfoward Neural Network.",
"However, a thorough comparison of various DNN architectures is lacking and any applications to dynamic pricing or energy efficiency is absent."
],
[
"Conclusion",
"In this paper, we focused on energy wastage in the electrical grid.",
"To achieve this, we first needed to have an accurate algorithm for STLF.",
"With the advent of many deep learning algorithms, we compared the accuracy of a number of deep learning methods and traditional methods.",
"The results indicate that most DNN architectures achieve greater accuracy than traditional methods even when the data is split into weekdays and weekends.",
"However such algorithms have longer runtimes.",
"We also discussed how these algorithms can have a significant impact in conserving energy at both the producer and consumer levels."
]
] | 1606.05018 |
[
[
"Analytic solutions for links and triangles distributions in finite\n Barab\\'asi-Albert networks"
],
[
"Abstract Barab\\'asi-Albert model describes many different natural networks, often yielding sensible explanations to the subjacent dynamics.",
"However, finite size effects may prevent from discerning among different underlying physical mechanisms and from determining whether a particular finite system is driven by Barab\\'asi-Albert dynamics.",
"Here we propose master equations for the evolution of the degrees, links and triangles distributions, solve them both analytically and by numerical iteration, and compare with numerical simulations.",
"The analytic solutions for all these distributions predict the network evolution for systems as small as 100 nodes.",
"The analytic method we developed is applicable for other classes of networks, representing a powerful tool to investigate the evolution of natural networks."
],
[
"Acknowledgements",
"We acknowledge support from the Centro de Física Computacional, Universidade Federal do Rio Grande do Sul.",
"This work has been partially supported by Brazilian agencies FAPERGS, CAPES, and CNPq.",
"It is part of the project PRONEX-FAPERGS 10/0008-0."
]
] | 1606.04913 |
[
[
"Note on the classification of super-resolution in far-field microscopy\n and information theory"
],
[
"Abstract In recent years several far-field microscopy techniques have been developed which manage to overcome the diffraction limit of resolution.",
"A unifying classification scheme for them is clearly desirable.",
"We argue that existing schemes based on the information capacity of the optical system can not easily be extended to cover e.g., STED microscopy or techniques based on single molecule imaging.",
"We suggest a classification based on a reconstruction of the Abbe limit."
],
[
"Introduction",
"In 1873 Ernst Abbe established his celebrated diffraction limit and showed that the resolution of an imaging system is limited by the wave-length $\\lambda $ and the numerical aperture NA according to [1]: $d_{\\mathrm {Abbe}}=\\frac{\\lambda }{c\\cdot {\\mathrm {NA}}}, $ with $c=1$ (coherent illumination) or $c=2$ (incoherent illumination).",
"Abbe's derivation was embeded in a model of the image formation which involves a double Fourier transformation [2], [3].",
"This is equivalent to an image formation described by a convolution of the object intensity (or amplitude), $O(x,y)$ , with the PSF of the system (We restrict ourselves to the lateral plane): $I(x,y) &=& O(x,y) \\otimes \\mathrm {PSF}(x,y).",
"$ Applying a Fourier transformation and the convolution theorem yields: $\\tilde{I}(k_x,k_y) = \\widetilde{O}(k_x,k_y)\\cdot \\mathrm {OTF}(k_x,k_y).",
"$ Here, the image formation is described by the Fourier transform of the PSF, i.e., the optical transfer function (OTF).",
"The Abbe resolution limit then amounts to the fact, that the cut-off of the optical transfer function (OTF) is given by $k_0=\\frac{c\\cdot NA}{\\lambda }$ .",
"Super-resolution has become the collective term for techniques to break this limit and we will summarize some of these developments briefly in Sec. 2.",
"Our focus will be put on applications in far-field microscopy.",
"Given the diversity of different schemes a classification of them is desirable.",
"In Sec.",
"3 we review briefly the existing suggestions based on information theory and point to their shortcomings in the current formulation.",
"A novel classification scheme will be proposed in Sec.",
"4."
],
[
"Super-resolution imaging",
"Already in the 1950s several researchers challenged this alleged fundamental Abbe limit.",
"As pointed out by Wolter and Harris [4], [5], [6] the diffraction limit (Eq.",
"REF ) assumes an infinitely extended object.",
"These authors noted, that the Fourier transform of a finite function [5] is analytic, i.e., given the OTF on a finite interval the transfer function can be recovered uniquely (and beyond the cut-off) by analytic continuation.",
"This opens the possibility of super-resolution by deconvolution.",
"However, noise renders the corresponding inverse problem ill-posed and practical applications of this “computational super-resolution\" achieve only a minor resolution improvement [7], [8].",
"But given that the finite extent of the probe is a generic property all this demonstrates that the resolution is only limited by noise – contrary to Abbe's claim [9].",
"Another early attempt to break the diffraction limit was made by Giuliano Toraldo di Francia.",
"In [10] it is shown that the width of the point-spread function can be reduced arbitrarily without increasing the side-lobs in the field-of-view.",
"The idea is to apply a filtering or masking technique (called apodization).",
"A further elaboration was given by Frieden [11].",
"However, the resulting central peak is very weak which limits the practical application of the idea [12].",
"A recent implementation of this scheme is developed in [13].",
"Even in confocal microscopy (developed in the 1950s by the cognitive scientist Marvin Minsky and later re-discovered, compare [14]) the band-width of the OTF is extended by a factor of 2, given that its effective PSF is the square of the illumination- and detection-PSF [15].",
"Again, noise and the finite pin-hole size render the actual resolution gain smaller or even vanishing.",
"Confocal microscopy is a scanning technique, i.e., applies a non-uniform illumination of the sample.",
"The idea to enhance resolution by non-uniform illumination (often called “structured illumination\") was suggested already in 1952 by the french physicist Maurice Françon.",
"Some simple applications are discussed in [4] and [16] contains further developments of the concept (still restricted to coherent illumination).",
"The idea of structured illumination microscopy is to apply an illumination which contains a spatial frequency $k_1$ and gives rise to the Moiré effect with fringes of frequency $|k-k_1|$ (with $k$ a sample frequency).",
"For $|k-k_1|<k_0$ these fringes will be observable in the miscroscope, i.e., effectively the passband is extended by $k_1$ .",
"Given that the highest frequency in the illumination pattern is as much diffraction limited as the detection passband the maximum value is $k_1=k_0$ , hence the resolution can be extended by a factor of 2 (as in confocal microscopy; at least theoretically).",
"However, a resolution beyond the Abbe limit needs incoherent illumination and the first discussion in the context of fluorescence microscopy (i.e., incoherent illumination) was given by Heintzmann and Cremer [17].",
"The successful implementation and measurement results are reported in [18].",
"Structured illumination microscopy (SIM) can be improved further if the effects of non-linearity between excitation and emission are exploited [19], [20].",
"As in linear SIM, higher frequency contributions can be moved into the passband of the original system.",
"If the non-linearity is non-polynomial the passband is (theoretically) even unbounded.",
"Heintzmann et al.",
"[19] suggest to use the saturation of the fluorophore excitation as such a non-linearity (called SSIM for “saturated structured illumination microscopy\").",
"[21] describes the practical implementation of this scheme and reports an experimental resolution of $<$ 50nm (i.e., roughly a four-fold improvement compared to the Abbe limit).",
"Fluorescence microscopy is also the arena for the latest developments.",
"Two families can be distinguished which utilize the ability to switch fluorophores between different states (“on – off\" or “bright – dark\").",
"Stimulated emission depletion (STED) microscopy excites the fluophores in a diffraction limited spot at first.",
"However, an additional STED beam de-excites all fluorophores but those in a small region close to the zero-point of the doughnut-shaped STED beam [22], [23].",
"This leads to a reduced area of potential emittance.",
"The width of the effective PSF depends on the intensity of the STED beam, $I_{\\mathrm {max}}$ , and the saturation intensity of the corresponding fluophores, $I_{\\mathrm {sat}}$ , according to [24]: $d_{\\mathrm {STED}}=\\frac{d_{\\mathrm {Abbe}}}{\\sqrt{1+\\frac{I_{\\mathrm {max}}}{I_{\\mathrm {sat}}}}}.",
"$ Stimulated emission is just one process to distinguish markers and the generalized class of microscopy techniques which exploit similar effects has been labeled RESOLFT (reversible saturable optical fluorescence transitions), [25].",
"Finally, we should mention the family of microscopy techniques based on single molecule imaging (SMI).",
"As suggested by Betzig the basic idea is the separation of nearby fluorophores through “unique optical characteristics\" [26].",
"To achieve this separation through the time domain was accomplished for the first time by Lidke et al.",
"[27] through the “blinking\" of quantum dots.",
"For biological imaging the successful application was reported in 2006 independently by three groups.",
"Their methods were named STORM (Stochastic Optical Reconstruction Microscopy) [28], PALM (Photo Activated localization Microscopy) [29] and FPALM (Fluorescence Photo Activated localization Microscopy) [30].",
"In the mean time other modified schemes of localization microscopy have been developed like dSTORM (direct STORM) or PALMIRA (PALM with independently running acquisition).",
"All of them apply a similar strategy: The probe is labeled with photo-switchable fluorescent markers and a weak light pulse activates a random, sparse subset of these fluorophores.",
"Ideally each of these sources is separated by more than the Abbe limit.",
"However, for an emitter known to be isolated the localization precision is not restricted by diffraction.",
"Given the shape of the PSF its mean can be estimated from a fit to the data with a precision limited by the signal intensity and SNR only.",
"For the detection of a full image a strong “bleaching\" pulse is applied to make the active molecules permanently (or temporary) dark.",
"Another activation pulse then turns on a different sparse subset, which is again localized.",
"This cycle is repeated until sufficient image details have been acquired or all the dye molecules have been switched.",
"This results into a list of emitter positions, localization precisions, noise and background.",
"This data can be used to render an image which shows details with a resolution between 10 and 50nm [31]."
],
[
"Classification based on information theory",
"Our brief summary of far-field microscopy techniques to break the Abbe limit illustrated the diversity of approaches.",
"However, it has been noted by Testorf and Fiddy [32] that while most of them have been labeled as “super-resolution\", this has been done “ [...] frequently independent of adherence to any of the conditions that would define a meaningful resolution limit in classical terms.",
"[...] This, in turn, has created a culture of reporting on new superresolution schemes in terms of the achievable resolution rather than in terms of the relationships to preexisting methods or to fundamental underlying assumptions.\"",
"They conclude, that the comparison of different schemes is hindered and that a proper and systematic classification of existing methods would even aid the development of new methods.",
"In a similar vein Sheppard [33] has noted that the concept of super-resolution is “somewhat confused\".",
"In [32] and [33] a classification based on information theory is applied to arrive at a unified framework for the discussion of super-resolution and we will briefly review this work.",
"According to Cox and Sheppard [34] the information capacity of an optical system is given by: $N=(2L_x B_x+1)(2L_y B_y+1)(2L_z B_z+1) (2TB_T+1) \\cdot \\log (1+\\mathrm {SNR}) $ Here, $B_x$ , $B_y$ and $B_z$ denote the spatial band-width in the corresponding direction, $B_T$ the temporal band-width, $T$ the observation time, $L_xL_y$ the field-of-view and $L_z$ the depth-of-field of the system.",
"SNR denotes the signal-to-noise ratio.",
"While this information capacity can not be exceeded, which is called the “Theorem of Invariance of Information Capacity\" [34], the spatial band-width can be increased at the expense of e.g., the temporal band-width or the field-of-view.",
"In [32], [34], [33] this approach is used to make the trade-offs transparent which underly different super-resolution schemes.",
"The super-resolving pupils suggested in [10] increase the resolution at the expense of the reduced field-of-view since they produce side-lobes.",
"Thus, in themselves (i.e., without restricting the field-of-view) these pupils do not increase the spatial frequency band-width, i.e., they are not super-resolving per se.",
"This is why Cox and Sheppard [34] suggest the term “ultraresolution\" for them.",
"In contrast, do confocal microscopy or SIM (briefly touched uppon in Sec.",
"2) increase the spatial frequency band-width at the expense of the temporal band-width, i.e., both methods need several images to be taken (confocal microscopy is a scanning technique and SIM needs several images with rotated geometry of the structured illumination).",
"Note, that the latter methods assume some prior knowldege about the object, namely that it does not vary in time.",
"Only then the temporal band-width can be traded against the spatial band-width.",
"We note in passing that also confocal or structured illumination microscopy are not super-resolving per se until the additional images have been recorded.",
"Why a similar argument qualifies Toraldo di Francia's pupils an instance of “ultraresolution\" only (i.e., no super-resolution proper) appears questionabe to us.",
"Cox and Sheppard [34] apply the information capacity approach also in the case of unrestricted super-resolution provided by analytic continuation.",
"Here, the trade-off is with respect to the SNR, given that the spectrum within the passband is not know with arbitrary precision and the limit is set by noise.",
"Summing up, Sheppard [33] suggests three types of super-resolution: (i) improved spatial frequency response with unchanged cut-off (as with superresolving pupils introduced by Toraldo di Francia) (ii) techniques with increased cut-off (like e.g., confocal scanning laser microscopy and SIM) and (iii) unrestricted super-resolution by fundamentally unconfined increase of the cut-off of the OTF (e.g., by analytic continuation).",
"However, how do the more recent techniques like STED, SSIM or single molecule imaging (SMI) fit into this picture?",
"While STED and SSIM are briefly mentioned in [33] this work was apparently written before the advent of STORM, PALM and FPALM.",
"SSIM and STED may be related to this classification by noting that also here the unrestricted resolution needs a trade-off with the SNR.",
"Given that the photo-chemical and spectral properties of the fluorophores provide a piece of information about the object they apparently fit well into the conceptual framework of information theory.",
"This holds at least if one views the fluorophores as the “object\" of fluorescence microscopy – and not the structure which has been labeled.",
"Otherwise the fluorophores are actually part of the imaging system.",
"However, how to quantify this information in the current framework is rather unclear and one may conjecture that an additional term needs to be included into Eq.",
"REF .",
"Key to the novel superresolving techniques in fluorescence microscopy is to exploit the (non-linear) interaction with the contrast generating agent.",
"One may also argue that the concept of an imaging system as a (passive) information channel is not sufficient to capture this novel aspect of imaging.",
"Be this as it may, we will leave this question aside and turn to single molecule imaging (SMI) which is more complicated still.",
"Note, that the discussion so far was framed in terms of isoplanatic systems in which the PSF does not depend on the position.",
"Only then the image formation can be described by a convolution (Eq.",
"REF ).",
"However, this assumption does not apply in single molecule imaging.",
"In the first place, these methods produce initially no image at all but a data set of emitter positions, localization precisions and intensities (signal and background).",
"Agreed, all image-data acquisition systems produce “data\" in the first place, but here there is no natural way to display this data and they need to be rendered according to some user specified method (See e.g., [35] for a discussion of several visualization methods and their interrelation to the resolution issue).",
"A natural candidate for the effective PSF in SMI microscopy is apparently a point-spread function (e.g., in the Gaussian approximation) with the localization precision as its width.",
"However, this ignores that in general SMI microscopy yields a different precision for each fitted emitter.",
"Thus, the image formation can not be described by a convolution with a PSF, since there is no single (effective) PSF to convolve with.",
"Only if one neglects the position dependence of the effective PSF, SMI can be described by a convolution (see e.g., [36] where PALMIRA is treated).",
"However, the quality of this approximation needs to be tested case by case.",
"Hence, the strategy of SMI microscopy to break the Abbe limit is not covered by the classification suggested in [33].",
"One may wonder how the resolution in SMI microscopy is defined, given that the common approach via the OTF cut-off is blocked.",
"In fact, usually the mean localization precision (e.g., in terms of the full width at half maximum [28]), the resolution derived from the labeling density by the Nyquist criterion [37] or a combination of both [38] are quoted.",
"However, all these definitions are heuristic only.",
"Given these deep conceptual problems other integral resolution measures have been suggested.",
"For example [39], [40] base their different approaches on estimation theory while in [41], [42] the Fourier ring correlation (FRC) is proposed as a resolution measure for SMI microscopy."
],
[
"Novel classification by reconsidering the Abbe limit",
"In order to classify the strategies for breaking the Abbe limit properly we therefore suggest to spell out its content more carefully.",
"It is certainly well known, that in its current Fourier optical formulation the Abbe limit can be construed as a conjunction of three hierarchically related claims: Convolution assumption (CONV), whereby the image formation can be represented as a convolution of the object distribution with a point-spread function (PSF).",
"Resolution-cut-off relation (RCR), whereby the principle resolution limit is given by the cut-off frequency of the Fourier transform of the PSF (i.e., the OTF).",
"Abbe Cut-off (AC), whereby the cut-off frequency of an optical system is given by $k_0={\\mathrm {NA}}/\\lambda $ (coherent case) or $k_0=2{\\mathrm {NA}}/\\lambda $ (incoherent case).",
"Now it becomes evident that “breaking the Abbe limit\" can mean quite different things.",
"E. g. confocal microscopy, linear structured illumination, but also STED microscopy or SSIM expand the band-width explicitly or effectively.",
"Here the claim AC is refuted while RCR (and also CONV) remain the underlying assumption and motivation.",
"One might describe the situation by saying that this breaking or “bypassing\" of the Abbe limit is still guided by Abbe's original reasoning.",
"In contrast, the observation that by analytic continuation the OTF can (in principle) be extrapolated beyond the cut-off frequency refutes the claim RCR.",
"Note, that the cut-off is not altered by this procedure and that the actual passband provided by any specific optical system (e.g., according to AC) plays no role whatsoever.",
"Finally, in SMI microscopy (e.g., STORM, PALM or FPALM) the underlying convolution assumption (CONV) regarding the image formation does not apply.",
"Expressed pointedly, if there is no OTF, it can neither be used to define the resolution nor can its cut-off be extended."
],
[
"Summary and conclusion",
"We have argued that it is difficult to apply the current information theoretic framework to deal with the recent developments in super-resolving fluorescense microscopy.",
"It remains true that all kinds of super-resolution exploit some prior knowledge about the sample but to incorporate this into the information theoretic framework needs to be an objective of future developments in this field.",
"E.g., STED microscopy utilizes properties of the fluorophores which can not be quantified in the current information theoretical formalism.",
"In SMI microscopy the very definition of the band-width of the imaging system is intricate.",
"Thus, also here does the current information theoretic framework not help to make the corresponding trade-offs transparent.",
"Our simple reconstruction of the Abbe limit as the conjunction of the (i) convolution assumption, (ii) the OTF-based resolution definition and (iii) the specific OTF cut-off values as derived by Abbe allows for a more differentiated description of the strategies to break this limit.",
"This perspective may provide a complementary view on the classification in super-resolution microscopy."
],
[
"Acknowledgement",
"We gratefully acknowledge the illuminating email exchange with Marcel Lauterbach, discussions with Marc Müller and helpfull comments by two anonymous referees."
]
] | 1606.05081 |
[
[
"Stellar counter-rotation in lenticular galaxy NGC 448"
],
[
"Abstract The counter-rotation phenomenon in disc galaxies directly indicates a complex galaxy assembly history which is crucial for our understanding of galaxy physics.",
"Here we present the complex data analysis for a lenticular galaxy NGC 448, which has been recently suspected to host a counter-rotating stellar component.",
"We collected deep long-slit spectroscopic observations using the Russian 6-m telescope and performed the photometric decomposition of Sloan Digital Sky Survey (SDSS) archival images.",
"We exploited (i) a non-parametric approach in order to recover stellar line-of-sight velocity distributions and (ii) a parametric spectral decomposition technique in order to disentangle stellar population properties of both main and counter-rotating stellar discs.",
"Our spectral decomposition stays in perfect agreement with the photometric analysis.",
"The counter-rotating component contributes $\\approx$30 per cent to the total galaxy light.",
"We estimated its stellar mass to be $9.0^{+2.7}_{-1.8}\\cdot10^{9}M_\\odot$.",
"The radial scale length of counter-rotating disc is $\\approx$3 times smaller than that of the main disc.",
"Both discs harbour old stars but the counter-rotating components reveals a detectable negative age gradient that might suggest an extended inside-out formation during $3\\dots4$ Gyrs.",
"The counter-rotating disc hosts more metal-rich stars and possesses a shallower metallicity gradient with respect to the main disc.",
"Our findings rule out cosmological filaments as a source of external accretion which is considered as a potential mechanism of the counter-rotating component formation in NGC 448, and favour the satellite merger event with the consequent slow gas accretion."
],
[
"Introduction",
"The kinematical appearance, as well as the stellar population properties keep a fossil record about the process of galaxy assembly which is crucial for our understanding of galaxy physics.",
"Most disc galaxies possess regular rotation but systems with peculiar kinematics are also observed.",
"proposed a term “multi-spin galaxies” for objects possessing embedded kinematically decoupled components, the angular momentum of which differs from that of the host galaxy.",
"The class of multi-spin galaxies includes objects with various kinematically distinct components containing gas and/or stars having different spatial extent and inclination with respect to the main galaxy disc: inner polar discs, extended polar rings/discs, moderately inclined rings, kinematically decoupled cores, and extended counter-rotating components.",
"Studies of multi-spin components shed the light on the process of gas accretion and merging history which are thought to be responsible for shaping disc galaxies.",
"The stellar counter-rotation confined to the main galactic plane can be considered as a final product of evolution of externally acquired gas that has been processed through at least two major steps: (i) settling of the gas into the equatorial galactic plane due to, for instance, dynamical influence of the gravitational potential of the main disc ; (ii) and subsequent star formation .",
"Potentially, studies of counter-rotating components may allow us to date accretion events and the following star formation that leads to understanding of the galaxy assembly.",
"The counter-rotation phenomena were deeply studied by means of numerical simulations, , investigated various mechanisms which are able to produce a counter-rotating gas component, such as episodic and continuous gas infall or merger with a gas-rich dwarf satellite.",
"In their simulations, only small counter-rotating stellar discs with radial density profiles different from exponential were generated.",
"Binary major mergers of giant galaxies usually destroy discs and form ellipticals [5].",
"However, a strictly coplanar merger of two gas-rich giant progenitor galaxies is able to build up a massive counter-rotating disc , .",
"Otherwise, recent numerical cosmological simulation by [4] naturally predicts the formation of counter-rotating discs as a consequence of gas accretion from two distinct filamentary structures.",
"Presently, several dozens of galaxies with counter-rotating components are known , .",
"The majority of them have been only suspected to host counter-rotating components.",
"Particular progress in the studies of counter-rotating discs has been achieved upon development of the spectral decomposition approaches.",
"For the first time, such an approach was applied by in order to recover a spectrum of an ultra-compact dwarf galaxy contaminated by the host galaxy light.",
"Independently, firstly proposed the spectral decomposition approach based on the spectral pixel fitting technique ppxf for the investigation of the counter-rotation in NGC 5719.",
"The spectral decomposition allows one to disentangle contributions of both counter-rotating and main components to the observed spectrum and to determine stellar population properties and, therefore, constrain their formation mechanisms.",
"simultaneously measured kinematics and stellar population properties and found that less massive counter-rotating component is younger, less metal abundant and $\\alpha $ -element enhanced with respect to the main stellar disc.",
"Later, Coccato et el.",
"and other teams, including ours, analysed spectra of a few counter-rotating galaxies by using the same or similar approaches.",
"The disc galaxies with counter-rotation, NGC 3593, NGC 4138, NGC 4191, NGC 4550, NGC 5719, and IC 719, have been studied in this manner , , , , , .",
"The results of these studies indicate that the counter-rotating components in all those objects being less massive than the main stellar discs, have younger stellar populations than the main discs while their metallicities and $\\alpha $ -element abundances can deviate in both directions.",
"Those findings are in agreement with the formation scenario where counter-rotating stars emerge from the externally accreted gas .",
"In this paper we expand the sample of well-studied counter-rotating galaxies by one more object.",
"We present an analysis of new deep long-slit spectroscopic data for a counter-rotating lenticular galaxy NGC 448."
],
[
"NGC 448: general description",
"NGC 448 is a lenticular galaxy located at a distance of 29.5 Mpc with the total luminosity $M_B=-19.17$ mag (according to the HyperLeda databasehttp://leda.univ-lyon1.fr/ ) and $M_K=-23.02$ mag .",
"In the RC3 NGC 448 is classified as S0⌃- edge-on.",
"Nevertheless, the direct SDSS image (Fig.",
"REF ) does not explicitly indicate a strongly edge-on galaxy orientation.",
"Figure: Coloured SDSS (DR12 ) image of NGC 448 and itssatellite GALEXMSC J011516.31-013456.8.",
"North is up, east is left.",
"The image wasextracted using the cutout image service.SDSS image cutout tool http://skyserver.sdss.org/dr12/en/help/docs/api.aspx#cutout NGC 448 was included into the ATLAS3D integral-field spectroscopic survey .",
"For the first time, NGC 448 was noticed as a galaxy possessing two large-scale counter-rotating disc-like stellar components by basing on ATLAS3D stellar line-of-sight velocity and velocity dispersion maps which indicate a counter-rotating core and double peak features, correspondingly.",
"NGC 448 has a companion galaxy 2.6 arcmin to the North which corresponds to the projected separation of 22 kpc for the adopted galaxy distance of 29.5 Mpc.",
"According to the NASA/IPAC Extragalactic Database (NED)http://ned.ipac.caltech.edu/, this galaxy is cross-identified with GALEXMSC J011516.31-013456.8, which radial velocity $v_r=22105$ $\\textrm {km~s$ -1$}$ was determined in the CAIRNS (Cluster and Infall Region Nearby Survey) project .",
"However, extremely deep optical imaging with the MegaCam camera at the Canada-France-Hawaii Telescope (CFHT) unambiguously shows a tidal interaction between NGC 448 and a disturbed companion (see Fig.",
"16 in The deep image obtained with the MegaCam camera indicating the tidal interaction between NGC 448 and companion can be seen at the URL: http://irfu.cea.fr/Projets/matlas/public/Atlas3D/NGC0448_meg.html).",
"One can see a faint tidal tail around GALEXMSC J011516.31-013456.8 even at the contrast-enhanced SDSS image (Fig.",
"REF ).",
"This points to the wrong redshift measurement for GALEXMSC J011516.31-013456.8 and suggests the physical interaction between this galaxy and NGC 448.",
"Investigation of stellar population properties by using optical long-slit spectra revealed the old stellar population ($\\sim $ 9 Gyr) and slightly sub-solar metallicity in NGC 448 .",
"Similar values were determined by based on the SAURON Lick index measurements within one effective radius ($R_{eff}=11.2^{\\prime \\prime }$ ): $T_{SSP}=8.0\\pm 1.5$ Gyr, $Z_{SSP}=-0.21\\pm 0.05$ dex, [$\\alpha $ /Fe]$=0.1\\pm 0.06$ dex.",
"Observations of the CO emission provides only upper limits on the mass of molecular hydrogen, $M($ H$_2)<5.5\\cdot 10^7$ M$_\\odot $ , .",
"We do not find any data on the Hi content of NGC 448."
],
[
"Observations and data reduction",
"We obtained new deep spectroscopic data for NGC 448 on October 25, 2013 by using the SCORPIO universal spectrograph [1] mounted at the prime focus of the Russian 6-m BTA telescope operated by the Special Astrophysical Observatory.",
"We used the long-slit mode with the slit width of 1.0 arcsec placed along the major axis of the galaxy.",
"The volume phase holographic grism VPHG2300G gives the spectral resolving power $R\\approx 2000$ (instrumental velocity dispersion of $\\sigma _{inst}\\approx 65$ $\\textrm {km~s$ -1$}$ ) in the wavelength range 4800$\\dots $ 5600 Å which includes strong absorption (Mg$b$ , Fe5270, Fe5335, etc.)",
"and emission lines (H$\\beta $ , [Oiii], [Ni]).",
"We collected 16 exposures of 15 min each (4 hours in total) under good atmospheric transparency with the average seeing of 1.5 arcsec FWHM.",
"The detector CCD EEV42-40 (2048$\\times $ 2048 pixels) provides the spectral sampling of 0.37 Å pix$^{-1}$ and the slit plate scale of 0.357 arcsec pix$^{-1}$ in the $1\\times 2$ binning mode.",
"In addition to the science spectra, we obtained internal flat field, arcs (He-Ne-Ar) and twilight spectra.",
"We reduced our spectroscopic data with our own idl-based pipeline which consists of standard procedures such as bias subtraction, flat fielding, cosmic ray hit rejection by using Laplacian filtering technique L.A.Cosmic , the wavelength calibration, the sky background subtraction and correction for the spectral sensitivity inhomogeneity.",
"In order to build the wavelength solution, we fitted the arc line positions using bivariate polynomial of the 5th degree along dispersion and the 4th degree across dispersion in order to take into account the slit image curvature.",
"A typical value of the RMS in every line is 0.05 Å.",
"Due to optical distortions, the spectral line spread function (LSF) of the spectrograph varies along the slit as well as within the wavelength range.",
"The LSF variations affect the night-sky spectrum which is derived by using outer areas of the frame where LSF shape differs from that in the regions of the galaxy close to the slit center.",
"In order to take into account LSF variations, we used the reconstruction method in the Fourier space described in ; a detailed discussion on the sky subtraction for a changing LSF is provided in , .",
"Finally, we linearized the spectra and integrated all separate exposures controlling the position of the bright galaxy center in order to take into account the atmospheric refraction.",
"We computed error frames from the photon statistics and proceeded them through all reduction step.",
"We exploited the twilight spectrum in order to extract the LSF along the slit by fitting it against a high-resolution solar spectrum.",
"Fig.",
"REF demonstrates a fragment of the co-added reduced spectrum of NGC 448 near the strong Mg$b$ absorption feature.",
"The spectrum is normalized by mean values in every row along the slit for presentation purposes in order to achieve the higher contrast.",
"The complex velocity structure of the absorption lines which corresponds to kinematically separated stellar components is clearly seen.",
"Figure: Fragment of the long-slit spectrum around the Mgbb absorption feature.In order to achieve a better contrast, the spectrum was normalized by the average lightprofile.",
"Two kinematically distinguished components are clearly seen."
],
[
"The Non-parametric LOSVD Reconstruction",
"A visual inspection of strong absorption lines in the spectrum (see Fig.",
"REF ) reveals a complex, multi-component structure of the line-of-sight velocity distribution (LOSVD) of stars which cannot be successfully described by a single Gaussian or a Gaussian-Hermite function .",
"We applied a non-parametric reconstruction approach in order to determine the stellar LOSVD of NGC 448.",
"First, we adaptively bin the long-slit spectrum in the spatial direction in order to reach a minimal signal-to-noise ratio of $S/N=30$ per bin per spectral pixel in the middle of the spectral range.",
"After that, we determine a template spectrum by using the nbursts package , .",
"This package implements a pixel-to-pixel $\\chi ^2$ minimization fitting algorithm where an observed spectrum is approximated by a stellar population model broadened with a parametric LOSVD (the Gaussi-Hermite shape is used at this step) and multiplied by some polynomial continuum in order to take into account dust attenuation and/or possible flux calibration errors in both observations and models.",
"We used a grid of pegase.hr high resolution simple stellar population (SSP) models based on the ELODIE3.1 empirical stellar library , the Salpeter initial mass function and pre-convolved with the SCORPIO LSF determined from a twilight spectrum as explained above.",
"Then, we used the stellar population model broadened with the LSF only (without the LOSVD) as a template spectrum for the non-parametric LOSVD reconstruction.",
"The reconstuction technique does not require any a priori knowledge about the LOSVD shape and searches the solution of the convolution problem as a linear inverse ill-conditioned problem by using a smoothing regularization.",
"For more details, see and where we applied the same approach to recover a counter-rotating stellar disc in the lenticular galaxy IC 719 and a complex LOSVD in NGC 524.",
"Figure: Our result of the non-parametric reconstruction of the stellar LOSVD presented as aposition-velocity diagram.",
"Black and blue circles show Gaussian peak positions fora double Gaussian decomposition.Fig.",
"REF shows the reconstructed non-parametric stellar LOSVD of NGC 448 for each spatial bin.",
"Again a complex two-component peaked structure is clearly visible.",
"Then we fitted a non-parametric LOSVD by a combination of two Gaussians in every bin and estimated the line-of-sight velocities and stellar velocity dispersions for the two kinematically distinct components."
],
[
"The Parametric Spectral Decomposition",
"In order to study stellar population properties of the counter-rotating and main stellar components, we separated their contributions to the integrated spectrum by using the two-component mode of the nbursts package.",
"In general, we fitted an observed spectrum by a linear combination of two SSP templates, each characterized by its own age and metallicity, and broadened by different Gaussian-shaped LOSVDs.",
"The multiplicative polynomial continuum was the same for both components.",
"The Gaussian parameters of the previous non-parametric LOSVD decomposition were used as an initial guess for the kinematics of components.",
"Finally, the free parameters of our model are LOS velocity, velocity dispersion, age, metallicity for every component, relative weights and the 15th order the polynomial continuum, hence, 24 parameters in total.",
"Thanks to the high signal-to-noise ratio and relatively high spectral resolution, the fitting procedure was stable enough so we did not have to fix any parameters of the model.",
"In order to estimate the parameter uncertainties, we ran a Monte-Carlo simulations for a hundred realization of synthetic spectra for each spatial bin which were created by adding a random noise corresponding to the signal-noise ratio in the bin to the best-fitting model.",
"Figure: Fragment of the spectrum at a radius r=-12r=-12 arcsec.",
"The red line correspondsto the best-fitting model overplotted on top of the observed spectrum.",
"Green and blue linesshow the main disc and the counter-rotating component, correspondingly.Figure: The spectral decomposition results for NGC 448.",
"Black symbolscorrespond to the main stellar component, magenta symbols to thecounter-rotating disc.",
"Left panels show radial profiles of the line-of-sightsvelocity (top) and the stellar velocity dispersion (bottom).",
"Middle panels:SSP equivalent stellar population ages (top) and metallicities (bottom).Right panels: relative contribution of every component to the integratedspectrum (top); the bottom panel shows the light profile which was obtained byintegrating the spectra along the slit (filled symbols) andlight profiles of each component (open symbols).Fig.",
"REF shows an example of a spectrum with the overplotted best-fitting model in a region where the complex two-component structure of absorption lines is clearly seen.",
"The decomposed radial profiles of the kinematics and stellar populations are shown in the Fig.",
"REF ."
],
[
"The Photometric Decomposition",
"As a complementary approach to the spectroscopic decomposition, we performed a two-dimensional (2D) decomposition of a SDSS $g$ -band image of NGC 448.",
"We used the $g$ -band in order to compare the photometric decomposition results with those obtain from the spectra since the spectral range of our observations is close to the SDSS $g$ -band.",
"In order to construct a photometric model of the galaxy, we used the galfit software.",
"An accurate treatment of uncertainties, both for the galaxy image and for the point spread function (PSF), is required in order to obtain reliable component parameters of the photometric model when using galfit.",
"We took into account the image uncertainties following the SDSS documentation http://data.sdss3.org/datamodel/files/BOSS_PHOTOOBJ /frames/RERUN/RUN/CAMCOL/frame.html.",
"The PSF function was determined by the SExtractor/PSFEx packages [7], [6].",
"We masked background/foreground interlopers (stars, galaxies) superimposed on the galaxy by using the fitting residuals map and then repeated the fitting process.",
"The field around NGC 448 is crowded because of Galactic foreground stars.",
"That could potentially lead to the imperfect sky subtraction and, consequently, bias the photometric decomposition.",
"We re-analyzed the sky unsubtracted SDSS image (constructed according to the published documentation[]) and estimated the sky level by using azimuthally averaged surface brightness profile determined by fitting ellipses with constant ellipticity and position angle at large radii from the galaxy center (150–200 arcsec).",
"We found that our sky background estimate differs from the SDSS pipeline valueThe SDSS pipeline produces bivariate sky map.",
"To compare it with our sky level estimate we averaged this map within the galaxy area, which is used for the photometric decomposition.",
"by 0.2 per cent which stays within uncertainties of our background estimate.",
"Table: Statistical tests of photometric models.We performed numerous tests and found that the galaxy image can be successfully decomposed into four components: a Sérsic nuclear core, two exponential discs and a halo described also by a Sérsic function.",
"Beside this model, we tested other photometric decompositions: (i) 2 Sérsic + exponential disc + edge-on disc; and (ii) 2 Sérsic + exponential disc.",
"In order to choose between models and to measure quantitatively how well those models fit the same data, we compared normalized $\\chi ^2$ values, the Akaike and Bayesian information criteria (AIC, [2]; BIC, ) which we present in the Table REF .",
"The model with the lowest AIC or BIC is preferred, though a difference $\\Delta $ AIC or $\\Delta $ BIC of at least $\\sim $ 6 is usually required before one model can be deemed clearly superior .",
"Finally we adopted a four-component model that consists of a Sérsic nuclear core, two exponential discs, and a Sérsic halo.",
"Hereafter, the more centrally concentrated exponential disc is referred to as “CR” assuming its association with the counter-rotating stellar component.",
"Fig.",
"REF graphically demonstrates the results of the decomposition, and Fig.",
"REF displays the comparison between the spectroscopic and photometric results.",
"Table REF presents the best-fitting parameters of the photometric model as well as parameter uncertainties.",
"We estimated the parameter uncertainties from the covariance matrix as a standard output of the Levenberg–Marquardt minimization algorithm used in the galfit routine and also by Monte-Carlo simulations for a hundred realizations of synthetic galaxy images in the same manner as for the spectroscopic decomposition.",
"Error estimates from both approaches are in good agreement.",
"We emphasize that our uncertainty estimates should probably be considered as lower limits of the true parameter uncertainties , and should be used with caution.",
"Table: Structural component parameters of NGC 448 from the photometricdecomposition.",
"Column (2) corresponds to the central surface brightness μ 0 \\mu _0,(3) is the disc scalelength hh or effective radius r e r_e for Sérsiccomponent, (4) is the Sérsic index nn, (5) is the positional angle PAPA, (6)is the axis ratio q=b/aq=b/a, (5) is the diskiness/boxiness C 0 C_0, (6) is thecontribution to the total image (C/TC/T).",
"The parameter uncertaintiesoutcome from galfit routine, while errors in the parentheses areextracted by means of Monte-Carlo simulations.Figure: The 2D photometric decomposition results of the SDSS gg-band image.",
"Thetop panel in the left block presents the galaxy image superimposed by contoursaccording to the surface brightness of 18, 19, 20, 21, 22, 23, 24 mag arcsec -2 ^ {-2}.Blue lines correspond to the contours of the model image.",
"The bottompanel in the left block presents a residual map in magnitudes with the modelcontours as a reference.",
"The middle block shows images of the model subcomponents(from top to bottom): core, disc (CR), main disc, halo.",
"The right block presents aone-dimensional light profile along the major axis of the galaxy (black line)with the overplotted total 4-component model (in red) and the subcomponentprofiles (in colours).",
"Dotted lines show the level of uncertainties.",
"The bottompanel in this block corresponds to the fitting residuals.Figure: Comparison of the photometric and spectroscopic decompositions.Filled circles show the light profile extracted from our long-slit spectrum,open circles are the results of the spectroscopic decomposition on the main (in black) andcounter-rotating (in magenta) components.",
"Solid lines display the result of thephotometric decomposition.",
"The red solid line corresponds to the total photometricmodel, the blue solid line is a sum of disc (CR) and core componentsthat is assumed to be associated with the counter-rotating component, the green solidline is a main component represented by the sum of the main disc and halocomponents.",
"Here we took into account the atmospheric seeing differencebetween SCORPIO observations (FWHM=1.5 arcsec) and SDSS gg-band image(FWHM=1.0 arcsec) as well as sampling effects.",
"This comparison clearly demonstratesthe successful agreement between spectroscopic and photometric approach."
],
[
"Results",
"It is clearly seen on the position-velocity diagram (Fig.",
"REF ) as well as on the radial profiles of the light contribution of the individual components extracted from the long-slit data (the right panel of Fig.",
"REF ) that the counter-rotating component dominates in the integrated spectrum in the central region of NGC 448.",
"The total light fraction of the counter-rotating disc within one effective radius ($R_{eff}=11.2$ arcsec) is 60 per cent and decreases to 45 per cent within $2R_{eff}$ (see Fig.",
"REF ).",
"Figure: Total light fraction of the counter-rotating component (CR) withinthe galactocentric (not projected) radius RR, which is calculated based onthe light profiles extracted along the major axis from the long-slitspectroscopic decomposition (magenta) and from the two-dimensional photometricdecomposition (blue).",
"The dashed blue line corresponds to the fraction of the Disc CR _{CR}component only, while the solid blue line shows Disc CR _{CR} + Core.",
"The blue dottedhorizontal line presents a total contribution of the CR component (Disc CR _{CR} + Core)to the galaxy image from the galfit modeling.",
"Note that wecalculated the light fraction from one-dimensional light profiles assuming anaxisymmetric galaxy light distribution.Our photometric results completely agree with the spectral decomposition.",
"Fig.",
"REF demonstrates the comparison of the major axis light profiles extracted from the spectrum and from the image.",
"Note, that in the central region we cannot separate the contributions of the kinematical components due to the very small difference of their line-of-sight velocities.",
"Therefore, it is more reasonable to compare the total contribution of the CR disc and the core component together.",
"Fig.",
"REF shows the comparison of the total light fraction of the counter-rotating component within a given galactocentric radius.",
"We assume that the counter-rotating component has a disc morphology.",
"This assumption is supported by a relatively low velocity dispersion ($V/\\sigma \\sim 2$ ) and by the exponential shape of the light profile distribution (see the right panel of Fig.",
"REF ).",
"Our two-dimensional photometric model of NGC 448 contains two exponential discs.",
"In our photometric model, the counter-rotating disc (CR) has a significant diskiness of isophotes which is characterized by $C_0=0.2$ .",
"The nearly edge-on disc orientation can produce such values of the $C_0$ coefficient.",
"The galfit package contains the edge-on disc model.",
"We tested it and found that it yields higher values of $\\chi ^2$ as well as the statistical criteria AIC and BIC which may be connected to the disc thickness changing along the radius.",
"We cannot determine the exact orientation of the counter-rotating disc with respect to the main disc by relying only on our long-slit kinematical data because we observed NGC 448 only in one slit position.",
"However, there are several arguments which support the discs to be settled in the same plane: (i) the equal amplitudes of the line-of-sight velocity variations along the radius; (ii) the co-alignment of the kinematical major axes of the central counter-rotating region and the outermost main disc component (see ATLAS3D velocity maps in ); (iii) similar values of the isophote major axis position angles and inclinationsIndeed, assuming flatnesses of the discs the axis ratios $q_{CR}=0.280$ and $q=0.367$ produce inclinations $i_{CR}=\\arccos (q_{CR})=74^\\circ $ and $i=68^\\circ $ , respectively.",
"We found that in the galaxy center both discs contain old stars ($T_{SSP}\\approx 9$ Gyr) having sub-solar metallicity putting our results in agreement with the literature , .",
"Our deep spectroscopic data allowed us to determine a noticeable age gradient $\\Delta \\log T_{CR}=-0.087\\pm 0.026$ dex per dexThe gradient calculated as a slope of a linear fit of the form $a + b \\log (r/R_{eff})$ .",
"The uncertainties of age and metallicity measurements are invoked to the $\\chi ^2$ minimization routine to estimate 1-$\\sigma $ error of the gradient.",
"in the counter-rotating stellar disc, while in the main stellar disc the age profile is flat ($\\Delta \\log T_{MD}=-0.009\\pm 0.024$ dex per dex).",
"The stellar metallicity gradients are also noticeably different: $\\Delta \\log Z_{CR}=-0.09\\pm 0.04$ dex per dex while $\\Delta \\log Z_{MD}=-0.33\\pm 0.05$ dex per dex.",
"We have subtracted the best-fitting absorption-line model from the observed spectrum in order to obtain a pure emission-line spectrum.",
"Our long-slit spectrum covers the positions of the H$\\beta $ and [Oiii] emission lines.",
"We have not detected any significant signal in these lines.",
"Although we have not clearly detected emission lines in the spectra of NGC 448, we found signs of possible star formation in infrared (IR) images.",
"We analysed available archival Spitzer images of the galaxy at 3.6$\\mu $ and 8$\\mu $ .",
"While emission in the first band originates almost exclusively from the stellar light, the dominate sources of emission at 8$\\mu $ are polycyclic aromatic hydrocarbons (PAH), being excited by UV photons from massive and/or hot stars.",
"Hence, if bright emission is seen in an 8$\\mu $ image after the subtraction of the stellar population contribution, this should indicate the presence of gas excited either by the star formation in a galaxy or by hot evolved blue horizontal branch stars.",
"In order to subtract the stellar continuum in the 8$\\mu $ image, we used the calibration by that estimated the 8$\\mu $ stellar contribution to be about 24.9 per cent of the total flux at 3.6$\\mu $ for early-type galaxies.",
"The subtracted image corresponding to the non-stellar emission at 8$\\mu $ is shown in Fig.",
"REF with overlaid contours from the optical SDSS-$g$ image.",
"Figure: A stellar continuum subtracted Spitzer image at 8μ\\mu with overlaidcontours according to the surface brightness values of 18, 19, 20, 21, 22, 23, 24mag arcsec -2 ^{-2}.",
"Some non-stellar signal is seen in the central part, R<10 '' R<10^{\\prime \\prime }.We calculated the stellar mass of the counter-rotating component by using mass-to-light ratios ($M/L$ ) determined from spectral fitting based on the pegase.hr models with the Salpeter IMF and the light profile which we recovered from the photometry.",
"As we described above the counter-rotating component is assumed to be associated with both the core and the CR disc components in the photometric decomposition.",
"Since the core is very centrally concentrated, we used a single stellar mass-to-light ratio of $(M/L_g)_{core}=6.3$ that corresponds to the central stellar population properties.",
"For the disc (CR) component, we estimated $(M/L_g)_ {disc}$ pixel-by-pixel by using the gradients of the stellar population parameters retrieved from the long-slit spectrum.",
"The total mass of the counter-rotating component is excpected to be $M_{CR}=9.0^{+2.7}_{-1.8}\\cdot 10^{9}M_\\odot $ including $M_{disc}=7.1^{+2.6}_{-1.8}\\cdot 10^ 9M_\\odot $ and $M_{core}=1.8^{+0.2}_{-0.2}\\cdot 10^{9}M_\\odot $ .",
"Mass uncertainties have been calculated by using uncertainties of the stellar population properties."
],
[
"Discussion and Conclusions",
"The main challenge in the studies of galaxies possessing counter-rotating components is to establish a possible source of material with a different direction of the angular momentum and to clarify at least some details of the counter-rotating disc formation.",
"It is generally accepted that the presence of counter-rotating stars within the main stellar disc is a result of external material acquisition .",
"Nevertheless, suggested a scenario of internal origin of counter-rotating stars where stars take retrograde orbits during the bar dissolution process (separatrix crossing).",
"From the stellar population point of view, only counter-rotating stellar discs with identical stellar population properties can be produced in the framework of this scenario.",
"Hence, we decline it in the case of NGC 448 because the two discs have significantly different stellar population properties.",
"Moreover, suggested the separatrix crossing as a natural mechanism for building identical (with the same ages and scale lengths) counter-rotating discs in NGC 4550.",
"However, the applicability of such mechanism for counter-rotating discs with very different scale lengths is not obvious.",
"Many recent detailed studies of disc galaxies with large-scale counter-rotating components support the external origin of counter-rotating stars.",
"In all studied galaxies, stellar population properties derived from the spectra play an important role.",
"It has been shown that the counter-rotating components detected in NGC 3593, NGC 5719, NGC 4191, NGC 4550, and IC 719 , , , , have younger stellar populations compared to the main stellar discs, and their ionized gas rotates in the same direction as the secondary stellar components, i.e.",
"it also counter-rotates with respect to the main disc.",
"These findings favour the scenario where counter-rotating stars have been formed in-situ from the externally accreted gas.",
"Cosmological filaments and gas-rich satellites are considered as main candidate sources of external cold gas.",
"However, it is often difficult to unambiguously disentangle between them.",
"In both scenarios, one can expect to find either metal-poor or metal-rich counter-rotating stellar population with respect to the main disc, depending on the star formation history while the brief duration of the subsequent star formation event results in the $\\alpha $ -element enhancement.",
"The diversity of properties is observed.",
"For instance, NGC 3593 and NGC 5719 indicate lower stellar metallicity in their counter-rotating discs with respect to the main discs while for NGC 4550 and NGC 4191 both discs have similar populations; and the counter-rotating stars in IC 719 and NGC 448 are more metal-rich than their main stellar discs.",
"The duration of the accretion events can be also various that results in the super-solar $\\alpha $ -element to iron ratios in the counter-rotating stellar populations for short timescales while prolonged formation of a counter-rotating component should produce a solar $\\alpha $ /Fe abundance ratio and/or significant age gradient.",
"Nevertheless, for some targets the scenario of accretion from gas-rich neighbouring galaxies looks more plausible.",
"For instance, no doubt that NGC 5719 and IC 719 have accreted their counter-rotating stellar and gaseous discs from the closest neighbours because of the Hi bridge between the galaxy and its neighbour, NGC 5713, in the case of NGC 5719, and the common Hi envelope including both galaxies, in the case of IC 719 and its neighbour IC 718.",
"Recently, have investigated the structure of the Virgo cluster S0 galaxy NGC 4191 by using the IFU spectroscopy obtained with the VIRUS-W spectrograph.",
"They have interpreted the results of photometric as well as spectroscopic decomposition of NGC 4191 in the context of the recent cosmological simulation by [4] where it is demonstrated how distinct cosmological filamentary structures providing material with the opposite spins can finally produce two coplanar counter-rotating stellar discs.",
"found significant negative age gradients in both components of NGC 4191 correlating at least in the central 20 arcsec that indicates the inside-out formation of both components.",
"In our case of NGC 448, we observe a different situation.",
"First, we show that the two independent approaches, the spectral and the photometric 2D decompositions, are in perfect agreement.",
"The secondary counter-rotating disc reveals a detectable negative age gradient ($\\Delta \\log T_{CR}=-0.087\\pm 0.026$ dex per dex) that also gives an evidence for a prolonged inside-out formation process during approximately $3\\dots 4$ Gyrs.",
"At the same time, the main stellar disc does not show any significant age gradient ($\\Delta \\log T_{MD}=-0.009\\pm 0.024$ dex per dex) that indicates that it was probably formed independently, by another mechanism than the secondary one.",
"This is also supported by different metallicity gradients ($\\Delta \\log Z_{CR}=-0.09\\pm 0.04$ dex per dex, $\\Delta \\log Z_{MD}=-0.33\\pm 0.05$ dex per dex).",
"Moreover, based on our stellar metallicity measurements, we suggest that the accreted gas was pre-enriched by metals in the companion galaxy and that the scenario of the counter-rotating disc formation from cosmological filaments is implausible for NGC 448.",
"Taking into account the large mass of the counter-rotating stellar population $M_{CR}=9.0^{+2.7}_{-1.8}\\cdot 10^{9}M_\\odot $ and indistinguishable properties of the stellar populations in the central region of the galaxy, we conclude that a significant fraction of accreted material had been already transformed into stars prior to the moment when the disc was acquired by NGC 448.",
"All these observational facts indicate that the most probable formation scenario for the counter-rotating disc in NGC 448 is a merger event with a consequent prolonged gaseous accretion.",
"Remarkably, neither the search for emission lines of ionized gas in our optical spectra, nor attempts of radio observation of CO emission and Hi 21-cm line yielded any detection of gas in NGC 448.",
"Only IR image indirectly points to the probable presence of gas.",
"The formation of its counter-rotating disc occurred approximately $6\\dots 7$ Gyrs ago since the remnants of the gas could be swept away by a recent tidal interaction between the galaxy and its satellite GALEXMSC J011516.31-013456.8 supported by the presence of a low surface brightness bridge connecting the two galaxies.",
"In conclusion, several recent studies, including ours, demonstrate that various scenarios to form counter-rotating stellar components can take place in real galaxies.",
"It is important to expand the sample of disc galaxies with counter-rotation studied in detail, in order to produce quantitative conclusions on the probability of different formation scenarios supported by the statistics.",
"The on-going large spectroscopic surveys of galaxies such as CALIFA , MANGA , SAMI would help to compile a list of good candidates for subsequent detailed studies.",
"However, dedicated follow-up deep spectroscopic observations (IFU or long-slit) are required in order to obtain high signal-to-noise ratios and also the appropriate spectral resolution in order for the spectral decomposition techniques to be successful."
],
[
"Acknowledgments",
"I.K.",
"thanks Lodovico Coccato and Sergey Khoperskov for useful discussions.",
"The work was supported by the Russian Science Foundation project 14-22-00041 “VOLGA – A View On the Life of GAlaxies”.",
"The final interpretation of the data and paper writing was performed during visits to Chamonix workshop and The Research Institute in Astrophysics and Planetology in Toulouse which were supported by MD-7355.2015.2, RFBR 15-32-21062, and RFBR-CNRS 15-52-15050 grants.",
"The project used computational resources funded by the M.V.",
"Lomonosov Moscow State University Program of Development.",
"The Russian 6-m telescope is exploited under the financial support by the Russian Federation Ministry of Education and Science (agreement No14.619.21.0004, project ID RFMEFI61914X0004).",
"This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration, and of the Lyon Extragalactic Database (LEDA).",
"In this study, we used the SDSS DR12 data.",
"Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England.",
"The SDSS Web site is http://www.sdss.org/."
]
] | 1606.04862 |
[
[
"Simulating disk galaxies and interactions in Milgromian dynamics"
],
[
"Abstract Since its publication 1983, Milgromian dynamics (aka MOND) has been very successful in modeling the gravitational potential of galaxies from baryonic matter alone.",
"However, the dynamical modeling has long been an unsolved issue.",
"In particular, the setup of a stable galaxy for Milgromian N-body calculations has been a major challenge.",
"Here, we show a way to set up disc galaxies in MOND for calculations in the PHANTOM OF RAMSES (PoR) code by L\\\"ughausen (2015) and Teyssier (2002).",
"The method is done by solving the QUMOND Poisson equations based on a baryonic and a phantom dark matter component.",
"The resulting galaxy models are stable after a brief settling period for a large mass and size range.",
"Simulations of single galaxies as well as colliding galaxies are shown."
],
[
"INTRODUCTION",
"Since its invention by Milgrom (1983) modified Newtonian dynamics (MOND) has been successfully implemented in numerical codes for simulations of galaxies and galaxy systems (see Famaey & McGaugh 2012 for a review).",
"The most recent implementation of MOND is the PHANTOM OF RAMSES (PoR) code by Lüghausen et al.",
"(2015), based on RAMSES by Teyssier (2002).",
"In this contribution the setup of stable disk galaxies is described and applied to the simulation of interacting galaxies.",
"For the first time, the rotation curve of a galaxy in MOND is calculated."
],
[
"NUMERICAL METHODS",
"The setup of an N-body system in MOND is not as straight-forward as in Newtonian dynamics since the Milgromian gravitation depends on the Newtonian acceleration and thus two iterating Poisson equations need to be solved.",
"The first step is the setup of the desired density profile, $\\rho _\\text{b}(\\mathbf {x})$ .",
"In this contribution exponential disk profiles with a finite truncation radius are used.",
"Next the Newtonian Poisson equation is solved to obtain the Newtonian accelerations, ${\\mathbf {{\\bf \\nabla }}} \\phi $ .",
"Using the quasi-linear formulation of MOND (QUMOND, Milgrom 2010), in which the Milgromian modification term of gravity is represented by a “phantom dark matter” density distribution, the combined Newtonian and Milgromian Poisson equation can be written as $\\nabla ^2 \\Phi (\\mathbf {x})= 4 \\pi G \\rho _\\text{b}(\\mathbf {x}) +\\nabla \\cdot \\left[ \\nu \\left(|{\\mathbf {{\\bf \\nabla }}} \\phi |/a_0\\right) {\\mathbf {{\\bf \\nabla }}} \\phi (\\mathbf {x}) \\right]$ or $\\nabla ^2 \\Phi (\\mathbf {x})= 4 \\pi G \\left( \\rho _\\text{b}(x) + \\rho _\\text{ph}(x) \\right) \\,.$ Here, $ \\rho _\\text{b}(x)$ is the density distribution of the baryonic (real) matter, and $ \\rho _\\text{ph}(x)$ describes the distribution of the phantom dark matter.",
"By solving the QUMOND Poisson equation the accelerations and thus the circular velocities of particles in a stable disk galaxy can be obtained.",
"The setup is done via the MKGALAXY script by McMillan & Dehnen (2007), modified by Lüghausen in 2015 for MOND.",
"Besides an installation of PoR the script requires the NEMO stellar dynamics library and the PNBODY Python library for N-body calculations.",
"Gas initial conditions require to patch the PoR code by editing the CONDINIT subroutine accordingly.",
"This has been done by I. Thies in 2015 based on the MERGER patch by D. Chapon in 2010 which is readily available in the RAMSES installation.",
"The new patch is currently available on request from the author."
],
[
"Stable isolated galaxies",
"A disk galaxy with a stellar mass of $80\\cdot 10^9\\,\\text{M}_\\odot $ and gas mass of $10\\cdot 10^9\\,\\text{M}_\\odot $ is set with an exponential radial profile and a scale radius of 2 kpc.",
"There is no initial bulge.",
"In the initial configuration as well as after 5 Gyr the rotation curves are calculated as the circular velocities from the radial accelerations of the particles.",
"In agreement with observations the rotation curves are relatively flat in the outer parts, as can be seen in Fig.",
"REF ."
],
[
"Interacting galaxies",
"A pair of galaxies, each with the parameters described above, has been set for a grazing collision.",
"As can be seen in Fig.",
"REF , the interaction causes prominent tidal arms immediately after collision.",
"The galaxy cores then orbit each other several times before eventually merging after about 5.5 Gyr.",
"After about 5 Gyr satellite galaxies are visible as gaseous clumps orbiting in the plane of the encounter orbit.",
"Figure: Snapshots of the gas component of two interacting galaxies.",
"Notethat it takes about 5500 Myr, measured from the time of collision at timestamp500 Myr, for the galaxies to merge.",
"This corresponds to six perigalactic passages.Note the three substantial tidal dwarf galaxies in the lower central panel(5200 Myr)In Fig.",
"REF the stellar component of the interacting galaxies is plotted.",
"More precisely, the stars which formed since the begin of the simulation are shown in order to emphasise the regions of star formation.",
"As in the gas plot at timestamp 5200 Myr a few satellite galaxies are visible around the almost merged original disk galaxies.",
"It has to be noted that due to the resolution limits of both the gas mesh and the star-representing particles only the most massive satellites can form and remain stable.",
"In higher resolution simulations more satellites of lower masses are expected to form.",
"Figure: Snapshot of the stellar component of two interacting galaxiesat timestamp 5200 Myr.",
"The satellites correspond to the gas clumps in the lower middlepanel in Fig.",
"."
],
[
"Summary and future perspectives",
"In this contribution the method of setting up stable disk galaxies in Milgromian dynamics (MOND) has been introduced.",
"For the first time, the rotation curve of such a galaxy model, which qualitativels matches observed rotation curves of existing disk galaxies, has been shown.",
"Furthermore, it has been demonstrated that interacting galaxies tend to merge relatively late (here after six perigalactic passages within about 6 Gyr) in contradiction to the $\\Lambda $ CDM model which predicts quick mergers due to the dynamical friction of the dark matter halos (Privon, Barnes et al.",
"2013).",
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"The predictions of these models may then be tested with observations."
],
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]
] | 1606.04942 |
[
[
"Hysteresis of nanocylinders with Dzyaloshinskii-Moriya interaction"
],
[
"Abstract The potential for application of magnetic skyrmions in high density storage devices provides a strong drive to investigate and exploit their stability and manipulability.",
"Through a three-dimensional micromagnetic hysteresis study, we investigate the question of existence of skyrmions in cylindrical nanostructures of variable thickness.",
"We quantify the applied field and thickness dependence of skyrmion states, and show that these states can be accessed through relevant practical hysteresis loop measurement protocols.",
"As skyrmionic states have yet to be observed experimentally in confined helimagnetic geometries, our work opens prospects for developing viable hysteresis process-based methodologies to access and observe skyrmionic states."
],
[
"Hysteresis of nanocylinders with Dzyaloshinskii-Moriya interaction Rebecca Carey Faculty of Engineering and the Environment, University of Southampton, Southampton, SO17 1BJ, United Kingdom Marijan Beg Faculty of Engineering and the Environment, University of Southampton, Southampton, SO17 1BJ, United Kingdom Maximilian Albert Faculty of Engineering and the Environment, University of Southampton, Southampton, SO17 1BJ, United Kingdom Marc-Antonio Bisotti Faculty of Engineering and the Environment, University of Southampton, Southampton, SO17 1BJ, United Kingdom David Cortés-Ortuño Faculty of Engineering and the Environment, University of Southampton, Southampton, SO17 1BJ, United Kingdom Mark Vousden Faculty of Engineering and the Environment, University of Southampton, Southampton, SO17 1BJ, United Kingdom Weiwei Wang Department of Physics, Ningbo University, Ningbo 315211, China Ondrej Hovorka Faculty of Engineering and the Environment, University of Southampton, Southampton, SO17 1BJ, United Kingdom Hans Fangohr h.fangohr@soton.ac.uk Faculty of Engineering and the Environment, University of Southampton, Southampton, SO17 1BJ, United Kingdom The potential for application of magnetic skyrmions in high density storage devices provides a strong drive to investigate and exploit their stability and manipulability.",
"Through a three-dimensional micromagnetic hysteresis study, we investigate the question of existence of skyrmions in cylindrical nanostructures of variable thickness.",
"We quantify the applied field and thickness dependence of skyrmion states, and show that these states can be accessed through relevant practical hysteresis loop measurement protocols.",
"As skyrmionic states have yet to be observed experimentally in confined helimagnetic geometries, our work opens prospects for developing viable hysteresis process-based methodologies to access and observe skyrmionic states.",
"There is a continuous demand for developing magnetic data storage devices with higher recording density and improved reliability and robustness.",
"Recent research demonstrates that magnetic skyrmions show great potential to meet such demands,[1], [2], [3] which drives vigorous research activity to understand the fundamental aspects of their emergence, stability, and manipulability.",
"Magnetic skyrmions are topologically stable quasi-particles, which have been found to exist[4] with diameters as small as $1\\,$ nm.",
"They arise in magnetic systems that lack inversion symmetry in the crystal lattice, which gives rise to the chiral Dzyaloshinskii-Moriya interaction (DMI),[5], [6] such as in bulk helimagnetic materials with a non-centrosymmetric crystal lattice,[5], [6] or at the interface between two dissimilar materials.",
"[7], [8] Along with skyrmions, DMI may give rise to different types of magnetic spin configurations, including helical and conical structures, all of which have been observed through theory,[9], [10], [11], [12] simulation,[1], [13] and experiment.",
"[14], [15], [16], [17], [4] Sustaining stable skyrmion states in continuous magnetic films requires the application of a significant external field,[15], [16] which has been seen as a disadvantage for developing applications in information storage.",
"As shown recently, however, zero-field isolated skyrmions can be sustained in magnetic nanostructures with confined geometries,[13], [18] and created through several standard techniques including spin-polarized current injection.",
"[19], [20] The research so far has focused predominantly on two dimensional confined nanostructures with negligible thickness $-$ the influence of which is not yet understood.",
"Indeed, the recent evidence suggests that modulations of magnetization through the thickness of a nanostructure is an important factor determining the stability of skyrmions.",
"[21], [13], [22] To explore such thickness dependent magnetization modulations in confined geometries, in this paper we study the hysteresis behavior of nanocylinder structures through full three dimensional micromagnetic simulations.",
"The external field is varied starting from a well defined saturated state to produce a hysteresis loop, and the magnetization patterns recorded along the loop are analysed and classified into accessible states.",
"Such hysteresis loop measurements are a standard laboratory magnetometry protocol, which highlights the practical aspect of our work.",
"We study hysteresis behavior of nanocylinders of a non-centrosymmetric lattice material FeGe.",
"Such systems of 150 nm diameter and 10 nm thickness were demonstrated earlier to sustain stable quasi-uniform states, isolated skyrmions and target states,[13] and here we investigate how the diversity and structure of these states evolves with the increased sample thickness.",
"We demonstrate the role of the magnetostatic (demagnetizing) field, which is often neglected in simulations, in determining the overall hysteresis behavior of nanocylinders and show that it aids the skyrmion stability and extends the applied field and thickness range supporting their existence.",
"Figure: Hysteresis loops and magnetization states of FeGe nanocylinders for different thickness.",
"(i) Hysteresis curves for nanocylinders with thicknesses t=20t=20, 35 and 5555\\,nm.",
"The external magnetic field of strength HH was applied in the direction perpendicular to the cylinder base and swept between ±4M s \\pm 4 M_{\\mathrm {s}}.",
"The different highlighted regions (a)-(h) indicate the range of field values for which different states occurred along the increasing hysteresis loop branch.",
"These states are marked by letters (a)-(h) in (ii), which shows 3D plots of the thickness-averaged zz component of the magnetization, 〈m z 〉\\langle m_z\\rangle , of typical representative states (a)-(h) and the underlying magnetization profile m z (x)m_z(x) along the dotted line highlighted across each cylinder.",
"The states (a) and (g) are categorized as incomplete skyrmions, (c) and (f) are isolated skyrmions with cores up and down, (e) is a target state and states (b), (d) and (h), the only states not to exhibit rotational symmetry, are defined as transition states.",
"(iii) Demonstrating how skyrmion and incomplete skyrmion profiles differ at different external field strengths.",
"The figure shows an incomplete skyrmion state (a) at H=-3M s H=-3M_{\\mathrm {s}}, -1M s -1M_{\\mathrm {s}}, 0M s 0M_{\\mathrm {s}} and a skyrmion state (f) at H=0.4M s H=0.4M_{\\mathrm {s}}, 1M s 1M_{\\mathrm {s}} in a t=20t=20\\,nm sample.",
"A key feature is decreased tilting at the edges in stronger fields.",
"(iv) Simulated geometry.The 3D micromagnetic simulations were performed using a finite element approach outlined in Nmag [23] and extended to include the DMI term.",
"The simulations integrated the Landau-Lifshitz-Gilbert (LLG) equation using the effective field framework consistent with the following micromagnetic energy: $W[\\mathbf {m}] = \\int _V [\\omega _{\\mathrm {ex}}(\\mathbf {m}) + \\omega _{\\mathrm {dmi}}(\\mathbf {m}) + \\omega _{\\mathrm {d}}(\\mathbf {m}) + \\omega _{\\mathrm {z}}(\\mathbf {m})] \\mathrm {d}V$ where $\\mathbf {m} = \\mathbf {M} / M_{\\mathrm {s}}$ is the normalised magnetization vector, with $M_{\\mathrm {s}} = |\\mathbf {M}|$ being the saturation magnetization.",
"The micromagnetic energy density includes the symmetric exchange $\\omega _{\\mathrm {ex}} = A\\left[(\\nabla m_x)^2 + (\\nabla m_y)^2 + (\\nabla m_z)^2 \\right]$ and DMI energy density $\\omega _{\\mathrm {dmi}} = D \\mathbf {m} \\cdot (\\nabla \\times \\mathbf {m})$ , with $A$ and $D$ being the exchange stiffness and DMI strength, respectively; magnetostatic energy density $\\omega _{\\mathrm {d}}$ giving rise to demagnetizing field; and the Zeeman term $\\omega _{\\mathrm {z}} = -\\mu _0M_{\\mathrm {s}}\\mathbf {H} \\cdot \\mathbf {m}$ , with $\\mathbf {H}$ denoting the external field vector.",
"We considered FeGe nanocylinders with a diameter of $150\\,$ nm and a variable thickness $t$ (Fig.",
"REF (iv)).",
"The finite element mesh discretization was set to $3\\,$ nm and guaranteed to be smaller than any of the relevant micromagnetic length scales.",
"The micromagnetic material parameters of FeGe[13] used were $M_{\\mathrm {s}}=384 \\, \\mathrm {kAm^{-1}}$ , $A=8.78 \\, \\mathrm {pJm^{-1}}$ and $D=1.58 \\, \\mathrm {mJm^{-2}}$ .",
"The thickness of the nanocylinders was varied between 10-$80\\,$ nm in $5\\,$ nm increments and a hysteresis loop was computed for each thickness.",
"The system was first initialised by equilibrating the magnetic state in a negative saturating external field oriented perpendicular to the cylinder base ($z$ -axis).",
"The hysteresis behavior was then simulated with a consistent procedure by starting from a well defined negative saturation state, increasing the external field $H$ in fine steps $\\Delta H$ , equilibrating the system at every field step, and using the previous magnetization state as an initialization in the subsequent field step.",
"In all simulations, the $H$ was applied perpendicular to the cylinder base and swept between the saturating values $H = \\pm 4M_\\mathrm {s}$ .",
"We used $\\Delta H=0.02M_{\\mathrm {s}}$ , after confirming that this step size was sufficient to guarantee reproducible simulations which were not affected by its further reduction.",
"The examples of hysteresis loops for thicknesses $t = 20$ , 35 and $55\\,$ nm are shown in Fig.",
"REF (i) as plots of the spatially averaged $z$ -component of magnetization, $\\left< m_z \\right>$ , for different external fields $H$ .",
"The highlighted regions (a)-(h) indicate the external field intervals where distinct classes of magnetization patterns were observed during the hysteresis process.",
"Thus overall we found 8 different magnetization configurations (a)-(h), examples of which are shown in Fig.",
"REF (ii) (a)-(h) and (iii).",
"Below, we classify these configurations into the following four categories.",
"1.",
"Isolated skyrmions (Fig.",
"REF (ii) (c), (f)).",
"Isolated skyrmions are defined as axially symmetric states that contain one full spin rotation along the diameter in the sample, i.e.",
"the modulations of the magnetization along the diameter are seen to rotate by at least $2\\pi $ .",
"Figs.",
"REF (ii) (c) and (f) show isolated skyrmion states with core up and down, respectively, as typically observed in our simulations along a hysteresis loop in the low and high field regions (c) and (f) (Fig.",
"REF (i)).",
"Fig.",
"REF (iii) shows the associated magnetization vs. radial position profiles of the isolated skyrmions (f) with core down, at fields $H=0.4M_{\\mathrm {s}}$ and $1M_{\\mathrm {s}}$ for a cylindrical sample of thickness $t = 20\\,$ nm.",
"As can be seen, in contrast to theoretical predictions made for skyrmions in infinite systems, the finite size calculations display additional magnetization tilting at the edge, which results from specific boundary conditions consistent with the boundary DMI interaction in isolated geometries.",
"[18] The extent of this tilting depends on the strength of the external field along the hysteresis loop.",
"2.",
"Incomplete skyrmions (Fig.",
"REF (ii) (a), (g)).",
"Incomplete skyrmions is a terminology used to refer collectively to axially symmetric states that do not sustain full spin rotation along the diameter.",
"These states have also been termed as quasi-ferromagnetic or edged vortex states.",
"[13] Fig.",
"REF (iii) demonstrates how the magnetisation profiles of these incomplete skyrmions (at fields $H=-3M_{\\mathrm {s}}$ , $-1M_{\\mathrm {s}}$ , $0M_{\\mathrm {s}}$ ) differ from isolated skyrmions (at fields $H=0.4M_{\\mathrm {s}}$ , $1M_{\\mathrm {s}}$ ).",
"The tilting at the edges of the sample in incomplete skyrmions, prevalent even at strong external fields, is again due to the DMI.",
"It is strongly dependent on the external field strength and penetrates into the sample as the field strength decreases.",
"[18], [24] 3.",
"Target states (Fig.",
"REF (ii) (e)).",
"Target states are also radially symmetric states and in contrast to isolated skyrmions contain two or more full spin rotations along the diameter, depending on the diameter of the sample and the external field.",
"[25] In our examples in Fig.",
"REF (i) the target states are not seen for samples with low thickness $t = 20$ and $35\\,$ nm, but are observed at thickness $t = 55\\,$ nm.",
"As can be seen in Fig.",
"REF (ii) (e), the target state associated with this thickness sustains two full spin rotations.",
"As will be discussed below, target states appear in our samples when the thickness $t \\ge 45\\,$ nm.",
"4.",
"Transition states (Fig.",
"REF (ii) (b), (d), (h)).",
"We refer to transition states as the states that do not exhibit rotational symmetry.",
"These include the states which are evident precursors to the isolated skyrmion and target states emerging during the hysteresis loop process (Figs.",
"REF (ii) (b), (d)), as well as the states resembling the regular `striped' helical state (Fig.",
"REF (ii) (h)).",
"The latter of these, the helical-like state, does not exhibit the one-dimensional helical structure with uniform propagation direction of regular helices found in infinite geometries and instead, due to the boundary conditions of the confined geometry, partially follows the curvature of the cylinder.",
"[18] Figure: Magnetization state `phase diagrams' showing the type of states sustained in the nanocylinders of different thicknesses tt and at different external field strengths HH along the increasing branch of hysteresis loop between the saturating fields H=±4M s H=\\pm 4M_{\\mathrm {s}}.",
"(i) full magnetostatics calculations, (ii) no magnetostatics, i.e.",
"without any demagnetizing field effects.Having defined a terminology for all observed states, we now systematically investigate their occurrence at different sample thicknesses $t$ and external field strengths $H$ , in consistency with the major hysteresis loop process run from negative to positive saturation.",
"Such a $t-H$ `state phase' plot is shown in Fig.",
"REF (i) and demonstrates that the magnetization states from the four categories introduced above form well-defined regions.",
"In the thin geometry range for $t<20\\,$ nm, the simulated samples support the radially symmetric down and up incomplete skyrmions (a, g) and the isolated skyrmion with core down (f), which are separated by well defined threshold field values.",
"The incomplete skyrmion down state (a) remains into positive fields before switching to the isolated skyrmion (f).",
"The thickness range $20\\,$ nm$\\,<t<40\\,$ nm behaves similarly, with the addition of a small field interval with transition states (h) separating the incomplete and isolated skyrmion state phases at smaller and larger fields, respectively.",
"The behavior becomes enriched at the threshold thickness $t = 45\\,$ nm, which now allows a sustained isolated skyrmion with core up (c), in addition to the isolated skyrmion with the core down preferred at higher fields, and also the target state (e).",
"The isolated skyrmion with core up (c) is stabilised at negative fields and is seen to exist into the positive field region, and this trend seems to be preserved for higher sample thicknesses.",
"Similarly, the emergence of the target state (e) seems to be preserved when the sample thickness increases.",
"Thus in cylindrical structures with thickness $t > 45\\,$ nm the `polarity' of the isolated skyrmion can be switched solely via the hysteresis process.",
"Furthermore, given that the isolated skyrmion and target states are yet to be observed experimentally, our calculations demonstrate that well-controlled hysteresis loop measurements combined with appropriate high resolution magnetic domain imaging techniques might provide a way for the observation of them.",
"Fig.",
"REF (ii) shows an identical $t-H$ state phase plot to that in Fig.",
"REF (i) but with the magnetostatic field neglected, i.e.",
"any demagnetizing field effects not present.",
"Overall, the differences between the state phase plots (i) and (ii) in Fig.",
"REF indicate that the demagnetizing field plays a subtle but important role in governing the behavior along a hysteresis loop.",
"In particular, the transition states are also seen in the low thickness range $t\\le 20\\,$ nm, indicating they become suppressed under the action of demagnetizing fields; the region of fields around $H=0$ supporting the isolated skyrmions with core up (c) becomes narrower; and the ranges of fields supporting the isolated skyrmion with core down (f) and target states (e) are also reduced.",
"From this we deduce that once in a specific state, the effects of a demagnetizing field act as an augmenting stabilising factor extending the state phase regions associated with the different states.",
"The enriched behavior at thickness $t = 45\\,$ nm remains present in both Fig.",
"REF (i) and (ii) and so does not seem to be driven magnetostatically and the resulting tendencies towards the flux closure formation.",
"We note, that the apparent discontinuity in the behavior at thickness around $70\\,$ nm is due to the similarity of the states, which could not be resolved using our image detection approach based on quantifying the symmetry of magnetization patterns.",
"Figure: Demonstrating the distorted geometries used to determine the robustness of the results with respect to irregularities which may arise during fabrication processes, here shown with state (f).",
"The distortion methods were introduced through (i) surface roughness and (ii) tapering the cylinder to form a truncated cone.Additional hysteresis simulations were performed for samples with thicknesses of 40, 45 and $70\\,$ nm, corresponding to the critical regions of the phase diagram, to determine the robustness of the results with respect to irregularities which may arise during fabrication processes.",
"Specifically, we (1) introduced surface roughness, demonstrated in Fig.",
"REF (i), by generating randomly distributed surface regions with maximum waviness depth equal to 1% of the original cylinder diameter, and (2) tapered the cylinder to form a truncated cone geometry (Fig.",
"REF (ii)) with a maximum base-to-top radius difference of 7%.",
"The tapering is typical of the etching fabrication process.",
"We found that the impact of these geometry distortions, in comparison the previous phase plots corresponding to the perfect cylindrical geometry, was a shift to the threshold field values by a maximum of $\\pm 0.06M_{\\mathrm {s}}$ , which can be considered negligible to the external field scales of the hysteresis loop.",
"In conclusion, we have investigated the magnetization behavior in non-centrosymmetric material nanocylinders through micromagnetic simulations of hysteresis loop processes.",
"For the specific choice of 150 nm diameter nanocylinders, the isolated skyrmions and target states emerge in the specific well-defined field intervals along the hysteresis loop if the sample thickness is greater than 45 nm.",
"As these states have yet to be observed experimentally our calculations demonstrate that well-controlled hysteresis loop measurements achievable through standard laboratory magnetometry protocols, and combined with appropriate high resolution magnetic domain imaging techniques, might provide a practical way for their observation.",
"The $t-H$ state phase diagram shown in Fig.",
"REF (i) gives a guide for targeting the individual states.",
"We also demonstrate the subtle effects of the demagnetizing magnetostatic fields in nanocylindrical samples, which are important for the stabilisation of the isolated skyrmion and target states.",
"The raw data for the figures in this paper, as well Jupyter notebooks [26] which reproduce the figures are available online for this paper.",
"[27] We acknowledge financial support from EPSRC's DTC grant EP/G03690X/1."
]
] | 1606.05181 |
[
[
"Guarded Cubical Type Theory: Path Equality for Guarded Recursion"
],
[
"Abstract This paper improves the treatment of equality in guarded dependent type theory (GDTT), by combining it with cubical type theory (CTT).",
"GDTT is an extensional type theory with guarded recursive types, which are useful for building models of program logics, and for programming and reasoning with coinductive types.",
"We wish to implement GDTT with decidable type-checking, while still supporting non-trivial equality proofs that reason about the extensions of guarded recursive constructions.",
"CTT is a variation of Martin-L\\\"of type theory in which the identity type is replaced by abstract paths between terms.",
"CTT provides a computational interpretation of functional extensionality, is conjectured to have decidable type checking, and has an implemented type-checker.",
"Our new type theory, called guarded cubical type theory, provides a computational interpretation of extensionality for guarded recursive types.",
"This further expands the foundations of CTT as a basis for formalisation in mathematics and computer science.",
"We present examples to demonstrate the expressivity of our type theory, all of which have been checked using a prototype type-checker implementation, and present semantics in a presheaf category."
],
[
"Introduction",
"Guarded recursion is a technique for defining and reasoning about infinite objects.",
"Its applications include the definition of productive operations on data structures more commonly defined via coinduction, such as streams, and the construction of models of program logics for modern programming languages with features such as higher-order store and concurrency [6].",
"This is done via the type-former $$ , called `later', which distinguishes data which is available immediately from data only available after some computation, such as the unfolding of a fixed-point.",
"For example, guarded recursive streams are defined by the equation ${A} \\;=\\; A \\times {{A}}$ rather than the more standard ${A} = A \\times {A}$ , to specify that the head is available now but the tail only later.",
"The type for fixed-point combinators is then $(A\\rightarrow A)\\rightarrow A$ , rather than the logically inconsistent $(A\\rightarrow A)\\rightarrow A$ , disallowing unproductive definitions such as taking the fixed-point of the identity function.",
"Guarded recursive types were developed in a simply-typed setting by Clouston et al.",
"[9], following earlier work [21], [3], [1], alongside a logic for reasoning about such programs.",
"For large examples such as models of program logics, we would like to be able to formalise such reasoning.",
"A major approach to formalisation is via dependent types, used for example in the proof assistants Coq [18] and Agda [22].",
"Bizjak et al.",
"[8], following earlier work [5], [20], introduced guarded dependent type theory ($\\mathsf {GDTT}$ ), integrating the $$ type-former into a dependently typed calculus, and supporting the definition of guarded recursive types as fixed-points of functions on universes, and guarded recursive operations on these types.",
"We wish to formalise non-trivial theorems about equality between guarded recursive constructions, but such arguments often cannot be accommodated within intensional Martin-Löf type theory.",
"For example, we may need to be able to reason about the extensions of streams in order to prove the equality of different stream functions.",
"Hence $\\mathsf {GDTT}$ includes an equality reflection rule, which is well known to make type checking undecidable.",
"This problem is close to well-known problems with functional extensionality [13], and indeed this analogy can be developed.",
"Just as functional extensionality involves mapping terms of type $(x:A)\\rightarrow \\operatorname{\\mathsf {Id}}B\\, (fx)\\, (gx)$ to proofs of $\\operatorname{\\mathsf {Id}}\\,(A\\rightarrow B)\\, f\\, g$ , extensionality for guarded recursion requires an extensionality principle for later types, namely the ability to map terms of type $\\operatorname{\\mathsf {Id}}A\\,t\\,u$ to proofs of $\\operatorname{\\mathsf {Id}}\\,(A)\\,({t})\\,({u})$ , where ${}$ is the constructor for $$ .",
"These types are isomorphic in the intended model, the presheaf category $\\widehat{\\omega }$ known as the topos of trees, and so in $\\mathsf {GDTT}$ their equality was asserted as an axiom.",
"But in a calculus without equality reflection we cannot merely assert such axioms without losing canonicity.",
"Cubical type theory ($\\mathsf {CTT}$ ) [10] is a new type theory with a computational interpretation of functional extensionality but without equality reflection, and hence is a candidate for extension with guarded recursion, so that we may formalise our arguments without incurring the disadvantages of fully extensional identity types.",
"$\\mathsf {CTT}$ was developed primarily to provide a computational interpretation of the univalence axiom of Homotopy Type Theory [26].",
"The most important novelty of $\\mathsf {CTT}$ is the replacement of inductively defined identity types by paths, which can be seen as maps from an abstract interval $I$ , and are introduced and eliminated much like functions.",
"$\\mathsf {CTT}$ can be extended with identity types which model all rules of standard Martin-Löf type theory [10], but these are equivalent to path types, and in our paper it suffices to work with path types only.",
"$\\mathsf {CTT}$ has sound denotational semantics in (fibrations in) cubical sets, a presheaf category that is used to model homotopy types.",
"Many basic syntactic properties of $\\mathsf {CTT}$ , such as the decidability of type checking, and canonicity for base types, are yet to be proved, but a type checker has been implementedhttps://github.com/mortberg/cubicaltt that confers some confidence in such properties.",
"In Sec.",
"of this paper we propose guarded cubical type theory ($\\mathsf {GCTT}$ ), a combination of the two type theorieswith the exception of the clock quantification of $\\mathsf {GDTT}$ , which we leave to future work.",
"which supports non-trivial proofs about guarded recursive types via path equality, while retaining the potential for good syntactic properties such as decidable type-checking and canonicity.",
"In particular, just as a term can be defined in $\\mathsf {CTT}$ to witness functional extensionality, a term can be defined in $\\mathsf {GCTT}$ to witness extensionality for later types.",
"Further, we use elements of the interval of $\\mathsf {CTT}$ to annotate fixed-points, and hence control their unfoldings.",
"This ensures that fixed-points are path equal, but not judgementally equal, to their unfoldings, and hence prevents infinite unfoldings, an obvious source of non-termination in any calculus with infinite constructions.",
"The resulting calculus is shown via examples to be useful for reasoning about guarded recursive operations; we also view it as potentially significant from the point of view of $\\mathsf {CTT}$ , extending its expressivity as a basis for formalisation.",
"In Sec.",
"we give sound semantics to this type theory via the presheaf category over the product of the categories used to define semantics for $\\mathsf {GDTT}$ and $\\mathsf {CTT}$ .",
"This requires considerable work to ensure that the constructions of the two type theories remain sound in the new category, particularly the glueing and universe of $\\mathsf {CTT}$ .",
"The key technical challenge is to ensure that the $$ type-former supports the compositions that all types must carry in the semantics of $\\mathsf {CTT}$ .",
"We have implemented a prototype type-checker for this extended type theoryhttp://github.com/hansbugge/cubicaltt/tree/gcubical, which provides confidence in the type theory's syntactic properties.",
"All examples in this paper, and many others, have been formalised in this type checker.",
"For reasons of space many details and proofs are omitted from this paper, but are included in a technical appendixhttp://cs.au.dk/~birke/papers/gdtt-cubical-technical-appendix.pdf."
],
[
"Guarded Cubical Type Theory",
"This section introduces guarded cubical type theory ($\\mathsf {GCTT}$ ), and presents examples of how it can be used to prove properties of guarded recursive constructions."
],
[
"Cubical Type Theory",
"We first give a brief overview of cubical type theoryhttp://www.cse.chalmers.se/~coquand/selfcontained.pdf is a self-contained presentation of $\\mathsf {CTT}$ .",
"($\\mathsf {CTT}$ ) [10].",
"We start with a standard dependent type theory with $\\Pi $ , $\\Sigma $ , natural numbers, and a Russell-style universe: $\\begin{array}{lcl@{\\hspace{85.3987pt}}l}\\Gamma , \\Delta & ::=& () ~ | ~ \\Gamma , x : A & \\text{Contexts} \\\\[1ex]t,u,A,B & ::=& x ~|~ \\lambda x : A .",
"t ~|~ t\\, u ~|~ (x : A) \\rightarrow B &\\text{$\\Pi $-types} \\\\& | & (t,u) ~|~ t.1 ~|~ t.2 ~|~ (x:A) \\times B &\\text{$\\Sigma $-types} \\\\& | & \\operatorname{\\mathsf {0}}~|~ \\operatorname{\\mathsf {s}}t ~|~ \\operatorname{\\mathsf {natrec}}t \\, u ~|~ \\operatorname{\\mathsf {N}}& \\text{Natural numbers} \\\\& | & \\operatorname{\\mathsf {U}}& \\text{Universe}\\end{array}$ We adhere to the usual conventions of considering terms and types up to $\\alpha $ -equality, and writing $A \\rightarrow B$ , respectively $A \\times B$ , for non-dependent $\\Pi $ and $\\Sigma $ -types.",
"We use the symbol `$=$ ' for judgemental equality.",
"The central novelty of $\\mathsf {CTT}$ is its treatment of equality.",
"Instead of the inductively defined identity types of intensional Martin-Löf type theory [17], $\\mathsf {CTT}$ has paths.",
"The paths between two terms $t,u$ of type $A$ form a sort of function space, intuitively that of continuous maps from some interval $I$ to $A$ , with endpoints $t$ and $u$ .",
"Rather than defining the interval $I$ concretely as the unit interval $[0,1] \\subseteq \\mathbb {R}$ , it is defined as the free De Morgan algebra on a discrete infinite set of names $\\lbrace i, j, k, \\dots \\rbrace $ .",
"A De Morgan algebra is a bounded distributive lattice with an involution $1-\\cdot $ satisfying the De Morgan laws $1-(i\\wedge j) &= (1-i) \\vee (1-j), & 1-(i\\vee j) &= (1-i) \\wedge (1-j).$ The interval $[0,1] \\subseteq \\mathbb {R}$ , with $\\operatorname{\\mathsf {min}}$ , $\\operatorname{\\mathsf {max}}$ and $1-\\cdot $ , is an example of a De Morgan algebra.",
"The syntax for elements of $I$ is: $r, s ~ ::=~ 0 ~|~ 1 ~|~ i ~|~ 1-r ~|~ r \\wedge s ~|~ r \\vee s.$ 0 and 1 represent the endpoints of the interval.",
"We extend the definition of contexts to allow introduction of a new name: $\\Gamma , \\Delta ~::=~ \\cdots ~|~ \\Gamma , i:I.$ The judgement $\\Gamma \\vdash r : I$ means that $r$ draws its names from $\\Gamma $ .",
"Despite this notation, $I$ is not a first-class type.",
"Path types and their elements are defined by the rules in Fig.",
"REF .",
"Path abstraction, $\\mathop {\\langle i \\rangle } t$ , and path application, $t \\, r$ , are analogous to $\\lambda $ -abstraction and function application, and support the familiar $\\beta $ -equality $(\\mathop {\\langle i \\rangle } t)\\, r = t[r/i]$ and $\\eta $ -equality $\\mathop {\\langle i \\rangle } t\\, i = t$ .",
"There are two additional judgemental equalities for paths, regarding their endpoints: given $p : \\operatorname{\\mathsf {Path}}A ~ t ~ u$ we have $p\\, 0 = t$ and $p\\, 1 = u$ .",
"Figure: Typing rules for path types.Paths provide a notion of identity which is more extensional than that of intensional Martin-Löf identity types, as exemplified by the proof term for functional extensionality: $\\operatorname{\\mathsf {funext}} \\, f\\, g \\triangleq \\lambda p .",
"\\mathop {\\langle i \\rangle } \\lambda x .\\, p \\, x \\, i~ : ~\\left((x : A) \\rightarrow \\operatorname{\\mathsf {Path}}B ~ (f\\, x) ~ (g\\, x)\\right) \\rightarrow \\operatorname{\\mathsf {Path}}~ (A\\rightarrow B) ~ f ~ g.$ The rules above suffice to ensure that path equality is reflexive, symmetric, and a congruence, but we also need it to be transitive and, where the underlying type is the universe, to support a notion of transport.",
"This is done via (Kan) composition operations.",
"To define these we need the face lattice, $\\mathbb {F}$ , defined as the free distributive lattice on the symbols $(i=0)$ and $(i=1)$ for all names $i$ , quotiented by the relation $(i=0) \\wedge (i=1) = 0_{\\mathbb {F}}$ .",
"The syntax for elements of $\\mathbb {F}$ is: $\\varphi ,\\psi ~::=~0_{\\mathbb {F}}~|~ 1_{\\mathbb {F}}~|~ (i=0) ~|~ (i=1) ~|~ \\varphi \\wedge \\psi ~|~ \\varphi \\vee \\psi .$ As with the interval, $\\mathbb {F}$ is not a first-class type, but the judgement $\\Gamma \\vdash \\varphi : \\mathbb {F}$ asserts that $\\varphi $ draws its names from $\\Gamma $ .",
"We also have the judgement ${\\Gamma }{\\varphi }{\\psi }[\\mathbb {F}]$ which asserts the equality of $\\varphi $ and $\\psi $ in the face lattice.",
"Contexts can be restricted by elements of $\\mathbb {F}$ : $\\Gamma , \\Delta ~::=~ \\cdots ~|~ \\Gamma ,\\varphi .$ Such a restriction affects equality judgements so that, for example, ${\\Gamma ,\\varphi }{\\psi _1}{\\psi _2}[\\mathbb {F}]$ is equivalent to ${\\Gamma }{\\varphi \\wedge \\psi _1}{\\varphi \\wedge \\psi _2}[\\mathbb {F}]$ We write $\\Gamma \\vdash t : A[\\varphi \\mapsto u]$ as an abbreviation for the two judgements $\\Gamma \\vdash t : A$ and ${\\Gamma ,\\varphi }{t}{u}[A]$ , noting the restriction with $\\varphi $ in the equality judgement.",
"Now the composition operator is defined by the typing and equality rule $\\Gamma \\vdash \\varphi : \\mathbb {F}$ $\\Gamma , i:I \\vdash A$ $\\Gamma , \\varphi , i : I \\vdash u : A$ $\\Gamma \\vdash a_0 : A[0/i][\\varphi \\mapsto u[0/i]]$ $\\Gamma \\vdash \\operatorname{\\mathsf {comp}}^i\\,A~[\\varphi \\mapsto u]~a_0 : A[1/i][\\varphi \\mapsto u[1/i]]$ .",
"A simple use of composition is to implement the transport operation for $\\operatorname{\\mathsf {Path}}$ types $\\operatorname{\\mathsf {transp}}^i \\, A ~ a~\\triangleq ~\\operatorname{\\mathsf {comp}}^i\\,A~[0_{\\mathbb {F}}\\mapsto []]~a~:~A[1/i],$ where $a$ has type $A[0/i]$ .",
"The notation $[]$ stands for an empty system.",
"In general a system is a list of pairs of faces and terms, and it defines an element of a type by giving the individual components at each face.",
"We extend the syntax as follows: $t,u,A,B ~::=~ \\cdots ~|~ [ \\varphi _1~t_1,\\ldots ,\\varphi _n~t_n ].$ Below we see two of the rules for systems; they ensure that the components of a system agree where the faces overlap, and that all the cases possible in the current context are covered: $\\Gamma \\vdash A$ 1...n$1_{\\mathbb {F}}$ [$\\mathbb {F}$ ] $\\Gamma , \\varphi _i \\vdash t_i : A$ ,ijtitj[A] i,j=1 ...n $\\Gamma \\vdash [ \\varphi _1~t_1,\\ldots ,\\varphi _n~t_n ] : A$ $\\Gamma \\vdash [ \\varphi _1~t_1,\\ldots ,\\varphi _n~t_n ] : A$ i$1_{\\mathbb {F}}$ [$\\mathbb {F}$ ] [ 1 t1,...,n tn ]ti[A] We will shorten $[\\varphi _1\\vee \\ldots \\vee \\varphi _n \\mapsto [ \\varphi _1~t_1,\\ldots ,\\varphi _n~t_n ]]$ to $[ \\varphi _1\\mapsto t_1,\\ldots ,\\varphi _n\\mapsto t_n ]$ .",
"A non-trivial example of the use of systems is the proof that $\\operatorname{\\mathsf {Path}}$ is transitive; given $p\\,:\\,\\operatorname{\\mathsf {Path}}A~a~b$ and $q\\,:\\,\\operatorname{\\mathsf {Path}}A~b~c$ we can define $\\operatorname{\\mathsf {transitivity}}\\,p\\,q \\triangleq \\mathop {\\langle i \\rangle } \\operatorname{\\mathsf {comp}}^j\\,A~[(i=0) \\mapsto a, (i=1) \\mapsto q\\,j]~(p \\, i) \\, : \\, \\operatorname{\\mathsf {Path}}A~a~c.$ This builds a path between the appropriate endpoints because we have the equalities $\\operatorname{\\mathsf {comp}}^j\\,A~[1_{\\mathbb {F}}\\mapsto a]~(p \\, 0) = a$ and $\\operatorname{\\mathsf {comp}}^j\\,A~[1_{\\mathbb {F}}\\mapsto q\\,j]~(p \\, 1) = q\\,1 = c$ .",
"For reasons of space we have omitted the descriptions of some features of $\\mathsf {CTT}$ , such as glueing, and the further judgemental equalities for terms of the form $\\operatorname{\\mathsf {comp}}^i\\,A~[\\varphi \\mapsto u]~a_0$ that depend on the structure of $A$ ."
],
[
"Later Types",
"In Fig.",
"REF we present the `later' types of guarded dependent type theory ($\\mathsf {GDTT}$ ) [8], with judgemental equalities in Figs.",
"REF and REF .",
"Note that we do not add any new equation for the interaction of compositions with $$ ; such an equation would be necessary if we were to add the eliminator $\\operatorname{prev}$ for $$ , but this extension (which involves clock quantifiers) is left to further work.",
"We delay the presentation of the fixed-point operation until the next section.",
"The typing rules use the delayed substitutions of $\\mathsf {GDTT}$ , as defined in Fig.",
"REF .",
"Delayed substitutions resemble Haskell-style do-notation, or a delayed form of let-binding.",
"If we have a term $t:{A}$ , we cannot access its contents `now', but if we are defining a type or term that itself has some part that is available `later', then this part should be able to use the contents of $t$ .",
"Therefore delayed substitutions allow terms of type ${A}$ to be unwrapped by $$ and $$ .",
"As observed by Bizjak et al.",
"[8] these constructions generalise the applicative functor [19] structure of `later' types, by the definitions $\\operatorname{\\mathsf {pure}} \\, t \\triangleq {t}$ , and $f \\circledast t \\triangleq [\\left[ f^{\\prime } \\leftarrow f, t^{\\prime } \\leftarrow t \\right]]{f^{\\prime } \\, t^{\\prime }}$ , as well as a generalisation of the $\\circledast $ operation from simple functions to $\\Pi $ -types.",
"We here make the new observation that delayed substitutions can express the function $\\widehat{\\triangleright }:{\\operatorname{\\mathsf {U}}}\\rightarrow \\operatorname{\\mathsf {U}}$ , introduced by Birkedal and Møgelberg [4] to express guarded recursive types as fixed-points on universes, as $\\lambda u.",
"[[u^{\\prime }\\leftarrow u]]{u^{\\prime }}$ ; see for example the definition of streams in Sec.",
"REF .",
"Figure: Formation rules for delayed substitutions.Figure: Typing rules for later types.Figure: Type equality rules for later types (congruence and equivalence rules are omitted).Figure: Term equality rules for later types.",
"We omit congruence and equivalence rules, and the rules for terms of typeerror\\operatorname{\\mathsf {U}}, which reflect the type equality rules of Fig.",
".Example 1 In $\\mathsf {GDTT}$ it is essential that we can convert terms of type $[\\xi ]{\\operatorname{\\mathsf {Id}}_A \\, t~u}$ into terms of type $\\operatorname{\\mathsf {Id}}_{[\\xi ]{A}} \\, ([\\xi ]{t})~([\\xi ]{u})$ , as it is essential for Löb induction, the technique of proof by guarded recursion where we assume ${p}$ , deduce $p$ , and hence may conclude $p$ with no assumptions.",
"This is achieved in $\\mathsf {GDTT}$ by postulating as an axiom the following judgemental equality: $\\operatorname{\\mathsf {Id}}_{[\\xi ]{A}} \\, ([\\xi ]{t})~([\\xi ]{u})\\;=\\;[\\xi ]{\\operatorname{\\mathsf {Id}}_A \\, t~u}$ A term from left-to-right of (REF ) can be defined using the $\\operatorname{\\mathsf {J}}$ -eliminator for identity types, but the more useful direction is right-to-left, as proofs of equality by Löb induction involve assuming that we later have a path, then converting this into a path on later types.",
"In fact in $\\mathsf {GCTT}$ we can define a term with the desired type: $\\lambda p.\\langle i \\rangle [\\xi [p^{\\prime }\\leftarrow p]]{p^{\\prime }\\, i} \\;:\\; ([\\xi ]{\\operatorname{\\mathsf {Path}}A\\, t\\, u})\\rightarrow \\operatorname{\\mathsf {Path}}\\,([\\xi ]{A})\\,([\\xi ]{t})\\,([\\xi ]{u}).$ Note the similarity of this term and type with that of $\\operatorname{\\mathsf {funext}}$ , for functional extensionality, presented on page REF .",
"Indeed we claim that (REF ) provides a computational interpretation of extensionality for later types."
],
[
"Fixed Points",
"In this section we complete the presentation of $\\mathsf {GCTT}$ by addressing fixed points.",
"In $\\mathsf {GDTT}$ there are fixed-point constructions ${x}{t}$ with the judgemental equality ${x}{t} =t[{{x}{t}}/x]$ .",
"In $\\mathsf {GCTT}$ we want decidable type checking, including decidable judgemental equality, and so we cannot admit such an unrestricted unfolding rule.",
"Our solution it that fixed points should not be judgementally equal to their unfoldings, but merely path equal.",
"We achieve this by decorating the fixed-point combinator with an interval element which specifies the position on this path.",
"The 0-endpoint of the path is the stuck fixed-point term, while the 1-endpoint is the same term unfolded once.",
"However this threatens canonicity for base types: if we allow stuck fixed-points in our calculus, we could have stuck closed terms $[i]{x}{t}$ inhabiting $\\operatorname{\\mathsf {N}}$ .",
"To avoid this, we introduce the delayed fixed-point combinator $\\operatorname{\\mathsf {dfix}}$ , which produces a term `later' instead of a term `now'.",
"Its typing rule, and notion of equality, is given in Fig.",
"REF .",
"We will write $[r]{x}{t}$ for $t[[r]{x}{t}/x]$ , ${x}{t}$ for $[0]{x}{t}$ , and ${x}{t}$ for $[0]{x}{t}$ .",
"Figure: Typing and equality rules for the delayed fixed-pointLemma 2 (Canonical unfold lemma) For any term $\\Gamma , x : {A} \\vdash t : A$ there is a path between ${x}{t}$ and $t[{{x}{t}}/x]$ , given by the term $\\mathop {\\langle i \\rangle } [i]{x}{t}$ .",
"Transitivity of paths (via compositions) ensures that ${x}{t}$ is path equal to any number of fixed-point unfoldings of itself.",
"A term $a$ of type $A$ is said to be a guarded fixed point of a function $f:{A}\\rightarrow A$ if there is a path from $a$ to $f({a})$ .",
"Proposition 3 (Unique guarded fixed points) Any guarded fixed-point $a$ of a term $f : {A} \\rightarrow A$ is path equal to ${x}{f\\, x}$ .",
"Given $p : \\operatorname{\\mathsf {Path}}A ~ a ~ (f \\, ({a}))$ , we proceed by Löb induction, i.e., by assuming $\\operatorname{\\mathsf {ih}} : {(\\operatorname{\\mathsf {Path}}A ~ a ~ ({x}{f\\, x}))}$ .",
"We can define a path $s\\triangleq \\mathop {\\langle i \\rangle } f ([\\left[ q \\leftarrow \\operatorname{\\mathsf {ih}} \\right]]{q\\, i})~:~\\operatorname{\\mathsf {Path}}A ~ (f ({a})) ~ (f ({{x}{f\\, x}})),$ which is well-typed because the type of the variable $q$ ensures that $q\\, 0$ is judgementally equal to $a$ , resp.",
"$q\\,1$ and ${x}{f\\, x}$ .",
"Note that we here implicitly use the extensionality principle for later (REF ).",
"We compose $s$ with $p$ , and then with the inverse of the canonical unfold lemma of Lem.",
"REF , to obtain our path from $a$ to ${x}{f\\, x}$ .",
"We can write out our full proof term, where $p^{-1}$ is the inverse path of $p$ , as ${\\operatorname{\\mathsf {ih}}}{\\mathop {\\langle i \\rangle } \\operatorname{\\mathsf {comp}}^j\\,A~[(i=0) \\mapsto p^{-1}, (i=1) \\mapsto f ([1-j]{x}{f\\, x})]~(f ([\\left[ q \\leftarrow \\operatorname{\\mathsf {ih}} \\right]]{q\\, i}) )}.$"
],
[
"Programming and Proving with Guarded Recursive Types",
"In this section we show some simple examples of programming with guarded recursion, and prove properties of our programs using Löb induction."
],
[
"Streams.",
"The type of guarded recursive streams in $\\mathsf {GCTT}$ , as with $\\mathsf {GDTT}$ , are defined as fixed points on the universe: ${A} \\;\\triangleq \\; {x}{A\\times [[y\\leftarrow x]]{y}}$ Note the use of a delayed substitution to transform a term of type ${\\operatorname{\\mathsf {U}}}$ to one of type $\\operatorname{\\mathsf {U}}$ , as discussed at the start of Sec.",
"REF .",
"Desugaring to restate this in terms of $\\operatorname{\\mathsf {dfix}}$ , we have ${A} \\;=\\;A\\times [[y\\leftarrow [0]{x}{A\\times [[y\\leftarrow x]]{y}}]]{y}$ The head function $\\operatorname{\\mathsf {hd}}:{A}\\rightarrow A$ is the first projection.",
"The tail function, however, cannot be the second projection, since this yields a term of type $[\\left[ y \\leftarrow [0]{x}{A\\times [\\left[ y\\leftarrow x \\right]]{y}} \\right]]{y}$ rather than the desired ${{A}}$ .",
"However we are not far off; ${{A}}$ is judgementally equal to $[\\left[ y \\leftarrow [1]{x}{A\\times [\\left[ y\\leftarrow x \\right]]{y}} \\right]]{y}$ , which is the same term as (REF ), apart from endpoint 1 replacing 0.",
"The canonical unfold lemma (Lem.",
"REF ) tells us that we can build a path in $\\operatorname{\\mathsf {U}}$ from ${A}$ to $A \\times {{A}}$ ; call this path $\\langle i \\rangle {A}^i$ .",
"Then we can transport between these types: $\\operatorname{\\mathsf {unfold}} \\, s \\triangleq \\operatorname{\\mathsf {transp}}^i \\, {A}^i \\, s\\qquad \\qquad \\qquad \\operatorname{\\mathsf {fold}} \\, s \\triangleq \\operatorname{\\mathsf {transp}}^i \\, {A}^{1-i} \\, s$ Note that the compositions of these two operations are path equal to identity functions, but not judgementally equal.",
"We can now obtain the desired tail function $\\operatorname{\\mathsf {tl}} : {A} \\rightarrow {{A}}$ by composing the second projection with $\\operatorname{\\mathsf {unfold}}$ , so $\\operatorname{\\mathsf {tl}} \\, s \\triangleq (\\operatorname{\\mathsf {unfold}} \\, s).2$ .",
"Similarly we can define the stream constructor $\\operatorname{\\mathsf {cons}}$ (written infix as $::$ ) by using $\\operatorname{\\mathsf {fold}}$ : $\\operatorname{\\mathsf {cons}} \\triangleq \\lambda a, s .",
"\\operatorname{\\mathsf {fold}} \\, (a, s)~:~A \\rightarrow {{A}} \\rightarrow {A}.$ We now turn to higher order functions on streams.",
"We define $\\operatorname{\\mathsf {zipWith}} : (A \\rightarrow B \\rightarrow C) \\rightarrow {A} \\rightarrow {B} \\rightarrow {C}$ , the stream function which maps a binary function on two input streams to produce an output stream, as $\\operatorname{\\mathsf {zipWith}} \\, f \\triangleq {z}{\\lambda s_1, s_2 .",
"f \\, (\\operatorname{\\mathsf {hd}} \\, s_1) \\, (\\operatorname{\\mathsf {hd}} \\, s_2) \\,::\\,[\\left[\\begin{array}{l} z^{\\prime } \\leftarrow z \\\\ t_1 \\leftarrow \\operatorname{\\mathsf {tl}} \\, s_1 \\\\ t_2 \\leftarrow \\operatorname{\\mathsf {tl}} \\, s_2 \\end{array}\\right]]{z^{\\prime } \\, t_1 \\, t_2}}.$ Of course $\\operatorname{\\mathsf {zipWith}}$ is definable even with simple types and $$ , but in $\\mathsf {GCTT}$ we can go further and prove properties about the function: Proposition 4 ($\\operatorname{\\mathsf {zipWith}}$ preserves commutativity) If $f : A \\rightarrow A \\rightarrow B$ is commutative, then $\\operatorname{\\mathsf {zipWith}}\\, f : {A} \\rightarrow {A} \\rightarrow {B}$ is commutative.",
"Let $\\operatorname{\\mathsf {c}} : (a_1 : A) \\rightarrow (a_2 : A) \\rightarrow \\operatorname{\\mathsf {Path}}B ~ (f\\, a_1 \\, a_2) ~ (f\\, a_2 \\, a_1)$ witness commutativity of $f$ .",
"We proceed by Löb induction, i.e., by assuming $\\operatorname{\\mathsf {ih}} : {\\left((s_1:{A})\\rightarrow (s_2:{A})\\rightarrow \\operatorname{\\mathsf {Path}}B~ (\\operatorname{\\mathsf {zipWith}}\\, f \\, s_1 \\, s_2)~ (\\operatorname{\\mathsf {zipWith}}\\, f \\, s_2 \\, s_1)\\right)}.$ Let $i:I$ be a fresh name, and $s_1, s_2 : {A}$ .",
"Our aim is to construct a stream $v$ which is $\\operatorname{\\mathsf {zipWith}}\\, f \\, s_1 \\, s_2$ when substituting 0 for $i$ , and $\\operatorname{\\mathsf {zipWith}}\\, f \\, s_2 \\, s_1$ when substituting 1 for $i$ .",
"An initial attempt at this proof is the term $v \\,\\triangleq \\,\\operatorname{\\mathsf {c}} \\, (\\operatorname{\\mathsf {hd}}\\, s_1) \\, (\\operatorname{\\mathsf {hd}}\\, s_2) \\, i ~::~[\\left[\\begin{array}{l} q \\leftarrow \\operatorname{\\mathsf {ih}} \\\\t_1 \\leftarrow \\operatorname{\\mathsf {tl}}\\, s_1 \\\\t_2 \\leftarrow \\operatorname{\\mathsf {tl}} \\, s_2 \\end{array}\\right]]{q \\, t_1 \\, t_2 \\, i}~:~{B},$ which is equal to $f \\, (\\operatorname{\\mathsf {hd}} \\, s_1) \\, (\\operatorname{\\mathsf {hd}} \\, s_2) ~::~[\\left[\\begin{array}{l} t_1 \\leftarrow \\operatorname{\\mathsf {tl}} \\, s_1 \\\\ t_2 \\leftarrow \\operatorname{\\mathsf {tl}} \\, s_2 \\end{array}\\right]]{\\operatorname{\\mathsf {zipWith}} \\, f \\, t_1 \\, t_2}$ when substituting 0 for $i$ , which is $\\operatorname{\\mathsf {zipWith}}\\, f\\, s_1\\, s_2$ , but unfolded once.",
"Similarly, $v[1/i]$ is $\\operatorname{\\mathsf {zipWith}}\\, f\\, s_2\\, s_1$ unfolded once.",
"Let $\\langle j \\rangle \\operatorname{\\mathsf {zipWith}}^j$ be the canonical unfold lemma associated with $\\operatorname{\\mathsf {zipWith}}$ (see Lem.",
"REF ).",
"We can now finish the proof by composing $v$ with (the inverse of) the canonical unfold lemma.",
"Diagrammatically, with $i$ along the horizontal axis and $j$ along the vertical: Figure: NO_CAPTION The complete proof term, in the language of the type checker, can be found in Appendix .",
"A key feature of guarded recursive types are that they support negative occurrences of recursion variables.",
"This is important for applications to models of program logics [6].",
"Here we consider a simple example of a negative variance recursive type, namely $ \\operatorname{\\mathsf {Rec}}_A \\triangleq {x}{([[x^{\\prime }\\leftarrow x]]{x^{\\prime }})\\rightarrow A} $ , which is path equal to ${\\operatorname{\\mathsf {Rec}}_A} \\rightarrow A$ .",
"As a simple demonstration of the expressiveness we gain from negative guarded recursive types, we define a guarded variant of Curry's Y combinator: $\\begin{array}{lclcl}\\Delta &\\triangleq & \\lambda x.f([[x^{\\prime }\\leftarrow x]]{((\\operatorname{\\mathsf {unfold}}x^{\\prime }) x})) &:& \\operatorname{\\mathsf {Rec}}_A\\rightarrow A \\\\\\operatorname{\\mathsf {Y}}&\\triangleq & \\lambda f.\\Delta (\\operatorname{\\mathsf {fold}}\\Delta ) &:& (A\\rightarrow A)\\rightarrow A,\\end{array}$ where $\\operatorname{\\mathsf {fold}}$ and $\\operatorname{\\mathsf {unfold}}$ are the transports along the path between $\\operatorname{\\mathsf {Rec}}_A$ and ${\\operatorname{\\mathsf {Rec}}_A} \\rightarrow A$ .",
"As with $\\operatorname{\\mathsf {zipWith}}$ , $\\operatorname{\\mathsf {Y}}$ can be defined with simple types and $$ [1]; what is new to $\\mathsf {GCTT}$ is that we can also prove properties about it: Proposition 5 ($\\operatorname{\\mathsf {Y}}$ is a guarded fixed-point combinator) $\\operatorname{\\mathsf {Y}} f$ is path equal to $f \\, (({\\operatorname{\\mathsf {Y}} f}))$ , for any $f : {A} \\rightarrow A$ .",
"Therefore, by Prop.",
"REF , $\\operatorname{\\mathsf {Y}}$ is path equal to $\\operatorname{\\mathsf {fix}}$ .",
"$\\operatorname{\\mathsf {Y}} f$ simplifies to $f\\, ({(\\operatorname{\\mathsf {unfold}} \\, (\\operatorname{\\mathsf {fold}} \\Delta ) \\, ({\\operatorname{\\mathsf {fold}} \\Delta }))})$ , and $\\operatorname{\\mathsf {unfold}}\\, (\\operatorname{\\mathsf {fold}} \\Delta )$ is path equal to $\\Delta $ .",
"A congruence over this path yields our path between $\\operatorname{\\mathsf {Y}} f$ and $f ({(\\operatorname{\\mathsf {Y}} f)})$ ."
],
[
"Semantics",
"In this section we sketch the semantics of $\\mathsf {GCTT}$ .",
"The semantics is based on the category ${\\widehat{\\mathcal {C}\\times \\omega }}$ of presheaves on the category $\\mathcal {C}\\times \\omega $ , where $\\mathcal {C}$ is the category of cubes [10] and $\\omega $ is the poset of natural numbers.",
"The category of cubes is the opposite of the Kleisli category of the free De Morgan algebra monad on finite sets.",
"More concretely, given a countably infinite set of names $i, j, k, \\ldots $ , $\\mathcal {C}$ has as objects finite sets of names $I$ , $J$ .",
"A morphism $I \\rightarrow J \\in \\mathcal {C}$ is a function $J \\rightarrow \\mathbf {DM}\\left(I\\right)$ , where $\\mathbf {DM}\\left(I\\right)$ is the free De Morgan algebra with generators $I$ .",
"Following the approach of Cohen et al.",
"[10], contexts of $\\mathsf {GCTT}$ will be interpreted as objects of ${\\widehat{\\mathcal {C}\\times \\omega }}$ .",
"Types in context $\\Gamma $ will be interpreted as pairs $(A, c_A)$ of a presheaf $A$ on the category of elements of $\\Gamma $ and a composition structure $c_A$ .",
"We call such a pair a fibrant type.",
"To aid in defining what a composition structure is, and in showing that composition structure is preserved by all the necessary type constructions, we will make use of the internal language of ${\\widehat{\\mathcal {C}\\times \\omega }}$ in the form of dependent predicate logic; see for example Phoa [24].",
"A type of $\\mathsf {GCTT}$ in context $\\Gamma $ will then be interpreted as a pair of a type $\\Gamma \\vdash A$ in the internal language of ${\\widehat{\\mathcal {C}\\times \\omega }}$ , and a composition structure $c_A$ , where $c_A$ is a term in the internal language of a specific type $\\Phi (\\Gamma ;A)$ , which we define below after introducing the necessary constructs.",
"Terms of $\\mathsf {GCTT}$ will be interpreted as terms of the internal language.",
"We use categories with families [12] as our notion of a model.",
"Due to space limits we omit the precise definition of the category with families here, and refer to the online technical appendix.",
"The semantics is split into several parts, which provide semantics at different levels of generality.",
"We first show that every presheaf topos with a non-trivial internal De Morgan algebra $I$ satisfying the disjunction property can be used to give semantics to the subset of the cubical type theory $\\mathsf {CTT}$ without glueing and the universe.",
"We further show that, for any category $\\mathbb {D}$ , the category of presheaves on $\\mathcal {C}\\times \\mathbb {D}$ has an interval $I$ , which is the inclusion of the interval in presheaves over the category of cubes $\\mathcal {C}$ .",
"We then extend the semantics to include glueing and universes.",
"We show that the topos of presheaves $\\mathcal {C}\\times \\mathbb {D}$ for any category $\\mathbb {D}$ with an initial object can be used to give semantics to the entire cubical type theory.",
"Finally, we show that the category of presheaves on $\\mathcal {C}\\times \\omega $ gives semantics to delayed substitutions and fixed points.",
"Using these and some additional properties of the delayed substitutions we show in the internal language of ${\\widehat{\\mathcal {C}\\times \\omega }}$ that $[\\xi ]{A}$ has composition whenever $A$ has composition.",
"Combining all three, we give semantics to $\\mathsf {GCTT}$ in ${\\widehat{\\mathcal {C}\\times \\omega }}$ ."
],
[
"Model of $\\mathsf {CTT}$ Without Glueing and the Universe",
"Let $\\mathcal {E}$ be a topos with a natural numbers object, and let $I$ be a De Morgan algebra internal to $\\mathcal {E}$ which satisfies the finitary disjunction property, i.e., $(i \\vee j) = 1 \\Rightarrow (i = 1) \\vee (j = 1),\\quad \\text{and}\\quad \\lnot (0 = 1).$"
],
[
"Faces.",
"Using the interval $I$ we define the type $\\mathbb {F}$ as the image of the function $\\cdot = 1 : I\\rightarrow \\Omega $ , where $\\Omega $ is the subobject classifier.",
"More precisely, $\\mathbb {F}$ is the subset type $\\mathbb {F}\\triangleq \\left\\lbrace p : \\Omega \\;|\\;\\exists (i : I), p = (i = 1) \\right\\rbrace $ We will implicitly use the inclusion $\\mathbb {F}\\rightarrow \\Omega $ .",
"The following lemma states in particular that the inclusion is compatible with all the lattice operations, so omitting it is justified.",
"The disjunction property is crucial for validity of this lemma.",
"Lemma 6 $\\mathbb {F}$ is a lattice for operations inherited from $\\Omega $ .",
"The corestriction $\\cdot = 1 : I\\rightarrow \\mathbb {F}$ is a lattice homomorphism.",
"It is not injective in general.",
"Given $\\Gamma \\vdash \\varphi : \\mathbb {F}$ , we write $\\left[\\varphi \\right] \\triangleq \\operatorname{Id}_{\\mathbb {F}} (\\varphi , \\top )$ .",
"Given $\\Gamma \\vdash A$ and $\\Gamma \\vdash \\varphi : \\mathbb {F}$ a partial element of type $A$ of extent $\\varphi $ is a term $t$ of type $\\Gamma \\vdash t : \\Pi (p : \\left[\\varphi \\right]).A$ .",
"If we are in a context with $p : \\left[\\varphi \\right]$ , then we will treat such a partial element $t$ as a term of type $A$ , leaving implicit the application to the proof $p$ , i.e., we will treat $t$ as $t\\,p$ .",
"We will often write $\\Gamma , \\left[\\varphi \\right]$ instead of $\\Gamma , p : \\left[\\varphi \\right]$ when we do not mention the proof term $p$ explicitly in the rest of the judgement.",
"This is justified since inhabitants of $\\left[\\varphi \\right]$ are unique up to judgemental equality (recall that dependent predicate logic is a logic over an extensional dependent type theory).",
"Given $\\Gamma , p : \\left[\\varphi \\right] \\vdash B$ we write $B^{\\varphi }$ for the dependent function space $\\Pi (p : \\left[\\varphi \\right]).B$ and again leave the proof $p$ implicit.",
"For a term $\\Gamma , p:\\left[\\varphi \\right] \\vdash u : A$ we define $A[\\varphi \\mapsto u] \\triangleq \\Sigma (a:A).\\left(\\operatorname{Id}_{A} (a, u )\\right)^{\\varphi }$ .",
"Faces allow us to define the type of compositions $\\Phi (\\Gamma ;A)$ .",
"Homotopically, compositions allow us to put a lid on a box [10].",
"Given $\\Gamma \\vdash A$ we define the corresponding type of compositions as $\\Phi (\\Gamma ;A) \\triangleq \\Pi &(\\gamma : I\\rightarrow \\Gamma )(\\varphi : \\mathbb {F})\\left(u : \\Pi (i:I).",
"\\left(A(\\gamma (i))\\right)^{\\varphi }\\right) .",
"\\\\& A(\\gamma (0))[ \\varphi \\mapsto u(0) ] \\rightarrow A(\\gamma (1))[ \\varphi \\mapsto u(1) ].$ Here we treat the context $\\Gamma $ as a closed type.",
"This is justified because there is a canonical bijection between contexts and closed types of the internal language.",
"The notation $A(\\gamma (i))$ means substitution along the (uncurried) $\\gamma $ .",
"Due to lack of space we do not show how the standard constructs of the type theory are interpreted.",
"We only sketch how the following composition term is interpreted in terms of the composition in the model.",
"$\\Gamma \\vdash \\varphi : \\mathbb {F}$ $\\Gamma , i:I \\vdash A$ $\\Gamma , \\varphi , i : I \\vdash u : A$ $\\Gamma \\vdash a_0 : A[0/i][\\varphi \\mapsto u[0/i]]$ $\\Gamma \\vdash \\operatorname{\\mathsf {comp}}^i\\,A~[\\varphi \\mapsto u]~a_0 : A[1/i][\\varphi \\mapsto u[1/i]]$ .",
"By assumption we have $c_A$ of type $\\Phi (\\Gamma , i : I;A)$ and $u$ and $a_0$ are interpreted as terms in the internal language of the corresponding types.",
"The interpretation of composition is the term $\\gamma : \\Gamma \\vdash c_A \\left(\\lambda (i : I) .",
"(\\gamma , i)\\right)\\varphi \\left(\\lambda (i : I) (p : \\left[\\varphi \\right]) .",
"u\\right)a_0 : A(\\gamma (1))[ \\varphi \\mapsto u(1)]$ where we have omitted writing the proof $u(0) = a_0$ on $\\left[\\varphi \\right]$ .",
"The category of cubical sets has an internal interval type satisfying the disjunction property [10].",
"It is the functor mapping $I \\in \\mathcal {C}$ to $\\mathbf {DM}\\left(I\\right)$ .",
"Since the theory of a De Morgan algebra with $0 \\ne 1$ and the disjunction property is geometric [16] we have that for any topos $\\mathcal {F}$ and geometric morphism $\\varphi :\\mathcal {F}\\rightarrow \\widehat{\\mathcal {C}}$ , $\\varphi ^*(I) \\in \\mathcal {F}$ is a De Morgan algebra with the disjunction propertyA statement very close to this can be used as a characterisation of $\\widehat{\\mathcal {C}}$ : this topos classifies the geometric theory of flat De Morgan algebras [25]..",
"In particular, given any category $\\mathbb {D}$ there is a projection functor $\\pi : \\mathcal {C}\\times \\mathbb {D}\\rightarrow \\mathcal {C}$ which induces the (essential) geometric morphism $\\pi ^* \\dashv \\pi _* : \\widehat{\\mathcal {C}\\times \\mathbb {D}} \\rightarrow \\widehat{\\mathcal {C}}$ , where $\\pi ^*$ is precomposition with $\\pi $ , and $\\pi _*$ takes limits along $\\mathbb {D}$ .",
"With the semantic structures developed thus far we can give semantics to the subset of $\\mathsf {CTT}$ without glueing and the universe."
],
[
"Adding Glueing and the Universe",
"The glueing construction [10] is used to prove both fibrancy and, subsequently, univalence of the universe of fibrant types.",
"Concretely, given $\\Gamma \\vdash \\varphi : I\\qquad \\Gamma , [\\varphi ] \\vdash T\\qquad \\Gamma \\vdash A\\qquad \\Gamma \\vdash w : (T \\rightarrow A)^{\\varphi }$ we define the type $\\operatorname{Glue}\\, \\left[\\varphi \\mapsto (T,w)\\right] \\, A$ in two steps.",
"First we define the typeThis type is already present in Kapulkin at al. [15].",
"$Glue^{\\prime }_\\Gamma (\\varphi , T, A, w) \\triangleq \\sum _{a : A}\\sum _{t : T^{\\varphi }}\\prod _{p : [\\varphi ]} w p (t p) = a.$ For this type we have the following property $\\Gamma , [\\varphi ] \\vdash T \\cong Glue^{\\prime }_\\Gamma (\\varphi , T, A, w)$ .",
"However, we need an equality, not an isomorphism, to obtain the correct typing rules.",
"The technical appendix provides a general strictification lemma which allows us to define the type $Glue$ .",
"To show that the type $\\operatorname{Glue}\\, \\left[\\varphi \\mapsto (T,w)\\right] \\, A$ is fibrant we need to additionally assume that the map $\\varphi \\mapsto \\lambda \\_.\\varphi : \\mathbb {F}\\rightarrow (I\\rightarrow \\mathbb {F})$ has an internal right adjoint $\\forall $ .",
"Such a right adjoint exists in all toposes $\\widehat{\\mathcal {C}\\times \\mathbb {D}}$ , for any small category $\\mathbb {D}$ with an initial object."
],
[
"Universe of fibrant types.",
"Given a (Grothendieck) universe $\\mathfrak {U}$ in the meta-theory, the Hofmann-Streicher universe [14] $\\mathcal {U}^\\omega $ in ${\\widehat{\\mathcal {C}\\times \\omega }}$ maps $(I,n)$ to the set of functors valued in $\\mathfrak {U}$ on the category of elements of $y(I,n)$ , where $y$ is the Yoneda embedding.",
"As in Cohen et al.",
"[10] we define the universe of fibrant types $\\mathcal {U}_{f}^\\omega $ by setting $\\mathcal {U}_{f}^\\omega (I,n)$ to be the set of fibrant types in context $y(I,n)$ .",
"The universe $\\mathcal {U}_{f}^\\omega $ satisfies the rules $\\Gamma \\vdash a : \\mathcal {U}$ $ \\vdash \\mathbf {c} : \\Phi (\\Gamma ;\\operatorname{El}(a))$$\\Gamma \\vdash \\llparenthesis a, \\mathbf {c}\\rrparenthesis : \\mathcal {U}_{f}$ $\\Gamma \\vdash a : \\mathcal {U}_{f}$$\\Gamma \\vdash \\operatorname{El}(a)$ $\\Gamma \\vdash a : \\mathcal {U}_{f}$$ \\vdash \\operatorname{Comp}(a) : \\Phi (\\Gamma ;\\operatorname{El}(a))$ Using the glueing operation, one shows that the universe of fibrant types is itself fibrant and, moreover, that it is univalent."
],
[
"Adding the Later Type-Former",
"We now fix the site to be $\\mathcal {C}\\times \\omega $ .",
"From the previous sections we know that ${\\widehat{\\mathcal {C}\\times \\omega }}$ gives semantics to $\\mathsf {CTT}$ .",
"The new constructs of $\\mathsf {GDTT}$ are the $\\operatorname{\\triangleright }$ type-former and its delayed substitutions, and guarded fixed points.",
"Continuing to work in the internal language, we first show that the internal language of ${\\widehat{\\mathcal {C}\\times \\omega }}$ can be extended with these constructions, allowing interpretation of the subset of the type theory $\\mathsf {GDTT}$ without clock quantification [8].",
"Due to lack of space we omit the details of this part, but do remark that $\\operatorname{\\triangleright }$ is defined as $(\\operatorname{\\triangleright }(X))(I,n){\\left\\lbrace \\begin{array}{ll}\\lbrace \\star \\rbrace & \\text{if } n = 0\\\\X(I,m) & \\text{if } n = m+1\\end{array}\\right.",
"}$ The essence of this definition is that $\\operatorname{\\triangleright }$ depends only on the “$\\omega $ component” and ignores the “$\\mathcal {C}$ component”.",
"Verification that all the rules of $\\mathsf {GDTT}$ are satisfied is therefore very similar to the verification that the topos $\\widehat{\\omega }$ is a model of the same subset of $\\mathsf {GDTT}$ .",
"The only additional property we need now is that $\\operatorname{\\triangleright }$ preserves compositions, in the sense that if we have a delayed substitution $\\vdash \\xi : \\Gamma \\rightarrowtriangle \\Gamma ^{\\prime }$ and a type $\\Gamma ,\\Gamma ^{\\prime } \\vdash A$ together with a closed term $\\mathbf {c}_A$ of type $\\Phi (\\Gamma ,\\Gamma ^{\\prime };A)$ then we can construct $\\mathbf {c}^{\\prime }_{[\\xi ]{A}}$ of type $\\Phi (\\Gamma ;[\\xi ]{A})$ .",
"The following lemma uses the notion of a type $\\Gamma \\vdash A$ being constant with respect to $\\omega $.",
"This notion is a natural generalisation to types-in-context of the property that a presheaf is in the image of the functor $\\pi ^*$ .",
"We refer to the online technical appendix for the precise definition.",
"Here we only remark that the interval type $I$ is constant with respect to $\\omega $ , as is the type $\\Gamma \\vdash \\left[\\varphi \\right]$ for any term $\\Gamma \\vdash \\varphi : \\mathbb {F}$ .",
"Lemma 7 Assume $\\Gamma \\vdash A$ , $\\Gamma ,\\Gamma ^{\\prime },x : A \\vdash B$ and $\\vdash \\xi : \\Gamma \\rightarrowtriangle \\Gamma ^{\\prime }$ , and further that $A$ is constant with respect to $\\omega $ .",
"Then the following two types are isomorphic $\\Gamma \\vdash [\\xi ]{\\Pi (x : A).B} \\cong \\Pi (x : A).",
"[\\xi ]B$ and the canonical morphism $\\lambda f .",
"\\lambda x .",
"[\\left[ \\xi ,f^{\\prime } \\leftarrow f \\right]]{f^{\\prime }\\,x}$ from left to right is an isomorphism.",
"Corollary 8 If $\\Gamma \\vdash \\varphi : \\mathbb {F}$ then we have an isomorphism of types $\\Gamma \\vdash [\\xi ]{\\Pi (p : \\left[\\varphi \\right]).B} \\cong \\Pi (x : \\left[\\varphi \\right]).",
"[\\xi ]{B}.$ Lemma 9 (${\\xi }$ -types preserve compositions) If $[\\xi ]{A}$ is a well-formed type in context $\\Gamma $ and we have a composition term $\\mathbf {c}_A : \\Phi (\\Gamma ,\\Gamma ^{\\prime };A)$ , then there is a composition term $\\mathbf {c}: \\Phi (\\Gamma ;[\\xi ]{A})$ .",
"We show the special case with an empty delayed substitution.",
"For the more general proof we refer to the technical appendix.",
"Assume we have a composition $\\mathbf {c}_A : \\Phi (\\Gamma ;A)$ .",
"Our goal is to find a term $\\mathbf {c}: \\Phi (\\Gamma ;{A})$ , so we first introduce some variables: : I: $\\mathbb {F}$ u : (i:I).",
"((A)( i)) a0 : (A)( 0)[u 0].",
"Using the isomorphisms from Cor.",
"REF and Lem.",
"REF we obtain a term $\\tilde{u} : {(\\Pi (i : I).",
"(A(\\gamma \\, i))^{\\varphi })}$ isomorphic to $u$ .",
"We can now – almost – write the term $[\\left[\\begin{array}{l} u^{\\prime } \\leftarrow \\tilde{u} \\\\ a_0^{\\prime } \\leftarrow a_0 \\end{array}\\right]]{\\mathbf {c}_A \\, \\gamma \\, \\varphi \\, u^{\\prime } \\, a_0^{\\prime }}~:~ {(A(\\gamma \\, 1))},\\qquad \\mathrm {(*)}$ what is missing is to check that $a_0^{\\prime } = u^{\\prime }\\, 0$ on the extent $\\varphi $ , so that we can legally apply $\\mathbf {c}_A$ ; this is equivalent to saying that the type $[\\left[ u^{\\prime } \\leftarrow \\tilde{u}, a_0^{\\prime } \\leftarrow a_0 \\right]]{\\operatorname{Id}_{A(\\gamma \\, 0)} (a_0^{\\prime }, u^{\\prime }\\, 0 )^{\\varphi }}$ is inhabited.",
"We transform this type as follows: $[\\left[\\begin{array}{l} u^{\\prime } \\leftarrow \\tilde{u} \\\\ a_0^{\\prime } \\leftarrow a_0 \\end{array}\\right]]{\\operatorname{Id}_{} (a_0^{\\prime }, u^{\\prime } \\, 0 )^{\\varphi }}& \\cong \\left( [\\left[\\begin{array}{l} u^{\\prime } \\leftarrow \\tilde{u} \\\\ a_0^{\\prime } \\leftarrow a_0 \\end{array}\\right]]{\\operatorname{Id}_{} (a_0^{\\prime }, u^{\\prime } \\, 0 )}\\right)^{\\varphi } \\\\& =\\left( \\operatorname{Id}_{} ([\\left[\\begin{array}{l} u^{\\prime } \\leftarrow \\tilde{u} \\\\ a_0^{\\prime } \\leftarrow a_0 \\end{array}\\right]]{a_0^{\\prime }}, [\\left[\\begin{array}{l} u^{\\prime } \\leftarrow \\tilde{u} \\\\ a_0^{\\prime } \\leftarrow a_0 \\end{array}\\right]]{u^{\\prime }\\, 0} ) \\right)^{\\varphi } \\\\& =\\left(\\operatorname{Id}_{} (a_0, u\\, 0 )\\right)^{\\varphi },$ where the last equality uses that $\\tilde{u}$ is defined using the inverse of $\\lambda f \\lambda x .",
"[\\left[ f^{\\prime } \\leftarrow f \\right]]{f^{\\prime } \\, x}$ (Lem.",
"REF ).",
"By assumption it is the case that $\\left(\\operatorname{Id}_{} (a_0, u\\, 0 )\\right)^{\\varphi }$ is inhabited, and therefore (REF ) is well-defined.",
"It remains only to check that (REF ) is equal to $u \\, 1$ on the extent $\\varphi $ , but this follows from the equalities of $\\mathbf {c}_A$ and by the definition of $\\tilde{u}$ (Lem.",
"REF ).",
"Assuming $\\varphi $ , we have $[\\left[\\begin{array}{l} u^{\\prime } \\leftarrow \\tilde{u} \\\\ a_0^{\\prime } \\leftarrow a_0 \\end{array}\\right]]{\\mathbf {c}_A \\, \\gamma \\, \\varphi \\, u^{\\prime } \\, a_0^{\\prime }}=[\\left[\\begin{array}{l} u^{\\prime } \\leftarrow \\tilde{u} \\\\ a_0^{\\prime } \\leftarrow a_0 \\end{array}\\right]]{u^{\\prime } \\, 1}=u \\, 1.$"
],
[
"Summary.",
"In this section we have highlighted the key ingredients that go into a sound interpretation of $\\mathsf {GCTT}$ in ${\\widehat{\\mathcal {C}\\times \\omega }}$ .",
"For the precise statement of the interpretation of all the constructs, and the soundness theorem, we refer to the online technical appendix."
],
[
"Conclusion",
"In this paper we have made the following contributions: We introduce guarded cubical type theory ($\\mathsf {GCTT}$ ), which combines features of cubical type theory ($\\mathsf {CTT}$ ) and guarded dependent type theory ($\\mathsf {GDTT}$ ).",
"The path equality of $\\mathsf {CTT}$ is shown to support reasoning about extensional properties of guarded recursive operations, and we use the interval of $\\mathsf {CTT}$ to constrain the unfolding of fixed-points.",
"We show that $\\mathsf {CTT}$ can be modelled in any presheaf topos with an internal non-trivial De Morgan algebra with the disjunction property, an operator $\\forall $ , and a universe of fibrant types.",
"Most of these constructions are done via the internal logic.",
"We then show that a class of presheaf models of the form $\\widehat{\\mathcal {C}\\times \\mathbb {D}}$ , for any category $\\mathbb {D}$ with an initial object, satisfy the above axioms and hence gives rise to a model of $\\mathsf {CTT}$ .",
"We give semantics to $\\mathsf {GCTT}$ in the topos of presheaves over $\\mathcal {C}\\times \\omega $ ."
],
[
"Further work.",
"We wish to establish key syntactic properties of $\\mathsf {GCTT}$ , namely decidable type-checking and canonicity for base types.",
"Our prototype implementation establishes some confidence in these properties.",
"We wish to further extend $\\mathsf {GCTT}$ with clock quantification [3], such as is present in $\\mathsf {GDTT}$ .",
"Clock quantification allows for the controlled elimination of the later type-former, and hence the encoding of first-class coinductive types via guarded recursive types.",
"The generality of our approach to semantics in this paper should allow us to build a model by combining cubical sets with the presheaf model of $\\mathsf {GDTT}$ with multiple clocks [7].",
"The main challenges lie in ensuring decidable type checking ($\\mathsf {GDTT}$ relies on certain rules involving clock quantifiers which seem difficult to implement), and solving the coherence problem for clock substitution.",
"Finally, some higher inductive types, like the truncation, can be added to $\\mathsf {CTT}$ .",
"We would like to understand how these interact with $$ ."
],
[
"Related work.",
"Another type theory with a computational interpretation of functional extensionality, but without equality reflection, is observational type theory ($\\mathsf {OTT}$ ) [2].",
"We found $\\mathsf {CTT}$ 's prototype implementation, its presheaf semantics, and its interval as a tool for controlling unfoldings, most convenient for developing our combination with $\\mathsf {GDTT}$ , but extending $\\mathsf {OTT}$ similarly would provide an interesting comparison.",
"Spitters [25] used the interval of the internal logic of cubical sets to model identity types.",
"Coquand [11] defined the composition operation internally to obtain a model of type theory.",
"We have extended both these ideas to a full model of $\\mathsf {CTT}$ .",
"Recent independent work by Orton and Pitts [23] axiomatises a model for $\\mathsf {CTT}$ without a universe, again building on Coquand [11].",
"With the exception of the absence of the universe, their development is more general than ours.",
"Our semantic developments are sufficiently general to support the sound addition of guarded recursive types to $\\mathsf {CTT}$ ."
],
[
"Acknowledgements.",
"We gratefully acknowledge our discussions with Thierry Coquand, and the comments of our reviewers.",
"This research was supported in part by the ModuRes Sapere Aude Advanced Grant from The Danish Council for Independent Research for the Natural Sciences (FNU).",
"Aleš Bizjak was supported in part by a Microsoft Research PhD grant."
],
[
"$\\operatorname{\\mathsf {zipWith}}$ Preserves Commutativity",
"We provide a formalisation of Sec.",
"REF which can be verified by our type checker.",
"This file, among other examples, is available in the gctt-examples folder in the type-checker repository."
]
] | 1606.05223 |
[
[
"No Need to Pay Attention: Simple Recurrent Neural Networks Work! (for\n Answering \"Simple\" Questions)"
],
[
"Abstract First-order factoid question answering assumes that the question can be answered by a single fact in a knowledge base (KB).",
"While this does not seem like a challenging task, many recent attempts that apply either complex linguistic reasoning or deep neural networks achieve 65%-76% accuracy on benchmark sets.",
"Our approach formulates the task as two machine learning problems: detecting the entities in the question, and classifying the question as one of the relation types in the KB.",
"We train a recurrent neural network to solve each problem.",
"On the SimpleQuestions dataset, our approach yields substantial improvements over previously published results --- even neural networks based on much more complex architectures.",
"The simplicity of our approach also has practical advantages, such as efficiency and modularity, that are valuable especially in an industry setting.",
"In fact, we present a preliminary analysis of the performance of our model on real queries from Comcast's X1 entertainment platform with millions of users every day."
],
[
"Introduction",
"First-order factoid question answering (QA) assumes that the question can be answered by a single fact in a knowledge base (KB).",
"For example, “How old is Tom Hanks” is about the [age] of [Tom Hanks].",
"Also referred to as simple questions by Bordes et al.",
"Bordes:2015aa, recent attempts that apply either complex linguistic reasoning or attention-based complex neural network architectures achieve up to 76% accuracy on benchmark sets [11], [27].",
"While it is tempting to study QA systems that can handle more complicated questions, it is hard to reach reasonably high precision for unrestricted questions.",
"For more than a decade, successful industry applications of QA have focused on first-order questions.",
"This bears the question: are users even interested in asking questions beyond first-order (or are these use cases more suitable for interactive dialogue)?",
"Based on voice logs from a major entertainment platform with millions of users every day, Comcast X1, we find that most existing use cases of QA fall into the first-order category.",
"Our strategy is to tailor our approach to first-order QA by making strong assumptions about the problem structure.",
"In particular, we assume that the answer to a first-order question is a single property of a single entity in the KB, and decompose the task into two subproblems: (a) detecting entities in the question and (b) classifying the question as one of the relation types in the KB.",
"We simply train a vanilla recurrent neural network (RNN) to solve each subproblem [10].",
"Despite its simplicity, our approach (RNN-QA) achieves the highest reported accuracy on the SimpleQuestions dataset.",
"While recent literature has focused on building more complex neural network architectures with attention mechanisms, attempting to generalize to broader QA, we enforce stricter assumptions on the problem structure, thereby reducing complexity.",
"This also means that our solution is efficient, another critical requirement for real-time QA applications.",
"In fact, we present a performance analysis of RNN-QA on Comcast's X1 entertainment system, used by millions of customers every day."
],
[
"Related work",
"If knowledge is presented in a structured form (e.g., knowledge base (KB)), the standard approach to QA is to transform the question and knowledge into a compatible form, and perform reasoning to determine which fact in the KB answers a given question.",
"Examples of this approach include pattern-based question analyzers [6], combination of syntactic parsing and semantic role labeling [3], [2], as well as lambda calculus [1] and combinatory categorical grammars (CCG) [23].",
"A downside of these approaches is the reliance on linguistic resources/heuristics, making them language- and/or domain-specific.",
"Even though Reddy et al.",
"Reddy:2014aa claim that their approach requires less supervision than prior work, it still relies on many English-specific heuristics and hand-crafted features.",
"Also, their most accurate model uses a corpus of paraphrases to generalize to linguistic diversity.",
"Linguistic parsers can also be too slow for real-time applications.",
"In contrast, an RNN can detect entities in the question with high accuracy and low latency.",
"The only required resources are word embeddings and a set of questions with entity words tagged.",
"The former can be easily trained for any language/domain in an unsupervised fashion, given a large text corpus without annotations [19], [21].",
"The latter is a relatively simple annotation task that exists for many languages and domains, and it can also be synthetically generated.",
"Many researchers have explored similar techniques for general NLP tasks [8], such as named entity recognition [18], [13], sequence labeling [12], [7], part-of-speech tagging [16], [24], chunking [16].",
"Deep learning techniques have been studied extensively for constructing parallel neural networks for modeling a joint probability distribution for question-answer pairs [15], [26], [14], [20] and re-ranking answers output by a retrieval engine [22], [25].",
"These more complex approaches might be needed for general-purpose QA and sentence similarity, where one cannot make assumptions about the structure of the input or knowledge.",
"However, as noted in Section , first-order factoid questions can be represented by an entity and a relation type, and the answer is usually stored in a structured knowledge base.",
"Dong et al.",
"Dong:2015ab similarly assume that the answer to a question is at most two hops away from the target entity.",
"However, they do not propose how to obtain the target entity, since it is provided as part of their dataset.",
"Bordes et al.",
"Bordes:2014aa take advantage of the KB structure by projecting entities, relations, and subgraphs into the same latent space.",
"In addition to finding the target entity, the other key information to first-order QA is the relation type corresponding to the question.",
"Many researchers have shown that classifying the question into one of the pre-defined types (e.g., based on patterns [28] or support vector machines [6]) improves QA accuracy."
],
[
"Approach",
"(a) From Question to Structured Query.",
"Our approach relies on a knowledge base, containing a large set of facts, each one representing a binary [subject, relation, object] relationship.",
"Since we assume first-order questions, the answer can be retrieved from a single fact.",
"For instance, “How old is Sarah Michelle Gellar?” can be answered by the fact: [Sarah Michelle Gellar,bornOn,4/14/1977] The main idea is to dissect a first-order factoid natural-language question by converting it into a structured query: {entity “Sarah Michelle Gellar”, relation: bornOn}.",
"The process can be modularized into two machine learning problems, namely entity detection and relation prediction.",
"In the former, the objective is to tag each question word as either entity or not.",
"In the latter, the objective is to classify the question into one of the $K$ relation types.",
"We modeled both using an RNN.",
"We use a standard RNN architecture: Each word in the question passes through an embedding lookup layer $E$ , projecting the one-hot vector into a $d$ -dimensional vector $x_t$ .",
"A recurrent layer combines this input representation with the hidden layer representation from the previous word and applies a non-linear transformation to compute the hidden layer representation for the current word.",
"The hidden representation of the final recurrent layer is projected to the output space of $k$ dimensions and normalized into a probability distribution via soft-max.",
"In relation prediction, the question is classified into one of the 1837 classes (i.e., relation types in Freebase).",
"In the entity detection task, each word is classified as either entity or context (i.e., $k=2$ ).",
"Given a new question, we run the two RNN models to construct the structured query.",
"Once every question word is classified as entity (denoted by E) or context (denoted by C), we can extract entity phrase(s) by grouping consecutive entity words.",
"For example, for question “How old is Michelle Gellar”, the output of entity detection is [C C C E E], from which we can extract a single entity “Michelle Gellar”.",
"The output of relation prediction is bornOn.",
"The inferred structured query $q$ becomes the following: $\\lbrace \\textrm {\\emph {entityText}: } \\textrm {``michelle gellar^{\\prime \\prime }},\\textrm {\\emph {relation}: } \\texttt {bornOn}\\rbrace $ (b) Entity Linking.",
"The textual reference to the entity (entityText in $q$ ) needs to be linked to an actual entity node in our KB.",
"In order to achieve that, we build an inverted index $I_{\\textrm {entity}}$ that maps all $n$ -grams of an entity ($n\\in \\lbrace 1,2,3\\rbrace $ ) to the entity's alias text (e.g., name or title), each with an associated $TF$ -$IDF$ score.",
"We also map the exact text ($n=\\infty $ ) to be able to prioritize exact matches.",
"Following our running example, let us demonstrate how we construct $I_{\\textrm {entity}}$ .",
"Let us assume there is a node $e_i$ in our KB that refers to the actress “Sarah Michelle Gellar”.",
"The alias of this entity node is the name, which has three unigrams (“sarah”, “michelle”, “gellar”), two bigrams (“sarah michelle”, “michelle gellar”) and a single trigram (i.e., the entire name).",
"Each one of these $n$ -grams gets indexed in $I_{\\textrm {entity}}$ with $TF$ -$IDF$ weights.",
"Here is how the weights would be computed for unigram “sarah” and bigram “michelle gellar” ($\\Rightarrow $ denotes mapping): $&I_{\\textrm {entity}}(\\textrm {``sarah^{\\prime \\prime }}) \\Rightarrow \\lbrace \\textrm {node}: e_i, \\\\& \\textrm {score}: TF\\textrm {-}IDF(\\textrm {``sarah^{\\prime \\prime }}, \\textrm {``sarah michelle gellar^{\\prime \\prime }})\\rbrace \\\\&I_{\\textrm {entity}}(\\textrm {``michelle gellar^{\\prime \\prime }}) \\Rightarrow \\lbrace \\textrm {node}: e_i, \\\\&\\textrm {score}: TF\\textrm {-}IDF(\\textrm {``michelle gellar^{\\prime \\prime }}, \\\\& \\; \\;\\; \\;\\; \\;\\; \\;\\; \\;\\; \\;\\; \\;\\; \\;\\; \\;\\; \\;\\; \\;\\; \\;\\; \\;\\textrm {``sarah michelle gellar^{\\prime \\prime }})\\rbrace $ This is performed for every $n$ -gram ($n\\in \\lbrace 1,2,3,\\infty \\rbrace $ ) of every entity node in the KB.",
"Assuming there is an entity node, say $e_j$ , for the actress “Sarah Jessica Parker”, we would end up creating a second mapping from unigram “sarah”: $&I_{\\textrm {entity}}(\\textrm {``sarah^{\\prime \\prime }}) \\Rightarrow \\lbrace \\textrm {node}: e_j,\\\\& \\textrm {score}: TF\\textrm {-}IDF(\\textrm {``sarah^{\\prime \\prime }}, \\textrm {``sarah jessica parker^{\\prime \\prime }})\\rbrace $ In other words, “sarah” would be linked to both $e_i$ and $e_j$ , with corresponding $TF$ -$IDF$ weights.",
"Once the index $I_{\\textrm {entity}}$ is built, we can link entityText from the structured query (e.g., “michelle gellar”) to the intended entity in the KB (e.g., $e_i$ ).",
"Starting with $n=\\infty $ , we iterate over $n$ -grams of entityText and query $I_{\\textrm {entity}}$ , which returns all matching entities in the KB with associated $TF$ -$IDF$ relevance scores.",
"For each $n$ -gram, retrieved entities are appended to the candidate set $C$ .",
"We continue this process with decreasing value of $n$ (i.e., $n\\in \\lbrace \\infty ,3,2,1\\rbrace $ ) Early termination happens if $C$ is non-empty and $n$ is less than or equal to the number of tokens in entityText.",
"The latter criterion is to avoid cases where we find an exact match but there are also partial matches that might be more relevant: For “jurassic park”, for $n=\\infty $ , we get an exact match to the original movie “Jurassic Park”.",
"But we would also like to retrieve “Jurassic Park II” as a candidate entity, which is only possible if we keep processing until $n=2$ .",
"(c) Answer Selection.",
"Once we have a list of candidate entities $C$ , we use each candidate node $e_{\\textrm {cand}}$ as a starting point to reach candidate answers.",
"A graph reachability index $I_{\\textrm {reach}}$ is built for mapping each entity node $e$ to all nodes $e^{\\prime }$ that are reachable, with the associated path $p(e,e^{\\prime })$ .",
"For the purpose of the current approach, we limit our search to a single hop away, but this index can be easily expanded to support a wider search.",
"$&I_{reach}(e_i) \\Rightarrow \\\\&\\lbrace \\textrm {node:} e_{i_1}, \\textrm {text:} \\textrm {The Grudge}, \\textrm {path:} [\\texttt {actedIn}]\\rbrace \\\\&\\lbrace \\textrm {node:} e_{i_2}, \\textrm {text:} \\textrm {4/14/1977}, \\textrm {path:} [\\texttt {bornOn}]\\rbrace \\\\&\\lbrace \\textrm {node:} e_{i_3}, \\textrm {text:} \\textrm {F. Prinze}, \\textrm {path:} [\\texttt {marriedTo}]\\rbrace $ We use $I_{\\textrm {reach}}$ to retrieve all nodes $e^\\prime $ that are reachable from $e_{\\textrm {cand}}$ , where the path from is consistent with the predicted relation $r$ (i.e., $r\\in p(e_{\\textrm {cand}},e^\\prime $ )).",
"These are added to the candidate answer set $A$ .",
"For example, in the example above, node $e_{i_2}$ would have been added to the answer set $A$ , since the path [bornOn] matches the predicted relation in the structured query.",
"After repeating this process for each entity in $C$ , the highest-scored node in $A$ is our best answer to the question."
],
[
"Experimental Setup",
"Data.",
"Evaluation of RNN-QA was carried out on SimpleQuestions, which uses a subset of Freebase containing 17.8M million facts, 4M unique entities, and 7523 relation types.",
"Indexes $I_\\textrm {entity}$ and $I_\\textrm {reach}$ are built based on this knowledge base.",
"SimpleQuestions was built by [4] to serve as a larger and more diverse factoid QA dataset.75910/10845/21687 question-answer pairs for training/validation/test is an order of magnitude larger than comparable datasets.",
"Vocabulary size is 55K as opposed to around 3K for WebQuestions [1].",
"Freebase facts are sampled in a way to ensure a diverse set of questions, then given to human annotators to create questions from, and get labeled with corresponding entity and relation type.",
"There are a total of 1837 unique relation types that appear in SimpleQuestions.",
"Training.",
"We fixed the embedding layer based on the pre-trained 300-dimensional Google News embedding,word2vec.googlecode.com since the data size is too small for training embeddings.",
"Out-of-vocabulary words were assigned to a random vector (sampled from uniform distribution).",
"Parameters were learned via stochastic gradient descent, using categorical cross-entropy as objective.",
"In order to handle variable-length input, we limit the input to $N$ tokens and prepend a special pad word if input has fewer.Input length ($N$ ) was set to 36, the maximum number of tokens across training and validation splits.",
"We tried a variety of configurations for the RNN: four choices for the type of RNN layer (GRU or LSTM, bidirectional or not); depth from 1 to 3; and drop-out ratio from 0 to 0.5, yielding a total of 48 possible configurations.",
"For each possible setting, we trained the model on the training portion and used the validation portion to avoid over-fitting.",
"After running all 48 experiments, the most optimal setting was selected by micro-averaged F-score of predicted entities (entity detection) or accuracy (relation prediction) on the validation set.",
"We concluded that the optimal model is a 2-layer bidirectional LSTM (BiLSTM2) for entity detection and BiGRU2 for relation prediction.",
"Drop-out was 10% in both cases."
],
[
"Results",
"End-to-End QA.",
"For evaluation, we apply the relation prediction and entity detection models on each test question, yielding a structured query $q=\\lbrace entityText\\textrm {: }t_e, relation\\textrm {: }r\\rbrace $ (Section a).",
"Entity linking gives us a list of candidate entity nodes (Section b).",
"For each candidate entity $e_{\\textrm {cand}}$ , we can limit our relation choices to the set of unique relation types that some candidate entity $e_{\\textrm {cand}}$ is associated with.",
"This helps eliminate the artificial ambiguity due to overlapping relation types as well as the spurious ambiguity due to redundancies in a typical knowledge base.",
"Even though there are 1837 relation types in Freebase, the number of relation types that we need to consider per question (on average) drops to 36.",
"The highest-scored answer node is selected by finding the highest scored entity node $e$ that has an outward edge of type $r$ (Section c).",
"We follow Bordes et al.",
"Bordes:2015aa in comparing the predicted entity-relation pair to the ground truth.",
"A question is counted as correct if and only if the entity we select (i.e., $e$ ) and the relation we predict (i.e, $r$ ) match the ground truth.",
"Table REF summarizes end-to-end experimental results.",
"We use the best models based on validation set accuracy and compare it to three prior approaches: a specialized network architecture that explicitly memorizes facts [5], a network that learns how to convolve sequence of characters in the question [11], and a complex network with attention mechanisms to learn most important parts of the question [27].",
"Our approach outperforms the state of the art in accuracy (i.e., precision at top 1) by 11.9 points (15.6% relative).",
"Table: Top-1 accuracy on test portion of SimpleQuestions.",
"Ablation study on last three rows.Last three rows quantify the impact of each component via an ablation study, in which we replace either entity detection (ED) or relation prediction (RP) models with a naive baseline: (i) we assign the relation that appears most frequently in training data (i.e., bornOn), and/or (ii) we tag the entire question as an entity (and then perform the $n$ -gram entity linking).",
"Results confirm that RP is absolutely critical, since both datasets include a diverse and well-balanced set of relation types.",
"When we applied the naive ED baseline, our results drop significantly, but they are still comparable to prior results.",
"Given that most prior work do not use the network to detect entities, we can deduce that our RNN-based entity detection is the reason our approach performs so well.",
"Error Analysis.",
"In order to better understand the weaknesses of our approach, we performed a blame analysis: Among 2537 errors in the test set, 15% can be blamed on entity detection — the relation type was correctly predicted, but the detected entity did not match the ground truth.",
"The reverse is true for 48% cases.In remaining 37% incorrect answers, both models fail, so the blame is shared.",
"We manually labeled a sample of 50 instances from each blame scenario.",
"When entity detection is to blame, 20% was due to spelling inconsistencies between question and KB, which can be resolved with better text normalization during indexing (e.g., “la kings” refers to “Los Angeles Kings”).",
"We found 16% of the detected entities to be correct, even though it was not the same as the ground truth (e.g., either “New York” or “New York City” is correct in “what can do in new york?”); 18% are inherently ambiguous and need clarification (e.g., “where bin laden got killed?” might mean “Osama” or “Salem”).",
"When blame is on relation prediction, we found that the predicted relation is reasonable (albeit different than ground truth) 29% of the time (e.g., “what was nikola tesla known for” can be classified as profession or notable_for).",
"RNN-QA in Practice.",
"In addition to matching the state of the art in effectiveness, we also claimed that our simple architecture provides an efficient and modular solution.",
"We demonstrate this by applying our model (without any modifications) to the entertainment domain and deploying it to the Comcast X1 platform serving millions of customers every day.",
"Training data was generated synthetically based on an internal entertainment KB.",
"For evaluation, 295 unique question-answer pairs were randomly sampled from real usage logs of the platform.",
"We can draw two important conclusions from Table REF : First of all, we find that almost all of the user-generated natural-language questions (278/295$\\sim $ 95%) are first-order questions, supporting the significance of first-order QA as a task.",
"Second, we show that even if we simply use an open-sourced deep learning toolkit (keras.io) for implementation and limit the computational resources to 2 CPU cores per thread, RNN-QA answers 75% of questions correctly with very reasonable latency.",
"Table: Evaluation of RNN-QA on real questions from X platform."
],
[
"Conclusions and Future work",
"We described a simple yet effective approach for QA, focusing primarily on first-order factual questions.",
"Although we understand the benefit of exploring task-agnostic approaches that aim to capture semantics in a more general way (e.g., [17]), it is also important to acknowledge that there is no “one-size-fits-all” solution as of yet.",
"One of the main novelties of our work is to decompose the task into two subproblems, entity detection and relation prediction, and provide solutions for each in the form of a RNN.",
"In both cases, we have found that bidirectional networks are beneficial, and that two layers are sufficiently deep to balance the model's ability to fit versus its ability to generalize.",
"While an ablation study revealed the importance of both entity detection and relation prediction, we are hoping to further study the degree of which improvements in either component affect QA accuracy.",
"Drop-out was tuned to 10% based on validation accuracy.",
"While we have not implemented attention directly on our model, we can compare our results side by side on the same benchmark task against prior work with complex attention mechanisms (e.g., [27]).",
"Given the proven strength of attention mechanisms, we were surprised to find our simple approach to be clearly superior on SimpleQuestions.",
"Even though deep learning has opened the potential for more generic solutions, we believe that taking advantage of problem structure yields a more accurate and efficient solution.",
"While first-order QA might seem like a solved problem, there is clearly still room for improvement.",
"By revealing that 95% of real use cases fit into this paradigm, we hope to convince the reader that this is a valuable problem that requires more attention."
]
] | 1606.05029 |
[
[
"Evaluation of the Convolution Sums $\\underset{\\substack{\n {(l,m)\\in\\mathbb{N}_{0}^{2}} {\\alpha\\,l+\\beta\\,m=n} }\n }{\\sum}\\sigma(l)\\sigma(m)$, where $\\alpha\\beta=44,52$"
],
[
"Abstract The convolution sum, $\\underset{\\substack{ {(l,m)\\in\\mathbb{N}_{0}^{2}} {\\alpha\\,l+\\beta\\,m=n} } }{\\sum}\\sigma(l)\\sigma(m)$, where $\\alpha\\beta=44,52$, is evaluated for all natural numbers $n$.",
"We then use these convolution sums to determine formulae for the number of representations of a natural number by the octonary quadratic forms $a\\,(x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2})+ b\\,(x_{5}^{2} + x_{6}^{2} + x_{7}^{2} + x_{8}^{2})$, where $(a,b)= (1,11),(1,13)$."
],
[
"Introduction",
"The sets of natural numbers, non-negative integers, integers, rational numbers, real numbers and complex numbers, are denoted by $\\mathbb {N}$ , $\\mathbb {N}_{0}$ , $\\mathbb {Z}$ , $\\mathbb {Q}$ , $\\mathbb {R}$ and $\\mathbb {C}$ , respectively.",
"Suppose that $k,n\\in \\mathbb {N}$ .",
"We define the sum of positive divisors of $n$ to the power of $k$ , $\\sigma _{k}(n)$ , by $ \\sigma _{k}(n) = \\sum _{0<d|n}d^{k}.$ We write $\\sigma (n)$ as a synonym for $\\sigma _{1}(n)$ and we set $\\sigma _{k}(m)=0$ if $m\\notin \\mathbb {N}$ .",
"The convolution sum, $W_{(\\alpha ,\\beta )}(n)$ , is defined for all $\\alpha ,\\beta \\in \\mathbb {N}$ such that $\\alpha \\le \\beta $ as follows: $ W_{(\\alpha , \\beta )}(n) = \\sum _{\\begin{array}{c}{(l,m)\\in \\mathbb {N}_{0}^{2}} \\\\ {\\alpha \\,l+\\beta \\,m=n}\\end{array} }\\sigma (l)\\sigma (m).$ We write $W_{\\beta }(n)$ as a short hand for $W_{(1,\\beta )}(n)$ .",
"For those convolution sums $W_{(\\alpha , \\beta )}(n)$ that have so far been evaluated, the values of $(\\alpha ,\\beta )$ are given in introduction-table-1.",
"We evaluate the convolution sums for $(\\alpha ,\\beta )=(1,44)$ , $(4,11)$ , $(1,52)$ , $(4,13)$ , i.e., $\\alpha \\beta =44$ and $\\alpha \\beta =52$ .",
"The evaluation of these convolution sums have not been done yet according to introduction-table-1.",
"Let $a,b,c,d\\in \\mathbb {N}$ be such that $\\gcd (a,b)=1$ and $\\gcd (c,d)=1$ .",
"The convolution sums are generally used to determine explicit formulae for the number of representations of a positive integer $n$ by the octonary quadratic forms $ a\\,(x_{1}^{2} +x_{2}^{2} + x_{3}^{2} + x_{4}^{2})+ b\\,(x_{5}^{2} + x_{6}^{2} +x_{7}^{2} + x_{8}^{2}),$ and $ c\\,(x_{1}^{2} + x_{1}x_{2} + x_{2}^{2} + x_{3}^{2} + x_{3}x_{4} + x_{4}^{2})+ d\\,(x_{5}^{2} + x_{5}x_{6}+ x_{6}^{2} + x_{7}^{2} + x_{7}x_{8}+x_{8}^{2}),$ respectively.",
"We use the evaluated convolution sums and other known convolution sums to determine formulae for the number of representations of a positive integer $n$ by the octonary quadratic form introduction-eq-1 for which $(a,b)=(1,11),(1,13)$ .",
"These number of representations are also new according to introduction-table-rep2 which displays known explicit formulae for the number of representations of $n$ by the octonary form introduction-eq-1.",
"We have organized this paper as follows.",
"In [modularForms]Section modularForms we briefly discuss modular forms and define eta functions and convolution sums.",
"Then in [convolution4452]Section convolution4452 we discuss our main results on the evaluation of the convolution sums; the main results on the formulae for the number of representations of a positive integer $n$ are given in [representations4452]Section representations4452.",
"We use a software for symbolic scientific computation to obtain the results of this paper.",
"The open source software packages GiNaC, Maxima, REDUCE, SAGE and the commercial software package MAPLE build this software."
],
[
"Preliminaries",
"We consider the upper half-plane, $\\mathbb {H}=\\lbrace z\\in \\mathbb {C}~ | ~\\text{Im}(z)>0\\rbrace $ , and the group $G=\\text{SL}_{2}(\\mathbb {R})$ of $2\\times 2$ -matrices $\\left({\\begin{matrix} a & b \\\\ c &d \\end{matrix}}\\right)$ such that $a,b,c,d\\in \\mathbb {R}$ and $ad-bc=1$ .",
"Let $\\Gamma =\\text{SL}_{2}(\\mathbb {Z})$ be a subset of $G$ and let $N\\in \\mathbb {N}$ .",
"Then $\\Gamma (N) & = \\bigl \\lbrace ~\\left({\\begin{matrix} a & b \\\\ c &d \\end{matrix}}\\right)\\in \\text{SL}_{2}(\\mathbb {Z})~ |~\\left({\\begin{matrix} a & b \\\\ c &d\\end{matrix}}\\right)\\equiv \\left({\\begin{matrix} 1 & 0 \\\\ 0 & 1\\end{matrix}}\\right) \\pmod {N} ~\\bigr \\rbrace $ is a subgroup of $\\Gamma $ .",
"The subgroup $\\Gamma (N)$ is called a principal congruence subgroup of level N. If a subgroup $H$ of $G$ contains $\\Gamma (N)$ , then it is a congruence subgroup of level N. For our purpose we consider the congruence subgroup $\\Gamma _{0}(N) & = \\bigl \\lbrace ~\\left({\\begin{matrix} a & b \\\\ c &d \\end{matrix}}\\right)\\in \\text{SL}_{2}(\\mathbb {Z})~ | ~c\\equiv 0 \\pmod {N} ~\\bigr \\rbrace .$ Let $k\\in \\mathbb {Z}, \\gamma \\in \\Gamma $ and $f^{[\\gamma ]_{k}} :\\mathbb {H}\\cup \\mathbb {Q}\\cup \\lbrace \\infty \\rbrace \\rightarrow \\mathbb {C}\\cup \\lbrace \\infty \\rbrace $ be the function whose value at $z$ is $f^{[\\gamma ]_{k}}(z)=(cz+d)^{-k}f(\\gamma (z))$ .",
"The following definition is based on N. Koblitz's textbook [15].",
"Definition 2.1 Let $N\\in \\mathbb {N}$ , $k\\in \\mathbb {Z}$ , $f$ be a meromorphic function on $\\mathbb {H}$ and $\\Gamma ^{\\prime }\\subset \\Gamma $ be a congruence subgroup of level $N$ .",
"$f$ is a modular function of weight $k$ for $\\Gamma ^{\\prime }$ if $f^{[\\gamma ]_{k}}=f$ for all $\\gamma \\in \\Gamma ^{\\prime }$ , for any $\\delta \\in \\Gamma $ it holds that $f^{[\\delta ]_{k}}(z)$ can be expressed in the form $\\underset{n\\in \\mathbb {Z}}{\\sum }a_{n}e^{\\frac{2\\pi i z n}{N}}$ , wherein $a_{n}\\ne 0$ for finitely many $n\\in \\mathbb {Z}$ such that $n<0$ .",
"$f$ is a modular form of weight $k$ for $\\Gamma ^{\\prime }$ if $f$ is a modular function of weight $k$ for $\\Gamma ^{\\prime }$ , $f$ is holomorphic on $\\mathbb {H}$ , $a_{n}=0$ for all $\\delta \\in \\Gamma $ and for all $n\\in \\mathbb {Z}$ such that $n<0$ .",
"$f$ is a cusp form of weight $k$ for $\\Gamma ^{\\prime }$ if $f$ is a modular form of weight $k$ for $\\Gamma ^{\\prime }$ , $a_{0}=0$ for all $\\delta \\in \\Gamma $ .",
"Let $k,N\\in \\mathbb {N}$ .",
"We denote by $\\mbox{$ I\\!\\!",
"M$}_{k}(\\Gamma _{0}(N))$ be the space of modular forms of weight $k$ for $\\Gamma _{0}(N)$ , ${S}_{k}(\\Gamma _{0}(N))$ the subspace of cusp forms of weight $k$ for $\\Gamma _{0}(N)$ , and $\\mbox{$ I\\!\\!",
"E$}_{k}(\\Gamma _{0}(N))$ the subspace of Eisenstein forms of weight $k$ for $\\Gamma _{0}(N)$ .",
"In W. A. Stein's book (online version) [25] it is shown that $\\mbox{$ I\\!\\!",
"M$}_{k}(\\Gamma _{0}(N)) =\\mbox{$ I\\!\\!",
"E$}_{k}(\\Gamma _{0}(N))\\oplus {S}_{k}(\\Gamma _{0}(N))$ .",
"According to Section 5.3 of W. A. Stein's book [25] $E_{k}(q) = 1 - \\frac{2k}{B_{k}}\\,\\underset{n=1}{\\overset{\\infty }{\\sum }}\\,\\sigma _{k-1}(n)\\,q^{n}$ , where $B_{k}$ are the Bernoulli numbers, if the primitive Dirichlet characters are trivial and $2\\le k$ is even.",
"We only consider trivial primitive Dirichlet characters and $4\\le k$ even in the sequel.",
"Based on this consideration Theorems 5.8 and 5.9 in Section 5.3 of [25] also hold."
],
[
"Eta Functions",
"On the upper half-plane $\\mathbb {H}$ the Dedekind eta function, $\\eta (z)$ , is defined by $\\eta (z) = e^{\\frac{2\\pi i z}{24}}\\overset{\\infty }{\\underset{n=1}{\\prod }}(1 -e^{2\\pi i n z})$ .",
"When we set $q=e^{2\\pi i z}$ , then $\\eta (z) = q^{\\frac{1}{24}}\\overset{\\infty }{\\underset{n=1}{\\prod }}(1 - q^{n})= q^{\\frac{1}{24}} F(q),\\qquad \\text{ where } F(q)=\\overset{\\infty }{\\underset{n=1}{\\prod }}(1 - q^{n}).$ L. J. P. Kilford's book [14] and G. Köhler's book [16] have a proof of the following theorem which we will apply to determine eta functions which belong to $\\mbox{$ I\\!\\!",
"M$}_{k}(\\Gamma _{0}(N))$ , and particularly those eta functions that belong to ${S}_{k}(\\Gamma _{0}(N))$ .",
"As noted by A. Alaca et al.",
"[1] credit to this theorem also goes to M. Newman [20], [21].",
"Theorem 2.2 (M. Newman and G. Ligozat) Let $N\\in \\mathbb {N}$ and let $f(z)=\\overset{}{\\underset{1\\le \\delta |N}{\\prod }}\\eta ^{r_{\\delta }}(\\delta z)$ be an eta function which satisfies the following conditions: Table: NO_CAPTION(v) for each positive divisor $d$ of $N$ , the inequality $\\overset{}{\\underset{1\\le \\delta |N}{\\sum }}\\frac{\\text{gcd}(\\delta ,d)^{2}}{\\delta } r_{\\delta } \\ge 0$ holds.",
"Then $f(z)\\in \\mbox{$ I\\!\\!",
"M$}_{k}(\\Gamma _{0}(N))$ .",
"If (v) is replaced by (v') for each positive divisor $d$ of $N$ the inequality $\\overset{}{\\underset{1\\le \\delta |N}{\\sum }}\\frac{\\text{gcd}(\\delta ,d)^{2}}{\\delta } r_{\\delta } > 0$ holds then $f(z)\\in {S}_{k}(\\Gamma _{0}(N))$ ."
],
[
"Evaluating $W_{(\\alpha , \\beta )}(n)$",
"Let $\\alpha ,\\beta \\in \\mathbb {N}$ be such that $\\alpha \\le \\beta $ .",
"We let the convolution sum, $W_{(\\alpha ,\\beta )}(n)$ , be defined as in def-convolutionsum.",
"Following the observation by A. Alaca et al.",
"[2], we assume that $\\text{gcd}(\\alpha ,\\beta )=1$ .",
"Suppose that $q\\in \\mathbb {C}$ is such that $|q|<1$ .",
"We define the Eisenstein series $L(q)$ and $M(q)$ by $L(q) = E_{2}(q) = 1-24\\,\\sum _{n=1}^{\\infty }\\sigma (n)q^{n}, \\\\M(q) = E_{4}(q) = 1 + 240\\,\\sum _{n=1}^{\\infty }\\sigma _{3}(n)q^{n}.$ The following two results whose proofs are given by A. Alaca et al.",
"[1] are essential for the sequel of this work Lemma 2.3 Let $\\alpha , \\beta \\in \\mathbb {N}$ .",
"Then $( \\alpha \\, L(q^{\\alpha }) - \\beta \\, L(q^{\\beta }) )^{2}\\in \\mbox{$ I\\!\\!",
"M$}_{4}(\\Gamma _{0}(\\alpha \\beta )).$ Theorem 2.4 Let $\\alpha ,\\beta \\in \\mathbb {N}$ be such that $\\alpha $ and $\\beta $ are relatively prime and $\\alpha < \\beta $ .",
"Then $( \\alpha \\, L(q^{\\alpha }) - \\beta \\, L(q^{\\beta }) )^{2} = &(\\alpha - \\beta )^{2}+ \\sum _{n=1}^{\\infty }\\biggl (\\ 240\\,\\alpha ^{2}\\,\\sigma _{3}(\\frac{n}{\\alpha })+ 240\\,\\beta ^{2}\\,\\sigma _{3}(\\frac{n}{\\beta }) \\\\ &+ 48\\,\\alpha \\,(\\beta -6n)\\,\\sigma (\\frac{n}{\\alpha })+ 48\\,\\beta \\,(\\alpha -6n)\\,\\sigma (\\frac{n}{\\beta }) \\\\ &- 1152\\,\\alpha \\beta \\,W_{(\\alpha ,\\beta )}(n)\\,\\biggr )q^{n}.$"
],
[
"Evaluation of the convolution sums $W_{(\\alpha ,\\beta )}(n)$ , where\n{{formula:cee1ff64-c1bb-4480-ad5f-eefe4b9834a7}}",
"We give explicit formulae for the convolution sums $W_{(1,44)}(n)$ , $W_{(4,11)}(n)$ , $W_{(1,52}(n)$ and $W_{(4,13}(n)$ ."
],
[
"Bases for $\\mbox{$ I\\!\\! E$}_{4}(\\Gamma _{0}(\\alpha \\beta ))$ and\n{{formula:b93257aa-f7e4-4954-8c32-4d0e7dca37f0}} with {{formula:ca6a6fa6-2978-44bf-8230-f9deab3f9fff}}",
"We apply the dimension formulae for the space of Eisenstein forms and the space of cusp forms in T. Miyake's book [19] or W. A. Stein's book [25] to compute $\\text{dim}(\\mbox{$ I\\!\\!",
"E$}_{4}(\\Gamma _{0}(44)))=\\text{dim}(\\mbox{$ I\\!\\!",
"E$}_{4}(\\Gamma _{0}(52)))=6$ , $\\text{dim}({S}_{4}(\\Gamma _{0}(44))=15$ and $\\text{dim}({S}_{4}(\\Gamma _{0}(52))=18$ .",
"Let $D(44)=\\lbrace 1,2,4,11,22,44\\rbrace $ and $D(52)=\\lbrace 1,2,4,13,26,52\\rbrace $ be the sets of positive divisors of 44 and 52, respectively.",
"We apply ligozattheorem $(i)-(v^{\\prime })$ to determine as many elements of ${S}_{4}(\\Gamma _{0}(44))$ and ${S}_{4}(\\Gamma _{0}(52))$ as possible.",
"From these elements we then determine the basis elements.",
"Theorem 3.1 The sets ${B}_{E,44}=\\lbrace \\,M(q^{t})\\,\\mid ~t\\in D(44)\\,\\rbrace $ and ${B}_{E,52}=\\lbrace \\, M(q^{t})\\,\\mid ~ t\\in D(52)\\,\\rbrace $ are bases of $\\mbox{$ I\\!\\!",
"E$}_{4}(\\Gamma _{0}(44))$ and $\\mbox{$ I\\!\\!",
"E$}_{4}(\\Gamma _{0}(52))$ , respectively.",
"Let $1\\le i\\le 15$ and $1\\le j\\le 18$ be positive integers.",
"Let $\\delta _{1}\\in D(44)$ and $(r(i,\\delta _{1}))_{i,\\delta _{1}}$ be the convolutionSums-411-table of the powers of $\\eta (\\delta _{1} z)$ .",
"Let $\\delta _{2}\\in D(52)$ and $(r(j,\\delta _{2}))_{j,\\delta _{2}}$ be the convolutionSums-413-table of the powers of $\\eta (\\delta _{2} z)$ .",
"Let furthermore $A_{i}(q)=\\underset{\\delta _{1}|44}{\\prod }\\eta ^{r(i,\\delta _{1})}(\\delta _{1}z)$ and $B_{j}(q)=\\underset{\\delta _{2}|52}{\\prod }\\eta ^{r(j,\\delta _{2})}(\\delta _{2}z)$ be selected elements of ${S}_{4}(\\Gamma _{0}(44))$ and ${S}_{4}(\\Gamma _{0}(52))$ , respectively.",
"Then the sets ${B}_{S,44}=\\lbrace \\,A_{i}(q)\\,\\mid ~ 1\\le i\\le 15\\,\\rbrace $ and ${B}_{S,52}=\\lbrace \\,B_{j}(q)\\,\\mid ~ 1\\le j\\le 18\\,\\rbrace $ are bases of ${S}_{4}(\\Gamma _{0}(44))$ and ${S}_{4}(\\Gamma _{0}(52))$ , repectively.",
"The sets ${B}_{M,44}={B}_{E,44}\\cup {B}_{S,44}$ and ${B}_{M,52}={B}_{E,52}\\cup {B}_{S,52}$ constitute bases of $\\mbox{$ I\\!\\!",
"M$}_{4}(\\Gamma _{0}(44))$ and $\\mbox{$ I\\!\\!",
"M$}_{4}(\\Gamma _{0}(52))$ , respectively.",
"For $1\\le i\\le 15$ and $1\\le j\\le 18$ the eta quotients $A_{i}(q)$ and $B_{j}(q)$ can be expressed in the form $\\underset{n=1}{\\overset{\\infty }{\\sum }}a_{i}(n)q^{n}$ and $\\underset{n=1}{\\overset{\\infty }{\\sum }}b_{j}(n)q^{n}$ , respectively.",
"We only prove the case $\\alpha \\beta =44$ .",
"The case $\\alpha \\beta =52$ is proved similarly.",
"When we apply Theorem 5.8 in Section 5.3 of W. A. Stein [25], it follows that $M(q^{t})$ belongs to $\\mbox{$ I\\!\\!",
"M$}_{4}(\\Gamma _{0}(t))$ for each $t\\in D(44)$ .",
"Since $\\mbox{$ I\\!\\!",
"E$}_{4}(\\Gamma _{0}(44))$ has a finite dimension, it is sufficient to show that the set of $M(q^{t})$ such that $t\\in D(44)$ is linearly independent.",
"Suppose that $x_{t}\\in \\mathbb {C}$ with $t|44$ .",
"Then $\\underset{t|44}{\\sum }x_{t}\\,M(q^{t})=\\underset{t|44}{\\sum }x_{t}+240\\,\\underset{n\\ge 1}{\\sum }\\biggl (\\underset{t|44}{\\sum }x_{t}\\sigma _{3}(\\frac{n}{t})\\biggr )q^{n}=0.$ We compare the coefficients of $q^{n}$ for $n\\in D(44)$ to obtain the following homogeneous system of 6 equations in 6 unknowns: $\\underset{u|44}{\\sum }\\sigma _{3}(\\frac{t}{u})x_{u}=0,\\qquad t\\in D(44).$ The matrix of this homogeneous system of equations is triangular with positive integer values, 1, on the diagonal.",
"Hence, the solution is $x_{t}=0$ for all $t\\in D(44)$ .",
"Therefore, the set ${B}_{E}$ is linearly independent and hence is a basis of $\\mbox{$ I\\!\\!",
"E$}_{4}(\\Gamma _{0}(44))$ .",
"As mentioned above, the $A_{i}(q)$ with $1\\le i\\le 15$ are obtained from an exhaustive search using ligozattheorem $(i)-(v^{\\prime })$ .",
"Hence, each $A_{i}(q)$ is in the space ${S}_{4}(\\Gamma _{0}(44))$ .",
"Since the dimension of ${S}_{4}(\\Gamma _{0}(44))$ is 15, it is sufficient to show that the set $\\lbrace \\,A_{i}(q)\\mid 1\\le i\\le 15\\rbrace $ is linearly independent.",
"For that suppose that $x_{i}\\in \\mathbb {C}$ and $\\underset{i=1}{\\overset{15}{\\sum }}x_{i}\\,A_{i}(q)=0$ .",
"Then $\\underset{i=1}{\\overset{15}{\\sum }}x_{i}\\,A_{i}(q)= \\underset{n=1}{\\overset{\\infty }{\\sum }}(\\,\\underset{i=1}{\\overset{15}{\\sum }}x_{i}\\,a_{i}(n)\\,)q^{n} = 0$ which gives the following homogeneous system of equations in 15 unknowns $ \\underset{i=1}{\\overset{15}{\\sum }}\\,a_{i}(n)\\,x_{i}= 0,\\qquad 1\\le n\\le 15.$ A computation using a software package for (symbolic) scientific computation shows that the determinant of the matrix of this homogeneous system of equations is non-zero.",
"So, $x_{i}=0$ for all $1\\le i\\le 15$ .",
"Hence, the set $\\lbrace \\, A_{i}(q)\\mid 1\\le i\\le 15\\,\\rbrace $ is linearly independent and therefore a basis of ${S}_{4}(\\Gamma _{0}(44))$ .",
"Since $\\mbox{$ I\\!\\!",
"M$}_{4}(\\Gamma _{0}(44))=\\mbox{$ I\\!\\!",
"E$}_{4}(\\Gamma _{0}(44))\\oplus {S}_{4}(\\Gamma _{0}(44))$ , the result follows from (a) and (b).",
"We observe that the basis elements $A_{i}(q)$ , $1\\le i\\le 5$ , come from ${S}_{4}(\\Gamma _{0}(22))$ which is the space of cusp forms necessary for the evaluation of the convolution sums $W_{22}(n)$ and $W_{(2,11)}(n)$ given by A. Alaca et al.",
"[1].",
"The element $A_{2}(q)$ is inherited from ${S}_{4}(\\Gamma _{0}(11))$ which is part of $\\mbox{$ I\\!\\!",
"M$}_{4}(\\Gamma _{0}(11))$ ; the convolution sum $W_{11}(n)$ is evatuated by E. Royer [24].",
"$A_{2i}(q)=A_{i}(q^{2})$ , for $i = 2,3,4,5$ .",
"Therefore, $a_{2i}(n)=a_{i}(\\frac{n}{2})$ , for $i = 2,3,4,5$ .",
"$B_{j}(q)$ , $1\\le j\\le 7$ , $B_{15}(q)$ and $B_{17}(q)$ are imported from ${S}_{4}(\\Gamma _{0}(26))$ which is the space of cusp forms required for the evaluation of the convolution sums $W_{26}(n)$ and $W_{(2,13)}(n)$ given by A. Alaca et al.",
"[1].",
"$B_{2j}(q)=B_{j}(q^{2})$ , for $4\\le j\\le 7$ , $B_{16}(q)=B_{15}(q^{2})$ and $B_{18}(q)=B_{17}(q^{2})$ .",
"Consequently, $b_{2j}(n)=b_{j}(\\frac{n}{2})$ , for $4\\le j\\le 7$ , $b_{16}(n)=b_{15}(\\frac{n}{2})$ and $b_{18}(n)=b_{17}(\\frac{n}{2})$ .",
"The above observation is based on the fact that $\\mbox{$ I\\!\\!",
"M$}_{4}(\\Gamma _{0}(11)) \\subset \\mbox{$ I\\!\\!",
"M$}_{4}(\\Gamma _{0}(22)) \\subset \\mbox{$ I\\!\\!",
"M$}_{4}(\\Gamma _{0}(44))\\quad \\text{and} \\\\\\mbox{$ I\\!\\!",
"M$}_{4}(\\Gamma _{0}(13)) \\subset \\mbox{$ I\\!\\!",
"M$}_{4}(\\Gamma _{0}(26)) \\subset \\mbox{$ I\\!\\!",
"M$}_{4}(\\Gamma _{0}(52)).$ As mentioned in (o1) above, the eta quotient $A_{2}(q)$ is a basis element of ${S}_{4}(\\Gamma _{0}(11))$ .",
"Hence, basis elements of ${S}_{4}(\\Gamma _{0}(11))$ can be determined using ligozattheorem.",
"There is no basis element of ${S}_{4}(\\Gamma _{0}(13))$ in the space of cusp forms ${S}_{4}(\\Gamma _{0}(26))$ ; this is an indication that there is no basis element of ${S}_{4}(\\Gamma _{0}(13))$ that can be determined using ligozattheorem."
],
[
"Evaluation of $W_{(\\alpha ,\\beta )}(n)$ when {{formula:aa567ae7-02ee-4f20-b872-26086a2d6851}}",
"Lemma 3.2 We have $( L(q) - 44\\, L(q^{44}))^{2}= 1849 + \\sum _{n=1}^{\\infty }\\biggl (\\,\\frac{124464}{61}\\,\\sigma _{3}(n)- \\frac{577662336}{40565}\\,\\sigma _{3}(\\frac{n}{2}) \\\\+ \\frac{68986368}{5795}\\,\\sigma _{3}(\\frac{n}{4})- \\frac{174240}{61}\\,\\sigma _{3}(\\frac{n}{11})+ \\frac{62064288}{5795}\\,\\sigma _{3}(\\frac{n}{22})+ \\frac{2525690112}{5795}\\,\\sigma _{3}(\\frac{n}{44}) \\\\+ \\frac{1440}{61}\\,a_{1}(n)- \\frac{82927872}{5795}\\,a_{2}(n)- \\frac{887345568}{5795}\\,a_{3}(n)- \\frac{1676429568}{5795}\\,a_{4}(n) \\\\- \\frac{2804007168}{5795}\\,a_{5}(n)+ \\frac{3753380736}{5795}\\,a_{6}(n)- \\frac{13356288}{19}\\,a_{7}(n)+ \\frac{4226609664}{5795}\\,a_{8}(n) \\\\- \\frac{633600}{19}\\,a_{9}(n)- \\frac{527332608}{1159}\\,a_{10}(n)+ \\frac{7679232}{19}\\,a_{11}(n)- \\frac{15231744}{95}\\,a_{12}(n) \\\\- \\frac{131079168}{95}\\,a_{13}(n)+ \\frac{317952}{19}\\,a_{14}(n)- \\frac{12595968}{95}\\,a_{15}(n)\\biggr )q^{n}, $ $( 4\\,L(q^{4}) - 11\\, L(q^{11}))^{2}= 49 + \\sum _{n=1}^{\\infty }\\biggl (\\,- \\frac{110880}{61}\\,\\sigma _{3}(n)+ \\frac{80121888}{5795}\\,\\sigma _{3}(\\frac{n}{2}) \\\\- \\frac{48338688}{5795}\\,\\sigma _{3}(\\frac{n}{4})+ \\frac{1817904}{61}\\,\\sigma _{3}(\\frac{n}{11})- \\frac{98480448}{5795}\\,\\sigma _{3}(\\frac{n}{22})- \\frac{27320832}{5795}\\,\\sigma _{3}(\\frac{n}{44}) \\\\+ \\frac{110880}{61}\\,a_{1}(n)+ \\frac{174857472}{5795}\\,a_{2}(n)+ \\frac{1169427168}{5795}\\,a_{3}(n)+ \\frac{2114189568}{5795}\\,a_{4}(n) \\\\+ \\frac{3025513728}{5795}\\,a_{5}(n)- \\frac{3511080576}{5795}\\,a_{6}(n)+ \\frac{13318272}{19}\\,a_{7}(n)- \\frac{3641762304}{5795}\\,a_{8}(n) \\\\+ \\frac{633600}{19}\\,a_{9}(n)+ \\frac{663913728}{1159}\\,a_{10}(n)- \\frac{7679232}{19}\\,a_{11}(n)+ \\frac{15231744}{95}\\,a_{12}(n) \\\\+ \\frac{131079168}{95}\\,a_{13}(n)- \\frac{317952}{19}\\,a_{14}(n)+ \\frac{12595968}{95}\\,a_{15}(n)\\,\\biggr )q^{n},$ $( L(q) - 52\\, L(q^{52}))^{2}= 2601 + \\sum _{n=1}^{\\infty }\\biggl (\\,\\frac{6109008}{1243}\\,\\sigma _{3}(n)- \\frac{456504084816}{6064597}\\,\\sigma _{3}(\\frac{n}{2}) \\\\+ \\frac{254592}{41}\\,\\sigma _{3}(\\frac{n}{4})- \\frac{7361952}{1243}\\,\\sigma _{3}(\\frac{n}{13})- \\frac{4829528827344}{6064597}\\,\\sigma _{3}(\\frac{n}{26}) \\\\+ \\frac{434738304}{41}\\,\\sigma _{3}(\\frac{n}{52})- \\frac{3066144}{1243}\\,b_{1}(n)+ \\frac{498157179048}{6064597}\\,b_{2}(n) \\\\+ \\frac{927327070704}{6064597}\\,b_{3}(n)- \\frac{442577500560}{6064597}\\,b_{4}(n)- \\frac{8530413669648}{6064597}\\,b_{5}(n) \\\\- \\frac{10161699732288}{6064597}\\,b_{6}(n)- \\frac{10388366352}{1243}\\,b_{7}(n)+ \\frac{1040832}{41}\\,b_{8}(n) \\\\+ 7488\\,b_{9}(n)+ \\frac{329100929664}{147917}\\,b_{10}(n)+ 27456\\,b_{11}(n) \\\\- \\frac{15249288510144}{6064597}\\,b_{12}(n)+ 17472\\,b_{13}(n)+ \\frac{47009664}{41}\\,b_{14}(n) \\\\- \\frac{25166713896}{551327}\\,b_{15}(n)+ \\frac{4167031826880}{6064597}\\,b_{16}(n)- \\frac{126425023920}{6064597}\\,b_{17}(n) \\\\+ \\frac{868608}{41}\\,b_{18}(n)\\, \\biggr )q^{n}, $ $(4\\, L(q^{4}) - 13\\, L(q^{13}))^{2}= 81 + \\sum _{n=1}^{\\infty }\\biggl (\\,\\frac{3066144}{1243}\\,\\sigma _{3}(n)- \\frac{240061230672}{6064597}\\,\\sigma _{3}(\\frac{n}{2}) \\\\+ \\frac{139392}{41}\\,\\sigma _{3}(\\frac{n}{4})+ \\frac{45798672}{1243}\\,\\sigma _{3}(\\frac{n}{13})- \\frac{53922031824}{6064597}\\,\\sigma _{3}(\\frac{n}{26}) \\\\+ \\frac{20290176}{41}\\,\\sigma _{3}(\\frac{n}{52})- \\frac{3066144}{1243}\\,b_{1}(n)+ \\frac{212735819880}{6064597}\\,b_{2}(n) \\\\+ \\frac{251848851024}{6064597}\\,b_{3}(n)- \\frac{400561037808}{6064597}\\,b_{4}(n)- \\frac{5152459820400}{6064597}\\,b_{5}(n) \\\\- \\frac{5408748312192}{6064597}\\,b_{6}(n)- \\frac{5489355312}{1243}\\,b_{7}(n)+ \\frac{150336}{41}\\,b_{8}(n) \\\\- 7488\\,b_{9}(n)+ \\frac{151016538432}{147917}\\,b_{10}(n)- 27456\\,b_{11}(n) \\\\- \\frac{8224832431680}{6064597}\\,b_{12}(n)- 17472\\,b_{13}(n)- \\frac{544896}{41}\\,b_{14}(n) \\\\- \\frac{11115614088}{551327}\\,b_{15}(n)+ \\frac{2056953609600}{6064597}\\,b_{16}(n)- \\frac{64745693328}{6064597}\\,b_{17}(n) \\\\- \\frac{2304}{41}\\,b_{18}(n)\\,\\biggr )q^{n}.$ We just prove the case $( 4\\,L(q^{4}) - 11\\,L(q^{11}))^{2}$ .",
"The other cases are proved similarly.",
"From [evalConvolClass-lema-1]Lemma evalConvolClass-lema-1 it follows that $( 4\\, L(q^{4}) - 11\\, L(q^{11}) )^{2}\\in \\mbox{$ I\\!\\!",
"M$}_{4}(\\Gamma _{0}(44))$ .",
"Hence, by basisCusp4452 (c), there exist $X_{\\delta },Y_{j}\\in \\mathbb {C},1\\le j\\le 15\\text{ and } \\delta \\in D(44)$ , such that $( 4\\, L(q^{4}) - 11\\, L(q^{11}) )^{2} & =\\sum _{\\delta |44}X_{\\delta }\\,M(q^{\\delta }) + \\sum _{j=1}^{15}\\,Y_{j}\\,A_{j}(q) \\\\~ & = \\sum _{\\delta |44}X_{\\delta } + \\sum _{n=1}^{\\infty }\\biggl (\\,240\\,\\sum _{\\delta |44}\\,\\sigma _{3}(\\frac{n}{\\delta })\\,X_{\\delta } +\\sum _{j=1}^{m_{S}}\\,a_{j}(n)\\,Y_{j}\\, \\biggr )q^{n}.",
"$ We compare the right hand side of convolution4452-eqn-0a with that of evalConvolClass-eqn-11 when we have set $(\\alpha ,\\beta )=(4,11)$ to obtain $\\sum _{n=1}^{\\infty }\\biggl (\\,240\\,\\sum _{\\delta |44}\\,\\sigma _{3}(\\frac{n}{\\delta })\\,X_{\\delta } +\\sum _{j=1}^{15}\\,a_{j}(n)\\,Y_{j}\\, \\biggr )q^{n} = \\sum _{n=1}^{\\infty }\\biggl (\\ 3840\\,\\sigma _{3}(\\frac{n}{4}) + 29040\\,\\sigma _{3}(\\frac{n}{11}) \\\\+ 192\\,(11 - 6\\,n)\\,\\sigma (\\frac{n}{4}) + 528\\,(4 - 6\\,n)\\,\\sigma (\\frac{n}{11})- 50688\\,W_{(4,11)}(n)\\,\\biggr )q^{n}.$ We then take the coefficients of $q^{n}$ for which $n$ is in $\\lbrace \\,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,20,22,44\\, \\rbrace .$ to obtain a system of linear equations whose resolution using a software package for symbolic scientific computation yields the unique solution which determines the values of the unknowns $X_{\\delta }$ for all $\\delta \\in D(44)$ and the values of the unkowns $Y_{j}$ for all $1\\le j\\le 15$ .",
"Therefore, we get the stated result.",
"Our main result of this section will now be stated and proved.",
"Theorem 3.3 Let $n$ be a positive integer.",
"Then $W_{(1,44)}(n) = &- \\frac{13}{366}\\,\\sigma _{3}(n)+ \\frac{501443}{1784860}\\,\\sigma _{3}(\\frac{n}{2})- \\frac{1361}{5795}\\,\\sigma _{3}(\\frac{n}{4})+ \\frac{55}{976}\\,\\sigma _{3}(\\frac{n}{11}) \\\\ &- \\frac{19591}{92720}\\,\\sigma _{3}(\\frac{n}{22})+ \\frac{9878}{17385}\\,\\sigma _{3}(\\frac{n}{44})+ (\\frac{1}{24}-\\frac{1}{176}n)\\sigma (n) \\\\ &+ (\\frac{1}{24}-\\frac{1}{4}n)\\sigma (\\frac{n}{44})- \\frac{5}{10736}\\,a_{1}(n)+ \\frac{35993}{127490}\\,a_{2}(n) \\\\ &+ \\frac{3081061}{1019920}\\,a_{3}(n)+ \\frac{66147}{11590}\\,a_{4}(n)+ \\frac{1217017}{127490}\\,a_{5}(n)- \\frac{3258143}{254980}\\,a_{6}(n) \\\\ &+ \\frac{527}{38}\\,a_{7}(n)- \\frac{917233}{63745}\\,a_{8}(n)+ \\frac{25}{38}\\,a_{9}(n)+ \\frac{20807}{2318}\\,a_{10}(n) \\\\ &- \\frac{303}{38}\\,a_{11}(n)+ \\frac{601}{190}\\,a_{12}(n)+ \\frac{2586}{95}\\,a_{13}(n)- \\frac{69}{209}\\,a_{14}(n) + \\frac{497}{190}\\,a_{15}(n),$ $W_{(4,11)}(n) = &\\frac{35}{976}\\,\\sigma _{3}(n)- \\frac{25291}{92720}\\,\\sigma _{3}(\\frac{n}{2})+ \\frac{4178}{17385}\\,\\sigma _{3}(\\frac{n}{4})- \\frac{11}{732}\\,\\sigma _{3}(\\frac{n}{11}) \\\\ &+ \\frac{15543}{46360}\\,\\sigma _{3}(\\frac{n}{22})+ \\frac{539}{5795}\\,\\sigma _{3}(\\frac{n}{44})+ (\\frac{1}{24}-\\frac{1}{44}n)\\sigma (\\frac{n}{4}) \\\\ &+ (\\frac{1}{24}-\\frac{1}{16}n)\\sigma (\\frac{n}{11})- \\frac{35}{976}\\,a_{1}(n)- \\frac{75893}{127490}\\,a_{2}(n) \\\\ &- \\frac{4060511}{1019920}\\,a_{3}(n)- \\frac{917617}{127490}\\,a_{4}(n)- \\frac{1313157}{127490}\\,a_{5}(n)+ \\frac{3047813}{254980}\\,a_{6}(n) \\\\ &- \\frac{1051}{76}\\,a_{7}(n)+ \\frac{790313}{63745}\\,a_{8}(n)- \\frac{25}{38}\\,a_{9}(n)- \\frac{288157}{25498}\\,a_{10}(n) \\\\ &+ \\frac{303}{38}\\,a_{11}(n)- \\frac{601}{190}\\,a_{12}(n)- \\frac{2586}{95}\\,a_{13}(n)+ \\frac{69}{209}\\,a_{14}(n) - \\frac{497}{190}\\,a_{15}(n),$ $W_{(1,52)}(n) = &- \\frac{97}{1243}\\,\\sigma _{3}(n)+ \\frac{731577059}{582201312}\\,\\sigma _{3}(\\frac{n}{2})- \\frac{17}{164}\\,\\sigma _{3}(\\frac{n}{4})+ \\frac{5899}{59664}\\,\\sigma _{3}(\\frac{n}{13}) \\\\ &+ \\frac{7739629531}{582201312}\\,\\sigma _{3}(\\frac{n}{26})- \\frac{81757}{492}\\,\\sigma _{3}(\\frac{n}{52})+ (\\frac{1}{24}-\\frac{1}{208}n)\\sigma (n) \\\\ &+ (\\frac{1}{24}-\\frac{1}{4}n)\\sigma (\\frac{n}{52})+ \\frac{31939}{775632}\\,b_{1}(n)- \\frac{6918849709}{5045744704}\\,b_{2}(n) \\\\ &- \\frac{19319313973}{7568617056}\\,b_{3}(n)+ \\frac{236419605}{194067104}\\,b_{4}(n)+ \\frac{4556844909}{194067104}\\,b_{5}(n) \\\\ &+ \\frac{1357064601}{48516776}\\,b_{6}(n)+ \\frac{5549341}{39776}\\,b_{7}(n)- \\frac{139}{328}\\,b_{8}(n) \\\\ &- \\frac{1}{8}\\,b_{9}(n)- \\frac{65925667}{1775004}\\,b_{10}(n)- \\frac{11}{24}\\,b_{11}(n)+ \\frac{2036496863}{48516776}\\,b_{12}(n) \\\\ &- \\frac{7}{24}\\,b_{13}(n)- \\frac{3139}{164}\\,b_{14}(n)+ \\frac{349537693}{458704064}\\,b_{15}(n) \\\\ &- \\frac{556494635}{48516776}\\,b_{16}(n)+ \\frac{67534735}{194067104}\\,b_{17}(n)- \\frac{29}{82}\\,b_{18}(n),$ $W_{(4,13)}(n) = &- \\frac{31939}{775632}\\,\\sigma _{3}(n)+ \\frac{5001275639}{7568617056}\\,\\sigma _{3}(\\frac{n}{2})+ \\frac{47}{6396}\\,\\sigma _{3}(\\frac{n}{4})+ \\frac{24049}{387816}\\,\\sigma _{3}(\\frac{n}{13}) \\\\ &+ \\frac{1123375663}{7568617056}\\,\\sigma _{3}(\\frac{n}{26})- \\frac{17613}{2132}\\,\\sigma _{3}(\\frac{n}{52})+ (\\frac{1}{24}-\\frac{1}{52}n)\\sigma (\\frac{n}{4}) \\\\ &+ (\\frac{1}{24}-\\frac{1}{16}n)\\sigma (\\frac{n}{13})+ \\frac{31939}{775632}\\,b_{1}(n)- \\frac{2954664165}{5045744704}\\,b_{2}(n) \\\\ &- \\frac{5246851063}{7568617056}\\,b_{3}(n)+ \\frac{8345021621}{7568617056}\\,b_{4}(n)+ \\frac{35780970975}{2522872352}\\,b_{5}(n) \\\\ &+ \\frac{4695094021}{315359044}\\,b_{6}(n)+ \\frac{38120523}{517088}\\,b_{7}(n)- \\frac{261}{4264}\\,b_{8}(n) \\\\ &+ \\frac{1}{8}\\,b_{9}(n)- \\frac{786544471}{46150104}\\,b_{10}(n)+ \\frac{11}{24}\\,b_{11}(n)+ \\frac{42837668915}{1892154264}\\,b_{12}(n) \\\\ &+ \\frac{7}{24}\\,b_{13}(n)+ \\frac{473}{2132}\\,b_{14}(n)+ \\frac{154383529}{458704064}\\,b_{15}(n) \\\\ &- \\frac{5356650025}{946077132}\\,b_{16}(n)+ \\frac{1348868611}{7568617056}\\,b_{17}(n)+ \\frac{1}{1066}\\,b_{18}(n).$ We prove the case $W_{(4,13)}(n)$ as the other cases are proved similarly.",
"We compare the right hand side of convolSum-eqn-413 with that of evalConvolClass-eqn-11 when we have set $(\\alpha ,\\beta )=(4,13)$ , namely $\\sum _{n=1}^{\\infty }\\biggl (\\ 3840\\,\\sigma _{3}(\\frac{n}{4})+ 40560\\,\\sigma _{3}(\\frac{n}{13})+ 192\\,(13 - 6\\,n)\\,\\sigma (\\frac{n}{4}) \\\\+ 624\\,(4 - 6\\,n)\\,\\sigma (\\frac{n}{13})- 59904\\,W_{(4,13)}(n)\\,\\biggr )q^{n} = \\\\\\sum _{n=1}^{\\infty }\\biggl (\\,\\frac{3066144}{1243}\\,\\sigma _{3}(n)- \\frac{240061230672}{6064597}\\,\\sigma _{3}(\\frac{n}{2})+ \\frac{139392}{41}\\,\\sigma _{3}(\\frac{n}{4}) \\\\+ \\frac{45798672}{1243}\\,\\sigma _{3}(\\frac{n}{13})- \\frac{53922031824}{6064597}\\,\\sigma _{3}(\\frac{n}{26})+ \\frac{20290176}{41}\\,\\sigma _{3}(\\frac{n}{52}) \\\\- \\frac{3066144}{1243}\\,b_{1}(n)+ \\frac{212735819880}{6064597}\\,b_{2}(n)+ \\frac{251848851024}{6064597}\\,b_{3}(n)- \\frac{400561037808}{6064597}\\,b_{4}(n) \\\\- \\frac{5152459820400}{6064597}\\,b_{5}(n)- \\frac{5408748312192}{6064597}\\,b_{6}(n)- \\frac{5489355312}{1243}\\,b_{7}(n) \\\\+ \\frac{150336}{41}\\,b_{8}(n)- 7488\\,b_{9}(n)+ \\frac{151016538432}{147917}\\,b_{10}(n)- 27456\\,b_{11}(n) \\\\- \\frac{8224832431680}{6064597}\\,b_{12}(n)- 17472\\,b_{13}(n)- \\frac{544896}{41}\\,b_{14}(n)- \\frac{11115614088}{551327}\\,b_{15}(n) \\\\+ \\frac{2056953609600}{6064597}\\,b_{16}(n)- \\frac{64745693328}{6064597}\\,b_{17}(n)- \\frac{2304}{41}\\,b_{18}(n)\\,\\biggr )q^{n}$ We obtain the stated result when we solve for $W_{(4,13)}(n)$ ."
],
[
"Number of Representations of a positive Integer $n$ by the Octonary\nQuadratic Form using {{formula:6d1a58e2-c155-4ccb-acb3-84ab4d9452f8}} when {{formula:0d32e4d6-df9a-4182-ab20-de49dd956f0e}} \n",
"Let $n\\in \\mathbb {N}_{0}$ and then assune that $r_{4}(n)$ denote the number of representations of $n$ by the quaternary quadratic form $x_{1}^{2} +x_{2}^{2}+x_{3}^{2} +x_{4}^{2}$ which is defined by $r_{4}(n)=\\text{card}(\\lbrace (x_{1},x_{2},x_{3},x_{4})\\in \\mathbb {Z}^{4}~|~ n = x_{1}^{2}+x_{2}^{2} + x_{3}^{2} + x_{4}^{2}\\rbrace ).$ Obviously $r_{4}(0) = 1$ .",
"The Jacobi's identity $\\forall n\\in \\mathbb {N}\\qquad r_{4}(n) = 8\\sigma (n) - 32\\sigma (\\frac{n}{4}).$ is proved in K. S. Williams' book [28]; it will be very useful in the following.",
"Let $a,b\\in \\mathbb {N}$ and let $N_{(a,b)}(n)$ denote the number of representations of $n$ by the octonary quadratic form $a\\,(x_{1}^{2} +x_{2}^{2} + x_{3}^{2} + x_{4}^{2})+ b\\,(x_{5}^{2} + x_{6}^{2} + x_{7}^{2} + x_{8}^{2})$ which is defined by $N_{(a,b)}(n)=\\text{card}(\\lbrace (x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8})\\in \\mathbb {Z}^{8}~|~n = a\\,( x_{1}^{2} +x_{2}^{2} \\\\+ x_{3}^{2} + x_{4}^{2} ) +b\\,( x_{5}^{2} +x_{6}^{2} + x_{7}^{2} + x_{8}^{2}) \\rbrace ).$ We then deduce the following result.",
"Theorem 4.1 Let $n\\in \\mathbb {N}$ and $(a,b)=(1,11),(1,13)$ .",
"Then $N_{(1,11)}(n) = &8\\sigma (n) - 32\\sigma (\\frac{n}{4}) + 8\\sigma (\\frac{n}{11}) -32\\sigma (\\frac{n}{44}) \\\\ &+ 64\\, W_{(1,11)}(n) + 1024\\, W_{(1,11)}(\\frac{n}{4})- 256\\, \\biggl ( W_{(4,11)}(n) + W_{(1,44)}(n) \\biggr ), \\\\N_{(1,13)}(n) = &8\\sigma (n) - 32\\sigma (\\frac{n}{4}) + 8\\sigma (\\frac{n}{13}) -32\\sigma (\\frac{n}{52}) \\\\ &+ 64\\, W_{(1,13)}(n) + 1024\\, W_{(1,13)}(\\frac{n}{4})- 256\\, \\biggl ( W_{(4,13)}(n) + W_{(1,52)}(n) \\biggr ).$ We only prove $N_{(1,11)}(n)$ since that for $N_{(1,13)}(n)$ is done similarly.",
"From the definition of $N_{(1,11)}(n)$ it follows that $N_{(1,11)}(n) =\\sum _{\\begin{array}{c}{(l,m)\\in \\mathbb {N}_{0}^{2}} \\\\ {l+11\\,m=n}\\end{array}}r_{4}(l)r_{4}(m)= r_{4}(n)r_{4}(0) + r_{4}(0)r_{4}(\\frac{n}{11})+ \\sum _{\\begin{array}{c}{(l,m)\\in \\mathbb {N}^{2}} \\\\ {l+11\\,m=n}\\end{array}}r_{4}(l)r_{4}(m).$ We apply representations-eqn-4-1 to derive $N_{(1,11)}(n) = &8\\sigma (n) - 32\\sigma (\\frac{n}{4}) + 8\\sigma (\\frac{n}{11}) -32\\sigma (\\frac{n}{52}) \\\\ &+ \\sum _{\\begin{array}{c}{(l,m)\\in \\mathbb {N}^{2}} \\\\ {l+11\\,m=n}\\end{array}} (8\\sigma (l) - 32\\sigma (\\frac{l}{4}))(8\\sigma (m) - 32\\sigma (\\frac{m}{4})).$ From this previous identity we observe that $(8\\sigma (l) - 32\\sigma (\\frac{l}{4}))(8\\sigma (m) - 32\\sigma (\\frac{m}{4})) = &64\\sigma (l)\\sigma (m) - 256\\sigma (\\frac{l}{4})\\sigma (m) \\\\ &- 256\\sigma (l)\\sigma (\\frac{m}{4}) + 1024\\sigma (\\frac{l}{4})\\sigma (\\frac{m}{4}).$ E. Royer [24] has shown the evaluation of $W_{(1,11)}(n) = \\sum _{\\begin{array}{c}{(l,m)\\in \\mathbb {N}^{2}} \\\\ {l+11\\,m=n}\\end{array}}\\sigma (l)\\sigma (m).$ When we assign $4l$ to $l$ , then we infer $W_{(4,11)}(n) = \\sum _{\\begin{array}{c}{(l,m)\\in \\mathbb {N}^{2}} \\\\ {l+11\\,m=n}\\end{array}}\\sigma (\\frac{l}{4})\\sigma (m)= \\sum _{\\begin{array}{c}{(l,m)\\in \\mathbb {N}^{2}} \\\\ {4\\,l+11\\,m=n}\\end{array}}\\sigma (l)\\sigma (m).$ The evaluation of $W_{(4,11)}(n)$ is given in convolSum-theor-w411.",
"When we next assign $4m$ to $m$ , we conclude that $W_{(1,44)}(n) = \\sum _{\\begin{array}{c}{(l,m)\\in \\mathbb {N}^{2}} \\\\ {l+11\\,m=n}\\end{array}}\\sigma (l)\\sigma (\\frac{m}{4}) = \\sum _{\\begin{array}{c}{(l,m)\\in \\mathbb {N}^{2}} \\\\ {l+44\\,m=n}\\end{array}}\\sigma (l)\\sigma (m).$ The evaluation of $W_{(1,44)}(n)$ is provided by convolSum-theor-w144.",
"When we simultaneously assign $4l$ to $l$ and $4m$ to $m$ , we deduce that $\\sum _{\\begin{array}{c}{(l,m)\\in \\mathbb {N}^{2}} \\\\ {l+11\\,m=n}\\end{array}}\\sigma (\\frac{l}{4})\\sigma (\\frac{m}{4})= \\sum _{\\begin{array}{c}{(l,m)\\in \\mathbb {N}^{2}} \\\\ {l+11\\,m=\\frac{n}{4}}\\end{array}}\\sigma (l)\\sigma (m)= W_{(1,11)}(\\frac{n}{4}).$ Again, E. Royer [24] has proved the evaluation of $W_{(1,11)}(n)$ .",
"We then bring these evaluations together to obtain the stated result for $N_{(1,11)}(n)$ ."
],
[
"Tables",
"|r|r|r| $(\\alpha ,\\beta )$ Authors References (1,1) M. Besge, J. W. L. Glaisher, S. Ramanujan [8], [12], [23] (1,2),(1,3),(1,4) J. G. Huard & Z. M. Ou & B. K. Spearman & K. S. Williams [13] (1,5),(1,7) M. Lemire & K. S. Williams, S. Cooper & P. C. Toh [17], [10] (1,6),(2,3) S. Alaca & K. S. Williams [7] (1,8), (1,9) K. S. Williams [27], [26] (1,10), (1,11),(1,13), (1,14) E. Royer [24] (1,12),(1,16),(1,18), (1,24),(2,9),(3,4), A. Alaca & S. Alaca & K. S. Williams [2], [3], [4], [5] (3,8) (1,15),(3,5) B. Ramakrishman & B. Sahu [22] (1,20),(2,5),(4,5) S. Cooper & D. Ye [11] (1,23) H. H. Chan & S. Cooper [9] (1,25) E. X. W. Xia & X. L. Tian & O. X. M. Yao [29] (1,27),(1,32) S. Alaca & Y. Kesicio$\\check{g}$ lu [6] (1,36),(4,9) D. Ye [30] (1,22),(1,26),(2,7), (2,11),(2,13) A. Alaca & Ş. Alaca & E. Ntienjem [1] Known convolution sums $W_{(\\alpha , \\beta )}(n)$ |r|r|r| $(a,b)$ Authors References (1,1),(1,3), (1,9),(2,3) A. Alaca & Ş. Alaca & E. Ntienjem [1] (1,2) K. S. Williams [27] (1,4) A. Alaca & S. Alaca & K. S. Williams [3] (1,5) S. Cooper & D. Ye [11] (1,6) B. Ramakrishman & B. Sahu [22] (1,8) S. Alaca & Y. Kesicio$\\check{g}$ lu [6] Known representations of $n$ by the form introduction-eq-1 |c|cccccc| 1 2 4 13 26 52 1 1 5 0 3 -1 0 2 3 3 0 1 1 0 3 1 3 0 3 1 0 4 3 1 0 1 3 0 5 1 1 0 3 3 0 6 3 -1 0 1 5 0 7 1 -1 0 3 5 0 8 0 3 1 0 1 3 9 2 1 1 -2 3 3 10 0 1 1 0 3 3 11 2 -1 1 -2 5 3 12 0 3 -1 0 1 5 13 2 1 -1 -2 3 5 14 0 1 -1 0 3 5 15 -1 5 0 5 -1 0 16 0 -1 5 0 5 -1 17 7 -3 0 -3 7 0 18 0 7 -3 0 -3 7 Power of $\\eta $ -functions being basis elements for $S_{4}(\\Gamma _{0}(52))$ |c|cccccc| 1 2 4 11 22 44 1 6 -2 0 6 -2 0 2 4 0 0 4 0 0 3 2 2 0 2 2 0 4 0 4 0 0 4 0 5 -2 6 0 -2 6 0 6 0 2 2 0 2 2 7 0 -3 5 0 5 1 8 0 0 4 0 0 4 9 3 0 1 -1 0 5 10 0 -2 6 0 -2 6 11 1 -3 4 -3 5 4 12 2 0 0 2 -4 8 13 0 2 0 0 -2 8 14 -3 9 0 1 1 0 15 0 0 2 0 -4 10 Power of $\\eta $ -functions being basis elements for $S_{4}(\\Gamma _{0}(44))$"
]
] | 1606.05155 |
[
[
"Geodesic completeness and the lack of strong singularities in effective\n loop quantum Kantowski-Sachs spacetime"
],
[
"Abstract Resolution of singularities in the Kantowski-Sachs model due to non-perturbative quantum gravity effects is investigated.",
"Using the effective spacetime description for the improved dynamics version of loop quantum Kantowski-Sachs spacetimes, we show that even though expansion and shear scalars are universally bounded, there can exist events where curvature invariants can diverge.",
"However, such events can occur only for very exotic equations of state when pressure or derivatives of energy density with respect to triads become infinite at a finite energy density.",
"In all other cases curvature invariants are proved to remain finite for any evolution in finite proper time.",
"We find the novel result that all strong singularities are resolved for arbitrary matter.",
"Weak singularities pertaining to above potential curvature divergence events can exist.",
"The effective spacetime is found to be geodesically complete for particle and null geodesics in finite time evolution.",
"Our results add to a growing evidence for generic resolution of strong singularities using effective dynamics in loop quantum cosmology by generalizing earlier results on isotropic and Bianchi-I spacetimes."
],
[
"In general relativity (GR), occurrence of singularities brings forth the underlying limitations of the classical continuum spacetime.",
"A singular event in classical theory has many characteristics, generally captured via curvature divergences and break down of geodesic evolution in a finite proper time.",
"However, not all singularities are necessarily the boundaries of classical spacetime.",
"Their strength matters.",
"It is believed that strong singularities [1], [2], [3], which cause inevitable complete destruction of arbitrarily strong in-falling detectors, are the true boundaries of the classical continuum spacetime.",
"Strong singular events, such as the big bang or a central singularity inside a black hole, are conjectured to be associated with geodesic incompleteness [2], [3].",
"In contrast, weak singularities can be considered harmless.",
"Even though some curvature components may diverge at such events, a sufficiently strong detector survives such a singularity.",
"Geodesics can be extended beyond such singularities in the classical spacetime.For examples in cosmological spacetimes, see Ref.",
"[4].",
"Thus, not all space-like singularities are necessarily harmful.",
"A fundamental challenge for any theory going beyond Einsteinian gravity is whether it can resolve all of the strong singularities.",
"In the absence of quantum gravitational effects which profoundly modify the structure of the underlying spacetime, singularity resolution has been elusive.",
"In the last decade, applications of loop quantum gravity (LQG) to cosmological spacetimes in loop quantum cosmology (LQC) indicate that non-perturbative quantum gravitational modifications play an important role in singularity resolution [5].",
"Various examples of cosmological spacetimes, including isotropic [6], [7] and anisotropic models [8], [9], and also hybrid Gowdy models [10], [11], [12], have been thoroughly studied at a rigorous quantum level.",
"Extensive numerical simulations of quantum cosmological models have been performed [6], [13], which show that the singularities such as the big bang and big crunch are resolved and replaced by a non-singular bounce.",
"Singularity resolution has also been understood in terms of quantum probabilities using consistent histories approach [14].",
"At the fundamental level, geometry in LQC is discrete resulting in a classical continuum quickly below the Planck curvature scale.",
"In fact, the quantum Hamiltonian constraint in LQC on quantum geometry can be very well approximated by the Wheeler-DeWitt Hamiltonian constraint on classical continuum as soon as the spacetime curvature becomes less than a percent of the Planck curvature.",
"Interestingly, numerical simulations show that an effective quantum continuum description exists which captures the quantum dynamics in LQC at all scales.",
"This effective spacetime description, or effective dynamics, has been used in a plenty of investigations.",
"In relation to the objectives of this manuscript, notable results include the following.",
"Using effective dynamics, expansion and shear scalars have been found to be generically bounded for isotropic and anisotropic spacetimes [15], [16], [17], [19], [9], [18], [20].",
"Strong singularities are shown to be absent and effective spacetime is found to be geodesically complete for loop quantized isotropic cosmological and Bianchi-I spacetimes for matter with a vanishing anisotropic stress [16], [17], [18].",
"However, weak and non-curvature singularities can exist [16], [17], [18], of which various examples have been studied in isotropic and Bianchi-I spacetimes in LQC [21].",
"The goal of this manuscript is to investigate the resolution of strong singularities and geodesic completeness in the Kantowski-Sachs model in LQC using the effective spacetime description.",
"We wish to understand under what conditions, for arbitrary non-viscous matter, quantum geometry effects as understood in LQC lead to a generic resolution of strong singularities, and whether there exist any weak singularities.",
"Essentially, our aim is to generalize the results of geodesic completeness and generic resolution of strong singularities in isotropic and Bianchi-I spacetime in LQC to the Kantowski-Sachs model.",
"The Kantowski-Sachs spacetime is an interesting avenue to study for various reasons.",
"In the absence of matter, it captures the interior $(r < 2m)$ of the Schwarzschild black hole.",
"In the presence of matter, it is an anisotropic cosmological model with a spatial curvature.",
"This spacetime has been loop quantized with different regularizations of the Hamiltonian constraint, given by Ashtekar-Bojowald [22] (see also Refs.",
"[23], [24]), Corichi-Singh [25], and Boehmer-Vandersloot [26] (see also Ref.",
"[27]).",
"The first quantization prescription is a reminiscent of `fixed area of the loop' procedure in LQC [28], [29] which is known to have various phenomenological issues due to fiducial cell dependence [30].",
"The second and third quantization prescriptions overcome this limitation in their unique ways, and give qualitatively different physics of singularity resolution.",
"In this manuscript, we study the Kantowski-Sachs spacetime with Boehmer-Vandersloot prescription which is an avatar of the improved dynamics prescription in LQC [6].",
"It should be noted that resolution of strong singularities and geodesic completeness for isotropic and Bianchi-I spacetime in LQC have been achieved for this particular prescription.",
"Interestingly, a loop quantization of this spacetime, in absence of matter and also in presence of cosmological constant, results in a singularity resolution with a pre-bounce spacetime which is a product of two constant curvature spaces and with an almost Planckian curvature [31].",
"An important result pertinent to our investigations is that the expansion and shear scalars in this quantization turn out to be universally bounded [32].",
"In the following, our reference to loop quantized Kantowski-Sachs model will imply Boehmer-Vandersloot prescription.",
"Our analysis assumes the validity of the effective dynamics in LQC at all scales.",
"In LQC, effective Hamiltonian is obtained using a geometrical formulation of quantum theory, and, as noted earlier, turns out to be an excellent approximation for isotropic and anisotropic models.",
"This is true at least for states which correspond to macroscopic spacetimes at late times.",
"A specific example is the case of homogeneous and isotropic spatially flat spacetimes where effective Hamiltonian has been derived explicitly using coherent states for the case of the massless scalar [33].",
"The resulting effective dynamics has been tested rigourously using numerical simulations [6], [13], which validate the analytical derivation of the effective Hamiltonian for the above family of physical states.",
"In terms of the gravitational phase space variables, Ashtekar-Barbero connection components and conjugate triads, the effective Hamiltonian contains trigonometric terms of the connections, apart from the triads and the matter variables.",
"These trigonometric or the polymerized terms arise from expressing field strength of the connection in terms of holonomies over a minimum physical area in the improved dynamics.",
"The resulting Hamilton's equations from the effective Hamiltonian encode the quantum gravitational repulsiveness and result in singularity resolution.",
"In the effective Hamiltonian, there can also be modifications coming from expressing inverse powers of triads in terms of Poisson brackets between holonomies and triads.",
"The role of these modifications in singularity resolution is generally found to be negligible when compared to the effects originating from the polymerized terms.These modifications in absence of polymerized terms can also lead to singularity resolution and interesting phenomenology in spatially curved models, see e.g.",
"[34], [35].",
"In our analysis, we ignore the latter modifications.It should be noted that in models where these terms have been argued to become significant, their overall effect is to strengthen the singularity resolution effects [9], [19], [38].",
"We will find that the effective Hamiltonian even in the absence of these terms suffices to obtain generic resolution of strong singularities.",
"Interestingly, the polymerized Hamiltonian of LQC can also be obtained using an inverse procedure without any prior hints of the canonical structure or assuming a Lagrangian, just by demanding a form of repulsive nature of gravity at high curvature scales and general covariance [36].",
"The main results from our investigation are the following.",
"Considering non-viscous minimally coupled matter with a general equation of state and anisotropic stress, we show that for any finite proper time, energy density is always finite in the loop quantized Kantowski-Sachs model.",
"This is the first novel result of our analysis.",
"In previous works, such as in Refs.",
"[26], [32], [31], energy density was found not to diverge dynamically using numerical simulations.",
"Since such simulations do not cover the entire set of solutions, an analytical understanding of the behavior of energy density was very much needed.",
"Our second result is to show that the physical volume remains non-zero and finite throughout the finite time evolution.",
"Along with our result on energy density finiteness, this rules out big bang/crunch, big rip and big freeze singularities.Unlike big bang/crunch, big rip singularity occurs at infinite volume with an infinite energy density (see e.g.",
"[37]).",
"A big freeze singularity occurs at finite volume but has infinite energy density (see e.g.",
"[16]).",
"Our third result is to show that even though expansion and shear scalars are universally bounded in the effective spacetime, curvature invariants can still diverge.",
"Albeit, this only happens when pressure or derivatives of energy density with respect to the triads diverge while energy density remains finite.",
"Such exotic equations of state are known to cause sudden singularities in cosmological models (see e.g.",
"[16]).",
"It turns out that the divergence in curvature invariants in finite proper time correspond to weak curvature singularities.",
"Our fourth result is to show that all strong singularities are absent for any finite time evolution.",
"Finally, analysis of the time-like and null geodesics shows that the effective spacetime is geodesically complete.",
"There is no breakdown of geodesics for any finite proper time in the effective dynamics evolution in loop quantized Kantowski-Sachs spacetime.",
"The effective quantum spacetime in loop quantized Kantowski-Sachs model is geodesically complete.",
"Our manuscript is organized in the following way.",
"In Sec.",
"II, we provide a brief review of the classical Hamiltonian formulation of the Kantowski-Sachs spacetime in Ashtekar variables and obtain the dynamical equations.",
"We calculate the expressions for the expansion and shear scalar, and for curvature invariants in terms of connection and triad variables.",
"The effective Hamiltonian from LQC based on Boehmer-Vandersloot quantization is studied in Sec.",
"III where we obtain the quantum gravitational modified dynamical equations and obtain the bounded behavior of expansion and shear scalars.",
"We obtain the analytical bounds on the triad variables from the dynamical equations for finite time evolution, which then imply that the energy density is finite for any finite proper time.",
"We show that for any finite time evolution, the physical volume is non-zero and that the curvature invariants are non-divergent except for singular events where pressure and energy density derivatives with respect to triads diverge at finite energy density.",
"In Section IV, we consider the special case of matter with vanishing anisotropic stress, and show that the above results turn out to be true using a simpler argument.",
"Behavior of the geodesics is investigated in Sec.",
"VA, where they are shown to not break down in any finite proper time evolution.",
"In Sec.",
"VB, we show that the Kantowski-Sachs spacetime in effective dynamics does not satisfy the necessary conditions for the existence of strong curvature singularities.",
"Hence we conclude that the above mentioned pressure singularities are weak singularities.",
"We summarize with conclusions in Sec.",
"VI."
],
[
"Classical Dynamics of Kantowski-Sachs space-time",
"In this section, we summarize the basic features of classical Kantowski-Sachs spacetime in Ashtekar variables.",
"The homogeneity of the Kantowski-Sachs spacetime leads to a simple diagonal form for the Ashtekar-Barbero connection components and conjugate triads [22]: $A_a^i \\tau _i {\\rm d}x^a &=& \\tilde{c} \\tau _3 {\\rm d}x + \\tilde{b} \\tau _2 {\\rm d}\\theta - \\tilde{b} \\tau _1 \\sin \\theta {\\rm d}\\phi + \\tau _3 \\cos \\theta {\\rm d}\\phi ~,\\\\\\tilde{E}_i^a \\tau _i \\partial _a &=& \\tilde{p}_c \\tau _3 \\sin \\theta \\partial _x + \\tilde{p}_b \\tau _2 \\sin \\theta \\partial _\\theta - \\tilde{p}_b \\tau _1 \\partial _\\phi ~,$ where $\\tau _i = - i \\sigma _i/2$ , and $\\sigma _i$ are the Pauli spin matrices.",
"The symmetry reduced triads $\\tilde{p_b}$ and $\\tilde{p_c}$ are related to the metric components of the spacetime line element: $ds^2=-N^2{\\rm d}t^2+g_{xx}{\\rm d}x^2+g_{\\Omega \\Omega }\\left({\\rm d}\\theta ^2+\\sin ^2{\\theta }{\\rm d}\\phi ^2\\right).",
"$ as $g_{xx}=\\frac{\\tilde{p_b}^2}{\\tilde{p_c}}, ~~~~ \\mathrm {and} ~~~~ g_{\\Omega \\Omega }=|\\tilde{p_c}|.",
"$ The modulus sign arises due to two orientations of the triad.",
"Since the matter considered in this analysis is non-fermionic, we can fix one orientation.",
"In the following, the orientation of the triads is chosen to be positive without any loss of generality.",
"The Kantowski-Sachs spacetime is naturally foliated with spatial slices of topology $\\mathbb {R} \\times \\mathbb {S}^2$ .",
"The spatial slices are non-compact in $x$ -direction.",
"In order to define a symplectic structure on the spatial slices, we need to restrict the integration along the $x$ -direction to a fiducial length, say $L_o$ .",
"The resulting symplectic structure is: ${\\bf \\Omega }=\\frac{L_o}{2G\\gamma }\\left(2 {\\rm d}\\tilde{b}\\wedge {\\rm d}\\tilde{p_b}+d\\tilde{c}\\wedge {\\rm d}\\tilde{p_c}\\right).$ where $\\gamma $ is the Barbero-Immirzi parameter, its value is set to 0.2375 from black hole entropy calculations in loop quantum gravity.",
"The fiducial length is a non-physical parameter in our theory, and can be arbitrarily re-scaled.",
"In order to make the symplectic structure independent of $L_o$ , we introduce the new triad and connection variables $p_b$ , $p_c$ , and $b$ , $c$ obtained by re-scaling the symmetry reduced triad and connection variables : $p_b=L_o \\tilde{p_b}, \\text{ }p_c=\\tilde{p_c}, \\text{ }b=\\tilde{b}, \\text{ }c=L_o \\tilde{c}.",
"~$ The non-vanishing Poisson brackets between these new variables are given by, $\\left\\lbrace b,p_b \\right\\rbrace = G \\gamma , \\text{ } \\left\\lbrace c,p_c \\right\\rbrace = 2G \\gamma .$ In terms of these phase space variables, the classical Hamiltonian constraint is the following for lapse $N = 1$ : $\\mathcal {H}_{\\rm {cl}}=\\frac{-1}{2G\\gamma ^2}\\left[2bc \\sqrt{p_c}+\\left(b^2+\\gamma ^2\\right)\\frac{p_b}{\\sqrt{p_c}}\\right]\\, + \\, 4\\pi p_b \\sqrt{p_c} \\rho ~.$ Here $\\rho $ is the energy density, related to matter Hamiltonian as $\\rho = {\\cal H}_m/V$ , and $V$ is the physical volume of the fiducial cell: $V = 4 \\pi p_b \\sqrt{p_c}$ .",
"The energy density is taken to depend only on triad variables, and not on connection variables.",
"The Hamilton's equations for the triad and connection variables are: $\\dot{p_b}&=&-G\\gamma \\frac{\\partial \\mathcal {H}_{\\rm {cl}}}{\\partial b}=\\frac{1}{\\gamma }\\left(c\\sqrt{p_c}+\\frac{bp_b}{\\sqrt{p_c}}\\right) ,\\\\\\dot{p_c}&=&-2G\\gamma \\frac{\\partial \\mathcal {H}_{\\rm {cl}}}{\\partial c}=\\frac{1}{\\gamma }2b\\sqrt{p_c} ,\\\\\\dot{b}&=&G\\gamma \\frac{\\partial \\mathcal {H}_{\\rm {cl}}}{\\partial p_b}=\\frac{-1}{2\\gamma \\sqrt{p_c}}\\left( b^2+\\gamma ^2 \\right) + 4 \\pi G\\gamma \\sqrt{p_c} \\bigg (\\rho + p_b \\frac{\\partial \\rho }{\\partial p_b} \\bigg ) , \\\\\\dot{c}&=&2G\\gamma \\frac{\\partial \\mathcal {H}_{\\rm {cl}}}{\\partial p_c}=\\frac{-1}{\\gamma \\sqrt{p_c}}\\left(bc-\\left( b^2+\\gamma ^2\\right)\\frac{p_b}{2p_c} \\right) + 4 \\pi G\\gamma \\frac{p_b}{\\sqrt{p_c}} \\bigg (\\rho + 2 p_c \\frac{\\partial \\rho }{\\partial p_c} \\bigg )~.$ Here `dot' refers to derivative with respect to proper time.",
"Using the above equations, a useful result follows: $\\frac{d}{dt}(c p_c - b p_b)=\\frac{\\gamma p_b}{\\sqrt{p_c}} + G \\gamma V \\bigg (2 p_c \\frac{\\partial \\rho }{\\partial p_c} -p_b \\frac{\\partial \\rho }{\\partial p_b}\\bigg ) ~.",
"$ As will be proved in Sec.",
"III, it turns out that the same expression also holds in the presence of quantum gravitational modifications in LQC.",
"Let us now find some useful expressions to understand singularities in Kantowski-Sachs spacetime.",
"The simplest to obtain is the expression for energy density in terms of the gravitational phase space variables, by imposing the vanishing of the Hamiltonian constraint, $\\mathcal {H}_{\\rm {cl}} \\approx 0$ : $\\rho = {8 \\pi G}\\bigg ({\\gamma ^2 p_b} + {\\gamma ^2 p_c} + {p_c}\\bigg ) ~.$ Two useful quantities of interest to understand the behavior of geodesics as singularities are approached are the expansion and shear scalars.",
"The expansion scalar $\\theta $ is given by $\\theta = \\frac{\\dot{V}}{V} = \\frac{\\dot{p_b}}{p_b}+\\frac{\\dot{p_c}}{2p_c}.",
"$ The shear scalar $\\sigma ^2$ expressed in terms of the directional Hubble rates $H_i=\\dot{\\sqrt{g_{ii}}}/\\sqrt{g_{ii}}$ is given by $\\sigma ^2= \\frac{1}{2}\\displaystyle \\sum \\limits _{i=1}^3 \\left(H_i-\\frac{1}{3}\\theta \\right)^2 =\\frac{1}{3}\\left(\\frac{\\dot{p_c}}{p_c}-\\frac{\\dot{p_b}}{p_b}\\right)^2 .",
"~$ Next we find the expressions for curvature invariants, which when diverge signal singularities (though not necessarily strong ones).",
"In terms of the reduced triad and connection variables, the expressions for the Ricci scalar, the square of the Weyl scalar and the Kretschmann scalar are respectively as follows: $R &=&2\\frac{\\ddot{p}_b}{p_b}+\\frac{\\ddot{p}_c}{p_c}+\\frac{2}{p_c} , \\\\C_{abcd}C^{abcd} &=&\\frac{1}{3}\\left[3\\frac{\\dot{p}_c}{p_c}\\left(\\frac{\\dot{p}_b}{p_b}-\\frac{\\dot{p}_c}{p_c}\\right)-2\\left(\\frac{\\ddot{p}_b}{p_b}-\\frac{\\ddot{p}_c}{p_c}\\right)-\\frac{2}{p_c}\\right]^2 $ and $K &=& 6\\left(\\frac{\\dot{p}_b}{p_b}\\frac{\\dot{p}_c}{p_c}\\right)^2-8\\frac{\\dot{p}_b}{p_b}\\frac{\\dot{p}_c}{p_c}\\frac{\\ddot{p}_b}{p_b}+4\\left(\\frac{\\ddot{p}_b}{p_b}\\right)^2+6\\frac{\\ddot{p}_b}{p_b}\\left(\\frac{\\dot{p}_c}{p_c}\\right)^2-4\\frac{\\ddot{p}_b}{p_b}\\frac{\\ddot{p}_c}{p_c} \\\\\\nonumber & & -8\\frac{\\dot{p}_b}{p_b}\\left(\\frac{\\dot{p}_c}{p_c}\\right)^3+4\\frac{\\dot{p}_b}{p_b}\\frac{\\dot{p}_c}{p_c}\\frac{\\ddot{p}_c}{p_c}+\\frac{7}{2}\\left(\\frac{\\dot{p}_c}{p_c}\\right)^4+2\\left(\\frac{\\dot{p}_c}{p_c}\\right)^2\\frac{1}{p_c}\\\\\\nonumber & & -5\\left(\\frac{\\dot{p}_c}{p_c}\\right)^2\\frac{\\ddot{p}_c}{p_c}+\\frac{4}{p_c^2}+3\\left(\\frac{\\ddot{p}_c}{p_c}\\right)^2 ~.$ We notice that the expansion and shear scalar, and all the curvature invariants depend on the following five quantities : $\\frac{\\ddot{p_b}}{p_b}$ , $\\frac{\\ddot{p_c}}{p_c}$ , $\\frac{\\dot{p_b}}{p_b}$ , $\\frac{\\dot{p_c}}{p_c}$ and $\\frac{1}{p_c}$ .",
"The behavior of $\\dot{p}_b/p_b$ and $\\dot{p}_c/p_c$ is obtained from the Hamilton's equations.",
"Taking their time derivatives, we obtain $\\frac{\\ddot{p_b}}{p_b} &=& \\frac{bc}{\\gamma ^2 p_b} + 8 \\pi G \\rho + 4\\pi G \\bigg (p_b \\frac{\\partial \\rho }{\\partial p_b} + 2 p_c \\frac{\\partial \\rho }{\\partial p_c} \\bigg ) ~, \\\\\\frac{\\ddot{p_c}}{p_c} &=& -\\frac{1}{p_c} + \\frac{b^2}{\\gamma ^2 p_c} + 8 \\pi G \\rho + 8\\pi G p_b \\frac{\\partial \\rho }{\\partial p_b}~.",
"$ It is clear from the classical Hamilton's equations (REF –) and the above equations that the expansion scalar, shear scalar, curvature invariants and energy density all diverge as the triad components vanish, and/or the connection components diverge and/or the terms $\\frac{\\partial \\rho }{\\partial p_b}$ and $\\frac{\\partial \\rho }{\\partial p_c}$ diverge.",
"Generic physical solutions obtained from the classical Hamiltonian constraint (REF ) turn out to be of this form and are singular."
],
[
"Effective loop quantum cosmological dynamics",
"In the previous section, we obtained the classical singular dynamical equations from the classical Hamiltonian constraint of the Kantowski-Sachs spacetime.",
"Let us now see the way quantum gravitational modifications result in non-singular dynamics.",
"Our starting point is the effective Hamiltonian constraint [26]: $\\mathcal {H}=-\\frac{p_b\\sqrt{p_c}}{2G\\gamma ^2\\Delta }\\left[2\\sin (b\\delta _b)\\sin (c\\delta _c)+\\sin ^2(b\\delta _b)+\\frac{\\gamma ^2\\Delta }{p_c}\\right]+4\\pi p_b\\sqrt{p_c}\\rho $ where $\\Delta $ denotes the minimum non-zero eigenvalue of the area operator in loop quantum gravity: $\\Delta = 4 \\sqrt{3} \\pi \\gamma l_{\\mathrm {Pl}}^2$ , and $\\delta _b= \\sqrt{\\frac{\\Delta }{p_c}}, ~~~\\quad ~~~\\delta _c=\\frac{\\sqrt{\\Delta p_c}}{p_b} ~.$ It should be noted that the above effective Hamiltonian corresponds to the improved dynamics prescription in LQC [6].",
"For $\\delta _b$ and $\\delta _c$ which are arbitrary functions of phase space variables, this prescription turns out to be unique in the sense that it yields physics independent of the fiducial length $L_o$ and as discussed below universal bounds on expansion and shear scalars [32].",
"Using the Hamilton's equations, we obtain the modified dynamical equations for the gravitational phase space variables (assuming that the energy density depends only on triad variables, and not on connection variables): $\\dot{p}_b & = & \\frac{p_b\\cos (b\\delta _b)}{\\gamma \\sqrt{\\Delta }} \\left(\\sin (c\\delta _c)+ \\sin (b\\delta _b)\\right), \\\\\\dot{p}_c & = & \\frac{2p_c}{\\gamma \\sqrt{\\Delta }} \\sin (b\\delta _b) \\cos (c\\delta _c), \\\\\\dot{b} & = & -\\frac{\\sqrt{p_c}}{2 \\gamma \\Delta }\\left[2\\sin (b \\delta _b)\\sin (c \\delta _c)+\\sin ^2(b \\delta _b)+\\frac{\\gamma ^2\\Delta }{p_c}\\right] \\nonumber \\\\& & + \\frac{c p_c}{\\gamma \\sqrt{\\Delta } p_b}\\sin (b \\delta _b)\\cos (c \\delta _c) + 4 \\pi G\\gamma \\sqrt{p_c} \\bigg (\\rho + p_b \\frac{\\partial \\rho }{\\partial p_b} \\bigg ) $ and $\\dot{c} & = & -\\frac{p_b}{2 \\gamma \\Delta \\sqrt{p_c}}\\left[2\\sin (b\\delta _b)\\sin (c\\delta _c)+\\sin ^2(b\\delta _b)+\\frac{\\gamma ^2\\Delta }{p_c}\\right] \\nonumber \\\\& & - \\frac{c}{\\gamma \\sqrt{\\Delta }}\\sin (b \\delta _b)\\cos (c \\delta _c)+\\frac{b p_b}{\\gamma \\sqrt{\\Delta } p_c} \\cos (b\\delta _b)\\left(\\sin (c\\delta _c)+ \\sin (b\\delta _b)\\right) \\nonumber \\\\& & +\\frac{\\gamma p_b}{\\sqrt{p_c}} + 4 \\pi G\\gamma \\frac{p_b}{\\sqrt{p_c}} \\bigg (\\rho + 2 p_c \\frac{\\partial \\rho }{\\partial p_c} \\bigg ).$ As in the classical theory (see eq.",
"REF ), it turns out that time derivative of $(c p_c - b p_b)$ is given by $\\frac{d}{dt}(c p_c - b p_b)= \\nonumber \\frac{\\gamma p_b}{\\sqrt{p_c}} + G \\gamma V \\bigg (2 p_c \\frac{\\partial \\rho }{\\partial p_c} -p_b\\frac{\\partial \\rho }{\\partial p_b}\\bigg )~.",
"$ The change in the Hamiltonian evolution from classical theory to LQC, results in $\\dot{p}_c$ /$p_c$ and $\\dot{p}_b$ /$p_b$ as bounded functions.",
"This in turn yields a non-divergent behavior of expansion and shear scalars.",
"The expansion scalar is given by [32] $\\theta &=& \\frac{1}{\\gamma \\sqrt{\\Delta }} \\left( \\sin {(b\\delta _b)} \\cos {(c\\delta _c)}+\\cos {(b\\delta _b)} \\sin {(c\\delta _c)} +\\sin {(b\\delta _b)} \\cos {(b \\delta _b)} \\right),$ which is bounded above due to discrete quantum geometric effects inherited via area gap $\\Delta $ : $ |\\theta | \\le 2.78/l_{\\mathrm {Pl}}$ .",
"The shear scalar becomes, $\\sigma ^2 = \\frac{1}{3\\gamma ^2 \\Delta } \\left(2\\sin {(b\\delta _b)}\\cos {(c\\delta _c)}-\\cos {(b\\delta _b)}\\left(\\sin {(c\\delta _c)}+\\sin {(b\\delta _b)}\\right) \\right)^2 ~,$ which is also universally bounded [32]: $\\sigma ^2 \\le 5.76/l_{\\mathrm {Pl}}^{2}$ .",
"From (REF ) and () an important result follows on the permitted values of $p_b$ and $p_c$ .",
"Let $t_0$ be some time in the evolution at which $p_c$ and $p_b$ have some given non-zero finite values $p_c^0$ and $p_b^0$ .",
"Then from () we have $\\int _{p_c^0}^{p_c(t)}\\frac{dp_c}{p_c}= \\int _{t_0}^t\\frac{2}{\\gamma \\sqrt{\\Delta }} \\sin (b\\delta _b) \\cos (c\\delta _c) {\\rm d}t$ which implies $p_c(t) = p_c^0 \\exp \\left\\lbrace \\frac{1}{\\gamma \\sqrt{\\Delta }} \\int _{t_0}^t\\bigg (\\sin (b\\delta _b + c\\delta _c) + \\sin (b\\delta _b -c\\delta _c)\\bigg ) {\\rm d}t\\right\\rbrace ~.$ Since $ | \\sin (b\\delta _b + c\\delta _c) + \\sin (b\\delta _b - c\\delta _c)|\\le 2$ , the integration (inside the exponential) over a finite time is finite.",
"Hence, at any finite proper time, in the past or in the future : $0<p_c(t)<\\infty ~.$ Similarly using (REF ), we get $p_b(t) = p_b^0 \\exp \\left\\lbrace \\frac{1}{\\gamma \\sqrt{\\Delta }} \\int _{t_0}^t\\cos (b\\delta _b)\\bigg (\\sin (c\\delta _c)+ \\sin (b\\delta _b)\\right){\\rm d}t\\bigg \\rbrace ~.$ And since the integration inside the exponential is again over a bounded function, we obtain a finite integral over finite range of time.",
"Hence, we obtain $0<p_b(t)<\\infty $ for any given finite time in past or future.",
"Therefore, we reach an important result that $p_b$ , $p_c$ and consequently the volume $V$ ($V=4 \\pi p_b\\sqrt{p_c})$ are finite, positive and non-zero at any finite time.",
"Note that a similar argument was used in Ref.",
"[12] to show the finiteness of the triad variables for any finite time in the effective dynamics of the Gowdy model.",
"From the vanishing of the Hamiltonian constraint, we can get the energy density in terms of dynamical variables: $\\rho = \\frac{1}{8\\pi G\\gamma ^2\\Delta }\\left[2\\sin (b\\delta _b)\\sin (c\\delta _c)+\\sin ^2(b\\delta _b)+\\frac{\\gamma ^2\\Delta }{p_c}\\right] ~.$ Hence the energy density remains finite by virtue of (REF ) and (REF ) for any finite proper time.",
"So far we have seen that the expansion and shear scalars are generically bounded for all time.",
"Energy density $\\rho $ remains finite under evolution over a finite proper time.",
"And $p_b$ , $p_c$ and $V$ remain non-zero, positive and finite under evolution over a finite proper time.",
"The terms $\\dot{p}_c$ /$p_c$ and $\\dot{p}_b$ /$p_b$ are bounded due to () and (REF ).",
"Hence the divergence in curvature invariants given by (REF ), () and (REF ) may only come from divergence in $\\frac{\\ddot{p}_c}{p_c}$ and $\\frac{\\ddot{p}_b}{p_b}$ .",
"After a straightforward calculation, the expressions for $\\ddot{p}_b$ and $\\ddot{p}_c$ turn out to be the following: $\\ddot{p}_b & = & p_b \\Bigg [\\frac{\\cos (b\\delta _b)\\cos (c\\delta _c)}{p_c} + \\frac{\\cos ^2(b\\delta _b)}{\\gamma ^2 \\Delta }(\\sin (b\\delta _b)+\\sin (c\\delta _c))^2 \\nonumber \\\\& & - \\frac{4 \\pi }{\\gamma ^2 \\sqrt{\\Delta }}\\frac{(cp_c-bp_b)}{V}\\cos (c\\delta _c)\\bigg (\\sin (c\\delta _c)+\\sin ^3(b\\delta _b)\\bigg ) \\nonumber \\\\& & +4\\pi G \\bigg (2p_c\\frac{\\partial \\rho }{\\partial p_c}\\cos (b\\delta _b)\\cos (c\\delta _c)-p_b\\frac{\\partial \\rho }{\\partial p_b}\\sin (b\\delta _b)\\sin (c\\delta _c) + p_b\\frac{\\partial \\rho }{\\partial p_b}\\cos (2b\\delta _b)\\bigg )\\Bigg ] $ and $\\ddot{p}_c & = & p_c \\Bigg [ -2\\frac{\\sin (b\\delta _b)\\sin (c\\delta _c)}{p_c} + \\frac{4\\sin ^2(b\\delta _b)\\cos ^2(c\\delta _c)}{\\gamma ^2 \\Delta }\\nonumber \\\\& & + \\frac{4 \\pi }{\\gamma ^2 \\sqrt{\\Delta }}\\frac{(cp_c-bp_b)}{V}\\sin (2b\\delta _b)\\bigg (1+\\sin (b\\delta _b)\\sin (c\\delta _c)\\bigg )\\nonumber \\\\& & + 8\\pi G\\bigg (p_b\\frac{\\partial \\rho }{\\partial p_b}\\cos (b\\delta _b)\\cos (c\\delta _c)-2p_c\\frac{\\partial \\rho }{\\partial p_c}\\sin (b\\delta _b)\\sin (c\\delta _c)\\bigg ) \\Bigg ] ~.",
"$ The unboundedness in above terms can arise from terms containing $(cp_c-bp_b)$ and/or from terms with $\\frac{\\partial \\rho }{\\partial p_b}$ and $\\frac{\\partial \\rho }{\\partial p_c}$ .",
"It turns out that any potential divergences from the first type are tied to the second type in the following way.",
"We have earlier found, below eq.",
"(REF ), that the time derivative of this difference is given by the same expression in the classical theory and LQC (eq.",
"(REF )).",
"In eq.",
"(REF ), first term on the R.H.S is finite due to (REF ) and (REF ).",
"Integrating the right hand side, we see that the quantity $(cp_c-bp_b)$ may diverge if the derivatives of the energy density with respect to triads diverge.",
"Otherwise $(cp_c-bp_b)$ will be finite at any finite past or future time.",
"In conclusion, any divergences in $\\frac{\\ddot{p}_c}{p_c} $ and $\\frac{\\ddot{p}_b}{p_b}$ , and consequently in the curvature invariants come from the terms $\\frac{\\partial \\rho }{\\partial p_b}$ and $\\frac{\\partial \\rho }{\\partial p_c}$ .",
"Since energy density is always finite for any finite time, we need matter with an equation of state which has divergent triad derivatives of energy density with energy density being finite.",
"Only then $\\ddot{p}_b$ and $\\ddot{p}_c$ can diverge in finite time in this loop quantized Kantowski-Sachs model.",
"And only then the curvature invariants can diverge.",
"These divergences in curvature invariants in case of special matter types lead to weak curvature singularities as will be shown in section V. For all other types of matter the curvature invariants are non-divergent for all finite times indicating the absence of any singularities."
],
[
"Effective Dynamics : matter with vanishing anisotropic stress and pressure singularities",
"In the previous section we found that for effective spacetime description in LQC, the only way curvature invariants can diverge is when derivatives of energy density with respect to triads diverge at finite energy density.",
"We will now show that for the special case of matter having a vanishing anisotropic stress, these divergences are related to the pressure divergences.",
"For such a matter, the energy density is a function of volume only, i.e.",
"$\\rho (p_b,p_c)=\\rho (p_b\\sqrt{p_c})$ .",
"Then the pressure $P$ can be written as, $P= -\\frac{\\partial \\mathcal {H}_{\\rm {matt}}}{\\partial V} = -\\rho - V\\frac{\\partial \\rho }{\\partial V} ~.$ The derivatives of the energy density with respect to triads can be written in terms of the energy density and pressure as: $p_b\\frac{\\partial \\rho }{\\partial p_b}=2p_c\\frac{\\partial \\rho }{\\partial p_c}=-\\rho -P , $ Using the above equations, the expression for $\\ddot{p}_b$ in the case of vanishing anisotropic stress can be obtained from eq.",
"(REF ).",
"It turns out to be: $\\ddot{p}_b & = & p_b \\Bigg [\\frac{\\cos (b\\delta _b)\\cos (c\\delta _c)}{p_c} + \\frac{\\cos ^2(b\\delta _b)}{\\gamma ^2 \\Delta }(\\sin (b\\delta _b)+\\sin (c\\delta _c))^2 \\nonumber \\\\& & - \\frac{4 \\pi }{\\gamma ^2 \\sqrt{\\Delta }}\\frac{(cp_c-bp_b)}{V}\\cos (c\\delta _c)\\bigg (\\sin (c\\delta _c)+\\sin ^3(b\\delta _b)\\bigg ) \\nonumber \\\\& & -4\\pi G \\bigg (\\cos (b\\delta _b+c\\delta _c)+\\cos (2b\\delta _b)\\bigg )(\\rho +P)\\Bigg ] ~.$ Similarly, eq.",
"(REF ) yields, $\\ddot{p}_c & = & p_c \\Bigg [ -2\\frac{\\sin (b\\delta _b)\\sin (c\\delta _c)}{p_c} + \\frac{4\\sin ^2(b\\delta _b)\\cos ^2(c\\delta _c)}{\\gamma ^2 \\Delta }\\nonumber \\\\& & + \\frac{4 \\pi }{\\gamma ^2 \\sqrt{\\Delta }}\\frac{(cp_c-bp_c)}{V}\\sin (2b\\delta _b)\\bigg (1+\\sin (b\\delta _b)\\sin (c\\delta _c)\\bigg )\\nonumber \\\\& & - 8\\pi G\\bigg (\\cos (b\\delta _b+c\\delta _c)\\bigg )(\\rho +P)\\Bigg ] ~.$ Before we analyze the nature of potential singularities, it is interesting to note that the time derivative of $(c p_c - b p_b)$ in case of matter with vanishing anisotropic stress is given by $\\frac{d}{dt}(cp_c-bp_b)=\\frac{\\gamma p_b}{\\sqrt{p_c}} ~.$ This is easily checked by using eq.",
"(REF ) in eq.",
"(REF ).",
"The right hand side of (REF ) is bounded by virtue of (REF ) and (REF ), which implies that the quantity $(cp_c-bp_b)$ is also bounded at any finite past or future time.",
"We have shown earlier in Sec.",
"III that $p_b$ , $p_c$ and $V$ remain non-zero, positive and finite under evolution over a finite proper time.",
"Equation (REF ) implies that $(cp_c-bp_c)$ is finite upon evolution over a finite time.",
"Hence in effective dynamics in LQC for matter with a vanishing anisotropic stress, both $\\frac{\\ddot{p}_c}{p_c}$ and $\\frac{\\ddot{p}_b}{p_b}$ , and consequently the curvature invariants given by (REF ), () and (REF ) are non-divergent except when the pressure diverges at a finite value of energy density, shear scalar and expansion scalar and non-zero volume.",
"In the next section, we would show that such pressure singularities are weak singularities and geodesics evolution does not break down at such events."
],
[
"Analysis of geodesics and strength of possible singularities",
"In this section, we analyze whether the effective spacetime description of the Kantowski-Sachs model in LQC results in geodesic evolution which breaks down in finite time, and the strength of potential singularities.",
"We start with an analysis of geodesics, followed by the strength of singularities using Królak's condition [3]."
],
[
"Geodesics",
"We noted in the previous section that the curvature invariants are generically bounded in effective dynamics except for very specific type of pressure singularities.",
"This means that there may be potential singularities in the effective spacetime description of Kantowski-Sachs spacetime.",
"A commonly used criterion to characterize singularities is that all the geodesics that go into the singularity must end at the singularity, i.e.",
"the geodesics must not be extendible beyond the singularity.",
"However, if geodesics can be extended beyond the point where the curvature invariants diverge, then it may not be a strong enough singularity to be physically significant.For examples in GR and LQC, see Refs.",
"[4] and [16] respectively.",
"The spacetime may be extendable in such a case.",
"Geodesic (in)completeness analysis is therefore important to understand the exact nature of singularities or lack thereof.",
"For the metric of the Kantowski-Sachs spacetime (REF ), the geodesic equations yield the following second order equations in the affine parameter $\\tau $ : $\\left(g_{xx}x^{\\prime }\\right)^{\\prime }= 0, ~~~~~~ \\left(g_{\\Omega \\Omega }\\sin ^2(\\theta )\\phi ^{\\prime }\\right)^{\\prime } = 0,$ $\\left(g_{\\Omega \\Omega }\\theta ^{\\prime }\\right)^{\\prime } = g_{\\Omega \\Omega }\\sin \\theta \\cos \\theta \\phi ^{\\prime 2},$ and $-2t^{\\prime }t^{\\prime \\prime }=g_{xx}^{\\prime }x^{\\prime 2}+g_{\\Omega \\Omega }^{\\prime }(\\theta ^{\\prime 2} + \\sin ^2\\theta \\phi ^{\\prime 2}) ~.$ Here prime denotes derivative with respect to the affine parameter.",
"And, we recall that the metric components, $g_{xx}$ and $g_{\\Omega \\Omega }$ are related to the triads $p_b$ and $p_c$ via eq.",
"(REF ).",
"To find the solutions, we notice that one can rotate angular coordinates in such a way that initially when affine parameter $\\tau =0$ , $\\theta (0)=\\pi /2$ and $\\theta ^{\\prime }(0)=0$ .",
"Then $\\theta (\\tau )=\\pi /2$ for all $\\tau $ is a solution of the above $\\theta $ geodesic equation with these initial conditions.",
"Due to the uniqueness of solutions of second order differential equations with given initial conditions, this is the unique solution.",
"Therefore, we will assume that $\\theta =\\pi /2$ hereafter.",
"Using this result, the solutions to the remaining geodesic equations in $x$ , $\\phi $ and $t$ are: $x^{\\prime }=\\frac{C_x}{g_{xx}}, ~~~~~ \\phi ^{\\prime }=\\frac{C_{\\phi }}{g_{\\Omega \\Omega }} ~, $ and $t^{\\prime 2}=\\epsilon + \\frac{C_x^2}{g_{xx}}+ \\frac{C_{\\phi }^2}{g_{\\Omega \\Omega }} ~.",
"$ Here $C_x$ and $C_{\\phi }$ are constants of integration, and $\\epsilon =1$ for timelike geodesics and $\\epsilon =0$ for null geodesics.",
"In classical GR, the geodesic equations break down if either $g_{xx}$ or $g_{\\Omega \\Omega }$ vanishes at a finite value of the affine parameter.",
"This is certainly the case for the classical singularity in the Kantowski-Sachs spacetime which results in geodesic incompleteness.",
"The situation changes dramatically, when quantum gravitational effects in LQC are in play.",
"Due to the bounds on the values of $p_b$ and $p_c$ given in (REF ) and (REF ), both $g_{xx}$ and $g_{\\Omega \\Omega }$ , as defined in equation (REF ), are finite, non-zero, positive functions for any finite time.",
"This implies that the geodesic evolution never breaks down in effective dynamics in loop quantized Kantowski-Sachs model.",
"For any finite time evolution, effective spacetime is geodesically complete."
],
[
"Strength of Singularities",
"Apart from the analysis of geodesics, important information about the nature of singularities can be found by analyzing their strength.",
"This is determined by considering what happens to an object as it falls into the singularity.",
"A strong curvature singularity is defined as one that crushes any in-falling objects to zero volume irrespective of the properties or composition of the objects [1], [2].",
"Basically the curvature squeezes any in-falling objects to infinite density.",
"Infinite tidal forces completely destroy any arbitrary in-falling object.",
"In contrast to the strong singularities, weak singularities do not imply a complete destruction of the in-falling objects.",
"Even though some curvature components or curvature invariants may diverge, it is possible to construct a sufficiently strong detector which survives large tidal forces and escapes the singular event.",
"These qualitative notions has been put in precise mathematical terms by Tipler [2] and Królak [3].",
"It has been conjectured that the physical singularities in the sense of geodesic incompleteness are those which are also strong curvature type [2], [3].",
"It has been argued that if the conjecture is satisfied then a weak form of Penrose's cosmic censorship hypothesis can be proved [3].",
"The necessary conditions for a strong curvature singularity derived by Królak are broader than Tipler's conditions.",
"Any singularity which is weak by Królak's conditions will be weak by Tipler criteria, but the converse is not true.",
"So we use the Królak conditions in order to search for the signs of strong singularities in the broadest sense.",
"According to necessary conditions due to Królak [3], if a singularity is a strong curvature singularity, then for some non-spacelike geodesic running into the singularity, the following integral diverges as the singularity is approached: $K^{i}_{j} =\\int _0^{\\tau } d\\tilde{\\tau }|R^{i}_{4j4} (\\tilde{\\tau })| ~.",
"$ That is, if there is a strong curvature singularity in the region, then for a non-spacelike geodesic running into the singularity the following necessary condition is satisfied: $\\lim _{\\tau \\rightarrow \\tau _o}K^{i}_{j} \\rightarrow \\infty $ where $\\tau _o$ is the value of the affine parameter at which the singularity is located.",
"Considering the behavior of the integrand, i.e.",
"components of Riemann tensor, can lead us to understand which terms may potentially diverge and result in strong singularities.",
"The non-zero components of the Riemann tensor for the Kantowski-Sachs metric in terms of the triads are: $R^1_{212}&=&g_{xx}R^2_{121} \\nonumber \\\\&=&\\bigg (\\frac{p_b^2}{L_o^2 p_c} \\bigg ) \\bigg [ -\\frac{3}{4} \\bigg (\\frac{\\dot{p}_c}{p_c} \\bigg )^2 - \\bigg (\\frac{\\ddot{p}_b}{p_b} \\bigg )+ \\bigg (\\frac{\\dot{p}_b}{p_b} \\bigg )\\bigg (\\frac{\\dot{p}_c}{p_c} \\bigg ) + \\frac{1}{2}\\bigg (\\frac{\\ddot{p}_c}{p_c} \\bigg ) \\bigg ] ,\\\\R^1_{441}&=& \\sin ^2\\theta R^1_{331} = p_c\\sin ^2\\theta R^3_{131} = p_c\\sin ^2\\theta R^4_{141} \\nonumber \\\\&=& p_c\\sin ^2\\theta \\bigg [ \\frac{1}{4}\\bigg (\\frac{\\dot{p}_c}{p_c} \\bigg )^2 - \\frac{1}{2}\\bigg (\\frac{\\ddot{p}_c}{p_c} \\bigg ) \\bigg ], \\\\R^2_{442}&=&\\sin ^2\\theta R^2_{332}=-\\frac{p_c}{g_{xx}}R^3_{232}=-\\frac{p_c}{g_{xx}}R^4_{242}\\nonumber \\\\&=&-p_c \\sin ^2\\theta \\bigg [\\frac{1}{2}\\bigg (\\frac{\\dot{p}_b}{p_b} \\bigg )\\bigg (\\frac{\\dot{p}_c}{p_c} \\bigg ) - \\frac{1}{4}\\bigg (\\frac{\\dot{p}_c}{p_c} \\bigg )^2 \\bigg ],$ and, $R^3_{443}=-\\sin ^2\\theta R^4_{343}= -\\sin ^2\\theta \\bigg [ 1+\\frac{p_c}{4}\\bigg (\\frac{\\dot{p}_c}{p_c} \\bigg )^2 \\bigg ] ~.$ Note that the factors of $\\sin ^2\\theta $ can be ignored in this analysis, as we can always choose $\\theta =\\pi /2$ along our geodesics as discussed in Sec.",
"VA.",
"Most of the terms in all the Riemann tensor components are of the type $\\bigg (\\frac{\\dot{p}_1}{p_1}\\bigg )^m \\bigg (\\frac{\\dot{p}_2}{p_2}\\bigg )^n \\bigg (\\frac{\\dot{p}_3}{p_3}\\bigg )^q f(p_1,p_2,p_3)$ , which are made out of products of powers of $p_b, p_c, \\frac{\\dot{p}_c}{p_c}$ and $\\frac{\\dot{p}_b}{p_b}$ , which are functions of the affine parameter.",
"The other type of terms are $g(p_b,p_c)\\frac{\\ddot{p}_c}{p_c}$ or $g(p_b,p_c)\\frac{\\ddot{p}_b}{p_b}$ , where $g(p_b,p_c)$ is a function only of $p_b$ and $p_c$ without involving any of their derivatives.",
"First note that the integral in (REF ) involves an integral of the absolute value of Riemann tensor components, and in turn each Riemann tensor component is a sum of several terms.",
"Since the integral of the absolute value of a sum is always less than or equal to the integral of the sum of the absolute value of each term, for our purposes it would suffice to look individually at the integrals of the absolute value of each term separately.",
"So we consider the different types of terms present in the Riemann tensor components mentioned above one by one.",
"We first look at terms of type $\\bigg (\\frac{\\dot{p}_1}{p_1}\\bigg )^m \\bigg (\\frac{\\dot{p}_2}{p_2}\\bigg )^n \\bigg (\\frac{\\dot{p}_3}{p_3}\\bigg )^q f(p_1,p_2,p_3)$ .",
"We can split the integral from 0 to $\\tau _o$ into pieces where the integrand takes a definite sign (positive or negative), e.g.",
"$\\int _0^{\\tau _o} d\\tau \\bigg | \\bigg (\\frac{\\dot{p}_1}{p_1}\\bigg )^m \\bigg (\\frac{\\dot{p}_2}{p_2}\\bigg )^n \\bigg (\\frac{\\dot{p}_3}{p_3}\\bigg )^q f(p_1,p_2,p_3)\\bigg | &=& \\int _0^{\\tau _1} d\\tau \\bigg (\\frac{\\dot{p}_1}{p_1}\\bigg )^m \\bigg (\\frac{\\dot{p}_2}{p_2}\\bigg )^n \\bigg (\\frac{\\dot{p}_3}{p_3}\\bigg )^q f(p_1,p_2,p_3) \\nonumber \\\\& & - \\int _{\\tau _1}^{\\tau _2} d\\tau \\bigg (\\frac{\\dot{p}_1}{p_1}\\bigg )^m \\bigg (\\frac{\\dot{p}_2}{p_2}\\bigg )^n \\bigg (\\frac{\\dot{p}_3}{p_3}\\bigg )^q f(p_1,p_2,p_3) \\nonumber \\\\& & + \\int _{\\tau _2}^{\\tau _3} d\\tau \\bigg (\\frac{\\dot{p}_1}{p_1}\\bigg )^m \\bigg (\\frac{\\dot{p}_2}{p_2}\\bigg )^n \\bigg (\\frac{\\dot{p}_3}{p_3}\\bigg )^q f(p_1,p_2,p_3) \\nonumber \\\\& & .",
"\\nonumber \\\\& & .",
"\\nonumber \\\\& & .",
"\\nonumber \\\\& & + \\int _{\\tau _n}^{\\tau _0} d\\tau \\bigg (\\frac{\\dot{p}_1}{p_1}\\bigg )^m \\bigg (\\frac{\\dot{p}_2}{p_2}\\bigg )^n \\bigg (\\frac{\\dot{p}_3}{p_3}\\bigg )^q f(p_1,p_2,p_3) \\nonumber $ Now focus on any one of the terms from the above expression, say $\\tau _k$ to $\\tau _{k+1}$ , $\\int _{\\tau _k}^{\\tau _{k+1}} d\\tau \\bigg (\\frac{\\dot{p}_1}{p_1}\\bigg )^m \\bigg (\\frac{\\dot{p}_2}{p_2}\\bigg )^n \\bigg (\\frac{\\dot{p}_3}{p_3}\\bigg )^q f(p_1,p_2,p_3)&=&\\int _{t_k}^{t_{k+1}} dt \\frac{d\\tau }{dt} \\bigg (\\frac{\\dot{p}_1}{p_1}\\bigg )^m \\bigg (\\frac{\\dot{p}_2}{p_2}\\bigg )^n \\bigg (\\frac{\\dot{p}_3}{p_3}\\bigg )^q f(p_1,p_2,p_3) \\nonumber \\\\[10pt]&=&\\int _{t_k}^{t_{k+1}} dt \\frac{\\bigg (\\frac{\\dot{p}_1}{p_1}\\bigg )^m \\bigg (\\frac{\\dot{p}_2}{p_2}\\bigg )^n \\bigg (\\frac{\\dot{p}_3}{p_3}\\bigg )^q f(p_1,p_2,p_3)}{\\sqrt{\\epsilon + \\frac{C_x^2 L_o^2 p_c}{p_b^2} + \\frac{C_{\\phi }^2}{p_c}}} ~.$ Here we have used eq.",
"(REF ), and note that $\\epsilon $ is 1 for timelike geodesics and 0 for null geodesics.",
"We have shown earlier in Sec.",
"III, particularly equations (), (REF ), (REF ) and (REF ) that $\\frac{\\dot{p}_b}{p_b}$ and $\\frac{\\dot{p}_c}{p_c}$ are bounded functions, and $p_b, p_c$ are non-zero and finite as well for all finite values of the coordinate time $t$ .",
"The quantity under the square root in the denominator of equation (REF ) is therefore positive definite because of the bounds on $p_b$ and $p_c$ .",
"In the following, let us consider the special case when both of the integration constants of the geodesic equations, $C_x$ and $C_{\\phi }$ , happen to be simultaneously zero.",
"In the timelike case, since $\\epsilon $ is equal to unity, we see from (REF ) and (REF ) that it represents the worldline of a massive particle sitting at rest at a location in space, and the denominator in (REF ) becomes unity.",
"However, in the case of null geodesics (photons), since $\\epsilon $ is zero it seems that the denominator in the R.H.S.",
"of (REF ) could be zero if both $C_x$ and $C_{\\phi }$ vanish simultaneously.",
"But we find from (REF ) and (REF ) that if both $C_x$ and $C_{\\phi }$ are simultaneously zero, then we have the peculiar situation with coordinates $x,\\phi $ and $t$ being constant as a function of the affine parameter.",
"This means that the whole geodesic will be just one event in the spacetime manifold, i.e.",
"the photon appears for one moment at some location and disappears immediately.",
"Such a case is hence not relevant for our discussion of the strength of singularities as it does not correspond to a physically suitable null geodesic.",
"Thus, we show that the integrand in R.H.S.",
"of (REF ) is well-defined, real and finite for all finite values of the time $t$ .",
"That means that the integral will be finite if the upper limit $t_o$ is finite.",
"If initially both $\\tau $ and $t$ start from zero, then $\\tau _0=\\int _0^{\\tau _o}d\\tau =\\int _0^{t_o} dt \\frac{d\\tau }{dt}=\\int _0^{t_o} \\frac{dt}{\\sqrt{\\epsilon + \\frac{C_x^2 L_o^2 p_c}{p_b^2} +\\frac{C_{\\phi }^2}{p_c}}} .$ Note that for observers comoving with respect to the matter world lines (the fundamental observers), the proper time is given by $t$ , hence $\\tau _0 = t_0$ .",
"Hence for a finite $\\tau _0$ , $t_0$ is always finite for such observers.",
"The integral in (REF ) is then finite for finite $\\tau _0$ .",
"In general, the integrand on the R.H.S.",
"of (REF ) is positive definite and finite for finite $t$ .",
"It is possible that for certain geodesics there can be potential cases where the upper limit $t_0$ is infinite even when $\\tau _0$ is finite.",
"If such a case exists and if the integral (REF ) diverges in such a case, then this divergence occurs at an infinite proper time for fundamental observers.",
"Further, for such a potential divergence the energy density is still finite in the finite time evolution for the matter world lines (using the results from Sec.",
"III).",
"Hence, such a potential divergence would not correspond to any known strong singularity such as big bang/crunch, big rip and big freeze singularities which are characterized by divergence in energy density in finite proper time for fundamental observers.",
"Thus, we can conclude that terms in the Riemann curvature components of the type $\\bigg (\\frac{\\dot{p}_1}{p_1}\\bigg )^m \\bigg (\\frac{\\dot{p}_2}{p_2}\\bigg )^n \\bigg (\\frac{\\dot{p}_3}{p_3}\\bigg )^q f(p_1,p_2,p_3)$ in (REF ) will not contribute to any potential divergences in finite proper time measured by comoving observers.",
"Let us now consider the other type of terms in the Riemann tensor components.",
"These are of the type $g(p_b,p_c)\\frac{\\ddot{p}_c}{p_c}$ or $g(p_b,p_c)\\frac{\\ddot{p}_b}{p_b}$ with $g(p_b,p_c)$ independent of any derivatives.",
"Again as before, we split the integral of the absolute value into pieces where the integrand has a definite sign, and then look at one of those pieces.",
"These terms will be integrated at least once in (REF ), and it can be seen that on integrating by parts, there are no terms left with double derivatives of $p_b$ or $p_c$ .",
"For example, $\\int g(p_b,p_c)\\frac{\\ddot{p}_c}{p_c}d\\tau &=& \\int \\frac{g(p_b,p_c)}{\\sqrt{\\epsilon + \\frac{C_x^2 L_o^2 p_c}{p_b^2} + \\frac{C_{\\phi }^2}{p_c}}}\\frac{\\ddot{p}_c}{p_c}dt =: \\int g_1(p_b,p_c)\\ddot{p}_c dt \\nonumber \\\\&=& g_1(p_b,p_c)\\dot{p}_c - \\int \\dot{p}_c \\bigg (\\frac{d g_1(p_b,p_c)}{dt}\\bigg )dt \\nonumber \\\\&=& g_2(p_b, p_c)\\frac{\\dot{p}_c}{p_c} - \\int f_2(p_b, p_c, \\frac{\\dot{p}_c}{p_c},\\frac{\\dot{p}_b}{p_b}) dt .$ Here in the last line, we have defined $g_2(p_b, p_c) := p_c g_1(p_b,p_c)$ , and $f_2(p_b, p_c, \\frac{\\dot{p}_c}{p_c},\\frac{\\dot{p}_b}{p_b}) := \\dot{p}_c \\dot{g}_1(p_b,p_c)$ .",
"Note that $f_2(p_b, p_c, \\frac{\\dot{p}_c}{p_c},\\frac{\\dot{p}_b}{p_b})$ is a term of type $\\bigg (\\frac{\\dot{p}_1}{p_1}\\bigg )^m \\bigg (\\frac{\\dot{p}_2}{p_2}\\bigg )^n \\bigg (\\frac{\\dot{p}_3}{p_3}\\bigg )^q f(p_1,p_2,p_3)$ .",
"We have already shown in (REF ) that integrals of terms like $f_2(p_b, p_c, \\frac{\\dot{p}_c}{p_c},\\frac{\\dot{p}_b}{p_b})$ over a finite range of proper time for fundamental observers are non-divergent.",
"And $g_2(p_b, p_c)\\frac{\\dot{p}_c}{p_c}$ is finite for finite values of time $t$ .",
"As for the case of (REF ), integral (REF ) does not result in a divergence occuring in a finite proper time for comoving observers.",
"The terms containing quantities like $\\frac{\\ddot{p}_c}{p_c}$ or $\\frac{\\ddot{p}_b}{p_b}$ , which could lead to potential divergences arising from pressure or derivative of energy density with respect to the triads, as mentioned in Secs.",
"III and IV, are removed upon integrating once.",
"Hence we have established that the necessary condition (REF ) for the presence of strong curvature singularity, i.e equation (REF ), is not satisfied in effective dynamics of Kantowski-Sachs spacetime for any finite proper time measured by fundamental observers.",
"Therefore, all the potential curvature divergent events associated with pressure and derivatives of energy density with triads discussed in Secs.",
"III and IV turn out to be weak singularities."
],
[
"Conclusions",
"A key question for any quantum theory of gravity is whether it can successfully resolve various spacelike singularities.",
"As classical singularities are generic features of the classical continuum spacetime, the analogous question is whether non-existence of singularities is a generic result of quantum spacetime.",
"As there are singularity theorems in classical GR, is there an analogous non-singularity theorem in quantum gravity?",
"Since we do not have a full theory of quantum gravity, these questions can not be fully answered at the present stage.",
"Yet, valuable insights can be gained by understanding whether and how quantum gravitational effects lead to singularity resolution in spacetimes which can be quantized.",
"By systematically studying such spacetimes with increasing complexity, one expects that key features of singularity resolution in general in quantum gravity can be uncovered.",
"Loop quantum cosmology provides a very useful avenue for these studies.",
"In recent years, a rigorous quantization of various spacetimes has been performed, and resolution of singularities in different models has been found [5].",
"The effective spacetime description of LQC enables us to understand singularity resolution in considerable detail.",
"In previous works, using this description above questions on generic resolution of singularities have been addressed in isotropic and Bianchi-I spacetimes [16], [17], [18].",
"In these works, it was found that in the effective spacetime description of LQC all strong singularities are resolved and spacetime is geodesically complete.",
"Our goal in this manuscript was to probe these issues in Kantowski-Sachs spacetime in LQC using Boehmer-Vandersloot prescription [26].",
"In contrast to the previous investigations on this topic, Kantowski-Sachs spacetime is additionally non-trivial.",
"Unlike the isotropic and Bianchi-I spacetime in LQC, energy density is not universally bounded because of the presence of inverse power of a triad component $(p_c)$ .",
"Note that universal bound on energy density played an important role in proving geodesic completeness and resolution of strong singularities in isotropic and Bianchi-I spacetimes [16], [17], [18].",
"It was recently found using numerical simulations that dynamical bounds exist on energy density in loop quantized Kantowski-Sachs model [32], [39].",
"However, an analytical proof was needed to reach general conclusions about singularity resolution.",
"A novel result in our analysis is that in any finite time range, energy density remains finite.",
"Coupled with another result from our present analysis, that volume never becomes zero or infinite in finite time evolution, we find that singularities such as big bang/crunch which occur at zero volume with infinite energy density, big rip singularities occurring at infinite volume with infinite energy density, and big freeze singularities occurring at finite volume but infinite energy density are avoided.",
"The finiteness of energy density, expansion and shear scalars does not imply that curvature invariants are also finite.",
"Investigating their behavior, we find that the curvature invariants remain bounded for all finite times except under certain circumstances.",
"If the matter present is such that the derivatives of the energy density with respect to the triad variables can diverge even though the energy density is finite, then the curvature invariants diverge at these events.",
"By considering the case of matter with vanishing anisotropic stress we show that these triad-derivatives of energy density are related to the pressure.",
"In other words these divergences occur due to divergences in pressure at finite value of energy density.",
"Do these events where curvature invariants diverge imply strong singularities and geodesic incompleteness?",
"The answer turns out to be negative.",
"Analyzing geodesics to understand the nature of the potential singularities indicated by divergences in curvature invariants, we find that the Kantowski-Sachs spacetime is geodesically complete in the effective dynamics of LQC.",
"That means geodesics can be extended beyond the potential singularities where pressure or triad derivatives of energy density diverge at finite energy density, expansion and shear scalars.",
"Using Królak's condition of the strength of the singularities, these potential singularities turn out to be weak.",
"We find that all known strong curvature singularities are non-existent in finite time evolution in effective spacetime.",
"Thus, the only possible singularities in effective spacetime of Kantowski-Sachs model in LQC are weak singularities beyond which geodesic can be extended.",
"Our analysis, thus generalizes previous results on geodesic completeness and strong singularity resolution in LQC to Kantowski-Sachs spacetime providing useful insights on singularity resolution in black hole interior and in presence of anisotropies and spatial curvature.",
"Note that our analytical results though show strong singularity avoidance in any finite time evolution, the question of how exactly the singularity is resolved for a specific matter can be answered only using numerical simulations.",
"Such numerical investigations carried out for scalar fields, massless and in presence of potentials, show that classical singularity is replaced by bounces of triads [26], [27], [32], [39].",
"All these results obtained in different spacetimes imply robust signs of quantum geometric effects as understood in loop quantum gravity yielding a generic resolution of strong singularities.",
"Future investigations with more complex and richer spacetimes are important in this direction.",
"We thank an anonymous referee for useful comments and suggestions on the manuscript which led to its improvement.",
"This work is supported by NSF grants PHYS1404240 and PHYS1454832."
]
] | 1606.04932 |
[
[
"On algorithmization of Janashia-Lagvilava matrix spectral factorization\n method"
],
[
"Abstract We consider three different ways of algorithmization of the Janashia-Lagvilava spectral factorization method.",
"The first algorithm is faster than the second one, however, it is only suitable for matrices of low dimension.",
"The second algorithm, on the other hand, can be applied to matrices of substantially larger dimension.",
"The third algorithm is a superfast implementation of the method, but only works in the polynomial case under the additional restriction that the zeros of the determinant are not too close to the boundary.",
"All three algorithms fully utilize the advantage of the method which carries out spectral factorization of leading principal submatrices step-by-step.",
"The corresponding results of numerical simulations are reported in order to describe the characteristic features of each algorithm and compare them to other existing algorithms."
],
[
"Introduction",
"The Matrix Spectral Factorization (MSF) theorem [22],[10] asserts that if $S=\\begin{pmatrix} s_{11}(t)& s_{12}(t)& \\cdots &s_{1r}(t)\\\\s_{21}(t)& s_{22}(t)& \\cdots &s_{2r}(t)\\\\\\vdots &\\vdots &\\vdots &\\vdots \\\\s_{r1}(t)& s_{r2}(t)&\\cdots &s_{rr}(t)\\end{pmatrix},$ $|t|=1$ , is a positive definite $($ a.e.$)$ matrix function with integrable entries defined on the unit circle ${\\mathbb {T}}$ in the complex plane, $s_{ij}(t)\\in L_1({\\mathbb {T}})$ , and if the Paley-Wiener condition $\\log \\det S(t)\\in L_1({\\mathbb {T}})$ is satisfied, then $(1)$ admits a spectral factorization $S(t)=S^+(t)\\big (S^+(t)\\big )^*.$ Here the entries of $S^+$ are square integrable functions, $s^+_{ij}\\in L_2(\\mathbb {T})$ , which can be extended analytically inside $\\mathbb {T}$ , i.e.",
"$s^+_{ij}$ belongs to the Hardy space $H_2$ .",
"Furthermore a spectral factor $S^+$ can be selected such that $\\det S^+$ is an outer analytic function (see, e.g.",
"[6]) and factorization (REF ) is unique (up to a constant right unitary multiplier) under these conditions.",
"$S^+$ is unique if we require $S^+(0)$ to be positive definite, and we always assume that it satisfies this condition as well.",
"In the scalar case, $r=1$ , the spectral factor $S^+\\in H_2$ can be explicitly written by the formula $S^+(z)=\\exp \\left(\\frac{1}{4\\pi }\\int \\nolimits _0^{2\\pi }\\frac{e^{i\\theta }+z}{e^{i\\theta }-z}\\log S(e^{i\\theta })\\,d\\theta \\right).$ If (REF ) is a Laurent polynomial matrix $S(t)=\\sum _{k=-n}^n C_kt^k,\\;\\; C_k\\in {\\mathbb {C}}^{r\\times r},$ then the spectral factor $S^+(t)=\\sum _{k=0}^n A_kt^k,\\;\\; A_k\\in {\\mathbb {C}}^{r\\times r},$ is a polynomial matrix of the same degree $n$ (see e.g.",
"[4] for an elementary proof).",
"Factorization (REF ) was first used in linear prediction theory of multidimensional stationary processes.",
"Nowadays, it is widely known that MSF plays a crucial role in the solution of various applied problems for multiple-input and multiple-output systems in Communications and Control Engineering [14].",
"Recently MSF became an important step in non-parametric estimations of Granger causality used in Neuroscience [2],[21].",
"These applications require the matrix coefficients of analytic $S^+$ to be determined, at least approximately, for a given matrix function $S$ .",
"Therefore, starting with Wiener's original efforts [23] to create a sound computational method of MSF, dozens of different algorithms have appeared in the literature (see the survey papers [16], [18] and the references therein, and also [1], [11] for more recent results).",
"A novel approach to the solution of the MSF problem, without imposing any additional restriction on $S$ besides the necessary and sufficient condition (REF ) for the existence of spectral factorization, was originally developed by Janashia and Lagvilava in [12] for $2\\times 2$ matrices.",
"This approach was subsequently extended to matrices of arbitrary dimension in [13]This method obtained USPTO patent recently: No.",
"9,318,232; issued April 19, 2016..",
"Results of preliminary numerical simulations based on the proposed method were presented in the same paper [13].",
"However, a closer look at possible algorithmization ways of this method revealed further advantages.",
"In fact, numerical simulations carried out by the improved algorithms produced much better results than it was reported in [13].",
"That this development required additional investigations is not surprising, as all methods of MSF are quite demanding and, as it is mentioned in [16]: “the numerical properties of each method strongly depend on the way it is algorithmized\".",
"In the present paper, after a general description of the Janashia-Lagvilava method (Sections III and IV), we describe three different algorithms of MSF based on this method: JLE-1 (Section VI), JLE-2 (Section VII), and JLE-3 (Section VIII).",
"As it was mentioned above, the method is general and also suitable for non-rational matrices.",
"However, since in practical applications the data is finite, we concentrate our attention on the polynomial case.",
"Furthermore, JLE-algorithm 3 is designed only for polynomial matrices (REF ) with the additional restriction that $\\det S(t)\\ne 0$ for $t\\in \\mathbb {T}$ (the so-called non-singular case).",
"Its theoretical justification is not yet completed.",
"Nevertheless, due to its superfast speed, we present JLE-3 in the current form.",
"The JLE-algorithm 1 is faster than JLE-2 and it can deal with singular case as well, but it is only suitable for low dimensional matrices.",
"JLE-algorithm 2 can be applied for much larger matrices, depending on available time and accuracy.",
"In Section IX, we demonstrate the ability of the method to factorize singular matrices.",
"In Section X, we compare with Wilson's MSF method.",
"The results of provided numerical simulations are presented in Section XI and concluding remarks are given in Section XII.",
"We emphasize that the proposed MSF method uses the existing scalar spectral factorization algorithms, whenever they are called for, and does not attempt to improve upon these."
],
[
"Notation",
"Let $\\mathbb {D}=\\lbrace z\\in \\mathbb {C}:|z|<1\\rbrace $ be the open unit disk, and $\\mathbb {T}=\\partial \\mathbb {D}$ be the unit circle.",
"As usual, $L_p=L_p(\\mathbb {T})$ , $0<p<\\infty $ , denotes the Lebesgue space of $p$ -integrable complex functions defined on $\\mathbb {T}$ ($L_\\infty $ is the space of essentially bounded functions).",
"For $p\\ge 1$ , $\\Vert f\\Vert _p$ is the usual norm.",
"$H_p=H_p(\\mathbb {D})$ , $0<p\\le \\infty $ , is the Hardy space of analytic functions in $\\mathbb {D}$ , $H_p:=\\left\\lbrace f\\in \\mathcal {A}(\\mathbb {D}):\\sup \\limits _{r<1}\\int \\nolimits _0^{2\\pi }|f(re^{i\\theta })|^p\\,d\\theta <\\infty \\right\\rbrace $ ($H_\\infty $ is the space of bounded analytic functions), and $L_p^+=L_p^+(\\mathbb {T})$ denotes the class of their boundary functions.",
"A function $f\\in H_p$ is called outer, denoted $f\\in H_p^O$ , if $f(z)=c\\cdot \\exp \\left(\\frac{1}{2\\pi }\\int \\nolimits _0^{2\\pi }\\frac{e^{i\\theta }+z}{e^{i\\theta }-z}\\log \\big |f(e^{i\\theta })\\big |\\,d\\theta \\right),\\;\\;\\;\\;\\;|c|=1.$ The $n$ th Fourier coefficient of an integrable function $f\\in L_1(\\mathbb {T})$ is denoted by $c_k\\lbrace f\\rbrace $ .",
"For $p\\ge 1$ , $L_p^+(\\mathbb {T})$ coincides with the class of functions from $L_p(\\mathbb {T})$ whose Fourier coefficients with negative indices are equal to zero.",
"The set of trigonometric polynomials is denoted by $\\mathcal {P}$ , i.e.",
"$f\\in \\mathcal {P}$ if $f$ has only a finite number of nonzero Fourier coefficients.",
"In particular, for integers $m\\le n$ , let $\\mathcal {P}_{\\lbrace m,n\\rbrace }:=\\lbrace f\\in \\mathcal {P}:c_k\\lbrace f\\rbrace =0\\text{ whenever } k<m \\text{ or } k>n\\rbrace $ and, for a non-negative integer $N$ , let $\\mathcal {P}_N^+:=\\mathcal {P}_{\\lbrace 0,N\\rbrace }$ , $\\mathcal {P}_N^-:=\\mathcal {P}_{\\lbrace -N,0\\rbrace }$ .",
"Obviously, $f\\in \\mathcal {P}_N^+ \\Leftrightarrow \\overline{f}\\in \\mathcal {P}_N^-$ .",
"For a function $f\\in L_1$ with Fourier expansion $f\\sim \\sum _{n\\in \\mathbb {Z}}c_kt^k$ (or for a formal Fourier series) and positive integer $N$ , let $\\mathbb {P}_N^+$ , $\\mathbb {P}_N^-$ , and $\\mathbb {Q}_N^+$ be the following projection operators: $\\mathbb {P}_N^+[f]=\\sum _{k=0}^N c_kt^k, \\mathbb {P}_N^-[f]=\\sum _{k=0}^N c_{-k}t^{-k},\\text{ and }\\mathbb {Q}_N^-[f]=\\sum _{k=1}^N c_{-k}t^{-k}$ If $M $ is a matrix, then $\\overline{M}$ denotes the matrix with complex conjugate entries and $M^*:=\\overline{M}^T$ .",
"Furthermore, $\\mathbb {C}^{m\\times m}$ , $L_p(\\mathbb {T})^{m\\times m}$ , etc., denote the set of $m\\times m$ matrices with the entries from $\\mathbb {C}$ , $L_p(\\mathbb {T})$ , etc.",
"If $S\\in \\mathbb {C}^{r\\times r}$ is a matrix (function) and $m\\le r$ , then $S_{[m]}$ stands for the upper-left $m\\times m$ submatrix of $S$ ($S_{[0]}$ is assumed to be 1) and $S_{[1:\\,r,m]}$ stands for $m$ th column of $S$ .",
"Matrices like $S_{[1:\\,r-1,m]}$ or $S_{[1:\\,r-1,1:\\,m]}$ are defined accordingly.",
"The matrix $S_{]i,j[}$ is obtained from $S$ by deleting the $i$ th row and $j$ th column.",
"For a polynomial $p(t)=\\sum _{k=0}^k p_kt^k$ , let $\\Vert p\\Vert =\\sup _{0\\le k\\le n}|p_k|$ , and for a polynomial matrix $P=\\big (P_{ij}\\big )_{i,j=1}^r$ , let $\\Vert P\\Vert =\\sup _{1\\le i,j\\le r}\\Vert P_{ij}\\Vert $ .",
"A matrix $M\\in \\mathbb {C}^{r\\times r}$ is called positive definite if $X^*MX>0$ for all $0\\ne X\\in \\mathbb {C}^{r\\times 1}$ , and $S\\in L_1(\\mathbb {T})^{r\\times r}$ is called positive definite if it is positive definite for a.a. $t\\in \\mathbb {T}$ .",
"A matrix function $U\\in L^\\infty (\\mathbb {T})^{r\\times r}$ is called unitary if $U(t)U^*(t)=I_r \\;\\;\\text{a.e.",
"},$ where $I_r$ stands for $r\\times r$ identity matrix.",
"$\\mathbf {0}_{r\\times m}$ and $\\mathbf {1}_{r\\times m}$ stand for $r\\times m$ matrices with all entries equal to 0 and 1, respectively.",
"Using Matlab's notation, if $A\\in \\mathbb {C}^{r\\times m_1}$ and $B\\in \\mathbb {C}^{r\\times m_2}$ , then $[A\\;B]$ is $r\\times (m_1+m_2)$ matrix, while if $A\\in \\mathbb {C}^{r_1\\times m}$ and $B\\in \\mathbb {C}^{r_2\\times m}$ , then $[A\\,;\\;B]=[A^T\\;B^T]^T$ is $(r_1+r_2)\\times m$ matrix.",
"For a column vector $\\mathbf {a}=[a_0\\,a_1\\,\\cdots \\,a_{l}]^T\\in \\mathbb {C}^{(l+1)\\times 1}$ and a positive integer $m\\in \\mathbb {N}$ , let $T(\\mathbf {a};m)$ be the $(l+m+1)\\times (m+1)$ Toeplitz matrix with the first column $[\\mathbf {a}\\,; \\mathbf {0}_{m\\times 1}]\\in \\mathbb {C}^{l+m+1}$ and the first row $[a_0\\,\\mathbf {0}_{1\\times m}]\\in \\mathbb {C}^{1\\times (m+1)}$ .",
"We say that a sequence of matrix functions $S_n$ , $n=1,2,\\ldots $ is convergent to a matrix function $S$ (in some sense) if the entries of $S_n$ are convergent to the corresponding entries of $S$ (in this sense).",
"Finally, $\\delta _{ij}$ stands for the Kronecker delta, i.e.",
"$\\delta _{ij}=1$ if $i=j$ and $\\delta _{ij}=0$ otherwise."
],
[
"General description of the method ",
"The first step of the MSF method proposed in [13] is the triangular factorization of (REF ) $S(t)=M(t)M^*(t),$ where $M(t)$ is the lower triangular matrix $M(t)=\\begin{pmatrix}f^+_1(t)&0&\\cdots &0&0\\\\\\xi _{21}(t)&f^+_2(t)&\\cdots &0&0\\\\\\vdots &\\vdots &\\vdots &\\vdots &\\vdots \\\\\\xi _{r-1,1}(t)&\\xi _{r-1,2}(t)&\\cdots &f^+_{r-1}(t)&0\\\\\\xi _{r1}(t)&\\xi _{r2}(t)&\\cdots &\\xi _{r,r-1}(t)&f^+_r(t)\\end{pmatrix},$ $\\xi _{ij}\\in L_2(\\mathbb {T})$ , $f_i^+\\in H_2^O$ .",
"The spectral factor $S^+$ is represented in the form $S^+(t)=M(t)\\mathbf {U}_2(t)\\mathbf {U}_3(t)\\ldots \\mathbf {U}_r(t)\\cdot U.$ Here each $\\mathbf {U}_m$ is a block matrix function $\\mathbf {U}_m(t)=\\begin{pmatrix}U_{m}(t)&\\mathbf {0}_{m\\times (r-m)}\\\\\\mathbf {0}_{(r-m)\\times m}&I_{r-m}\\end{pmatrix},$ where $U_m(t)$ is a special unitary matrix function of the form $U_m(t)=\\begin{pmatrix}u_{11}(t)&u_{12}(t)&\\cdots &u_{1,m-1}(t)&u_{1m}(t)\\\\u_{21}(t)&u_{22}(t)&\\cdots &u_{2,m-1}(t)&u_{2m}(t)\\\\\\vdots &\\vdots &\\vdots &\\vdots &\\vdots \\\\u_{m-1,1}(t)&u_{m-1,2}(t)&\\cdots &u_{m-1,m-1}(t)&u_{m-1,m}(t)\\\\[3mm]\\overline{u_{m1}(t)}&\\overline{u_{m2}(t)}&\\cdots &\\overline{u_{m,m-1}(t)}&\\overline{u_{mm}(t)}\\\\\\end{pmatrix},$ with $ u_{ij}\\in L^\\infty _+,\\;\\; \\text{ and }\\;\\; \\det U(t)=1 \\text{ a.e.",
"}$ (for reasons explained in [7] such matrices can as well be called “wavelet matrices\").",
"Furthermore, for each $m=2,3,\\ldots ,r$ , $S_{[m]}^+=\\big (M\\mathbf {U}_2\\mathbf {U}_3\\ldots \\mathbf {U}_m\\big )_{[m]}$ is a spectral factor of $S_{[m]}$ .",
"In particular, $S^+_0:=M\\mathbf {U}_2\\mathbf {U}_3\\ldots \\mathbf {U}_r$ is a spectral factor of (REF ), and the constant unitary matrix $U$ in (REF ) makes $S^+$ positive definite in the origin, namely (see [5]) $U=\\big (S^+_0(0)\\big )^{-1}\\sqrt{S^+_0(0)(S^+_0(0))^*}.$ To obtain unitary matrix function (REF ) for each $m=2,3,\\ldots ,r$ recurrently, we consider a matrix function $F_m(t)=\\begin{pmatrix}1&0&0&\\cdots &0&0\\\\0&1&0&\\cdots &0&0\\\\0&0&1&\\cdots &0&0\\\\\\vdots &\\vdots &\\vdots &\\vdots &\\vdots &\\vdots \\\\0&0&0&\\cdots &1&0\\\\\\zeta _{1}(t)&\\zeta _{2}(t)&\\zeta _{3}(t)&\\cdots &\\zeta _{m-1}(t)&f^+_m(t)\\end{pmatrix},$ where the last row of (REF ) consists of the first $m$ entries of the $m$ th row of the product $M_{m-1}:=M\\mathbf {U}_2\\mathbf {U}_3\\ldots \\mathbf {U}_{m-1},$ and then obtain a matrix function (REF ), (REF ) such that (see [5]) $F_mU_m\\in L_2^+(\\mathbb {T})^{m\\times m}.$ Particularly, we have $\\big (M_{m-1}\\big )_{[m]}=\\left[\\begin{matrix}& S_{[m-1]}^+(t)& & \\begin{matrix}0\\\\0\\\\ \\vdots \\\\0\\end{matrix}\\\\\\zeta _1(t) & \\ldots & \\zeta _{m-1}(t)& f^+_m(t)\\end{matrix}\\right]=\\left[\\begin{matrix}& S_{[m-1]}^+(t)& & \\begin{matrix}0\\\\0\\\\ \\vdots \\\\0\\end{matrix}\\\\0 & \\ldots & 0& 1\\end{matrix}\\right] F_m(t)$ and $S_{[m]}^+(t)=\\left[\\begin{matrix}& & S_{[m-1]}^+(t)& & \\begin{matrix}0\\\\0\\\\ \\vdots \\\\0\\end{matrix}\\\\0 & 0 & \\ldots & 0& 1\\end{matrix}\\right] F_m(t) U_m(t).$ In order to achieve (REF ), one needs to consider the following system of conditions (see [13]) ${\\left\\lbrace \\begin{array}{ll} \\zeta _1(t)x^+_m(t)-f_m^+(t)\\overline{x^+_1(t)}\\in L_2^+,\\\\\\zeta _2(t)x^+_m(t)-f_m^+(t)\\overline{x^+_2(t)}\\in L_2^+,\\\\\\vdots \\\\\\zeta _{m-1}(t)x^+_m(t)-f_m^+(t)\\overline{x^+_{m-1}(t)}\\in L_2^+,\\\\\\zeta _1(t)x^+_1(t)+\\zeta _2(t)x^+_2(t)+\\ldots +\\zeta _{m-1}(t)x^+_{m-1}(t)+f_m^+(t)\\overline{x^+_m(t)}\\in L_2^+,\\end{array}\\right.",
"}$ and columns of (REF ) are $m$ independent solutions of (REF ).",
"To construct (REF ) approximately the following procedures should be performed: For a large positive $N$ , let $F_m^{\\lbrace N\\rbrace }$ be the matrix function (REF ) with the last row replaced by $(\\zeta _1^{\\lbrace N\\rbrace },\\zeta _2^{\\lbrace N\\rbrace },\\ldots ,\\zeta _{m-1}^{\\lbrace N\\rbrace },f_m^+),$ where $\\zeta _j^{\\lbrace N\\rbrace }(t):=\\sum _{k=-N}^\\infty c_k\\lbrace \\zeta _j\\rbrace t^{k},\\;\\;\\;j=1,2,\\ldots ,m-1.$ Then one can find the unitary matrix function $U_m^{\\lbrace N\\rbrace }$ of the form (REF ) such that $\\det U_m^{\\lbrace N\\rbrace }(t)=1$ , $U_m^{\\lbrace N\\rbrace }(1)=I_m$ , $u_{ij}\\in \\mathcal {P}_N^+$ and $F_m^{\\lbrace N\\rbrace }U_m^{\\lbrace N\\rbrace }\\in \\mathcal {P}_N^+$ (see [13]).",
"In particular, the columns of $U_m^{\\lbrace N\\rbrace }$ are $m$ independent solutions of the system (REF ) where $\\zeta _1,\\zeta _2,\\ldots ,\\zeta _{m-1}$ are replaced by $\\zeta _1^{\\lbrace N\\rbrace },\\zeta _2^{\\lbrace N\\rbrace },\\ldots ,\\zeta _{m-1}^{\\lbrace N\\rbrace }$ , and they can be actually found by solving a single system of $(N+1)\\times (N+1)$ linear algebraic equations with $m$ different right-hand sides (see the proof of Theorem 1 in [13]).",
"Details of the computation are given in Section IV.",
"One can prove that $U_m^{\\lbrace N\\rbrace }\\rightarrow U_m$ at least in measure as $N\\rightarrow \\infty $ , which guarantees that (see [5]) $M\\mathbf {U}_2\\mathbf {U}_3\\ldots \\mathbf {U}_{m}^{\\lbrace N\\rbrace }\\rightarrow M\\mathbf {U}_2\\mathbf {U}_3\\ldots \\mathbf {U}_{m}\\;\\;\\text{ in } L_2.$"
],
[
"Construction of wavelet matrices",
"In this section we provide the details of computation of the unitary matrix function $U_N:=U_m^{\\lbrace N\\rbrace }$ for a given matrix function (REF ).",
"$N$ and $m$ are assumed fixed throughout this section.",
"Let $\\mathbb {P}_N^+[f_m^+](t)=\\sum _{k=0}^N d_kt^k,\\;\\;\\;\\mathbb {Q}_N^-[\\zeta _i](t)=\\sum _{k=1}^N\\gamma _{in}t^{-k},\\;\\;\\text{ and } \\;\\;\\mathbb {P}_N^+[1/f_m^+](t)=\\sum _{k=0}^Nb_kt^k.$ (Note that the knowledge of $\\mathbb {P}_N^+[f_m^+]$ is sufficient to determine $\\mathbb {P}_N^+[1/f_m^+]$ .)",
"Suppose $D^{-1}$ is the upper triangular Toeplitz matrix with the first row $(b_0,b_1,\\ldots ,b_N),$ and $\\Gamma _i$ , $i=1,2,\\ldots ,m-1$ is the upper triangular Hankel matrix withe the first row $(0,\\gamma _{i,1},\\gamma _{i,2},\\ldots ,\\gamma _{iN})$ (see [13]) and let $\\Theta _i=D^{-1}\\,\\Gamma _i\\,,\\;\\;i=1,2,\\ldots ,m-1.$ Note that $\\Theta _i$ is the upper triangular Hankel matrix (see [13]) with the first row $\\Lambda _i:=(\\eta _{i0}, \\eta _{i1},\\ldots ,\\eta _{iN}),$ where $\\sum _{k=0}^N \\eta _{in}t^{-k}=\\mathbb {P}_N^-\\left[\\sum _{k=0}^Nb_kt^k\\cdot \\sum _{k=1}^N\\gamma _{in}t^{-k}\\right]$ .",
"Take $\\Delta =\\sum _{i=1}^{m-1}\\Theta _i\\Theta _i^*+I_{N+1},$ which is a positive definite matrix (with all eigenvalues $\\ge 1$ ), and solve the same system of equations (see (REF )) $\\Delta X=\\Lambda _i^T$ with $m$ different right hand sides corresponding to $i=1,2,\\ldots ,m$ .",
"Here it is assumed that $\\Lambda _m=(1,0,0,\\ldots ,0)$ .",
"The matrix (REF ) has a displacement structure of rank $m$ , namely $\\Delta -Z\\Delta Z^*=\\sum _{i=1}^{m-1}\\Lambda _i\\Lambda _i^*+\\mathcal {E}\\mathcal {E}^*$ has rank $m$ , where $Z$ is the upper triangular $(N+1)\\times (N+1)$ matrix with 1's on the first up-diagonal and 0's elsewhere (i.e.",
"a Jordan block with eigenvalue 0) and ${\\mathcal {E}}=(0,0,\\ldots ,0,1)^T \\in \\mathbb {C}^{N+1,1}$ (see [13]).",
"Therefore its triangular factorization $\\Delta =LDL^*$ can be achieved in $O(mN^2)$ operations instead of $O(N^3)$ as explained e.g.",
"in [14] without even constructing the matrix $\\Delta $ (just using the $(N+1)\\times m$ matrix $[\\Lambda _1,\\Lambda _2,\\ldots ,\\Lambda _{m-1},\\mathcal {E}]$ ).",
"Let the solution of (REF ) be $X_i=(a_{i0},a_{i1},\\ldots , a_{iN})^T$ , and denote $v_{mi}(t):=\\sum _{k=0}^N a_{in}t^k,\\;\\;\\;\\;i=1,2,\\ldots ,m,$ Suppose also $v_{ij}(t)=\\mathbb {P}_N^+\\left[\\sum _{k=0}^N \\overline{\\eta }_{n}t^k\\cdot \\sum _{k=0}^N \\overline{a_{in}}t^{-k}\\right]-\\delta _{ij},$ $1\\le i\\le m-1$ , $1\\le j\\le m$ , and let $V(t)=\\begin{pmatrix}v_{11}(t)&v_{12}(t)&\\cdots &v_{1,m-1}(t)&v_{1m}(t)\\\\v_{21}(t)&v_{22}(t)&\\cdots &v_{2,m-1}(t)&v_{2m}(t)\\\\\\vdots &\\vdots &\\vdots &\\vdots &\\vdots \\\\v_{m-1,1}(t)&v_{m-1,2}(t)&\\cdots &v_{m-1,m-1}(t)&v_{m-1,m}(t)\\\\[3mm]\\overline{v_{m1}(t)}&\\overline{v_{m2}(t)}&\\cdots &\\overline{v_{m,m-1}(t)}&\\overline{v_{mm}(t)}\\\\\\end{pmatrix}.$ Then (see [13]) $U_N(t)=V(t)\\cdot V^{-1}(0).$ It is proved in [13] that $V(0)$ is nonsingular and the condition number of this matrix is estimated in [8]."
],
[
"A shortcut in the recursive step",
"As it was mentioned in Section III, in order to perform $m$ th recursive step in the proposed MSF method, we need only to consider $S_{[m-1]}^+=\\big (M_{m-1}\\big )_{[m-1]}$ (see (REF ) and (REF )), which has already been constructed (at least approximately) and the first $m$ entries in the $m$ th row of $M_{m-1}$ $\\big (M_{m-1}\\big )_{[m,1:m]}=(\\zeta _1,\\zeta _2,\\ldots ,\\zeta _{m-1},f^+_m)$ (see (REF )).",
"Because of the block structure of matrices in (REF ), the entry $f^+_m$ is the same as in (REF ).",
"Thus it can be computed by the formula (see [13]) $f^+_m=\\frac{\\big (\\det S_{[m]}\\big )^+}{\\big (\\det S_{[m-1]}\\big )^+}$ ( $(\\cdot )^+$ stands for the scalar spectral factorization (REF ) ).",
"Since $S=M_{m-1}M_{m-1}^*$ (see (REF ), (REF ), and (REF )) and particularly $(S)_{[m]}=\\big (M_{m-1}\\big )_{[m]}\\big (M_{m-1}\\big )_{[m]}^*$ (see (REF )), we have $S_{[m-1]}^+\\cdot \\big (\\zeta _1,\\zeta _2,\\ldots ,\\zeta _{m-1}\\big )^*=S_{[1:\\,m-1,\\,m]}.$ Therefore, instead of computing matrices $M_m$ for each $m=2,3,\\ldots ,r-1$ by (REF ), we can directly compute the entries $\\zeta _1,\\zeta _2,\\ldots ,\\zeta _{m-1}$ from (REF ).",
"Having computed the functions $\\zeta _1,\\zeta _2,\\ldots ,\\zeta _{m-1}$ , one can find $|f^+_m|^2$ from the formula (see (REF )) $\\sum _{j=1}^{m-1}|\\zeta _j|^2+|f_m^+|^2=s_{mm}.$ Therefore, an alternative way of computing (REF ) is the scalar spectral factorization of $s_{mm}-\\sum _{j=1}^{m-1}|\\zeta _j|^2$ .",
"In the next three sections we present three different implementations of the described algorithm for polynomial data (REF ), followed by the results of corresponding numerical simulations."
],
[
"JLE-algorithm 1",
"This algorithm relies on computation of polynomial matrix determinant.",
"Namely, for a polynomial matrix of order $n$ $P(t)=\\sum _{k=0}^n B_kt^k,\\;\\; B_k\\in {\\mathbb {C}}^{m\\times m},$ $\\det P$ is a polynomial of order $mn$ .",
"Therefore, having evaluated $\\det P(t)$ at $mn+1$ DFT nodes $t_l=\\exp \\left(\\frac{2\\pi il}{mn+1}\\right)$ , $l=0,1,\\ldots ,mn$ , the coefficients of $\\det P$ can be computed by interpolation, namely computing the inverse DFT of $[\\det P(t_0), \\ldots ,\\det P(t_{mn})]$ .",
"This algorithm of polynomial matrix determinant computation is fast and accurate for matrices of small dimension.",
"However, the algorithm suffers from severe round-off errors and the accuracy is destroyed for large dimensional matrices.",
"For example, with a standard double precision in Matlab, we have found a computation error in the formula $\\Vert \\det (P_1P_2)-\\det P_1\\,\\det P_2\\Vert $ as small as $10^{-8}$ for randomly selected polynomial matrices $P_1$ and $P_2$ of degree $n=10$ and dimension $m=10$ , and as large as $10^9$ for ones with $n=20$ and $m=15$ .",
"The reason of such increase is that the coefficients of $\\det P$ become very large (at least for randomly selected coefficients $B_k$ in (REF )) and floating point machine arithmetic loses significant digits.",
"Therefore JLE-algorithm 1 (with input (REF ) and output (REF )) is suitable for small dimensional matrices ($r<20$ and $n<25$ ).",
"Its basic computational procedures are described below.",
"Procedure 1.",
"Compute the diagonal entries of the triangular factor (REF ) by the formula (REF ), where $m=1,2,\\ldots ,r$ .",
"Each $f_m^+$ can be represented as a rational function $p_m/q_m$ , where $p_m\\in \\mathcal {P}_{mn}^+$ and $q_m\\in \\mathcal {P}_{(m-1)n}^+$ .",
"In addition, the denominator is free of zeros inside $\\mathbb {T}$ , and $f_m^+$ is free of poles on $\\mathbb {T}$ (since $f_m^+\\in L_2^+(\\mathbb {T})$ ).",
"For the scalar spectral factorization of $\\det S_{[m]}$ , we first apply exp-log implementation by using FFT [9] and then we improve the accuracy by using 4-5 iterations of Wilson's scalar factorization algorithm [24].",
"Procedure 2.",
"For $m=2,3,\\ldots ,r$ , assume that $S_{[m-1]}^+$ has already been (approximately) constructed as an $(m-1)\\times (m-1)$ polynomial matrix of degree $n$ and perform the following steps.",
"Step 1.",
"Compute $\\zeta _j$ , $j=1,2,\\ldots ,m-1$ , by the Cramer's rule from equation (REF ).",
"In particular, each $\\overline{\\zeta _j}$ will be of the form $p/q$ , where $p\\in \\mathcal {P}_{\\lbrace -n,\\,(m-1)n\\rbrace }$ and $q\\in \\mathcal {P}_{(m-1)n}^+$ , again with $q$ free of zeros inside $\\mathbb {T}$ and $\\zeta _j$ free of poles on $\\mathbb {T}$ .",
"Note that $\\zeta _j$ -s can be computed in parallel.",
"Step 2.",
"Select a large positive integer $N$ .",
"Theoretically, as $N\\rightarrow \\infty $ , the computed spectral factor $\\hat{S}_{[m]}^+$ converges to exact ${S}_{[m]}^+$ (assuming that all previous factors including $S_{[m-1]}^+$ are computed exactly).",
"However, in practise we never achieve an exact result.",
"Nevertheless, the accuracy $\\Vert {S}_{[m]}-\\hat{S}_{[m]}^+\\big (\\hat{S}_{[m]}^+\\big )^*\\Vert $ can be controlled and the value of $N$ can be increased, if necessary, at each intermediate stage, in order to achieve a satisfactory approximation in the final result.",
"Step 3.",
"From obtained representations of $\\zeta _j$ , $j=1,2,\\ldots ,m-1$ , and $f_m^+$ as rational functions, find $\\zeta _j^{\\lbrace N\\rbrace }:=\\mathbb {Q}_N^-[\\zeta _j]+\\mathbb {P}_n^+[\\zeta _j]=\\sum _{k=-N}^n c_k\\lbrace \\zeta _j\\rbrace t^{k}$ and $f_m^{\\lbrace N\\rbrace }:=\\mathbb {P}_{N+n}^+[f_m^+]=\\sum _{k=0}^{N+n}c_k\\lbrace f_m^+\\rbrace t^k.$ We do this by the standard division algorithm of two polynomials, utilizing the advantages of denominator being free from zeros inside $\\mathbb {T}$ and function having no poles on $\\mathbb {T}$ .",
"Step 4.",
"Using $(\\zeta _1^{\\lbrace N\\rbrace },\\zeta _2^{\\lbrace N\\rbrace },\\ldots ,\\zeta _{m-1}^{\\lbrace N\\rbrace }, f_m^{\\lbrace N\\rbrace })$ as the last row of (REF ), construct a unitary matrix function $U_N:=U_m^{\\lbrace N\\rbrace }$ as it is described in Section IV.",
"Step 5.",
"Consider the product $S_{[m]}^+\\approx \\begin{pmatrix}&&&&0\\\\&&S_{[m-1]}^+&&\\vdots \\\\&&&&0\\\\{\\zeta }_{1}^{\\lbrace N\\rbrace }& {\\zeta }_{2}^{\\lbrace N\\rbrace }&\\ldots & {\\zeta }_{m-1}^{\\lbrace N\\rbrace }& {f}^{\\lbrace N\\rbrace }_m\\end{pmatrix}\\begin{pmatrix}u_{11}&u_{12}&\\cdots &u_{1m}\\\\\\vdots &\\vdots &\\vdots &\\vdots \\\\u_{m-1,1}&u_{m-1,2}&\\cdots &u_{m-1,m}\\\\[3mm]\\overline{u_{m1}}&\\overline{u_{m2}}&\\cdots &\\overline{u_{mm}}\\\\\\end{pmatrix}$ (the last matrix is $U_m^{\\lbrace N\\rbrace }$ ), where all coefficients of polynomials in the right-hand side product with indices outside the range $[0,n]$ are neglected (since we know that the exact $S_{[m]}^+$ is matrix polynomial of degree $n$ ).",
"Therefore, $S_{[m-1]}^+$ can be separately multiplied by the first $m-1$ rows of $U_m^{\\lbrace N\\rbrace }$ and then its last row can be multiplied by $U_m^{\\lbrace N\\rbrace }$ .",
"Procedure 3.",
"For $m=r$ , $S_{[r]}^+$ is an approximate spectral factor of $S$ .",
"We can multiply $S_{[r]}^+$ by the constant unitary matrix $U$ defined by (REF ) (taking $S_{[r]}^+$ instead of $S_{0}^+$ ) to obtain $S^+$ ."
],
[
"JLE-algorithm 2",
"In this implementation, computations of polynomial matrix determinants are avoided.",
"Consequently much higher dimensional matrices can be factorized accurately by this algorithm at the expense of large computer memory usage.",
"Procedure 1.",
"Compute a scalar spectral factor $f_1^+$ of $s_{11}$ by using the same exp-log and Wilson's methods as in Procedure 1 of JLE-algorithm 1.",
"Procedure 2.",
"For $m=2,3,\\ldots ,r$ , assume that $S_{[m-1]}^+$ has already been (approximately) constructed as an $(m-1)\\times (m-1)$ polynomial matrix of degree $n$ and perform the following steps.",
"Step 1.",
"Take a large number of DFT nodes, usually $2^\\kappa $ , where $10\\le \\kappa \\le 23$ : $t_l=\\exp \\left(\\frac{2\\pi il}{2^\\kappa }\\right)$ , $l=0,1,\\ldots ,2^\\kappa -1$ .",
"This $\\kappa $ becomes another tuning parameter in the algorithm (along with $N$ ), which can be selected and changed during recursive steps in order to improve the accuracy (REF ).",
"Step 2.",
"For each node $t_l$ , $l=0,1,\\ldots ,2^\\kappa -1$ , evaluate the matrices $S_{[m-1]}^+(t_l)$ and $S_{[1:\\,m-1,\\,m]}(t_l)$ , and solve the following system of linear equations (see (REF )): $S_{[m-1]}^+(t_l)\\cdot X=S_{[1:\\,m-1,\\,m]}(t_l).$ We have $\\big (\\zeta _1(t_l),\\zeta _2(t_l),\\ldots ,\\zeta _{m-1}(t_l),\\big )=X_l^*$ , where $X_l$ is the solution of (REF ).",
"If it happens that the system (REF ) is singular or ill conditioned, then we can apply the continuity of functions $\\zeta _j$ and assume that $X_l=X_{l-1}$ .",
"When standard routines are well optimized (as it is in Matlab), this step is not as time-consuming as it might appear at the first glance.",
"Step 3.",
"Compute $|f_m^+(t_l)|^2$ , $l=0,1,\\ldots ,2^\\kappa -1$ , from the formula (REF ) Step 4.",
"Select a large positive integer $N$ , and using the values of $|f_m^+|^2$ at DFT nodes, perform an approximate scalar spectral factorization to reconstruct $f_m^{\\lbrace N\\rbrace }:=\\sum _{k=0}^{N+n}c_k\\lbrace f_m^+\\rbrace t^k.$ For this step, one can use the exp-log method of scalar spectral factorization which utilizes the boundary values of a spectral density.",
"The integer $N$ has a natural bound $2^\\kappa -n$ in this situation, however an optimal ratio (from 1/10 to 1/50) of $N/2^\\kappa $ should be selected in order to achieve a good accuracy.",
"Step 5.",
"From the values of $\\zeta _j$ at DFT nodes $t_l$ , $l=0,1,\\ldots ,2^\\kappa -1$ , reconstruct (approximately) $\\zeta _j^{\\lbrace N\\rbrace }:=\\sum _{k=-N}^n c_k\\lbrace \\zeta _j\\rbrace t^{k}$ by using the inverse FFT and selecting corresponding coefficients.",
"The remaining steps are the same as Steps 4 and 5 in JLE-algorithm 1, including Procedure 3."
],
[
"JLE-algorithm 3",
"This implementation utilizes formulas (REF ), (REF ), and (REF ) for $m=r$ : $ S^+(t)=\\left[\\begin{matrix}& & S_{[r-1]}^+(t)& & \\begin{matrix}0\\\\0\\\\ \\vdots \\\\0\\end{matrix}\\\\\\zeta _1(t) & \\zeta _2(t) & \\ldots & \\zeta _{r-1}(t)& f_r^+(t)\\end{matrix}\\right]\\left[\\begin{matrix}u_{11}(t)&u_{12}(t)&\\cdots &u_{1r}(t)\\\\u_{21}(t)&u_{22}(t)&\\cdots &u_{2r}(t)\\\\\\vdots &\\vdots &\\vdots &\\vdots \\\\u_{r-1,1}(t)&u_{r-1,2}(t)&\\cdots &u_{r-1,r}(t)\\\\[3mm]\\overline{u_{r1}(t)}&\\overline{u_{r2}(t)}&\\cdots &\\overline{u_{rr}(t)}\\\\\\end{matrix}\\right],$ $ f_r^+(t)=\\det S^+(t)/\\det S^+_{[r-1]}(t),$ and $ [\\zeta _1(t), \\zeta _2(t), \\ldots , \\zeta _{r-1}(t)]\\cdot \\big (S^+_{[r-1]}(t)\\big )^*=S_{[r,1:\\,r-1]}(t).$ Let $U(t)=U_r(t)$ be the last matrix in (REF ).",
"Then, for $j\\le r$ , it follows from (REF ) that $ S^+_{[r-1]}(t)\\cdot U_{[1:\\,r-1,\\,j]}(t)=S^+_{[1:\\,r-1,\\,j]}(t),$ and furthermore $ S^+_{[r-1]}(t)\\cdot U_{]r,j[}(t)=S^+_{]r,j[}(t).$ Since $U(t)$ is a unitary matrix ($U^{-1}(t)=U^*(t)$ ) and $\\det U(t)=1$ , it follows that $u_{r,j}(t)=\\det U_{]r,j[}(t)$ and, taking into account (REF ), we get $ u_{r,j}(t)=\\det S^+_{]r,j[}(t)/\\det S^+_{[r-1]}(t).$ It also follows from (REF ) that $ [\\zeta _1(t), \\zeta _2(t), \\ldots , \\zeta _{r-1}(t)]\\cdot U_{[1:\\,r-1,\\,j]}(t)+f_r^+(t)\\overline{u_{r,j}(t)}=S^+_{rj}(t).$ Substituting into (REF ) $[\\zeta _1, \\ldots , \\zeta _{r-1}]=S_{[r,1:\\,r-1]}\\cdot \\big (S^+_{[r-1]}\\big )^{-*}$ (see (REF ) ), $ U_{[1:\\,r-1,\\,j]}=\\big (S^+_{[r-1]}\\big )^{-1}\\cdot S^+_{[1:\\,r-1,\\,j]}$ (see (REF ) ), (REF ), and (REF ), and taking into account that $S_{[r-1]}=S^+_{[r-1]}\\big (S^+_{[r-1]}\\big )^*$ , we get $S_{[r,1:\\,r-1]}(t)\\cdot \\big (S_{[r-1]}(t)\\big )^{-1}\\cdot S^+_{[1:\\,r-1,\\,j]}(t)+\\frac{\\det S^+(t)\\overline{\\det S^+_{]r,j[}(t)}}{\\det S_{[r-1]}(t)}=S^+_{rj}(t).$ Consequently, $ S_{[r,1:\\,r-1]}(t)\\cdot \\mathop {\\rm Cof}\\big \\lbrace S_{[r-1]}(t)\\big \\rbrace ^{T}\\cdot S^+_{[1:\\,r-1,\\,j]}(t)+{\\big ({\\det S(t)\\big )^+}\\cdot \\overline{\\det S^+_{]r,j[}(t)}}=S^+_{rj}(t){\\det S_{[r-1]}(t)},$ where it is assumed that $\\big (\\det S(t)\\big )^+$ can be found from ${\\det S(t)}$ , as the problem is reduced to the scalar spectral factorization.",
"In the equation (REF ), $S_{[r,1:\\,r-1]}$ , $\\mathop {\\rm Cof}\\big \\lbrace S_{[r-1]}\\big \\rbrace ^{T}$ , $\\big (\\det S(t)\\big )^+$ and ${\\det S_{[r-1]}}$ are assumed to be the known (matrix) functions, and $S^+_{[1:\\,r-1,\\,j]}$ , ${\\det S^+_{]r,j[}}$ , and $S^+_{rj}$ are unknown (matrix) functions.",
"Assume now that $S$ is a matrix polynomial of degree $n$ (see (REF )), i.e.",
"$S\\in (\\mathcal {P}_{\\lbrace -n,n\\rbrace })^{r\\times r}$ .",
"Let us observe that for functions in (REF ) we have: ${S}_{[r,1:\\,r-1]}\\in (\\mathcal {P}_{\\lbrace -n,n\\rbrace })^{1\\times (r-1)}; {\\mathop {\\rm Cof}\\big \\lbrace S_{[r-1]}\\big \\rbrace }^{T}\\in (\\mathcal {P}_{\\lbrace -n(r-2), n(r-2)\\rbrace })^{(r-1)\\times (r-1)};\\\\ {S^+_{[1:\\,r-1,\\,j]}}\\in (\\mathcal {P}_{\\lbrace 0,n\\rbrace })^{(r-1)\\times 1}; {(\\det S)^+}\\in \\mathcal {P}_{\\lbrace 0,rn\\rbrace }; {\\overline{\\det S^+_{]r,j[}}}\\in \\mathcal {P}_{\\lbrace -(r-1)n,0\\rbrace }; {S^+_{rj}}\\in \\mathcal {P}_{\\lbrace 0,n\\rbrace },$ and ${\\det S_{[r-1]}}\\in \\mathcal {P}_{\\lbrace -(r-1)n, (r-1)n\\rbrace }$ .",
"Thus all products in (REF ) have the range of indices of (nonzero) Fourier coefficients in $[-(r-1)n,rn]$ .",
"If we equate the corresponding coefficients in these products, we get $2rn-n+1$ linear algebraic equations with respect to coefficients of unknown (matrix) polynomials $S^+_{[1:\\,r-1,\\,j]}$ , ${\\det S^+_{]r,j[}}$ , and $S^+_{rj}$ .",
"The total number of these coefficients is $(r-1)(n+1)+\\lbrace (r-1)n+1\\rbrace +(n+1)=2rn-n+r+1$ .",
"We can factorize $S(t)$ at a single point on the unit circle, say $t=1$ , and getting the representation $S(1)=S^+(1)\\big (S^+(1)\\big )^*$ , we can assume that $[S^+_{[1:\\,r-1,\\,j]}(1)\\,S^+_{r,j}(1)]^T$ is the $j$ -th column of $S^+(1)$ .",
"This gives the additional $r$ conditions on coefficients of (matrix) polynomials $S^+_{[1:\\,r-1,\\,j]}$ and $S^+_{r,j}$ , and thus additional $r$ equations.",
"In the end we get the same number of linear equations and unknowns $2rn-n+r+1$ .",
"The basic computational procedures of the algorithm are described below.",
"Step 1.",
"Compute the polynomial determinants $\\det S(t)$ and $\\det S_{[r-1]}(t)$ by the method described in JLE-1.",
"Step 2.",
"Compute the scalar spectral factor $\\big (\\det S(t)\\big )^+$ by the method described in Procedure 1 of JLE-1.",
"Step 3.",
"Compute $\\mathop {\\rm Cof}\\big \\lbrace S_{[r-1]}(t)\\big \\rbrace ^{T}$ by evaluating it at $N=2n(r-1)+1$ DFT nodes $t_l=\\exp \\left(\\frac{2\\pi il}{N}\\right)$ , $l=0,1,2,\\ldots ,N-1$ , by the formula $\\mathop {\\rm Cof}\\big \\lbrace S_{[r-1]}(t_l)\\big \\rbrace ^{T}=\\det S_{[r-1]}(t_l)\\big (S_{[r-1]}(t_l)\\big )^{-1}$ and then use the inverse Fourier transform.",
"Step 4.",
"Multiply matrix polynomials $S_{[r,1:\\,r-1]}$ and $\\mathop {\\rm Cof}\\big \\lbrace S_{[r-1]}\\big \\rbrace ^T$ .",
"Let $\\big (\\det S(t)\\big )^+=\\sum _{k=0}^{rn}a_kt^k$ , $t^{(r-1)n}\\det S_{[r-1]}(t)=\\sum _{k=0}^{2(r-1)n}b_kt^k$ , and $t^{(r-1)n}S_{[r,1:\\,r-1]}(t)\\mathop {\\rm Cof}\\big \\lbrace S_{[r-1]}\\big \\rbrace ^T(t)=\\sum _{k=0}^{2(r-1)n}C_kt^k=\\big [\\sum _{k=0}^{2(r-1)n}c_k^{\\lbrace 1\\rbrace }t^k\\cdots \\sum _{k=0}^{2(r-1)n}c_k^{\\lbrace r-1\\rbrace }t^k\\big ],$ $C_k\\in \\mathbb {C}^{1\\times (r-1)}$ , $c_k^{\\lbrace j\\rbrace }\\in \\mathbb {C}$ .",
"Introduce also the notation: $\\mathbf {a}=[a_0\\,a_1\\,\\cdots \\,a_{rn}]^T\\in \\mathbb {C}^{(2rn+1)\\times 1}$ ; $\\mathbf {b}=[b_0\\,b_1\\,\\cdots b_{2(r-1)n}]^T\\in \\mathbb {C}^{(2(r-1)n+1)\\times 1}$ ; $\\mathbf {c}^{\\lbrace j\\rbrace }=[c_0^{\\lbrace j\\rbrace }\\,c_1^{\\lbrace j\\rbrace }\\,\\cdots c_{2(r-1)n}^{\\lbrace j\\rbrace }]^T\\!\\in \\!",
"\\mathbb {C}^{(2(r-1)n+1)\\times 1}$ , $j=1,2,\\ldots , r-1$ .",
"Step 5.",
"Construct the $(2rn-n+1)\\times (2rn-n+r+1)$ matrix $\\Delta _0=[\\Delta _1\\;\\Delta _2\\;\\Delta _3]$ , where $\\Delta _1=[T(\\mathbf {c}^{\\lbrace 1\\rbrace }\\,;n)\\;T(\\mathbf {c}^{\\lbrace 2\\rbrace }\\,;n)\\;\\cdots \\;T(\\mathbf {c}^{\\lbrace r-1\\rbrace }\\,;n)]\\in \\mathbb {C}^{(2rn-n+1)\\times (r-1)(n+1)}$ ,$\\Delta _2=-T(\\mathbf {b}\\,; n)\\in \\mathbb {C}^{(2rn-n+1)\\times (n+1)}$ , and $\\Delta _3=T(\\mathbf {a}\\,; (r-1)n)\\in \\mathbb {C}^{(2rn-n+1)\\times ((r-1)n+1)}$ and then the $(2rn-n+r+1)\\times (2rn-n+r+1)$ matrix $\\Delta =[\\Delta _1\\;\\Delta _2\\;\\Delta _3\\,; \\mathbf {I}\\;\\mathbf {0}_{r\\times ((r-1)n+1)}]$ , where $\\mathbf {I}\\in \\mathbb {C}^{r\\times r(n+1)}$ is the $r\\times r$ block identity matrix with entries $\\mathbf {1}_{1\\times (n+1)}$ on the block diagonal and $\\mathbf {0}_{1\\times (n+1)}$ elsewhere.",
"Step 6.",
"Perform the Cholesky factorization of the positive definite matrix $S(1)=S^+(1)\\big (S^+(1)\\big )^*$ and assume that $S^+(1)=[h_1\\;h_2\\cdots h_r]$ , where $h_j\\in \\mathbb {C}^{r\\times 1}$ .",
"Step 7.",
"For each $j=1,2,\\ldots , r$ , solve the $(2rn-n+r+1)\\times (2rn-n+r+1)$ system of equations $\\Delta X=\\Lambda _j,$ with right-hand sides $\\Lambda _j=[\\mathbf {0}_{(2rn-n+1)\\times 1}\\;; h_j]$ , and denote the respective solution by $X_j=[x_0^{\\lbrace j\\rbrace }\\,x_1^{\\lbrace j\\rbrace }\\,\\cdots x_{2rn-n+r}^{\\lbrace j\\rbrace }]^T$ .",
"Step 8.",
"Set a spectral factor $S_0^+=\\big (s^+_{ij}\\big )_{i,j=1}^r$ , where $s^+_{ij}(t)=\\sum _{k=0}^n x_{(n+1)(i-1)+k}^{\\lbrace j\\rbrace } t^k$ Step 9.",
"Find $S^+$ by $S^+_0U$ , where $U$ is defined by the formula (REF ).",
"Since we know the existence of decomposition (REF ), the solution to equation (REF ) exists for each $j$ .",
"However it might happen that $\\det \\Delta =0$ .",
"Furthermore, computer simulations suggest that $\\Delta $ is nonsingular whenever $\\det S(t)\\ne 0$ for each $t\\in \\mathbb {T}$ and $\\Delta $ is singular whenever $\\det S(t)= 0$ for some $t\\in \\mathbb {T}$ .",
"Therefore JLE-3 works under the additional condition $\\det S(t)> 0$ for $t\\in \\mathbb {T}$ .",
"If this condition holds, but zeros of $\\det S$ are rather close to the boundary, the matrix $\\Delta $ might become ill-conditioned.",
"In such situations, the solutions of (REF ) are inaccurate and approximation to $S^+$ is lost.",
"The techniques of solution of ill-conditioned systems might be useful, however we have not investigated this question yet.",
"As numerical simulations show in Section IX, JLE-algorithm 3 can satisfactory factorize random matrices with $r=6$ and $n=20$ , which might be useful in certain applications to Mobile Communications [19]."
],
[
"Factorization of singular matrices",
"Symmetric positive matrix polynomials which are chosen randomly or obtained by channel estimation in wireless communication are usually non-singular, i.e.",
"their determinants do not vanish on $\\mathbb {T}$ .",
"However, in certain optimal control and wavelet design problems, one encounters a need to factorize singular matrices.",
"It is well known that all MSF methods have difficulties in this situation and some of them cannot handle zeros on the unit circle at all.",
"Obviously, convergence of JLE algorithms also slows down in singular cases.",
"However, if we fully utilize the ability of Janashia-Lagvilava's method to decompose a large scale problem into smaller parts and deal with any arising difficulties by intermediate interventions, in number of cases we can substantially improve the performance of the algorithm.",
"In this section we demonstrate this advantage by factorizing specific singular matrices.",
"First, consider a test matrix from [13] whose spectral factorization is known beforehand: $\\begin{pmatrix}2z^{-1}+6+2z&11z^{-1}+22+7z\\\\7z^{-1}+22+11z&38z^{-1}+84+38z\\end{pmatrix}=\\begin{pmatrix}2+z&1\\\\7+5z&3+z\\end{pmatrix}\\begin{pmatrix}2+z^{-1}&7+5z^{-1}\\\\1&3+z^{-1}\\end{pmatrix}$ This matrix is very simple, but its determinant, $-z^{-2}+2-z^2$ , has two double zeros on the boundary.",
"When data was fed into \"standard\" JLE-algorithm 1 with 5 iterations in scalar spectral factorization of $\\det S$ by Wilson's algorithm (see Sect.",
"6, Procedure 1), we get 4 correct digits.",
"When we increase the number of the iterations up to 45, the maximum optimum value, we get 7 correct digits.",
"If we compute the determinant by the direct formula $\\det S=s_{11}s_{22}-s_{12}s_{21}$ , avoiding the minimal round-off errors introduced with computation of the determinant by FFT (see Section 6), then we get 14 correct digits.",
"All these computations take less than 0.01 seconds as the matrix is very small and and it suffices to select the parameter $N$ as small as 20.",
"We observed that Wilson's MSF algorithm (see the next section) can perform factorization (REF ) with no more than 6 correct digits (with optimum parameter $\\kappa =19$ ) which takes around 3 minutes.",
"Next we factorize a small size $2\\times 2$ matrix $S(z)=\\sum _{k=-3}^3 C_k z^k=\\begin{pmatrix} s_{11}(z)& s_{12}(z)\\\\s_{21}(z)& s_{22}(z)\\end{pmatrix},$ where $s_{11}(z)=-\\frac{1-4\\overline{\\alpha }}{64}z^{-3}+\\frac{1+4{\\alpha }}{64}z^{-1}+1 +\\frac{1+4{\\alpha }}{64}z -\\frac{1-4\\overline{\\alpha }}{64}z^{3}$ ; $s_{12}(z)=\\frac{\\overline{\\alpha }}{16}z^{-3}-\\frac{{\\alpha }}{16}z^{-1}+\\frac{{\\alpha }}{16}z -\\frac{\\overline{\\alpha }}{16}z^{3}$ ; $s_{21}(z)=s_{12}(1/z)$ ; and $s_{22}(z)=\\frac{1-4\\overline{\\alpha }}{64}z^{-3}-\\frac{1+4{\\alpha }}{64}z^{-1}+1 -\\frac{1+4{\\alpha }}{64}z +\\frac{1-4\\overline{\\alpha }}{64}z^{3}$ ; with $\\alpha =4+\\sqrt{15}$ and $\\overline{\\alpha }=4-\\sqrt{15}$ .",
"This matrix is singular and, furthermore, its determinant has an explicit form $\\det S(z)=\\frac{8\\overline{\\alpha }-1}{4096}(z+1)^4(z-1)^4(z+i)^2(z-i)^2$ .",
"Its spectral factorization $S(z)=\\sum _{k=0}^3 A_k z^k \\sum _{k=0}^3 A_k^T z^{-k}$ .",
"is required for construction of the so called SA4 multiwavelet [20] which possess certain nice properties.",
"The realization of these properties depends on the accuracy by which the coefficients $A_k$ are computed.",
"The efforts to factorize (REF ) with a maximal possible accuracy by the Youla-Kazanjian method [27] is described in [15], where the error $err_1=\\Vert S(z)-\\sum \\nolimits _{k=0}^3 \\hat{A}_k z^k \\sum \\nolimits _{k=0}^3 \\hat{A}_k^T z^{-k}\\Vert =4.086\\cdot 10^{-8}$ is achieved.",
"(As the exact values of $A_k$ are unknown in this situation, this error is used to estimate the accuracy $\\Vert A_k-\\hat{A_k}\\Vert $ .)",
"As we checked, this performance cannot be improved by the Wilson MSF method either.",
"In fact, the error cannot be reduced to lower than $10^{-5}$ by the method (with optimal tuning parameter $\\kappa =18$ : see Section 10).",
"When we ran JLE-1 with the matrix $S$ and increase the number of iterations in the scalar factorization step up to 60 (see Procedure 1), we obtain the error $err_2=4.373\\cdot 10^{-5}$ .",
"However, if we cancel out the common roots in the triangular factorization (REF ) and factorize the determinant $\\det S$ manually we achieve the error $err_3=1.843\\cdot 10^{-14}$ .",
"In these computations, it is sufficient to take the tuning parameter $N=100$ and so the consumed time is very small (less than 0.1 seconds).",
"In general, when a singular polynomial (with a zero on $\\mathbb {T}$ ) is factorized in the scalar case, the best way to deal with the singularity is to factor out the zeros with unit modulus.",
"This procedure is more demanding in the matrix case (see [17]).",
"The above examples demonstrate that Janashia-Lagvilava method is capable of reducing a problem of the singularity of a spectral matrix density to the level of scalar factorization.",
"In fact, the method has already been used to improve the coefficients of other well-known multiwavelets as well by effective factorization of related singular matrices which will be the topic of another paper."
],
[
"Comparison with Wilson's algorithm ",
"Wilson's method of MSF appeared in the 70's of the last century [25], [26].",
"Since then, several authors claimed that they obtained MSF algorithms with reduced computational complexity (see [16], [14]).",
"These are algorithms based on the solution of algebraic Riccati equations and some of them are implemented in Matlab.",
"As a consequence, in our attempts to compare Janashia-Lagvilava algorithm with other existing methods of MSF, we did not originally consider the Wilson method and only concentrated our attention on those methods which were implemented in Matlab (see [13]).",
"However, recently we learned that Prof. Rangarajan and his collaborators, who apply MSF in Neuroscience [2], [3], developed an efficient implementation of Wilson's method which works rather fast.",
"This implementation takes data matrix in frequency domain.",
"Nevertheless, this idea can be easily translated for matrices given in time domain.",
"In particular, for a matrix (REF ) with given coefficients $C_k$ , $k=0,1,\\ldots ,N$ , we select $\\kappa $ as a tuning parameter and find $2^\\kappa $ values of the matrix function $S$ in DFT nodes: $S(t_0),\\dots ,S(t_{2^\\kappa })$ , where $t_j=\\exp \\left(\\frac{2\\pi ij}{2^\\kappa }\\right)$ .",
"Then we use the Wilson's recurrent formula $S^+_{k+1}=S^+_k\\left[(S^+_k)^{-1}S(S^+_k)^{-*}+I\\right]^+$ with initial data $S_0=\\sqrt{C_0}$ .",
"After performing sufficient iterations, we return back to the time domain and approximately compute the coefficients $A_k$ of (REF ).",
"Here, like other minor improvements we introduced in the implementation of Wilson's method, we empirically observed that the upper triangular constant matrix $S_\\tau $ in formula (3.2) in [26] can be omitted in (REF ).",
"Such implementation of Wilson's algorithm essentially works as efficient as JLE-1 and frequently better than JLE-2.",
"In addition, a flexible combination of Janashia-Lagvilava and Wilson methods can be sometimes useful."
],
[
"Numerical simulations",
"The computer code for implementation of JLE-algorithms was written in Matlab in order to test them numerically.",
"A laptop with characteristics Intel(R) Core(TM) i7-4600U CPU (2 cores, 4 threads), 2.40GHz, RAM 8.00Gb was used and some of the tests were performed on the HPC cluster “Dalma\" at NYUAD.",
"For all numerical simulations of MSF algorithms randomly selected polynomial matrices have been used.",
"Namely, for given matrix dimension $r$ and polynomial degree $n$ , a random polynomial matrix $\\sum _{k=0}^n A_kt^k$ , $A_k\\in [-1,1]^{r\\times r}$ , has been chosen, and positive definite (on $\\mathbb {T}$ ) matrix polynomial $S(t)=\\sum _{k=0}^n A_kt^k\\sum _{k=0}^n A_k^*t^{-k}$ has been approximately factorized.",
"In rare occasions, which are emphasized below, some deterministic efforts have been introduced in order to artificially improve the properties of $S$ .",
"The error $err=\\Vert S-\\hat{S}^+(\\hat{S}^+)^*\\Vert $ is used to estimate the accuracy of the factorization since there is no other way to decide how close is $\\hat{S}^+$ ro $S^+$ .",
"The basic problem in order to demonstrate the most effective performance of the constructed algorithms was an empirical selection of tuning parameters ($N$ for JLE-1, $N$ and $\\kappa $ for JLE-2, and $\\kappa $ and the number of iterations for Wilson's algorithm) which would make an optimal trade-off between the available memory, the computation time and the accuracy.",
"For realistic applications, automatic selection of the optimal tuning parameters during the factorization remains a challenging problem.",
"When different algorithms are compared, it is assumed that they were run with the same data.",
"We start with JLE-3 which has the advantage that it contains no tuning parameters.",
"Below we demonstrate its performance within the range of polynomial matrices for which it is applicable.",
"The tuning parameters in JLE-1 and Wilson have been selected so as to achieve the same accuracy as in JLE-3.",
"Beyond the indicated range of matrix dimension $m$ and polynomial degree $n$ the accuracy (REF ) of JLE-3 becomes unsatisfactory.",
"(In all tables below, $r\\times n$ indicates that a $r\\times r$ test matrix was selected with Laurent polynomial entries of degree $n$ having nonzero coefficients indexed from $-n$ to $n$ ).",
"Table I Performance of JLE-3 Table: NO_CAPTIONNext we compare JLE-1 and Wilson within the range of matrices where JLE-1 operates well.",
"The tuning parameter $N=5mn$ has been taken for $m$ th recursion in JLE-1 and $\\kappa $ has been selected in Wilson so as to achieve the same accuracy as in JLE-1.",
"Table II Comparision of JLE-1 and Wilson Table: NO_CAPTIONNext we factorize random $100\\times 100$ matrices (with polynomial degree $n=30$ ) by JLE-2 and Wilson.",
"We tried to factorize such matrices with accuracy that is acceptable in practice, namely $error=10^{-4}$ , and selected the tuning parameters accordingly.",
"A substantial drop in the accuracy has been observed at the final step of recursion $m=100$ in JLE-2 and it was observed that Wilson can factorize the $99\\times 99$ leading submatrix of $S$ much more easily than $S$ itself.",
"We empirically explain this phenomenon by the following reason: the probability for zeros of $\\det S_{[m]}$ to be very close to $\\mathbb {T}$ (in which case all spectral factorization algorithms become slowly convergent) is higher for $m=r$ than for $m<r$ (however no theoretical proofs has been attempted).",
"Therefore, in a variant of our implementation, we have combined JLE-2 by Wilson which resulted in certain improvements.",
"Table III Comparision of JLE-2 and Wilson Table: NO_CAPTIONWhen we added artificially $I_r$ to a random matrix $S$ in order to avoid zeros close to $\\mathbb {T}$ , we achieved the same accuracy within improved computation time.",
"We display the results below.",
"Table IV Comparision of JLE-2 and Wilson Table: NO_CAPTIONIn the end we demonstrate that “good\" matrices of dimension as large as $700\\times 700$ can be factorized with accuracy $error=10^{-3}$ which is acceptable in practice and within the available computer memory (120GB of one node at “Dalma\" in our situation).",
"With respect to time usage, the advantage of Wilson's MSF method is evident in this case.",
"The reason is that JLE-2 requires the tuning parameter $N$ to be selected very large at the last recursive steps in order to achieve the given accuracy.",
"However, JLE-2 algorithm still can be invoked to analyze and overcome the problem when Wilson's method is unable to factorize a matrix obtained from real applications.",
"Table V Factorization of large matrices Table: NO_CAPTION"
],
[
"Conclusions",
"Matrix spectral factorization is widely used in modern control theory and wireless communications.",
"Furthermore, improved algorithms of MSF may lead to new areas to which they could be successfully applied.",
"In the present paper, we consider three different algorithms based on Janashia-Lagvilava method, which may be competitive with other existing MSF algorithms.",
"A general description of their computational capabilities, as well as a comparison to Wilson's MSF algorithm, are provided by means of numerical simulations."
],
[
"Acknowledgments",
"The authors are thankful for an opportunity to run part of the tests using the High Performance Computing resources at New York University Abu Dhabi."
]
] | 1606.04909 |
[
[
"Learning feed-forward one-shot learners"
],
[
"Abstract One-shot learning is usually tackled by using generative models or discriminative embeddings.",
"Discriminative methods based on deep learning, which are very effective in other learning scenarios, are ill-suited for one-shot learning as they need large amounts of training data.",
"In this paper, we propose a method to learn the parameters of a deep model in one shot.",
"We construct the learner as a second deep network, called a learnet, which predicts the parameters of a pupil network from a single exemplar.",
"In this manner we obtain an efficient feed-forward one-shot learner, trained end-to-end by minimizing a one-shot classification objective in a learning to learn formulation.",
"In order to make the construction feasible, we propose a number of factorizations of the parameters of the pupil network.",
"We demonstrate encouraging results by learning characters from single exemplars in Omniglot, and by tracking visual objects from a single initial exemplar in the Visual Object Tracking benchmark."
],
[
"Introduction",
"Deep learning methods have taken by storm areas such as computer vision, natural language processing, and speech recognition.",
"One of their key strengths is the ability to leverage large quantities of labelled data and extract meaningful and powerful representations from it.",
"However, this capability is also one of their most significant limitations since using large datasets to train deep neural network is not just an option, but a necessity.",
"It is well known, in fact, that these models are prone to overfitting.",
"Thus, deep networks seem less useful when the goal is to learn a new concept on the fly, from a few or even a single example as in one shot learning.",
"These problems are usually tackled by using generative models [18], [12] or, in a discriminative setting, using ad-hoc solutions such as exemplar support vector machines (SVMs) [14].",
"Perhaps the most common discriminative approach to one-shot learning is to learn off-line a deep embedding function and then to define on-line simple classification rules such as nearest neighbors in the embedding space [4], [16], [13].",
"However, computing an embedding is a far cry from learning a model of the new object.",
"In this paper, we take a very different approach and ask whether we can induce, from a single supervised example, a full, deep discriminative model to recognize other instances of the same object class.",
"Furthermore, we do not want our solution to require a lengthy optimization process, but to be computable on-the-fly, efficiently and in one go.",
"We formulate this problem as the one of learning a deep neural network, called a learnet, that, given a single exemplar of a new object class, predicts the parameters of a second network that can recognize other objects of the same type.",
"Our model has several elements of interest.",
"Firstly, if we consider learning to be any process that maps a set of images to the parameters of a model, then it can be seen as a “learning to learn” approach.",
"Clearly, learning from a single exemplar is only possible given sufficient prior knowledge on the learning domain.",
"This prior knowledge is incorporated in the learnet in an off-line phase by solving millions of small one-shot learning tasks and back-propagating errors end-to-end.",
"Secondly, our learnet provides a feed-forward learning algorithm that extracts from the available exemplar the final model parameters in one go.",
"This is different from iterative approaches such as exemplar SVMs or complex inference processes in generative modeling.",
"It also demonstrates that deep neural networks can learn at the “meta-level” of predicting filter parameters for a second network, which we consider to be an interesting result in its own right.",
"Thirdly, our method provides a competitive, efficient, and practical way of performing one-shot learning using discriminative methods.",
"The rest of the paper is organized as follows.",
"s:related discusses the works most related to our.",
"s:method describes the learnet approaches and nuances in its implementation.",
"s:experiments demonstrates empirically the potential of the method in image classification and visual tracking tasks.",
"Finally, s:conc summarizes our findings."
],
[
"Related work",
"Our work is related to several others in the literature.",
"However, we believe to be the first to look at methods that can learn the parameters of complex discriminative models in one shot.",
"One-shot learning has been widely studied in the context of generative modeling, which unlike our work is often not focused on solving discriminative tasks.",
"One very recent example is by Rezende et al.",
"[18], which uses a recurrent spatial attention model to generate images, and learns by optimizing a measure of reconstruction error using variational inference [8].",
"They demonstrate results by sampling images of novel classes from this generative model, not by solving discriminative tasks.",
"Another notable work is by Lake et al.",
"[12], which instead uses a probabilistic program as a generative model.",
"This model constructs written characters as compositions of pen strokes, so although more general programs can be envisioned, they demonstrate it only on Optical Character Recognition (OCR) applications.",
"A different approach to one-shot-learning is to learn an embedding space, which is typically done with a siamese network [1].",
"Given an exemplar of a novel category, classification is performed in the embedding space by a simple rule such as nearest-neighbor.",
"Training is usually performed by classifying pairs according to distance [4], or by enforcing a distance ranking with a triplet loss [16].",
"A variant is to combine embeddings using the outer-product, which yields a bilinear classification rule [13].",
"The literature on zero-shot learning (as opposed to one-shot learning) has a different focus, and thus different methodologies.",
"It consists of learning a new object class without any example image, but based solely on a description such as binary attributes or text.",
"It is usually framed as a modality transfer problem and solved through transfer learning [20].",
"The general idea of predicting parameters has been explored before by Denil et al.",
"[3], who showed that it is possible to linearly predict as much as 95% of the parameters in a layer given the remaining 5%.",
"This is a very different proposition from ours, which is to predict all of the parameters of a layer given an external exemplar image, and to do so non-linearly.",
"Our proposal allows generating all the parameters from scratch, generalizing across tasks defined by different exemplars, and can be seen as a network that effectively “learns to learn”."
],
[
"One-shot learning as dynamic parameter prediction",
"Since we consider one-shot learning as a discriminative task, our starting point is standard discriminative learning.",
"It generally consists of finding the parameters $W$ that minimize the average loss $\\mathcal {L}$ of a predictor function $\\varphi (x;\\, W)$ , computed over a dataset of $n$ samples $x_{i}$ and corresponding labels $\\ell _{i}$ : $\\min _{W}\\frac{1}{n}\\sum _{i=1}^n\\mathcal {L}(\\varphi (x_{i};\\, W),\\,\\ell _{i}).$ Unless the model space is very small, generalization also requires constraining the choice of model, usually via regularization.",
"However, in the extreme case in which the goal is to learn $W$ from a single exemplar $z$ of the class of interest, called one-shot learning, even regularization may be insufficient and additional prior information must be injected into the learning process.",
"The main challenge in discriminative one-shot learning is to find a mechanism to incorporate domain-specific information in the learner, i.e.",
"learning to learn.",
"Another challenge, which is of practical importance in applications of one-shot learning, is to avoid a lengthy optimization process such as eq:std-optim.",
"We propose to address both challenges by learning the parameters $W$ of the predictor from a single exemplar $z$ using a meta-prediction process, i.e.",
"a non-iterative feed-forward function $\\omega $ that maps $(z;\\, W^{\\prime })$ to $W$ .",
"Since in practice this function will be implemented using a deep neural network, we call it a learnet.",
"The learnet depends on the exemplar $z$ , which is a single representative of the class of interest, and contains parameters $W^{\\prime }$ of its own.",
"Learning to learn can now be posed as the problem of optimizing the learnet meta-parameters $W^{\\prime }$ using an objective function defined below.",
"Furthermore, the feed-forward learnet evaluation is much faster than solving the optimization problem (REF ).",
"In order to train the learnet, we require the latter to produce good predictors given any possible exemplar $z$ , which is empirically evaluated as an average over $n$ training samples $z_{i}$ : $\\min _{W^{\\prime }}\\frac{1}{n}\\sum _{i=1}^n\\mathcal {L}(\\varphi (x_{i};\\,\\omega (z_{i};\\, W^{\\prime })),\\,\\ell _{i}).$ In this expression, the performance of the predictor extracted by the learnet from the exemplar $z_i$ is assessed on a single “validation” pair $(x_i,\\ell _i)$ , comprising another exemplar and its label $\\ell _i$ .",
"Hence, the training data consists of triplets $(x_{i},z_{i},\\ell _{i})$ .",
"Notice that the meaning of the label $\\ell _{i}$ is subtly different from eq:std-optim since the class of interest changes depending on the exemplar $z_{i}$ : $\\ell _{i}$ is positive when $x_i$ and $z_i$ belong to the same class and negative otherwise.",
"Triplets are sampled uniformly with respect to these two cases.",
"Importantly, the parameters of the original predictor $\\varphi $ of eq:std-optim now change dynamically with each exemplar $z_{i}$ .",
"Note that the training data is reminiscent of that of siamese networks [1], which also learn from labeled sample pairs.",
"However, siamese networks apply the same model $\\varphi (x;\\, W)$ with shared weights $W$ to both $x_{i}$ and $z_{i}$ , and compute their inner-product to produce a similarity score: $\\min _{W}\\frac{1}{n}\\sum _{i=1}^n\\mathcal {L}(\\langle \\varphi (x_{i};\\, W),\\,\\varphi (z_{i};\\, W)\\rangle ,\\,\\ell _{i}).$ There are two key differences with our model.",
"First, we treat $x_{i}$ and $z_{i}$ asymmetrically, which results in a different objective function.",
"Second, and most importantly, the output of $\\omega (z;\\, W^{\\prime })$ is used to parametrize linear layers that determine the intermediate representations in the network $\\varphi $ .",
"This is significantly different to computing a single inner product in the last layer (eq:siamese-optim).",
"A similar argument can be made of bilinear networks [13].",
"eq:optim specifies the optimization objective of one-shot learning as dynamic parameter prediction.",
"By application of the chain rule, backpropagating derivatives through the computational blocks of $\\varphi (x;\\, W)$ and $\\omega (z;\\, W^{\\prime })$ is no more difficult than through any other standard deep network.",
"Nevertheless, when we dive into concrete implementations of such models we face a peculiar challenge, discussed next."
],
[
"The challenge of naive parameter prediction",
"In order to analyse the practical difficulties of implementing a learnet, we will begin with one-shot prediction of a fully-connected layer, as it is simpler to analyse.",
"This is given by $y=Wx+b,$ given an input $x\\in \\mathbb {R}^{d}$ , output $y\\in \\mathbb {R}^{k}$ , weights $W\\in \\mathbb {R}^{d\\times k}$ and biases $b\\in \\mathbb {R}^{k}$ .",
"We now replace the weights and biases with their functional counterparts, $w(z)$ and $b(z)$ , representing two outputs of the learnet $\\omega (z;\\, W^{\\prime })$ given the exemplar $z\\in \\mathbb {R}^{m}$ as input (to avoid clutter, we omit the implicit dependence on $W^{\\prime }$ ): $y=w(z)x+b(z).$ While eq:fc seems to be a drop-in replacement for linear layers, careful analysis reveals that it scales extremely poorly.",
"The main cause is the unusually large output space of the learnet $w:\\mathbb {R}^{m}\\rightarrow \\mathbb {R}^{d\\times k}$ .",
"For a comparable number of input and output units in a linear layer ($d\\simeq k$ ), the output space of the learnet grows quadratically with the number of units.",
"While this may seem to be a concern only for large networks, it is actually extremely difficult also for networks with few units.",
"Consider a simple linear learnet $w(z)=W^{\\prime }z$ .",
"Even for a very small fully-connected layer of only 100 units ($d=k=100$ ), and an exemplar $z$ with 100 features ($m=100$ ), the learnet already contains 1M parameters that must be learned.",
"Overfitting and space and time costs make learning such a regressor infeasible.",
"Furthermore, reducing the number of features in the exemplar can only achieve a small constant-size reduction on the total number of parameters.",
"The bottleneck is the quadratic size of the output space $dk$ , not the size of the input space $m$ ."
],
[
"Factorized linear layers",
"A simple way to reduce the size of the output space is to consider a factorized set of weights, by replacing eq:fc with: $y=M^{\\prime }\\operatorname{diag}\\left(w(z)\\right)Mx+b(z).$ The product $M^{\\prime }\\mathrm {diag}\\left(w(z)\\right)M$ can be seen as a factorized representation of the weights, analogous to the Singular Value Decomposition.",
"The matrix $M\\in \\mathbb {R}^{d\\times d}$ projects $x$ into a space where the elements of $w(z)$ represent disentangled factors of variation.",
"The second projection $M^{\\prime }\\in \\mathbb {R}^{d\\times k}$ maps the result back from this space.",
"Both $M$ and $M^{\\prime }$ contain additional parameters to be learned, but they are modest in size compared to the case discussed in sub:naive.",
"Importantly, the one-shot branch $w(z)$ now only has to predict a set of diagonal elements (see eq:fc-fac), so its output space grows linearly with the number of units in the layer (i.e.",
"$w(z)$ : $\\mathbb {R}^{m}\\rightarrow \\mathbb {R}^{d}$ )."
],
[
"Factorized convolutional layers",
"The factorization of eq:fc-fac can be generalized to convolutional layers as follows.",
"Given an input tensor $x\\in \\mathbb {R}^{r\\times c\\times d}$ , weights $W\\in \\mathbb {R}^{f\\times f\\times d\\times k}$ (where $f$ is the filter support size), and biases $b\\in \\mathbb {R}^{k}$ , the output $y\\in \\mathbb {R}^{r^{\\prime }\\times c^{\\prime }\\times k}$ of the convolutional layer is given by $y=W*x+b,$ where $*$ denotes convolution, and the biases $b$ are applied to each of the $k$ channels.",
"Projections analogous to $M$ and $M^{\\prime }$ in eq:fc-fac can be incorporated in the filter bank in different ways and it is not obvious which one to pick.",
"Here we take the view that $M$ and $M^{\\prime }$ should disentangle the feature channels (i.e.",
"third dimension of $x$ ), allowing $w(z)$ to choose which filter to apply to each channel.",
"As such, we consider the following factorization: $y=M^{\\prime }*w(z) *_{d}M*x+b(z),$ where $M\\in \\mathbb {R}^{1\\times 1\\times d\\times d}$ , $M^{\\prime }\\in \\mathbb {R}^{1\\times 1\\times d\\times k}$ , and $w(z)\\in \\mathbb {R}^{f\\times f\\times d}$ .",
"Convolution with subscript $d$ denotes independent filtering of $d$ channels, i.e.",
"each channel of $x *_{d}y$ is simply the convolution of the corresponding channel in $x$ and $y$ .",
"In practice, this can be achieved with filter tensors that are diagonal in the third and fourth dimensions, or using $d$ filter groups [11], each group containing a single filter.",
"An illustration is given in fig:one-shot-conv.",
"The predicted filters $w(z)$ can be interpreted as a filter basis, as described in the supplementary material (sec. A).",
"Notice that, under this factorization, the number of elements to be predicted by the one-shot branch $w(z)$ is only $f^{2}d$ (the filter size $f$ is typically very small, e.g.",
"3 or 5 [4], [21]).",
"Without the factorization, it would be $f^{2}dk$ (the number of elements of $W$ in eq:conv-layer).",
"Similarly to the case of fully-connected layers (sub:fc-fac), when $d\\simeq k$ this keeps the number of predicted elements from growing quadratically with the number of channels, allowing them to grow only linearly.",
"Examples of filters that are predicted by learnets are shown in figs.",
"REF and REF .",
"The resulting activations confirm that the networks induced by different exemplars do indeed possess different internal representations of the same input."
],
[
"Experiments",
"We evaluate learnets against baseline one-shot architectures (s:arch) on two one-shot learning problems in Optical Character Recognition (OCR; sec:omniglot) and visual object tracking (sec:tracking)."
],
[
"Architectures",
"As noted in s:method, the closest competitors to our method in discriminative one-shot learning are embedding learning using siamese architectures.",
"Therefore, we structure the experiments to compare against this baseline.",
"In particular, we choose to implement learnets using similar network topologies for a fairer comparison.",
"The baseline siamese architecture comprises two parallel streams $\\varphi (x;W)$ and $\\varphi (z;W)$ composed of a number of layers, such as convolution, max-pooling, and ReLU, sharing parameters $W$ (fig:archs.a).",
"The outputs of the two streams are compared by a layer $\\Gamma (\\varphi (x;W),\\varphi (z;W))$ computing a measure of similarity or dissimilarity.",
"We consider in particular: the dot product $\\langle a,b\\rangle $ between vectors $a$ and $b$ , the Euclidean distance $\\Vert a-b\\Vert $ , and the weighted $l^1$ -norm $\\Vert w \\odot a - w \\odot b\\Vert _1$ where $w$ is a vector of learnable weights and $\\odot $ the Hadamard product).",
"The first modification to the siamese baseline is to use a learnet to predict some of the intermediate shared stream parameters (fig:archs.b).",
"In this case $W = \\omega (z;W^{\\prime })$ and the siamese architecture writes $\\Gamma (\\varphi (x;\\omega (z;W^{\\prime })),\\varphi (z;\\omega (z;W^{\\prime })))$ .",
"Note that the siamese parameters are still the same in the two streams, whereas the learnet is an entirely new subnetwork whose purpose is to map the exemplar image to the shared weights.",
"We call this model the siamese learnet.",
"The second modification is a single-stream learnet configuration, using only one stream $\\varphi $ of the siamese architecture and predicting its parameter using the learnet $\\omega $ .",
"In this case, the comparison block $\\Gamma $ is reinterpreted as the last layer of the stream $\\varphi $ (fig:archs.c).",
"Note that: i) the single predicted stream and learnet are asymmetric and with different parameters and ii) the learnet predicts both the final comparison layer parameters $\\Gamma $ as well as intermediate filter parameters.",
"The single-stream learnet architecture can be understood to predict a discriminant function from one example, and the siamese learnet architecture to predict an embedding function for the comparison of two images.",
"These two variants demonstrate the versatility of the dynamic convolutional layer from eq:fc-fac.",
"Finally, in order to ensure that any difference in performance is not simply due to the asymmetry of the learnet architecture or to the induced filter factorizations (sub:fc-fac and sub:c-fac), we also compare unshared siamese nets, which use distinct parameters for each stream, and factorized siamese nets, where convolutions are replaced by factorized convolutions as in learnet."
],
[
"Character recognition in foreign alphabets",
"This section describes our experiments in one-shot learning on OCR.",
"For this, we use the Omniglot dataset [12], which contains images of handwritten characters from 50 different alphabets.",
"These alphabets are divided into 30 background and 20 evaluation alphabets.",
"The associated one-shot learning problem is to develop a method for determining whether, given any single exemplar of a character in an evaluation alphabet, any other image in that alphabet represents the same character or not.",
"Importantly, all methods are trained using only background alphabets and tested on the evaluation alphabets.",
"Dataset and evaluation protocol.",
"Character images are resized to $28 \\times 28$ pixels in order to be able to explore efficiently several variants of the proposed architectures.",
"There are exactly 20 sample images for each character, and an average of 32 characters per alphabet.",
"The dataset contains a total of 19,280 images in the background alphabets and 13,180 in the evaluation alphabets.",
"Algorithms are evaluated on a series of recognition problems.",
"Each recognition problem involves identifying the image in a set of 20 that shows the same character as an exemplar image (there is always exactly one match).",
"All of the characters in a single problem belong to the same alphabet.",
"At test time, given a collection of characters $(x_{1}, \\dots , x_{m})$ , the function is evaluated on each pair $(z, x_{i})$ and the candidate with the highest score is declared the match.",
"In the case of the learnet architectures, this can be interpreted as obtaining the parameters $W = \\omega (z; W^{\\prime })$ and then evaluating a static network $\\varphi (x_{i}; W)$ for each $x_{i}$ .",
"Architecture.",
"The baseline stream $\\varphi $ for the siamese, siamese learnet, and single-stream learnet architecture consists of 3 convolutional layers, with $2\\times 2$ max-pooling layers of stride 2 between them.",
"The filter sizes are $5 \\times 5 \\times 1 \\times 16$ , $5 \\times 5 \\times 16 \\times 64$ and $4 \\times 4 \\times 64 \\times 512$ .",
"For both the siamese learnet and the single-stream learnet, $\\omega $ consists of the same layers as $\\varphi $ , except the number of outputs is 1600 – one for each element of the 64 predicted filters (of size $5 \\times 5$ ).",
"To keep the experiments simple, we only predict the parameters of one convolutional layer.",
"We conducted cross-validation to choose the predicted layer and found that the second convolutional layer yields the best results for both of the proposed variants.",
"Siamese nets have previously been applied to this problem by Koch et al.",
"[9] using much deeper networks applied to images of size $105 \\times 105$ .",
"However, we have restricted this investigation to relatively shallow networks to enable a thorough exploration of the parameter space.",
"A more powerful algorithm for one-shot learning, Hierarchical Bayesian Program Learning [12], is able to achieve human-level performance.",
"However, this approach involves computationally expensive inference at test time, and leverages extra information at training time that describes the strokes drawn by the human author.",
"Learning.",
"Learning involves minimizing the objective function specific to each method (e.g.",
"eq:optim for learnet and eq:siamese-optim for siamese architectures) and uses stochastic gradient descent (SGD) in all cases.",
"As noted in s:method, the objective is obtained by sampling triplets $(z_i,x_i,\\ell _i)$ where exemplars $z_i$ and $x_i$ are congruous ($\\ell _i=+1$ ) or incongruous ($\\ell _i=-1$ ) with 50% probability.",
"We consider 100,000 random pairs for training per epoch, and train for 60 epochs.",
"We conducted a random search to find the best hyper-parameters for each algorithm (initial learning rate and geometric decay, standard deviation of Gaussian parameter initialization, and weight decay).",
"Results and discussion.",
"tab:omniglot-error shows the classification error obtained using variants of each architecture.",
"A dash indicates a failure to converge given a large range of hyper-parameters.",
"The two learnet architectures combined with the weighted $\\ell ^{1}$ distance are able to achieve significantly better results than other methods.",
"The best architecture reduced the error from 37.3% for a siamese network with shared parameters to 28.6% for a single-stream learnet.",
"While the Euclidean distance gave the best results for siamese networks with shared parameters, better results were achieved by learnets (and siamese networks with unshared parameters) using a weighted $\\ell ^{1}$ distance.",
"In fact, none of the alternative architectures are able to achieve lower error under the Euclidean distance than the shared siamese net.",
"The dot product was, in general, less effective than the other two metrics.",
"The introduction of the factorization in the convolutional layer might be expected to improve the quality of the estimated model by reducing the number of parameters, or to worsen it by diminishing the capacity of the hypothesis space.",
"For this relatively simple task of character recognition, the factorization did not seem to have a large effect."
],
[
"Object tracking",
"The task of single-target object tracking requires to locate an object of interest in a sequence of video frames.",
"A video frame can be seen as a collection $\\mathcal {F} =\\lbrace w_1,\\dots ,w_K\\rbrace $ of image windows; then, in a one-shot setting, given an exemplar $z\\in \\mathcal {F}_1$ of the object in the first frame $\\mathcal {F}_1$ , the goal is to identify the same window in the other frames $\\mathcal {F}_2,\\dots ,\\mathcal {F}_M$ .",
"Datasets.",
"The method is trained using the ImageNet Large Scale Visual Recognition Challenge 2015 [19], with 3,862 videos totalling more than one million annotated frames.",
"Instances of objects of thirty different classes (mostly vehicles and animals) are annotated throughout each video with bounding boxes.",
"For tracking, instance labels are retained but object class labels are ignored.",
"We use 90% of the videos for training, while the other 10% are held-out to monitor validation error during network training.",
"Testing uses the VOT 2015 benchmark [10].",
"Architecture.",
"We experiment with siamese and siamese learnet architectures (fig:archs) where the learnet $\\omega $ predicts the parameters of the second (dynamic) convolutional layer of the siamese streams.",
"Each siamese stream has five convolutional layers and we test three variants of those: variant (A) has the same configuration as AlexNet [11] but with stride 2 in the first layer, and variants (B) and (C) reduce to 50% the number of filters in the first two convolutional layers and, respectively, to 25% and 12.5% the number of filters in the last layer.",
"Training.",
"In order to train the architecture efficiently from many windows, the data is prepared as follows.",
"Given an object bounding box sampled at random, a crop $z$ double the size of that is extracted from the corresponding frame, padding with the average image color when needed.",
"The border is included in order to incorporate some visual context around the exemplar object.",
"Next, $\\ell \\in \\lbrace +1,-1\\rbrace $ is sampled at random with 75% probability of being positive.",
"If $\\ell =-1$ , an image $x$ is extracted by choosing at random a frame that does not contain the object.",
"Otherwise, a second frame containing the same object and within 50 temporal samples of the first is selected at random.",
"From that, a patch $x$ centered around the object and four times bigger is extracted.",
"In this way, $x$ contains both subwindows that do and do not match $z$ .",
"Images $z$ and $x$ are resized to $127 \\times 127$ and $255 \\times 255$ pixels, respectively, and the triplet $(z,x,\\ell )$ is formed.",
"All $127 \\times 127$ subwindows in $x$ are considered to not match $z$ except for the central $2\\times 2$ ones when $\\ell =+1$ .",
"All networks are trained from scratch using SGD for 50 epoch of 50,000 sample triplets $(z_i,x_i,\\ell _i)$ .",
"The multiple windows contained in $x$ are compared to $z$ efficiently by making the comparison layer $\\Gamma $ convolutional (fig:archs), accumulating a logistic loss across spatial locations.",
"The same hyper-parameters (learning rate of $10^{-2}$ geometrically decaying to $10^{-5}$ , weight decay of 0.005, and small mini-batches of size 8) are used for all experiments, which we found to work well for both the baseline and proposed architectures.",
"The weights are initialized using the improved Xavier [5] method, and we use batch normalization [7] after all linear layers.",
"Testing.",
"Adopting the initial crop as exemplar, the object is sought in a new frame within a radius of the previous position, proceeding sequentially.",
"This is done by evaluating the pupil net convolutionally, as well as searching at five possible scales in order to track the object through scale space.",
"Figure: The predicted filters and the output of a dynamic convolutional layer in a siamese learnet trained for the object tracking task.Best viewed in colour.Table: Tracking accuracy and number of tracking failures in the VOT 2015 Benchmark, as reported by the toolkit .",
"Architectures are grouped by size of the main network (see text).",
"For each group, the best entry for each column is in bold.We also report the scores of 5 recent trackers.Results and discussion.",
"tab:tracking-error compares the methods in terms of the official metrics (accuracy and number of failures) for the VOT 2015 benchmark [10].",
"The ranking plot produced by the VOT toolkit is provided in the supplementary material (fig. B.1).",
"From tab:tracking-error, it can be observed that factorizing the filters in the siamese architecture significantly diminishes its performance, but using a learnet to predict the filters in the factorization recovers this gap and in fact achieves better performance than the original siamese net.",
"The performance of the learnet architectures is not adversely affected by using the slimmer prediction networks B and C (with less channels).",
"An elementary tracker based on learnet compares favourably against recent tracking systems, which make use of different features and online model update strategies: DAT [17], DSST [2], MEEM [22], MUSTer [6] and SO-DLT [21].",
"SO-DLT in particular is a good example of direct adaptation of standard batch deep learning methodology to online learning, as it uses SGD during tracking to fine-tune an ensemble of deep convolutional networks.",
"However, the online adaptation of the model comes at a big computational cost and affects the speed of the method, which runs at 5 frames-per-second (FPS) on a GPU.",
"Due to the feed-forward nature of our one-shot learnets, they can track objects in real-time at framerates in excess of 60 FPS, while achieving less tracking failures.",
"We consider, however, that our implementation serves mostly as a proof-of-concept, using tracking as an interesting demonstration of one-shot-learning, and is orthogonal to many technical improvements found in the tracking literature [10]."
],
[
"Conclusions",
"In this work, we have shown that it is possible to obtain the parameters of a deep neural network using a single, feed-forward prediction from a second network.",
"This approach is desirable when iterative methods are too slow, and when large sets of annotated training samples are not available.",
"We have demonstrated the feasibility of feed-forward parameter prediction in two demanding one-shot learning tasks in OCR and visual tracking.",
"Our results hint at a promising avenue of research in “learning to learn” by solving millions of small discriminative problems in an offline phase.",
"Possible extensions include domain adaptation and sharing a single learnet between different pupil networks."
],
[
"Basis filters",
"This appendix provides an additional interpretation for the role of the predicted filters in a factorized convolutional layer (Section REF ).",
"To make the presentation succint, we will use a notation that is slightly different from the main text.",
"Let $x$ be a tensor of activations, then $x_{i}$ denotes channel $i$ of $x$ .",
"If $a$ is a multi-channel filter, then $a_{i j}$ denotes the filter for output channel $i$ and input channel $j$ .",
"That is, if $a$ is $m \\times n \\times p \\times q$ then $a_{i j}$ is $m \\times n$ for $i \\in [p]$ , $j \\in [q]$ .",
"The set $\\lbrace 1, \\dots , n\\rbrace $ is denoted $[n]$ .",
"The factorised convolution is $y = A x = M^{\\prime } W M x \\hspace{5.0pt}.$ where $M$ and $M^{\\prime }$ are pixel-wise projections and $W$ is a diagonal convolution.",
"While a general convolution computes $ \\textstyle (A v)_{i} = \\sum _{j} a_{i j} * v_{j}$ where each $a_{i j}$ is a single-channel filter, a diagonal convolution computes $ \\textstyle (W v)_{i} = w_{i} * v_{i}$ where each $w_{i}$ is a single-channel filter, and a pixel-wise projection computes $ \\textstyle (M v)_{i} = \\sum _{j} m_{i j} v_{j}$ where each $m_{i j}$ is a scalar.",
"Let $q$ be the number of channels of $x$ , let $p$ be the number of channels of $y$ and let $r$ be the number of channels of the intermediate activations.",
"Combining the above gives $ \\textstyle (W M x)_{k} & \\textstyle = w_{k} * \\left(\\sum _{j \\in [q]} m_{k j} x_{j}\\right)= \\sum _{j \\in [q]} m_{k j} w_{k} * x_{j} \\\\(M^{\\prime } W M x)_{i} & \\textstyle = \\sum _{k \\in [r]} m^{\\prime }_{i k} \\left(\\sum _{j \\in [q]} m_{k j} w_{k} * x_{j}\\right)= \\sum _{j \\in [q]} \\left(\\sum _{k \\in [r]} m^{\\prime }_{i k} m_{k j} w_{k}\\right) * x_{j} \\hspace{5.0pt}.$ This is therefore equivalent to a general convolution $y = A x$ where each filter $a_{i j}$ is a combination of $r$ single-channel basis filters $w_{k}$ $ \\textstyle a_{i j} = \\sum _{k \\in [r]} m^{\\prime }_{i k} m_{k j} w_{k}$ The predictions used in the dynamic convolution (Section REF ) essentially modify these $r$ basis filters."
]
] | 1606.05233 |
[
[
"Search for heavy long-lived charged $R$-hadrons with the ATLAS detector\n in 3.2 fb$^{-1}$ of proton--proton collision data at $\\sqrt{s} = 13$ TeV"
],
[
"Abstract A search for heavy long-lived charged $R$-hadrons is reported using a data sample corresponding to 3.2$^{-1}$ of proton--proton collisions at $\\sqrt{s} = 13$ TeV collected by the ATLAS experiment at the Large Hadron Collider at CERN.",
"The search is based on observables related to large ionisation losses and slow propagation velocities, which are signatures of heavy charged particles travelling significantly slower than the speed of light.",
"No significant deviations from the expected background are observed.",
"Upper limits at 95% confidence level are provided on the production cross section of long-lived $R$-hadrons in the mass range from 600 GeV to 2000 GeV and gluino, bottom and top squark masses are excluded up to 1580 GeV, 805 GeV and 890 GeV, respectively."
],
[
"Introduction",
"Heavy long-lived particles (LLP) are predicted in a range of theories extending the Standard Model (SM) in an attempt to address the hierarchy problem .",
"These theories include supersymmetry (SUSY) , , , , , , which allows for long-lived charged sleptons ($\\tilde{\\ell }$ ), squarks ($\\tilde{q}$ ), gluinos ($\\tilde{g}$ ) and charginos ($\\tilde{\\chi }^{\\pm }_{1}$ ) in models that either violate , , or conserve , , , , , , $R$ -parity.",
"Heavy long-lived charged particles can be produced at the Large Hadron Collider (LHC).",
"A search for composite colourless states of squarks or gluinos together with SM quarks or gluons, called $R$ -hadrons , is presented in this Letter.",
"The search exploits the fact that these particles are expected to propagate with a velocity, $\\beta =v/c$ , substantially lower than one and to exhibit a specific ionisation energy loss, $\\mathrm {d}E/\\mathrm {d}x$ , larger than that for any charged SM particle.",
"Similar searches have been performed previously by the ATLAS and CMS Collaborations , using data samples from Run 1 at the LHC.",
"No excesses of events above the expected backgrounds were observed, and lower mass limits were set at 95% confidence level (CL) around 1300 for gluino $R$ -hadrons.",
"$R$ -hadrons can be produced in $pp$ collision as either charged or neutral states, and can be modified to a state with different charge by interactions with the detector material , , arriving as neutral, charged or doubly charged particles in the muon spectrometer (MS) of the ATLAS detector.",
"This search does not use information from the MS and follows the “MS-agnostic” $R$ -hadron search approach in Ref. .",
"This strategy avoids assumptions about $R$ -hadron interactions with the detector, especially in the calorimeters, and is sensitive to scenarios in which $R$ -hadrons decay or become neutral (via parton exchange with the detector material in hadronic interactions) before reaching the MS."
],
[
"ATLAS detector",
"The ATLAS detector is a multi-purpose particle-physics detector consisting of an inner detector (ID) immersed in an axial magnetic field to reconstruct trajectories of charged particles, calorimeters to measure the energy of particles that interact electromagnetically or hadronically and a MS within a toroidal magnetic field to provide tracking for muons.",
"With near $4\\pi $ coverage in solid angle,ATLAS uses a right-handed coordinate system with its origin at the nominal interaction point (IP) in the centre of the detector and the $z$ -axis coinciding with the axis of the beam pipe.",
"The $x$ -axis points from the IP to the centre of the LHC ring, and the $y$ -axis points upward.",
"Cylindrical coordinates ($r$ , $\\phi $ ) are used in the transverse plane, $\\phi $ being the azimuthal angle around the beam pipe.",
"The pseudorapidity is defined in terms of the polar angle $\\theta $ as $\\eta = - \\ln \\tan (\\theta /2)$ .",
"the ATLAS detector is able to deduce the missing transverse momentum, $\\vec{p}_{\\mathrm {T}}^\\mathrm {~miss}$ , associated with each event.",
"The components of particular importance to this search are described in more detail below.",
"The ID consists of two distinct silicon detectors and a straw tracker, which jointly provide good momentum measurements for charged tracks.",
"The innermost part of the ID, a silicon pixel detector, typically provides four or more precision measurements for each track in the region $|\\eta |<2.5$ at radial distances $3.4<r<13$ cm from the LHC beam line.",
"All pixel layers are similar, except the innermost Insertable B-Layer (IBL) , which has a smaller pixel size and a reduced thickness, but also 4-bit instead of 8-bit encoding and hence poorer charge resolution than the other pixel layers.",
"The charge released by the passage of a charged particle is rarely contained within a single silicon pixel, and a neural network algorithm is used to form clusters from the single pixel charges.",
"For each cluster in the pixel detector a $\\mathrm {d}E/\\mathrm {d}x$ estimate can be provided, from which an overall $\\mathrm {d}E/\\mathrm {d}x$ measurement is calculated as a truncated mean to reduce the effect of the tail of the Landau distribution, by disregarding the one or two largest measurements .",
"Radiation sensitivity of the IBL electronics results in the measured $\\mathrm {d}E/\\mathrm {d}x$ drifting over time.",
"This effect is corrected by applying a dedicated time-dependent ionisation correction of 1.2% on average.",
"The mean and RMS of the $\\mathrm {d}E/\\mathrm {d}x$ measurement for a minimum-ionising particle are 1.12 $\\mathrm {g}^{-1}\\mathrm {cm}^{2}$ and 0.13 $\\mathrm {g}^{-1}\\mathrm {cm}^{2}$ , respectively, while the distribution extends to higher $\\mathrm {d}E/\\mathrm {d}x$ values, due to the remnants of the Landau tail.",
"The ATLAS calorimeter in the central detector region consists of an electromagnetic liquid-argon calorimeter followed by a hadronic tile calorimeter.",
"The estimation of $\\beta $ from time-of-flight measurements relies on timing and distance information from tile-calorimeter cells crossed by the extrapolated candidate track in three radial layers in the central barrel as well as an extended barrel on each side, as illustrated in Figure REF .",
"To reduce effects of detector noise, only cells in which the associated particle has deposited a minimum energy $E_\\text{min,cell}=500~$ are taken into account.",
"The time resolution depends on the energy deposited in the cell and also the layer type and thickness of the cell.",
"A series of calibration techniques is applied to achieve optimal performance, using a $Z \\rightarrow \\mu \\mu $ control sample.",
"Muons on average deposit slightly less energy than expected from signal, but variations sufficiently cover the relevant range.",
"First, a common time shift is applied for each short period of data taking (run) followed by five additional cell-by-cell $\\beta $ corrections.",
"A geometry-based cell correction is introduced to minimise the $\\eta $ dependence of $\\beta $ within each individual cell.",
"This is done by taking into account the actual trajectory ($\\eta $ and path length) of the extrapolated track in each calorimeter cell, to recalculate the distance-of-flight, instead of using the centre of the cell, as done in previous ATLAS searches (e.g.",
"in ).",
"The effect is most prominent at the edges of the largest cells at high $|\\eta |$ with shifts of up to 0.05 in $\\beta $ , and almost negligible for the cells at low $|\\eta |$ .",
"An additional correction, linear in $|\\eta |$ and only applied in simulation, is added to account for a timing mismodelling due to an imperfect simulation.",
"This correction is again most prominent for the cells at high $|\\eta |$ with shifts up to 0.1 in $\\beta $ .",
"The Optimal Filtering Algorithm (OFA) used for the readout of the tile calorimeter cells is optimised for in-time signals and introduces a bias towards lower values of $\\beta $ in the measured cell time of late-arriving particles.",
"To compensate for this bias for late-arriving particles, a correction is estimated from a fit to simulated late signals.",
"Cell times larger than 25 ns are discarded, to limit the size of the required correction.",
"The size of the correction is up to 0.05 in $\\beta $ .",
"A cell-time smearing is applied to adjust the cell-time resolution in simulation to that observed in data.",
"The uncertainty in the single $\\beta $ measurements is scaled up by about 12%, based on the requirement that the pull distribution $(\\beta - \\beta _{\\mathrm {true}}) / \\sigma _{\\beta }$ be a unit Gaussian.",
"Finally the $\\beta $ associated with the particle is estimated as a weighted average, using the $\\beta $ measurement in each traversed cell and its uncertainty, $\\sigma _{\\beta }$ .",
"Figure: Resolution of β\\beta for different cells in the ATLAS tile calorimeter obtained from a Z→μμZ \\rightarrow \\mu \\mu selection in data.",
"The final β\\beta measurement is a weighted average of the β\\beta measurements in the cells traversed by the candidate.After all calibrations, the single cell-time resolution ranges from 1.3 ns in cells at large radii to 2.5 ns in cells at small radii.",
"The distances from the nominal interaction point (IP) to the cell centres are 2.4 m to 3.6 m (4.2 m to 5.7 m) at $|\\eta | \\sim 0$ ($|\\eta | \\sim 1.25$ ).",
"This in turn results in a resolution of 0.06 to 0.23 in $\\beta $ , as shown in Figure REF .",
"The larger cells at large radii have a better resolution due to the higher energy deposits and their increased distance from the IP.",
"As described in Section , the expected $\\beta $ distribution for the background is determined from data.",
"However, the $\\beta $ distribution for the $R$ -hadron signal is obtained from simulation.",
"Figure REF shows the $\\beta $ distributions obtained for both data and simulation for a control sample of $Z \\rightarrow \\mu \\mu $ events that is used to validate the $\\beta $ measurement.",
"Good agreement between data and simulation supports the use of the simulation to predict the behaviour expected for the $R$ -hadron signal.",
"Figure: Distributions of β\\beta for data and simulation after a Z→μμZ \\rightarrow \\mu \\mu selection.",
"The values given for the mean and width are taken from Gaussian functions matched to data and simulation."
],
[
"Data and simulated events",
"The work presented in this Letter is based on 3.2 of $pp$ collision data collected in 2015 at a centre-of-mass energy $\\sqrt{s}=13~$ .",
"Reconstructed $Z \\rightarrow \\mu \\mu $ events in data and simulation are used for timing resolution studies.",
"Simulated signal events are used to study the expected signal behaviour.",
"$R$ -hadron signal events are generated with gluino (bottom-squark and top-squark) masses from 600 to 2000 (600 to 1400 ).",
"Pair production of gluinos and squarks is simulated in Pythia 6.427 with the AUET2B set of tuned parameters for the underlying event and the CTEQ6L1 parton distribution function (PDF) set, incorporating Pythia-specific specialised hadronisation routines , , to produce final states containing $R$ -hadrons.",
"The masses of the other SUSY particles are set to very high values to ensure that their contribution to the production cross section is negligible.",
"For a given sparticle mass the production cross section for gluino $R$ -hadrons is typically an order of magnitude higher than for bottom-squark and top-squark $R$ -hadrons.",
"The probability for a gluino to form a gluon–gluino bound state is assumed, based on a colour-octet model, to be 10% .",
"The associated hadronic activity produced by the colour field of the sparticle typically only possesses a small fraction of the initial energy of the sparticle , which should therefore be reasonably isolated.",
"To achieve a more accurate description of QCD radiative effects, the Pythia events are reweighted to match the transverse-momentum distribution of the gluino–gluino or squark–squark system to that obtained in dedicated MG5_aMC@NLO v2.2.3.p0 events, as MG5_aMC@NLO can produce additional QCD initial-state radiation (ISR) jets as part of the hard process, while Pythia only includes showering to add jets to the event.",
"All events pass through a full detector simulation , where interactions with matter are handled by dedicated Geant4 routines based on different scattering models: the model used to describe gluino (squark) $R$ -hadron interactions is referred to as the generic (Regge) model .",
"The $R$ -hadrons interact only moderately with the detector material, as most of the $R$ -hadron momentum is carried by the heavy gluino or squark, which has little interaction cross section.",
"Typically, the energy deposit in the calorimeters is less than 10 .",
"All simulated events include a modelling of contributions from pile-up by overlaying minimum-bias $pp$ interactions from the same (in-time pile-up) and nearby (out-of-time pile-up) bunch crossings, and are reconstructed using the same software used for collision data.",
"Simulated events are reweighted so that the distribution of the expected number of collisions per bunch crossing matches that of the data."
],
[
"Event selection",
"Events are selected online via a trigger based on the magnitude of the missing transverse momentum, $E_{\\mathrm {T}}^\\mathrm {miss}$ .",
"Large $E_{\\mathrm {T}}^\\mathrm {miss}$ values are produced mainly when QCD initial-state radiation (ISR) boosts the $R$ -hadron system, resulting in an imbalance between ISR and $R$ -hadrons whose momenta are not fully accounted for in the $E_{\\mathrm {T}}^\\mathrm {miss}$ calculation.",
"In particular, the adopted trigger imposes a threshold of 70 on $E_{\\mathrm {T}}^\\mathrm {miss}$ calculated solely from energy deposits in the calorimeters .",
"The signal efficiency of the $E_{\\mathrm {T}}^\\mathrm {miss}$ trigger varies between 32% and 50%, depending on the mass and type of the $R$ -hadron.",
"The offline event selection requires all relevant detector components to be fully operational; a primary vertex (PV) built from at least two well-reconstructed charged-particle tracks, each with a transverse momentum, , above 400 ; and at least one $R$ -hadron candidate track that meets the criteria specified below.",
"$R$ -hadron candidates are based on ID tracks with $>50~$ and $|\\eta |<1.65$ .",
"Candidates must not be within $\\Delta R = \\sqrt{\\left( \\Delta \\eta \\right)^2 + \\left(\\Delta \\phi \\right)^2} = 0.3$ of any jet with $> 50~$ , reconstructed using the anti-$k_t$ jet algorithm with radius parameter set to 0.4.",
"Furthermore, the candidates must not have any additional nearby ($\\Delta R < 0.2$ ) tracks with $> 10~$ .",
"Tracks reconstructed with $p>6.5~$ are rejected as unphysical.",
"To ensure a well reconstructed track, a minimum number of seven hits in the silicon detectors is required.",
"Of these, at least two clusters used to measure $\\mathrm {d}E/\\mathrm {d}x$ in the pixel detector are required, to ensure a good $\\mathrm {d}E/\\mathrm {d}x$ measurement.",
"Candidates with $|z_0^\\text{PV}\\sin (\\theta )| > 0.5~\\text{mm}$ or $|d_0|>2.0~\\text{mm}$ are removed, where $d_{0}$ is the transverse impact parameter at the candidate track's point of closest approach to the IP and $z_0^\\text{PV}$ is the $z$ coordinate of this point relative to the PV.",
"To suppress background muons stemming from cosmic-ray interactions, candidates with direction $\\left(\\eta , \\phi \\right)$ are rejected if an oppositely-charged track with almost specular direction, i.e.",
"with $|\\Delta \\eta | < 0.005$ and $|\\Delta \\phi | < 0.005$ with respect to $\\left( -\\eta , \\pi -\\phi \\right)$ , is identified on the opposite side of the detector.",
"In order to minimise the background from $Z \\rightarrow \\mu \\mu $ decays, candidates are rejected if they result in an invariant mass closer than 10 to the mass of the $Z$ boson when combined with the highest-muon candidate in the event.",
"In addition to the above mentioned track-quality criteria, candidates must also satisfy observable-quality criteria, defined by an unambiguous $\\beta \\gamma $ determination from the $\\mathrm {d}E/\\mathrm {d}x$ value, estimated using an empirical relation (more details can be found in Ref.",
"), determined from low-momentum pions, kaons and protons , and a $\\beta $ measurement, with an uncertainty $\\sigma _{\\beta }$ of less than 0.12.",
"In the following, $\\beta \\gamma $ refers to quantities derived from the $\\mathrm {d}E/\\mathrm {d}x$ measurement in the silicon pixel detector and $\\beta $ refers to the time-of-flight-based measurement in the tile calorimeter.",
"After this initial selection, 226107 of the approximately 36 million initially triggered data events as well as 10% to 15% of simulated signal events (the percentage increases with hypothesised mass) remain.",
"Only the candidate with the largest is used in events with multiple $R$ -hadron candidates.",
"The final signal selection, requiring a momentum above 200 as well as criteria summarised in Table REF , is based on $\\beta \\gamma $ and $\\beta $ , requiring $\\beta \\gamma <1.35$ ($<1.15$ ) for $R$ -hadron masses up to (greater than) 1.4 and $\\beta < 0.75$ in all cases.",
"The signal region is defined in the $m_{\\beta \\gamma }$ –$m_\\beta $ plane for each $R$ -hadron mass point, where $m_{\\beta \\gamma }$ and $m_\\beta $ are extracted independently from the measurement of the momentum as well as $\\beta \\gamma $ and $\\beta $ , respectively, via $m=p/{\\beta \\gamma }$ .",
"The minimum mass requirements, $m_{\\beta \\gamma }^\\text{min}$ and $m_\\beta ^\\text{min}$ , are set to correspond to a value about $2\\sigma $ below the nominal $R$ -hadron mass value, given the mass resolution expected for the signal.",
"Table: Final selection requirements as a function of the simulated RR-hadron mass.The total selection efficiency depends on the sparticle mass and varies between 9% and 15% for gluino and top-squark $R$ -hadrons and 6% to 8% for bottom-squark $R$ -hadrons.",
"The lower efficiency for bottom squarks is expected, as $R$ -hadrons are most likely produced in mesonic states, where those with down-type squarks tend to be neutral more often than those with up-type squarks, due to light-quark production ratios of $u:d:s \\approx 1:1:0.3$ during hadronisation.",
"The expected signal yield and efficiency, estimated background and observed number of events in data for the full mass range after the final selection are summarised in Table REF ."
],
[
"Background estimation",
"The background is evaluated in a data-driven manner.",
"First, probability distribution functions (pdf) in the momentum, and also in the $\\beta $ and $\\beta \\gamma $ values, are determined from data.",
"These pdfs are produced from candidates in data, which have passed the initial selection mentioned earlier, but fall in sidebands of the signal region, as described below.",
"Background distributions in $m_\\beta $ and $m_{\\beta \\gamma }$ are obtained by randomly sampling the pdfs derived above and then using the equation $m=p/{\\beta \\gamma }$ .",
"These mass distributions, which are normalised to the data events outside the signal region (i.e.",
"not passing both mass requirements of the hypothesis in question), are shown in Figure REF along with the data and expected signal for the 1000 gluino $R$ -hadron mass hypothesis.",
"Each $R$ -hadron mass hypothesis has a different selection, and therefore corresponding individual background estimates are produced accordingly.",
"The momentum pdf is produced from events that pass the momentum cut, but fail the $\\beta $ and $\\beta \\gamma $ requirements in Table REF for the chosen $R$ -hadron mass hypothesis, but nonetheless have $\\beta < 1$ and $\\beta \\gamma < 2.5$ .",
"The $\\beta $ and $\\beta \\gamma $ pdfs are produced by selecting events which pass the respective $\\beta $ and $\\beta \\gamma $ selection and have momentum in the range $50~< p < 200~$ .",
"Since momentum is correlated with $|\\eta |$ , any correlation between $|\\eta |$ and $\\beta $ ($\\beta \\gamma $ ) will lead to a correlation between momentum and $\\beta $ ($\\beta \\gamma $ ), invalidating the background estimate.",
"The size and impact of such correlations are reduced by determining the three pdfs in five equal-width bins of $|\\eta |$ .",
"This procedure also ensures that different detector regions are treated separately.",
"Figure: Data (black dots) and background estimates (red solid line) for m β m_{\\beta } (left) and m βγ m_{\\beta \\gamma } (right) for the gluino RR-hadron search (1000 ).",
"The green shaded band illustrates the statistical uncertainty of the background estimate.",
"The blue dashed lines illustrate the expected signal (on top of background) for the given RR-hadron mass hypothesis.",
"The black dashed vertical lines at 500 show the mass selection and the last bin includes all entries/masses above.",
"(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)"
],
[
"Systematic uncertainties",
"The systematic uncertainties are obtained from data, whenever possible.",
"The two major uncertainties for which this is not the case are cross sections and ISR, the latter being folded with the trigger efficiency curve obtained from data to produce the overall $E_{\\mathrm {T}}^\\mathrm {miss}$ trigger efficiency.",
"The individual contributions are outlined below and summarised in Table REF .",
"Theoretical cross sections Signal cross sections are calculated to next-to-leading order in the strong coupling constant, including the resummation of soft gluon emission at next-to-leading-logarithmic accuracy (NLO+NLL) , , .",
"The nominal cross section and the uncertainty are taken from an envelope of cross-section predictions using different PDF sets and factorisation and renormalisation scales, as described in Ref. .",
"This prescription results in an uncertainty of $14\\%$ (at 600 ) rising to $24\\%$ (at 1600 ) and to $32\\%$ (at 2000 ) for gluino $R$ -hadrons and marginally larger values for squark $R$ -hadrons.",
"Signal efficiency The $E_{\\mathrm {T}}^\\mathrm {miss}$ trigger uses only calorimeter information to calculate $E_{\\mathrm {T}}^\\mathrm {miss}$ , and has very low sensitivity to muons.",
"Hence, $Z \\rightarrow \\mu \\mu $ events can be used for calibration and to study systematic errors.",
"To evaluate the trigger efficiency, the trigger turn-on curve is obtained by fitting the measured efficiency vs. $E_{\\mathrm {T}}^\\mathrm {miss}$ in $Z \\rightarrow \\mu \\mu $ events, in both data and simulation.",
"These efficiency turn-on curves are then applied to the $E_{\\mathrm {T}}^\\mathrm {miss}$ spectrum from simulated $R$ -hadron events.",
"The total uncertainty is estimated from four contributions: the relative difference between the efficiencies obtained using the fitted threshold curves from $Z \\rightarrow \\mu \\mu $ data and simulation, the differences in efficiency obtained from independent $\\pm 1\\sigma $ variations in fit parameters relative to the unchanged turn-on-curve fit for both $Z \\rightarrow \\mu \\mu $ data and simulation and a $10\\%$ variation of the $E_{\\mathrm {T}}^\\mathrm {miss}$ to assess the scale uncertainty.",
"The $E_{\\mathrm {T}}^\\mathrm {miss}$ trigger is estimated to contribute a total uncertainty of 2% to the signal efficiency.",
"To address a possible mismodelling of ISR, and hence $E_{\\mathrm {T}}^\\mathrm {miss}$ in the signal events, half of the difference between the selection efficiency for the Pythia events and those reweighted with MG5_aMC@NLO is taken as an uncertainty in the expected signal and found to be below 14% in all cases.",
"The uncertainty in the pile-up modelling in simulation is found to affect the signal efficiency by between 7% and 1%, decreasing as a function of the simulated $R$ -hadron mass.",
"The systematic uncertainty in the $\\beta $ estimation is assessed by scaling the calorimeter-cell-time smearing of simulated events by $\\pm $ 10%, varying by $\\pm 1\\sigma $ the parameters of the linear fit to correct the remaining $\\eta $ dependence of the measured calorimeter time and by removing or doubling the cell-time correction introduced to correct the bias due to the OFA.",
"The uncertainty is calculated as half the maximum variation in signal efficiency in all combinations divided by the average signal efficiency and is found to be between $10\\%$ and $2\\%$ , decreasing with simulated $R$ -hadron mass.",
"The systematic uncertainty of the pixel $\\beta \\gamma $ measurement is assessed by taking into account the differences between simulation and data, the remaining variation in the reconstruction of reference masses after a run-by-run correction of an observed drift of $\\mathrm {d}E/\\mathrm {d}x$ , due to radiation sensitivity of the IBL electronics, and the stability of the $\\mathrm {d}E/\\mathrm {d}x$ -based proton mass estimate over time.",
"The impact on the signal efficiency is obtained by applying the variations corresponding to the above-listed uncertainties independently and the overall size of these effects is found to be below $3\\%$ for any simulated $R$ -hadron mass.",
"The uncertainty of the integrated luminosity is $5\\% $ , as derived following a methodology similar to that detailed in Ref.",
", from a calibration of the luminosity scale using $x$ –$y$ beam-separation scans performed in August 2015.",
"Background estimation The uncertainty in the background estimate is evaluated by varying both the number of $|\\eta |$ bins used when creating the $p$ , $\\beta $ and $\\beta \\gamma $ pdfs and the requirements on the background selection region.",
"The nominal number of $|\\eta |$ bins is varied between three and eight, while the requirements on observables are set to a medium ($\\beta < 0.975$ , $\\beta \\gamma < 2.45$ and $60~< p < 190~$ ) and a tight ($\\beta < 0.95$ , $\\beta \\gamma < 2.4$ and $70~< p < 180~$ ) selection.",
"Uncertainties introduced by statistical fluctuations in the pdfs are estimated by repeating $O(100)$ times the background estimation using pdfs with Poisson variations of the content in each bin.",
"The correction applied to high values of calorimeter cell times measured with the OFA is found to affect the background estimate by between 5% and 14%.",
"Effects arising from the $\\mathrm {d}E/\\mathrm {d}x$ measurement are assessed by using an analytical description to vary the shape of the high-ionisation tail and by changing the IBL ionisation correction by $\\pm 1\\sigma $ and are found to be between $17\\%$ and $6\\%$ , decreasing with simulated $R$ -hadron mass.",
"The effect of signal contamination in the background estimation is studied by introducing the expected number of signal events into the data before building the background estimate and is found to be $10\\%$ at a simulated mass of 600 , while negligible for higher masses, and is included in the overall uncertainty in the background estimate.",
"The overall uncertainty in the background estimate is found to be $30\\%$ to $43\\%$ , rising with simulated $R$ -hadron mass.",
"Since the background is very small for high $R$ -hadron masses ($\\ge $ 1400 ) the relatively large uncertainty does not affect the sensitivity in this region.",
"Table: Summary of all studied systematic uncertainties.",
"Ranges indicate a dependency on the RR-hadron mass hypothesis (from low to high masses)."
],
[
"Results",
"The resulting mass distributions of events for the 1000 gluino $R$ -hadron mass hypothesis can be seen in Figure REF .",
"Two events with masses above 500 pass the event selection for the 1000 mass hypothesis, while only one of these events passes the event selection for the 1600 mass hypothesis.",
"However, as can be seen in Table REF , at no point in the examined mass range does this search exhibit any statistically significant excess of events above the expected background, which is $1.23\\pm 0.37$ and $0.185\\pm 0.071$ for the two above-mentioned mass hypotheses, respectively.",
"Therefore, 95% CL upper limits are placed on the $R$ -hadron production cross section, as shown in Figure REF .",
"These limits are obtained from the expected signal and the estimated background in the signal region and using a one-bin counting experiment applying the $CL_s$ prescription .",
"Given the predicted theoretical cross sections, also shown in Figure REF , the cross-section limits are translated into lower limits on $R$ -hadron masses.",
"Expected lower limits at 95% CL on the $R$ -hadron masses of 1655 , 865 and 945 for the production of long-lived gluino, bottom-squark and top-squark $R$ -hadrons are derived, respectively.",
"Corresponding observed lower mass limits at 95% CL for gluino, bottom-squark and top-squark $R$ -hadrons are found to be 1580 , 805 and 890 , respectively.",
"For comparison, the corresponding ATLAS Run-1 8 lower limits at 95% CL on the mass of gluino, bottom-squark and top-squark $R$ -hadrons are also shown in Figure REF .",
"Figure: Expected (dashed black line) and observed (solid red line) 95% CL upper limits on the cross section as a function of mass for the production of long-lived gluino (top), bottom-squark (bottom-left) and top-squark (bottom-right) RR-hadrons.",
"The theory prediction along with its ±1σ\\pm 1\\sigma uncertainty is show as a black line and a blue band, respectively.",
"The observed 8 Run-1 limit and theory prediction are shown in dash-dotted and dotted lines, respectively.",
"(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)"
],
[
"Conclusion",
"A search for heavy long-lived particles in the form of composite colourless states of squarks or gluinos together with SM quarks and gluons, called $R$ -hadrons, and taking advantage of both ionisation and time-of-flight measurements is presented in this Letter.",
"The search uses 3.2 fb$^{-1}$ of $pp$ collisions at $\\sqrt{s}$ = 13 collected by the ATLAS experiment at the LHC.",
"No statistically significant excess of events above the expected background is found for any $R$ -hadron mass hypothesis.",
"Long-lived $R$ -hadrons containing a gluino, bottom or top squark are excluded at 95% CL for masses up to 1580 , 805 and 890 , respectively.",
"These results substantially extend previous ATLAS and CMS limits from 8 Run-1 data in case of gluino $R$ -hadrons and are complementary to searches for SUSY particles which decay promptly."
],
[
"Acknowledgements",
"We thank CERN for the very successful operation of the LHC, as well as the support staff from our institutions without whom ATLAS could not be operated efficiently.",
"We acknowledge the support of ANPCyT, Argentina; YerPhI, Armenia; ARC, Australia; BMWFW and FWF, Austria; ANAS, Azerbaijan; SSTC, Belarus; CNPq and FAPESP, Brazil; NSERC, NRC and CFI, Canada; CERN; CONICYT, Chile; CAS, MOST and NSFC, China; COLCIENCIAS, Colombia; MSMT CR, MPO CR and VSC CR, Czech Republic; DNRF and DNSRC, Denmark; IN2P3-CNRS, CEA-DSM/IRFU, France; GNSF, Georgia; BMBF, HGF, and MPG, Germany; GSRT, Greece; RGC, Hong Kong SAR, China; ISF, I-CORE and Benoziyo Center, Israel; INFN, Italy; MEXT and JSPS, Japan; CNRST, Morocco; FOM and NWO, Netherlands; RCN, Norway; MNiSW and NCN, Poland; FCT, Portugal; MNE/IFA, Romania; MES of Russia and NRC KI, Russian Federation; JINR; MESTD, Serbia; MSSR, Slovakia; ARRS and MIZŠ, Slovenia; DST/NRF, South Africa; MINECO, Spain; SRC and Wallenberg Foundation, Sweden; SERI, SNSF and Cantons of Bern and Geneva, Switzerland; MOST, Taiwan; TAEK, Turkey; STFC, United Kingdom; DOE and NSF, United States of America.",
"In addition, individual groups and members have received support from BCKDF, the Canada Council, CANARIE, CRC, Compute Canada, FQRNT, and the Ontario Innovation Trust, Canada; EPLANET, ERC, FP7, Horizon 2020 and Marie Skłodowska-Curie Actions, European Union; Investissements d'Avenir Labex and Idex, ANR, Région Auvergne and Fondation Partager le Savoir, France; DFG and AvH Foundation, Germany; Herakleitos, Thales and Aristeia programmes co-financed by EU-ESF and the Greek NSRF; BSF, GIF and Minerva, Israel; BRF, Norway; Generalitat de Catalunya, Generalitat Valenciana, Spain; the Royal Society and Leverhulme Trust, United Kingdom.",
"The crucial computing support from all WLCG partners is acknowledged gratefully, in particular from CERN and the ATLAS Tier-1 facilities at TRIUMF (Canada), NDGF (Denmark, Norway, Sweden), CC-IN2P3 (France), KIT/GridKA (Germany), INFN-CNAF (Italy), NL-T1 (Netherlands), PIC (Spain), ASGC (Taiwan), RAL (UK) and BNL (USA) and in the Tier-2 facilities worldwide.",
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"Baines$^\\textrm {\\scriptsize 131}$ , O.K.",
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"Bannoura$^\\textrm {\\scriptsize 174}$ , L. Barak$^\\textrm {\\scriptsize 32}$ , E.L. Barberio$^\\textrm {\\scriptsize 89}$ , D. Barberis$^\\textrm {\\scriptsize 52a,52b}$ , M. Barbero$^\\textrm {\\scriptsize 86}$ , T. Barillari$^\\textrm {\\scriptsize 101}$ , M-S Barisits$^\\textrm {\\scriptsize 32}$ , T. Barklow$^\\textrm {\\scriptsize 143}$ , N. Barlow$^\\textrm {\\scriptsize 30}$ , S.L.",
"Barnes$^\\textrm {\\scriptsize 85}$ , B.M.",
"Barnett$^\\textrm {\\scriptsize 131}$ , R.M.",
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"Barr$^\\textrm {\\scriptsize 120}$ , L. Barranco Navarro$^\\textrm {\\scriptsize 166}$ , F. Barreiro$^\\textrm {\\scriptsize 83}$ , J. Barreiro Guimarães da Costa$^\\textrm {\\scriptsize 35a}$ , R. Bartoldus$^\\textrm {\\scriptsize 143}$ , A.E.",
"Barton$^\\textrm {\\scriptsize 73}$ , P. Bartos$^\\textrm {\\scriptsize 144a}$ , A. Basalaev$^\\textrm {\\scriptsize 123}$ , A. Bassalat$^\\textrm {\\scriptsize 117}$ , R.L.",
"Bates$^\\textrm {\\scriptsize 55}$ , S.J.",
"Batista$^\\textrm {\\scriptsize 158}$ , J.R. Batley$^\\textrm {\\scriptsize 30}$ , M. Battaglia$^\\textrm {\\scriptsize 137}$ , M. Bauce$^\\textrm {\\scriptsize 132a,132b}$ , F. Bauer$^\\textrm {\\scriptsize 136}$ , H.S.",
"Bawa$^\\textrm {\\scriptsize 143}$$^{,f}$ , J.B. Beacham$^\\textrm {\\scriptsize 111}$ , M.D.",
"Beattie$^\\textrm {\\scriptsize 73}$ , T. Beau$^\\textrm {\\scriptsize 81}$ , P.H.",
"Beauchemin$^\\textrm {\\scriptsize 161}$ , P. Bechtle$^\\textrm {\\scriptsize 23}$ , H.P.",
"Beck$^\\textrm {\\scriptsize 18}$$^{,g}$ , K. Becker$^\\textrm {\\scriptsize 120}$ , M. Becker$^\\textrm {\\scriptsize 84}$ , M. Beckingham$^\\textrm {\\scriptsize 169}$ , C. Becot$^\\textrm {\\scriptsize 110}$ , A.J.",
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"Bednyakov$^\\textrm {\\scriptsize 66}$ , M. Bedognetti$^\\textrm {\\scriptsize 107}$ , C.P.",
"Bee$^\\textrm {\\scriptsize 148}$ , L.J.",
"Beemster$^\\textrm {\\scriptsize 107}$ , T.A.",
"Beermann$^\\textrm {\\scriptsize 32}$ , M. Begel$^\\textrm {\\scriptsize 27}$ , J.K. Behr$^\\textrm {\\scriptsize 44}$ , C. Belanger-Champagne$^\\textrm {\\scriptsize 88}$ , A.S. Bell$^\\textrm {\\scriptsize 79}$ , G. Bella$^\\textrm {\\scriptsize 153}$ , L. Bellagamba$^\\textrm {\\scriptsize 22a}$ , A. Bellerive$^\\textrm {\\scriptsize 31}$ , M. Bellomo$^\\textrm {\\scriptsize 87}$ , K. Belotskiy$^\\textrm {\\scriptsize 98}$ , O. Beltramello$^\\textrm {\\scriptsize 32}$ , N.L.",
"Belyaev$^\\textrm {\\scriptsize 98}$ , O. Benary$^\\textrm {\\scriptsize 153}$ , D. Benchekroun$^\\textrm {\\scriptsize 135a}$ , M. Bender$^\\textrm {\\scriptsize 100}$ , K. Bendtz$^\\textrm {\\scriptsize 146a,146b}$ , N. Benekos$^\\textrm {\\scriptsize 10}$ , Y. Benhammou$^\\textrm {\\scriptsize 153}$ , E. Benhar Noccioli$^\\textrm {\\scriptsize 175}$ , J. Benitez$^\\textrm {\\scriptsize 64}$ , D.P.",
"Benjamin$^\\textrm {\\scriptsize 47}$ , J.R. Bensinger$^\\textrm {\\scriptsize 25}$ , S. Bentvelsen$^\\textrm {\\scriptsize 107}$ , L. Beresford$^\\textrm {\\scriptsize 120}$ , M. Beretta$^\\textrm {\\scriptsize 49}$ , D. Berge$^\\textrm {\\scriptsize 107}$ , E. Bergeaas Kuutmann$^\\textrm {\\scriptsize 164}$ , N. Berger$^\\textrm {\\scriptsize 5}$ , J. Beringer$^\\textrm {\\scriptsize 16}$ , S. Berlendis$^\\textrm {\\scriptsize 57}$ , N.R.",
"Bernard$^\\textrm {\\scriptsize 87}$ , C. Bernius$^\\textrm {\\scriptsize 110}$ , F.U.",
"Bernlochner$^\\textrm {\\scriptsize 23}$ , T. Berry$^\\textrm {\\scriptsize 78}$ , P. Berta$^\\textrm {\\scriptsize 129}$ , C. Bertella$^\\textrm {\\scriptsize 84}$ , G. Bertoli$^\\textrm {\\scriptsize 146a,146b}$ , F. Bertolucci$^\\textrm {\\scriptsize 124a,124b}$ , I.A.",
"Bertram$^\\textrm {\\scriptsize 73}$ , C. Bertsche$^\\textrm {\\scriptsize 44}$ , D. Bertsche$^\\textrm {\\scriptsize 113}$ , G.J.",
"Besjes$^\\textrm {\\scriptsize 38}$ , O. Bessidskaia Bylund$^\\textrm {\\scriptsize 146a,146b}$ , M. Bessner$^\\textrm {\\scriptsize 44}$ , N. Besson$^\\textrm {\\scriptsize 136}$ , C. Betancourt$^\\textrm {\\scriptsize 50}$ , S. Bethke$^\\textrm {\\scriptsize 101}$ , A.J.",
"Bevan$^\\textrm {\\scriptsize 77}$ , R.M.",
"Bianchi$^\\textrm {\\scriptsize 125}$ , L. Bianchini$^\\textrm {\\scriptsize 25}$ , M. Bianco$^\\textrm {\\scriptsize 32}$ , O. Biebel$^\\textrm {\\scriptsize 100}$ , D. Biedermann$^\\textrm {\\scriptsize 17}$ , R. Bielski$^\\textrm {\\scriptsize 85}$ , N.V. Biesuz$^\\textrm {\\scriptsize 124a,124b}$ , M. Biglietti$^\\textrm {\\scriptsize 134a}$ , J. Bilbao De Mendizabal$^\\textrm {\\scriptsize 51}$ , T.R.V.",
"Billoud$^\\textrm {\\scriptsize 95}$ , H. Bilokon$^\\textrm {\\scriptsize 49}$ , M. Bindi$^\\textrm {\\scriptsize 56}$ , S. Binet$^\\textrm {\\scriptsize 117}$ , A. Bingul$^\\textrm {\\scriptsize 20b}$ , C. Bini$^\\textrm {\\scriptsize 132a,132b}$ , S. Biondi$^\\textrm {\\scriptsize 22a,22b}$ , D.M.",
"Bjergaard$^\\textrm {\\scriptsize 47}$ , C.W.",
"Black$^\\textrm {\\scriptsize 150}$ , J.E.",
"Black$^\\textrm {\\scriptsize 143}$ , K.M.",
"Black$^\\textrm {\\scriptsize 24}$ , D. Blackburn$^\\textrm {\\scriptsize 138}$ , R.E.",
"Blair$^\\textrm {\\scriptsize 6}$ , J.-B.",
"Blanchard$^\\textrm {\\scriptsize 136}$ , T. Blazek$^\\textrm {\\scriptsize 144a}$ , I. Bloch$^\\textrm {\\scriptsize 44}$ , C. Blocker$^\\textrm {\\scriptsize 25}$ , W. Blum$^\\textrm {\\scriptsize 84}$$^{,*}$ , U. Blumenschein$^\\textrm {\\scriptsize 56}$ , S. Blunier$^\\textrm {\\scriptsize 34a}$ , G.J.",
"Bobbink$^\\textrm {\\scriptsize 107}$ , V.S.",
"Bobrovnikov$^\\textrm {\\scriptsize 109}$$^{,c}$ , S.S. Bocchetta$^\\textrm {\\scriptsize 82}$ , A. Bocci$^\\textrm {\\scriptsize 47}$ , C. Bock$^\\textrm {\\scriptsize 100}$ , M. Boehler$^\\textrm {\\scriptsize 50}$ , D. Boerner$^\\textrm {\\scriptsize 174}$ , J.A.",
"Bogaerts$^\\textrm {\\scriptsize 32}$ , D. Bogavac$^\\textrm {\\scriptsize 14}$ , A.G. Bogdanchikov$^\\textrm {\\scriptsize 109}$ , C. Bohm$^\\textrm {\\scriptsize 146a}$ , V. Boisvert$^\\textrm {\\scriptsize 78}$ , P. Bokan$^\\textrm {\\scriptsize 14}$ , T. Bold$^\\textrm {\\scriptsize 40a}$ , A.S. Boldyrev$^\\textrm {\\scriptsize 163a,163c}$ , M. Bomben$^\\textrm {\\scriptsize 81}$ , M. Bona$^\\textrm {\\scriptsize 77}$ , M. Boonekamp$^\\textrm {\\scriptsize 136}$ , A. Borisov$^\\textrm {\\scriptsize 130}$ , G. Borissov$^\\textrm {\\scriptsize 73}$ , J. Bortfeldt$^\\textrm {\\scriptsize 32}$ , D. Bortoletto$^\\textrm {\\scriptsize 120}$ , V. Bortolotto$^\\textrm {\\scriptsize 61a,61b,61c}$ , K. Bos$^\\textrm {\\scriptsize 107}$ , D. Boscherini$^\\textrm {\\scriptsize 22a}$ , M. Bosman$^\\textrm {\\scriptsize 13}$ , J.D.",
"Bossio Sola$^\\textrm {\\scriptsize 29}$ , J. Boudreau$^\\textrm {\\scriptsize 125}$ , J. Bouffard$^\\textrm {\\scriptsize 2}$ , E.V.",
"Bouhova-Thacker$^\\textrm {\\scriptsize 73}$ , D. Boumediene$^\\textrm {\\scriptsize 36}$ , C. Bourdarios$^\\textrm {\\scriptsize 117}$ , S.K.",
"Boutle$^\\textrm {\\scriptsize 55}$ , A. Boveia$^\\textrm {\\scriptsize 32}$ , J. Boyd$^\\textrm {\\scriptsize 32}$ , I.R.",
"Boyko$^\\textrm {\\scriptsize 66}$ , J. Bracinik$^\\textrm {\\scriptsize 19}$ , A. Brandt$^\\textrm {\\scriptsize 8}$ , G. Brandt$^\\textrm {\\scriptsize 56}$ , O. Brandt$^\\textrm {\\scriptsize 59a}$ , U. Bratzler$^\\textrm {\\scriptsize 156}$ , B. Brau$^\\textrm {\\scriptsize 87}$ , J.E.",
"Brau$^\\textrm {\\scriptsize 116}$ , H.M. Braun$^\\textrm {\\scriptsize 174}$$^{,*}$ , W.D.",
"Breaden Madden$^\\textrm {\\scriptsize 55}$ , K. Brendlinger$^\\textrm {\\scriptsize 122}$ , A.J.",
"Brennan$^\\textrm {\\scriptsize 89}$ , L. Brenner$^\\textrm {\\scriptsize 107}$ , R. Brenner$^\\textrm {\\scriptsize 164}$ , S. Bressler$^\\textrm {\\scriptsize 171}$ , T.M.",
"Bristow$^\\textrm {\\scriptsize 48}$ , D. Britton$^\\textrm {\\scriptsize 55}$ , D. Britzger$^\\textrm {\\scriptsize 44}$ , F.M.",
"Brochu$^\\textrm {\\scriptsize 30}$ , I. Brock$^\\textrm {\\scriptsize 23}$ , R. Brock$^\\textrm {\\scriptsize 91}$ , G. Brooijmans$^\\textrm {\\scriptsize 37}$ , T. Brooks$^\\textrm {\\scriptsize 78}$ , W.K.",
"Brooks$^\\textrm {\\scriptsize 34b}$ , J. Brosamer$^\\textrm {\\scriptsize 16}$ , E. Brost$^\\textrm {\\scriptsize 108}$ , J.H Broughton$^\\textrm {\\scriptsize 19}$ , P.A.",
"Bruckman de Renstrom$^\\textrm {\\scriptsize 41}$ , D. Bruncko$^\\textrm {\\scriptsize 144b}$ , R. Bruneliere$^\\textrm {\\scriptsize 50}$ , A. Bruni$^\\textrm {\\scriptsize 22a}$ , G. Bruni$^\\textrm {\\scriptsize 22a}$ , L.S.",
"Bruni$^\\textrm {\\scriptsize 107}$ , BH Brunt$^\\textrm {\\scriptsize 30}$ , M. Bruschi$^\\textrm {\\scriptsize 22a}$ , N. Bruscino$^\\textrm {\\scriptsize 23}$ , P. Bryant$^\\textrm {\\scriptsize 33}$ , L. Bryngemark$^\\textrm {\\scriptsize 82}$ , T. Buanes$^\\textrm {\\scriptsize 15}$ , Q. Buat$^\\textrm {\\scriptsize 142}$ , P. Buchholz$^\\textrm {\\scriptsize 141}$ , A.G. Buckley$^\\textrm {\\scriptsize 55}$ , I.A.",
"Budagov$^\\textrm {\\scriptsize 66}$ , F. Buehrer$^\\textrm {\\scriptsize 50}$ , M.K.",
"Bugge$^\\textrm {\\scriptsize 119}$ , O. Bulekov$^\\textrm {\\scriptsize 98}$ , D. Bullock$^\\textrm {\\scriptsize 8}$ , H. Burckhart$^\\textrm {\\scriptsize 32}$ , S. Burdin$^\\textrm {\\scriptsize 75}$ , C.D.",
"Burgard$^\\textrm {\\scriptsize 50}$ , B. Burghgrave$^\\textrm {\\scriptsize 108}$ , K. Burka$^\\textrm {\\scriptsize 41}$ , S. Burke$^\\textrm {\\scriptsize 131}$ , I. Burmeister$^\\textrm {\\scriptsize 45}$ , J.T.P.",
"Burr$^\\textrm {\\scriptsize 120}$ , E. Busato$^\\textrm {\\scriptsize 36}$ , D. Büscher$^\\textrm {\\scriptsize 50}$ , V. Büscher$^\\textrm {\\scriptsize 84}$ , P. Bussey$^\\textrm {\\scriptsize 55}$ , J.M.",
"Butler$^\\textrm {\\scriptsize 24}$ , C.M.",
"Buttar$^\\textrm {\\scriptsize 55}$ , J.M.",
"Butterworth$^\\textrm {\\scriptsize 79}$ , P. Butti$^\\textrm {\\scriptsize 107}$ , W. Buttinger$^\\textrm {\\scriptsize 27}$ , A. Buzatu$^\\textrm {\\scriptsize 55}$ , A.R.",
"Buzykaev$^\\textrm {\\scriptsize 109}$$^{,c}$ , S. Cabrera Urbán$^\\textrm {\\scriptsize 166}$ , D. Caforio$^\\textrm {\\scriptsize 128}$ , V.M.",
"Cairo$^\\textrm {\\scriptsize 39a,39b}$ , O. Cakir$^\\textrm {\\scriptsize 4a}$ , N. Calace$^\\textrm {\\scriptsize 51}$ , P. Calafiura$^\\textrm {\\scriptsize 16}$ , A. Calandri$^\\textrm {\\scriptsize 86}$ , G. Calderini$^\\textrm {\\scriptsize 81}$ , P. Calfayan$^\\textrm {\\scriptsize 100}$ , G. Callea$^\\textrm {\\scriptsize 39a,39b}$ , L.P. Caloba$^\\textrm {\\scriptsize 26a}$ , S. Calvente Lopez$^\\textrm {\\scriptsize 83}$ , D. Calvet$^\\textrm {\\scriptsize 36}$ , S. Calvet$^\\textrm {\\scriptsize 36}$ , T.P.",
"Calvet$^\\textrm {\\scriptsize 86}$ , R. Camacho Toro$^\\textrm {\\scriptsize 33}$ , S. Camarda$^\\textrm {\\scriptsize 32}$ , P. Camarri$^\\textrm {\\scriptsize 133a,133b}$ , D. Cameron$^\\textrm {\\scriptsize 119}$ , R. Caminal Armadans$^\\textrm {\\scriptsize 165}$ , C. Camincher$^\\textrm {\\scriptsize 57}$ , S. Campana$^\\textrm {\\scriptsize 32}$ , M. Campanelli$^\\textrm {\\scriptsize 79}$ , A. Camplani$^\\textrm {\\scriptsize 92a,92b}$ , A. Campoverde$^\\textrm {\\scriptsize 141}$ , V. Canale$^\\textrm {\\scriptsize 104a,104b}$ , A. Canepa$^\\textrm {\\scriptsize 159a}$ , M. Cano Bret$^\\textrm {\\scriptsize 35e}$ , J. Cantero$^\\textrm {\\scriptsize 114}$ , R. Cantrill$^\\textrm {\\scriptsize 126a}$ , T. Cao$^\\textrm {\\scriptsize 42}$ , M.D.M.",
"Capeans Garrido$^\\textrm {\\scriptsize 32}$ , I. Caprini$^\\textrm {\\scriptsize 28b}$ , M. Caprini$^\\textrm {\\scriptsize 28b}$ , M. Capua$^\\textrm {\\scriptsize 39a,39b}$ , R. Caputo$^\\textrm {\\scriptsize 84}$ , R.M.",
"Carbone$^\\textrm {\\scriptsize 37}$ , R. Cardarelli$^\\textrm {\\scriptsize 133a}$ , F. Cardillo$^\\textrm {\\scriptsize 50}$ , I. Carli$^\\textrm {\\scriptsize 129}$ , T. Carli$^\\textrm {\\scriptsize 32}$ , G. Carlino$^\\textrm {\\scriptsize 104a}$ , L. Carminati$^\\textrm {\\scriptsize 92a,92b}$ , S. Caron$^\\textrm {\\scriptsize 106}$ , E. Carquin$^\\textrm {\\scriptsize 34b}$ , G.D. Carrillo-Montoya$^\\textrm {\\scriptsize 32}$ , J.R. Carter$^\\textrm {\\scriptsize 30}$ , J. Carvalho$^\\textrm {\\scriptsize 126a,126c}$ , D. Casadei$^\\textrm {\\scriptsize 19}$ , M.P.",
"Casado$^\\textrm {\\scriptsize 13}$$^{,h}$ , M. Casolino$^\\textrm {\\scriptsize 13}$ , D.W. Casper$^\\textrm {\\scriptsize 162}$ , E. Castaneda-Miranda$^\\textrm {\\scriptsize 145a}$ , R. Castelijn$^\\textrm {\\scriptsize 107}$ , A. Castelli$^\\textrm {\\scriptsize 107}$ , V. Castillo Gimenez$^\\textrm {\\scriptsize 166}$ , N.F.",
"Castro$^\\textrm {\\scriptsize 126a}$$^{,i}$ , A. Catinaccio$^\\textrm {\\scriptsize 32}$ , J.R. Catmore$^\\textrm {\\scriptsize 119}$ , A. Cattai$^\\textrm {\\scriptsize 32}$ , J. Caudron$^\\textrm {\\scriptsize 23}$ , V. Cavaliere$^\\textrm {\\scriptsize 165}$ , E. Cavallaro$^\\textrm {\\scriptsize 13}$ , D. Cavalli$^\\textrm {\\scriptsize 92a}$ , M. Cavalli-Sforza$^\\textrm {\\scriptsize 13}$ , V. Cavasinni$^\\textrm {\\scriptsize 124a,124b}$ , F. Ceradini$^\\textrm {\\scriptsize 134a,134b}$ , L. Cerda Alberich$^\\textrm {\\scriptsize 166}$ , B.C.",
"Cerio$^\\textrm {\\scriptsize 47}$ , A.S. Cerqueira$^\\textrm {\\scriptsize 26b}$ , A. Cerri$^\\textrm {\\scriptsize 149}$ , L. Cerrito$^\\textrm {\\scriptsize 133a,133b}$ , F. Cerutti$^\\textrm {\\scriptsize 16}$ , M. Cerv$^\\textrm {\\scriptsize 32}$ , A. Cervelli$^\\textrm {\\scriptsize 18}$ , S.A. Cetin$^\\textrm {\\scriptsize 20d}$ , A. Chafaq$^\\textrm {\\scriptsize 135a}$ , D. Chakraborty$^\\textrm {\\scriptsize 108}$ , S.K.",
"Chan$^\\textrm {\\scriptsize 58}$ , Y.L.",
"Chan$^\\textrm {\\scriptsize 61a}$ , P. Chang$^\\textrm {\\scriptsize 165}$ , J.D.",
"Chapman$^\\textrm {\\scriptsize 30}$ , D.G.",
"Charlton$^\\textrm {\\scriptsize 19}$ , A. Chatterjee$^\\textrm {\\scriptsize 51}$ , C.C.",
"Chau$^\\textrm {\\scriptsize 158}$ , C.A.",
"Chavez Barajas$^\\textrm {\\scriptsize 149}$ , S. Che$^\\textrm {\\scriptsize 111}$ , S. Cheatham$^\\textrm {\\scriptsize 73}$ , A. Chegwidden$^\\textrm {\\scriptsize 91}$ , S. Chekanov$^\\textrm {\\scriptsize 6}$ , S.V.",
"Chekulaev$^\\textrm {\\scriptsize 159a}$ , G.A.",
"Chelkov$^\\textrm {\\scriptsize 66}$$^{,j}$ , M.A.",
"Chelstowska$^\\textrm {\\scriptsize 90}$ , C. Chen$^\\textrm {\\scriptsize 65}$ , H. Chen$^\\textrm {\\scriptsize 27}$ , K. Chen$^\\textrm {\\scriptsize 148}$ , S. Chen$^\\textrm {\\scriptsize 35c}$ , S. Chen$^\\textrm {\\scriptsize 155}$ , X. Chen$^\\textrm {\\scriptsize 35f}$ , Y. Chen$^\\textrm {\\scriptsize 68}$ , H.C. Cheng$^\\textrm {\\scriptsize 90}$ , H.J Cheng$^\\textrm {\\scriptsize 35a}$ , Y. Cheng$^\\textrm {\\scriptsize 33}$ , A. Cheplakov$^\\textrm {\\scriptsize 66}$ , E. Cheremushkina$^\\textrm {\\scriptsize 130}$ , R. Cherkaoui El Moursli$^\\textrm {\\scriptsize 135e}$ , V. Chernyatin$^\\textrm {\\scriptsize 27}$$^{,*}$ , E. Cheu$^\\textrm {\\scriptsize 7}$ , L. Chevalier$^\\textrm {\\scriptsize 136}$ , V. Chiarella$^\\textrm {\\scriptsize 49}$ , G. Chiarelli$^\\textrm {\\scriptsize 124a,124b}$ , G. Chiodini$^\\textrm {\\scriptsize 74a}$ , A.S. Chisholm$^\\textrm {\\scriptsize 19}$ , A. Chitan$^\\textrm {\\scriptsize 28b}$ , M.V.",
"Chizhov$^\\textrm {\\scriptsize 66}$ , K. Choi$^\\textrm {\\scriptsize 62}$ , A.R.",
"Chomont$^\\textrm {\\scriptsize 36}$ , S. Chouridou$^\\textrm {\\scriptsize 9}$ , B.K.B.",
"Chow$^\\textrm {\\scriptsize 100}$ , V. Christodoulou$^\\textrm {\\scriptsize 79}$ , D. Chromek-Burckhart$^\\textrm {\\scriptsize 32}$ , J. Chudoba$^\\textrm {\\scriptsize 127}$ , A.J.",
"Chuinard$^\\textrm {\\scriptsize 88}$ , J.J. Chwastowski$^\\textrm {\\scriptsize 41}$ , L. Chytka$^\\textrm {\\scriptsize 115}$ , G. Ciapetti$^\\textrm {\\scriptsize 132a,132b}$ , A.K.",
"Ciftci$^\\textrm {\\scriptsize 4a}$ , D. Cinca$^\\textrm {\\scriptsize 45}$ , V. Cindro$^\\textrm {\\scriptsize 76}$ , I.A.",
"Cioara$^\\textrm {\\scriptsize 23}$ , C. Ciocca$^\\textrm {\\scriptsize 22a,22b}$ , A. Ciocio$^\\textrm {\\scriptsize 16}$ , F. Cirotto$^\\textrm {\\scriptsize 104a,104b}$ , Z.H.",
"Citron$^\\textrm {\\scriptsize 171}$ , M. Citterio$^\\textrm {\\scriptsize 92a}$ , M. Ciubancan$^\\textrm {\\scriptsize 28b}$ , A. Clark$^\\textrm {\\scriptsize 51}$ , B.L.",
"Clark$^\\textrm {\\scriptsize 58}$ , M.R.",
"Clark$^\\textrm {\\scriptsize 37}$ , P.J.",
"Clark$^\\textrm {\\scriptsize 48}$ , R.N.",
"Clarke$^\\textrm {\\scriptsize 16}$ , C. Clement$^\\textrm {\\scriptsize 146a,146b}$ , Y. Coadou$^\\textrm {\\scriptsize 86}$ , M. Cobal$^\\textrm {\\scriptsize 163a,163c}$ , A. Coccaro$^\\textrm {\\scriptsize 51}$ , J. Cochran$^\\textrm {\\scriptsize 65}$ , L. Colasurdo$^\\textrm {\\scriptsize 106}$ , B. Cole$^\\textrm {\\scriptsize 37}$ , A.P.",
"Colijn$^\\textrm {\\scriptsize 107}$ , J. Collot$^\\textrm {\\scriptsize 57}$ , T. Colombo$^\\textrm {\\scriptsize 32}$ , G. Compostella$^\\textrm {\\scriptsize 101}$ , P. Conde Muiño$^\\textrm {\\scriptsize 126a,126b}$ , E. Coniavitis$^\\textrm {\\scriptsize 50}$ , S.H.",
"Connell$^\\textrm {\\scriptsize 145b}$ , I.A.",
"Connelly$^\\textrm {\\scriptsize 78}$ , V. Consorti$^\\textrm {\\scriptsize 50}$ , S. Constantinescu$^\\textrm {\\scriptsize 28b}$ , G. Conti$^\\textrm {\\scriptsize 32}$ , F. Conventi$^\\textrm {\\scriptsize 104a}$$^{,k}$ , M. Cooke$^\\textrm {\\scriptsize 16}$ , B.D.",
"Cooper$^\\textrm {\\scriptsize 79}$ , A.M. Cooper-Sarkar$^\\textrm {\\scriptsize 120}$ , K.J.R.",
"Cormier$^\\textrm {\\scriptsize 158}$ , T. Cornelissen$^\\textrm {\\scriptsize 174}$ , M. Corradi$^\\textrm {\\scriptsize 132a,132b}$ , F. Corriveau$^\\textrm {\\scriptsize 88}$$^{,l}$ , A. Corso-Radu$^\\textrm {\\scriptsize 162}$ , A. Cortes-Gonzalez$^\\textrm {\\scriptsize 32}$ , G. Cortiana$^\\textrm {\\scriptsize 101}$ , G. Costa$^\\textrm {\\scriptsize 92a}$ , M.J. Costa$^\\textrm {\\scriptsize 166}$ , D. Costanzo$^\\textrm {\\scriptsize 139}$ , G. Cottin$^\\textrm {\\scriptsize 30}$ , G. Cowan$^\\textrm {\\scriptsize 78}$ , B.E.",
"Cox$^\\textrm {\\scriptsize 85}$ , K. Cranmer$^\\textrm {\\scriptsize 110}$ , S.J.",
"Crawley$^\\textrm {\\scriptsize 55}$ , G. Cree$^\\textrm {\\scriptsize 31}$ , S. Crépé-Renaudin$^\\textrm {\\scriptsize 57}$ , F. Crescioli$^\\textrm {\\scriptsize 81}$ , W.A.",
"Cribbs$^\\textrm {\\scriptsize 146a,146b}$ , M. Crispin Ortuzar$^\\textrm {\\scriptsize 120}$ , M. Cristinziani$^\\textrm {\\scriptsize 23}$ , V. Croft$^\\textrm {\\scriptsize 106}$ , G. Crosetti$^\\textrm {\\scriptsize 39a,39b}$ , A. Cueto$^\\textrm {\\scriptsize 83}$ , T. Cuhadar Donszelmann$^\\textrm {\\scriptsize 139}$ , J. Cummings$^\\textrm {\\scriptsize 175}$ , M. Curatolo$^\\textrm {\\scriptsize 49}$ , J. Cúth$^\\textrm {\\scriptsize 84}$ , H. Czirr$^\\textrm {\\scriptsize 141}$ , P. Czodrowski$^\\textrm {\\scriptsize 3}$ , G. D'amen$^\\textrm {\\scriptsize 22a,22b}$ , S. D'Auria$^\\textrm {\\scriptsize 55}$ , M. D'Onofrio$^\\textrm {\\scriptsize 75}$ , M.J. Da Cunha Sargedas De Sousa$^\\textrm {\\scriptsize 126a,126b}$ , C. Da Via$^\\textrm {\\scriptsize 85}$ , W. Dabrowski$^\\textrm {\\scriptsize 40a}$ , T. Dado$^\\textrm {\\scriptsize 144a}$ , T. Dai$^\\textrm {\\scriptsize 90}$ , O. Dale$^\\textrm {\\scriptsize 15}$ , F. Dallaire$^\\textrm {\\scriptsize 95}$ , C. Dallapiccola$^\\textrm {\\scriptsize 87}$ , M. Dam$^\\textrm {\\scriptsize 38}$ , J.R. Dandoy$^\\textrm {\\scriptsize 33}$ , N.P.",
"Dang$^\\textrm {\\scriptsize 50}$ , A.C. Daniells$^\\textrm {\\scriptsize 19}$ , N.S.",
"Dann$^\\textrm {\\scriptsize 85}$ , M. Danninger$^\\textrm {\\scriptsize 167}$ , M. Dano Hoffmann$^\\textrm {\\scriptsize 136}$ , V. Dao$^\\textrm {\\scriptsize 50}$ , G. Darbo$^\\textrm {\\scriptsize 52a}$ , S. Darmora$^\\textrm {\\scriptsize 8}$ , J. Dassoulas$^\\textrm {\\scriptsize 3}$ , A. Dattagupta$^\\textrm {\\scriptsize 62}$ , W. Davey$^\\textrm {\\scriptsize 23}$ , C. David$^\\textrm {\\scriptsize 168}$ , T. Davidek$^\\textrm {\\scriptsize 129}$ , M. Davies$^\\textrm {\\scriptsize 153}$ , P. Davison$^\\textrm {\\scriptsize 79}$ , E. Dawe$^\\textrm {\\scriptsize 89}$ , I. Dawson$^\\textrm {\\scriptsize 139}$ , R.K. Daya-Ishmukhametova$^\\textrm {\\scriptsize 87}$ , K. De$^\\textrm {\\scriptsize 8}$ , R. de Asmundis$^\\textrm {\\scriptsize 104a}$ , A.",
"De Benedetti$^\\textrm {\\scriptsize 113}$ , S. De Castro$^\\textrm {\\scriptsize 22a,22b}$ , S. De Cecco$^\\textrm {\\scriptsize 81}$ , N. De Groot$^\\textrm {\\scriptsize 106}$ , P. de Jong$^\\textrm {\\scriptsize 107}$ , H. De la Torre$^\\textrm {\\scriptsize 83}$ , F. De Lorenzi$^\\textrm {\\scriptsize 65}$ , A.",
"De Maria$^\\textrm {\\scriptsize 56}$ , D. De Pedis$^\\textrm {\\scriptsize 132a}$ , A.",
"De Salvo$^\\textrm {\\scriptsize 132a}$ , U.",
"De Sanctis$^\\textrm {\\scriptsize 149}$ , A.",
"De Santo$^\\textrm {\\scriptsize 149}$ , J.B. De Vivie De Regie$^\\textrm {\\scriptsize 117}$ , W.J.",
"Dearnaley$^\\textrm {\\scriptsize 73}$ , R. Debbe$^\\textrm {\\scriptsize 27}$ , C. Debenedetti$^\\textrm {\\scriptsize 137}$ , D.V.",
"Dedovich$^\\textrm {\\scriptsize 66}$ , N. Dehghanian$^\\textrm {\\scriptsize 3}$ , I. Deigaard$^\\textrm {\\scriptsize 107}$ , M. Del Gaudio$^\\textrm {\\scriptsize 39a,39b}$ , J. Del Peso$^\\textrm {\\scriptsize 83}$ , T. Del Prete$^\\textrm {\\scriptsize 124a,124b}$ , D. Delgove$^\\textrm {\\scriptsize 117}$ , F. Deliot$^\\textrm {\\scriptsize 136}$ , C.M.",
"Delitzsch$^\\textrm {\\scriptsize 51}$ , M. Deliyergiyev$^\\textrm {\\scriptsize 76}$ , A. Dell'Acqua$^\\textrm {\\scriptsize 32}$ , L. Dell'Asta$^\\textrm {\\scriptsize 24}$ , M. Dell'Orso$^\\textrm {\\scriptsize 124a,124b}$ , M. Della Pietra$^\\textrm {\\scriptsize 104a}$$^{,k}$ , D. della Volpe$^\\textrm {\\scriptsize 51}$ , M. Delmastro$^\\textrm {\\scriptsize 5}$ , P.A.",
"Delsart$^\\textrm {\\scriptsize 57}$ , D.A.",
"DeMarco$^\\textrm {\\scriptsize 158}$ , S. Demers$^\\textrm {\\scriptsize 175}$ , M. Demichev$^\\textrm {\\scriptsize 66}$ , A. Demilly$^\\textrm {\\scriptsize 81}$ , S.P.",
"Denisov$^\\textrm {\\scriptsize 130}$ , D. Denysiuk$^\\textrm {\\scriptsize 136}$ , D. Derendarz$^\\textrm {\\scriptsize 41}$ , J.E.",
"Derkaoui$^\\textrm {\\scriptsize 135d}$ , F. Derue$^\\textrm {\\scriptsize 81}$ , P. Dervan$^\\textrm {\\scriptsize 75}$ , K. Desch$^\\textrm {\\scriptsize 23}$ , C. Deterre$^\\textrm {\\scriptsize 44}$ , K. Dette$^\\textrm {\\scriptsize 45}$ , P.O.",
"Deviveiros$^\\textrm {\\scriptsize 32}$ , A. Dewhurst$^\\textrm {\\scriptsize 131}$ , S. Dhaliwal$^\\textrm {\\scriptsize 25}$ , A.",
"Di Ciaccio$^\\textrm {\\scriptsize 133a,133b}$ , L. Di Ciaccio$^\\textrm {\\scriptsize 5}$ , W.K.",
"Di Clemente$^\\textrm {\\scriptsize 122}$ , C. Di Donato$^\\textrm {\\scriptsize 132a,132b}$ , A.",
"Di Girolamo$^\\textrm {\\scriptsize 32}$ , B.",
"Di Girolamo$^\\textrm {\\scriptsize 32}$ , B.",
"Di Micco$^\\textrm {\\scriptsize 134a,134b}$ , R. Di Nardo$^\\textrm {\\scriptsize 32}$ , A.",
"Di Simone$^\\textrm {\\scriptsize 50}$ , R. Di Sipio$^\\textrm {\\scriptsize 158}$ , D. Di Valentino$^\\textrm {\\scriptsize 31}$ , C. Diaconu$^\\textrm {\\scriptsize 86}$ , M. Diamond$^\\textrm {\\scriptsize 158}$ , F.A.",
"Dias$^\\textrm {\\scriptsize 48}$ , M.A.",
"Diaz$^\\textrm {\\scriptsize 34a}$ , E.B.",
"Diehl$^\\textrm {\\scriptsize 90}$ , J. Dietrich$^\\textrm {\\scriptsize 17}$ , S. Diglio$^\\textrm {\\scriptsize 86}$ , A. Dimitrievska$^\\textrm {\\scriptsize 14}$ , J. Dingfelder$^\\textrm {\\scriptsize 23}$ , P. Dita$^\\textrm {\\scriptsize 28b}$ , S. Dita$^\\textrm {\\scriptsize 28b}$ , F. Dittus$^\\textrm {\\scriptsize 32}$ , F. Djama$^\\textrm {\\scriptsize 86}$ , T. Djobava$^\\textrm {\\scriptsize 53b}$ , J.I.",
"Djuvsland$^\\textrm {\\scriptsize 59a}$ , M.A.B.",
"do Vale$^\\textrm {\\scriptsize 26c}$ , D. Dobos$^\\textrm {\\scriptsize 32}$ , M. Dobre$^\\textrm {\\scriptsize 28b}$ , C. Doglioni$^\\textrm {\\scriptsize 82}$ , J. Dolejsi$^\\textrm {\\scriptsize 129}$ , Z. Dolezal$^\\textrm {\\scriptsize 129}$ , B.A.",
"Dolgoshein$^\\textrm {\\scriptsize 98}$$^{,*}$ , M. Donadelli$^\\textrm {\\scriptsize 26d}$ , S. Donati$^\\textrm {\\scriptsize 124a,124b}$ , P. Dondero$^\\textrm {\\scriptsize 121a,121b}$ , J. Donini$^\\textrm {\\scriptsize 36}$ , J. Dopke$^\\textrm {\\scriptsize 131}$ , A. Doria$^\\textrm {\\scriptsize 104a}$ , M.T.",
"Dova$^\\textrm {\\scriptsize 72}$ , A.T. Doyle$^\\textrm {\\scriptsize 55}$ , E. Drechsler$^\\textrm {\\scriptsize 56}$ , M. Dris$^\\textrm {\\scriptsize 10}$ , Y. Du$^\\textrm {\\scriptsize 35d}$ , J. Duarte-Campderros$^\\textrm {\\scriptsize 153}$ , E. Duchovni$^\\textrm {\\scriptsize 171}$ , G. Duckeck$^\\textrm {\\scriptsize 100}$ , O.A.",
"Ducu$^\\textrm {\\scriptsize 95}$$^{,m}$ , D. Duda$^\\textrm {\\scriptsize 107}$ , A. Dudarev$^\\textrm {\\scriptsize 32}$ , E.M. Duffield$^\\textrm {\\scriptsize 16}$ , L. Duflot$^\\textrm {\\scriptsize 117}$ , M. Dührssen$^\\textrm {\\scriptsize 32}$ , M. Dumancic$^\\textrm {\\scriptsize 171}$ , M. Dunford$^\\textrm {\\scriptsize 59a}$ , H. Duran Yildiz$^\\textrm {\\scriptsize 4a}$ , M. Düren$^\\textrm {\\scriptsize 54}$ , A. Durglishvili$^\\textrm {\\scriptsize 53b}$ , D. Duschinger$^\\textrm {\\scriptsize 46}$ , B. Dutta$^\\textrm {\\scriptsize 44}$ , M. Dyndal$^\\textrm {\\scriptsize 44}$ , C. Eckardt$^\\textrm {\\scriptsize 44}$ , K.M.",
"Ecker$^\\textrm {\\scriptsize 101}$ , R.C.",
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"Fawcett$^\\textrm {\\scriptsize 120}$ , L. Fayard$^\\textrm {\\scriptsize 117}$ , O.L.",
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"Feng$^\\textrm {\\scriptsize 32}$ , H. Feng$^\\textrm {\\scriptsize 90}$ , A.B.",
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"Ferreira de Lima$^\\textrm {\\scriptsize 59b}$ , A. Ferrer$^\\textrm {\\scriptsize 166}$ , D. Ferrere$^\\textrm {\\scriptsize 51}$ , C. Ferretti$^\\textrm {\\scriptsize 90}$ , A. Ferretto Parodi$^\\textrm {\\scriptsize 52a,52b}$ , F. Fiedler$^\\textrm {\\scriptsize 84}$ , A. Filipčič$^\\textrm {\\scriptsize 76}$ , M. Filipuzzi$^\\textrm {\\scriptsize 44}$ , F. Filthaut$^\\textrm {\\scriptsize 106}$ , M. Fincke-Keeler$^\\textrm {\\scriptsize 168}$ , K.D.",
"Finelli$^\\textrm {\\scriptsize 150}$ , M.C.N.",
"Fiolhais$^\\textrm {\\scriptsize 126a,126c}$ , L. Fiorini$^\\textrm {\\scriptsize 166}$ , A. Firan$^\\textrm {\\scriptsize 42}$ , A. Fischer$^\\textrm {\\scriptsize 2}$ , C. Fischer$^\\textrm {\\scriptsize 13}$ , J. Fischer$^\\textrm {\\scriptsize 174}$ , W.C. Fisher$^\\textrm {\\scriptsize 91}$ , N. Flaschel$^\\textrm {\\scriptsize 44}$ , I. Fleck$^\\textrm {\\scriptsize 141}$ , P. Fleischmann$^\\textrm {\\scriptsize 90}$ , G.T.",
"Fletcher$^\\textrm {\\scriptsize 139}$ , R.R.M.",
"Fletcher$^\\textrm {\\scriptsize 122}$ , T. Flick$^\\textrm {\\scriptsize 174}$ , A. Floderus$^\\textrm {\\scriptsize 82}$ , L.R.",
"Flores Castillo$^\\textrm {\\scriptsize 61a}$ , M.J. Flowerdew$^\\textrm {\\scriptsize 101}$ , G.T.",
"Forcolin$^\\textrm {\\scriptsize 85}$ , A. Formica$^\\textrm {\\scriptsize 136}$ , A. Forti$^\\textrm {\\scriptsize 85}$ , A.G. Foster$^\\textrm {\\scriptsize 19}$ , D. Fournier$^\\textrm {\\scriptsize 117}$ , H. Fox$^\\textrm {\\scriptsize 73}$ , S. Fracchia$^\\textrm {\\scriptsize 13}$ , P. Francavilla$^\\textrm {\\scriptsize 81}$ , M. Franchini$^\\textrm {\\scriptsize 22a,22b}$ , D. Francis$^\\textrm {\\scriptsize 32}$ , L. Franconi$^\\textrm {\\scriptsize 119}$ , M. Franklin$^\\textrm {\\scriptsize 58}$ , M. Frate$^\\textrm {\\scriptsize 162}$ , M. Fraternali$^\\textrm {\\scriptsize 121a,121b}$ , D. Freeborn$^\\textrm {\\scriptsize 79}$ , S.M.",
"Fressard-Batraneanu$^\\textrm {\\scriptsize 32}$ , F. Friedrich$^\\textrm {\\scriptsize 46}$ , D. Froidevaux$^\\textrm {\\scriptsize 32}$ , J.A.",
"Frost$^\\textrm {\\scriptsize 120}$ , C. Fukunaga$^\\textrm {\\scriptsize 156}$ , E. Fullana Torregrosa$^\\textrm {\\scriptsize 84}$ , T. Fusayasu$^\\textrm {\\scriptsize 102}$ , J. Fuster$^\\textrm {\\scriptsize 166}$ , C. Gabaldon$^\\textrm {\\scriptsize 57}$ , O. Gabizon$^\\textrm {\\scriptsize 174}$ , A. Gabrielli$^\\textrm {\\scriptsize 22a,22b}$ , A. Gabrielli$^\\textrm {\\scriptsize 16}$ , G.P.",
"Gach$^\\textrm {\\scriptsize 40a}$ , S. Gadatsch$^\\textrm {\\scriptsize 32}$ , S. Gadomski$^\\textrm {\\scriptsize 51}$ , G. Gagliardi$^\\textrm {\\scriptsize 52a,52b}$ , L.G.",
"Gagnon$^\\textrm {\\scriptsize 95}$ , P. Gagnon$^\\textrm {\\scriptsize 62}$ , C. Galea$^\\textrm {\\scriptsize 106}$ , B. Galhardo$^\\textrm {\\scriptsize 126a,126c}$ , E.J.",
"Gallas$^\\textrm {\\scriptsize 120}$ , B.J.",
"Gallop$^\\textrm {\\scriptsize 131}$ , P. Gallus$^\\textrm {\\scriptsize 128}$ , G. Galster$^\\textrm {\\scriptsize 38}$ , K.K.",
"Gan$^\\textrm {\\scriptsize 111}$ , J. Gao$^\\textrm {\\scriptsize 35b,86}$ , Y. Gao$^\\textrm {\\scriptsize 48}$ , Y.S.",
"Gao$^\\textrm {\\scriptsize 143}$$^{,f}$ , F.M.",
"Garay Walls$^\\textrm {\\scriptsize 48}$ , C. García$^\\textrm {\\scriptsize 166}$ , J.E.",
"García Navarro$^\\textrm {\\scriptsize 166}$ , M. Garcia-Sciveres$^\\textrm {\\scriptsize 16}$ , R.W.",
"Gardner$^\\textrm {\\scriptsize 33}$ , N. Garelli$^\\textrm {\\scriptsize 143}$ , V. Garonne$^\\textrm {\\scriptsize 119}$ , A. Gascon Bravo$^\\textrm {\\scriptsize 44}$ , K. Gasnikova$^\\textrm {\\scriptsize 44}$ , C. Gatti$^\\textrm {\\scriptsize 49}$ , A. Gaudiello$^\\textrm {\\scriptsize 52a,52b}$ , G. Gaudio$^\\textrm {\\scriptsize 121a}$ , L. Gauthier$^\\textrm {\\scriptsize 95}$ , I.L.",
"Gavrilenko$^\\textrm {\\scriptsize 96}$ , C. Gay$^\\textrm {\\scriptsize 167}$ , G. Gaycken$^\\textrm {\\scriptsize 23}$ , E.N.",
"Gazis$^\\textrm {\\scriptsize 10}$ , Z. Gecse$^\\textrm {\\scriptsize 167}$ , C.N.P.",
"Gee$^\\textrm {\\scriptsize 131}$ , Ch.",
"Geich-Gimbel$^\\textrm {\\scriptsize 23}$ , M. Geisen$^\\textrm {\\scriptsize 84}$ , M.P.",
"Geisler$^\\textrm {\\scriptsize 59a}$ , C. Gemme$^\\textrm {\\scriptsize 52a}$ , M.H.",
"Genest$^\\textrm {\\scriptsize 57}$ , C. Geng$^\\textrm {\\scriptsize 35b}$$^{,o}$ , S. Gentile$^\\textrm {\\scriptsize 132a,132b}$ , C. Gentsos$^\\textrm {\\scriptsize 154}$ , S. George$^\\textrm {\\scriptsize 78}$ , D. Gerbaudo$^\\textrm {\\scriptsize 13}$ , A. Gershon$^\\textrm {\\scriptsize 153}$ , S. Ghasemi$^\\textrm {\\scriptsize 141}$ , H. Ghazlane$^\\textrm {\\scriptsize 135b}$ , M. Ghneimat$^\\textrm {\\scriptsize 23}$ , B. Giacobbe$^\\textrm {\\scriptsize 22a}$ , S. Giagu$^\\textrm {\\scriptsize 132a,132b}$ , P. Giannetti$^\\textrm {\\scriptsize 124a,124b}$ , B. Gibbard$^\\textrm {\\scriptsize 27}$ , S.M.",
"Gibson$^\\textrm {\\scriptsize 78}$ , M. Gignac$^\\textrm {\\scriptsize 167}$ , M. Gilchriese$^\\textrm {\\scriptsize 16}$ , T.P.S.",
"Gillam$^\\textrm {\\scriptsize 30}$ , D. Gillberg$^\\textrm {\\scriptsize 31}$ , G. Gilles$^\\textrm {\\scriptsize 174}$ , D.M.",
"Gingrich$^\\textrm {\\scriptsize 3}$$^{,d}$ , N. Giokaris$^\\textrm {\\scriptsize 9}$ , M.P.",
"Giordani$^\\textrm {\\scriptsize 163a,163c}$ , F.M.",
"Giorgi$^\\textrm {\\scriptsize 22a}$ , F.M.",
"Giorgi$^\\textrm {\\scriptsize 17}$ , P.F.",
"Giraud$^\\textrm {\\scriptsize 136}$ , P. Giromini$^\\textrm {\\scriptsize 58}$ , D. Giugni$^\\textrm {\\scriptsize 92a}$ , F. Giuli$^\\textrm {\\scriptsize 120}$ , C. Giuliani$^\\textrm {\\scriptsize 101}$ , M. Giulini$^\\textrm {\\scriptsize 59b}$ , B.K.",
"Gjelsten$^\\textrm {\\scriptsize 119}$ , S. Gkaitatzis$^\\textrm {\\scriptsize 154}$ , I. Gkialas$^\\textrm {\\scriptsize 154}$ , E.L. Gkougkousis$^\\textrm {\\scriptsize 117}$ , L.K.",
"Gladilin$^\\textrm {\\scriptsize 99}$ , C. Glasman$^\\textrm {\\scriptsize 83}$ , J. Glatzer$^\\textrm {\\scriptsize 50}$ , P.C.F.",
"Glaysher$^\\textrm {\\scriptsize 48}$ , A. Glazov$^\\textrm {\\scriptsize 44}$ , M. Goblirsch-Kolb$^\\textrm {\\scriptsize 25}$ , J. Godlewski$^\\textrm {\\scriptsize 41}$ , S. Goldfarb$^\\textrm {\\scriptsize 89}$ , T. Golling$^\\textrm {\\scriptsize 51}$ , D. Golubkov$^\\textrm {\\scriptsize 130}$ , A. Gomes$^\\textrm {\\scriptsize 126a,126b,126d}$ , R. Gonçalo$^\\textrm {\\scriptsize 126a}$ , J. Goncalves Pinto Firmino Da Costa$^\\textrm {\\scriptsize 136}$ , G. Gonella$^\\textrm {\\scriptsize 50}$ , L. Gonella$^\\textrm {\\scriptsize 19}$ , A. Gongadze$^\\textrm {\\scriptsize 66}$ , S. González de la Hoz$^\\textrm {\\scriptsize 166}$ , G. Gonzalez Parra$^\\textrm {\\scriptsize 13}$ , S. Gonzalez-Sevilla$^\\textrm {\\scriptsize 51}$ , L. Goossens$^\\textrm {\\scriptsize 32}$ , P.A.",
"Gorbounov$^\\textrm {\\scriptsize 97}$ , H.A.",
"Gordon$^\\textrm {\\scriptsize 27}$ , I. Gorelov$^\\textrm {\\scriptsize 105}$ , B. Gorini$^\\textrm {\\scriptsize 32}$ , E. Gorini$^\\textrm {\\scriptsize 74a,74b}$ , A. Gorišek$^\\textrm {\\scriptsize 76}$ , E. Gornicki$^\\textrm {\\scriptsize 41}$ , A.T. Goshaw$^\\textrm {\\scriptsize 47}$ , C. Gössling$^\\textrm {\\scriptsize 45}$ , M.I.",
"Gostkin$^\\textrm {\\scriptsize 66}$ , C.R.",
"Goudet$^\\textrm {\\scriptsize 117}$ , D. Goujdami$^\\textrm {\\scriptsize 135c}$ , A.G. Goussiou$^\\textrm {\\scriptsize 138}$ , N. Govender$^\\textrm {\\scriptsize 145b}$$^{,p}$ , E. Gozani$^\\textrm {\\scriptsize 152}$ , L. Graber$^\\textrm {\\scriptsize 56}$ , I. Grabowska-Bold$^\\textrm {\\scriptsize 40a}$ , P.O.J.",
"Gradin$^\\textrm {\\scriptsize 57}$ , P. Grafström$^\\textrm {\\scriptsize 22a,22b}$ , J. Gramling$^\\textrm {\\scriptsize 51}$ , E. Gramstad$^\\textrm {\\scriptsize 119}$ , S. Grancagnolo$^\\textrm {\\scriptsize 17}$ , V. Gratchev$^\\textrm {\\scriptsize 123}$ , P.M. Gravila$^\\textrm {\\scriptsize 28e}$ , H.M. Gray$^\\textrm {\\scriptsize 32}$ , E. Graziani$^\\textrm {\\scriptsize 134a}$ , Z.D.",
"Greenwood$^\\textrm {\\scriptsize 80}$$^{,q}$ , C. Grefe$^\\textrm {\\scriptsize 23}$ , K. Gregersen$^\\textrm {\\scriptsize 79}$ , I.M.",
"Gregor$^\\textrm {\\scriptsize 44}$ , P. Grenier$^\\textrm {\\scriptsize 143}$ , K. Grevtsov$^\\textrm {\\scriptsize 5}$ , J. Griffiths$^\\textrm {\\scriptsize 8}$ , A.A. Grillo$^\\textrm {\\scriptsize 137}$ , K. Grimm$^\\textrm {\\scriptsize 73}$ , S. Grinstein$^\\textrm {\\scriptsize 13}$$^{,r}$ , Ph.",
"Gris$^\\textrm {\\scriptsize 36}$ , J.-F. Grivaz$^\\textrm {\\scriptsize 117}$ , S. Groh$^\\textrm {\\scriptsize 84}$ , J.P. Grohs$^\\textrm {\\scriptsize 46}$ , E. Gross$^\\textrm {\\scriptsize 171}$ , J. Grosse-Knetter$^\\textrm {\\scriptsize 56}$ , G.C.",
"Grossi$^\\textrm {\\scriptsize 80}$ , Z.J.",
"Grout$^\\textrm {\\scriptsize 79}$ , L. Guan$^\\textrm {\\scriptsize 90}$ , W. Guan$^\\textrm {\\scriptsize 172}$ , J. Guenther$^\\textrm {\\scriptsize 63}$ , F. Guescini$^\\textrm {\\scriptsize 51}$ , D. Guest$^\\textrm {\\scriptsize 162}$ , O. Gueta$^\\textrm {\\scriptsize 153}$ , E. Guido$^\\textrm {\\scriptsize 52a,52b}$ , T. Guillemin$^\\textrm {\\scriptsize 5}$ , S. Guindon$^\\textrm {\\scriptsize 2}$ , U. Gul$^\\textrm {\\scriptsize 55}$ , C. Gumpert$^\\textrm {\\scriptsize 32}$ , J. Guo$^\\textrm {\\scriptsize 35e}$ , Y. Guo$^\\textrm {\\scriptsize 35b}$$^{,o}$ , R. Gupta$^\\textrm {\\scriptsize 42}$ , S. Gupta$^\\textrm {\\scriptsize 120}$ , G. Gustavino$^\\textrm {\\scriptsize 132a,132b}$ , P. Gutierrez$^\\textrm {\\scriptsize 113}$ , N.G.",
"Gutierrez Ortiz$^\\textrm {\\scriptsize 79}$ , C. Gutschow$^\\textrm {\\scriptsize 46}$ , C. Guyot$^\\textrm {\\scriptsize 136}$ , C. Gwenlan$^\\textrm {\\scriptsize 120}$ , C.B.",
"Gwilliam$^\\textrm {\\scriptsize 75}$ , A. Haas$^\\textrm {\\scriptsize 110}$ , C. Haber$^\\textrm {\\scriptsize 16}$ , H.K.",
"Hadavand$^\\textrm {\\scriptsize 8}$ , N. Haddad$^\\textrm {\\scriptsize 135e}$ , A. Hadef$^\\textrm {\\scriptsize 86}$ , P. Haefner$^\\textrm {\\scriptsize 23}$ , S. Hageböck$^\\textrm {\\scriptsize 23}$ , Z. Hajduk$^\\textrm {\\scriptsize 41}$ , H. Hakobyan$^\\textrm {\\scriptsize 176}$$^{,*}$ , M. Haleem$^\\textrm {\\scriptsize 44}$ , J. Haley$^\\textrm {\\scriptsize 114}$ , G. Halladjian$^\\textrm {\\scriptsize 91}$ , G.D. Hallewell$^\\textrm {\\scriptsize 86}$ , K. Hamacher$^\\textrm {\\scriptsize 174}$ , P. Hamal$^\\textrm {\\scriptsize 115}$ , K. Hamano$^\\textrm {\\scriptsize 168}$ , A. Hamilton$^\\textrm {\\scriptsize 145a}$ , G.N.",
"Hamity$^\\textrm {\\scriptsize 139}$ , P.G.",
"Hamnett$^\\textrm {\\scriptsize 44}$ , L. Han$^\\textrm {\\scriptsize 35b}$ , K. Hanagaki$^\\textrm {\\scriptsize 67}$$^{,s}$ , K. Hanawa$^\\textrm {\\scriptsize 155}$ , M. Hance$^\\textrm {\\scriptsize 137}$ , B. Haney$^\\textrm {\\scriptsize 122}$ , S. Hanisch$^\\textrm {\\scriptsize 32}$ , P. Hanke$^\\textrm {\\scriptsize 59a}$ , R. Hanna$^\\textrm {\\scriptsize 136}$ , J.B. Hansen$^\\textrm {\\scriptsize 38}$ , J.D.",
"Hansen$^\\textrm {\\scriptsize 38}$ , M.C.",
"Hansen$^\\textrm {\\scriptsize 23}$ , P.H.",
"Hansen$^\\textrm {\\scriptsize 38}$ , K. Hara$^\\textrm {\\scriptsize 160}$ , A.S. Hard$^\\textrm {\\scriptsize 172}$ , T. Harenberg$^\\textrm {\\scriptsize 174}$ , F. Hariri$^\\textrm {\\scriptsize 117}$ , S. Harkusha$^\\textrm {\\scriptsize 93}$ , R.D.",
"Harrington$^\\textrm {\\scriptsize 48}$ , P.F.",
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"Hawkes$^\\textrm {\\scriptsize 19}$ , R.J. Hawkings$^\\textrm {\\scriptsize 32}$ , D. Hayakawa$^\\textrm {\\scriptsize 157}$ , D. Hayden$^\\textrm {\\scriptsize 91}$ , C.P.",
"Hays$^\\textrm {\\scriptsize 120}$ , J.M.",
"Hays$^\\textrm {\\scriptsize 77}$ , H.S.",
"Hayward$^\\textrm {\\scriptsize 75}$ , S.J.",
"Haywood$^\\textrm {\\scriptsize 131}$ , S.J.",
"Head$^\\textrm {\\scriptsize 19}$ , T. Heck$^\\textrm {\\scriptsize 84}$ , V. Hedberg$^\\textrm {\\scriptsize 82}$ , L. Heelan$^\\textrm {\\scriptsize 8}$ , S. Heim$^\\textrm {\\scriptsize 122}$ , T. Heim$^\\textrm {\\scriptsize 16}$ , B. Heinemann$^\\textrm {\\scriptsize 16}$ , J.J. Heinrich$^\\textrm {\\scriptsize 100}$ , L. Heinrich$^\\textrm {\\scriptsize 110}$ , C. Heinz$^\\textrm {\\scriptsize 54}$ , J. Hejbal$^\\textrm {\\scriptsize 127}$ , L. Helary$^\\textrm {\\scriptsize 32}$ , S. Hellman$^\\textrm {\\scriptsize 146a,146b}$ , C. Helsens$^\\textrm {\\scriptsize 32}$ , J. Henderson$^\\textrm {\\scriptsize 120}$ , R.C.W.",
"Henderson$^\\textrm {\\scriptsize 73}$ , Y. Heng$^\\textrm {\\scriptsize 172}$ , S. Henkelmann$^\\textrm {\\scriptsize 167}$ , A.M. Henriques Correia$^\\textrm {\\scriptsize 32}$ , S. Henrot-Versille$^\\textrm {\\scriptsize 117}$ , G.H.",
"Herbert$^\\textrm {\\scriptsize 17}$ , Y. Hernández Jiménez$^\\textrm {\\scriptsize 166}$ , G. Herten$^\\textrm {\\scriptsize 50}$ , R. Hertenberger$^\\textrm {\\scriptsize 100}$ , L. Hervas$^\\textrm {\\scriptsize 32}$ , G.G.",
"Hesketh$^\\textrm {\\scriptsize 79}$ , N.P.",
"Hessey$^\\textrm {\\scriptsize 107}$ , J.W.",
"Hetherly$^\\textrm {\\scriptsize 42}$ , R. Hickling$^\\textrm {\\scriptsize 77}$ , E. Higón-Rodriguez$^\\textrm {\\scriptsize 166}$ , E. Hill$^\\textrm {\\scriptsize 168}$ , J.C. Hill$^\\textrm {\\scriptsize 30}$ , K.H.",
"Hiller$^\\textrm {\\scriptsize 44}$ , S.J.",
"Hillier$^\\textrm {\\scriptsize 19}$ , I. Hinchliffe$^\\textrm {\\scriptsize 16}$ , E. Hines$^\\textrm {\\scriptsize 122}$ , R.R.",
"Hinman$^\\textrm {\\scriptsize 16}$ , M. Hirose$^\\textrm {\\scriptsize 50}$ , D. Hirschbuehl$^\\textrm {\\scriptsize 174}$ , J. Hobbs$^\\textrm {\\scriptsize 148}$ , N. Hod$^\\textrm {\\scriptsize 159a}$ , M.C.",
"Hodgkinson$^\\textrm {\\scriptsize 139}$ , P. Hodgson$^\\textrm {\\scriptsize 139}$ , A. Hoecker$^\\textrm {\\scriptsize 32}$ , M.R.",
"Hoeferkamp$^\\textrm {\\scriptsize 105}$ , F. Hoenig$^\\textrm {\\scriptsize 100}$ , D. Hohn$^\\textrm {\\scriptsize 23}$ , T.R.",
"Holmes$^\\textrm {\\scriptsize 16}$ , M. Homann$^\\textrm {\\scriptsize 45}$ , T.M.",
"Hong$^\\textrm {\\scriptsize 125}$ , B.H.",
"Hooberman$^\\textrm {\\scriptsize 165}$ , W.H.",
"Hopkins$^\\textrm {\\scriptsize 116}$ , Y. Horii$^\\textrm {\\scriptsize 103}$ , A.J.",
"Horton$^\\textrm {\\scriptsize 142}$ , J-Y.",
"Hostachy$^\\textrm {\\scriptsize 57}$ , S. Hou$^\\textrm {\\scriptsize 151}$ , A. Hoummada$^\\textrm {\\scriptsize 135a}$ , J. Howarth$^\\textrm {\\scriptsize 44}$ , M. Hrabovsky$^\\textrm {\\scriptsize 115}$ , I. Hristova$^\\textrm {\\scriptsize 17}$ , J. Hrivnac$^\\textrm {\\scriptsize 117}$ , T. Hryn'ova$^\\textrm {\\scriptsize 5}$ , A. Hrynevich$^\\textrm {\\scriptsize 94}$ , C. Hsu$^\\textrm {\\scriptsize 145c}$ , P.J.",
"Hsu$^\\textrm {\\scriptsize 151}$$^{,t}$ , S.-C. Hsu$^\\textrm {\\scriptsize 138}$ , D. Hu$^\\textrm {\\scriptsize 37}$ , Q. Hu$^\\textrm {\\scriptsize 35b}$ , S. Hu$^\\textrm {\\scriptsize 35e}$ , Y. Huang$^\\textrm {\\scriptsize 44}$ , Z. Hubacek$^\\textrm {\\scriptsize 128}$ , F. Hubaut$^\\textrm {\\scriptsize 86}$ , F. Huegging$^\\textrm {\\scriptsize 23}$ , T.B.",
"Huffman$^\\textrm {\\scriptsize 120}$ , E.W.",
"Hughes$^\\textrm {\\scriptsize 37}$ , G. Hughes$^\\textrm {\\scriptsize 73}$ , M. Huhtinen$^\\textrm {\\scriptsize 32}$ , P. Huo$^\\textrm {\\scriptsize 148}$ , N. Huseynov$^\\textrm {\\scriptsize 66}$$^{,b}$ , J. Huston$^\\textrm {\\scriptsize 91}$ , J. Huth$^\\textrm {\\scriptsize 58}$ , G. Iacobucci$^\\textrm {\\scriptsize 51}$ , G. Iakovidis$^\\textrm {\\scriptsize 27}$ , I. Ibragimov$^\\textrm {\\scriptsize 141}$ , L. Iconomidou-Fayard$^\\textrm {\\scriptsize 117}$ , E. Ideal$^\\textrm {\\scriptsize 175}$ , Z. Idrissi$^\\textrm {\\scriptsize 135e}$ , P. Iengo$^\\textrm {\\scriptsize 32}$ , O. Igonkina$^\\textrm {\\scriptsize 107}$$^{,u}$ , T. Iizawa$^\\textrm {\\scriptsize 170}$ , Y. Ikegami$^\\textrm {\\scriptsize 67}$ , M. Ikeno$^\\textrm {\\scriptsize 67}$ , Y. Ilchenko$^\\textrm {\\scriptsize 11}$$^{,v}$ , D. Iliadis$^\\textrm {\\scriptsize 154}$ , N. Ilic$^\\textrm {\\scriptsize 143}$ , T. Ince$^\\textrm {\\scriptsize 101}$ , G. Introzzi$^\\textrm {\\scriptsize 121a,121b}$ , P. Ioannou$^\\textrm {\\scriptsize 9}$$^{,*}$ , M. Iodice$^\\textrm {\\scriptsize 134a}$ , K. Iordanidou$^\\textrm {\\scriptsize 37}$ , V. Ippolito$^\\textrm {\\scriptsize 58}$ , N. Ishijima$^\\textrm {\\scriptsize 118}$ , M. Ishino$^\\textrm {\\scriptsize 155}$ , M. Ishitsuka$^\\textrm {\\scriptsize 157}$ , R. Ishmukhametov$^\\textrm {\\scriptsize 111}$ , C. Issever$^\\textrm {\\scriptsize 120}$ , S. Istin$^\\textrm {\\scriptsize 20a}$ , F. Ito$^\\textrm {\\scriptsize 160}$ , J.M.",
"Iturbe Ponce$^\\textrm {\\scriptsize 85}$ , R. Iuppa$^\\textrm {\\scriptsize -308}$ , W. Iwanski$^\\textrm {\\scriptsize 41}$ , H. Iwasaki$^\\textrm {\\scriptsize 67}$ , J.M.",
"Izen$^\\textrm {\\scriptsize 43}$ , V. Izzo$^\\textrm {\\scriptsize 104a}$ , S. Jabbar$^\\textrm {\\scriptsize 3}$ , B. Jackson$^\\textrm {\\scriptsize 122}$ , P. Jackson$^\\textrm {\\scriptsize 1}$ , V. Jain$^\\textrm {\\scriptsize 2}$ , K.B.",
"Jakobi$^\\textrm {\\scriptsize 84}$ , K. Jakobs$^\\textrm {\\scriptsize 50}$ , S. Jakobsen$^\\textrm {\\scriptsize 32}$ , T. Jakoubek$^\\textrm {\\scriptsize 127}$ , D.O.",
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"Johns$^\\textrm {\\scriptsize 7}$ , W.J.",
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"Kahn$^\\textrm {\\scriptsize 86}$ , T. Kaji$^\\textrm {\\scriptsize 170}$ , E. Kajomovitz$^\\textrm {\\scriptsize 47}$ , C.W.",
"Kalderon$^\\textrm {\\scriptsize 120}$ , A. Kaluza$^\\textrm {\\scriptsize 84}$ , S. Kama$^\\textrm {\\scriptsize 42}$ , A. Kamenshchikov$^\\textrm {\\scriptsize 130}$ , N. Kanaya$^\\textrm {\\scriptsize 155}$ , S. Kaneti$^\\textrm {\\scriptsize 30}$ , L. Kanjir$^\\textrm {\\scriptsize 76}$ , V.A.",
"Kantserov$^\\textrm {\\scriptsize 98}$ , J. Kanzaki$^\\textrm {\\scriptsize 67}$ , B. Kaplan$^\\textrm {\\scriptsize 110}$ , L.S.",
"Kaplan$^\\textrm {\\scriptsize 172}$ , A. Kapliy$^\\textrm {\\scriptsize 33}$ , D. Kar$^\\textrm {\\scriptsize 145c}$ , K. Karakostas$^\\textrm {\\scriptsize 10}$ , A. Karamaoun$^\\textrm {\\scriptsize 3}$ , N. Karastathis$^\\textrm {\\scriptsize 10}$ , M.J. Kareem$^\\textrm {\\scriptsize 56}$ , E. Karentzos$^\\textrm {\\scriptsize 10}$ , M. Karnevskiy$^\\textrm {\\scriptsize 84}$ , S.N.",
"Karpov$^\\textrm {\\scriptsize 66}$ , Z.M.",
"Karpova$^\\textrm {\\scriptsize 66}$ , K. Karthik$^\\textrm {\\scriptsize 110}$ , V. Kartvelishvili$^\\textrm {\\scriptsize 73}$ , A.N.",
"Karyukhin$^\\textrm {\\scriptsize 130}$ , K. Kasahara$^\\textrm {\\scriptsize 160}$ , L. Kashif$^\\textrm {\\scriptsize 172}$ , R.D.",
"Kass$^\\textrm {\\scriptsize 111}$ , A. Kastanas$^\\textrm {\\scriptsize 15}$ , Y. Kataoka$^\\textrm {\\scriptsize 155}$ , C. Kato$^\\textrm {\\scriptsize 155}$ , A. Katre$^\\textrm {\\scriptsize 51}$ , J. Katzy$^\\textrm {\\scriptsize 44}$ , K. Kawagoe$^\\textrm {\\scriptsize 71}$ , T. Kawamoto$^\\textrm {\\scriptsize 155}$ , G. Kawamura$^\\textrm {\\scriptsize 56}$ , V.F.",
"Kazanin$^\\textrm {\\scriptsize 109}$$^{,c}$ , R. Keeler$^\\textrm {\\scriptsize 168}$ , R. Kehoe$^\\textrm {\\scriptsize 42}$ , J.S.",
"Keller$^\\textrm {\\scriptsize 44}$ , J.J. Kempster$^\\textrm {\\scriptsize 78}$ , K Kentaro$^\\textrm {\\scriptsize 103}$ , H. Keoshkerian$^\\textrm {\\scriptsize 158}$ , O. Kepka$^\\textrm {\\scriptsize 127}$ , B.P.",
"Kerševan$^\\textrm {\\scriptsize 76}$ , S. Kersten$^\\textrm {\\scriptsize 174}$ , R.A. Keyes$^\\textrm {\\scriptsize 88}$ , M. Khader$^\\textrm {\\scriptsize 165}$ , F. Khalil-zada$^\\textrm {\\scriptsize 12}$ , A. Khanov$^\\textrm {\\scriptsize 114}$ , A.G. Kharlamov$^\\textrm {\\scriptsize 109}$$^{,c}$ , T.J. Khoo$^\\textrm {\\scriptsize 51}$ , V. Khovanskiy$^\\textrm {\\scriptsize 97}$ , E. Khramov$^\\textrm {\\scriptsize 66}$ , J. Khubua$^\\textrm {\\scriptsize 53b}$$^{,y}$ , S. Kido$^\\textrm {\\scriptsize 68}$ , C.R.",
"Kilby$^\\textrm {\\scriptsize 78}$ , H.Y.",
"Kim$^\\textrm {\\scriptsize 8}$ , S.H.",
"Kim$^\\textrm {\\scriptsize 160}$ , Y.K.",
"Kim$^\\textrm {\\scriptsize 33}$ , N. Kimura$^\\textrm {\\scriptsize 154}$ , O.M.",
"Kind$^\\textrm {\\scriptsize 17}$ , B.T.",
"King$^\\textrm {\\scriptsize 75}$ , M. King$^\\textrm {\\scriptsize 166}$ , S.B.",
"King$^\\textrm {\\scriptsize 167}$ , J. Kirk$^\\textrm {\\scriptsize 131}$ , A.E.",
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"Klein$^\\textrm {\\scriptsize 37}$ , M. Klein$^\\textrm {\\scriptsize 75}$ , U. Klein$^\\textrm {\\scriptsize 75}$ , K. Kleinknecht$^\\textrm {\\scriptsize 84}$ , P. Klimek$^\\textrm {\\scriptsize 108}$ , A. Klimentov$^\\textrm {\\scriptsize 27}$ , R. Klingenberg$^\\textrm {\\scriptsize 45}$ , J.A.",
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"Knoops$^\\textrm {\\scriptsize 86}$ , A. Knue$^\\textrm {\\scriptsize 55}$ , A. Kobayashi$^\\textrm {\\scriptsize 155}$ , D. Kobayashi$^\\textrm {\\scriptsize 157}$ , T. Kobayashi$^\\textrm {\\scriptsize 155}$ , M. Kobel$^\\textrm {\\scriptsize 46}$ , M. Kocian$^\\textrm {\\scriptsize 143}$ , P. Kodys$^\\textrm {\\scriptsize 129}$ , N.M. Koehler$^\\textrm {\\scriptsize 101}$ , T. Koffas$^\\textrm {\\scriptsize 31}$ , E. Koffeman$^\\textrm {\\scriptsize 107}$ , T. Koi$^\\textrm {\\scriptsize 143}$ , H. Kolanoski$^\\textrm {\\scriptsize 17}$ , M. Kolb$^\\textrm {\\scriptsize 59b}$ , I. Koletsou$^\\textrm {\\scriptsize 5}$ , A.A. Komar$^\\textrm {\\scriptsize 96}$$^{,*}$ , Y. Komori$^\\textrm {\\scriptsize 155}$ , T. Kondo$^\\textrm {\\scriptsize 67}$ , N. Kondrashova$^\\textrm {\\scriptsize 44}$ , K. Köneke$^\\textrm {\\scriptsize 50}$ , A.C. König$^\\textrm {\\scriptsize 106}$ , T. Kono$^\\textrm {\\scriptsize 67}$$^{,z}$ , R. Konoplich$^\\textrm {\\scriptsize 110}$$^{,aa}$ , N. Konstantinidis$^\\textrm {\\scriptsize 79}$ , R. Kopeliansky$^\\textrm {\\scriptsize 62}$ , S. Koperny$^\\textrm {\\scriptsize 40a}$ , L. Köpke$^\\textrm {\\scriptsize 84}$ , A.K.",
"Kopp$^\\textrm {\\scriptsize 50}$ , K. Korcyl$^\\textrm {\\scriptsize 41}$ , K. Kordas$^\\textrm {\\scriptsize 154}$ , A. Korn$^\\textrm {\\scriptsize 79}$ , A.A. Korol$^\\textrm {\\scriptsize 109}$$^{,c}$ , I. Korolkov$^\\textrm {\\scriptsize 13}$ , E.V.",
"Korolkova$^\\textrm {\\scriptsize 139}$ , O. Kortner$^\\textrm {\\scriptsize 101}$ , S. Kortner$^\\textrm {\\scriptsize 101}$ , T. Kosek$^\\textrm {\\scriptsize 129}$ , V.V.",
"Kostyukhin$^\\textrm {\\scriptsize 23}$ , A. Kotwal$^\\textrm {\\scriptsize 47}$ , A. Kourkoumeli-Charalampidi$^\\textrm {\\scriptsize 121a,121b}$ , C. Kourkoumelis$^\\textrm {\\scriptsize 9}$ , V. Kouskoura$^\\textrm {\\scriptsize 27}$ , A.B.",
"Kowalewska$^\\textrm {\\scriptsize 41}$ , R. Kowalewski$^\\textrm {\\scriptsize 168}$ , T.Z.",
"Kowalski$^\\textrm {\\scriptsize 40a}$ , C. Kozakai$^\\textrm {\\scriptsize 155}$ , W. Kozanecki$^\\textrm {\\scriptsize 136}$ , A.S. Kozhin$^\\textrm {\\scriptsize 130}$ , V.A.",
"Kramarenko$^\\textrm {\\scriptsize 99}$ , G. Kramberger$^\\textrm {\\scriptsize 76}$ , D. Krasnopevtsev$^\\textrm {\\scriptsize 98}$ , M.W.",
"Krasny$^\\textrm {\\scriptsize 81}$ , A. Krasznahorkay$^\\textrm {\\scriptsize 32}$ , A. Kravchenko$^\\textrm {\\scriptsize 27}$ , M. Kretz$^\\textrm {\\scriptsize 59c}$ , J. Kretzschmar$^\\textrm {\\scriptsize 75}$ , K. Kreutzfeldt$^\\textrm {\\scriptsize 54}$ , P. Krieger$^\\textrm {\\scriptsize 158}$ , K. Krizka$^\\textrm {\\scriptsize 33}$ , K. Kroeninger$^\\textrm {\\scriptsize 45}$ , H. Kroha$^\\textrm {\\scriptsize 101}$ , J. Kroll$^\\textrm {\\scriptsize 122}$ , J. Kroseberg$^\\textrm {\\scriptsize 23}$ , J. Krstic$^\\textrm {\\scriptsize 14}$ , U. Kruchonak$^\\textrm {\\scriptsize 66}$ , H. Krüger$^\\textrm {\\scriptsize 23}$ , N. Krumnack$^\\textrm {\\scriptsize 65}$ , A. Kruse$^\\textrm {\\scriptsize 172}$ , M.C.",
"Kruse$^\\textrm {\\scriptsize 47}$ , M. Kruskal$^\\textrm {\\scriptsize 24}$ , T. Kubota$^\\textrm {\\scriptsize 89}$ , H. Kucuk$^\\textrm {\\scriptsize 79}$ , S. Kuday$^\\textrm {\\scriptsize 4b}$ , J.T.",
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"Kurochkin$^\\textrm {\\scriptsize 93}$ , V. Kus$^\\textrm {\\scriptsize 127}$ , E.S.",
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"La Rosa$^\\textrm {\\scriptsize 101}$ , J.L.",
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"Landon$^\\textrm {\\scriptsize 77}$ , M.C.",
"Lanfermann$^\\textrm {\\scriptsize 51}$ , V.S.",
"Lang$^\\textrm {\\scriptsize 59a}$ , J.C. Lange$^\\textrm {\\scriptsize 13}$ , A.J.",
"Lankford$^\\textrm {\\scriptsize 162}$ , F. Lanni$^\\textrm {\\scriptsize 27}$ , K. Lantzsch$^\\textrm {\\scriptsize 23}$ , A. Lanza$^\\textrm {\\scriptsize 121a}$ , S. Laplace$^\\textrm {\\scriptsize 81}$ , C. Lapoire$^\\textrm {\\scriptsize 32}$ , J.F.",
"Laporte$^\\textrm {\\scriptsize 136}$ , T. Lari$^\\textrm {\\scriptsize 92a}$ , F. Lasagni Manghi$^\\textrm {\\scriptsize 22a,22b}$ , M. Lassnig$^\\textrm {\\scriptsize 32}$ , P. Laurelli$^\\textrm {\\scriptsize 49}$ , W. Lavrijsen$^\\textrm {\\scriptsize 16}$ , A.T. Law$^\\textrm {\\scriptsize 137}$ , P. Laycock$^\\textrm {\\scriptsize 75}$ , T. Lazovich$^\\textrm {\\scriptsize 58}$ , M. Lazzaroni$^\\textrm {\\scriptsize 92a,92b}$ , B. Le$^\\textrm {\\scriptsize 89}$ , O.",
"Le Dortz$^\\textrm {\\scriptsize 81}$ , E. Le Guirriec$^\\textrm {\\scriptsize 86}$ , E.P.",
"Le Quilleuc$^\\textrm {\\scriptsize 136}$ , M. LeBlanc$^\\textrm {\\scriptsize 168}$ , T. LeCompte$^\\textrm {\\scriptsize 6}$ , F. Ledroit-Guillon$^\\textrm {\\scriptsize 57}$ , C.A.",
"Lee$^\\textrm {\\scriptsize 27}$ , S.C. Lee$^\\textrm {\\scriptsize 151}$ , L. Lee$^\\textrm {\\scriptsize 1}$ , B. Lefebvre$^\\textrm {\\scriptsize 88}$ , G. Lefebvre$^\\textrm {\\scriptsize 81}$ , M. Lefebvre$^\\textrm {\\scriptsize 168}$ , F. Legger$^\\textrm {\\scriptsize 100}$ , C. Leggett$^\\textrm {\\scriptsize 16}$ , A. Lehan$^\\textrm {\\scriptsize 75}$ , G. Lehmann Miotto$^\\textrm {\\scriptsize 32}$ , X. Lei$^\\textrm {\\scriptsize 7}$ , W.A.",
"Leight$^\\textrm {\\scriptsize 31}$ , A. Leisos$^\\textrm {\\scriptsize 154}$$^{,ab}$ , A.G. Leister$^\\textrm {\\scriptsize 175}$ , M.A.L.",
"Leite$^\\textrm {\\scriptsize 26d}$ , R. Leitner$^\\textrm {\\scriptsize 129}$ , D. Lellouch$^\\textrm {\\scriptsize 171}$ , B. Lemmer$^\\textrm {\\scriptsize 56}$ , K.J.C.",
"Leney$^\\textrm {\\scriptsize 79}$ , T. Lenz$^\\textrm {\\scriptsize 23}$ , B. Lenzi$^\\textrm {\\scriptsize 32}$ , R. Leone$^\\textrm {\\scriptsize 7}$ , S. Leone$^\\textrm {\\scriptsize 124a,124b}$ , C. Leonidopoulos$^\\textrm {\\scriptsize 48}$ , S. Leontsinis$^\\textrm {\\scriptsize 10}$ , G. Lerner$^\\textrm {\\scriptsize 149}$ , C. Leroy$^\\textrm {\\scriptsize 95}$ , A.A.J.",
"Lesage$^\\textrm {\\scriptsize 136}$ , C.G.",
"Lester$^\\textrm {\\scriptsize 30}$ , M. Levchenko$^\\textrm {\\scriptsize 123}$ , J. Levêque$^\\textrm {\\scriptsize 5}$ , D. Levin$^\\textrm {\\scriptsize 90}$ , L.J.",
"Levinson$^\\textrm {\\scriptsize 171}$ , M. Levy$^\\textrm {\\scriptsize 19}$ , D. Lewis$^\\textrm {\\scriptsize 77}$ , A.M. Leyko$^\\textrm {\\scriptsize 23}$ , M. Leyton$^\\textrm {\\scriptsize 43}$ , B. Li$^\\textrm {\\scriptsize 35b}$$^{,o}$ , H. Li$^\\textrm {\\scriptsize 148}$ , H.L.",
"Li$^\\textrm {\\scriptsize 33}$ , L. Li$^\\textrm {\\scriptsize 47}$ , L. Li$^\\textrm {\\scriptsize 35e}$ , Q. Li$^\\textrm {\\scriptsize 35a}$ , S. Li$^\\textrm {\\scriptsize 47}$ , X. Li$^\\textrm {\\scriptsize 85}$ , Y. Li$^\\textrm {\\scriptsize 141}$ , Z. Liang$^\\textrm {\\scriptsize 35a}$ , B. Liberti$^\\textrm {\\scriptsize 133a}$ , A. Liblong$^\\textrm {\\scriptsize 158}$ , P. Lichard$^\\textrm {\\scriptsize 32}$ , K. Lie$^\\textrm {\\scriptsize 165}$ , J. Liebal$^\\textrm {\\scriptsize 23}$ , W. Liebig$^\\textrm {\\scriptsize 15}$ , A. Limosani$^\\textrm {\\scriptsize 150}$ , S.C. Lin$^\\textrm {\\scriptsize 151}$$^{,ac}$ , T.H.",
"Lin$^\\textrm {\\scriptsize 84}$ , B.E.",
"Lindquist$^\\textrm {\\scriptsize 148}$ , A.E.",
"Lionti$^\\textrm {\\scriptsize 51}$ , E. Lipeles$^\\textrm {\\scriptsize 122}$ , A. Lipniacka$^\\textrm {\\scriptsize 15}$ , M. Lisovyi$^\\textrm {\\scriptsize 59b}$ , T.M.",
"Liss$^\\textrm {\\scriptsize 165}$ , A. Lister$^\\textrm {\\scriptsize 167}$ , A.M. Litke$^\\textrm {\\scriptsize 137}$ , B. Liu$^\\textrm {\\scriptsize 151}$$^{,ad}$ , D. Liu$^\\textrm {\\scriptsize 151}$ , H. Liu$^\\textrm {\\scriptsize 90}$ , H. Liu$^\\textrm {\\scriptsize 27}$ , J. Liu$^\\textrm {\\scriptsize 86}$ , J.B. Liu$^\\textrm {\\scriptsize 35b}$ , K. Liu$^\\textrm {\\scriptsize 86}$ , L. Liu$^\\textrm {\\scriptsize 165}$ , M. Liu$^\\textrm {\\scriptsize 47}$ , M. Liu$^\\textrm {\\scriptsize 35b}$ , Y.L.",
"Liu$^\\textrm {\\scriptsize 35b}$ , Y. Liu$^\\textrm {\\scriptsize 35b}$ , M. Livan$^\\textrm {\\scriptsize 121a,121b}$ , A. Lleres$^\\textrm {\\scriptsize 57}$ , J. Llorente Merino$^\\textrm {\\scriptsize 35a}$ , S.L.",
"Lloyd$^\\textrm {\\scriptsize 77}$ , F. Lo Sterzo$^\\textrm {\\scriptsize 151}$ , E. Lobodzinska$^\\textrm {\\scriptsize 44}$ , P. Loch$^\\textrm {\\scriptsize 7}$ , W.S.",
"Lockman$^\\textrm {\\scriptsize 137}$ , F.K.",
"Loebinger$^\\textrm {\\scriptsize 85}$ , A.E.",
"Loevschall-Jensen$^\\textrm {\\scriptsize 38}$ , K.M.",
"Loew$^\\textrm {\\scriptsize 25}$ , A. Loginov$^\\textrm {\\scriptsize 175}$$^{,*}$ , T. Lohse$^\\textrm {\\scriptsize 17}$ , K. Lohwasser$^\\textrm {\\scriptsize 44}$ , M. Lokajicek$^\\textrm {\\scriptsize 127}$ , B.A.",
"Long$^\\textrm {\\scriptsize 24}$ , J.D.",
"Long$^\\textrm {\\scriptsize 165}$ , R.E.",
"Long$^\\textrm {\\scriptsize 73}$ , L. Longo$^\\textrm {\\scriptsize 74a,74b}$ , K.A.",
"Looper$^\\textrm {\\scriptsize 111}$ , L. Lopes$^\\textrm {\\scriptsize 126a}$ , D. Lopez Mateos$^\\textrm {\\scriptsize 58}$ , B. Lopez Paredes$^\\textrm {\\scriptsize 139}$ , I. Lopez Paz$^\\textrm {\\scriptsize 13}$ , A. Lopez Solis$^\\textrm {\\scriptsize 81}$ , J. Lorenz$^\\textrm {\\scriptsize 100}$ , N. Lorenzo Martinez$^\\textrm {\\scriptsize 62}$ , M. Losada$^\\textrm {\\scriptsize 21}$ , P.J.",
"Lösel$^\\textrm {\\scriptsize 100}$ , X. Lou$^\\textrm {\\scriptsize 35a}$ , A. Lounis$^\\textrm {\\scriptsize 117}$ , J. Love$^\\textrm {\\scriptsize 6}$ , P.A.",
"Love$^\\textrm {\\scriptsize 73}$ , H. Lu$^\\textrm {\\scriptsize 61a}$ , N. Lu$^\\textrm {\\scriptsize 90}$ , H.J.",
"Lubatti$^\\textrm {\\scriptsize 138}$ , C. Luci$^\\textrm {\\scriptsize 132a,132b}$ , A. Lucotte$^\\textrm {\\scriptsize 57}$ , C. Luedtke$^\\textrm {\\scriptsize 50}$ , F. Luehring$^\\textrm {\\scriptsize 62}$ , W. Lukas$^\\textrm {\\scriptsize 63}$ , L. Luminari$^\\textrm {\\scriptsize 132a}$ , O. Lundberg$^\\textrm {\\scriptsize 146a,146b}$ , B. Lund-Jensen$^\\textrm {\\scriptsize 147}$ , P.M. Luzi$^\\textrm {\\scriptsize 81}$ , D. Lynn$^\\textrm {\\scriptsize 27}$ , R. Lysak$^\\textrm {\\scriptsize 127}$ , E. Lytken$^\\textrm {\\scriptsize 82}$ , V. Lyubushkin$^\\textrm {\\scriptsize 66}$ , H. Ma$^\\textrm {\\scriptsize 27}$ , L.L.",
"Ma$^\\textrm {\\scriptsize 35d}$ , Y. Ma$^\\textrm {\\scriptsize 35d}$ , G. Maccarrone$^\\textrm {\\scriptsize 49}$ , A. Macchiolo$^\\textrm {\\scriptsize 101}$ , C.M.",
"Macdonald$^\\textrm {\\scriptsize 139}$ , B. Maček$^\\textrm {\\scriptsize 76}$ , J. Machado Miguens$^\\textrm {\\scriptsize 122,126b}$ , D. Madaffari$^\\textrm {\\scriptsize 86}$ , R. Madar$^\\textrm {\\scriptsize 36}$ , H.J.",
"Maddocks$^\\textrm {\\scriptsize 164}$ , W.F.",
"Mader$^\\textrm {\\scriptsize 46}$ , A. Madsen$^\\textrm {\\scriptsize 44}$ , J. Maeda$^\\textrm {\\scriptsize 68}$ , S. Maeland$^\\textrm {\\scriptsize 15}$ , T. Maeno$^\\textrm {\\scriptsize 27}$ , A. Maevskiy$^\\textrm {\\scriptsize 99}$ , E. Magradze$^\\textrm {\\scriptsize 56}$ , J. Mahlstedt$^\\textrm {\\scriptsize 107}$ , C. Maiani$^\\textrm {\\scriptsize 117}$ , C. Maidantchik$^\\textrm {\\scriptsize 26a}$ , A.A. Maier$^\\textrm {\\scriptsize 101}$ , T. Maier$^\\textrm {\\scriptsize 100}$ , A. Maio$^\\textrm {\\scriptsize 126a,126b,126d}$ , S. Majewski$^\\textrm {\\scriptsize 116}$ , Y. Makida$^\\textrm {\\scriptsize 67}$ , N. Makovec$^\\textrm {\\scriptsize 117}$ , B. Malaescu$^\\textrm {\\scriptsize 81}$ , Pa. Malecki$^\\textrm {\\scriptsize 41}$ , V.P.",
"Maleev$^\\textrm {\\scriptsize 123}$ , F. Malek$^\\textrm {\\scriptsize 57}$ , U. Mallik$^\\textrm {\\scriptsize 64}$ , D. Malon$^\\textrm {\\scriptsize 6}$ , C. Malone$^\\textrm {\\scriptsize 143}$ , S. Maltezos$^\\textrm {\\scriptsize 10}$ , S. Malyukov$^\\textrm {\\scriptsize 32}$ , J. Mamuzic$^\\textrm {\\scriptsize 166}$ , G. Mancini$^\\textrm {\\scriptsize 49}$ , B. Mandelli$^\\textrm {\\scriptsize 32}$ , L. Mandelli$^\\textrm {\\scriptsize 92a}$ , I. Mandić$^\\textrm {\\scriptsize 76}$ , J. Maneira$^\\textrm {\\scriptsize 126a,126b}$ , L. Manhaes de Andrade Filho$^\\textrm {\\scriptsize 26b}$ , J. Manjarres Ramos$^\\textrm {\\scriptsize 159b}$ , A. Mann$^\\textrm {\\scriptsize 100}$ , A. Manousos$^\\textrm {\\scriptsize 32}$ , B. Mansoulie$^\\textrm {\\scriptsize 136}$ , J.D.",
"Mansour$^\\textrm {\\scriptsize 35a}$ , R. Mantifel$^\\textrm {\\scriptsize 88}$ , M. Mantoani$^\\textrm {\\scriptsize 56}$ , S. Manzoni$^\\textrm {\\scriptsize 92a,92b}$ , L. Mapelli$^\\textrm {\\scriptsize 32}$ , G. Marceca$^\\textrm {\\scriptsize 29}$ , L. March$^\\textrm {\\scriptsize 51}$ , G. Marchiori$^\\textrm {\\scriptsize 81}$ , M. Marcisovsky$^\\textrm {\\scriptsize 127}$ , M. Marjanovic$^\\textrm {\\scriptsize 14}$ , D.E.",
"Marley$^\\textrm {\\scriptsize 90}$ , F. Marroquim$^\\textrm {\\scriptsize 26a}$ , S.P.",
"Marsden$^\\textrm {\\scriptsize 85}$ , Z. Marshall$^\\textrm {\\scriptsize 16}$ , S. Marti-Garcia$^\\textrm {\\scriptsize 166}$ , B. Martin$^\\textrm {\\scriptsize 91}$ , T.A.",
"Martin$^\\textrm {\\scriptsize 169}$ , V.J.",
"Martin$^\\textrm {\\scriptsize 48}$ , B. Martin dit Latour$^\\textrm {\\scriptsize 15}$ , M. Martinez$^\\textrm {\\scriptsize 13}$$^{,r}$ , V.I.",
"Martinez Outschoorn$^\\textrm {\\scriptsize 165}$ , S. Martin-Haugh$^\\textrm {\\scriptsize 131}$ , V.S.",
"Martoiu$^\\textrm {\\scriptsize 28b}$ , A.C. Martyniuk$^\\textrm {\\scriptsize 79}$ , M. Marx$^\\textrm {\\scriptsize 138}$ , A. Marzin$^\\textrm {\\scriptsize 32}$ , L. Masetti$^\\textrm {\\scriptsize 84}$ , T. Mashimo$^\\textrm {\\scriptsize 155}$ , R. Mashinistov$^\\textrm {\\scriptsize 96}$ , J. Masik$^\\textrm {\\scriptsize 85}$ , A.L.",
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"Maxfield$^\\textrm {\\scriptsize 75}$ , D.A.",
"Maximov$^\\textrm {\\scriptsize 109}$$^{,c}$ , R. Mazini$^\\textrm {\\scriptsize 151}$ , S.M.",
"Mazza$^\\textrm {\\scriptsize 92a,92b}$ , N.C. Mc Fadden$^\\textrm {\\scriptsize 105}$ , G. Mc Goldrick$^\\textrm {\\scriptsize 158}$ , S.P.",
"Mc Kee$^\\textrm {\\scriptsize 90}$ , A. McCarn$^\\textrm {\\scriptsize 90}$ , R.L.",
"McCarthy$^\\textrm {\\scriptsize 148}$ , T.G.",
"McCarthy$^\\textrm {\\scriptsize 101}$ , L.I.",
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"Mcfayden$^\\textrm {\\scriptsize 79}$ , G. Mchedlidze$^\\textrm {\\scriptsize 56}$ , S.J.",
"McMahon$^\\textrm {\\scriptsize 131}$ , R.A. McPherson$^\\textrm {\\scriptsize 168}$$^{,l}$ , M. Medinnis$^\\textrm {\\scriptsize 44}$ , S. Meehan$^\\textrm {\\scriptsize 138}$ , S. Mehlhase$^\\textrm {\\scriptsize 100}$ , A. Mehta$^\\textrm {\\scriptsize 75}$ , K. Meier$^\\textrm {\\scriptsize 59a}$ , C. Meineck$^\\textrm {\\scriptsize 100}$ , B. Meirose$^\\textrm {\\scriptsize 43}$ , D. Melini$^\\textrm {\\scriptsize 166}$ , B.R.",
"Mellado Garcia$^\\textrm {\\scriptsize 145c}$ , M. Melo$^\\textrm {\\scriptsize 144a}$ , F. Meloni$^\\textrm {\\scriptsize 18}$ , A. Mengarelli$^\\textrm {\\scriptsize 22a,22b}$ , S. Menke$^\\textrm {\\scriptsize 101}$ , E. Meoni$^\\textrm {\\scriptsize 161}$ , S. Mergelmeyer$^\\textrm {\\scriptsize 17}$ , P. Mermod$^\\textrm {\\scriptsize 51}$ , L. Merola$^\\textrm {\\scriptsize 104a,104b}$ , C. Meroni$^\\textrm {\\scriptsize 92a}$ , F.S.",
"Merritt$^\\textrm {\\scriptsize 33}$ , A. Messina$^\\textrm {\\scriptsize 132a,132b}$ , J. Metcalfe$^\\textrm {\\scriptsize 6}$ , A.S. Mete$^\\textrm {\\scriptsize 162}$ , C. Meyer$^\\textrm {\\scriptsize 84}$ , C. Meyer$^\\textrm {\\scriptsize 122}$ , J-P. Meyer$^\\textrm {\\scriptsize 136}$ , J. Meyer$^\\textrm {\\scriptsize 107}$ , H. Meyer Zu Theenhausen$^\\textrm {\\scriptsize 59a}$ , F. Miano$^\\textrm {\\scriptsize 149}$ , R.P.",
"Middleton$^\\textrm {\\scriptsize 131}$ , S. Miglioranzi$^\\textrm {\\scriptsize 52a,52b}$ , L. Mijović$^\\textrm {\\scriptsize 48}$ , G. Mikenberg$^\\textrm {\\scriptsize 171}$ , M. Mikestikova$^\\textrm {\\scriptsize 127}$ , M. Mikuž$^\\textrm {\\scriptsize 76}$ , M. Milesi$^\\textrm {\\scriptsize 89}$ , A. Milic$^\\textrm {\\scriptsize 63}$ , D.W. Miller$^\\textrm {\\scriptsize 33}$ , C. Mills$^\\textrm {\\scriptsize 48}$ , A. Milov$^\\textrm {\\scriptsize 171}$ , D.A.",
"Milstead$^\\textrm {\\scriptsize 146a,146b}$ , A.A. Minaenko$^\\textrm {\\scriptsize 130}$ , Y. Minami$^\\textrm {\\scriptsize 155}$ , I.A.",
"Minashvili$^\\textrm {\\scriptsize 66}$ , A.I.",
"Mincer$^\\textrm {\\scriptsize 110}$ , B. Mindur$^\\textrm {\\scriptsize 40a}$ , M. Mineev$^\\textrm {\\scriptsize 66}$ , Y. Ming$^\\textrm {\\scriptsize 172}$ , L.M.",
"Mir$^\\textrm {\\scriptsize 13}$ , K.P.",
"Mistry$^\\textrm {\\scriptsize 122}$ , T. Mitani$^\\textrm {\\scriptsize 170}$ , J. Mitrevski$^\\textrm {\\scriptsize 100}$ , V.A.",
"Mitsou$^\\textrm {\\scriptsize 166}$ , A. Miucci$^\\textrm {\\scriptsize 18}$ , P.S.",
"Miyagawa$^\\textrm {\\scriptsize 139}$ , J.U.",
"Mjörnmark$^\\textrm {\\scriptsize 82}$ , T. Moa$^\\textrm {\\scriptsize 146a,146b}$ , K. Mochizuki$^\\textrm {\\scriptsize 95}$ , S. Mohapatra$^\\textrm {\\scriptsize 37}$ , S. Molander$^\\textrm {\\scriptsize 146a,146b}$ , R. Moles-Valls$^\\textrm {\\scriptsize 23}$ , R. Monden$^\\textrm {\\scriptsize 69}$ , M.C.",
"Mondragon$^\\textrm {\\scriptsize 91}$ , K. Mönig$^\\textrm {\\scriptsize 44}$ , J. Monk$^\\textrm {\\scriptsize 38}$ , E. Monnier$^\\textrm {\\scriptsize 86}$ , A. Montalbano$^\\textrm {\\scriptsize 148}$ , J. Montejo Berlingen$^\\textrm {\\scriptsize 32}$ , F. Monticelli$^\\textrm {\\scriptsize 72}$ , S. Monzani$^\\textrm {\\scriptsize 92a,92b}$ , R.W.",
"Moore$^\\textrm {\\scriptsize 3}$ , N. Morange$^\\textrm {\\scriptsize 117}$ , D. Moreno$^\\textrm {\\scriptsize 21}$ , M. Moreno Llácer$^\\textrm {\\scriptsize 56}$ , P. Morettini$^\\textrm {\\scriptsize 52a}$ , D. Mori$^\\textrm {\\scriptsize 142}$ , T. Mori$^\\textrm {\\scriptsize 155}$ , M. Morii$^\\textrm {\\scriptsize 58}$ , M. Morinaga$^\\textrm {\\scriptsize 155}$ , V. Morisbak$^\\textrm {\\scriptsize 119}$ , S. Moritz$^\\textrm {\\scriptsize 84}$ , A.K.",
"Morley$^\\textrm {\\scriptsize 150}$ , G. Mornacchi$^\\textrm {\\scriptsize 32}$ , J.D.",
"Morris$^\\textrm {\\scriptsize 77}$ , S.S. Mortensen$^\\textrm {\\scriptsize 38}$ , L. Morvaj$^\\textrm {\\scriptsize 148}$ , M. Mosidze$^\\textrm {\\scriptsize 53b}$ , J. Moss$^\\textrm {\\scriptsize 143}$ , K. Motohashi$^\\textrm {\\scriptsize 157}$ , R. Mount$^\\textrm {\\scriptsize 143}$ , E. Mountricha$^\\textrm {\\scriptsize 27}$ , S.V.",
"Mouraviev$^\\textrm {\\scriptsize 96}$$^{,*}$ , E.J.W.",
"Moyse$^\\textrm {\\scriptsize 87}$ , S. Muanza$^\\textrm {\\scriptsize 86}$ , R.D.",
"Mudd$^\\textrm {\\scriptsize 19}$ , F. Mueller$^\\textrm {\\scriptsize 101}$ , J. Mueller$^\\textrm {\\scriptsize 125}$ , R.S.P.",
"Mueller$^\\textrm {\\scriptsize 100}$ , T. Mueller$^\\textrm {\\scriptsize 30}$ , D. Muenstermann$^\\textrm {\\scriptsize 73}$ , P. Mullen$^\\textrm {\\scriptsize 55}$ , G.A.",
"Mullier$^\\textrm {\\scriptsize 18}$ , F.J. Munoz Sanchez$^\\textrm {\\scriptsize 85}$ , J.A.",
"Murillo Quijada$^\\textrm {\\scriptsize 19}$ , W.J.",
"Murray$^\\textrm {\\scriptsize 169,131}$ , H. Musheghyan$^\\textrm {\\scriptsize 56}$ , M. Muškinja$^\\textrm {\\scriptsize 76}$ , A.G. Myagkov$^\\textrm {\\scriptsize 130}$$^{,ae}$ , M. Myska$^\\textrm {\\scriptsize 128}$ , B.P.",
"Nachman$^\\textrm {\\scriptsize 143}$ , O. Nackenhorst$^\\textrm {\\scriptsize 51}$ , K. Nagai$^\\textrm {\\scriptsize 120}$ , R. Nagai$^\\textrm {\\scriptsize 67}$$^{,z}$ , K. Nagano$^\\textrm {\\scriptsize 67}$ , Y. Nagasaka$^\\textrm {\\scriptsize 60}$ , K. Nagata$^\\textrm {\\scriptsize 160}$ , M. Nagel$^\\textrm {\\scriptsize 50}$ , E. Nagy$^\\textrm {\\scriptsize 86}$ , A.M. Nairz$^\\textrm {\\scriptsize 32}$ , Y. Nakahama$^\\textrm {\\scriptsize 103}$ , K. Nakamura$^\\textrm {\\scriptsize 67}$ , T. Nakamura$^\\textrm {\\scriptsize 155}$ , I. Nakano$^\\textrm {\\scriptsize 112}$ , H. Namasivayam$^\\textrm {\\scriptsize 43}$ , R.F.",
"Naranjo Garcia$^\\textrm {\\scriptsize 44}$ , R. Narayan$^\\textrm {\\scriptsize 11}$ , D.I.",
"Narrias Villar$^\\textrm {\\scriptsize 59a}$ , I. Naryshkin$^\\textrm {\\scriptsize 123}$ , T. Naumann$^\\textrm {\\scriptsize 44}$ , G. Navarro$^\\textrm {\\scriptsize 21}$ , R. Nayyar$^\\textrm {\\scriptsize 7}$ , H.A.",
"Neal$^\\textrm {\\scriptsize 90}$ , P.Yu.",
"Nechaeva$^\\textrm {\\scriptsize 96}$ , T.J. Neep$^\\textrm {\\scriptsize 85}$ , A. Negri$^\\textrm {\\scriptsize 121a,121b}$ , M. Negrini$^\\textrm {\\scriptsize 22a}$ , S. Nektarijevic$^\\textrm {\\scriptsize 106}$ , C. Nellist$^\\textrm {\\scriptsize 117}$ , A. Nelson$^\\textrm {\\scriptsize 162}$ , S. Nemecek$^\\textrm {\\scriptsize 127}$ , P. Nemethy$^\\textrm {\\scriptsize 110}$ , A.A. Nepomuceno$^\\textrm {\\scriptsize 26a}$ , M. Nessi$^\\textrm {\\scriptsize 32}$$^{,af}$ , M.S.",
"Neubauer$^\\textrm {\\scriptsize 165}$ , M. Neumann$^\\textrm {\\scriptsize 174}$ , R.M.",
"Neves$^\\textrm {\\scriptsize 110}$ , P. Nevski$^\\textrm {\\scriptsize 27}$ , P.R.",
"Newman$^\\textrm {\\scriptsize 19}$ , D.H. Nguyen$^\\textrm {\\scriptsize 6}$ , T. Nguyen Manh$^\\textrm {\\scriptsize 95}$ , R.B.",
"Nickerson$^\\textrm {\\scriptsize 120}$ , R. Nicolaidou$^\\textrm {\\scriptsize 136}$ , J. Nielsen$^\\textrm {\\scriptsize 137}$ , A. Nikiforov$^\\textrm {\\scriptsize 17}$ , V. Nikolaenko$^\\textrm {\\scriptsize 130}$$^{,ae}$ , I. Nikolic-Audit$^\\textrm {\\scriptsize 81}$ , K. Nikolopoulos$^\\textrm {\\scriptsize 19}$ , J.K. Nilsen$^\\textrm {\\scriptsize 119}$ , P. Nilsson$^\\textrm {\\scriptsize 27}$ , Y. Ninomiya$^\\textrm {\\scriptsize 155}$ , A. Nisati$^\\textrm {\\scriptsize 132a}$ , R. Nisius$^\\textrm {\\scriptsize 101}$ , T. Nobe$^\\textrm {\\scriptsize 155}$ , M. Nomachi$^\\textrm {\\scriptsize 118}$ , I. Nomidis$^\\textrm {\\scriptsize 31}$ , T. Nooney$^\\textrm {\\scriptsize 77}$ , S. Norberg$^\\textrm {\\scriptsize 113}$ , M. Nordberg$^\\textrm {\\scriptsize 32}$ , N. Norjoharuddeen$^\\textrm {\\scriptsize 120}$ , O. Novgorodova$^\\textrm {\\scriptsize 46}$ , S. Nowak$^\\textrm {\\scriptsize 101}$ , M. Nozaki$^\\textrm {\\scriptsize 67}$ , L. Nozka$^\\textrm {\\scriptsize 115}$ , K. Ntekas$^\\textrm {\\scriptsize 10}$ , E. Nurse$^\\textrm {\\scriptsize 79}$ , F. Nuti$^\\textrm {\\scriptsize 89}$ , F. O'grady$^\\textrm {\\scriptsize 7}$ , D.C. O'Neil$^\\textrm {\\scriptsize 142}$ , A.A. O'Rourke$^\\textrm {\\scriptsize 44}$ , V. O'Shea$^\\textrm {\\scriptsize 55}$ , F.G. Oakham$^\\textrm {\\scriptsize 31}$$^{,d}$ , H. Oberlack$^\\textrm {\\scriptsize 101}$ , T. Obermann$^\\textrm {\\scriptsize 23}$ , J. Ocariz$^\\textrm {\\scriptsize 81}$ , A. Ochi$^\\textrm {\\scriptsize 68}$ , I. Ochoa$^\\textrm {\\scriptsize 37}$ , J.P. Ochoa-Ricoux$^\\textrm {\\scriptsize 34a}$ , S. Oda$^\\textrm {\\scriptsize 71}$ , S. Odaka$^\\textrm {\\scriptsize 67}$ , H. Ogren$^\\textrm {\\scriptsize 62}$ , A. Oh$^\\textrm {\\scriptsize 85}$ , S.H.",
"Oh$^\\textrm {\\scriptsize 47}$ , C.C.",
"Ohm$^\\textrm {\\scriptsize 16}$ , H. Ohman$^\\textrm {\\scriptsize 164}$ , H. Oide$^\\textrm {\\scriptsize 32}$ , H. Okawa$^\\textrm {\\scriptsize 160}$ , Y. Okumura$^\\textrm {\\scriptsize 155}$ , T. Okuyama$^\\textrm {\\scriptsize 67}$ , A. Olariu$^\\textrm {\\scriptsize 28b}$ , L.F. Oleiro Seabra$^\\textrm {\\scriptsize 126a}$ , S.A. Olivares Pino$^\\textrm {\\scriptsize 48}$ , D. Oliveira Damazio$^\\textrm {\\scriptsize 27}$ , A. Olszewski$^\\textrm {\\scriptsize 41}$ , J. Olszowska$^\\textrm {\\scriptsize 41}$ , A. Onofre$^\\textrm {\\scriptsize 126a,126e}$ , K. Onogi$^\\textrm {\\scriptsize 103}$ , P.U.E.",
"Onyisi$^\\textrm {\\scriptsize 11}$$^{,v}$ , M.J. Oreglia$^\\textrm {\\scriptsize 33}$ , Y. Oren$^\\textrm {\\scriptsize 153}$ , D. Orestano$^\\textrm {\\scriptsize 134a,134b}$ , N. Orlando$^\\textrm {\\scriptsize 61b}$ , R.S.",
"Orr$^\\textrm {\\scriptsize 158}$ , B. Osculati$^\\textrm {\\scriptsize 52a,52b}$ , R. Ospanov$^\\textrm {\\scriptsize 85}$ , G. Otero y Garzon$^\\textrm {\\scriptsize 29}$ , H. Otono$^\\textrm {\\scriptsize 71}$ , M. Ouchrif$^\\textrm {\\scriptsize 135d}$ , F. Ould-Saada$^\\textrm {\\scriptsize 119}$ , A. Ouraou$^\\textrm {\\scriptsize 136}$ , K.P.",
"Oussoren$^\\textrm {\\scriptsize 107}$ , Q. Ouyang$^\\textrm {\\scriptsize 35a}$ , M. Owen$^\\textrm {\\scriptsize 55}$ , R.E.",
"Owen$^\\textrm {\\scriptsize 19}$ , V.E.",
"Ozcan$^\\textrm {\\scriptsize 20a}$ , N. Ozturk$^\\textrm {\\scriptsize 8}$ , K. Pachal$^\\textrm {\\scriptsize 142}$ , A. Pacheco Pages$^\\textrm {\\scriptsize 13}$ , L. Pacheco Rodriguez$^\\textrm {\\scriptsize 136}$ , C. Padilla Aranda$^\\textrm {\\scriptsize 13}$ , M. Pagáčová$^\\textrm {\\scriptsize 50}$ , S. Pagan Griso$^\\textrm {\\scriptsize 16}$ , F. Paige$^\\textrm {\\scriptsize 27}$ , P. Pais$^\\textrm {\\scriptsize 87}$ , K. Pajchel$^\\textrm {\\scriptsize 119}$ , G. Palacino$^\\textrm {\\scriptsize 159b}$ , S. Palazzo$^\\textrm {\\scriptsize 39a,39b}$ , S. Palestini$^\\textrm {\\scriptsize 32}$ , M. Palka$^\\textrm {\\scriptsize 40b}$ , D. Pallin$^\\textrm {\\scriptsize 36}$ , E.St.",
"Panagiotopoulou$^\\textrm {\\scriptsize 10}$ , C.E.",
"Pandini$^\\textrm {\\scriptsize 81}$ , J.G.",
"Panduro Vazquez$^\\textrm {\\scriptsize 78}$ , P. Pani$^\\textrm {\\scriptsize 146a,146b}$ , S. Panitkin$^\\textrm {\\scriptsize 27}$ , D. Pantea$^\\textrm {\\scriptsize 28b}$ , L. Paolozzi$^\\textrm {\\scriptsize 51}$ , Th.D.",
"Papadopoulou$^\\textrm {\\scriptsize 10}$ , K. Papageorgiou$^\\textrm {\\scriptsize 154}$ , A. Paramonov$^\\textrm {\\scriptsize 6}$ , D. Paredes Hernandez$^\\textrm {\\scriptsize 175}$ , A.J.",
"Parker$^\\textrm {\\scriptsize 73}$ , M.A.",
"Parker$^\\textrm {\\scriptsize 30}$ , K.A.",
"Parker$^\\textrm {\\scriptsize 139}$ , F. Parodi$^\\textrm {\\scriptsize 52a,52b}$ , J.A.",
"Parsons$^\\textrm {\\scriptsize 37}$ , U. Parzefall$^\\textrm {\\scriptsize 50}$ , V.R.",
"Pascuzzi$^\\textrm {\\scriptsize 158}$ , E. Pasqualucci$^\\textrm {\\scriptsize 132a}$ , S. Passaggio$^\\textrm {\\scriptsize 52a}$ , Fr.",
"Pastore$^\\textrm {\\scriptsize 78}$ , G. Pásztor$^\\textrm {\\scriptsize 31}$$^{,ag}$ , S. Pataraia$^\\textrm {\\scriptsize 174}$ , J.R. Pater$^\\textrm {\\scriptsize 85}$ , T. Pauly$^\\textrm {\\scriptsize 32}$ , J. Pearce$^\\textrm {\\scriptsize 168}$ , B. Pearson$^\\textrm {\\scriptsize 113}$ , L.E.",
"Pedersen$^\\textrm {\\scriptsize 38}$ , M. Pedersen$^\\textrm {\\scriptsize 119}$ , S. Pedraza Lopez$^\\textrm {\\scriptsize 166}$ , R. Pedro$^\\textrm {\\scriptsize 126a,126b}$ , S.V.",
"Peleganchuk$^\\textrm {\\scriptsize 109}$$^{,c}$ , O. Penc$^\\textrm {\\scriptsize 127}$ , C. Peng$^\\textrm {\\scriptsize 35a}$ , H. Peng$^\\textrm {\\scriptsize 35b}$ , J. Penwell$^\\textrm {\\scriptsize 62}$ , B.S.",
"Peralva$^\\textrm {\\scriptsize 26b}$ , M.M.",
"Perego$^\\textrm {\\scriptsize 136}$ , D.V.",
"Perepelitsa$^\\textrm {\\scriptsize 27}$ , E. Perez Codina$^\\textrm {\\scriptsize 159a}$ , L. Perini$^\\textrm {\\scriptsize 92a,92b}$ , H. Pernegger$^\\textrm {\\scriptsize 32}$ , S. Perrella$^\\textrm {\\scriptsize 104a,104b}$ , R. Peschke$^\\textrm {\\scriptsize 44}$ , V.D.",
"Peshekhonov$^\\textrm {\\scriptsize 66}$ , K. Peters$^\\textrm {\\scriptsize 44}$ , R.F.Y.",
"Peters$^\\textrm {\\scriptsize 85}$ , B.A.",
"Petersen$^\\textrm {\\scriptsize 32}$ , T.C.",
"Petersen$^\\textrm {\\scriptsize 38}$ , E. Petit$^\\textrm {\\scriptsize 57}$ , A. Petridis$^\\textrm {\\scriptsize 1}$ , C. Petridou$^\\textrm {\\scriptsize 154}$ , P. Petroff$^\\textrm {\\scriptsize 117}$ , E. Petrolo$^\\textrm {\\scriptsize 132a}$ , M. Petrov$^\\textrm {\\scriptsize 120}$ , F. Petrucci$^\\textrm {\\scriptsize 134a,134b}$ , N.E.",
"Pettersson$^\\textrm {\\scriptsize 87}$ , A. Peyaud$^\\textrm {\\scriptsize 136}$ , R. Pezoa$^\\textrm {\\scriptsize 34b}$ , P.W.",
"Phillips$^\\textrm {\\scriptsize 131}$ , G. Piacquadio$^\\textrm {\\scriptsize 143}$$^{,ah}$ , E. Pianori$^\\textrm {\\scriptsize 169}$ , A. Picazio$^\\textrm {\\scriptsize 87}$ , E. Piccaro$^\\textrm {\\scriptsize 77}$ , M. Piccinini$^\\textrm {\\scriptsize 22a,22b}$ , M.A.",
"Pickering$^\\textrm {\\scriptsize 120}$ , R. Piegaia$^\\textrm {\\scriptsize 29}$ , J.E.",
"Pilcher$^\\textrm {\\scriptsize 33}$ , A.D. Pilkington$^\\textrm {\\scriptsize 85}$ , A.W.J.",
"Pin$^\\textrm {\\scriptsize 85}$ , M. Pinamonti$^\\textrm {\\scriptsize 163a,163c}$$^{,ai}$ , J.L.",
"Pinfold$^\\textrm {\\scriptsize 3}$ , A. Pingel$^\\textrm {\\scriptsize 38}$ , S. Pires$^\\textrm {\\scriptsize 81}$ , H. Pirumov$^\\textrm {\\scriptsize 44}$ , M. Pitt$^\\textrm {\\scriptsize 171}$ , L. Plazak$^\\textrm {\\scriptsize 144a}$ , M.-A.",
"Pleier$^\\textrm {\\scriptsize 27}$ , V. Pleskot$^\\textrm {\\scriptsize 84}$ , E. Plotnikova$^\\textrm {\\scriptsize 66}$ , P. Plucinski$^\\textrm {\\scriptsize 91}$ , D. Pluth$^\\textrm {\\scriptsize 65}$ , R. Poettgen$^\\textrm {\\scriptsize 146a,146b}$ , L. Poggioli$^\\textrm {\\scriptsize 117}$ , D. Pohl$^\\textrm {\\scriptsize 23}$ , G. Polesello$^\\textrm {\\scriptsize 121a}$ , A. Poley$^\\textrm {\\scriptsize 44}$ , A. Policicchio$^\\textrm {\\scriptsize 39a,39b}$ , R. Polifka$^\\textrm {\\scriptsize 158}$ , A. Polini$^\\textrm {\\scriptsize 22a}$ , C.S.",
"Pollard$^\\textrm {\\scriptsize 55}$ , V. Polychronakos$^\\textrm {\\scriptsize 27}$ , K. Pommès$^\\textrm {\\scriptsize 32}$ , L. Pontecorvo$^\\textrm {\\scriptsize 132a}$ , B.G.",
"Pope$^\\textrm {\\scriptsize 91}$ , G.A.",
"Popeneciu$^\\textrm {\\scriptsize 28c}$ , D.S.",
"Popovic$^\\textrm {\\scriptsize 14}$ , A. Poppleton$^\\textrm {\\scriptsize 32}$ , S. Pospisil$^\\textrm {\\scriptsize 128}$ , K. Potamianos$^\\textrm {\\scriptsize 16}$ , I.N.",
"Potrap$^\\textrm {\\scriptsize 66}$ , C.J.",
"Potter$^\\textrm {\\scriptsize 30}$ , C.T.",
"Potter$^\\textrm {\\scriptsize 116}$ , G. Poulard$^\\textrm {\\scriptsize 32}$ , J. Poveda$^\\textrm {\\scriptsize 32}$ , V. Pozdnyakov$^\\textrm {\\scriptsize 66}$ , M.E.",
"Pozo Astigarraga$^\\textrm {\\scriptsize 32}$ , P. Pralavorio$^\\textrm {\\scriptsize 86}$ , A. Pranko$^\\textrm {\\scriptsize 16}$ , S. Prell$^\\textrm {\\scriptsize 65}$ , D. Price$^\\textrm {\\scriptsize 85}$ , L.E.",
"Price$^\\textrm {\\scriptsize 6}$ , M. Primavera$^\\textrm {\\scriptsize 74a}$ , S. Prince$^\\textrm {\\scriptsize 88}$ , K. Prokofiev$^\\textrm {\\scriptsize 61c}$ , F. Prokoshin$^\\textrm {\\scriptsize 34b}$ , S. Protopopescu$^\\textrm {\\scriptsize 27}$ , J. Proudfoot$^\\textrm {\\scriptsize 6}$ , M. Przybycien$^\\textrm {\\scriptsize 40a}$ , D. Puddu$^\\textrm {\\scriptsize 134a,134b}$ , M. Purohit$^\\textrm {\\scriptsize 27}$$^{,aj}$ , P. Puzo$^\\textrm {\\scriptsize 117}$ , J. Qian$^\\textrm {\\scriptsize 90}$ , G. Qin$^\\textrm {\\scriptsize 55}$ , Y. Qin$^\\textrm {\\scriptsize 85}$ , A. Quadt$^\\textrm {\\scriptsize 56}$ , W.B.",
"Quayle$^\\textrm {\\scriptsize 163a,163b}$ , M. Queitsch-Maitland$^\\textrm {\\scriptsize 85}$ , D. Quilty$^\\textrm {\\scriptsize 55}$ , S. Raddum$^\\textrm {\\scriptsize 119}$ , V. Radeka$^\\textrm {\\scriptsize 27}$ , V. Radescu$^\\textrm {\\scriptsize 120}$ , S.K.",
"Radhakrishnan$^\\textrm {\\scriptsize 148}$ , P. Radloff$^\\textrm {\\scriptsize 116}$ , P. Rados$^\\textrm {\\scriptsize 89}$ , F. Ragusa$^\\textrm {\\scriptsize 92a,92b}$ , G. Rahal$^\\textrm {\\scriptsize 177}$ , J.A.",
"Raine$^\\textrm {\\scriptsize 85}$ , S. Rajagopalan$^\\textrm {\\scriptsize 27}$ , M. Rammensee$^\\textrm {\\scriptsize 32}$ , C. Rangel-Smith$^\\textrm {\\scriptsize 164}$ , M.G.",
"Ratti$^\\textrm {\\scriptsize 92a,92b}$ , F. Rauscher$^\\textrm {\\scriptsize 100}$ , S. Rave$^\\textrm {\\scriptsize 84}$ , T. Ravenscroft$^\\textrm {\\scriptsize 55}$ , I. Ravinovich$^\\textrm {\\scriptsize 171}$ , M. Raymond$^\\textrm {\\scriptsize 32}$ , A.L.",
"Read$^\\textrm {\\scriptsize 119}$ , N.P.",
"Readioff$^\\textrm {\\scriptsize 75}$ , M. Reale$^\\textrm {\\scriptsize 74a,74b}$ , D.M.",
"Rebuzzi$^\\textrm {\\scriptsize 121a,121b}$ , A. Redelbach$^\\textrm {\\scriptsize 173}$ , G. Redlinger$^\\textrm {\\scriptsize 27}$ , R. Reece$^\\textrm {\\scriptsize 137}$ , K. Reeves$^\\textrm {\\scriptsize 43}$ , L. Rehnisch$^\\textrm {\\scriptsize 17}$ , J. Reichert$^\\textrm {\\scriptsize 122}$ , H. Reisin$^\\textrm {\\scriptsize 29}$ , C. Rembser$^\\textrm {\\scriptsize 32}$ , H. Ren$^\\textrm {\\scriptsize 35a}$ , M. Rescigno$^\\textrm {\\scriptsize 132a}$ , S. Resconi$^\\textrm {\\scriptsize 92a}$ , O.L.",
"Rezanova$^\\textrm {\\scriptsize 109}$$^{,c}$ , P. Reznicek$^\\textrm {\\scriptsize 129}$ , R. Rezvani$^\\textrm {\\scriptsize 95}$ , R. Richter$^\\textrm {\\scriptsize 101}$ , S. Richter$^\\textrm {\\scriptsize 79}$ , E. Richter-Was$^\\textrm {\\scriptsize 40b}$ , O. Ricken$^\\textrm {\\scriptsize 23}$ , M. Ridel$^\\textrm {\\scriptsize 81}$ , P. Rieck$^\\textrm {\\scriptsize 17}$ , C.J.",
"Riegel$^\\textrm {\\scriptsize 174}$ , J. Rieger$^\\textrm {\\scriptsize 56}$ , O. Rifki$^\\textrm {\\scriptsize 113}$ , M. Rijssenbeek$^\\textrm {\\scriptsize 148}$ , A. Rimoldi$^\\textrm {\\scriptsize 121a,121b}$ , M. Rimoldi$^\\textrm {\\scriptsize 18}$ , L. Rinaldi$^\\textrm {\\scriptsize 22a}$ , B. Ristić$^\\textrm {\\scriptsize 51}$ , E. Ritsch$^\\textrm {\\scriptsize 32}$ , I. Riu$^\\textrm {\\scriptsize 13}$ , F. Rizatdinova$^\\textrm {\\scriptsize 114}$ , E. Rizvi$^\\textrm {\\scriptsize 77}$ , C. Rizzi$^\\textrm {\\scriptsize 13}$ , S.H.",
"Robertson$^\\textrm {\\scriptsize 88}$$^{,l}$ , A. Robichaud-Veronneau$^\\textrm {\\scriptsize 88}$ , D. Robinson$^\\textrm {\\scriptsize 30}$ , J.E.M.",
"Robinson$^\\textrm {\\scriptsize 44}$ , A. Robson$^\\textrm {\\scriptsize 55}$ , C. Roda$^\\textrm {\\scriptsize 124a,124b}$ , Y. Rodina$^\\textrm {\\scriptsize 86}$ , A. Rodriguez Perez$^\\textrm {\\scriptsize 13}$ , D. Rodriguez Rodriguez$^\\textrm {\\scriptsize 166}$ , S. Roe$^\\textrm {\\scriptsize 32}$ , C.S.",
"Rogan$^\\textrm {\\scriptsize 58}$ , O. Røhne$^\\textrm {\\scriptsize 119}$ , A. Romaniouk$^\\textrm {\\scriptsize 98}$ , M. Romano$^\\textrm {\\scriptsize 22a,22b}$ , S.M.",
"Romano Saez$^\\textrm {\\scriptsize 36}$ , E. Romero Adam$^\\textrm {\\scriptsize 166}$ , N. Rompotis$^\\textrm {\\scriptsize 138}$ , M. Ronzani$^\\textrm {\\scriptsize 50}$ , L. Roos$^\\textrm {\\scriptsize 81}$ , E. Ros$^\\textrm {\\scriptsize 166}$ , S. Rosati$^\\textrm {\\scriptsize 132a}$ , K. Rosbach$^\\textrm {\\scriptsize 50}$ , P. Rose$^\\textrm {\\scriptsize 137}$ , O. Rosenthal$^\\textrm {\\scriptsize 141}$ , N.-A.",
"Rosien$^\\textrm {\\scriptsize 56}$ , V. Rossetti$^\\textrm {\\scriptsize 146a,146b}$ , E. Rossi$^\\textrm {\\scriptsize 104a,104b}$ , L.P. Rossi$^\\textrm {\\scriptsize 52a}$ , J.H.N.",
"Rosten$^\\textrm {\\scriptsize 30}$ , R. Rosten$^\\textrm {\\scriptsize 138}$ , M. Rotaru$^\\textrm {\\scriptsize 28b}$ , I. Roth$^\\textrm {\\scriptsize 171}$ , J. Rothberg$^\\textrm {\\scriptsize 138}$ , D. Rousseau$^\\textrm {\\scriptsize 117}$ , C.R.",
"Royon$^\\textrm {\\scriptsize 136}$ , A. Rozanov$^\\textrm {\\scriptsize 86}$ , Y. Rozen$^\\textrm {\\scriptsize 152}$ , X. Ruan$^\\textrm {\\scriptsize 145c}$ , F. Rubbo$^\\textrm {\\scriptsize 143}$ , M.S.",
"Rudolph$^\\textrm {\\scriptsize 158}$ , F. Rühr$^\\textrm {\\scriptsize 50}$ , A. Ruiz-Martinez$^\\textrm {\\scriptsize 31}$ , Z. Rurikova$^\\textrm {\\scriptsize 50}$ , N.A.",
"Rusakovich$^\\textrm {\\scriptsize 66}$ , A. Ruschke$^\\textrm {\\scriptsize 100}$ , H.L.",
"Russell$^\\textrm {\\scriptsize 138}$ , J.P. Rutherfoord$^\\textrm {\\scriptsize 7}$ , N. Ruthmann$^\\textrm {\\scriptsize 32}$ , Y.F.",
"Ryabov$^\\textrm {\\scriptsize 123}$ , M. Rybar$^\\textrm {\\scriptsize 165}$ , G. Rybkin$^\\textrm {\\scriptsize 117}$ , S. Ryu$^\\textrm {\\scriptsize 6}$ , A. Ryzhov$^\\textrm {\\scriptsize 130}$ , G.F. Rzehorz$^\\textrm {\\scriptsize 56}$ , A.F.",
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"Schachtner$^\\textrm {\\scriptsize 100}$ , D. Schaefer$^\\textrm {\\scriptsize 32}$ , R. Schaefer$^\\textrm {\\scriptsize 44}$ , J. Schaeffer$^\\textrm {\\scriptsize 84}$ , S. Schaepe$^\\textrm {\\scriptsize 23}$ , S. Schaetzel$^\\textrm {\\scriptsize 59b}$ , U. Schäfer$^\\textrm {\\scriptsize 84}$ , A.C. Schaffer$^\\textrm {\\scriptsize 117}$ , D. Schaile$^\\textrm {\\scriptsize 100}$ , R.D.",
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"Schoenrock$^\\textrm {\\scriptsize 91}$ , E. Schopf$^\\textrm {\\scriptsize 23}$ , M. Schott$^\\textrm {\\scriptsize 84}$ , J. Schovancova$^\\textrm {\\scriptsize 8}$ , S. Schramm$^\\textrm {\\scriptsize 51}$ , M. Schreyer$^\\textrm {\\scriptsize 173}$ , N. Schuh$^\\textrm {\\scriptsize 84}$ , A. Schulte$^\\textrm {\\scriptsize 84}$ , M.J. Schultens$^\\textrm {\\scriptsize 23}$ , H.-C. Schultz-Coulon$^\\textrm {\\scriptsize 59a}$ , H. Schulz$^\\textrm {\\scriptsize 17}$ , M. Schumacher$^\\textrm {\\scriptsize 50}$ , B.A.",
"Schumm$^\\textrm {\\scriptsize 137}$ , Ph.",
"Schune$^\\textrm {\\scriptsize 136}$ , A. Schwartzman$^\\textrm {\\scriptsize 143}$ , T.A.",
"Schwarz$^\\textrm {\\scriptsize 90}$ , H. Schweiger$^\\textrm {\\scriptsize 85}$ , Ph.",
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"Seixas$^\\textrm {\\scriptsize 26a}$ , G. Sekhniaidze$^\\textrm {\\scriptsize 104a}$ , K. Sekhon$^\\textrm {\\scriptsize 90}$ , S.J.",
"Sekula$^\\textrm {\\scriptsize 42}$ , D.M.",
"Seliverstov$^\\textrm {\\scriptsize 123}$$^{,*}$ , N. Semprini-Cesari$^\\textrm {\\scriptsize 22a,22b}$ , C. Serfon$^\\textrm {\\scriptsize 119}$ , L. Serin$^\\textrm {\\scriptsize 117}$ , L. Serkin$^\\textrm {\\scriptsize 163a,163b}$ , M. Sessa$^\\textrm {\\scriptsize 134a,134b}$ , R. Seuster$^\\textrm {\\scriptsize 168}$ , H. Severini$^\\textrm {\\scriptsize 113}$ , T. Sfiligoj$^\\textrm {\\scriptsize 76}$ , F. Sforza$^\\textrm {\\scriptsize 32}$ , A. Sfyrla$^\\textrm {\\scriptsize 51}$ , E. Shabalina$^\\textrm {\\scriptsize 56}$ , N.W.",
"Shaikh$^\\textrm {\\scriptsize 146a,146b}$ , L.Y.",
"Shan$^\\textrm {\\scriptsize 35a}$ , R. Shang$^\\textrm {\\scriptsize 165}$ , J.T.",
"Shank$^\\textrm {\\scriptsize 24}$ , M. Shapiro$^\\textrm {\\scriptsize 16}$ , P.B.",
"Shatalov$^\\textrm {\\scriptsize 97}$ , K. Shaw$^\\textrm {\\scriptsize 163a,163b}$ , S.M.",
"Shaw$^\\textrm {\\scriptsize 85}$ , A. Shcherbakova$^\\textrm {\\scriptsize 146a,146b}$ , C.Y.",
"Shehu$^\\textrm {\\scriptsize 149}$ , P. Sherwood$^\\textrm {\\scriptsize 79}$ , L. Shi$^\\textrm {\\scriptsize 151}$$^{,ak}$ , S. Shimizu$^\\textrm {\\scriptsize 68}$ , C.O.",
"Shimmin$^\\textrm {\\scriptsize 162}$ , M. Shimojima$^\\textrm {\\scriptsize 102}$ , M. Shiyakova$^\\textrm {\\scriptsize 66}$$^{,al}$ , A. Shmeleva$^\\textrm {\\scriptsize 96}$ , D. Shoaleh Saadi$^\\textrm {\\scriptsize 95}$ , M.J. Shochet$^\\textrm {\\scriptsize 33}$ , S. Shojaii$^\\textrm {\\scriptsize 92a,92b}$ , S. Shrestha$^\\textrm {\\scriptsize 111}$ , E. Shulga$^\\textrm {\\scriptsize 98}$ , M.A.",
"Shupe$^\\textrm {\\scriptsize 7}$ , P. Sicho$^\\textrm {\\scriptsize 127}$ , A.M. Sickles$^\\textrm {\\scriptsize 165}$ , P.E.",
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"Sijacki$^\\textrm {\\scriptsize 14}$ , J. Silva$^\\textrm {\\scriptsize 126a,126d}$ , S.B.",
"Silverstein$^\\textrm {\\scriptsize 146a}$ , V. Simak$^\\textrm {\\scriptsize 128}$ , Lj.",
"Simic$^\\textrm {\\scriptsize 14}$ , S. Simion$^\\textrm {\\scriptsize 117}$ , E. Simioni$^\\textrm {\\scriptsize 84}$ , B. Simmons$^\\textrm {\\scriptsize 79}$ , D. Simon$^\\textrm {\\scriptsize 36}$ , M. Simon$^\\textrm {\\scriptsize 84}$ , P. Sinervo$^\\textrm {\\scriptsize 158}$ , N.B.",
"Sinev$^\\textrm {\\scriptsize 116}$ , M. Sioli$^\\textrm {\\scriptsize 22a,22b}$ , G. Siragusa$^\\textrm {\\scriptsize 173}$ , S.Yu.",
"Sivoklokov$^\\textrm {\\scriptsize 99}$ , J. Sjölin$^\\textrm {\\scriptsize 146a,146b}$ , M.B.",
"Skinner$^\\textrm {\\scriptsize 73}$ , H.P.",
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"Smart$^\\textrm {\\scriptsize 5}$ , L. Smestad$^\\textrm {\\scriptsize 15}$ , J. Smiesko$^\\textrm {\\scriptsize 144a}$ , S.Yu.",
"Smirnov$^\\textrm {\\scriptsize 98}$ , Y. Smirnov$^\\textrm {\\scriptsize 98}$ , L.N.",
"Smirnova$^\\textrm {\\scriptsize 99}$$^{,am}$ , O. Smirnova$^\\textrm {\\scriptsize 82}$ , M.N.K.",
"Smith$^\\textrm {\\scriptsize 37}$ , R.W.",
"Smith$^\\textrm {\\scriptsize 37}$ , M. Smizanska$^\\textrm {\\scriptsize 73}$ , K. Smolek$^\\textrm {\\scriptsize 128}$ , A.A. Snesarev$^\\textrm {\\scriptsize 96}$ , S. Snyder$^\\textrm {\\scriptsize 27}$ , R. Sobie$^\\textrm {\\scriptsize 168}$$^{,l}$ , F. Socher$^\\textrm {\\scriptsize 46}$ , A. Soffer$^\\textrm {\\scriptsize 153}$ , D.A.",
"Soh$^\\textrm {\\scriptsize 151}$ , G. Sokhrannyi$^\\textrm {\\scriptsize 76}$ , C.A.",
"Solans Sanchez$^\\textrm {\\scriptsize 32}$ , M. Solar$^\\textrm {\\scriptsize 128}$ , E.Yu.",
"Soldatov$^\\textrm {\\scriptsize 98}$ , U. Soldevila$^\\textrm {\\scriptsize 166}$ , A.A. Solodkov$^\\textrm {\\scriptsize 130}$ , A. Soloshenko$^\\textrm {\\scriptsize 66}$ , O.V.",
"Solovyanov$^\\textrm {\\scriptsize 130}$ , V. Solovyev$^\\textrm {\\scriptsize 123}$ , P. Sommer$^\\textrm {\\scriptsize 50}$ , H. Son$^\\textrm {\\scriptsize 161}$ , H.Y.",
"Song$^\\textrm {\\scriptsize 35b}$$^{,an}$ , A. Sood$^\\textrm {\\scriptsize 16}$ , A. Sopczak$^\\textrm {\\scriptsize 128}$ , V. Sopko$^\\textrm {\\scriptsize 128}$ , V. Sorin$^\\textrm {\\scriptsize 13}$ , D. Sosa$^\\textrm {\\scriptsize 59b}$ , C.L.",
"Sotiropoulou$^\\textrm {\\scriptsize 124a,124b}$ , R. Soualah$^\\textrm {\\scriptsize 163a,163c}$ , A.M. Soukharev$^\\textrm {\\scriptsize 109}$$^{,c}$ , D. South$^\\textrm {\\scriptsize 44}$ , B.C.",
"Sowden$^\\textrm {\\scriptsize 78}$ , S. Spagnolo$^\\textrm {\\scriptsize 74a,74b}$ , M. Spalla$^\\textrm {\\scriptsize 124a,124b}$ , M. Spangenberg$^\\textrm {\\scriptsize 169}$ , F. Spanò$^\\textrm {\\scriptsize 78}$ , D. Sperlich$^\\textrm {\\scriptsize 17}$ , F. Spettel$^\\textrm {\\scriptsize 101}$ , R. Spighi$^\\textrm {\\scriptsize 22a}$ , G. Spigo$^\\textrm {\\scriptsize 32}$ , L.A. Spiller$^\\textrm {\\scriptsize 89}$ , M. Spousta$^\\textrm {\\scriptsize 129}$ , R.D. St.",
"Denis$^\\textrm {\\scriptsize 55}$$^{,*}$ , A. Stabile$^\\textrm {\\scriptsize 92a}$ , R. Stamen$^\\textrm {\\scriptsize 59a}$ , S. Stamm$^\\textrm {\\scriptsize 17}$ , E. Stanecka$^\\textrm {\\scriptsize 41}$ , R.W.",
"Stanek$^\\textrm {\\scriptsize 6}$ , C. Stanescu$^\\textrm {\\scriptsize 134a}$ , M. Stanescu-Bellu$^\\textrm {\\scriptsize 44}$ , M.M.",
"Stanitzki$^\\textrm {\\scriptsize 44}$ , S. Stapnes$^\\textrm {\\scriptsize 119}$ , E.A.",
"Starchenko$^\\textrm {\\scriptsize 130}$ , G.H.",
"Stark$^\\textrm {\\scriptsize 33}$ , J. Stark$^\\textrm {\\scriptsize 57}$ , P. Staroba$^\\textrm {\\scriptsize 127}$ , P. Starovoitov$^\\textrm {\\scriptsize 59a}$ , S. Stärz$^\\textrm {\\scriptsize 32}$ , R. Staszewski$^\\textrm {\\scriptsize 41}$ , P. Steinberg$^\\textrm {\\scriptsize 27}$ , B. Stelzer$^\\textrm {\\scriptsize 142}$ , H.J.",
"Stelzer$^\\textrm {\\scriptsize 32}$ , O. Stelzer-Chilton$^\\textrm {\\scriptsize 159a}$ , H. Stenzel$^\\textrm {\\scriptsize 54}$ , G.A.",
"Stewart$^\\textrm {\\scriptsize 55}$ , J.A.",
"Stillings$^\\textrm {\\scriptsize 23}$ , M.C.",
"Stockton$^\\textrm {\\scriptsize 88}$ , M. Stoebe$^\\textrm {\\scriptsize 88}$ , G. Stoicea$^\\textrm {\\scriptsize 28b}$ , P. Stolte$^\\textrm {\\scriptsize 56}$ , S. Stonjek$^\\textrm {\\scriptsize 101}$ , A.R.",
"Stradling$^\\textrm {\\scriptsize 8}$ , A. Straessner$^\\textrm {\\scriptsize 46}$ , M.E.",
"Stramaglia$^\\textrm {\\scriptsize 18}$ , J. Strandberg$^\\textrm {\\scriptsize 147}$ , S. Strandberg$^\\textrm {\\scriptsize 146a,146b}$ , A. Strandlie$^\\textrm {\\scriptsize 119}$ , M. Strauss$^\\textrm {\\scriptsize 113}$ , P. Strizenec$^\\textrm {\\scriptsize 144b}$ , R. Ströhmer$^\\textrm {\\scriptsize 173}$ , D.M.",
"Strom$^\\textrm {\\scriptsize 116}$ , R. Stroynowski$^\\textrm {\\scriptsize 42}$ , A. Strubig$^\\textrm {\\scriptsize 106}$ , S.A. Stucci$^\\textrm {\\scriptsize 18}$ , B. Stugu$^\\textrm {\\scriptsize 15}$ , N.A.",
"Styles$^\\textrm {\\scriptsize 44}$ , D. Su$^\\textrm {\\scriptsize 143}$ , J. Su$^\\textrm {\\scriptsize 125}$ , S. Suchek$^\\textrm {\\scriptsize 59a}$ , Y. Sugaya$^\\textrm {\\scriptsize 118}$ , M. Suk$^\\textrm {\\scriptsize 128}$ , V.V.",
"Sulin$^\\textrm {\\scriptsize 96}$ , S. Sultansoy$^\\textrm {\\scriptsize 4c}$ , T. Sumida$^\\textrm {\\scriptsize 69}$ , S. Sun$^\\textrm {\\scriptsize 58}$ , X. Sun$^\\textrm {\\scriptsize 35a}$ , J.E.",
"Sundermann$^\\textrm {\\scriptsize 50}$ , K. Suruliz$^\\textrm {\\scriptsize 149}$ , G. Susinno$^\\textrm {\\scriptsize 39a,39b}$ , M.R.",
"Sutton$^\\textrm {\\scriptsize 149}$ , S. Suzuki$^\\textrm {\\scriptsize 67}$ , M. Svatos$^\\textrm {\\scriptsize 127}$ , M. Swiatlowski$^\\textrm {\\scriptsize 33}$ , I. Sykora$^\\textrm {\\scriptsize 144a}$ , T. Sykora$^\\textrm {\\scriptsize 129}$ , D. Ta$^\\textrm {\\scriptsize 50}$ , C. Taccini$^\\textrm {\\scriptsize 134a,134b}$ , K. Tackmann$^\\textrm {\\scriptsize 44}$ , J. Taenzer$^\\textrm {\\scriptsize 158}$ , A. Taffard$^\\textrm {\\scriptsize 162}$ , R. Tafirout$^\\textrm {\\scriptsize 159a}$ , N. Taiblum$^\\textrm {\\scriptsize 153}$ , H. Takai$^\\textrm {\\scriptsize 27}$ , R. Takashima$^\\textrm {\\scriptsize 70}$ , T. Takeshita$^\\textrm {\\scriptsize 140}$ , Y. Takubo$^\\textrm {\\scriptsize 67}$ , M. Talby$^\\textrm {\\scriptsize 86}$ , A.A. Talyshev$^\\textrm {\\scriptsize 109}$$^{,c}$ , K.G.",
"Tan$^\\textrm {\\scriptsize 89}$ , J. Tanaka$^\\textrm {\\scriptsize 155}$ , M. Tanaka$^\\textrm {\\scriptsize 157}$ , R. Tanaka$^\\textrm {\\scriptsize 117}$ , S. Tanaka$^\\textrm {\\scriptsize 67}$ , B.B.",
"Tannenwald$^\\textrm {\\scriptsize 111}$ , S. Tapia Araya$^\\textrm {\\scriptsize 34b}$ , S. Tapprogge$^\\textrm {\\scriptsize 84}$ , S. Tarem$^\\textrm {\\scriptsize 152}$ , G.F. Tartarelli$^\\textrm {\\scriptsize 92a}$ , P. Tas$^\\textrm {\\scriptsize 129}$ , M. Tasevsky$^\\textrm {\\scriptsize 127}$ , T. Tashiro$^\\textrm {\\scriptsize 69}$ , E. Tassi$^\\textrm {\\scriptsize 39a,39b}$ , A. Tavares Delgado$^\\textrm {\\scriptsize 126a,126b}$ , Y. Tayalati$^\\textrm {\\scriptsize 135e}$ , A.C. Taylor$^\\textrm {\\scriptsize 105}$ , G.N.",
"Taylor$^\\textrm {\\scriptsize 89}$ , P.T.E.",
"Taylor$^\\textrm {\\scriptsize 89}$ , W. Taylor$^\\textrm {\\scriptsize 159b}$ , F.A.",
"Teischinger$^\\textrm {\\scriptsize 32}$ , P. Teixeira-Dias$^\\textrm {\\scriptsize 78}$ , K.K.",
"Temming$^\\textrm {\\scriptsize 50}$ , D. Temple$^\\textrm {\\scriptsize 142}$ , H. Ten Kate$^\\textrm {\\scriptsize 32}$ , P.K.",
"Teng$^\\textrm {\\scriptsize 151}$ , J.J. Teoh$^\\textrm {\\scriptsize 118}$ , F. Tepel$^\\textrm {\\scriptsize 174}$ , S. Terada$^\\textrm {\\scriptsize 67}$ , K. Terashi$^\\textrm {\\scriptsize 155}$ , J. Terron$^\\textrm {\\scriptsize 83}$ , S. Terzo$^\\textrm {\\scriptsize 101}$ , M. Testa$^\\textrm {\\scriptsize 49}$ , R.J. Teuscher$^\\textrm {\\scriptsize 158}$$^{,l}$ , T. Theveneaux-Pelzer$^\\textrm {\\scriptsize 86}$ , J.P. Thomas$^\\textrm {\\scriptsize 19}$ , J. Thomas-Wilsker$^\\textrm {\\scriptsize 78}$ , E.N.",
"Thompson$^\\textrm {\\scriptsize 37}$ , P.D.",
"Thompson$^\\textrm {\\scriptsize 19}$ , A.S. Thompson$^\\textrm {\\scriptsize 55}$ , L.A. Thomsen$^\\textrm {\\scriptsize 175}$ , E. Thomson$^\\textrm {\\scriptsize 122}$ , M. Thomson$^\\textrm {\\scriptsize 30}$ , M.J. Tibbetts$^\\textrm {\\scriptsize 16}$ , R.E.",
"Ticse Torres$^\\textrm {\\scriptsize 86}$ , V.O.",
"Tikhomirov$^\\textrm {\\scriptsize 96}$$^{,ao}$ , Yu.A.",
"Tikhonov$^\\textrm {\\scriptsize 109}$$^{,c}$ , S. Timoshenko$^\\textrm {\\scriptsize 98}$ , P. Tipton$^\\textrm {\\scriptsize 175}$ , S. Tisserant$^\\textrm {\\scriptsize 86}$ , K. Todome$^\\textrm {\\scriptsize 157}$ , T. Todorov$^\\textrm {\\scriptsize 5}$$^{,*}$ , S. Todorova-Nova$^\\textrm {\\scriptsize 129}$ , J. Tojo$^\\textrm {\\scriptsize 71}$ , S. Tokár$^\\textrm {\\scriptsize 144a}$ , K. Tokushuku$^\\textrm {\\scriptsize 67}$ , E. Tolley$^\\textrm {\\scriptsize 58}$ , L. Tomlinson$^\\textrm {\\scriptsize 85}$ , M. Tomoto$^\\textrm {\\scriptsize 103}$ , L. Tompkins$^\\textrm {\\scriptsize 143}$$^{,ap}$ , K. Toms$^\\textrm {\\scriptsize 105}$ , B. Tong$^\\textrm {\\scriptsize 58}$ , E. Torrence$^\\textrm {\\scriptsize 116}$ , H. Torres$^\\textrm {\\scriptsize 142}$ , E. Torró Pastor$^\\textrm {\\scriptsize 138}$ , J. Toth$^\\textrm {\\scriptsize 86}$$^{,aq}$ , F. Touchard$^\\textrm {\\scriptsize 86}$ , D.R.",
"Tovey$^\\textrm {\\scriptsize 139}$ , T. Trefzger$^\\textrm {\\scriptsize 173}$ , A. Tricoli$^\\textrm {\\scriptsize 27}$ , I.M.",
"Trigger$^\\textrm {\\scriptsize 159a}$ , S. Trincaz-Duvoid$^\\textrm {\\scriptsize 81}$ , M.F.",
"Tripiana$^\\textrm {\\scriptsize 13}$ , W. Trischuk$^\\textrm {\\scriptsize 158}$ , B. Trocmé$^\\textrm {\\scriptsize 57}$ , A. Trofymov$^\\textrm {\\scriptsize 44}$ , C. Troncon$^\\textrm {\\scriptsize 92a}$ , M. Trottier-McDonald$^\\textrm {\\scriptsize 16}$ , M. Trovatelli$^\\textrm {\\scriptsize 168}$ , L. Truong$^\\textrm {\\scriptsize 163a,163c}$ , M. Trzebinski$^\\textrm {\\scriptsize 41}$ , A. Trzupek$^\\textrm {\\scriptsize 41}$ , J.C-L. Tseng$^\\textrm {\\scriptsize 120}$ , P.V.",
"Tsiareshka$^\\textrm {\\scriptsize 93}$ , G. Tsipolitis$^\\textrm {\\scriptsize 10}$ , N. Tsirintanis$^\\textrm {\\scriptsize 9}$ , S. Tsiskaridze$^\\textrm {\\scriptsize 13}$ , V. Tsiskaridze$^\\textrm {\\scriptsize 50}$ , E.G.",
"Tskhadadze$^\\textrm {\\scriptsize 53a}$ , K.M.",
"Tsui$^\\textrm {\\scriptsize 61a}$ , I.I.",
"Tsukerman$^\\textrm {\\scriptsize 97}$ , V. Tsulaia$^\\textrm {\\scriptsize 16}$ , S. Tsuno$^\\textrm {\\scriptsize 67}$ , D. Tsybychev$^\\textrm {\\scriptsize 148}$ , Y. Tu$^\\textrm {\\scriptsize 61b}$ , A. Tudorache$^\\textrm {\\scriptsize 28b}$ , V. Tudorache$^\\textrm {\\scriptsize 28b}$ , A.N.",
"Tuna$^\\textrm {\\scriptsize 58}$ , S.A. Tupputi$^\\textrm {\\scriptsize 22a,22b}$ , S. Turchikhin$^\\textrm {\\scriptsize 66}$ , D. Turecek$^\\textrm {\\scriptsize 128}$ , D. Turgeman$^\\textrm {\\scriptsize 171}$ , R. Turra$^\\textrm {\\scriptsize 92a,92b}$ , A.J.",
"Turvey$^\\textrm {\\scriptsize 42}$ , P.M. Tuts$^\\textrm {\\scriptsize 37}$ , M. Tyndel$^\\textrm {\\scriptsize 131}$ , G. Ucchielli$^\\textrm {\\scriptsize 22a,22b}$ , I. Ueda$^\\textrm {\\scriptsize 155}$ , M. Ughetto$^\\textrm {\\scriptsize 146a,146b}$ , F. Ukegawa$^\\textrm {\\scriptsize 160}$ , G. Unal$^\\textrm {\\scriptsize 32}$ , A. Undrus$^\\textrm {\\scriptsize 27}$ , G. Unel$^\\textrm {\\scriptsize 162}$ , F.C.",
"Ungaro$^\\textrm {\\scriptsize 89}$ , Y. Unno$^\\textrm {\\scriptsize 67}$ , C. Unverdorben$^\\textrm {\\scriptsize 100}$ , J. Urban$^\\textrm {\\scriptsize 144b}$ , P. Urquijo$^\\textrm {\\scriptsize 89}$ , P. Urrejola$^\\textrm {\\scriptsize 84}$ , G. Usai$^\\textrm {\\scriptsize 8}$ , A. Usanova$^\\textrm {\\scriptsize 63}$ , L. Vacavant$^\\textrm {\\scriptsize 86}$ , V. Vacek$^\\textrm {\\scriptsize 128}$ , B. Vachon$^\\textrm {\\scriptsize 88}$ , C. Valderanis$^\\textrm {\\scriptsize 100}$ , E. Valdes Santurio$^\\textrm {\\scriptsize 146a,146b}$ , N. Valencic$^\\textrm {\\scriptsize 107}$ , S. Valentinetti$^\\textrm {\\scriptsize 22a,22b}$ , A. Valero$^\\textrm {\\scriptsize 166}$ , L. Valery$^\\textrm {\\scriptsize 13}$ , S. Valkar$^\\textrm {\\scriptsize 129}$ , J.A.",
"Valls Ferrer$^\\textrm {\\scriptsize 166}$ , W. Van Den Wollenberg$^\\textrm {\\scriptsize 107}$ , P.C.",
"Van Der Deijl$^\\textrm {\\scriptsize 107}$ , H. van der Graaf$^\\textrm {\\scriptsize 107}$ , N. van Eldik$^\\textrm {\\scriptsize 152}$ , P. van Gemmeren$^\\textrm {\\scriptsize 6}$ , J.",
"Van Nieuwkoop$^\\textrm {\\scriptsize 142}$ , I. van Vulpen$^\\textrm {\\scriptsize 107}$ , M.C.",
"van Woerden$^\\textrm {\\scriptsize 32}$ , M. Vanadia$^\\textrm {\\scriptsize 132a,132b}$ , W. Vandelli$^\\textrm {\\scriptsize 32}$ , R. Vanguri$^\\textrm {\\scriptsize 122}$ , A. Vaniachine$^\\textrm {\\scriptsize 130}$ , P. Vankov$^\\textrm {\\scriptsize 107}$ , G. Vardanyan$^\\textrm {\\scriptsize 176}$ , R. Vari$^\\textrm {\\scriptsize 132a}$ , E.W.",
"Varnes$^\\textrm {\\scriptsize 7}$ , T. Varol$^\\textrm {\\scriptsize 42}$ , D. Varouchas$^\\textrm {\\scriptsize 81}$ , A. Vartapetian$^\\textrm {\\scriptsize 8}$ , K.E.",
"Varvell$^\\textrm {\\scriptsize 150}$ , J.G.",
"Vasquez$^\\textrm {\\scriptsize 175}$ , F. Vazeille$^\\textrm {\\scriptsize 36}$ , T. Vazquez Schroeder$^\\textrm {\\scriptsize 88}$ , J. Veatch$^\\textrm {\\scriptsize 56}$ , V. Veeraraghavan$^\\textrm {\\scriptsize 7}$ , L.M.",
"Veloce$^\\textrm {\\scriptsize 158}$ , F. Veloso$^\\textrm {\\scriptsize 126a,126c}$ , S. Veneziano$^\\textrm {\\scriptsize 132a}$ , A. Ventura$^\\textrm {\\scriptsize 74a,74b}$ , M. Venturi$^\\textrm {\\scriptsize 168}$ , N. Venturi$^\\textrm {\\scriptsize 158}$ , A. Venturini$^\\textrm {\\scriptsize 25}$ , V. Vercesi$^\\textrm {\\scriptsize 121a}$ , M. Verducci$^\\textrm {\\scriptsize 132a,132b}$ , W. Verkerke$^\\textrm {\\scriptsize 107}$ , J.C. Vermeulen$^\\textrm {\\scriptsize 107}$ , A. Vest$^\\textrm {\\scriptsize 46}$$^{,ar}$ , M.C.",
"Vetterli$^\\textrm {\\scriptsize 142}$$^{,d}$ , O. Viazlo$^\\textrm {\\scriptsize 82}$ , I. Vichou$^\\textrm {\\scriptsize 165}$$^{,*}$ , T. Vickey$^\\textrm {\\scriptsize 139}$ , O.E.",
"Vickey Boeriu$^\\textrm {\\scriptsize 139}$ , G.H.A.",
"Viehhauser$^\\textrm {\\scriptsize 120}$ , S. Viel$^\\textrm {\\scriptsize 16}$ , L. Vigani$^\\textrm {\\scriptsize 120}$ , M. Villa$^\\textrm {\\scriptsize 22a,22b}$ , M. Villaplana Perez$^\\textrm {\\scriptsize 92a,92b}$ , E. Vilucchi$^\\textrm {\\scriptsize 49}$ , M.G.",
"Vincter$^\\textrm {\\scriptsize 31}$ , V.B.",
"Vinogradov$^\\textrm {\\scriptsize 66}$ , C. Vittori$^\\textrm {\\scriptsize 22a,22b}$ , I. Vivarelli$^\\textrm {\\scriptsize 149}$ , S. Vlachos$^\\textrm {\\scriptsize 10}$ , M. Vlasak$^\\textrm {\\scriptsize 128}$ , M. Vogel$^\\textrm {\\scriptsize 174}$ , P. Vokac$^\\textrm {\\scriptsize 128}$ , G. Volpi$^\\textrm {\\scriptsize 124a,124b}$ , M. Volpi$^\\textrm {\\scriptsize 89}$ , H. von der Schmitt$^\\textrm {\\scriptsize 101}$ , E. von Toerne$^\\textrm {\\scriptsize 23}$ , V. Vorobel$^\\textrm {\\scriptsize 129}$ , K. Vorobev$^\\textrm {\\scriptsize 98}$ , M. Vos$^\\textrm {\\scriptsize 166}$ , R. Voss$^\\textrm {\\scriptsize 32}$ , J.H.",
"Vossebeld$^\\textrm {\\scriptsize 75}$ , N. Vranjes$^\\textrm {\\scriptsize 14}$ , M. Vranjes Milosavljevic$^\\textrm {\\scriptsize 14}$ , V. Vrba$^\\textrm {\\scriptsize 127}$ , M. Vreeswijk$^\\textrm {\\scriptsize 107}$ , R. Vuillermet$^\\textrm {\\scriptsize 32}$ , I. Vukotic$^\\textrm {\\scriptsize 33}$ , Z. Vykydal$^\\textrm {\\scriptsize 128}$ , P. Wagner$^\\textrm {\\scriptsize 23}$ , W. Wagner$^\\textrm {\\scriptsize 174}$ , H. Wahlberg$^\\textrm {\\scriptsize 72}$ , S. Wahrmund$^\\textrm {\\scriptsize 46}$ , J. Wakabayashi$^\\textrm {\\scriptsize 103}$ , J. Walder$^\\textrm {\\scriptsize 73}$ , R. Walker$^\\textrm {\\scriptsize 100}$ , W. Walkowiak$^\\textrm {\\scriptsize 141}$ , V. Wallangen$^\\textrm {\\scriptsize 146a,146b}$ , C. Wang$^\\textrm {\\scriptsize 35c}$ , C. Wang$^\\textrm {\\scriptsize 35d,86}$ , F. Wang$^\\textrm {\\scriptsize 172}$ , H. Wang$^\\textrm {\\scriptsize 16}$ , H. Wang$^\\textrm {\\scriptsize 42}$ , J. Wang$^\\textrm {\\scriptsize 44}$ , J. Wang$^\\textrm {\\scriptsize 150}$ , K. Wang$^\\textrm {\\scriptsize 88}$ , R. Wang$^\\textrm {\\scriptsize 6}$ , S.M.",
"Wang$^\\textrm {\\scriptsize 151}$ , T. Wang$^\\textrm {\\scriptsize 23}$ , T. Wang$^\\textrm {\\scriptsize 37}$ , W. Wang$^\\textrm {\\scriptsize 35b}$ , X. Wang$^\\textrm {\\scriptsize 175}$ , C. Wanotayaroj$^\\textrm {\\scriptsize 116}$ , A. Warburton$^\\textrm {\\scriptsize 88}$ , C.P.",
"Ward$^\\textrm {\\scriptsize 30}$ , D.R.",
"Wardrope$^\\textrm {\\scriptsize 79}$ , A. Washbrook$^\\textrm {\\scriptsize 48}$ , P.M. Watkins$^\\textrm {\\scriptsize 19}$ , A.T. Watson$^\\textrm {\\scriptsize 19}$ , M.F.",
"Watson$^\\textrm {\\scriptsize 19}$ , G. Watts$^\\textrm {\\scriptsize 138}$ , S. Watts$^\\textrm {\\scriptsize 85}$ , B.M.",
"Waugh$^\\textrm {\\scriptsize 79}$ , S. Webb$^\\textrm {\\scriptsize 84}$ , M.S.",
"Weber$^\\textrm {\\scriptsize 18}$ , S.W.",
"Weber$^\\textrm {\\scriptsize 173}$ , J.S.",
"Webster$^\\textrm {\\scriptsize 6}$ , A.R.",
"Weidberg$^\\textrm {\\scriptsize 120}$ , B. Weinert$^\\textrm {\\scriptsize 62}$ , J. Weingarten$^\\textrm {\\scriptsize 56}$ , C. Weiser$^\\textrm {\\scriptsize 50}$ , H. Weits$^\\textrm {\\scriptsize 107}$ , P.S.",
"Wells$^\\textrm {\\scriptsize 32}$ , T. Wenaus$^\\textrm {\\scriptsize 27}$ , T. Wengler$^\\textrm {\\scriptsize 32}$ , S. Wenig$^\\textrm {\\scriptsize 32}$ , N. Wermes$^\\textrm {\\scriptsize 23}$ , M. Werner$^\\textrm {\\scriptsize 50}$ , M.D.",
"Werner$^\\textrm {\\scriptsize 65}$ , P. Werner$^\\textrm {\\scriptsize 32}$ , M. Wessels$^\\textrm {\\scriptsize 59a}$ , J. Wetter$^\\textrm {\\scriptsize 161}$ , K. Whalen$^\\textrm {\\scriptsize 116}$ , N.L.",
"Whallon$^\\textrm {\\scriptsize 138}$ , A.M. Wharton$^\\textrm {\\scriptsize 73}$ , A. White$^\\textrm {\\scriptsize 8}$ , M.J. White$^\\textrm {\\scriptsize 1}$ , R. White$^\\textrm {\\scriptsize 34b}$ , D. Whiteson$^\\textrm {\\scriptsize 162}$ , F.J. Wickens$^\\textrm {\\scriptsize 131}$ , W. Wiedenmann$^\\textrm {\\scriptsize 172}$ , M. Wielers$^\\textrm {\\scriptsize 131}$ , P. Wienemann$^\\textrm {\\scriptsize 23}$ , C. Wiglesworth$^\\textrm {\\scriptsize 38}$ , L.A.M.",
"Wiik-Fuchs$^\\textrm {\\scriptsize 23}$ , A. Wildauer$^\\textrm {\\scriptsize 101}$ , F. Wilk$^\\textrm {\\scriptsize 85}$ , H.G.",
"Wilkens$^\\textrm {\\scriptsize 32}$ , H.H.",
"Williams$^\\textrm {\\scriptsize 122}$ , S. Williams$^\\textrm {\\scriptsize 107}$ , C. Willis$^\\textrm {\\scriptsize 91}$ , S. Willocq$^\\textrm {\\scriptsize 87}$ , J.A.",
"Wilson$^\\textrm {\\scriptsize 19}$ , I. Wingerter-Seez$^\\textrm {\\scriptsize 5}$ , F. Winklmeier$^\\textrm {\\scriptsize 116}$ , O.J.",
"Winston$^\\textrm {\\scriptsize 149}$ , B.T.",
"Winter$^\\textrm {\\scriptsize 23}$ , M. Wittgen$^\\textrm {\\scriptsize 143}$ , J. Wittkowski$^\\textrm {\\scriptsize 100}$ , T.M.H.",
"Wolf$^\\textrm {\\scriptsize 107}$ , M.W.",
"Wolter$^\\textrm {\\scriptsize 41}$ , H. Wolters$^\\textrm {\\scriptsize 126a,126c}$ , S.D.",
"Worm$^\\textrm {\\scriptsize 131}$ , B.K.",
"Wosiek$^\\textrm {\\scriptsize 41}$ , J. Wotschack$^\\textrm {\\scriptsize 32}$ , M.J. Woudstra$^\\textrm {\\scriptsize 85}$ , K.W.",
"Wozniak$^\\textrm {\\scriptsize 41}$ , M. Wu$^\\textrm {\\scriptsize 57}$ , M. Wu$^\\textrm {\\scriptsize 33}$ , S.L.",
"Wu$^\\textrm {\\scriptsize 172}$ , X. Wu$^\\textrm {\\scriptsize 51}$ , Y. Wu$^\\textrm {\\scriptsize 90}$ , T.R.",
"Wyatt$^\\textrm {\\scriptsize 85}$ , B.M.",
"Wynne$^\\textrm {\\scriptsize 48}$ , S. Xella$^\\textrm {\\scriptsize 38}$ , D. Xu$^\\textrm {\\scriptsize 35a}$ , L. Xu$^\\textrm {\\scriptsize 27}$ , B. Yabsley$^\\textrm {\\scriptsize 150}$ , S. Yacoob$^\\textrm {\\scriptsize 145a}$ , D. Yamaguchi$^\\textrm {\\scriptsize 157}$ , Y. Yamaguchi$^\\textrm {\\scriptsize 118}$ , A. Yamamoto$^\\textrm {\\scriptsize 67}$ , S. Yamamoto$^\\textrm {\\scriptsize 155}$ , T. Yamanaka$^\\textrm {\\scriptsize 155}$ , K. Yamauchi$^\\textrm {\\scriptsize 103}$ , Y. Yamazaki$^\\textrm {\\scriptsize 68}$ , Z. Yan$^\\textrm {\\scriptsize 24}$ , H. Yang$^\\textrm {\\scriptsize 35e}$ , H. Yang$^\\textrm {\\scriptsize 172}$ , Y. Yang$^\\textrm {\\scriptsize 151}$ , Z. Yang$^\\textrm {\\scriptsize 15}$ , W-M. Yao$^\\textrm {\\scriptsize 16}$ , Y.C.",
"Yap$^\\textrm {\\scriptsize 81}$ , Y. Yasu$^\\textrm {\\scriptsize 67}$ , E. Yatsenko$^\\textrm {\\scriptsize 5}$ , K.H.",
"Yau Wong$^\\textrm {\\scriptsize 23}$ , J. Ye$^\\textrm {\\scriptsize 42}$ , S. Ye$^\\textrm {\\scriptsize 27}$ , I. Yeletskikh$^\\textrm {\\scriptsize 66}$ , A.L.",
"Yen$^\\textrm {\\scriptsize 58}$ , E. Yildirim$^\\textrm {\\scriptsize 84}$ , K. Yorita$^\\textrm {\\scriptsize 170}$ , R. Yoshida$^\\textrm {\\scriptsize 6}$ , K. Yoshihara$^\\textrm {\\scriptsize 122}$ , C. Young$^\\textrm {\\scriptsize 143}$ , C.J.S.",
"Young$^\\textrm {\\scriptsize 32}$ , S. Youssef$^\\textrm {\\scriptsize 24}$ , D.R.",
"Yu$^\\textrm {\\scriptsize 16}$ , J. Yu$^\\textrm {\\scriptsize 8}$ , J.M.",
"Yu$^\\textrm {\\scriptsize 90}$ , J. Yu$^\\textrm {\\scriptsize 65}$ , L. Yuan$^\\textrm {\\scriptsize 68}$ , S.P.Y.",
"Yuen$^\\textrm {\\scriptsize 23}$ , I. Yusuff$^\\textrm {\\scriptsize 30}$$^{,as}$ , B. Zabinski$^\\textrm {\\scriptsize 41}$ , R. Zaidan$^\\textrm {\\scriptsize 35d}$ , A.M. Zaitsev$^\\textrm {\\scriptsize 130}$$^{,ae}$ , N. Zakharchuk$^\\textrm {\\scriptsize 44}$ , J. Zalieckas$^\\textrm {\\scriptsize 15}$ , A. Zaman$^\\textrm {\\scriptsize 148}$ , S. Zambito$^\\textrm {\\scriptsize 58}$ , L. Zanello$^\\textrm {\\scriptsize 132a,132b}$ , D. Zanzi$^\\textrm {\\scriptsize 89}$ , C. Zeitnitz$^\\textrm {\\scriptsize 174}$ , M. Zeman$^\\textrm {\\scriptsize 128}$ , A. Zemla$^\\textrm {\\scriptsize 40a}$ , J.C. Zeng$^\\textrm {\\scriptsize 165}$ , Q. Zeng$^\\textrm {\\scriptsize 143}$ , K. Zengel$^\\textrm {\\scriptsize 25}$ , O. Zenin$^\\textrm {\\scriptsize 130}$ , T. Ženiš$^\\textrm {\\scriptsize 144a}$ , D. Zerwas$^\\textrm {\\scriptsize 117}$ , D. Zhang$^\\textrm {\\scriptsize 90}$ , F. Zhang$^\\textrm {\\scriptsize 172}$ , G. Zhang$^\\textrm {\\scriptsize 35b}$$^{,an}$ , H. Zhang$^\\textrm {\\scriptsize 35c}$ , J. Zhang$^\\textrm {\\scriptsize 6}$ , L. Zhang$^\\textrm {\\scriptsize 50}$ , R. Zhang$^\\textrm {\\scriptsize 23}$ , R. Zhang$^\\textrm {\\scriptsize 35b}$$^{,at}$ , X. Zhang$^\\textrm {\\scriptsize 35d}$ , Z. Zhang$^\\textrm {\\scriptsize 117}$ , X. Zhao$^\\textrm {\\scriptsize 42}$ , Y. Zhao$^\\textrm {\\scriptsize 35d}$ , Z. Zhao$^\\textrm {\\scriptsize 35b}$ , A. Zhemchugov$^\\textrm {\\scriptsize 66}$ , J. Zhong$^\\textrm {\\scriptsize 120}$ , B. Zhou$^\\textrm {\\scriptsize 90}$ , C. Zhou$^\\textrm {\\scriptsize 47}$ , L. Zhou$^\\textrm {\\scriptsize 37}$ , L. Zhou$^\\textrm {\\scriptsize 42}$ , M. Zhou$^\\textrm {\\scriptsize 148}$ , N. Zhou$^\\textrm {\\scriptsize 35f}$ , C.G.",
"Zhu$^\\textrm {\\scriptsize 35d}$ , H. Zhu$^\\textrm {\\scriptsize 35a}$ , J. Zhu$^\\textrm {\\scriptsize 90}$ , Y. Zhu$^\\textrm {\\scriptsize 35b}$ , X. Zhuang$^\\textrm {\\scriptsize 35a}$ , K. Zhukov$^\\textrm {\\scriptsize 96}$ , A. Zibell$^\\textrm {\\scriptsize 173}$ , D. Zieminska$^\\textrm {\\scriptsize 62}$ , N.I.",
"Zimine$^\\textrm {\\scriptsize 66}$ , C. Zimmermann$^\\textrm {\\scriptsize 84}$ , S. Zimmermann$^\\textrm {\\scriptsize 50}$ , Z. Zinonos$^\\textrm {\\scriptsize 56}$ , M. Zinser$^\\textrm {\\scriptsize 84}$ , M. Ziolkowski$^\\textrm {\\scriptsize 141}$ , L. Živković$^\\textrm {\\scriptsize 14}$ , G. Zobernig$^\\textrm {\\scriptsize 172}$ , A. Zoccoli$^\\textrm {\\scriptsize 22a,22b}$ , M. zur Nedden$^\\textrm {\\scriptsize 17}$ , L. Zwalinski$^\\textrm {\\scriptsize 32}$ .",
"$^{1}$ Department of Physics, University of Adelaide, Adelaide, Australia $^{2}$ Physics Department, SUNY Albany, Albany NY, United States of America $^{3}$ Department of Physics, University of Alberta, Edmonton AB, Canada $^{4}$ $^{(a)}$ Department of Physics, Ankara University, Ankara; $^{(b)}$ Istanbul Aydin University, Istanbul; $^{(c)}$ Division of Physics, TOBB University of Economics and Technology, Ankara, Turkey $^{5}$ LAPP, CNRS/IN2P3 and Université Savoie Mont Blanc, Annecy-le-Vieux, France $^{6}$ High Energy Physics Division, Argonne National Laboratory, Argonne IL, United States of America $^{7}$ Department of Physics, University of Arizona, Tucson AZ, United States of America $^{8}$ Department of Physics, The University of Texas at Arlington, Arlington TX, United States of America $^{9}$ Physics Department, University of Athens, Athens, Greece $^{10}$ Physics Department, National Technical University of Athens, Zografou, Greece $^{11}$ Department of Physics, The University of Texas at Austin, Austin TX, United States of America $^{12}$ Institute of Physics, Azerbaijan Academy of Sciences, Baku, Azerbaijan $^{13}$ Institut de Física d'Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Barcelona, Spain, Spain $^{14}$ Institute of Physics, University of Belgrade, Belgrade, Serbia $^{15}$ Department for Physics and Technology, University of Bergen, Bergen, Norway $^{16}$ Physics Division, Lawrence Berkeley National Laboratory and University of California, Berkeley CA, United States of America $^{17}$ Department of Physics, Humboldt University, Berlin, Germany $^{18}$ Albert Einstein Center for Fundamental Physics and Laboratory for High Energy Physics, University of Bern, Bern, Switzerland $^{19}$ School of Physics and Astronomy, University of Birmingham, Birmingham, United Kingdom $^{20}$ $^{(a)}$ Department of Physics, Bogazici University, Istanbul; $^{(b)}$ Department of Physics Engineering, Gaziantep University, Gaziantep; $^{(d)}$ Istanbul Bilgi University, Faculty of Engineering and Natural Sciences, Istanbul,Turkey; $^{(e)}$ Bahcesehir University, Faculty of Engineering and Natural Sciences, Istanbul, Turkey, Turkey $^{21}$ Centro de Investigaciones, Universidad Antonio Narino, Bogota, Colombia $^{22}$ $^{(a)}$ INFN Sezione di Bologna; $^{(b)}$ Dipartimento di Fisica e Astronomia, Università di Bologna, Bologna, Italy $^{23}$ Physikalisches Institut, University of Bonn, Bonn, Germany $^{24}$ Department of Physics, Boston University, Boston MA, United States of America $^{25}$ Department of Physics, Brandeis University, Waltham MA, United States of America $^{26}$ $^{(a)}$ Universidade Federal do Rio De Janeiro COPPE/EE/IF, Rio de Janeiro; $^{(b)}$ Electrical Circuits Department, Federal University of Juiz de Fora (UFJF), Juiz de Fora; $^{(c)}$ Federal University of Sao Joao del Rei (UFSJ), Sao Joao del Rei; $^{(d)}$ Instituto de Fisica, Universidade de Sao Paulo, Sao Paulo, Brazil $^{27}$ Physics Department, Brookhaven National Laboratory, Upton NY, United States of America $^{28}$ $^{(a)}$ Transilvania University of Brasov, Brasov, Romania; $^{(b)}$ National Institute of Physics and Nuclear Engineering, Bucharest; $^{(c)}$ National Institute for Research and Development of Isotopic and Molecular Technologies, Physics Department, Cluj Napoca; $^{(d)}$ University Politehnica Bucharest, Bucharest; $^{(e)}$ West University in Timisoara, Timisoara, Romania $^{29}$ Departamento de Física, Universidad de Buenos Aires, Buenos Aires, Argentina $^{30}$ Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom $^{31}$ Department of Physics, Carleton University, Ottawa ON, Canada $^{32}$ CERN, Geneva, Switzerland $^{33}$ Enrico Fermi Institute, University of Chicago, Chicago IL, United States of America $^{34}$ $^{(a)}$ Departamento de Física, Pontificia Universidad Católica de Chile, Santiago; $^{(b)}$ Departamento de Física, Universidad Técnica Federico Santa María, Valparaíso, Chile $^{35}$ $^{(a)}$ Institute of High Energy Physics, Chinese Academy of Sciences, Beijing; $^{(b)}$ Department of Modern Physics, University of Science and Technology of China, Anhui; $^{(c)}$ Department of Physics, Nanjing University, Jiangsu; $^{(d)}$ School of Physics, Shandong University, Shandong; $^{(e)}$ Department of Physics and Astronomy, Shanghai Key Laboratory for Particle Physics and Cosmology, Shanghai Jiao Tong University, Shanghai; (also affiliated with PKU-CHEP); $^{(f)}$ Physics Department, Tsinghua University, Beijing 100084, China $^{36}$ Laboratoire de Physique Corpusculaire, Clermont Université and Université Blaise Pascal and CNRS/IN2P3, Clermont-Ferrand, France $^{37}$ Nevis Laboratory, Columbia University, Irvington NY, United States of America $^{38}$ Niels Bohr Institute, University of Copenhagen, Kobenhavn, Denmark $^{39}$ $^{(a)}$ INFN Gruppo Collegato di Cosenza, Laboratori Nazionali di Frascati; $^{(b)}$ Dipartimento di Fisica, Università della Calabria, Rende, Italy $^{40}$ $^{(a)}$ AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, Krakow; $^{(b)}$ Marian Smoluchowski Institute of Physics, Jagiellonian University, Krakow, Poland $^{41}$ Institute of Nuclear Physics Polish Academy of Sciences, Krakow, Poland $^{42}$ Physics Department, Southern Methodist University, Dallas TX, United States of America $^{43}$ Physics Department, University of Texas at Dallas, Richardson TX, United States of America $^{44}$ DESY, Hamburg and Zeuthen, Germany $^{45}$ Lehrstuhl für Experimentelle Physik IV, Technische Universität Dortmund, Dortmund, Germany $^{46}$ Institut für Kern- und Teilchenphysik, Technische Universität Dresden, Dresden, Germany $^{47}$ Department of Physics, Duke University, Durham NC, United States of America $^{48}$ SUPA - School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom $^{49}$ INFN Laboratori Nazionali di Frascati, Frascati, Italy $^{50}$ Fakultät für Mathematik und Physik, Albert-Ludwigs-Universität, Freiburg, Germany $^{51}$ Section de Physique, Université de Genève, Geneva, Switzerland $^{52}$ $^{(a)}$ INFN Sezione di Genova; $^{(b)}$ Dipartimento di Fisica, Università di Genova, Genova, Italy $^{53}$ $^{(a)}$ E. Andronikashvili Institute of Physics, Iv.",
"Javakhishvili Tbilisi State University, Tbilisi; $^{(b)}$ High Energy Physics Institute, Tbilisi State University, Tbilisi, Georgia $^{54}$ II Physikalisches Institut, Justus-Liebig-Universität Giessen, Giessen, Germany $^{55}$ SUPA - School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom $^{56}$ II Physikalisches Institut, Georg-August-Universität, Göttingen, Germany $^{57}$ Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS/IN2P3, Grenoble, France $^{58}$ Laboratory for Particle Physics and Cosmology, Harvard University, Cambridge MA, United States of America $^{59}$ $^{(a)}$ Kirchhoff-Institut für Physik, Ruprecht-Karls-Universität Heidelberg, Heidelberg; $^{(b)}$ Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg; $^{(c)}$ ZITI Institut für technische Informatik, Ruprecht-Karls-Universität Heidelberg, Mannheim, Germany $^{60}$ Faculty of Applied Information Science, Hiroshima Institute of Technology, Hiroshima, Japan $^{61}$ $^{(a)}$ Department of Physics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong; $^{(b)}$ Department of Physics, The University of Hong Kong, Hong Kong; $^{(c)}$ Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China $^{62}$ Department of Physics, Indiana University, Bloomington IN, United States of America $^{63}$ Institut für Astro- und Teilchenphysik, Leopold-Franzens-Universität, Innsbruck, Austria $^{64}$ University of Iowa, Iowa City IA, United States of America $^{65}$ Department of Physics and Astronomy, Iowa State University, Ames IA, United States of America $^{66}$ Joint Institute for Nuclear Research, JINR Dubna, Dubna, Russia $^{67}$ KEK, High Energy Accelerator Research Organization, Tsukuba, Japan $^{68}$ Graduate School of Science, Kobe University, Kobe, Japan $^{69}$ Faculty of Science, Kyoto University, Kyoto, Japan $^{70}$ Kyoto University of Education, Kyoto, Japan $^{71}$ Department of Physics, Kyushu University, Fukuoka, Japan $^{72}$ Instituto de Física La Plata, Universidad Nacional de La Plata and CONICET, La Plata, Argentina $^{73}$ Physics Department, Lancaster University, Lancaster, United Kingdom $^{74}$ $^{(a)}$ INFN Sezione di Lecce; $^{(b)}$ Dipartimento di Matematica e Fisica, Università del Salento, Lecce, Italy $^{75}$ Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom $^{76}$ Department of Physics, Jožef Stefan Institute and University of Ljubljana, Ljubljana, Slovenia $^{77}$ School of Physics and Astronomy, Queen Mary University of London, London, United Kingdom $^{78}$ Department of Physics, Royal Holloway University of London, Surrey, United Kingdom $^{79}$ Department of Physics and Astronomy, University College London, London, United Kingdom $^{80}$ Louisiana Tech University, Ruston LA, United States of America $^{81}$ Laboratoire de Physique Nucléaire et de Hautes Energies, UPMC and Université Paris-Diderot and CNRS/IN2P3, Paris, France $^{82}$ Fysiska institutionen, Lunds universitet, Lund, Sweden $^{83}$ Departamento de Fisica Teorica C-15, Universidad Autonoma de Madrid, Madrid, Spain $^{84}$ Institut für Physik, Universität Mainz, Mainz, Germany $^{85}$ School of Physics and Astronomy, University of Manchester, Manchester, United Kingdom $^{86}$ CPPM, Aix-Marseille Université and CNRS/IN2P3, Marseille, France $^{87}$ Department of Physics, University of Massachusetts, Amherst MA, United States of America $^{88}$ Department of Physics, McGill University, Montreal QC, Canada $^{89}$ School of Physics, University of Melbourne, Victoria, Australia $^{90}$ Department of Physics, The University of Michigan, Ann Arbor MI, United States of America $^{91}$ Department of Physics and Astronomy, Michigan State University, East Lansing MI, United States of America $^{92}$ $^{(a)}$ INFN Sezione di Milano; $^{(b)}$ Dipartimento di Fisica, Università di Milano, Milano, Italy $^{93}$ B.I.",
"Stepanov Institute of Physics, National Academy of Sciences of Belarus, Minsk, Republic of Belarus $^{94}$ National Scientific and Educational Centre for Particle and High Energy Physics, Minsk, Republic of Belarus $^{95}$ Group of Particle Physics, University of Montreal, Montreal QC, Canada $^{96}$ P.N.",
"Lebedev Physical Institute of the Russian Academy of Sciences, Moscow, Russia $^{97}$ Institute for Theoretical and Experimental Physics (ITEP), Moscow, Russia $^{98}$ National Research Nuclear University MEPhI, Moscow, Russia $^{99}$ D.V.",
"Skobeltsyn Institute of Nuclear Physics, M.V.",
"Lomonosov Moscow State University, Moscow, Russia $^{100}$ Fakultät für Physik, Ludwig-Maximilians-Universität München, München, Germany $^{101}$ Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), München, Germany $^{102}$ Nagasaki Institute of Applied Science, Nagasaki, Japan $^{103}$ Graduate School of Science and Kobayashi-Maskawa Institute, Nagoya University, Nagoya, Japan $^{104}$ $^{(a)}$ INFN Sezione di Napoli; $^{(b)}$ Dipartimento di Fisica, Università di Napoli, Napoli, Italy $^{105}$ Department of Physics and Astronomy, University of New Mexico, Albuquerque NM, United States of America $^{106}$ Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen/Nikhef, Nijmegen, Netherlands $^{107}$ Nikhef National Institute for Subatomic Physics and University of Amsterdam, Amsterdam, Netherlands $^{108}$ Department of Physics, Northern Illinois University, DeKalb IL, United States of America $^{109}$ Budker Institute of Nuclear Physics, SB RAS, Novosibirsk, Russia $^{110}$ Department of Physics, New York University, New York NY, United States of America $^{111}$ Ohio State University, Columbus OH, United States of America $^{112}$ Faculty of Science, Okayama University, Okayama, Japan $^{113}$ Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma, Norman OK, United States of America $^{114}$ Department of Physics, Oklahoma State University, Stillwater OK, United States of America $^{115}$ Palacký University, RCPTM, Olomouc, Czech Republic $^{116}$ Center for High Energy Physics, University of Oregon, Eugene OR, United States of America $^{117}$ LAL, Univ.",
"Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay, France $^{118}$ Graduate School of Science, Osaka University, Osaka, Japan $^{119}$ Department of Physics, University of Oslo, Oslo, Norway $^{120}$ Department of Physics, Oxford University, Oxford, United Kingdom $^{121}$ $^{(a)}$ INFN Sezione di Pavia; $^{(b)}$ Dipartimento di Fisica, Università di Pavia, Pavia, Italy $^{122}$ Department of Physics, University of Pennsylvania, Philadelphia PA, United States of America $^{123}$ National Research Centre \"Kurchatov Institute\" B.P.Konstantinov Petersburg Nuclear Physics Institute, St. Petersburg, Russia $^{124}$ $^{(a)}$ INFN Sezione di Pisa; $^{(b)}$ Dipartimento di Fisica E. Fermi, Università di Pisa, Pisa, Italy $^{125}$ Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh PA, United States of America $^{126}$ $^{(a)}$ Laboratório de Instrumentação e Física Experimental de Partículas - LIP, Lisboa; $^{(b)}$ Faculdade de Ciências, Universidade de Lisboa, Lisboa; $^{(c)}$ Department of Physics, University of Coimbra, Coimbra; $^{(d)}$ Centro de Física Nuclear da Universidade de Lisboa, Lisboa; $^{(e)}$ Departamento de Fisica, Universidade do Minho, Braga; $^{(f)}$ Departamento de Fisica Teorica y del Cosmos and CAFPE, Universidad de Granada, Granada (Spain); $^{(g)}$ Dep Fisica and CEFITEC of Faculdade de Ciencias e Tecnologia, Universidade Nova de Lisboa, Caparica, Portugal $^{127}$ Institute of Physics, Academy of Sciences of the Czech Republic, Praha, Czech Republic $^{128}$ Czech Technical University in Prague, Praha, Czech Republic $^{129}$ Faculty of Mathematics and Physics, Charles University in Prague, Praha, Czech Republic $^{130}$ State Research Center Institute for High Energy Physics (Protvino), NRC KI, Russia $^{131}$ Particle Physics Department, Rutherford Appleton Laboratory, Didcot, United Kingdom $^{132}$ $^{(a)}$ INFN Sezione di Roma; $^{(b)}$ Dipartimento di Fisica, Sapienza Università di Roma, Roma, Italy $^{133}$ $^{(a)}$ INFN Sezione di Roma Tor Vergata; $^{(b)}$ Dipartimento di Fisica, Università di Roma Tor Vergata, Roma, Italy $^{134}$ $^{(a)}$ INFN Sezione di Roma Tre; $^{(b)}$ Dipartimento di Matematica e Fisica, Università Roma Tre, Roma, Italy $^{135}$ $^{(a)}$ Faculté des Sciences Ain Chock, Réseau Universitaire de Physique des Hautes Energies - Université Hassan II, Casablanca; $^{(b)}$ Centre National de l'Energie des Sciences Techniques Nucleaires, Rabat; $^{(c)}$ Faculté des Sciences Semlalia, Université Cadi Ayyad, LPHEA-Marrakech; $^{(d)}$ Faculté des Sciences, Université Mohamed Premier and LPTPM, Oujda; $^{(e)}$ Faculté des sciences, Université Mohammed V, Rabat, Morocco $^{136}$ DSM/IRFU (Institut de Recherches sur les Lois Fondamentales de l'Univers), CEA Saclay (Commissariat à l'Energie Atomique et aux Energies Alternatives), Gif-sur-Yvette, France $^{137}$ Santa Cruz Institute for Particle Physics, University of California Santa Cruz, Santa Cruz CA, United States of America $^{138}$ Department of Physics, University of Washington, Seattle WA, United States of America $^{139}$ Department of Physics and Astronomy, University of Sheffield, Sheffield, United Kingdom $^{140}$ Department of Physics, Shinshu University, Nagano, Japan $^{141}$ Fachbereich Physik, Universität Siegen, Siegen, Germany $^{142}$ Department of Physics, Simon Fraser University, Burnaby BC, Canada $^{143}$ SLAC National Accelerator Laboratory, Stanford CA, United States of America $^{144}$ $^{(a)}$ Faculty of Mathematics, Physics & Informatics, Comenius University, Bratislava; $^{(b)}$ Department of Subnuclear Physics, Institute of Experimental Physics of the Slovak Academy of Sciences, Kosice, Slovak Republic $^{145}$ $^{(a)}$ Department of Physics, University of Cape Town, Cape Town; $^{(b)}$ Department of Physics, University of Johannesburg, Johannesburg; $^{(c)}$ School of Physics, University of the Witwatersrand, Johannesburg, South Africa $^{146}$ $^{(a)}$ Department of Physics, Stockholm University; $^{(b)}$ The Oskar Klein Centre, Stockholm, Sweden $^{147}$ Physics Department, Royal Institute of Technology, Stockholm, Sweden $^{148}$ Departments of Physics & Astronomy and Chemistry, Stony Brook University, Stony Brook NY, United States of America $^{149}$ Department of Physics and Astronomy, University of Sussex, Brighton, United Kingdom $^{150}$ School of Physics, University of Sydney, Sydney, Australia $^{151}$ Institute of Physics, Academia Sinica, Taipei, Taiwan $^{152}$ Department of Physics, Technion: Israel Institute of Technology, Haifa, Israel $^{153}$ Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv University, Tel Aviv, Israel $^{154}$ Department of Physics, Aristotle University of Thessaloniki, Thessaloniki, Greece $^{155}$ International Center for Elementary Particle Physics and Department of Physics, The University of Tokyo, Tokyo, Japan $^{156}$ Graduate School of Science and Technology, Tokyo Metropolitan University, Tokyo, Japan $^{157}$ Department of Physics, Tokyo Institute of Technology, Tokyo, Japan $^{158}$ Department of Physics, University of Toronto, Toronto ON, Canada $^{159}$ $^{(a)}$ TRIUMF, Vancouver BC; $^{(b)}$ Department of Physics and Astronomy, York University, Toronto ON, Canada $^{160}$ Faculty of Pure and Applied Sciences, and Center for Integrated Research in Fundamental Science and Engineering, University of Tsukuba, Tsukuba, Japan $^{161}$ Department of Physics and Astronomy, Tufts University, Medford MA, United States of America $^{162}$ Department of Physics and Astronomy, University of California Irvine, Irvine CA, United States of America $^{163}$ $^{(a)}$ INFN Gruppo Collegato di Udine, Sezione di Trieste, Udine; $^{(b)}$ ICTP, Trieste; $^{(c)}$ Dipartimento di Chimica, Fisica e Ambiente, Università di Udine, Udine, Italy $^{164}$ Department of Physics and Astronomy, University of Uppsala, Uppsala, Sweden $^{165}$ Department of Physics, University of Illinois, Urbana IL, United States of America $^{166}$ Instituto de Fisica Corpuscular (IFIC) and Departamento de Fisica Atomica, Molecular y Nuclear and Departamento de Ingeniería Electrónica and Instituto de Microelectrónica de Barcelona (IMB-CNM), University of Valencia and CSIC, Valencia, Spain $^{167}$ Department of Physics, University of British Columbia, Vancouver BC, Canada $^{168}$ Department of Physics and Astronomy, University of Victoria, Victoria BC, Canada $^{169}$ Department of Physics, University of Warwick, Coventry, United Kingdom $^{170}$ Waseda University, Tokyo, Japan $^{171}$ Department of Particle Physics, The Weizmann Institute of Science, Rehovot, Israel $^{172}$ Department of Physics, University of Wisconsin, Madison WI, United States of America $^{173}$ Fakultät für Physik und Astronomie, Julius-Maximilians-Universität, Würzburg, Germany $^{174}$ Fakultät für Mathematik und Naturwissenschaften, Fachgruppe Physik, Bergische Universität Wuppertal, Wuppertal, Germany $^{175}$ Department of Physics, Yale University, New Haven CT, United States of America $^{176}$ Yerevan Physics Institute, Yerevan, Armenia $^{177}$ Centre de Calcul de l'Institut National de Physique Nucléaire et de Physique des Particules (IN2P3), Villeurbanne, France $^{a}$ Also at Department of Physics, King's College London, London, United Kingdom $^{b}$ Also at Institute of Physics, Azerbaijan Academy of Sciences, Baku, Azerbaijan $^{c}$ Also at Novosibirsk State University, Novosibirsk, Russia $^{d}$ Also at TRIUMF, Vancouver BC, Canada e Also at Department of Physics & Astronomy, University of Louisville, Louisville, KY, United States of America $^{f}$ Also at Department of Physics, California State University, Fresno CA, United States of America $^{g}$ Also at Department of Physics, University of Fribourg, Fribourg, Switzerland $^{h}$ Also at Departament de Fisica de la Universitat Autonoma de Barcelona, Barcelona, Spain $^{i}$ Also at Departamento de Fisica e Astronomia, Faculdade de Ciencias, Universidade do Porto, Portugal $^{j}$ Also at Tomsk State University, Tomsk, Russia $^{k}$ Also at Universita di Napoli Parthenope, Napoli, Italy $^{l}$ Also at Institute of Particle Physics (IPP), Canada $^{m}$ Also at National Institute of Physics and Nuclear Engineering, Bucharest, Romania $^{n}$ Also at Department of Physics, St. Petersburg State Polytechnical University, St. Petersburg, Russia $^{o}$ Also at Department of Physics, The University of Michigan, Ann Arbor MI, United States of America $^{p}$ Also at Centre for High Performance Computing, CSIR Campus, Rosebank, Cape Town, South Africa $^{q}$ Also at Louisiana Tech University, Ruston LA, United States of America $^{r}$ Also at Institucio Catalana de Recerca i Estudis Avancats, ICREA, Barcelona, Spain $^{s}$ Also at Graduate School of Science, Osaka University, Osaka, Japan $^{t}$ Also at Department of Physics, National Tsing Hua University, Taiwan $^{u}$ Also at Institute for Mathematics, Astrophysics and Particle Physics, Radboud University Nijmegen/Nikhef, Nijmegen, Netherlands $^{v}$ Also at Department of Physics, The University of Texas at Austin, Austin TX, United States of America $^{w}$ Also at Institute of Theoretical Physics, Ilia State University, Tbilisi, Georgia $^{x}$ Also at CERN, Geneva, Switzerland $^{y}$ Also at Georgian Technical University (GTU),Tbilisi, Georgia $^{z}$ Also at Ochadai Academic Production, Ochanomizu University, Tokyo, Japan $^{aa}$ Also at Manhattan College, New York NY, United States of America $^{ab}$ Also at Hellenic Open University, Patras, Greece $^{ac}$ Also at Academia Sinica Grid Computing, Institute of Physics, Academia Sinica, Taipei, Taiwan $^{ad}$ Also at School of Physics, Shandong University, Shandong, China $^{ae}$ Also at Moscow Institute of Physics and Technology State University, Dolgoprudny, Russia $^{af}$ Also at Section de Physique, Université de Genève, Geneva, Switzerland $^{ag}$ Also at Eotvos Lorand University, Budapest, Hungary $^{ah}$ Also at Departments of Physics & Astronomy and Chemistry, Stony Brook University, Stony Brook NY, United States of America $^{ai}$ Also at International School for Advanced Studies (SISSA), Trieste, Italy $^{aj}$ Also at Department of Physics and Astronomy, University of South Carolina, Columbia SC, United States of America $^{ak}$ Also at School of Physics and Engineering, Sun Yat-sen University, Guangzhou, China $^{al}$ Also at Institute for Nuclear Research and Nuclear Energy (INRNE) of the Bulgarian Academy of Sciences, Sofia, Bulgaria $^{am}$ Also at Faculty of Physics, M.V.Lomonosov Moscow State University, Moscow, Russia $^{an}$ Also at Institute of Physics, Academia Sinica, Taipei, Taiwan $^{ao}$ Also at National Research Nuclear University MEPhI, Moscow, Russia $^{ap}$ Also at Department of Physics, Stanford University, Stanford CA, United States of America $^{aq}$ Also at Institute for Particle and Nuclear Physics, Wigner Research Centre for Physics, Budapest, Hungary $^{ar}$ Also at Flensburg University of Applied Sciences, Flensburg, Germany $^{as}$ Also at University of Malaya, Department of Physics, Kuala Lumpur, Malaysia $^{at}$ Also at CPPM, Aix-Marseille Université and CNRS/IN2P3, Marseille, France $^{*}$ Deceased"
]
] | 1606.05129 |
[
[
"Strong analog classical simulation of coherent quantum dynamics"
],
[
"Abstract A strong analog classical simulation of general quantum evolution is proposed, which serves as a novel scheme in quantum computation and simulation.",
"The scheme employs the approach of geometric quantum mechanics and quantum informational technique of quantum tomography, which applies broadly to cases of mixed states, nonunitary evolution, and infinite dimensional systems.",
"The simulation provides an intriguing classical picture to probe quantum phenomena, namely, a coherent quantum dynamics can be viewed as a globally constrained classical Hamiltonian dynamics of a collection of coupled particles or strings.",
"Efficiency analysis reveals a fundamental difference between the locality in real space and locality in Hilbert space, the latter enables efficient strong analog classical simulations.",
"Examples are also studied to highlight the differences and gaps among various simulation methods."
],
[
"Introduction",
"To properly understand as well as utilize quantum resources, e.g.",
"quantum coherence, are the main theme of modern quantum science.",
"Quantum coherence and its dynamics, namely, decoherence, which essentially leads to entanglement [1], play central roles in many studies such as measurement and quantum-to-classical transition [2], quantum computation [3], quantum resource theory [4], as well as strongly correlated many-body systems.",
"Besides, a more primary quest is to understand and seek the mechanism, or origin, of quantum coherence itself.",
"Efforts have been made in the very early days to understand Schrödinger equation that describes the quantum behavior of particles in space using classical pictures [5], [6], [7], [8], and many approaches prove to be significant for various applications, such as quantum trajectories and hydrodynamics [9], and the efforts continue till nowadays.",
"The emerging field of quantum computation [3], especially quantum simulation and classical simulation of quantum processes [10], [11] provide new perspective to understand quantum coherence and various quantum properties.",
"In this work, we raise the question whether it is possible to simulate (or simply put, reproduce) general quantum evolution by classical mechanics in the spirit of quantum simulation, and our result shows that the answer is yes.",
"However, it may seem impossible at first look since quantum mechanics (QM) is well known as a generalization of classical mechanics, as well as probability theory, the latter two can be reached from QM through the mechanism of decoherence [2].",
"However, our simulation reveals that a quantum dynamics can be viewed as a set of coupled classical dynamics with global constraints.",
"Furthermore, we study the problem of whether and when such classical simulation can be efficient, i.e., the cost of simulation scales polynomially with the size of the quantum target, quantified by a proper measure.",
"Our result shows that quantum dynamics and classical (also statistical) dynamics can be described in a unified way in terms of Hamiltonian dynamics and symplectic geometry, however, a general quantum dynamics cannot be efficiently simulated classically.",
"Despite this, there are cases of practical interest that are classically tractable, such as linear optics [12], [13] and (discrete-time coined) quantum walk [14] studied in this work.",
"In general, classifications of quantum processes regarding classical simulation efficiencies would be of broad implications, e.g., for complexity studies.",
"Our simulation of quantum dynamics is classical, analog, as well as strong, explained as follows.",
"In quantum simulation literature, there are many notions of simulation [10], [11], [15], [16], and a simulation can be conveniently classified by a set of independent binary features, notably here, digital/analog, classical/quantum, and weak/strong At least eight types of simulation exist, while further classifications are surely possible in various contexts..",
"In general, digital/analog refers to whether a simulation is performed on a universal computer or some dedicated-purpose devices specified by analog parameters that can be mapped to those of the simulated objects [10], classical/quantum refers to whether the simulator is classical or quantum, and weak/strong, based on the weak/strong operator topology [16], refers to what the simulation is about, namely, whether it simulates partly or completely the properties of the simulated objects.",
"Different simulations apply naturally in different contexts.",
"Classical digital simulation, e.g., to compute measurement results on a quantum process Note there are many kinds of classical simulations of quantum processes, such as a “weak” simulation [15], which is a sampling simulation that samples from a probability distribution instead of computing it., refers to numerical simulation on classical computers (computational physics), while quantum digital and analog simulations aim to find quantum speedup and learn complicated quantum systems [10].",
"Furthermore, strong simulation, which requires the approximation of an object itself, hence a “white box,” is natural for quantum simulation since quantum simulators can produce the desired quantum process itself, such as local Hamiltonian-evolution [19] and quantum channel [20], [21] simulations.",
"On the contrary, weak simulation is common in classical simulation since it only aims to yield partial information of an object, e.g., the action of an operator on a state, without the requirement to simulate the process itself or the process as a white box.",
"In the landscape of quantum simulation, a strong analog classical (SAC) simulation, which is suitable for probing the quantum-classical distinction, is largely unexplored and overlooked.",
"Our work presents a SAC simulation of quantum evolution, which is, on the one hand, novel in the field of quantum computation and simulation, and, on the other hand, serves as an approach for the understanding of quantum dynamics in terms of classical pictures.",
"In details, the SAC simulation problem and scheme is: given a quantum process to be simulated (may include state preparation, evolution, final state verification, and measurement), named as simulatee, a procedure that employs the methods of geometric quantum mechanics and quantum tomography designs a simulator, which is a classical Hamiltonian system that reproduces the simulatee.",
"Both the simulation quality (accuracy) and cost can be precisely assessed.",
"Also there exist extensions and variations of the main simulation scheme.",
"The theory of geometric quantum mechanics (GQM) [22], [23], [24], [25] is employed to construct the SAC simulation scheme, which provides a unique viewpoint to reveal the quantum-classical distinction and connection, e.g., in the study of geometric phase [26], [27].",
"In this work we find that efficient SAC simulations can be ensured by a locality in Hilbert space (see the study in section ), which is the notion of locality employed in the definition of quantum Turing machine [28], instead of the locality in the so-called real space, e.g., the locality in local Hamiltonian many-body systems.",
"By comparison, in the framework of matrix product states [29], [30] ground state properties of local Hamiltonian can be efficiently simulated on classical computers, and the simulation cost scales with the amount of (bipartite) entanglement (or the bond dimension).",
"Another widely explored algebraic and computational approach is the stabilizer formalism, and Wigner function negativity also contextuality are identified as the quantum notions that determine the classical simulation efficiency [31], [32], which employs a weak (or sampling) simulation scheme, not the same as our SAC simulation.",
"Also interference, long been known as a quantum-classical distinction, was recently identified as a resource for quantum speedup and a classical sampling simulation scheme was employed [33].",
"Generally speaking, the understanding of quantum-classical distinction largely depends on the simulation methods involved, and the SAC simulation is more restrictive, hence more accurate for the description of quantum dynamics in terms of classical pictures.",
"Our study also shows that the linear optics simulation of quantum computation [12], [13] (without nonlinear effects) can be viewed as an example of SAC simulation, hence our work can provide an angle to reveal the connections and differences of various simulations.",
"In the following, section develops the strong analog classical simulation framework based on geometric quantum mechanics and quantum tomography.",
"Afterwards, the simulation efficiency issue is considered in section , where we also analyze several practical examples and differences from other types of simulations.",
"In section various extensions of the main simulation scheme are developed, including the cases for nonunitary evolution and infinite-dimensional systems.",
"We conclude in section with a brief summary and discussion."
],
[
"Simulation framework",
"We start from finite-dimensional unitary evolution of pure states.",
"The generalizations to mixed states, nonunitary evolution as well as infinite-dimensional cases are discussed later.",
"Quantum states live in projective Hilbert space $\\mathcal {P}\\mathcal {H}$ since they are normalized vectors with any global phase physically trivial.",
"Distance between any two states $|\\psi \\rangle $ and $|\\phi \\rangle $ is based on the overlap function $\\langle \\phi |\\psi \\rangle $ .",
"Geometric quantum mechanics (GQM) [22], [23], [24], [25] shows that the space $\\mathcal {P}\\mathcal {H}$ is a Kähler manifold, which is specified by a symmetric Riemannian form, $\\text{Re}(\\langle \\phi |\\psi \\rangle )$ , and a skew-symmetric symplectic form, $\\text{Im}(\\langle \\phi |\\psi \\rangle )$ .",
"The symplectic form implies that a Hamiltonian dynamics exists and the space $\\mathcal {P}\\mathcal {H}$ can be viewed as a phase space.",
"For an orthonormal basis $\\lbrace |i\\rangle \\rbrace $ , a state $|\\psi \\rangle \\in \\mathcal {P}\\mathcal {H}$ is mapped to a set of coefficients $\\psi _i:=\\langle i|\\psi \\rangle $ .",
"Name the real part $q_i:= \\text{Re} (\\psi _i)$ as “position” and imaginary part $p_i:= \\text{Im} (\\psi _i)$ as “momentum” Note one can also use $\\psi _i$ and $\\pi _i:=i \\psi _i^*$ instead.",
"Position can be denoted by either $q$ or $x$ .",
"such that the normalization condition becomes $\\langle \\psi |\\psi \\rangle =\\sum _i |\\psi _i|^2=-i \\sum _i \\pi _i \\psi _i =\\sum _i (p_i^2+q_i^2) =1.$ As a result, the unitary evolution driven by a Hamiltonian $\\hat{H}$ of pure state $i |\\dot{\\psi }\\rangle =\\hat{H} |\\psi \\rangle $ can be equivalently written as Hamilton's equations $\\frac{\\partial H }{\\partial q_i } =-\\dot{p}_i, \\;\\frac{\\partial H }{\\partial p_i } = \\dot{q}_i, \\; \\forall i,$ with classical Hamiltonian (or energy) $H=\\langle \\psi |\\hat{H}|\\psi \\rangle $ The classical Hamiltonian is quadratic of the dynamical variables, there is no higher-order terms, which may appear for nonlinear modifications..",
"Note the above equations are equivalent to $\\frac{\\partial H }{\\partial \\psi _i } =-\\dot{\\pi }_i$ , $\\frac{\\partial H }{\\partial \\pi _i } = \\dot{\\psi }_i, \\; \\forall i.$ Also the Hamilton's equations hold for time-dependent Hamiltonian $\\hat{H}(t)$ .",
"This shows that the unitary dynamics of a quantum state $|\\psi \\rangle $ is equivalent to the Hamiltonian dynamics of a set of $d:=\\text{dim} \\mathcal {P}\\mathcal {H}$ coupled “particles” $(q_i,p_i)$ in phase space with the normalization condition (REF ).",
"The GQM above builds a close connection between QM and classical mechanics in phase space, which provides a hidden classical picture of quantum dynamics in terms of constrained Hamiltonian dynamics of coupled classical particles.",
"However, there also exist many other bases hence other collection of hidden particles dynamics, which are equivalent to each other via unitary basis transformations.",
"This is due to the extra Riemannian form for QM, which is absent for classical case, and related to the non-commutativity (or complementarity) of quantum operators.",
"This also implies that a quantum dynamics may arise from a set of Hamiltonian dynamics such that the Riemannian form is respected.",
"In order to construct a SAC simulation, the central problem is how many sets of Hamiltonian dynamics are inevitable.",
"It could be infinite, which turns out not to be necessary due to the geometry of the set of quantum states.",
"From the information theoretic viewpoint, especially quantum tomography [3], which is to reconstruct a quantum state or operation using finite number of operations, a finite number of bases is sufficient (as long as it is complete).",
"The co-existence hence co-simulation of Hamiltonian dynamics in different bases is a manifestation and also requirement of complementarity of general quantum operators.",
"Quantum tomography (QT), including quantum state tomography (QST) and quantum process tomography (QPT) [3], is a computation process that takes an unknown quantum object (state or process) as input and outputs its classical mathematical description, denoted by $[\\cdot ]$ .",
"For instance, QST is a map $\\mathcal {QST}: \\mathcal {P}\\mathcal {H}^{\\otimes n}\\rightarrow \\mathbb {R}^m : |\\psi \\rangle ^{\\otimes n}\\mapsto [\\psi ],$ for $n$ samples of the unknown state $|\\psi \\rangle $ and $[\\psi ]\\in \\mathbb {R}^m$ the description of it, with some integers $n,m\\in \\mathbb {Z}^+$ .",
"QPT is a generalization of QST that it requires to inject a complete set of input states $\\lbrace |\\phi _i\\rangle \\rbrace $ into the unknown process.",
"The quantum-to-classical part in QT is performed by measurement, represented by POVM $\\mathcal {M}:=\\lbrace M_i\\rbrace $ such that $\\sum _i M_i={1}$ and each $M_i$ corresponds to an operation $\\mathcal {M}_i:\\rho \\mapsto \\text{tr}(\\rho E_i) \\ge 0.$ For QST of a qudit, an informationally complete measurement can be realized by a projective measurement $\\mathcal {P}$ along an operator basis $\\lbrace |\\psi _i\\rangle \\langle \\psi _i|\\rbrace $ , with cardinality up to $d^2$ .",
"Another way is to measure a complete set of operators such that their eigenvectors contain an operator basis.",
"Further, QPT of a general qudit process requires in general $d^4$ measurement operations since a QST is needed for each of the $d^2$ inputs $\\lbrace |\\phi _i\\rangle \\rbrace $ .",
"Now, to build the SAC simulation, the simulator has to pass a verification test in the spirit of quantum prover interactive proof system [36] such that a verifier, for whom the simulator is a black box, cannot tell the simulator from the simulatee.",
"Here we employ QT as the verification scheme of our simulator, that given an input state the simulator is able to yield the correct output state.",
"Before we can build the simulator, we first need a SAC simulation of QT, described as follows.",
"Figure: Schematic illustrations for the simulations of QST.",
"(a) QST of an unknown state (the black boxes) by measurements ℳ i \\mathcal {M}_iwith results m i m_i that are processed into a classical computer to yield the information [ψ][\\psi ] of the unknown state.",
"(b) Quantum simulation of QST by starting from samples of the state |ψ〉|\\psi \\rangle .",
"(c) Classical digital simulation of QST with many copies of [ψ][\\psi ] and the information for each measurement [ℳ i ][\\mathcal {M}_i].",
"(d) SAC simulation of QST with fixed-basis states |ψ〉 i |\\psi \\rangle _iand classical measurements ℳ i \\mathcal {M}_i of position and momentum.The classification of simulation also applies to the simulation of QT, see Fig.",
"REF for the case of QST.",
"Note that QT, Fig.",
"REF (a), is to reveal an unknown object while a simulation of QT requires that the object is known.",
"First, a quantum simulation of QST is rather straightforward by simply performing the measurements on physical states, see Fig.",
"REF (b).",
"Also a classical (digital) simulation (Fig.",
"REF (c)) is to evaluate measurement results given the information of the input quantum state, that is, it realizes $[\\psi ]^{n}\\mapsto [\\psi ] $ , which trivially simulates the behavior of QST without computing unknown quantities.",
"However, SAC simulation (Fig.",
"REF (d)) requires the input can indeed represent the physical state.",
"This is done by using a set of fixed-basis classical states in different bases $\\lbrace \\mathcal {B}_\\alpha \\rbrace $ ; that is, for each basis $\\mathcal {B}_\\alpha $ , the initial input state can be represented by a classical system (see Eq.",
"(REF )), and each measurement in the same basis is simulated by a classical measurement on each classical system, which is simply to measure the position $q$ and momentum $p$ of each hidden particle Note it is reasonable to require that the classical measurement result needs to be directly extractable from the classical system, i.e., no other non-trivial computation is allowed during the measurement.. A precise scheme is as follows.",
"We use the method of measuring a complete set of operators to perform QST.",
"A convenient choice is a traceless orthonormal operator basis $\\lbrace \\sigma _i\\rbrace $ such that $\\sigma _0={1}$ , $\\text{tr}\\sigma _i=0$ , $\\text{tr}(\\sigma _i^\\dagger \\sigma _j)=d \\delta _{ij}$ .",
"For any state $\\rho =\\left( {1} + \\vec{n}\\cdot \\vec{\\sigma } \\right)/d$ with $\\vec{\\sigma }:=(\\sigma _i)$ and Bloch vector $\\vec{n}:=(n_i)$ for $n_i=\\text{tr}(\\sigma _i^\\dagger \\rho )$ and $|\\vec{n}|\\le \\sqrt{d-1}$ , the measurement of $\\lbrace \\sigma _i\\rbrace $ generates $\\vec{n}$ that determines $\\rho $ .",
"A well-known construction is the Heisenberg-Weyl (HW) basis $\\lbrace M_{jk}\\rbrace $ [38], [39], [40] $X_j=\\sum _{i=0}^{d-1}|i\\rangle \\langle i+j|, \\quad Z_k=\\sum _{l=0}^{d-1}\\omega ^{lk}|l\\rangle \\langle l| \\quad (\\mathrm {mod}\\; d),$ for $M_{jk}=X_jZ_k$ , and $\\omega =e^{i2\\pi /d}$ , $j, k\\in \\lbrace 0,\\dots , d-1\\rbrace $ .",
"The eigenvectors of each operator $M_{jk}$ provide a complete basis, denoted by $\\mathcal {B}_{jk}:=\\lbrace |b_i\\rangle _{jk},i\\in \\mathbb {Z}_d\\rbrace $ .",
"A nice property is that for each fixed $j$ the set of operators $M_{jk}$ have similar eigenvectors.",
"Given a state $|\\psi \\rangle $ , QST can be done by projective measurements along each basis $\\mathcal {B}_{jk}$ on different samples of the given state.",
"The SAC simulation of QST is to prepare up to $d^2$ classical systems, each in a basis $\\mathcal {B}_{jk}$ , the corresponding state is denoted by $|\\psi \\rangle _{jk}$ , and then each fixed-basis state $|\\psi \\rangle _{jk}$ is classically measured in the basis $\\mathcal {B}_{jk}$ , respectively.",
"The measurement results can then be programmed by a classical computer to yield $[\\psi ]$ , same with QST.",
"To simulate QPT, each input state $|\\phi \\rangle $ is first substituted by the set of fixed-basis state $\\lbrace |\\phi \\rangle _{jk}\\rbrace $ , and the SAC simulation of QST can be done on each input.",
"For $d^2$ input, one has to perform $d^4$ runs of the SAC simulation.",
"Now we can build the SAC simulator of a general quantum evolution based on the simulation of QT.",
"Given a simulatee, the algorithm to construct the simulator is specified by: the input is a unitary process $U$ specified by a Hamiltonian $\\hat{H}$ and time $t$ , and a verification scheme specified by QT (e.g., a complete set $\\lbrace |\\phi _i\\rangle \\rbrace $ for QPT and the set of bases $\\lbrace \\mathcal {B}_{jk}\\rbrace $ for QST), and the output is the simulator.",
"For the simulator the input states of the classical hidden particles can be obtained from QT, the set of Hamiltonian dynamics can be obtained from QT and $U$ .",
"To simulate an evolution $|\\psi \\rangle \\mapsto U|\\psi \\rangle $ , the simulator runs the set of complementary classical hidden systems under Hamiltonian dynamics with the normalization constraint, and yields the final state.",
"If QT is employed to verify the simulator, the simulation scheme of QT can ensure that our simulator passes the QT verification and serves a valid simulator.",
"A schematic diagram is shown in Fig.",
"REF .",
"Figure: SAC simulation of a unitary evolution U=e -itH ^ U=e^{-it\\hat{H}}.",
"(a) For an arbitrary initial state |ψ〉|\\psi \\rangle the final state is |ψ f 〉=U|ψ〉|\\psi ^f\\rangle =U|\\psi \\rangle .",
"(b) A single run of the SAC simulator is to input a fixed-basis state |ψ〉 jk |\\psi \\rangle _{jk},represented by the position and momentum (q i ,p i )(q_i,p_i) of dd hidden particles,and a constrained Hamiltonian dynamics with H=〈ψ|H ^|ψ〉H=\\langle \\psi | \\hat{H}|\\psi \\rangle generates final values of (q i ,p i )(q_i,p_i),which represents |ψ f 〉 jk |\\psi ^f\\rangle _{jk}.When QT verification is performed, many runs of the simulator as well as simulatee are required.Finally, the simulation accuracy can also be quantified.",
"The SAC simulator can be implemented by classical point particles moving in real space, and even also quantum particles (see subsection REF ).",
"In practice there could be many sources of simulation errors depending on the physical implementations.",
"This leads to the notion of approximate simulation with an accuracy.",
"In this context, the distinction between weak and strong simulations becomes apparent.",
"Given an object $O$ and a simulation accuracy bound $\\epsilon $ , strong simulation can be defined as $\\sup _i \\Vert O(i)-\\tilde{O}(i) \\Vert \\le \\epsilon ,$ while for weak simulation, the simulatee is a property of $O$ , denoted as $f_O$ , and the accuracy condition is $\\sup _i \\Vert f_O(i)-f_{\\tilde{O}}(i) \\Vert \\le \\epsilon ,$ for $\\Vert \\cdot \\Vert $ denoting a certain distance measure [16].",
"Also note that the supreme over all input ($i$ ) may not be necessary for some simulation tasks.",
"Now, for the SAC simulation of a unitary evolution the simulation error can be quantified by the distance on the final state.",
"A good simulator should yield approximate final state given arbitrary input state.",
"This is the operator norm distance $\\Vert U-\\tilde{U}\\Vert :=\\sup _{|\\psi \\rangle }\\Vert (U-\\tilde{U})|\\psi \\rangle \\Vert $ for $U$ as the quantum simulatee and $\\tilde{U}$ as the SAC simulator, and $|\\psi \\rangle $ as input state.",
"The simulator may have a different input state from the simulatee due to inaccurate initialization, which yet does not destroy the simulation.",
"If the distance on the initial state is upper bounded by $\\epsilon _0$ , and the distance on the evolution is upper bounded by $\\epsilon $ , then the total distance is upper bounded by $\\epsilon +\\epsilon _0$ , following simply from the property of operator norm."
],
[
"Simulation efficiency and examples",
"After establishing the primary framework of SAC simulation, in this section we further study another important issue: the simulation efficiency, and highlight the differences from other simulation methods by several practical examples.",
"We find that, different from other simulations, the SAC simulation efficiency depends on the notions of locality: whether it is the locality in real space (coordinate space), which is a classical notion, or it is the locality in Hilbert space, which is a genuine quantum notion.",
"Here locality in Hilbert space means that, given an order (e.g., by eigenvalues) on a basis $\\lbrace |i\\rangle \\rbrace $ of $\\mathcal {H}$ , and starting with $|i\\rangle $ , the dynamics acting on it only lead to a finite number of states close to it.",
"This notion has been implicitly used in the model of quantum Turing machine [28].",
"We term this type of locality as Hilbert locality for simplicity, different from the locality in the model of local Hamiltonian, which is still classical.",
"Please note that this is also not the same as the notion of nonlocality defined by Bell's inequality.",
"Before we proceed, let us first extend our simulation method based on Hamiltonian dynamics to discrete-time case.",
"In phase space, the dynamics specified by Eq.",
"(REF ) is symplectic, which preserves the area defined in phase space [41].",
"The Jacobian $J$ defined by $J_{ij}=\\frac{\\partial y_i}{\\partial \\tilde{y}_i}$ for $y_i$ denoting each input variables $p_1,\\dots ,p_n$ , $q_1,\\dots ,q_n$ , and $\\tilde{y}_i$ denoting each output variables $\\tilde{p}_1,\\dots ,\\tilde{p}_n$ , $\\tilde{q}_1,\\dots ,\\tilde{q}_n$ preserves the symplectic form $\\Delta =\\begin{pmatrix}0 & -{1} \\\\ {1} & 0\\end{pmatrix}$ such that $J\\Delta J^t= \\Delta $ .",
"In order to consider the simulation of circuits and algorithms in quantum computation, which often involve discrete-time execution of a sequence of gates, we also allow symplectic matrix $S\\in Sp(2n,\\mathbb {R})$ in our SAC simulation.",
"As both continuous-time and discrete-time evolution are common in reality, we allow both continuous-time and discrete-time types SAC simulation, similar with the case of quantum simulation [19], [14], [20], [21], e.g., there are both continuous-time and discrete-time quantum walks.",
"In the following we study several examples, including linear optics, quantum walk, local Hamiltonian evolution, stabilizer circuits, and matrix product states to reveal the main features of SAC simulation.",
"For simplicity, we ignore the verification part by tomography of the simulation and the study of simulation accuracy, while we focus on the efficiency of SAC simulations and differences with other simulation methods."
],
[
"Hilbert locality: Linear optical quantum computation",
"In this section we study a nontrivial setup that can be thought of as an efficient SAC simulator that benefits from Hilbert locality.",
"There are many different approaches for using linear (and also nonlinear) quantum optics for quantum computation, here we analyze the approach using the so-called dual-rail encoding [12], [13].",
"As illustrated in Fig.",
"REF , to simulate a unitary gate $U\\in SU(N)$ , $N$ paths (or called modes) are employed to represent the $N$ levels of the input and output system, and single photon can be injected into each mode.",
"The unitary $U$ can be built up by $O(N^2)$ beam splitters and phase shifters [42].",
"It's clear to see that if $N=2^m$ for $m$ as the number of qubits, the simulation of an $m$ -qubit unitary by such linear optics setup is not efficient.",
"The unitary gate $U$ is usually understood in terms of the mapping $U: (\\hat{a}^\\dagger _1,\\dots , \\hat{a}^\\dagger _N)\\mapsto (\\hat{b}^\\dagger _1,\\dots , \\hat{b}^\\dagger _N)$ for input (output) creation operator $\\hat{a}^\\dagger _i$ ($\\hat{b}^\\dagger _i$ ).",
"Note that the photon in each mode has trivial self-evolution, and the dynamics on all the photons are driven by external optical elements.",
"As $\\hat{a}^\\dagger _i=\\hat{q}_i+i \\hat{p}_i$ , let $U=V+iW$ , then it is equivalent to the symplectic matrix $S_U:=\\begin{pmatrix}V & -W \\\\ W & V\\end{pmatrix}\\in Sp(2N,\\mathbb {R})$ acting on $\\hat{q}_1,\\dots ,\\hat{q}_N,\\hat{p}_1,\\dots ,\\hat{p}_N$ .",
"This linear optics setting can perform small-scale efficient simulation of quantum circuits on qubits and also local Hamiltonian evolution [13].",
"The simulation is strong since it simulates the process itself instead of observable effects, and is classical in the sense that the dynamics can be understood in the phase space picture.",
"However, it is not totally analog, instead it is digital since it uses universal elements (beam splitter and phase shifter) to represent the simulatee.",
"Furthermore, the linear optics simulation is even not totally classical since the photon space is actually second-quantized.",
"That is, the space of the input photons is not $\\mathbb {C}^N=\\mathbb {C}\\oplus \\mathbb {C} \\oplus \\cdots \\oplus \\mathbb {C},$ instead, each mode $\\mathbb {C}$ is second-quantized to a Fock space $L_2(\\mathbb {R})$ , which is acted upon by $\\hat{q}_i$ and $\\hat{p}_i$ .",
"The unitary $U$ is equivalent to another unitary $\\mathbb {W}$ , the so-called metaplectic representation [43], acting on $L_2(\\mathbb {R}^N)$ .",
"This second-quantization feature goes beyond the phase space framework in this paper (yet, see subsection REF for a further analysis), which is still a classical space, and it benefits the linear optics simulation for other tasks, e.g., notably, the boson sampling algorithm [44].",
"However, we can still interpret the linear optics simulation using the dual-rail encoding as a SAC simulation if the second-quantization feature is simply dismissed, and the simulation of $U$ can also be understood as the simulation of a continuous-time evolution for $U=e^{-it\\hat{H}}$ with a Hamiltonian $\\hat{H}$ .",
"As a result, the linear optics setup serves as an example of SAC simulation in the generalized sense, and the photon in each mode can be viewed as the hidden particle in our terminology.",
"The merit of the optics setup is that the optical elements (beam splitters and phase shifters) are local operations in Hilbert space, hence efficient simulation can be built up as long as the total dimension $N$ does not scale exponentially with the problem size, e.g., the number of qubits $m$ of the simulatee."
],
[
"Hilbert locality: Discrete-time quantum walk",
"Besides linear optics, another model that employs the Hilbert locality is the discrete-time coined quantum walk [14], which has been proven to be a universal model for quantum computing.",
"It can be viewed as a simplified version of a quantum Turing machine [28] so that the state of the walker specifies the register tape and the state of the coin specifies the control processor.",
"Here we analyze the standard setting of quantum walk from the viewpoint of SAC simulation, we find that a SAC simulation of quantum walk is efficient.",
"In coined quantum walk, each step is specified by $U=S(H\\otimes {1})$ for Hadamard $H$ acting on the coin qubit and $S$ a uniformly-controlled shift operator $S=P_0\\otimes X^\\dagger + P_1\\otimes X$ with the coin as control and the walker as target, and the shift operator $X=\\sum _{x=-d}^d |x\\rangle \\langle x+1|$ is a generator of the Heisenberg-Weyl group (see Eq.",
"(REF )) and the walker space dimension is $2d+1$ .",
"Then the system (c$\\otimes $ w) starting from a product state $ |\\psi (0)\\rangle =|\\psi (0)\\rangle _c |0\\rangle _w$ evolves after $T$ (for $d\\ge T$ ) steps and yields $ |\\psi (T)\\rangle =(U)^T |\\psi (0)\\rangle .$ The mixing time evaluated from the final probability distribution $\\mathcal {P}(x)$ of the walker position is $O(T)$ , which shows a quadratic speedup than the classical random walk, which is $O(T^2)$ .",
"For SAC simulation of this model, indeed the linear optics setup discussed above can be employed, which has also been realized in practice.",
"The shift operator $X$ (REF ) is two-local in Hilbert space and the operator $S$ is also local, hence each step can be efficiently simulated, and the total simulation cost scales linearly with $T$ .",
"In particular, in SAC simulation the evolution of each mode $x(t)$ is primary and can reveal more information of the algorithm than the final probability distribution $\\mathcal {P}(x)$ .",
"In Fig.",
"REF we simulated the evolution of several modes in phase space for $d=T=100$ and a randomly chosen coin qubit state.",
"The phase space behavior shows that each mode (hidden particle) undergoes peculiar non-stationary dynamics, and the hidden particles interact with each other.",
"The particle at the center ($x=0$ ) decays in a choppy way to the `equilibrium point' $(0,0)$ in phase space, while the particle at $x=40$ starts to oscillate at a later time.",
"In other words, those particles form the media for the propagation of the input wave packet, which spreads out with a linear mixing time.",
"On the contrary, there is no such phase space dynamics and a set of hidden particles for the classical random walk model, which shows a fundamental difference between the classical and quantum cases.",
"However, the classical case can be achieved when there exists significant decoherence [45], and a generalized SAC simulation involving random bits can be developed (see Sec.",
"REF ), which would not be analyzed explicitly here.",
"Figure: The SAC simulation of coined quantum walk for step d=T=100d=T=100 in phase spacefor mode |0〉 c |x=0〉 w |0\\rangle _c|x=0\\rangle _w (a), |1〉 c |x=0〉 w |1\\rangle _c|x=0\\rangle _w (c),|0〉 c |x=40〉 w |0\\rangle _c|x=40\\rangle _w (b), and |1〉 c |x=40〉 w |1\\rangle _c|x=40\\rangle _w (d).The initial state is the product of a random |ψ(0)〉 c |\\psi (0)\\rangle _c and |0〉 w |0\\rangle _w.All other modes also show similar trajectories.Insert panel in (a) is a top view showing clearly the `equilibrium point' (0,0)(0,0),which is the same for all other modes."
],
[
"Separation between SAC simulation and quantum simulation",
"In this subsection we analyze the simulation scheme in the multipartite setting when the system contains several physically local subsystems.",
"This serves as an example that allows efficient quantum simulation [19] yet does not allow efficient SAC simulation, and also as an example with physical locality in the real space but not the Hilbert locality.",
"A quantum many-body system is usually described by a local Hamiltonian, which is a sum of local terms, each of which only acts on a small portion of the whole system.",
"To be precise, an $n$ -particle $k$ -local Hamiltonian $\\hat{H}=\\sum _{\\lambda =1}^{\\Lambda } \\hat{H}_{\\lambda }$ is a sum of terms $\\hat{H}_{\\lambda }$ which acts on at most $k$ particles, with $\\Lambda \\in O(n^k)$ .",
"A standard quantum simulation problem is to simulate the unitary evolution $U=e^{-it\\hat{H}}$ for a given time $t$ [19].",
"Now a SAC simulation of a local Hamiltonian evolution $U$ can be constructed as follows.",
"First, due to the non-commutability of the local terms, Trotter-Suzuki formula [46], [47] is employed to approximate $U$ by $\\tilde{U}=[U_{\\chi }(\\tau )]^r$ , with $t=r\\tau $ and $s=1/(4-4^{1/(2p-1)})$ , $1<p\\le \\chi $ , $U_1(\\tau )=\\prod _{\\lambda =1}^{\\Lambda } U_{\\lambda }(\\tau /2)\\prod ^1_{\\lambda =\\Lambda }U_{\\lambda }(\\tau /2),$ $U_p(\\tau )=\\left[U_{p-1}\\left(s\\tau \\right)\\right]^2U_{p-1}\\left((1-4s)\\tau \\right)\\left[U_{p-1}\\left(s\\tau \\right)\\right]^2,$ and $U_{\\lambda }(\\tau /2)=e^{-i\\hat{H}_{\\lambda }\\tau /2}$ .",
"The error of this approximation is the operator-norm distance $\\Vert U-\\tilde{U}\\Vert \\in O\\left(t^{2\\chi +1}/r^{2\\chi }\\right)$ , better than $O(t^2)$ if Trotter's formula is employed [46].",
"There is a tradeoff between the approximation error and the number of gates: the error decreases as $\\chi $ increases, while there is a number of gates exponential with $\\chi $ in the sequence.",
"As a result, in practice an optimal value of $\\chi $ can be chosen for one's purpose.",
"Then the given local Hamiltonian evolution $U$ is formally expressed as $U\\approx \\tilde{U}=U_1U_2\\cdots U_m$ with each unitary specified by a local term $U_i=e^{-it_i \\hat{H}_\\lambda }$ and a corresponding time interval $t_i$ .",
"Hence the simulation of $U$ can be reduced to the simulation of each $U_i$ , which can be simulated by a Hamiltonian dynamics given $t_i$ and $\\hat{H}_\\lambda $ .",
"Given a basis $\\lbrace |{\\bf i}\\rangle :=|i_1,i_2,\\dots ,i_n\\rangle \\rbrace $ of an $n$ -partite system, if each local dimension is $d$ , there will be $d^n$ pair of variables $(q_{{\\bf i}}=\\text{Re}(\\psi _{{\\bf i}}),p_{{\\bf i}}=\\text{Im}(\\psi _{{\\bf i}}))$ for a state $|\\psi \\rangle =\\sum _{{\\bf i}} \\psi _{{\\bf i}}|{\\bf i}\\rangle $ .",
"Note that here the position $q_{{\\bf i}}$ and momentum $p_{{\\bf i}}$ do not belong to any of the local particles.",
"That is, $(q_{{\\bf i}}, p_{{\\bf i}})$ are not variables of real particles, instead they describe hidden particles for each basis state $|{\\bf i}\\rangle $ of the whole Hilbert space.",
"Also all the hidden particles $(q_{{\\bf i}}, p_{{\\bf i}})$ participate in each local Hamiltonian dynamics driven by $H_\\lambda =\\langle \\psi | \\hat{H}_\\lambda |\\psi \\rangle $ .",
"This means that although each term $\\hat{H}_\\lambda $ is local in the sense that it acts on a limited number of locally connected particles, the action of $\\hat{H}_\\lambda $ is globally on the whole Hilbert space, hence on all hidden particles $(q_{{\\bf i}}, p_{{\\bf i}})$ .",
"Finally, the SAC simulation composes a sequence of Hamiltonian dynamics corresponding to the sequence (REF ).",
"We can see that although there are physically separable subsystems, the global entanglement among them, as well as the global action of each term $U_i=e^{-it_i \\hat{H}_\\lambda }$ on all hidden particles require the system to be treated as a whole.",
"This contrasts with classical systems that allow classical correlations but no entanglement, which is a quantum global constraint.",
"The dimension of the space of $n$ quantum particles grows exponentially with $n$ due to entanglement, while the dimension of the phase space of $n$ classical particles only grows linearly with $n$ ."
],
[
"Separation between SAC simulation and classical digital simulation",
"Further, we analyze two examples to illustrate the differences between SAC simulation and common simulation methods on classical (digital) computers.",
"We find that the previously claimed efficiencies do not hold for SAC simulation, which manifests that our SAC simulations assess the simulation cost in a more complete way.",
"Consider the computation on a matrix product state (MPS) [30] that takes the form $|\\Psi \\rangle =\\sum _{i_1\\cdots i_N} \\langle R| A(i_N)\\cdots A(i_1) |L\\rangle |i_1 \\cdots i_N\\rangle ,$ which can be prepared by an ancilla-driven sequential quantum circuit [48], see Fig.",
"REF .",
"Each set of local tensors $\\lbrace A(i_n)\\rbrace $ defines a channel $\\mathcal {E}_n$ acting on the ancilla, the dimension of which is called the bond dimension $\\chi $ , which can vary from site to site yet usually assumed as the largest one.",
"It is clear to see that the generation of an MPS on a quantum computer is efficient as long as the bond dimension does not scale exponentially with the number of qudits in the system.",
"Also the (digital) simulation on classical computer is efficient since it does not contain exponential number of parameters, instead, it is $O(\\chi ^2 dN)$ .",
"However, it turns out this generation process does not permit efficient SAC simulation.",
"Although each gate $U_n$ is local, namely, acting on the ancilla and one qudit, its effect is on all other qudits before it since they are entangled together, i.e., it has nonlocal effects in the Hilbert space.",
"Furthermore, it has been known that [29] a polynomial circuit with at most two-local gates on an MPS can be efficiently simulated on classical computers such that the bond dimension at each stage does not blow up.",
"The effects of gates can be efficiently updated on the local tensors, and the whole circuit can be described as a sequence of MPS with small bond dimensions.",
"On the contrary, for similar reasons, it is straightforward to see that the SAC simulation of such computation cannot be efficient since the local gates do not enjoy Hilbert locality.",
"Also it should be pointed out that classical simulation is usually weak simulation since often merely some dominant observable effects (after a process) are concerned.",
"With the MPS form, the properties of the system can be expressed in terms of the properties of the ancilla (e.g., the channels $\\lbrace \\mathcal {E}_n\\rbrace $ ).",
"But strong simulation of a process is not designed for particular observable effects [16], which is also a feature of SAC simulation and serves as a difference from classical digital simulation.",
"Besides the MPS formalism for efficient description of states with local structures, another widely studied formalism is the stabilizer states.",
"In quantum computing the Gottesman-Knill theorem [3] shows that stabilizer circuits can be efficiently simulated on classical computers.",
"The reason is that in a stabilizer circuit the state at each stage can be specified by its set of stabilizers, which is an efficient description, and then a quantum computational outcome can be simulated by analyzing the stabilizers.",
"In our notation, the description (or representation) $[\\psi ]$ of a stabilizer state $|\\psi \\rangle $ is efficient, while by comparison, a general state needs an exponential number of parameters for its description.",
"In the stabilizer formalism, even highly entangled states, such as cluster states [49] can be efficiently simulated.",
"In SAC simulation, however, it is not only required an efficient description, instead it requires to use classical systems to mimic or reproduce the actual quantum simulatee, which is much more stronger than the requirement of a description.",
"To simulate an $n$ -qubit stabilizer circuit, if there exists global entanglement on all the qubits, the simulation cannot be efficient.",
"For instance, consider the generation of a linear cluster state and action of gates from the Clifford group and Pauli observable measurements on it.",
"In the MPS form (REF ) a linear cluster state has bond dimension $\\chi =2$ and on-site tensors $H$ and $HZ$ , while in stabilizer form its stabilizers take the simple form $ZXZ$ , hence the digital simulation of the generation process is efficient, either due to its small bond dimension or the simple stabilizer description.",
"However, generically as the cluster grows the number of terms in the expansion of the state grows exponentially [49], which means, for SAC simulation, exponential number of hidden particles are required.",
"Furthermore, for gates and measurements, the digital simulation is efficient since it is still within the stabilizer formalism, while SAC simulation is not efficient since Clifford gates on a local qubit do not possess Hilbert locality, similar with the case studied in subsection REF .",
"As a result, the two examples above demonstrate that SAC simulation is a more restrictive method than the common simulation methods on classical computers, and an efficiency gap between them exists.",
"In this section we further consider several extensions of the SAC simulation scheme, namely, to the cases of nonunitary evolution and infinite-dimensional system.",
"We find that these generalizations can be properly achieved."
],
[
"Mixed state and nonunitary evolution",
"The generalization to mixed state is straightforward, while different methods are available depending on various decompositions of mixed state.",
"In this subsection we provide two different methods.",
"First, if one interprets $\\rho $ as a convex mixture of several pure states $\\lbrace p_i, |\\psi _i\\rangle \\rbrace $ , then the only extra resource is a classical random number to generate the probability distribution $\\lbrace p_i\\rbrace $ , and then an average is taken over the set of evolution.",
"Along with the mixture form of states, the way to handle nonunitary evolution, described as completely positive trace-preserving mappings [50], i.e.",
"channels, is to employ dilation method to convert a given channel into a unitary evolution acting on a bigger space formed by the system and an ancilla with initial state $|0\\rangle $ such that $\\mathcal {E}(\\rho ) \\mapsto \\text{tr}_\\text{A} \\left( U (\\rho \\otimes |0\\rangle \\langle 0|)U^\\dagger \\right),$ and $\\text{tr}_\\text{A}$ represents the trace of the ancilla, which can be realized by a projective measurement along an ancillary basis $\\lbrace |i\\rangle \\rbrace $ .",
"A mixed system input state $\\rho $ can be viewed as a mixture of several pure states $\\lbrace p_i,|\\psi _i\\rangle \\rbrace $ , hence the dynamics (REF ) can be simplified as a mixture of $\\mathcal {E}(|\\psi _i\\rangle )$ with pure state input.",
"The trace operation can also be simulated statistically: for each projection $|j\\rangle \\langle j|$ , the resulting system state is $K_j|\\psi _i\\rangle $ (up to renormalization) for Kraus operator $K_j:=\\langle j|U|0\\rangle $ with probability $q_{ij}:=\\langle \\psi _i|K_j^\\dagger K_j|\\psi _i\\rangle $ .",
"As a result, the SAC simulator can be constructed as a mixture, according to $\\lbrace p_i\\rbrace $ and $\\lbrace q_{ij}\\rbrace $ , of the simulation for each pure system state input and each projective operation on the ancilla.",
"The evolution $U$ can be simulated by Hamiltonian dynamics, and the simulation accuracy is quantified by distance on channels [21].",
"In addition, this simulation scheme also highlights the feature of quantum mechanics compared with classical mechanics and statistical mechanics: if a general quantum state $\\rho $ is treated as a mixture of pure states $|\\psi _i\\rangle $ with corresponding probabilities $p_i$ , probability theory captures the part of quantum dynamics specified by $p_i$ , ignoring the details of each state $|\\psi _i\\rangle $ , while classical mechanics captures the part of quantum dynamics of each state $|\\psi _i\\rangle $ , ignoring the statistics specified by $p_i$ , and quantum theory as a whole is a consistent combination of them.",
"Second, we propose another scheme that does not consume classical random numbers.",
"According to channel-state duality [50], a quantum channel $\\mathcal {E}$ can be equivalently represented as a state, the so-called Choi state $\\mathcal {C}=\\mathcal {E}\\otimes {1} (\\eta )$ , for $\\eta =|\\eta \\rangle \\langle \\eta |$ and $|\\eta \\rangle =\\sum _i |ii\\rangle $ .",
"For example, a unitary operator $U=(u_{ij})$ can be represented by a vector $|U\\rangle =\\sum _{ij}u_{ij}|ij\\rangle $ , which is actually the reshaping of $U$ .",
"Also a state $\\rho =(\\rho _{ij})$ is represented by $|\\rho \\rangle =\\sum _{ij}\\rho _{ij}|ij\\rangle $ .",
"The inner product of any two operators $A$ and $B$ is $\\text{tr}(A^\\dagger B)=\\langle A|B\\rangle $ .",
"Now Eq.",
"(REF ) can be equivalently written as $|\\rho \\rangle =(|\\eta \\rangle +\\sum _i n_i |\\sigma _i\\rangle )/d$ for $|\\eta \\rangle =|{1}\\rangle =|\\sigma _0\\rangle $ , and in general, $n_i\\in \\mathbb {C}$ , e.g., in HW basis.",
"This is the expansion of $|\\rho \\rangle $ in the basis $\\lbrace |\\sigma _i\\rangle \\rbrace $ .",
"Let $n_0=1$ , ${Q}_i:=\\text{Re}(n_i)$ , ${P}_i:=\\text{Im}(n_i)$ , then a state $\\rho $ can be treated as a set of $d^2$ classical hidden particles $({Q}_i, {P}_i)$ as in the pure state case.",
"Also now the normalization is $\\langle \\rho |\\rho \\rangle =\\text{tr}\\rho ^2=\\frac{1}{d}(1+|\\vec{n}|^2)=\\frac{1}{d} \\sum _i ({Q}_i^2 + {P}_i^2) .$ We can say that the mixed state case is “second order” and pure state case is “first order” due to their definitions of hidden particles and normalization conditions (higher-order states can also be constructed in the framework of superchannel [51]).",
"Now the unitary dynamics $i\\dot{\\rho }=[\\hat{H},\\rho ]$ is mapped to $i|\\dot{\\rho }\\rangle ={\\hat{H}}|\\rho \\rangle $ for Hamiltonian ${\\hat{H}}:=\\hat{H}\\otimes {1}-{1}\\otimes \\hat{H}^*$ .",
"The unitary evolution $U=e^{-it\\hat{H}}$ is mapped to ${U}=U\\otimes U^*$ .",
"Note the above forms reduce to the case when $\\rho $ is pure: a state $|\\psi \\rangle \\langle \\psi |$ is mapped to $|\\psi \\rangle |\\psi ^*\\rangle $ , and evolution (REF ) reduces to Eq.",
"(REF ).",
"Defining the classical Hamiltonian as ${H}=\\langle \\rho |{\\hat{H}}|\\rho \\rangle $ , now it is direct to find that Eq.",
"(REF ) can be written as $\\frac{\\partial {H} }{\\partial {Q}_i } =-\\dot{{P}}_i, \\;\\frac{\\partial {H} }{\\partial {P}_i } = \\dot{{Q}}_i, \\; \\forall i,$ as a generalization of (REF ).",
"There is a crucial difference between them: the $d^2$ hidden particles are coupled together for the mixed state case (REF ), and the variables $({Q}_i, {P}_i)$ are “second order,” while only $d$ hidden particles are coupled for the pure state case (REF ) but $d$ of them are needed to form a complementary complete set, so in total $d^2$ , and the variables $(q_i, p_i)$ are “first order.” A SAC simulator for the mixed state case can be built straightforwardly.",
"Next, for nonunitary evolution we consider Lindblad equation [52] $i \\dot{\\rho }=\\mathcal {\\hat{L}}\\rho $ for $\\mathcal {L}\\rho =[\\hat{H},\\rho ]+i\\sum _i\\gamma _i (L_i \\rho L_i^\\dagger -\\frac{1}{2}L_i^\\dagger L_i\\rho -\\frac{1}{2}\\rho L_i^\\dagger L_i)$ , which can be equivalently written as $i|\\dot{\\rho }\\rangle ={\\hat{L}}|\\rho \\rangle $ for an effective Hamiltonian ${\\hat{L}}:={\\hat{H}}+ i\\sum _i\\gamma _i ( L_i\\otimes L_i^*- \\frac{1}{2} L_i^\\dagger L_i \\otimes {1}- \\frac{1}{2} {1} \\otimes L_i^t L_i^* ).$ Its classical version can be defined as ${L}=\\langle \\rho |\\hat{{L}}|\\rho \\rangle $ , which still drives a Hamiltonian dynamics, similar with (REF ), but now the normalization (REF ) becomes time-dependent since the evolution changes the Bloch vector length $|\\vec{n}|$ .",
"By comparison, the first scheme, “mixture plus dilation,” requires a larger space hence the trace operation and classical bits, but the evolution can be simulated the same as pure state unitary evolution.",
"While the “second order” vector method does not require a larger space, but the evolution becomes more complicated, e.g., the normalization condition can be time-dependent.",
"This shows a tradeoff between them."
],
[
"Infinite dimensional cases",
"Finally, we show that the simulation scheme can also be generalized to infinite dimensional cases.",
"As expected, the dynamics is not for a collection of discrete particles, instead it is for a field.",
"The finite case above can be viewed as a discretization of a field.",
"A state $|\\psi \\rangle \\in L_2(\\mathbb {R})$ can be expanded in the position basis $\\lbrace |x\\rangle \\rbrace $ or momentum basis $\\lbrace |p\\rangle \\rbrace $ for $x,p\\in \\mathbb {R}$ as $|\\psi \\rangle = \\int dx \\psi (x) |x\\rangle $ or $|\\psi \\rangle = \\int dp \\phi (p) |p\\rangle $ .",
"The unitary evolution now becomes $i \\dot{\\psi }(x)= H(x) \\psi (x),$ for $H(x)=\\langle x|\\hat{H}|x\\rangle $ .",
"In the position basis, the corresponding “momentum” variable of $\\psi (x)$ is $\\pi (x)\\equiv i\\psi ^*(x)$ , also we can use $\\text{Re}\\psi (x)$ and $\\text{Im} \\psi (x)$ as in the discrete case.",
"With energy $H=\\int dx \\psi ^*(x) H(x) \\psi (x)$ , the Hamilton dynamics is $\\frac{\\partial H }{\\partial \\psi (x) } =-\\dot{\\pi }(x),\\frac{\\partial H }{\\partial \\pi (x) } = \\dot{\\psi }(x).$ Different from a true classical field dynamics, here both the dynamics of $\\psi (x)$ and $\\phi (p)$ are needed, which necessarily leads to the uncertainty relation on position operator $\\hat{x}$ and momentum operator $\\hat{p}$ .",
"Further, the information contained by $\\psi (x)$ and $\\phi (p)$ is just equivalent to the well-known Wigner function $W(x,p)$ .",
"A SAC simulator can be simply built by the Hamiltonian dynamics for both $\\psi (x)$ and $\\phi (p)$ , and tomography can be performed by technique to measure Wigner function.",
"This scheme can be further generalized to cases when there are both discrete and continuous degree of freedoms.",
"The Hamilton's equations then describe the dynamics of several coupled fields $\\frac{\\partial H }{\\partial \\psi _i({\\bf x}) } =-\\dot{\\pi }_i({\\bf x}),\\frac{\\partial H }{\\partial \\pi _i({\\bf x}) } = \\dot{\\psi }_i({\\bf x})$ for energy $H=\\sum _{i=1}^N\\int d{\\bf x} \\psi ^*_i({\\bf x}) H_{ij}({\\bf x}) \\psi _j({\\bf x}),$ and ${\\bf x}=x_1\\cdots x_N$ , $H_{ij}({\\bf x})=\\langle i|\\langle {\\bf x}| \\hat{\\mathbb {H}}|{\\bf x}\\rangle |j\\rangle $ , and a Hamiltonian operator $\\hat{\\mathbb {H}}$ acting on space $L_2(\\mathbb {R}^N)$ .",
"In fact, this can be employed to simulate the evolution of photons in the linear optics setup studied in subsection REF when the second-quantization feature of photons are considered.",
"The metaplectic representation $\\mathbb {W}$ relates to $\\hat{\\mathbb {H}}$ such that $\\mathbb {W}=e^{-it \\hat{\\mathbb {H}}}.$ This shows that a second-quantized system can also be described in phase space.",
"However, a further analysis of nonunitary dynamics, e.g., Gaussian channels, would go beyond the current scope of this work."
],
[
"Conclusion",
"In this work two central problems have been studied: first, whether it is possible to simulate quantum evolution by classical means in a stronger sense, which is formalized as strong analog classical (SAC) simulation; second, whether such simulation can be efficient for specific quantum processes.",
"Our study shows that indeed quantum coherent dynamics can be described in phase space, which is a standard framework for classical mechanics, hence the quantum-classical distinction can be revealed by SAC simulation.",
"More important issue is efficiency, which is a central concept in computer science rather than physics, and we find that the locality in Hilbert space serves as a sufficient condition for efficient SAC simulation.",
"We have constructed a SAC simulation scheme mainly for continuous-time Hamiltonian evolution with verification requirement and analysis of simulation accuracy.",
"The scheme is generalizable to cover cases of discrete-time evolution, nonunitary evolution, and infinite-dimensional systems.",
"Our simulation shows that quantum dynamics can be treated as a set of complicated, but geometrically concise, coupled classical dynamics; roughly, a bunch of complementary strings (set of particles for the discrete case) driven by Hamiltonian dynamics.",
"Compared with other frameworks, e.g., the theory of contextuality [31], [32] or entanglement [1], our approach is dynamical rather than kinematical or algebraic.",
"However, our studies do not intend to make any connection with the hidden variable theories, instead our simulation is based on geometric quantum mechanics, which has been a novel approach for many studies including geometric phase.",
"The study of simulation efficiency manifests the role of locality in quantum coherent dynamics.",
"Two distinct notions of locality can be separated by the SAC simulation: dynamics with locality in real space cause problems for efficient SAC simulations, while dynamics with locality in Hilbert space permit efficient SAC simulations.",
"The locality in real space is more common in physics, yet the Hilbert locality is less understood.",
"The examples of linear optical quantum computing and quantum walk demonstrate the central roles of Hilbert locality in quantum computing and quantum algorithms, which may benefit the designs of new quantum algorithms.",
"As well, at present it is not clear whether Hilbert locality is necessary for SAC simulation.",
"The relation between these two kinds of localities is also a worthy topic by itself.",
"We also made the efficiency separation between the SAC simulation and other simulation methods, including quantum simulation and classical digital simulation.",
"These comparison shows that the SAC simulation accounts for simulation costs more completely, hence revealing the quantum-classical distinction more faithfully.",
"Efficiency of classical simulations depends on the simulatee and the involved simulation methods.",
"In the spirit of resource theory, our study highlights the role of quantum entanglement, especially global entanglement, due to which a multipartite quantum system is preferred to be treated as a whole instead of separable parts.",
"As well, our results also trigger the following question: can the simulator be treated as a model of the true physical reality?",
"This should be examined with care since, e.g., the concept of spin does not really mean a particle is spinning.",
"However, this quests further investigation to understand what quantum coherence actually is.",
"In addition, it would also be interesting and important to explore other related problems, such as other verification methods besides tomography.",
"The author would like to thank R. Cockett, P. Høyer, and B. Sanders for the discussions on the notion of simulation at an early stage of this work, and R. Raussendorf for comments.",
"Funding support from NSERC of Canada and a research fellowship at Department of Physics and Astronomy, University of British Columbia are acknowledged."
]
] | 1606.04998 |
[
[
"Supergravity from D0-brane Quantum Mechanics"
],
[
"Abstract The gauge/gravity duality conjecture claims the equivalence between gauge theory and superstring/M-theory.",
"In particular, the one-dimensional gauge theory of D0-branes and type IIA string theory should agree on properties of hot black holes.",
"Type IIA superstring theory predicts the leading $N^2$ behavior of the black hole internal energy to be $E/N^2=a_0T^{14/5}+ a_1T^{23/5}+a_2T^{29/5}+\\cdots$ with the supergravity prediction $a_0=7.41$ and unknown coefficients $a_1$, $a_2$, $\\ldots$ associated with stringy corrections.",
"In order to test this duality we perform a lattice study of the gauge theory and extract a continuum, large-$N$ value of $a_0=7.4\\pm 0.5$---the first direct confirmation of the supergravity prediction at finite temperature---and constrain the stringy corrections ($a_1=-9.7\\pm2.2$ and $a_2=5.6\\pm1.8$).",
"We also study the sub-leading $1/N^2$ corrections to the internal energy."
],
[
"=1 Supergravity from D0-brane Quantum Mechanics E. Berkowitz Nuclear and Chemical Sciences Division, Lawrence Livermore National Laboratory, Livermore, California 94550, USA E. Rinaldi Nuclear and Chemical Sciences Division, Lawrence Livermore National Laboratory, Livermore, California 94550, USA M. Hanada Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305, USA Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan The Hakubi Center for Advanced Research, Kyoto University, Yoshida Ushinomiyacho, Sakyo-ku, Kyoto 606-8501, Japan G. Ishiki Center for Integrated Research in Fundamental Science and Engineering (CiRfSE), University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan S. Shimasaki Research and Education Center for Natural Sciences, Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan KEK Theory Center, High Energy Accelerator Research Organization, Tsukuba 305-0801, Japan P. Vranas Nuclear and Chemical Sciences Division, Lawrence Livermore National Laboratory, Livermore, California 94550, USA Monte Carlo String/M-theory Collaboration (MCSMC) LLNL-JRNL-694846, UTHEP-690, YITP-16-74 The gauge/gravity duality conjecture claims the equivalence between gauge theory and superstring/M-theory.",
"In particular, the one-dimensional gauge theory of D0-branes and type IIA string theory should agree on properties of hot black holes.",
"Type IIA superstring theory predicts the leading $N^2$ behavior of the black hole internal energy to be $E/N^2=a_0T^{14/5}+ a_1T^{23/5}+a_2T^{29/5}+\\cdots $ with the supergravity prediction $a_0=7.41$ and unknown coefficients $a_1$ , $a_2$ , $\\ldots $ associated with stringy corrections.",
"In order to test this duality we perform a lattice study of the gauge theory and extract a continuum, large-$N$ value of $a_0=7.4\\pm 0.5$ —the first direct confirmation of the supergravity prediction at finite temperature—and constrain the stringy corrections ($a_1 = −9.7\\pm 2.2$ and $a_2=5.6\\pm 1.8$ ).",
"We also study the sub-leading $1/N^2$ corrections to the internal energy.",
"Introduction: The gauge/gravity duality conjecture [1] has played a central role in theoretical high energy physics for almost two decades.",
"If the duality is correct, then superstring theory is described by manifestly unitary supersymmetric gauge theories, which provide us with an important key to solve the black hole information loss paradox.",
"Furthermore, the duality can translate hard problems in strongly coupled field theories in the large-$N$ limit to easier, classical gravity problems.",
"Given such interesting consequences, it is of crucial importance to provide evidence that this duality holds, by explicitly solving the gauge theory in a regime where non-perturbative effects are dominant.",
"In a dynamical setup, e.g.",
"at finite temperature, this is extremely difficult.",
"It is well known that Monte Carlo calculations, analogous to the ones of lattice QCD, are the best tool which can accomplish this task, and provide accurate and improvable results.",
"Historically, it had been widely believed that the Monte Carlo approach does not work for supersymmetric gauge theories.",
"The situation has changed in the last fifteen years; various supersymmetric theories relevant for the gauge/gravity duality can now be studied [2].",
"However, the calculation can be very expensive.",
"In this paper we concentrate on the gauge theory of D0-brane quantum mechanics [3], [4], [5], [6]—this is still expensive, but it is possible to take the continuum and large-$N$ limit using state-of-the-art simulation techniques and supercomputers.",
"D0-brane quantum mechanics is defined on a Euclidean circle with circumference $\\beta $ .",
"With antiperiodic boundary conditions for the fermions and periodic boundary conditions for the bosons, $\\beta $ is identified with the inverse temperature $1/T$ of the system.",
"This model consists of nine $N\\times N$ bosonic hermitian matrices $X_M$ ($M=1,2,\\cdots ,9$ ), sixteen fermionic matrices $\\psi _\\alpha $ ($\\alpha =1,2,\\cdots ,16$ ) and the gauge field $A_t$ .",
"Both $X_M$ and $\\psi _\\alpha $ are in the adjoint representation of the $U(N)$ gauge group.",
"The covariant derivative acts as $D_t\\cdot = \\partial _t \\cdot +i[A_t,\\cdot ]$ .",
"The continuum Euclidean action is given by $S=S_b+S_f$ , where the bosonic part $S_b$ and the fermionic part $S_f$ are given by Sb = N0dt $\\text{Tr}$ { 12(Dt XM)2 - 14[XM,XN]2 }, Sf = N0dt $\\text{Tr}$ { i10Dt- M[XM,] }.",
"while $\\gamma ^M$ ($M=1,\\cdots ,10$ ) are $16\\times 16$ the left-handed part of the (9+1)-dimensional gamma matrices.",
"This model is obtained by dimensionally reducing the ten-dimensional ${\\cal N}=1$ super Yang–Mills theory to one dimension.",
"The index $\\alpha $ of the fermionic matrices $\\psi _\\alpha $ corresponds to the spinor index in ten dimension, and $\\psi _\\alpha $ is Majorana-Weyl in the ten-dimensional sense.",
"The 't Hooft coupling $\\lambda $ is related to the Yang–Mills coupling by $\\lambda = g_{YM}^2N$ .",
"It has the dimension of $({\\rm mass})^3$ , and sets the typical energy scale of the theory.",
"All dimensionful quantities are measured in units of $\\lambda $ —the dimensionless effective temperature and internal energy are $\\lambda ^{-1/3}T$ and $\\lambda ^{-1/3}E$ , respectively.",
"The 't Hooft limit is $N\\rightarrow \\infty $ with $\\lambda ^{-1/3}T$ fixed, and $\\lambda ^{-1/3}E$ scales as $N^2$ there.",
"In the following, we set $\\lambda =1$ for simplicity and without loss of generality.",
"According to the gauge/gravity duality conjecture, the internal energy in D0-brane quantum mechanics should agree with the mass of the black zero-brane in type IIA superstring theory [6] $\\frac{E}{N^2}=\\frac{a_0 T^{14/5} + a_1 T^{23/5} + a_2 T^{29/5} + \\cdots }{N^0}+ \\frac{b_0 T^{2/5} + b_1 T^{11/5}+\\cdots }{N^2}+ \\mathcal {O}(\\frac{1}{N^4})=\\frac{E_0(T)}{N^0}+ \\frac{E_1(T)}{N^2}+ \\mathcal {O}(\\frac{1}{N^4}).$ The leading term $a_0 T^{14/5}$ , with $a_0=7.41$ , is determined by supergravity.",
"Other terms are stringy $\\alpha ^{\\prime }$ - and $g_s$ -corrections due to finite string length and virtual string loops, respectively, with $\\alpha ^{\\prime }\\sim T^{3/5}$ and $g_s\\sim N^{-2}T^{-21/5}$ .",
"The first term in the $O(N^{-2})$ sector is known to be $b_0=-5.77$ based on an analytic study [13].",
"D0-brane quantum mechanics has been investigated using Monte Carlo methods starting with Ref. [9].",
"Although existing results suggest a consistency with those expected from the supergravity, the simulations used in these previous tests of the duality were not extrapolated to the continuum limit and the $N \\rightarrow \\infty $ limit—both these limits are of paramount importance to confirm the duality.",
"In particular, the results were not precise enough to confirm the coefficient $a_0=7.41$ predicted by supergravity (SUGRA).",
"In order to obtain this precision, the discretization errors and corrections due to finite $N$ need to be correctly estimated.",
"This is achieved for the first time in our study.",
"Moreover, the high accuracy of our large-scale numerical simulations allows us to robustly determine the first $\\alpha ^{\\prime }$ correction, resolving a slight tension present in previous studies, and to estimate quantum string corrections.",
"Lattice setup: To compute observables in D0-brane quantum mechanics using the path integral formulation, we discretize the $0+1$ -dimensional spacetime on a linear lattice with $L$ sites.",
"The length of the circle is $\\beta =aL$ , where $a$ is the lattice spacing.",
"For numerical efficiency, we adopt the static diagonal gauge [7], $A_t=\\frac{1}{\\beta }\\cdot {\\rm diag}(\\alpha _1,\\cdots ,\\alpha _N),\\qquad -\\pi <\\alpha _i\\le \\pi .$ The corresponding Faddeev-Popov term $S_{F.P.",
"}=- \\sum _{i<j}2\\log \\left|\\sin \\left(\\frac{\\alpha _i-\\alpha _j}{2}\\right)\\right|$ is added to the action to compensate for the gauge-fixing.",
"Our lattice action is $S_{F.P.",
"}+S_b+S_f$ where $S_b&=&\\frac{N}{2a}\\sum _{t,M}\\text{Tr}\\left\\lbrace \\left(D_+X_M(t)\\right)^2\\right\\rbrace \\nonumber \\\\&&-\\frac{Na}{4}\\sum _{t,M,N}\\text{Tr}\\left\\lbrace [X_M(t),X_N(t)]^2 \\right\\rbrace ,$ $S_f&=&\\sum _{t}\\text{Tr}\\Bigl \\lbrace iN\\bar{\\psi }(t)\\left(\\begin{array}{cc}0 & D_+\\\\D_- & 0\\end{array}\\right)\\psi (t)\\nonumber \\\\&&-aN\\sum _{t,M}\\bar{\\psi }(t)\\gamma ^M[X_M(t),\\psi (t)] \\Bigl \\rbrace ,$ where the $t$ -independent gauge links are $U=\\exp (i a A_t)$ .",
"We improve the covariant lattice derivative so that it is related to the derivative in the continuum theory by $D_\\pm \\psi (t)= aD_t\\psi (t) +\\mathcal {O}(a^3)$ [17].",
"At finite lattice spacing the theory loses most of its symmetries.",
"In particular, supersymmetry is broken by the finite lattice spacing and by the boundary conditions, but it is recovered in the continuum limit.",
"We simulate this theory with the RHMC algorithm with MPI parallelization [8].",
"We have neglected the complex phase of the Pfaffian, by using the phase-quenched approximation.",
"For an argument justifying this procedure, see the longer companion paper Ref. [17].",
"We have studied $N=16,24$ and 32, $T=0.4$ to $1.0$ in steps of $0.1$ , with lattice size $L=8, 12, 16, 24$ and 32.",
"At each $T$ , we performed two kinds of extrapolations, (i) $L\\rightarrow \\infty $ first via a quadratic extrapolation in $L^{-1}$ and a subsequent $N\\rightarrow \\infty $ via a linear extrapolation in $N^{-2}$ , and (ii) $L\\rightarrow \\infty $ and $N\\rightarrow \\infty $ together, via $\\frac{E}{N^2}= e_{00} + \\frac{e_{01}}{L} + \\frac{e_{02}}{L^2} + \\frac{e_{10}}{N^2}.$ The central values of the continuum large-$N$ energy $e_{00}$ are consistent across the two procedures.",
"An example extrapolation is shown in Fig.",
"REF .",
"Henceforth we discuss the results of (ii), which has systematically smaller uncertainties.",
"For more details about the lattice setup and the continuum large-$N$ extrapolations see Ref. [17].",
"Figure: A simultaneous continuum- and large-NN extrapolation for T=0.5T=0.5 via the surface given by Eq.",
"().In the right panel, we show all the data points and the N=∞N=\\infty slice of the best-fit surface.The black diamond represents the continuum and large-NN corner of the best-fit surface, e 00 e_{00}.The black circle is the result of first performing a continuum extrapolation at each NN followed by an extrapolation to large-NN.The continuum extrapolations at each NN are shown as black symbols in the left panel.We show the large-NN extrapolation of those values as a dashed line with dotted bands.We also show fixed-LL slices of the best-fit surface as solid lines with error bands.Large-$N$ results: The $N=\\infty , L=\\infty $ extrapolated values of the energy $e_{00}$ coming from the fit of the measurements of $E/N^2$ at fixed temperature to Eq.",
"(REF ) are shown in Fig.",
"REF .",
"The $1/N^2$ corrections in the continuum limit $e_{10}$ are also obtained from the fit.",
"Our results are the first of this kind: no other numerical study of D0-brane quantum mechanics has ever computed both the continuum limit and the large-$N$ limit.",
"Figure: Our large-NN continuum data e 00 e_{00} are shown as black diamonds.The solid blue/dotted cyan lines are different fit forms for E 0 (T)E_0(T) described in the text.We also show the results from Ref. /Ref.",
"as red dot-dashed/green dashed line.The SUGRA result is in black.To test the gauge/gravity correspondence of D0-brane quantum mechanics and supergravity, we want to be able to reproduce the analytical expectation for the leading order term $E_0(T)$ in Eq.",
"(REF ).",
"The high accuracy of our extrapolated results $e_{00}$ at several temperatures allows us to perform this test with great precision.",
"We do a three-parameter fit to $e_{00}$ using $E_0(T)=a_0T^{14/5}+a_1T^{23/5}+a_2T^{29/5}$ , scanning different ranges of temperature $0.4\\le T\\le 0.9$ .",
"Our best fit includes all the data points.",
"We obtain $a_0=7.4\\pm 0.5$ , $a_1=-9.7\\pm 2.2$ and $a_2=5.6\\pm 1.8$ with $\\chi ^2/{\\rm DOF}=2.6/3$ .",
"This result very nicely matches the dual gravity theory expectation $a_0=7.41$ and has a very small uncertainty of about $7\\%$ .",
"To test the stability of our fit procedure, we set $a_0$ to 7.41 (its known value) and perform a two-parameter fit to $a_1$ and $a_2$ .",
"We obtain $a_1=-10.0\\pm 0.4$ and $a_2=5.8\\pm 0.5$ , in perfect agreement with the previous fit, increasing our confidence in those results.",
"These values are also consistent with results of a similar fit at finite-$N$ [12].",
"In order to compare with existing results for $a_1$ , we perform a different fit based on the function $E_0(T)=7.41T^{14/5}+a_1T^{p_1}$ .",
"This would allow us to also predict the next-to-leading temperature behavior $p_1$ , which is expected to be $p_1=23/5=4.6$ .",
"Previously, two results at finite $N$ and without a continuum limit provided slightly different values by fitting to this form: $a_1=-5.55(7)$ , $p_1=4.58(3)$ [10] and $a_1=-9(2)$ , $p_1=4.7(3)$ [11].",
"With our data, we cannot fit that form successfully.",
"However, a fit to $E_0(T)=7.41T^{14/5}+a_1T^{p_1}+a_2T^{p_1+6/5}$ , as motivated from string theory, produces $a_1=-10.2\\pm 2.4$ , $a_2=6.2\\pm 2.6$ and $p_1=4.6\\pm 0.3$ .",
"Our results indicate that the previous 2$\\sigma $ tension on $a_1$ arose because the next $\\alpha ^{\\prime }$ correction was not taken into account at temperatures where it is important.",
"$1/N^2$ correction: We also consider the corrections of order $1/N^2$ to the internal energy, which correspond to the quantum effects arising from virtual loops of strings.",
"The dual gravity calculation predicts the $1/N^2$ functional form to be $E_1(T)=b_0T^{2/5}+b_1T^{11/5}+\\cdots $ [13], with $b_0=-5.77$ .",
"The first term should become dominant at very low temperature.",
"This regime $T<0.1$ has been studied with small $N=3,4,5$ finding good agreement with the gravity prediction [14].",
"Our results at $N=\\infty $ , where the continuum $1/N^2$ correction ($e_{10}$ in Eq.",
"(REF )) is extracted directly from the lattice data, are shown as black diamonds in Fig.",
"REF .",
"We also show the result of a two-parameter fit $b_0T^{2/5}+b_1T^{11/5}$ and a fit of $b_1$ with $b_0$ fixed to its known value.",
"Although the data is not good enough to extract precision values, a general consistency can be observed.",
"Figure: Two fits of E 1 (T)E_1(T) to our values for e 10 e_{10}.We show our measurements as black diamonds with 1σ\\sigma error bars.A fit with fixed/free b 0 b_0 is shown as a dotted blue/dashed cyan line with a 1σ\\sigma error band.The two curves lie on top of one another.The known low-temperature behavior b 0 =-5.77b_0=-5.77 is shown as a black solid line.The one-parameter fit with b 0 b_0 fixed to be -5.77-5.77 gives b 1 =-3.5±2.0b_1=-3.5 \\pm 2.0, while a two-parameter fit gives b 0 =-5.8±3.0b_0=-5.8 \\pm 3.0 and b 1 =-3.4±5.7b_1=-3.4 \\pm 5.7.Discussion: The current data suggests that the string $\\alpha ^{\\prime }$ corrections become negligible at $T\\lesssim 0.3$ , so that the leading supergravity part, including the power $14/5$ , can be determined there.",
"In this parameter region, the problem of the flat direction becomes more severe [9], [14], but it should be possible to overcome this difficulty by going to very large $N$ with large-scale parallel simulations.",
"Note also that by further improving the precision at $T\\gtrsim 0.5$ , an accurate test of the quantum ($1/N^2$ ) string correction is possible, though the low-temperature region studied in Ref.",
"[14] may be more cost-effective.",
"An even more interesting direction is the study of M-theory.",
"The black zero-brane is expected to turn to the Schwarzschild black hole in M-theory at very low temperatures, where the temperature scales as a negative power of $N$ [5], [6].",
"Also, the plane-wave matrix model [15], which is a supersymmetric deformation of the D0-brane quantum mechanics, is conjectured to describe $M2$ - and $M5$ -branes [16].",
"Large-scale lattice simulation is the only practical tool to verify these conjectures and reveal the dynamical properties of M-theory.",
"We believe that strengthening the connection between string theory and lattice gauge theory is an important key for furthering the study of superstring/M-theory and quantum gravity.",
"Acknowledgments: Numerical calculations were performed on the Vulcan BlueGene/Q at LLNL, supported by the LLNL Multiprogrammatic and Institutional Computing program through a Tier 1 Grand Challenge award, and on the RIKEN K supercomputer.",
"E.B., E.R., and P.V.",
"acknowledge the support of the DOE under contract DE-AC52-07NA27344 (LLNL).",
"The work of M.H.",
"is supported in part by the Grant-in-Aid of the Japanese Ministry of Education, Sciences and Technology, Sports and Culture (MEXT) for Scientific Research (No.",
"25287046).",
"The work of G.I.",
"was supported, in part, by Program to Disseminate Tenure Tracking System, MEXT, Japan and by KAKENHI (16K17679).",
"S.S. was supported by the MEXT-Supported Program for the Strategic Research Foundation at Private Universities “Topological Science” (Grant No.",
"S1511006)."
]
] | 1606.04948 |
[
[
"Distributed Beam Scheduling for Multi-RAT Coexistence in mm-Wave 5G\n Networks"
],
[
"Abstract Millimetre-wave communication (licensed or unlicensed) is envisaged to be an important part of the fifth generation (5G) multi-RAT ecosystem.",
"In this paper, we consider the spectrum bands shared by 5G cellular base stations and some existing networks, such as WiGig.",
"Sharing the same band among such multiple radio access technologies (RATs) is very challenging due to the lack of centralized coordination and demands novel and efficient interference mitigation and coexistence mechanisms to reduce the mutual interference.",
"To address this important challenge, we propose in this paper a novel multi-RAT coexistence mechanism where neighbouring 5G and WiGig base stations, each serving their own associated UEs, schedule their beam configurations in a distributed manner such that their own utility function, e.g.",
"spectral efficiency, is maximized.",
"We formulate the problem as a combinatorial optimization problem and show via simulations that our proposed distributed algorithms yield a comparable spectral efficiency for the entire networks as that using an exhaustive search, which requires global coordination among coexisting RATs and also has a much higher algorithmic complexity."
],
[
"Introduction",
"One of the primary contributors to global mobile traffic growth is the increasing number of wireless devices that are accessing mobile networks.",
"Over half a billion (526 million) mobile devices and connections were added in 2013 and the overall mobile data traffic is expected to grow to 15.9 exabytes per month by 2018, nearly an 11-fold increase over 2013 [1].",
"In order to address this issue, a recent trend in 3GPP is to utilize both the licensed and unlicensed spectrum simultaneously for extending available system bandwidth.",
"In this context, LTE in unlicensed spectrum, referred to as LTE-U, is proposed to enable mobile operators to offload data traffic onto unlicensed frequencies more efficiently and effectively, and provides high performance and seamless user experience [2].",
"Integration of unlicensed bands is also considered as one of the key enablers for 5G cellular systems [3].",
"However, unlike the typical operation in licensed bands, where operating base stations (BS) have exclusive access to spectrum and therefore are able to coordinate by exchanging of signaling to mitigate mutual interference, such a multi-standard and multi-operator spectrum sharing scenario imposes significant challenges on coexistence in terms of interference mitigation.",
"Figure: Coexistence scenario in 60 GHz deployment with beamformingLicensed Assisted Access (LAA) with listen-before-talk (LBT) protocol has been proposed for the current coexistence mechanism of LTE-U [4].",
"For coexistence in 5G, one of the major issues is that the use of highly directional antennas as one of the key enablers for 5G networks [7], [8] becomes problematic for the current coexistence mechanisms where omni-directional antennas were mostly assumed.",
"For example, transmission by a different nearby 5G BS or WiGig access Point (AP) may not be detected due to the narrow beam that has been used, resulting a transmission that causes excessive interference to the victim user equipment (UE), e.g., UEs in the central area as illustrated in Fig.",
"REF .",
"In this regard, the simultaneous transmission should be coordinated and exploited fully to greatly enhance the network capacity.",
"Such a mechanism is referred as beam scheduling.",
"There has been some related work on this topic based on TDMA [9], [10] and a concept of exclusive region is introduced in [11] to enable concurrent transmission with significant interference reduction.",
"However, the effect of interference aggregation is not captured.",
"Other approaches are proposed based on centralized coordination in [12], [13], where the access points are coordinated in a centralized manner to reduce interferences and improve network capacity.",
"In this paper, we consider a multi-RAT deployment where 5G BSs and other 60GHz APs, e.g., WiGig APs, co-exist, all transmitting via beamforming.",
"We form a scheduled beam sequence containing the indices of the beams used at different time slots, and formulate an optimization problem to find the optimal scheduled beam sequence to maximize the spectral efficiency of the entire network.",
"It is known that such a combinatorial problem is NP-hard and highly computationally costly when using exhaustive search [14].",
"We therefore further propose a novel distributed learning algorithm where different BSs cooperatively and iteratively update the beam sequences such that near maximum spectral efficiency is achieved.",
"It is shown that the proposed algorithm almost achieves comparable spectral efficiency to that using the exhaustive search, while at the same time having a much reduced complexity and signaling overhead.",
"It should be noted that the proposed algorithm and analysis are general and can be easily applied to other bands, such as 28 GHz.",
"The rest of the paper is organized as follows.",
"System model will be introduced in the next section and the optimization problem is formulated in section III.",
"In section IV, we detail the proposed distributed learning algorithm and compare its complexity with that using exhaustive search and distributed greedy scheduling.",
"Simulation results are presented in section V and Section VI concludes the paper."
],
[
"System Model",
"In this paper, we assume that both 5G BSs and co-existing APs employ beamforming to tackle the increased path loss in 60 GHz band.",
"From now on, we do not differentiate 5G BS and co-existing AP and refer to them as 5G AP for simplicity.",
"We also assume that each 5G AP is only able to transmit data using a single beam at a time for simplicity and the mechanism considered here can be extended to the multi-beam case.",
"We assume a coexistence deployment scenario with $N$ 5G APs, which could either be 5G BSs or co-existing APs or a mixture of both, and $M$ associated UEs for every AP.",
"Each 5G AP has $N_t$ transmit antennas, whilst each UE has one receive antenna.",
"At a given time, the $n$ th ($n=1,\\dots , N$ ) 5G AP transmits to the $m$ th ($m=1,\\cdots , M$ ) UE using a beam $\\mathbf {w}_{nm}$ , where $\\mathbf {w}_{nm}$ is vector with length $N_t$ .",
"To obtain the beamforming vector $\\mathbf {w}_{nm}$ , we assume that the 5G AP selects the beam configuration within a predefined beam codebook with cardinality $C$ that uniformly covers the azimuth directions around the AP.",
"In particular, the codebooks at the transmitter are formed by vectors $\\lbrace \\mathbf {v}_{1}, \\cdots , \\mathbf {v}_{C}\\rbrace $ , with the $i$ th length-$N_t$ vector $\\mathbf {v}_i$ denoting the beam for the $i$ th codebook entry.",
"The $n$ th AP selects the $\\hat{i}$ th entry in the codebook according to $\\hat{i} = \\arg \\!\\max _{i=1,\\cdots , C}\\left|\\mathbf {v}_i^T\\mathbf {h}_{mn}\\right|^2$ and $\\mathbf {w}_{nm} = \\mathbf {v}_{\\hat{i}}.$ We define a scheduling cycle with duration of $M$ time slots.",
"Within each scheduling cycle, we consider scheduling the beams for $M$ UEs associated with a particular 5G AP, for example, the $n$ th AP.",
"Suppose at a given time slot $m$ ($m=1,\\cdots , M$ ), this AP is transmitting to only one of the UEs via one beam in a round robin manner, which could be any one of the beams from $\\mathbf {w}_{n1}, \\cdots , \\mathbf {w}_{nM}$ , indexed as beam $1, \\cdots , M$ .",
"During one scheduling cycle, the indices of the transmitted beams therefore form a beam sequence vector with a length $M$ , denoted as $\\mathbf {b}_n(t) = [b_{n1}(t), \\cdots , b_{nM}(t)]^T$ , where $b_{nm}(t)\\in [1,\\cdots , M]$ .",
"It is known that there are $\\prod _{m=1}^Mm = M!$ permutations of such beam sequences, whereas there may exist only one optimal sequence given particular network criterion.",
"This paper therefore aims at finding an optimal $\\mathbf {b}_n(t)$ for the $n$ th 5G AP, such that certain utility function is optimized in every scheduling cycle.",
"In particular, we consider using the spectral efficiency as the utility function and aim to find an optimal beam sequence that maximizes the spectral efficiency."
],
[
"Problem Formulation",
"Let $\\mathcal {B}$ denote the set that contains all $M!$ possible beam sequences.",
"The mathematical description of the problem is given by $\\hat{\\mathbf {b}}_n = \\arg \\!\\max _{\\mathbf {b}_n\\in \\mathcal {B}}U(\\mathbf {b}_n)$ where $U(\\mathbf {b}_n)$ is a utility function obtained when the sequence $\\mathbf {b}_n$ is chosen as the beam sequence within one scheduling cycle with duration of $M$ time slot.",
"When spectral efficiency is considered, the utility function for the entire scheduling cycle is given by $U(\\mathbf {b}_n) = \\frac{1}{M}\\sum _{m=1}^MU(b_{nm})$ where $U(b_{nm})$ is the utility function for the $m$ th user.",
"We then consider the problem of finding the optimal $\\hat{\\mathbf {b}}_n$ such that the average spectral efficiency is maximized."
],
[
"Derivation of Spectral Efficiency",
"We now show the derivation of spectral efficiency $U(b_{nm})$ .",
"Suppose at a given time slot, the $m$ th UE is scheduled and the 5G AP is transmitting via beam $\\mathbf {w}_{nm}$ .",
"The spectral efficiency for the given time slot can be expressed as [15] $U(b_{nm}) = \\log _2\\left(1+\\frac{P_r(n,m)}{I(m)+N(m)}\\right)$ where $P_r(n, m)$ is the received signal power at the scheduled UE $m$ , and $I(m)$ and $N(m)$ are the interference and noise term, respectively.",
"The received signal power is given by $P_r(n,m)(dB) = P_{n} + G_{n}(m) - PL(d)$ where $P_{n}$ and $G_{n}(m)$ are the transmission power and beamforming gain at the $n$ th 5G AP.",
"In this paper we consider a constant transmission power, given by $P_{n} = \\frac{P_{total}}{B}$ , where $B$ is the bandwidth.",
"In addition, the beamforming gain at the $n$ th 5G AP $G_{n}(m)$ is calculated as $G_{n}(m) = \\left|\\mathbf {w}^H_{nm}\\mathbf {h}_{nm}\\right|^2$ where $\\mathbf {h}_{nm}$ is the channel between the $n$ th base station to the scheduled UE given in [15] as $\\mathbf {h}_{nm} = \\sqrt{\\frac{N}{L}}\\sum _{l=1}^L\\alpha _l\\mathbf {a}_{UE}\\left(\\gamma _{l}^{UE}\\right)\\mathbf {a}^*_{AP}\\left(\\gamma _{l}^{AP}\\right).$ In (REF ), $\\alpha _l$ is the complex gain of the $l$ th path, $\\gamma _{l}^{UE}$ and $\\gamma _{l}^{AP} \\in [0, 2\\pi ]$ are the uniformly distributed random variables representing the angles of arrival and departure, respectively, and $\\mathbf {a}_{UE}$ and $\\mathbf {a}_{AP}$ are the antenna array responses at the UEs and 5G APs, respectively.",
"Assuming uniform linear arrays, $\\mathbf {a}_{AP}$ can be written as $\\mathbf {a}_{AP} = \\frac{1}{\\sqrt{N_{AP}}}\\left[1, \\cdots , e^{j(N_{AP}-1)\\frac{2\\pi }{\\lambda }D\\sin (\\gamma _l^{AP})}\\right]^T\\nonumber .$ For single antenna UE, we have $\\mathbf {a}_{UE} = 1\\nonumber .$ Lastly, $PL(d)$ is the path loss component between the $n$ th 5G AP and the $m$ th user, which is a function of the distance $d$ between two nodes.",
"We now look at the interference term given in (5), which is given by $I(m) = \\sum _{n^{\\prime }=1\\atop n^{\\prime }\\ne n}^{N}P_r(n^{\\prime },m)$ where $P_r(n^{\\prime },m)$ is calculated in the same manner as $P_r(n,m)$ .",
"The noise term $N(m)$ in (5) is simply white Gaussian noise, given by $N(m) = K_BTB$ where $K_B$ is the Boltzmann constant and $T$ is the noise temperature.",
"Having obtained the spectral efficiency, the optimization problem given in (REF ) can then be solved and the optimal beam sequence can be found.",
"One could perform an exhaustive search in the finite set of possible beam sequences, known to yield a high computational complexity.",
"In the following section, we propose a novel distributed learning algorithm to solve the optimization problem, which is shown to yield comparable performance than that using exhaustive search, while achieving a much reduced complexity."
],
[
"Beam Scheduling Algorithms",
"We first present a distributed greedy scheduling mechanism, followed by a detailed description of the distributed learning algorithm."
],
[
"Distributed Greedy Scheduling",
"In the distributed greedy scheduling algorithm, at the beginning of each scheduling cycle, $\\mathbf {b}_n$ is randomly chosen from the $M!$ possible permutation sequences for each 5G AP.",
"A block-coordinate optimization algorithm is then applied to maximize the individual utility function of each 5G AP sequentially [16].",
"Different from exhaustive search, where a global optimization is reached and maximum spectral efficiency is achieved for all BSs, the distributed greedy scheduling mechanism maximizes the utility function with respect to $\\mathbf {b}_n$ while keeping other $\\mathbf {b}_i (i\\ne n)$ unchanged.",
"In other words, the $n$ th 5G AP computes the utility functions for all possible permutations of $\\mathbf {b}_n$ , and then greedily selects the sequence that yields the maximum utility value, i.e., spectrum efficiency, for itself, assuming the first $(n-1)$ 5G APs are using the optimal sequences obtained in the previous selection process.",
"The process continues until it reaches the last 5G AP, which completes one iteration of greedy selection.",
"The same iteration will be repeated $N_{DG}$ times until a scheduling decision is made."
],
[
"Distributed Learning Scheduling",
"In this section, we propose a distributed learning algorithm for beam scheduling.",
"In the proposed learning algorithm, we allocate each sequence a probability at the beginning of each scheduling cycle and then update the probability and utility functions of the sequences iteratively.",
"The optimal beam sequence is then selected at the end of the learning procedure according to such a probability.",
"Such a learning algorithm is detailed as follows.",
"Suppose the $k$ th ($k\\in [1,\\cdots , M!",
"]$ ) beam sequence is selected for 5G AP $n$ at iteration $t$ , which we denote as $\\mathbf {b}_{nk}^{(t)}\\in \\mathcal {B}$ .",
"At the beginning, i.e., $t=1$ , each sequence is assigned with the same probability $p(U(\\mathbf {b}_{nk}^{(1)}))=\\frac{1}{M!",
"}$ , and one sequence $\\mathbf {b}_{nk}^{(1)}\\in \\mathcal {B}$ is randomly selected for the $n$ th 5G AP according to this probability.",
"The utility functions are then calculated for each 5G AP.",
"At the end of the $t$ th iteration, the probability $p(U(\\mathbf {b}_{ni}^{(t+1)}))$ ($1 \\leqslant i \\leqslant M!$ ) is updated for the $n$ th 5G AP according to [17] as $p(U(\\mathbf {b}_{ni}^{(t+1)})) = p(U(\\mathbf {b}_{ni}^{(t)}))-w\\frac{U(\\mathbf {b}_{ni}^{(t)})}{U^{max}(t)}p(U(\\mathbf {b}_{ni}^{(t)}))$ subject to $\\sum _{i=1}^{M!",
"}p(U(\\mathbf {b}_{ni}^{(t+1)}))=1$ , where $i \\ne k$ , $w$ is a weighting factor, and $U^{max}(t)$ is the maximum utility function obtained up to iteration $t$ , given by $U^{max}(t) = \\max \\lbrace U(\\mathbf {b}_n^{(1)}), \\cdots , U(\\mathbf {b}_n^{(t)})\\rbrace .$ For $i=k$ , $p(U(\\mathbf {b}_{nk}^{(t+1)}))$ is updated as $p(U(\\mathbf {b}_{nk}^{(t+1)})) = p(U(\\mathbf {b}_{nk}^{(t)}))+w\\frac{U(\\mathbf {b}_{nk}^{(t)})}{U^{max}(t)}P_n^{sum}$ where $P_n^{sum} = \\sum _{i=1,i \\ne k}^{M!",
"}p(U(\\mathbf {b}_{ni}^{(t)}))$ The similar learning procedure is applied to the next 5G AP until it reaches the last one and then the $(t+1)$ th iteration starts.",
"Such a learning process continues until the maximum number of iteration $T$ is hit and then the training phase stops.",
"The final probabilities used to choose the optimal sequence for the $n$ th 5G AP among all permutations is given by $\\hat{k}_n(M) = \\arg \\!\\max _{k\\in \\lbrace 1,\\cdots ,M!\\rbrace }\\lbrace p(U(\\mathbf {b}_{n}^{(1)})), \\cdots , p(U(\\mathbf {b}_{n}^{(M!",
")}))\\rbrace .$ Figure: Flow chart of the distributed learning scheduling algorithmA flow chart of the distributed learning scheduling algorithm is given in Fig.",
"REF ."
],
[
"Complexity and Signaling Overhead Analysis",
"It is known that for exhaustive search, the statistical utility functions need to be computed for all APs and all possible beam sequences, yielding a complexity of $O(\\left(M!\\right)^N)$ , which becomes prohibitive especially with a large number of APs.",
"For the distributed greedy scheduling algorithm, for $N$ APs and $N_{DG}$ iterations, we need to calculate $N_{DG}NM!$ utility functions in total, having a computational complexity of $O(N_{DG}NM!",
")$ .",
"The proposed distributed learning algorithm, on the contrary, computes only one utility function for each 5G AP at a given iteration, yielding a complexity of $O(N_{LE}N)$ , which is much smaller than the exhaustive search as well as the distributed greedy scheduling.",
"In addition, as illustrated in the next section, the number of iterations required by the proposed leaning algorithm is also less than that of the greedy ones, i.e., $N_{LE} < N_{DG}$ , leading to even less calculations.",
"In terms of signaling overhead, exhaustive search requires global utility function information to be exchanged among all 5G APs, whilst the signaling overhead of the proposed distributed learning scheduling algorithm is similar to the distributed greedy scheduling algorithm since there is no need for exchanging utility function globally.",
"Table: Main System Parameters"
],
[
"Simulations",
"In this section, we present simulation results obtained based on the scheduling algorithms proposed in the previous section.",
"We assume a total transmission power of 30 dBm and a total bandwidth of 500 MHz and the 5G APs distribute the power uniformly over the entire bandwidth.",
"The pathloss model used here is given in [7].",
"The noise temperature $T$ is taken as the room temperature of $300K$ .",
"The detailed system parameters are presented in Table REF .",
"Figure: Deployment of 2 5G APs (5 UEs per AP)Fig.",
"REF shows an example of a deployment scenario with 2 5G APs, each covering 5 UEs.",
"In the figure, UE$_{ij}$ denotes the $j$ th UE associated with the $i$ th 5G AP.",
"It can be seen that if UE$_{13}$ and UE$_{22}$ are scheduled at the same time, the transmission beam of AP$_{1}$ to UE$_{13}$ will cause interference to UE$_{22}$ .",
"Fig.",
"REF illustrates the fluctuation of the utility functions of two APs obtained during the entire learning procedure with weighting factor $w=0.15$ .",
"As illustrated, the utility functions rapidly converge to the optimal values in less than 55 learning iterations.",
"The average utility function of two also converges to the maximum at the same pace.",
"Figure: Convergence behavior of the distributed learning scheduling algorithmThe cumulative distribution functions (CDFs) of utility functions for different scheduling algorithms are illustrated in Fig.",
"REF .",
"For comparison purpose, we present the results obtained by using random selection of permutation sequences.",
"As expected, the exhaustive search algorithm achieves the maximum overall utility function and the performance of the distributed greedy scheduling algorithm is slightly worse than the exhaustive one, which also serves as a performance boundary for all distributed scheduling algorithms.",
"The performance of the proposed distributed learning algorithm is very close to the greed one but with significantly reduced complexity and signaling overhead as aforementioned.",
"When the cell size is changed from 400m to 200m, the performance gap between the distributed greedy scheduling algorithm and the learning algorithm becomes even smaller.",
"Figure: CDF of spectrum efficiency (inter-cell distance = 400m)Figure: CDF of spectrum efficiency (inter-cell distance = 200m)Figure: Convergence behavior of two scheduling schemesThen we increase the number of deployed 5G APs to 10, each covering 3 UEs, and clearly such deployment will result in a higher probability for a UE to receive interferences from other 5G APs.",
"In Fig.",
"REF , the convergence speed of the distributed greedy algorithm and the proposed learning algorithm is compared.",
"Even though the greedy algorithm achieves a higher utility value, the convergence speed of the proposed learning algorithm is much higher (more than 150 iterations less), therefore leading to further reduced complexity.",
"The CDFs of the utility function are illustrated in Fig.",
"REF .",
"The proposed learning algorithm outperforms the random one and is close to the greedy one."
],
[
"Conclusion and Future Works",
"In this paper, we propose a novel multi-RAT coexistence mechanism where neighboring 5G and WiGig APs, each serving their own associated UEs, schedule their beams in a distributed manner such that their own utility function, e.g., spectral efficiency, is maximized.",
"The proposed distributed algorithm yields a comparable spectral efficiency for the entire networks as that using exhaustive search, which requires centralized coordination among multi-RAT networks with much higher algorithmic complexity.",
"Our future work will focus on game theoretical analysis of our proposed algorithm with respect to fairness and its convergence properties."
],
[
"Acknowledgements",
"The authors would like to thank Francesco Guidolin for his support and assistance with the simulations.",
"The research leading to these results received funding from the European Commission H2020 programme under grant agreement n°671650 (5G PPP mmMAGIC project).",
"Figure: CDF of spectrum efficiency (inter-cell distance = 400m)"
]
] | 1606.05135 |
[
[
"On a nonisothermal ideal gas Navier-Stokes-Fourier equations"
],
[
"Abstract In this paper we are concerned with a non-isothermal compressible Navier-Stokes-Fourier model with density dependent viscosity that vanish on the vacuum.",
"We prove sequential stability of variational weak solutions in periodic domain \\Omega= T3.",
"The main point is that the pressure is given by P = R\\rho\\theta."
],
[
"Introduction",
" A compressible and heat-conducting fluid governed by the Navier-Stokes-Fourier equations satisfies the following system in $R_{+}\\times \\Omega $ : $\\partial _{t}\\rho +{\\rm div}(\\rho u) =0, \\\\\\partial _{t}(\\rho u)+{\\rm div}(\\rho u\\otimes u)+\\nabla P={\\rm div}\\mathbb {S}, \\\\\\partial _{t} (\\rho E) + {\\rm div}(\\rho E u)+{\\rm div} q+ {\\rm div}(P u)={\\rm div} (\\mathbb {S}u),$ where the functions $\\rho , u,\\theta $ represent the density,the velocity field, the absolute temperature.",
"$P$ stands for the pressure, $\\mathbb {S}$ denotes the viscous stress tensor.",
"$\\rho E=\\rho e+ \\frac{\\rho |u|^{2}}{2}$ the total energy, $e$ the internal energy.",
"$q$ the heat flux.",
"Eqs.",
"(REF ), (), () respectively express the conservation of mass, momentum and total energy.",
"Our analysis is based on the following physically grounded assumptions: The viscosity stress tensor $\\mathbb {S}$ is determined by the Newton's rheological law $\\mathbb {S}=2\\mu (\\rho ) D(u)+\\lambda (\\rho ) {\\rm div}_{x} u \\mathbb {I},$ where $3\\lambda + 2\\mu \\ge 0$ and $D(u)= \\frac{1}{2}(\\nabla u+ \\nabla ^{T} u)$ denotes the strain rate tensor, we require $\\lambda (\\rho )= 2(\\rho \\mu ^{\\prime }(\\rho )- \\mu (\\rho ))$ .",
"For simplicity, we only consider a particular case $\\mu (\\rho )=\\rho , \\lambda (\\rho )=0$ .",
"A key element of the system (REF )-() is pressure $P$ , which obeys the following equation of state: $P(\\rho ,\\theta )=R\\rho \\theta ,$ where R is the perfect gas constant, for simplicity, we set $R=1$ .",
"This assumption means ideal gas given by Boyle's law.",
"In accordance with the second thermodynamics law, the form of the internal energy reads: $e=C_{\\nu }\\theta ,$ where $C_{\\nu }$ is termed the specific heat at constant volume, for simplicity, we set $C_{\\nu }=1$ .",
"The heat flux $q$ is expressed through the classical Fourier's law: $q=-\\kappa \\nabla \\theta ,$ where the heat conducting coefficient $\\kappa $ is assumed to satisfy: $\\kappa (\\rho ,\\theta )= \\kappa _{0}(\\rho ,\\theta )(1+\\rho )(1+\\theta ^{a}),$ where $a\\ge 2$ , $\\kappa _{0}$ is a continuous function of temperature and density satisfying: $ C_{1} \\le \\kappa _{0} (\\rho , \\theta ) \\le \\frac{1}{C_{1}}$ , for some positive $C_{1}$ .",
"To complete the system (REF )-(), the initial conditions are given by $\\rho (0,\\cdot )=\\rho _{0}, (\\rho u)(0,\\cdot )=m_{0}, \\theta (0,\\cdot )=\\theta _{0},$ together with the compatibility condition: $m_{0}=0~~on~the~set~~\\lbrace x\\in \\Omega |\\rho _{0}(x)=0\\rbrace .$ If the solutions are smooth, the temperature equation is easier to deduce as follows: $\\partial _{t} (\\rho \\theta )+ {\\rm div} (\\rho \\theta u) =\\mathbb {S}: \\nabla u+ {\\rm div} (\\kappa \\nabla \\theta )-\\rho \\theta {\\rm div} u .$ Finally, assuming the pressure and internal energy, we can define the specific entropy through Gibbs relationship.",
"Accordingly, the temperature equation may be put into an equivalent form of the entropy equation: $\\partial _{t} (\\rho s)+ {\\rm div} (\\rho s u)-{\\rm div}\\bigg (\\frac{\\kappa \\nabla \\theta }{\\theta }\\bigg ) =\\frac{\\mathbb {S}: \\nabla u}{\\theta }+ \\frac{\\kappa |\\nabla \\theta |^{2}}{\\theta ^{2}}.$ where $s= \\ln \\theta -\\ln \\rho $ .",
"In the follows we give the definition of a variational solution to (REF )-(REF ).",
"Definition 1.1 We call $(\\rho , u, \\theta )$ is as a varational weak solution to the problem (REF )-(REF ), if the following is satisfied.",
"(1)the density $\\rho $ is a non-negative function satisfying the internal identity $\\int _{0}^{T} \\int _{\\Omega } \\rho \\partial _{t} \\phi + \\rho u\\cdot \\nabla \\phi dx dt+ \\int _{\\Omega } \\rho _{0} \\phi (0)dx=0,$ for any test function $\\phi \\in \\mathcal {D}([0,T)\\times \\overline{\\Omega })$ .",
"(2) The momentum equation holds in $D^{\\prime }((0,T)\\times \\Omega )$ , that means, $\\begin{aligned}&\\int _{\\Omega }m_{0} \\phi (0) dx+\\int _{0}^{T} \\int _{\\Omega } \\rho u \\cdot \\partial _{t} \\phi + \\rho (u\\otimes u): \\nabla \\phi + P {\\rm div} \\phi dx dt\\\\&= \\int _{0}^{T} \\int _{\\Omega } \\mathbb {S}: \\nabla \\phi dx dt,~for~any ~\\phi \\in \\mathcal {D}([0,T)\\times \\overline{\\Omega }),\\end{aligned}$ (3) If we admit there are variational solutions that may fail to satisfy the weak formulation of the total energy balance, thus this can be expressed by the variational principle of entropy production: $\\int _{\\Omega } \\rho _{0} s_{0} \\phi (0)dx +\\int _{0}^{T} \\int _{\\Omega } \\rho s \\partial _{t} \\phi + \\rho s u \\cdot \\nabla \\phi - \\frac{\\kappa \\nabla \\theta }{\\theta } \\nabla \\phi dx dt+ <\\sigma ,\\phi >=0$ It is satisfied for any smooth function $\\phi (x,t)$ , such that $\\phi \\ge 0$ and $\\phi (T,\\cdot )=0$ , where $\\sigma \\in \\mathcal {M}^{+}((0,T)\\times \\Omega )$ is a nonnegative measure such that $\\sigma \\ge \\frac{\\mathbb {S}: \\nabla u}{\\theta }+ \\frac{\\kappa |\\nabla \\theta |^{2}}{\\theta ^{2}}$ (4)The global balance of total energy $\\int _{\\Omega } (\\rho E)^{0} \\phi (0) dx +\\int _{0}^{T}\\int _{\\Omega } \\rho E \\partial _{t} \\phi dxdt=0$ holds for any smooth function $\\phi (t)$ , such that $\\phi (T)=0$ .",
"Now, we are ready to formulate the main result of this paper.",
"Theorem 1.2 (Stability) Let $\\Omega $ be the periodic box $T^{3}$ .",
"Assume that the pressure $P$ , the conductivity coefficient $\\kappa $ and the viscosity coefficient $\\mu $ satisfy the condition (REF )-(REF ).",
"Let $(\\rho _{n}, u_{n},\\theta _{n})_{n\\in \\mathbb {N}}$ be sequence of weak solutions of (REF )-() satisfying entropy inequalities (REF ), (REF ) and (REF ), with initial data $\\rho _{n}|_{t=0}=\\rho _{0}^{n}(x), ~~\\rho _{n} u_{n}|_{t=0}=m_{0}^{n}(x)= \\rho _{0}^{n}(x) u_{0}^{n}(x),~~\\theta _{n}|_{t=0}= \\theta _{0}^{n}(x),$ where $\\theta _{0}^{n}$ , $u_{0}^{n}$ and $\\theta _{0}^{n}$ are such that $\\rho _{0}^{n}\\ge 0,~~\\rho _{0}^{n} \\rightarrow \\rho _{0}~~~\\in ~ L^{1}(\\Omega ),~~\\rho _{0}^{n}u_{0}^{n}\\rightarrow \\rho _{0}u_{0}~~~\\in ~ L^{1}(\\Omega ),~~\\theta _{0}^{n} \\rightarrow \\theta _{0}~~~\\in ~ L^{1}(\\Omega ),$ and satisfying the following bounds (with $C$ constant independent on $n$ ): $\\int _{\\Omega }\\frac{|m_{0}|^{2}}{\\rho _{0}}+ \\rho _{0} \\theta _{0}dx <C,~~\\int _{\\Omega }\\nabla \\sqrt{\\rho _{0}} dx<C, ~~~ \\int _{\\Omega }\\rho _{0} s_{0}<C ,$ Then, up to a subsequence, $(\\rho _{n},\\sqrt{\\rho _{n}}u_{n},\\theta _{n})$ converges strongly to a weak solution of (REF )-() satisfying entropy inequalities (REF ), (REF ) and (REF ).",
"For this full compressible Navier-Stokes system in the constant viscosity case, Lions [6] and Feireisl[4],[5] proved the global existence of variational weak solutions.",
"Such an existence result is obtained for specific pressure laws, given by general pressure equation $P(\\rho ,\\theta )= P_{b}(\\rho )+ \\theta P_{\\theta }(\\rho ).$ Unfortunately, the perfect gas equation of state is not covered by this result.",
"Namely the dominant role of the first, barotropic pressure $P_{b}$ is one of the key argument to obtain such an existence result.",
"As for the density depending viscosities case, the existence of global weal solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids was proved by D. Bresch and B. Desjardins [3].",
"The equation of state is ideal polytropic gas type: $P =R\\rho \\theta + P_{c}(\\rho ),$ However, they still need additional cold pressure assumption $P_{c}$ .",
"Therefore, Our aim in this work is to remove additional assumption on the equation of state.",
"For related B-D entropy inequality, we refer the paper [1],[2],[8].",
"In order to prove the stability of variational weak solutions, the first step is to obtain suitable a priori bounds on $(\\rho _{n},u_{n},\\theta _{n})$ .",
"The next step is to obtain compactness on $(\\rho _{n},u_{n},\\theta _{n})$ in suitably strong topologies and prove that the limit $(\\rho ,u,\\theta )$ satisfies Eqs.",
"(REF )-(REF ) in the variational sense.",
"In the forthcoming we will use the idea of Li and Xin [7] to construct approximate solution, thus we can complete the global existence of variational weak solution for non-isothermal Navier-Stokes-Fouier system.",
"This paper is organized as follows.",
"In section 2, we deduce a priori estimates from (REF ).",
"In section 3, we establish the compactness of solutions $(\\rho _{n},u_{n},\\theta _{n})$ .",
"In section 4, we will prove the proof of theorem 1.2 using Aubin-Lions Lemma."
],
[
"A priori bounds",
" In this section, we collect the available a priori estimates for sequence of smooth functions $\\lbrace \\rho _{n}, u_{n}, \\theta _{n} \\rbrace $ solving (REF )-().",
"As mentioned above, assuming smoothness of solutions, we will deduce enough estimates to establish the compactness of solutions.",
"The following estimates are valid for each $n=1,2,...$ but we skip the subindex it does not lead to any confusion."
],
[
"First, it is easier to know that the total mass of the fluid is as constant of motion, i.e.",
"$\\int _{\\Omega } \\rho (x,t) dx= \\int _{\\Omega } \\rho _{0}dx=M_{0},~~~for~~t\\in [0,T].$ Moreover maximum principle can be applied to the continuity equation in order to show that $\\rho _{n}>c(n)\\ge 0$ , more precisely $\\rho _{n}(x,t)\\ge \\inf _{x\\in \\Omega } \\rho _{n}^{0} exp(-\\int _{0}^{T}\\Vert {\\rm div} u_{n}\\Vert _{L^{\\infty }(\\Omega )}dt),$ in particular $\\rho >0$ .",
"Next, by a similar reasoning we can prove non-negatively of $\\theta $ on $[0,T]\\times \\Omega $ .",
"Lemma 2.1 Assume that $\\theta =\\theta _{n}$ is a smooth solution of (1.1), then $\\theta (t,x)>c(N)\\ge 0,~~~for ~(t,x)\\in [0,T]\\times \\Omega .$"
],
[
"The physical energy inequality (involving the internal and kinetic energy) is classical in the full compressible Navier-Stokes equations which is shown in the following: Lemma 2.2 (Physical energy estimates) $ \\int _{\\Omega } \\rho (\\frac{|u|^{2}}{2}+ \\theta )(t,x) dx \\le \\int _{\\Omega }(\\frac{|m_{0}|^{2}}{\\rho _{0}}+ \\rho _{0} \\theta _{0})dx$ Integrate () with respect to the space variable and employ the periodic boundary conditions.",
"Assume the initial total energy is finite, we immediately obtain the following bounds: $\\Vert \\sqrt{\\rho } u\\Vert _{L^{\\infty }(0,T;L^{2}(\\Omega ))} \\le C, \\Vert \\rho \\theta \\Vert _{L^{\\infty }(0,T;L^{1}(\\Omega ))}\\le C,$ Next, we give some temperature estimates from the entropy equation: Lemma 2.3 (Entropy estimates) Assume that $\\rho _{0} s_{0} \\in L^{1}(\\Omega )$ , Then for all $T \\ge 0$ , one has $ \\int _{0}^{T}\\int _{\\Omega } \\frac{\\kappa |\\nabla \\theta |^{2}}{\\theta ^{2}}+ \\frac{2\\rho |D(u)|^{2}}{\\theta } dx dt\\le \\int _{\\Omega }\\rho s + |\\rho _{0} s_{0}|dx$ Integrate (REF ) with respect to the space variable and employ the periodic boundary conditions.",
"The first term on the right hand of (REF ) can be estimated by: $\\int _{\\Omega } \\rho s dx \\le \\int _{\\Omega }\\rho \\ln \\theta dx- \\int _{\\Omega } \\rho \\ln \\rho dx,$ Multiplying the mass equation by $1+\\ln \\rho $ , we get $\\partial _{t}(\\rho \\ln \\rho ) + {\\rm div}(\\rho \\ln \\rho u)+ \\rho {\\rm div} u=0,$ Therefore, the right hand of (REF ) can be estimated by $\\begin{aligned}\\int _{\\Omega } \\rho s dx &\\le \\int _{\\Omega } \\rho \\theta dx + \\int _{\\Omega } |\\rho _{0} \\ln \\rho _{0}| dx + \\int _{0}^{T} \\int _{\\Omega } \\rho {\\rm div} u dx dt \\\\& \\le C+ \\int _{0}^{T} \\int _{\\Omega } \\sqrt{\\frac{\\rho }{\\theta }}|{\\rm div} u| \\sqrt{\\rho \\theta } dx dt\\\\& \\le C + \\varepsilon \\int _{0}^{T} \\int _{\\Omega }\\frac{\\rho |D(u)|^{2}}{\\theta } dx dt + C(\\varepsilon )\\int _{0}^{T} \\int _{\\Omega }\\rho \\theta dx dt,\\end{aligned}$ Hence if $\\rho _{0} s_{0}$ and $\\rho _{0} \\ln \\rho _{0}$ belong to $L^{1}(\\Omega )$ , then the component of following quantities $\\sqrt{\\rho } D(u) / \\sqrt{\\rho }$ , $(\\sqrt{\\rho }+1) \\nabla \\theta ^{a/2}$ , $(\\sqrt{\\rho }+1) \\nabla \\ln \\theta $ are bounded in $L^{2}(\\Omega \\times (0,T))$ .",
"We note that the last two bounds involving the temperature gradient provide the following useful estimates: $(\\sqrt{\\rho }+1) \\nabla \\theta ^{\\alpha } \\in L^{2}(\\Omega \\times (0,T)),~for~all~\\alpha ~such~ that ~0\\le \\alpha \\le a/2.$ Now, we will derive some estimates on velocity and associated effective B-D entropy energy.",
"Lemma 2.4 (The kinetic energy estimates) $ \\frac{d}{dt}\\int _{\\Omega } \\frac{1}{2}\\rho |u|^{2}dx+ \\int _{\\Omega } 2\\rho |D(u)|^{2} dx=\\int _{\\Omega } P {\\rm div} u dx,$ Multiply the momentum equation () by $u$ and integrate over $\\Omega $ .",
"Lemma 2.5 (B-D effective energy estimates) $ \\begin{aligned}&\\frac{d}{dt}\\int _{\\Omega } \\frac{1}{2}\\rho |u+ 2\\nabla \\ln \\rho |^{2} dx + \\int _{\\Omega } 2\\rho |A(u)|^{2}dx+ 2\\int _{\\Omega } \\frac{|\\nabla \\rho |^{2} \\theta }{\\rho } dx\\\\& = \\int _{\\Omega } P {\\rm div} u dx- 2 \\int _{\\Omega } \\nabla \\rho \\cdot \\nabla \\theta dx,\\end{aligned}$ The idea of the proof is from the original work of Bresch and Desjaedins.",
"For more detail we refer to [3].",
"In order to get enough a priori estimates from Lemma 2.4 and 2.5, we have to control the right-hand side terms of (REF ) and (REF ).",
"Lemma 2.6 ($\\int _{\\Omega } P {\\rm div} u dx$ ) $ \\begin{aligned}&\\int _{\\Omega } P {\\rm div} u dx \\\\\\le & \\varepsilon \\int _{\\Omega } \\rho |{\\rm div} u|^{2}dx + C(\\varepsilon ) \\int _{\\Omega } \\int _{\\Omega } \\rho \\theta ^{2} dx\\\\\\le &\\varepsilon \\int _{\\Omega } \\rho |{\\rm div} u|^{2}dx + C(\\varepsilon ) \\Vert \\theta \\Vert _{L^{3}}^{2} \\Vert \\nabla \\sqrt{\\rho }\\Vert _{L^{2}}^{2},\\end{aligned}$ Lemma 2.7 ($\\int _{\\Omega } \\nabla \\rho \\cdot \\nabla \\theta dx$ ) $ \\begin{aligned}&\\int _{\\Omega } \\nabla \\rho \\cdot \\nabla \\theta dx \\\\\\le & C\\int _{\\Omega } \\frac{\\rho \\theta ^{2}}{\\kappa } |\\nabla \\sqrt{\\rho }|^{2}dx + C \\int _{\\Omega } \\frac{\\kappa |\\nabla \\theta |^{2}}{\\theta ^{2}}\\\\\\le & C \\int _{\\Omega } |\\nabla \\sqrt{\\rho }|^{2}dx + C,\\end{aligned}$ Thus, by taking $\\varepsilon $ small enough, (REF ) and Sobolev inequality, $\\theta \\in L^{2}([0,T]; L^{6}(\\Omega ))\\cap L^{2}([0,T]; L^{6}(\\Omega ))$ , it is possible to get some a priori estimates via Gronwall's inequality.",
"From above we get the space compactness of the density and temperature, therefore the strong convergence of the density and temperature can be derived.",
"But there still is lack of information about the velocity.",
"To this goal, in the following we can prove the so-called Mellet-Vasseur type estimate.",
"Lemma 2.8 (Mellet-Vasseur type estimate) $ \\sup _{0\\le t\\le T} \\int _{\\Omega } \\rho (1+|u|^{2}) \\ln (1+|u|^{2}) dx \\le C,$ Multiplying momentum equation () by $(1+\\ln (1+|u|^{2}))u$ and integrating over $\\Omega $ lead to $\\begin{aligned}&\\frac{1}{2} \\frac{d}{dt} \\int _{\\Omega } \\rho (1+|u|^{2}) \\ln (1+|u|^{2}) dx + \\int _{\\Omega } (1+\\ln (1+|u|^{2}) \\rho |D(u)|^{2}dx\\\\& \\le C\\int _{\\Omega } \\rho |D(u)|^{2} dx - \\int _{\\Omega } (1+\\ln (1+|u|^{2})u\\cdot \\nabla (\\rho \\theta ) dx,\\end{aligned}$ where the last term on the right side can be estimated as follows: $\\begin{aligned}&| \\int _{\\Omega } (1+\\ln (1+|u|^{2})u\\cdot \\nabla (\\rho \\theta ) dx|\\\\\\le & \\int _{\\Omega } (1+\\ln (1+|u|^{2}) {\\rm div}u \\rho \\theta dx + \\int _{\\Omega } \\frac{2u_{i}u_{k}}{1+|u|^{2}} \\partial _{i} u_{k} \\rho \\theta dx \\\\\\le &\\varepsilon \\int _{\\Omega } (1+\\ln (1+|u|^{2}) \\rho |Du|^{2} dx+ C \\int _{\\Omega } (1+\\ln (1+|u|^{2}) \\rho \\theta ^{2} dx\\\\+& C\\Vert \\sqrt{\\rho } \\nabla u \\Vert _{L^{2}(\\Omega )}\\Vert \\theta \\Vert _{L^{3}(\\Omega )} \\Vert \\sqrt{\\rho } \\Vert _{L^{6}(\\Omega )}, \\\\\\le &C + C \\int _{\\Omega } (1+|u|) \\rho \\theta ^{2}dx\\\\\\le &C + C \\int _{\\Omega }\\rho |u|^{2} dx + C\\int _{\\Omega } \\rho \\theta ^{4}dx\\\\\\le & C\\end{aligned}$"
],
[
"Compactness of $\\rho _{n}, \\sqrt{\\rho _{n}}u_{n}, \\theta _{n}$",
" We recall that the initial data must satisfy (REF ), (REF ) and (REF ) to make use of all the inequalities presented in the previous section.",
"More precisely, we take $\\rho _{0}^{n}~~ is~~ bounded~~ in~~ L^{1}(\\Omega ), \\rho _{0}^{n}\\ge 0~~ a.e.~~ in~~ \\Omega ,$ $\\rho _{0}^{n}|u_{0}^{n}|^{2}= |m_{0}^{n}|^{2}/\\rho _{0}^{n} ~~is~~ bounded~~ in~~ L^{1}(\\Omega ),$ $\\rho _{0}s_{0}~~and~~\\rho _{0} \\ln \\rho _{0}~~is~~ bounded~~ in~~ L^{1}(\\Omega ),$ $\\nabla \\sqrt{\\rho _{0}^{n}} ~~is~~ bounded~~ in~~ L^{2}(\\Omega ),$ $\\int _{\\Omega } \\rho _{0}^{n} \\frac{|u_{0}^{n}|^{2}}{2}\\ln (1+ |u_{0}^{n}|^{2})dx< C ,$ Using inequalities (REF ), (REF ),(REF ),(REF ) and (REF ), we deduce the following estimates, which we shall use throughout the proof of Theorem 1.2: $\\Vert \\sqrt{\\rho _{n}} u_{n}\\Vert _{L^{\\infty }(0,T;L^{2}(\\Omega ))} \\le C,$ $\\Vert \\sqrt{\\rho _{n}} \\nabla u_{n}\\Vert _{L^{2}(0,T;L^{2}(\\Omega ))} \\le C,$ $\\Vert \\rho _{n}\\Vert _{L^{\\infty }(0,T;L^{1}(\\Omega ))} \\le C,$ $\\Vert \\nabla \\sqrt{\\rho _{n}}\\Vert _{L^{\\infty }(0,T;L^{2}(\\Omega ))} \\le C,$ $\\Vert (1+\\sqrt{\\rho _{n}}) \\nabla \\theta _{n}^{\\alpha }\\Vert _{L^{\\infty }(0,T;L^{2}(\\Omega ))} \\le C,$ $\\int _{\\Omega } \\rho _{n} \\frac{|u_{n}|^{2}}{2}\\ln (1+ |u_{n}|^{2})dx< C ,$ Given above a priori bounds, we now intend to study the compactness of sequences of approximate solutions $\\rho _{n}$ , $\\sqrt{\\rho _{n}}u_{n}$ and $\\theta _{n}$ and pass the limit in the nonlinear terms."
],
[
"Lemma 3.1 Up to a subsequence, $\\sqrt{\\rho _{n}} \\rightarrow \\sqrt{\\rho } ~a.e.~~ and~~~L^{2}_{loc}((0,T)\\times \\Omega )~~strong.$ In particular, $\\rho _{n}\\rightarrow \\rho ~~~~in~ C^{0}(0,T;L^{3/2}_{loc}(\\Omega )),$ The proof is refer to Lemma 4.1 in [8]."
],
[
"Lemma 3.2 Up to a subsequence, $\\rho _{n} u_{n}\\rightarrow \\rho u ~~strongly~in~~ L^{2}(0,T; L^{p}_{loc}(\\Omega )),~~for~p\\in ~[1,3/2).$ $\\sqrt{\\rho _{n}} u_{n}\\rightarrow \\sqrt{\\rho } u ~~strongly~in~~ L^{2}_{loc}((0,T)\\times \\Omega ),$ The proof is refer to Lemma 4.6 in [8]."
],
[
"By (REF ) and Sobolev imbedding gives the estimate of the norm of $\\theta $ in $L^{2}(0,T;L^{6}(\\Omega ))$ , and so, due to the boundedness of $\\nabla \\theta ^{\\frac{\\alpha }{2}}$ in $L^{2}((0,T)\\times \\Omega )$ , one gets $\\theta ^{\\frac{\\alpha }{2}}\\in L^{2}(0,T;W^{1,2}(\\Omega )),$ Therefore we deduce existence of a subsequence such that $\\theta _{n}\\rightarrow \\theta \\ weakly \\ in \\ L^{2}(0,T;W^{1,2}(\\Omega )),$ however, time-compactness cannot proved directly from the internal energy equation (REF ).",
"The reason for this is lack of control over a part of the heat flux proportional to $\\rho _{n}\\theta _{n}^{a}\\nabla \\theta _{n}$ .",
"This obstacle can be overcome by deducing analogous information from the entropy equation (REF ).",
"We will first show that all of the terms appearing in the entropy balance (REF ) are nonnegative or belong to $W^{-1,p}((0,T)\\times \\Omega )$ , for some $p>1$ .",
"Indeed, first we recall that due to (2.8) $|\\rho _{N}s_{n}|\\le C (\\rho _{n} |\\ln \\theta _{n}|+ \\rho _{n}|\\ln \\rho _{n}|)$ and $|\\rho _{n}s_{n}u_{n}|\\le C ( |\\rho _{n}\\ln \\theta _{n}u_{n}|+ |\\rho _{n}\\ln \\rho _{n}u_{n}|)$ whence due to (REF ),(REF ) and (REF ) we deduce that $\\lbrace \\rho _{n}s_{n}\\rbrace _{n=1}^{\\infty } \\ in \\ bounded \\ in \\ L^{2}((0,T)\\times \\Omega ),$ moreover $\\lbrace \\rho _{n}s_{n}u_{n}\\rbrace _{n=1}^{\\infty } \\ in \\ bounded \\ in \\ L^{2}(0,T;L^{\\frac{6}{5}} (\\Omega )),$ The entropy flux can be estimated as follows $|\\frac{\\kappa (\\rho _{n},\\theta _{n})\\nabla \\rho _{n}}{\\theta _{n}}|\\le |\\nabla \\ln \\theta _{n}|+|\\rho _{n}\\nabla \\ln \\theta _{n}|+|\\theta _{n}^{\\alpha -1}\\nabla \\theta _{n}|+|\\rho _{n}\\theta _{n}^{\\alpha -1}\\nabla \\theta _{n}|$ where the most restrictive can be controlled as follows $|\\rho _{n}\\theta _{n}^{\\alpha -1}\\nabla \\theta _{n}|\\le |\\sqrt{\\rho _{n}}\\theta _{n}^{\\frac{\\alpha }{2}}||\\sqrt{\\rho _{n}}\\nabla \\theta _{n}^{\\frac{\\alpha }{2}}|$ , which is bounded on account of (REF ) provided $\\rho _{n}\\theta _{n}^{\\alpha }$ is bounded in $L^{p}((0,T)\\times \\Omega )$ for $p>1$ , uniformly with respect to n. Note that for $0\\le \\beta \\le 1$ we have $\\rho _{n}\\theta _{n}^{\\alpha }=(\\rho _{n}\\theta _{n})^{\\beta }\\rho _{n}^{1-\\beta }\\theta _{n}^{\\alpha -\\beta }$ , where $(\\rho _{n}\\theta _{n})^{\\beta },\\rho _{n}^{1-\\beta },\\theta _{n}^{\\alpha -\\beta }$ are uniformly bounded in $L^{\\infty }(0,T;L^{\\frac{1}{\\beta }}(\\Omega )),L^{\\infty }(0,T;L^{\\frac{3}{1-\\beta }}(\\Omega )),L^{\\frac{\\alpha }{\\alpha -\\beta }}(0,T;L^{\\frac{3\\alpha }{\\alpha -\\beta }}(\\Omega ))$ , respectively.",
"Therefore $\\lbrace \\frac{\\kappa (\\rho _{n},\\theta _{n})\\nabla \\rho _{n}}{\\theta _{n}}\\rbrace _{n=1}^{\\infty } \\ is \\ bounded \\ in \\ L^{p}(0,T;L^{q}(\\Omega )).$ for p and q satisfying $\\frac{1}{p}=\\frac{\\alpha }{\\alpha -\\beta },\\frac{1}{q}=\\beta +\\frac{1-\\beta }{3}+\\frac{\\alpha -\\beta }{3\\alpha }$ .",
"In particular $p,q>1$ provided $0<\\beta <\\frac{2\\alpha }{2\\alpha -1}$ .",
"We are now ready to proceed with the proof of the strong convergence of the temperature.",
"To this end we will need the following variant of the Aubin-Lions Lemma.",
"Lemma 3.3 Let $g^{n}$ converges weakly to g in $L^{p_{1}}(0,T;L^{p_{2}}(\\Omega ))$ and let $h^{n}$ converges weakly to h in $L^{q_{1}}(0,T;L^{q_{2}}(\\Omega ))$ , where $1\\le p_{1},p_{2}\\le \\infty $ and $\\frac{1}{p_{1}}+\\frac{1}{q_{1}}=\\frac{1}{p_{2}}+\\frac{1}{q_{2}}=1$ Let us assume in addition that $\\frac{\\partial g^{n}}{\\partial t} \\ is \\ bounded \\ in \\ L^{1}(0,T;W^{-m,1}(\\Omega )) \\ for \\ some \\ m\\ge 0 \\ independent\\ of \\ n.$ $\\Vert h^{n}-h^{n}(\\cdot +\\xi ,t)\\Vert _{L^{q_{1}}(0,T;L^{q_{2}}(\\Omega ))}\\rightarrow 0,\\ uniformly \\ in \\ n.$ Then $g^{n}h^{n}$ converges to gh in the sense of distributions on $\\Omega \\times (0,T)$ .",
"For the proof see [4], Lemm5.1.",
"Taking $g^{n}=\\rho _{n}s_{n}$ and $h^{n}=\\theta _{n}$ we verify, due to (REF ), that conditions (REF ) and (REF ) are satisfied with $p_{1},p_{2},q_{1},q_{2}=2$ .",
"Moreover for m sufficiently large $L^{1}(\\Omega )$ is imbedded into $W^{-m,1}$ , thus by the previous considerations, condition (REF ) is also fulfilled.",
"Therefore, passing to the subsequences we may deduce that $\\lim _{n\\rightarrow \\infty } \\rho _{n}s(\\rho _{n},\\theta _{n})\\theta _{n}=\\overline{\\rho s(\\rho ,\\theta )}\\theta .$ On the other hand, $\\rho _{n}$ converges to $\\rho $ a.e.",
"on $(0,T)\\times \\Omega $ , hence $\\overline{\\rho s(\\rho ,\\theta )}\\theta =\\rho \\overline{ s(\\rho ,\\theta )}\\theta $ , in particular, we have that $C_{\\mu }\\overline{\\rho \\ln \\theta }\\theta -R\\overline{\\rho \\ln \\rho }=C_{\\mu }\\rho \\overline{\\ln \\theta }\\theta -R\\rho \\overline{\\ln \\rho }\\theta $ Combining weak convergence of the temperature with strong convergence of the density we identify $R\\rho \\overline{\\ln \\rho }\\theta =R\\rho \\ln \\rho \\theta $ so (REF ) implies that $\\rho \\overline{\\ln \\theta \\theta }=\\rho \\overline{\\ln \\theta }\\theta $ .",
"This in return yields that $\\overline{\\ln \\theta \\theta }=\\ln \\theta \\theta $ a.e.",
"on $(0,T)\\times \\Omega $ ,since $\\rho >0$ a.e.",
"on $(0,T)\\times \\Omega $ , which, due to convexity of function $x \\ln x$ , gives rise to $\\theta _{n}\\rightarrow \\theta \\ a.e.\\ on \\ (0,T)\\times \\Omega $"
],
[
"Proof of the Theorem 1.2",
" To finish the proof of Theorem 2.1, we need to check that the limit $\\rho ,u,\\theta $ are indeed the weak solutions, as defined in the introduction.",
"We will complete this proof by several steps.",
"Step 1.",
"Convergence of the mass conservation equation.",
"For the mass conservation, by the strong convergences of $\\rho _{n}$ to $\\rho $ in $C([0,T];L^{3/2}(\\Omega ))$ and the strong convergence of $\\sqrt{\\rho }u$ in $L^{2}(0,T; L^{2}_{loc}(\\Omega ))$ , the mass conservation equation (REF ) is satisfied in the sense of distribution.",
"Step 2.",
"Convergence of the momentum conservation equation.",
"For the momentum equation, the strong convergence of $\\rho _{n}u_{n}$ and $\\rho _{n}u_{n}\\otimes u_{n}$ $L^{1}((0,T)\\times \\Omega )$ can ensure the passing to limit in the sense of distribution for the two corresponding term in the momentum conservation equation ().",
"On the other hand, since $\\rho _{n}$ and $\\theta _{n}$ respectively converge strongly in $C(0,T;L^{3/2}(\\Omega ))$ and $L^{2}((0,T)\\times \\Omega )$ , the term $\\nabla (\\rho _{n}\\theta _{n})$ converges to the limit $\\nabla (\\rho \\theta )$ in the sense of distribution.",
"As for the convergence of the viscous term, we rewrite this term as follows: $\\int _{0}^{T}\\int _{\\Omega } \\rho _{n} \\nabla u_{n} \\phi dx dt=- \\int _{0}^{T}\\int _{\\Omega } \\sqrt{\\rho _{n}} \\sqrt{\\rho _{n}} u_{n} \\nabla \\phi dx dt- \\int _{0}^{T}\\int _{\\Omega } \\sqrt{\\rho _{n}}u_{n} \\nabla \\sqrt{\\rho _{n}}\\phi dx dt,$ where $\\phi $ be a test function.",
"Since $\\sqrt{\\rho _{n}}$ converges strongly to $\\sqrt{\\rho }$ in $L^{\\infty }(0,T;L^{2}(\\Omega ))$ and $\\sqrt{\\rho _{n}} u_{n}$ converges strongly to $\\sqrt{\\rho }u$ in $L^{2}(0,T;L^{2}(\\Omega ))$ , the first term on the right-hand of (REF ) converges to the corresponding term in the sense of distribution.",
"The converges of the second term on the right-hand side of (REF ) in the sense of distribution can be shown by the weak convergence $\\nabla {\\sqrt{\\rho _{n}}}$ and the strong converges in $L^{2}(0,T;L^{2}(\\Omega ))$ .",
"Step 3.",
"Convergence of the entropy equation.",
"In view of (REF )-(REF ), it is easy to pass to the limit $n\\rightarrow \\infty $ in all terms appearing in (REF ), except the entropy production rate $\\sigma $ .",
"However, in accordance with (REF ) we still have that $\\lbrace \\sqrt{\\frac{\\rho _{n}}{\\theta _{n}}}D(u_{n}) \\rbrace _{n=1}^{\\infty }.$ is bounded in $L^{2}((0,T)\\times \\Omega )$ .",
"Moreover, by virtue of (REF ), (REF ) and () we deduce $\\sqrt{\\frac{\\rho _{n}}{\\theta _{n}}}D(u_{n})\\rightarrow \\sqrt{\\frac{\\rho }{\\theta }}D(u) .$ Evidently, we may treat all the remaining terms $\\lbrace \\frac{\\sqrt{\\kappa (\\rho _{n},\\theta _{n})}}{\\theta _{n}} \\nabla \\theta _{n} \\rbrace _{n=1}^{\\infty }.$ in the similar way using the fact that they are linear with respect to the weakly convergent sequences of gradients of $\\rho _{n},u_{n},\\theta _{n}$ .",
"Thus, preserving the sign of the entropy inequality (REF ) in the limit $n\\rightarrow \\infty $ follows by the lower semicontinuity of convex superposition of operators.",
"Step 4.",
"Convergence of the total energy balance.",
"It is straight to pass the limit $n\\rightarrow \\infty $ in the total energy balance.",
"tocsectionBibliography"
]
] | 1606.05013 |
[
[
"Radio Galaxy Zoo: discovery of a poor cluster through a giant wide-angle\n tail radio galaxy"
],
[
"Abstract We have discovered a previously unreported poor cluster of galaxies (RGZ-CL J0823.2+0333) through an unusual giant wide-angle tail radio galaxy found in the Radio Galaxy Zoo project.",
"We obtained a spectroscopic redshift of $z=0.0897$ for the E0-type host galaxy, 2MASX J08231289+0333016, leading to M$_r = -22.6$ and a $1.4\\,$GHz radio luminosity density of $L_{\\rm 1.4} = 5.5\\times10^{24}$ W Hz$^{-1}$.",
"These radio and optical luminosities are typical for wide-angle tailed radio galaxies near the borderline between Fanaroff-Riley (FR) classes I and II.",
"The projected largest angular size of $\\approx8\\,$arcmin corresponds to $800\\,$kpc and the full length of the source along the curved jets/trails is $1.1\\,$Mpc in projection.",
"X-ray data from the XMM-Newton archive yield an upper limit on the X-ray luminosity of the thermal emission surrounding RGZ J082312.9+033301,at $1.2-2.6\\times10^{43}$ erg s$^{-1}$ for assumed intra-cluster medium temperatures of $1.0-5.0\\,$keV.",
"Our analysis of the environment surrounding RGZ J082312.9+033301 indicates that RGZ J082312.9+033301 lies within a poor cluster.",
"The observed radio morphology suggests that (a) the host galaxy is moving at a significant velocity with respect to an ambient medium like that of at least a poor cluster, and that (b) the source may have had two ignition events of the active galactic nucleus with $10^7\\,$yrs in between.",
"This reinforces the idea that an association between RGZ J082312.9+033301, and the newly discovered poor cluster exists."
],
[
"Introduction",
"High-resolution radio surveys performed over the past decades have shown the wide variety of radio morphologies of galaxies illustrating the complexity of the underlying physics.",
"The majority of radio sources have compact morphology , while the extended radio-loud sources tend to be Fanaroff and Riley (FR) type I and II .",
"However, there are some extended radio-loud sources that do not fit the standard FRI or FRII classification.",
"The division of tailed radio galaxies into narrow-angle tails (NAT, head) and wide-angle tails (WATs) was introduced by and .",
"Tailed radio galaxies have provided evidence that in both dense and sparse environments, the bending and distortions of radio galaxies are the result of motions with respect to the thermal plasma.",
"WATs and straight FRI sources are often associated with the brightest galaxies in clusters (BCGs).",
"Their radio morphologies reflect both the initial jet momentum and the mild effects of motions and pressure gradients in the intracluster medium , .",
"WATs generally have C-shaped morphologies and have radio luminosities near the FRI and FRII luminosity transition .",
"WATs are found in both merging and relaxed clusters at, or near, the centre and display highly collimated jets.",
"Early models of WATs suggested that ICM ram pressure resulting from velocities $> 1000\\,$ km s$^{-1}$ was required to produce the observed bends , but models using light jets by , , and show that bulk velocities around $100\\,$ km s$^{-1}$ are sufficient to produce WATs.",
"FR II radio sources have their size and shape dominated by the momentum of the overpressured jet.",
"As the jet expands it develops a cocoon and a more collimated jet flows out to power the hot spots.",
"The jets in FRI-NATs undergo a complete disruption, after which they are carried back by external motions alone with no surrounding cocoon.",
"Intermediate between these extremes, FRI-WATs and straight FRIs have cocoons surrounding the outer portions of their jets , , so both jet momentum and motions in the surrounding thermal medium influence the subsequent flow.",
"Questions remain unanswered from both the models and the observations.",
"Are any bright radio spots or knots in the jets powered by an impinging jet?",
"Has the energy supply shut off, so that the knots are in the process of fading due to radiative losses and mechanical dissipation?",
"Is the jet outflow exposed to the external plasma or shielded by a stationary or slower-moving cocoon of relativistic plasma as observed around FRII and some FRI jets?",
"The Radio Galaxy Zoo discovery of a giant WAT (RGZ J082312.9+033301) shows the power of using bent radio sources as tracers of clusters.",
"Upcoming wide-area radio surveys the Evolutionary Map of the Universe , the WODAN survey , and the deeper radio survey MeerKAT MIGHTEE are expected to detect over 100,000 bent radio sources .",
"Radio Galaxy Zoo will allow us to locate these bent radio sources and to investigate the physics that allows jets to be tightly collimated while undergoing significant bending.",
"The present paper is organized as follows.",
"Section describes the discovery of the WAT while in Section we discuss the implications with respect to environment, dynamics, and the central AGN.",
"Section presents our conclusions.",
"Throughout this paper we adopt a $\\Lambda $ CDM cosmology of $\\Omega _{M} = 0.3, \\,\\Omega _{\\Lambda } = 0.7$ with a Hubble constant of $H_{\\rm 0} = 70\\,$ km s$^{-1}$ Mpc$^{-1}$ .",
"With $z=0.0897$ , the luminosity distance is $D_{L}=410\\,$ Mpc and the angular size distance is $D_{A}=345.3\\,$ Mpc giving a scale of $1.674\\,$ kpc arcsec$^{-1}$ .",
"We define the radio spectral index as $S_{\\nu } \\propto \\nu ^{\\alpha }$ ."
],
[
"Wide Angle Tail RGZ J082312.9+033301",
"The discovery of RGZ J082312.9+033301 was made in the citizen science project Radio Galaxy Zoohttp://radio.galaxyzoo.org .",
"Radio Galaxy Zoo offers overlays of the $3.4\\,\\mu $ m image from the Wide-Field Infrared Survey Explorer with the $1.4\\,$ GHz image from the Faint Images of the Radio Sky at Twenty Centimeters , .",
"The $3 \\times 3$ arcmin$^2$ Radio Galaxy Zoo images are centred on the position of the radio sources listed as extended in the FIRST catalogue (version 14 March 2004) and then overlaid on the infrared image as we show in Fig.",
"REF .",
"The radio images are illustrated with radio brightness contours, overlaid on a WISE $3.4\\,\\mu $ m image in a heat scale.",
"RGZ J082312+033301 was identified as an unusual object in December 2013 by two citizen scientists (T. Matorny & I. Terentev) after examining the Radio Galaxy Zoo $3 \\times 3$ arcmin$^2$ cutout (Fig.",
"REF ) of a section of the radio galaxy (N2 in Fig.",
"REF ).",
"Matorny first suggested that the radio emission pointed towards another object by way of the radio extension towards the S.W.. A further investigation of RGZ J082312+033301 by Terentev and Rudnick was completed in RadioTalkhttp://radiotalk.galaxyzoo.org by examining the larger cutout of both the FIRST and WISE images along with images from the NRAO VLA Sky Survey and the Sloan Digital Sky Survey (SDSS) Data Release 10 [3] and Data Release 12 [4].",
"It was then realized that the isolated component seen in Fig.",
"REF was part of a much more extended radio source which could be classified as a WAT.",
"In Fig.",
"REF we show the FIRST image in a color scale and the NVSS image with contours.",
"We found that the core (C in Fig.",
"REF ) is coincident with 2MASX J08231289+0333016 (SDSS J082312.91+033301.3) and has 175 morphology votes from Galaxy Zoo 2 indicating that the host galaxy has morphological features consistent with type E0 with M$_r = -22.6$ and M$_V = -22.3$ .",
"There is no spectrum of SDSS J082312.91+033301.3 in SDSS DR12.",
"We used the Oxford Short Wavelength Integral Field Spectrograph for the Large Telescope (SWIFT) Integral Field Unit (IFU) spectrograph on the Palomar 5-m Hale telescope to obtain an optical spectrum of SDSS J082312.91+033301.3 on the night of 2013 December 29 UT.",
"The target was observed with the large (0.235 arcsec) plate scale in natural seeing.",
"We took two 300s exposures, offset from each other by 40 arcsec along the long axis of the detector to allow background subtraction.",
"We present the one-dimensional spectrum, extracted from a 7 arcec diameter circular aperture, in Fig.",
"REF .",
"Only one secure non-telluric feature is detected in this spectrum: a narrow line centered at 7151.18Å.",
"We identify the line as H$\\alpha $ (rest wavelength of 6562.81Å), providing a redshift of $z = 0.0897\\,\\pm \\,0.0001$ .",
"The uncertainty in the redshift is dominated by the subjective choice of baseline when fitting the continuum level interactively in the IRAF software; the intrinsic resolution of the spectrograph is $R \\approx 4000$ .",
"We note that there is no clear evidence for [N ii] $\\lambda $ 6583 or [S ii] $\\lambda \\lambda $ 6716,31 emission.",
"Archival VLA data at $8\\,$ GHz (project ID AM0593) allow us to determine that the core ($S_{\\rm 8.4} = 12.6\\,\\pm \\,0.2\\,$ mJy) is a flat-spectrum object with $\\alpha = -0.10 \\pm 0.01$ .",
"Table REF lists the possible components of RGZ J082312.9+033301 and the corresponding NVSS and FIRST identifications.",
"Using radio components S2 to N3, we estimate the luminosity density to be $L_{\\rm 1.4}=5.5\\times 10^{24}$ W Hz$^{-1}$ .",
"This places RGZ J082312.9+033301 below the FR I/II boundary in a radio-versus-optical luminosity diagram like Fig.",
"4 of .",
"However, RGZ J082312.9+033301 is still inside the rectangular area where FR Is and FR IIs occur with almost equal frequency, making it an analogue of local FRI radio galaxies like 3C 31.",
"The lack of strong lines in the spectrum suggests that RGZ J082312.9+033301 is a low-excitation radio galaxy (LERG).",
"The Northern section of the radio complex is marked with the labels N1, N2, N3, and N.E.",
"bridge in Fig.",
"REF .",
"In Fig.",
"REF (a) we show the diffuse emission connecting components N2 to N3.",
"The N.E.",
"bridge is detected in the lower-resolution NVSS image at a peak brightness of $\\approx 4$ mJy beam$^{-1}$ with a total flux density of 23 mJy spread over 5 NVSS beams, and is not detected in FIRST.",
"We note that N3 may originate from a faint galaxy SDSS J082328.28+033733.2 (Fig.",
"REF b) at a spectroscopic redshift of $z = 0.2601$ [2].",
"Deeper radio observations are required to determine if component N3 is connected to the rest of RGZ J082312.9+033301.",
"Given the presently available data, there is no hint of diffuse emission connecting components S2 and S3.",
"However, preliminary analysis of the radio structure from DnC configuration Karl G. Jansky Very Large Array observations (Heywood et al.",
"in preparation) indicate a diffuse radio structure to the south of S2 as marginally detected in the NVSS data (Fig.",
"REF ).",
"This structure is not included in the analysis of this current work.",
"The WAT has a projected largest angular size of LAS $\\approx 8\\,$ arcmin, corresponding to $800\\,$ kpc, and the total length along the curved ridge of jets/trails is $\\approx 1.1\\,$ Mpc in projection.",
"This makes this WAT comparable in size to 4C+47.51, which, to our knowledge, is still the largest WAT reported by ."
],
[
"Environment of RGZ J082312+033301",
"As radio galaxies of WAT morphology tend to trace rich environments, in this section we assess the environment of RGZ J082312.9+033301 using the optical and X-ray data available to us.",
"Fig.",
"REF shows the SDSS colour composite image of the environment surrounding the host galaxy of RGZ J082312.9+033301.",
"The dashed circle represents a radius of 1.0 Mpc.",
"Table REF lists all of the galaxies within 31 arcmin of RGZ J082312+033301 and with spectroscopically measured redshifts from SDSS DR12 [4] between $0.08 < z < 0.09$ .",
"We found no reported galaxies with a spectroscopic redshift between $0.09 < z < 0.10$ within our search radius and none between 0.053 and 0.080 within 20$^{\\prime }$ radius.",
"To determine the richness of the cluster environment surrounding RGZJ082312+033301, we use the parameter $N_{1.0}^{-19}$ as described in .",
"This is a background subtracted count of all galaxies brighter than $M_r = -19$ at the redshift of the radio source and within a radius of $1.0\\,$ Mpc around the radio source.",
"The background count rate is determined locally using an annulus centred on the radio source with a radius from 2.7 to $3.0\\,$ Mpc.",
"We find RGZ J082312.9+033301 to be located in an environment with $N_{1.0}^{-19} =42\\,\\pm \\,1$ .",
"This implies that the cluster environment surrounding RGZ J082312.9+033301 is near the poor end of the cluster richness spectrum; the vast majority of Abell clusters analysed by have $N_{1.0}^{-19} > 40$ .",
"However, do not detect a cluster near RGZ J082312.9+033301 in their redMaPPer catalog.",
"The galaxy group which we propose as a possible counterpart of radio component N3 was identified with by as the brightest galaxy of the cluster GMBCG J125.86785+03.62590 with $z_{phot}=0.288$ , as well as by as the brightest cluster galaxy of redMaPPer cluster RM J082328.3+033733.2 with $z_{phot}=0.2647$ and richness $=28\\pm 3$ (as of v5.10 of the cataloguehttp://risa.stanford.edu/redmapper).",
"However, the RGZ WAT host cluster proposed in the present paper is not listed in this cluster catalog, possibly indicating that its richness is $\\lambda \\lesssim 20$ ."
],
[
"Velocity dispersion",
"The velocity dispersion of the environment gives an indication of its virial mass and therefore of its richness.",
"The radial velocity distribution of the 14 galaxies with spectroscopic redshifts within 20 arcmin (2 Mpc) of RGZ J082312.9+033301 is strongly peaked around 25000 km s$^{-1}$ .",
"We ran these velocities through the ROBUST software to estimate the mean velocity and velocity dispersion, finding $C_{BI} =25743\\,\\pm \\,100\\,$ km s$^{-1}$ and $S_{BI} = 373\\,\\pm \\,65\\,$ km s$^{-1}$ .",
"This velocity dispersion would again be consistent with a rich group or poor cluster of galaxies (e.g., ).",
"Interestingly, with a redshift of $z=0.0897$ , RGZ J082312.9+033301 is an outlier in the velocity distribution, with a radial velocity of $cz = 26900$ km s$^{-1}$ , so that it has a relative (or “peculiar”) velocity with respect to the 13 remaining galaxies of $(26900 - 25700)/(1+z) = +1100 \\pm 100$ km s$^{-1}$ .",
"We return to this point below, Section REF ."
],
[
"X-ray emission",
"RGZ J082312.9+033301 lies at the extreme edge of an archival XMM-Newton dataset (Project ID 0721900101).",
"At 14 arcmin from the detector centre, the target is out of the field of view of the two MOS cameras and barely in the pn field of view.",
"The pn data were moderately affected by soft proton flaring and were filtered to an exposure time of 13.8 ks before analysis.",
"RGZ J082312.9+033301 is clearly detected as a point-like source in the pn data coincident with the radio core (C).",
"In the 0.3–0.8 Kev range there are approximately 40 counts after background subtraction in a 30-arcsec radius centred on the detection, just enough to fit a rough spectrum on the assumption of a power-law model with a fixed photon index of 1.8 and Galactic absorption ($N_{\\rm H} = 3.77 \\times 10^{20}$ cm$^{-2}$ , from colden).",
"This gives a background-subtracted 2–10-keV luminosity of $(3 \\pm 1) \\times 10^{41}$ erg s$^{-1}$ , which is entirely consistent with what we might expect from jet-related emission from the unresolved core, given its 1.4-GHz flux density .",
"There is no visual evidence for additional thermal X-ray emission directly surrounding RGZ J082312.9+033301, which in itself rules out a rich cluster environment at this redshift.",
"To make this quantitative we measured counts in a 60$^\\circ $ pie-slice to the NE of RGZ J082312.9+033301 , excluding the AGN and extending out to an AGN-centric radius of 280 arcsec (480 kpc).",
"We find a $3\\sigma $ upper limit on the 0.3–8.0-keV counts in this region of 195, leading to a limit on the counts from an assumed circularly symmetric X-ray environment of $<1170$ counts.",
"The temperature of the non-detected environment is unknown but we convert this limit to a luminosity on the assumption of various temperatures in the range $kT= 1.0$ keV (appropriate for a reasonably rich group) to $kT = 5.0$ keV (a rich cluster), assuming 0.3 solar abundance and the redshift of RGZ J082312.9+033301.",
"The bolometric luminosity upper limits implied by this are between $1.2 \\times 10^{43}$ erg s$^{-1}$ for $kT = 1.0$ keV and $2.6 \\times 10^{43}$ erg s$^{-1}$ for $kT=5.0$ keV, which is certainly not consistent with a rich cluster environment given the well-known temperature-luminosity relation for groups and clusters.",
"It is, however, consistent with the measured velocity dispersion: find that one may expect a luminosity $\\sim 10^{43}$ erg s$^{-1}$ for $S_{BI} \\approx 400$ km s$^{-1}$ .",
"We note that this is also the typical luminosity for a low-excitation radio galaxy of this luminosity found in the study of .",
"It is worth noting that there is a marginally significant detection of extended emission with $90 \\pm 30$ background-subtracted 0.3–8.0-keV counts in a 1-arcmin source circle centred at RA=08h 23m 07.0s, Dec=+03$^\\circ $ 33$^{\\prime }$ 53$^{\\prime \\prime }$ , 1.7 arcmin (170 kpc) to the NW of RGZ J082312.9+033301.",
"Given the signal to noise it is impossible to confirm that this is thermal emission, but it is perfectly possible that it represents the peak of the thermal emission from a group of galaxies with the properties estimated above from the optical data, possibly associated with the bright nearby galaxy SDSS J082306.16+033412.1.",
"The limits we describe above on emission from around RGZ J082312.9+033301 would clearly be consistent with a detection at this level.",
"If so, this would reinforce the idea that the radio galaxy host is somewhat dynamically and physically offset from the rest of the environment.",
"Deeper X-ray data are required to investigate this further.",
"We can conclude, based on the optical and X-ray constraints we have, that the environment of RGZ J082312.9+033301 is consistent with being a rich group or poor cluster of galaxies.",
"We designate this poor cluster as the “Matorny-Terentev Cluster” RGZ-CL J0823.2+0333.",
"Indications of such a cluster have appeared only indirectly in a few references that refer to some of its members as galaxy pairs , , , or to galaxy groups , ."
],
[
"Dynamics of RGZ J082312.9+033301",
"We want to determine if there is a plausible set of jet and medium parameters that could explain the large size and bent radio morphology of RGZ J082312.9+033301.",
"The parameters include the radius of curvature $r_c$ , the radio jet radius $r_r$ , density ratios of the radio jets $\\rho _r$ to that of the cluster environment $\\rho _{\\rm ICM}$ , the velocity of the radio jets $v_r$ , and the velocity of the galaxy with respect to the cluster's barycentre $v_g$ .",
"The high ratio of the velocity of the WAT host to the velocity dispersion of the cluster discussed above (1100 km s$^{-1}$ / 373 km s$^{-1}$ = 2.9, for the galaxies within 20 arcmin) raises the question of whether it could be a background object, and not bound to the cluster.",
"However, as a background object it would not have a significant local thermal plasma to bend the radio structure.",
"Some perspective on this issue comes from the study of the dynamical distribution of X-ray AGN in 26 LoCuSS clusters .",
"They show that AGN have velocities between 1 and 3 times the velocity dispersion of their clusters, with a mean velocity dispersion 1.5 times that of the non-active galaxies.",
"This distribution is indicative of an infalling population, rather than a virialized one.",
"The WAT host is consistent with this behaviour, and might thus be recently encountering the cluster.",
"The spatial offset between the AGN host and the peak of the X-ray emission, if real, would also be consistent with such a picture.",
"Fig.",
"REF allows us to place an upper limit on the radius of the radio jets of $r_r \\le 15\\,$ kpc (9 arcsec) if we use the projected size of the knots or hotspots in the FIRST image.",
"We can also estimate the radius of curvature of the entire WAT to $r_c = 345\\,$ kpc by fitting a circle to connect N3 to S2.",
"In order to determine the densities required to produce the observed bending we use a range of density ratios $\\rho _r/\\rho _{\\rm ICM} = 10^{-6} - 10^{-2}$ based on our observation of RGZ J082312.9+033301 living in a cluster (Section REF ).",
"Using Euler's equation , : $\\frac{\\rho _rv^2_r}{r_c} = \\frac{\\rho _{\\rm ICM}v^2_g}{r_r}\\, ,$ and the values above we find that the velocity of the radio jets is in the range $0.005{\\rm c} < v_r < 0.5{\\rm c}$ .",
"While this could be suggestive of explaining its bent shape, we note that these are only radial velocities, and to see a significant bend, an appreciable transverse (plane-of-sky) velocity is required.",
"In addition, if the equation of state is relativistic then the relativistic enthalpy density is more relevant.",
"The bending depends on the ratio of the jet momentum flux and the ram pressure in the putative cross flow (or, more properly, the pressure change across the jet).",
"If we know or can estimate the ratio of the pressures in the jet and the ambient medium, we can write the bending formula (Eqn.",
"REF ) in terms of that ratio and the ratio of the jet and cross flow Mach numbers.",
"In this form one finds that NATs require a relatively small Mach number for the jet flow, of typically only a few if the jet and ambient pressures are comparable.",
"In a WAT the jet Mach number would be higher relative to the cross flow Mach number.",
"Future radio and X-ray observations will provide the necessary data to constrain these parameters."
],
[
"Possible re-ignition of RGZ J082312.9+033301",
"We find that RGZ J082312.9+033301 displays an unusual radio structure extending over a large linear size.",
"Typical features of WAT radio galaxies are regions brightening along the jet trails, especially around the bends.",
"The brightening regions extend farther into bright diffuse components that slowly fade away in brightness.",
"However, RGZ J082312.9+033301 shows tightly collimated radio structure throughout its extent (Fig.",
"REF ).",
"Here, we speculate if it is possible that the observed bright regions are hotspots typical of FR II radio galaxies rather than knots along the jet paths (Fig.",
"REF ).",
"The existence of hotspots could suggest that the WAT is a re-started (double-double) radio galaxy having had two or three episodes of activity during its lifetime.",
"Assuming the head jet speed is $v=0.05{\\rm c}$ from the velocity range in Section REF , we evaluate the minimum age of the observed radio structures simply as $t_{\\rm min} = d/v_r$ .",
"In Table REF we list the separation between components and the estimated timescale for $z=0.0897$ .",
"We calculate the minimum age of the Northern arm (C – N2) as 19 Myr and of the Southern arm (C – S2) as 20 Myr.",
"It typically takes between $0.1\\,$ Myr and $100\\,$ Myr for the radio structures to dissipate once the central AGN engine switches off , , .",
"With these timescales one would expect hotspots from previous activity to almost entirely fade away leaving behind diffuse emission of the remnant lobes (N2, S2).",
"Component N2 appears to have features consistent with FRII hotspots while component S2 does not, suggesting that the AGN engine has switched off and the hotspots are beginning to dissipate.",
"Radio galaxies have been shown to undergo multiple active phases separated by periods of quiescence time when the jet production is shutdown , , .",
"The dormant/quiescent phase for the AGN may last between $1000\\,$ yr and $100\\,$ Myr , , .",
"The inner components (second phase of AGN activity) may be as young as 5 Myr (C – N1) and 8 Myr (C – S1).",
"The implied dormant phase of the order of $10\\,$ Myr would be consistent with typical timescales for the quiescent phase of radio activity.",
"Therefore, RGZ J082312.9+033301 could have had two episodes of AGN activity.",
"However, the speculative N3 and S3 components would only work in such a scenario if one considers a much rarer case of an inverted triple-double source.",
"A triple-double is a radio galaxy that displays three episodes of activity; at least one such example is known .",
"An inverted double-double is a re-started radio galaxy in which the hotspots from the new activity episode were formed farther away from the radio core than the previous activity older material.",
"This may happen if the density of the previous activity plasma has decreased enough for the re-started jets to pass easily through.",
"This scenario would require the dormant stage of the order of $100\\,$ Myr and no hotspots formed within components N1, S1, N2 and S2 , .",
"We find no evidence of extended relic radio emission around components N3 and S3, but instead rather compact emission.",
"Therefore, RGZ J082312.9+033301 does not show characteristics of an inverted triple-double source.",
"All previously discovered double-double radio galaxies display classical straight radio structures (FR II morphology).",
"If RGZ J082312.9+033301 is indeed a re-started radio source, it would pose significant questions as well as constraints on the evolutionary models of re-started radio galaxies , .",
"To resolve this issue both high resolution high radio frequency, and low radio frequency observations are required, which will allow one to investigate the existence of hotspots and determine the spectral ages of the components.",
"We are currently pursuing observations to address these issues and results will be presented in forthcoming publications."
],
[
"Conclusions",
"We presented evidence for a previously unreported poor cluster of galaxies (Matorny-Terentev Cluster, or RGZ-CL J0823.2+0333) discovered through an unusual giant WAT radio galaxy found with the citizen science project Radio Galaxy Zoo .",
"The host of RGZ J082312.9+033301 is also known as 2MASX J08231289+0333016 and has been classified by volunteers of Galaxy Zoo 2 to be of Hubble type E0.",
"We estimate the $1.4\\,$ GHz luminosity density to be $L_{\\rm 1.4}=5.5\\times 10^{24}\\,$ W Hz$^{-1}$ .",
"Using the Oxford Swift IFU spectrograph on the Palomar 5m telescope we found RGZ J082312.9+033301 to have a redshift $z=0.0897\\,\\pm \\,0.0001$ .",
"At this redshift, we find the largest linear size of RGZ J082312.9+033301 to be $0.8\\,$ Mpc, and measuring $1.1\\,$ Mpc along its curved tails making this giant WAT comparable in size to the largest known WAT 4C+47.51.",
"Investigation of the surrounding environment through the $N^{-19}_{1.0}$ measurement by and through Abell's cluster richness [1] indicate that RGZ J082312.9+033301 lives in a cluster near the poor end of the cluster richness spectrum.",
"Investigation of the surround environment through the $N_{1.0}^{-19}$ measurement by indicates that RGZ J082312.9+033301 is located in a cluster near the poor end of the cluster richness spectrum, consistent with Abell's cluster richness classification [1].",
"However, RGZ-CL J0823.2+0333 was not found in the redMaPPer catalog and our X-ray analysis indicates that RGZ J082312.9+033301 lives within a rich group rather than a cluster.",
"Thus, combining the radio, optical, and X-ray data we conclude that RGZ J082312.9+033301 lives in an unreported poor cluster of galaxies.",
"In order to understand the radio morphology of RGZ J082312.9+033301, we have placed limits on the dynamics of the system.",
"We estimate the velocity of RGZ J082312.9+033301 to be $v_g \\ge 1100\\,$ km s$^{-1}$ with a jet velocity of $0.005{\\rm c} \\le v_r \\le 0.5{\\rm c}$ .",
"Using these values we have shown that RGZ J082312.9+033301 could have multiple re-triggering episodes of active phases with a dormant phase on the order of $10\\,$ Myr.",
"The discovery of RGZ J082312.9+033301 shows the benefit of using bent-tail radio sources as beacons of clusters of galaxies.",
"However, the difficulty lies in detecting them in the data of the upcoming radio surveys like EMU, WODAN and MeerKAT MIGHTEE.",
"Expanding on further citizen science projects building on Radio Galaxy Zoo, or on the future development of machine-learning techniques will be key to locating these bent-tail radio sources to find and study clusters of galaxies."
],
[
"Acknowledgements",
"This publication has been made possible by the participation of more than 8700 volunteers in the Radio Galaxy Zoo project.",
"Their contributions are individually acknowledged at http://rgzauthors.galaxyzoo.org.",
"We thank M. Chow-Martínez for extracting dynamical parameters using the ROBUST software and Nathan Secrest and Nora Loiseau for information on the XMM-Newton data.",
"Parts of this research were conducted by the Australian Research Council Centre of Excellence for All-sky Astrophysics (CAASTRO), through project number CE110001020.",
"OIW acknowledges a Super Science Fellowship from the Australian Research Council.",
"LR, KW and TWJ acknowledge partial support from the U.S. National Science Foundation under grant AST-1211595 to the University of Minnesota.",
"MJH acknowledges support from the UK's Science and Technology Facilities Council [ST/M001008/1].",
"GC acknowledges support from STFC grant ST/K005596/1 and SV acknowledges an doctoral studentship supported by STFC grant ST/N504233/1.",
"SS thanks the Australian Research Council for an Early Career Fellowship DE130101399.",
"NS is the recipient of an Australian Research Council Future Fellowship.",
"This publication makes use of data products from the Wide-field Infrared Survey Explorer and the Very Large Array.",
"The Wide-field Infrared Survey Explorer is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration.",
"The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.",
"This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.",
"The figures in this work made use of Astropy, a community-developed core Python package for Astronomy [5]."
]
] | 1606.05016 |
[
[
"Generating Functions for Inverted Semistandard Young Tableaux and\n Generalized Ballot Numbers"
],
[
"Abstract An inverted semistandard Young tableau is a row-standard tableau along with a collection of inversion pairs that quantify how far the tableau is from being column semistandard.",
"Such a tableau with precisely $k$ inversion pairs is said to be a $k$-inverted semistandard Young tableau.",
"Building upon earlier work by Fresse and the author, this paper develops generating functions for the numbers of $k$-inverted semistandard Young tableau of various shapes $\\lambda$ and contents $\\mu$.",
"An easily-calculable generating function is given for the number of $k$-inverted semistandard Young tableau that \"standardize\" to a fixed semistandard Young tableau.",
"For $m$-row shapes $\\lambda$ and standard content $\\mu$, the total number of $k$-inverted standard Young tableau of shape $\\lambda$ are then enumerated by relating such tableaux to $m$-dimensional generalizations of Dyck paths and counting the numbers of \"returns to ground\" in those paths.",
"In the rectangular specialization of $\\lambda = n^m$ this yields a generating function that involves $m$-dimensional analogues of the famed Ballot numbers.",
"Our various results are then used to directly enumerate all $k$-inverted semistandard Young tableaux with arbitrary content and two-row shape $\\lambda = a^1 b^1$, as well as all $k$-inverted standard Young tableaux with two-column shape $\\lambda=2^n$."
],
[
"Introduction",
"For any non-increasing strong partition $\\lambda = (\\lambda _1,\\lambda _2,\\hdots ,\\lambda _m)$ of the positive integer $N$ , a Young diagram $Y$ of shape $\\lambda $ is a left-justified array of $N$ boxes such that there are precisely $\\lambda _i$ boxes in the $i^{th}$ row of $Y$ .",
"A filling of a Young diagram $Y$ is an assignment of positive integers (possibly repeated) to the boxes of $Y$ such that no integer is skipped.",
"If a filling uses precisely $\\mu _i$ copies of $i$ for each integer $1 \\le i \\le M$ , where $M \\le N$ and $\\mu _1 + \\hdots \\mu _M = N$ , we say that the filling has content $\\mu = (\\mu _1,\\mu _2,\\hdots , \\mu _M)$ .",
"We will oftentimes use the shorthand notation $\\mu = 1^{\\mu _1} 2^{\\mu _2} \\hdots M^{\\mu _M}$ .",
"This paper will utilize several distinct types of fillings.",
"A filling is said to be row-standard if entries increase from left-to-right across each row, and to be column-standard (resp.",
"column-semistandard) if entries increase (resp.",
"weakly increase) from top-to-bottom down each column.",
"If a filling is both row-standard and column-standard, as well as if $\\mu = 1^1 2^1 \\hdots N^1$ , the resulting array $T$ is said to be a standard Young tableau.",
"If a filling is merely row-standard and column-semistandard, the resulting array is said to be a semistandard Young tableau.",
"We denote the set of all standard Young tableaux of shape $\\lambda $ by $S(\\lambda )$ , and the set of all semistandard Young tableaux of shape $\\lambda $ and content $\\mu $ by $S(\\lambda ,\\mu )$ .",
"For a comprehensive introduction to Young tableaux, see Fulton [7].",
"Our primary focus are inversions of Young tableaux, as first introduced by Fresse [5] to calculate the Betti numbers of Springer fibers in type A.An entirely distinct notion also referred to as “tableau inversions\" has been presented by Shynar [10].",
"Adopting the terminology of Beagley and the author [2], [3], let $\\tau $ be a row-standard filling and let $i,j$ be a pair of entries from the same column of $\\tau $ .",
"Let $i_k$ denote the entry precisely $k$ boxes to the right of $i$ in $\\tau $ , and let $j_k$ denote the entry precisely $k$ boxes to the right of $j$ in $\\tau $ .",
"Then $(i,j)$ form an inversion pair of $\\tau $ if $i < j$ and one of the following holds: Either $i_1$ or $j_1$ doesn't exist, and $i$ appears below $j$ .",
"$i_1 > j_1$ .",
"$i_k = j_k$ for all $1 \\le k \\le n$ , either $i_{n+1}$ or $j_{n+1}$ doesn't exist, and $i$ appears below $j$ .",
"$i_k = j_k$ for all $1 \\le k \\le n$ , and $i_{n+1} > j_{n+1}$ .",
"Taken together, the four conditions above identity an instance where the column containing $i,j$ is not in the correct (non-decreasing) order relative to the first column on its right where an appropriate order may be discerned.",
"As such, inversion pairs quantify how far a row-standard tableau is from also being column semistandard.",
"Notice that the absolute vertical placement of $i$ and $j$ within $\\tau $ doesn't necessarily determine whether $(i,j)$ constitutes an inversion pair; this is an entirely “local\" phenomenon that only concerns the ordering of a column relative to the ordering of more rightward columns.",
"See Figure REF for examples of row-standard fillings along with their inversion pairs.",
"Figure: A row-standard filling (left) with inversion pairs (1,2)(1,2), (5,7)(5,7), (6,8)(6,8), and a row-standard filling (right) with inversion pairs (1,2)(1,2), (1,3)(1,3), (4,5)(4,5), (4,5)(4,5).Observe that if $\\tau $ possesses repeated entries, then it is possible for a single inversion pair to appear multiple times within $\\tau $ .",
"The existence of repeated inversion pairs only requires that the entries involved in the repeated inversion pairs $(i,j)$ aren't the same instances of $i$ and $j$ .",
"As such, we will always list the inversion pairs of $\\tau $ with multiplicity.",
"If it ever becomes necessary to specify that an inversion pair $(i,j)$ involves entries from the $k^{th}$ column of $\\tau $ , we adopt the notation $(i,j)^k$ .",
"Furthermore, if we need to specify which instances of a repeated entry are involved in an inversion pair, we will use alphabetic subscripts $i_a$ , $i_b$ , etc.",
"that have been indexed from top-to-bottom in the $k^{th}$ column of $\\tau $ .",
"If the row-standard filling $\\tau $ contains precisely $k$ inversion pairs, we write $\\operatorname{n_{inv}}(\\tau ) = k$ and say that $\\tau $ is a k-inverted semistandard Young tableau.",
"More precisely, if $\\operatorname{n_{inv}}(\\tau ) = k$ and $\\tau $ lacks repeated entries we say that $\\tau $ is a k-inverted standard Young tableau.",
"We denote the set of all $k$ -inverted standard Young tableaux of shape $\\lambda $ by $S_k(\\lambda )$ , and the set of all inverted standard Young tableaux of shape $\\lambda $ by $I(\\lambda ) = \\bigcup _k S_k(\\lambda )$ .",
"Similarly, we denote the set of all $k$ -inverted semistandard Young tableaux with given $\\lambda $ and $\\mu $ by $S_k(\\lambda ,\\mu )$ , and the set of all inverted semistandard Young tableaux of given $\\lambda $ and $\\mu $ by $I(\\lambda ,\\mu ) = \\bigcup _k S_k(\\lambda ,\\mu )$ .",
"Observe that the traditional notions of standard and semistandard Young tableaux correspond to $S_0(\\lambda ) = S(\\lambda )$ and $S_0(\\lambda ,\\mu ) = S(\\lambda ,\\mu )$ , as a row-standard filling is also column-(semi)standard if and only if it has exactly zero inversion pairs.",
"Fresse [5] argued that the columns of any inverted standard Young tableau $\\tau $ may be independently reordered to produce a unique column-standard Young tableau that we refer to as the standardization $\\operatorname{st}(\\tau )$ of $\\tau $ .",
"Fresse's work [5] also allows us to conclude that every inverted standard Young tableau is uniquely determined by its standardization and its collection of inversion pairs.",
"Fixing a standardization $T \\in S(\\lambda )$ , we define $S_k^T(\\lambda ) = \\lbrace \\tau \\in S_k(\\lambda ) \\ \\vert \\ \\operatorname{st}(\\tau ) = T \\rbrace $ and $I^T(\\lambda ) = \\lbrace \\tau \\in I(\\lambda ) \\ \\vert \\ \\operatorname{st}(\\tau ) = T \\rbrace $ .",
"The author [3] later argued that unique standardizations $\\operatorname{st}(\\tau )$ also exist for inverted semistandard Young tableau.",
"Fixing semistandard $T \\in S(\\lambda ,\\mu )$ , we adopt analogous notations $S_k^T(\\lambda ,\\mu ) = \\lbrace \\tau \\in S_k(\\lambda ,\\mu ) \\ \\vert \\ \\operatorname{st}(\\tau ) = T \\rbrace $ and $I^T(\\lambda ,\\mu ) = \\lbrace \\tau \\in I(\\lambda ,\\mu ) \\ \\vert \\ \\operatorname{st}(\\tau ) = T \\rbrace $ .",
"One final piece of terminology that we will repeatedly use is the height order on entries in an inverted Young tableau.",
"As originally introduced by the author [3], you may define a complete order $\\blacktriangleleft $ on the entries $\\lbrace a_i \\rbrace $ in each column of an inverted Young tableau $\\tau $ by working through $\\tau $ one column at a time, from right to left, according to the following rules: If either $a_i$ or $a_j$ lacks a entry directly to its right and $a_i$ appears above $a_j$ , then $a_i \\blacktriangleleft a_j$ .",
"If $b_i$ is directly to the right of $a_i$ , $b_j$ is directly right of $a_j$ , and $b_i < b_j$ , then $a_i \\blacktriangleleft a_j$ .",
"If $b_i$ is directly to the right of $a_i$ , $b_j$ is directly right of $a_j$ , and $b_i = b_j$ with $b_i \\blacktriangleleft b_j$ , then $a_i \\blacktriangleleft a_j$ .",
"If $c \\in \\tau $ is the $k^{th}$ smallest element in its column relative to the height order, we say that $c$ has height $k$ and write $\\operatorname{ht}(c)=k$ .",
"The height order is constructed to tell us how the entries in a column need to be ordered (relative to the columns on its right) if one wants to completely avoid inversion pairs in that column.",
"This observation allows us to succinctly recast the definition of inversion pair: Proposition 1.1 (Proposition 5 of [3]) Let $\\tau $ be an inverted Young tableau and let $i,j$ be distinct entries from the same column of $\\tau $ .",
"Then $(i,j)$ is an inversion pair of $\\tau $ if and only if $i<j$ and $j \\blacktriangleleft i$ ."
],
[
"Outline of Results",
"This paper is composed of three distinct yet interrelated pieces.",
"In Section we present a series of foundational results that formalize the theory of inverted semistandard Young tableau beyond what was already presented by Fresse [5] and the author [3].",
"This includes propositions describing what information is necessary to uniquely identify an inverted semistandard tableau from its standardization (Proposition REF ), and how any inverted semistandard tableau may be obtained from its standardization via a finite series of “partial row transpositions\" (Proposition REF ).",
"Section culminates with Theorem REF , which gives a generating function $\\chi ^T(q)$ for the numbers $\\vert S_k^T (\\lambda ,\\mu ) \\vert $ of $k$ -inverted semistandard Young tableaux with fixed standardization $T \\in S(\\lambda ,\\mu )$ .",
"The $\\operatorname{dp}$ and $\\operatorname{dp^{\\hspace{-1.0pt}*} \\hspace{-1.0pt}}$ operators below represent positive integers known as “inversions depths\" that are easily read from $T$ .",
"Theorem 1.2 (Theorem REF ) Let $T \\in S(\\lambda ,\\mu )$ be a semistandard Young tableaux.",
"Then: $\\chi ^T(q) \\ = \\ \\displaystyle {\\sum _k \\vert S_k^T(\\lambda ,\\mu ) \\vert q^k \\ = \\ \\frac{\\prod _{i,j} [\\operatorname{dp^{\\hspace{-1.0pt}*} \\hspace{-1.0pt}}(a_{ij})+1]_q}{\\prod _{i,j} [\\operatorname{dp^{\\hspace{-1.0pt}*} \\hspace{-1.0pt}}(a_{ij}) - \\operatorname{dp}(a_{ij}) + 1]_q}}$ Section explores the relationship between inverted standard Young tableau and various properties of higher-dimensional Dyck paths, expanding upon the well-known bijection between standard Young tableau $S(\\lambda )$ of shape $\\lambda $ and Dyck paths $\\mathcal {D}_\\lambda $ of the same shape.",
"In the two-row rectangular case, this allows us to give a non-$T$ -specific generating function $\\xi (q)$ for the $\\vert S_k(\\lambda ) \\vert $ in terms of the celebrated Ballot numbers $B(a,b)$ , a result that eventually reappears as Corollary REF : Corollary 1.3 (Corollary REF ) Let $\\lambda = (n,n)$ .",
"Then the $S_k(\\lambda )$ have generating function: $\\xi (q) \\ = \\ \\sum _k \\vert S_k(\\lambda ) \\vert q^k \\ = \\ \\sum _i B(n-1,n-i) (1+q)^i \\ = \\ \\sum _i \\frac{i}{n} \\binom{2n - i - 1}{n-i} (1+q)^i$ That result is generalized to arbitrary $m$ -row tableaux shapes in Theorem REF by relating our generating function $\\xi (q)$ to “generalized ballot numbers\" $\\vert \\mathcal {D}_\\lambda (k_1,\\hdots ,k_{m-1}) \\vert $ , which represent the number of Dyck paths in $\\mathcal {D}_\\lambda $ possessing various numbers of “higher-dimensional returns-to-ground\": Theorem 1.4 (Theorem REF ) Let $\\lambda = (\\lambda _1,\\hdots ,\\lambda _m)$ .",
"Then the $S_k(\\lambda )$ have generating function: $\\xi (q) \\ = \\ \\sum _k \\vert S_k(\\lambda ) \\vert q^k \\ = \\ \\sum _{(i_1,\\hdots ,i_{m-1})} \\left( \\vert \\mathcal {D}_\\lambda (i_1,\\hdots ,i_{m-1}) \\vert \\prod _{j=1}^{m-1} [j+1]_q^{i_j} \\right)$ In Section we utilize the results of earlier sections to directly enumerate $k$ -inverted tableau in several cases that have not previously been attempted.",
"Theorem REF explicitly enumerates $k$ -inverted semistandard Young tableaux for an arbitrary two-row shape $\\lambda =a^1 b^1$ , generalizing earlier enumerations [5], [3] of two-row inverted tableaux in the standard specialization.",
"Theorem REF then provides a generating function for the $\\vert S_k(\\lambda ) \\vert $ in the case of two-column rectangular shapes $\\lambda = 2^n$ .",
"As a whole, notice that this paper follows the purely combinatorial approach of Beagley and the author [2], [3], and will only briefly mention several potential geometric implications for Springer varieties (and Spaltenstein varieties) that may follow from the original work of Fresse [5]."
],
[
"Generating Functions for Inverted Semistandard Young Tableaux",
"Before proceeding to our central enumerative results, we pause to lay the necessary combinatorial groundwork.",
"We begin this foundational section with a thorough treatment of “formal\" results that were not previously addressed by the author [3].",
"This formalism accomplishes for inverted semistandard tableaux what Fresse has already established for inverted standard tableaux [5].",
"More importantly, it provides us with the vocabulary needed to precisely define an explicit generating function $\\chi ^T(q) = \\sum _k \\vert S_k^T(\\lambda ,\\mu )\\vert $ for the number of $k$ -inverted semistandard Young tableaux with a fixed standardization $T \\in S(\\lambda ,\\mu )$ .",
"This section closes with a series of corollaries demonstrating the usefulness of our generating function.",
"It should be noted that, although generating functions for the $\\vert S_k^T(\\lambda ,\\mu ) \\vert $ have yet to appear anywhere in the literature, the specialization of our $\\chi ^T(q)$ to the case of standard Young tableaux recovers Fresse's generating function for the $\\vert S_k^T(\\lambda ) \\vert $ (see Propositions 2.3b and 4.2 of [5]).",
"Yet even in the case of non-repeated entries, our terminology is more direct (avoiding the reference of appropriately defined sub-tableau) and thus allows for more quickly calculable generating functions.",
"As another brief aside, for the rest of this section we will often need to specify which instance of a repeated entry we are dealing with in a tableau.",
"We reserve the term “entry\" if we want to refer to a (potentially repeated) integer in a specific cell, and will use the notation $a_{ij}$ if we want to emphasize that the entry is located in the $i^{th}$ row and $j^{th}$ column of our inverted tableau.",
"If $a_{ij}$ is an instance of the integer $k$ , we will call it an “entry of value $k$ \" and write $a_{ij} = k$ .",
"Our first foundational result follows most directly from the work of Fresse [5].",
"It should be noted that the subsequent proposition may be independently derived via a careful reworking of the “admissible transposition\" procedure that appears in Proposition REF .",
"Proposition 2.1 Take any inverted semistandard Young tableau $\\tau \\in I(\\lambda ,\\mu )$ .",
"$\\tau $ is uniquely identified by its standardization $st(\\tau ) = T$ alongside a collection of inversion pairs $(a_{ij},b_{ij})$ that specify the entries of $T$ involved in each inversion.",
"Take any inverted tableau $\\tau \\in I^T(\\lambda ,\\mu )$ with standardization $\\operatorname{st}(\\tau ) = T$ , and assume $\\sum _k \\mu _k = N$ .",
"For each value $\\alpha $ , we simultaneously place a complete order $\\prec $ on the $\\mu _\\alpha $ copies of $\\alpha $ in both $T$ and $\\tau $ .",
"Let $\\prec $ be the unique complete order such that $\\alpha _k \\prec \\alpha _l$ if $\\alpha _k$ appears in a more leftward column than $\\alpha _l$ , and $\\alpha _k \\prec \\alpha _l$ if $\\alpha _k,\\alpha _l$ appear in the same column and $\\alpha _k \\blacktriangleleft \\alpha _l$ .",
"Label the copies of $\\alpha $ in both $T$ and $\\tau $ according to this complete order, so that $\\alpha _1 \\prec \\hdots \\prec \\alpha _{\\mu _\\alpha }$ .",
"Then re-index all $N$ entries in both $T$ and $\\tau $ according to the map $\\phi (\\alpha _k) = \\mu _1 + \\hdots + \\mu _{(\\alpha -1)}+k$ .",
"This re-indexing replaces $T$ with a standard Young tableau $\\widetilde{T} \\in S(\\lambda )$ and replaces $\\tau $ with an inverted standard Young tableau $\\widetilde{\\tau } \\in I^T(\\lambda )$ .",
"By construction, $\\tau \\mapsto \\widetilde{\\tau }$ represents a bijection between $I^T(\\lambda ,\\mu )$ and all inverted tableau in $I^T(\\lambda )$ that do not possess an inversion pair of the form $(a,b)$ with $\\phi ^{-1}(a) = \\phi ^{-1}(b)$ .",
"In particular, $\\operatorname{n_{inv}}(\\tau ) = \\operatorname{n_{inv}}(\\widetilde{\\tau })$ and $(a,b)^j$ is an inversion pair of $\\tau $ if and only if $(\\phi (a),\\phi (b))^j$ is an inversion pair of $\\widetilde{\\tau }$ .",
"Citing Proposition 2.3a of Fresse [5], we know that $\\widetilde{\\tau }$ is uniquely identified from $\\widetilde{T}$ via its collection of inversion pairs.",
"If we specify which entries with a repeated value are involved in each inversion pair, the aforementioned bijection allows us to conclude that $\\tau $ is uniquely identified from $T$ via a collection of entry-specific inversion pairs.",
"Recall that, with standard tableaux, uniquely identifying an inverted tableaux required only a standardization and a list of the values involved in each inversion pairs.",
"In the semistandard case, as repeated inversion pairs $(a,b)$ are possible it was obviously necessary to be column-specific and to count inversion pairs with multiplicity.",
"Less obvious is the requirement that we must identify the exact location of entries involved in repeated inversion pairs.",
"See Figure REF for an example showing that a specific standardization as well as a collection of inversion pairs, even with column-specific information and accounting for multiplicity, is not enough to specify a unique inverted tableau.",
"Figure: A pair of distinct tableaux in the same set I(λ,μ)I(\\lambda ,\\mu ) with identical standardizations TT and identical inversion pairs (1,2) 1 ,(1,2) 1 (1,2)^1,(1,2)^1.",
"The two tableau are uniquely identified via the entry-specific inversion pairs sets (1 a ,2 a ) 1 ,(1 b ,2 a ) 1 (1_a,2_a)^1,(1_b,2_a)^1 and (1 b ,2 a ) 1 ,(1 b ,2 b ) 1 (1_b,2_a)^1,(1_b,2_b)^1We now define an operation on inverted tableaux whose repeated application may be used to obtain any inverted tableau from its standardization.",
"So take $\\tau \\in I^T(\\lambda ,\\mu )$ , and let $a_{i_1j},a_{i_2j}$ be entries from the $j^{th}$ column of $\\tau $ such that $a_{i_1j} < a_{i_2j}$ and $\\vert \\operatorname{ht}(a_{i_1j}) - \\operatorname{ht}(a_{i_2j}) \\vert = 1$ .",
"A partial row transposition at $(a_{i_1j},a_{i_2j})$ , denoted $a_{i_1j} \\leftrightarrow a_{i_2j}$ , is a transposition of the $i_1^{th}$ and $i_2^{th}$ rows of $\\tau $ from the $j^{th}$ column leftward.",
"Now let $b_1$ denote the entry directly to the right of $a_1$ and let $b_2$ denote the entry directly to the right of $a_2$ , if those entries in fact exist.",
"If $a_{i_1j}$ and $a_{i_2j}$ are both smaller than $b_1$ and $b_2$ (if they exist), the partial row transposition $a_{i_1j} \\leftrightarrow a_{i_2j}$ preserves row-standardness and we refer to the operation as an admissible (partial row) transposition.",
"If $\\tau \\in I^T(\\lambda ,\\mu )$ , the fact that an admissible transposition $a_{i_1j} \\leftrightarrow a_{i_2j}$ doesn't permute entries between columns means that the resulting tableau $\\tau ^{\\prime }$ is also an element of $I^T(\\lambda ,\\mu )$ .",
"$\\tau ^{\\prime }$ is related to $\\tau $ in that it flips $\\operatorname{ht}(a_{i_1j})$ and $\\operatorname{ht}(a_{i_2j})$ while fixing the height of every other element.",
"As $a_{i_1j}$ and $a_{i_2j}$ are consecutive elements in the height order on the $j^{th}$ column of $\\tau $ , it follows that the admissible transposition $a_{i_1j} \\leftrightarrow a_{i_2j}$ either adds or removes the inversion pair $(a_{i_1j},a_{i_2j})$ while leaving other inversion pairs unchanged.",
"These observations allow us to assert the following: Proposition 2.2 Every inverted tableau $\\tau \\in I^T(\\lambda ,\\mu )$ may be obtained from its standardization $\\operatorname{st}(\\tau ) = T$ via a finite sequence of admissible transpositions.",
"Furthermore, the minimum number of admissible transpositions needed to obtain $\\tau $ from $T$ is $\\operatorname{n_{inv}}(\\tau )$ .",
"Taking $\\tau \\in I^T(\\lambda ,\\mu )$ , we utilize Proposition REF to uniquely identify $\\tau $ via its collection of entry-specific inversion pairs $S = \\lbrace (a_\\alpha ,b_\\alpha )^{j_\\alpha } \\rbrace $ .",
"As each admissible transposition adds precisely one inversion pair, obtaining $\\tau $ from $T$ with fewer than $\\operatorname{n_{inv}}(\\tau )$ transpositions is clearly impossible.",
"To obtain $\\tau $ with precisely $\\operatorname{n_{inv}}(\\tau )$ transpositions, we add inversion pairs one column at a time, from right to left.",
"Within each column, we work through entries from top-to-bottom via their placement in $T$ , assuming that the relative ordering of repeated instances with a fixed value in the $j^{th}$ column is unchanged as we pass from $T$ through various elements of $I^T(\\lambda ,\\mu )$ .",
"So assume that we have followed this procedure up to the $j^{th}$ column of $T$ , whose entries we denote $a_{1j} \\le a_{2j} \\le \\hdots a_{hj}$ from top-to-bottom, and that we are ready to add inversions whose larger element is the entry $a_{ij}$ .",
"Let $\\widetilde{\\tau } \\in I^T(\\lambda ,\\mu )$ denote the intermediate tableau that immediately precedes the addition of these inversion pairs.",
"Notice that, as $a_{ij}$ has yet to be the site of a partial row transposition, $a_{ij}$ still possesses its original height of $\\operatorname{ht}(a_{ij}) = i$ in $\\widetilde{\\tau }$ .",
"Furthermore, the entries of lower height in the $j^{th}$ column of $\\widetilde{\\tau }$ are $a_{1j},\\hdots ,a_{(i-1)j}$ , in some (possibly permuted) order.",
"Reverse index those entries according to their height order in $\\widetilde{\\tau }$ as $c_{i-1} \\blacktriangleleft \\hdots \\blacktriangleleft c_1$ .",
"If $a_{ij}$ is the larger entry in precisely $m$ inversion pairs in $\\tau $ , notice that those inversion pairs must be $(c_1,a_{ij})^j,\\hdots ,(c_m,a_{ij})^j$ .",
"Otherwise, $\\tau $ would have possessed an additional inversion pair of the form $(c_\\beta ,c_\\gamma )$ that we had failed to account for at an earlier step.",
"Then perform the $m$ admissible transpositions $c_1 \\leftrightarrow a_{ij}, \\hdots , c_m \\leftrightarrow a_{ij}$ in the stated order.",
"Continuing this procedure through all entries $a_{ij}$ of $T$ yields the desired inverted tableau $\\tau $ .",
"See Figure REF for an example of the procedure from the proof of Proposition REF .",
"Notice that this procedure provides one minimal sequence of admissible transpositions that yields $\\tau $ from $\\operatorname{st}(\\tau ) = T$ .",
"In general, there are many such sequences, even if one performs only the minimum possible number of $\\operatorname{n_{inv}}(\\tau )$ transpositions.",
"The author suspects that there exists a rich theory of composition patterns for admissible row transpositions of row-standard tableau.",
"As ordinary permutations $\\sigma \\in S_n$ correspond to inverted tableau of shape $\\lambda = 1^n$ , this topic would directly generalize the existing theory of reduced word decompositions for permutations.",
"Figure: Obtaining τ\\tau with inversion pairs (1,2) 1 ,(1,3) 1 ,(4,5 a ) 2 ,(4,5 b ) 2 (1,2)^1,(1,3)^1,(4,5_a)^2,(4,5_b)^2 from st(τ)=T\\operatorname{st}(\\tau )=T via a sequence of admissible transpositions.",
"5 a 5_a and 5 b 5_b denote the two entries of value 5 in second column, with 5 a ◂5 b 5_a \\blacktriangleleft 5_b.We are now set to develop the language needed for our generating function $\\chi ^T(q) = \\sum _k \\vert S_k^T(\\lambda ,\\mu )\\vert $ .",
"So take a semistandard Young tableau $T \\in S(\\lambda ,\\mu )$ , and let $a_{ij}$ be an entry of value $k$ from the $j^{th}$ column of $T$ .",
"We define the (standard) inversion depth of $a_{ij}$ , denoted $\\operatorname{dp}(a_{ij})$ , to equal the number of entries $b$ in the $j^{th}$ column of $T$ such that $b<k$ minus the number of entries $c$ in the $(j+1)^{st}$ column of $T$ such that $c<k$ (and this second number is taken to be zero if $T$ has no $(j+1)^{st}$ column).",
"We similarly define the modified inversion depth of $a_{ij}$ , denoted $\\operatorname{dp^{\\hspace{-1.0pt}*} \\hspace{-1.0pt}}(a_{ij})$ to equal the total number of entries above $a_{ij}$ in the $j^{th}$ column of $T$ minus the number of entries $c$ in the $(j+1)^{st}$ column of $T$ such that $c<k$ (once again taken to be zero if $T$ has no $(j+1)^{st}$ column).",
"For an example of a semistandard Young tableau alongside the inversion depths of each of its entries, standard and modified, see Figure REF .",
"Notice that, if $a_{ij}$ is the only entry of value $k$ in the $j^{th}$ column of of $T$ , then $\\operatorname{dp}(a_{ij}) = \\operatorname{dp^{\\hspace{-1.0pt}*} \\hspace{-1.0pt}}(a_{ij})$ .",
"In particular, if $T$ is a standard Young tableaux, then $\\operatorname{dp}(a_{ij}) = \\operatorname{dp^{\\hspace{-1.0pt}*} \\hspace{-1.0pt}}(a_{ij})$ for all entries $a_{ij}$ .",
"In the case of $T$ a standard Young tableaux, also notice that our $\\operatorname{dp}(a_{ij})$ correspond with the $p_k$ defined by Fresse [5] as the number of appropriately long rows in the sub-tableau $T[1,\\hdots ,k]$ .",
"Figure: A semistandard Young tableaux with the inversion depth (center) and modified inversion depth (right) of each of its entries.",
"Entries where dp(a ij )≠error(a ij )\\operatorname{dp}(a_{ij}) \\ne \\operatorname{dp^{\\hspace{-1.0pt}*} \\hspace{-1.0pt}}(a_{ij}) have been highlighted.Standard inversion depth $\\operatorname{dp}(a_{ij})$ has been defined to equal the number of distinct admissible partial row-transpositions possible in the $j^{th}$ column of $T$ where $a_{ij}$ is the larger entry being transposed.",
"This is equivalent to saying that $\\tau \\in I^T(\\lambda ,\\mu )$ may possess at most $\\operatorname{dp}(a_{ij})$ distinct inversion pairs of the form $(a_{i^{\\prime }j},a_{ij})^j$ .",
"Despite placing a quick upper bound on $\\operatorname{n_{inv}}(\\tau )$ for $\\tau \\in I^T(\\lambda ,\\mu )$ , knowing the standard inversion depth of each element in $T$ will not be sufficient if one wishes to determine the generating function $\\chi ^T(q)$ .",
"This is because standard inversion depth doesn't easily account for inversion pairs when the $j^{th}$ column of $T$ contains multiple entries of a fixed value $k$ .",
"Luckily, modified inversion depth is perfectly suited to this task and immediately proves useful in the development of generating functions $\\chi ^T(q)$ for one-column semistandard Young tableaux.",
"As presented below, Lemma REF actually represents little more than a rewording of Theorem 13 from [3], which gave a generating function for the $\\vert S_k(\\lambda ,\\mu ) \\vert $ when $\\lambda = 1^n$ .",
"In what follows we use the standard notation for the $q$ -number $[p]_q = 1 + q + \\hdots + q^{p-1}$ , the $q$ -factorial $[p]_q!",
"= [1]_q [2]_q \\hdots [p]_q$ , and the $q$ -binomial coefficients $\\binom{a}{b}_q = \\frac{[a]_q!}{[b]_q![a-b]_q!",
"}$ .",
"Proposition 2.3 Let $T \\in S(\\lambda ,\\mu )$ be a one-column semistandard Young tableau of shape $\\lambda = 1^n$ .",
"Then: $\\chi ^T(q) = \\displaystyle {\\sum _k \\vert S_k^T(\\lambda ,\\mu ) \\vert q^k \\ = \\ \\frac{\\prod _i [\\operatorname{dp^{\\hspace{-1.0pt}*} \\hspace{-1.0pt}}(a_{i1})+1]_q}{\\prod _i [\\mu _i]_q!",
"}}$ Notice that when $\\lambda = 1^n$ that $S(\\lambda ,\\mu )$ consists of a single column-semistandard tableau, which we label $T$ .",
"Theorem 13 of [3] showed that $\\sum _k \\vert S_k^T(\\lambda ,\\mu ) \\vert q^k = \\sum _k \\vert S_k(\\lambda ,\\mu ) \\vert q^k = \\frac{[n]_q!",
"}{\\prod _i [\\mu _i]_q!",
"}$ .",
"For a one-column tableau, the lack of rightward entries ensures $\\operatorname{dp^{\\hspace{-1.0pt}*} \\hspace{-1.0pt}}(a_{i1}) = i-1$ for all $a_{i1}$ .",
"Thus $[n]_q!",
"= \\prod _i [\\operatorname{dp^{\\hspace{-1.0pt}*} \\hspace{-1.0pt}}(a_{i1}) + 1]_q$ and the result directly follows.",
"Before generalizing Lemma REF to the multi-column case we pause to observe that, if $a_{ij}$ is the $m^{th}$ occurrence of value $k$ in the $j^{th}$ column of $T$ , then $\\operatorname{dp}(a_{ij}) + m - 1 = \\operatorname{dp^{\\hspace{-1.0pt}*} \\hspace{-1.0pt}}(a_{ij})$ .",
"It follows that $\\operatorname{dp^{\\hspace{-1.0pt}*} \\hspace{-1.0pt}}(a_{ij}) - \\operatorname{dp}(a_{ij})$ measures the number of instances of $k$ that lie above $a_{ij}$ in the $j^{th}$ column of $T$ .",
"Theorem 2.4 Let $T \\in S(\\lambda ,\\mu )$ be a semistandard Young tableaux.",
"Then: $\\chi ^T(q) \\ = \\ \\displaystyle {\\sum _k \\vert S_k^T(\\lambda ,\\mu ) \\vert q^k \\ = \\ \\frac{\\prod _{i,j} [\\operatorname{dp^{\\hspace{-1.0pt}*} \\hspace{-1.0pt}}(a_{ij})+1]_q}{\\prod _{i,j} [\\operatorname{dp^{\\hspace{-1.0pt}*} \\hspace{-1.0pt}}(a_{ij}) - \\operatorname{dp}(a_{ij}) + 1]_q}}$ Proposition REF has already shown that every inverted semistandard Young tableau is uniquely determined by its standardization and a collection of entry-specific inversion pairs.",
"Fixing $T \\in S(\\lambda ,\\mu )$ , we merely need to determine which collections of inversion pairs are valid in the sense that they actually describe a tableau $\\tau \\in I^T(\\lambda ,\\mu )$ that is row-standard.",
"When reordering the columns of $T$ to produce an arbitrary inverted tableau $\\tau \\in I^T(\\lambda ,\\mu )$ , notice that a rearrangement of entries in a leftward column in no way effects inversion pairs in rightward columns.",
"As inversion pairs are determined by height order (as opposed to vertical placement), we also know that a particular ordering of a rightward column does not effect whether a specific inversion pair is possible in a more leftward column.",
"These observations allow us to independently determine valid collections of inversion pairs for $I^T(\\lambda ,\\mu )$ one column at a time.",
"So consider the $j^{th}$ column of $T$ , and assume that the $j^{th}$ column contains precisely $c_k$ entries of value $k$ .",
"Modifying the technique used in Theorem 13 of [3], we consider the number of different ways to “build up\" a valid height order on the $j^{th}$ column by recursively inserting all $c_k$ copies of $k$ into the height order of an “intermediate\" column with content $1^{c_1}2^{c_2} \\hdots (k-1)^{c_{k-1}}$ .",
"This yields a sequence $\\lbrace \\rho _1,\\rho _2,\\hdots ,\\rho _m \\rbrace $ of partial columns such that $\\rho _m$ describes the height order of the entire $j^{th}$ column of a row-standard tableau.",
"Notice that distinct insertions at any step in this procedure result in distinct height orders on the resulting column.",
"Also notice that a valid placement of the $c_k$ copies of $k$ is not subsequently made invalid upon the insertion of larger entries, as the placement of later entries may only increase the height order of a specific entry.",
"Lastly, note that the number of inversion pairs in the $j^{th}$ column whose larger element if $k$ is completely determined by the step in our procedure where we insert the $c_k$ copies of $k$ into $\\rho _{k-1}$ to produce $\\rho _k$ .",
"So assume that we have a valid intermediate column $\\rho _{k-1}$ and that we are prepared to insert the $c_k$ copies of $k$ in a way that preserves row-standardness.",
"By the definition of standard inversion depth, each instance $a_{ij}$ of $k$ may be involved in up to (a fixed number) of $\\operatorname{dp}(a_{ij}) = \\operatorname{dp}(k)$ inversion pairs where it is the larger element.",
"The number of such inversion pairs involving a specific instance $a_{ij}$ is determined by the number of entries in $\\rho _{k-1}$ that it is moved ahead of in the height order.",
"If we want the resulting column $\\rho _k$ to possess precisely $l$ inversion pairs whose larger entry is some instance $k$ , it follows that such arrangements are in bijection with partitions of $l$ into at most $c_k$ parts where each part has size at most $\\operatorname{dp}(k)$ .",
"Recall that the coefficient of $q^l$ in $\\binom{\\alpha + \\beta }{\\alpha }_q$ equals the number of partitions of $l$ into at most $\\alpha $ parts, each of which has size at most $\\beta $ .",
"If $k_1,\\hdots ,k_{c_k}$ denote the $c_k$ instances of $k$ in the $j^{th}$ column (read from top to bottom), this means that the number of different ways to produce a row-standard intermediate clumn $\\rho _k$ with various numbers of inversions whose larger entry is $k$ has generating function: $f_{j,k}(q) \\ = \\ \\binom{\\operatorname{dp}(k) + c_k}{c_k}_q \\ = \\ \\frac{[\\operatorname{dp}(k)+1]_q \\hdots [\\operatorname{dp}(k)+c_k]_q}{[c_k]_q!}",
"\\ = \\ \\frac{\\prod _{i=1}^{c_k}[\\operatorname{dp^{\\hspace{-1.0pt}*} \\hspace{-1.0pt}}(k_i)+1]_q}{\\prod _{i=1}^{c_k}[\\operatorname{dp^{\\hspace{-1.0pt}*} \\hspace{-1.0pt}}(a_{ij}) - \\operatorname{dp}(a_{ij}) + 1]_q}$ The last equality above uses the aforementioned fact that the $m^{th}$ instance $k_m$ of value $k$ in the $j^{th}$ column satisfies $\\operatorname{dp}(k_m) + m - 1 = \\operatorname{dp^{\\hspace{-1.0pt}*} \\hspace{-1.0pt}}(k_m)$ .",
"Ranging over all values in the $j^{th}$ column of $T$ , and then over all columns in $T$ , gives the formula from the theorem.",
"Example 2.5 If $T$ is the tableau of Figure REF , Theorem REF gives generating function: $\\chi ^T(q) \\ = \\ \\sum _k \\vert S_k^T(\\lambda ,\\mu ) \\vert q^k \\ = \\ \\frac{(1+q)^4 (1+q+q^2)^4}{(1+q)^2}$ $= \\ 1 + 6q + 19q^2 + 40q^3 + 61q^4 + 70q^5 + 61q^6 + 40q^7 + 19q^8 + 6q^9 + q^{10}$ We close this section by drawing a number of quick corollaries from Theorem REF .",
"First notice that, if $T$ lacks repeated entries, then $\\operatorname{dp^{\\hspace{-1.0pt}*} \\hspace{-1.0pt}}(a_{ij}) = \\operatorname{dp}(a_{ij})$ for all $a_{ij}$ and the expression of Theorem REF may be reduced to a generating function that is equivalent to the one introduced by Fresse [5]: Corollary 2.6 Let $T \\in S(\\lambda )$ be a standard Young tableaux.",
"Then: $ \\chi ^T(q) \\ = \\ \\sum _k \\vert S_k^T(\\lambda ) \\vert q^k \\ = \\ \\prod _{i,j} [\\operatorname{dp}(a_{ij})+1]_q$ Now notice that our generating function $\\chi ^T(q)$ is always monic, meaning that there is precisely one “maximally inverted\" tableau $\\tau \\in I^T(\\lambda ,\\mu )$ for each choice of $T \\in S(\\lambda ,\\mu )$ .",
"This maximally inverted tableau always has degree $m_T = \\prod _{i,j} \\operatorname{dp^{\\hspace{-1.0pt}*} \\hspace{-1.0pt}}(a_{ij})/\\prod _{i,j} (\\operatorname{dp^{\\hspace{-1.0pt}*} \\hspace{-1.0pt}}(a_{ij}) - \\operatorname{dp}(a_{ij}))$ .",
"Corollary 2.7 Let $T \\in S(\\lambda ,\\mu )$ be any semistandard Young tableau, and define $m_T$ as above.",
"Then $\\vert S_k^T(\\lambda ,\\mu ) \\vert = 0$ for all $k > m_T$ and $\\vert S_{m_T}^T(\\lambda ,\\mu ) \\vert = 1$ .",
"Compare Corollary REF to Theorem 7 of [3], in which the author determined a maximal inversion number $M_{\\lambda ,\\mu }$ when ranging over all standardizations $T \\in S(\\lambda ,\\mu )$ and showed that precisely one $\\tau _{max} \\in I(\\lambda ,\\mu )$ actually realized that number of inversions.In [3], it is shown that $M_{\\lambda ,\\mu } = \\sum _j T_{(h_j -1)} - \\sum _i T_{(\\mu _i - 1)}$ , where $T_k$ is the triangle number and $h_j$ is the height of the $j^{th}$ column in any tableau of shape $\\lambda $ .",
"The unique element of $I(\\lambda ,\\mu )$ obtaining $M_{\\lambda ,\\mu }$ inversions was labelled $\\tau _{max}$ , and its standardization was denoted $\\operatorname{st}(\\tau _{max}) = T^*$ .",
"Applying Corollary REF to the language of Theorem 7 from [3] allows us to conclude that the standardization $T^* = \\operatorname{st}(\\tau _{max})$ is the unique element of $S(\\lambda ,\\mu )$ whose generating function $\\chi ^T(q)$ obtains the maximal possible degree of $M_{\\lambda ,\\mu }$ .",
"In order to draw our final corollaries, notice that $\\chi ^T(q)$ is always a product of palindromic unimodal polynomials of the form $\\binom{\\alpha }{\\beta }_q$ .",
"As the product of two palindromic unimodal polynomial is itself palindromic and unimodal (see Proposition 1 of Stanley [13]), we have both of the following: Corollary 2.8 Let $T \\in S(\\lambda ,\\mu )$ be any semistandard Young tableaux.",
"Then $\\chi ^T(q)$ is a palindromic polynomial.",
"In particular, if $m_T = \\deg (\\chi ^T(q))$ then $\\displaystyle {\\vert S_k^T(\\lambda ,\\mu ) \\vert = \\vert S_{m_T - k}^T(\\lambda ,\\mu ) \\vert }$ for all $k$ .",
"Corollary 2.9 Let $T \\in S(\\lambda ,\\mu )$ be any semistandard Young tableaux.",
"Then $\\chi ^T(q)$ is unimodal.",
"Pause to observe that the notion of unimodality addressed in Corollary REF is distinct from the unimodality proven by Fresse, Mansour and Melnikov [6].",
"Corollary REF proves the unimodality of $\\chi ^T(q) = \\sum _k \\vert S_k^T(\\lambda ,\\mu ) \\vert $ for fixed standardization $T$ yet for arbitrary $\\lambda ,\\mu $ .",
"Fresse, Mansour and Melnikov restrict themselves to standard tableau and prove the unimodality of $\\xi (q) = \\sum _T \\chi ^T(q) = \\sum _k \\vert S_k(\\lambda ) \\vert $ for specific (relatively) easy choices of $\\lambda $ , such as two-row and “hook-shaped\" cases.",
"Although the unimodality of $\\chi ^T(q)$ is nearly immediate, log-convexity of $\\chi ^T(q)$ does not necessarily follow because the $q$ -binomial coefficients that constitute $\\chi ^T(q)$ need not be log-concave.",
"For example, take the sole semistandard tableau $T$ of $S(\\lambda ,\\mu )$ for $\\lambda = 1^4$ and $\\mu = 1^2 2^2$ (i.e.- the standardization of both tableaux from Figure REF ).",
"This $T$ has $\\chi ^T(q) = \\binom{4}{2}_q = 1 + q + 2q^2 + q^3 + q^4$ , which is palindromic and unimodal but not log-concave.",
"If however we restrict our attention to standard Young tableau, from Corollary REF we see that $\\chi ^T(q)$ is a product of log-concave polynomials of the form $[\\alpha ]_q$ and hence is itself log-concave: Corollary 2.10 Let $T \\in S(\\lambda )$ be any standard Young tableaux.",
"Then $\\chi ^T(q)$ is log-concave."
],
[
"Inverted Young Tableaux & Generalized Ballot Numbers",
"In Section we fixed a semistandard Young tableau $T \\in S(\\lambda ,\\mu )$ and developed a generating function for the numbers of $k$ -inverted tableau $\\vert S_k^T(\\lambda ,\\mu ) \\vert $ whose standardization was $T$ .",
"It is natural to ask whether our results for the $\\vert S_k^T(\\lambda ,\\mu ) \\vert $ aid in the enumeration of the entire sets $S_k(\\lambda ,\\mu )$ or in the development of a non-$T$ -specific generating function $\\xi (q) = \\sum _T \\chi ^T(q) = \\sum _k \\vert S_k(\\lambda ,\\mu ) \\vert $ ?",
"As noted by Beagley and the author [2], [3], this widening of scope is extremely difficult because it requires specific knowledge of every $T \\in S(\\lambda ,\\mu )$ .",
"In particular, a direct generalization would require knowledge of how many $T \\in S(\\lambda ,\\mu )$ possess a fixed generating function $\\chi ^T(q)$ .",
"It is because of these difficulties that previous attempts at enumerating the $S_k(\\lambda ,\\mu )$ have disregarded the $S_k^T(\\lambda ,\\mu )$ and utilized more direct techniques.",
"Even in the standard case, such techniques have only been previously applied to calculate $S_k(\\lambda )$ for specific easy choices of $k$ and $\\lambda $ .",
"See Fresse, Mansour and Melnikov [6] for calculations of $\\vert S_k (\\lambda ) \\vert $ when $\\lambda $ is a two-row or “hook\" shape, as well as Beagley and the author [2] for an independent verification of the two-row case along with the $k=1$ case for an arbitrary shape $\\lambda $ .",
"The situation for semistandard tableaux is even more daunting, as there does not even exist a generalized “hook-length formula\" to enumerate semistandard Young tableaux $\\vert S(\\lambda ,\\mu ) \\vert $ with fixed shape and content.",
"The author [3] was still able to determine $\\vert S_k(\\lambda ,\\mu ) \\vert $ for arbitrary $\\mu $ when $\\lambda $ was a two-row or one-column shape, conveniently corresponding to the cases where calculating $\\vert S(\\lambda ,\\mu ) \\vert $ is relatively straightforward.",
"For an arbitrary shape $\\lambda $ , the author [3] also placed $S_1(\\lambda ,\\mu )$ in bijection with $\\bigcup _{\\lambda _i} S_0(\\lambda _i,\\mu )$ for a certain finite collection of related shapes $\\lambda _i$ .",
"For the remainder of this paper, we present two new approaches for enumerating the $S_k(\\lambda ,\\mu )$ .",
"In this section, we restrict ourselves to the standard case and utilize the bijection between $m$ -row standard Young tableaux and $m$ -dimensional Dyck paths to explicitly determine how many $T \\in S(\\lambda )$ have a fixed generating function $\\chi ^T(q)$ .",
"This is done first in the familiar two-dimensional case, which motivates the general $m$ -row case.",
"In all that follows, let $S(\\lambda ,\\mu ) |_\\chi \\subset S(\\lambda ,\\mu )$ denote the set of semistandard Young tableau that have a fixed generating function $\\chi ^T(q)$ , a notation that we predictably adapt to the standard case as $S(\\lambda ) |_\\chi $ .",
"If $\\lambda $ and $\\mu $ are understood, we significantly shorten notation for the cardinalities of these sets to $\\phi (\\chi ) = \\vert S(\\lambda ,\\mu ) |_\\chi \\vert $ .",
"Clearly $\\xi (q) = \\sum _\\chi \\phi (\\chi ) \\chi ^T(q)$ ."
],
[
"2-Dimensional Dyck Paths and Two-Row Inverted Young Tableaux",
"We begin by recapping basic results about traditional (two-dimensional) Dyck paths and how they relate to standard Young tableaux.",
"Let $\\lambda = (a,b)$ , where $a,b$ are positive integers.",
"A Dyck path of shape $\\mathbf {\\lambda }$ is an integer lattice path from $(0,0)$ to $(a,b)$ utilizing only “East\" $E = (1,0)$ and “North\" $N=(0,1)$ steps such that every point $(x,y)$ along the path satisfies $y \\le x$ .",
"We denote the set of all Dyck paths of shape $\\lambda $ by $\\mathcal {D}_\\lambda = \\mathcal {D}_{(a,b)}$ .",
"A particular Dyck path $P \\in \\mathcal {D}_\\lambda $ will often be specified by $P = \\lbrace v_0,\\hdots ,v_{a+b} \\rbrace $ , where the $v_i$ are the integer lattice points along the path.",
"Thus $v_0 = (0,0)$ , $v_{a+b} = (a,b)$ , and $v_i - v_{i-1} = (1,0)$ or $(0,1)$ for all $1 \\le i \\le a+b$ .",
"Obviously $\\mathcal {D}_{(a,b)} = \\emptyset $ if $a>b$ .",
"If $a=b=n$ , it is well-known that the cardinality of $\\mathcal {D}_{(n,n)} = \\mathcal {D}_n$ equals the $n^{th}$ Catalan number $C_n = \\frac{1}{n+1} \\binom{2n}{n}$ .",
"More generally, there is a well-studied bijection between $\\mathcal {D}_\\lambda $ and standard Young tableaux of shape $\\lambda $ .",
"See Figure REF for an example of this straightforward bijection, in which $E$ steps of the Dyck path correspond to values in the first row of the associated tableau and $N$ steps of the Dyck path correspond to values in the second row of the tableau.",
"Figure: A standard Young tableau of shape λ=(4,3)\\lambda = (4,3) and the corresponding Dyck path in 𝒟 λ \\mathcal {D}_\\lambda .There are many important statistics on Dyck paths.",
"The statistic that will prove useful here is a Dyck path's number of “returns to ground\".",
"Formally, a path $P = \\lbrace v_0,\\hdots , v_{a+b} \\rbrace $ in $\\mathcal {D}_{(a,b)}$ has a return to ground at $v_i$ if $v_i = (x,x)$ for some positive integer $x$ .",
"Observe that $v_0 = (0,0)$ does not qualify as a return to ground.",
"If $P \\in \\mathcal {D}_{\\lambda }$ has precisely $k$ returns to returns to ground, we write $\\operatorname{ret}(P)=k$ .",
"We also establish the notation $\\mathcal {D}_\\lambda (k) = \\lbrace P \\in \\mathcal {D}_\\lambda \\ \\vert \\ \\operatorname{ret}(P) = k \\rbrace $ .",
"When $\\lambda = (n,n)$ , the sets $\\mathcal {D}_n(k)$ are related to the famed ballot numbers $B(\\alpha ,\\beta ) = \\frac{\\alpha - \\beta + 1}{\\alpha + 1} \\binom{\\alpha + \\beta }{\\alpha }$ as $\\vert \\mathcal {D}_n(k) \\vert = B(n-1,n-k) = \\frac{k}{n} \\binom{2n-k-1}{n-1}$ .",
"These ballot numbers are frequently presented as entries in the so-called Catalan triangle of Figure REF , whose rows sum to the Catalan numbers as $\\sum _\\beta B(\\alpha ,\\beta ) = C_\\alpha $ .",
"This means that our $\\vert \\mathcal {D}_n(k) \\vert = B(n-1,n-k)$ may be read by proceeding from right-to-left along the appropriate row of Catalan's triangle.",
"See sequence A009766 on OEIS [11] for a thorough overview of Catalan triangle's, as well as Forder [4] or Barcucci and Verri [1] for earlier investigations of the Ballot numbers.",
"Figure: Rows 0 through 5 of the Catalan triangle, whose (α,β)(\\alpha ,\\beta ) entry is the Ballot number B(α,β)=α-β+1 α+1α+β αB(\\alpha ,\\beta ) = \\frac{\\alpha -\\beta +1}{\\alpha +1} \\binom{\\alpha +\\beta }{\\alpha }.",
"Note that the leftmost column of the triangle corresponds to β=0\\beta = 0.Yet how does all of this relate to inverted standard Young tableaux?",
"In the example of Figure REF , notice that the associated tableau $T$ has $\\operatorname{dp}(4)=\\operatorname{dp}(6)=1$ and $\\operatorname{dp}(i)=0$ for all other entries $i \\in T$ .",
"This suggests that a return to ground at $v_i$ in $P \\in \\mathcal {D}_\\lambda $ is related to the inversion depth of $i$ in the associated standard Young tableau $T \\in S(\\lambda )$ .",
"This proves to be a general phenomenon for any standard Young tableau of any two-row shape $\\lambda $ : Proposition 3.1 Take $T \\in S(\\lambda )$ for some two-row shape $\\lambda = (a,b)$ , and let $P = \\lbrace v_0,\\hdots , v_{a+b} \\rbrace $ be the associated Dyck path in $\\mathcal {D}_\\lambda $ .",
"Then an entry $i$ of $T$ has $\\operatorname{dp}(i) = 1$ if and only if $P$ has a return to ground at $v_i$ .",
"Observe that an entry $i$ of $T$ has $\\operatorname{dp}(i) = 1$ if and only if $i = 2m$ for some positive integer $m$ , meaning that the first $m$ columns of $T$ coincide with some $\\widetilde{T} \\in S(\\widetilde{\\lambda })$ for $\\widetilde{\\lambda } = m^2$ .",
"This occurs if and only if the first $2m$ steps in the associated $P \\in \\mathcal {D}_\\lambda $ correspond to a Dyck path from $(0,0)$ to $(m,m)$ .",
"Hence we may conclude that $\\operatorname{dp}(i) = 1$ if and only if $v_i= (m,m)$ for some $m > 0$ .",
"When $\\lambda $ is a two-row shape, $\\operatorname{dp}(i)=0$ and $\\operatorname{dp}(i)=1$ are the only options for each entry $i$ of any $T \\in S(\\lambda )$ .",
"Proposition REF then implies that an entry $i$ of $T$ has $\\operatorname{dp}(i) = 1$ if the associated Dyck path $P \\in \\mathcal {D}_\\lambda $ has a return to ground at $v_i$ , while $\\operatorname{dp}(i) = 0$ if $P$ does not have a return to ground at $v_i$ .",
"This directly prompts our primary result of this subsection: Theorem 3.2 Take any two-column shape $\\lambda = (a,b)$ .",
"Then the $S_k(\\lambda )$ have generating function: $\\xi (q) \\ = \\ \\sum _k \\vert S_k(\\lambda ) \\vert q^k \\ = \\ \\sum _i \\vert \\mathcal {D}_\\lambda (i) \\vert (1+q)^i$ Take any $T \\in S(\\lambda )$ and let $P \\in \\mathcal {D}_\\lambda $ be the corresponding Dyck path.",
"By Corollary REF and Proposition REF , $\\chi ^T(q) = [2]_q^i = (1+q)^i$ if and only if $P \\in \\mathcal {D}_\\lambda (i)$ .",
"Ranging over all $T \\in S(\\lambda )$ , for each $i \\ge 1$ we have precisely $\\vert D_\\lambda (i) \\vert $ contributions of the form $(1+q)^i$ to our overall generating function $\\xi (q) = \\sum _T \\chi ^T(q)$ .",
"The stated result immediately follows.",
"It should be noted that Fresse, Mansour and Melnikov [6] have already derived a closed formula for each $\\vert S_k(\\lambda ) \\vert $ in the case of an arbitrary two-row shape $\\lambda = (a,b)$ .",
"Skip forward to Subsection REF to see our generalization of that formula to the semistandard case.",
"The reason that we still take time to derive the generating function of Theorem REF , apart from the fact that it highlights an interesting relationship between inverted tableau and Dyck paths, is that it easily generalizes to larger tableau for which closed formulas do not currently exist.",
"When $a=b=n$ , Theorem REF also admits the following specialization that directly relates the $\\vert S_k(\\lambda ) \\vert $ to the Ballot numbers: Corollary 3.3 Let $\\lambda = (n,n)$ .",
"Then the $S_k(\\lambda )$ have generating function: $\\xi (q) \\ = \\ \\sum _k \\vert S_k(\\lambda ) \\vert q^k \\ = \\ \\sum _i B(n-1,n-i) (1+q)^i \\ = \\ \\sum _i \\frac{i}{n} \\binom{2n - i - 1}{n-i} (1+q)^i$ Example 3.4 For $\\lambda = (4,4)$ the $S_k(\\lambda )$ have generating function: $\\xi (q) \\ = \\ B(3,3)(1+q) + B(3,2)(1+q)^2 + B(3,1)(1+q)^3 + B(3,0)(1+q)^4$ $= \\ 5(1+q) + 5(1+q)^2 + 3(1+q)^3 + 1(1+q)^4 \\ = \\ 14 + 28q + 20q^2 + 7q^3 + q^4$ As one final comment about the two-row rectangular case, observe that the $\\vert S_k(n,n) \\vert $ resulting from Corollary REF may be arranged into an integer triangle of their own.",
"This triangle is already known under a variety of different contexts as OEIS sequence A039599 [11].",
"The relationship between the $\\vert S_k(n,n) \\vert $ and the other combinatorial interpretations of A039599 is an interesting topic for future investigation."
],
[
"m-Dimensional Dyck Paths and Inverted Young Tableaux of Shape $\\lambda = n^m$",
"Generalizing the methodology of Subsection REF to $m$ -row standard Young tableaux requires $m$ -dimensional analogues of Dyck paths.",
"So let $\\lambda = (\\lambda _1,\\hdots ,\\lambda _m)$ , where the $\\lambda _i$ are positive integers, and set $N = \\lambda _1 + \\hdots \\lambda _m$ .",
"We adopt the standard notation of $\\hat{e}_j$ as the unit vector in the $j^{th}$ coordinate of $\\mathbb {R}^m$ .",
"By an m-dimensional Dyck path of shape $\\mathbf {\\lambda }$ we mean a lattice path $P = \\lbrace v_1,\\hdots , v_N \\rbrace $ with vertices $v_i \\in \\mathbb {Z}^m$ such that: $v_0 = (0,\\hdots ,0)$ and $v_N = (\\lambda _1,\\hdots ,\\lambda _m)$ .",
"For each $1 \\le i \\le N$ we have $v_i - v_{i-1} = \\hat{e}_j$ for some $1 \\le j \\le m$ .",
"NOTE: If $v_i - v_{i-1} = \\hat{e}_j$ we say that $P$ has a “$j$ -step\" at $v_i$ .",
"$v_i=(x_1,\\hdots ,x_m)$ satisfies $x_1 \\le x_2 \\le \\hdots \\le x_m$ for all $1 \\le i \\le N$ .",
"By analogy with two-dimensional Dyck paths, we denote the set of all $m$ -dimensional Dyck paths of shape $\\lambda $ by $\\mathcal {D}_\\lambda = \\mathcal {D}_{(\\lambda _1,\\hdots ,\\lambda _m)}$ .",
"Also in direct analogy with two-dimensional Dyck paths, there is a well-known bijection between $\\mathcal {D}_\\lambda $ and $m$ -row standard Young tableaux of shape $\\lambda $ .",
"Under this bijection, $P \\in \\mathcal {D}_\\lambda $ has a $j$ -step at $v_i$ if and only if $i$ appears in the $j^{th}$ row of the corresponding tableau $T$ .",
"The fact that every point $v_i = (x_1,\\hdots ,x_m)$ along $P$ satisfies $x_1 \\le \\hdots \\le x_m$ ensures that the corresponding tableau $T$ is in fact column standard.",
"Notice that, if $\\lambda = n^m$ is a rectangular two-row shape, this bijection implies that $\\vert \\mathcal {D}_\\lambda \\vert $ is the so-called $m$ -dimensional Catalan number $C_{d,n} = \\left( \\prod _{i=0}^{d-1} \\frac{i!}{(n+i)!}",
"\\right) (dn)!$ .",
"See OEIS sequence A060854 [11] or Gorska and Penson [8] for treatments of these numbers.",
"Before focusing upon the necessary properties of $m$ -dimensional Dyck paths, we pause to establish a fact that follows quickly from the aforementioned bijection between $\\mathcal {D}_\\lambda $ and $S(\\lambda )$ .",
"Here we use the standard notation of $T[1,\\hdots ,i]$ to denote the sub-tableau of $T \\in S(\\lambda )$ that retains the cells of $T$ with values $1,\\hdots ,i$ .",
"Proposition 3.5 Take an $m$ -row tableau $T \\in S(\\lambda )$ and let $P = \\lbrace v_0,v_1,\\hdots \\rbrace $ be the corresponding $m$ -dimensional Dyck path in $\\mathcal {D}_\\lambda $ .",
"Then $T[1,\\hdots ,i]$ has shape $\\lambda _i = (x_1,\\hdots ,x_m)$ if and only if $v_i = (x_1,\\hdots ,x_m)$ .",
"$v_i = (x_1,\\hdots ,x_m)$ if and only if the first $i$ steps of $P$ contain $x_1$ 1-steps, $x_2$ 2-steps, etc.",
"This corresponds to the situation where precisely $x_1$ elements of $[i] = \\lbrace 1,\\hdots ,i \\rbrace $ have been assigned to the first row of $T$ , precisely $x_2$ elements of $[i]$ have been assigned to the second row of $T$ , etc.",
"The relevant statistic on $m$ -dimensional Dyck paths is a higher-dimensional analogue of our two-dimensional “returns to ground\".",
"These “returns\" will be points $v_i = (x_1,\\hdots ,x_m)$ along $P \\in \\mathcal {D}_\\lambda $ at which we have some sort of equality within $x_1 \\le \\hdots \\le x_m$ .",
"The difficulty here is that, even in the three-dimensional case, different lengths of equalities are possible within $x_1 \\le \\hdots \\le x_m$ and some of these lengths are possible with different collections of coordinates.",
"Take an $m$ -dimensional Dyck path $P = \\lbrace v_0,\\hdots v_N \\rbrace $ in $\\mathcal {D}_\\lambda $ , and let $v_{i-1} = (x_1,\\hdots ,x_m)$ , $v_i = (x_1^{\\prime },\\hdots ,x_m^{\\prime })$ be two consecutive lattice points along $P$ .",
"By definition, $x_\\gamma ^{\\prime } = x_\\gamma + 1$ for precisely one $1 \\le \\gamma \\le m$ while $x_i^{\\prime } = x_i$ for all other $1 \\le i \\le m$ .",
"We say that $P$ has a d-degree return to ground (in the $\\mathbf {\\gamma }$ coordinate) at $v_i$ if $x_{\\gamma -d}^{\\prime } = x_{\\gamma -d+1}^{\\prime } = \\hdots = x_{\\gamma }^{\\prime }$ yet $x_{\\gamma -d-1} \\ne x_{\\gamma }$ .",
"If $P$ has an $(m-1)$ -degree return to ground in the $m^{th}$ coordinate at $v_i$ , meaning that $v_i = (x,x,\\hdots ,x)$ for some positive integer $x$ , we say that $P$ has a full return to ground at $v_i$ .",
"As the situation described above ensures $x_{\\gamma + 1}^{\\prime } > x_{\\gamma }^{\\prime }$ , the degree $d$ is one less than the length of the longest string of equalities in the coordinates of $v_i$ that did not already exist in the coordinates of $v_{i-1}$ .",
"Clearly, $d$ -degree returns to ground are only possible in the $\\mathbf {\\gamma }$ coordinate if $\\gamma > d$ .",
"In the case of a two-dimensional Dyck path, the only valid returns to ground are 1-degree returns to ground in the second coordinate, which qualify as full returns to ground.",
"In this sense, our definition is a natural extension of the pre-existing notion of returns to ground for two-dimensional Dyck paths.",
"If $P \\in \\mathcal {D}_\\lambda $ has precisely $k_i$ $i$ -degree returns to ground for each $i \\ge 1$ (disregarding the coordinates in which those returns appear), we write $\\operatorname{\\overrightarrow{\\text{Ret}}}(P) = (k_1,k_2,\\hdots ) = \\vec{k}$ .",
"If $P \\in \\mathcal {D}_\\lambda $ possesses precisely $k$ full returns to ground, we write $\\operatorname{ret}(P) = k$ in direct analogy with the two-dimensional case.",
"Define $\\mathcal {D}_\\lambda (\\vec{k}) = \\lbrace P \\in \\mathcal {D}_\\lambda \\ \\vert \\ \\operatorname{\\overrightarrow{\\text{Ret}}}(P) = \\vec{k} \\rbrace $ and $\\mathcal {D}_\\lambda (k) = \\lbrace P \\in \\mathcal {D}_\\lambda \\ \\vert \\ \\operatorname{ret}(P) = k \\rbrace $ .",
"Even in the case of $\\lambda = n^m$ , there has been no previous attempt to enumerate the $\\mathcal {D}_\\lambda (\\vec{k})$ or the $\\mathcal {D}_\\lambda (k)$ .",
"Such an attempt is clearly outside the scope of this paper.",
"We do pause to note that, if $\\lambda = n^m$ , ranging over $n \\ge 1$ allows one to compile both the $\\vert \\mathcal {D}_\\lambda (\\vec{k}) \\vert $ and the $\\vert \\mathcal {D}_\\lambda (k) \\vert $ into analogues of the Catalan triangle.",
"For the $\\vert \\mathcal {D}_\\lambda (\\vec{k}) \\vert $ this would take the form an $(m+1)$ -dimensional array such that the set of all integers in each $m$ -dimensional “tier\" sums to the $m$ -dimensional Catalan number $C_{d,n}$ .",
"For the $\\vert \\mathcal {D}_\\lambda (k) \\vert $ we have a two-dimensional array that presumably has much more in common with the original Catalan triangle, with each row in the array summing to $C_{d,n}$ .",
"One could define the entries in this second array to be “$m$ -dimensional ballot numbers\", namely $B^m(n-1,n-k) = \\vert \\mathcal {D}_\\lambda (k) \\vert $ when $\\lambda = n^m$ .",
"As in the two-dimensional case of Subsection REF , the interest of this paper lies not in the inherent enumerative properties of the $ \\mathcal {D}_\\lambda (\\vec{k})$ but in how the sizes of these sets determine generating functions for inverted standard Young tableaux.",
"The most general result, which holds for any $m$ -row shape, is the following generalization of Proposition REF : Theorem 3.6 For an $m$ -row shape $\\lambda = (\\lambda _1,\\hdots \\lambda _m)$ , take $T \\in S(\\lambda )$ and let $P = \\lbrace v_0,v_1,\\hdots \\rbrace $ be the associated $m$ -dimensional Dyck path in $\\mathcal {D}_\\lambda $ .",
"Then the entry $a_{ij} = k$ of $T$ has $\\operatorname{dp}(k) = d$ if and only if $P$ has a $d$ -degree return to ground at $v_k$ .",
"Take an entry $a_{ij} = k$ in the $i^{th}$ row and $j^{th}$ column of $T$ , and let $v_k = (x_1,\\hdots ,x_m)$ be the associated lattice point in $P \\in \\mathcal {D}_\\lambda $ .",
"By definition, $\\operatorname{dp}(k) = i - 1 - \\beta _k$ , where $\\beta _k$ is the number of entries in the $(j+1)^{st}$ column of $T$ that are smaller than $k$ .",
"By Proposition REF , $\\beta _k$ corresponds to the number of rows in $T[1,\\hdots ,k]$ whose length is strictly greater than $j$ .",
"It follows that $\\operatorname{dp}(k) = \\gamma _k - 1$ , where $\\gamma _k$ equals the number of rows in $T[1,\\hdots ,k]$ of length precisely $j$ .",
"Once again citing Proposition REF , $\\gamma _j$ equals the longest string of equalities in the coordinates of $v_k$ whose final coordinate is $x_i$ .",
"By definition, we have that $P$ possesses a $(\\gamma _k - 1)$ -degree return to ground at $v_k$ .",
"Both directions of the theorem statement immediately follow.",
"Those familiar with the work of Fresse will notice that the notation of the preceding proof falls far closer to that of [5] than our own notation of Section .",
"Theorem REF lends an unexpected convenience to his sub-tableau approach if one wants to invoke facts about the related Dyck paths, despite the fact that his work is entirely unconcerned with Dyck paths analogues.",
"Our usage of the subtableau $T[1,\\hdots ,k]$ actually doesn't extend beyond Theorem REF , as the approach of Section once again becomes much more convenient in applying that result toward a generating function $\\xi (q)$ : Theorem 3.7 Let $\\lambda = (\\lambda _1,\\hdots ,\\lambda _m)$ .",
"Then the $S_k(\\lambda )$ have generating function: $\\xi (q) \\ = \\ \\sum _k \\vert S_k(\\lambda ) \\vert q^k \\ = \\ \\sum _{(i_1,\\hdots ,i_{m-1})} \\left( \\vert \\mathcal {D}_\\lambda (i_1,\\hdots ,i_{m-1}) \\vert \\prod _{j=1}^{m-1} [j+1]_q^{i_j} \\right)$ Take $T \\in S(\\lambda )$ and let $P \\in \\mathcal {D}_\\lambda $ be the associated Dyck path.",
"Corollary REF and Theorem REF together imply that $\\chi ^T(q) = \\prod _{j=1}^{m-1} [j+1]_q^{i_j}$ if and only if $P \\in \\mathcal {D}_\\lambda (i_1,\\hdots ,i_{m-1})$ .",
"Ranging over all $T \\in S(\\lambda )$ , there are precisely $\\vert \\mathcal {D}_\\lambda (i_1,\\hdots ,i_{m-1}) \\vert $ contributions of the form $\\prod _{j=1}^{m-1} [j+1]_q^{i_j}$ to $\\xi (q) = \\sum _T \\chi ^T(q)$ .",
"The theorem immediately follows."
],
[
"Direct Enumerations of Inverted Young Tableaux",
"In this final section, we present direct enumerations of the $\\vert S_k(\\lambda ,\\mu ) \\vert $ whose methodologies are completely distinct from the generalized Dyck path approach of Section .",
"All of these methods make direct use of our generating function $\\chi ^T(q)$ from Theorem REF or our notions of inversion depth.",
"This allows us to tackle a somewhat different collection of cases than previously attempted, and to generalize some pre-existing results to the semistandard case for the very first time.",
"In particular, we develop an explicit formula for the $\\vert S_k(\\lambda ,\\mu ) \\vert $ in the general two-row case of $\\lambda = a^1 b^1$ .",
"Opening up a wider range of tractable shapes by specializing to standard tableau, we close with an explicit enumeration of $\\vert S_k(\\lambda ) \\vert $ for the two-column rectangular case of $\\lambda = 2^n$ ."
],
[
"Enumeration of $S_k(\\lambda ,\\mu )$ for {{formula:ef9839a8-0228-4122-b194-fe1d46a2e400}}",
"In the standard specialization, Fresse, Mansour and Melnikov [6] have already explicitly enumerated the $S_k(\\lambda )$ for a general two-row shape $\\lambda = (a,b)$ .",
"Theorem 2.1 of [6] gives: $\\vert S_k(\\lambda ) \\vert \\ = \\ \\beta _{b-k}^\\lambda \\ = \\ \\frac{a-b+1+2k}{a+1+k} \\binom{a+b}{b-k}$ In this equation, $\\beta _j^\\lambda $ denotes the Betti number of the associated Springer variety $F_\\lambda $ , an algebraic interpretation that we do not make use of here.",
"The shift in subscript follows from the fact that $\\dim (f_\\lambda ) = b$ , a quantity that equals the maximal inversion number $M_\\lambda $ for $\\tau \\in I(\\lambda )$ .",
"The $S_k(\\lambda ,\\mu )$ were independently enumerated in the semistandard generalization by the author [3], but only in the rectangular case of $a=b=n$ .",
"Theorem 14 of [3] equated $\\vert S_k(\\lambda ,\\mu ) \\vert $ with a product of various Catalan numbers whose subscripts strongly partition $n$ .",
"One can verify that this enumeration agrees with Equation REF as $\\vert S_k(\\lambda ) \\vert = \\frac{1+2k}{n+1+k} \\binom{2n}{n-k}$ for standard content $\\mu $ if one rewrites the Catalan products of the author [3] as an appropriate two-parameter Fuss-Catalan number (Raney number) via a modification of the results from Hilton and Pedersen [9].",
"Here we directly extend Equation REF to the semistandard case in a way that recovers the formula of the author [3] when $a=b=n$ .",
"Essential to our technique is the invariance of $\\vert S_k(\\lambda ,\\mu ) \\vert $ under permutation of content, as originally presented by the author [3]: Theorem 4.1 (Theorem 11 of [3]) For any shape $\\lambda $ , compatible content $\\mu = 1^{\\mu _1} 2^{\\mu _2} \\hdots M^{\\mu _M}$ , and permutation $\\sigma \\in S_M$ , $\\vert S_k(\\lambda ,\\mu ) \\vert = \\vert S_k (\\lambda ,\\sigma (\\mu )) \\vert $ for all $k \\ge 0$ .",
"For any two-row shape $\\lambda = (a,b)$ , clearly the only contents $\\mu $ that allow for row-standard tableau are those where each value appears at most twice.",
"Using Theorem REF , we are justified in restricting our attention to $\\mu $ where $1,2,\\hdots ,m$ each appear twice and $m+1,m+2,\\hdots ,a+b-2m$ each appear once.",
"This allows for the following: Theorem 4.2 Let $\\lambda =(a,b)$ for any $a \\ge b \\ge 1$ , and let $\\mu = 1^{\\mu _1} 2^{\\mu _2} \\hdots M^{\\mu _M}$ be any content with $\\sum _i \\mu _i = a+b$ .",
"If $\\mu _i = 2$ for $m$ choices of $i$ and $\\mu _i = 1$ for the remaining $a+b-2m$ choices of $i$ , then: $\\vert S_k(\\lambda ,\\mu ) \\vert \\ = \\ \\frac{a-b+1+2k}{a+1+k-m} \\binom{a+b-2m}{b-k-m}$ As previously argued, Theorem REF allows us to restrict our attention to the specific content $\\mu $ where $1,2,\\hdots ,m$ each appear twice and $m+1,m+2,\\hdots ,a+b-2m$ each appear once.",
"Any tableau with this $\\lambda $ and $\\mu $ must take the form shown in Figure REF : Such tableaux are clearly in bijection with standard Young tableaux of shape $\\widetilde{\\lambda } = (a-m,b-m)$ , under the map that re-indexes entries in the final $b-m$ columns by $x \\mapsto x-2m$ .",
"For each $T \\in S(\\lambda ,\\mu )$ , this bijection preserves the associated generating function $\\chi ^T(q)$ .",
"This is due to the fact that all entries $a_{ij}$ in the first $m$ columns of $T$ have $\\operatorname{dp}(a_{ij}) = \\operatorname{dp^{\\hspace{-1.0pt}*} \\hspace{-1.0pt}}(a_{ij}) = 0$ and hence do not contribute to $\\chi ^T(q)$ by Theorem REF .",
"It follows that $\\xi (q) = \\sum _k \\vert S_k(\\lambda ,\\mu ) \\vert = \\sum _k \\vert S_k(\\widetilde{\\lambda }) \\vert $ .",
"For any $k \\ge 0$ , directly applying Equation REF then gives: $\\vert S_k (\\lambda ,\\mu ) \\vert \\ = \\ \\vert S_k (\\widetilde{\\lambda }) \\vert \\ = \\ \\frac{(a-m) - (b-m) + 1 + 2k}{(a-m)+1+k} \\binom{(a-m) + (b-m)}{(b-m) - k}$ Figure: An arbitrary T∈S(λ,μ)T \\in S(\\lambda ,\\mu ) with λ=(a,b)\\lambda = (a,b) and μ=1 2 ⋯m 2 (m+1) 1 ⋯(a+b-2m) 1 \\mu = 1^2 \\hdots m^2 (m+1)^1 \\hdots (a+b-2m)^1.",
"Here the shaded cells correspond to some standard Young tableau under the re-indexing x↦x-2mx \\mapsto x-2m."
],
[
"Enumeration of $S_k(\\lambda )$ for {{formula:a723dc08-6eff-4c10-a697-6d4b71c07c16}}",
"Enumeration of the $\\vert S_k(\\lambda ,\\mu ) \\vert $ in the two-column rectangular case of $\\lambda = 2^n$ has yet to appear anywhere in the literature, even for standard tableaux.",
"In this final subsection, we tackle the two-column case in the standard specialization.",
"Pause to observe that this entire subsection is made necessary because inversion depths, and hence the $\\vert S_k(\\lambda ) \\vert $ , are not preserved under transposition.",
"If $\\bar{\\lambda }$ denotes the conjugate partition of $\\lambda $ (obtained by transposing of the underlying Young diagram), it is well known that for standard Young tableaux we have $\\vert S(\\lambda ) \\vert = \\vert S(\\bar{\\lambda }) \\vert $ .",
"However, it is typically not true that $\\vert S_k(\\lambda ) \\vert = \\vert S_k(\\bar{\\lambda }) \\vert $ for $k \\ge 1$ if $\\bar{\\lambda }$ is in fact distinct from $\\lambda $ .",
"For a basic example of this fact, consider any one-row shape $\\lambda = m^1$ , so that $\\bar{\\lambda } = 1^m$ .",
"Clearly $\\vert S_k(\\lambda ) \\vert = 0$ for all $k > 0$ , yet $\\vert S_k(\\bar{\\lambda }) \\vert \\ne 0$ for all $k \\le \\binom{m}{2}$ via an application of Corollary REF .",
"To obtain a generating function $\\xi (q)$ for the $\\vert S_k (\\lambda ) \\vert $ , we determine which $T$ -specific generating functions $\\chi ^T(q)$ are possible for $T \\in S(\\lambda )$ , and then develop a closed formula for the number $\\phi (\\chi )$ of tableaux with a fixed $\\chi ^T(q)$ .",
"As in Section , this allows us to conclude that $\\xi (q) = \\sum _\\chi \\phi (\\chi ) \\chi ^T(q)$ .",
"The first step in this process is a recognition of the fact that, for two-column tableaux, there is at most one $T \\in S(\\lambda )$ with a particular arrangement of inversion depths across its entries: Lemma 4.3 Let $\\lambda = 2^m 1^{n-m}$ , and take $T \\in S(\\lambda )$ .",
"If $a_1,\\hdots ,a_n$ are the entries in the first column of $T$ , read from top to bottom, then $T$ is uniquely identified by the collection of inversion depths $\\operatorname{dp}(a_i) = c_i$ .",
"Pause to note that, if $b_1,\\hdots ,b_m$ are the entries in the second (rightmost) column of $T$ , read from top to bottom, then $\\operatorname{dp}(b_i) = i-1$ no matter our choice of $T$ .",
"This is why we require only the inversion depths of the $a_i$ to distinguish tableaux in $S(\\lambda )$ .",
"So fix $T \\in S(\\lambda )$ and let $a_1,\\hdots ,a_n$ be the entries in the first column of $T$ .",
"By definition, $\\operatorname{dp}(a_i) = i - 1 - \\beta _i$ , where $\\beta _i$ is the number of entries in the second column of $T$ that are smaller than $a_i$ .",
"As $T$ possesses only two-columns, we also know that $a_i = i + \\beta _i$ and hence that $\\operatorname{dp}(a_i) = i - 1 - (a_i - i) = 2i - 1 - a_i$ .",
"Given this direct relationship between the $a_i$ and the $\\operatorname{dp}(a_i) = c_i$ , it is clear that distinct tableaux in $S(\\lambda )$ must possess distinct values of $c_i$ for at least one $i$ .",
"Using Lemma REF still requires us to characterize which sequences of inversion depths $c_1,c_2,\\hdots $ are possible.",
"The necessary conditions on the $c_1,c_2,\\hdots $ actually prove to be fairly straightforward: Lemma 4.4 Set $\\lambda = 2^n$ , and let $c_1,\\hdots ,c_n$ be a sequence of non-negative integers.",
"Then there exists (unique) $T \\in S(\\lambda )$ whose first column entries $a_1,\\hdots ,a_n$ satisfy $\\operatorname{dp}(a_i) = c_i$ for all $1 \\le i \\le n$ if and only if $c_1 = 0$ and $c_i \\le c_{i-1} + 1$ for all $i > 1$ .",
"($\\Rightarrow $ ) Take any $T \\in S(\\lambda )$ .",
"Clearly, the top entry $a_1$ in the first column of $T$ must have an inversion depth of 0, requiring $c_1 = 0$ .",
"By the definition of inversion depth, for any $i>1$ there exist $\\beta _{i-1} = (i-1)-1-c_{i-1}$ entries in the second column of $T$ that are smaller than $a_{i-1}$ and $\\beta _i = i-1-c_i$ entries in the second column of $T$ smaller than $a_i$ .",
"As $a_{i-1} < a_i$ we have $\\beta _{i-1} \\le \\beta _i$ and thus $c_i \\le c_{i-1} + 1$ .",
"($\\Leftarrow $ ) Now take a sequence of non-negative integers $c_1,\\hdots ,c_n$ satisfying the stated conditions.",
"We use that sequence to construct the tableau $T$ below, where $b_1,\\hdots ,b_n$ are the $n$ remaining integers not used in the first column (arranged in increasing order).",
"Table: NO_CAPTIONVia equivalent reasoning to the proof of Lemma REF , if $a_i = 2i - 1 - c_i$ then $\\operatorname{dp}(a_i) = 2i - 1 - (2i - 1 - c_i) = c_i$ .",
"The assumptions that $c_1 = 0$ and $c_i \\le c_{i-1} + 1$ imply $c_i \\le i-1$ and hence $a_i \\ge i$ .",
"This ensures that the entries in the first column of $T$ are always positive integers between 1 and $2n-1$ .",
"By construction, $a_i \\le 2i-1$ and there are always enough larger entries leftover for the $b_i$ to ensure that $T$ is row-standard.",
"The fact that $c_i \\le c_{i-1} + 1$ also guarantees that $a_i \\ge a_{i-1}$ , allowing us to conclude that $T$ is column-standard.",
"Hence $T \\in S(\\lambda )$ is our standard Young tableau with the required inversion depths down its first column.",
"Combining Lemmas REF and REF , we may conclude that there is precisely one $T \\in S(\\lambda )$ for every sequence of non-negative integers $c_1,\\hdots ,c_n$ such that $c_1 = 0$ and $c_i \\le c_{i-1} + 1$ for all $1 \\le i \\le n$ .",
"One may use a generating tree to verify that the total number of such sequences is the Catalan number $C_n$ , as expected from the hook-length formula applied to $\\lambda = 2^n$ .",
"Notice that if $T \\in S(\\lambda )$ is associated with the sequence $c_1,\\hdots ,c_n$ , then Corollary REF gives generating function $\\chi ^T(q) = [n]_q!",
"\\prod _i [c_i + 1]_q$ , where the leading $[n]_q!$ comes from the inversion depths of entries in the second column of $T$ .",
"To identify the number $\\phi (\\chi )$ of $T \\in S(\\lambda )$ with a fixed generating function $\\chi ^T(q)$ , we thus only need to determine how many valid sequences $c_1,\\hdots ,c_n$ have a particular number of zeroes, ones, twos, etc.",
"Once again pause to observe that the conditions on our sequences $c_1,\\hdots ,c_n$ guarantee that $c_i \\le i-1$ and hence that we only need to count occurrences of $0,1,\\hdots ,n-1$ .",
"Henceforth let $\\psi (\\alpha _0,\\hdots ,\\alpha _{n-1})$ denote the number of non-negative integer sequences $c_1,\\hdots ,c_n$ satisfying both $c_1 = 0$ and $c_i \\le c_{i-1} + 1$ that possess precisely $\\alpha _k$ instances of $k$ .",
"Clearly we must have $\\sum _i \\alpha _i = n$ in order for $\\psi (\\alpha _0,\\hdots ,\\alpha _{n-1}) \\ne 0$ .",
"To achieve $\\psi (\\alpha _0,\\hdots ,\\alpha _{n-1}) \\ne 0$ , the $c_i \\le c_{i-1} + 1$ condition also requires that a zero value $\\alpha _k = 0$ never be followed by a nonzero value $\\alpha _{k+1} \\ne 0$ .",
"One can show that these are the only two conditions needed to guarantee that $\\psi (\\alpha _0,\\hdots ,\\alpha _{n-1}) \\ne 0$ , although we do not require this fact in the subsequent proof.",
"Theorem 4.5 Let $\\lambda = 2^n$ .",
"Then the $\\vert S_k(\\lambda ) \\vert $ have generating function: $\\xi (q) \\ = \\ \\sum _k \\vert S_k(\\lambda ) \\vert q^k \\ = \\ [n]_q!",
"\\left( \\sum _{\\alpha _0 + \\hdots + \\alpha _{n-1} = n} \\left( \\prod _{i=1}^{n-1} \\binom{\\alpha _i + \\alpha _{i-1} - 1}{\\alpha _i} [i+1]_q^{\\alpha _i} \\right) \\right)$ Our primary goal is to show that $\\psi (\\alpha _0,\\hdots ,\\alpha _{n-1}) = \\prod _{i=1}^{n-1} \\binom{\\alpha _i + \\alpha _{i-1} - 1}{\\alpha _i}$ .",
"So assume that we want to construct an arbitrary sequence of non-negative integers $c_1,\\hdots ,c_n$ satisfying both $c_1=0$ and $c_i \\le c_{i-1} + 1$ that possesses precisely $\\alpha _k$ instances of $k$ for each $k \\ge 0$ .",
"We recursively build up this sequence by simultaneously inserting all $\\alpha _k$ copies of $k$ into a partial sequence that contains $\\alpha _j$ copies of $j$ for all $0 \\le j \\le k-1$ , making sure that our conditions on the $c_i$ are preserved after each step.",
"We begin by inserting the $\\alpha _0$ copies of 0 into an empty sequence, noticing that $c_1 = 0$ implies that we always have $\\alpha _0 > 0$ .",
"There is obviously only one way to perform this insertion.",
"Now assume that we have continued this process up to the insertion of the $\\alpha _k$ copies of $k$ .",
"Preserving the $c_i \\le c_{i-1} + 1$ condition requires that these instances of $k$ may only appear directly after an instance of $(k-1)$ or another instance of $k$ .",
"This means that our previous placement of all instances of $0,1,\\hdots ,(k-2)$ is irrelevant to the validity of our placement at the step, and that we only need to consider the relative placement of the $\\alpha _k + \\alpha _{k-1}$ instances of $k$ and $(k-1)$ .",
"It follows that there are precisely $\\binom{\\alpha _k + \\alpha _{k-1} - 1}{\\alpha _k}$ valid ways to insert our $\\alpha _k$ copies of $k$ .",
"Ranging over all $k$ , we may conclude that there are precisely $\\psi (\\alpha _0,\\hdots ,\\alpha _{n-1}) = \\prod _{i=1}^{n-1} \\binom{\\alpha _i + \\alpha _{i-1} - 1}{\\alpha _i}$ valid sequences $c_1,\\hdots ,c_n$ that contain $\\alpha _0$ copies of 0, $\\alpha _1$ copies of 1, etc.",
"Notice that, if $\\alpha _{k+1} \\ne 0$ follows $\\alpha _k = 0$ , then $\\binom{\\alpha _{k+1} + \\alpha _k - 1}{\\alpha _{k+1}} = \\binom{\\alpha _{k+1} - 1}{\\alpha _{k+1}} = 0$ and $\\psi (\\alpha _0,\\hdots ,\\alpha _{n-1}) = 0$ , as expected.",
"So assume that $c_1,\\hdots , c_n$ possesses $\\alpha _k$ copies of $k$ for all $0 \\le k \\le (n-1)$ , and that $T \\in S(\\lambda )$ is any tableau associated to $c_1,\\hdots ,c_n$ via the bijection of Lemma REF .",
"Corollary REF then gives generating function $\\chi ^T(q) = [n]_q!\\prod _{i=1}^{n-1} [i+1]_q^{\\alpha _i}$ .",
"The $\\phi (\\alpha _0,\\hdots ,\\alpha _{n-1})$ tableaux with that generating function contribute a total of $\\phi (\\alpha _0,\\hdots ,\\alpha _{n-1}) [n]_q!\\prod _{i=1}^{n-1} [i+1]_q^{\\alpha _i} = [n]_q!",
"\\prod _{i=1}^{n-1} \\binom{\\alpha _i + \\alpha _{i-1} - 1}{\\alpha _i} [i+1]_q^{\\alpha _i}$ to the overall generating function $\\xi (q) = \\sum _k \\vert S_k(\\lambda ) \\vert q^k$ .",
"Ranging over all combinations of the $\\alpha _i$ such that $\\alpha _0 + \\hdots + \\alpha _{n-1} = n$ then gives the desired result.",
"Example 4.6 For $\\lambda = 2^3$ , we have $\\operatorname{dp}(a_{ij}) \\le 2$ for all entries $a_{ij}$ of $T \\in S(\\lambda )$ .",
"The only non-zero values for $\\psi (\\alpha _0,\\alpha _1,\\alpha _2)$ are shown below, followed by the generating function for the $\\vert S_k(\\lambda ) \\vert $ : $\\psi (3,0,0) = 1 \\hspace{21.68121pt} \\psi (2,1,0) = 2 \\hspace{21.68121pt} \\psi (1,2,0) = 1 \\hspace{21.68121pt} \\psi (1,1,1) = 1$ $\\xi (q) \\ = \\ [3]_q!",
"\\left( 1[1]_q^3 + 2[1]_q^2 [2]_q + 1 [1]_q [2]_q^2 + 1 [1]_q [2]_q [3]_q \\right)$ $= \\ 5 + 16x + 25x^2 + 24x^3 + 14x^4 + 5x^5 + x^6$ A methodology similar to the one above may be applied to the non-rectangular two-column case of $\\lambda = 2^m 1^{n-m}$ , as tableaux of this shape are still uniquely identified by the inversion depths of entries in their first column.",
"Sadly, a generalization of the generating function from Theorem REF becomes so convoluted that we do not attempt a full derivation here.",
"The difficulty in generalizing to $\\lambda = 2^m 1^{n-m}$ is that a different (and more complicated) collection of inversion depths $c_1,\\hdots ,c_n$ are now possible.",
"In particular, entries $a_i$ in the one-column “tail\" of $T \\in S(\\lambda )$ now have a nonzero lower bound on $\\operatorname{dp}(a_i) = c_i$ .",
"One may quickly verify that inversion depths in the first column of $T \\in S(\\lambda )$ fall within the ranges shown in Figure REF .",
"Figure: Possible inversion depths for entries in an arbitrary T∈S(λ)T \\in S(\\lambda ) of shape λ=2 m 1 n-m \\lambda = 2^m 1^{n-m}.With the ranges of Figure REF in mind, we close this paper with the following generalization of Lemma REF .",
"The following proposition would be the first step in developing a closed generating function $\\xi (q)$ for the $\\vert S^k(\\lambda ) \\vert $ when $\\lambda = 1^m 2^{n-m}$ .",
"Proposition 4.7 Set $\\lambda =2^m 1^{n-m}$ , and let $c_1,\\hdots ,c_n$ be a sequence of non-negative integers.",
"Then there exists (unique) $T \\in S(\\lambda )$ whose first column entries $a_1,\\hdots ,a_n$ satisfy $\\operatorname{dp}(a_i) = c_i$ for all $1 \\le i \\le n$ if and only if $c_1 =0$ , $c_i \\le c_{i-1} + 1$ for all $i > 1$ , and $c_i>i-m-1$ for all $i>m$ .",
"The method of proof is directly equivalent to that of Lemma REF .",
"For the ($\\Leftarrow $ ) direction, a valid sequence $c_1,\\hdots ,c_n$ of integers is now associated with the unique tableau shown below, which is a minor modification of the tableau from Lemma REF .",
"Table: NO_CAPTION"
]
] | 1606.04869 |
[
[
"Assessing Human Error Against a Benchmark of Perfection"
],
[
"Abstract An increasing number of domains are providing us with detailed trace data on human decisions in settings where we can evaluate the quality of these decisions via an algorithm.",
"Motivated by this development, an emerging line of work has begun to consider whether we can characterize and predict the kinds of decisions where people are likely to make errors.",
"To investigate what a general framework for human error prediction might look like, we focus on a model system with a rich history in the behavioral sciences: the decisions made by chess players as they select moves in a game.",
"We carry out our analysis at a large scale, employing datasets with several million recorded games, and using chess tablebases to acquire a form of ground truth for a subset of chess positions that have been completely solved by computers but remain challenging even for the best players in the world.",
"We organize our analysis around three categories of features that we argue are present in most settings where the analysis of human error is applicable: the skill of the decision-maker, the time available to make the decision, and the inherent difficulty of the decision.",
"We identify rich structure in all three of these categories of features, and find strong evidence that in our domain, features describing the inherent difficulty of an instance are significantly more powerful than features based on skill or time."
],
[
"Introduction",
"Several rich strands of work in the behavioral sciences have been concerned with characterizing the nature and sources of human error.",
"These include the broad of notion of bounded rationality [25] and the subsequent research beginning with Kahneman and Tversky on heuristics and biases [26].",
"With the growing availability of large datasets containing millions of human decisions on fixed, well-defined, real-world tasks, there is an interesting opportunity to add a new style of inquiry to this research — given a large stream of decisions, with rich information about the context of each decision, can we algorithmically characterize and predict the instances on which people are likely to make errors?",
"This genre of question — analyzing human errors from large traces of decisions on a fixed task — also has an interesting relation to the canonical set-up in machine learning applications.",
"Typically, using instances of decision problems together with “ground truth” showing the correct decision, an algorithm is trained to produce the correct decisions in a large fraction of instances.",
"The analysis of human error, on the other hand, represents a twist on this formulation: given instances of a task in which we have both the correct decision and a human's decision, the algorithm is trained to recognize future instances on which the human is likely to make a mistake.",
"Predicting human error from this type of trace data has a history in human factors research [14], [24], and a nascent line of work has begun to apply current machine-learning methods to the question [16], [17]."
],
[
"As the investigation of human error using large datasets grows increasingly feasible, it becomes useful to understand which styles of analysis will be most effective.",
"For this purpose, as in other settings, there is enormous value in focusing on model systems where one has exactly the data necessary to ask the basic questions in their most natural formulations.",
"What might we want from such a model system?",
"(i) It should consist of a task for which the context of the human decisions has been measured as thoroughly as possible, and in a very large number of instances, to provide the training data for an algorithm to analyze errors.",
"(ii) So that the task is non-trivial, it should be challenging even for highly skilled human decision-makers.",
"(iii) Notwithstanding the previous point (ii), the “ground truth” — the correctness of each candidate decision — should be feasibly computable by an algorithm.",
"Guided by these desiderata, we focus in this paper on chess as a model system for our analysis.",
"In doing so, we are proceeding by analogy with a long line of work in behavioral science using chess as a model for human decision-making [4], [5], [6].",
"Chess is a natural domain for such investigations, since it presents a human player with a sequence of concrete decisions — which move to play next — with the property that some choices are better than others.",
"Indeed, because chess provides data on hard decision problems in such a pure fashion, it has been described as the “drosophila of psychology” [7], [10].",
"(It is worth noting our focus here on human decisions in chess, rather than on designing algorithms to play chess [22].",
"This latter problem has also, of course, generated a rich literature, along with a closely related tag-line as the “drosophila of artificial intelligence” [19].)",
"Despite the clean formulation of the decisions made by human chess players, we still must resolve a set of conceptual challenges if our goal is to assemble a large corpus of chess moves with ground-truth labels that classify certain moves as errors.",
"Let us consider three initial ideas for how we might go about this, each of which is lacking in some crucial respect for our purposes.",
"First, for most of the history of human decision-making research on chess, the emphasis has been on focused laboratory studies at small scales in which the correct decision could be controlled by design [4].",
"In our list of desiderata, this means that point (iii), the availability of ground truth, is well under control, but a significant aspect of point (i) — the availability of a vast number of instances — is problematic due to the necessarily small scales of the studies.",
"A second alternative would be to make use of two important computational developments in chess — the availability of databases with millions of recorded chess games by strong players; and the fact that the strongest chess programs — generally referred to as chess engines — now greatly outperform even the best human players in the world.",
"This makes it possible to analyze the moves of strong human players, in a large-scale fashion, comparing their choices to those of an engine.",
"This has been pursued very effectively in the last several years by Biswas and Regan [2], [3], [23]; they have used the approach to derive interesting insights including proposals for how to estimate the depth at which human players are analyzing a position.",
"For the current purpose of assembling a corpus with ground-truth error labels, however, engines present a set of challenges.",
"The basic difficulty is that even current chess engines are far from being able to provide guarantees regarding the best move(s) in a given position.",
"In particular, an engine may prefer move $m$ to $m^{\\prime }$ in a given position, supplementing this preference with a heuristic numerical evaluation, but $m^{\\prime }$ may ultimately lead to the same result in the game, both under best play and under typical play.",
"In these cases, it is hard to say that choosing $m^{\\prime }$ should be labeled an error.",
"More broadly, it is difficult to find a clear-cut rule mapping an engine's evaluations to a determination of human error, and efforts to label errors this way would represent a complex mixture of the human player's mistakes and the nature of the engine's evaluations.",
"Finally, a third possibility is to go back to the definition of chess as a deterministic game with two players (White and Black) who engage in alternating moves, and with a game outcome that is either (a) a win for White, (b) a win for Black, or (c) a draw.",
"This means that from any position, there is a well-defined notion of the outcome with respect to optimal play by both sides — in game-theoretic terms, this is the minimax value of the position.",
"In each position, it is the case that White wins with best play, or Black wins with best play, or it is a draw with best play, and these are the three possible minimax values for the position.",
"This perspective provide us with a clean option for formulating the notion of an error, namely the direct game-theoretic definition: a player has committed an error if their move worsens the minimax value from their perspective.",
"That is, the player had a forced win before making their move but now they don't; or the player had a forced draw before making their move but now they don't.",
"But there's an obvious difficulty with this route, and it's a computational one: for most chess positions, determining the minimax value is hopelessly beyond the power of both human players and chess engines alike.",
"We now discuss the approach we take here.",
"In our work, we use minimax values by leveraging a further development in computer chess — the fact that chess has been solved for all positions with at most $k$ pieces on the board, for small values of $k$ [18], [21], [15].",
"(We will refer to such positions as $\\le $$k$ -piece positions.)",
"Solving these positions has been accomplished not by forward construction of the chess game tree, but instead by simply working backward from terminal positions with a concrete outcome present on the board and filling in all other minimax values by dynamic programming until all possible $\\le $$k$ -piece positions have been enumerated.",
"The resulting solution for all $\\le $$k$ -piece positions is compiled into an object called a $k$ -piece tablebase, which lists the game outcome with best play for each of these positions.",
"The construction of tablebases has been a topic of interest since the early days of computer chess [15], but only with recent developments in computing and storage have truly large tablebases been feasible.",
"Proprietary tablebases with $k = 7$ have been built, requiring in excess of a hundred terabytes of storage [18]; tablebases for $k = 6$ are much more manageable, though still very large [21], and we focus on the case of $k = 6$ in what follows.There are some intricacies in how tablebases interact with certain rules for draws in chess, particularly threefold-repetition and the 50-move rule, but since these have essentially negligible impact on our use of tablebases in the present work, we do not go into further details here.",
"Tablebases and traditional chess engines are thus very different objects.",
"Chess engines produce strong moves for arbitrary positions, but with no absolute guarantees on move quality in most cases; tablebases, on the other hand, play perfectly with respect to the game tree — indeed, effortlessly, via table lookup — for the subset of chess containing at most $k$ pieces on the board.",
"Thus, for arbitrary $\\le $$k$ -piece positions, we can determine minimax values, and so we can obtain a large corpus of chess moves with ground-truth error labels: Starting with a large database of recorded chess games, we first restrict to the subset of $\\le $$k$ -piece positions, and then we label a move as an error if and only it worsens the minimax value from the perspective of the player making the move.",
"Adapting chess terminology to the current setting, we will refer to such an instance as a blunder.",
"This is our model system for analyzing human error; let us now check how it lines up with desiderata (i)-(iii) for a model system listed above.",
"Chess positions with at most $k = 6$ pieces arise relatively frequently in real games, so we are left with many instances even after filtering a database of games to restrict to only these positions (point (i)).",
"Crucially, despite their simple structure, they can induce high error rates by amateurs and non-trivial error rates even by the best players in the world; in recognition of the inherent challenge they contain, textbook-level treatments of chess devote a significant fraction of their attention to these positions [9] (point (ii)).",
"And they can be evaluated perfectly by tablebases (point (iii)).",
"Focusing on $\\le $$k$ -piece positions has an additional benefit, made possible by a combination of tablebases and the recent availability of databases with millions of recorded chess games.",
"The most frequently-occurring of these positions arise in our data thousands of times.",
"As we will see, this means that for some of our analyses, we can control for the exact position on the board and still have enough instances to observe meaningful variation.",
"Controlling for the exact position is not generally feasible with arbitrary positions arising in the middle of a chess game, but it becomes possible with the scale of data we now have, and we will see that in this case it yields interesting and in some cases surprising insights.",
"Finally, we note that our definition of blunders, while concrete and precisely aligned with the minimax value of the game tree, is not the only definition that could be considered even using tablebase evaluations.",
"In particular, it would also be possible to consider “softer” notions of blunders.",
"Suppose for example that a player is choosing between moves $m$ and $m^{\\prime }$ , each leading to a position whose minimax value is a draw, but suppose that the position arising after $m$ is more difficult for the opponent, and produces a much higher empirical probability that the opponent will make a mistake at some future point and lose.",
"Then it can be viewed as a kind of blunder, given these empirical probabilities, to play $m^{\\prime }$ rather than the more challenging $m$ .",
"This is sometimes termed speculative play [12], and it can be thought of primarily as a refinement of the coarser minimax value.",
"This is an interesting extension, but for our work here we focus on the purer notion of blunders based on the minimax value."
],
[
"Setting Up the Analysis",
"In formulating our analysis, we begin from the premise that for analyzing error in human decisions, three crucial types of features are the following: (a) the skill of the decision-maker; (b) the time available to make the decision; and (c) the inherent difficulty of the decision.",
"Any instance of the problem will implicitly or explicitly contain features of all three types: an individual of a particular level of skill is confronting a decision of a particular difficulty, with a given amount of time available to make the decision.",
"In our current domain, as in any other setting where the question of human error is relevant, there are a number of basic genres of question that we would like to ask.",
"These include the following.",
"For predicting whether an error will be committed in a given instance, which types of features (skill, time, or difficulty) yield the most predictive power?",
"In which kinds of instances does greater skill confer the largest relative benefit?",
"Is it for more difficult decisions (where skill is perhaps most essential) or for easier ones (where there is the greatest room to realize the benefit)?",
"Are there particular kinds of instances where skill does not in fact confer an appreciable benefit?",
"An analogous set of questions for time in place of skill: In which kinds of instances does greater time for the decision confer the largest benefit?",
"Is additional time more beneficial for hard decisions or easy ones?",
"And are there instances where additional time does not reduce the error rate?",
"Finally, there are natural questions about the interaction of skill and time: is it higher-skill or lower-skill decision-makers who benefit more from additional time?",
"These questions motivate our analyses in the subsequent sections.",
"We begin by discussing how features of all three types (skill, time, and difficulty) are well-represented in our domain.",
"Our data comes from two large databases of recorded chess games.",
"The first is a corpus of approximately 200 million games from the Free Internet Chess Server (FICS), where amateurs play each other on-lineThis data is publicly available at ficsgames.org..",
"The second is a corpus of approximately 1 million games played in international tournaments by the strongest players in the world.",
"We will refer to the first of these as the FICS dataset, and the second as the GM dataset.",
"(GM for “grandmaster,” the highest title a chess player can hold.)",
"For each corpus, we extract all occurrences of $\\le $ 6-piece positions from all of the games; we record the move made in the game from each occurrence of each position, and use a tablebase to evaluate all possible moves from the position (including the move that was made).",
"This forms a single instance for our analysis.",
"Since we are interested in studying errors, we exclude all instances in which the player to move is in a theoretically losing position — where the opponent has a direct path to checkmate — because there are no blunders in losing positions (the minimax value of the position is already as bad as possible for the player to move).",
"There are 24.6 million (non-losing) instances in the FICS dataset, and 880,000 in the GM dataset.",
"We now consider how feature types (a), (b), and (c) are associated with each instance.",
"First, for skill, each chess player in the data has a numerical rating, termed the Elo rating, based on their performance in the games they've played [8], [11].",
"Higher numbers indicate stronger players, and to get a rough sense of the range: most amateurs have ratings in the range 1000-2000, with extremely strong amateurs getting up to 2200-2400; players above 2500-2600 belong to a rarefied group of the world's best; and at any time there are generally about fewer than five people in the world above 2800.",
"If we think of a game outcome in terms of points, with 1 point for a win and 0.5 points for a draw, then the Elo rating system has the property that when a player is paired with someone 400 Elo points lower, their expected game outcome is approximately $0.91$ points — an enormous advantage.In general, the system is designed so that when the rating difference is $400d$ , the expected score for the higher-ranked player under the Elo system is $1 / (1 + 10^{-d})$ .",
"For our purposes, an important feature of Elo ratings is the fact that a single number has empirically proven so powerful at predicting performance in chess games.",
"While ratings clearly cannot contain all the information about players' strengths and weaknesses, their effectiveness in practice argues that we can reasonably use a player's rating as a single numerical feature that approximately represents their skill.",
"With respect to temporal information, chess games are generally played under time limits of the form, “play $x$ moves in $y$ minutes” or “play the whole game in $y$ minutes.” Players can choose how they use this time, so on each move they face a genuine decision about how much of their remaining allotted time to spend.",
"The FICS dataset contains the amount of time remaining in the game when each move was played (and hence the amount of time spent on each move as well); most of the games in the FICS dataset are played under extremely rapid time limits, with a large fraction of them requiring that the whole game be played in 3 minutes for each player.",
"To avoid variation arising from the game duration, we focus on this large subset of the FICS data consisting exclusively of games with 3 minutes allocated to each side.",
"Our final set of features will be designed to quantify the difficulty of the position on the board — i.e.",
"the extent to which it is hard to avoid selecting a move that constitutes a blunder.",
"There are many ways in which one could do this, and we are guided in part by the goal of developing features that are less domain-specific and more applicable to decision tasks in general.",
"We begin with perhaps the two most basic parameters, analogues of which would be present in any setting with discrete choices and a discrete notion of error — these are the number of legal moves in the position, and the number of these moves that constitute blunders.",
"Later, we will also consider a general family of parameters that involve looking more deeply into the search tree, at moves beyond the immediate move the player is facing.",
"To summarize, in a single instance in our data, a player of a given rating, with a given amount of time remaining in the game, faces a specific position on the board, and we ask whether the move they select is a blunder.",
"We now explore how our different types of features provide information about this question, before turning to the general problem of prediction."
],
[
"Difficulty",
"We begin by considering a set of basic features that help quantify the difficulty inherent in a position.",
"There are many features we could imagine employing that are highly domain-specific to chess, but our primary interest is in whether a set of relatively generic features can provide non-trivial predictive value.",
"Above we noted that in any setting with discrete choices, one can always consider the total number of available choices, and partition these into the number that constitute blunders and the number that do not constitute blunders.",
"In particular, let's say that in a given chess position $P$ , there are $n(P)$ legal moves available — these are the possible choices — and of these, $b(P)$ are blunders, in that they lead to a position with a strictly worse minimax value.",
"Note that it is possible to have $b(P) = 0$ , but we exclude these positions because it is impossible to blunder.",
"Also, by the definition of the minimax value, we must have $b(P) \\le n(P) - 1$ ; that is, there is always at least one move that preserves the minimax value.",
"Figure: A heat map showing the empirical blunder rate as afunction of the two variables (n(P),b(P))(n(P),b(P)), for the FICS dataset.A global check of the data reveals an interesting bimodality in both the FICS and GM datasets: positions with $b(P) = 1$ and positions with $b(P) = n(P) - 1$ are both heavily represented.",
"The former correspond to positions in which there is a unique blunder, and the latter correspond to positions in which there is a unique correct move to preserve the minimax value.",
"Our results will cover the full range of $(n(P),b(P))$ values, but it is useful to know that both of these extremes are well-represented.",
"Now, let us ask what the empirical blunder rate looks like as a bivariate function of this pair of variables $(n(P),b(P))$ .",
"Over all instances in which the underlying position $P$ satisfies $n(P) = n$ and $b(P) = b$ , we define $r(n,b)$ to be the fraction of those instances in which the player blunders.",
"How does the empirical blunder rate vary in $n(P)$ and $b(P)$ ?",
"It seems natural to suppose that for fixed $n(P)$ , it should generally increase in $b(P)$ , since there are more possible blunders to make.",
"On the other hand, instances with $b(P) = n(P) - 1$ often correspond to chess positions in which the only non-blunder is “obvious” (for example, if there is only one way to recapture a piece), and so one might conjecture that the empirical blunder rate will be lower for this case.",
"In fact, the empirical blunder rate is generally monotone in $b(P)$ , as shown by the heatmap representation of $r(n,b)$ in Figure REF .",
"(We show the function for the FICS data; the function for the GM data is similar.)",
"Moreover, if we look at the heavily-populated line $b(P) = n(P) - 1$ , the blunder rate is increasing in $n(P)$ ; as there are more blunders to compete with the unique non-blunder, it becomes correspondingly harder to make the right choice.",
"Figure: The empirical blunder rate as a function of the blunder potential,shown for both the GM and the FICS data.",
"On the left are standard axes,on the right are logarithmic y-axes.",
"The plots on the right also showan approximate fit to the γ\\gamma -value defined inSection ."
],
[
"Given the monotonicity we observe, there is an informative way to combine $n(P)$ and $b(P)$ : by simply taking their ratio $b(P) / n(P)$ .",
"This quantity, which we term the blunder potential of a position $P$ and denote $\\beta (P)$ , is the answer to the question, “If the player selects a move uniformly at random, what is the probability that they will blunder?”.",
"This definition will prove useful in many of the analyses to follow.",
"Intuitively, we can think of it as a direct measure of the danger inherent in a position, since it captures the relative abundance of ways to go wrong.",
"In Figure REF we plot the function $y = r(x)$ , the proportion of blunders in instances with $\\beta (P) = x$ , for both our GM and FICS datasets on linear as well as logarithmic $y$ -axes.",
"The striking regularity of the $r(x)$ curves shows how strongly the availability of potential mistakes translates into actual errors.",
"One natural starting point for interpreting this relationship is to note that if players were truly selecting their moves uniformly at random, then these curves would lie along the line $y = x$ .",
"The fact that they lie below this line indicates that in aggregate players are preferentially selecting non-blunders, as one would expect.",
"And the fact that the curve for the GM data lies much further below $y = x$ is a reflection of the much greater skill of the players in this dataset, a point that we will return to shortly.",
"We find that a surprisingly simple model qualitatively captures the shapes of the curves in Figure REF quite well.",
"Suppose that instead of selecting a move uniformly at random, a player selected from a biased distribution in which they were preferentially $c$ times more likely to select a non-blunder than a blunder, for a parameter $c > 1$ .",
"Figure: The empirical blunder rate as a function of player rating,shown for both the (top) GM and (bottom) FICS data.If this were the true process for move selection, then the empirical blunder rate of a position $P$ would be $\\gamma _c(P) = \\frac{b(P)}{c (n(P) - b(P)) + b(P)}.$ We will refer to this as the $\\gamma $ -value of the position $P$ , with parameter $c$ .",
"Using the definition of the blunder potential $\\beta (P)$ to write $b(P) = \\beta (P) n(P)$ , we can express the $\\gamma $ -value directly as a function of the blunder potential: $\\gamma _c(P)= \\frac{\\beta (P) n(P)}{c (n(P) - \\beta (P) n(P)) + \\beta (P) n(P)}= \\frac{\\beta (P)}{c - (c-1) \\beta (P)}.$ We can now find the value of $c$ for which $\\gamma _c(P)$ best approximates the empirical curves in Figure REF .",
"The best-fit values of $c$ are $c \\approx 15$ for the FICS data and $c \\approx 100$ for the GM data, again reflecting the skill difference between the two domains.",
"These curves are shown superimposed on the empirical plot in the figure (on the right, with logarithmic $y$ -axes).",
"We note that in game-theoretic terms the $\\gamma $ -value can be viewed as a kind of quantal response [20], in which players in a game select among alternatives with a probability that decreases according to a particular function of the alternative's payoff.",
"Since the minimax value of the position corresponds to the game-theoretic payoff of the game in our case, a selection rule that probabilistically favors non-blunders over blunders can be viewed as following this principle.",
"(We note that our functional form cannot be directly mapped onto standard quantal response formulations.",
"The standard formulations are strictly monotonically decreasing in payoff, whereas we have cases where two different blunders can move the minimax value by different amounts — in particular, when a win changes to a draw versus a win changes to a loss — and we treat these the same in our simple formulation of the $\\gamma $ -value.)"
],
[
"Skill",
"A key focus in the previous subsection was to understand how the empirical blunder rate varies as a function of parameters of the instance.",
"Here we continue this line of inquiry, with respect to the skill of the player in addition to the difficulty of the position.",
"Recall that a player's Elo rating is a function of the outcomes of the games they've played, and is effective in practice for predicting the outcomes of a game between two rated players [8].",
"It is for this reason that we use a player's rating as a proxy for their skill.",
"However, given that ratings are determined by which games a player wins, draws, or loses, rather than by the extent to which they blunder in $\\le $ 6-piece positions, a first question is whether the empirical blunder rate in our data shows a clean dependence on rating.",
"In fact it does.",
"Figure REF shows the empirical blunder rate $f(x)$ averaged over all instances in which the player has rating $x$ .",
"The blunder rate declines smoothly with rating for both the GM and FICS data, with a flattening of the curve at higher ratings."
],
[
"We can think of the downward slope in Figure REF as a kind of skill gradient, showing the reduction in blunder rate as skill increases.",
"The steeper this reduction is in a given setting, the higher the empirical benefit of skill in reducing error.",
"It is therefore natural to ask how the skill gradient varies across different conditions in our data.",
"As a first way to address this, we take each possible value of the blunder potential $\\beta $ (rounded to the nearest multiple of $0.1$ ), and define the function $f_\\beta (x)$ to be the empirical error rate of players of rating $x$ in positions of blunder potential $\\beta $ .",
"Figure REF shows plots of these curves for $\\beta $ equal to each multiple of $0.1$ , for both the GM and FICS datasets.",
"Figure: The empirical blunder rate as a function of Elo rating inthe FICS data, for positions with fixed values of (n(P),b(P))(n(P),b(P)).Figure: The empirical blunder rate as a function of Elo rating,for a set of frequently-occurring positions.We observe two properties of these curves.",
"First, there is remarkably little variation among the curves.",
"When viewed on a logarithmic y-axis the curves are almost completely parallel, indicating the same rate of proportional decrease across all blunder potentials.",
"A second, arguably more striking, property is how little the curves overlap in their ranges of $y$ -values.",
"In effect, the curves form a kind of “ladder” based on blunder potential: for every value of the discretized blunder potential, every rating in 1200-1800 range on FICS has a lower empirical blunder rate at blunder potential $\\beta $ than the best of these ratings at blunder potential $\\beta + 0.2$ .",
"In effect, each additional $0.2$ increment in blunder potential contributes more, averaging over all instances, to the aggregate empirical blunder rate than an additional 600 rating points, despite the fact that 600 rating points represent a vast difference in chess performance.",
"We see a similar effect for the GM data, where small increases in blunder potential have a greater effect on blunder rate than the enormous difference between a rating of 2300 and a rating of 2700.",
"(Indeed, players rated 2700 are making errors at a greater rate in positions of blunder potential 0.9 than players rated 1200 are making in positions of blunder potential 0.3.)",
"And we see the same effects when we separately fix the numerator and denominator that constitute the blunder potential, $b(P)$ and $n(P)$ , as shown in Figure REF .",
"To the extent that this finding runs counter to our intuition, it bears an interesting relation to the fundamental attribution error — the tendency to attribute differences in people's performance to differences in their individual attributes, rather than to differences in the situations they face [13].",
"What we are uncovering here is that a basic measure of the situation — the blunder potential, which as we noted above corresponds to a measure of the danger inherent in the underlying chess position — is arguably playing a larger role than the players' skill.",
"This finding also relates to work of Abelson on quantitative measures in a different competitive domain, baseball, where he found that a player's batting average accounts for very little of the variance in their performance in any single at-bat [1].",
"We should emphasize, however, that despite the strong effect of blunder potential, skill does play a fundamental role role in our domain, as the analysis of this section has shown.",
"And in general it is important to take multiple types of features into account in any analysis of decision-making, since only certain features may be under our control in any given application.",
"For example, we may be able to control the quality of the people we recruit to a decision, even if we can't control the difficulty of the decision itself.",
"Grouping positions together by common $(n(P),b(P))$ values gives us a rough sense for how the skill gradient behaves in positions of varying difficulty.",
"But this analysis still aggregates together a large number of different positions, each with their own particular properties, and so it becomes interesting to ask — how does the empirical blunder rate vary with Elo rating when we fix the exact position on the board?",
"The fact that we are able to meaningfully ask this question is based on a fact noted in Section , that many non-trivial $\\le $ 6-piece positions recur in the FICS data, exactly, several thousand times.To increase the amount of data we have on each position, we cluster together positions that are equivalent by symmetry: we can apply a left-to-right reflection of the board, or we can apply a top-bottom reflection of the board (also reversing the colors of the pieces and the side to move), or we can do both.",
"Each of the four resulting positions is equivalent under the rules of chess.",
"For each such position $P$ , we have enough instances to plot the function $f_P(x)$ , the rate of blunders committed by players of rating $x$ in position $P$ .",
"Let us say that the function $f_P(x)$ is skill-monotone if it is decreasing in $x$ — that is, if players of higher rating have a lower blunder rate in position $P$ .",
"A natural conjecture would be that every position $P$ is skill-monotone, but in fact this is not the case.",
"Among the most frequent positions, we find several that we term skill-neutral, with $f_P(x)$ remaining approximately constant in $x$ , as well as several that we term skill-anomalous, with $f_P(x)$ increasing in $x$ .",
"Figure REF shows a subset of the most frequently occurring positions in the FICS data that contains examples of each of these three types: skill-monotone, skill-neutral, and skill-anomalous.For readers interested in looking at the exact positions in question, each position in Figure REF is described in Forsyth-Edwards notation (FEN) above the panel in which its plot appears.",
"The existence of skill-anomalous positions is surprising, since there is a no a priori reason to believe that chess as a domain should contain common situations in which stronger players make more errors than weaker players.",
"Moreover, the behavior of players in these particular positions does not seem explainable by a strategy in which they are deliberately making a one-move blunder for the sake of the overall game outcome.",
"In each of the skill-anomalous examples in Figure REF , the player to move has a forced win, and the position is reduced enough that the worst possible game outcome for them is a draw under any sequence of moves, so there is no long-term value in blundering away the win on their present move."
],
[
"Time",
"Finally, we consider our third category of features, the time that players have available to make their moves.",
"Recall that players have to make their own decisions about how to allocate a fixed budget of time across a given number of moves or the rest of the game.",
"The FICS data has information about the time remaining associated with each move in each game, so we focus our analysis on FICS in this subsection.",
"Specifically, as noted in Section , FICS games are generally played under extremely rapid conditions, and for uniformity in the analysis we focus on the most commonly-occurring FICS time constraint — the large subset of games in which each player is allocated 3 minutes for the whole game.",
"As a first object of study, let's define the function $g(t)$ to be the empirical blunder rate in positions where the player begins considering their move with $t$ seconds left in the game.",
"Figure REF shows a plot of $g(t)$ ; it is natural that the blunder rate increases sharply as $t$ approaches 0, though it is notable how flat the value of $g(t)$ becomes once $t$ exceeds roughly 10 seconds.",
"Figure: The empirical blunder rate as a function of time remaining.Figure: The empirical blunder rate as a function of the time remaining,for positions with fixed blunder potential values.Figure: The empirical blunder rate as a function of time remaining,fixing the blunder potential and player rating."
],
[
"This plot in Figure REF can be viewed as a basic kind of time gradient, analogous to the skill gradient, showing the overall improvement in empirical blunder rate that arises from having extra time available.",
"Here too we can look at how the time gradient restricted to positions with fixed blunder potential, or fixed blunder potential and player rating.",
"We start with Figure REF , which shows $g_\\beta (t)$ , the blunder rate for players within a narrow skill range (1500-1599 Elo) with $t$ seconds remaining in positions with blunder potential $\\beta $ .",
"In this sense, it is a close analogue of Figure REF , which plotted $f_\\beta (x)$ , and for values of $t$ above 8 seconds, it shows a very similar “ladder” structure in which the role of blunder potential is dominant.",
"Specifically, for every $\\beta $ , players are blundering at a lower rate with 8 to 12 seconds remaining at blunder potential $\\beta $ than they are with over a minute remaining at blunder potential $\\beta + 0.2$ .",
"A small increase in blunder potential has a more extensive effect on blunder rate than a large increase in available time.",
"We can separate the instances further both by blunder potential and by the rating of the player, via the function $g_{\\beta ,x}(t)$ which gives the empirical blunder rate with $t$ seconds remaining when restricted to players of rating $x$ in positions of blunder potential $\\beta $ .",
"Figure REF plots these functions, with a fixed value of $\\beta $ in each panel.",
"We can compare curves for players of different rating, observing that for higher ratings the curves are steeper: extra time confers a greater relative empirical benefit on higher-rated players.",
"Across panels, we see that for higher blunder potential the curves become somewhat shallower: more time provides less relative improvement as the density of possible blunders proliferates.",
"But equally or more striking is the fact that all curves retain a roughly constant shape, even as the empirical blunder rate climbs by an order of magnitude from the low ranges of blunder potential to the highest.",
"Comparing across points in different panels helps drive home the role of blunder potential even when considering skill and time simultaneously.",
"Consider for example (a) instances in which players rated 1200 (at the low end of the FICS data) with 5-8 seconds remaining face a position of blunder potential 0.4, contrasted with (b) instances in which players rated 1800 (at the high end of the FICS data) with 42-58 seconds remaining face a position of blunder potential 0.8.",
"As the figure shows, the empirical blunder rate is lower in instances of type (a) — a weak player in extreme time pressure is making blunders at a lower rate because they're dealing with positions that contain less danger.",
"Thus far we've looked at how the empirical blunder rate depends on the amount of time remaining in the game.",
"However, we can also ask how the probability of a blunder varies with the amount of time the player actually spends considering their move before playing it.",
"When a player spends more time on a move, should we predict they're less likely to blunder (because they gave the move more consideration) or more likely to blunder (because the extra time suggests they didn't know what do)?",
"The data turns out to be strongly consistent with the latter view: the empirical blunder rate is higher in aggregate for players who spend more time playing a move.",
"We find that this property holds across the range of possible values for the time remaining and the blunder potential, as well as when we fix the specific position."
],
[
"Prediction",
"We've now seen how the empirical blunder rate depends on our three fundamental dimensions: difficulty, the skill of the player, and the time available to them.",
"We now turn to a set of tasks that allow us to further study the predictive power of these dimensions."
],
[
"Greater Tree Depth",
"In order to formulate our prediction methods for blunders, we first extend the set of features available for studying the difficulty of a position.",
"Once we have these additional features, we will be prepared to develop the predictions themselves.",
"Thus far, when we've considered a position's difficulty, we've used information about the player's immediate moves, and then invoked a tablebase to determine the outcome after these immediate moves.",
"We now ask whether it is useful for our task to consider longer sequences of moves beginning at the current position.",
"Specifically, if we consider all $d$ -move sequences beginning at the current position, we can organize these into a game tree of depth $d$ with the current position $P$ as the root, and nodes representing the states of the game after each possible sequence of $j \\le d$ moves.",
"Chess engines use this type of tree as their central structure in determining which moves to make; it is less obvious, however, how to make use of these trees in analyzing blunders by human players, given players' imperfect selection of moves even at depth 1.",
"Let us introduce some notation to describe how we use this information.",
"Suppose our instance consists of position $P$ , with $n$ legal moves, of which $b$ are blunders.",
"We will denote the moves by $m_1, m_2, ..., m_n$ , leading to positions $P_1, P_2, \\ldots , P_n$ respectively, and we'll suppose they are indexed so that $m_1, m_2, \\ldots , m_{n-b}$ are the non-blunders, and $m_{n-b+1}, \\ldots , m_n$ are the blunders.",
"We write $T_0$ for the indices of the non-blunders $\\lbrace 1, 2, \\ldots , n-b\\rbrace $ and $T_1$ for the indices of the blunders $\\lbrace n-b+1, \\ldots , n\\rbrace $ .",
"Finally, from each position $P_i$ , there are $n_i$ legal moves, of which $b_i$ are blunders.",
"The set of all pairs $(n_i,b_i)$ for $i = 1, 2, \\ldots , n$ constitutes a potentially useful source of information in the depth-2 game tree from the current position.",
"What might it tell us?",
"First, suppose that position $P_i$ , for $i \\in T_1$ , is a position reachable via a blunder $m_i$ .",
"Then if the blunder potential $\\beta (P_i) = b_i/n_i$ is large, this means that it may be challenging for the opposing player to select a move that capitalizes on the blunder $m_i$ made at the root position $P$ ; there is a reasonable chance that the opposing will instead blunder, restoring the minimax value to something larger.",
"This, in turn, means that it may be harder for the player in the root position of our instance to see that move $m_i$ , leading to position $P_i$ , is in fact a blunder.",
"The conclusion from this reasoning is that when the blunder potentials of positions $P_i$ for $i \\in T_1$ are large, it suggests a larger empirical blunder rate at $P$ .",
"It is less clear what to conclude when there are large blunder potentials at positions $P_i$ for $i \\in T_0$ — positions reachable by non-blunders.",
"Again, it suggests that player at the root may have a harder time correctly evaluating the positions $P_i$ for $i \\in T_0$ ; if they appear better than they are, it could lead the player to favor these non-blunders.",
"On the other hand, the fact that these positions are hard to evaluate could also suggest a general level of difficulty in evaluating $P$ , which could elevate the empirical blunder rate.",
"There is also a useful aggregation of this information, as follows.",
"If we define $b(T_1) = \\sum _{i \\in T_1} b_i$ and $n(T_1) = \\sum _{i \\in T_1} n_i$ , and analogously for $b(T_0)$ and $n(T_0)$ , then the ratio $\\beta _1 = b(T_1)/n(T_1)$ is a kind of aggregate blunder potential for all positions reachable by blunders, and analogously for $\\beta _0 = b(T_0)/n(T_0)$ with respect to positions reachable by non-blunders.",
"In the next subsection, we will see that the four quantities $b(T_1)$ , $n(T_1)$ , $b(T_0)$ , $n(T_0)$ indeed contain useful information for prediction, particularly when looking at families of instances that have the same blunder potential at the root position $P$ .",
"We note that one can construct analogous information at greater depths in the game tree, by similar means, but we find in the next subsection that these do not currently provide improvements in prediction performance, so we do not discuss greater depths further here."
],
[
"Prediction Results",
"We develop three nested prediction tasks: in the first task we make predictions about an unconstrained set of instances; in the second we fix the blunder potential at the root position; and in the third we control for the exact position."
],
[
"In our first task we formulate the basic error-prediction problem: we have a large collection of human decisions for which we know the correct answer, and we want to predict whether the decision-maker will err or not.",
"In our context, we predict whether the player to move will blunder, given the position they face and the various features of it we have derived, how much time they have to think, and their skill level.",
"In the process, we seek to understand the relative value of these features for prediction in our domain.",
"We restrict our attention to the 6.6 million instances that occurred in the 320,000 empirically most frequent positions in the FICS dataset.",
"Since the rate of blundering is low in general, we down-sample the non-blunders so that half of our remaining instances are blunders and the other half are non-blunders.",
"This results in a balanced dataset with 600,000 instances, and we evaluate model performance with accuracy.",
"For ease of interpretation, we use both logistic regression and decision trees.",
"Since the relative performance of these two classifiers is virtually identical, but decision trees perform slightly better, we only report the results using decision trees here.",
"Table REF defines the features we use for prediction.",
"In addition the notation defined thus far, we define: $S=\\lbrace \\textrm {Elo},\\textrm {Opp-elo}\\rbrace $ to be the skill features consisting of the rating of the player and the opponent; $a(P)=n(P)-b(P)$ for the number of non-blunders in position $P$ ; $D_1=\\lbrace a(P),b(P),\\beta (P)\\rbrace $ to be the difficulty features at depth 1; $D_2=\\lbrace a(T_0), b(T_0), a(T_1),b(T_1), \\beta _0(P), \\beta _1(P)\\rbrace $ as the difficulty features at depth 2 defined in the previous subsection; and $t$ as the time remaining.",
"Table: Features for blunder prediction.In Table REF , we show the performance of various combinations of our features.",
"The most striking result is how dominant the difficulty features are.",
"Using all of them together gives 0.75 accuracy on this balanced dataset, halfway between random guessing and perfect performance.",
"In comparison, skill and time are much less informative on this task.",
"The skill features $S$ only give 55% accuracy, time left $t$ yields 53% correct predictions, and neither adds predictive value once position difficulty features are in the model.",
"The weakness of the skill and time features is consistent with our findings in Section , but still striking given the large ranges over which the Elo ratings and time remaining can extend.",
"In particular, a player rated 1800 will almost always defeat a player rated 1200, yet knowledge of rating is not providing much predictive power in determining blunders on any individual move.",
"Similarly, a player with 10 seconds remaining in the entire game is at an enormous disadvantage compared to a player with two minutes remaining, but this too is not providing much leverage for blunder prediction at the move level.",
"While these results only apply to our particular domain, it suggests a genre of question that can be asked by analogy in many domains.",
"(To take one of many possible examples, one could similarly ask about the error rate of highly skilled drivers in difficult conditions versus bad drivers in safe conditions.)",
"Another important result is that most of the predictive power comes from depth 1 features of the tree.",
"This tells us the immediate situation facing the player is by far the most informative feature.",
"Finally, we note that the prediction results for the GM data (where we do not have time information available) are closely analogous; we get a slightly higher accuracy of $0.77$ , and again it comes entirely from our basic set of difficulty features for the position.",
"Table: Accuracy results on Task 1.Given the accuracy of algorithms for Task 1, it is natural to consider how this compares to the performance of human chess players on such a task.",
"To investigate this question, we developed a version of Task 1 as a web app quiz and promoted it on two popular Internet chess forums.",
"Each quiz question provided a pair of $\\le $ 6-piece instances with White to move, each showing the exact position on the board, the ratings of the two players, and the time remaining for each.",
"The two instances were chosen from the FICS data with the property that White blundered in one of them and not the other, and the quiz question was to determine in which instance White blundered.",
"In this sense, the quiz is a different type of chess problem from the typical style, reflecting the focus of our work here: rather than “White to play and win,” it asked “Did White blunder in this position?”.",
"Averaging over approximately 6000 responses to the quiz from 720 participants, we find an accuracy of $0.69$ , non-trivially better than random guessing but also non-trivially below our model's performance of $0.79$ .Note that the model performs slightly better here than in our basic formulation of Task 1, since there instances were not presented in pairs but simply as single instances drawn from a balanced distribution of positive and negative cases.",
"The relative performance of the prediction algorithm and the human forum participants forms an interesting contrast, given that the human participants were able to use domain knowledge about properties of the exact chess position while the algorithm is achieving almost its full performance from a single number – the blunder potential — that draws on a tablebase for its computation.",
"We also investigated the extent to which the guesses made by human participants could be predicted by an algorithm; our accuracy on this was in fact lower than for the blunder-prediction task itself, with the blunder potential again serving as the most important feature for predicting human guesses on the task.",
"Given how powerful the depth 1 features are, we now control for $b(P)$ and $n(P)$ and investigate the predictive performance of our features once blunder potential has been fixed.",
"Our strategy on this task is very similar to before: we compare different groups of features on a binary classification task and use accuracy as our measure.",
"These groups of features are: $D_2$ , $S$ , $S \\cup D_2$ , $\\lbrace t\\rbrace $ , $\\lbrace t\\rbrace \\cup D_2$ , and the full set $S\\cup \\lbrace t\\rbrace \\cup D_2$ .",
"For each of these models, we have an accuracy score for every $(b(P),n(P))$ pair.",
"The relative performances of the models are qualitatively similar across all $(b(P),n(P))$ pairs: again, position difficulty dominates time and rating, this time at depth 2 instead of depth 1.",
"In all cases, the performance of the full feature set is best (the mean accuracy is 0.71), but $D_2$ alone achieves 0.70 accuracy on average.",
"This further underscores the importance of position difficulty.",
"Additionally, inspecting the decision tree models reveals a very interesting dependence of the blunder rate on the depth 1 structure of the game tree.",
"First, recall that the most frequently occurring positions in our datasets have either $b(P)=1$ or $b(P) =n(P)-1$ .",
"In so-called “only-move” situations, where there is only one move that is not a blunder, the dependence of blunder rate on $D_2$ is as one would expect: the higher the $b(T_1)$ ratio, the more likely the player is to blunder.",
"But for positions with only one blunder, the dependence reverses: blunders are less likely with higher $b(T_1)$ ratios.",
"Understanding this latter effect is an interesting open question.",
"Our final prediction question is about the degree to which time and skill are informative once the position has been fully controlled for.",
"In other words, once we understand everything we can about a position's difficulty, what can we learn from the other dimensions?",
"To answer this question, we set up a final task where we fix the position completely, create a balanced dataset of blunders and non-blunders, and consider how well time and skill predict whether a player will blunder in the position or not.",
"We do this for all 25 instances of positions for which there are over 500 blunders in our data.",
"On average, knowing the rating of the player alone results in an accuracy of 0.62, knowing the times available to the player and his opponent yields 0.54, and together they give 0.63.",
"Thus once difficulty has been completely controlled for, there is still substantive predictive power in skill and time, consistent with the notion that all three dimensions are important."
],
[
"Discussion",
"We have used chess as a model system to investigate the types of features that help in analyzing and predicting error in human decision-making.",
"Chess provides us with a highly instrumented domain in which the time available to and skill of a decision-maker are often recorded, and, for positions with few pieces, the set of optimal decisions can be determined computationally.",
"Through our analysis we have seen that the inherent difficulty of the decision, even approximated simply by the proportion of available blunders in the underlying position, can be a more powerful source of information than the skill or time available.",
"We have also identified a number of other phenomena, including the ways in which players of different skill levels benefit differently, in aggregate, from easier instances or more time.",
"And we have found, surprisingly, that there exist skill-anomalous positions in which weaker players commit fewer errors than stronger players.",
"We believe there are natural opportunities to apply the paper's framework of skill, time, and difficulty to a range of settings in which human experts make a sequence of decisions, some of which turn out to be in error.",
"In doing so, we may be able to differentiate between domains in which skill, time, or difficulty emerge as the dominant source of predictive information.",
"Many questions in this style can be asked.",
"For a setting such as medicine, is the experience of the physician or the difficulty of the case a more important feature for predicting errors in diagnosis?",
"Or to recall an analogy raised in the previous section, for micro-level mistakes in a human task such as driving, we think of inexperienced and distracted drivers as a major source of risk, but how do these effects compare to the presence of dangerous road conditions?",
"Finally, there are a number of interesting further avenues for exploring our current model domain of chess positions via tablebases.",
"One is to more fully treat the domain as a competitive activity between two parties.",
"For example, is there evidence in the kinds of positions we study that stronger players are not only avoiding blunders, but also steering the game toward positions that have higher blunder potential for their opponent?",
"More generally, the interaction of competitive effects with principles of error-prone decision-making can lead to a rich collection of further questions."
],
[
"We thank Tommy Ashmore for valuable discussions on chess engines and human chess performance, the ficsgames.org team for providing the FICS data, Bob West for help with web development, and Ken Rogoff, Dan Goldstein, and Sébastien Lahaie for their very helpful feedback.",
"This work has been supported in part by a Simons Investigator Award, an ARO MURI grant, a Google Research Grant, and a Facebook Faculty Research Grant."
]
] | 1606.04956 |
[
[
"Waterfront on the Martian Planitia: Algorithmic emergent catchments on\n disordered terrain"
],
[
"Abstract Under a terraforming scenario, a reactivated hydrological cycle on Mars will result in upwards movement of water due to evaporation and precipitation.",
"If Mars' embryonic fossilized catchments provide inadequate drainage, Mars' limited supplies of water may be absorbed entirely by crater lakes and glaciers, with negative consequences for the terraforming effort.",
"We demonstrate a stable, convergent algorithm for the efficient modeling of water flow over disordered terrain.",
"This model is applied to Mars Orbital Laser Altimeter data and successfully predicts the formation of fossilized waterways and canyons visible only at much higher resolution.",
"This exploratory study suggests that despite its impossibly rugged appearance, ancient water flows have carved channels that provide effective drainage over the majority of Mars' surface.",
"We also provide one possible reconstruction of a terraformed surface water distribution."
],
[
"Mars used to be wet, Mars may be wet again",
"The product of robotic NASA and ESA Mars exploration campaigns over the last two decades has returned a cornucopia of data on Martian paleohydrology[1].",
"Publicly available, this data has been used to show that Mars once possessed an atmosphere and environmental conditions substantially more favorable to life.",
"While large scale flow features had been discovered by the Viking spacecraft[2], science performed in situ by the Mars Exploration Rovers (Spirit and Opportunity) and the Mars Science Laboratory (Curiosity) have made a convincing case that Mars once had sustained periods of surface water flow[3], [4].",
"Meanwhile, orbiting satellites such as Mars Global Surveyor and Mars Reconaissance Orbiter have provided us with high resolution imagery and topographic data.",
"This work in particular leverages data from the Mars Orbital Laser Altimeter (MOLA) [5], produced between 1999 and 2001.",
"MOLA provides altimetric data and a gravitational datum with a resolution of up to 1/128$^{\\circ }$ , or 460m, and a vertical accuracy of around 3m.",
"Terraforming other planets, or even exoplanets, is a common trope of science fiction, and some authors have gone so far as to speculate on the geographic effects of a nascent hydrological cycle.",
"Perhaps most notably, in his (pre-MOLA) 1994 novel “Blue Mars”, (p. 324) Kim Stanley Robinson wrote: The southern highlands were everywhere lumpy, shattered, pocked, cracked, hillocky, scarped, slumped, fissured, and fractured; when analyzed as potential watersheds, they were hopeless.",
"Nothing led anywhere; there was no downhill for long.",
"The entire south was a plateau three to four kilometers above the old datum, with only local bumps and dips.",
"Never had Nadia seen more clearly the difference between this highland and any continent on Earth.",
"On Earth, tectonic movement had pushed up mountains every few-score million years, and then water had run down these fresh slopes, following the paths of least resistance back to the sea, carving the fractal vein patterns of watersheds everywhere.",
"Even the dry basin regions on Earth were seamed with arroyos and dotted with playas.",
"In the Martian south, however, the meteoric bombardment of the Noachian had hammered the land ferociously, leaving craters and ejecta everywhere; and then the battered irregular wasteland had lain there for two billion years under the ceaseless scouring of the dusty winds, tearing at every flaw.",
"If they poured water onto this pummeled land they would end up with a crazy quilt of short streams, running down local inclines to the nearest rimless crater.",
"Hardly any streams would make it to the sea in the north, or even into the Hellas or Argyre basins, both of which were ringed by mountain ranges of their own ejecta.",
"Robinson had to employ a large degree of imagination, as high resolution surface data was not available in 1994.",
"All other investigations of global scale exohydrology are similarly speculative, because the necessary data only exists for the Earth (which already has rivers), the Moon (which will never have rivers) and Mars (which once had rivers of some sort).",
"Mars is thus the only case where such literary speculation can now be approached with some degree of rigor.",
"While the terraforming of Mars is one or possibly two steps beyond NASA's existing plans for Mars exploration, the sun has gotten a lot warmer since the collapse of Mars' greenhouse effect billions of years ago[6].",
"More recently, humans have developed the sort of integrated heavy industry without which crewed space exploration was previously, and may yet be again, impossible.",
"In other words, now is the best and perhaps the only time to warm the planet back up and once again run rivers on its surface.",
"This research attempts to evaluate what proportion of, and at what scales, Mars' surface is hydrologically disordered or merely deranged[7].",
"On Earth there are a handful of deserts where geotectonic (or volcanic) orogenic processes are more rapid than water or wind erosion, leading to deranged drainage and fractal endorheic basins.",
"If this were generally the case on Mars, a reactivated hydrological cycle would have to cut new watercourses before it (literally) ran out of steam.",
"In one particularly grim scenario, a brief warm/wet period results in mass migration of water to high altitude glaciers.",
"In this case, planetary albedo increases and crashes the climate, and what little water Mars has is lost via sublimation and rapid solar atmospheric stripping.",
"Martian paleohydrology has been studied intensively since Viking data first revealed massive flow outburst channels[8], [9], [10].",
"These (possibly ice-covered) flows, derived from destabilized aquifers rather than meteoric precipitation, carved deep channels into the bedrock and fed a putative northern ocean[11], [12].",
"Smaller scale integrated valley networks link regions of intermittent glaciation and lower level drainage[13], according to the cold/wet ancient climate models[14], [15].",
"This paper is a novel quantitative attempt to predict global hydrology for a terraformed planet.",
"Here, we describe the algorithm used to predict water flow, present an example data set, compare it to high resolution topographic data, and then examine the potential future of watercourses on Mars."
],
[
"Algorithm",
"While commercial software packages are available to simulate terrestrial watersheds for, e.g., agricultural runoff or stormwater drain system design[16], in this section we outline our extensible model algorithm for simple surface runoff and precipitation, to illustrate how surface water flow can be handled computationally.",
"Considering a 1D discretized test case, the change in water level at any particular point is due to flow from adjacent points to that point, which is in turn dependent on the relative heights of water at each point.",
"A back of the envelope calculation shows that water flow is governed by a parabolic equation ($h,_t = k h,_{xx}$ ), wherein local variations diffuse outwards through the surrounding area.",
"In more detail, given a water level height $h(x,t)$ at time $t$ and position $x$ , its change over time is governed by how much water flows to or away from that point from immediately adjacent points.",
"This can be expressed: $h(x_1,t_1)=h(x_1,t_0) + (t_1-t_0) \\frac{k}{x_1-x_0} \\bigg (\\frac{h(x_2,t_0)-h(x_1,t_0)}{x_2-x_1}-\\frac{h(x_1,t_0)-h(x_0,t_0)}{x_1-x_0}\\bigg )\\;,$ or $\\frac{h(x_1,t_1)-h(x_1,t_0)}{t_1-t_0} = k \\frac{h(x_0,t_0)-2h(x_1,t_0)+h(x_2,t_0)}{(x_1-x_0)^2}\\;,$ from which the continuous case is trivially derived.",
"The diffusion equation, however, is inappropriate for solving this problem as it does not impose a positive constraint on water depth; as a result, the depth parameter can become unphysically negative.",
"The approach we used to overcome this limitation is the imposition, between successive applications of the finite forward difference, of an intersticial normalization step to ensure that at no point would the depth parameter drop below zero.",
"Since we intend to use this algorithm to predict erosion rates of the entire primeval Martian surface, we include it here for a more thorough exposition of a few of its subtleties.",
"In particular, more sophisticated algorithms leverage primarily well-ordered catchments, an assumption explicitly excluded within the framework of this study.",
"Below, the algorithm is presented in a pointwise frame of reference, although in practice an array-oriented implementation is more efficient.",
"Our reference implementation was written in Mathematica and can be downloaded at [17].",
"Initialize datasets $\\mathbf {T}$ for topographic data and $\\mathbf {D}$ for water depth.",
"We used a greyscale MOLA image (no relief shading) with a resolution of 720$\\times $ 1440, or 0.25$^\\circ $ , to derive $\\mathbf {T}$ .",
"$\\mathbf {D}$ was initialized at 147m everywhere, although the algorithm can use any initial value.",
"As well as being close to upper estimates of total reserves of water remaining on Mars (isotope analysis indicates perhaps 85% of water has escaped since the Noachian[18]), 147m is also equivalent to 0.5% of the total planetary topographic relief of 29429m.",
"Extend the domain using ghost zones to ensure continuity in periodic domains and consistant array sizes.",
"Obtain the (negative) forward difference as a proxy for the induced flow in the positive direction.",
"$\\mathbf {F}_i = (\\mathbf {T}+\\mathbf {D})_{i}-(\\mathbf {T}+\\mathbf {D})_{i+1}\\;.$ Split $\\mathbf {F}$ into positive $\\mathbf {F}^+$ and negative $\\mathbf {F}^-$ parts, and assign each flow condition to the cell from which the water will flow.",
"Multiply by a notional step size $t$ to determine the total depth of water that will be subtracted from each cell in the subsequent step in each direction $\\mathbf {L}$ .",
"$\\mathbf {L}_i = t \\big (\\mathbf {F}^-_i ,\\mathbf {F}^+_{i+1} \\big )\\;.$ Reduce $\\mathbf {L}$ if necessary to ensure the total water lost is less than depth.",
"$\\tilde{\\mathbf {L}}_i = \\mathrm {min}(1,\\frac{\\mathbf {D}_i}{\\Sigma \\mathbf {L}_i})\\mathbf {L}_i \\;.$ Calculate a metric factor to convert depths to volumes and vice versa.",
"This allows the algorithm to work between adjacent lines of latitude and remain conservative.",
"Some finessing around the poles is necessary to retain a sensible Courant (or CFL) condition[19].",
"$\\mathbf {M} = \\sin \\theta \\;.$ Reallocate water accordingly.",
"$\\mathbf {\\Delta D}_i = \\mathbf {M}_{i-1} \\tilde{\\mathbf {L}}^+_{i-1} + \\mathbf {M}_{i+1} \\tilde{\\mathbf {L}}^-_{i+1} - \\mathbf {M}_i \\Sigma \\tilde{\\mathbf {L}}_i\\;.$ Apply algorithm alternately in each direction, updating the depth each time.",
"We used the explicit Euler method, which is stable under the usual conditions.",
"As applied this method will quickly converge to a series of puddles, lakes, seas, and oceans, with steadily decreasing flow between them as the algorithm flattens their respective surfaces and shunts the difference over the lip.",
"To ensure continuous recharge, we added a precipitation function to each time step.",
"A constant quantity of depth was subtracted from every suffiently deep body of water and then redistributed with a proportionality that varied with the underlying altitude, mimicking the rain-making properties of mountains due to adiabatic cooling under orographic lifting.",
"The model does not include any atmospheric effects such as prevailing wind, rain shadowing, or seasonal variation which, by adding complexity without contributing additional insight to the central question of emergent watershed formation, are beyond the scope of this study."
],
[
"Results",
"The primary concern of this work was not to create in any sense an accurate climate-driven hydrological model, but to determine whether, if precipitation occurred on the potentially poorly drained and high altitude southern hemisphere, water would pool and gradually redistribute itself to a higher average altitude.",
"In this case, one might expect glaciation, a raised albedo, and counterproductive terraforming effects, not to mention a general slowing destabilization of the hydrological cycle.",
"The precipitation algorithm has one parameter, namely the quantity of evaporation $E$ per time step.",
"$E$ is best interpreted in comparison with the rate of water flow over the surface, or the chance that any given water molecule has of making it to the ocean versus evaporating on the way.",
"Systems with a relatively large $E$ have behaviour like the Antarctic dry valleys, where water tends to be redistributed upwards until all the mountains are buried under water (or ice) and the valleys and sinks are relatively dry.",
"Conversely, systems with a low value of $E$ represent rapid motion of water across the ground relative to evaporation and lead to the formation of large, integrated drainage systems and much drier mountain peaks, such as the American southwest.",
"Figure: (a) MOLA altimetry dataset and its key (left).",
"(b) Converged water depth data with its respective key (right), showing a shallow northern ocean and lake-strewn south pole region.Here, I present a sample converged set of results, shown in Figures REF and REF .",
"Initial depth was 147m, evaporation was a factor $10^{-5}$ smaller than this, or about 1.5mm, per time step.",
"A rectangular equi-angular projection of altitude data, shown in Figure REF , was used.",
"Converged depth is shown in Figure REF , consisting of flooded impact craters of various sizes, some of the more prominent channels feeding Chryse Planitia, and a large, shallow northern ocean, with substantial exposed `sea bed' compared to the putative ancient oceans.",
"The location of these, and other regions, are labelled in Figure REF .",
"It is important to note that geoid/areoid changes, , true polar wander, , crustal subsidence and uplift, and post-Hesperian volcanism have altered the ancient water courses, in some places substantially.",
"One particularly salient example is the basalt flow-modified western edge of the Echus Chaos.",
"These changes underscore the risk of assuming that ancient watercourses will automatically resume their function without performing a surface runoff calculation.",
"Figure: (a) Flux (brightness) and topographic relief (hue) show both watercourses and regions of high erosion (bright red), and its key (left).",
"(b) Flux (brightness) and flow direction (hue) show directionality of flow, stagnation, and positions of watershed features, together with its key (right).",
"Conical features such as volcanoes show radial flow.In analysing the data, we have used both hue and brightness to convey a sample of what this dataset can describe.",
"Surface water flow volume, or flux, is shown with brightness.",
"Brighter areas represent regions of concentrated and rapid surface flow, or gradual flow in deep bodies of water.",
"Colour is used in Figure REF to encode the relative gradient of the underlying topography, showing waterfalls, rapids, and regions of relatively fast incision in bright red.",
"If drainage channels incise headwards rapidly enough, they may be able to capture and drain high altitude lakes before they freeze or sequester all available water.",
"In the next section, we analyse one particular case study in detail.",
"In Figure REF , we use hue instead to encode flow direction.",
"This is useful for determining the ultimate path of watercourses, and the demarcation of watersheds.",
"Olympus Mons, whose location is shown in Figure REF , has a particularly nice radial flow pattern.",
"One interesting case is the ponderous course taken by water on the eastern rim of Noctis Labyrinthus, which flows east across Syria Planum, Solis Planum, Melas Fossae, to Nirgal Vallis, before heading northeast to join Ares Vallis draining into Chryse, with locations shown in Figure REF .",
"Figure: Figures showing flow directions as given by the key on Figure : green up, red right, light blue left, violet down.",
"(a) South pole region flux shows that a relatively small area around the pole seems to pool, with some outflow to the catchments to the north.",
"(b) North pole region flux shows a series of drainage outflows feeding a large, triangular ocean.In Figure REF , I show flow strength and direction at the north and south pole, highlighting the network of relatively poorly drained basins at the south pole.",
"Nevertheless, they mostly appear to have natural drainage in some stage of development.",
"In summary, these data show that, contrary to some expectations, the re-activated Martian hydrosphere has catchments that are mostly not deranged.",
"Figure REF shows that channelized flow would incise and drain lakes, with Mars' enhanced vertical relief compensating for its commensurately reduced surface gravity.",
"With sufficient mountainous precipitation, several large river networks would form along the surface.",
"Figure REF shows the division of the surface into three primary watersheds focused on Argyre, Hellas, and the northern basin.",
"A relative handful of isolated craters (e.g.",
"Darwin, discussed below) have their own small endorheic drainage systems, but under our precipitation model never overcome evaporation to fill even partially, due to their relatively tiny catchment areas.",
"Finally, a small region around the south pole, extensively modified by the polar ice cap and poorly resolved by MOLA, has indeterminate drainage, as shown in Figure REF ."
],
[
"Validation",
"Figure REF shows areas in bright red where large volumes of water combined with steep underlying topography indicate candidate areas for rapid stream incision and notch regression.",
"In particular, the drainage pattern on the southeast rim of Argyre Planitia features a steep valley joining a shallow highland lake with the drowned crater floor, shown in Figure REF .",
"When compared to much (32$\\times $ ) higher resolution MOLA data, this flow has resulted in the ancient incision of an actual physical valley (Dzigai Vallis, Oceanidum Fossa) on Mars.",
"This valley's notch indicates recession to the floor of the highland lake (Doanus Vallis), indicating that sufficient water (by precipitation or aquifer outburst) flowed to cut this previously endorheic region completely, facilitating its drainage.",
"These locations are labelled in Figure REF .",
"The valley appears to have regressed towards the edge of another endorheic basin, between craters Sahr and Wegener.",
"Here, the drainage splinters as it crosses the Hellas-Argyre watershed, as shown in Figure REF .",
"Interestingly, some craters coopt nearby flows and fill with water (and sediment), while others (Darwin, Maraldi, Galle, Green), ringed completely by high walls and with limited catchment, remain relatively dry.",
"This demonstrates that, contrary to initial expectations, much of the Martian southern hemisphere has had sufficient erosion by ancient water flows to be well drained."
],
[
"Discussion – Looking to the future",
"Comparison of several other predicted flow regions with the relevant MOLA data shows, in particular, an integrated drainage system flowing from the mountain peaks northwest of Hellas to the shore of the putative northern ocean, a straight line distance of around 4000km with a fall of around 6km, equivalent to some of the longest rivers on Earth.",
"These results show that low resolution predictions of drainage and flow patterns are largely supported by high resolution images and altimetry data, which show that ancient flows have already carved drainage channels into the walls and floors of transitory paleolakes.",
"Additionally, global drainage patterns show that, contrary to our initial expectation, most of the Martian surface is well drained.",
"This means that a revived hydrological cycle could function with little to no modification of existing landforms by, e.g., dams, blasting of channels, or melting of crater walls with orbital mirrors.",
"While it is clear that drainage systems on Mars after the late heavy bombardment were insufficiently long-lived to significantly degrade the craters of the southern highlands, this exploratory study gives some measure of confidence that, in the event of a revived Martian hydrological cycle, most liberated liquid water would not be absorbed by endorheic high altitude lakes.",
"In other words, despite appearances, much of the region has established, if subtle, drainage patterns.",
"This study gives a first estimate of how Mars' current estimated water supply could recreate a hydrological cycle, though it is possible additional water sources exist, such as recharged subsurface aquifers."
],
[
"Looking to the more immediate future",
"While this study addresses fundamental questions about emergent watersheds during terraforming of another planet, it also raises further questions that could be addressed in future work.",
"Some of these include: Locating regions of maximal future erosion, by leveraging the entire MOLA dataset for a high resolution study that would use existing canyons not resolved in this study.",
"Estimating relative and absolute flow rates of new rivers, by implementing a more sophisticated climate model that could take into account solar heating, prevailing wind, rain shadowing, and perhaps seasonal variations.",
"This would provide quantitative estimates of relative flow rates in different catchments, as well as more precisely predicting relative levels of the northern ocean, and Hellas and Argyre seas, under a range of hypothesized climate and water availability estimates.",
"Predicting future evolution of catchments by simulating erosion of high volume, high relief flows to estimate basin drainage and stream capture.",
"Include glaciation and glacial erosion to more accurately estimate sediment flows and downstream deposition."
],
[
"Conclusion",
"Our initial assumptions about drainage patterns (or lack thereof) in the relatively primal and highly cratered southern terrain were found to be partly unfounded.",
"In particular, multiple predicted sites of rapid erosion and canyon incision were found to already exhibit these features, indicating that large regions of the southern hemisphere already have drainage systems.",
"While not obvious from a distance, a potential revived hydrological cycle would be less prone to mass uphill water migration and subsequent stability problems than first supposed.",
"Consequently, lower altitude lakes and seas would have a greater share of the water, ensuring that the terraformed Martian riviera has enough waterfront for everyone."
],
[
"Acknowledgements",
"The author wishes to thank Dr. Christine Corbett Moran for a close reading and numerous helpful suggestions during this paper's development."
],
[
"References",
"These references are a mix of academic journal articles (open access where possible), online datasets, and relevant Wikipedia articles intended to get casual readers up to speed on subject-specific terminology."
]
] | 1606.05224 |
[
[
"Inscribed Matter Communication: Part II"
],
[
"Abstract This paper is Part II of a two-paper set which develops a finest-grain per-molecule timing treatment of molecular communication.",
"We first consider a simple one-molecule timing channel with a molecule launch deadline, similar to but different from previous work (\"Bits Through Queues\") where the constraint was mean launch time.",
"We also derive a number of results related to the {\\em ordering entropy}, a key quantity which undergirds the capacity bounds for the molecular timing channel, both with and without token data payloads.",
"We then conclude with an upper bound on molecular timing-channel capacity."
],
[
"Introduction",
"In Part-I [1] of this two-paper set we defined a signaling model and developed an information theoretic framework for evaluating the capacity and efficiency of channels which use molecules (or “tokens”) as information carriers.",
"Here in Part-II we provide some necessary undergirding results which are interesting in their own right.",
"In particular, we consider a timing channel similar but not identical to Anantharam's and Verdú's “Bits Through Queues” channel [2], [3], [4] wherein a mean launch time constraint is replaced by a launch deadline constraint.",
"We derive closed forms for the optimizing distribution and the channel capacity for this older timing channel and then apply the results to the molecular communication problem.",
"We also derive analytic expressions and bounds for a key quantity in our analysis – the ordering entropy $H(\\Omega |\\vec{{\\bf S}}, {\\bf T})$ – first generally and then specifically for exponential first-passage time distribution.",
"These results support the capacity bounds of Part-I [1], provide capacity results for a timing channel with an emission deadline under exponential first-passage, and also establish that unlike the mean-constrained timing channel, the worst case corruption is not exponential first-passage.",
"Our analysis ends with the derivation of an upper bound on the token timing channel capacity."
],
[
"Brief Problem Description",
"A detailed discussion of the underlying molecular communication problem and its importance in both biology and engineering is provided in Part-I [1].",
"Here we assume basic familiarity with the concepts and provide only the mathematical description of the system.",
"As a reader aid, key quantities are provided in TABLE REF and in an identical table in Part-I.",
"Table: Glossary of useful termsThus, consider a communication system in which $M$ identical tokens are released/launched at times $T_1, T_2, \\cdots , T_M$ with no assumption that the $T_m$ are ordered in time.",
"The duration of each token's journey from transmitter to receiver is a random variable $D_m$ so that token $m$ arrives at time $S_m = T_m + D_m$ .",
"The $D_m$ are assumed independent and identically distributed (i.i.d.).",
"In vector notation, we have ${\\bf S}= {\\bf T}+ {\\bf D}$ .",
"We denote the density of each $D_m$ as $f_{D_m}(d) = g(d)$ , $d \\ge 0$ and the cumulative distribution function (CDF) as $F_{D_m}(d) =G(\\cdot )$ .",
"Likewise, the complementary CDF (CCDF) is ${\\bar{G}}(\\cdot )$ .",
"The channel output is the time-sorted version of the $\\lbrace S_m \\rbrace $ which we denote as $\\lbrace \\vec{S}_i \\rbrace $ , $\\vec{S}_i \\le \\vec{S}_{i+1}$ .",
"However, since the tokens are identical and their transit times are random, the receiver cannot unequivocally know which arrival, $\\vec{S}_i$ corresponds to which transmission $T_m$ .",
"That is, $\\vec{{\\bf S}}$ , the ordered arrival times are related to ${\\bf S}$ through a permutation operation, $P_{\\Omega }(\\vec{{\\bf S}}) = {\\bf S}$ and from the receiver's perspective, $\\Omega $ is a random variable, $\\Omega = 1, 2, \\cdots , M!$ .",
"In the next section, we provide a sampling of results from Part-I upon which we will expand here in Part-II."
],
[
"Key Results from the Companion Paper {{cite:c393509b0643fb9a472184b6f67145208249010a}}",
"A good deal of effort was expended in Part-I quantifying the relationships between ${\\bf T}$ , ${\\bf D}$ , ${\\bf S}$ and $\\vec{{\\bf S}}$ , and in developing a signaling discipline wherein the measure of communication efficacy is determined by the mutual information between $\\vec{{\\bf S}}$ and ${\\bf T}$ , $I(\\vec{{\\bf S}};{\\bf T})$ .",
"That is, we took care to make sure that channel coding theorem results [5] could be applied by deriving a model in which channel uses were (asymptotically) independent.",
"Specifically, we assume sequential finite signaling intervals/epochs of duration $\\tau $ and then define the token intensity as ${\\lambda }= \\frac{M}{\\tau }$ as a proxy for transmitter power (each emission “costs” some fixed energy).",
"In addition, we assume that the mean first-passage time exists with $E[D] = 1/{\\mu }$ so that tokens always (eventually) arrive at the receiver.",
"It is important to note that finite first-passage time is important for information-theoretic patency of the analysis.",
"As shown in Part-I [1], finite first-passage allows sequential signaling intervals (channel uses) to be derived which are, in the limit of long intervals, asymptotically independent.",
"Infinite first-passage does not allow such asymptotically independent sequential intervals to be constructed so that mutual information $I(\\vec{{\\bf S}};{\\bf T})$ is not necessarily the proper measure of information carriage for the system.",
"We note that transport processes such as free-space diffusion do not have finite first-passage.",
"However, any physical system is limited in extent and therefore does have finite (though perhaps long) first-passage under an ergodic transport model.",
"So, the analysis holds for situations where tokens eventually arrive at the receiver.",
"Of course, as discussed in Part-I, there are situations where a token might never arrive at any time.",
"Such situations include channels where the token “denatures” and becomes unrecognizable by the receiver or is “gettered” by agents in the channel which remove the token from circulation before detection [6], [7].",
"Such tokens do not contribute to intersymbol interference (earlier tokens corrupting a subsequent interval) so it is possible that slightly different first-passage time distributions could be used which still preserve asymptotically independent channel uses.",
"However, since any such model produces a first passage density, $g(d)$ with singularities, the specific analysis used in Part-I is not immediately applicable.",
"The implications (and shortcomings) of the finite first-passage assumption are discussed more carefully in the Discussion & Conclusion section of Part-I.",
"Now, as a prelude to deriving channel capacity, we recall from Part-I [1] that if $Q({\\bf x})$ is a hypersymmetric function, $Q({\\bf x}) = Q(P_k({\\bf x}))$ $\\forall k$ where $P_k(\\cdot )$ is a permutation operator and ${\\bf X}$ is a hypersymmetric random vector whose PDF obeys $f_{{\\bf X}}({\\bf x}) =f_{{\\bf X}}(P_k({\\bf x}))$ , then when $\\vec{{\\bf X}}$ is the ordered version of random vector ${\\bf X}$ we have c EX [Q(X) ] = EX [Q(X) ] This expression (Theorem 1 from Part-I) allows us to avoid deriving order distributions on potentially correlated random variables.",
"Next, the mutual information between the input ${\\bf T}$ and the output $\\vec{{\\bf S}}$ of the token timing channel is given by cI(S; T) = h(S) - h(S|T) Then, if we assume that $g(\\cdot )$ does not contain singularities, we observe that the set of all ${\\bf S}$ for which two or more elements are equal is of zero measure which allows us to “fold” the distribution on $f_{{\\bf S}}(\\cdot )$ to obtain a distribution on the ordered $\\vec{{\\bf S}}$ .",
"If we then in addition assume hypersymmetric ${\\bf X}$ , we can write equation () as cI(S; T) = h(S) - M!",
"- h(S|T) Hypersymmetry of ${\\bf X}$ and no singularity in $g(\\cdot )$ implies that we can ignore situations where one or more of the $S_i$ are equal, which then implies an equivalence c{ S, } S which leads to ch(S|T) = h(S, |T) = H(|S,T) + h(S|T) where $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ is the ordering entropy, a measure of the uncertainty about which $S_m$ correspond to which $\\vec{S}_i$ .",
"Equation () allows us to write the equation () as $I(\\vec{{\\bf S}}; {\\bf T})=I({\\bf S};{\\bf T})-\\left( \\log M!",
"- H(\\Omega |\\vec{{\\bf S}},{\\bf T}) \\right)$ And since we know asymptotically independent channel uses can be assured (Part-I Theorem 2, [1]), the channel capacity in bits/nats per channel use is cC = fT() [ I(S;T) - ( M!",
"- H(|S,T) ) ] We then derived an upper bound for the ordering entropy $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ in Part-I as c H(|S,t) H(t) and derived/defined ${H^{\\uparrow }}(\\cdot )$ as rCl3lH(t) = =1M-1 (1 + ) m=M-1 |x| = j=1m Gxj(tm+1 - tj) G1 -xj(tm+1 - tj) with $\\vec{{\\bf t}}$ the size-ordered version of ${\\bf t}$ and $\\bar{{\\bf x}}$ a binary $m$ -vector with $|\\bar{{\\bf x}}|$ defined as its number of non-zero entries.",
"Then, through hypersymmetry arguments as in equation (), we showed that cH(|S,T) = Et [ H(|S,t) ] Et [ H(t) ] = H( T) with equality iff the first passage time is exponential (Theorem 8, Part-I [1]).",
"Based on asymptotically independent channel uses, two key measures of channel capacity were derived.",
"The first, $C_q$ , is the asymptotic per token capacity: cCq = M 1M I(S;T) and the second is $C_t = {\\lambda }C_q$ , the asymptotic per unit time capacity (Theorem 4, Part-I [1]).",
"In what follows we will seek to maximize $h({\\bf S})$ under the deadline constraints on the ${\\bf T}$ , derive a variety of expressions for ${H^{\\uparrow }}({\\bf T})$ for general and then for exponential first-passage under both a deadline and also the mean launch constraint considered in “Bits Through Queues” [2] and elsewhere [3], [4].",
"We follow with asymptotic results for ${H^{\\uparrow }}({\\bf T})/M$ as $M \\rightarrow \\infty $ again assuming exponential first-passage and close by providing upper bounds for $C_q$ and $C_t$ ."
],
[
"Preliminaries",
"The award-winning paper, “Bits Through Queues” [2] and others [3], [4] derived capacity results for a timing channel under a mean launch time constraint.",
"In this section we derive results for a similar single-token timing channel where instead of a mean constraint, the launch time $T$ is limited to $[0,\\tau ]$ [8], [9] and first-passage is exponential with parameter ${\\mu }$ .",
"Here we provide closed forms for both the capacity and for the capacity-achieving input density.",
"However, unlike in [2] we show that exponential first-passage is not the worst case corruption for the launch deadline-constrained channel.",
"Since $T$ is independent of $D$ , the density of $S=T+D$ is given by cfS(s) = 0s fT(t) fD(s-t) dt 0 s and because $T$ is constrained to $[0,\\tau ]$ , we can divide $f_S(s)$ into two regions: region $I$ where $s \\in [0,\\tau ]$ and region $II$ where $s \\in (\\tau ,\\infty )$ .",
"We then have cfS(s)= { l l fS|I(s) 0 s (1-) fS|II(s) s > .",
"where c= 0 fS(s) ds with cfS|I(s) = 0s fT(t) fD(s-t) dt and c(1- ) fS|II(s) = 0 fT(t) fD(s-t) dt For $D$ exponential with parameter ${\\mu }$ we have c fS(s) = 0s fT(t) e-(s-t) dt 0 s and cfS|I(s) = 0s fT(t) e-(s-t) dt and c (1- ) fS|II(s) = e-s 0 fT(t) et dt The entropy of $S$ is then c $h(S) & = -\\int _0^{\\infty } f_{S}(s) \\log f_{S}(s) ds \\\\& = -\\int _0^{\\tau } \\sigma f_{S|I}(s) \\log \\left( \\sigma f_{S|I}(s) \\right) ds\\\\& - \\int _{\\tau }^{\\infty } (1-\\sigma )f_{S|II}(s) \\log \\left( (1-\\sigma )f_{S|II}(s)\\right) ds\\\\& = \\sigma h(S|I) + (1-\\sigma ) h(S|II) + H_B (\\sigma ) \\\\$ where $H_B(\\cdot )$ is the binary entropy function.",
"Notice that no particular care has to be taken with the integrals at $s=T$ because $f_S(s)$ cannot contain singularities – it is obtained by the convolution of two densities, one of which, $f_D(\\cdot ) = g(\\cdot )$ , contains no singularities."
],
[
"Maximization of $h(S)$",
"We observe of equation (REF ) that the shape of the conditional density for $s>\\tau $ is completely determined – an exponential with parameter ${\\mu }$ as depicted in FIGURE REF .",
"Figure: The shapes associated with f S (s)f_{S}(s): We assume arbitrary shape in region I and the requisite exponential shape in region II.Thus, selection of $f_T(\\cdot )$ does not affect $f_{S|II}(\\cdot )$ and we must have $h(S|II) = 1 - \\log {\\mu }$ .",
"This observation suggests a three-step approach to maximizing $h(S)$ .",
"In the first two steps, we completely ignore $f_T(\\cdot )$ and find the shape $f_{S|I}(\\cdot )$ and value of $\\sigma $ which maximize equation (REF ).",
"In step three, we determine that there indeed exists a density $f_T(\\cdot )$ which produces the optimizing $f_S(\\cdot )$ .",
"Step 1: For fixed $\\sigma $ we see from equation (REF ) that $h(S)$ is maximized solely by our choice of $f_{S|I}(\\cdot )$ .",
"The uniform density maximizes entropy on a finite interval [10].",
"Thus, $f_{S|I}(s) = \\frac{1}{\\tau }$ and $h(S|I) = \\log \\tau $ as depicted in FIGURE REF .",
"Figure: The updated shape of f S (s)f_{S}(s) after step 1: f S|I (s)f_{S|I}(s) is chosen as 1 τ\\frac{1}{\\tau }.Step 2: Since for any $\\sigma $ , $h(S|I) = \\log \\tau $ , we have $h(S)=\\sigma \\log \\tau +(1-\\sigma ) (1 - \\log {\\mu }) + H_B (\\sigma ) \\\\$ Taking the derivative of equation (REF ) with respect to $\\sigma $ yields c- (1-) - (1+) + (1+(1-)) which we set to zero to obtain c- 1- - 1 = 0 We rearrange to obtain c= e1- from which we deduce that the optimal $\\sigma $ is c * = e + Returning to the entropy maximization we have cfT() h(S) * + (1-*) (1 - ) + HB (*) which through substitution of $\\sigma ^*$ according to equation (REF ) yields cfT() h(S) ( e + ) with equality when cfS(s) = {ll e + 0 s < ee+ e- (s-) s .",
"Step 3: All that remains is to ascertain whether $\\exists f_T(\\cdot )$ which can generate the $f_S(s)$ of equation (REF ).",
"Since $f_S(\\cdot )$ is the convolution of $f_D(\\cdot )$ and $f_T(\\cdot )$ we can use Fourier transforms to obtain a candidate solution for $f_T(\\cdot )$ .",
"That is, the Fourier transform of $f_D(\\cdot )$ is $\\frac{{\\mu }}{{\\mu }+ j 2 \\pi f}$ so the Fourier transform of $f_T(\\cdot )$ is cF { fT() } = F { fS() } ( j 2 f + 1 ) Multiplication by $j 2 \\pi f$ implies differentiation so we must have cfT(t) = 1 ddt fS(t) + fS(t) which implies via equation (REF ) that c fT(t) = {ll e+ 0<t< (t) 1e + + (t-) e-1e + o.w.",
".",
"– a valid probability density function.",
"We can now state the maximum mutual information (capacity in bits per channel use) as c fT() I(S;T) = ( e ) - (1- ) = ( 1 + e ) which is achieved using the emission time density of equation (REF ).",
"We summarize the result as a theorem: Theorem 1 Maximum $I(S;T)$ Under a Deadline Constraint: If $S=T+D$ where $D$ is an exponential random variable with parameter ${\\mu }$ and $T \\in [0,\\tau ]$ , then the mutual information between $S$ and $T$ obeys cI(S;T) ( 1 + e ) with equality when cfT(t) = {ll e+ 0<t< (t) 1e + + (t-) e-1e + o.w.",
".",
"and cfS(s) = {ll e + 0 s < ee+ e- (s-) s .",
"Proof: Theorem (REF ) See the development leading to the statement of equation (REF ).",
"$\\bullet $ The only remaining question is whether for interval-limited inputs, the exponential first-passage time density, to quote [2] “plays the same role ... that Gaussian noise plays in additive noise channels.” Unfortunately the answer is no, a result we state as a theorem: Theorem 2 For $T$ Constrained to $[0,\\tau ]$ , the Minmax Mutual Information First-Passage Density Is NOT Exponential: If $g(\\cdot )$ is a first passage density with mean $1/\\mu $ and $f_T(\\cdot )$ can be nonzero only on $[0,\\tau ]$ , then c*arg ming() [ fT() I(S;T) ] = g*(s) e-s u(s) where $u(\\cdot )$ is the unit step function.",
"Proof: Theorem (REF ) Consider that cI(S;T) = fT(t) g(s-t) g(s-t)fS(s) dt ds is convex in $g(\\cdot )$ [5], [11].",
"Since we constrain $g(\\cdot )$ to be non-negative with mean $1/\\mu $ and unit integral, we can apply Euler-Lagrange variational techniques [12].",
"That is, we set $q(x) = g(x) + \\epsilon \\eta (x)$ where $\\eta (x)$ is any function defined on $[0,\\infty )$ , and look for the stationary point rCl3ldd [ fT(t) q(s-t) q(s-t) fT(x) q(s-x) dx dt ds .",
"+ .",
"a ( s q(s) ds - 1 ) + b (q(s) ds - 1 ) ]= 0 =0 where $a$ and $b$ are (Lagrange) multipliers.",
"Satisfaction of equation (REF ) for any possible $\\eta (\\cdot )$ requires (after expansion and a change of coordinate systems in the double integral) that cg(s) = 0fT(t) fS(s+t) dt + a s + b for the $g(\\cdot )$ that minimizes equation (REF ).",
"Now, from Theorem REF we know the form of the optimizing $f_T(t)$ , $t \\in [0,\\tau ]$ and the resulting $f_S(s)$ were $g(\\cdot )$ exponential with mean $1/\\mu $ .",
"We also know that $I(S;T)$ is concave in $f_T(t)$[5], [11].",
"Thus, were exponential $g(\\cdot )$ to minimize the maximum mutual information, the left hand side of equation (REF ) would be a linear function of $s$ .",
"Thus, the integral term on the right would also need to be a linear function of $s$ given $f_S(s)$ as in equation (REF ) and $f_T(t)$ as in equation (REF ).",
"For $s \\ge \\tau $ we have $f_S(s+t) = \\frac{\\mu e}{e + \\mu \\tau } e^{ - \\mu (s+t-\\tau )}$ and thence rCl3l0fT(t) fS(s+t) dt = 0 ( (t) + + (t-)(e-1) ) (-(s+t-))e + dt + ee + = - s ee + - 2e + .",
"(s+t - )22 |0+ ee + = ee + +- s ee + - 2e + ( s - 22 ) which is indeed a linear function of $s$ .",
"However, when $0 \\le s < \\tau $ we obtain rCl3l0fT(t) fS(s+t) dt = 0- s ((t)e + + e + ) e + dt + - s ( e + + (t-)(e-1)e + ) ee + dt - - s ( e + + (t-)(e-1)e + ) (s+t-) dt = 1 + (- s)e + e + + s +e - 1e + ee + - s (e-1)e + - 2e + .",
"(s+t-)22|-s = e + + 2s + e(1- s) - 1e + - 2e + s22 which does not have the requisite form owing to the term in $s^2$ .",
"Therefore, for $T$ constrained to $[0,\\tau ]$ , the minmax $I(S;T)$ first-passage density, $g(\\cdot )$ , is not exponential.",
"$\\bullet $ It is important to note that owing to a faulty proof [8], exponential first passage was previously claimed to maximally suppress capacity of the constrained-launch channel.",
"Theorem REF corrects this error."
],
[
"Ordering Entropy, $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$",
"In this section we derive a number of results for the ordering entropy, $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ , both generally and for exponential first-passage.",
"As a prelude, we recall from section and TABLE REF that $\\vec{{\\bf t}}= \\lbrace \\vec{t}_1, \\vec{t}_2, \\cdots , \\vec{t}_M \\rbrace $ is the ordered version of ${\\bf t}= \\lbrace t_1, t_2, \\cdots , t_M\\rbrace $ , the launch times, and that $G(\\cdot )$ is the cumulative distribution function (CDF) for the first-passage time $D$ (with ${\\bar{G}}(\\cdot )$ its complementary cumulative distribution function (CCDF).",
"We recall from Part-I [1] that $H(\\Omega |\\vec{{\\bf S}},{\\bf t}) \\le {H^{\\uparrow }}({\\bf t})$ from equation () with ${H^{\\uparrow }}({\\bf t})$ defined as in equation ().",
"We also recall from Part-I that $\\bar{{\\bf x}}$ is a binary vector of dimension $m$ and $\\sum _{|\\bar{{\\bf x}}| = \\ell }$ is a sum over all $\\bar{{\\bf x}}$ containing exactly $\\ell $ 1's.",
"The inequality in equation () is an equality iff first-passage is exponential with ${\\bar{G}}(x) =e^{-{\\mu }x} u(x)$ , where $u(x)$ is the unit step function (Theorem 9, Part-I)."
],
[
"General Calculation of ${H^{\\uparrow }}({\\bf t})$",
"To calculate ${H^{\\uparrow }}({\\bf t})$ we first define c m,(t) |x| = j=1m Gxj(tm+1 - tj) G1 -xj(tm+1 - tj) which implies via equation () that c H(t) = =1M-1 (1 + ) m=M-1 m,(t) In principle, we could derive ${H^{\\uparrow }}({\\bf T})$ by taking the expectation of equation (REF ) with respect to ordered emission times, $\\vec{{\\bf t}}$ .",
"However, direct analytic evaluation of ${H^{\\uparrow }}({\\bf T})$ requires we derive joint order densities on the underlying ${\\bf T}$ , a difficult task in general when the individual $\\lbrace T_m \\rbrace $ are not necessarily independent.",
"So, we take a different approach.",
"The sum over all permutations of binary vector $\\bar{{\\bf x}}$ in the definition of $\\Theta _{m,\\ell }(\\vec{{\\bf t}})$ (equation (REF )) renders it hypersymmetric in $\\vec{t}_1 ,\\cdots , \\vec{t}_m$ given the $(m+1)^{\\mbox{st}}$ smallest emission time $\\vec{t}_{m+1}$ .",
"That is, $\\Theta _{m,\\ell }(\\vec{{\\bf t}}) = \\Theta _{m,\\ell }(P_k(\\vec{t}_1, \\cdots ,\\vec{t}_m), \\vec{t}_{m+1})$ for any permutation function $k$ so long as $\\vec{t}_{m+1}$ is fixed.",
"In what follows we therefore drop the over-vector notation for the $t_1, t_2, \\cdots , t_m$ and assume all are less than $\\vec{t}_{m+1}$ .",
"Therefore, by equation () we can define $E[\\Theta _{m,\\ell }] = \\bar{\\Theta }_{m,\\ell }$ as cETm+1 [ ET1,,Tm|Tm+1 [ m,(T1,,Tm,Tm+1) ] ] Then, the CDF, $F_{\\vec{T}_{m+1}} (t_{m+1})$ , of the $(m+1)^{\\mbox{st}}$ smallest emission time is rCl3lFTm+1 (tm+1) = 1 - k=0m M ()k 0tm+1 $k$ tm+1 $M-k$ fT(t) dtM dt1 and likewise, the CDF, $F_{T_1,\\cdots ,T_m|{\\vec{T}}_{m+1}} (t_1,\\cdots ,t_m|\\vec{t}_{m+1})$ , of the smallest unordered $T_1,\\cdots ,T_m$ given $\\vec{T}_{m+1}$ is rCl3lFT1,,Tm|Tm+1 (t1,,tm|tm+1) = FT1,,Tm (t1,,tm)FT1,,Tm (tm+1,,tm+1) $\\forall t_j \\le \\vec{t}_{m+1}$ where $j=1,\\cdots ,m$ .",
"Therefore, by the hypersymmetry of ${\\Theta }_{m,\\ell }$ in $t_1,\\cdots ,t_m$ we may write $\\bar{\\Theta }_{m,\\ell }$ as c0 0tm+1 fTm+1(tm+1) fTm(tm)B(m,,t)FTm(tm+1,,tm+1) dtm dtm+1 where ${\\bf T_m} = \\lbrace T_1, \\cdots , T_M \\rbrace $ , ${\\bf t_m} = \\lbrace T_1, \\cdots , t_M \\rbrace $ and c B(m,,t) m () j=1 G(tm+1 - tj)k=+1m G(tm+1 - tk) and thence ${H^{\\uparrow }}({\\bf T})=\\sum _{\\ell =1}^{M-1}\\log (1 + \\ell )\\sum _{m=\\ell }^{M-1}\\bar{\\Theta }_{m,\\ell }$ In addition, if we define $\\Gamma _{M,\\ell }=\\sum _{m=\\ell }^{M-1}\\bar{\\Theta }_{m,\\ell }$ and $\\Delta \\Gamma _{M,\\ell }=\\Gamma _{M,\\ell } - \\Gamma _{M,\\ell +1}$ then we can also express ${H^{\\uparrow }}({\\bf T})$ as ${H^{\\uparrow }}({\\bf T})=\\sum _{\\ell =1}^{M-1}\\Delta \\Gamma _{M,\\ell }\\log (\\ell + 1)!$ The development starting in section REF proves the following theorem: Theorem 3 The General Form of ${H^{\\uparrow }}({\\bf T})$ : If we define $\\Gamma _{M,\\ell }=\\sum _{m=\\ell }^{M-1}\\bar{\\Theta }_{m,\\ell }$ and $\\Delta \\Gamma _{M,\\ell }=\\Gamma _{M,\\ell } - \\Gamma _{M,\\ell +1}$ where $\\bar{\\Theta }_{m,\\ell }$ is as defined by equations (REF ) and equation (REF ).",
"then we can express ${H^{\\uparrow }}({\\bf T})$ as cH(T) = =1M-1 M, (+ 1)!",
"Proof: Theorem (REF ) See the development starting in section REF leading to the statement of Theorem REF .",
"$\\bullet $ This concludes our calculation of ${H^{\\uparrow }}({\\bf T})$ for general input distributions $f_{{\\bf T}}(\\cdot )$ .",
"The key utility of our formulation is that it does not require joint order distributions for the $\\lbrace T_m\\rbrace $ , only the more easily calculable $m^{\\mbox{th}}$ order distribution for ${\\vec{T}}_m$ .",
"We now turn to the case where the ${\\bf T}$ are i.i.d.",
"– important because i.i.d.",
"${\\bf T}$ increases entropy $h({\\bf S})$ ."
],
[
"$H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ for General IID {{formula:e5bd77bc-4ba0-45bc-8eeb-b2a4b9dbfaf3}}",
"With i.i.d.",
"${\\bf T}$ , we can use the definition of $\\Theta _{m,\\ell }(\\cdot )$ in equation (REF ) and the hypersymmetric result of equation (REF ) to obtain cm, = ETm+1 [ c m () ET Tm+1 [ G(Tm+1 - T) ] $\\times $ ET Tm+1m- [ (1 - G(Tm+1 - T)) ] ] From the definition of $F_{{\\vec{T}}{m+1}}(\\cdot )$ in equation (REF ) we obtain cfTm+1(t) = ddt [ 1 - k=0m M ()k FTk(t)(1-FT(t))M-k ] which after rearranging as a telescoping sum simplifies to ck=0m (M-k) M ()k fT(t)FTk(t)(1-FT(t))M-k-1 - k=0m-1 (k+1) M ()k+1 fT(t)FTk(t)(1-FT(t))M-k-1 which further simplifies to c(m+1) M ()m+1 fT(t)FTm(t)(1-FT(t))M-m-1 We then define c (t) = 0t fT(x) G(t-x) dx and c0t fT(x) (1- G(t-x)) dx = FT(t) - (t) which allows us to write c(Tm+1) = ET Tm+1 [ G(Tm+1 - T) ] and cFT(t) - (Tm+1) = ET Tm+1 [ 1 - G(Tm+1 - T) ] which upon substitution into equation (REF ) allows us to write $\\bar{\\Theta }_{m,\\ell }$ as rCl3lm, = (m+1) M ()m+1 m () 0 [ c fT(t)(1-FT(t))M-m-1 (t) $\\times $ (FT(t)-(t))m- ] dt and then as rCl3lm, = M M -1 () M - - 1 ()m- c 0 [ c fT(t)(1-FT(t))M-m-1 (t) $\\times $ (FT(t)-(t))m- ] dt To evaluate ${H^{\\uparrow }}({\\bf T})$ in equation (REF ) we must first compute $\\Gamma _{M,\\ell } =\\sum _{m=\\ell }^{M-1}\\bar{\\Theta }_{m,\\ell }$ as rCl3lM, = M M -1 () 0 fT(t) [ c (1- FT(t))M-1 ( (t)FT(t) - (t) ) $\\times $ m=M-1 M - - 1 ()m- ( FT(t)-(t)1 - FT(t) )m ] dt which we rewrite as rCl3lM, = M M -1 () 0 fT(t) [ c (1- FT(t))M-1 ( (t)FT(t) - (t) ) $\\times $ m=0M-1- M - - 1 ()m ( FT(t)-(t)1 - FT(t) )m+ ] dt We consolidate the binomial sum to obtain rCl3lM, = M M -1 () 0 fT(t) [ c (1- FT(t))M-1 ( (t)FT(t) - (t) ) $\\times $ ( FT(t)-(t)1 - FT(t) ) ( 1-(t)1 - FT(t) )M-1- ] dt which reduces to c M, = 0 M M-1 () fT(t) (t) ( 1 - (t) )M-1- dt for $\\ell = 1,2,\\cdots ,M-1$ .",
"Now consider the integrand of the difference $\\Gamma _{M,\\ell } -\\Gamma _{M,\\ell +1}$ where we drop the $t$ dependence for notational convenience cM, - M,+1 = [ c M M-1 () ( 1 -)M--1 $-$ M M1 ()+1 +1 ( 1 -)M--2 ] We can rewrite this expression as cM [ M-1 () + r=1M- -1 (-1)r r [ c M-1 () M- - 1 ()r $+$ M-1 ()+1 M- - 2 ()r-1 ] ] which after consolidating terms becomes cM [ c M-1 () $+$ 1M M ()+1 r=1M--1 (-1)r M--1 ()r (+r+1) r ] Extending the sum to $r=0$ and subtracting the $r=0$ term produces cM ()+1 r=0M--1 (-1)r M--1 ()r (+r+1) r+ which can be recognized as cdd [ M ()+1 r=0M--1 (-1)r M--1 ()r r++1 ] and then reduced to cM ()+1 dd [ +1 (1-)M-- 1 ] so that we have $\\Delta \\Gamma _{M,\\ell }$ as c M ()+1 r=0M- -1 (-1)r M- -1 ()r (+r+1) E [ r+(t) ] where $E[\\cdot ]$ is the expectation using $f_T(t)$ .",
"The previous development of section REF proves the following theorem: Theorem 4 An Upper Bound for Ordering Entropy $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ with I.I.D.",
"${\\bf T}$ : If ${\\bf T}$ is i.i.d., then we can write $\\Delta \\Gamma _{M,\\ell }$ as cM ()+1 r=0M- -1 (-1)r M- -1 ()r (+r+1) E [ r+(t) ] where c(t) = 0t fT(x) G(t-x) dx so that cH(|S,T) H(T) = =1M-1 M, (+ 1)!",
"Proof: Theorem (REF ) See the development of section REF leading to the statement of Theorem REF .",
"$\\bullet $"
],
[
"$H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ Special Case IID {{formula:79097842-dc7e-4377-82f0-38ce2b2c7e7a}}",
"Here we derive expressions for $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ when the i.i.d.",
"input distribution is that which maximizes $I({\\bf S};{\\bf T})$ .",
"We consider the following cases: Exponential first-passage with $E[T] = \\tau $ Exponential first-passage with emission deadline, $\\tau $"
],
[
"Exponential Transit Times with a Mean Constraint",
"For exponential first-passage times with mean $1/{\\mu }$ , the probability density of ${\\bf T}$ that maximizes $h({\\bf S})$ subject to a mean constraint $E[\\sum _m T_m]\\le M\\tau $ is i.i.d.",
"with marginal c fTm(t) = a(t) + a(1-a) e-a t u(t) where $a = 1/({\\mu }\\tau + 1)$ [2] and $u(t)$ is the unit step function [2].",
"For exponential transit we have cG(t) = e-t u(t) and thereby c(t) = 0t fT(x) G(t-x) dx = a e-a t u(t) We then require an expression for $E_T[\\phi ^k(T)]$ .",
"Remembering that $\\int _{0^-}^{0^+} \\delta (t) u^k(t) dt = \\frac{1}{k+1}$ we obtain cET[k(T)] = 0 fT(t) ak e-ka t uk(t) dt = akk+1 so that equation (REF ) becomes cM, = M ()+1 r=0M--1 (-1)r M--1 ()r ar+ which reduces to c M, = M ()+1 a(1-a)M-- 1 for $\\ell = 1,2,\\cdots ,M-1$ .",
"With $a = \\frac{1}{{\\mu }\\tau + 1}$ we can write $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ as c (+ 1) k=0M (k!)",
"M ()k (+ 1 )M-k (1+ 1 )k which is the expectation of $({\\mu }\\tau + 1) \\log K!$ for a binomial random variable $K$ with parameters $M$ and $\\frac{1}{{\\mu }\\tau + 1}$ , or c H(|S,T) = (+ 1) EK [ K! ]",
"We restate this result as a theorem: Theorem 5 $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ for Exponential First-Passage with a Mean Constraint ($E[T] = \\tau $ ): For $T$ distributed as equation (REF ) and exponential first-passage with parameter ${\\mu }$ , we have cH(|S,T) = (+ 1) EK [ K! ]",
"where $K$ is a binomial random variable with parameters $M$ and $\\frac{1}{1+{\\mu }\\tau }$ .",
"Proof: Theorem (REF ) See the development leading to the statement of Theorem REF and direct application of Theorem REF .",
"$\\bullet $"
],
[
"Exponential Transit Times with a Deadline",
"Theorem REF states that if $T$ is constrained to $[0,\\tau ]$ then the $f_T(t)$ that maximizes $h(S)$ (and therefore $h({\\bf S})$ when in i.i.d.",
"form) is c fT(t) = 1e + (t) + e + + e-1e + (t-) for $t \\in [0,\\tau ]$ and zero otherwise.",
"To obtain the corresponding $H(\\Omega |\\vec{{\\bf S}},{\\bf T}) = {H^{\\uparrow }}({\\bf T})$ we calculate $\\phi (t)$ as c 0t fT(x) e-(t-x) dx = {ll 1 e + 0t ee + e-(t - ) t > 0 o.w.",
".",
"Once again, we require an expression for the integral $\\int _0^{\\infty } f_T(t)\\phi ^k(t)dt$ , and again remembering that $\\int _{0^-}^{0^+} \\delta (t) u^k(t) dt = \\frac{1}{k+1}$ we obtain $E_T \\left[ \\phi ^k(T) \\right]$ as c(1e+ )k+1 0-0+ (t) uk(t) dt + (1e+ )k+1 0 dt + (e-1)( 1e+ )k+1 -+ (t-) ( 1 + (e-1)u(t-) )k dt which reduces to c(1e+ )k+1 [ 1k+1 + + r=0k k ()r 1r+1 (e-1)r+1 ] which further reduces to c(1e+ )k+1 [ 1k+1 + + ek+1k+1 - 1k+1 ] and then finally, cET [ k(T) ] = (1e+ )k+1 [ + ek+1k+1 ] so that $\\Delta \\Gamma _{M,\\ell }$ in equation (REF ) becomes cM ()+1 r=0M- -1 [ c (-1)r M--1 ()r (+r+1) (1e+ )r++1 [ + er++1r++1 ] ] which reduces to cM ()+1 (ee+ )+1 (e+ )M--1 $+$ M ()+1 r=0M--1 [ c (-1)r M--1 ()r (+r+1) (1e+ )r++1 ] and then to cM ()+1 (ee+ )+1 (e+ )M--1 $+$ [ c M ()+1 (1 - 1e+ )M--2 (1e+ )+1 ( + 1 -Me+ ) ] If we define $k = \\ell +1$ and then cp1 = ee+ and cp2 = 1e+ we can then write c M,k-1 = [ c M ()k p1k (1-p1)M-k $+$ 1-p2 [ k - M+ e ] M ()k p2k (1-p2)M-k ] Now if we define random variables $K_i$ to be binomial with parameters $M$ and $p_i$ , the following theorem results from direct application of Theorem REF : Theorem 6 $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ for Exponential First-Passage with a Launch Deadline (${\\bf T}\\in [0,\\tau ]^M$ ): For $T$ distributed as equation (REF ) we have c H(|S,T) = [ c EK1 [ K1! ]",
"+ 1-p2 EK2 [ K2 K2! ]",
"$-$ M (1-p2)(+ e) EK2 [ K2! ]",
"] where $K_1$ is a binomial random variable with parameters $M$ and $\\frac{e}{e+{\\mu }\\tau }$ and $K_2$ is a binomial random variable with parameters $M$ and $\\frac{1}{e+{\\mu }\\tau }$ .",
"Proof: Theorem (REF ) See the development leading to the statement of Theorem REF and direct application of Theorem REF .",
"$\\bullet $"
],
[
"Asymptotic $H(\\Omega |\\vec{{\\bf S}},{\\bf T})/M$ For Exponential First-Passage",
"We are interested in asymptotic values of $H(\\Omega |\\vec{{\\bf S}},{\\bf T})/M$ owing to our definition of capacity per token in equation () (see also in Part-I [1]).",
"To that end, recall that ${\\lambda }\\tau = M$ and we define $\\rho = {\\lambda }/{\\mu }$ , a measure of system token “load” (also a proxy for power expenditure in units of energy per passage time), so that c11 + M/ = 11 + M/ and likewise cee + M/ = ee + M/ and c1e + M/ = 1e + M/ Now, remember the binomial distribution for fixed $k$ and large $M$ is approximated by cM ()k pk (1-p)M-k Mkk!",
"pk (1-p)M-k So, for any finite $k$ it is easily seen that for $M \\rightarrow \\infty $ c M ()k ( 11 + M )k (1 - 11 + M )M-k e- 1k!",
"k c M ()k ( 1e + M )k (1 - 1e + M )M-k e- 1k!",
"k and c M ()k ( ee + M )k (1 - ee + M )M-k e-e 1k!",
"k ek and we note that all these limiting distributions are Poisson.",
"Equation (REF ) and equation (REF ) can then be combined with equation (REF ), equation (REF ) and equation (REF ) to produce the following two theorems: Theorem 7 Asymptotic $H(\\Omega |\\vec{{\\bf S}},{\\bf T})/M$ for Exponential First-Passage with a Mean Constraint ($E[T] = \\tau $ ): For exponential first-passage with $E[T] = \\tau $ and $f_T(\\cdot )$ as given in equation (REF ), $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ is given by c M H(|S,T)M = e- k=2 k-1 k!k!",
"= E[k!]",
"where the final expectation is for $k$ a Poisson random variable with parameter $\\rho $ .",
"Proof: Theorem (REF ) See Theorem REF and the development leading up to the statement of Theorem REF .",
"$\\bullet $ Theorem 8 Asymptotic $H(\\Omega |\\vec{{\\bf S}},{\\bf T})/M$ for Exponential First-Passage with a Deadline Constraint ($T \\in [0,\\tau ]$ ) For exponential first-passage with ${\\bf T}\\in [0,M/\\rho ]^M$ and $f_T(\\cdot )$ as given in equation (REF ), $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ is given by c M H(|S,T)M = E[ (k - 1) k!]",
"where the final expectation is for $k$ a Poisson random variable with parameter $\\rho $ .",
"Proof: Theorem (REF ) See Theorem REF and the development leading up to the statement of Theorem REF .",
"$\\bullet $"
],
[
"Upper Bound for $I(\\vec{{\\bf S}};{\\bf T})$",
"With analytic bounds for $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ , we can now consider bounds on the mutual information, $I(\\vec{{\\bf S}};{\\bf T})$ .",
"In Part-I (using results from this, Part-II) lower bounds were derived.",
"Here we consider an upper bound.",
"To begin, however, we must find an upper bound for $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ ."
],
[
"A Useful Upper Bound On $H(\\Omega |\\vec{{\\bf S}},{\\bf T}$ )",
"We state the bound as a theorem with proof.",
"Theorem 9 An Upper Bound for $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ : Given cQ() = G(||) where ${\\bar{G}}(\\cdot )$ is the CCDF of the passage time, and defining c T = ET [ Q(T1 - T2) ] we have c H(|S,T) ET[ H(T) ] M ( 1 + M-12 T ) Proof: Theorem (REF ) ${H^{\\uparrow }}({\\bf t})$ , defined in equation () and derived in Part-I [1], [9] is an upper bound for $H(\\Omega |\\vec{{\\bf S}},\\vec{{\\bf t}})$ .",
"The bound is satisfied with equality iff the first-passage density is exponential [1], [9].",
"For a given $m$ , let us define ${\\bar{G}}_k = {\\bar{G}}(\\vec{t}_{m+1} - \\vec{t}_k)$ and $G_k$ in a corresponding way.",
"Then, consider the sum of the following $2^m$ terms cGm Gm-1 Gm-2 G3 G2 G1 + Gm Gm-1 Gm-2 G3 G2 G1 + $\\vdots $ + Gm Gm-1 Gm-2 G3 G2 G1 + Gm Gm-1 Gm-2 G3 G2 G1 Taken pairwise it is easy to see that this sum telescopes to 1 since ${\\bar{G}}_i + G_i = 1$ so that the ensemble of terms is a PMF.",
"Furthermore, since $m = 1,2, \\cdots , M$ , the complete ensemble of the terms, $\\prod _{j=1}^m {\\bar{G}}^{\\bar{x}_j}(\\vec{t}_{m+1} - \\vec{t}_j) G^{1-\\bar{x}_j}(\\vec{t}_{m+1} -\\vec{t}_j)$ , $m=1,2,\\cdots , M$ , sums to $M$ .",
"So, we can define c p|t,m = |x| = j=1m Gxj(tm+1 - tj) G1 -xj(tm+1 - tj) and then c p|t = m=M-1 |x| = j=1m Gxj(tm+1 - tj) G1 -xj(tm+1 - tj)M for $\\ell = 0, 1, \\cdots , M-1$ .",
"We can use Jensen's inequality to write c H(t) = E|t [(1+) ] M (E[|t] + 1) Now consider that cE[|t] = m=0M-1 1M E[|t,m] and the explicit expansion of $E[\\ell |\\vec{{\\bf t}},m]$ is c =0m $\\left({\\displaystyle \\sum _{|\\bar{{\\bf x}}| = \\ell }\\prod _{j=1}^m}{{{\\bar{G}}}^{\\bar{x}_j}(\\vec{t}_{m+1} - \\vec{t}_j){G}^{1- \\bar{x}_j}(\\vec{t}_{m+1} - \\vec{t}_j)}\\right)$ Then consider that $E[\\ell | \\vec{{\\bf t}},m]$ has the terms c .",
"0 [ c Gm Gm-1 Gm-2 G3 G3 G1 ] } 1 term c. 1 [ c Gm Gm-1 Gm-2 G3 G2 G1 + Gm Gm-1 Gm-2 G3 G2 G1 + $\\vdots $ + Gm Gm-1 Gm-2 G3 G2 G1 ] } $m$ terms c. 2 [ c Gm Gm-1 Gm-2 G3 G2 G1 + $\\vdots $ + Gm Gm-1 Gm-2 G3 G2 G1 ] }${m \\atopwithdelims ()2}$ terms with final term c. m [ c Gm Gm-1 Gm-2 G3 G2G1 ] }1 term Then consider the term $G_{m} G_{m-1} G_{m-2} \\cdots G_2 {\\bar{G}}_1$ and group together the other $2^{m-1} - 1$ different terms that contain ${\\bar{G}}_1$ .",
"The sum of all these terms is ${\\bar{G}}_1$ .",
"We can do a corresponding grouping for each of the $m$ terms in which ${\\bar{G}}_i$ appears exactly once.",
"Thus, by expanding and regrouping the inner product terms of equation (REF ) we can show that cE[|t,m] = j=1m G(tm+1 - tj) which results in cH(t) M ( 1 + 1M m=1M-1 j=1m G(tm+1 - tj) ) via equation (REF ) and equation (REF ), remembering that $E[\\ell |\\vec{{\\bf t}},m=0] = 0$ .",
"Taking the expectation in $\\vec{{\\bf T}}$ yields c H(T) M ( 1 + m=1M-1 j=1m E [ G(Tm+1 - Tj) ]M ) We then note that all ordered differences between the $T_i$ are accounted for in equation (REF ).",
"For any given ${\\bf T}$ there are $\\frac{M(M-1)}{2}$ ordered terms.",
"Thus, we can rewrite equation (REF ) as c ET[ H(T) ] M ( 1 + i,j, i jM E [ G(|Ti - Tj |) ]2M ) where the factor of $\\frac{1}{2}$ is introduced to account for terms $T_i < T_j$ which would not appear in the ordered case of equation (REF ).",
"Finally, hypersymmetry of ${\\bf T}$ requires that $E \\left[ {{\\bar{G}}}(\\left|T_i - T_j \\right|) \\right] = \\gamma _T$ , a constant for $i\\ne j$ so that cH(|S,T) ET[ H(T) ] M ( 1 + M-12 T ) which matches the result stated in Theorem (REF ) and thus proves the theorem.",
"$\\bullet $"
],
[
"Maximizing $h({\\bf S}) + M \\log \\left( 1 + \\gamma _S (M-1) \\right)$",
"We now have the rudiments of an upper bound for $I(\\vec{{\\bf S}};{\\bf T})$ in rCl3lfT() I(S;T)M h(S)M + ( 1 + S (M-1) ) - M!M - h(D) However, the upper bound equation (REF ) is in terms of $f_{{\\bf T}}(\\cdot )$ whereas $h({\\bf S})$ is a function(al) of $f_{{\\bf S}}(\\cdot )$ .",
"Therefore, we must develop a relationship between $\\gamma _T = E \\left[ Q(T_1 - T_2) \\right]$ and $\\gamma _S = E \\left[ Q(S_1 - S_2) \\right]$ .",
"This relationship allows us to fix $\\gamma _S$ and maximize $h({\\bf S})$ while still maintaining an upper bound on $H(\\Omega |\\vec{{\\bf S}},\\vec{{\\bf T}})$ .",
"From here onward we assume exponential first-passage of tokens.",
"Theorem 10 $\\gamma _T$ versus $\\gamma _S$ for Exponential First-Passage: If the first-passage density $f_D(\\cdot )$ is exponential then cE [ Q(S1 - S2) ] 12 E [ Q(T1 - T2) ] or cS 12 T Proof: Theorem (REF ) Let $\\Delta = T_1 - T_2$ and ${\\cal D} = D_2 - D_1$ .",
"Then $\\Delta +{\\cal D} =S_1 - S_2$ .",
"For the i.i.d.",
"$D_i$ exponential we have ${{\\bar{G}}}(d) = e^{-{\\lambda }d}$ , $d \\ge 0$ .",
"Thus, $Q(\\cdot )= e^{-{\\lambda }|\\cdot |}$ .",
"We then note that $|a+b| \\le |a| + |b|$ so that rClE[Q(+D)] = E[e-| +D |] E[e-|| -|D |] = E[Q()]E[Q(D)] because $\\Delta $ and ${\\cal D}$ are independent.",
"Then consider that the density of ${\\cal D}$ is $f_{\\cal D}({\\cdot }) = \\frac{{\\lambda }}{2}e^{-{\\lambda }|{\\cdot }|}$ so that $E[Q({\\cal D})] = \\int _{=\\infty }^\\infty \\frac{{\\lambda }}{2}e^{-{\\lambda }|z|} e^{-{\\lambda }|z|} dz =\\frac{1}{2}$ which completes the proof.",
"$\\bullet $ Now, suppose we fix $E \\left[ Q(S_1 - S_2) \\right] = \\gamma _S$ .",
"Then, owing to hypersymmetry we have $E \\left[ Q(S_i - S_j) \\right] = \\gamma _S$ $\\forall i,j,i \\ne j$ .",
"Using standard Euler-Lagrange optimization [12], we can find the density $f_{{\\bf S}}$ which maximizes $h({\\bf S})$ as cfS* (s) = 1A() ei,ji jQ(si - sj) where cA() = ei,jijQ(si - sj) dsand $\\beta $ is a constant chosen to satisfy $E[Q(S_1 - S_2)] = \\gamma _S$ .",
"The entropy of ${\\bf S}$ is then ch(S) = A() - M(M-1) S We note that for $\\beta =0$ , $f_{{\\bf S}}(\\cdot )$ is uniform.",
"Increasing $\\beta $ makes $f_{{\\bf S}}(\\cdot )$ more “peaky” in regions where $s_i \\approx s_j$ since $Q(0)=1$ and $Q(\\cdot )$ is monotonically decreasing away from zero.",
"Likewise, decreasing $\\beta $ reduces $f_{{\\bf S}}(\\cdot )$ in the vicinity of $s_i \\approx s_j$ .",
"Thus, $\\gamma _S$ increases monotonically with $\\beta $ .",
"The result is that $\\gamma _S^{\\prime }(\\cdot )$ is strictly positive.",
"More formally, we have from the definition of $\\gamma _S(\\beta )$ that cM(M-1)S() = E[i,jijQ(si - sj) ] S() Then c S() = E[ ( i,jijQ(si - sj) )2 ] - E2 [i,jijQ(si - sj) ] which is a variance and therefore greater than or equal to zero.",
"Thus, $\\gamma _S^{\\prime }(\\beta ) \\ge 0$ .",
"And since $0 \\le \\gamma _S(\\beta ) \\le 1$ , we must also have $\\gamma _S^{\\prime }(\\beta ) \\rightarrow 0$ in the limits $\\beta \\rightarrow \\pm \\infty $ .",
"Now, consider all terms as functions of $\\beta $ as in c rcl I(S;T) A() - M(M-1)S() + M ( 1 + S()(M-1) ) - h(S|T) - M!",
"We can find extremal points by differentiating equation (REF ) with respect to $\\beta $ to obtain the first derivative cM(M-1) S () ( -+ 11 + S()(M-1) ) and the second derivative cc M(M-1)S() ( -+ 11 + S()(M-1) ) + -M(M-1) S () ( 1 + (M-1)S()( 1 + S()(M-1) )2 ) which when the first derivative is zero reduces to c-M(M-1) S () ( 1 + (M-1) c S()( 1 + S()(M-1) )2 ) 0 We then have c S* = S(*) = 1 - *(M-1)* and note that equation (REF ) requires $\\frac{1}{M} \\le {\\beta ^*}\\le 1$ since $0\\le \\gamma _S(\\beta ) \\le 1$ .",
"In addition, there is at most one solution to equation (REF ) since $\\frac{1 - {\\beta ^*}}{(M-1){\\beta ^*}}$ monotonically decreases in $\\beta $ while $\\gamma _S(\\beta )$ monotonically increases in $\\beta $ .",
"Since the second derivative at the extremal is non-positive, the unique point defined by equation (REF ) is a maximum.",
"Unfortunately, solutions to equation (REF ) have no closed form and numerical solutions for asymptotically large $M$ are impractical.",
"Nonetheless, the constraints on ${\\beta ^*}$ will allow an oblique approach to deriving a bound.",
"We note again that $\\Gamma _S^{\\prime }(\\beta )$ , is the variance of $\\sum _{i\\ne j}Q(s_i- s_i)$ and must decrease monotonically in $\\beta $ since as previously discussed, increased $\\beta $ concentrates $f_{{\\bf S}}(\\cdot )$ around larger values of $\\sum _{i\\ne j}Q(s_i - s_i)$ .",
"Thus, cS() S(0) $\\forall \\beta > 0$ which in turn implies c S() S(0) + S(0) $\\forall \\beta \\in (0,1]$ .",
"Assuming exponential first-passage, $Q(x) = e^{-\\mu |x|}$ and remembering that $\\Gamma _S(\\beta ) = M(M-1)\\gamma _S(\\beta )$ , we can calculate both $\\gamma _S(0)$ and $\\gamma _S^{\\prime }(0)$ in closed form as c S(0) = Z() 2()2 ( + e- - 1 ) and c S(0) = [ c (M-2)(M-3) S2(0) + 2Z(2 ) + 24M-2()3 ( - 2 + e- (2 + ) ) - M(M-1) S2(0) ] respectively.",
"Defining $M = {\\lambda }\\tau $ and taking the limit for large $M$ yields c M M S(0) = 2 = 2and c M (M-1)S(0) = 8 22 + 2 = 82 + 2where once again $\\rho = \\frac{{\\lambda }}{{\\mu }}$ .",
"Remembering that $\\Gamma _S(\\beta ) = M(M-1)\\gamma _S(\\beta )$ and utilizing equation (REF ) we have c S(0) S(*) S(0) *+ S(0) Thus, the $\\gamma $ -intercept of the monotonically decreasing $\\frac{1-\\beta }{(M-1)\\beta }$ with the right hand side of equation (REF ) must yield a value at least as large as $\\gamma ({\\beta ^*})$ .",
"To solve for this intercept we set rCl1-(M-1) = S(0) + S(0) = 1M-1 ( 82 + 2 ) + 21M so that in the limit of large $M$ we have c= 1 + 12 +36 2 - (1 + 2)16 2 + 4 = 14 + 1 which results in c (M-1) (*) 14 + 1 (8 2 + 2 ) + 2 = 4 so that for large $M$ we have rClI(S;T) A(*) - *M (M-1)S(*) + M ( 1 + 4 ) - h(S|T) - M!",
"To complete the mutual information bound, we could then derive upper bounds on $A({\\beta ^*}) - {\\beta ^*}M (M-1)\\gamma _S({\\beta ^*})$ .",
"However, in the limit of large $M = \\tau /{\\lambda }$ , the density on ${\\bf S}$ is effectively constrained to $({\\bf 0},{\\tau })$ [13], [1] which constrains $h({\\bf S}) \\le M \\log \\tau $ .",
"Then, since $h({\\bf S}|{\\bf T}) = M (1-\\log \\mu )$ for exponential first-passage, equation (REF ) produces mutual information per token c I(S;T)M - (1-) + (1 + 4 ) - M!M Application of Stirling's approximation for large $M$ cM!M M - 1 in combination with equation (REF ) produces our main theorem: Theorem 11 An Upper Bound on the Asymptotic Capacity per Token, $C_q$ : For exponential passage with mean first-passage time $1/\\mu $ and token emission intensity ${\\lambda }$ , an upper bound for the asymptotic capacity per token is given by cCq = fT() M 1M I(S;T) (1 + 4 ) where $\\rho = \\frac{{\\lambda }}{{\\mu }}$ .",
"Proof: Theorem REF Substitution of equation (REF ) and $\\tau = M/{\\lambda }$ into equation (REF ) completes the proof.",
"$\\bullet $"
],
[
"Discussion & Conclusion",
"The timing channel [2], [3], [4], [14] is a building block upon which the information theory of the identical molecule/token timing channel is built.",
"In this paper we considered a version of the timing channel were a single emission is restricted to an interval $[0,\\tau ]$ and we derived closed form expressions for the channel capacity under exponential first-passage as well as the optimal input (emission) distribution.",
"We also established that unlike for the mean-constrained channel, exponential first-passage is not the worst case corruption.",
"Building block though the single emission channel is, the identical molecule timing channel differs from previous models because which emission corresponds to which arrival is ambiguous expressly because travel time from sender to receiver is random and the molecules are identical.",
"This ambiguity is captured by a quantity we define as the “ordering entropy” $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ and understanding its properties is critical to understanding the capacity of not only the molecular timing channel, but also channels where tokens/molecules may themselves carry information payloads – portions of messages to be strung together at the receiver [1].",
"In the Part-I companion to this paper [1], we carefully explored the information theory formulation of the problem to establish that the usual information $I({\\bf S};{\\bf T})$ is indeed the proper measure of information flow over this channel and its relationship to $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ .",
"In this paper, Part-II, we carefully explored the properties of $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ , showing how it can be calculated without deriving full order distributions and deriving closed form expressions for cases where the emission times ${\\bf T}$ are i.i.d.",
"random variables.",
"We then derived closed form expressions for the special cases of the input distribution being that which achieves capacity for the mean-constrained and the deadline-constrained timing channel with exponential first-passage and the asymptotic behavior $\\lim _{M \\rightarrow \\infty }H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ .",
"Our understanding of $H(\\Omega |\\vec{{\\bf S}},{\\bf T})$ then allowed derivation of lower bounds on timing channel capacity for exponential first passage (Part-I, Theorem 14) and here in Part-II, an upper bound for the molecular timing channel capacity.",
"Although the machinery necessary to consider a mean-constrained version of the identical token timing channel was derived, capacity results were not pursued owing to our inability to derive an appropriate sequential channel use model with asymptotic independence.",
"However, if physically parallel channels were used (so as to avoid corruption of one channel by arrivals from another), the results of [2] combined with Theorem REF might be used to derive upper and lower bounds analogous to those provided here and in Part-I [1].",
"This might prove interesting since the mean-constraint seems analytically simpler than the deadline constraint with respect to both the single-token entropy and capacity as well as the ordering entropy."
],
[
"Acknowledgments",
"Profound thanks are owed to A. Eckford, N. Farsad, S. Verdú and V. Poor for useful discussions and guidance.",
"We are also extremely grateful to the editorial staff and the raft of especially careful and helpful anonymous reviewers.",
"This work was supported in part by NSF Grant CDI-0835592."
]
] | 1606.05036 |
[
[
"Revisiting the Majority Problem: Average-Case Analysis with Arbitrarily\n Many Colours"
],
[
"Abstract The majority problem is a special case of the heavy hitters problem.",
"Given a collection of coloured balls, the task is to identify the majority colour or state that no such colour exists.",
"Whilst the special case of two-colours has been well studied, the average-case performance for arbitrarily many colours has not.",
"In this paper, we present heuristic analysis of the average-case performance of three deterministic algorithms that appear in the literature.",
"We empirically validate our analysis with large scale simulations."
],
[
"Introduction",
"The majority problem is a special case of the heavy hitters problem.",
"Indeed, research into the more general case was prompted by one of the solutions to the majority problem (see [1], Sec.",
"5.8 and [2]).",
"Given a collection of $n$ balls, $\\lbrace x_1, ... x_n\\rbrace $ , each of which is coloured with one of $m$ colours, the majority problem is to identify the colour that appears on more than $n/2$ of the balls, or to state that no such colour exists.",
"Boyer and Moore were the first to propose a solution which they called MJRTY [1].",
"Soon after, Fischer and Salzberg provided an algorithm with optimal worst-case performance [3].",
"Matula proposed an alternative algorithm with the same worst-case performance that we refer to as the Tournament algorithm [4].",
"All three algorithms are based on pairing balls of different colours.",
"When there is a majority, we are guaranteed that after such a process the only unpaired balls remaining will be of the majority colour.",
"The analysis of this problem has therefore focused on the number of comparisons.",
"Fischer and Salzberg proved that, when there are arbitrarily many colours, $\\lceil \\frac{3n}{2}\\rceil -2$ comparisons are necessary and sufficient [3].",
"Matula repeated the proof independently [4] Aside from the proofs for the worst-case, all the analysis of the majority problem has been for the special case where there are only two possible colours.",
"In that case, it has been shown that $n-\\nu (n)$ comparisons are necessary and sufficient, where $\\nu (n)$ is the number of 1s in the binary expansion of $n$ [5], [6], [7].",
"Assuming that all $2^n$ possible inputs are equally probable, the average-case complexity has been shown to be lower-bounded by $\\frac{2n}{3}-\\sqrt{\\frac{8n}{9\\pi }}+\\Theta (1)$ and that $\\frac{2n}{3}-\\sqrt{\\frac{8n}{9\\pi }}+O(\\log n)$ comparisons are necessary and sufficient [8].",
"Under the same assumption, it has been shown that the average-case complexity of Boyer and Moore's MJRTY algorithm is $n - \\sqrt{2n/\\pi } +O(1)$ [9].",
"In this paper, we consider the average-case complexity of the three deterministic algorithms.",
"We assume that there are an arbitrary number of colours and that all $n^m$ possible streams are equally likely.",
"Our analysis is heuristic but we empirically validate our analysis with large scale simulations.",
"Boyer and Moore's algorithm, MJRTY, works by simulating a process of pairing balls of different colours and discarding the pairs.",
"The algorithm uses two variables: One to store the candidate colour and one to store the number of unpaired balls of that colour.",
"Whenever the counter is zero, the colour of the next ball drawn becomes the candidate colour; otherwise the ball that is drawn is compared to the candidate colour.",
"If the new balls has the same colour as the candidate colour, then the counter is incremented because we have encountered another unpaired ball.",
"If the new ball has a different colour, then the counter is decrement which represents the pairing and discarding of two unmatched balls.",
"When this process completes, the candidate colour is the colour of the majority of balls, if a majority exists.",
"A second pass over the stream is required to verify whether the candidate colour appears on a majority of balls.",
"This is done by simply counting the number of times that colour is seen.",
"The pseudocode for the algorithm can be found in the Appendix."
],
[
"Analysis of the Algorithm",
"The number of comparisons required in the second (verification) phase is always $n$ because we have to compare every ball to the candidate without exception.",
"The number of comparisons required in the first phase is $n$ less the number of times the counter is zero, because that is the only situation in which a comparison is not performed.",
"The key to our analysis is to note that, although there may be an arbitrary number of colours, the problem reduces to a two-colour problem as soon as a candidate is selected.",
"This is because the candidate counter is incremented for one colour and decremented for all other colours.",
"When counting the number of comparisons, therefore, we only need to consider the number of comparisons needed in a two-colour problem.",
"Let $p$ be the probability of incrementing the counter and $q = 1-p$ be the probability of decrementing the counter.",
"We can model the algorithm's behaviour as a random walk in which we walk up with probability $p$ and down with probability $q$ .",
"The number of times this walk hits the horizon is equivalent to the number of times the candidate counter returns to zero.",
"To calculate the expected number of times the walk hits the horizon, we could consider Dyck paths, since a return to zero must start at zero and return to zero.",
"We could then create an indicator variable, $X_i$ which is 1 if the walk is at zero after $2i$ steps and 0 otherwise.",
"Finding the expected value of $X$ would give the expected number of times the counter is zero.",
"However, in our case the horizon is reflective such that whenever the walk is at the horizon the next step is an up, with probability 1.",
"Therefore, we would need to consider the Catalan Triangle numbers, $T(i,k)$ , which give the number of paths of length $2i$ having $k$ returns to zero [10].",
"To find the expected number of times the walk hits the horizon, we would need to sum over all values of $i$ up to $n/2$ and for each value of $i$ , another sum over all values of $k$ up to $i$ .",
"This has no closed form and is extremely expensive to calculate as $n$ increases.",
"An alternative, heuristic analysis is to consider the expected number of balls that are drawn between the counter being zero and returning to zero.",
"In fact, we must find the expected number of balls that are drawn between the counter being one and returning to zero because whenever the counter is zero it must move to one on the next draw.",
"This can again be modelled as a asymmetric random walk.",
"The question now becomes the expected number of steps to go from position 1 to 0.",
"Faris provides the solution in section 1.2 of his work Lectures on Stochastic Processes [11].",
"The solution is $1(q-p)$ .",
"In our case, we have $p = 1/m$ and $q = 1 - 1/m$ , which makes the expected number of steps required to go from one to zero equal to $m/(m-2)$ .",
"Let $t$ be the expected number of steps required to go from one to zero.",
"$1+t$ will therefore be the expected number of balls drawn when the counter moves from zero back to zero again.",
"Heuristically, we can say that the expected number of zeros is $n/(1+t) = n/(1+m/(m-2))$ .",
"The expected total number of comparisons is therefore: $E[C] = 2n - 1 - n/(1+\\frac{m}{m-2})$ We note that this equation has no solution when there are two colours, but the average number of comparisons required in that case has already been provided by Alonso and Reingold [9]."
],
[
"Experimental Validation",
"We validate our analysis with large scale simulations by running the MJRTY algorithm on synthetic data.",
"We vary the number of colours, selecting 3, 5 and 10 colours.",
"For each colour, we vary the stream length between 250 and 10,000 in steps of 250.",
"For each stream length we create 100 streams with colours represented as integers.",
"The streams are created so that every “colour” is assigned to each item uniformly at random.",
"For every stream, the number of comparisons used by the MJRTY algorithm is recorded and compared to the expected number, using equation (REF ).",
"The expected number of comparisons is found to always be within the 99% confidence interval of the mean number of observed comparisons.",
"Furthermore, the relative difference between the mean of the observed number of comparisons and the expected number is on average 0.05%, 0.04% and 0.03% for 3, 5 and 10 colours respectively.",
"Figure: A comparison of the expected and actual number of comparisons for the MJRTY algorithm for 3 and 10 colours - best viewed in colour.For a visual example we simulate 1,000 streams each of which is a random length between 10 and 1,000.",
"For each stream we calculate the actual number of comparisons used and compare to the expected number.",
"We repeat the experiment for 3 and 10 colours.",
"Fig.",
"REF shows the results."
],
[
"Brief Description of the Algorithm",
"The Tournament algorithm, first proposed by Matula, is also based on the pairing off of balls but it stores the results of those comparisons to speed up verification.",
"The algorithm maintains a set of lists where each list has an associated weight ($w = 0,1,2,3\\dots $ ) such that every item in the list is a shorthand for $2^w$ balls of the same colour.",
"There is also a list of discarded tuples of the form $\\lbrace v^{\\prime },v^{\\prime \\prime },j\\rbrace $ where $v^{\\prime }$ and $v^{\\prime \\prime }$ are balls of different colours both of which are shorthand for $2^j$ balls of that colour.",
"Initially all balls are in the list with weight 0 (this is just the initial stream).",
"Then, in each round, pairs of balls from the same list are compared and either merged or discarded.",
"If the two balls have the same colour, then they are merged and placed as a single entry in the next list.",
"If they do not have the same colour, then they are formed into a tuple and added to the list of discarded tuples.",
"The process continues until no list contains more than one ball.",
"At this point, the candidate ball is the one in the most heavily weighted list which contains a ball.",
"A verification phase is needed in which the lists are traversed in reverse order and balls (really representing collections of balls) are compared to the candidate.",
"A counter is maintained that is incremented by the appropriate amount for each list.",
"Once all the lists are traversed, the discarded tuples must be examined as well.",
"The pseudocode for the algorithm can be found in the Appendix."
],
[
"Analysis of the Algorithm",
"In the first round of the Tournament algorithm there are $n$ balls and we therefore need $n/2$ comparisons.",
"A ball is entered into the next round if, upon comparison, the second ball matches the first which happens with probability $1/m$ .",
"So we would expected to find $n/2m$ balls in the second round.",
"In the second round we perform $n/4m$ comparisons and expect $n/4m^2$ balls in the third round.",
"In general, we expect to perform $n/2^im^{i-1}$ comparisons in round $i$ , starting with round 1.",
"There are a maximum of $\\log _2(n)$ rounds but if we sum to infinity, the sum simplifies.",
"The expected number of comparisons in the first phase is therefore: $E[C_1] = \\sum _{i=1}^\\infty \\frac{n}{2^im^{(i-1)}} = \\frac{mn}{2m-1}$ During the second phase, we need to examine the balls in the lists as well as those in the discarded tuples.",
"Let us first consider the tuples.",
"The number of discarded tuples will depend on the number of comparisons during the first phase.",
"A tuple is created whenever a comparison is performed and the two balls being compared have different colours.",
"The expected number of comparisons during the first phase is given by equation (REF ) above, and the probability that two balls do not have the same colour is $1-1/m$ .",
"Heuristically, therefore, we can say that the expected number of discarded tuples is the product of equation (REF ) and $1-1/m$ .",
"But for each tuple we may have to perform one comparison (if the first ball in the tuple matches the candidate) or two comparisons (if the first ball in the tuple does not match the candidate).",
"Since the probability of the first ball not matching the candidate is $1-1/m$ , the expected number of comparisons per tuple is $2-1/m$ .",
"Putting these elements together, we can say that the expected number of comparisons in the second phase from the discarded tuples is: $E[C_{2D}] = \\frac{mn}{2m-1}(1-\\frac{1}{m})(2-\\frac{1}{m}) = \\frac{(1-2m)(1-m)n}{m(2m-1)}$ Finally, we also need to consider the comparisons for the lists.",
"Once the first phase of the algorithm is completed, the lists will be empty unless they initially had an odd number of balls in which case they will contain a single ball.",
"The expected size of list $i$ is $n/(2m)^i$ , but it is not possible to say whether this is odd or even in general.",
"However, we can iterate through the lists from $i=0$ to $i=\\log _2(n)$ and if the expected size is odd then add an additional comparison."
],
[
"Experimental Validation",
"We validate our analysis with experiments in the same way as for the MJRTY algorithm.",
"The only difference is that for this algorithm we also include the case of two colours.",
"The expected number of comparisons is found to always be within the 99% confidence interval of the mean number of observed comparisons.",
"The average relative error is 0.11%, 0.08%, 0.06% and 0.05% for 2, 3, 5 and 10 colours respectively.",
"Figure: A comparison of the expected and actual number of comparisons for the Tournament algorithm for a number of different colours (best viewed in colour).Fig.",
"REF shows the results of our second experiment with 2, 3 and 10 colours."
],
[
"Brief Description of the Algorithm",
"The Fischer-Salzberg algorithm is also based on the principle of pairing balls but with a slight twist.",
"Rather than pairing and discarding, the algorithm attempts to create an ordering of paired balls by having a condition that no two adjacent balls can be of the same colour.",
"We refer to this condition as the adjacency condition.",
"The idea is that, if there is a majority, it will be impossible to create such an ordering There is one exception, in the case that $n$ is odd and the first, last and every second ball is of the same colour.",
"We do not consider this case here explicitly for simplicity but the algorithm handles such a case with no additional comparison compared to a case of no majority.",
"The algorithm starts by creating a list which conforms to the adjacency condition.",
"This is achieved through two data-structures: the list and the bucket.",
"Balls are drawn from the stream and compared to the current last ball in the list.",
"If the two balls have the same colour then they cannot be placed next to each other, so instead the drawn ball is placed in the bucket.",
"If the two balls have different colours, the drawn ball is added to the end of the list and a ball from the bucket is added afterwards (if any exist).",
"By taking balls from the bucket and appending them to the list, it is guaranteed that all balls in the bucket have the same colour, which is also the colour of the last ball in the list.",
"It is also part of creating a list of all the balls without breaking the adjacency condition.",
"Once all the balls have been drawn, the last ball in the list is the candidate and will be of the majority, if one exists.",
"However, balls may remain in the bucket and it may be possible to insert them into the list without violating the adjacency condition.",
"If that is possible, then there is no majority.",
"Therefore, a second phase is required which attempts to insert the balls from the bucket into the list without violating the adjacency condition.",
"During the second phase, the list is traversed in reverse order, always comparing the last ball in the list to the candidate.",
"If the two are the same, then the last two balls in the list are discarded because balls from the bucket cannot be inserted either at the end of the list or between the last two balls (because all balls in the bucket will be the same colour as the candidate).",
"If they are not the same, then the last ball in the list and one from the bucket are discarded because a ball from the bucket can be inserted at the end of the list without breaking the adjacency condition.",
"The algorithm stops if a ball is ever needed from the bucket but the bucket is empty because then we have successfully found an ordering that does not break the adjacency condition and there is no majority colour.",
"If the entire list is traversed and at least one ball remains in the bucket, then that ball is of the majority colour.",
"The pseudocode for the algorithm can be found in the Appendix."
],
[
"Analysis of the Algorithm",
"The Fischer-Salzberg algorithm is extremely difficult to analyse because it is so sensitive to the order of the balls.",
"Therefore, here we present a heuristic analysis that we show, empirically, to be extremely accurate.",
"The first phase of the algorithm will always use $n-1$ comparisons.",
"The number of comparisons in the second phase depends on the size of the bucket and whether or not there is a majority.",
"During the first phase, balls are added to the bucket if they are the same colour as the last ball in the list.",
"Otherwise, a ball is removed from the bucket.",
"The size of the bucket can therefore be modelled as a one-dimensional random walk with probability $1/m$ of going up and probability $1-1/m$ of going down.",
"There is a barrier at 0 such that when the walk is at zero, down means staying at 0.",
"Unfortunately, we were unable to solve this random walk to provide an expected position after $n$ steps.",
"However, we can provide a heuristic analysis by treating the walk as if there was no barrier.",
"When there is a majority, the probability of going up dominates.",
"Let $\\rho $ be the proportion of the majority coloured ball, then the expected position of the walk would be $n\\rho - n(1-\\rho ) = n(2\\rho -1)$ which would then be the expected size of the bucket after the first phase.",
"The list would then have $n - n(2\\rho -1) = 2n(1-\\rho )$ balls in it.",
"We would expect to traverse the list two balls at a time for $n(1-\\rho )$ comparisons in the second phase.",
"To complete this part of the analysis we need to know the value of $\\rho $ - the expected proportion of balls of the majority colour.",
"$\\rho $ is the expected value of a Bernoulli distribution with probability of success $1/m$ conditioned on there being a majority of successes.",
"We found no closed form for this but it can be calculated directly from the definition of conditional expected value (taking into account that we do not care which colour is of the majority): $\\rho = m\\sum _{x=\\frac{n}{2}+1}^{n} x {n\\atopwithdelims ()x}(1/m)^x(1-1/m)^{n-x}$ When there is no majority, the expected position of the walk is negative, equating to an empty bucket.",
"The list length is therefore equal to $n$ .",
"We must traverse the list, two balls at a time, until we encounter a non-matching ball.",
"When there are only two colours and no majority we will have to traverse the entire list for $n/2$ comparisons.",
"When there are more than two colours, we can model the list as a binomial distribution with “success” defined as encountering a non-matching ball which happens with probability $1-1/m$ .",
"The expected number of trials before encountering the first success in a binomial distribution is the reciprocal of the probability of success so we would expect to have $1/(1-1/m) = m/(m-1)$ comparisons in the second phase.",
"Finally, to produce an overall expected number of comparisons, we must compute the probability of there being a majority.",
"Again we model our stream as a binomial distribution with probability of success $1/m$ .",
"The probability of having a minority of successes can be directly calculated from the cumulative distribution function and from that the probability of having a majority can be found.",
"Let $P(m)$ be the probability of having a majority and let $E_m[C]$ be the expected number of comparisons when there is a majority.",
"We can then say that the expected number of comparisons for the Fischer-Salzberg algorithm is: $E[C] = (n-1) + \\left(P(m)E_m[C] + (1-P(m)m/(m-1)\\right)$"
],
[
"Experimental Validation",
"We validate our analysis with experiments in the same way as for the MJRTY and Tournament algorithms.",
"The expected number of comparisons is found to always be within the 99% confidence interval of the mean number of observed comparisons.",
"The mean relative error was 0.03%, 0.23%, 0.04% and 0.01% for 2, 3, 5 and 10 colours respectively.",
"The relatively high mean relative error with 3 colours is because at small stream sizes (250 and 500) the relative error was 2.27% and 1.40% which brought up the average.",
"Figure: A comparison of the expected and actual number of comparisons for the Fischer-Salzberg algorithm for a number of different colours (best viewed in colour).",
"The results for 3 and 10 colours overlap.Fig.",
"REF shows the results of our second experiment.",
"The number of comparisons with 3 and 10 colours are very similar and cannot be discerned in the graph."
],
[
"Conclusion",
"The majority problem is an interesting problem that is a special case of the heavy hitters problem.",
"In this paper we have presented a heuristic analysis of the expected number of comparisons required by the three deterministic algorithms that appear in the literature, assuming an arbitrary number of colours.",
"Fig.",
"REF shows a direct comparison of the three algorithms when varying the stream length but keeping the number of colours constant at five colours.",
"Fig.",
"REF shows a direct comparison of the three algorithms with a constant stream size of 5,000 balls but varying the number of colours.",
"It is interesting to note that the most-widely discussed of the algorithms (Boyer and Moore's MJRTY) algorithm requires the largest number of comparisons.",
"In contrast, the Fischer-Salzberg algorithm, which has been described as being “essentially identical” [12], requires the fewest.",
"Both algorithms have the same number of comparisons in the first phase but Fischer-Salzberg does some extra work in the first phase to save comparisons in the second phase.",
"The Tournament algorithm also requires fewer comparisons than MJRTY.",
"This is in both the first and second phase.",
"The trade-off is the overhead of maintaining various lists.",
"These results suggest that measuring the number of comparisons is not sufficient to fully characterise the performance of these algorithms.",
"A fuller investigation is warranted which considers the actual execution times.",
"Nevertheless, our analysis shows that the Tournament algorithm has the fewest expected comparisons in the first phase.",
"It has an expected number of $mn/(2m-1)$ which, as the number of colours increases, tends to $n/2$ .",
"This compares to $n-1$ for both of the other algorithms.",
"Since much of the later work on the more general heavy hitters problem does not consider verification (i.e.",
"the second phase), this lower number of expected comparisons suggests that variations of the Tournament algorithm may outperform variants of the MJRTY algorithm.",
"Furthermore, there have been proposals for a parallel version of the MJRTY algorithm [13], [14].",
"The Tournament algorithm, however, seems better suited for a parallel implementation and it would be interesting to see whether the overheads of parallelism result in the Tournament algorithm being more efficient than MJRTY."
],
[
"Pseudocode for the MJRTY Algorithm",
"[h!]",
"Pseudocode for the MJRTY Algorithm [1] c $\\leftarrow 0$ $i \\leftarrow 1$ to $n$ $c = 0$ $j \\leftarrow i$ $c \\leftarrow 1$ $x_i = x_j$ $c \\leftarrow c + 1$ $c \\leftarrow c - 1$ $c = 0$ No majority $c \\leftarrow 0$ $i \\leftarrow 1$ to $n$ $x_i = x_j$ $c \\leftarrow c+1$ $c > n/2$ Majority is of colour $x_j$ No majority"
],
[
"Pseudocode for the Fischer-Salzberg Algorithm",
"[h!]",
"Pseudocode for the Fischer-Salzberg Algorithm [1] $l=0$ $ i \\leftarrow 1$ to $n$ $x_i = $ list[$l$ ] bucket.append($x_i$ ) list.append($x_i$ ) $l$ ++ bucket.empty() list.append(bucket.pop()) $l$ ++ $C=$ list[$l$ ] $!$ list.empty() list.pop()=$C$ list.pop() bucket.empty() No majority bucket.pop() Majority is $C$"
],
[
"Pseudocode for the Tournament Algorithm",
"[h!]",
"Pseudocode for the Tournament Algorithm [1] $i \\leftarrow 0$ !",
"list[$i$ ].empty() $j \\leftarrow 0$ $j < $ list[$i$ ].length() && list[$i$ ].length() $> 1$ list[$i$ ][$j$ ] == list[$i$ ][$j+1$ ] list[$i+1$ ].append(list[$i$ ][$j$ ]) tuples.add(list[$i$ ][$j$ ], list[$i$ ][$j+1$ ], $i$ ) $j \\leftarrow j+2$ $i++$ $i > 0$ list[$i$ ].length() $== 1$ Candidate == NULL Candidate = list[$i$ ][0] $c = 2^i$ list[$i$ ][0] == Candidate $c \\leftarrow c + 2^i$ $i--$ tuples tuple[0] == Candidate OR tuple[1] == Candidate $c \\leftarrow c + 2^{tuple[2]}$ $c > n/2$ Candidate is majority No majority"
]
] | 1606.05123 |
[
[
"From Frazier-Jawerth characterizations of Besov spaces to Wavelets and\n Decomposition spaces"
],
[
"Abstract This article describes how the ideas promoted by the fundamental papers published by M. Frazier and B. Jawerth in the eighties have influenced subsequent developments related to the theory of atomic decompositions and Banach frames for function spaces such as the modulation spaces and Besov-Triebel-Lizorkin spaces.",
"Both of these classes of spaces arise as special cases of two different, general constructions of function spaces: coorbit spaces and decomposition spaces.",
"Coorbit spaces are defined by imposing certain decay conditions on the so-called voice transform of the function/distribution under consideration.",
"As a concrete example, one might think of the wavelet transform, leading to the theory of Besov-Triebel-Lizorkin spaces.",
"Decomposition spaces, on the other hand, are defined using certain decompositions in the Fourier domain.",
"For Besov-Triebel-Lizorkin spaces, one uses a dyadic decomposition, while a uniform decomposition yields modulation spaces.",
"Only recently, the second author has established a fruitful connection between modern variants of wavelet theory with respect to general dilation groups (which can be treated in the context of coorbit theory) and a particular family of decomposition spaces.",
"In this way, optimal inclusion results and invariance properties for a variety of smoothness spaces can be established.",
"We will present an outline of these connections and comment on the basic results arising in this context."
],
[
"Introductory statement",
"This article is part of a series of notes (see e.g.",
"[28], [27], [26]), which describe the role of different function spaces, their various characterizations and their possible applications from a “postmodern viewpoint”, emphasizing the importance of the concept of Banach frames (first defined in [46]).",
"For this development, a series of papers by Michael Frazier and Bjoern Jawerth ([36], [37], [38]) have been of great relevance.",
"Hence we will try to demonstrate how the ideas of these papers lived on and expanded in the subsequent decades."
],
[
"Notation",
"We will employ the following notation: For a group $G$ and any function $f : G \\rightarrow S$ , for some set $S$ , we define Lx f : G S, y f(x-1y), Rx f : G S, y f(yx), x,y G. In the special case of the (abelian) group $\\mathbb {R}^d$ , we also write $T_x := L_x$ .",
"Furthermore, for $f : \\mathbb {R}^d \\rightarrow {\\mathbb {C}}$ and $\\xi \\in \\mathbb {R}^d$ , we define the modulation of $f$ by $\\xi $ as $M_\\xi f : \\mathbb {R}^d \\rightarrow {\\mathbb {C}}, x \\mapsto e^{2\\pi i x \\cdot \\xi } \\cdot f(x).$ Finally, for $h \\in \\operatorname{GL}(\\mathbb {R}^d)$ , we define the (${L}^2$ normalized) dilation of $f$ by $h$ as $D_h f : \\mathbb {R}^d \\rightarrow {\\mathbb {C}}, x \\mapsto |\\det h|^{-1/2} \\cdot f(h^{-1} x).$ For the special case $h = a \\cdot {\\mathrm {id}}$ with $a \\in \\mathbb {R}\\setminus \\lbrace 0\\rbrace $ , we also write $D_a := D_{a \\cdot {\\mathrm {i}d}}$ .",
"For the Fourier transform, we use the normalization $\\mathcal {F}f ( \\xi ) := \\widehat{f}(\\xi ) := \\int _{\\mathbb {R}^d} f(x) e^{-2\\pi i x\\cdot \\xi } {\\rm d}x, \\quad f \\in {L}^1(\\mathbb {R}^d).$ It is well-known that the Fourier transform extends to a unitary automorphism of ${L}^2(\\mathbb {R}^d)$ , where the inverse is the unique extension of the operator $\\mathcal {F}^{-1}$ , given by $\\mathcal {F}^{-1} h ( x) = \\int _{\\mathbb {R}^d} h(\\xi ) e^{2\\pi i x \\cdot \\xi } {\\rm d}\\xi ,\\quad h \\in {L}^1(\\mathbb {R}^d).$ $\\lambda (M)$ denotes the $d$ -dimensional Lebesgue measure of a (measurable) set $M \\subset \\mathbb {R}^d$ .",
"Finally, for $x \\in \\mathbb {R}$ , we write $x_+ := \\max \\lbrace x,0\\rbrace $ ."
],
[
"The Essence of the work of Frazier-Jawerth",
"To the best of our knowledge, the influential papers [36], [37], [38] have been the first to fully characterize two families of Banach spaces of tempered distributions, namely the Besov spaces ${ ({{{B}^{s}_{p,q}}({\\mathbb {R}}^d)}, \\Vert \\mbox{$\\,\\cdot \\,$}\\Vert _{{{B}^s_{p,q}}}) }$ and the Triebel-Lizorkin spaces ${ ({{{F}^{s}_{p,q}}({\\mathbb {R}}^d)}, \\Vert \\mbox{$\\,\\cdot \\,$}\\Vert _{{{F}^s_{p,q}}}) }$ , by corresponding growth and summability conditions on a suitably defined sequence of coefficients.",
"These coefficients depend linearly on the function/distribution under consideration.",
"Similar atomic representation theorems had been realized only a few years earlier in the context of harmonic function spaces (see e.g.",
"[9], [64]) In a more modern terminology (going back to the work of K. Gröchenig [46]), one could say that Frazier and Jawerth established specific, but rather concrete Banach frame expansions for these two families of function spaces, starting from the characterization of these spaces via dyadic partitions of unity on the Fourier transform side.",
"This characterization in turn is based on the description of Besov-Triebel-Lizorkin spaces via dyadic decompositions (see [62], [68], [69]).",
"In this general theory of Banach frames, one has—roughly speaking—the following situation: (i) Starting from a fixed family of (possibly non-orthogonal, but sufficiently rich) atoms $(g_i)_{i \\in I}$ , one obtains atomic representations of the form $f = \\sum _{i \\in I} c_i g_i$ , typically with convergence in the norm of $({B}, \\, \\Vert \\mbox{$\\,\\cdot \\,$}\\Vert _{B})$ (or at least $ w^{*} $ -convergence for the case of dual spaces).",
"(ii) The coefficients are obtained in a linear way, i.e., there are suitable bounded linear coefficient mappings $f \\mapsto c_i = c_i (f)$ satisfying $f = \\sum _{i \\in I} c_i g_i$ .",
"These mappings are often realized as scalar products with respect to a suitable family of atoms $(h_i)_{i \\in I}$ , forming a so-called dual frame.",
"Thus, $c_i = \\langle f, h_i \\rangle $ , where the “scalar product” can be viewed as an extension of the Hilbert-space scalar product.",
"(iii) Although the representation is by far not unique, there is a high degree of compatibility between the membership of $f$ in one of the spaces under consideration and membership of the coefficients $(c_i)_{i \\in I}$ in a suitable Banach space of sequences, in fact, in a suitable solid BK space.",
"These spaces have the property that sequences which are smaller in terms of absolute values—in a coordinate-wise sense—also have a smaller norm.",
"Precisely, it is claimed that for a function space ${X}$ (from a certain class), one can find a BK space ${Y}\\le {\\mathbb {C}}^I$ , a bounded coefficient map ${\\mathcal {C}}: {X}\\rightarrow {Y}, f \\mapsto (c_i(f))_{i \\in I}$ , and a synthesis mapping ${\\mathcal {R}}: {Y}\\rightarrow {X}, {\\bf c} \\mapsto \\sum _{i \\in I} c_i g_i$ which is a left inverse for ${\\mathcal {C}}$ .",
"More concretely, the $\\varphi $ -transform as introduced by Frazier and Jawerth in [36], [37], [38] is a fixed, linear transformation $S_\\varphi : f \\mapsto \\left((S_\\varphi f)_Q \\right)_{Q \\in \\mathcal {Q}}$ with the benefit that a variety of function spaces can be characterized in terms of the size of $S_\\varphi f$ , i.e.",
"by ${\\Vert S_\\varphi f\\Vert _{{Y}}}$ for suitable solid sequence spaces ${Y}$ .",
"Solidity of ${Y}$ formally encodes the requirement that only the size of the coefficients should be important, so that there is no cancellation between the different coefficients.",
"The class of spaces for which a characterization in terms of the $\\varphi $ -transform is possible includes the classes of Besov spaces ${{B}^s_{p,q}}$ and the Triebel-Lizorkin spaces ${{F}^s_{p,q}}$ .",
"Note that both of these classes of spaces come in two variants: homogeneous and inhomogeneous spaces.",
"The theory developed by Frazier and Jawerth applies to both of these subclasses.",
"For concreteness, we will concentrate in the sequel on the inhomogeneous spaces, which have the advantage of being modulation invariant.",
"To describe the $\\varphi $ -transform and the resulting decomposition results more precisely, we begin with the index set $\\mathcal {Q}$ , which is the set of all dyadic cubes with side-length $\\le 1$ .",
"Here, by definition (cf.",
"[38]), a dyadic cube is a set of the form $Q_{\\nu , k} = \\left\\lbrace x \\in \\mathbb {R}^d \\,|\\,\\forall i \\in \\lbrace 1, \\dots , d\\rbrace : 2^{-\\nu } k_i \\le x_i < 2^{-\\nu } (k_i + 1) \\right\\rbrace $ for arbitrary $\\nu \\in \\mathbb {Z}$ and $k \\in \\mathbb {Z}^d$ .",
"Given such a cube $Q = Q_{\\nu , k}$ , we define the lower left corner of $Q$ as $x_Q := 2^{-\\nu }k$ and the side length of $Q$ as $\\ell (Q) := 2^{-\\nu }$ .",
"To distinguish cubes with different side lengths, we also define for $\\nu \\in \\mathbb {Z}$ the set $\\mathcal {Q}_\\nu := \\left\\lbrace Q \\in \\mathcal {Q} \\,|\\,\\ell (Q) = 2^{-\\nu } \\right\\rbrace .$ Finally, for an arbitrary function $\\varphi : \\mathbb {R}^d \\rightarrow {\\mathbb {C}}$ and $Q = Q_{\\nu , k}$ , we let $\\varphi _Q (x) := 2^{\\nu d /2} \\cdot \\varphi (2^{\\nu } x - k) = 2^{-\\nu d/2} \\cdot \\varphi _{\\nu } (x - x_Q),$ where $\\varphi _\\nu (x) := 2^{\\nu d} \\cdot \\varphi (2^{\\nu } x)$ .",
"Note that if $\\varphi $ is “concentrated” in $[0,1]^d$ , then $\\varphi _Q$ is “concentrated” on $Q$ .",
"Furthermore, we have ${\\Vert \\varphi _Q\\Vert _{{L}^2}} = {\\Vert \\varphi \\Vert _{{L}^2}}$ .",
"Now, the $\\varphi $ -transform is defined using two analyzing windows $\\varphi , \\varphi ^0 \\in {{{\\mathcal {S}}}({\\mathbb {R}}^d)}$ .",
"The basic assumption regarding these windows is that there are “dual windows” $\\psi , \\psi ^0 \\in {{{\\mathcal {S}}}({\\mathbb {R}}^d)}$ satisfying the following (cf.",
"[38]) supp0, 0 { Rd | || 2 }, supp, { Rd | 1/2 || 2 }, 0() 0() + = 1 (2-) (2-) =1 Rd.",
"Under this assumption, the (inhomogeneous) $\\varphi $ -transform $S_\\varphi $ is the analysis operator of the frame, so that the $\\varphi $ -transform $S_\\varphi f = \\left( (S_\\varphi f)_Q \\right)_{Q \\in \\mathcal {Q}}$ of a tempered distribution $f \\in { { {{\\mathcal {S}}}}^{\\prime }({{{\\mathbb {R}}^d}})}$ is defined by $(S_\\varphi f)_Q := {\\left\\lbrace \\begin{array}{ll}\\langle f, \\varphi _Q\\rangle ,& \\text{if } \\ell (Q) < 1,\\\\\\langle f, \\varphi ^0_Q\\rangle ,& \\text{if } \\ell (Q) = 1.\\end{array}\\right.",
"}$ Finally, the (formal) inverse of $S_\\varphi $ —the corresponding synthesis operator—is $T_\\psi [(s_Q)_{Q \\in \\mathcal {Q}}] := \\sum _{Q \\in \\mathcal {Q}_0} s_Q \\psi ^0_Q + \\sum _{\\nu = 1}^{\\infty } \\sum _{Q \\in \\mathcal {Q}_{\\nu }} s_Q \\psi _Q.$ Now, as shown in [38], we have $f = T_\\psi S_\\varphi f$ for all $f \\in { { {{\\mathcal {S}}}}^{\\prime }({{{\\mathbb {R}}^d}})}$ , with convergence of the series defining $T_\\psi (S_\\varphi f)$ in the topology of ${ { {{\\mathcal {S}}}}^{\\prime }({{{\\mathbb {R}}^d}})}$ .",
"The main result of Frazier and Jawerth concerning inhomogeneous Triebel-Lizorkin spaces using the $\\varphi $ -transform reads as follows (cf.",
"[38]): Theorem 1 For $s \\in \\mathbb {R}$ , $0<p<\\infty $ and $0<q\\le \\infty $ , the operators $S_\\varphi : {{F}^s_{p,q}}\\rightarrow {{f} ^s_{p,q}}\\qquad \\text{ and } \\qquad T_\\psi : {{f} ^s_{p,q}}\\rightarrow {{F}^s_{p,q}}$ are bounded, with $T_\\psi \\circ S_\\varphi = \\mathrm {id}$ on ${{F}^s_{p,q}}$ .",
"Hence, ${\\Vert f\\Vert _{{{F}^s_{p,q}}}}\\asymp {\\Vert S_\\varphi f\\Vert _{{{f} ^s_{p,q}}}}$ and ${{F}^s_{p,q}}$ is a retract of ${{f} ^s_{p,q}}$ , i.e., it can be identified with a complemented subspace of ${{f} ^s_{p,q}}$ .",
"In the above theorem, the solid BK-space ${{f} ^s_{p,q}}$ is defined as the set of all sequences $c = (c_Q)_{Q \\in \\mathcal {Q}} \\in {\\mathbb {C}}^{\\mathcal {Q}}$ for which the (quasi)-norm cf sp,q := ([(Q)]-s|cQ|Q)QQq(Q)Lp( dx) is finite.",
"Here, we used the symbol $\\widetilde{\\chi _E} := (\\lambda (E))^{-1/2} \\cdot \\chi _E$ for the ${L}^2$ -normalized version of $\\chi _E$ , for any measurable set $E \\subset \\mathbb {R}^d$ of finite, positive measure.",
"In addition to the above result for Triebel-Lizorkin spaces, [36] provides the following analogue characterization of Besov spaces.",
"Theorem 2 For $0 < p,q \\le \\infty $ and $s \\in \\mathbb {R}$ , the operators $S_\\varphi : {{B}^s_{p,q}}\\rightarrow {{{b}^s_{p,q}}}\\qquad \\text{ and } \\qquad T_\\psi : {{{b}^s_{p,q}}}\\rightarrow {{B}^s_{p,q}}$ are bounded, with $T_\\psi \\circ S_\\varphi = \\mathrm {id}$ on ${{B}^s_{p,q}}$ .",
"Hence, ${\\Vert f\\Vert _{{{B}^s_{p,q}}}}\\asymp {\\Vert S_\\varphi f\\Vert _{{{{b}^s_{p,q}}}}}$ and ${{B}^s_{p,q}}$ is a retract of ${{{b}^s_{p,q}}}$ , and can be identified with a complemented subspace of ${{{b}^s_{p,q}}}$ .",
"The solid BK-space ${{{b}^s_{p,q}}}$ consists of all $c = (c_Q)_{Q \\in \\mathcal {Q}} \\in {\\mathbb {C}}^{\\mathcal {Q}}$ for which the following quasi-norm is finite: cbsp,q:= (([(Q)]d(1p-12) - s cQ)QQp(Q))N0q(N0).",
"Similar results have later been described in the books of H. Triebel, who showed that multivariate wavelet orthonormal bases are not just bases for the Hilbert space ${{{L}^2}({\\mathbb {R}}^d)}$ , but also (in a modern terminology) Riesz projection bases for the corresponding solid BK-spaces.",
"This is another way of saying that the unitary isomorphism between ${L}^2 (\\mathbb {R}^d)$ and ${\\ell }^2(I)$ —which is induced by the orthonormal basis $(\\varphi _i)_{i \\in I}$ —extends to an isomorphism between Banach spaces and their discrete counterparts, for a whole family of spaces and with uniform bounds.",
"But the existence of orthonormal wavelet bases was established (in [56], [57], [17]) only after the appearance of the paper [36] of Frazier and Jawerth.",
"In summary, these “decomposition” results of Frazier and Jawerth imply a variety of statements: (1) A consistency statement: Any two (admissible) window-families $\\varphi _1, \\varphi ^0_1$ and $\\varphi _2, \\varphi ^0_2$ yield the same space by imposing certain decay conditions on the associated $\\varphi $ -transform, i.e., we have for $ i = 1,2$ : $\\left\\lbrace f \\in { { {{\\mathcal {S}}}}^{\\prime }({{{\\mathbb {R}}^d}})}\\,|\\,S_{\\varphi _i} f\\in {{f} ^s_{p,q}}\\right\\rbrace = {{F}^s_{p,q}}.$ (2) A decomposition (or representation) statement: $f = \\sum _{Q \\in \\mathcal {Q}} s_Q (f) \\cdot \\psi _Q,\\quad f \\in {{F}^s_{p,q}},$ with $(s_Q (f))_{Q \\in \\mathcal {Q}} = S_\\varphi f \\in {{f} ^s_{p,q}}$ and atoms $\\psi _Q$ of the special form $\\qquad \\qquad \\psi _Q (x) = 2^{-\\nu d/2} \\cdot \\psi _{\\nu } (x - x_Q) = [\\pi (x_Q, 2^{-\\nu }) \\psi ](x) \\quad \\text{ for } \\quad \\ell (Q) = 2^{-\\nu },$ where $\\pi (x, a) := T_x D_a$ is the quasi-regular representation on ${L}^2 (\\mathbb {R}^d)$ of the $ax+b$ group.",
"This was already observed in [38].",
"(3) The two parts of the representation are all valid for a certain range of spaces, not just for a single Hilbert space or one individual Banach space.",
"It is also worth noting that, if one restricts the construction of Frazier and Jawerth to the setting of Hilbert spaces, one finds that they are implicitly establishing that the set of analyzing atoms forms a frame; but the elements used to perform the reconstruction are not the elements of the (canonical) dual frame.",
"(4) The concrete BK-spaces ${{f} ^s_{p,q}}$ arising in this context are not just abstract BK-spaces (Banach spaces of numerical sequences, whose coordinates depend continuously on the sequence).",
"In addition, they are Banach lattices (following the terminology of Luxemburg-Zaanen) resp.",
"so-called solid BK-spaces: With each sequence ${\\bf d} = (d_Q)_{Q \\in \\mathcal {Q}} \\in {{f} ^s_{p,q}}$ , also any sequence $ {\\bf b} = (b_Q)_{Q \\in \\mathcal {Q}}$ with $ |b_Q| \\le |d_Q|$ for all $Q \\in \\mathcal {Q}$ belongs to ${{f} ^s_{p,q}}$ , with $ \\Vert \\bf b\\Vert _{{{f} ^s_{p,q}}} \\le \\Vert \\bf d\\Vert _{{{f} ^s_{p,q}}}$ .",
"The same statements remain true with ${{B}^s_{p,q}}$ and ${{{b}^s_{p,q}}}$ instead of ${{F}^s_{p,q}}$ and ${{f} ^s_{p,q}}$ .",
"The above observations are similar to the main properties of coorbit spaces, as introduced by Feichtinger and Gröchenig in [30], [31], [32], see also Section .",
"Remark While the $\\varphi $ -transform and the resulting atomic decompositions for Besov (resp.",
"Triebel-Lizorkin) spaces grew out of the characterization of these spaces using Littlewood-Paley theory—essentially established by the earlier work of J. Peetre and H. Triebel—it turned out that they were also fore-runners for the characterization of these function spaces via wavelet frames."
],
[
"Modulation Spaces",
"One of the earliest variations of the atomic decomposition method proposed by Frazier and Jawerth arose in connection with so-called modulation spaces.",
"These spaces have been introduced with the idea of describing smoothness for functions over locally compact Abelian groups (LCA groups) via uniform (instead of dyadic) decompositions on the Fourier transform side.",
"The first report [20] issued in 1983 (see also [24] for an expanded form) came too early to be appreciated at the time, certainly also because it was formulated in the most general setting.",
"A large number of references concerning these spaces can be found in [25].",
"Nowadays, modulation spaces are viewed as natural domains for time-frequency analysis and certain families of pseudo-differential operators.",
"Although designed from the very beginning at the most general level, a specific subfamily of modulation spaces, namely the modulation spaces $\\big ( {{{M}^s_{p,q}}({\\mathbb {R}}^d)}, \\, \\Vert \\mbox{$\\,\\cdot \\,$}\\Vert _{{{M}^s_{p,q}}} \\big )$ , with $1 \\le p,q \\le \\infty , s \\in {\\mathbb {R}}$ , are most similar to the corresponding family of Besov spaces ${ ({{{B}^{s}_{p,q}}({\\mathbb {R}}^d)}, \\Vert \\mbox{$\\,\\cdot \\,$}\\Vert _{{{B}^s_{p,q}}}) }$ , with the same parameters.",
"In fact (only) for the case $p=q=2$ these spaces coincide, while otherwise they are different (cf.",
"the PhD thesis of P. Gröbner, [45]).",
"By now, there is an extensive literature concerning the inclusion relation between different types of modulation spaces resp.",
"more general decomposition spaces, see e.g.",
"[58], [66], [65], [52], [70].",
"A Frazier-Jawerth type atomic characterization of modulation spaces has first been given in [22] (received by the editors in Oct. 1986).",
"These results are quite similar to those of Frazier-Jawerth, although no direct reference is made to their papers, obviously because their results have not been used.",
"Instead, the argument is based on a combination of Shannon's theorem and techniques from the theory of Wiener amalgam spaces developed in the early 80s ([19], [23]).",
"These techniques provide refined variants of Poisson's formula, which are used to obtain what is nowadays called a Gabor characterization of modulation spaces.",
"To emphasize the analogy between [36], [37] and [22], we note the following: In each case, suitable partitions of unity are used, which are bounded families within the Fourier algebra $\\big ( { {\\mathcal {F}}{{{L}^1}}({{{\\mathbb {R}}^d}}) }, \\, \\Vert \\mbox{$\\,\\cdot \\,$}\\Vert _{{ {\\mathcal {F}}{{L}^1}}} \\big )$ .",
"Further, the building blocks are carefully chosen by the authors in a specific way, to achieve the goal of atomic representation.",
"In contrast to the last point, the typical question treated e.g.",
"in the context of Gabor analysis—or more generally coorbit theory—is the following: Given some (structured) family of atoms (so without chance of the user to match the atoms with the function spaces), can one still verify frame properties and compute a dual frame, or at least guarantee the existence of such a dual frame?",
"Compared to the situation in wavelet theory, one can say: The possibility of generating orthonormal wavelet bases came certainly as a surprise, especially to Yves Meyer, who tried to prove the converse, but ended up with his first construction of such a basis.",
"His negative expectations were probably based on his knowledge of the Balian-Low Theorem which excludes the existence of Riesz bases (and thus also of orthonormal bases) of Gaborian type with “good atoms” which are well concentrated in the time-frequency sense ([2], [3]).",
"The rapid development of wavelet theory, starting with [56], is in large parts due to the possibility of having orthonormal wavelet bases which may be difficult to construct, but which are easy to use.",
"The construction of compactly supported orthonormal wavelet bases with a given amount of smoothness by Ingrid Daubechies ([17]) was one of the main reasons why the Frazier-Jawerth decomposition method was not pursued too much for a while.",
"A crucial property of both orthonormal wavelets as well as the Schwartz building blocks for the Frazier-Jawerth decompositions was the fact that from the very beginning, the canonical isomorphism between the Hilbert space $\\big ( {{{L}^2}({\\mathbb {R}}^d)}, \\, \\Vert \\mbox{$\\,\\cdot \\,$}\\Vert _2 \\big )$ and ${\\big ( {{\\ell }^2}, \\, \\Vert \\mbox{$\\,\\cdot \\,$}\\Vert _2 \\big )}$ extends automatically to an isomorphism between the classical function spaces (those from the family of Besov-Triebel-Lizorkin type, including ${{H}^1 ({{{\\mathbb {R}}^d}})}$ and ${{BMO}}$ ) and (subspaces of) the corresponding solid BK-spaces (typically weighted mixed-norm sequence spaces).",
"It is interesting to note that the sequence spaces used for the Frazier-Jawerth characterizations and the orthonormal wavelet systems are more or less the same, once one identifies the collection of cubes with different centers with the unique affine transformation (from the $ax+b$ -group) needed to obtain it from a symmetric standard cube.",
"The common structure of all the different types of atomic characterizations going back to Frazier-Jawerth is the identification of a family of Banach spaces (of functions or distributions) via some coefficient mapping with suitable closed and complemented subspace of the corresponding family of solid BK-spaces."
],
[
"Unifying Approach to the CWT and the STFT",
"Once the theory of wavelets started to gain momentum, with the classical function spaces being characterized via the Frazier-Jawerth decompositions or via suitable (orthonormal) wavelet systems and once it was understood that there are similar characterizations for modulation spaces via Gabor expansions, it became a natural question to ask whether these two signal representation methods have anything further in common.",
"Group representation theory finally provided such a link: in both (as well as other) cases there is a (square) integrable group representation of some locally compact group, acting in an irreducible way (more or less) on a given Hilbert space.",
"In the language of group representation theory, the relevant group for the theory of wavelets is the affine group (also called “$ax+b$ ”-group), acting by dilations and translations on ${L}^2 (\\mathbb {R}^d)$ .",
"On the other hand, for the theory of modulation spaces or Gabor expansions, the group acting in the background is the (reduced) Heisenberg group ${{\\mathbb {H}}^d}$ , acting on ${L}^2 (\\mathbb {R}^d)$ by translations and modulations."
],
[
"The Essence of Coorbit Theory",
"The core of coorbit theory, as outlined in [31], [32], is to generate—from a given unitary group representation $\\pi $ of some locally compact group ${G}$ on a Hilbert space $\\mathcal {H}$ —a whole family of spaces ${{{\\mathcal {C}o}}({Y})}$ , the so-called coorbit spaces (for $({G},\\pi )$ ).",
"For large subfamilies of these spaces, coorbit theory provides atomic decompositions which allow to characterize elements of the space ${{{\\mathcal {C}o}}({Y})}$ by the fact that they can be described using coefficients in the solid BK-spaces ${Y}_d$ which is naturally associated with ${{{\\mathcal {C}o}}({Y})}$ .",
"The first step for describing ${{{\\mathcal {C}o}}({Y})}$ is to define the voice transform $V_g$ , by $(V_g f)(x) := \\langle f,\\pi (x) g\\rangle , \\qquad x \\in {G}, \\quad f,g \\in \\mathcal {H}.$ If $g$ is suitably chosen (i.e.",
"satisfying $\\Vert V_g g \\Vert _{{L}^2} = \\Vert g\\Vert _{\\mathcal {H}}$ ), the mapping $f \\mapsto V_g f$ maps $\\mathcal {H}$ isometrically[48] into $\\big ( {{{L}^2}({G})}, \\, \\Vert \\mbox{$\\,\\cdot \\,$}\\Vert _2 \\big )$ , so that $\\mathcal {H}_0 := V_g(\\mathcal {H})$ is a closed, left-invariant subspace of ${{{L}^2}({G})}$ .",
"Note that this crucially uses that the representation $\\pi $ is (square) integrable and irreducible.",
"The name “voice transform” goes back to the paper [50], where this name is used synonymously with the term “cycle-octave transform” for what later became the continuous wavelet transform (CWT).",
"Since in that paper both the Gabor and the wavelet case have been addressed, this terminology was taken into the general coorbit theory.",
"It was also used later on in the context of the Blaschke group by Margit Pap and Ferenc Schipp (see e.g.",
"[61], [59], [60]).",
"Given the voice transform $V_g$ and a translation invariant, solid BF-spaceThis means that $({Y}, \\, \\Vert \\mbox{$\\,\\cdot \\,$}\\Vert _{Y})$ is a space of measurable functions on ${G}$ such that if $f \\in {Y}$ and if $g : {G}\\rightarrow {\\mathbb {C}}$ is measurable with $|g|\\le |f|$ almost everywhere, then $g \\in {Y}$ with $\\Vert g \\Vert _{{Y}} \\le \\Vert f \\Vert _{{Y}}$ .",
"$({Y}, \\, \\Vert \\mbox{$\\,\\cdot \\,$}\\Vert _{Y})$ on ${G}$ , we would like to define the associated coorbit space ${{{\\mathcal {C}o}}({Y})}$ as ${{{\\mathcal {C}o}}({Y})}:= \\lbrace f \\in \\mathcal {H}{ \\, | \\, } V_g f \\in {Y}\\rbrace $ , with the norm ${\\Vert f\\Vert _{{{{\\mathcal {C}o}}({Y})}}}:= {\\Vert V_g f\\Vert _{{Y}}}$ .",
"But this will in general not yield a complete space.",
"Recalling that the classical function spaces (like Besov spaces) are defined as subsets of the space ${ { {{\\mathcal {S}}}}^{\\prime }({{{\\mathbb {R}}^d}})}$ of tempered distributions and not of ${L}^2 (\\mathbb {R}^d)$ , we thus have to find a suitable replacement for the class of Schwartz functions in the present generality.",
"In this setting, however, there will (in general) be no “universal” class of test functions like the Schwartz space.",
"Instead, given the solid BF-space $({Y}, \\, \\Vert \\mbox{$\\,\\cdot \\,$}\\Vert _{Y})$ , one has to choose a suitable class of test functions specific to $({Y}, \\, \\Vert \\mbox{$\\,\\cdot \\,$}\\Vert _{Y})$ .",
"For this, one first needs to choose a so-called control weight $w : {G}\\rightarrow (0,\\infty )$ for the space $({Y}, \\, \\Vert \\mbox{$\\,\\cdot \\,$}\\Vert _{Y})$ .",
"This weight depends in a certain way on the operator norms of left- and right translation on $({Y}, \\, \\Vert \\mbox{$\\,\\cdot \\,$}\\Vert _{Y})$ .",
"For the sake of brevity we omit these details.",
"Note, however, that $w$ is assumed to be submultiplicative (i.e., $w(xy) \\le w(x) \\cdot w(y)$ ).",
"Given such a control weight $w$ , we introduce the space ${\\mathcal {H}^1_{w}}:= \\lbrace f \\in \\mathcal {H}{ \\, | \\, } V_g f \\in {{L}^1_{w}}\\rbrace , \\qquad \\text{ where} \\qquad {{L}^1_{w}}= \\lbrace f { \\, | \\, } fw \\in {{L}^1}({G}) \\rbrace ,$ which will play the role of “test functions” in the present setting; the coorbit spaces will thus be subspaces of the (anti)dual space ${(\\mathcal {H}^1_{w})^{\\angle }}$ of ${\\mathcal {H}^1_{w}}$ .",
"Of course, the “analyzing window” $g$ —which is used to define the space ${\\mathcal {H}^1_{w}}$ —needs to be chosen carefully.",
"As shown in [30], [31], the space ${\\mathcal {H}^1_{w}}$ is independent of the precise choice of $g$ , as long as $0 \\ne g \\in \\mathcal {A}_w ({G}) := \\left\\lbrace g \\in \\mathcal {H}\\,|\\,V_g g \\in {{L}^1_{w}}\\right\\rbrace .$ The space $\\mathcal {A}_w$ is called the space of analyzing windows.",
"A crucial assumption for the applicability of coorbit theory is that $\\mathcal {A}_w \\supsetneq \\lbrace 0\\rbrace $ is nontrivial.",
"Later on we will need the class of better vectors $\\mathcal {B}_w \\subset \\mathcal {A}_w$ , whose precise definition we omit.",
"For the initiated reader, we note that $g \\in \\mathcal {B}_w$ needs to satisfy $V_g g \\in {W}^R ({L}^\\infty , {{L}^1_{w}})$ , where ${W}^R$ denotes a (right-sided) Wiener amalgam space.",
"Before defining coorbit spaces in full generality, we need to extend the voice transform to the reservoir ${(\\mathcal {H}^1_{w})^{\\angle }}$ : Due to the $\\pi $ -invariance of ${\\mathcal {H}^1_{w}}$ , the (generalized) voice transform $ (V_g f)(x) = \\langle f, \\pi (x)g \\rangle , \\quad x \\in {G},$ of $f \\in {(\\mathcal {H}^1_{w})^{\\angle }}$ with respect to $g \\in {\\mathcal {H}^1_{w}}$ is well-defined.",
"Here, the pairing is understood as $\\langle f, g\\rangle = f(g)$ , which extends the usual scalar product on $\\mathcal {H}$ .",
"With all this, we define the corresponding coorbit space $ {{{\\mathcal {C}o}}({Y})}:= \\lbrace f \\in {(\\mathcal {H}^1_{w})^{\\angle }}{ \\, | \\, } V_g f \\in {Y}\\rbrace ,$ with the usual natural norm.",
"Again one can show that the space is independent of the analyzing window $0 \\ne g \\in \\mathcal {A}_w$ .",
"Another main result of coorbit theory concerns the atomic decomposition claim: Theorem 3 For any $0 \\ne g \\in \\mathcal {B}_w$ there exists a neighborhood $U$ of the identity $e \\in {G}$ such that the following is true: For any well-separatedThis means that $(x_i)_{i \\in I}$ is the finite union of uniformly separated sets in ${G}$ .",
"family $(x_i)_{i \\in I}$ in ${G}$ which is $U$ -dense, the family $(\\pi (x_i) g)_{i \\in I}$ defines a Banach frame for ${{{\\mathcal {C}o}}({Y})}$ .",
"More precisely: There exists a solid BK-space ${Y}_d = {Y}_d ( (x_i)_{i \\in I} ) \\le {\\mathbb {C}}^I$ and a bounded linear mapping ${\\mathcal {C}}: f \\mapsto (c_i (f))_{i \\in I}$ from ${{{\\mathcal {C}o}}({Y})}$ to ${Y}_d$ , such that $ f = {\\mathcal {R}}({\\mathcal {C}}f) = \\sum _{i \\in I} c_i (f) \\pi (x_i)g, \\quad \\forall \\, f \\in {{{\\mathcal {C}o}}({Y})}, $ with (unconditional) convergence in ${({{{\\mathcal {C}o}}({Y})},\\Vert \\mbox{$\\,\\cdot \\,$}\\Vert _{{{{\\mathcal {C}o}}({Y})}})}$ , if ${\\mathcal {H}^1_{w}}$ is dense in ${{{\\mathcal {C}o}}({Y})}$ (and in the $ w^{*} $ -sense otherwise).",
"Furthermore, ${\\mathcal {R}}: {Y}_d \\rightarrow {{{\\mathcal {C}o}}({Y})}$ is bounded.",
"In comparison with the original Frazier-Jawerth approach we note the following: (a) In the FJ-approach both the analyzing and the synthesizing vector are specified by the construction; although there is some freedom in the construction, the two elements have to be chosen in a very specific way.",
"(b) In contrast, coorbit theory provides a much larger reservoir of atoms; in fact, coorbit theory shows that one can reconstruct the complete voice transform $V_g f$ (and hence the element $f \\in {{{\\mathcal {C}o}}({Y})}$ ) from its samples $(V_g f (x_i))_{i\\in I}$ , as long as $g \\in \\mathcal {B}_w$ and as long as the samples are taken densely enough.",
"(c) Both theories provide a retraction between a family of function spaces and the corresponding solid BK-spaces.",
"Since the $ax+b$ -group and the (reduced) Heisenberg group both have two unbounded variables, it is natural to use mixed-norm conditions on the variables.",
"The corresponding discrete variants are discrete mixed-norm spaces, as long as the sampling points $(x_i)_{i \\in I}$ form a product set.",
"For a rearrangement invariant space ${Y}$ , the sequence space ${{Y}_{{ }d}}$ is just the “natural” discrete variant; e.g.",
"$({L}^p)_d = \\ell ^p$ .",
"(d) In both the Frazier-Jawerth-theory and the coorbit approach, one has the choice to describe them in full generality or for a family of spaces.",
"The published versions of Frazier-Jawerth-theory describe the decompositions in the setting of tempered distributions; consequently they are able to characterize Besov spaces for all real parameters $s$ .",
"If one relaxes the conditions on the building blocks (e.g.",
"if one assumes only a certain number of vanishing moments instead of requiring the building blocks to be compactly supported away from the origin), then one would have a valid statement only for a certain range of spaces ${{{B}^{s}_{p,q}}({\\mathbb {R}}^d)}$ , with $|s| \\le s_0$ .",
"In the coorbit setting, the family of spaces ${{{\\mathcal {C}o}}({Y})}$ for which a certain window $g$ is suitable is determined by the weight $w$ : An analyzing window $g \\in \\mathcal {A}_w$ is only guaranteed to characterize ${{{\\mathcal {C}o}}({Y})}$ , i.e., to satisfy ${{{\\mathcal {C}o}}({Y})}= \\left\\lbrace f \\,|\\,V_g f \\in {Y}\\right\\rbrace ,$ if $w$ is a control weight for ${Y}$ .",
"Likewise, Theorem REF only guarantees that $g \\in \\mathcal {B}_w$ generates a Banach frame for ${{{\\mathcal {C}o}}({Y})}$ if $w$ is a control weight for ${Y}$ ."
],
[
"Shearlets and other constructions",
"The first aim of the theory of coorbit spaces as proposed in [30], [31], [32], [33] was certainly to provide a unified treatment of the two most important situations (see Subsection REF ) where painless non-orthogonal expansions arose ([18]).",
"But it also paved the way to consider other examples through the lens of the general theory.",
"Already early on, it was clear that certain Moebius invariant Banach spaces of analytic functions on the unit disk (see the work of Arazy-Fisher-Peetre [1]) fit into the framework of coorbit theory.",
"These spaces are described by the behaviour of their members, typically by imposing integrability properties, expressed by weighted mixed norm spaces, with a radial and a circular component.",
"In a way, they can be compared with Fock spaces, which consist of analytic functions over phase space, but can be identified with the set of voice transforms (i.e.",
"STFTs) with Gaussian window.",
"After shearlets were first introduced by Kutyniok, Labate, Lim, Guo and Weiss[55], [51], their group theoretic nature was realized in [13], see also [54].",
"Building upon that group-theoretic background of the shearlet transform, the theory of shearlet spaces was investigated in [14], [15], [10], [16], [11], [12].",
"In [14], applicability of coorbit theory (i.e.",
"$\\mathcal {B}_w \\ne \\lbrace 0\\rbrace $ ) in dimension $d=2$ was established and the associated Banach frames (as provided by Theorem REF ) were written down explicitly.",
"The generalization to higher dimensions (including the definition of a possible shearlet group in dimensions $d>2$ ) was obtained in [15].",
"Another generalization of the shearlet group to higher dimensions was considered in [12].",
"Moreover, the relation of shearlet coorbit spaces to more classical smoothness spaces (namely to (sums of) homogeneous Besov spaces) was investigated in [16] for $d=2$ and in [11] for higher dimensions.",
"Related results will be discussed in Section .",
"Yet another group, the so-called Blaschke group, is in the background of a series of papers by M. Pap and her coauthors ([61], [59], [60], [34]).",
"A natural starting point for the definition of decomposition spaces is the observation that modulation spaces, as well as Besov spaces, can be described by imposing certain decay conditions on the sequence of ${L}^p$ norms $\\left( \\Vert \\mathcal {F}^{-1} (\\varphi _i \\widehat{g})\\Vert _{{L}^p} \\right)_{i\\in I}$ for suitable families of functions $(\\varphi _i)_{i \\in I}$ .",
"In the case of (inhomogeneous) Besov spaces, the $(\\varphi _n)_{n \\in \\mathbb {N}_0}$ form a dyadic partition of unity, while for modulation spaces, a uniform partition of unity $(\\varphi _k)_{k \\in \\mathbb {Z}^d}$ is used.",
"By utilizing this observation, decomposition spaces, as introduced by Feichtinger and Gröbner in [29], [21], provide a common framework for Besov- and modulation spaces, as well as many other smoothness spaces occurring in harmonic analysis.",
"Indeed, we will see that decomposition spaces can be used to describe the $\\alpha $ -modulation spaces—a family of spaces intermediate to Besov- and modulation spaces.",
"Furthermore, they provide an alternative view on a large class of wavelet type coorbit spaces.",
"We will see that the decomposition space viewpoint makes many properties of these spaces transparent, while it is highly nontrivial (if not impossible) to obtain these properties directly using the coorbit viewpoint."
],
[
"Basic properties of decomposition spaces",
"The original—very general—definition of decomposition spaces[29], [21] starts with a covering $\\mathcal {Q} = (Q_i)_{i \\in I}$ of a given space $X$ .",
"Most of the time, one may think of $X$ as (a subset of) the frequency space $\\mathbb {R}^d$ ; but in principle, it could be some manifold or other domain.",
"Now, given a suitable function space ${B}$ , the idea is to define the decomposition space norm of a function/distribution $f$ by measuring the local behaviour of $f$ in the ${B}$ -norm, i.e.",
"by localizing $f$ to the different sets $Q_i$ (using a suitable partition of unity $(\\varphi _i)_{i \\in I}$ ) and by measuring the individual pieces using the ${B}$ -norm.",
"The global properties of this (generalized) sequence of ${B}$ norms are then restricted using a suitable sequence space ${Y}$ .",
"Formally, we define $\\Vert f \\Vert _{\\mathcal {D}(\\mathcal {Q}, {B}, {Y})} := \\left\\Vert \\left( \\Vert \\varphi _i \\cdot f\\Vert _{B}\\right)_{i \\in I} \\right\\Vert _{Y}\\, .$ Of course, one has to impose certain restrictions on the covering $\\mathcal {Q}$ , on the sequence space ${Y}$ and on the family $(\\varphi _i)_{i \\in I}$ to ensure that equation (REF ) yields a well-defined norm/space, independent of the partition of unity $(\\varphi _i)_{i \\in I}$ .",
"The most important assumptions are that the covering $\\mathcal {Q}$ has the finite overlap property (described below) and that the $(\\varphi _i)_{i \\in I}$ are uniformly bounded in the pointwise multiplier algebra of ${B}$ .",
"In this paper, however, we will not use the general framework of decomposition spaces from [29], [21].",
"Instead, we restrict ourselves to (a slight modification of) the more specialized setting from [6], which we describe now.",
"We start with an open subset $\\mathcal {O}$ of the frequency space $\\mathbb {R}^d$ and a covering $\\mathcal {Q} = (Q_i)_{i \\in I}$ of $\\mathcal {O}$ , which we assume to be of a certain regular form: Definition 4 (cf.",
"[29], [6] and [70]) For $J\\subset I$ we define the set of neighbors of $J$ as $J^{\\ast }:=\\left\\lbrace i\\in I\\,|\\,\\exists j\\in J:\\, Q_{i}\\cap Q_{j}\\ne \\varnothing \\right\\rbrace .$ Inductively, we set $J^{0\\ast }:=J \\qquad \\text{and} \\qquad J^{\\left(n+1\\right)\\ast }:=\\left(J^{n\\ast }\\right)^{\\ast } \\text{ for } n \\in \\mathbb {N}_0.$ We also define $i^{k\\ast }:=\\left\\lbrace i\\right\\rbrace ^{k\\ast }$ for $i\\in I$ and $k\\in \\mathbb {N}_{0}$ .",
"We say that $\\mathcal {Q}$ is an admissible covering of $\\mathcal {O}$ , if $\\mathcal {Q}$ is a covering of $\\mathcal {O}$ with $Q_i \\ne \\varnothing $ for all $i \\in I$ and if the following constant is finite: $N_{\\mathcal {Q}}:=\\sup _{i\\in I}\\left|i^{\\ast }\\right| \\, .$ If $N_\\mathcal {Q}< \\infty $ , we say that $\\mathcal {Q}$ has the finite overlap property.",
"We say that $\\mathcal {Q}$ is an almost structured covering (of $\\mathcal {O}$ ) if it is of the form $\\left(Q_{i}\\right)_{i \\in I} = \\left(T_{i} Q_i ^{\\prime } +b_{i}\\right)_{i\\in I}$ for certain $T_i \\in {\\rm GL}(\\mathbb {R}^d)$ , $b_i \\in \\mathbb {R}^d$ and certain open sets $Q_i ^{\\prime } \\subset \\mathbb {R}^d$ , satisfying the following properties: $\\mathcal {Q}$ is an admissible covering of $\\mathcal {O}$ , The set $\\bigcup _{i \\in I} Q_i ^{\\prime } \\subset \\mathbb {R}^d$ is bounded.",
"The following expression (then a constant) is finite: $C_{\\mathcal {Q}}:=\\sup _{i\\in I}\\sup _{j\\in i^{\\ast }}\\left\\Vert \\smash{T_{i}^{-1}}T_{j}\\right\\Vert .$ There is a family $(P_i ^{\\prime })_{i \\in I}$ of open sets $P_i ^{\\prime } \\subset \\mathbb {R}^d$ such that: (i) The sets $\\left\\lbrace P_i ^{\\prime } \\,|\\,i\\in I \\right\\rbrace $ and $\\left\\lbrace Q_i ^{\\prime } \\,|\\,i\\in I \\right\\rbrace $ are both finite.",
"(ii) We have $\\overline{P_i ^{\\prime }} \\subset Q_i ^{\\prime }$ for all $i \\in I$ and $\\mathcal {O}= \\bigcup _{i \\in I} (T_i P_i ^{\\prime } + b_i)$ .",
"In addition to these assumptions on $\\mathcal {Q}$ we require $\\Phi = (\\varphi _i)_{i \\in I}$ to be uniformly bounded as pointwise multipliers of ${B}$ , i.e., we impose $\\Vert \\varphi _i f\\Vert _{B}\\le C_\\Phi \\Vert f\\Vert _{B}$ , for all $f \\in {B}, i \\in I.$ For ${B}= \\mathcal {F}{L}^p$ , this boils down to the condition $ \\sup _{i \\in I} \\Vert \\hat{\\varphi _i}\\Vert _{{L}^1}< \\infty .$ The precise definition reads as follows: Definition 5 (cf.",
"[29] and [6]) Let $\\mathcal {Q} = (Q_i )_{i \\in I}$ be an almost structured covering of $\\mathcal {O}$ .",
"A family $\\Phi = (\\varphi _i)_{i \\in I}$ is called a bounded admissible partition of unity (BAPU) (subordinate to $\\mathcal {Q}$ ), if the following hold: $\\varphi _i \\in C_c^\\infty (\\mathcal {O})$ with $\\operatorname{supp}\\varphi _i \\subset Q_i$ for all $i \\in I$ , $\\sum _{i \\in I} \\varphi _i \\equiv 1$ on $\\mathcal {O}$ , $\\sup _{i\\in I} \\Vert \\mathcal {F}^{-1} \\varphi _i \\Vert _{{L}^1} <\\infty $ .",
"Remark In most concrete cases, the requirement $\\varphi _i \\in C_c^\\infty (\\mathcal {O})$ can be relaxed substantially, as long as all the involved expressions make sense.",
"In this case, however, the reservoirs $\\mathcal {D}^{\\prime } (\\mathcal {O})$ and $Z^{\\prime }(\\mathcal {O})$ —which will be used in Definitions REF and REF below—have to be replaced by suitable substitutes, cf.",
"[29].",
"The only remaining assumption which we need to define decomposition spaces pertains to the sequence space ${Y}$ from equation (REF ).",
"For the sake of simplicity, we restrict ourselves to the case of weighted sequence spacesWe use the convention ${\\ell }_u^q (I) = \\lbrace (c_i)_{i \\in I} { \\, | \\, } (u_i c_i)_{i \\in I} \\in {\\ell }^q (I)\\rbrace $ , with the obvious norm., i.e.",
"${Y}= {\\ell }_u^q (I)$ for a certain weight $u = (u_i)_{i \\in I}$ .",
"Now, the general theory of decomposition spaces requires ${Y}$ to be a so-called $\\mathcal {Q}$ -regular sequence space, cf.",
"[29].",
"In our setting, this leads to the following condition: Definition 6 Let $\\mathcal {Q}= (Q_i)_{i \\in I}$ be an admissible covering of $\\mathcal {O}$ .",
"We say that a weight $u = (u_i)_{i \\in I}$ is $\\mathcal {Q}$ -moderate, if the constant $C_{u, \\mathcal {Q}} := \\sup _{i \\in I}\\sup _{\\ell \\in i^\\ast } \\frac{u_i}{u_\\ell }$ is finite, i.e.",
"if $u_i \\asymp u_\\ell $ for $Q_i \\cap Q_\\ell \\ne \\varnothing $ (uniformly w.r.t.",
"$i,\\ell $ ).",
"Now that we have clarified all of our assumptions, we can finally give a formal definition of the decomposition spaces that we will consider in this paper.",
"Since the norm on these spaces requires to compute frequency localizations of the form $f_i = \\mathcal {F}^{-1} (\\varphi _i \\widehat{f})$ , it is often more convenient to directly work on the Fourier side, as in the following definition.",
"The usual space sided version of decomposition spaces will be introduced below.",
"Definition 7 Assume that $\\Phi =\\left(\\varphi _{i}\\right)_{i\\in I}$ is a BAPU subordinate to the almost structured covering $\\mathcal {Q}=\\left(Q_{i}\\right)_{i\\in I}$ of $\\mathcal {O}$ , let $p,q \\in [1,\\infty ]$ , and assume that $u = (u_i)_{i \\in I}$ is $\\mathcal {Q}$ -moderate.",
"Then we define for $f\\in \\mathcal {D}^{\\prime }\\left(\\mathcal {O}\\right)$ $\\left\\Vert f\\right\\Vert _{\\mathcal {D}_{\\mathcal {F}}\\left(\\mathcal {Q},{L}^{p},{\\ell }_u^q \\right)}:=\\left\\Vert f\\right\\Vert _{\\mathcal {D}_{\\mathcal {F},\\Phi }\\left(\\mathcal {Q},{L}^{p},{\\ell }_u^q\\right)}:=\\left\\Vert \\left(\\left\\Vert \\mathcal {F}^{-1}\\left(\\varphi _{i}f\\right)\\right\\Vert _{{L}^{p}}\\right)_{i\\in I}\\right\\Vert _{{\\ell }_u^q}\\in \\left[0,\\infty \\right],$ with the convention that for a family $c=\\left(c_{i}\\right)_{i\\in I}$ with $c_{i}\\in \\left[0,\\infty \\right]$ , the expression $\\left\\Vert c\\right\\Vert _{{\\ell }_u^q}$ is to be read as $\\infty $ if $c_{i}=\\infty $ for some $i\\in I$ or if $c\\notin {\\ell }_u^q (I)$ .",
"Define the Fourier-side decomposition space $\\mathcal {D}_{\\mathcal {F}}\\left(\\mathcal {Q},{L}^{p},{\\ell }_u^q\\right)$ with respect to the covering $\\mathcal {Q}$ , integrability exponent $p$ and global component ${\\ell }_u^q$ as $\\mathcal {D}_{\\mathcal {F}}\\left(\\mathcal {Q},{L}^{p},{\\ell }_u^q\\right):=\\left\\lbrace f\\in \\mathcal {D}^{\\prime }\\left(\\mathcal {O}\\right)\\,|\\,\\left\\Vert f\\right\\Vert _{\\mathcal {D}_{\\mathcal {F}}\\left(\\mathcal {Q},{L}^{p},{\\ell }_u^q\\right)}<\\infty \\right\\rbrace .$ Remark We observe that $\\varphi _{i}f$ is a distribution on $\\mathcal {O}$ with compact support in $\\mathcal {O}$ , which thus extends to a (tempered) distribution on $\\mathbb {R}^{d}$ .",
"By the Paley-Wiener theorem, this implies that $\\mathcal {F}^{-1}\\left(\\varphi _{i}f\\right)\\in \\mathcal {S}^{\\prime }\\left(\\mathbb {R}^{d}\\right)$ is given by (integration against) a smooth function.",
"Thus, it makes sense to write $\\left\\Vert \\mathcal {F}^{-1}\\left(\\varphi _{i}f\\right)\\right\\Vert _{{L}^{p}}$ , with the caveat that this expression could be infinite.",
"We finally observe that the notations $\\left\\Vert \\cdot \\right\\Vert _{\\mathcal {D}_{\\mathcal {F}}\\left(\\mathcal {Q},{L}^{p},{\\ell }_u^q\\right)}$ and $\\mathcal {D}_{\\mathcal {F}}\\left(\\mathcal {Q},{L}^{p},{\\ell }_u^q\\right)$ suppress the family $\\left(\\varphi _{i}\\right)_{i\\in I}$ used to define the norm above.",
"But the general theory of decomposition spaces from [29] implies that different choices yield the same spaces with equivalent norms.",
"Since it is more common to work with the proper, “space-sided” objects, rather than with their Fourier-side versions, we will now define the usual, space-sided version of decomposition spaces.",
"To this end, we first introduce the reservoir $Z^{\\prime }\\left(\\mathcal {O}\\right)$ which will be used for these spaces.",
"Our notation is inspired by Triebel's book [68].",
"Definition 8 For $\\varnothing \\ne \\mathcal {O}\\subset \\mathbb {R}^{d}$ open, we define $Z\\left(\\mathcal {O}\\right):=\\mathcal {F}\\left(C_{c}^{\\infty }\\left(\\mathcal {O}\\right)\\right):=\\left\\lbrace \\smash{\\widehat{f}}\\,|\\,f\\in C_{c}^{\\infty }\\left(\\mathcal {O}\\right)\\right\\rbrace \\le \\mathcal {S}\\left(\\smash{\\mathbb {R}^{d}}\\right)$ and endow this space with the unique topology that makes the Fourier transform $\\mathcal {F}:C_{c}^{\\infty }\\left(\\mathcal {O}\\right)\\rightarrow Z\\left(\\mathcal {O}\\right)$ a homeomorphism.",
"We equip the topological dual space $Z^{\\prime }\\left(\\mathcal {O}\\right):=\\left[Z\\left(\\mathcal {O}\\right)\\right]^{\\prime }$ of $Z\\left(\\mathcal {O}\\right)$ with the weak-$\\ast $ -topology, i.e., with the topology of pointwise convergence on $Z\\left(\\mathcal {O}\\right)$ .",
"Finally, as on the Schwartz space, we extend the Fourier transform by duality to $Z^{\\prime }\\left(\\mathcal {O}\\right)$ , i.e.",
"we define $\\mathcal {F}:Z^{\\prime }\\left(\\mathcal {O}\\right)\\rightarrow \\mathcal {D}^{\\prime }\\left(\\mathcal {O}\\right),f\\mapsto \\widehat{f} := f\\circ \\mathcal {F}.$ Remark Since $\\mathcal {F}:C_{c}^{\\infty }\\left(\\mathcal {O}\\right)\\rightarrow Z\\left(\\mathcal {O}\\right)$ is a linear homeomorphism, the Fourier transform as defined in equation (REF ) is a linear homeomorphism as well.",
"Finally, we can define the space-side decomposition spaces.",
"Definition 9 In the setting of Definition REF and for $f\\in Z^{\\prime }\\left(\\mathcal {O}\\right)$ , we define $\\left\\Vert f\\right\\Vert _{\\mathcal {D}\\left(\\mathcal {Q},{L}^{p},{\\ell }_u^q\\right)} :=\\left\\Vert \\smash{\\widehat{f}}\\right\\Vert _{\\mathcal {D}_{\\mathcal {F}}\\left(\\mathcal {Q},{L}^{p},{\\ell }_u^q\\right)}=\\left\\Vert \\left(\\left\\Vert \\mathcal {F}^{-1}\\left(\\varphi _{i}\\smash{\\widehat{f}}\\right)\\right\\Vert _{{L}^{p}}\\right)_{i\\in I}\\right\\Vert _{{\\ell }_u^q}\\in \\left[0,\\infty \\right]$ and define the (space-side) decomposition space $\\mathcal {D}\\left(\\mathcal {Q},{L}^{p},{\\ell }_u^q\\right)$ with respect to the covering $\\mathcal {Q}$ , integrability exponent $p$ and global component ${\\ell }_u^q$ by $\\mathcal {D}\\left(\\mathcal {Q},{L}^{p},{\\ell }_u^q\\right):=\\left\\lbrace f\\in Z^{\\prime }\\left(\\mathcal {O}\\right)\\,|\\,\\left\\Vert f\\right\\Vert _{\\mathcal {D}\\left(\\mathcal {Q},{L}^{p},{\\ell }_u^q\\right)}<\\infty \\right\\rbrace .$ Remark 10 Since the Fourier transform $\\mathcal {F}:Z^{\\prime }\\left(\\mathcal {O}\\right)\\rightarrow \\mathcal {D}^{\\prime }\\left(\\mathcal {O}\\right)$ is an isomorphism, it is clear that the Fourier transform restricts to an (isometric) isomorphism $\\mathcal {F}:\\mathcal {D}\\left(\\mathcal {Q},{L}^{p},{\\ell }_u^q\\right)\\rightarrow \\mathcal {D}_{\\mathcal {F}}\\left(\\mathcal {Q},{L}^{p},{\\ell }_u^q\\right).$ Hence, it does not really matter whether one considers the “space-side” or the “Fourier-side” version of these spaces.",
"At this point, one might ask why we use the spaces $\\mathcal {D}^{\\prime } (\\mathcal {O})$ and $Z^{\\prime }(\\mathcal {O})$ at all, instead of resorting to the more common reservoir ${ { {{\\mathcal {S}}}}^{\\prime }({{{\\mathbb {R}}^d}})}$ , which is for example used to define Besov spaces.",
"The main reasons for this are the following: (a) We want to allow the case $\\mathcal {O}\\subsetneq \\mathbb {R}^{d}$ .",
"If we were to take the space ${ { {{\\mathcal {S}}}}^{\\prime }({{{\\mathbb {R}}^d}})}$ , the decomposition space norm would not be positive definite, or we would have to factor out a certain subspace of ${ { {{\\mathcal {S}}}}^{\\prime }({{{\\mathbb {R}}^d}})}$ .",
"This is for example done in the usual definition of homogeneous Besov spaces, which are subspaces of ${ { {{\\mathcal {S}}}}^{\\prime }({{{\\mathbb {R}}^d}})}/ \\mathcal {P}$ , where $\\mathcal {P}$ is the space of polynomials.",
"Here, it seems more natural to use the space $\\mathcal {D}^{\\prime }\\left(\\mathcal {O}\\right)$ .",
"(b) In case of $\\mathcal {O}=\\mathbb {R}^{d}$ , one could use ${ { {{\\mathcal {S}}}}^{\\prime }({{{\\mathbb {R}}^d}})}$ as the reservoir, as e.g.",
"in [6].",
"But as shown in [70], this does in general not yield a complete space, even for the case ${Y}={\\ell }_{u}^{1}$ with a $\\mathcal {Q}$ -moderate weight $u$ .",
"The following theorem settles the issue of completeness: Theorem 11 (cf.",
"[70]) Under the assumptions of Definition REF , the (Fourier-side) decomposition space $\\mathcal {D}_{\\mathcal {F}}\\left(\\mathcal {Q},{L}^{p},{\\ell }_u^q\\right)$ is a Banach space which embeds continuously into $\\mathcal {D}^{\\prime }\\left(\\mathcal {O}\\right)$ .",
"Likewise, $\\mathcal {D}\\left(\\mathcal {Q},{L}^{p},{\\ell }_u^q\\right)$ is also complete and embeds continuously into $Z^{\\prime } (\\mathcal {O})$ .",
"Before we close this section on the basic properties of decomposition spaces, we note that we have always assumed that we are given some BAPU $\\Phi = (\\varphi _i)_{i \\in I}$ subordinate to the covering $\\mathcal {Q}$ .",
"It is thus important to know whether such a BAPU actually exists.",
"The next theorem shows that this is the case for every almost structured covering.",
"We remark that the result itself, and also the proof, are inspired heavily by the construction used in [6].",
"Theorem 12 (cf.",
"[71]) Let $\\mathcal {Q}$ be an almost structured covering.",
"Then there is a BAPU $\\Phi = (\\varphi _i)_{i \\in I}$ subordinate to $\\mathcal {Q}$ .",
"All in all, we now know how to obtain well-defined decomposition spaces with respect to reasonable coverings.",
"In the next subsection, we consider a special example of decomposition spaces—the $\\alpha $ modulation spaces—in greater detail."
],
[
"$\\alpha $ -modulation spaces",
"The starting point for the original definition of $\\alpha $ -modulation spaces in Gröbner's thesis[45] were the two parallel worlds of Besov-Triebel-Lizorkin spaces and modulation spaces.",
"Given these two types of spaces, it was natural to ask whether there is a way to connect these two families in a “continuous way”.",
"Although one could of course apply complex interpolation in order to construct spaces which are (for fixed parameters $p,q,s$ ) “in between” ${ ({{{B}^{s}_{p,q}}({\\mathbb {R}}^d)}, \\Vert \\mbox{$\\,\\cdot \\,$}\\Vert _{{{B}^s_{p,q}}}) }$ and $\\big ( {{{M}^s_{p,q}}({\\mathbb {R}}^d)}, \\, \\Vert \\mbox{$\\,\\cdot \\,$}\\Vert _{{{M}^s_{p,q}}} \\big )$ , this approach was—so far—not really successful, because it seems to be very difficult to provide constructive characterizations of the members of such spaces which could be used in practice.",
"In contrast, it appeared—after some reflection—quite natural to try to interpolate the two types of spaces in a geometric sense, i.e.",
"to consider decompositions of the Fourier domain which are “in between” the dyadic partitions of unity, playing a crucial role in the description of Besov-Triebel-Lizorkin-spaces, and the uniform partitions of unity, which are relevant for the description of modulation spaces.",
"The basic observation for this “geometric interpolation” approach is that the uniform covering $(Q_k)_{k \\in \\mathbb {Z}^d}$ and the dyadic covering $(P_n)_{n \\in \\mathbb {N}_0}$ satisfy [(Qk)]1/d 0 Qk k Zd, (Pn)1/d 1 Pn n N0, with $\\langle \\xi \\rangle := 1 + |\\xi |$ .",
"Thus, for $\\alpha \\in [0,1]$ , an $\\alpha $ -covering should be an (open, admissible) covering $(Q_i)_{i \\in I}$ of $\\mathbb {R}^d$ which satisfies $[\\lambda (Q_i)]^{1/d} \\asymp \\langle \\xi \\rangle ^{\\alpha } \\qquad \\forall \\xi \\in Q_i \\qquad \\forall i \\in I,$ with the implied constant independent of the precise choice of $\\xi $ and $i$ .",
"Apart from this natural condition, a certain further assumption (cf.",
"[5]) is imposed to rule out pathological coverings.",
"For brevity, we omit this condition.",
"Given this definition, one might wonder (at least for $\\alpha \\in (0,1)$ ), whether there exist $\\alpha $ -coverings.",
"This is indeed the case.",
"As shown in [5], there is an $\\alpha $ covering $\\mathcal {Q}^{(\\alpha )}$ of the form $\\mathcal {Q}^{(\\alpha )} = (B_k^r)_{k \\in \\mathbb {Z}^d \\setminus \\lbrace 0\\rbrace }$ , with $B_k^r := \\left\\lbrace \\xi \\in \\mathbb {R}^d \\,|\\,\\left| \\xi - |k|^{\\alpha _0} k\\right| < r \\cdot |k|^{\\alpha _0} \\right\\rbrace ,$ where $r = r_\\alpha >0$ is chosen suitably and where $\\alpha _0 := \\frac{\\alpha }{1-\\alpha }$ .",
"Furthermore, as shown in [70], $\\mathcal {Q}^{(\\alpha )}$ is an (almost) structured covering of $\\mathbb {R}^d$ and thus admits a BAPU $(\\varphi _i)_{i \\in I}$ .",
"Finally, in [5], it was shown that any two $\\alpha $ -coverings of $\\mathbb {R}^d$ are equivalent (in a certain sense).",
"Given the covering $\\mathcal {Q}^{(\\alpha )}$ , we need suitable $\\mathcal {Q}^{(\\alpha )}$ -moderate weights, in order to define the $\\alpha $ -modulation spaces.",
"By [70], moderateness holds—for arbitrary $\\gamma \\in \\mathbb {R}$ —for the weight $w^{(\\gamma )} := (\\langle k\\rangle ^{\\gamma })_{k \\in \\mathbb {Z}^d \\setminus \\lbrace 0\\rbrace }\\, ,$ so that the $\\alpha $ modulation spaces ${{M}^{s,\\alpha }_{p,q}}({\\mathbb {R}}^d) := \\mathcal {D}( {\\mathcal {Q}^{(\\alpha )}}, {L}^{p}, {{\\ell }^q_{w^{(s/(1-\\alpha ))}}})$ are well-defined Banach spaces.",
"Note that the weight $w^{(s/(1-\\alpha ))}$ satisfies $\\langle \\xi \\rangle ^{s} \\asymp w_k^{(s/(1-\\alpha ))} \\qquad \\forall k \\in \\mathbb {Z}^d \\setminus \\lbrace 0\\rbrace \\, \\forall \\xi \\in B_r^k,$ similar to the case of Besov spaces and modulation spaces.",
"In particular, for the “limit cases” $\\alpha = 0$ and $\\alpha = 1$ , we have ${M}^{s,0}_{p,q}(\\mathbb {R}^d) = {M}^{s}_{p,q}(\\mathbb {R}^d)$ , as well as ${M}^{s, 1}_{p,q}(\\mathbb {R}^d) = {B}^{s}_{p,q} (\\mathbb {R}^d)$ .",
"Since the $\\alpha $ modulation spaces are obtained using “geometric interpolation” between Besov spaces and modulation spaces, one could expect that ${{M}^{s,\\alpha }_{p,q}}(\\mathbb {R}^d)$ can also be obtained by “ordinary” means of interpolation (like complex interpolation) from Besov- and modulation spaces.",
"But at least for the case of complex interpolation, this is false: In [49], it is shown that $\\left[ {M}^{s_1, \\alpha _1}_{p_1, q_1} (\\mathbb {R}^d) , {M}^{s_2, \\alpha _2}_{p_2, q_2} (\\mathbb {R}^d) \\right]_\\theta = {M}^{s, \\alpha }_{p,q}(\\mathbb {R}^d)$ for certain $p, p_1, p_2, q, q_1, q_2 \\in [1,\\infty ]$ and $s, s_1, s_2 \\in \\mathbb {R}$ , as well as $\\alpha , \\alpha _1, \\alpha _2 \\in [0,1]$ and $\\theta \\in (0,1)$ can only hold if $\\alpha _1 = \\alpha _2 \\quad \\text{ or } \\quad p_1 = q_1 = 2 \\quad \\text{ or } \\quad p_2 = q_2 = 2.$ If one of these conditions hold, then interpolation works as expected.",
"This might be surprising for $\\alpha _1 \\ne \\alpha _2$ ; but in this case, we have $p_1 = q_1 = 2$ or $p_2 = q_2 = 2$ , which implies ${M}^{s_1,\\alpha _1}_{p_1,q_1} = {H}^{s_1} = {M}^{s_1, \\alpha _2}_{p_1, q_1}$ or ${M}^{s_2,\\alpha _2}_{p_2,q_2} = {H}^{s_2} = {M}^{s_2, \\alpha _1}_{p_2, q_2}$ , respectively.",
"But also apart from interpolation, it is natural to ask how the different $\\alpha $ modulation spaces are related.",
"Concretely, one might wonder under which conditions an embedding of the form ${M}^{s_1, \\alpha _1}_{p_1, q_1} (\\mathbb {R}^d) \\hookrightarrow {M}^{s_2, \\alpha _2}_{p_2, q_2}(\\mathbb {R}^d)$ holds.",
"For $\\alpha _1, \\alpha _2 \\in \\lbrace 0,1\\rbrace $ , this question was solved by Kobayashi and Sugimoto[53].",
"For general $\\alpha $ , it was considered by Toft and Wahlberg[67], shortly before it was solved completely—for $(p_1, q_1) = (p_2, q_2)$ —by Han and Wang[52].",
"Based in part on their ideas, the second named author of the present paper developed a general theory of embeddings between decomposition spaces (cf.",
"[70]), which we will outline in Subsection REF .",
"Using this theory, the above question can be solved easily—even for $(p_1, q_1) \\ne (p_2, q_2)$ —cf.",
"[70]: Theorem 13 Let $0 \\le \\alpha \\le \\beta \\le 1$ , $p_1, p_2, q_1, q_2 \\in [1,\\infty ]$ and $s_1, s_2 \\in \\mathbb {R}$ .",
"Define s(0) := ( 1p2 - 1p1 ) + (- ) ( 1p2 - 1q1 )+ , s(1) := ( 1p2 - 1p1 ) + (- ) ( 1q2 - 1p1 )+ , with $p^{\\triangledown } := \\min \\lbrace p, p^{\\prime }\\rbrace $ , as well as $p^{\\triangle } := \\max \\lbrace p, p^{\\prime }\\rbrace $ .",
"We have ${M}^{s_1, \\alpha }_{p_1, q_1} (\\mathbb {R}^d) \\hookrightarrow {M}^{s_2, \\beta }_{p_2, q_2}(\\mathbb {R}^d)$ if and only if $p_1 \\le q_1$ and ${\\left\\lbrace \\begin{array}{ll}s_2 \\le s_1 + d \\cdot s^{(0)} , & \\text{if } q_1 \\le q_2,\\\\s_2 < s_1 + d \\cdot \\left( s^{(0)} + (1-\\beta )\\left( q_1^{-1} - q_2^{-1} \\right) \\right), & \\text{if } q_1 > q_2.\\end{array}\\right.",
"}$ Conversely, we have ${M}^{s_1, \\beta }_{p_1, q_1} (\\mathbb {R}^d) \\hookrightarrow {M}^{s_2, \\alpha }_{p_2, q_2}(\\mathbb {R}^d)$ if and only if $p_1 \\le q_1$ and ${\\left\\lbrace \\begin{array}{ll}s_2 \\le s_1 + d \\cdot s^{(1)} , & \\text{if } q_1 \\le q_2, \\\\s_2 < s_1 + d \\cdot \\left( s^{(1)} + (1 - \\beta ) \\left( q_1^{-1} - q_2^{-1} \\right) \\right), & \\text{if } q_1 > q_2.\\end{array}\\right.",
"}$ Using the theory of embeddings between decomposition spaces from Subsection REF , one can explain the main geometric properties of the $\\alpha $ coverings $\\mathcal {Q}^{(\\alpha )}$ which lead to the preceding theorem: The main point is that the covering $\\mathcal {Q}^{(\\alpha )}$ is almost subordinate to (“finer than”, cf.",
"equation (REF )) the covering $\\mathcal {Q}^{(\\beta )}$ for $\\alpha \\le \\beta $ .",
"Furthermore, the precise conditions depend on the number of “smaller sets” that are needed to cover the “bigger” sets, cf.",
"[70] and Theorems REF and REF from below.",
"We finally remark that the theorems in [70] apply for the full range $(0,\\infty ]$ of the exponents.",
"But in the present paper, we restrict ourselves to the range $[1,\\infty ]$ for simplicity."
],
[
"Embeddings between different decomposition spaces",
"In this subsection, we consider embeddings between decomposition spaces with respect to different coverings, i.e.",
"of the form $\\mathcal {D}( {\\mathcal {Q}}, {L}^{p_1}, {{\\ell }_u^{q_1}}) \\hookrightarrow \\mathcal {D}( {\\mathcal {P}}, {L}^{p_2}, {{\\ell }_v^{q_2}})$ for $p_1, p_2, q_1, q_2 \\in [1,\\infty ]$ and two almost structured coverings $\\mathcal {Q}= (Q_i)_{i \\in I} = (T_i Q_i ^{\\prime } + b_i)_{i \\in I} \\qquad \\text{ and } \\qquad \\mathcal {P}= (P_j)_{j \\in J} = (S_j P_j ^{\\prime } + c_j)_{j \\in J}$ of two (possibly different) subsets $\\mathcal {O}, \\mathcal {O}^{\\prime }$ of the frequency space $\\mathbb {R}^d$ .",
"We assume the weights $u = (u_i)_{i \\in I}$ and $v = (v_j)_{j \\in J}$ to be moderate w.r.t.",
"$\\mathcal {Q}$ and $\\mathcal {P}$ , respectively.",
"As seen in the previous subsection, our setting includes embeddings between $\\alpha $ modulation spaces for different values of $\\alpha $ .",
"As the main standing requirement for this subsection, we assume that $\\mathcal {Q}$ is almost subordinate to $\\mathcal {P}$ .",
"Very roughly, this means that the sets $Q_i$ are “smaller” than the sets $P_j$ , or that $\\mathcal {Q}$ is “finer” then $\\mathcal {P}$ .",
"Rigorously, it means that there is some fixed $n \\in \\mathbb {N}_0$ such that for every $i \\in I$ , there is some $j_i \\in J$ satisfying $Q_i \\subset P_{j_i}^{n\\ast } := \\bigcup _{\\ell \\in j_i^{n\\ast }} P_\\ell .$ Note that this assumption implies $\\mathcal {O}\\subset \\mathcal {O}^{\\prime }$ .",
"Even more importantly, this assumption destroys the symmetry between $\\mathcal {Q}, \\mathcal {P}$ in equation (REF ), so that in addition to (REF ), we will also consider the “reverse” embedding $\\mathcal {D}( {\\mathcal {P}}, {L}^{p_2}, {{\\ell }_v^{q_2}}) \\hookrightarrow \\mathcal {D}( {\\mathcal {Q}}, {L}^{p_1}, {{\\ell }_u^{q_1}}).$ Under the assumption that $\\mathcal {Q}$ is almost subordinate to $\\mathcal {P}$ , the object which describes those features of the coverings $\\mathcal {Q}, \\mathcal {P}$ which are relevant for us is the family of intersection sets given by $I_j := \\left\\lbrace i \\in I \\,|\\,Q_i \\cap P_j \\ne \\varnothing \\right\\rbrace \\qquad \\text{ for } j \\in J.$ Of course, $I_j = \\varnothing $ if and only if $P_j \\cap \\mathcal {O}= \\varnothing $ .",
"Since these sets will be irrelevant for our purposes, we additionally define $J_{\\mathcal {O}} := \\left\\lbrace j \\in J \\,|\\,P_j \\cap \\mathcal {O}\\ne \\varnothing \\right\\rbrace .$ In the remainder of this subsection, we will state sufficient conditions and necessary conditions for the existence of the embeddings (REF ) and (REF ), respectively.",
"In general, these two conditions will only coincide for a certain range of $p_1$ or $p_2$ , while there is a gap between the two conditions outside of this range.",
"Under suitable additional assumptions, however, more strict necessary conditions can be derived; in fact, a complete characterization of the existence of the embeddings can be achieved.",
"For this to hold, we will (occasionally, but not always) assume that the following properties are fulfilled: Definition 14 We say that the weight $u = (u_i)_{i \\in I}$ is relatively $\\mathcal {P}$ -moderate, if $\\sup _{j \\in J} \\,\\, \\sup _{i,\\ell \\in I_j} \\frac{u_i}{u_\\ell } < \\infty .$ The (almost structured) covering $\\mathcal {Q}= (T_i Q_i ^{\\prime } + b_i)_{i \\in I}$ is called relatively $\\mathcal {P}$ -moderate, if the weight $(|\\det T_i|)_{i \\in I}$ is relatively $\\mathcal {P}$ -moderate.",
"Roughly speaking, this means that two (small) sets $Q_i, Q_\\ell $ have essentially the same measure if they intersect the same (large) set $P_j$ .",
"Although these assumptions might appear rather restrictive, they are fulfilled in many practical cases; in particular if $\\mathcal {Q}$ and $\\mathcal {P}$ are coverings associated to $\\alpha $ -modulation spaces, and if $u,v$ are the usual weights for these spaces.",
"We can now analyze existence of the embedding (REF ): Theorem 15 (cf.",
"[70]) Define $p_2^\\triangle := \\max \\lbrace p_2, p_2 ^{\\prime }\\rbrace $ and for $r \\in [1,\\infty ]$ , let C1(r) := ( ( |Ti|1p2 - 1p1 ui )i Ijq1 (r / q1)' (Ij) / vj )j Jq1 (q2 / q1)'(J) , C2 := ( uijvj |Tij|1p2 - ( 1q1 - 1p2 )+ - 1p1 |Sj|( 1q1 - 1p2 )+ )j JOq1 (q2 / q1)'(JO) , where for each $j \\in J_{\\mathcal {O}}$ some $i_j \\in I$ with $Q_{i_j} \\cap P_j \\ne \\varnothing $ can be chosen arbitrarily.",
"Then the following hold: If $C_1^{(p_2^\\triangle )} < \\infty $ and $p_2 \\le p_1$ , then the map $\\iota : \\mathcal {D}( {\\mathcal {P}}, {L}^{p_2}, {{\\ell }_v^{q_2}}) \\rightarrow \\mathcal {D}( {\\mathcal {Q}}, {L}^{p_1}, {{\\ell }_u^{q_1}}), f \\mapsto f|_{\\mathcal {F}(C_c^\\infty (\\mathcal {O}))}$ is well-defined and bounded.",
"Conversely, if $\\qquad \\qquad \\theta : \\left(\\mathcal {F}^{-1}(C_c^\\infty (\\mathcal {O})) , {\\Vert \\cdot \\Vert _{\\mathcal {D}( {\\mathcal {P}}, {L}^{p_2}, {{\\ell }_v^{q_2}})}}\\right) \\rightarrow \\mathcal {D}( {\\mathcal {Q}}, {L}^{p_1}, {{\\ell }_u^{q_1}}), f \\mapsto f$ is bounded, then $C_1^{(p_2)} < \\infty $ and $p_2 \\le p_1$ .",
"Finally, if $P_j \\subset \\mathcal {O}$ holdsThe main case in which this holds is if $\\mathcal {O}= \\mathcal {O}^{\\prime }$ .",
"for all $j \\in J_{\\mathcal {O}}$ and if additionally the covering $\\mathcal {Q}$ and the weight $u = (u_i)_{i \\in I}$ are both relatively $\\mathcal {P}$ -moderate, we have the following equivalence: is bounded is bounded (C1(p2) < and p2 p1 ) (C2 < and p2 p1 ).",
"Remark We achieve a complete characterization of the existence of the embedding (REF ) if $\\mathcal {Q}$ and $u$ are relatively $\\mathcal {P}$ -moderate, but also in case of $p_2 \\in [2,\\infty ]$ , since in this case, $p_2^{\\triangle } = p_2$ and hence $C_1^{(p_2^\\triangle )} = C_1^{(p_2)}$ .",
"Even for well-understood special cases like $\\alpha $ modulation spaces, the above theorem yields new results, since even in the most general previous result [52], only the case $(p_1, q_1) = (p_2, q_2)$ was studied.",
"Considering $\\iota $ as an embedding is not always justified.",
"For example, if $\\mathcal {O}^{\\prime } \\setminus \\mathcal {O}$ has nonempty interior, then every $f \\in \\mathcal {F}^{-1} ( C_c^\\infty (\\mathcal {O}^{\\prime } \\setminus \\mathcal {O}))$ satisfies $\\iota f = 0$ , although $f \\ne 0$ is possible.",
"The result for the embedding (REF ) is similar: Theorem 16 (cf.",
"[70]) Let $(\\varphi _i)_{i \\in I}$ be a BAPU for $\\mathcal {Q}$ .",
"Define $p_2^\\triangledown := \\min \\lbrace p_2, p_2 ^{\\prime }\\rbrace $ and for $r \\in [1,\\infty ]$ , let C1(r) := ( vj (ui-1 |Ti|1p1 - 1p2 )i Ijr (q1 / r)' (Ij) )j Jq2 (q1 / q2)'(J) , C2 := ( vjuij |Tij|1p1 - ( 1p2 - 1q1 ) + - 1p2 |Sj|( 1p2 - 1q1 ) + )j JOq2 (q1 / q2)'(JO) , where for each $j \\in J_{\\mathcal {O}}$ some $i_j \\in I$ with $Q_{i_j} \\cap P_j \\ne \\varnothing $ can be chosen arbitrarily.",
"Then the following hold: If $C_1^{(p_2^\\triangledown )} < \\infty $ and if $p_1 \\le p_2$ , then the map $\\quad \\qquad \\iota = \\iota _{\\Phi } : \\mathcal {D}( {\\mathcal {Q}}, {L}^{p_1}, {{\\ell }_u^{q_1}}) \\hookrightarrow \\mathcal {D}( {\\mathcal {P}}, {L}^{p_2}, {{\\ell }_v^{q_2}}), f \\mapsto \\sum _{i \\in I} \\mathcal {F}^{-1} ( \\varphi _i \\cdot \\widehat{f}\\,)$ is well-defined and bounded.",
"Here, $\\iota f$ acts as follows: $\\langle \\iota f, \\gamma \\rangle = \\sum _{i \\in I} \\langle f, \\mathcal {F} (\\varphi _i \\cdot \\mathcal {F}^{-1} \\gamma ) \\rangle \\quad \\text{ for } \\quad \\gamma \\in \\mathcal {F}(C_c^\\infty (\\mathcal {O}^{\\prime })),$ with absolute convergence of the series for $f \\in \\mathcal {D}( {\\mathcal {Q}}, {L}^{p_1}, {{\\ell }_u^{q_1}})$ .",
"Furthermore, $\\langle \\iota f, \\gamma \\rangle = \\langle f, \\gamma \\rangle $ holds for all $\\gamma \\in \\mathcal {F}(C_c^\\infty (\\mathcal {O}))$ , so that $\\iota f \\in \\mathcal {D}( {\\mathcal {P}}, {L}^{p_2}, {{\\ell }_v^{q_2}}) \\subset Z^{\\prime }(\\mathcal {O}^{\\prime })$ extends $f \\in \\mathcal {D}( {\\mathcal {Q}}, {L}^{p_1}, {{\\ell }_u^{q_1}}) \\subset Z^{\\prime }(\\mathcal {O})$ .",
"If the map $\\quad \\qquad \\theta : \\left( \\mathcal {F}^{-1}(C_c^\\infty (\\mathcal {O})), {\\Vert \\cdot \\Vert _{\\mathcal {D}( {\\mathcal {Q}}, {L}^{p_1}, {{\\ell }_u^{q_1}})}}\\right) \\rightarrow \\mathcal {D}( {\\mathcal {P}}, {L}^{p_2}, {{\\ell }_v^{q_2}}), f \\mapsto f$ is bounded, then $p_1 \\le p_2$ and $C_1^{(p_2)} < \\infty $ .",
"Finally, if $P_j \\subset \\mathcal {O}$ holds for all $j \\in J_{\\mathcal {O}}$ and if additionally the covering $\\mathcal {Q}$ and the weight $u = (u_i)_{i \\in I}$ are both relatively $\\mathcal {P}$ -moderate, we have the following equivalence: is bounded is bounded.",
"(C1(p2) < and p1 p2 ) (C2 < and p1 p2 ).",
"Remark We achieve a complete characterization of the existence of the embedding (REF ) for $p_2 \\in [1,2]$ .",
"If $\\mathcal {Q}$ and $u$ are relatively $\\mathcal {P}$ -moderate, we get a complete characterization for arbitrary $p_2 \\in [1,\\infty ]$ .",
"In contrast to Theorem REF , $\\iota $ is always injective in the present setting.",
"Note that the definition of $\\iota $ is independent of $p_1, p_2, q_1, q_2$ and $u,v$ .",
"In fact, if $\\mathcal {O}= \\mathcal {O}^{\\prime }$ , then $\\iota f = f$ for all $f \\in \\mathcal {D}( {\\mathcal {Q}}, {L}^{p_1}, {{\\ell }_u^{q_1}}) \\subset Z^{\\prime }(\\mathcal {O}) = Z^{\\prime }(\\mathcal {O}^{\\prime })$ .",
"For one concrete application of Theorems REF and REF , we refer the reader to the characterization of embeddings between different $\\alpha $ modulation spaces in Theorem REF .",
"Further applications will be given in Section .",
"There are also results which apply if neither $\\mathcal {Q}$ is almost subordinate to $\\mathcal {P}$ , nor vice versa.",
"In fact, it suffices if one can write $\\mathcal {O}\\cap \\mathcal {O}^{\\prime } = A \\cup B$ , such that $\\mathcal {Q}$ is almost subordinate to $\\mathcal {P}$ near $A$ and vice versa near $B$.",
"For a precise formulation of this condition, and the resulting embedding results, we refer to [70].",
"Finally, there are also results for embeddings of the form $\\mathcal {D}( {\\mathcal {Q}}, {L}^{p}, {{Y}}) \\hookrightarrow W^{k,q}(\\mathbb {R}^d)$ for the classical Sobolev spaces $W^{k,q}$ .",
"These results are easy to apply, since no subordinateness is required.",
"As shown in [71], existence of the embedding can be completely characterized for $q \\in [1,2] \\cup \\lbrace \\infty \\rbrace $ , while for $q \\in (2,\\infty )$ , certain sufficient and certain necessary criteria are given; but in general, these do not coincide."
],
[
"Banach frames for Decomposition spaces",
"Borup and Nielsen[6] gave a construction of Banach frames for decomposition spaces which applies in a very general setting.",
"In particular, their construction applies to ($\\alpha $ )-modulation spaces and Besov spaces.",
"Since this construction makes the power of Banach frames available for decomposition spaces, we could not resist discussing their results.",
"Furthermore, their results fit well into the present context: As Borup and Nielsen write themselves: “[our] frame expansion should perhaps be considered an adaptable variant of the $\\varphi $ -transform of Frazier and Jawerth” (cf.",
"[6]).",
"To describe their construction, we first introduce structured coverings.",
"An almost structured covering $\\mathcal {Q}= (Q_i)_{i \\in I} = (T_i Q_i ^{\\prime } + b_i)_{i \\in I}$ of $\\mathcal {O}= \\mathbb {R}^d$ is called structured if $Q_i ^{\\prime } = Q$ for all $ i\\in I$ , i.e., if all $Q_i$ are affine images of a fixed set.",
"The idea is to transfer the orthonormal basis $(e^{2 \\pi i \\langle k , \\cdot \\rangle })_{k \\in \\mathbb {Z}^d}$ of $L^2 ([-1/2 ,1/2]^d)$ to each of the sets $Q_i^{(a)} = T_i [-a,a]^d + b_i \\supset Q_i$ for certain $a > 0$ .",
"Then, one truncates these periodic functions using a certain (quadratic) partition of unity subordinate to $\\mathcal {Q}$ .",
"Thus, one obtains a tight frame for ${L}^2 (\\mathbb {R}^d)$ .",
"The nontrivial part is to show that one also obtains Banach frames for the whole range of decomposition spaces.",
"The construction proceeds as follows: By [6] on finds a family $(\\theta _i)_{i \\in I}$ such that: $\\operatorname{supp}\\theta _i \\subset Q_i$ for all $i \\in I$ , $\\sum _{i \\in I} \\theta _i^2 \\equiv 1$ on $\\mathcal {O}= \\mathbb {R}^d$ , $\\sup _{i \\in I} \\Vert \\mathcal {F}^{-1} \\theta _i \\Vert _{{L}^1} < \\infty $ , $\\sup _{i \\in I} \\Vert \\partial ^{\\alpha } [\\theta _i (T_i \\cdot + b_i)] \\Vert _{\\sup } <\\infty $ for all $\\alpha \\in \\mathbb {N}_0^d$ .",
"Given such a family $(\\theta _i)_{i \\in I}$ , we choose a cube $Q_a \\subset \\mathbb {R}^d$ of side-length $2a$ satisfying $Q \\subset Q_a$ .",
"Finally, for $i \\in I$ and $n \\in \\mathbb {Z}^d$ , define $e_{n,i} : \\mathbb {R}^d \\rightarrow {\\mathbb {C}}$ by $e_{n,i} (\\xi ) := (2a)^{-d/2} \\cdot |\\det T_i|^{-1/2} \\cdot \\chi _{Q_a} (T_i^{-1} (\\xi - b_i)) \\cdot e^{i \\frac{\\pi }{a} n \\cdot T_i^{-1}(\\xi - b_i)} \\text{ for } \\xi \\in \\mathbb {R}^d,$ and set $\\eta _{n,i} := \\mathcal {F}^{-1} (\\theta _i \\cdot e_{n,i}).$ Since the family $(e_{n,i})_{n \\in \\mathbb {Z}^d}$ forms an orthonormal basis of ${L}^2(T_i Q_a + b_i)$ and because of $\\sum _{i \\in I} \\theta _i^2 \\equiv 1$ , it follows (cf.",
"[6]) that the family $(\\eta _{n,i})_{n\\in \\mathbb {Z}^d, i\\in I}$ forms a tight frame for ${L}^2 (\\mathbb {R}^d)$ .",
"Of course, we are not simply interested in (tight) frames for ${L}^2 (\\mathbb {R}^d)$ with a given form of frequency localization—we want to obtain a Banach frame for the decomposition space $\\mathcal {D}( {\\mathcal {Q}}, {L}^{p}, {{\\ell }_u^q})$ .",
"Thus, we define the ${L}^p$ -normalized version $\\eta _{n,i}^{(p)} := |\\det T_i|^{\\frac{1}{2} - \\frac{1}{p}} \\cdot \\eta _{n,i} \\quad \\text{ for } i \\in I \\text{ and } n \\in \\mathbb {Z}^d.$ Then, Borup and Nielsen showed (cf.",
"[6]) that there is a suitable solid BK space $d(\\mathcal {Q}, {\\ell }^p, {\\ell }_u^q)$ such that the coefficient operator ${\\mathcal {C}}: \\mathcal {D}( {\\mathcal {Q}}, {L}^{p}, {{\\ell }_u^q}) \\rightarrow d(\\mathcal {Q}, {\\ell }^p, {\\ell }_u^q), f \\mapsto (\\langle f, \\eta _{n,i}^{(p)}\\rangle )_{n \\in \\mathbb {Z}^d, i\\in I}$ is bounded.",
"As familiar by now, there is also a bounded reconstruction operator ${\\mathcal {R}}: d(\\mathcal {Q}, {\\ell }^p, {\\ell }_u^q) \\rightarrow \\mathcal {D}( {\\mathcal {Q}}, {L}^{p}, {{\\ell }_u^q})$ which satisfies ${\\mathcal {R}}\\circ {\\mathcal {C}}= {\\rm {id}}_{\\mathcal {D}( {\\mathcal {Q}}, {L}^{p}, {{\\ell }_u^q})}$ .",
"Thus, the family $(\\eta _{n,i}^{(p)})_{i \\in I, n\\in \\mathbb {Z}^d}$ forms a Banach frame for $\\mathcal {D}( {\\mathcal {Q}}, {L}^{p}, {{\\ell }_u^q})$ .",
"As noted in Section , the description of the spaces ${{B}^s_{p,q}}$ and ${{F}^s_{p,q}}$ via the $\\varphi $ -transform—or via wavelets—can be viewed (at least in part) as an application of the general theory of coorbit spaces to the affine group which acts on ${L}^2(\\mathbb {R}^d)$ via translations and isotropic dilations.",
"To be precise, this group action yields the homogeneous Besov- and Triebel-Lizorkin spaces ${\\dot{{B}}^s_{p,q}}$ and ${\\dot{{F}}^s_{p,q}}$ , not the inhomogeneous ones.",
"One well known characterization of (homogeneous) Besov spaces shows that these spaces are obtained by placing certain integrability conditions on the (continuous) wavelet transform $W_{\\varphi } f : \\mathbb {R}^d \\times \\mathbb {R}^{\\ast } \\rightarrow {\\mathbb {C}}, (x,a) \\mapsto \\langle f, T_x D_a \\varphi \\rangle $ of a function or distribution $f$ and a certain analyzing window $\\varphi $ .",
"Other applications of the wavelet transform include the characterization of the wave-front set of a distribution using the decay of the transform [63] However, due to the isotropic nature of the dilations $D_a f(x) = a^{-d/2} \\cdot f(a^{-1}x),$ such a single wavelet characterization is only valid in dimension $d=1$ (cf.",
"[7] or [35]), where smoothness is an “undirected property”.",
"Even beyond this specific problem of characterizing the wave-front set, it was noted in recent years that the isotropic, directionless nature of the wavelet transform is a limitation for many applications.",
"To overcome this problem, a vast number of “directional” variants of wavelets were invented: In particular, curvelets[7], [8] and shearlets[54], [14].",
"Among these two systems, shearlets have the special property that there is—as in the case of wavelets—an underlying dilation group through which the family of shearlets can be generated from a single “mother wavelet”, see also Section REF .",
"In view of these two very different dilation groups—the affine group and the shearlet group—it becomes natural to consider the bigger picture: Given any (closed) subgroup $H \\subset \\operatorname{GL}(\\mathbb {R}^d)$ , one can form the group $G = \\mathbb {R}^d \\rtimes H$ of all affine mappings generated by arbitrary translations and all dilations in $H$ .",
"The multiplication on $G$ is given by $(x,h)(y,g) = (x+hy, hg)$ and the Haar measure is ${\\rm d}(x,h) = |\\det h|^{-1} {\\rm d}x \\, {\\rm d}h,$ where ${\\rm d}h$ is the Haar measure of $H$ .",
"The group $G$ from above acts unitarily on ${L}^2 (\\mathbb {R}^d)$ via translations and dilations, i.e., by the quasi-regular representation $\\pi : G \\rightarrow \\mathcal {U}({L}^2(\\mathbb {R}^d)), (x,h) \\mapsto T_x D_h.$ This representation comes with an associated (generalized) wavelet transform $W_\\varphi f : G \\rightarrow {\\mathbb {C}}, (x,h) \\mapsto \\langle f, \\pi (x,h) \\varphi \\rangle \\quad \\text{ for } \\quad f,\\varphi \\in {L}^2 (\\mathbb {R}^d),$ where the (fixed) function $\\varphi $ is called the analyzing window.",
"In the general description of coorbit theory in Section , this was called the voice transform.",
"Given this transform, it is natural to ask which properties of $f$ can be easily read off from $W_\\varphi f$ .",
"As for wavelets (for $d=1$ [63]) or for shearlets (for $d=2$ [54], [47]), it turns out[35] that for large classes of dilation groups, the wave-front set of a (tempered) distribution $f$ can be characterized via the decay of $W_\\varphi f$ .",
"Another important property of a generalized wavelet system (like shearlets) are its approximation theoretic properties.",
"Here, the question is: Which classes of functions can be approximated well by linear combinations of only a few elements of the wavelet system?",
"For “ordinary” wavelets, this question leads to the theory of Besov spaces and their atomic decompositions, as explored by Frazier and Jawerth.",
"For a general dilation group, these approximation theoretic properties are (at least in part) encoded by the associated wavelet type coorbit spaces, which we will now discuss in greater detail.",
"One particular problem which is of interest to us is the following: If a function/signal $f$ can be well approximated by one wavelet system, can it also be well approximated using a different wavelet system?",
"Of course, the answer to this question will depend on the precise nature of the two wavelet systems and on the way in which the statement “$f$ can be well approximated by ...” is made mathematically precise."
],
[
"General wavelet type coorbit spaces",
"As long as $\\pi $ acts irreducibly on ${L}^2$ (and if the representation is (square) integrable), we can apply the general coorbit theory as described in Section to form the coorbit spaces ${{\\mathcal {C}o}}(G, {Y}) = \\left\\lbrace f \\in \\mathcal {R}\\,|\\,W_\\varphi f \\in {Y}\\right\\rbrace ,$ where $\\mathcal {R}= \\mathcal {R}_{Y}$ is a suitable reservoir, which plays the role of $ { {{\\mathcal {S}}}}^{\\prime }({{{\\mathbb {R}}^d}})$ for the usual Besov- or Triebel-Lizorkin spaces.",
"Furthermore, $\\varphi \\in {L}^2(\\mathbb {R}^d)$ has to be a suitable analyzing window.",
"Formally, this means that $\\varphi $ must fulfill $\\varphi \\in \\mathcal {A}_{v_0}$ (cf.",
"equation (REF )) for a so-called control weight $v_0 : G \\rightarrow (0,\\infty )$ .",
"As we will see (cf.",
"Definition REF ), this condition is closely related to the usual “vanishing moments condition” for ordinary wavelets.",
"In this section, instead of the general coorbit spaces ${{\\mathcal {C}o}}({Y}) = {{\\mathcal {C}o}}(G, {Y})$ , we will consider the more restrictive case of the (weighted) mixed Lebesgue space ${Y}= {L}_m^{p,q}(G)$ for $p,q \\in [1,\\infty ]$ and a weight $m : H \\rightarrow (0,\\infty )$ .",
"Precisely, the space ${L}_m^{p,q}(G)$ is the space of all measurable functions $f : G \\rightarrow {\\mathbb {C}}$ for which the norm $\\Vert f\\Vert _{{L}_m^{p,q}} := \\left\\Vert h \\mapsto m(h) \\cdot \\Vert f(x,h)\\Vert _{{L}^p (\\mathbb {R}^d, \\,{\\rm d}x)} \\right\\Vert _{{L}^q (H, \\, {\\rm d}h / |\\det h|)}$ is finite.",
"This normalization—in particular the measure ${\\rm d}h / |\\det h|$ on $H$ —is chosen such that we have ${L}^{p,p}_m (G) = {L}_m^p (G)$ , cf.",
"equation (REF ).",
"Recall from Section that the space ${Y}$ needs to be a solid BF space on $G$ which is invariant under left- and right translations.",
"Clearly, ${Y}= {L}_m^{p,q}(G)$ satisfies all of these properties, except possibly for invariance under left- and right translations.",
"To ensure this, we assume that $m$ is $v$ -moderate for some (measurable, locally bounded, submultiplicative) weight $v : H \\rightarrow (0,\\infty )$ , i.e., we assume $m(xyz) \\le v(x) m(y) v(z) \\qquad \\forall x,y,z \\in H.$ Under these assumptions, it is shown in [44] that there is a control weight $v_0 : G \\rightarrow (0,\\infty )$ for ${Y}= {L}_m^{p,q}(G)$ which is (by abuse of notation) of the form $v_0 (x,h) = v_0 (h)$ and measurable, submultiplicative and locally bounded.",
"In the present setting, coorbit theory can be seen as a theory of nice wavelets and nice signals, cf.",
"[42].",
"Nice wavelets are those belonging to the class $\\mathcal {A}_{v_0}$ of analyzing windows, while a nice signal $f$ (with respect to an analyzing wavelet $\\varphi $ ) is one for which $W_\\varphi f \\in {L}_m^{p,q}$ , i.e., for which $f \\in {{\\mathcal {C}o}}({L}_m^{p,q})$ .",
"Now, coorbit theory—if it is applicable—yields two main properties: A consistency statement: Nice wavelets agree upon nice signals, i.e.",
"if $\\varphi , \\psi \\in \\mathcal {A}_{v_0} \\setminus \\lbrace 0\\rbrace $ (with $v_0$ depending on $p,q,m$ ), then $W_\\varphi f \\in {L}_m^{p,q} \\qquad \\Longleftrightarrow \\qquad W_\\psi f \\in {L}_m^{p,q}.$ We even get a norm-equivalence, so that the coorbit space from equation (REF ) with ${Y}= {L}_m^{p,q}(G)$ is well-defined.",
"An atomic decomposition result: As seen in Theorem REF , we can guarantee atomic decompositions of the form $f = \\sum _{g \\in G_0} \\alpha _g (f) \\cdot \\pi (g) \\psi $ for elements $f \\in {{\\mathcal {C}o}}({L}_m^{p,q})$ and coefficients $(\\alpha _g (f))_{g \\in G_0}$ lying in a suitable sequence space, if $\\psi $ is a better vector (in comparison to just being an analyzing vector), i.e.",
"if $\\psi \\in \\mathcal {B}_{v_0}$ .",
"Despite these pleasant features, the theory of (generalized) wavelet type coorbit spaces raises several questions: For which dilation groups $H$ is the quasi-regular representation from equation (REF ) irreducible and (square)-integrable, so that coorbit theory is applicable in principle?",
"Is coorbit theory applicable, i.e.",
"are there “nice wavelets”?",
"Precisely, do we have $\\mathcal {A}_{v_0} \\ne \\lbrace 0\\rbrace $ and $\\mathcal {B}_{v_0} \\ne \\lbrace 0\\rbrace $ and are there convenient sufficient criteria for a function $\\varphi \\in {L}^2 (\\mathbb {R}^d)$ to belong to $\\mathcal {A}_{v_0}$ or to $\\mathcal {B}_{v_0}$ ?",
"How are the resulting coorbit spaces ${{\\mathcal {C}o}}(\\mathbb {R}^d \\rtimes H, {L}_m^{p,q})$ related to classical function spaces like ${{B}^s_{p,q}}$ of ${{F}^s_{p,q}}$ (or their homogeneous counterparts)?",
"Furthermore, how are coorbit spaces with respect to different dilation groups related to each other?",
"This connects to the question posed above: If a given function/signal can be well approximated using one wavelet system, does the same also hold for a different system?",
"Even for the special case of the shearlet dilation group, these questions are nontrivial and triggered several papers [14], [10], [16], [11].",
"Nevertheless, they also admit satisfactory answers in the present generality: As we will see, each of these questions is linked to the dual action $\\varrho : H \\times \\mathbb {R}^d \\rightarrow \\mathbb {R}^d, (h,\\xi ) \\mapsto h^{-T} \\xi $ of the dilation group $H$ on the frequency space $\\mathbb {R}^d$ .",
"To see the relevance of the dual action, note that the Fourier transform of $W_\\varphi f(\\cdot , h)$ is given by $\\left(\\mathcal {F}[W_\\varphi f (\\cdot , h)]\\right) (\\xi ) = |\\det h|^{1/2} \\cdot \\widehat{f}(\\xi ) \\cdot \\overline{\\widehat{\\psi }(h^T \\xi )}.$ Thus, if $\\widehat{\\psi }$ has support in $U \\subset \\mathbb {R}^d$ , then $W_\\varphi f(\\cdot , h)$ is bandlimited to $h^{-T}U = \\varrho (h, U)$ .",
"The remaining subsections deal with the three questions listed above."
],
[
"Question 1: Irreducibility and square-integrability of $\\pi $",
"As shown in [40], [39], [4], the quasi-regular representation $\\pi $ is irreducible and square-integrable if and only if the following two properties are satisfied: There is some $\\xi _0 \\in \\mathbb {R}^d \\setminus \\lbrace 0\\rbrace $ such that the orbit $\\mathcal {O}:= H^T \\xi _0 = \\left\\lbrace h^T \\xi _0 \\,|\\,h \\in H \\right\\rbrace $ is open and of full measure (i.e.",
"$\\mathcal {O}^c$ is a Lebesgue null set).",
"The stabilizer $H_{\\xi _0} := \\left\\lbrace h \\in H \\,|\\,h^T \\xi _0 = \\xi _0 \\right\\rbrace $ is compact.",
"In this case, we call $\\mathcal {O}$ the (open) dual orbit of the dilation group $H$ , whereas the null-set $\\mathcal {O}^c$ is called the blind spot of $H$ .",
"Finally, a dilation group $H$ fulfilling the two properties above is called admissible.",
"In the following, we fix $\\xi _0 \\in \\mathcal {O}$ .",
"To give the reader an idea of the richness of admissible dilation groups, we mention the following admissible dilation groups in dimension $d=2$ : The diagonal group $H_1 := \\left\\lbrace {\\rm diag}(a,b) \\,|\\,a,b \\in \\mathbb {R}\\setminus \\lbrace 0\\rbrace \\right\\rbrace $ with dual orbit $\\mathcal {O}= \\left( \\mathbb {R}\\setminus \\lbrace 0\\rbrace \\right)^2$ .",
"The similitude group $H_2 := (0,\\infty ) \\cdot {\\mathrm {SO}} (\\mathbb {R}^2)$ , with $\\mathcal {O}= \\mathbb {R}^2 \\setminus \\lbrace 0\\rbrace $ .",
"The family of shearlet type groups $H_3^{(c)} := \\left\\lbrace \\varepsilon \\cdot \\left( \\begin{matrix}a & b \\\\ 0 & a^c \\end{matrix} \\right) \\,|\\,a \\in (0,\\infty ), b \\in \\mathbb {R}, \\varepsilon \\in \\lbrace 1, -1\\rbrace \\right\\rbrace .$ Here, the anisotropy parameter $c \\in \\mathbb {R}$ can be chosen arbitrarily.",
"Regardless of this choice, the dual orbit is always $\\mathcal {O}= (\\mathbb {R}\\setminus \\lbrace 0\\rbrace ) \\times \\mathbb {R}$ ."
],
[
"Question 2: Existence of “nice” wavelets",
"Here, we are given an admissible dilation group $H$ and we are interested in conditions which guarantee $\\mathcal {A}_{v_0} \\ne \\left\\lbrace 0 \\right\\rbrace $ or $\\mathcal {B}_{v_0} \\ne \\lbrace 0\\rbrace $ , where $v_0 : H \\rightarrow (0,\\infty )$ (interpreted by abuse of notation as a weight on $G = \\mathbb {R}^d \\rtimes H$ ) is a (locally bounded) control weight for ${Y}= {L}_m^{p,q}$ .",
"As we saw above, such a control weight always exists under our general assumptions on the weight $m : H \\rightarrow (0,\\infty )$ .",
"In the present setting, “nice wavelets” and “better wavelets” exist in abundance: Theorem 17 (cf.",
"[44]) Let $v_0 : H \\rightarrow (0,\\infty )$ be measurable and locally bounded.",
"Then, every function $\\varphi \\in \\mathcal {F}^{-1}(C_c^\\infty (\\mathcal {O})) \\subset {{{\\mathcal {S}}}({\\mathbb {R}}^d)}$ satisfies $\\varphi \\in \\mathcal {B}_{v_0} \\subset \\mathcal {A}_{v_0}$ and the map $\\varrho : C_c^\\infty (\\mathcal {O}) \\rightarrow {L}_{v_0}^{1}(G), g \\mapsto W_\\varphi (\\mathcal {F}^{-1} g)$ is well-defined and continuous.",
"This theorem, however, does not yield existence of compactly supported “nice wavelets”.",
"In the case of “traditional” wavelets, it is well-known that a certain amount of “vanishing moments” (i.e.",
"$\\partial ^\\alpha \\widehat{\\varphi }(0) = 0$ for all $|\\alpha |\\le r$ ) makes a wavelet “nice”.",
"This generalizes to the present setting: Definition 18 ([43]) Let $r \\in \\mathbb {N}$ .",
"We say that $\\varphi \\in {L}^1(\\mathbb {R}^d)$ has vanishing moments on $\\mathcal {O}^c$ of order $r$ if $\\widehat{\\varphi } \\in C^r (\\mathbb {R}^d)$ and if $(\\partial ^\\alpha \\widehat{\\varphi }) |_{\\mathcal {O}^c} \\equiv 0 \\qquad \\text{ for } |\\alpha | < r.$ As shown in [43], given the weight $v_0 : H \\rightarrow (0,\\infty )$ , one can explicitly compute a number $\\ell = \\ell (H, v_0) \\in \\mathbb {N}$ such that every function $\\psi \\in {L}^1 (\\mathbb {R}^d)$ with $\\Vert \\widehat{\\psi }\\Vert _{\\ell , \\ell }< \\infty $ and with vanishing moments of order $\\ell $ on $\\mathcal {O}^c$ satisfies $\\psi \\in \\mathcal {B}_{v_0} \\subset \\mathcal {A}_{v_0}$ .",
"Here, we employed the usual Schwartz-type norm $\\Vert f\\Vert _{\\ell ,\\ell } := \\max _{\\alpha \\in \\mathbb {N}_0^d, |\\alpha | \\le \\ell } \\,\\, \\sup _{x \\in \\mathbb {R}^d} (1+|x|)^\\ell \\cdot |\\partial ^{\\alpha } f(x)| \\in [0,\\infty ].$ Finally, in the setting of ordinary wavelets, one can obtain a wavelet with enough vanishing moments by taking derivatives of a given smooth function of sufficient decay.",
"In the present general setting, [41] yield a similar “algorithm” for obtaining functions with suitably many vanishing moments on $\\mathcal {O}^c$ .",
"For the sake of brevity, we refrain from giving more details and instead refer the interested reader to [41], [43]."
],
[
"Question 3: Relation of generalized wavelet coorbit spaces to other spaces",
"Here, we are interested in the relation of the coorbit space ${{\\mathcal {C}o}}({L}_m^{p,q})$ to the classical Besov- or Triebel-Lizorkin spaces, but also in the relation between ${{\\mathcal {C}o}}({L}_{m_1}^{p_1,q_1}, \\mathbb {R}^d \\rtimes H_1)$ and ${{\\mathcal {C}o}}({L}_{m_2}^{p_2,q_2}, \\mathbb {R}^d \\rtimes H_2)$ for two different admissible dilation groups $H_1, H_2 \\le \\operatorname{GL}(\\mathbb {R}^d)$ .",
"In view of the embedding theory for decomposition spaces (cf.",
"Subsection REF ), this question would be solved (at least to a significant extent) if we knew that the coorbit space ${{\\mathcal {C}o}}({L}_m^{p,q})$ is (canonically isomorphic to) a decomposition space.",
"But the main result of [44] is precisely such an isomorphism.",
"In more detail, [44] shows ${{\\mathcal {C}o}}({L}_m^{p,q}(\\mathbb {R}^d \\rtimes H)) \\cong \\mathcal {D}( {\\mathcal {Q}_H}, {L}^{p}, {{\\ell }_u^q})$ for a so-called induced covering $\\mathcal {Q}_H$ and a so-called decomposition weight $u$ .",
"Below, we provide a concrete example indicating how this isomorphism and the above embedding results can be used to obtain novel embedding results for shearlet coorbit spaces.",
"In the above isomorphism, the induced covering $\\mathcal {Q}_H$ of the dual orbit $\\mathcal {O}$ is given by $\\mathcal {Q}_H = (h_i^{-T}Q)_{i \\in I}$ for certain $h_i \\in H$ and a suitable $Q \\subset \\mathcal {O}$ .",
"Its precise construction is described in the following paragraph.",
"But once $\\mathcal {Q}= (h_i^{-T}Q)_{i \\in I}$ is known, the decomposition weight $u = (u_i)_{i \\in I}$ from above (cf.",
"[44]) is given by $u_i = |\\det h_i|^{\\frac{1}{2} - \\frac{1}{q}} \\cdot m(h_i) \\qquad \\forall i \\in I.$ Let us reconsider the induced covering $\\mathcal {Q}_H$ .",
"The family $(h_i)_{i \\in I}$ has to be well spread in $H$ , i.e.",
"there are compact unit neighborhoods $K_1, K_2 \\subset H$ such that $H = \\bigcup _{i \\in I} h_i K_1 \\qquad \\text{ and } \\qquad h_i K_2 \\cap h_j K_2 = \\varnothing \\text{ for } i,j \\in I \\text{ with } i\\ne j.$ Finally, the set $Q \\subset \\mathcal {O}$ is an arbitrary open, bounded set such that the closure $\\overline{Q} \\subset \\mathbb {R}^d$ is contained in $\\mathcal {O}$ and such that there is a smaller open set $P \\subset \\mathbb {R}^d$ with $\\overline{P} \\subset Q$ and $\\mathcal {O}= \\bigcup _{i \\in I}h_i^{-T}P$ .",
"As shown in [44], such sets $P, Q$ always exist if $(h_i)_{i \\in I}$ is well-spread in $H$ .",
"The same theorem also shows that the resulting covering $\\mathcal {Q}_H = (h_i^{-T} Q)_{i \\in I}$ is an (almost) structured admissible covering of $\\mathcal {O}$ .",
"In particular, there is a BAPU $(\\varphi _i)_{i \\in I}$ subordinate to $\\mathcal {Q}_H$ , cf.",
"Theorem REF .",
"Finally, [44] show that the decomposition weight from equation (REF ) is $\\mathcal {Q}_H$ -moderate.",
"This implies (cf.",
"Subsection REF ) that the decomposition space $\\mathcal {D}( {\\mathcal {Q}_H}, {L}^{p}, {{\\ell }_u^q})$ on the right-hand side of equation (REF ) is well-defined.",
"As an illustration of the concept of an induced covering, we consider the shearlet type group $H_3^{(c)}$ from equation (REF ).",
"A picture of (a part of) the associated induced covering for different values of the anisotropy parameter $c \\in \\mathbb {R}$ is shown in Fig.",
"REF .",
"Figure: The figure shows (a part of) the induced covering for the group H 3 (c) H_3^{(c)} for c=-1 2c= - \\frac{1}{2}, c=0c=0 and c=1 2c = \\frac{1}{2}.",
"These choices show the qualitatively different behaviour of the covering for different values of cc.",
"For c 1 <c 2 c_1 < c_2, the covering 𝒮 (c 1 ) \\mathcal {S}^{(c_1)} is “larger/coarser” near the yy-axis, whereas the covering 𝒮 (c 2 ) \\mathcal {S}^{(c_2)} is “larger/coarser” away from the yy-axis.Note: The images are taken from Figure 1.For the explicit description of the isomorphism from equation (REF ), we note that it is a (relatively) easy consequence of equation (REF ) that $\\iota : Z(\\mathcal {O}) = \\mathcal {F}(C_c^\\infty (\\mathcal {O})) \\rightarrow \\mathcal {H}^{1}_{v_0}, f \\mapsto \\overline{f},$ with $\\mathcal {H}_{v_0}^{1}$ as in equation (REF ) is well-defined, antilinear and continuous.",
"By duality, this implies that $\\iota ^T : \\left( \\mathcal {H}_{v_0}^1 \\right)^\\angle \\rightarrow Z^{\\prime }(\\mathcal {O}), \\theta \\mapsto \\theta \\circ \\iota $ is well-defined, continuous and linear.",
"Recall from Section that ${{\\mathcal {C}o}}({L}_m^{p,q})$ is a subspace of the reservoir $\\mathcal {R}= \\left( \\mathcal {H}_{v_0}^1 \\right)^\\angle $ for a certain control weight $v_0$ .",
"Thus, $\\iota ^T f \\in Z^{\\prime }(\\mathcal {O})$ is well-defined for every $f \\in {{\\mathcal {C}o}}({L}_m^{p,q})$ .",
"The claim of equation (REF ) is precisely that $\\iota ^T : {{\\mathcal {C}o}}({L}_m^{p,q}) \\rightarrow \\mathcal {D}( {\\mathcal {Q}}, {L}^{p}, {{\\ell }_u^q})$ is an isomorphism of Banach spaces.",
"A proof of this fact can be found in [44].",
"This representation of ${{\\mathcal {C}o}}({L}_m^{p,q}(G))$ as a decomposition space has several important consequences, both mathematically and conceptually: As we saw above, the coorbit space ${{\\mathcal {C}o}}({L}_m^{p,q}(G))$ with its original definition is heavily tied to the group $G = \\mathbb {R}^d \\times H$ and thus to the dilation group $H$ .",
"In particular, ${{\\mathcal {C}o}}({L}_m^{p,q}(G))$ is a subspace of the reservoir $\\mathcal {R}= (\\mathcal {H}_{v_0}^1 (G))^\\angle $ .",
"This makes it difficult to consider an element $f \\in {{\\mathcal {C}o}}({L}_{m_1}^{p_1,q_1}(\\mathbb {R}^d \\rtimes H_1))$ as an element of another coorbit space ${{\\mathcal {C}o}}({L}_{m_2}^{p_2,q_2}(\\mathbb {R}^d \\rtimes H_2))$ , or as an element of more classical function spaces like ${{B}^s_{p,q}}$ .",
"In contrast, as we saw in Section , it is (at least in principle) possible to compare decomposition spaces $\\mathcal {D}( {\\mathcal {Q}}, {L}^{p_1}, {{\\ell }_u^{q_1}})$ and $\\mathcal {D}( {\\mathcal {P}}, {L}^{p_2}, {{\\ell }_v^{q_2}})$ which are defined using two different coverings $\\mathcal {Q}, \\mathcal {P}$ of the sets $\\mathcal {O}, \\mathcal {O}^{\\prime } \\subset \\mathbb {R}^d$ .",
"Thus, using the decomposition space view, it becomes possible to compare wavelet coorbit spaces defined by different dilation groups.",
"Even ignoring the issue of the different reservoirs (e.g.",
"by restricting to Schwartz functions), it is not at all obvious how the decay or integrability condition $W_\\varphi f \\in {L}_{m_1}^{p_1,q_1}(\\mathbb {R}^d \\rtimes H_1)$ relates to another decay condition $W_\\varphi f \\in {L}_{m_2}^{p_2,q_2}(\\mathbb {R}^d \\rtimes H_2)$ , even if the same analyzing window is used in both cases.",
"One of the reasons is that it is difficult to compare the two actions of the dilation groups on $\\varphi $ , as well as the two distinct Haar measures.",
"In comparison, the decomposition space point of view translates these two elusive properties into (more or less) transparent quantities, namely The induced covering $\\mathcal {Q}_H = (h_i^{-T} Q)_{i \\in I}$ for some well-spread family $(h_i)_{i \\in I}$ in $H$ and a suitable set $Q \\subset \\mathcal {O}$ , The decomposition weight $u_i = |\\det h_i|^{\\frac{1}{2} - \\frac{1}{q}} \\cdot m(h_i)$ .",
"Using the methods from Subsection REF , it is then (comparatively) easy to establish embeddings $\\mathcal {D}( {\\mathcal {Q}_{H_1}}, {L}^{p_1}, {{\\ell }_{u_1}^{q_1}}) \\hookrightarrow \\mathcal {D}( {\\mathcal {Q}_{H_2}}, {L}^{p_2}, {{\\ell }_{u_2}^{q_2}})$ between the associated decomposition spaces and thus of the two coorbit spaces ${{\\mathcal {C}o}}({L}_{m_1}^{p_1, q_1}(\\mathbb {R}^d \\rtimes H_1))$ and ${{\\mathcal {C}o}}({L}_{m_2}^{p_2, q_2}(\\mathbb {R}^d \\rtimes H_2))$ .",
"Similarly, one can use the methods from Subsection REF to establish embeddings between generalized wavelet coorbit spaces and classical smoothness spaces like Sobolev- and Besov spaces.",
"Conceptually, all these considerations show that the approximation theoretic properties of the wavelet system generated by a dilation group $H$ are completely determined by the way in which (the dual action of) $H$ covers/partitions the frequencies.",
"Theorem REF below is an example of results that can be obtained by combining the embedding results from Subsection REF with the isomorphism ${{\\mathcal {C}o}}({L}_m^{p,q}(\\mathbb {R}^d \\rtimes H)) \\cong \\mathcal {D}( {\\mathcal {Q}_H}, {L}^{p}, {{\\ell }_u^q}).$ Precisely, we consider embeddings between shearlet coorbit spaces and inhomogeneous Besov spaces.",
"For the sake of brevity, we only consider embeddings of the shearlet coorbit space into inhomogeneous Besov spaces.",
"Results for the reverse direction are also available (cf.",
"[70]), but are omitted here.",
"We only consider the case $c \\in (0,1]$ .",
"This ensures that the induced covering $\\mathcal {S}^{(c)} = \\mathcal {Q}_{H_3^{(c)}}$ is almost subordinate to the inhomogeneous dyadic covering.",
"See [70] for a formal proof and Figure REF for a graphical illustration.",
"Note, however, that $\\mathcal {S}^{(c)}$ is not relatively moderate with respect to the inhomogeneous dyadic covering.",
"This limits sharpness of our results to a certain range of $p_2$ .",
"Theorem 19 ([70]) Let $c \\in (0,1]$ , $p_1, p_2, q_1, q_2 \\in [1,\\infty ]$ and $\\alpha , \\beta , \\gamma \\in \\mathbb {R}$ .",
"Set $p_2^{\\triangledown } := \\min \\lbrace p_2, p_2 ^{\\prime }\\rbrace $ , define the weight $u^{(\\alpha , \\beta )} : H_3^{(c)} \\rightarrow (0,\\infty ), h \\mapsto \\Vert h^{-1}\\Vert ^{\\alpha } \\cdot |\\det h|^{\\beta }$ and set (1) := 1+cc ( 1p1 - 1p2 - 1q1 + 12 + ), (1) := - (1+c) ( 1p1 - 1p2 - 1q1 + 12 + ) + (c-1) ( 1p2 - 1q1 )+.",
"If $p_1 \\le p_2$ as well as ${\\left\\lbrace \\begin{array}{ll}\\gamma \\le \\alpha + \\gamma ^{(1)}, &\\text{if } q_1 \\le q_2, \\\\\\gamma < \\alpha + \\gamma ^{(1)}, &\\text{if } q_1 > q_2\\end{array}\\right.",
"}\\quad \\text{ and } \\quad {\\left\\lbrace \\begin{array}{ll}\\max \\left\\lbrace \\frac{1}{p_2^\\triangledown } - \\frac{1}{q_1}, \\, \\alpha \\right\\rbrace < \\alpha ^{(1)}, & \\text{if } q_1 > p_2^{\\triangledown }, \\\\\\max \\left\\lbrace 0,\\alpha \\right\\rbrace \\le \\alpha ^{(1)}, & \\text{if } q_1 \\le p_2^{\\triangledown }\\end{array}\\right.",
"}$ hold, then ${{\\mathcal {C}o}}({L}_{u^{(\\alpha , \\beta )}}^{p_1, q_1}(\\mathbb {R}^2 \\rtimes H_3^{(c)})) \\hookrightarrow {B}^{\\gamma }_{p_2, q_2} (\\mathbb {R}^2).$ A necessary condition for existence of this embedding is obtained by replacing $p_2^{\\triangledown }$ by $p_2$ everywhere (also in the definition of $\\gamma ^{(1)}$ ).",
"Remark Existence of the embedding (REF ) has to be interpreted suitably.",
"Precisely, (REF ) means that there is a bounded linear map $\\iota : {{\\mathcal {C}o}}({L}_{u^{(\\alpha , \\beta )}}^{p_1, q_1}(\\mathbb {R}^2 \\rtimes H_3^{(c)})) \\rightarrow {B}_{p_2, q_2}^{\\gamma }(\\mathbb {R}^2)$ which satisfies $\\iota f = f$ for all $f \\in {L}^2 (\\mathbb {R}^2) \\cap {{\\mathcal {C}o}}({L}_{u^{(\\alpha , \\beta )}}^{p_1, q_1}(\\mathbb {R}^2 \\rtimes H_3^{(c)}))$ .",
"The preceding theorem is superficially similar to [16].",
"But the two results are very different, since Dahlke et al.",
"consider embeddings of the strict subspace $\\mathcal {SCC}_{p,r} \\lneq {{\\mathcal {C}o}}({L}_{u^{(0,-2r/3)}}^{p,p}(\\mathbb {R}^2 \\rtimes H_3^{(1/2)}))$ into a sum of homogeneous Besov spaces $\\dot{{B}}_{p,p}^{\\sigma _1}(\\mathbb {R}^2) + \\dot{{B}}_{p,p}^{\\sigma _2}(\\mathbb {R}^2)$ for certain $\\sigma _1, \\sigma _2$ .",
"In contrast, the preceding theorem investigates embeddings of the whole shearlet coorbit space into a single, inhomogeneous Besov space.",
"The preceding theorem achieves a complete characterization of the embedding (REF ) for $p_2 \\in [1,2]$ , since we have $p_2^{\\triangledown } = p_2$ in this range.",
"As a conclusion, we remark that the embedding results and the isomorphism between generalized wavelet coorbit spaces and decomposition spaces can also be used to derive embeddings between the coorbit spaces ${{\\mathcal {C}o}}({L}_m^{p,q}(\\mathbb {R}^2 \\rtimes H_3^{(c)}))$ for different values of $c$ , see [70].",
"They can also be used to derive (non)boundedness of certain operators—e.g.",
"dilation operators—acting on coorbit spaces.",
"For example, in [70], the set of matrices which act boundedly by dilation simultaneously on all coorbit spaces of the shearlet type group $H_3^{(c)}$ (for a fixed $c \\in (0,1)$ ) is characterized completely."
],
[
"Acknowledgments",
"Both authors want to thank HIM—the Hausdorff Institute of Mathematics—where we both spent some time during the preparation of this manuscript.",
"FV was funded by the Excellence Initiative of the German federal and state governments, and by the German Research Foundation (DFG), under the contract FU 402/5-1."
]
] | 1606.04924 |
[
[
"Zero-Shot Hashing via Transferring Supervised Knowledge"
],
[
"Abstract Hashing has shown its efficiency and effectiveness in facilitating large-scale multimedia applications.",
"Supervised knowledge e.g.",
"semantic labels or pair-wise relationship) associated to data is capable of significantly improving the quality of hash codes and hash functions.",
"However, confronted with the rapid growth of newly-emerging concepts and multimedia data on the Web, existing supervised hashing approaches may easily suffer from the scarcity and validity of supervised information due to the expensive cost of manual labelling.",
"In this paper, we propose a novel hashing scheme, termed \\emph{zero-shot hashing} (ZSH), which compresses images of \"unseen\" categories to binary codes with hash functions learned from limited training data of \"seen\" categories.",
"Specifically, we project independent data labels i.e.",
"0/1-form label vectors) into semantic embedding space, where semantic relationships among all the labels can be precisely characterized and thus seen supervised knowledge can be transferred to unseen classes.",
"Moreover, in order to cope with the semantic shift problem, we rotate the embedded space to more suitably align the embedded semantics with the low-level visual feature space, thereby alleviating the influence of semantic gap.",
"In the meantime, to exert positive effects on learning high-quality hash functions, we further propose to preserve local structural property and discrete nature in binary codes.",
"Besides, we develop an efficient alternating algorithm to solve the ZSH model.",
"Extensive experiments conducted on various real-life datasets show the superior zero-shot image retrieval performance of ZSH as compared to several state-of-the-art hashing methods."
],
[
"Introduction",
"Hashing [30] is a powerful indexing technique for enabling efficient retrieval on large-scale multimedia data, such as image [25], [24], [38] and video [1].",
"Specifically, in order to achieve shorter response time and less computational cost, hashing encodes high-dimensional data into compact binary codes (i.e., 0 or 1) substantially.",
"In this way, data can be compactly stored and Hamming distances can be efficiently calculated with bit-wise XOR operations.",
"Because of its impressive capacity of dealing with “curse of dimensionality” problem, hashing has been extensively employed in various real-world applications, ranging from multimedia indexing [8], [40], [37], [36] to multimedia event detection [23].",
"Figure: An illustration of newly-emerging concepts and images unseen to the existing learning systems.There are mainly two branches of hashing, i.e., data-independent hashing and data-dependent hashing.",
"For data-independent hashing, such as Locality Sensitive Hashing [6], no prior knowledge (e.g., supervised information) about data is available, and hash functions are randomly generated.",
"Nonetheless, huge storage and computational overhead might be cost since more than $1,000$ bits are usually required to achieve acceptable performance.",
"To address this problem, research directions turn to data-dependent hashing, which leverages information inside data itself.",
"Roughly, data-dependant hashing can be divided into two categories: unsupervised hashing (e.g., Iterative Quantization [7] and Sparse Mutli-Modal Hashing [33]), and (semi-)supervised hashing (e.g., Supervised Hashing with Kernels [17], Supervised Discrete Hashing [25], Discrete Graph Hahsing [16] and Semi-Supervised Hashing [29]).",
"In general, supervised hashing usually achieves better performance than unsupervised ones because supervised information (e.g., semantic labels and/or pair-wise data relationship) can help to better explore intrinsic data property, thereby generating superior hash codes and hash functions.",
"Along with the explosive growth of Web data, traditional supervised hashing methods have been facing an enormous challenge, i.e., the generation of reliable supervised knowledge cannot catch up with the rapid increasing speed of newly-emerging semantic concepts and multimedia data.",
"In other words, due to the expensive cost of manual labelling (time-consuming and labor-intensive), sufficient labelled training data is usually not timely available for learning new hash functions that can accurately encodes data of new concepts.",
"As illustrated in Figure REF , within the “seen” zone, where images are attached with known categories, existing supervised hashing algorithms may perform well because they are fed with correct guidance.",
"However, outside the seen area, supervised hashing algorithms may easily fail to generalize to data of new categories that they never observe, e.g., $segway$ , a two-wheeled, self-balancing, battery-powered electric vehicle.",
"Moreover, most of current approaches use supervised information in the form of either 0/1 semantic labels or pair-wise data relationship for guiding the learning process, which implies that precious correlation among label semantics are inevitably ignored.",
"One straightforward consequence of the semantic independency is that each category can neither learn from other relevant categories nor distribute its own supervised knowledge to other seen classes and/or even those unseen ones.",
"The aforementioned disadvantages motivate us to consider whether we can encode images of “unseen” categories into binary codes with hash functions learned from limited training samples of“seen” categories?",
"The key challenge of achieving this goal is how to set up a tunnel to transfer supervised knowledge between “seen” and “unseen” categories.",
"In recent years, zero-shot learning (ZSL) [21], [27], [2], [20] has been widely recognized as a way to deal with this problem.",
"The ZSL paradigm aims to learn a general mapping from the feature space to a high-level semantic space, which helps avoid rebuilding models for unseen categories with extra manually labelled data.",
"ZSL is mostly achieved by using class-attribute descriptors to bridge the semantic gaps between low-level features and high-level semantics, where new categories are thus learned using only the relationship between attributes and categories.",
"However, most of existing attribute based ZSL methods still suffer from: (1) erroneous guidance derived from imprecise or incomplete human-labelled attributes [10], which is usually due to the lack of expertise or mislabeling by annotators, etc.",
"; (2) diminishing of discrimination for pre-defined attributes when confronted with dataset shift [14], [22].",
"Recently, mining other auxiliary datasets has been shown to be helpful to tackle the zero-shot learning problem.",
"For instance, with a huge corpus such as Wikipedia, one can obtain word embeddings that capture distributional similarity in the text corpus [28], such that similar words can be located in similar place.",
"During the learning phase, visual modality can be grounded by the word vectors, and such knowledge can thus be transferred into the learned model.",
"Inspired by this, many approaches choose to utilize auxiliary modalities to help address the zero-shot tasks.",
"Socher et al.",
"[27] uses word embedding as supervision in order to detect novel categories and perform classification accordingly.",
"Frome et al.",
"[5] adopts a similar manner, which connects raw features and word embedding space using the dot-product similarity and hinge rank loss.",
"In the hashing domain, however, the zero-shot problem has rarely been studied.",
"As previously analyzed, with the newly-emerging concepts and multimedia data, we are in urgent demand of a reliable and flexible hash function that can be adopted to hash images of unseen categories.",
"In this work, we propose a novel hashing scheme, termed zero-shot approach (ZSH).",
"Inspired by the superior capacity of the word embedding for capturing the semantic correlations among concepts, we map mutually independent labels into a semantic-rich space, where supervised knowledge of both seen and unseen labels can be completely shared.",
"This strategy helps to encode images of unseen categories without any assistance of visual observation in those unknown classes.",
"Besides, even though we cannot retrieve images of exactly the same category, semantically related objects can be returned.",
"Moreover, we recognize the problem of semantic shift caused by off-the-shelf embedding.",
"The embedded space is then rotated to make the hash functions more generalized to images of unseen categories.",
"To further improve the quality of hash functions, we also preserve local structural property and discrete nature in binary codes.",
"We summarize our main contributions as below: We address the problem of employing training data of seen categories to learn reliable hash functions for transforming images of unseen categories into binary codes.",
"We propose a novel zero-shot hashing, which bridges gaps between originally independent labels through a semantic embedding space.",
"To the best of our knowledge, this is one of the first works that study the problem of hashing data from newly-emerging concepts with limited seen supervised knowledge.",
"Extensive experiments on various multimedia data collections validate the efficacy of our proposed ZSH.",
"We devise an effective strategy for transferring available supervised knowledge from seen classes to unseen classes.",
"In particular, we transform labels into a word embedding space, where semantic correlations among labels can be quantitatively measured and captured.",
"In this way, unseen labels can leverage the well-established mapping from its semantically close seen categories.",
"For instance, segway may learn from bicycle and automobile.",
"Since the initial semantic embedding is from an off-the-shelf word embedding space, which may bring in severe semantic shift between categories and the original visual feature.",
"To alleviate the potential influence, we propose to further rotate the embedding space to better fit the underlying feature characteristics, thereby narrowing down the semantic gap effectively.",
"In order to generate more reliable hash functions, we propose to improve the intermediate binary codes of training data by exploring underlying data properties.",
"Concretely, we impose discrete constraints on binary codes during the code learning process as well as preserve data local structure, i.e., if two datums share similar representations in the original space, they are supposed to be close to each other in the learned Hamming space.",
"The rest of this paper is organized as follows.",
"In Section 2, we briefly review some related work on hashing and zero-shot learning.",
"In Section 3, we will elaborate our approach with details, together with our optimization method and an analysis of the algorithm.",
"With extensive experiments, various results on various different datasets will be reported in Section 4, followed by the conclusion of this work in Section 5.",
"Figure: An illustration of the overall architecture of the proposed zero-shot hashing framework."
],
[
"Related work",
"In this section, we aim to clarify the relationship between our work and other researches, due to the constraints of space, we cannot completely elaborate every detail of previous literature."
],
[
"Zero-Shot Learning",
"Learning with no data, i.e., zero-shot learning has been proved to be an efficient approach to tackle the increasing difficulty posed by insufficient training examples.",
"Many approaches have been proposed to solve this problem by using an intermediate layer to represent an image.",
"Specifically, with visual attributes or other semantic abundant descriptors, a novel image can thus be defined as the relationship between category and intermediate representation.",
"In the work [4] by Farhadi, he leverages attributes as a way to classify unseen objects by describing them with attributes.",
"The work [15] by Larochelle named Zero-data Learning of New Tasks has also proven to be useful when predict categories that are not shown in the training dataset.",
"Recently, learning novel images with auxiliary datasets (e.g., leveraging textual relationship in a large corpus) has been shown to be powerful at doing zero-shot tasks.",
"Learning the correlations between concepts, the label of a novel example omitted from training set can be reasonably inferred.",
"Renown works include Socher's work [27] Zero-shot Learning Through Cross-Modal Transfer, which uses label embedding to detect unseen classes and makes semantically reasonable deduction.",
"DEVISE [5] also uses the same scheme as [27], but with a different language modal and a different loss function to connect two modalities.",
"However, all above methods are limited to classification or prediction scenario.",
"To our best knowledge, we are the first one to handle the zero-shot retrieving problem, i.e., hash novel images that were not observed.",
"By adopting a natural language model [9] pre-trained with a large corpus from Wikipedia, we precisely capture the correlations between different words, and thus hash unseen images into correct Hamming space."
],
[
"Hashing",
"This subsection overviews fast search with binary codes using hashing technique.",
"Similarity search is a challenge of pursuing data points of smallest distance in a large scale database.",
"The easiest hashing scheme is dubbed Local Sensitive Hashing [6], which designs hashing function with no prior knowledge of the data distribution.",
"However, such hashing methods require significantly large code length to achieve an acceptable performance, generating large overheads in a database.",
"To address this problem, learning to hash comes as a trend.",
"Unsupervised hashing methods mine the statistic distributional information in the database, generating an optimal hashing function to preserve the similarity in the original space.",
"Classical algorithms such as Spectral Hashing (SH) [31], solves binary codes to preserve the Euclidean distance in the database; Inductive Manifold Hashing (IMH) [26], adopts manifold learning techniques to better model the intrinsic structure embedded in the feature space; Iterative Quantization [7], focuses on minimizing quantization error during unsupervised training.",
"Considering a real-world database is commonly described by multiple modalities, such as visual features (e.g., Caffe [11]) or textual information (e.g., image captions, lyrics), Sparse Multi-Modal Hashing [33] utilizes information of at least two different resources to achieve promising performance.",
"Since the unsupervised way is guided with little human-level knowledge, supervised hashing have been proposed to use supervision information to learn binary codes.",
"Hashing techniques in this category have been emerging continuously in recent years, representative methods include Kernel Supervise Hashing (KSH) [17], Minimal Loss Hashing (MLH) [19], Supervise Discrete Hashing (SDH) [25], Latent Factor Hashing (LFH) [39] as well as the recently proposed Column Sampling Based Discrete Supervised Hashing (COSDISH) [12], etc.",
"Using supervision information, these hashing schemes perform better than unsupervised ones.",
"Recently, with the rising of deep learning, image hashing using large convolutional neural network has also be shown to be effective in hashing domain [35].",
"By using hidden layers to represent images as feature vectors that are optimal for binary codes generation, hashing performance can be augmented greatly.",
"Admittedly, hashing algorithms have successfully tackled the “curse of dimensionality” in terms of fast search, however, what if we want to achieve data-dependent performance while no training example is provided?",
"All above hashing methods fail to generalize to “unseen” categories, limiting in the “seen” area where every category correspond to at least one training image.",
"Besides, as the database changes everyday, re-training hashing function frequently can be expensive, further prevents their practical usage in large dynamic real-world databases.",
"Based on tabove analysis, a hashing method that can perform well on unseen data draws a strong need, thus the orientation of zero-shot hashing is quite obvious."
],
[
"Zero-Shot Hashing",
"In this section, we elaborate our proposed zero-shot hashing (ZSH).",
"We firstly present a formal definition of hashing in zero-shot scenario, and then depict the details of ZSH, including a brief introduction of overall framework, supervision transfer, semantic alignment as well as hashing model.",
"Finally, we introduce the optimization process and algorithm analysis."
],
[
"Problem Definition",
"Suppose we are given $n$ training images ${X} = \\left[ {x_1,x_2, \\ldots ,x_n} \\right] \\in {\\mathbb {R}^{d \\times n}}$ labeled with a seen visual concept set $\\mathcal {C}$ , where $d$ is the dimensionality of visual feature space.",
"Denote $Y = \\left[ {{y_1},{y_2}, \\ldots ,{y_n}} \\right] \\in {\\lbrace 0,1\\rbrace ^{c \\times n}}$ is the binary label matrix, where $y_i\\in \\lbrace 0,1\\rbrace ^{c\\times 1}$ is the label vector of the $i$ -th sample $x_i$ and $c$ is the number of seen classes in $\\mathcal {C}$ .",
"Different from conventional supervised hashing scenario, where both testing data and training data are associated with the same concept set, i.e., $\\mathcal {C}$ , we intend to cope with the situation where testing data and training data share no common concepts.",
"In other words, testing data (denoted as $X^{(u)}$ ) belongs to an “unseen” category set $\\mathcal {C}^{(u)}$ , i.e., $\\mathcal {C}^{(u)}\\cap \\mathcal {C}=\\emptyset $ .",
"Using only the training images $X$ where no training samples of the “unseen” categories in $\\mathcal {C}^{(u)}$ are available, our goal is to learn a hash function $f:\\mathbb {R}^d\\mapsto \\lbrace -1,1\\rbrace ^{l\\times 1}$ , which can map images belonging to both $\\mathcal {C}^{(u)}$ and $\\mathcal {C}$ from original visual feature space to $l$ -bit binary codes.",
"The learned hash function $f$ not only guarantees that the binary codes of semantically relevant objects have short Hamming distances, but also generalizes well to the testing data belonging to the unseen categories, even though no training data are utilized in the training phase."
],
[
"Overall Framework",
"The flowchart of our overall framework is illustrated in Figure REF .",
"As we can see, there are two stages: the offline phase and the online phase.",
"In the offline phase, suppose only images of a limited number of categories are visible to our system.",
"We firstly extract their visual feature features through a convolutional neural network.",
"At the same time, we use a NLP model to transform seen labels into a semantic-rich embedding space, where each label is represented by a real-valued vector.",
"With the embedded semantics, the relationships among both seen and unseen categories can be well captured and characterized.",
"Instead of $0/1$ -form label vector, ZSH supervises the learning of hash functions with the embedded semantic vectors to transfer supervised knowledge.",
"We further rotate the off-the-shelf embedding space to better align with the low-level visual feature space.",
"Meanwhile, ZSH preserves local structural information and discrete nature of the intermediate binary codes to improve hash functions.",
"Finally, we use the learned hash functions to transform all the images in the database into binary codes for subsequent retrieval.",
"In the online phase, when a new query image of any unseen category comes, we encode the new image into binary code following the same mapping and retrieve images that are close to this query in the Hamming space."
],
[
"Transferring Supervised Knowledge",
"In general, most of existing supervised hashing algorithms may retrieve relevant results of queries in the seen categories since there are supervised information for understanding the queries.",
"Nevertheless, when the hashing systems have no knowledge of certain unseen classes, query images from these classes will be probably be misunderstood, thereby leading to inaccurate search.",
"One of the main causes is that the supervised information is in the form of $0/1$ -form label vectors or pair-wise data relationship, which implicitly makes labels independent to each other and omits the inherent correlation among their high-level semantics (e.g., cat is as different from truck as from dog).",
"As illustrated in Figure REF , using independent labels, each object will be mapped to an independent vertex of a hypercube, and the distance between any two categories will be the same.",
"In order to address such disadvantage, we propose to connect label semantics by taking advantage of the superior ability endowed by neural language processing techniques.",
"Specifically, as illustrated in Figure REF , we map independent labels into a word embedding space, where semantic correlations among labels can be quantitatively measured and captured.",
"Therefore, unseen labels can leverage the well-established mapping from its semantically close seen categories.",
"For example, in the embedding space, cat and dog will be close to each other, hence even the hashing systems may never observe any cat images, they can still gain some useful clues from the supervised knowledge of dog.",
"We adopt the language model [9] pre-trained using free Wikipedia text.",
"This model leverages not only local information but also global document context, therefore shows superior performance over other competitive approaches.",
"Every category is embedded into a 50-d word vectorIn practice, we find that by setting word vector to unit length, retrieval performance can be augmented with no distortion of the cosine similarities, thus we empirically normalize word vector to be unit length..",
"In the subsequent part, we consistently denote the embedded label matrix as $Y$ for brevity.",
"Figure: An illustration of independent 0/10/1-form labels v.s.",
"word embedding."
],
[
"Semantic Alignment",
"Note that the transformed supervised knowledge from the off-the-shelf embedding space may potentially deviate from the underlying semantics of the image data due to the problems such as domain difference, semantic shift, semasiological variation.",
"This will inevitably jeopardize the whole learning process in our proposed model.",
"In order to prevent this issue, we propose to a semantic alignment strategy, which actively aligns the initial embedding space with the distributional properties of low-level visual feature.",
"In particular, we seek for certain transformation $R\\in \\mathbb {R}^{c\\times c}$ matrix with orthogonal constraint $R^TR=I_c$ to rotate the embedding space to $R^TY$ .",
"Recall that we intend to use the amendatory supervised knowledge to guide the learning of high-quality hash codes and hash functions, therefore, we minimize the following error: $\\left\\Vert {R^TY - {W^T}B} \\right\\Vert _{F}^{2},$ which $W^T\\in R^{c\\times l}$ is the mapping matrix from binary codes $B\\in \\lbrace -1,1\\rbrace ^{l\\times n}$ to the supervised information.",
"$l$ is the code length.",
"The benefit of the above formulation is that it can help to narrow down the semantic gap between binary codes and the supervised knowledge."
],
[
"Hashing Model",
"For convenience, we firstly recap some previous settings here.",
"Suppose we have $n$ training samples $X = \\left[ x_{1},x_{2},\\cdots ,x_{n} \\right]\\in \\mathbb {R}^{d\\times n}$ .",
"For brevity, we denote the corresponding embedded label knowledge as $Y=\\left[y_{1},y_{2},\\cdots ,y_{n} \\right]\\in \\mathbb {R}^{p\\times n}$ .",
"Our ultimate target is to learn a set of hash functions from “seen” training data $X$ supervised by $Y$ , enabling generating high-quality binary codes for data of “unseen” categories.",
"Meanwhile, the quality of hash functions may heavily rely on the reliability of the intermediate binary codes of training data.",
"In other words, the model is supposed to simultaneously well control both hash functions and hash codes.",
"To achieve the above goals, we propose the following model: $\\begin{split}\\min \\limits _{f,W,B,R}~&{\\left\\Vert {R^TY - {W^T}B} \\right\\Vert _{F}^{2}} + \\lambda {\\left\\Vert W \\right\\Vert _{F}^{2}} + \\alpha \\left\\Vert {f(X) - B} \\right\\Vert _F^2\\\\&+\\beta \\left\\Vert {P} \\right\\Vert _F^2+ \\gamma \\sum \\limits _{i = 1}^n {\\sum \\limits _{j = 1}^n {{S_{ij}}\\left\\Vert {f({x_i}) - f({x_j})} \\right\\Vert _F^2} } \\\\&\\textrm {s.t.",
"}~B\\in \\lbrace -1,1\\rbrace ^{l\\times n}\\wedge R^TR = I_c,\\end{split}$ where $R\\in \\mathbb {R}^{c\\times c}$ is the semantic alignment matrix.",
"$W\\in \\mathbb {R}^{l\\times c}$ is the mapping matrix from binary codes to supervisory information.",
"$B = \\left[b_{1},b_{2},\\cdots ,b_{n} \\right]\\in \\left\\lbrace -1,1 \\right\\rbrace ^{l\\times n}$ denotes the binary codes of $X$ , where $b_i$ is the binary codes of the $i$ -th sample $x_{i}$ .",
"$I_c$ is a diagonal matrix of size $c\\times c$ .",
"$\\Vert \\cdot \\Vert _F$ denotes the Frobenius norm of a matrix.",
"$\\lambda >0,\\alpha >0,\\beta >0$ and $\\gamma >0$ are balancing parameters.",
"$f:\\mathbb {R}^{m\\times 1}\\rightarrow \\mathbb {R}^{l\\times 1}$ define a hash function from a non-linear embedded feature space to the desired Hamming space: $f(x)=P^T\\phi (x),$ where $f(X)=[f(x_1),f(x_2),\\ldots ,f(x_n)]$ .",
"$P\\in \\mathbb {R}^{l\\times m}$ is the transformation matrix.",
"Following the successful practice for learning hash functions in [17], we employ kernel mapping to handle the potential problem of linear inseparability: $\\phi (x) \\!= \\!\\left[\\exp (-\\frac{\\Vert x\\!-\\!a_1\\Vert ^2}{\\delta }),\\ldots ,\\exp (-\\frac{\\Vert x\\!-\\!a_m\\Vert ^2}{\\delta })\\right]^T.$ where $\\lbrace a_i\\rbrace |_{i=1}^m$ are $m$ anchors randomly sampled from $X$ and $\\delta $ is the bandwidth parameter.",
"Note that we keep the discrete constraint on the variable $B$ to prevent information loss of binary codes to the greatest extent.",
"The term $\\gamma \\sum \\nolimits _{i = 1}^n {\\sum \\nolimits _{j = 1}^n {{S_{ij}}\\left\\Vert {f({x_i}) - f({x_j})} \\right\\Vert _F^2} } $ in Eq.",
"(REF ) preserves local structural information of training data, i.e., if two samples are similar in the original feature space (large $S_{ij}$ ), then they are enforced to share similar binary codes in the Hamming space.",
"In the next part, we introduce an efficient alternating algorithm to optimize our zero-shot hashing model."
],
[
"Optimization",
"We first rewrite the model in matrix form as follows: $\\begin{split}\\min \\limits _{P,W,B,R}~&{\\left\\Vert {R^TY - {W^T}B} \\right\\Vert _{F}^{2}} + \\lambda {\\left\\Vert W \\right\\Vert _{F}^{2}} + \\alpha \\left\\Vert {P^T\\phi (X) - B} \\right\\Vert _F^2\\\\&+\\beta \\left\\Vert {P} \\right\\Vert _F^2+ \\gamma Tr(P^T\\phi (X)L\\phi (X)^TP)\\\\ &\\textrm {s.t.",
"}~B\\in \\lbrace -1,1\\rbrace ^{l\\times n} \\wedge R^TR = I,\\end{split}$ where $\\phi (X)=[\\phi (x_1),\\phi (x_2),\\ldots ,\\phi (x_n)]$ and the Laplacian matrix $L$ is computed as: $L = D-S,$ where $D$ is a diagonal matrix with its $i$ -th diagonal element computed as $D_{ii}=\\sum \\nolimits _{j = 1}^n {{S_{ij}}} $ .",
"Next, we present an alternating algorithm to optimize the model in Eq.",
"(REF )."
],
[
"Update P",
"Fixing all variables except for $P$ , we get the quadratic problem as: $\\min \\limits _P \\alpha \\Vert {P^T\\phi (X) \\!- \\!B}\\Vert _F^2\\!+\\!\\beta \\left\\Vert {P} \\right\\Vert _F^2\\!+\\!\\gamma Tr(P^T\\phi (X)L\\phi (X)^TP).$ By setting its derivative with respect to $P$ to 0, we have the following solution $P = \\left(\\phi (X)\\phi (X)^T+\\frac{\\beta }{\\alpha }I+\\frac{\\gamma }{\\alpha }\\phi (X)L\\phi (X)^T\\right)^{-1}\\phi (X)B^T.$"
],
[
"Update B",
"In this step, we fix all other variables and learn binary codes $B$ with discrete constraint.",
"The objective function can be reduced to $\\begin{split}\\mathop {\\min }\\limits _{B}~&{\\left\\Vert {R^TY - {W^T}B} \\right\\Vert _{F}^{2}} + \\alpha \\left\\Vert {P^T\\phi (X) - B} \\right\\Vert _F^2\\\\\\textrm {s.t.",
"}~&B\\in \\lbrace -1,1\\rbrace ^{l\\times n}.\\end{split}$ The above equation can be further written as $\\begin{split}\\mathop {\\min }\\limits _{B}~&{\\left\\Vert W^TB \\right\\Vert _{F}^{2}} -2Tr(B^TH)\\\\\\textrm {s.t.",
"}~&B\\in \\lbrace -1,1\\rbrace ^{l\\times n},\\end{split}$ where $H = WR^TY + \\alpha P^T\\phi (X)$ .",
"Inspired by [34], we apply the discrete coordinate descent (DCC) algorithm to solve the above sub-problem.",
"Denote $B$ as $B = \\left[q_1^T,q_2^T,\\cdots ,q_l^T \\right]$ , $H = \\left[ h_1^T,h_2^T,\\cdots ,h_l^T \\right]$ and $W = \\left[ u_1^T,u_2^T,\\cdots ,u_l^T \\right]$ , where $q_i^T$ , $h_i^T$ and $u_i^T$ are the $i$ -th row of $B$ , $H$ and $W$ , respectively.",
"Furthermore, for convenience, we denote $\\left\\lbrace \\begin{array}{l}{{B}_{\\lnot i}} = \\left[ {{q_1}^T,...,{q_{i - 1}}^T,{q_{i + 1}}^T,...,{q_l}^T} \\right],\\\\\\\\{H_{\\lnot i}} = \\left[ {{h_1}^T,...,{h_{i - 1}}^T,{h_{i + 1}}^T,...,{h_l}^T} \\right],\\\\\\\\{{W}_{\\lnot i}} = \\left[ {{u_1}^T,...,{u_{i - 1}}^T,{u_{i + 1}}^T,...,{u_l}^T} \\right].\\end{array} \\right.$ Then we can have $\\begin{split}\\Vert W^TB \\Vert ^2 &= Tr(B^TWW^TB)\\\\&=const + \\Vert q_i u_i^T\\Vert _F^2+2u_i^TW_{\\lnot i}^TB_{\\lnot i}q_i\\\\&=const + 2u_i^TW_{\\lnot i}^TB_{\\lnot i}q_i.\\end{split}$ Here, $\\Vert q_i u_i^T\\Vert ^2 = Tr(u_i q_i^Tq_iw_i^T) = const$ .",
"Following the same rule, we also have the following conclusion $Tr(B^TH) = const + h_i^Tq_i.$ The sub-problem can be transformed to $\\begin{split}\\min \\limits _{q_i}~&(u_i^TW_{\\lnot i}^TB_{\\lnot i}-h_i)q_i\\\\\\textrm {s.t.",
"}~&h_i\\in \\lbrace -1, 1\\rbrace ^{n\\times 1},\\end{split}$ The optimal solution of above equation is $\\begin{split}q_i = \\textrm {sgn}(h_i-B_{\\lnot i}^TW_{\\lnot i}u_i).\\end{split}$ where $\\textrm {sgn}(\\cdot )$ is the sign function.",
"We can see that each bit of the desired binary code $B$ can be learned based on other $l-1$ bits.",
"Thus, we can use cyclic coordinate descent approach to generate the optimal codes until the entire procedure converges."
],
[
"Update R",
"With $B,W,P$ fixed, we then have $\\begin{split}\\min \\limits _{R}~&\\left\\Vert {R^TY - {W^T}B} \\right\\Vert _{F}^{2}\\\\\\textrm {s.t.",
"}~&R^TR = I,\\end{split}$ which can efficiently solved by the algorithm proposed in [32]."
],
[
"Update W",
"By fixing $P,B,R$ , we arrive at a classic ridge regression problem: $\\begin{split}\\mathop {\\min }\\limits _{W}~&{\\left\\Vert {R^TY - {W^T}B} \\right\\Vert _{F}^{2}} + \\lambda {\\left\\Vert W \\right\\Vert _{F}^{2}}.\\end{split}$ The above equation has an closed-form solution $\\begin{split}W = (BB^T+\\lambda I_l)^{-1}BY^TR.\\end{split}$ where $I_l$ is a diagonal matrix of size $l\\times l$ .",
"By iteratively updating $P,W,B,R$ until convergence, we can arrive at an optima.",
"The overall algorithm is summarised in Algorithm REF .",
"[!h] [1] Training data $X$ and the embedded label matrix $Y$ ; Binary code $B$ , alignment matrix $R$ , hash function $P$ and mapping matrix $W$ ; Randomly initialize $B,P$ and $W$ ; Randomly initialize $R$ to to be orthogonal; Map $X$ to $\\phi (X)$ using $m$ anchors randomly selected from $X$ ; Construct Laplacian matrix $L$ ; Update $P$ according to Eq.",
"(REF ); Update $B$ iteratively by using the solution of (REF ); Update $R$ according to Eq.",
"(REF ); Update $W$ according to Eq.",
"(REF ); there is no change to $P,B,R,W$ $B,P,W,R$ ; Algorithm for optimizing Zero-Shot Hashing"
],
[
"Algorithm Analysis",
"In this section, we analyze the convergence and time complexity of our algorithm."
],
[
"Convergence Study",
"As shown in Algorithm REF , in each iteration, the updates of all variables make the objective function value decreased.",
"We also conducted empirical study on the convergence property using ImageNet [3].",
"Specifically, we trained our zero-shot hashing model with $30,000$ seen images randomly sampled from the ImageNet dataset, with label embedding as supervision.",
"We selected $1,000$ anchors and set the code length to be 64 bits.",
"As Figure REF shows, our algorithm starts with cost function value roughly at $30,000$ , but descends dramatically within only 10 iterations, and reaches a stable local minima at the 20-th iteration.",
"This phenomenon clearly indicates the efficiency of our algorithm.",
"Figure: Convergence study on ImageNet."
],
[
"Computational Complexity",
"In each iteration (line 6-9), the time cost is analyzed as follows.",
"The computation of $P$ in Eq.",
"(REF ) is $O(m^2n+nml+m^3)$ .",
"The DCC algorithm for updating $B$ costs $O(cl^2+l^2n)$ .",
"As to the optimization of the sub-problem in Eq.",
"(REF ), the time cost is $O(c^3)$ .",
"Finally, the computational cost of updating $W$ is $O(l^2n+lnc+lc^2+l^3)$ .",
"Given that $m\\ll n$ , $l\\ll n$ , $c\\ll n$ and our algorithm converges within a few iterations (less than 10), the overall time cost of our algorithm is $O(n)$ .",
"It is worth noting that the dominant operation of our algorithm is matrix multiplication, which can be greatly speeded up by using parallel and/or distributed algorithms."
],
[
"Experimental Settings",
"In our experiments, we employ three real-life image datasets, including CIFAR-10https://www.cs.toronto.edu/ kriz/cifar.html, ImageNethttp://image-net.org/ and MIRFlickrhttp://press.liacs.nl/mirflickr/.",
"CIFAR-10 consists of $60,000$ images which are manually labelled with 10 classes including airplane, automobile, bird, cat, deer, dog, frog, horse, ship, truck, with $6,000$ samples in each class.",
"The classes are completely mutually exclusive, i.e., no overlap between classes (e.g., automobiles and trucks).",
"ImageNet is an image dataset organized according to the WordNet [18] hierarchy.",
"The subset of ImageNet for the Large Scale Visual Recognition Challenge 2012 (ILSVRC2012) is used for our experiments, consisting of over 1.2 million Web images, manually labeled with $1,000$ object categories.",
"MIRFlickr consists of $25,000$ images collected from the social photography site Flickr through its public API.",
"Firstly introduced in 2008, this dataset is wildly used in multimedia research.",
"MIRFlickr is a multi-label dataset with every image associating with 24 popular tags such as sky, river, etc.",
"For all image data, we adopted the winning model for the 1000-class ImageNet Large Scale Visual Recognition Challenge 2012 [13] to extract the fully connected layer fc-7 as visual feature.",
"Various metrics are employed for performance of different evaluation tasks.",
"For image retrieval, we used the two traditional metrics i.e., Precision and Mean Average Precision (MAP).",
"MAP focuses on the ranking of retrieval results and we reported the results over the top $5,000$ retrieved samples.",
"Precision mainly concentrates on the retrieval accuracy and we reported the results with Hamming radius $r\\le 2$ .",
"We compared our proposed ZSH with four state-of-the-art supervised hashing approaches, including COSDISH [12], SDH [25], KSH [17] and LFH [39].",
"For all anchor-based algorithms, we randomly sampled $1,000$ anchors from the training dataset.",
"Furthermore, we compared to one of the most representative unsupervised hashing method, i.e., Inductive Hashing on Manifolds (IMH) [26].",
"For all comparing approaches, we followed their suggested parameter settings.",
"For ZSH, we empirically set $\\alpha $ to $10^{-5}$ and $\\gamma $ to $10^{-6}$ .",
"For regularization parameters $\\lambda $ and $\\beta $ , we set them to $10^{-2}$ and $10^{-4}$ , respectively.",
"The number of iterations is set to 10.",
"We define the similarity matrix $S$ to be computed by $\\nonumber {S_{ij}} = \\left\\lbrace {\\begin{array}{*{20}{c}}{\\exp (\\frac{{{{\\left\\Vert {{x_i} - {x_j}} \\right\\Vert }_2}}}{{2{\\sigma ^2}}}),}&{{\\rm {if }}~{x_i} \\in {\\mathcal {N}_k}({x_j})~{\\rm { or }}~{x_i} \\in {\\mathcal {N}_k}({x_j})}\\\\\\\\{0,}&{{\\rm {otherwise,}}}\\end{array}} \\right.$ where $\\mathcal {N}_k(\\cdot )$ is the function of searching $k$ nearest neighbors.",
"In our experiment, we set $\\sigma = 1$ .",
"To evaluate the efficacy of retrieving images in unseen categories, we split CIFAR-10 into a “seen” training set and an “unseen” testing set.",
"In particular, we select truck as unseen testing category and leave the rest 9 categories as seen training set.",
"For all comparing algorithms, we randomly sample $10,000$ images for learning hash functions.",
"For testing purpose, we randomly select $1,000$ images from the unseen category as query images, and the remaining $5,000$ test images together with the $54,000$ images of seen categories are combined to form the retrieval database.",
"The performance of all comparing approaches w.r.t.",
"different codes lengths (i.e., $\\lbrace 16, 32, 64, 96, 128\\rbrace $ ) is illustrated in Figure REF .",
"As we can see, the proposed ZSH outperforms all the other hashing algorithms in terms of MAP at all code lengths.",
"As to Precision, ZSH still shows superior image retrieval performance in most cases.",
"The underlying principle is that our method not only utilizes inherent semantic relationship among labels to transfer supervisory knowledge, but also preserves discrete and structural properties of data in the learning of hash codes and hash functions.",
"An interesting observation is the performance of IMH, which is an unsupervised method, gains competitive even better retrieval results in terms of Precision as compared to some supervised methods such as KSH, SDH.",
"While unsupervised methods encode images solely with the distributional properties in the feature space, the supervised ones may be misled by independent semantic labels in the learning processing.",
"Besides, MAP increases rapidly for all methods when code length varies from 16 to 64, and then reaches a slow-growth stage from 64 bits to 128 bits.",
"When code length is short, more codes are required to guarantee the descriptive and discriminative power.",
"However, after encoding space is large enough (e.g.",
"64 bits), the expression ability saturated, providing more bits cannot significantly improve the performance.",
"As to Precision, hashing performance significantly deteriorates as code length is larger than 64.",
"Recall that our searching radius is empirically set to 2, forming a hyper-ball of radius 2 in Hamming space.",
"When the code length increases from 16 to 64, significant improvement in retrieval ability counteracts the searching difficulty.",
"However, as Hamming space becomes larger, searching difficulty grows linearly, thereby degrading the Precision performance.",
"Therefore, as a trade-off between efficiency and effectiveness, an eclectic code length should be chosen.",
"Figure: Performance (MAP and Precision) of different comparing methods on zero-shot image retrieval over CIFAR-10 dataset."
],
[
"Effect of Different Unseen Category",
"In this experiment, we aim to evaluate the performance of zero-shot image retrieval on different unseen categories.",
"The experimental settings are the same as that in the previous subsection.",
"Figure REF illustrates the MAP and Precision performance of ZSH using each individual label as unseen testing data.",
"Figure: Performance (Precision and MAP) of zero-shot image retrieval for each individual unseen category on CIFAR-10 dataset.We can observe that zero-shot image retrieval performance varies from one class to another, reaching peak at bird and bottom at automobile.",
"Intuitively, if an unseen class is semantically closer to other seen categories, more relevant supervisory knowledge can be transferred from word embedding space for boosting the retrieval performance.",
"To dig deeper about the reason behind the fluctuation of performance on different unseen objects, we compute the average cosine similarity between each unseen category and other seen categories, and list the corresponding MAP in Table REF .",
"Table: Average cosine similarity of each category and all other categories, together the corresponding MAP performance.We observe that the MAP performance is positively related to the average cosine similarity.",
"For instance, those of larger cosine similarity (e.g., dog, cat) performs relatively well, while those of smaller similarity (e.g., airplane, automobile) gain relatively poor performance.",
"This observation implies that in order to achieve satisfactory retrieval results, unseen classes should have sufficient correlation with seen ones.",
"As shown in Figure REF , we also compare the effects of embedded labels and binary labels.",
"The performance of embedded labels is obviously better than that of binary labels.",
"The underlying reason is that the embedding space can help to capture the relationship between seen and unseen categories for transferring supervisory knowledge.",
"In contrast, binary labels neglect semantic correlations, thereby leading to irregular fluctuations of retrieval performance."
],
[
"Effect of Seen Category Ratio",
"In this experiment, we evaluate the performance of our proposed ZSH w.r.t.",
"different numbers of seen categories.",
"Specifically, we vary the ratio of seen categories in the training set from $0.1$ to $0.9$ .",
"For each ratio, we randomly sample $10,000$ images from the seen categories for training.",
"Further, we randomly select $1,000$ images from the unseen set as queries to search in the remaining $59,000$ images.",
"Note that when the ratio of seen categories decreases to $0.1$ , we use all $6,000$ datums of that class as training set.",
"Figure: Effects of different ratios of seen categories on CIFAR-10 dataset.We report the experimental results in Figure REF , from which we have the following observations: (1) The performance of both MAP and Precision boosts as the ratio of the seen categories grows; (2) As the ratio increases from $0.1$ to $0.3$ , we see a dramatic leap of the retrieval performance, followed by a relatively slight performance improvement from $0.3$ to $0.9$ .",
"We analyze that by observing more “seen” categories, we have higher possibility to find relevant supervision for the unseen class, which guides to learn better intermediate hash codes, thereby simultaneously improving the quality of hash functions."
],
[
"Effect of Training Size",
"This part of experiment mainly focuses on evaluating the effect of training size on the searching quality of ZSH.",
"We select truck as the unseen object and varies the size of training data in the range of $\\lbrace 1000,2000,\\ldots ,10000,20000,\\ldots ,50000\\rbrace $ .",
"The results are demonstrated in igure REF .",
"As we can see, when the size increases from $1,000$ to $10,000$ , we observe a rapid rise of the Precision performance.",
"Nonetheless, when fed with more training data, ZSH does not gain noticeable performance boost.",
"For the balance of training efficiency and effectiveness, in the rest experiments, we consistently set the training size to $10,000$ .",
"Figure: Effects of training size on Precision performance over CIFAR-10 dataset."
],
[
"Overall Comparison of Zero-shot Image Retrieval",
"In this part, we evaluate our proposed ZSH on zero-shot image retrieval as compared to other state-of-the-art methods using the Large Scale Visual Recognition Challenge 2012 (ILSVRC2012) dataset.",
"Recall that the ILSVRC2012 dataset contains more than $1.2$ million images tagged with $1,000$ synsets without any overlap.",
"For evaluation purpose, we randomly choose 100 categories which have corresponding word embedding learned from Wikipedia text corpus, which gives us a set of roughly $130,000$ images.",
"We split the data into a training set (90 seen categories) and a testing set (10 unseen categories).",
"For all comparing algorithms, we randomly select $10,000$ images of seen categories for training.",
"As to image queries, we randomly sample $1,000$ images from the unseen categories.",
"We use the learned hash function to encode all the remaining images to form the retrieval database.",
"Figure: Performance (MAP and Precision) of different comparing methods on zero-shot image retrieval over ImageNet dataset.The performance of our proposed ZSH and other four state-of-the-art supervised hashing methods with different code lengths are reported in Figure REF .",
"As we can see, ZSH consistently outperforms all other competitors in most cases.",
"As code length varies from 16 to 128, we can observe the similar variation tendency of performance on ImageNet to that on CIFAR-10.",
"This phenomenon again implies that we should choose a trade-off code length to guarantee the retrieval performance."
],
[
"Image Retrieval in Related Categories",
"In zero-shot image retrieval scenario, we expect that even though we fail to retrieve relevant images of the same category, we can still obtain semantically related images.",
"For instance, if the query image describes a cat, we may prefer to retrieve images of dog rather than images of car.",
"Our proposed ZSH utilizes semantic embedding to set up connections between semantically similar labels in the embedded space.",
"In this way, the supervision knowledge of seen categories can be transferred into hash functions, which can effectively encode images of unseen categories.",
"Figure: Comparison of ZSH and other hashing approach on the capability of retrieving semantically similar images from related categories.Since we need to search more related categories, all remaining images of both seen and unseen categories are used to form retrieval database.",
"All the other settings are the same as that in Section 4.3.1.",
"In order to evaluate the performance of retrieving related categories, we use two modified metrics, named MAP$_{{related}}$ and Precision$_{{related}}$ , which are defined as $\\begin{split}\\textrm {MAP}_{related} = \\sum _{i=1}^{K}\\frac{\\textrm {AP} _{related}@i}{K},\\end{split}$ $\\begin{split}\\textrm {Precision}_{related} = \\frac{n_{related}}{n_{retrieved}} ,\\end{split}$ where MAP$_{{related}}$ is calculated based on the top $K$ retrieved results, $AP_{related}@i$ is the average precision based on the related results, calculated by $\\begin{split}\\textrm {AP}_{related}@i = \\frac{n_{related}^{(i)}}{i} ,\\end{split}$ where $n_{related}^{(i)}$ is the number of related images in top $i$ retrieved results.",
"$n_{related}$ and $n_{retrieved}$ are the related retrieval under Hamming radius 2 and total examples retrieved under Hamming radius 2, respectively.",
"Using WordNet [18], which is a lexical database for the English language, we define query $A$ and retrieved object $B$ are related if: 1) $A$ and $B$ are not of the same category.",
"2) $A$ can reach $B$ on WordNet within 5 hops.",
"In practice, we set $K=5,000$ and $R=2$ .",
"Figure REF shows the experimental results.",
"We can see that in terms of MAP$_{related}@K$ , our method always outperforms other methods at every code length.",
"When we look at Precision$_{related}$ , our proposed ZSH achieves $0.3262$ , $0.2636$ , $0.2129$ at 32 bits, 64 bits and 96 bits, which significantly outperforms the second best method.",
"This observation indicates that ZSH is capable of detecting the semantically similar images from the most related categories."
],
[
"Results on MIRFlickr",
"In real-world pictures, especially in user-generated photos, there often exists multiple tags belong to one picture.",
"To examine the practical efficacy of our proposed ZSH, in this part, we conduct an extra experiment on a real-life multi-label dataset, i.e., MIRFlickr, which contains $25,000$ images downloaded from the social photography site Flickr.",
"Each image is associated with 24 tags.",
"Since in multi-label image dataset, different categories share overlapping images, which makes it difficult to divide the dataset into training set and testing set.",
"Hence, we employ ImageNet as an auxiliary dataset to train our hash functions and evaluate the zero-shot image retrieval performance on MIRFlickr.",
"Specifically, from the ILSVRC2012 dataset we select 100 categories which does not overlap with the 24 tags in MIRFlickr.",
"For fair comparison, all hashing approaches use $10,000$ randomly sampled images for training.",
"After the hash function is learned, we directly apply them to transform the MIRFlickr images into binary codes.",
"We then sample $1,000$ datums as query images and search in the remaining $24,000$ images.",
"We regard the retrieval images sharing at least two tags with the query as the true neighbors, and compute MAP on the top $5,000$ retrieved results and Precision under Hamming distance 2.",
"Figure REF illustrates our results of our ZSH and other comparing algorithms on MIRFlickr.",
"In the left sub-figure, we can see that with different code lengths, our ZSH can consistently achieve the best MAP performance among all the comparing algorithms.",
"As the code length increases, the MAP performance of each algorithm keeps increasing, reaching $0.2488$ at 128 bits, which outperforms the second best hashing method COSDISH by $19\\%$ at the same length.",
"In terms of Precision, ZSH exceeds all other methods in most cases.",
"Similar to that of CIFAR-10 and ImageNet, we can see a variation pattern with an increasing trend from 16 to 64 and a performance drop from 64 to 128.",
"The promising performance on MIRFlickr demonstrates the potential of ZSH in indexing and searching real-life image data."
],
[
"CONCLUSION",
"With the explosion of newly-emerging concepts and multimedia data on the Web, it is impossible to supply existing supervised hashing methods with sufficient labeled data in time.",
"In this paper, we studied the problem of how to map images of unseen categories using hash functions learned from limited seen classes.",
"We proposed a novel hashing scheme, termed zero-shot hashing (ZSH), which is capable of transmit supervised knowledge from seen categories to unseen categories.",
"Independent $0/1$ -form labels were projected into an off-the-shelf embedding space with abundant semantics, where label semantic correlations can be fully characterized and quantified.",
"Considering the issues of domain difference and semantic shift, we further narrowed down the gap between binary codes and high-level semantics by a semantic alignment operation.",
"Specifically, we intentionally rotated the embedding space to adjust the supervised knowledge more suitable for learning high-quality hash codes.",
"Besides, we also preserved local structural property and discrete nature of hash codes in the ZSH model.",
"An effective algorithm was designed to optimize the model in an iterative manner and the empirical study showed the convergency and efficiency.",
"We evaluated our proposed ZSH hashing approach on three real-world image datasets, including CIFAR-10, ImageNet and MIRFlickr.",
"The experimental results demonstrated the superiority of ZSH as compared to several state-of-the-art hashing approaches on the zero-shot image retrieval task.",
"In the future, we plan to enhance the exploration of label semantic correlations by integrating knowledge from multiple sources, including textual corpus and visual clues.",
"We expect this will compensate the incomplete representation of each individual modality, thereby solving the problem of domain difference and semantic shift fundamentally."
]
] | 1606.05032 |
[
[
"Performance studies of MRPC prototypes for CBM"
],
[
"Abstract Multi-gap Resistive Plate Chambers (MRPCs) with multi-strip readout are considered to be the optimal detector candidate for the Time-of-Flight (ToF) wall in the Compressed Baryonic Matter (CBM) experiment.",
"In the R&D phase MRPCs with different granularities, low-resistive materials and high voltage stack configurations were developed and tested.",
"Here, we focus on two prototypes called HD-P2 and THU-strip, both with strips of 27 cm$^2$ length and low-resistive glass electrodes.",
"The HD-P2 prototype has a single-stack configuration with 8 gaps while the THU-strip prototype is constructed in a double-stack configuration with 2 $\\times$ 4 gaps.",
"The performance results of these counters in terms of efficiency and time resolution carried out in a test beam time with heavy-ion beam at GSI in 2014 are presented in this proceeding."
],
[
"Introduction",
"CBM [2] is a future fixed target heavy ion experiment located at the Facility for Antiproton and Ion Research (FAIR) in Darmstadt, Germany.",
"CBM aims to study the phase diagram of strongly interacting matter at the highest baryon densities with unprecedented accuracy.",
"This goal can be achieved by measuring particle yields and flow of very rare probes like multi-strange objects but also bulk particles like pions, kaons and protons.",
"Therefore, an excellent particle identification (PID) at a targeted interaction rate of 10 MHz is required.",
"The PID of charged hadrons at CBM is realized by a Time-Of-Flight (ToF)-wall [3] positioned at about 10 m from the interaction point which is composed of Multi-gap RPCs with a total active area of about 120 m$^2$ .",
"The requirements of the full system is a time resolution better than 80 ps at an efficiency higher than 95 %.",
"Since the charged particle flux drops exponentially from small angle (2.5$^\\circ $ ) to large angle (25$^\\circ $ ), ranging from 25 kHz/cm$^2$ to 1 kHz/cm$^2$ high rate capable MRPCs with low-resistive electrode material [4], [5], [6], [7], [8], [9], [10] but also with thin float glass [11], [12] and with different granularities were developed.",
"An overview of the conceptual design of the CBM ToF-wall is given in [13]."
],
[
"Counter and experimental setup configuration",
"For the intermediate rate region (between 1.5 and 10 kHz/cm$^2$ ) in the current conceptual design we consider differential MRPCs with low-resistive glass of 0.7 mm thickness as floating electrodes and multi-strip readout electrodes with a granularity of 27 cm$^2$ , i.e.",
"a strip length of 27 cm and a strip pitch of 1 cm (the strip width is 7 mm).",
"MRPCs with such a readout electrode geometry have typically an impedance of about 100 $\\Omega $ in a single-stack configuration with 8 gaps and about 50 $\\Omega $ impedance in a double-stack configuration with 2 $\\times $ 4 gaps.",
"In order to evaluate the importance of impedance matching to the front-end electronics which has in our case 100 $\\Omega $ a single-stack prototype called HD-P2 and a double-stack prototype called THU-strip with the properties listed in table REF were built.",
"As reference, a smaller counter called HD-P5 was used.",
"Figure REF shows a photograph of all three counters.",
"Table: Counter types and their properties.Figure: Photograph of the counters under test (HD-P2 and THU-strip) and thereference counter (HD-P5) embedded in their gas-tight box.",
"The preamplifiersare connected directly to the readout electrode.These counters were tested the first time in October 2014 at GSI with reaction products stemming from a samarium beam with a kinetic energy of 1.2 GeV impinging on a 5 mm thick lead target creating a mean particle flux of a few hundred Hz/cm$^2$ .",
"The particle flux was measured by two plastic scintillators arranged in front and behind a stack of two MRPCs containing the detector under test (Dut) and the reference counter.",
"The HD-P2 prototype was replaced after some testing time by the THU-strip.",
"This setup (see fig.",
"REF ) was part of a bigger setup including other MRPC counters which were tested in parallel [14].",
"Also part of the setup was a diamond detector placed in the beam a few cm in front of the target.",
"Figure: Low-rate part of the in-beam test setup.",
"The counters were placedbelow the beam.",
"The distance to the target is about 5 m.The setup was positioned about 5 m from the target ensuring a quite uniform illumination on the counter surfaces.",
"The electronic chain components consisted of a preamplifier discriminator stage called PADI6 [15], an FPGA based TDC [16] and an FPGA based readout system called TRB3 [17].",
"The gas mixture was composed of 85% $C_2H_2F_4$ , 10% $SF_6$ and 5% iso-$C_4H_{10}$ ."
],
[
"Results",
"The data are processed in three major steps.",
"In step 1, the data are unpacked and TDC values are calibrated with respect to the differential non-linearities.",
"In step 2, the time information of both strip ends is combined to form a hit with a given position and a mean time.",
"The hit position offsets (which are in fact time offsets) are obtained by shifting the mean of the hit distributions of all strips to zero (position alignment).",
"The same procedure is done for the time offset with respect to the reference counter.",
"Subsequently, the Time-over-Threshold (ToT) value is calibrated by shifting the mean of the ToT distribution to a constant value.",
"In addition, a gain is introduced in order to have a common spread in the ToT distributions.",
"After this procedure the slewing correction with the normalized ToT is applied.",
"The last action in step 2 is the clusterization where hits with similar positions and arrival times are grouped to a cluster.",
"The time and position of every cluster is calculated by taking the mean of the times and positions of the contributing hits and weighting them with their ToT.",
"The same calibration is done for the reference counter.",
"The clusters from both detectors of a single event are matched to proper pairs (again in position and time).",
"All procedures of step 2 are done iteratively until no changes in position and time occur.",
"All correction parameters are stored in a ROOT file.",
"Step 3 is the data analysis.",
"First, corrections on the velocity spread of particles are applied by taking the diamond counter into account.",
"Now cuts on the reference counter can be applied to the cluster position (in order to prevent edge effects), to the cluster multiplicity (multi-hit study), to the cluster matching in-between the counters (usually the best matching pair is used), to the velocity of the particles and so on."
],
[
"Efficiency",
"The efficiency is calculated by taking a track formed by the diamond and the reference counter and checking if the Dut has a cluster in the vicinity of the penetration point.",
"This can be done since the reference counter is smaller than the Dut.",
"In order to be consistent with the time resolution results, cuts were applied on the position of the clusters on the reference counter.",
"The efficiency vs. applied high voltage for the HP-P2 (left) and THU-strip (right) for two different preamplifier thresholds is shown in fig.",
"REF .",
"For the HD-P2 counter a successful voltage scan was performed.",
"At 10.5 kV (corresponding to an electric field of about 120 kV/cm) the counter is still not in the plateau.",
"The plateau is reached at about 11 kV (125 kV/cm) with an efficiency higher than 98 %.",
"The red data point at 12.5 kV is not understood yet since the error is smaller than 0.1 % and within the symbols.",
"For the double-stack THU-strip counter the efficiency scan was not successful since the data files were corrupt.",
"Only the value at 5.5 kV (corresponding to an electric field of 110 kV/cm) could be analyzed.",
"The efficiency is about 96 %.",
"Here we assume that the plateau is reached with the argument that the difference of the efficiency for two different thresholds is only about 0.4 % and in the same order as for the single-stack at 11 kV.",
"The reason for reaching the efficiency plateau at lower fields is a higher mean weighting field in the double-stack counter.",
"However, with only one data point it is not possible to claim that the maximal efficiency is already reached.",
"Figure: Efficiency vs. applied high voltage."
],
[
"Time resolution",
"The system time resolution versus the applied high voltage for HD-P2 (left) and THU-strip (right) at two different preamplifier threshold settings are shown in fig.",
"REF .",
"System time resolution means in this context that the time resolution of the reference counter and also the electronics contribution is still included.",
"For the HD-P2 counter we observe a minimum of about 61 ps at about 11 kV followed by a rise probably caused by the onset of streamers.",
"Assuming both counters (Dut and reference) have the same contribution to the system time resolution the single counter resolution is about 43 ps for the best value.",
"Apparently, there is a trend that the system time resolution at the lower threshold is slightly higher.",
"However, the discrepancy is still within errors which are for all data points about 1 ps.",
"In addition, the situation for the THU-strip counter is opposite.",
"The system time resolution for the THU-strip is slightly worse (65 ps) which could be explained by a bigger gap size.",
"However, this speculation could not be tested due to the lack of systematic data.",
"For the data shown in fig.",
"REF an additional cut is applied with respect to the efficiency plot.",
"The cut excludes the slowest 3 % of particles (velocity cut), i.e.",
"from a total particle arrival time spread of 10 ns the particles with more than 7 ns delay are excluded.",
"However, taking all particles into account the system time resolution grows only by about 3 ps.",
"Figure: System time resolution vs. applied high voltage.Figure REF shows on the left side the time distribution between HD-P2 and reference counter as function of the cluster size of the Dut.",
"The plot on the right side shows the RMS value (blue) and the sigma of a fitted Gaussian (red) denoting the system time resolution for each bin.",
"Figure: Left: time distribution between Dut and reference counter as function of the cluster size of the Dut.",
"Right: RMS value (blue) and the sigma of a fitted Gaussian (red) for each bin.The mean cluster size is the mean x value (red ellipse) of the 2d plot which is in this case (at a high voltage of 11 kV and a threshold of 200 mV) 1.92.",
"The system time resolution has a minimum at cluster size 2.",
"At cluster size 3 the resolution is still better than at 1 but suffers from lower statistics.",
"This trend can be explained by the fact that we use the cluster mean time to get the resolution and therefore every firing strip contributes to the measured time.",
"On the other hand the more strips fire the lower is the signal/charge on the individual strips diminishing the precision of the measurement.",
"These two effects counterbalance and give a minimum at some intermediate cluster size.",
"This trend was also observed in Petris2016.",
"Another interesting dependence is the time distribution between HD-P2 and reference counter and the system time resolution as function of the cluster multiplicity (see fig REF ).",
"Here, cluster multiplicities up to 9 were observed.",
"The left plot showing the RMS values (blue) and the sigma of a fitted Gaussian (red) indicate a constant rise towards the higher multiplicities.",
"However, at multiplicity 7 corresponding to a occupancy of 50 % a system time resolution of 72 ps ($ \\mathrel {\\hat{=}} $ single counter resolution below 50 ps) is observed.",
"Figure: Left: time distribution between Dut and reference counter as function of the cluster multiplicity.",
"Right: RMS value (blue) and the sigma of a fitted Gaussian (red) for each bin."
],
[
"Conclusions",
"The performance in terms of efficiency and time resolution is studied for a single-stack counter (HD-P2) and a double-stack counter (THU-strip) in a heavy-ion beam at GSI.",
"For the HD-P2 prototype an efficiency higher than 98 % and a time resolution below 45 ps as best values were reached.",
"The results of THU-strip are slightly worse even though only data of one high voltage setting are available.",
"In a triggered system with TDC dead times larger than 20 ns, no difference between single-stack and double-stack counters were observed.",
"This might change in a free-running mode where the TDC dead time is below 5 ns and therefore reflections generated by mismatched impedances become visible.",
"This work was partially funded by BMBF grant no.",
"05P12VHFC7, NASR/CAPACITATI-Modul III, contract nr.",
"179EU, NUCLEU Project PN09370103 and by the FuturePID work package (WP19) of the HadronPhysics3 activity within the EU's Seventh Framework Program (FP7)."
]
] | 1606.04917 |
[
[
"Sharp geometric requirements in the Wachspress interpolation error\n estimate"
],
[
"Abstract Geometric conditions on general polygons are given in [9] in order to guarantee the error estimate for interpolants built from generalized barycentric coordinates, and the question about identifying sharp geometric restrictions in this setting is proposed.",
"In this work, we address the question when the construction is made by using Wachspress coordinates.",
"We basically show that the imposed conditions: bounded aspect ratio property (barp), maximum angle condition (MAC) and minimum edge length property (melp) are actually equivalent to [MAC,melp], and if any of these conditions is not satisfied, then there is no guarantee that the error estimate is valid.",
"In this sense, MAC and melp can be regarded as sharp geometric requirements in the Wachspress interpolation error estimate."
],
[
"Introduction",
"Many and different conditions on the geometry of finite elements were required in order to guarantee optimal convergence in the interpolation error estimate.",
"Some of them deal with interior angles like the maximum angle condition (maximum interior angle bounded away from $\\pi $ ) and the minimum angle condition (minimum interior angle bounded away from 0), but others deal with some lengths of the element like the minimum edge length property (the diameter of the element is comparable to the length of the segment determined by any two vertices) and the bounded aspect ratio property often called regularity condition (the diameter of the element and the diameter of the largest ball inscribed are comparable).",
"Classical results on general Lagrange finite elements consider the regularity condition [6].",
"On triangular elements, the error estimate holds under the minimum angle condition [16], [17].",
"However, on triangles, the minimum angle condition and the regularity condition are equivalent.",
"From [4], [5], [11] we know that the weakest sufficient condition on triangular elements is the maximum angle condition.",
"Some examples can be constructed in order to show that if a family of triangles does not satisfy the maximum angle condition, then the error estimate on these elements does not hold.",
"Recently, it was proved [3] that, for quadrilaterals elements, the minimum angle condition ($mac$ ) is the weakest known geometric condition required to obtain the classical $W^{1,p}$ -error estimate, when $1 \\le p < 3$ , to any arbitrary order $k$ greater than 1.",
"Moreover, in this case, $mac$ is also necessary.",
"In [2], [3] it was proved that the double angle condition (any interior angle bounded away from zero and $\\pi $ ) is a sufficient requirement to obtain the error estimate for any order and any $p \\ge 1$ .",
"When $k=1$ and $1 \\le p < 3$ , a less restrictive condition ensures the error estimate [1], [2]: the regular decomposition property ($RDP$ ).",
"Property $RDP$ requires that after dividing the quadrilateral into two triangles along one of its diagonals, each resultant triangle verifies the maximum angle condition and the quotient between the length of the diagonals is uniformly bounded.",
"This brief picture intends to show that study of sharp geometric restrictions on finite elements under which the optimal error estimate remains valid is an interesting and active field of research.",
"In [9], [10], geometric conditions on general polygons are given in order to guarantee the error estimate for interpolants built from generalized barycentric coordinates, and the question about identifying sharp geometric restrictions in this setting is proposed.",
"In this work, we address the question for the first-order Wachspress interpolation operator.",
"We show that the three sufficient conditions considered in [9] (regularity condition, maximum angle condition and minimum edge length property) are actually equivalent to the last two since the regularity condition is a consequence of the maximum angle condition and the minimum edge length property.",
"Then we exhibit families of polygons satisfying only one of these conditions and show that the interpolation error estimate does not hold to adequate functions.",
"In this sense, the maximum angle condition and the minimum edge length property can be regarded as sharp geometric requirements to obtain the optimal error estimate.",
"This work is structured as follows: In Section , we introduce notation and exhibit some basic relationships between different geometric conditions on general convex polygons.",
"Section is devoted to recall Wachspress coordinates and some elementary results associated to them; a brief picture about error estimates for the first-order Wachspress interpolation operator is also given there.",
"Finally, in Section , we present two counterexamples to show that $MAC$ and $melp$ are sharp geometric requirements under which the optimal error estimate is valid."
],
[
"Geometric conditions",
"In order to introduce notation and formalize the requirements of each geometric condition, we give the following definitions.",
"From now on, $\\Omega $ will refer to a general convex polygon.",
"(Bounded aspect ratio property) We say that $\\Omega $ satisfies the bounded aspect ratio property (also called regularity condition) if there exists a constant $\\sigma >0$ such that $\\frac{diam(\\Omega )}{\\rho (\\Omega )} \\le \\sigma ,$ where $\\rho (\\Omega )$ is the diameter of the maximum ball inscribed in $\\Omega $ .",
"In this case, we write $barp(\\sigma )$ .",
"(Minimum edge length property) We say that $\\Omega $ satisfies the minimum edge length property if there exists a constant $d_m>0$ such that $0<d_m \\le \\frac{\\left\\Vert {\\bf v}_i-{\\bf v}_j \\right\\Vert }{diam(\\Omega )}$ for all $i \\ne j$ , where ${\\bf v}_1, {\\bf v}_2, \\dots , {\\bf v}_n$ are the vertices of $\\Omega $ .",
"In this case, we write $melp(d_m)$ .",
"(Maximum angle condition) We say that $\\Omega $ satisfies the maximum angle condition if there exists a constant $\\psi _M>0$ such that $\\beta \\le \\psi _M < \\pi $ for all interior angle $\\beta $ of $\\Omega $ .",
"In this case, we write $MAC(\\psi _M)$ , (Minimum angle condition) We say that $\\Omega $ satisfies the minimum angle condition if there exists a constant $\\psi _m>0$ such that $0 < \\psi _m \\le \\beta .$ for all interior angle $\\beta $ of $\\Omega $ .",
"In this case, we write $mac(\\psi _m)$ .",
"All along this work, when we say regular polygon, we refer to a polygon satisfying the regularity condition given by (REF )."
],
[
"Some basic relationships",
"It is well known that regularity assumption implies that the minimum interior angle is bounded away from zero.",
"We state this result in the following lemma Lemma 2.1 If $\\Omega $ is a convex polygon satisfying $barp(\\sigma )$ , then $\\Omega $ verifies $mac(\\psi _m)$ where $\\psi _m$ is a constant depending only on $\\sigma $ .",
"See for instance [9].",
"$\\begin{picture}(15,15)(-5,-5)\\framebox (15,15){}\\end{picture} $ Considering the rectangle $R=[0,1] \\times [0,s]$ , where $0<s<1$ , and taking $s \\rightarrow 0^+$ , we see that the converse statement of Lemma REF does not hold.",
"Indeed, $R$ verifies the $mac(\\pi /2)$ (independently of $s$ ), but, when $s$ tends to zero, $R$ is not regular in the sense given by (REF ).",
"However, on triangular elements, $barp$ and $mac$ are equivalent.",
"We use this fact to show that, on general polygons, the regularity condition is a consequence of the minimum edge length property and the maximum angle condition.",
"To our knowledge, this elementary result has not been established or demonstrated previously.",
"Figure: (A): A polygon with its diameter attained as the length of the straight line joining two non-consecutive vertices.",
"(B): A polygon with its diameter attained as the length of the straight line joining two consecutive vertices.Lemma 2.2 If $\\Omega $ is a convex polygon satisfying $MAC(\\psi _M)$ and $melp(d_m)$ , then $\\Omega $ verifies $barp(\\sigma )$ , where $\\sigma =\\sigma (\\psi _M,d_m)$ .",
"We prove this by induction on the number $n$ of vertices of $\\Omega $ .",
"If $n=3$ , i.e., $\\Omega $ is a triangle, the result follows from the law of sines.",
"Indeed, we only have to prove that $\\Omega $ has its minimum interior angle bounded away from zero.",
"Let $\\alpha $ be the minimum angle of $\\Omega $ (if there is more than one choice, we choose it arbitrarily) and let $l$ be the length of its opposite side.",
"Since $diam(\\Omega )$ is attained on one side of $\\Omega $ , we can assume, without loss of generality, $l \\ne diam(\\Omega )$ .",
"We call $\\beta $ the opposite angle to $diam(\\Omega )$ .",
"It is clear that $\\beta $ can not approach zero and since it is bounded above by $\\psi _M$ , we get that $1/\\sin (\\beta ) \\le C$ for some positive constant $C$ .",
"Then, from the law of sines and the assumption $melp(d_m)$ , we have $\\frac{\\sin (\\alpha )}{\\sin (\\beta )} = \\frac{l}{diam(\\Omega )} \\ge d_m.$ In consequence, $\\sin (\\alpha ) \\ge C^{-1} d_m$ which proves that $\\alpha $ is bounded away from zero.",
"Let $n>3$ .",
"Since the diameter of $\\Omega $ realizes as the length of its longest diagonal, i.e., the longest straight line joining two vertices of $\\Omega $ , we need to consider two cases depending if these vertices are consecutive or not.",
"Assume that $diam(\\Omega )$ is attained as the length of the line joining two non-consecutive vertices (these may not be unique, in this case we choose them arbitrarily).",
"We can divide $\\Omega $ by this diagonal into two convex polygons $\\Omega _1$ and $\\Omega _2$ with less number of vertices (see Figure REF (A)).",
"It is clear that both of them satisfy $MAC(\\psi _M)$ and, since $diam(\\Omega _i)=diam(\\Omega )$ and the set of vertices of $\\Omega _i$ is a subset of the vertices of $\\Omega $ , we conclude that $\\Omega _i$ also verifies $melp(d_m)$ .",
"Therefore, by the inductive hypothesis, $\\Omega _1$ and $\\Omega _2$ verify $barp(\\sigma _1)$ and $barp(\\sigma _2)$ , respectively, for some constants $\\sigma _1, \\sigma _2$ depending only on $\\psi _M$ and $d_m$ .",
"Then, since $\\rho (\\Omega ) \\ge \\rho (\\Omega _i)$ , $i=1,2$ , we have $\\displaystyle \\frac{diam(\\Omega )}{\\rho (\\Omega )} = \\frac{diam(\\Omega _i)}{\\rho (\\Omega )} \\le \\frac{diam(\\Omega _i)}{\\rho (\\Omega _i)} \\le \\sigma _i.$ Finally, if $diam(\\Omega )$ is attained on a side of $\\Omega $ , i.e., is the length of the line joining two consecutive vertices ${\\bf v}_{j-1}$ and ${\\bf v}_j$ (these may not be unique, in this case we choose them arbitrarily), we divide $\\Omega $ by the diagonal joining ${\\bf v}_{j-1}$ and ${\\bf v}_{j+1}$ into the triangle $T_1=\\Delta ({\\bf v}_{j-1}{\\bf v}_j{\\bf v}_{j+1})$ and a convex polygon $\\Omega _2$ (see Figure REF (B)).",
"It is clear that $T_1$ verifies $melp(d_m)$ and $MAC(\\psi _M)$ , so (by the case $n=3$ ) we have that $T_1$ satisfies $barp(\\sigma _1)$ for some positive constant $\\sigma _1$ .",
"Then, since $diam(T_1)=diam(\\Omega )$ and $\\rho (\\Omega ) \\ge \\rho (T_1)$ , we have $\\displaystyle \\frac{diam(\\Omega )}{\\rho (\\Omega )} = \\frac{diam(T_1)}{\\rho (\\Omega )} \\le \\frac{diam(T_1)}{\\rho (T_1)} \\le \\sigma _1.$ $\\begin{picture}(15,15)(-5,-5)\\framebox (15,15){}\\end{picture} $ Corollary 2.1 $[MAC, melp]$ and $[barp, MAC, melp]$ are equivalent conditions.",
"Finally, notice that reciprocal statement of Lemma REF is false.",
"Consider the following families of quadrilaterals: $\\mathcal {F}_1=\\lbrace K(1,1-s,s,1-s) \\rbrace _{0<s<1}$ where $K(1,1-s,s,1-s)$ denotes the convex quadrilateral with vertices $(0,0), (1,0), (s, 1-s)$ and $(0,1-s)$ , and $\\mathcal {F}_2=\\lbrace K(1,1,s,s) \\rbrace _{1/2<s<1}$ where $K(1,1,s,s)$ denotes the convex quadrilateral with vertices $(0,0), (1,0), (s, s)$ and $(0,1)$ .",
"Clearly, any quadrilateral belonging to $\\mathcal {F}_1 \\cup \\mathcal {F}_2$ is regular in the sense given by (REF ).",
"Each element of $\\mathcal {F}_1$ satisfies $MAC(3\\pi /4)$ , but taking $s \\rightarrow 0^+$ , we see that the minimum edge length property is violated.",
"On the other hand, each element of $\\mathcal {F}_2$ verifies $melp(1/2)$ ; but taking $s \\rightarrow 1/2^+$ , we see that the maximum angle condition is not satisfied.",
"Wachspress coordinates and the error estimate Wachspress coordinates We start this section by remembering the definition of Wachspress coordinates and some of their main properties [8], [15].",
"Henceforth, we denote by ${\\bf v}_1, {\\bf v}_2, \\dots , {\\bf v}_n$ the vertices of $\\Omega $ enumerated in counterclockwise order starting in an arbitrary vertex.",
"Let $\\bf x$ denote an interior point of $\\Omega $ and let $A_i(\\bf x)$ denote the area of the triangle with vertices $\\bf x$ , ${\\bf v}_i$ and ${\\bf v}_{i+1}$ , i.e., $A_i({\\bf x})=|\\Delta ({\\bf x} {\\bf v}_i {\\bf v}_{i+1})|$ , where, by convention, ${\\bf v}_0:= {\\bf v}_n$ and ${\\bf v}_{n+1}:={\\bf v}_1$ .",
"Let $B_i$ denote the area of the triangle with vertices ${\\bf v}_{i-1}$ , ${\\bf v}_i$ and ${\\bf v}_{i+1}$ , i.e., $B_i=|\\Delta ({\\bf v}_{i-1} {\\bf v}_i {\\bf v}_{i+1})|$ .",
"We summarize the notation in Figure REF .",
"Figure: (A): Notation for A i (𝐱)A_i({\\bf x}).",
"(B): Notation for B i B_i.Define the Wachspress weight function $w_i$ as the product of the area of the “boundary” triangle, formed by ${\\bf v}_i$ and its two adjacent vertices, and the areas of the $n-2$ interior triangles, formed by the point ${\\bf x}$ and the polygon's adjacent vertices (making sure to exclude the two interior triangles that contain the vertex ${\\bf v}_i$ ), i.e., $\\displaystyle w_i({\\bf x}) = B_i \\prod _{j \\ne i,i-1} A_j(\\bf x).$ After applying the standard normalization, Wachspress coordinates are then given by $\\displaystyle \\lambda _i({\\bf x}) = \\frac{w_i({\\bf x})}{\\sum _{j=1}^n w_j({\\bf x})}.$ An equivalent expression of (REF ) for $w_i$ is given in [12]; the main advantages of this alternative expression is that the result is easy to implement and it shows that only the edge $\\overline{{\\bf x} {\\bf v}_i}$ and its two adjacent angles $\\alpha _i$ and $\\delta _i$ are needed (see Figure REF (A)).",
"Indeed, $w_i$ can be written as $w_i({\\bf x}) = \\frac{\\cot (\\alpha _i)+\\cot (\\delta _i)}{\\left\\Vert {\\bf x}-{\\bf v}_i \\right\\Vert ^2}$ where $\\alpha _i=\\angle \\ {\\bf x} {\\bf v}_i {\\bf v}_{i+1}$ and $\\delta _i=\\beta _i-\\alpha _i$ with $\\beta _i$ being the inner angle of $\\Omega $ associated to ${\\bf v}_i$ (see Figure REF ).",
"The evaluation of the Wachspress basis function is carried out using elementary vector calculus operations.",
"The angles $\\alpha _i$ and $\\delta _i$ are not explicitly computed, as suggested in [12], vector cross product and vector dot product formulas are used to find the cotangents.",
"Wachspress coordinates have the well-known following properties: (I) (Non-negativeness) $\\lambda _i \\ge 0$ on $\\Omega $ .",
"(II) (Linear Completeness) for any linear function $\\ell :\\Omega \\rightarrow \\mathbb {R}$ , there holds $\\ell = \\sum _{i} \\ell ({\\bf v}_i) \\lambda _i$ .",
"(Considering the linear map $\\ell \\equiv 1$ yields $\\sum _{i} \\lambda _i = 1$ ; this property is usually named partition of unity).",
"(III) (Invariance) If $L:\\mathbb {R}^2 \\rightarrow \\mathbb {R}^2$ is a linear map and $S:\\mathbb {R}^2 \\rightarrow \\mathbb {R}^2$ is a composition of rotation, translation and uniform scaling transformations, then $\\lambda _i({\\bf x})=\\lambda _i^L(L({\\bf x}))=\\lambda _i^S(S({\\bf x}))$ , where $\\lambda _i^F(F({\\bf x}))$ denotes a set of barycentric coordinates on $F(\\Omega )$ .",
"(IV) (Linear precision) $\\sum _{i} {\\bf v}_i \\lambda _i({\\bf x})={\\bf x}$ , i.e., every point on $\\Omega $ can be written as a convex combination of the vertices ${\\bf v}_1, {\\bf v}_2, \\dots , {\\bf v}_n$ .",
"(V) (Interpolation) $\\lambda _i({\\bf v}_j)=\\delta _{ij}$ .",
"Error estimate to the first-order Wachspress interpolation operator We only give a brief overview of some definitions and results which are of interest to us; for more details we refer to [7], [9], [14], [13].",
"Let $\\lbrace \\lambda _i \\rbrace $ be the Wachspress coordinates associated to $\\Omega $ (see (REF )).",
"Then, we can consider the first-order interpolation operator $I:H^2(\\Omega ) \\rightarrow span \\lbrace \\lambda _i \\rbrace \\subset H^1(\\Omega )$ defined as $\\displaystyle I_{\\Omega }u=Iu := \\sum _{i} u({\\bf v}_i) \\lambda _i.$ Properties (I)-(V) of the Wachspress coordinates (more generally, generalized barycentric coordinates) guarantee that $I$ has the desirable properties of an interpolant.",
"For this interpolant, called here the first-order Wachspress interpolation operator, the optimal convergence estimate $\\left\\Vert u-Iu \\right\\Vert _{H^1(\\Omega )} \\le C diam(\\Omega ) |u|_{H^2(\\Omega )}$ on polygons satisfying $[barp, MAC, melp]$ was proved [9].",
"Remark 3.1 Thanks to Corollary REF , we can affirm that (REF ) holds on general convex polygons satisfying $[MAC, melp]$ .",
"About sharpness on geometric restrictions Since $[MAC, melp]$ are sufficient conditions to obtain (REF ), we wonder if some of these requirements can be relaxed in order to obtain the error estimate.",
"This question was partially answered in [9], where a counterexample, using pentagonal elements, is given in order to show that the $MAC$ can not be removed.",
"For the sake of completeness, in Counterexample REF , we give a family of quadrilateral elements which does not satisfy $MAC$ but it verifies $melp$ and (REF ) does not hold.",
"This example shows two things: $MAC$ is necessary in order to obtain the error estimate and, since every element in this family is regular in the sense given by (REF ), $barp$ is not enough to obtain (REF ).",
"On the other hand, in Counterexample REF , we present a family of quadrilaterals which does not satisfy $melp$ but it verifies $MAC$ and (REF ) does not hold.",
"Then, in order to obtain the interpolation error estimate, $melp$ is necessary.",
"In this sense, the question raised in [9] about identifying sharp geometric restrictions under which the error estimates for the first-order Wachspress interpolation operator holds can be considered as answered.",
"Figure: Schematic picture of K s K_s and T s T_s (hatched area) considered in Counterexample .Counterexample 4.1 Consider the convex quadrilateral $K_s$ with the vertices ${\\bf v}_1=(0,0), {\\bf v}_2=(1,0), {\\bf v}_3=(s,s)$ and ${\\bf v}_4=(0,1)$ , where $1/2<s<1$ .",
"We will be interested in the case when $s$ tends to $1/2$ since then the family of quadrilaterals $\\lbrace K_s \\rbrace $ does not satisfy the maximum angle condition although it satisfies $melp(1/2)$ .",
"Consider the function $u({\\bf x})=x(1-x)$ .",
"Since $u({\\bf v}_1)=0=u({\\bf v}_2)=u({\\bf v}_4)$ , we have $Iu({\\bf x})=u({\\bf v}_3) \\lambda _3({\\bf x})= s(1-s) \\lambda _3({\\bf x}).$ An straightforward computation yields $\\displaystyle \\lambda _3({\\bf x}) = \\frac{(2s-1)x}{s} \\frac{y}{(s-1)(x+y)+s},$ therefore $\\displaystyle \\frac{\\partial \\lambda _3}{\\partial y} = \\frac{(2s-1)x}{s} \\frac{(s-1)x+s}{[(s-1)(x+y)+s]^2}.$ Consider the triangle $T_s$ with vertices $(1/4,3/4)$ , $(1/2,1/2)$ and $(1/2, (3s-1)/(2s))$ (see Figure REF ).",
"Then, on $T_s$ , we have $1/4 \\le x \\le 1/2$ , $1/2 \\le y \\le (3s-1)/(2s)$ and $x+y \\ge 1$ , so it follows that $0<(s-1)(x+y)+s \\le 2s-1\\quad \\text{and} \\quad (s-1)x+s \\ge (3s-1)/2$ and hence $\\displaystyle \\frac{\\partial \\lambda _3}{\\partial y} \\ge \\frac{(2s-1)}{4s} \\frac{3s-1}{2(2s-1)^2}=\\frac{3s-1}{8s(2s-1)}.$ Then $|u-Iu|_{H^1(K_s)} \\ge \\left\\Vert \\frac{\\partial (u-Iu)}{\\partial y} \\right\\Vert _{L^2(K_s)} = \\left\\Vert \\frac{\\partial Iu}{\\partial y} \\right\\Vert _{L^2(K_s)} = s(1-s)\\left\\Vert \\frac{\\partial \\lambda _3}{\\partial y} \\right\\Vert _{L^2(K_s)}$ and, consequently, $|u-Iu|_{H^1(K_s)} \\ge s(1-s)\\left\\Vert \\frac{\\partial \\lambda _3}{\\partial y} \\right\\Vert _{L^2(T_s)}.$ Since $|T_s|=(2s-1)/(2^4s)$ , we have $\\left\\Vert \\frac{\\partial \\lambda _3}{\\partial y} \\right\\Vert _{L^2(T_s)}^2 \\ge \\frac{(3s-1)^2}{(8s)^2(2s-1)^2}|T_s|=\\frac{(3s-1)^2}{2^{10}s^3(2s-1)} \\rightarrow \\infty $ when $s \\rightarrow 1/2^+$ .",
"Finally, as $|u|_{H^2(K_s)} = 2 |K_s|^{1/2} \\le 2$ and $diam(K_s)=\\sqrt{2}$ , we conclude that (REF ) can not hold.",
"Figure: Schematic picture of K s K_s and D s D_s (hatched area) considered in Counterexample .Counterexample 4.2 Consider now the convex quadrilateral $K_s$ with the vertices ${\\bf v}_1=(0,0), {\\bf v}_2=(1,0), {\\bf v}_3=(1-\\@root 4 \\of {s},s)$ and ${\\bf v}_4=(0,s)$ , where $0 < s < (1/2)^4$ .",
"Note that the family of quadrilaterals $\\lbrace K_s \\rbrace $ satisfies $MAC(\\pi /2+\\tan ^{-1}(2^3))$ $($ independently of $s)$ but it does not satisfy the minimum edge length property when $s$ tends to zero since $\\left\\Vert {\\bf v}_1-{\\bf v}_4 \\right\\Vert = s \\rightarrow 0^+$ and $diam(K_s) \\sim 1$ .",
"Consider the function $u({\\bf x})=x^2$ .",
"Since $u({\\bf v}_1)=0=u({\\bf v}_4)$ , we have, calling $a := 1-\\@root 4 \\of {s}$ , $Iu({\\bf x})=u({\\bf v}_2) \\lambda _2({\\bf x})+u({\\bf v}_3) \\lambda _3({\\bf x}) = \\lambda _2({\\bf x})+ a^2 \\lambda _3({\\bf x})$ where $\\lambda _2({\\bf x})=\\frac{x(s-y)}{s+y(a-1)} \\quad \\text{and} \\quad \\lambda _3({\\bf x})=\\frac{xy}{s+y(a-1)}.$ A simple computation yields $\\frac{\\partial (Iu-u)}{\\partial y} = \\frac{\\partial Iu}{\\partial y} = \\frac{xsa(a-1)}{(s+y(a-1))^2}.$ Let $D_s = K_s \\cap \\lbrace x \\ge 1/2 \\rbrace $ $($ see Figure REF ).",
"Since $a-1 <0$ , we get $s+y(a-1) \\le s$ and then, on $D_s$ , we have $\\left| \\frac{\\partial (Iu-u)}{\\partial y} \\right| \\ge \\frac{xa(1-a)}{s} \\ge \\frac{a(1-a)}{2s}.$ Therefore, $|Iu-u|_{H^1(K_s)}^2 \\ge \\left\\Vert \\frac{\\partial (Iu-u)}{\\partial y} \\right\\Vert _{L^2(K_s)}^2 \\ge \\left\\Vert \\frac{\\partial (Iu-u)}{\\partial y} \\right\\Vert _{L^2(D_s)}^2 \\ge \\frac{a^2(1-a)^2}{4s^2} |D_s|,$ and since $|D_s|=as/2$ , we conclude that $|Iu-u|_{H^1(K_s)}^2 \\ge \\frac{a^3(1-a)^2}{8s} =\\frac{(1-\\@root 4 \\of {s})^3}{8\\sqrt{s}}$ which tends to infinity when $s$ tends to zero.",
"Finally, since $|u|_{H^2(K_s)} = 2 |K_s|^{1/2} \\le 2$ and $diam(K_s) \\sim 1$ , we conclude that (REF ) can not hold."
],
[
"Wachspress coordinates",
"We start this section by remembering the definition of Wachspress coordinates and some of their main properties [8], [15].",
"Henceforth, we denote by ${\\bf v}_1, {\\bf v}_2, \\dots , {\\bf v}_n$ the vertices of $\\Omega $ enumerated in counterclockwise order starting in an arbitrary vertex.",
"Let $\\bf x$ denote an interior point of $\\Omega $ and let $A_i(\\bf x)$ denote the area of the triangle with vertices $\\bf x$ , ${\\bf v}_i$ and ${\\bf v}_{i+1}$ , i.e., $A_i({\\bf x})=|\\Delta ({\\bf x} {\\bf v}_i {\\bf v}_{i+1})|$ , where, by convention, ${\\bf v}_0:= {\\bf v}_n$ and ${\\bf v}_{n+1}:={\\bf v}_1$ .",
"Let $B_i$ denote the area of the triangle with vertices ${\\bf v}_{i-1}$ , ${\\bf v}_i$ and ${\\bf v}_{i+1}$ , i.e., $B_i=|\\Delta ({\\bf v}_{i-1} {\\bf v}_i {\\bf v}_{i+1})|$ .",
"We summarize the notation in Figure REF .",
"Figure: (A): Notation for A i (𝐱)A_i({\\bf x}).",
"(B): Notation for B i B_i.Define the Wachspress weight function $w_i$ as the product of the area of the “boundary” triangle, formed by ${\\bf v}_i$ and its two adjacent vertices, and the areas of the $n-2$ interior triangles, formed by the point ${\\bf x}$ and the polygon's adjacent vertices (making sure to exclude the two interior triangles that contain the vertex ${\\bf v}_i$ ), i.e., $\\displaystyle w_i({\\bf x}) = B_i \\prod _{j \\ne i,i-1} A_j(\\bf x).$ After applying the standard normalization, Wachspress coordinates are then given by $\\displaystyle \\lambda _i({\\bf x}) = \\frac{w_i({\\bf x})}{\\sum _{j=1}^n w_j({\\bf x})}.$ An equivalent expression of (REF ) for $w_i$ is given in [12]; the main advantages of this alternative expression is that the result is easy to implement and it shows that only the edge $\\overline{{\\bf x} {\\bf v}_i}$ and its two adjacent angles $\\alpha _i$ and $\\delta _i$ are needed (see Figure REF (A)).",
"Indeed, $w_i$ can be written as $w_i({\\bf x}) = \\frac{\\cot (\\alpha _i)+\\cot (\\delta _i)}{\\left\\Vert {\\bf x}-{\\bf v}_i \\right\\Vert ^2}$ where $\\alpha _i=\\angle \\ {\\bf x} {\\bf v}_i {\\bf v}_{i+1}$ and $\\delta _i=\\beta _i-\\alpha _i$ with $\\beta _i$ being the inner angle of $\\Omega $ associated to ${\\bf v}_i$ (see Figure REF ).",
"The evaluation of the Wachspress basis function is carried out using elementary vector calculus operations.",
"The angles $\\alpha _i$ and $\\delta _i$ are not explicitly computed, as suggested in [12], vector cross product and vector dot product formulas are used to find the cotangents.",
"Wachspress coordinates have the well-known following properties: (I) (Non-negativeness) $\\lambda _i \\ge 0$ on $\\Omega $ .",
"(II) (Linear Completeness) for any linear function $\\ell :\\Omega \\rightarrow \\mathbb {R}$ , there holds $\\ell = \\sum _{i} \\ell ({\\bf v}_i) \\lambda _i$ .",
"(Considering the linear map $\\ell \\equiv 1$ yields $\\sum _{i} \\lambda _i = 1$ ; this property is usually named partition of unity).",
"(III) (Invariance) If $L:\\mathbb {R}^2 \\rightarrow \\mathbb {R}^2$ is a linear map and $S:\\mathbb {R}^2 \\rightarrow \\mathbb {R}^2$ is a composition of rotation, translation and uniform scaling transformations, then $\\lambda _i({\\bf x})=\\lambda _i^L(L({\\bf x}))=\\lambda _i^S(S({\\bf x}))$ , where $\\lambda _i^F(F({\\bf x}))$ denotes a set of barycentric coordinates on $F(\\Omega )$ .",
"(IV) (Linear precision) $\\sum _{i} {\\bf v}_i \\lambda _i({\\bf x})={\\bf x}$ , i.e., every point on $\\Omega $ can be written as a convex combination of the vertices ${\\bf v}_1, {\\bf v}_2, \\dots , {\\bf v}_n$ .",
"(V) (Interpolation) $\\lambda _i({\\bf v}_j)=\\delta _{ij}$ ."
],
[
"Error estimate to the first-order Wachspress interpolation operator",
"We only give a brief overview of some definitions and results which are of interest to us; for more details we refer to [7], [9], [14], [13].",
"Let $\\lbrace \\lambda _i \\rbrace $ be the Wachspress coordinates associated to $\\Omega $ (see (REF )).",
"Then, we can consider the first-order interpolation operator $I:H^2(\\Omega ) \\rightarrow span \\lbrace \\lambda _i \\rbrace \\subset H^1(\\Omega )$ defined as $\\displaystyle I_{\\Omega }u=Iu := \\sum _{i} u({\\bf v}_i) \\lambda _i.$ Properties (I)-(V) of the Wachspress coordinates (more generally, generalized barycentric coordinates) guarantee that $I$ has the desirable properties of an interpolant.",
"For this interpolant, called here the first-order Wachspress interpolation operator, the optimal convergence estimate $\\left\\Vert u-Iu \\right\\Vert _{H^1(\\Omega )} \\le C diam(\\Omega ) |u|_{H^2(\\Omega )}$ on polygons satisfying $[barp, MAC, melp]$ was proved [9].",
"Remark 3.1 Thanks to Corollary REF , we can affirm that (REF ) holds on general convex polygons satisfying $[MAC, melp]$ ."
],
[
"About sharpness on geometric restrictions",
"Since $[MAC, melp]$ are sufficient conditions to obtain (REF ), we wonder if some of these requirements can be relaxed in order to obtain the error estimate.",
"This question was partially answered in [9], where a counterexample, using pentagonal elements, is given in order to show that the $MAC$ can not be removed.",
"For the sake of completeness, in Counterexample REF , we give a family of quadrilateral elements which does not satisfy $MAC$ but it verifies $melp$ and (REF ) does not hold.",
"This example shows two things: $MAC$ is necessary in order to obtain the error estimate and, since every element in this family is regular in the sense given by (REF ), $barp$ is not enough to obtain (REF ).",
"On the other hand, in Counterexample REF , we present a family of quadrilaterals which does not satisfy $melp$ but it verifies $MAC$ and (REF ) does not hold.",
"Then, in order to obtain the interpolation error estimate, $melp$ is necessary.",
"In this sense, the question raised in [9] about identifying sharp geometric restrictions under which the error estimates for the first-order Wachspress interpolation operator holds can be considered as answered.",
"Figure: Schematic picture of K s K_s and T s T_s (hatched area) considered in Counterexample .Counterexample 4.1 Consider the convex quadrilateral $K_s$ with the vertices ${\\bf v}_1=(0,0), {\\bf v}_2=(1,0), {\\bf v}_3=(s,s)$ and ${\\bf v}_4=(0,1)$ , where $1/2<s<1$ .",
"We will be interested in the case when $s$ tends to $1/2$ since then the family of quadrilaterals $\\lbrace K_s \\rbrace $ does not satisfy the maximum angle condition although it satisfies $melp(1/2)$ .",
"Consider the function $u({\\bf x})=x(1-x)$ .",
"Since $u({\\bf v}_1)=0=u({\\bf v}_2)=u({\\bf v}_4)$ , we have $Iu({\\bf x})=u({\\bf v}_3) \\lambda _3({\\bf x})= s(1-s) \\lambda _3({\\bf x}).$ An straightforward computation yields $\\displaystyle \\lambda _3({\\bf x}) = \\frac{(2s-1)x}{s} \\frac{y}{(s-1)(x+y)+s},$ therefore $\\displaystyle \\frac{\\partial \\lambda _3}{\\partial y} = \\frac{(2s-1)x}{s} \\frac{(s-1)x+s}{[(s-1)(x+y)+s]^2}.$ Consider the triangle $T_s$ with vertices $(1/4,3/4)$ , $(1/2,1/2)$ and $(1/2, (3s-1)/(2s))$ (see Figure REF ).",
"Then, on $T_s$ , we have $1/4 \\le x \\le 1/2$ , $1/2 \\le y \\le (3s-1)/(2s)$ and $x+y \\ge 1$ , so it follows that $0<(s-1)(x+y)+s \\le 2s-1\\quad \\text{and} \\quad (s-1)x+s \\ge (3s-1)/2$ and hence $\\displaystyle \\frac{\\partial \\lambda _3}{\\partial y} \\ge \\frac{(2s-1)}{4s} \\frac{3s-1}{2(2s-1)^2}=\\frac{3s-1}{8s(2s-1)}.$ Then $|u-Iu|_{H^1(K_s)} \\ge \\left\\Vert \\frac{\\partial (u-Iu)}{\\partial y} \\right\\Vert _{L^2(K_s)} = \\left\\Vert \\frac{\\partial Iu}{\\partial y} \\right\\Vert _{L^2(K_s)} = s(1-s)\\left\\Vert \\frac{\\partial \\lambda _3}{\\partial y} \\right\\Vert _{L^2(K_s)}$ and, consequently, $|u-Iu|_{H^1(K_s)} \\ge s(1-s)\\left\\Vert \\frac{\\partial \\lambda _3}{\\partial y} \\right\\Vert _{L^2(T_s)}.$ Since $|T_s|=(2s-1)/(2^4s)$ , we have $\\left\\Vert \\frac{\\partial \\lambda _3}{\\partial y} \\right\\Vert _{L^2(T_s)}^2 \\ge \\frac{(3s-1)^2}{(8s)^2(2s-1)^2}|T_s|=\\frac{(3s-1)^2}{2^{10}s^3(2s-1)} \\rightarrow \\infty $ when $s \\rightarrow 1/2^+$ .",
"Finally, as $|u|_{H^2(K_s)} = 2 |K_s|^{1/2} \\le 2$ and $diam(K_s)=\\sqrt{2}$ , we conclude that (REF ) can not hold.",
"Figure: Schematic picture of K s K_s and D s D_s (hatched area) considered in Counterexample .Counterexample 4.2 Consider now the convex quadrilateral $K_s$ with the vertices ${\\bf v}_1=(0,0), {\\bf v}_2=(1,0), {\\bf v}_3=(1-\\@root 4 \\of {s},s)$ and ${\\bf v}_4=(0,s)$ , where $0 < s < (1/2)^4$ .",
"Note that the family of quadrilaterals $\\lbrace K_s \\rbrace $ satisfies $MAC(\\pi /2+\\tan ^{-1}(2^3))$ $($ independently of $s)$ but it does not satisfy the minimum edge length property when $s$ tends to zero since $\\left\\Vert {\\bf v}_1-{\\bf v}_4 \\right\\Vert = s \\rightarrow 0^+$ and $diam(K_s) \\sim 1$ .",
"Consider the function $u({\\bf x})=x^2$ .",
"Since $u({\\bf v}_1)=0=u({\\bf v}_4)$ , we have, calling $a := 1-\\@root 4 \\of {s}$ , $Iu({\\bf x})=u({\\bf v}_2) \\lambda _2({\\bf x})+u({\\bf v}_3) \\lambda _3({\\bf x}) = \\lambda _2({\\bf x})+ a^2 \\lambda _3({\\bf x})$ where $\\lambda _2({\\bf x})=\\frac{x(s-y)}{s+y(a-1)} \\quad \\text{and} \\quad \\lambda _3({\\bf x})=\\frac{xy}{s+y(a-1)}.$ A simple computation yields $\\frac{\\partial (Iu-u)}{\\partial y} = \\frac{\\partial Iu}{\\partial y} = \\frac{xsa(a-1)}{(s+y(a-1))^2}.$ Let $D_s = K_s \\cap \\lbrace x \\ge 1/2 \\rbrace $ $($ see Figure REF ).",
"Since $a-1 <0$ , we get $s+y(a-1) \\le s$ and then, on $D_s$ , we have $\\left| \\frac{\\partial (Iu-u)}{\\partial y} \\right| \\ge \\frac{xa(1-a)}{s} \\ge \\frac{a(1-a)}{2s}.$ Therefore, $|Iu-u|_{H^1(K_s)}^2 \\ge \\left\\Vert \\frac{\\partial (Iu-u)}{\\partial y} \\right\\Vert _{L^2(K_s)}^2 \\ge \\left\\Vert \\frac{\\partial (Iu-u)}{\\partial y} \\right\\Vert _{L^2(D_s)}^2 \\ge \\frac{a^2(1-a)^2}{4s^2} |D_s|,$ and since $|D_s|=as/2$ , we conclude that $|Iu-u|_{H^1(K_s)}^2 \\ge \\frac{a^3(1-a)^2}{8s} =\\frac{(1-\\@root 4 \\of {s})^3}{8\\sqrt{s}}$ which tends to infinity when $s$ tends to zero.",
"Finally, since $|u|_{H^2(K_s)} = 2 |K_s|^{1/2} \\le 2$ and $diam(K_s) \\sim 1$ , we conclude that (REF ) can not hold."
]
] | 1606.04975 |
[
[
"The Picard group of motivic A(1)"
],
[
"Abstract We show that the Picard group $Pic(A(1))$ of the stable category of modules over $\\mathbb{C}$-motivic $A(1)$ is isomorphic to $\\mathbb{Z}^4$.",
"By comparison, the Picard group of classical $A(1)$ is $\\mathbb{Z}^2 \\oplus \\mathbb{Z}/2$.",
"One extra copy of $\\mathbb{Z}$ arises from the motivic bigrading.",
"The joker is a well-known exotic element of order $2$ in the Picard group of classical $A(1)$.",
"The $\\mathbb{C}$-motivic joker has infinite order."
],
[
"The Picard group of $\\mathcal {A}(1)$",
"Let $\\mathcal {A}(1)^{\\mathrm {cl}}$ be the subalgebra of the classical mod 2 Steenrod algebra generated by $\\operatorname{Sq}^1$ and $\\operatorname{Sq}^2$ .",
"The stable module category $\\mathrm {Stab}(\\mathcal {A}(1)^{\\mathrm {cl}})$ is the category whose objects are the finitely generated graded left $\\mathcal {A}(1)^{\\mathrm {cl}}$ -modules, and whose morphisms are the usual $\\mathcal {A}(1)^{\\mathrm {cl}}$ -module maps, modulo maps that factor through projective $\\mathcal {A}(1)^{\\mathrm {cl}}$ -modules.",
"The stable module category $\\mathrm {Stab}(\\mathcal {A}(1)^{\\mathrm {cl}})$ is equipped with a tensor product over $\\mathbb {F}_2$ .",
"The unit of this pairing is $\\mathbb {F}_2$ , and an object $M$ of $\\mathrm {Stab}(\\mathcal {A}(1)^{\\mathrm {cl}})$ is invertible if there exists another $\\mathcal {A}(1)^{\\mathrm {cl}}$ -module $N$ such that $M \\otimes _{\\mathbb {F}_2} N$ is stably isomorphic to $\\mathbb {F}_2$ .",
"The Picard group $\\operatorname{Pic}(\\mathcal {A}(1)^{\\mathrm {cl}})$ is the set of invertible stable isomorphism classes, with group operation given by tensor product over $\\mathbb {F}_2$ .",
"$\\operatorname{Ext}$ groups over $\\mathcal {A}(1)^{\\mathrm {cl}}$ are invariants of stable isomorphism classes of $\\mathcal {A}(1)^{\\mathrm {cl}}$ -modules.",
"Thus, $\\mathrm {Stab}(\\mathcal {A}(1)^{\\mathrm {cl}})$ is the natural category on which $\\operatorname{Ext}$ groups over $\\mathcal {A}(1)^{\\mathrm {cl}}$ are defined.",
"These $\\operatorname{Ext}$ groups are of topological interest because of the Adams spectral sequence $ E_2 = \\operatorname{Ext}_{\\mathcal {A}(1)^{\\mathrm {cl}}}( H\\mathbb {F}_2^{*}(X), \\mathbb {F}_2) \\Rightarrow \\mathit {ko}_{*}(X)^{\\wedge }_2$ converging to 2-completed $\\mathit {ko}$ -homology.",
"Adams and Priddy computed $\\operatorname{Pic}(\\mathcal {A}(1)^{\\mathrm {cl}})$ while studying infinite loop space structures on the classifying space $BSO$ [3].",
"They found that the Picard group is isomorphic to $\\mathbb {Z}^2 \\oplus \\mathbb {Z}/2$ .",
"One copy of $\\mathbb {Z}$ comes from the grading; one can shift the grading on $A(1)^{\\mathrm {cl}}$ -modules to obtain “new” $\\mathcal {A}(1)^{\\mathrm {cl}}$ -modules.",
"The other copy of $\\mathbb {Z}$ comes from the algebraic loop functor that is a formal part of the stable module category; see Definition REF below for more details.",
"The copy of $\\mathbb {Z}/2$ in $\\operatorname{Pic}(\\mathcal {A}(1)^{\\mathrm {cl}})$ is the most interesting part of the calculation.",
"It is exotic in the sense that it doesn't follow from the formal theory of stable module categories and Picard groups.",
"The copy of $\\mathbb {Z}/2$ is generated by the joker $J$ shown in Figure REF .",
"It turns out that $J \\otimes _{\\mathbb {F}_2} J$ is stably isomorphic to $\\mathbb {F}_2$ , so $J$ has order 2 in $\\operatorname{Pic}(\\mathcal {A}(1)^{\\mathrm {cl}})$ ."
],
[
"The motivic setting",
"There has been much recent work on the computational side of motivic homotopy theory.",
"In particular, the algebraic properties of the motivic Steenrod algebra have come under close scrutiny.",
"As part of this program, it is natural to ask about the Picard group of the motivic version of $\\mathcal {A}(1)$ .",
"The goal of this article is to carry out this computation for $-motivic $ A(1)$, which is the simplest motivic case.$ The fundamental difficulty in the motivic situation is that the ground ring $\\mathbb {M}_2$ is not a field.",
"Rather, it is a graded polynomial ring $\\mathbb {F}_2[\\tau ]$ .",
"Therefore, we must be careful to insert $\\mathbb {M}_2$ -freeness hypotheses at the appropriate places.",
"We will show that $\\operatorname{Pic}( \\mathcal {A}(1) )$ is isomorphic to $\\mathbb {Z}^4$ .",
"Two copies of $\\mathbb {Z}$ arise from the motivic bigrading, and one copy of $\\mathbb {Z}$ comes from the algebraic loop functor.",
"This leaves one copy of $\\mathbb {Z}$ , which is generated by the motivic joker $J$ (see Figure REF ).",
"It turns out that the motivic joker has infinite order.",
"The order of the motivic joker is the essential new aspect of the motivic calculation.",
"There are two main ideas in the proof.",
"First, the Hopf algebra $\\mathcal {A}(1)/\\tau $ is isomorphic to the group algebra of the dihedral group $D_8$ of order 8, so $\\mathcal {A}(1)/\\tau $ is well-understood.",
"In particular, the Picard group of $\\mathcal {A}(1)/\\tau $ is known.",
"Second, consider the functor that takes an $\\mathcal {A}(1)$ -module $M$ to its quotient $M/\\tau $ .",
"In general, quotienting is not an exact functor.",
"However, it turns out to be well-behaved for $\\mathcal {A}(1)$ -modules that are $\\mathbb {M}_2$ -free.",
"Using this well-behaved functor, we can pull back information about the Picard group of $\\mathcal {A}(1)/\\tau $ to information about the Picard group of $\\mathcal {A}(1)$ .",
"The difference between the $-motivic and classical Picard groupsis a familiar one.",
"Frequently, motivic computations are larger thanclassical ones.",
"However, they are also often more regular.",
"Thissituation is clearly displayed in our work, where the motivic Picardgroup is free, while the classical Picard group has torsion.$ We do not consider the Picard group of motivic $\\mathcal {A}(1)$ over other base fields.",
"The $-motivic phenomena described in thispaper will occur over other base fields, but it is possible that additionalcomplications arise.$ Our computation of the Picard group of motivic $\\mathcal {A}(1)$ is potentially useful for the following problem.",
"From our perspective, the most essential property of the $-motivic spectrum $ ko$is that its cohomology is isomorphic to $ A/ /A(1)$ \\cite {Is10}.One might ask whether such a $ -motivic spectrum is unique.",
"Suppose that $X$ and $Y$ are $-motivic spectra whose cohomology modules are both isomorphicto $ A/ /A(1)$.",
"In order to construct an equivalence between $ X$ and $ Y$,one could compute the maps between $ X$ and $ Y$ viathe motivic Adams spectral sequence, whose $ E2$-page takes the form$ ExtA (A/ /A(1), A/ /A(1) )$.",
"By a standard change of ringstheorem, this $ E2$-page is equal to$ ExtA(1) (M2, A/ /A(1) )$.It is possible that this Adams spectral sequence is analyzable,because $ A/ /A(1)$ probably splits as an $ A(1)$-module intosummands that belong to the Picard group.We leave the details for future work.$"
],
[
"Finite motivic Hopf algebras",
"We use the same notation and framework as in [9].",
"We work in the $-motivic setting at the prime 2.",
"The base ring is the motivic cohomology$ H,( S0,0; F2) $of the sphere spectrum.We write $ M2$ for this ring; it is isomorphic to$ F2[]$ with $$ in bidegree $ (0,1)$.Objects are bigraded in the form $ (s,w)$,where $ s$ corresponds to the classical internal degree and $ w$ is the motivic weight.$ Let $\\mathcal {A}$ be the $-motivic Steenrod algebra at the prime 2.",
"This Hopf algebra over $ M2$ was first computed in \\cite {Voe03}, and its structure is thoroughly understood.$ A fundamental difference between the $-motivic and the classicalsituations is that the base ring $ M2$ is not a field.Therefore, we must add freeness over $ M2$ as a hypothesisin Definition \\ref {defn:mot-Hopf} below.$ Definition 2.1 A finite motivic Hopf algebra is a cocommutative bigraded Hopf algebra over $\\mathbb {M}_2$ that is finitely generated and free as an $\\mathbb {M}_2$ -module.",
"Example 2.2 Recall the subalgebras $\\mathcal {A}(n)^{\\mathrm {cl}}$ and $\\mathcal {E}(n)^{\\mathrm {cl}}$ of the classical Steenrod algebra [2].",
"These subalgebras have $-motivic analogues,and they are finite motivic Hopf algebras.$ Throughout the article, $A$ will represent an arbitrary finite motivic Hopf algebra, while $\\mathcal {A}$ represents the $-motivic Steenrod algebra.Note that $ A$ is not finitely generated as an $ M2$-module.However, we are primarily interested in the subalgebra $ A(1)$ of$ A$ generated by $ Sq1$ and $ Sq2$, and $ A(1)$ isa finitely generated $ M2$-module.$ Lemma 2.3 Suppose that $A$ is a finite motivic Hopf algebra.",
"If $M$ is a finitely generated projective $A$ -module, then it is a finitely generated free $\\mathbb {M}_2$ -module.",
"Suppose that $M$ is a finitely generated projective $A$ -module.",
"Then $M$ is a summand of free $A$ -module $F$ .",
"The module $F$ is free and finitely generated as an $\\mathbb {M}_2$ -module, since $A$ is free and finitely generated as an $\\mathbb {M}_2$ -module.",
"Therefore, as an $\\mathbb {M}_2$ -module, $M$ is a summand of a free $\\mathbb {M}_2$ -module.",
"This shows that $M$ is a finitely generated projective $\\mathbb {M}_2$ -module.",
"It remains to show that finitely generated projective $\\mathbb {M}_2$ -modules are free.",
"The ring $\\mathbb {M}_2$ is a graded principal ideal domain whose graded ideals are of the form $(\\tau ^k)$ .",
"Therefore, a finitely generated $\\mathbb {M}_2$ -module is a direct sum of a free module and cyclic modules of the form $\\mathbb {M}_2/ \\tau ^k$ .",
"It follows that finitely generated projective $\\mathbb {M}_2$ -modules are the same as finitely generated free $\\mathbb {M}_2$ -modules."
],
[
"The stable category",
"We now recall the basic framework of stable module categories, as applied to a finite motivic Hopf algebra $A$ .",
"The stable category of modules over a group algebra is a classical construction in group representation theory [6].",
"In the case of a finite motivic Hopf algebra $A$ , the theory is similar to the case when $A$ is a finite dimensional graded connected Hopf algebra over a field, for which a good reference is [10].",
"However, since the base ring $\\mathbb {M}_2$ of a finite motivic Hopf algebra is not a field, one has to pay attention to the underlying theory of $\\mathbb {M}_2$ -modules, and add $\\mathbb {M}_2$ -freeness hypotheses when appropriate.",
"Definition 2.4 Let ${ }_{A}\\mathbf {Mod}$ be the category of bigraded finitely generated left $A$ -modules, and let ${ }_{A}\\mathbf {Mod}^{\\textnormal {f}}$ be the full subcategory of ${ }_{A}\\mathbf {Mod}$ consisting of left $A$ -modules that are free over $\\mathbb {M}_2$ .",
"Definition 2.5 Let $\\mathrm {Stab}(A)$ be the category whose objects are the same as in ${ }_{A}\\mathbf {Mod}^{\\textnormal {f}}$ , and whose morphisms are given by $\\operatorname{Hom}_{\\mathrm {Stab}(A)}(M,N) = {\\raisebox {.2em}{\\operatorname{Hom}_{A}(M,N)}\\left\\bad.\\raisebox {-.2em}{ \\sim }\\right.",
"},$ where two morphisms $f$ and $g$ are equivalent if their difference factors through a projective $A$ -module.",
"If $M$ and $N$ are objects of ${ }_{A}\\mathbf {Mod}^{\\textnormal {f}}$ , then we write $M \\cong N$ if $M$ and $N$ are stably equivalent, i.e., if they are isomorphic in the stable category $\\mathrm {Stab}(A)$ .",
"Our main interest is the Picard group $\\operatorname{Pic}(A)$ of the stable category of some finite motivic Hopf algebra $A$ .",
"We will see below in Remark REF that all representatives of every element in $\\operatorname{Pic}(A)$ are actually free over $\\mathbb {M}_2$ and thus captured by $\\mathrm {Stab}(A)$ .",
"In other words, the assumptions about $\\mathbb {M}_2$ -freeness in Definitions REF and REF are no loss of generality.",
"In the same vein, it is essential that we use constructions that preserve $\\mathbb {M}_2$ -free $A$ -modules.",
"For example for any finitely generated $A$ -module $M$ (not necessarily $\\mathbb {M}_2$ -free), the algebraic loop $\\Omega M$ (defined below in Definition REF ) is free over $\\mathbb {M}_2$ , as it is the kernel of a map from a finitely generated free $\\mathbb {M}_2$ -module.",
"The stable category $\\mathrm {Stab}(A)$ is naturally enriched over $A$ -modules, since the equivalence relation on morphisms is $A$ -linear.",
"The category $\\mathrm {Stab}(A)$ has additional structure that we describe next.",
"Proposition 2.6 The category $\\mathrm {Stab}(A)$ is a closed symmetric monoidal category.",
"This is a standard result from the theory of stable modules; see [10] for example.",
"The only additional observation is that the tensor product of $\\mathbb {M}_2$ -free modules is $\\mathbb {M}_2$ -free."
],
[
"Picard groups",
"Definition 2.7 Let $A$ be a finite motivic Hopf algebra.",
"The Picard group $\\operatorname{Pic}(A)$ is the group (of isomorphism classes) of invertible objects of $\\mathrm {Stab}(A)$ under the monoidal structure, i.e., the group of stably invertible modules with the tensor product as group law.",
"Note that $\\operatorname{Pic}(A)$ is an abelian group because $\\mathrm {Stab}(A)$ is symmetric monoidal.",
"Remark 2.8 In Definition REF , we are only considering finitely generated $A$ -modules.",
"This is no loss of generality because every invertible object must be finitely generated.",
"This follows from [11], for example.",
"Remark 2.9 In Definition REF , we have defined the Picard group using only $A$ -modules that are $\\mathbb {M}_2$ -free.",
"In fact, if $M$ and $N$ are arbitrary finitely generated $A$ -modules such that $M \\otimes N$ is stably equivalent to $\\mathbb {M}_2$ , then $M$ and $N$ must in fact be $\\mathbb {M}_2$ -free.",
"In other words, there is no harm in considering only $\\mathbb {M}_2$ -free modules in the Picard group.",
"For if $M \\otimes N$ is isomorphic to $\\mathbb {M}_2\\oplus P$ for some projective $A$ -module $P$ , then $P$ is $\\mathbb {M}_2$ -free by Lemma REF .",
"Therefore, $M \\otimes N$ is $\\mathbb {M}_2$ -free, and $M$ and $N$ are $\\mathbb {M}_2$ -free as well.",
"Definition 2.10 Denote the $\\mathbb {M}_2$ -linear dual functor by $D : { }_{A}\\mathbf {Mod}^{\\mathrm {op}} \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} { }_{A}\\mathbf {Mod}\\colon M \\mathrel { \\hspace{1.88889pt}\\hspace{-3.33328pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {G} \\hspace{-2.22214pt}\\textrm {A} } \\hspace{0.55542pt} DM = \\operatorname{Hom}_{\\mathbb {M}_2}(M, \\mathbb {M}_2).$ Lemma 2.11 The $\\mathbb {M}_2$ -linear dual functor $D$ induces a functor $D : \\mathrm {Stab}(A)^{\\mathrm {op}} \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\mathrm {Stab}(A).$ The dual functor $D$ preserves $\\mathbb {M}_2$ -freeness because $D$ is defined as $\\operatorname{Hom}$ over $\\mathbb {M}_2$ .",
"It suffices to check that if $P$ is $A$ -projective, then $DP$ is $A$ -projective.",
"Since the dual respects direct sums, it is enough to show that $DA$ is projective.",
"This follows as in [10] by considering a retraction $DA \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} DA \\otimes A \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} DA$ and observing that the “shearing map\" [10] makes $DA \\otimes A$ into a free $A$ -module.",
"Lemma REF shows that the dual functor $D$ corresponds to inversion in the Picard group.",
"Lemma 2.12 Let $M$ be an $A$ -module.",
"The evaluation morphism $DM \\otimes M \\stackrel{ev}{ \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} } \\mathbb {M}_2$ is a stable equivalence if and only if $M$ is invertible.",
"In particular, the inverse of any element $[M]$ in $\\operatorname{Pic}(A)$ is its dual $[DM]$ .",
"This fact is standard in stable module theory; see [7].",
"We next describe the “algebraic loop” functor that is part of the structure of a stable module category.",
"Definition 2.13 Let $\\Omega $ be the endo-functor of $\\mathrm {Stab}(A)$ given by $\\Omega M = \\ker ( P \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} M)$ where $P \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} M$ is any projective cover of $M$ .",
"For $k \\ge 0$ , define $\\Omega ^k M$ inductively to be $\\Omega (\\Omega ^{k-1} M)$ .",
"For $k < 0$ , define $\\Omega ^k M$ to be $D(\\Omega ^{-k} D M)$ .",
"An immediate application of Schanuel's lemma shows that $\\Omega M$ is independent of the choice of $P$ .",
"Note that $\\Omega M$ is $\\mathbb {M}_2$ -free because it is a subobject of $P$ , and $P$ is $\\mathbb {M}_2$ -free by Lemma REF .",
"Lemma 2.14 If $M$ is stably invertible, then so is $\\Omega M$ .",
"This is a standard part of the theory of stable modules.",
"It follows from the fact that $\\Omega M \\cong \\Omega \\mathbb {M}_2\\otimes M$ , and that $\\Omega \\mathbb {M}_2$ is stably invertible; see [4] for example.",
"Lemma REF implies that there is a group homomorphism $\\eta : \\mathbb {Z}^3 \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\operatorname{Pic}(A)$ sending $(m,n,s)$ to the stable class of $ \\Sigma ^{m,n} \\Omega ^s \\mathbb {M}_2$ .",
"This homomorphism constructs many elements in the Picard group of $A$ .",
"Such elements exist for essentially formal reasons and do not really reflect on the structure of the underlying algebra $A$ .",
"In a sense, the image of $\\eta $ consists of “uninteresting\" invertible elements."
],
[
"$\\tau $ quotients",
"Suppose that $A$ is a finite motivic Hopf algebra.",
"Then $A / \\tau = \\mathbb {F}_2\\otimes _{\\mathbb {M}_2} A$ is a Hopf algebra.",
"Since $A / \\tau $ is defined over a field $\\mathbb {F}_2$ , it is generally easier to understand than $A$ itself.",
"We shall use a change of basis functor that relates our finite motivic Hopf algebra $A$ to the Hopf algebra $A /\\tau $ .",
"Proposition 3.1 Tensoring with the $\\mathbb {M}_2$ -module $\\mathbb {F}_2$ induces a strongly monoidal functor ${ }_{A}\\mathbf {Mod}^{\\textnormal {f}}\\stackrel{(-)/\\tau }{ \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} } { }_{A/\\tau }\\mathbf {Mod}^{\\textnormal {f}}$ that preserves exact sequences.",
"This functor passes to the stable category and thus induces a strongly monoidal functor $\\mathrm {Stab}(A) \\stackrel{(-)/\\tau }{ \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} } \\mathrm {Stab}(A/\\tau ).$ The unit $\\mathbb {M}_2$ of the monoidal structure of ${ }_{A}\\mathbf {Mod}^{\\textnormal {f}}$ is sent to the unit $\\mathbb {F}_2$ .",
"The functor is strongly monoidal since ${\\raisebox {.2em}{M}\\left\\bad.\\raisebox {-.2em}{\\tau }\\right.}",
"\\otimes {\\raisebox {.2em}{N}\\left\\bad.\\raisebox {-.2em}{\\tau }\\right.}",
"\\cong {\\raisebox {.2em}{M \\otimes N}\\left\\bad.\\raisebox {-.2em}{\\tau }\\right.",
"};$ this is just an application of commuting colimits.",
"Consider a short exact sequence in ${ }_{A}\\mathbf {Mod}^{\\textnormal {f}}$ .",
"The sequence is split exact on the underlying free $\\mathbb {M}_2$ -modules.",
"It is still split exact as a sequence of $\\mathbb {F}_2$ -modules after tensoring with $\\mathbb {F}_2$ .",
"This shows that $(-)/\\tau $ is exact.",
"The functor sends free $A$ -modules to free $A/\\tau $ -modules.",
"By additivity, we conclude that it sends projective $A$ -modules to projective $A/\\tau $ -modules and thus descends to the stable categories.",
"Remark 3.2 Note that ${ }_{A/\\tau }\\mathbf {Mod}^{\\textnormal {f}} = { }_{A/\\tau }\\mathbf {Mod}$ , since the ground ring $\\mathbb {F}_2$ is a field.",
"Remark 3.3 The functor ${ }_{A}\\mathbf {Mod}^{\\textnormal {f}}\\stackrel{(-)/\\tau }{ \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} } { }_{A/\\tau }\\mathbf {Mod}^{\\textnormal {f}}$ of Proposition REF preserves exact sequences, but it is not an “exact functor\" in the usual sense because ${ }_{A}\\mathbf {Mod}^{\\textnormal {f}}$ is not an abelian category.",
"Namely, the cokernel of a map in ${ }_{A}\\mathbf {Mod}^{\\textnormal {f}}$ need not be $\\mathbb {M}_2$ -free.",
"We now come to the first major result that will allow us to understand the stable module category of a finite motivic Hopf algebra $A$ .",
"Lemma REF identifies projective $A$ -modules in terms of their quotients by $\\tau $ .",
"Lemma 3.4 Let $A$ be a finite motivic Hopf algebra, and let $M$ be a finitely generated $A$ -module that is $\\mathbb {M}_2$ -free.",
"The following conditions are equivalent: $M$ is projective as an $A$ -module.",
"$M/\\tau $ is projective as an $A/\\tau $ -module.",
"$M/\\tau $ is free as an $A/\\tau $ -module.",
"Note that $A/\\tau $ is a Frobenius algebra since it is a finite dimensional Hopf algebra over the field $\\mathbb {F}_2$ [10].",
"In particular, projective $A/\\tau $ -modules and free $A/\\tau $ -modules are the same.",
"This shows that conditions (2) and (3) are equivalent.",
"Now suppose that $M$ is a projective $A$ -module.",
"Then $M/\\tau $ is a projective $A/\\tau $ -module by Proposition REF .",
"This shows that condition (1) implies condition (2).",
"To show that condition (3) implies condition (1), suppose that $M/\\tau $ is a free $A/\\tau $ -module.",
"We will show that $\\operatorname{Ext}^i_A(M,N)$ vanishes for all $A$ -modules $N$ .",
"In fact, it suffices to assume that $N$ is finitely generated, for $\\operatorname{Hom}_A(M, -)$ commutes with filtered colimits since $M$ is finitely generated, and filtered colimits are exact.",
"Since $M/\\tau $ is free over $A/\\tau $ , we have an $A$ -free resolution of $M / \\tau $ of the form $\\cdots \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} 0 \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\bigoplus A \\stackrel{\\tau }{ \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} } \\bigoplus A \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} M/\\tau .$ Therefore, $\\operatorname{Ext}^{i}_A (M/\\tau ,N)$ vanishes whenever $i \\ge 2$ and $N$ is any $A$ -module.",
"Since $M$ is $\\mathbb {M}_2$ -free, we have a short exact sequence $0 \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} M \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} M \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} M/\\tau \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} 0.$ This sequence induces a long exact sequence $\\cdots \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\operatorname{Ext}^{i}_A (M,N) \\stackrel{\\tau }{ \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} }\\operatorname{Ext}^{i}_A (M,N) \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\operatorname{Ext}^{i+1}_A (M/\\tau ,N) \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\cdots $ for all $A$ -modules $N$ .",
"Since $\\operatorname{Ext}^{i+1}_A (M/\\tau ,N)$ is zero for $i \\ge 1$ by the previous paragraph, we conclude that the map $\\operatorname{Ext}^{i}_A(M,N) \\stackrel{\\tau }{ \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} } \\operatorname{Ext}^{i}_A(M,N)$ is surjective for $i \\ge 1$ .",
"Note that $M$ and $N$ are finitely generated as $\\mathbb {M}_2$ -modules since they are finitely generated as $A$ -modules, and $A$ is a finitely generated $\\mathbb {M}_2$ -module.",
"This implies that $\\operatorname{Ext}^{i}(M,N)$ vanishes in sufficiently low motivic weights.",
"The surjectivity of multiplication by $\\tau $ then implies that $\\operatorname{Ext}^i (M,N)$ vanishes in all weights.",
"This means that $M$ is a projective $A$ -module.",
"Lemma 3.5 Let $M$ and $N$ be finitely generated $A$ -modules that are also $\\mathbb {M}_2$ -free, and let $f: M \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} N$ be a map such that $f/\\tau : M/\\tau \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} N/\\tau $ is injective.",
"Then $f$ is also injective, and the cokernel of $f$ is $\\mathbb {M}_2$ -free.",
"Suppose that $x$ is an element of $M$ such that $f(x) = 0$ , and let $\\overline{x}$ be the corresponding element in $M/\\tau $ .",
"Then $(f/\\tau ) (\\overline{x})$ is zero, so $\\overline{x}$ is also zero because $f/\\tau $ is injective.",
"Therefore, $x$ equals $\\tau y$ for some $y$ .",
"Now $\\tau f(y) = f(\\tau y) = f(x) = 0$ , so $f(y)$ is also zero since $N$ is $\\mathbb {M}_2$ -free.",
"This shows that the kernel of $f$ consists of elements that are infinitely divisible by $\\tau $ .",
"Since $M$ is finitely generated, the kernel must be zero.",
"Now consider the cokernel $N/M$ of $f$ .",
"Since $N/M$ is finitely generated, it suffices to consider the annihilator of $\\tau $ in $N/M$ .",
"We will show that this annihilator is zero.",
"Let $x$ be an element of $N$ , and let $\\overline{x}$ be the element of $N/M$ that it represents.",
"Suppose that $\\tau \\overline{x}$ is zero.",
"Then $\\tau x$ belongs to $M$ .",
"Since $f/\\tau $ is injective and $(f/\\tau ) (\\tau x)$ is zero, we conclude as in the first paragraph that $\\tau x$ equals $\\tau y$ for some $y$ in $M$ .",
"Since $N$ is $\\mathbb {M}_2$ -free, it follows that $x$ equals $y$ .",
"In particular, $x$ belongs to $M$ .",
"In other words, $\\overline{x}$ is zero.",
"The strong monoidal exact functor $-/\\tau : \\mathrm {Stab}(A) \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\mathrm {Stab}(A/\\tau )$ of Proposition REF induces a group homomorphism $V : \\operatorname{Pic}(A) \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\operatorname{Pic}(A/\\tau ).$ Proposition 3.6 The map $V : \\operatorname{Pic}(A) \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\operatorname{Pic}(A/\\tau )$ is injective.",
"Let $M$ be a finitely generated $A$ -module such that $M$ is $\\mathbb {M}_2$ -free, and suppose that $[M]$ in $\\operatorname{Pic}(A)$ belongs to the kernel of $V$ .",
"Equivalently, $M/\\tau $ and $\\mathbb {F}_2$ are stably equivalent $A/\\tau $ -modules.",
"Since $A/\\tau $ is a finite dimensional Frobenius algebra over $\\mathbb {F}_2$ , we can use [10] to see that $M/\\tau $ is isomorphic to the direct sum of $\\mathbb {F}_2$ and a free $A/\\tau $ -module.",
"In other words, $M/\\tau $ is isomorphic to $\\mathbb {F}_2\\oplus F/\\tau $ , where $F$ is a free $A$ -module.",
"Let $j$ be the injection $F/\\tau \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} M/\\tau $ .",
"There is a commutative diagram $ { M @{->>}[r] & M/\\tau \\\\F @{-->}[u]^i @{->>}[r] & F/\\tau , @{^(->}[u]_j }$ in which the dashed arrow exists because $F$ is $A$ -projective and $M \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} M/\\tau $ is a surjection.",
"By Lemma REF , $i$ is injective because $j$ is injective.",
"We now compute the cokernel $C$ of $i$ .",
"Lemma REF implies that $C$ is $\\mathbb {M}_2$ -free.",
"Then Proposition REF says that $C/\\tau $ is isomorphic to the cokernel of $j$ , which is $\\mathbb {F}_2$ by inspection.",
"We conclude that $C$ is isomorphic to $\\mathbb {M}_2$ .",
"Thus, there is a short exact sequence $F M \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {G} \\hspace{-4.44443pt}\\textrm {A} \\hspace{-6.111pt}\\textrm {A} } \\hspace{0.55542pt} \\mathbb {M}_2,$ so $M \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {G} \\hspace{-4.44443pt}\\textrm {A} \\hspace{-6.111pt}\\textrm {A} } \\hspace{0.55542pt} \\mathbb {M}_2$ is a stable equivalence and $[M]$ is trivial in $\\operatorname{Pic}(A)$ ."
],
[
"The finite motivic Hopf algebra $\\mathcal {A}(1)$",
"In this section, we introduce the specific finite motivic Hopf algebra $\\mathcal {A}(1)$ whose Picard group we will compute.",
"Definition 4.1 The finite motivic Hopf algebra $\\mathcal {A}(1)$ is the $\\mathbb {M}_2$ -subalgebra of the motivic Steenrod algebra generated by $\\operatorname{Sq}^1$ and $\\operatorname{Sq}^2$ .",
"Lemma 4.2 The finite motivic Hopf algebra $\\mathcal {A}(1)$ is isomorphic to $\\frac{\\mathbb {M}_2[\\operatorname{Sq}^1,\\operatorname{Sq}^2]}{\\operatorname{Sq}^1 \\operatorname{Sq}^1, \\operatorname{Sq}^2\\operatorname{Sq}^2 + \\tau \\operatorname{Sq}^1\\operatorname{Sq}^2\\operatorname{Sq}^1, \\operatorname{Sq}^1\\operatorname{Sq}^2\\operatorname{Sq}^1\\operatorname{Sq}^2 + \\operatorname{Sq}^2\\operatorname{Sq}^1\\operatorname{Sq}^2\\operatorname{Sq}^1}.$ The element $\\operatorname{Sq}^1$ is primitive, and $\\Delta (\\operatorname{Sq}^2) = \\operatorname{Sq}^2 \\otimes 1 + \\tau \\operatorname{Sq}^1 \\otimes \\operatorname{Sq}^1 + 1 \\otimes \\operatorname{Sq}^2$ .",
"This follows immediately from Voevodsky's description of the motivic Steenrod algebra [14].",
"See Figure REF for a picture of $\\mathcal {A}(1)$ .",
"When writing $\\mathcal {A}(1)$ -modules we use the following conventions.",
"A straight line represents the action of $\\operatorname{Sq}^1$ , a curved line represents the action of $\\operatorname{Sq}^2$ , and a line is dotted if a squaring operation hits $\\tau $ times a generator.",
"For example, the dotted line in Figure REF shows the relation $\\operatorname{Sq}^2\\operatorname{Sq}^2 = \\tau \\operatorname{Sq}^1\\operatorname{Sq}^2\\operatorname{Sq}^1.$ Figure: The finite motivic Hopf algebra 𝒜(1)\\mathcal {A}(1)Lemma 4.3 As ungraded Hopf algebras, $\\mathcal {A}(1)/\\tau $ is isomorphic to the group algebra $\\mathbb {F}_2[D_8]$ of the dihedral group $D_8$ of order 8.",
"Lemma REF implies that $\\mathcal {A}(1) / \\tau $ is isomorphic to $\\frac{\\mathbb {F}_2[\\operatorname{Sq}^1,\\operatorname{Sq}^2]}{\\operatorname{Sq}^1 \\operatorname{Sq}^1, \\operatorname{Sq}^2\\operatorname{Sq}^2,\\operatorname{Sq}^1\\operatorname{Sq}^2\\operatorname{Sq}^1\\operatorname{Sq}^2 + \\operatorname{Sq}^2\\operatorname{Sq}^1\\operatorname{Sq}^2\\operatorname{Sq}^1}.$ For our purposes, a convenient presentation of $D_8$ consists of two generators $x$ and $y$ with the relations $x^2$ , $y^2$ , and $(xy)^4$ .",
"The isomorphism from $\\mathcal {A}(1)/\\tau $ to $\\mathbb {F}_2[D_8]$ takes $\\operatorname{Sq}^1$ to $1 +x$ and $\\operatorname{Sq}^2$ to $1+y$ .",
"Recall that a sub-Hopf algebra $B$ of a Hopf $\\mathbb {F}_2$ -algebra $A$ is elementary if it is isomorphic to an exterior algebra.",
"Note that $Q_0 = \\operatorname{Sq}^1$ and $Q_1 = \\operatorname{Sq}^2 \\operatorname{Sq}^1 + \\operatorname{Sq}^1 \\operatorname{Sq}^2$ are elements of $\\mathcal {A}(1)$ whose squares are zero.",
"Lemma 4.4 The maximal elementary sub-Hopf algebras of $\\mathcal {A}(1)/\\tau $ are the exterior algebras $E(Q_0, Q_1)$ and $E(\\operatorname{Sq}^2, Q_1)$ .",
"Lemma REF says that $A/\\tau $ is isomorphic to the group algebra $\\mathbb {F}_2[D_8]$ of the dihedral group of order 8.",
"The elementary sub-Hopf algebras of $\\mathbb {F}_2[D_8]$ correspond to the elementary abelian 2-subgroups of $D_8$ .",
"The group $D_8$ has two maximal elementary abelian subgroups.",
"Tracing back through the isomorphism of Lemma REF , one can identify the two maximal elementary sub-Hopf algebras of $\\mathcal {A}(1)/\\tau $ ."
],
[
"Margolis homology",
"We now turn to an algebraic invariant detecting projectivity of $\\mathcal {A}(1)$ -modules, analogous to Margolis's techniques using $P^s_t$ -homology [10].",
"Definition 4.5 Let $x$ be an element of $A$ such that $x^2$ is zero.",
"For any $A$ -module $M$ , define the Margolis homology $H(M;x)$ to be the annihilator of $x$ modulo the submodule $x M$ .",
"Recall that classically, an $\\mathcal {A}(1)^{\\mathrm {cl}}$ -module $M$ is projective if and only if $H(M; Q_0)$ and $H(M;Q_1)$ are both zero [1], which is a direct consequence of a more general result [12].",
"Our goal is to generalize this result to the motivic situation.",
"Unfortunately, the motivic situation is more complicated.",
"If $M$ is an $\\mathcal {A}(1)$ -module and $H(M; Q_0)$ and $H(M;Q_1)$ both vanish, then $M$ is not necessarily projective.",
"Example 4.6 Let $\\widetilde{\\mathcal {A}}(1)$ be the $\\mathcal {A}(1)$ -module on two generators $x$ and $y$ of degrees $(0,0)$ and $(2,0)$ respectively, subject to the relations $\\operatorname{Sq}^2 x = \\tau y$ and $\\operatorname{Sq}^1 \\operatorname{Sq}^2 \\operatorname{Sq}^1 x = \\operatorname{Sq}^2 y$ .",
"Figure REF represents $\\widetilde{\\mathcal {A}}(1)$ as an $\\mathcal {A}(1)$ -module.",
"The Margolis homology groups $H(\\widetilde{\\mathcal {A}}(1); Q_0)$ and $H(\\widetilde{\\mathcal {A}}(1); Q_1)$ both vanish.",
"However, $\\widetilde{\\mathcal {A}}(1)$ is not a projective $\\mathcal {A}(1)$ -module.",
"Figure: The 𝒜(1)\\mathcal {A}(1)-module 𝒜 ˜(1)\\widetilde{\\mathcal {A}}(1)It turns out that we need two additional criteria for projectivity beyond $Q_0$ -homology and Margolis $Q_1$ -homology.",
"Proposition 4.7 Let $M$ be a finitely generated $\\mathcal {A}(1)$ -module.",
"Then $M$ is projective if and only if: $M$ is free over $\\mathbb {M}_2$ ; and $H(M/\\tau ;Q_0)=0$ ; and $H(M/\\tau ;Q_1)=0$ ; and $H(M/\\tau ;\\operatorname{Sq}^2)=0$ .",
"First suppose that $M$ is projective.",
"By inspection, conditions (2) through (4) are satisfied when $M$ is $\\mathcal {A}(1)$ .",
"Therefore, these conditions are satisfied when $M$ is free.",
"Using that a projective module is a summand of a free module, conditions (2) through (4) are also satisfied for any projective $M$ .",
"Finally, Lemma REF shows that condition (1) is satisfied.",
"Now suppose that conditions (1) through (4) are satisfied.",
"By Lemma REF , it suffices to show that $M/\\tau $ is $A/\\tau $ -projective.",
"Note that $A/\\tau $ -projectivity is detected by restriction to the quasi-elementary sub-Hopf algebras of $A/\\tau $ [12].",
"See [12] for the definition of quasi-elementary sub-Hopf algebras.",
"For group algebras, quasi-elementary sub-Hopf algebras coincide with elementary sub-Hopf algebras [13] (as observed in [12]).",
"Since $A/\\tau $ is isomorphic to the group algebra $\\mathbb {F}_2[D_8]$ by Lemma REF , Lemma REF shows that the quasi-elementary sub-Hopf algebras of $A/\\tau $ are the exterior algebra $E(Q_0, Q_1)$ and the exterior algebra $E(\\operatorname{Sq}^2, Q_1)$ .",
"Conditions (2) and (3) imply that $M/\\tau $ is $E(Q_0, Q_1)$ -projective, and conditions (3) and (4) imply that $M/\\tau $ is $E(\\operatorname{Sq}^2, Q_1)$ -projective.",
"Remark 4.8 The exterior algebra $E(Q_0, Q_1)$ is the unique maximal quasi-elementary sub-Hopf algebra of the classial Hopf algebra $\\mathcal {A}(1)^{\\mathrm {cl}}$ .",
"This explains why condition (4) of Proposition REF is absent from the classification of projective $\\mathcal {A}(1)^{\\mathrm {cl}}$ -modules.",
"Corollary 4.9 Let $M$ and $N$ be finitely generated $\\mathcal {A}(1)$ -modules that are $\\mathbb {M}_2$ -free, and let $f: M \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} N$ be an $\\mathcal {A}(1)$ -module map.",
"Then $f$ is a stable equivalence if and only if $f/\\tau : M/\\tau \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} N/\\tau $ induces an isomorphism in Margolis homologies with respect to $Q_0$ , $Q_1$ , and $\\operatorname{Sq}^2$ .",
"We may choose a free $\\mathcal {A}(1)$ -module $F$ and a surjective map $g: M \\oplus F \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} N$ that restricts to $f$ on $M$ .",
"Then $f$ is a stable equivalence if and only if $g$ is a stable equivalence, and $f/\\tau $ induces isomorphisms in Margolis homologies if and only if $g/\\tau $ induces isomorphisms in Margolis homologies.",
"In other words, we may assume that $f$ is surjective.",
"(From a model categorical perspective, we have replaced $f$ by an equivalent fibration.)",
"Let $K$ be the kernel of $f$ .",
"The short exact sequence $0 \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} K \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} M \\stackrel{f}{ \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} } N \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} 0$ induces a short exact sequence $0 \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} K/\\tau \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} M/\\tau \\stackrel{f/\\tau }{ \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} } N/\\tau \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} 0$ by Proposition REF .",
"This last short exact sequence induces long exact sequences in Margolis homologies with respect to $Q_0$ , $Q_1$ and $\\operatorname{Sq}^2$ .",
"The long exact sequence shows that $f/\\tau $ is an isomorphism in Margolis homologies if and only if $K/\\tau $ has vanishing Margolis homologies.",
"Finally, Proposition REF implies that $K/\\tau $ has vanishing Margolis homologies if and only if $K$ is projective.",
"Note that $K$ is finitely generated and $\\mathbb {M}_2$ -free because it is a subobject of the finitely generated $\\mathbb {M}_2$ -free module $M$ .",
"Finally, $K$ is projective if and only if $f$ is a stable equivalence.",
"We establish a Künneth theorem for Margolis homology.",
"Proposition 4.10 Let $M$ and $N$ be $\\mathcal {A}(1)$ -modules that are free over $\\mathbb {M}_2$ .",
"Then $H(M/\\tau \\otimes N/\\tau ;x) \\cong H(M/\\tau ;x) \\otimes H(N/\\tau ;x)$ when $x$ is $Q_0$ , $Q_1$ , or $\\operatorname{Sq}^2$ .",
"Lemma REF gives the coproduct formula $\\Delta (\\operatorname{Sq}^2) =\\operatorname{Sq}^2 \\otimes 1 + \\tau \\operatorname{Sq}^1 \\otimes \\operatorname{Sq}^1 + 1 \\otimes \\operatorname{Sq}^2.$ Therefore, $\\operatorname{Sq}^2$ is primitive modulo $\\tau $ .",
"In particular, it acts as a derivation on $M/\\tau \\otimes N/\\tau $ .",
"The isomorphism in $\\operatorname{Sq}^2$ -homology follows from the classical Künneth formula for chain complexes over $\\mathbb {F}_2$ .",
"The arguments for $Q_0$ and $Q_1$ are the same, except slightly easier because these elements are primitive even before quotienting by $\\tau $ .",
"Proposition 4.11 Let $M$ be a finitely generated $\\mathcal {A}(1)$ -module that is $\\mathbb {M}_2$ -free.",
"Then $M$ is invertible if and only if $M/\\tau $ has one-dimensional Margolis homologies with respect to $Q_0$ , $Q_1$ , and $\\operatorname{Sq}^2$ .",
"First suppose that $M$ is invertible.",
"In other words, there exists an $\\mathcal {A}(1)$ -module $N$ and a stable equivalence $ M \\otimes N \\stackrel{\\simeq }{ \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} } \\mathbb {M}_2.$ Proposition REF implies that there is a stable equivalence $ (M \\otimes N)/\\tau \\stackrel{\\simeq }{ \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} } \\mathbb {F}_2$ of $\\mathcal {A}(1)/\\tau $ -modules.",
"Corollary REF shows that $ H ((M \\otimes N) / \\tau ; x) \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} H(\\mathbb {F}_2; x)$ is an isomorphism when $x$ is $Q_0$ , $Q_1$ , or $\\operatorname{Sq}^2$ .",
"Now use Proposition REF to deduce that $H(M/\\tau ;x) \\otimes H(N/\\tau ;x)$ is isomorphic to $\\mathbb {F}_2$ .",
"It follows that $H(M/\\tau ;x)$ is one-dimensional.",
"Now assume that $M/\\tau $ has one-dimensional Margolis homologies.",
"Note that $H(D(M/\\tau );x) \\cong \\operatorname{Hom}_{\\mathbb {F}_2}(H(M/\\tau ;x);\\mathbb {F}_2)$ when $x$ is $Q_0$ , $Q_1$ , or $\\operatorname{Sq}^2$ .",
"Therefore, $D(M/\\tau )$ also has one-dimensional Margolis homologies.",
"By Proposition REF , $M/\\tau \\otimes D(M/\\tau )$ also has one-dimensional Margolis homologies.",
"Hence the evaluation map $ M/\\tau \\otimes D(M/\\tau ) \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\mathbb {F}_2$ induces an isomorphism in Margolis homologies because both sides are one-dimensional.",
"Note that $M/\\tau \\otimes D(M/\\tau )$ is isomorphic to $(M \\otimes DM)/\\tau $ by Proposition REF .",
"Finally, Corollary REF shows that the evaluation map $ M \\otimes DM \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\mathbb {M}_2$ is a stable equivalence.",
"This shows that $M$ is invertible with inverse $DM$ ."
],
[
"The Picard group of $\\mathcal {A}(1)$",
"Definition 5.1 Let $J$ be the $\\mathcal {A}(1)$ -module on two generators $x$ and $y$ of degrees $(0,0)$ and $(2,0)$ respectively, subject to the relations $\\operatorname{Sq}^2 x = \\tau y$ , $\\operatorname{Sq}^1 \\operatorname{Sq}^2 \\operatorname{Sq}^1 x = \\operatorname{Sq}^2 y$ , and $\\operatorname{Sq}^1 y = 0$ .",
"Figure REF represents $J$ as an $\\mathcal {A}(1)$ -module.",
"Figure: The 𝒜(1)\\mathcal {A}(1)-module JJLemma 5.2 The $\\mathcal {A}(1)$ -module $J$ is invertible, and the order of $[J]$ in $\\operatorname{Pic}(\\mathcal {A}(1))$ is infinite.",
"Proposition REF implies that $J$ is invertible.",
"The $Q_0$ -homology and $Q_1$ -homology of $J/\\tau $ are generated by $x$ , while the $\\operatorname{Sq}^2$ -homology of $J/\\tau $ is generated by $y$ .",
"The degrees of $x$ and $y$ are different.",
"Therefore, the $\\operatorname{Sq}^2$ -homology and the $Q_0$ -homology of any tensor power $J^{\\otimes n}$ of $J$ are in different degrees.",
"On the other hand, the $\\operatorname{Sq}^2$ -homology and the $Q_0$ -homology of $\\mathbb {M}_2$ are in the same degree.",
"This shows that $J^{\\otimes n}$ is not stably equivalent to $\\mathbb {M}_2$ .",
"Remark 5.3 The classical joker is self-dual as an $\\mathcal {A}(1)^{\\mathrm {cl}}$ -module.",
"Therefore, it represents an element of order two in $\\operatorname{Pic}(\\mathcal {A}(1)^{\\mathrm {cl}}$ .",
"On the other hand, Figure REF shows that the motivic joker is not self-dual.",
"Theorem 5.4 There is an isomorphism $ \\mathbb {Z}^4 \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\operatorname{Pic}(\\mathcal {A}(1))$ sending $(a,b,c,d)$ to the class of $\\Sigma ^{a,b} \\Omega ^c J^d$ .",
"Recall the homomorphism $V : \\operatorname{Pic}(\\mathcal {A}(1)) \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\operatorname{Pic}(A/\\tau )$ from Proposition REF .",
"Consider the composition $\\mathbb {Z}^4 \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\operatorname{Pic}(\\mathcal {A}(1)) \\stackrel{V}{ \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} } \\operatorname{Pic}(\\mathcal {A}(1)/\\tau )\\stackrel{\\cong }{ \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} } \\operatorname{Pic}(\\mathbb {F}_2[D_8]),$ where the last isomorphism comes from Lemma REF .",
"Recall from [5] that the ungraded Picard group of $\\mathbb {F}_2[D_8]$ is isomorphic to $\\mathbb {Z}^2$ , generated by $\\Omega \\mathbb {F}_2$ and a module $L$ .",
"If we add the motivic bigrading, then we obtain that the graded Picard group $\\operatorname{Pic}(\\mathbb {F}_2[D_8])$ is isomorphic to $\\mathbb {Z}^4$ .",
"By direct computation, the composition sends the joker $J$ to $\\Omega L$ .",
"Therefore, the composition is an isomorphism.",
"This shows that $V$ is surjective.",
"We already know that $V$ is injective from Proposition REF .",
"Therefore, $V$ is an isomorphism, so the map $ \\mathbb {Z}^4 \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\operatorname{Pic}(\\mathcal {A}(1))$ is an isomorphism as well."
],
[
"$\\tau $ quotients",
"Suppose that $A$ is a finite motivic Hopf algebra.",
"Then $A / \\tau = \\mathbb {F}_2\\otimes _{\\mathbb {M}_2} A$ is a Hopf algebra.",
"Since $A / \\tau $ is defined over a field $\\mathbb {F}_2$ , it is generally easier to understand than $A$ itself.",
"We shall use a change of basis functor that relates our finite motivic Hopf algebra $A$ to the Hopf algebra $A /\\tau $ .",
"Proposition 3.1 Tensoring with the $\\mathbb {M}_2$ -module $\\mathbb {F}_2$ induces a strongly monoidal functor ${ }_{A}\\mathbf {Mod}^{\\textnormal {f}}\\stackrel{(-)/\\tau }{ \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} } { }_{A/\\tau }\\mathbf {Mod}^{\\textnormal {f}}$ that preserves exact sequences.",
"This functor passes to the stable category and thus induces a strongly monoidal functor $\\mathrm {Stab}(A) \\stackrel{(-)/\\tau }{ \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} } \\mathrm {Stab}(A/\\tau ).$ The unit $\\mathbb {M}_2$ of the monoidal structure of ${ }_{A}\\mathbf {Mod}^{\\textnormal {f}}$ is sent to the unit $\\mathbb {F}_2$ .",
"The functor is strongly monoidal since ${\\raisebox {.2em}{M}\\left\\bad.\\raisebox {-.2em}{\\tau }\\right.}",
"\\otimes {\\raisebox {.2em}{N}\\left\\bad.\\raisebox {-.2em}{\\tau }\\right.}",
"\\cong {\\raisebox {.2em}{M \\otimes N}\\left\\bad.\\raisebox {-.2em}{\\tau }\\right.",
"};$ this is just an application of commuting colimits.",
"Consider a short exact sequence in ${ }_{A}\\mathbf {Mod}^{\\textnormal {f}}$ .",
"The sequence is split exact on the underlying free $\\mathbb {M}_2$ -modules.",
"It is still split exact as a sequence of $\\mathbb {F}_2$ -modules after tensoring with $\\mathbb {F}_2$ .",
"This shows that $(-)/\\tau $ is exact.",
"The functor sends free $A$ -modules to free $A/\\tau $ -modules.",
"By additivity, we conclude that it sends projective $A$ -modules to projective $A/\\tau $ -modules and thus descends to the stable categories.",
"Remark 3.2 Note that ${ }_{A/\\tau }\\mathbf {Mod}^{\\textnormal {f}} = { }_{A/\\tau }\\mathbf {Mod}$ , since the ground ring $\\mathbb {F}_2$ is a field.",
"Remark 3.3 The functor ${ }_{A}\\mathbf {Mod}^{\\textnormal {f}}\\stackrel{(-)/\\tau }{ \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} } { }_{A/\\tau }\\mathbf {Mod}^{\\textnormal {f}}$ of Proposition REF preserves exact sequences, but it is not an “exact functor\" in the usual sense because ${ }_{A}\\mathbf {Mod}^{\\textnormal {f}}$ is not an abelian category.",
"Namely, the cokernel of a map in ${ }_{A}\\mathbf {Mod}^{\\textnormal {f}}$ need not be $\\mathbb {M}_2$ -free.",
"We now come to the first major result that will allow us to understand the stable module category of a finite motivic Hopf algebra $A$ .",
"Lemma REF identifies projective $A$ -modules in terms of their quotients by $\\tau $ .",
"Lemma 3.4 Let $A$ be a finite motivic Hopf algebra, and let $M$ be a finitely generated $A$ -module that is $\\mathbb {M}_2$ -free.",
"The following conditions are equivalent: $M$ is projective as an $A$ -module.",
"$M/\\tau $ is projective as an $A/\\tau $ -module.",
"$M/\\tau $ is free as an $A/\\tau $ -module.",
"Note that $A/\\tau $ is a Frobenius algebra since it is a finite dimensional Hopf algebra over the field $\\mathbb {F}_2$ [10].",
"In particular, projective $A/\\tau $ -modules and free $A/\\tau $ -modules are the same.",
"This shows that conditions (2) and (3) are equivalent.",
"Now suppose that $M$ is a projective $A$ -module.",
"Then $M/\\tau $ is a projective $A/\\tau $ -module by Proposition REF .",
"This shows that condition (1) implies condition (2).",
"To show that condition (3) implies condition (1), suppose that $M/\\tau $ is a free $A/\\tau $ -module.",
"We will show that $\\operatorname{Ext}^i_A(M,N)$ vanishes for all $A$ -modules $N$ .",
"In fact, it suffices to assume that $N$ is finitely generated, for $\\operatorname{Hom}_A(M, -)$ commutes with filtered colimits since $M$ is finitely generated, and filtered colimits are exact.",
"Since $M/\\tau $ is free over $A/\\tau $ , we have an $A$ -free resolution of $M / \\tau $ of the form $\\cdots \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} 0 \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\bigoplus A \\stackrel{\\tau }{ \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} } \\bigoplus A \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} M/\\tau .$ Therefore, $\\operatorname{Ext}^{i}_A (M/\\tau ,N)$ vanishes whenever $i \\ge 2$ and $N$ is any $A$ -module.",
"Since $M$ is $\\mathbb {M}_2$ -free, we have a short exact sequence $0 \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} M \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} M \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} M/\\tau \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} 0.$ This sequence induces a long exact sequence $\\cdots \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\operatorname{Ext}^{i}_A (M,N) \\stackrel{\\tau }{ \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} }\\operatorname{Ext}^{i}_A (M,N) \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\operatorname{Ext}^{i+1}_A (M/\\tau ,N) \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\cdots $ for all $A$ -modules $N$ .",
"Since $\\operatorname{Ext}^{i+1}_A (M/\\tau ,N)$ is zero for $i \\ge 1$ by the previous paragraph, we conclude that the map $\\operatorname{Ext}^{i}_A(M,N) \\stackrel{\\tau }{ \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} } \\operatorname{Ext}^{i}_A(M,N)$ is surjective for $i \\ge 1$ .",
"Note that $M$ and $N$ are finitely generated as $\\mathbb {M}_2$ -modules since they are finitely generated as $A$ -modules, and $A$ is a finitely generated $\\mathbb {M}_2$ -module.",
"This implies that $\\operatorname{Ext}^{i}(M,N)$ vanishes in sufficiently low motivic weights.",
"The surjectivity of multiplication by $\\tau $ then implies that $\\operatorname{Ext}^i (M,N)$ vanishes in all weights.",
"This means that $M$ is a projective $A$ -module.",
"Lemma 3.5 Let $M$ and $N$ be finitely generated $A$ -modules that are also $\\mathbb {M}_2$ -free, and let $f: M \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} N$ be a map such that $f/\\tau : M/\\tau \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} N/\\tau $ is injective.",
"Then $f$ is also injective, and the cokernel of $f$ is $\\mathbb {M}_2$ -free.",
"Suppose that $x$ is an element of $M$ such that $f(x) = 0$ , and let $\\overline{x}$ be the corresponding element in $M/\\tau $ .",
"Then $(f/\\tau ) (\\overline{x})$ is zero, so $\\overline{x}$ is also zero because $f/\\tau $ is injective.",
"Therefore, $x$ equals $\\tau y$ for some $y$ .",
"Now $\\tau f(y) = f(\\tau y) = f(x) = 0$ , so $f(y)$ is also zero since $N$ is $\\mathbb {M}_2$ -free.",
"This shows that the kernel of $f$ consists of elements that are infinitely divisible by $\\tau $ .",
"Since $M$ is finitely generated, the kernel must be zero.",
"Now consider the cokernel $N/M$ of $f$ .",
"Since $N/M$ is finitely generated, it suffices to consider the annihilator of $\\tau $ in $N/M$ .",
"We will show that this annihilator is zero.",
"Let $x$ be an element of $N$ , and let $\\overline{x}$ be the element of $N/M$ that it represents.",
"Suppose that $\\tau \\overline{x}$ is zero.",
"Then $\\tau x$ belongs to $M$ .",
"Since $f/\\tau $ is injective and $(f/\\tau ) (\\tau x)$ is zero, we conclude as in the first paragraph that $\\tau x$ equals $\\tau y$ for some $y$ in $M$ .",
"Since $N$ is $\\mathbb {M}_2$ -free, it follows that $x$ equals $y$ .",
"In particular, $x$ belongs to $M$ .",
"In other words, $\\overline{x}$ is zero.",
"The strong monoidal exact functor $-/\\tau : \\mathrm {Stab}(A) \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\mathrm {Stab}(A/\\tau )$ of Proposition REF induces a group homomorphism $V : \\operatorname{Pic}(A) \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\operatorname{Pic}(A/\\tau ).$ Proposition 3.6 The map $V : \\operatorname{Pic}(A) \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\operatorname{Pic}(A/\\tau )$ is injective.",
"Let $M$ be a finitely generated $A$ -module such that $M$ is $\\mathbb {M}_2$ -free, and suppose that $[M]$ in $\\operatorname{Pic}(A)$ belongs to the kernel of $V$ .",
"Equivalently, $M/\\tau $ and $\\mathbb {F}_2$ are stably equivalent $A/\\tau $ -modules.",
"Since $A/\\tau $ is a finite dimensional Frobenius algebra over $\\mathbb {F}_2$ , we can use [10] to see that $M/\\tau $ is isomorphic to the direct sum of $\\mathbb {F}_2$ and a free $A/\\tau $ -module.",
"In other words, $M/\\tau $ is isomorphic to $\\mathbb {F}_2\\oplus F/\\tau $ , where $F$ is a free $A$ -module.",
"Let $j$ be the injection $F/\\tau \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} M/\\tau $ .",
"There is a commutative diagram $ { M @{->>}[r] & M/\\tau \\\\F @{-->}[u]^i @{->>}[r] & F/\\tau , @{^(->}[u]_j }$ in which the dashed arrow exists because $F$ is $A$ -projective and $M \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} M/\\tau $ is a surjection.",
"By Lemma REF , $i$ is injective because $j$ is injective.",
"We now compute the cokernel $C$ of $i$ .",
"Lemma REF implies that $C$ is $\\mathbb {M}_2$ -free.",
"Then Proposition REF says that $C/\\tau $ is isomorphic to the cokernel of $j$ , which is $\\mathbb {F}_2$ by inspection.",
"We conclude that $C$ is isomorphic to $\\mathbb {M}_2$ .",
"Thus, there is a short exact sequence $F M \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {G} \\hspace{-4.44443pt}\\textrm {A} \\hspace{-6.111pt}\\textrm {A} } \\hspace{0.55542pt} \\mathbb {M}_2,$ so $M \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {G} \\hspace{-4.44443pt}\\textrm {A} \\hspace{-6.111pt}\\textrm {A} } \\hspace{0.55542pt} \\mathbb {M}_2$ is a stable equivalence and $[M]$ is trivial in $\\operatorname{Pic}(A)$ ."
],
[
"The finite motivic Hopf algebra $\\mathcal {A}(1)$",
"In this section, we introduce the specific finite motivic Hopf algebra $\\mathcal {A}(1)$ whose Picard group we will compute.",
"Definition 4.1 The finite motivic Hopf algebra $\\mathcal {A}(1)$ is the $\\mathbb {M}_2$ -subalgebra of the motivic Steenrod algebra generated by $\\operatorname{Sq}^1$ and $\\operatorname{Sq}^2$ .",
"Lemma 4.2 The finite motivic Hopf algebra $\\mathcal {A}(1)$ is isomorphic to $\\frac{\\mathbb {M}_2[\\operatorname{Sq}^1,\\operatorname{Sq}^2]}{\\operatorname{Sq}^1 \\operatorname{Sq}^1, \\operatorname{Sq}^2\\operatorname{Sq}^2 + \\tau \\operatorname{Sq}^1\\operatorname{Sq}^2\\operatorname{Sq}^1, \\operatorname{Sq}^1\\operatorname{Sq}^2\\operatorname{Sq}^1\\operatorname{Sq}^2 + \\operatorname{Sq}^2\\operatorname{Sq}^1\\operatorname{Sq}^2\\operatorname{Sq}^1}.$ The element $\\operatorname{Sq}^1$ is primitive, and $\\Delta (\\operatorname{Sq}^2) = \\operatorname{Sq}^2 \\otimes 1 + \\tau \\operatorname{Sq}^1 \\otimes \\operatorname{Sq}^1 + 1 \\otimes \\operatorname{Sq}^2$ .",
"This follows immediately from Voevodsky's description of the motivic Steenrod algebra [14].",
"See Figure REF for a picture of $\\mathcal {A}(1)$ .",
"When writing $\\mathcal {A}(1)$ -modules we use the following conventions.",
"A straight line represents the action of $\\operatorname{Sq}^1$ , a curved line represents the action of $\\operatorname{Sq}^2$ , and a line is dotted if a squaring operation hits $\\tau $ times a generator.",
"For example, the dotted line in Figure REF shows the relation $\\operatorname{Sq}^2\\operatorname{Sq}^2 = \\tau \\operatorname{Sq}^1\\operatorname{Sq}^2\\operatorname{Sq}^1.$ Figure: The finite motivic Hopf algebra 𝒜(1)\\mathcal {A}(1)Lemma 4.3 As ungraded Hopf algebras, $\\mathcal {A}(1)/\\tau $ is isomorphic to the group algebra $\\mathbb {F}_2[D_8]$ of the dihedral group $D_8$ of order 8.",
"Lemma REF implies that $\\mathcal {A}(1) / \\tau $ is isomorphic to $\\frac{\\mathbb {F}_2[\\operatorname{Sq}^1,\\operatorname{Sq}^2]}{\\operatorname{Sq}^1 \\operatorname{Sq}^1, \\operatorname{Sq}^2\\operatorname{Sq}^2,\\operatorname{Sq}^1\\operatorname{Sq}^2\\operatorname{Sq}^1\\operatorname{Sq}^2 + \\operatorname{Sq}^2\\operatorname{Sq}^1\\operatorname{Sq}^2\\operatorname{Sq}^1}.$ For our purposes, a convenient presentation of $D_8$ consists of two generators $x$ and $y$ with the relations $x^2$ , $y^2$ , and $(xy)^4$ .",
"The isomorphism from $\\mathcal {A}(1)/\\tau $ to $\\mathbb {F}_2[D_8]$ takes $\\operatorname{Sq}^1$ to $1 +x$ and $\\operatorname{Sq}^2$ to $1+y$ .",
"Recall that a sub-Hopf algebra $B$ of a Hopf $\\mathbb {F}_2$ -algebra $A$ is elementary if it is isomorphic to an exterior algebra.",
"Note that $Q_0 = \\operatorname{Sq}^1$ and $Q_1 = \\operatorname{Sq}^2 \\operatorname{Sq}^1 + \\operatorname{Sq}^1 \\operatorname{Sq}^2$ are elements of $\\mathcal {A}(1)$ whose squares are zero.",
"Lemma 4.4 The maximal elementary sub-Hopf algebras of $\\mathcal {A}(1)/\\tau $ are the exterior algebras $E(Q_0, Q_1)$ and $E(\\operatorname{Sq}^2, Q_1)$ .",
"Lemma REF says that $A/\\tau $ is isomorphic to the group algebra $\\mathbb {F}_2[D_8]$ of the dihedral group of order 8.",
"The elementary sub-Hopf algebras of $\\mathbb {F}_2[D_8]$ correspond to the elementary abelian 2-subgroups of $D_8$ .",
"The group $D_8$ has two maximal elementary abelian subgroups.",
"Tracing back through the isomorphism of Lemma REF , one can identify the two maximal elementary sub-Hopf algebras of $\\mathcal {A}(1)/\\tau $ ."
],
[
"Margolis homology",
"We now turn to an algebraic invariant detecting projectivity of $\\mathcal {A}(1)$ -modules, analogous to Margolis's techniques using $P^s_t$ -homology [10].",
"Definition 4.5 Let $x$ be an element of $A$ such that $x^2$ is zero.",
"For any $A$ -module $M$ , define the Margolis homology $H(M;x)$ to be the annihilator of $x$ modulo the submodule $x M$ .",
"Recall that classically, an $\\mathcal {A}(1)^{\\mathrm {cl}}$ -module $M$ is projective if and only if $H(M; Q_0)$ and $H(M;Q_1)$ are both zero [1], which is a direct consequence of a more general result [12].",
"Our goal is to generalize this result to the motivic situation.",
"Unfortunately, the motivic situation is more complicated.",
"If $M$ is an $\\mathcal {A}(1)$ -module and $H(M; Q_0)$ and $H(M;Q_1)$ both vanish, then $M$ is not necessarily projective.",
"Example 4.6 Let $\\widetilde{\\mathcal {A}}(1)$ be the $\\mathcal {A}(1)$ -module on two generators $x$ and $y$ of degrees $(0,0)$ and $(2,0)$ respectively, subject to the relations $\\operatorname{Sq}^2 x = \\tau y$ and $\\operatorname{Sq}^1 \\operatorname{Sq}^2 \\operatorname{Sq}^1 x = \\operatorname{Sq}^2 y$ .",
"Figure REF represents $\\widetilde{\\mathcal {A}}(1)$ as an $\\mathcal {A}(1)$ -module.",
"The Margolis homology groups $H(\\widetilde{\\mathcal {A}}(1); Q_0)$ and $H(\\widetilde{\\mathcal {A}}(1); Q_1)$ both vanish.",
"However, $\\widetilde{\\mathcal {A}}(1)$ is not a projective $\\mathcal {A}(1)$ -module.",
"Figure: The 𝒜(1)\\mathcal {A}(1)-module 𝒜 ˜(1)\\widetilde{\\mathcal {A}}(1)It turns out that we need two additional criteria for projectivity beyond $Q_0$ -homology and Margolis $Q_1$ -homology.",
"Proposition 4.7 Let $M$ be a finitely generated $\\mathcal {A}(1)$ -module.",
"Then $M$ is projective if and only if: $M$ is free over $\\mathbb {M}_2$ ; and $H(M/\\tau ;Q_0)=0$ ; and $H(M/\\tau ;Q_1)=0$ ; and $H(M/\\tau ;\\operatorname{Sq}^2)=0$ .",
"First suppose that $M$ is projective.",
"By inspection, conditions (2) through (4) are satisfied when $M$ is $\\mathcal {A}(1)$ .",
"Therefore, these conditions are satisfied when $M$ is free.",
"Using that a projective module is a summand of a free module, conditions (2) through (4) are also satisfied for any projective $M$ .",
"Finally, Lemma REF shows that condition (1) is satisfied.",
"Now suppose that conditions (1) through (4) are satisfied.",
"By Lemma REF , it suffices to show that $M/\\tau $ is $A/\\tau $ -projective.",
"Note that $A/\\tau $ -projectivity is detected by restriction to the quasi-elementary sub-Hopf algebras of $A/\\tau $ [12].",
"See [12] for the definition of quasi-elementary sub-Hopf algebras.",
"For group algebras, quasi-elementary sub-Hopf algebras coincide with elementary sub-Hopf algebras [13] (as observed in [12]).",
"Since $A/\\tau $ is isomorphic to the group algebra $\\mathbb {F}_2[D_8]$ by Lemma REF , Lemma REF shows that the quasi-elementary sub-Hopf algebras of $A/\\tau $ are the exterior algebra $E(Q_0, Q_1)$ and the exterior algebra $E(\\operatorname{Sq}^2, Q_1)$ .",
"Conditions (2) and (3) imply that $M/\\tau $ is $E(Q_0, Q_1)$ -projective, and conditions (3) and (4) imply that $M/\\tau $ is $E(\\operatorname{Sq}^2, Q_1)$ -projective.",
"Remark 4.8 The exterior algebra $E(Q_0, Q_1)$ is the unique maximal quasi-elementary sub-Hopf algebra of the classial Hopf algebra $\\mathcal {A}(1)^{\\mathrm {cl}}$ .",
"This explains why condition (4) of Proposition REF is absent from the classification of projective $\\mathcal {A}(1)^{\\mathrm {cl}}$ -modules.",
"Corollary 4.9 Let $M$ and $N$ be finitely generated $\\mathcal {A}(1)$ -modules that are $\\mathbb {M}_2$ -free, and let $f: M \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} N$ be an $\\mathcal {A}(1)$ -module map.",
"Then $f$ is a stable equivalence if and only if $f/\\tau : M/\\tau \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} N/\\tau $ induces an isomorphism in Margolis homologies with respect to $Q_0$ , $Q_1$ , and $\\operatorname{Sq}^2$ .",
"We may choose a free $\\mathcal {A}(1)$ -module $F$ and a surjective map $g: M \\oplus F \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} N$ that restricts to $f$ on $M$ .",
"Then $f$ is a stable equivalence if and only if $g$ is a stable equivalence, and $f/\\tau $ induces isomorphisms in Margolis homologies if and only if $g/\\tau $ induces isomorphisms in Margolis homologies.",
"In other words, we may assume that $f$ is surjective.",
"(From a model categorical perspective, we have replaced $f$ by an equivalent fibration.)",
"Let $K$ be the kernel of $f$ .",
"The short exact sequence $0 \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} K \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} M \\stackrel{f}{ \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} } N \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} 0$ induces a short exact sequence $0 \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} K/\\tau \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} M/\\tau \\stackrel{f/\\tau }{ \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} } N/\\tau \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} 0$ by Proposition REF .",
"This last short exact sequence induces long exact sequences in Margolis homologies with respect to $Q_0$ , $Q_1$ and $\\operatorname{Sq}^2$ .",
"The long exact sequence shows that $f/\\tau $ is an isomorphism in Margolis homologies if and only if $K/\\tau $ has vanishing Margolis homologies.",
"Finally, Proposition REF implies that $K/\\tau $ has vanishing Margolis homologies if and only if $K$ is projective.",
"Note that $K$ is finitely generated and $\\mathbb {M}_2$ -free because it is a subobject of the finitely generated $\\mathbb {M}_2$ -free module $M$ .",
"Finally, $K$ is projective if and only if $f$ is a stable equivalence.",
"We establish a Künneth theorem for Margolis homology.",
"Proposition 4.10 Let $M$ and $N$ be $\\mathcal {A}(1)$ -modules that are free over $\\mathbb {M}_2$ .",
"Then $H(M/\\tau \\otimes N/\\tau ;x) \\cong H(M/\\tau ;x) \\otimes H(N/\\tau ;x)$ when $x$ is $Q_0$ , $Q_1$ , or $\\operatorname{Sq}^2$ .",
"Lemma REF gives the coproduct formula $\\Delta (\\operatorname{Sq}^2) =\\operatorname{Sq}^2 \\otimes 1 + \\tau \\operatorname{Sq}^1 \\otimes \\operatorname{Sq}^1 + 1 \\otimes \\operatorname{Sq}^2.$ Therefore, $\\operatorname{Sq}^2$ is primitive modulo $\\tau $ .",
"In particular, it acts as a derivation on $M/\\tau \\otimes N/\\tau $ .",
"The isomorphism in $\\operatorname{Sq}^2$ -homology follows from the classical Künneth formula for chain complexes over $\\mathbb {F}_2$ .",
"The arguments for $Q_0$ and $Q_1$ are the same, except slightly easier because these elements are primitive even before quotienting by $\\tau $ .",
"Proposition 4.11 Let $M$ be a finitely generated $\\mathcal {A}(1)$ -module that is $\\mathbb {M}_2$ -free.",
"Then $M$ is invertible if and only if $M/\\tau $ has one-dimensional Margolis homologies with respect to $Q_0$ , $Q_1$ , and $\\operatorname{Sq}^2$ .",
"First suppose that $M$ is invertible.",
"In other words, there exists an $\\mathcal {A}(1)$ -module $N$ and a stable equivalence $ M \\otimes N \\stackrel{\\simeq }{ \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} } \\mathbb {M}_2.$ Proposition REF implies that there is a stable equivalence $ (M \\otimes N)/\\tau \\stackrel{\\simeq }{ \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} } \\mathbb {F}_2$ of $\\mathcal {A}(1)/\\tau $ -modules.",
"Corollary REF shows that $ H ((M \\otimes N) / \\tau ; x) \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} H(\\mathbb {F}_2; x)$ is an isomorphism when $x$ is $Q_0$ , $Q_1$ , or $\\operatorname{Sq}^2$ .",
"Now use Proposition REF to deduce that $H(M/\\tau ;x) \\otimes H(N/\\tau ;x)$ is isomorphic to $\\mathbb {F}_2$ .",
"It follows that $H(M/\\tau ;x)$ is one-dimensional.",
"Now assume that $M/\\tau $ has one-dimensional Margolis homologies.",
"Note that $H(D(M/\\tau );x) \\cong \\operatorname{Hom}_{\\mathbb {F}_2}(H(M/\\tau ;x);\\mathbb {F}_2)$ when $x$ is $Q_0$ , $Q_1$ , or $\\operatorname{Sq}^2$ .",
"Therefore, $D(M/\\tau )$ also has one-dimensional Margolis homologies.",
"By Proposition REF , $M/\\tau \\otimes D(M/\\tau )$ also has one-dimensional Margolis homologies.",
"Hence the evaluation map $ M/\\tau \\otimes D(M/\\tau ) \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\mathbb {F}_2$ induces an isomorphism in Margolis homologies because both sides are one-dimensional.",
"Note that $M/\\tau \\otimes D(M/\\tau )$ is isomorphic to $(M \\otimes DM)/\\tau $ by Proposition REF .",
"Finally, Corollary REF shows that the evaluation map $ M \\otimes DM \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\mathbb {M}_2$ is a stable equivalence.",
"This shows that $M$ is invertible with inverse $DM$ ."
],
[
"The Picard group of $\\mathcal {A}(1)$",
"Definition 5.1 Let $J$ be the $\\mathcal {A}(1)$ -module on two generators $x$ and $y$ of degrees $(0,0)$ and $(2,0)$ respectively, subject to the relations $\\operatorname{Sq}^2 x = \\tau y$ , $\\operatorname{Sq}^1 \\operatorname{Sq}^2 \\operatorname{Sq}^1 x = \\operatorname{Sq}^2 y$ , and $\\operatorname{Sq}^1 y = 0$ .",
"Figure REF represents $J$ as an $\\mathcal {A}(1)$ -module.",
"Figure: The 𝒜(1)\\mathcal {A}(1)-module JJLemma 5.2 The $\\mathcal {A}(1)$ -module $J$ is invertible, and the order of $[J]$ in $\\operatorname{Pic}(\\mathcal {A}(1))$ is infinite.",
"Proposition REF implies that $J$ is invertible.",
"The $Q_0$ -homology and $Q_1$ -homology of $J/\\tau $ are generated by $x$ , while the $\\operatorname{Sq}^2$ -homology of $J/\\tau $ is generated by $y$ .",
"The degrees of $x$ and $y$ are different.",
"Therefore, the $\\operatorname{Sq}^2$ -homology and the $Q_0$ -homology of any tensor power $J^{\\otimes n}$ of $J$ are in different degrees.",
"On the other hand, the $\\operatorname{Sq}^2$ -homology and the $Q_0$ -homology of $\\mathbb {M}_2$ are in the same degree.",
"This shows that $J^{\\otimes n}$ is not stably equivalent to $\\mathbb {M}_2$ .",
"Remark 5.3 The classical joker is self-dual as an $\\mathcal {A}(1)^{\\mathrm {cl}}$ -module.",
"Therefore, it represents an element of order two in $\\operatorname{Pic}(\\mathcal {A}(1)^{\\mathrm {cl}}$ .",
"On the other hand, Figure REF shows that the motivic joker is not self-dual.",
"Theorem 5.4 There is an isomorphism $ \\mathbb {Z}^4 \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\operatorname{Pic}(\\mathcal {A}(1))$ sending $(a,b,c,d)$ to the class of $\\Sigma ^{a,b} \\Omega ^c J^d$ .",
"Recall the homomorphism $V : \\operatorname{Pic}(\\mathcal {A}(1)) \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\operatorname{Pic}(A/\\tau )$ from Proposition REF .",
"Consider the composition $\\mathbb {Z}^4 \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\operatorname{Pic}(\\mathcal {A}(1)) \\stackrel{V}{ \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} } \\operatorname{Pic}(\\mathcal {A}(1)/\\tau )\\stackrel{\\cong }{ \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} } \\operatorname{Pic}(\\mathbb {F}_2[D_8]),$ where the last isomorphism comes from Lemma REF .",
"Recall from [5] that the ungraded Picard group of $\\mathbb {F}_2[D_8]$ is isomorphic to $\\mathbb {Z}^2$ , generated by $\\Omega \\mathbb {F}_2$ and a module $L$ .",
"If we add the motivic bigrading, then we obtain that the graded Picard group $\\operatorname{Pic}(\\mathbb {F}_2[D_8])$ is isomorphic to $\\mathbb {Z}^4$ .",
"By direct computation, the composition sends the joker $J$ to $\\Omega L$ .",
"Therefore, the composition is an isomorphism.",
"This shows that $V$ is surjective.",
"We already know that $V$ is injective from Proposition REF .",
"Therefore, $V$ is an isomorphism, so the map $ \\mathbb {Z}^4 \\mathrel { \\hspace{0.83328pt}\\textrm {G} \\hspace{-1.38885pt}\\textrm {G} \\hspace{-0.61111pt}\\textrm {A} \\hspace{0.55542pt}} \\operatorname{Pic}(\\mathcal {A}(1))$ is an isomorphism as well."
]
] | 1606.05191 |
[
[
"On the Group of Almost-Riordan Arrays"
],
[
"Abstract We study a super group of the group of Riordan arrays, where the elements of the group are given by a triple of power series.",
"We show that certain subsets are subgroups, and we identify a normal subgroup whose cosets correspond to Riordan arrays.",
"We give an example of an almost-Riordan array that has been studied in the context of Hankel and Hankel plus Toepliz matrices, and we show that suitably chosen almost-Riordan arrays can lead to transformations that have interesting Hankel transform properties."
],
[
"Introduction",
"The group of Riordan arrays $\\mathcal {R}$ [10] was first introduced by Shapiro, Getu, Woan, and Woodson in the early 1990's.",
"Since then, they have been extensively studied and applied in a number of different fields.",
"At its simplest, a Riordan array is formally defined by a pair of power series, say $g(x)$ and $f(x)$ , where $g(0)=1$ and $f(x)=x+a_2 x^2+a_3x^2+\\ldots $ , with integer coefficients (such Riordan arrays are called “proper” Riordan arrays).",
"The pair $(g, f)$ is then associated to the lower-triangular invertible matrix whose $(n,k)$ -th element $T_{n,k}$ is given by $T_{n,k}=[x^n] g(x)f(x)^k.$ We sometimes write $(g(x), f(x))$ although the variable “$x$ ” here is a dummy variable, in that $T_{n,k}=[x^n] g(x)f(x)^k = [t^n] g(t)f(t)^k.$ In this paper, we shall define a group of matrices defined by a triple of power series, and we shall demonstrate that the Riordan group is a factor group of this new group.",
"Having defined products and inverses in this new group, we look at some additional properties, and give some examples.",
"Our first example in this section is based on a special transformation that has been studied in the context of Hankel plus Toeplitz matrices [2], [3] By looking at examples closely related to the Catalan numbers, we arrive at transformations of sequences that have interesting Hankel transform properties.",
"Here we recall that for a sequence $a_n$ we define its Hankel transform to be the sequence of determinants $h_n=|a_{i+j}|_{0 \\le i,j \\le n}$ .",
"All the power series and matrices that we shall look at are assumed to have integer coefficients.",
"Thus power series are elements of $\\mathbb {Z}[[x]]$ .",
"The generating function 1 generates the sequence that we denote by $0^n$ , which begins $1,0,0,0,\\ldots $ .",
"All matrices are assumed to begin at the $(0,0)$ position, and to extend infinitely to the right and downwards.",
"Thus matrices in this article are elements of $\\mathbb {Z}^{\\mathbb {N}_0 \\times \\mathbb {N}_0}$ .",
"When examples are given, an obvious truncation is applied.",
"The Fundamental Theorem of Riordan arrays [11] says that the action of a Riordan array on a power series, namely $ (g(x), f(x))\\cdot a(x)= g(x)a(f(x)),$ is realised in matrix form by $\\left(T_{n,k}\\right)\\left( \\begin{array}{c}a_0\\\\a_1\\\\a_2\\\\a_3\\\\ \\vdots \\end{array}\\right)=\\left( \\begin{array}{c}b_0\\\\b_1\\\\b_2\\\\b_3\\\\ \\vdots \\end{array}\\right),$ where the power series $a(x)$ expands to give the sequence $a_0, a_1, a_2, \\ldots $ , and the image sequence $b_0, b_1, b_2, \\ldots $ has generating function $g(x)a(f(x))$ .",
"An important feature of Riordan arrays is that they have a number of sequence characterizations [4], [7].",
"The simplest of these is as follows.",
"Proposition 1 [7] Let $D=[d_{n,k}]$ be an infinite triangular matrix.",
"Then $D$ is a Riordan array if and only if there exist two sequences $A=[a_0,a_1,a_2,\\ldots ]$ and $Z=[z_0,z_1,z_2,\\ldots ]$ with $a_0 \\ne 0$ , $z_0 \\ne 0$ such that $d_{n+1,k+1}=\\sum _{j=0}^{\\infty } a_j d_{n,k+j}, \\quad (k,n=0,1,\\ldots )$ $d_{n+1,0}=\\sum _{j=0}^{\\infty } z_j d_{n,j}, \\quad (n=0,1,\\ldots )$ .",
"The coefficients $a_0,a_1,a_2,\\ldots $ and $z_0,z_1,z_2,\\ldots $ are called the $A$ -sequence and the $Z$ -sequence of the Riordan array $D=(g(x),f(x))$ , respectively.",
"Letting $A(x)$ be the generating function of the $A$ -sequence and $Z(x)$ be the generating function of the $Z$ -sequence, we have $ A(x)=\\frac{x}{\\bar{f}(x)}, \\quad Z(x)=\\frac{1}{\\bar{f}(x)}\\left(1-\\frac{1}{g(\\bar{f}(x))}\\right).$ Here, $\\bar{f}(x)$ is the series reversion of $f(x)$ , defined as the solution $u(x)$ of the equation $f(u)=x$ that satisfies $u(0)=0$ .",
"The inverse of the Riordan array $(g, f)$ is given by $(g(x), f(x))^{-1}=\\left(\\frac{1}{g(\\bar{f}(x))}, \\bar{f}(x)\\right).$ For a Riordan array $D$ , the matrix $P=D^{-1}\\cdot \\overline{D}$ is called its production matrix, where $\\overline{D}$ is the matrix $D$ with its top row removed.",
"The concept of a production matrix [5], [6] is a general one, but for this work we find it convenient to review it in the context of Riordan arrays.",
"Thus let $P$ be an infinite matrix (most often it will have integer entries).",
"Letting $\\mathbf {r}_0$ be the row vector $\\mathbf {r}_0=(1,0,0,0,\\ldots ),$ we define $\\mathbf {r}_i=\\mathbf {r}_{i-1}P$ , $i \\ge 1$ .",
"Stacking these rows leads to another infinite matrix which we denote by $A_P$ .",
"Then $P$ is said to be the production matrix for $A_P$ .",
"If we let $u^T=(1,0,0,0,\\ldots ,0,\\ldots )$ then we have $A_P=\\left(\\begin{array}{c}u^T\\\\u^TP\\\\u^TP^2\\\\\\vdots \\end{array}\\right)$ and $\\bar{I}A_P=A_PP$ where $\\bar{I}=(\\delta _{i+1,j})_{i,j \\ge 0}$ (where $\\delta $ is the usual Kronecker symbol): $ \\bar{I}=\\left(\\begin{array}{ccccccc} 0 & 1& 0 & 0 & 0 & 0 & \\ldots \\\\0 & 0 & 1 & 0 & 0 & 0 & \\ldots \\\\0 & 0 & 0 & 1& 0 & 0 & \\ldots \\\\ 0 & 0 & 0 & 0 & 1 & 0 & \\ldots \\\\ 0 & 0 &0& 0 & 0 & 1 & \\ldots \\\\0 & 0 & 0 & 0 & 0 & 0 &\\ldots \\\\ \\vdots & \\vdots &\\vdots & \\vdots & \\vdots & \\vdots &\\ddots \\end{array}\\right).$ We have $P=A_P^{-1}\\bar{I}A_P.$ Writing $\\overline{A_P}=\\bar{I}A_P$ , we can write this equation as $P=A_P^{-1}\\overline{A_P}.$ Note that $\\overline{A_P}$ is $A_P$ with the first row removed.",
"The production matrix $P$ is sometimes [9], [12] called the Stieltjes matrix $S_{A_P}$ associated to $A_P$ .",
"Other examples of the use of production matrices can be found in [1], for instance.",
"The sequence formed by the row sums of $A_P$ often has combinatorial significance and is called the sequence associated to $P$ .",
"Its general term $a_n$ is given by $a_n = u^T P^n e$ where $e=\\left(\\begin{array}{c}1\\\\1\\\\1\\\\\\vdots \\end{array}\\right).$ In the context of Riordan arrays, the production matrix associated to a proper Riordan array takes on a special form : Proposition 2 [6] Let $P$ be an infinite production matrix and let $A_P$ be the matrix induced by $P$ .",
"Then $A_P$ is an (ordinary) Riordan matrix if and only if $P$ is of the form $ P=\\left(\\begin{array}{ccccccc}\\xi _0 & \\alpha _0 & 0 & 0 & 0 & 0 & \\ldots \\\\\\xi _1 & \\alpha _1 &\\alpha _0 & 0 &0 & 0 & \\ldots \\\\ \\xi _2 & \\alpha _2 & \\alpha _1 & \\alpha _0 & 0 &0 & \\ldots \\\\ \\xi _3 & \\alpha _3 & \\alpha _2 & \\alpha _1 &\\alpha _0& 0 & \\ldots \\\\ \\xi _4 & \\alpha _4 & \\alpha _3 & \\alpha _2 & \\alpha _1 &\\alpha _0& \\ldots \\\\\\xi _5 & \\alpha _5 & \\alpha _4 & \\alpha _3 & \\alpha _2&\\alpha _1&\\ldots \\\\ \\vdots & \\vdots & \\vdots & \\vdots & \\vdots & \\vdots &\\ddots \\end{array}\\right),$ where $\\xi _0 \\ne 0$ , $\\alpha _0 \\ne 0$ .",
"Moreover, columns 0 and 1 of the matrix $P$ are the $Z$ - and $A$ -sequences, respectively, of the Riordan array $A_P$ .",
"We shall use the notation $\\tilde{a}(x)=\\sum _{n=1}a_n x^n=\\frac{a(x)-a_0}{x}$ in the sequel, where $a(x)=\\sum _{n=0}^n a_n x^n$ .",
"Where possible, we shall refer to known sequences and triangles by their OEIS numbers [13], [14].",
"For instance, the Catalan numbers $C_n=\\frac{1}{n+1}\\binom{2n}{n}$ with g.f. $c(x)=\\frac{1-\\sqrt{1-4x}}{2x}$ is the sequence A000108, the Fibonacci numbers are A000045, and the Motzkin numbers $M_n=\\sum _{k=0}^{\\lfloor \\frac{n}{2} \\rfloor }\\binom{n}{2k}C_k$ are A001006.",
"The binomial matrix $B=\\left(\\binom{n}{k}\\right)$ is A007318.",
"As a Riordan array, this is given by $B= \\left(\\frac{1}{1-x}, \\frac{x}{1-x}\\right).$ Note that in this article all sequences $a_n$ that have $a_0 \\ne 0$ are assumed to have $a_0=1$ .",
"Likewise for sequences $b_n$ with $b_0=0$ and $b_1 \\ne 0$ , we assume that $b_1=1$ ."
],
[
"Definitions and Properties",
"An almost-Riordan array is defined by an ordered triple $(a,g,f)$ of power series where $a(x)=\\sum _{n=0}^{\\infty }a_nx^n ,$ with $a_0=1$ , $g(x)=\\sum _{n=0}^{\\infty } g_n x^n$ , with $g_0=1$ , and $f(x)=\\sum _{n=0}^{\\infty } f_n x^n$ , with $f_0=0,\\,f_1=1$ .",
"The array is identified with the lower-triangular matrix defined as follows: its first column is given by the expansion of $a(x)$ , while its first row is the expansion of 1.",
"The remaining elements of the infinite tri-diagonal matrix (starting at the $(1,1)$ position) coincide with the Riordan array $(g,f)$ .",
"Here, we address the first element of the matrix as the $(0,0)$ -th element.",
"We shall denote by $a\\mathcal {R}$ the set of almost-Riordan arrays.",
"Formally this is the set of ordered triples $(a,g,f)$ as described above.",
"We identify these triples with lower-triangular matrices as in the example that follows.",
"We define an action of the element $(a, g, f)$ on the power series $b(x)$ by looking at the action of the corresponding matrix on the column vector given by the expansion of $b(x)$ .",
"The result $(a, g, f)\\cdot b$ is then the generating function of the sequence encapsulated in the column vector that arises by applying the matrix to the column vector whose elements are given by the expansion of $b(x)$ .",
"Example 3 We consider the almost-Riordan array defined by $\\left(\\frac{1}{1-2x}, \\frac{1}{1-x}, \\frac{x}{1-x}\\right)$ .",
"This matrix begins $\\left(\\begin{array}{ccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\2 & 1 & 0 & 0 & 0 & 0 & 0 \\\\4 & 1 & 1 & 0 & 0 & 0 & 0 \\\\8 & 1 & 2 & 1 & 0 & 0 & 0 \\\\16 & 1 & 3 & 3 & 1 & 0 & 0 \\\\32 & 1 & 4 & 6 & 4 & 1 & 0 \\\\64 & 1 & 5 & 10 & 10 & 5 & 1 \\\\\\end{array}\\right).$ Then $\\left(\\frac{1}{1-2x}, \\frac{1}{1-x}, \\frac{x}{1-x}\\right)\\cdot \\frac{1}{1-x-x^2}=\\frac{1-2x-x^2}{1-5x+7x^2-2x^3}$ is realised in matrix form by $\\left(\\begin{array}{ccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\2 & 1 & 0 & 0 & 0 & 0 & 0 \\\\4 & 1 & 1 & 0 & 0 & 0 & 0 \\\\8 & 1 & 2 & 1 & 0 & 0 & 0 \\\\16 & 1 & 3 & 3 & 1 & 0 & 0 \\\\32 & 1 & 4 & 6 & 4 & 1 & 0 \\\\64 & 1 & 5 & 10 & 10 & 5 & 1 \\\\\\end{array}\\right)\\cdot \\left(\\begin{array}{c}1\\\\1\\\\2\\\\3\\\\5\\\\8\\\\13\\\\ \\end{array}\\right)=\\left(\\begin{array}{c}1\\\\3\\\\7\\\\16\\\\37\\\\87\\\\208\\\\ \\end{array}\\right), $ where the expansion of the image $\\frac{1-2x-x^2}{1-5x+7x^2-2x^3}$ begins $1,3,7,16,37,\\ldots $ .",
"We write $(a,0,0)$ for the matrix whose first column is generated by $a(x)$ , with zeros elsewhere.",
"We then have $(a,g,f)=(a,0,0)+(xg, f)$ as a matrix equality.",
"The elements of the matrix $M=(a, g,f)$ are easily described.",
"Letting the $(n,k)$ -th element of $M$ be denoted by $M_{n,k}$ , we have $M_{n,k}=[x^{n-1}] g f^{k-1}, \\quad \\textrm {for\\,} n,k \\ge 1,\\quad M_{n,0}=a_n,\\quad M_{0,k}=0^k.$ Example 4 The almost-Riordan array $\\left(1, \\frac{1}{1-x}, \\frac{x}{1-x}\\right)$ has general term $\\binom{n-1}{n-k}$ .",
"In this case, this matrix coincides with the Riordan array $\\left(1, \\frac{x}{1-x}\\right)$ .",
"Our first result is the Fundamental Theorem of almost-Riordan arrays.",
"Proposition 5 Let $(a, g, f)$ define an almost-Riordan array, and consider a power series $h(x)=\\sum _{n=0}^{\\infty }h_n x^n$ .",
"We have $ (a, g, f) \\cdot h(x)=h_0 a(x)+x g(x) \\tilde{h}(f(x)),$ where $\\tilde{h}(x)=\\frac{h(x)-h_0}{x}.$ We have $(a, g, f) \\cdot h(x)&=&(a, xg, xgf, xgf^2,\\cdots )\\cdot \\left(\\begin{array}{c} h_0\\\\h_1\\\\h_2\\\\h_3\\\\ \\vdots \\end{array}\\right)\\\\&=& h_0 a + h_1 x g+ h_2 xgf + h_3 x g f^2+\\cdots \\\\&=& h_0 a + x g(h_1+h_2 f + h_3 f^2+ \\cdots ) \\\\&=& h_o a + xg \\tilde{h}(f).$ Corollary 6 We have $(a, g, f)\\cdot 1 = a.$ This follows since we have $(a, g, f)\\cdot 1=1.a+xg \\tilde{1}(f)=a,$ since $\\tilde{1}=0$ .",
"We next define the product of two almost-Riordan arrays.",
"Thus consider the almost-Riordan arrays $(a, g, f)$ and $(b, u, v)$ .",
"We define their product by $ (a, g, f) \\cdot (b,u, v)= ( (a, g, f) b, g u(f), v(f)).$ By construction, the product of two almost-Riordan arrays is again an almost-Riordan array.",
"We define $I=(1,1,x)$ .",
"We have Proposition 7 $ I \\cdot (b, u, v)=(b,u,v), \\quad \\quad (a,f,g) \\cdot I = (a, f, g).$ $I \\cdot (b, u, v)=(1,1,x)\\cdot (b,u,v)=((1,1,x) b, 1\\, u(x), v(x))=((1,1,x) b, u, v).$ Now $(1,1,x) b= b_0 \\,1 + x\\,1\\,\\tilde{b}(x)=1 + x \\frac{b-1}{x}=b.$ Thus $ I \\cdot (b, u ,v) = (b, u, v).$ Now $(a, g, f) \\cdot I = (a,g,f) \\cdot (1,1,x)=((a,g,f)\\cdot 1, g\\,1(f), x(f))=((a,g,f)\\cdot 1, g, f).$ We have $(a, g,f)\\cdot 1=1.",
"a+ x g \\tilde{1}(f)=a \\quad \\textrm {since\\,} \\tilde{1}=0.$ Thus $(a, g, f) \\cdot I = (a, g, f).$ Thus $I=(1,1,x)$ is an identity element for the set of almost-Riordan arrays.",
"We have elaborated the above proposition to show that the formalism works.",
"A more direct proof is to notice that $I=(1,1,x)$ is of course the normal (infinite) identity matrix with 1's on the diagonal and zeros elsewhere.",
"We now turn to look at the inverse of an almost-Riordan array.",
"Example 8 We consider the almost-Riordan array given by $(1, g, f)$ .",
"Its inverse is given by $(1, g, f)^{-1}=\\left(1, \\frac{1}{g(\\bar{f})}, \\bar{f}\\right).$ In other words, it is the matrix with first row and columns generated by 1, and starting at the $(1,1)$ -position, it coincides with the inverse Riordan array $(g, f)^{-1}$ .",
"This result is an immediate consequence of standard matrix partitioning.",
"Proposition 9 The inverse of the almost-Riordan array $(a, g, f)$ is the almost-Riordan array $ (a, g, f)^{-1}=\\left(a^*, \\frac{1}{g(\\bar{f})}, \\bar{f}\\right),$ where $ a^*(x)=(1, -g, f)^{-1} \\cdot a(x).$ We need to show that $ (a, g, f)\\cdot \\left(a^*, \\frac{1}{g(\\bar{f})}, \\bar{f}\\right)=I,$ and $ \\left(a^*, \\frac{1}{g(\\bar{f})}, \\bar{f}\\right) \\cdot (a, g, f)=I.$ Clearly, for the first expression, we only need to show that $(a, g, f)\\cdot a^*=1,$ since the result then follows by matrix partitioning.",
"We have $(a, f, g)\\cdot a^*&=& (a,g,f)\\cdot \\left(1, -\\frac{1}{g(\\bar{f})}, \\bar{f}\\right)\\cdot a(x)\\\\&=& \\left((a,g,f)\\cdot 1, g.-\\frac{1}{g(\\bar{f}(f))}, \\bar{f}(f)\\right)\\cdot a(x)\\\\&=& (a, -1, x) \\cdot a\\\\&=& a_0 a - x\\tilde{a}(x)\\\\&=& a-x \\frac{a(x)-1}{x}\\\\&=& a- a+1\\\\&=& 1 $ We must now show that $ \\left(a^*, \\frac{1}{g(\\bar{f})}, \\bar{f}\\right) \\cdot (a, g, f)=I.$ Again, by matrix partitioning, we need to show that $ \\left(a^*, \\frac{1}{g(\\bar{f})}, \\bar{f}\\right)\\cdot a =1.$ We have $\\left(a^*, \\frac{1}{g(\\bar{f})}, \\bar{f}\\right)\\cdot a&=&a_0 a^*+\\frac{x}{g(\\bar{f})}\\tilde{a}(\\bar{f})\\\\&=& (1, -g, f)^{-1}\\cdot a(x)+\\frac{x}{g(\\bar{f})}\\tilde{a}(\\bar{f})\\\\&=& \\left(1, -\\frac{1}{g(\\bar{f})}, \\bar{f}\\right)\\cdot a(x)+\\frac{x}{g(\\bar{f})}\\tilde{a}(\\bar{f})\\\\&=& a_0\\,1-\\frac{x}{g(\\bar{f})}\\tilde{a}(\\bar{f})+\\frac{x}{g(\\bar{f})}\\tilde{a}(\\bar{f})\\\\&=& 1.$ Thus the set of almost-Riordan arrays is in fact a group.",
"We denote this group by $a\\mathcal {R}$ .",
"The group of Riordan arrays $\\mathcal {R}$ is a subgroup of this group, identified as the subgroup of almost-Riordan arrays of the form $\\left(g, g\\frac{f}{x}, f\\right).$ Let us verify that the subset of $a\\mathcal {R}$ consisting of arrays of the form $\\left(g, g\\frac{f}{x}, f\\right)$ is closed under the product of $a\\mathcal {R}$ .",
"Thus let $\\left(g, g\\frac{f}{x}, f\\right)$ and $\\left(u, u \\frac{v}{x}, v\\right)$ be two elements of this subset.",
"We have $\\left(g, g\\frac{f}{x}, f\\right) \\cdot \\left(u, u \\frac{v}{x}, v\\right)&=&\\left(\\left(g, g\\frac{f}{x}, f\\right) \\cdot u, g\\frac{f}{x} \\frac{u(f) v(f)}{f}, v(f)\\right)\\\\&=& \\left(u_0g+xg \\frac{f}{x} \\tilde{u}(f), gu(f) \\frac{v(f)}{x}, v(f)\\right)\\\\&=& \\left(u_0 g+gf\\left(\\frac{u(f)-u_0}{f}\\right), gu(f) \\frac{v(f)}{x}, v(f)\\right)\\\\&=& \\left(u_0 g + g(u(f)-u_0), gu(f) \\frac{v(f)}{x}, v(f)\\right)\\\\&=& \\left(gu(f), gu(f) \\frac{v(f)}{x}, v(f)\\right).$ Thus the subset is closed under products.",
"We next show that this subset is closed under inverses.",
"We have $\\left(g, g\\frac{f}{x}, f\\right)^{-1}&=& \\left(g^*(x), \\frac{1}{\\frac{gf}{x} \\circ \\bar{f}}(x),\\bar{f}(x)\\right)\\\\&=&\\left(\\left(1,-g \\frac{f}{x}, f\\right)^{-1}\\cdot g, \\frac{1}{\\frac{g(\\bar{f}(x))f(\\bar{f}(x))}{\\bar{f}(x)}}, \\bar{f}\\right)\\\\&=& \\left(\\left(1, -\\frac{1}{\\frac{gf}{x}\\circ \\bar{f}}, \\bar{f}\\right)\\cdot g, \\frac{1}{g(\\bar{f}(x))}\\frac{\\bar{f}(x)}{x}, \\bar{f}\\right)\\\\&=& \\left(g_0.1-x \\frac{1}{\\frac{gf}{x} \\circ \\bar{f}} \\tilde{g}(\\bar{f}), \\frac{1}{g(\\bar{f}(x))}\\frac{\\bar{f}(x)}{x}, \\bar{f}\\right)\\\\&=& \\left(g_0-x \\frac{1}{g(\\bar{f}(x))}\\frac{\\bar{f}(x)}{x} \\left(\\frac{g(\\bar{f})-g_0}{\\bar{f}}\\right), \\frac{1}{g(\\bar{f}(x))}\\frac{\\bar{f}(x)}{x}, \\bar{f}\\right)\\\\&=& \\left(g_0-\\frac{1}{g(\\bar{f}(x))}(g(\\bar{f}(x))-g_0), \\frac{1}{g(\\bar{f}(x))}\\frac{\\bar{f}(x)}{x}, \\bar{f}\\right)\\\\&=& \\left(\\frac{1}{g(\\bar{f}(x))}, \\frac{1}{g(\\bar{f}(x))}\\frac{\\bar{f}(x)}{x}, \\bar{f}\\right).$ We thus have Proposition 10 The subset of $a\\mathcal {R}$ of almost-Riordan arrays of the form $\\left(g, g\\frac{f}{x}, f\\right)$ is a subgroup of $a\\mathcal {R}$ , isomorphic to the group $\\mathcal {R}$ of Riordan arrys.",
"We have just shown that this subset is a subgroup (with identity $(1, 1, x)=\\left(1, 1 \\frac{x}{x}, x\\right)$ ).",
"There is an obvious $1-1$ correspondence between elements $\\left(g, g\\frac{f}{x}, f\\right)$ of this subgroup and the corresponding element $(g, f) \\in \\mathcal {R}$ .",
"We write $\\phi (.",
")$ for this correspondence so that $ \\phi \\left(\\left(g, g \\frac{f}{x}, f\\right)\\right)=(g, f).$ It remains to show that this is a homomorphism.",
"We have $\\phi \\left( \\left(g, g\\frac{f}{x}, f\\right) \\cdot \\left(u, u \\frac{v}{x}, v\\right)\\right)&=&\\phi \\left( \\left(gu(f), gu(f)\\frac{v(f)}{x}, v(f)\\right)\\right)\\\\&=& (gu(f), v(f)).$ On the other hand, we have $\\phi \\left(\\left(g, g \\frac{f}{x}, f\\right)\\right)\\cdot \\phi \\left(\\left(u, u \\frac{v}{x}, v\\right)\\right)&=&(g, f) \\cdot (u, v) \\\\&=& (g u(f), v(f)).$ Similarly, we have $\\phi \\left(\\left(g, g\\frac{f}{x}, f\\right)^{-1}\\right)=\\phi \\left(\\left(\\frac{1}{g(\\bar{f}(x))}, \\frac{1}{g(\\bar{f}(x))} \\frac{\\bar{f}}{x}, \\bar{f}\\right)\\right)=\\left(\\frac{1}{g(\\bar{f}(x))}, \\bar{f}(x)\\right)=(g, f)^{-1}.$ This is not the only subgroup of $a\\mathcal {R}$ that is isomorphic to $\\mathcal {R}$ .",
"We have Proposition 11 The map $\\psi : (1, g, f) \\mapsto (g, f)$ is an isomorphism from the subset of $a\\mathcal {R}$ comprised of elements of the form $(1, g, f)$ to $\\mathcal {R}$ .",
"Clearly, we have $\\psi \\left( (1, g, f)\\cdot (1, u, v)\\right)=\\psi \\left((1, g u(f), v(f))\\right)=(gu(f), v(f))= (g, f)\\cdot (u, v),$ and $ \\psi \\left((1, g, f)^{-1}\\right)=\\psi \\left(\\left(1, \\frac{1}{g\\circ \\bar{f}}, \\bar{f}\\right)\\right)=\\left(\\frac{1}{g\\circ \\bar{f}}, \\bar{f}\\right)=(g, f)^{-1}=(\\psi (1, g, f))^{-1}.$ Another subgroup of $a\\mathcal {R}$ is the group of almost-Riordan arrays of the form $(a, 1, x).$ We have $(a, 1, x)\\cdot (b, 1,x)=((a,1,x) b, 1, x),$ showing that these elements are closed under multiplication.",
"In fact, we have $ (a,1,x) \\cdot (b,1,x)&=& ((a,1,x) b, 1, x)\\\\&=& (b_0 a+x \\tilde{b}(x),1,x) \\\\&=& \\left(b_0 a+x \\frac{b(x)-1}{x},1,x\\right)\\\\&=& (a+b-1, 1,x).$ Turning to inverses, we have $(a,1,x)^{-1}=(a^*, 1,x),$ where $a^*=(1,-1,x)\\cdot a=a_0.1 -x\\tilde{a}(x).$ Thus the first column of $(a,1,x)^{-1}$ is given by the sequence $a_0, -a_1, -a_2, -a_3, \\ldots .$ Proposition 12 The subset $\\mathcal {N}$ of the group of almost-Riordan arrays $a\\mathcal {R}$ of matrices of the form $(a, 1, x)$ is a subgroup, where products are defined by $(a, 1, x) \\cdot (b, 1, x)= (a+b-1, 1,x), $ and inverses are defined by $(a, 1, x)^{-1} =(a^*, 1,x)=(a_0.1 -x\\tilde{a}(x),1,x).$ Example 13 The almost-Riordan array $\\left(\\frac{1}{1-3x}, 1,x\\right)$ begins $\\left(\\begin{array}{ccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\3 & 1 & 0 & 0 & 0 & 0 & 0 \\\\9 & 0 & 1 & 0 & 0 & 0 & 0 \\\\27 & 0 & 0 & 1 & 0 & 0 & 0 \\\\81 & 0 & 0 & 0 & 1 & 0 & 0 \\\\243 & 0 & 0 & 0 & 0 & 1 & 0 \\\\729 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\\end{array}\\right).$ Its inverse begins $\\left(\\begin{array}{ccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\-3 & 1 & 0 & 0 & 0 & 0 & 0 \\\\-9 & 0 & 1 & 0 & 0 & 0 & 0 \\\\-27 & 0 & 0 & 1 & 0 & 0 & 0 \\\\-81 & 0 & 0 & 0 & 1 & 0 & 0 \\\\-243 & 0 & 0 & 0 & 0 & 1 & 0 \\\\-729 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\\end{array}\\right).$ Proposition 14 The subgroup $\\mathcal {N}$ of $a\\mathcal {R}$ of almost-Riordan arrays of the form $(a, 1,x)$ is a normal subgroup of $a\\mathcal {R}$ .",
"We must show that for an arbitrary element $(a, g, f)$ of $a\\mathcal {R}$ , the element $ (a, g, f)\\cdot (b, 1,x)\\cdot (a, g, f)^{-1}$ is of the form $(b^{\\prime }, 1,x)$ for an appropriate power series $b^{\\prime }$ .",
"We have $(a, g, f)\\cdot (b, 1,x)\\cdot (a, g, f)^{-1}&=&((a,g,f)\\cdot b, g, f)\\cdot \\left(a^*, \\frac{1}{g(\\bar{f})}, \\bar{f}\\right)\\\\&=&\\left(((a,g,f)\\cdot b, g, f)\\cdot a^*, g. \\frac{1}{g(\\bar{f}(f))}, \\bar{f}(f)\\right)\\\\&=& (((a,g,f) \\cdot b, g, f)\\cdot a^*, 1, x)$ as required.",
"It is instructive to continue the above calculation.",
"Thus we have $(a,g,f) \\cdot (b,1,x) \\cdot (a, g, f)^{-1}=(((a,g,f) \\cdot b, g, f)\\cdot a^*, 1, x).$ We simplify the first element of the latter matrix.",
"$((a,g,f) \\cdot b, g, f)\\cdot a^*&=&((a,g,f)\\cdot b, g,f)\\cdot \\left(1,-\\frac{1}{g(\\bar{f})}, \\bar{f}\\right)\\cdot a(x)\\\\&=& \\left(((a,g,f)\\cdot b, g,f)\\cdot 1, - g.\\frac{1}{g(\\bar{f}(f))}, \\bar{f}(f)\\right)\\cdot a(x)\\\\&=& ((a,g, f) \\cdot b, -1, x)\\cdot a(x)\\\\&=& a_0 (a, g,f) \\cdot b-x.1.\\tilde{a}(x)\\\\&=& (a,g,f)\\cdot b -x \\tilde{a}(x)\\\\&=& b_0 a+xg \\tilde{b}(f)-x \\tilde{a}(x)\\\\&=& a + x g \\tilde{b}(f)-x \\frac{a(x)-1}{x}\\\\&=& a+xg \\tilde{b}(f)-a(x)+1\\\\&=& 1+xg \\tilde{b}(f).$ Finally, we have $(a,g,f) \\cdot (b,1,x) \\cdot (a, g, f)^{-1}=(1+xg \\tilde{b}(f), 1, x).$ We have the following canonical factorization.",
"$(a, g, f)=(a,1,x) \\cdot (1, g, f).$ This follows since $(a, 1, x)\\cdot (1, g, f)=((a,1,x)\\cdot 1, g, f)=(a, g, f).$ Now let $(a, g, f) \\in a\\mathcal {R}$ .",
"We have $(a, g, f)\\mathcal {N} = (1, g, f)\\mathcal {N} \\Leftrightarrow (1,g,f)^{-1}\\cdot (a, g,f) \\in \\mathcal {N}.$ Now $(1, g, f)^{-1} \\cdot (a,g,f)&=& \\left(1, \\frac{1}{g \\circ \\bar{f}}, \\bar{f}\\right)\\cdot (a, g, f)\\\\&=& \\left((a, g, f).1, \\frac{1}{g \\circ \\bar{f}}.g(\\bar{f}), f(\\bar{f})\\right)\\\\&=& (a, 1, x) \\in \\mathcal {N}.$ Hence modulo $\\mathcal {N}$ , we have $ (a, g,f) \\backsim (1, g, f).$ It is clear that we have a $1-1$ correspondence between almost-Riordan arrays of the form $(1, g, f)$ and Riordan arrays $(g,f)$ .",
"Hence $ a\\mathcal {R}/\\mathcal {N} = \\mathcal {R}.$ Proposition 15 Let $a\\mathcal {R}$ be the group of almost-Riordan arrays, $\\mathcal {R}$ be the group of Riordan arrays, and $\\mathcal {N}$ be the normal subgroup of $a\\mathcal {R}$ consisting of arrays of the form $(a, 1, x)$ where $a_0=1$ .",
"Then $a\\mathcal {R}/\\mathcal {N} = \\mathcal {R}.$ Proposition 16 $\\mathcal {R}$ is not a normal subgroup of $a\\mathcal {R}$ .",
"We consider an element $\\left(u, u\\frac{v}{x}, v\\right)$ of the subgroup $\\mathcal {R}$ .",
"If $\\mathcal {R}$ were a normal subgroup, then for an arbitrary element $(a, g, f) \\in a\\mathcal {R}$ , we would have $ (a, g, f)\\cdot \\left(u, u\\frac{v}{x}, v\\right) \\cdot (a, g, f)^{-1}=\\left(U, U \\frac{V}{x}, V\\right),$ for appropriate power series $U(x)$ and $V(x)$ .",
"Now we have $(a, g, f)\\cdot \\left(u, u\\frac{v}{x}, v\\right)&=& \\left((a,g,f)u, g u(f)\\frac{v(f)}{f}, v(f)\\right)\\\\&=&\\left(u_0 a+xg\\tilde{u}(f),g u(f)\\frac{v(f)}{f}, v(f)\\right).$ Hence $(a, g, f)\\cdot \\left(u, u\\frac{v}{x}, v\\right) \\cdot (a, g, f)^{-1}&=&\\left(u_0 a+xg\\tilde{u}(f),g u(f)\\frac{v(f)}{f}, v(f)\\right)\\cdot (a, g, f)^{-1}\\\\&=&\\left(u_0 a+xg\\tilde{u}(f),g u(f)\\frac{v(f)}{f}, v(f)\\right)\\cdot \\left(a^*,\\frac{1}{g(\\bar{f})},\\bar{f}\\right).$ This last expression is equal to $\\left(\\left(u_0 a+xg\\tilde{u}(f),g u(f)\\frac{v(f)}{f}, v(f)\\right)\\cdot a^*,gu(f)\\frac{v(f)}{f} \\frac{1}{g\\circ \\bar{f}(v(f))}, \\bar{f}(v(f))\\right).$ We must therefore simplify $\\left(u_0 a+xg\\tilde{u}(f),g u(f)\\frac{v(f)}{f}, v(f)\\right)\\cdot a^*,$ where $a^*=\\left(1, -\\frac{1}{g(\\bar{f})}, \\bar{f}\\right)\\cdot a.$ Now $\\left(u_0 a+xg\\tilde{u}(f),g u(f)\\frac{v(f)}{f}, v(f)\\right)\\cdot a^*&=&\\left(u_0 a+xg\\tilde{u}(f),g u(f)\\frac{v(f)}{f}, v(f)\\right) \\cdot \\left(1, -\\frac{1}{g(\\bar{f})}, \\bar{f}\\right)\\cdot a\\\\&=& \\left(u_0 a+xg\\tilde{u}(f), -gu(f)\\frac{v(f)}{f} \\frac{1}{g\\circ \\bar{f}(v(f))}, \\bar{f}(v(f))\\right)\\cdot a\\\\&=& a_0(u_0 a+xg\\tilde{u}(f))-xgu(f)\\frac{v(f)}{f} \\frac{1}{g\\circ \\bar{f}(v(f))} \\tilde{a}(\\bar{f}(v(f)))\\\\&=& a+xg\\tilde{u}(f)-xgu(f)\\frac{v(f)}{f} \\frac{1}{g\\circ \\bar{f}(v(f))} \\tilde{a}(\\bar{f}(v(f))).$ We have thus arrived at $(a, g, f)\\cdot \\left(u, u\\frac{v}{x}, v\\right) \\cdot (a, g, f)^{-1}=$ $\\left(a+xg\\tilde{u}(f)-xgu(f)\\frac{v(f)}{f} \\frac{1}{g\\circ \\bar{f}(v(f))} \\tilde{a}(\\bar{f}(v(f))),gu(f)\\frac{v(f)}{f} \\frac{1}{g\\circ \\bar{f}(v(f))}, \\bar{f}(v(f))\\right)=$ $\\left(a+xg\\tilde{u}(f)-xgu(f)\\frac{v(f)}{f} \\frac{1}{g\\circ \\bar{f}(v(f))} \\frac{a(\\bar{f}(v(f)))-1}{\\bar{f}(v(f))},gu(f)\\frac{v(f)}{f} \\frac{1}{g\\circ \\bar{f}(v(f))}, \\bar{f}(v(f))\\right).$ The first element expands to give $a+xg\\tilde{u}(f)-xgu(f)\\frac{v(f)}{f} \\frac{1}{g\\circ \\bar{f}(v(f))} \\frac{a(\\bar{f}(v(f)))-1}{\\bar{f}(v(f))}=$ $a+xg\\tilde{u}(f)-xgu(f)\\frac{v(f)}{f} \\frac{1}{g\\circ \\bar{f}(v(f))}\\frac{a(\\bar{f}(v(f)))}{\\bar{f}(v(f))}+xgu(f)\\frac{v(f)}{f} \\frac{1}{g\\circ \\bar{f}(v(f))}\\frac{1}{\\bar{f}(v(f))}.$ Thus the obstruction to achieving normality is the expression $a+xg\\tilde{u}(f)-xgu(f)\\frac{v(f)}{f} \\frac{1}{g\\circ \\bar{f}(v(f))}\\frac{a(\\bar{f}(v(f)))}{\\bar{f}(v(f))}.$ The production matrix of $(a, g, f)$ is as follows.",
"Proposition 17 The production matrix of the almost-Riordan array $(a, g, f)$ is given as follows.",
"Its first column is $(a, g, f)^{-1}\\cdot \\tilde{a}(x)$ .",
"Its second column is given by $(a, g, f)^{-1} \\cdot g(x)$ .",
"Subsequent columns coincide with the $A$ -sequence of the Riordan array $(g, f)$ .",
"This follows immediately from the definition of the production matrix $(a, g, f)^{-1} \\cdot \\overline{(a, g, f)}.$ The matrix $\\overline{(a, g, f)}$ has a first column generated by $\\tilde{a}$ , alongside the Riordan array $(g, f)$ .",
"Thus the production matrix consists of the result of applying $(a, g, f)^{-1}$ to $\\tilde{a}$ , alongside the result of applying the inverse $(g, f)^{-1}$ to $(g, f)$ without its first row (since in $(a, g, f)$ , the component $(g, f)$ starts in the column that is one column in from the left).",
"We call the first (0-th) column of the production matrix of $(a, g, f)$ the $\\omega $ sequence, the second column the $Z$ -sequence and the third column the $A$ -sequence (with respective entries $\\omega _n$ , $Z_n$ and $A_n$ ).",
"Then we have the following.",
"$T_{n,0}=\\sum _{j=0}^n T_{n-1,j} \\omega _k.$ $T_{n,1}=\\sum _{j=0}^n T_{n-1,j} Z_j.$ $T_{n,k}=\\sum _{j=k-1}^n T_{n-1, j} A_j, \\quad k > 1.$ We have $\\omega (x)&=&(a, g, f)^{-1}\\cdot \\tilde{a}(x)\\\\&=& \\left(a^*, \\frac{1}{g(\\bar{f})}, \\bar{f}\\right)\\cdot \\tilde{a}(x) \\\\&=& \\tilde{a}_0 a^*+x\\frac{1}{g(\\bar{f})}\\tilde{a}(\\bar{f})\\\\&=& a_1 a^*+x\\frac{1}{g(\\bar{f})}\\tilde{a}(\\bar{f})\\\\&=& a_1 (a_0-x \\frac{1}{g(\\bar{f})} a(\\bar{f}))+x\\frac{1}{g(\\bar{f})}\\tilde{a}(\\bar{f})\\\\&=& a_1 + \\frac{x}{g(\\bar{f})}\\left(\\tilde{a}(\\bar{f})-a_1 a(\\bar{f})\\right).$"
],
[
"Iterating the process",
"A natural question that arises is whether the process of adding a new column on the left can be iterated to assemble a hierarchy of higher groups?",
"The following considerations indicate that this is indeed possible.",
"We define a set of matrices $\\mathcal {R}^{(2)}$ as follows.",
"Its elements are 4-tuples of power series $(a, b, g, f)$ where $a$ , $b$ , $g$ and $f$ satisfy $a_0=1$ , $b_0=1$ , $g_0=1$ and $f_0=0, f_1=1$ .",
"We define a product of such elements by $(a, b, g, f) \\cdot (h, k, u, v)=((a, b, g, f)\\cdot h, (b, g, f)\\cdot k, g u(f), v(f)).$ Here, the term $(b, g, f) \\cdot k$ is to be taken in the sense of $a\\mathcal {R}=\\mathcal {R}^{(1)}$ , while the term $g u(f)=(g, u)\\cdot f$ in the sense of $\\mathcal {R}=\\mathcal {R}^{(0)}$ .",
"Thus $(a,b,g,f) \\cdot (h, k,u, v)=((a, b,g,f,)\\cdot h, (b,g,f)\\cdot k, (g, u)\\cdot f, v(f)).$ It remains to say what is $(a, b, g, f) \\cdot h$ .",
"We define ${}(a, b, g, f) \\cdot h = h_0 a + h_1 xb + x^2 g \\tilde{\\tilde{h}}(f),$ where $\\tilde{\\tilde{h}}=\\frac{h(x)-h_0-h_1 x}{x^2}.$ The 4-tuple $(a, b, g, f)$ is identified with the following lower-triangular matrix: its first column (the 0-th column) is given by the expansion of $a(x)$ ; the second column begins with a 0, and from the $(1,1)$ -position downwards coincides with the expansion of $b(x)$ (that is, the second column coincides with the expansion of $xb(x)$ ).",
"Starting from the $(2,2)$ position, the matrix coincides with the Riordan array $(g,f)$ .",
"Other elements are zero.",
"Matrix multiplication of a vector (when that vector's elements coincide with the expansion of a generating function $h(x)$ ) then corresponds to the rule given by Equation (REF ).",
"This can then be called the Fundamental Theorem for $\\mathcal {R}^{(2)}$ .",
"We can define the inverse of a 4-tuple as follows.",
"$(a, b, g, f)^{-1}=\\left(a^{**}, b^*, \\frac{1}{g \\circ \\bar{f}}, \\bar{f}\\right),$ where $b^*=(1, - g, f)^{-1}\\cdot b = \\left(1, -\\frac{1}{g \\circ \\bar{f}}, \\bar{f}\\right)\\cdot b,$ and where $a^{**} = (1, -b, -g, f)^{-1} \\cdot a=\\left(1, -b^*, -\\frac{1}{g \\circ \\bar{f}}, \\bar{f}\\right) \\cdot a.$ Example 18 We consider the element $(a,b, g, f)=\\left(\\frac{1-2}{1-3x}, \\frac{1-x}{1-2}, \\frac{1}{1-x}, \\frac{x}{1-x}\\right) \\in \\mathcal {R}^{(2)}.$ We have $b^*&=&\\left(1, -\\frac{1}{1-x}, \\frac{x}{1-x}\\right)^{-1}\\cdot \\frac{1-x}{1-2x}\\\\&=& \\left(1, -\\frac{1}{1+x}, \\frac{x}{1+x}\\right)\\cdot \\frac{1-x}{1-2x}\\\\&=& \\frac{1-2x}{1-x}.$ Then $a^{**}&=& (1, -b,-g, f)^{-1} \\cdot a\\\\&=& \\left(1, -\\frac{1-x}{1-2x}, -\\frac{1}{1-x}, \\frac{x}{1-x}\\right)^{-1}\\cdot a\\\\&=& \\left(1, -\\frac{1-2x}{1-x}, -\\frac{1}{1+x}, \\frac{x}{1+x}\\right) \\cdot \\frac{1-2x}{1-3x}\\\\&=& 1 - x \\frac{1-2x}{1-x}-x^2 \\frac{1}{1+x} \\widetilde{\\widetilde{\\left(\\frac{1-2x}{1-3x}\\right)}}\\left(\\frac{x}{1+x}\\right)\\\\&=& \\frac{1-4x+3x^2-x^3}{(1-x)(1-2x)}.$ Here, we have $\\widetilde{\\widetilde{\\left(\\frac{1-2x}{1-3x}\\right)}}=\\frac{3}{1-3x}$ and hence $\\widetilde{\\widetilde{\\left(\\frac{1-2x}{1-3x}\\right)}}\\left(\\frac{x}{1+x}\\right)=\\frac{3}{1-3\\frac{x}{1+x}}=\\frac{3(1+x)}{1-2x}.$ Thus we have $\\left(\\frac{1-2}{1-3x}, \\frac{1-x}{1-2}, \\frac{1}{1-x}, \\frac{x}{1-x}\\right)^{-1}=\\left(\\frac{1-4x+3x^2-x^3}{(1-x)(1-2x)}, \\frac{1-2x}{1-x}, \\frac{1}{1+x}, \\frac{x}{1+x}\\right).$ Example 19 There are many ways of constructing elements of $\\mathcal {R}^{(2)}$ .",
"For instance, we can start with a Riordan array and pre-pend two columns appropriately.",
"Alternatively, we could start with a Riordan array and multiply it by an element of the form $(a,b,1,x)$ .",
"The following shows another method of defining an element of $\\mathcal {R}^{(2)}$ , starting with an element of $\\mathcal {R}^{(0)}=\\mathcal {R}$ .",
"We take the matrix $\\left(\\frac{1}{1+x}, \\frac{x}{(1+x)^2}\\right)^{-1}=(c(x), c(x)-1), $ which begins $\\left(\\begin{array}{ccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\1 & 1 & 0 & 0 & 0 & 0 & 0 \\\\2 & 3 & 1 & 0 & 0 & 0 & 0 \\\\5 & 9 & 5 & 1 & 0 & 0 & 0 \\\\14 & 28 & 20 & 7 & 1 & 0 & 0 \\\\42 & 90 & 75 & 35 & 9 & 1 & 0 \\\\132 & 297 & 275 & 154 & 54 & 11 & 1 \\\\\\end{array}\\right).$ We then form the product $\\left(\\begin{array}{ccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\1 & 1 & 0 & 0 & 0 & 0 & 0 \\\\2 & 3 & 1 & 0 & 0 & 0 & 0 \\\\5 & 9 & 5 & 1 & 0 & 0 & 0 \\\\14 & 28 & 20 & 7 & 1 & 0 & 0 \\\\42 & 90 & 75 & 35 & 9 & 1 & 0 \\\\132 & 297 & 275 & 154 & 54 & 11 & 1 \\\\\\end{array}\\right)\\cdot \\left(\\begin{array}{ccccccccc}1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 \\\\0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\\\\\end{array}\\right),$ to obtain the matrix that begins $\\left(\\begin{array}{ccccccccc}1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\1 & 2 & 2 & 1 & 0 & 0 & 0 & 0 & 0 \\\\2 & 5 & 6 & 4 & 1 & 0 & 0 & 0 & 0 \\\\5 & 14 & 19 & 15 & 6 & 1 & 0 & 0 & 0 \\\\14 & 42 & 62 & 55 & 28 & 8 & 1 & 0 & 0 \\\\42 & 132 & 207 & 200 & 119 & 45 & 10 & 1 & 0\\\\132 & 429 & 704 & 726 & 483 & 219 & 66 & 12 &1 \\\\\\end{array}\\right).$ We now complete this matrix to be lower-triangular as follows $M=\\left(\\begin{array}{ccccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\1 & 2 & 2 & 1 & 0 & 0 & 0 & 0 & 0 \\\\2 & 5 & 6 & 4 & 1 & 0 & 0 & 0 & 0 \\\\5 & 14 & 19 & 15 & 6 & 1 & 0 & 0 & 0 \\\\14 & 42 & 62 & 55 & 28 & 8 & 1 & 0 & 0 \\\\42 & 132 & 207 & 200 & 119 & 45 & 10 & 1 & 0\\\\132 & 429 & 704 & 726 & 483 & 219 & 66 & 12 &1 \\\\\\end{array}\\right).$ The production matrix of this array then begins $\\left(\\begin{array}{cccccccc}1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 1 & 1 & 1 & 0 & 0 & 0 & 0 \\\\1 & 2 & 2 & 2 & 1 & 0 & 0 & 0 \\\\-1 & -2 & 0 & 1 & 2 & 1 & 0 & 0 \\\\0 & 0 & -1 & 0 & 1 & 2 & 1 & 0 \\\\1 & 2 & 1 & 0 & 0 & 1 & 2 & 1 \\\\-1 & -2 & 0 & 0 & 0 & 0 & 1 & 2 \\\\\\end{array}\\right),$ indicating that the matrix $M$ is an element of $\\mathcal {R}^{(2)}$ .",
"We note for instance that the transform $b_n$ of the Fibonacci numbers $F_n$ by this matrix has a Hankel transform with generating function $\\frac{-x(1-10x+24x^2+64x^3)}{(1-4x)^4},$ while the Hankel transform of $b_{n+1}$ has generating function $\\frac{1-6x}{(1-4x)^2}.$ We have the following proposition.",
"Proposition 20 The set of 4-tuples $\\mathcal {R}^{(2)}$ defined above is a group, with identity $I=(1,1,1,x)$ .",
"The subset $\\mathcal {N}^{(2)}$ of 4-tuples of the form $(a, b, 1, x)$ is a normal subgroup of $\\mathcal {R}^{(2)}$ and we have $ \\mathcal {R}^{(2)}/\\mathcal {N}^{(2)} = \\mathcal {R}^{(0)}.$ In similar fashion, we may define a hierarchy of sets of $n$ -tuples of power series $\\mathcal {R}^{(n-2)}$ , where $\\mathcal {R}^{(0)}=\\mathcal {R},$ the Riordan group."
],
[
"Example 1: Almost-Riordan arrays and a special transformation",
"In [3], the authors consider a transformation on sequences $a_n$ with the property $a_{-n}=a_n$ , defined by $b_n=\\sum _{k=0}^{n-1} \\binom{n-1}{k}(a_{1-n+2k}+a_{2-n+k}).$ For the special sequence $a_n=x^n$ for $n \\ge 0$ , $a_n=x^{-n}$ for $n<0$ (i.e.",
"$a_n=x^{|n|}$ ), we obtain that the images $b_0, b_1, b_2, b_3, b_4,b_5, \\ldots $ are given by $0, x + 1, x^2 + 2x + 1, x^3 + 2x^2 + 3x + 2, x^4 + 2x^3 + 4x^2 + 6x + 3, x^5 + 2x^4 + 5x^3 + 8x^2 + 10x + 6, \\ldots ,$ with a coefficient array which begins $\\left(\\begin{array}{ccccccc}0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\1 & 1 & 0 & 0 & 0 & 0 & 0 \\\\1 & 2 & 1 & 0 & 0 & 0 & 0 \\\\2 & 3 & 2 & 1 & 0 & 0 & 0 \\\\3 & 6 & 4 & 2 & 1 & 0 & 0 \\\\6 & 10 & 8 & 5 & 2 & 1 & 0 \\\\10 & 20 & 15 & 10 & 6 & 2 & 1 \\\\\\end{array}\\right).$ Since the sequences $a_n$ that are of interest in this case all have $a_0=0$ , we can equivalently use the array that begins $M=\\left(\\begin{array}{ccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\1 & 1 & 0 & 0 & 0 & 0 & 0 \\\\1 & 2 & 1 & 0 & 0 & 0 & 0 \\\\2 & 3 & 2 & 1 & 0 & 0 & 0 \\\\3 & 6 & 4 & 2 & 1 & 0 & 0 \\\\6 & 10 & 8 & 5 & 2 & 1 & 0 \\\\10 & 20 & 15 & 10 & 6 & 2 & 1 \\\\\\end{array}\\right).$ This is an almost-Riordan array, defined by $\\left(\\frac{1+2x+\\sqrt{1-4x^2}}{2 \\sqrt{1-4x^2}}, \\frac{(1+2x)c(x^2)}{\\sqrt{1-4x^2}}, xc(x^2)\\right)=\\left(\\frac{1}{1+x}, \\frac{1-x}{1+x+x^2+x^3}, \\frac{x}{1+x^2}\\right)^{-1},$ where $c(x)=\\frac{1-\\sqrt{1-4x^2}}{2x}$ is the generating function of the Catalan numbers.",
"Its first column is given by $\\binom{n-1}{\\lfloor \\frac{n}{2} \\rfloor }$ .",
"Other than for the first column, this coincides with the Riordan array $R=\\left(\\frac{1+2x}{\\sqrt{1-4x^2}}, xc(x^2)\\right)=\\left(\\frac{1-x}{1+x}, \\frac{x}{1+x^2}\\right)^{-1},$ which has first column $1,2,2,4,6,10,20,\\ldots .$ In fact, we have $M = R \\cdot \\left(\\begin{array}{ccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\-1 & 1 & 0 & 0 & 0 & 0 & 0 \\\\1 & 0 & 1 & 0 & 0 & 0 & 0 \\\\-1 & 0 & 0 & 1 & 0 & 0 & 0 \\\\1 & 0 & 0 & 0 & 1 & 0 & 0 \\\\-1 & 0 & 0 & 0 & 0 & 1 & 0 \\\\1 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\\end{array}\\right)=R \\cdot \\left(\\frac{1}{1+x}, 1, x\\right).$ The production array of the almost-Riordan array $M$ begins $\\left(\\begin{array}{ccccccc}1 & 1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 1 & 1 & 0 & 0 & 0 & 0 \\\\1 & 0 & 0 & 1 & 0 & 0 & 0 \\\\-1 & 1 & 1 & 0 & 1 & 0 & 0 \\\\1 & -1 & 0 & 1 & 0 & 1 & 0 \\\\-1 & 1 & 0 & 0 & 0 & 1 & 0 \\\\1 & -1 & 0 & 0 & 0 & 0 & 1 \\\\\\end{array}\\right),$ where we can see the $\\omega -$ , $Z-$ and $A-$ sequences.",
"We note that the matrix $\\left(\\begin{array}{ccccccc}1 & 1 & 0 & 0 & 0 & 0 & 0 \\\\1 & 2 & 1 & 0 & 0 & 0 & 0 \\\\2 & 3 & 2 & 1 & 0 & 0 & 0 \\\\3 & 6 & 4 & 2 & 1 & 0 & 0 \\\\6 & 10 & 8 & 5 & 2 & 1 & 0 \\\\10 & 20 & 15 & 10 & 6 & 2 & 1 \\\\20 & 35 & 30 & 21 & 12 & 7 & 2 \\\\\\end{array}\\right)$ is equal to $\\left(\\begin{array}{ccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\1 & 1 & 0 & 0 & 0 & 0 & 0 \\\\2 & 1 & 1 & 0 & 0 & 0 & 0 \\\\3 & 3 & 1 & 1 & 0 & 0 & 0 \\\\6 & 4 & 4 & 1 & 1 & 0 & 0 \\\\10 & 10 & 5 & 5 & 1 & 1 & 0 \\\\20 & 15 & 15 & 6 & 6 & 1 & 1 \\\\\\end{array}\\right)\\cdot \\left(\\begin{array}{ccccccc}1 & 1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 1 & 1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 1 & 1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 1 & 1 & 0 & 0 \\\\0 & 0 & 0 & 0 & 1 & 1 & 0 \\\\0 & 0 & 0 & 0 & 0 & 1 & 1 \\\\0 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\\end{array}\\right),$ where the first matrix in this product is the Riordan array $\\left(\\frac{1-x}{1+x^2}, \\frac{x}{1+x^2}\\right)^{-1}=\\left(\\frac{1+xc(x^2)}{\\sqrt{1-4x^2}}, xc(x^2)\\right).$ Thus the product is given by $\\left(\\frac{1+xc(x^2)}{\\sqrt{1-4x^2}}, xc(x^2)\\right) \\cdot (1+x,x)^t.$ We next multiply the coefficient array by the binomial matrix $B=\\left(\\binom{n}{k}\\right)$ to obtain the almost-Riordan array that begins $B\\cdot M=\\left(\\begin{array}{ccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\2 & 1 & 0 & 0 & 0 & 0 & 0 \\\\4 & 4 & 1 & 0 & 0 & 0 & 0 \\\\9 & 12 & 5 & 1 & 0 & 0 & 0 \\\\22 & 34 & 18 & 6 & 1 & 0 & 0 \\\\57 & 95 & 58 & 25 & 7 & 1 & 0 \\\\153 & 266 & 178 & 90 & 33 & 8 & 1 \\\\\\end{array}\\right).$ This is the almost-Riordan array $\\left(\\frac{1+x+\\sqrt{1-2x-3x^2}}{2(1-x)\\sqrt{1-2x-3x^2}}, \\frac{(1-x)\\sqrt{1-2x-3x^2}-(1-2x-3x^2)}{2(1-4x+3x^2)}, \\frac{1-x-\\sqrt{1-2x-3x^2}}{2x^2}\\right),$ where the last entry is the g.f. of the Motzkin numbers.",
"This almost-Riordan array has an inverse given by $\\left(\\frac{1+2x^2}{1+2x+2x^2+x^3}, \\frac{1-x+x^2-x^3}{(1+x)(1+x+x^2)^2}, \\frac{x}{1+x+x^2}\\right).$ The production array of the above almost-Riordan array begins $\\left(\\begin{array}{ccccccc}2 & 1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 2 & 1 & 0 & 0 & 0 & 0 \\\\1 & 0 & 1 & 1 & 0 & 0 & 0 \\\\-1 & 1 & 1 & 1 & 1 & 0 & 0 \\\\1 & -1 & 0 & 1 & 1 & 1 & 0 \\\\-1 & 1 & 0 & 0 & 0 & 1 & 1 \\\\1 & -1 & 0 & 0 & 0 & 0 & 1 \\\\\\end{array}\\right),$ where we can see the usual effect of the binomial transform on the diagonal elements (namely, we increment each diagonal element by 1).",
"We finally multiply the almost-Riordan array $M$ by $\\left(\\begin{array}{ccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\-1 & 1 & 0 & 0 & 0 & 0 & 0 \\\\-1 & 0 & 1 & 0 & 0 & 0 & 0 \\\\-2 & 0 & 0 & 1 & 0 & 0 & 0 \\\\-3 & 0 & 0 & 0 & 1 & 0 & 0 \\\\-6 & 0 & 0 & 0 & 0 & 1 & 0 \\\\-10 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\\end{array}\\right)=\\left(\\begin{array}{ccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\1 & 1 & 0 & 0 & 0 & 0 & 0 \\\\1 & 0 & 1 & 0 & 0 & 0 & 0 \\\\2 & 0 & 0 & 1 & 0 & 0 & 0 \\\\3 & 0 & 0 & 0 & 1 & 0 & 0 \\\\6 & 0 & 0 & 0 & 0 & 1 & 0 \\\\10 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\\end{array}\\right)^{-1}$ to obtain $\\left(\\begin{array}{ccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\-1 & 1 & 0 & 0 & 0 & 0 & 0 \\\\-1 & 0 & 1 & 0 & 0 & 0 & 0 \\\\-2 & 0 & 0 & 1 & 0 & 0 & 0 \\\\-3 & 0 & 0 & 0 & 1 & 0 & 0 \\\\-6 & 0 & 0 & 0 & 0 & 1 & 0 \\\\-10 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\\end{array}\\right)\\cdot \\left(\\begin{array}{ccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\1 & 1 & 0 & 0 & 0 & 0 & 0 \\\\1 & 2 & 1 & 0 & 0 & 0 & 0 \\\\2 & 3 & 2 & 1 & 0 & 0 & 0 \\\\3 & 6 & 4 & 2 & 1 & 0 & 0 \\\\6 & 10 & 8 & 5 & 2 & 1 & 0 \\\\10 & 20 & 15 & 10 & 6 & 2 & 1 \\\\\\end{array}\\right)$ $=\\left(\\begin{array}{ccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 2 & 1 & 0 & 0 & 0 & 0 \\\\0 & 3 & 2 & 1 & 0 & 0 & 0 \\\\0 & 6 & 4 & 2 & 1 & 0 & 0 \\\\0 & 10 & 8 & 5 & 2 & 1 & 0 \\\\0 & 20 & 15 & 10 & 6 & 2 & 1 \\\\\\end{array}\\right).$"
],
[
"Example 2: Some Catalan related almost-Riordan arrays and Hankel transforms",
"We base this section on the almost-Riordan array given by $T = \\left(\\begin{array}{ccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\-1 & 1 & 0 & 0 & 0 & 0 & 0 \\\\1 & 0 & 1 & 0 & 0 & 0 & 0 \\\\-1 & 0 & 0 & 1 & 0 & 0 & 0 \\\\1 & 0 & 0 & 0 & 1 & 0 & 0 \\\\-1 & 0 & 0 & 0 & 0 & 1 & 0 \\\\1 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\\end{array}\\right) \\cdot R,$ where $R$ is the Riordan array $R=\\left(\\frac{1+2x}{\\sqrt{1-4x^2}}, xc(x^2)\\right)=\\left(\\frac{1-x}{1+x}, \\frac{x}{1+x^2}\\right)^{-1}$ seen in the previous section.",
"We obtain that $T$ is the almost-Riordan array that begins $T=\\left(\\begin{array}{ccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\1 & 1 & 0 & 0 & 0 & 0 & 0 \\\\3 & 2 & 1 & 0 & 0 & 0 & 0 \\\\3 & 3 & 2 & 1 & 0 & 0 & 0 \\\\7 & 6 & 4 & 2 & 1 & 0 & 0 \\\\11 & 10 & 8 & 5 & 2 & 1 & 0 \\\\21 & 20 & 15 & 10 & 6 & 2 & 1 \\\\\\end{array}\\right),$ where the first column $1,1,3,3,7,11,21,\\ldots $ is given by $a_n=(-1)^n+2 \\binom{n-1}{\\lfloor \\frac{n-1}{2} \\rfloor },$ with generating function $\\frac{1}{1+x}+\\frac{1+2x-\\sqrt{1-4x^2}}{\\sqrt{1-4x^2}}.$ We note that this sequence has a Hankel transform that begins $1, 2, -4, -24, 64, 352, -64, -1664, 256, 7680, -4096,\\ldots $ with a (conjectured) generating function $\\frac{1+2x-16x^3+48x^4+256x^5+256x^6-128x^7}{(1+4x^2)^2(1-4x^2+16x^4)}.$ In fact, $T$ is the almost-Riordan array $\\left(\\frac{1}{1+x}+\\frac{1+2x-\\sqrt{1-4x^2}}{\\sqrt{1-4x^2}}, \\frac{(1+2x)c(x^2)}{ \\sqrt{1-4x^2}},xc(x^2)\\right),$ where the Riordan array $\\left(\\frac{(1+2x)c(x^2)}{ \\sqrt{1-4x^2}},xc(x^2)\\right)$ has the inverse $\\left(\\frac{1-x}{(1+x)(1+x^2)}, \\frac{x}{1+x^2}\\right).$ We are interested in the effect of the matrix $T$ on the Fibonacci polynomials $F_n(y)=\\sum _{k=0}^{n-1} \\binom{n-k-1}{k}y^k,$ which have generating function $\\frac{x}{1-x-yx^2}.$ We find that the Hankel transform of the image $b_n(y)$ of $F_n(y)$ by $T$ has generating function $-\\frac{x(1-2(y-2)x+(y-1)^2 x^2)}{1-2(y^2-2y-1)x^2+(y-1)^4x^4}.$ Regarded as a bi-variate generating function, this generates the array that begins $G=\\left(\\begin{array}{ccccccccccc}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\4 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\-1 & -6 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\-8 & -12 & 16 & -4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\1 & 20 & 10 & -20 & 5 & 0 & 0 & 0 & 0 & 0 & 0 \\\\12 & 74 & -32 & -52 & 36 & -6 & 0 & 0 & 0 & 0 & 0\\\\-1 & -42 & -119 & 84 & 49 & -42 & 7 & 0 & 0 & 0 & 0\\\\-16 & -216 & -224 & 488 & -80 & -136 & 64 & -8 & 0& 0 & 0 \\\\1 & 72 & 468 & 168 & -738 & 216 & 132 & -72 & 9 & 0& 0 \\\\20 & 470 & 1536 & -984 & -2008 & 1828 & -160 & -280& 100 & -10 & 0 \\\\\\end{array}\\right).$ Thus for instance $G$ applied to $2^n$ (i.e.",
"to the vector $<1,2,4,8,\\ldots >$ ), returns the Hankel transform of the image by $T$ of the Jacobsthal numbers, namely the sequence $0,-1,0,1,0,-1,0,1,0,-1,0,\\ldots .$ The Hankel transform of the once-shifted sequence $b_{n+1}(y)$ is also of interest.",
"The sequence $b_{n+1}(y)$ is the image of the shifted Fibonacci polynomial $\\sum _{k=0}^n \\binom{n-k}{k}y^k$ by the Riordan array $\\left(\\frac{(1+2x)c(x^2)}{ \\sqrt{1-4x^2}},xc(x^2)\\right).$ We find that the Hankel transform of $b_{n+1}(y)$ is generated by $\\frac{1+(y-1)x}{1+2x+(y-1)x^2}$ with coefficient array $\\left(\\begin{array}{ccccccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\-3 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\5 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\-7 & -7 & 7 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\9 & 24 & -18 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\-11 & -55 & 22 & 22 & -11 & 1 & 0 & 0 & 0 & 0 & 0\\\\13 & 104 & 13 & -104 & 39 & 0 & -1 & 0 & 0 & 0 & 0\\\\-15 & -175 & -147 & 285 & -45 & -45 & 15 & -1 & 0 &0 & 0 \\\\17 & 272 & 476 & -544 & -170 & 272 & -68 & 0 & 1 &0 & 0 \\\\-19 & -399 & -1140 & 684 & 1102 & -874 & 76 & 76 &-19 & 1 & 0 \\\\21 & 560 & 2331 & -144 & -3598 & 1680 & 630 & -560& 105 & 0 & -1 \\\\\\end{array}\\right).$ The Hankel transform of the shifted image of the Fibonacci numbers (case of $y=1$ ) is then given by $(-2)^n$ while that of the Jacobsthal numbers is $(-1)^n$ .",
"We note an interesting property of the previous coefficient array.",
"If we multiply it (on the left) by the binomial matrix $B$ , we obtain the array that begins $\\left(\\begin{array}{ccccccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\-2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & -4 & 4 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 4 & -4 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & -8 & 12 & -6 & 1 & 0 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & 8 & -12 & 6 & -1 & 0 & 0 & 0 & 0 \\\\0 & 0 & 0 & -16 & 32 & -24 & 8 & -1 & 0 & 0 & 0 \\\\0 & 0 & 0 & 0 & 16 & -32 & 24 & -8 & 1 & 0 & 0 \\\\0 & 0 & 0 & 0 & -32 & 80 & -80 & 40 & -10 & 1 & 0\\\\0 & 0 & 0 & 0 & 0 & 32 & -80 & 80 & -40 & 10 & -1\\\\\\end{array}\\right).$ We see embedded in this the square $B^2$ of the binomial matrix, which begins $\\left(\\begin{array}{ccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\2 & 1 & 0 & 0 & 0 & 0 & 0 \\\\4 & 4 & 1 & 0 & 0 & 0 & 0 \\\\8 & 12 & 6 & 1 & 0 & 0 & 0 \\\\16 & 32 & 24 & 8 & 1 & 0 & 0 \\\\32 & 80 & 80 & 40 & 10 & 1 & 0 \\\\64 & 192 & 240 & 160 & 60 & 12 & 1 \\\\\\end{array}\\right).$ Reading the columns of the transformed coefficient array in reverse order (from the bottom up, left to right), we obtain an array associated to the Chebyshev polynomials of the fourth kind (see A228565 and A180870).",
"We finish by looking at the almost-Riordan array $(2-c(x), c(x), xc(x)) = \\left(\\frac{1}{1-x}, (1-x), x(1-x)\\right)^{-1}, $ which begins $\\left(\\begin{array}{ccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\-1 & 1 & 0 & 0 & 0 & 0 & 0 \\\\-2 & 1 & 1 & 0 & 0 & 0 & 0 \\\\-5 & 2 & 2 & 1 & 0 & 0 & 0 \\\\-14 & 5 & 5 & 3 & 1 & 0 & 0 \\\\-42 & 14 & 14 & 9 & 4 & 1 & 0 \\\\-132 & 42 & 42 & 28 & 14 & 5 & 1 \\\\\\end{array}\\right).$ Applying this to the shifted Fibonacci polynomials $F_{n+1}(y)=\\sum _{k=0}^{\\lfloor \\frac{n}{2} \\rfloor } \\binom{n-k}{k}y^k,$ we get a sequence that begins $1, 0, y, 4y, y^2 + 14y, 7y^2 + 48y, y^3 + 35y^2 + 165y, 10y^3 + 154y^2 + 572y, \\ldots .$ Taking the Hankel transform of this sequence, we arrive at a Hankel transform $H_n(y)$ that has the property that $\\frac{H_n(y)}{y^n}$ begins $1, 1, -2, 2(y - 1) - y^2, 2y^3 - 7y^2 + y + 3, - 3y^4 + 16y^3 - 4y^2 - 5y + 3, 4y^5 - 29y^4 + 25y^3 + 34y^2 - 5y - 4,\\ldots .$ This sequence has coefficient array that begins $\\left(\\begin{array}{ccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\-2 & 0 & 0 & 0 & 0 & 0 & 0 \\\\-2 & 2 & -1 & 0 & 0 & 0 & 0 \\\\3 & 1 & -7 & 2 & 0 & 0 & 0 \\\\3 & -5 & -4 & 16 & -3 & 0 & 0 \\\\-4 & -5 & 34 & 25 & -29 & 4 & 0 \\\\\\end{array}\\right),$ and has a (conjectured) generating function of $\\frac{1+(2y+1)x+(y^2+6y+2)x^2+2(2y^2+4y+1)x^3+(5y^2+3y+1)x^4+(y+1)(2y+1)x^5-yx^6-yx^7}{(1+(y+1)x+(2y+2)x^2+(y+1)x^3+x^4)^2}.$"
],
[
"Conclusion",
"We have shown that the Riordan group of invertible lower-triangular matrices is isomorphic to a subgroup of another group of lower-triangular invertible matrices.",
"These matrics, called “almost-Riordan” arrays in this note, appear to be worthy of study in their own right.",
"We have identified one normal subgroup of this new group.",
"We have shown instances where elements of the new group can produce interesting transformations on sequences, and in particular the images of such transformations, for suitable starting sequences, may have significant Hankel determinants.",
"2010 Mathematics Subject Classification: Primary 15B36; Secondary 11B83, 11C20.",
"Keywords: Riordan group, Riordan array, almost-Riordan group, almost-Riordan array."
]
] | 1606.05077 |
[
[
"Polarimetry of comets 67P/Churyumov-Gerasimenko, 74P/Smirnova-Chernykh,\n and 152P/Helin-Lawrence"
],
[
"Abstract Aims.",
"Polarimetric characteristics of comets at large heliocentric distances is a relatively unexplored area; we extend the idea by adding and analysing the data for three Jupiter family comets (JFCs).",
"Methods.",
"With the FORS2 instrument of the ESO VLT, we performed quasi-simultaneous photometry and polarimetry of three active JFCs 67P/Churyumov-Gerasimenko, 74P/Smirnova-Chernykh, and 152P/Helin-Lawrence.",
"Results.",
"We obtained in total 23 polarimetric measurements at different epochs, covering a phase-angle range ~1 -16 degrees and heliocentric distances from 3 to 4.6 au.",
"From our observations we obtained both colour and polarimetric maps to look for structures in the comae and tails of the comets.",
"Conclusions.",
"74P/Smirnova-Chernykh and 152P/Helin-Lawrence exhibit enough activity at large heliocentric distances to be detectable in polarimetric measurements.",
"Polarimetric and colour maps indicate no evidence of dust particle"
],
[
"Introduction",
"The light scattered by dust particles is linearly polarised by a small amount depending on the properties of the scattering media.",
"By studying the polarised light reflected by cometary dust we can obtain information on the size, shape, and optical properties of particles in the cometary comae [44], [32] The method we use to study, from Earth, the scattered light from a solar system body is via photometric and linear polarimetric measurements obtained at different phase angles (the phase angle is angle between the Sun, the target, and the observer).",
"The way the photometric and polarimetric measurements change as a function of phase angle can help us to determine the properties of the scattering medium, be it from a solid surface like an asteroid or the ejected dust from a comet [35], [34], [21].",
"Comets are of particular interest as they are believed to be some of the most primitive objects in the solar system and can give us information about the conditions in the early solar system when they were formed.",
"Since comets have eccentric orbits, the observable phase angle range is between 0 - 157$^\\circ $ .",
"At phase angles $$ 20$^\\circ $ the linear polarisation of cometary dust is usually negative.",
"Negative polarisation means that the preferred direction of oscillation of the electric field vector is in the direction parallel to the scattering plane.",
"This is in contrast to what is expected from the simple single Rayleigh scattering and Fresnel reflection model, which predicts polarisation perpendicular to the scattering plane.",
"Additionally, at very small phase angles $$ 2$^\\circ $ there may be an intrinsic increase in brightness above the linear brightening with phase angle.",
"Both these phenomena are consequences of the micro-structure of the scattering media.",
"Polarimetry has been used to classify active comets into two main categories: high and low polarisation comets [26].",
"The distinction between these two categories occurs at phase angles beyond 40 degrees.",
"At this point a fork in the polarimetric phase function occurs where dust-rich comets tend to show a much higher amount of linear polarisation and gas-rich comets tend to exhibit a low amount of linear polarisation.",
"However it has been shown by [22] that the classification of comets by their polarisation needs to be considered very carefully as the size of the aperture used and the role of molecular emission in the observed wavelength can play a crucial role in the amount of polarisation measured.",
"At phase angle ranges $$ 20$^\\circ $ the bulk of observational data consists of well-sampled data for comets 1P/Halley and C/1995 O1 Hale-Bopp.",
"From the database of comet aperture polarimetry [23] only comets 22P/Kopff [36], 47P/Ashbrook–Jackson (observations carried out by Jockers et al 1993, unpublished), 67P/Churyumov–Gerasimenko [37] and more recently by [20], 81P/Wild 2 [18], and C/1990 K1 Levy , have been observed within this phase angle range.",
"In addition, low phase-angle measurements of the nucleus of comet 2P/Encke have been presented by [12].",
"Furthermore, numerous polarimetric maps have been obtained for comets e.g.",
"1P/Halley [14], C/1995 O1 Hale-Bopp , , 22P/Kopff, and 81P/Wild 2 [18].",
"Almost all polarimetric observations of comets have been taken at heliocentric distances $<$ 2 au when comets are more active, and hence brighter meaning they are much easier to observe.",
"Beyond this heliocentric distance they become much harder to observe and it becomes difficult to investigate the properties of the dust.",
"The exceptions to this are observations of comets C/1995 O1 Hale-Bopp at a heliocentric distance between 2.7 and 3.9 au [31] and 47P/Ashbrook-Jackson at a distance of 2.3 au [40].",
"In this paper we present photometric and linear polarimetric observations of three Jupiter family comets (JFCs): 67P/Churyumov–Gerasimenko (hereafter 67P), 74P/Smirnova–Chernykh (hereafter 74P), and 152P/Helin–Lawrence (hereafter 152P) at large heliocentric distances.",
"Only 67P has been polarimetrically observed previously by [37] and more recently by [20].",
"All three comets have been photometrically observed to varying extents.",
"67P has been observed and modelled by numerous authors in recent years as it is the target of the European Space Agency (ESA) Rosetta spacecraft.",
"Photometry of 74P has been carried out by [27] and [25].",
"[25] observations were carried out with the Hubble Space Telescope as the comet was travelling outbound at a heliocentric distance of 3.56 au.",
"From these measurements [25] were able to derive a nucleus radius of 2.25 $\\pm $ 0.1 km, which exhibited an axis ratio $a/b = 1.14$ and a rotational period of 28 $\\pm $ 6 hours.",
"On the other hand, 152P has not been observed in as much detail; there is only one publication mentioning photometric observations [28].",
"From these observations [28] were able to find an upper limit on the size of the nucleus of 3.3 $\\pm $ 0.9 km assuming a standard albedo of 0.04.",
"The rotational period for 152P is unconstrained.",
"The comets were observed in service mode in February - March 2010 for comet 67P and April-September 2012 for comets 74P and 152P using the FORS2http://www.eso.org/sci/facilities/paranal/instruments/fors/ instrument installed on Unit Telescope 4 (UT4) of the ESO Very Large Telescope (VLT) [5].",
"The visual UV FOcal Reducer and low dispersion Spectrograph (FORS) is a multi-purpose instrument capable of imaging and spectroscopy, equipped with polarimetric optics that follow the design of [4].",
"The observations for comets 67P, 74P, and 152P consisted of both quasi-simultaneous photometric and linear polarimetric measurements.",
"The photometric observations for 67P consisted of two 60 s exposures in the R-Special filter ($\\lambda _0$ = 655 nm, FWHM = 165 nm), whereas for 74P and 152P it consisted of four 60 s exposures, using both the R-Special and v-high filters ($\\lambda _0$ = 557 nm, FWHM = 123.5 nm).",
"Owing to the exposure time, differential autoguiding of the telescope at the apparent velocity of the objects was applied to the observations.",
"Between each exposure a different offset was applied to the telescope to ensure that the image of the comet did not fall on the same pixels during each exposure.",
"The photometric observations were immediately followed by the linear polarisation measurements using the R-Special filter only.",
"In our initial set of polarimetric observations we obtained a series of frames with the half-wave plate set at eight different position angles 0 - 157.5$^\\circ $ in steps of 22.5$^\\circ $ each with an exposure time of 380 s for 67P, 270 s for 74P, and 300 s for 152P.",
"All of the comets' photocentres were found to be brighter than expected, and for 152P we eventually reduced the exposure time and increased the number of exposures to avoid saturation.",
"67P was observed at seven different epochs giving us access to a phase angle range of 2-15$^\\circ $ .",
"74P was photometrically observed at six different epochs and polarimetrically observed at eight different epochs giving us access to a phase angle range of 2-11$^\\circ $ .",
"152P was photometrically observed at six different epochs and polarimetrically observed at seven different epochs giving us access to a phase angle range of 3-15$^\\circ $ ."
],
[
"Infrared integral field spectroscopy",
"In addition to the photometric and polarimetric observations of comets 74P and 152P we also obtained infrared spectroscopic observations using the integral field spectrometer, SINFONI, installed at UT4 of the VLT [15], [13].",
"The spectra were obtained over the whole night of 28 June 2012 in visitor mode (see Table REF for the observing log).",
"For these observations we used the J and H+K grating corresponding to the wavelength range 1.1 - 2.45 $\\mu $ m. We also chose the largest field of view of 8$\\times $ 8 arcseconds, which gave us a spatial resolution of 125$\\times $ 250mas per pixel.",
"Adaptive optics were not used for these observations as it is not an option for a moving target.",
"The science observations were carried out using a fixed offset template, that shifted between the sky and the comet in an ABBA pattern where A is the sky and B is the target position of the telescope.",
"The exposure time used for both comets and the on-sky observations was 300 s. A total of 30 exposures were obtained for comet 152P, 16 in the J grating and 14 in the H+K grating.",
"For comet 74P a total of 19 exposures were obtained, 12 with the J grating and 7 with the H+K grating.",
"A number of telluric standard stars prescribed by the SINFONI calibration plan and our chosen solar analogue, Land (SA) 110-361, were observed using the same instrumental settings and at an airmass as close as possible to the comet's airmass to provide the necessary calibration."
],
[
"FORS data pre-processing",
"The polarimetric and photometric images were reduced in a similar way.",
"All images were bias-subtracted by removing a master bias image obtained from a series of five frames taken the morning after the observations.",
"The science images were then flat-field corrected by dividing the images by a flat-field obtained from a series of five sky flat-fields taken during twilight.",
"This process outlines the reduction steps that were applied to all the science images; we detail the different techniques used to analyse the data using this as the starting point."
],
[
"FORS photometry",
"The comets displayed coma and dust tails in the images indicating recent or current activity; this activity made it impossible to distinguish between the signals from the coma and nuclei of the comets.",
"We used an aperture of between 7 - 13 pixels which covers a diameter of 10,000 km around the coma.",
"This aperture included flux from most of the coma and some of the tail of the comet.",
"The background sky was estimated using a detached annulus at a location close to the comet but free from contamination from the coma, tail, and background stars.",
"In general, it is not possible to give accurate night-by-night values for the photometric zero point or extinction coefficients because photometric standard stars are not taken by default under the FORS calibration plan for polarimetric observations.",
"Therefore, we used the nightly zero point and extinction coefficient available on the ESO quality control and data processing web pagehttp://www.eso.org/observing/dfo/quality/index$\\_$ fors2.html.",
"These were calculated using all the photometric standard stars observed over a period of about 28 nights centred at each night under consideration.",
"We assigned an error of 0.05 mag to the magnitude measurements, which is consistent with the uncertainties of the zero points.",
"The errors due to photon noise and background subtractions are negligible in comparison with those of the zero points.",
"Finally the apparent magnitudes for different epochs were magnitude corrected for Sun and Earth distances of the comet using $H=m-5\\log _{10}(r \\times \\Delta )\\; ,$ where $m$ is the apparent magnitude, $r$ is the heliocentric distance, and $\\Delta $ is the geocentric distance.",
"This corrected magnitude $H$ allows us to investigate how the brightness of the comet changes independently of its distance from the observer while it is still dependent on the phase angle.",
"We can also use the apparent magnitudes in the V and R filters to calculate a V-R colour magnitude for the comets.",
"This allows us to see whether there are any fluctuations in the colour that would suggest a change in the dust particles emitted.",
"In addition to colour fluctuations, we can use the flux from the comets to measure the relative dust production rate.",
"This is done using the quantity $Af\\rho $ first defined by [3]; $Af\\rho $ is a slightly aperture-dependent quantity roughly proportional to the dust production rate of a comet.",
"It is typically measured in cm and is determined from the observations using $Af\\rho = \\frac{\\left(2\\Delta r\\right)^{2}}{\\rho }\\, \\frac{F_c}{F_\\odot }\\; ,$ where $A$ is the geometric albedo of the cometary dust grains, $f$ is the filling factor (i.e.",
"the percentage area covered by the dust), $\\rho $ the radius of the field of view used to measure the flux from the comet measured in cm, $r$ is the heliocentric distance measured in au, $\\Delta $ is the geocentric distance measured in cm, $F_C$ is the flux measured from the comet, and $F_\\odot $ is the solar flux taken using the same filter as the flux measured for the comet.",
"Since we are using a fixed aperture our value for $\\rho $ is $5 \\times 10^8$ cm.",
"The photometric results for comets 152P, 74P, and 67P are presented in Table REF and the results will be discussed in Sect.",
"REF , REF , and REF ."
],
[
"FORS intensity maps",
"FORS intensity maps are constructed by stacking all the photometric images taken using the same filter together.",
"We can also use the polarimetric images to create intensity maps; however, owing to the limited field of view of a FORS polarimetric strip, it is usually better to use photometric images if available.",
"Analysis of the coma and tail to search for structure was performed using a combination of numerical techniques and visual inspection.",
"The first of these techniques was the Laplace filtering, which highlights regions of intensity change that can be used to search for localised activity such as jets [11].",
"The second technique is radial renormalisation which highlights deviations in the mean coma brightness [2].",
"The analysis of the structure of each comet is discussed in Sects.",
"REF , REF , and REF ."
],
[
"Colour maps",
"Aperture photometry gives us information over a large portion of the active region of the comet.",
"To inspect the colour of the coma and search for small-scale features such as jets in the coma we created V-R colour maps using the photometric frames analysed in Sects.",
"REF and REF .",
"Here we give details on how these colour maps were obtained.",
"To reach the highest S/N, for each filter we combined all four photometric maps.",
"We note that each image was obtained with a slightly different telescope offset.",
"Owing to a residual second-order flat-field effect present in all FORS images (a common feature in all focal reducer instruments; [33]), the background spatial behaviour is more regular on each individual image than in the combined image.",
"Therefore, we preferred performing background subtraction on each individual frame prior to image stacking rather than removing the background from the combined image.",
"To remove the background sky from the photometric images we created a `full resolution' background map using the Source Extractorhttp://www.astromatic.net/software/sextractor (SE) software.",
"Not only does this background map estimate the background sky, but it also fits the second-order flat-field effect.",
"We prefer this method of background removal over a simple offset annulus estimation as the annulus only gives an estimation on a very localised spot on the CCD, and the colour map will extend for a much larger distance.",
"The background maps were carefully checked to ensure they contained no contribution from the coma and tail of the comets and then they were simply subtracted from the photometric images.",
"The comet in each photometric image was then shifted to the same position using the photometric centre of the comet.",
"This was done with both the v-high and R-Special which gave us four R-Special and four v-high images all centred on the same location.",
"At this point we carefully checked that both the V-high and R-Special images had similar seeing conditions so as not create artificial structures in the coma.",
"Once this check was made we simply added each set of images together and converted each pixel into an apparent magnitude.",
"Then the V and R magnitude images were subtracted from each other to create the final V-R colour maps which are discussed in Sects.",
"REF and REF ."
],
[
"Polarimetry",
"Linear polarimetric images were typically taken after the photometric images.",
"These polarimetric images are the same as the photometric ones except that a retarder wave plate and Wollaston prism are introduced into the optical path, and the light beam is split into two components.",
"Beam overlapping is avoided by introducing a Wollaston mask consisting of nine 22 strips.",
"The resulting image consists of nine pairs of 22 strips.",
"To calculate Stokes Q and U from these images we employed the beam swapping technique which significantly reduces the impact of instrumental polarisation [8].",
"Reduced Stokes parameters $P_X$ (where X=Q or X=U) are obtained using $P_X =\\frac{1}{2N}\\displaystyle \\sum \\limits _{j=1}^N\\left[\\left(\\frac{f^\\parallel - f^\\perp }{f^\\parallel + f^\\perp } \\right)_{\\alpha _j} -\\left(\\frac{f^\\parallel - f^\\perp }{f^\\parallel + f^\\perp }\\right)_{\\alpha _{j+45}}\\right]\\; ,$ where $f_\\parallel $ is the flux in the parallel beam, $f_\\perp $ the flux in the perpendicular beam, N the number of pairs of exposures, and $\\alpha _j$ denotes the position angle of the retarder waveplate.",
"For $\\alpha $ = 0$^\\circ $ , 90$^\\circ $ , 180$^\\circ $ , and 270$^\\circ $ Eq.",
"(REF ) gives $P_Q$ ; for $\\alpha $ = 22.5$^\\circ $ , 112.5$^\\circ $ , 202.5$^\\circ $ , 295.5$^\\circ $ Eq.",
"(REF ) gives $P_U$ .",
"Normally during the observations the vertical direction of the instrument field of view is aligned with the north celestial meridian.",
"We thus transform the Stokes parameters into a reference system whose reference direction is perpendicular to the scattering plane, using $P^{\\prime }_Q = \\rm {cos}\\left(2\\left(\\varphi +\\frac{\\pi }{2}\\right)\\right)\\mathit {P_Q} +\\rm {sin}\\left(2\\left(\\varphi +\\frac{\\pi }{2}\\right)\\right)\\mathit {P_U}$ $P^{\\prime }_U = -\\rm {sin}\\left(2\\left(\\varphi +\\frac{\\pi }{2}\\right)\\right)\\mathit {P_Q} +\\rm {cos}\\left(2\\left(\\varphi +\\frac{\\pi }{2}\\right)\\right)\\mathit {P_U}\\; ,$ where $\\varphi $ is the angle between the direction object-north pole and the object Sun direction [7].",
"In a system composed of randomly oriented particles $P_U$ should be zero for symmetry reasons.",
"However, if the particles are aligned along a certain direction, $P_U$ could deviate from zero by a small amount.",
"Otherwise, the consistency of $P_U$ with zero should be used as a good quality check."
],
[
"Aperture polarimetry",
"Aperture polarimetry is carried out in a similar way to that described in Sect REF .",
"The $f_\\parallel $ and $f_\\perp $ from Eq.",
"(REF ) are measured from the polarimetric FORS images by using aperture photometry.",
"As mentioned in Sect.",
"REF , the background sky level was estimated using an offset annulus that was close to the comet but far enough away so that influence from the coma, tail, and background stars was at a minimum.",
"The size of the aperture chosen to measure the flux on a given night was based on the error on the measured $P_Q$ and $P_U$ , and if the values of $P_Q$ and $P_U$ were not varying with size of aperture used [6].",
"The polarimetric results for comets 67P, 74P, and 152P are discussed in Sect.",
"REF ."
],
[
"Polarimetric map",
"Any change in polarimetric characteristics would be due to changes in characteristics of the scattering medium, in our case, in size or composition of the dust particles in the coma.",
"These polarimetric changes can then be compared to the V-R colour maps where variations are also related to variations in size or composition of the dust particles.",
"Creating the polarimetric maps is a much more laborious task than creating the colour maps of Sect.",
"REF ; we cannot easily remove the background sky from an entire polarimetric FORS image because the background sky is polarised causing a discontinuity of the flux counts in the parallel and perpendicular beam.",
"The SE interpolation algorithm cannot create an accurate estimation of the background sky owing to this striped nature.",
"Depending on the settings used, the interpolation will either contain an averaged out flux count from both strips or will create a transition area where the background flux gradually changes from one extreme to another.",
"Even if we separate the strips into parallel and perpendicular strips the SE interpolation algorithm has trouble creating a background map without containing a contribution from the coma.",
"Hence we have used an offset annulus to calculate the background sky.",
"This annulus should ideally be placed at a pixel location that is free of coma contribution.",
"However FORS suffers from instrumental errors the further you travel from the centre to the edge of the CCD due to a stressed element in the optical train.",
"For this reason the annulus was placed in a location that minimised the instrumental effects and the contribution of the coma.",
"The location of the annulus varied from epoch to epoch due to changes in the extent of the coma.",
"Ideally, we only need the strips in which the target appears; however, we use all the strips to see if the coma extends beyond the strip in which the comet primarily resides.",
"Once the background sky has been removed from these strips, we use Eq.",
"(REF ) and (REF ) to create $P_Q$ and $P_U$ maps for each comet.",
"The disadvantage of this method is that in order to utilise all frames obtained at each retarder wave plate position we have to remove the background sky 16 times.",
"If we assume that we incur a small error each time we create a background estimate, many of these small errors could become significant to the accuracy of our results.",
"A better method is to combine the images using Eq.",
"(REF ) and remove the background sky at the end.",
"This is outlined in the equation $P_X = \\frac{\\left(P_{X}^{\\text{tot}} \\times I_{X}^\\text{{tot}}\\right) - \\left(P_{X}^\\text{{sky}} \\times I_{X}^\\text{{sky}}\\right)}{\\left(I_{X}^\\text{{tot}} - I_{X}^{\\text{sky}}\\right)}\\; ,$ where $P_{X}^{\\text{tot}}$ is the total Stokes parameter without any background sky subtraction, $I_{X}^{\\text{tot}}$ the total flux counts of the images used to calculate the Stokes parameter, $P_{X}^{\\text{sky}}$ the background estimate of $P_{X}^{\\text{tot}}$ , and $I_{X}^{\\text{sky}}$ the background estimate of $I_{X}^{\\text{tot}}$ .",
"This method is numerically the same as Eq.",
"(REF ), but here we only have to create 2 background maps instead of 16 as was done previously, hence the error is reduced.",
"In the case of comet 74P we applied a $3 \\times 3$ boxcar to our original flux images to increase the S/N of the polarimetric maps.",
"The spatial resolution for 152P is 1668 km per arcsecond (417 km per pixel) and 2640 km per arcsecond (660 km per pixel) for 74P.",
"We present the polarisation maps of comets 152P and 74P in Sects.",
"REF and REF .",
"We note that we examined the other strips above and below the target and found no evidence of the coma or tail so we only show the strips containing the photometric centre of the comets."
],
[
"IR spectrophotometry",
"The infrared spectra obtained by the SINFONI instrument were reduced using the ESO SINFONI reduction pipeline (version 2.3.2), with all the relevant calibration files provided by ESO.",
"The pipeline was also used to extract all the spectra from the data cubes using a six-pixel aperture centred around the approximate photometric centre of the comet.",
"The individual spectra extracted in the J- and H+K-bands were corrected for the exposure time and combined performing a resistant mean with a threshold of 2.5 $\\sigma $ .",
"The same data reduction steps were applied to the solar analogue spectra.",
"The target spectrum was divided by that of the solar analogue Land (SA) 110-361 observed at similar airmass to correct for telluric lines and remove the Sun's contribution, obtaining in this way the comet relative reflectance spectrum.",
"The latter was normalised to unity at 1.2 micron.",
"Unfortunately, we do not have near-infrared photometric data to verify the adjustment of the separate spectra taken in the J and H+K wavelength bands.",
"We therefore need to rely on our data processing.",
"In Sect.",
"REF we present the relative reflectance spectrum of 152P only, since the signal-to-noise ratio of the comet 74P spectrum was too low to yield any useful information."
],
[
"Results",
"The results of the photometry and polarimetry for the comets are listed in Table REF .",
"Section REF is dedicated to the results of comet 152P, Sect.",
"REF to comet 74P, and Sect REF to comet 67P.",
"In Sects.",
"REF , REF , and REF we present the results of aperture photometry, $Af\\rho $ , colour maps and polarimetric maps.",
"In Sect.",
"REF we present the aperture polarimetry for 152P, 74P, and 67P."
],
[
"Aperture photometry",
"In Fig.",
"REF we plot the magnitude corrected for the Sun and Earth distances of comet 152P as a function of phase angle.",
"If we ignore the v-high result on the night of 5 April 2012 and use a straight line fit, the extrapolated average brightness at zero phase angle are 12.79 $\\pm $ 0.13 and 13.32 $\\pm $ 0.12 in the R and V filters assuming no opposition surge.",
"This results in an average V-R colour of 0.53$\\pm $ 0.18, which is equivalent to a spectral gradient of 18.03 %/100 nm.",
"[28] photometrically observed 152P in the V and R filters and found that 152P had a very red colour of 0.77 $\\pm $ 0.12 (47 %/100 nm) when the comet was at 4.6 au from the Sun.",
"Within the error our results are consistent with both measurements.",
"Using Eq.",
"(REF ), the flux extrapolated back to zero phase in the R-Special filter and the average $r$ and $\\Delta $ distances to the comet yields an $Af\\rho $ value of 194.9 $\\pm $ 22.0 cm.",
"This compares to the value of 56.4 $\\pm $ 3.6 cm measured by [28], which suggests the comet exhibits less activity beyond 4 au."
],
[
"Intensity maps",
"The intensity images for comet 152P are presented in Fig.",
"REF .",
"Apart from slightly asymmetric coma extensions in the northern and north-western directions and from the main tail axis no distinct coma structure (jet, fan, or shell) was found in the processed images.",
"Given the wide wavelength range of the R-Special and v-high filters used, the coma and tail are mainly represent the dust distribution around the nucleus.",
"Features that could be attributed to gas and plasma, for instance tail rays, are not seen in the images.",
"Tail-like extensions of the coma pointed westward in April and May 2012 and appeared in two parts during the second half of July, one towards west-north-west and one towards the east.",
"Finson-Probstein calculations [16], [10] (Table REF ) for the dust distribution show that the dust tail is orientated westward during April and May 2012.",
"The appearance of two dust tail features in the second half of July 2012 indicates that young dust grains, i.e.",
"typically released less than 2 months before the observing epoch, project into the eastern sector between angles of 90 and 100$^\\circ $ , while much older dust, typically released more than 8 months before the observing date, is found in the west-north-western coma region.",
"Dust produced by the nucleus between 2 and 8 months projects into the northern coma hemisphere (as seen from Earth) and creates the asymmetric appearance in the coma.",
"Figure: V-R colour map of comet 152P.",
"The green arrow points in the direction of the negative target velocity as seen in the observer's plane of sky.",
"The cyan arrow is the direction of the anti-solar direction.",
"North is up and east is to the left."
],
[
"Colour maps",
"Figure REF shows the colour maps of comet 152P.",
"The colour scale is centred about the solar reflectance colour of 0.35 for V-R. Any feature redder or bluer than this value appears in the figure as red or blue, respectively.",
"We note the occurrence of some red-blue “dipole” features, especially in the background.",
"These are artefacts due to the presence of background stars that, owing to the differential tracking of the telescope, appear offset in the R and V images.",
"The exceptions are the top two maps in Fig.",
"REF where the colour scale has been extended to accommodate their slightly redder colour compared to the other data; however, the colour scale is still centred about 0.35.",
"The colour maps created show little to no structure apart from the nights of 30 April, 21 and 23 May 2012.",
"On 30 April 2012 the photometric centre of the comet is bluer than that 5 April 2012.",
"This suggests either the comet was producing a redder material or larger particles on 5 April 2012 than on 30 April 2012 or was producing bluer material or particles of a smaller size on the 30 April 2012.",
"On 21 May 2012 the average coma colour is $\\sim $ 0.54; however, there is a small spot at the photometric centre with a colour of $\\sim $ 0.43, and north-west of this there is a much redder feature with a colour of $\\sim $ 0.63 (marked by a black arrow).",
"This could be an indication of activity, jet or outburst, in the inner coma.",
"Two days later on 23 May 2012 the average colour of the coma increased to $\\sim $ 0.60, perhaps indicating the results of the activity noticed on 21 May 2012.",
"On 23 May 2012 there is a feature north-west of the photometric centre of the coma as was the case on 21 May; however, the colour of this feature has changed to a value of $\\sim $ 0.43 (marked with a black arrow), which could be due to particle movement and nucleus rotation.",
"These features are not a simple case of image misalignment as this would result in the dipole feature seen in the background of all these colour maps which are due to stars moving in the background with respect to the comet.",
"In addition, the images used to calculate these colour maps were carefully selected to ensure that they had very similar seeing conditions.",
"After 23 May 2012 the coma continues to increase in colour to an average colour of $\\sim $ 0.66, which is fairly uniform across the coma.",
"Unfortunately the V and R images are taken under different seeing conditions making it impossible to see if there was any activity to cause the colour to drop to $\\sim $ 0.54 in the coma."
],
[
"Polarimetric maps",
"The polarimetric maps for comet 152P are presented in Fig.",
"REF ; on the left side are the $P^{\\prime }_Q$ maps and on the right side are the $P^{\\prime }_U$ maps.",
"On the night of April 2012 the background sky was highly polarised, which made getting an accurate background sky estimate difficult, as can be seen in the background features in the polarimetric maps.",
"In spite of this difficulty, there are no unusual polarimetric values that coincide with the red colour seen in the colour maps in Fig.",
"REF .",
"The polarimetric maps created for the nights of 21 and 23 May 2012 can be compared to the colour maps created in Fig.",
"REF to determine whether the same features are present.",
"Figure REF shows that on the night of 21 May 2012 the amount of polarisation becomes more negative $\\sim $ $-3\\%$ in the direction of the outburst seen in the colour map in Fig.",
"REF .",
"In the $P^{\\prime }_U$ map for the same night we see an increase in $P_U$ in the location of the outburst region of $\\sim $ 0.5$\\%$ .",
"Again, this suggests that the outburst is composed of a different material or has a different morphology to the surrounding coma.",
"On 23 May 2012 there is a slight hint of structure in the coma in the $P^{\\prime }_Q$ map for this night.",
"In the north-western direction from the photometric centre there is a slightly more negative polarisation, $\\sim $ $-1.4\\%$ , compared to $\\sim $ $0.7\\%$ in the south-eastern direction.",
"This polarisation difference occurs in the same location as the colour feature seen in the colour maps of Fig.",
"REF .",
"In the last two maps in July as seen in Fig.",
"REF both maps in $P^{\\prime }_Q$ show very few features of note.",
"In the $P^{\\prime }_U$ maps on the same nights there are small fluctuations in polarisation of around $\\pm $ 0.1$\\%$ , consistent with noise.",
"Figure: Relative reflectance spectrum of comet 152P (grey dots).",
"For comparison, a synthetic spectrum of amorphous carbon (AC) grains (particle diameter of 5 μ\\mu m, blue line) and the synthetic spectrum used to represent the Hartley 2 coma (solid red line) composed of 1 μ\\mu m water-ice grains and dust not in thermal equilibrium .Figure: P Q ' P^{\\prime }_Q (left) and P U ' P^{\\prime }_U (right) polarimetric maps for comet 152P.",
"Both P Q ' P^{\\prime }_Q and P U ' P^{\\prime }_U are measured in per cent.",
"The green arrow points in the direction of the negative target velocity as seen in the observer's plane of sky.",
"The cyan arrow is the direction of the anti-solar direction.",
"North is up and east is to the left."
],
[
"SINFONI infrared spectra",
"Figure REF shows the relative reflectance spectrum of comet 152P.",
"The target spectrum presents a positive spectral slope of 13.51 $\\pm $ 0.7 $\\%$ /100 nm similar to the spectral slope from the V-R measurements.",
"We looked for the presence of water ice absorption features at 1.5 and 2.0 $\\mu $ m, displayed by other JFC spectra such as that of Hartley 2 [1] and the outbursting comet P/2010 H2 (Vales) [43].",
"However, the 152P spectrum does not display clear water-ice absorptions and resembles, at first order, the spectrum of a dark and featureless refractory component (e.g.",
"amorphous carbon, dashed blue line).",
"The comparison of our spectral slope to other active comets is difficult as they are influenced by the presence of water ice.",
"Water ice is blue in the near infrared spectrum which tends to make the spectral slope shallower."
],
[
"Aperture photometry",
"74P shows a constant solar V-R colour throughout all our observations.",
"The only exception is the night of 21 June 2012, which was contaminated by two nearby saturated stars particularly when the R filter observations were carried out.",
"In Fig.",
"REF we plot the magnitude corrected for the Sun and Earth distances of comet 74P as a function of phase angle.",
"If we ignore the result on the night of 21 June 2012 and use a straight line fit, the extrapolated average brightnesses at zero phase angle are 12.76 $\\pm $ 0.13 and 13.43 $\\pm $ 0.15 in the R and V filters assuming no opposition surge.",
"This results in an average V-R colour of 0.67 $\\pm $ 0.20, which is equivalent to a spectral gradient of 34.28 $\\%$ /100 nm.",
"[27] observed 74P and found that it had a V-R colour of 0.44 $\\pm $ 0.10, which is not quite as red as suggested by our findings.",
"Using Eq.",
"(REF ) and the flux extrapolated back to zero phase in the R-Special filter and the average $r$ and $\\Delta $ distances to the comet yields an $Af\\rho $ value of 200.8 $\\pm $ 22.7 cm.",
"This compares to the values of 228.8 $\\pm $ 11.4 cm and 298.9 $\\pm $ 11.3 cm measured by [28] and [27] at a heliocentric distance of 4.2 and 4.6 au.",
"The large amount of activity shown by 74P at heliocentric distances beyond 4 au could suggest that this comet constantly shows signs of activity throughout its orbit."
],
[
"Intensity maps",
"The intensity images for comet 74P are presented in Fig.",
"REF .",
"No distinct coma structure was found in the processed images indicating no localised activity on the nucleus.",
"Again given the broad wavelength range of the filters used, the coma and tail are mostly representing the dust distribution around the nucleus.",
"Features that could be attributed to gas and plasma are not seen in the images.",
"A dust tail is present throughout the observing period in a westward direction.",
"At the beginning of the observing period this dust tail was also slightly curved towards the south.",
"In late September 2012, and possibly in August 2012, there is a noticeable additional coma extension into the north-western quadrant.",
"The Finson-Probstein analysis of the dust tail geometry (Table REF ) suggests that there is a coma asymmetry in the north-western coma section in September 2012 maybe due to dust grains released during the previous two months before the observations and projected into that coma quadrant as seen from Earth.",
"The westward pointing tail at this time consists of very old dust emitted by the nucleus about a year earlier.",
"The old dust overlaps with the more recent grains during the June and July observing epochs forming a brighter and wider slightly curved dust tail as seen in the images.",
"Figure: V-R colour map of comet 74P.",
"The green arrow points in the direction of the negative target velocity as seen in the observer's plane of sky.",
"The cyan arrow is the direction of the anti-solar direction.",
"North is up and east is to the left."
],
[
"Colour map",
"The colour maps produced for comet 74P are shown in Fig.",
"REF .",
"The blue and red spots exhibited on the nights of 21 June and 24 July 2012 are artificial features caused by a large seeing difference between the V and R images.",
"On the other nights not affected by seeing changes there is no clear evidence of colour variation within the coma or tail region of the comet."
],
[
"Polarimetric maps",
"In Fig.",
"REF we present the polarisation maps for 74P.",
"Since 74P was fainter and exhibited a narrow coma and tail the signal-to-noise ratio is lower than for 152P.",
"Additionally the comet passes close to background stars which makes it difficult to search for features in the coma and tail region especially when the $3 \\times 3$ box car is applied.",
"The quality of the observations for 74P is lower as our photometric and polarimetric measurements did not always occur on the same night and the presence of background stars changed throughout our polarimetric observations.",
"On the nights of 26 June, 19 August, and 10 September 2012, the comet passes close to or in front of background stars, which affects our search for structures within the coma and tail of the comet.",
"The least affected nights are the two nights in July 2012.",
"In the $P^{\\prime }_Q$ maps for the observation on 17 and 24 July 2012 (Fig.",
"REF ) we see no clear evidence of structure or change in the amount of polarisation across the coma.",
"In the $P^{\\prime }_U$ maps on both nights there is a small residual polarisation, which, however, shows no definite structure that suggests a jet or an outburst is present.",
"On the night of 16 September 2012 there was a tracking issue that caused the target to drift toward the edge of the strip during the exposures.",
"Nothing clear can be seen on this night, most likely due to the low signal-to-noise ratio from the data.",
"Figure: Magnitude corrected for the Sun and Earth distances of comet 67P as a function of phase angle.",
"We note that the points at phase angles 2.7 ∘ ^\\circ and 10.3 ∘ ^\\circ are contaminated by background sources and are ignored.Figure: Intensity map of comet 67P.",
"The white arrow points in the direction of the negative target velocity as seen in the observer's plane of sky.",
"The black arrow is the direction of the anti-solar direction.",
"North is up and east is to the left."
],
[
"Comet 67P/Churyumov–Gerasimenko",
"Since 67P was only observed in the R filter we were not able to create V-R colour maps.",
"Additionally 67P appeared 2-3 magnitudes fainter than the other comets meaning that the signal-to-noise ratio of the polarimetric measurements was so low that the polarimetric maps generated showed no clear structure.",
"Figure: P Q ' P^{\\prime }_Q (left) and P U ' P^{\\prime }_U (right) polarimetric maps for the comet 74P.",
"Both P Q ' P^{\\prime }_Q and P U ' P^{\\prime }_U are measured in per cent.",
"The green arrow points in the direction of the negative target velocity as seen in the observer's plane of sky.",
"The cyan arrow is the direction of the anti-solar direction.",
"North is up and east is to the left.Figure: P Q ' P^{\\prime }_Q as a function of phase angle for comets 67P, 74P, and 152P.",
"The solid black line is a best fit of previously observed comets.",
"Data points shown are 1P/Halley, C/1995 O1 Hale-Bopp, and other Jupiter family comets, which include 22P/Kopff, 49P/Ashbrook-Jackson, observations of 67P/Churyumov-Gerasimenko from 1984, and 81P/Wild 2.",
"These data points can all be found in the comet polarimetric database ."
],
[
"Aperture photometry",
"The results for comet 67P are shown in Table 1.",
"In Fig.",
"REF we plot the magnitude corrected for the Sun and Earth distances of comet 67P in the R-Special filter as a function of phase angle.",
"In this figure there is no evidence of an opposition surge at small phase angles.",
"We note that the points at phase angles 2.7$^\\circ $ and 10.3$^\\circ $ are contaminated by background sources and are ignored.",
"If we extrapolate, the average brightness at zero phase angle is 15.33 $\\pm $ 0.11 in the R filter.",
"Using Eq.",
"REF and the flux extrapolated back to zero phase in the R-Special filter and the average $r$ and $\\Delta $ distances to the comet yields an $Af\\rho $ value of 18.71 $\\pm $ 1.80 cm.",
"This compares to the value of $\\le $ 17.1 cm measured by [29] while the comet was at aphelion."
],
[
"Intensity maps",
"An intensity image for comet 67P is presented in Fig.",
"REF ; we only present a single intensity image as all the exposures look quite similar.",
"The coma of 67P does not show any sign of structure.",
"However, we note that the dust coma extends asymmetrically and is larger in the southern direction suggesting ongoing activity in the southern part of the nucleus.",
"There is also a significant but weak $\\sim $ 25\" coma peak in the direction of the Sun.",
"The overall appearance of the coma and tail does not change over the course of our observations.",
"The tail orientation is very constant at position angle $\\sim $ 296-297$^\\circ $ ; there is a slight trend to lower position angles with time.",
"Finson Probstein calculations of the dust tail (Table REF ) indicate that the material defining the main tail axis appears to be old and may have been produced by the nucleus about 1/2 to 1 year before observation."
],
[
"Aperture polarimetry of 152P, 74P, and 67P",
"In Fig.",
"REF we present the aperture polarimetry for 67P, 74P, and 152P.",
"The solid black line is a best fit representation of previously observed R-band polarimetric data of active comets taken from the polarimetric comet database [23].",
"The best fit we used is a trigonometric function that was introduced by [30] and outlined by [38] and is defined as $P(\\alpha ) = b\\left(\\rm {sin}\\, \\alpha \\right)^{c_1}[\\rm {cos} \\,(\\alpha /2)]^{c_2}\\, \\rm {sin}\\left(\\alpha - \\alpha _0\\right)\\; ,$ where $b$ , $c_1$ , $c_2$ , and $\\alpha _0$ are free parameters.",
"Each of these four parameters has an effect on the shape of the fitted phase curve as stated in [38].",
"The parameter b is mainly connected to the amplitude of polarisation with a physically acceptable range of values between 0 and 100 if we express the polarisation P($\\alpha $ ) in per cent.",
"The parameter $\\alpha _{0}$ is the inversion angle where negative polarisation turns into positive polarisation.",
"This parameter can range between 0 and 180$^\\circ $ but typically it is less than 30$^\\circ $ .",
"The two powers c$_1$ and c$_2$ influence the shape of the phase curve.",
"The parameter c$_1$ mainly affects the position of the minimum, while c$_2$ has an influence on the maximum and the asymmetry of the curve.",
"Both these parameters should have positive values.",
"This equation can be used for extrapolation only within a phase angle range where well-distributed data points are available.",
"As we can see from Fig.",
"REF all three comets show a very similar polarimetric phase relationship.",
"However polarimetric measurements of comet 67P have a large error owing to a poor signal-to-noise ratio, so it is difficult to make any firm conclusions from this data.",
"However both comets 74P and 152P show a slightly different polarimetric behaviour compared to the best fit curve.",
"They both exhibit an excess in negative polarisation at phase angles < 3$^\\circ $ .",
"The deviation from the best fit at the small phase angles for comet 152P corresponds to the data points on the 21 and 23 May 2012, which showed the presence of activity in the colour maps (see Fig.",
"REF ).",
"We are unable to state the cause for the deviation at small phase angles for comet 74P as we lack both good polarimetric and colour data for these nights, but it is likely due to activity or statistical scatter around the best fit.",
"Also presented in this graph are other JFCs observed in wavelength range that is similar to our observations.",
"These observations are within the error of both our observations and the best fit line.",
"However there are points that deviate from the best fit and this is likely due to the observations being in slightly different wavelengths or at different heliocentric distances.",
"Therefore, when we compare comet polarimetry in broadband filters we must take the values with a pinch of salt as there are many factors that can influence the amount of polarisation measured (gas contamination, outbursts of activity, etc.",
")."
],
[
"Discussion and conclusions",
"To investigate whether we see any polarimetric and colour trends along the solar anti-solar direction we have taken scans through the photometric centre of the comets.",
"Since comet 74P has few simultaneous photometric and polarimetric observations and 67P has no colour information, we only carry out this analysis for comet 152P.",
"The scans are presented in Fig.",
"REF along with a contour plot showing exactly which region of the coma is being scanned.",
"For 152P we do not see any colour or polarisation trends with cometocentric distance (see Fig.",
"REF ).",
"For the majority of the observations of 152P we see an average colour $\\sim $ 30-40$\\% / 100\\,\\rm {nm}$ , while the polarisation although it does vary from epoch to epoch due to changing phase angle remains constant across the coma.",
"We fail to see any trend from blue to red that would suggest the sublimation of water ice in our observations which supports our near-infrared observations where we see no water absorption [24].",
"The exception is possibly the scan on the 23 May 2012 where we see a change in colour about the photometric centre of the comet; however, we do not see a corresponding trend in polarisation, which instead we expect to see.",
"The reddish colour and the lack of water ice in the coma suggest that the dust is possibly made up of dirty ice or organic particles.",
"Again, if this is the case we do not see any trends that suggest the decomposition or fragmentation of the dust that would show itself as a change in colour from red to blue as the particles get smaller and become more efficient scatterers.",
"The lack of any trends could be a special feature of distant Jupiter family comets.",
"At these large heliocentric distances, sublimation and fragmentation of the dust particles are very slow because the solar radiation is less intense, and this may be the reason why we do not see any gradual cometocentric changes in colour and polarisation.",
"[9] modelled the lifetime of dust particles at a similar heliocentric distance and found that dirty ice particles live a maximum of 2-3 hours before the ice sublimates.",
"On the other hand, [9] showed that grains of pure water ice can survive many years before sublimation; however, since the probability of getting pure water ice grains is very small and the fact that we have found no water ice in the near-infrared spectrum suggests that this is not the cause of our lack of variation.",
"If we consider how the polarisation and colour properties change as a function of phase angle (Fig.",
"REF ) we can see two interesting anomalies for comet 152P.",
"The first occurs at the two small phase angles of 2.9$^\\circ $ and 3.4$^\\circ $ where we see both an increase in $P^{\\prime }_Q$ in absolute terms and an increase in colour.",
"The increase in $P^{\\prime }_Q$ is the opposite behaviour that we expect to see at these phase angles, indicating that something unusual happened on this night.",
"The combination of this with an increase in colour suggest that either at the phase angle 2.9$^\\circ $ the comet produced more ice than usual, or that at a phase angle of 3.4$^\\circ $ it produces less ice than usual.",
"The second anomaly occurs at large phase angles between 15 and 24 July 2012 when the colour shows a large decrease.",
"We note that the data point at phase angle 15.4$^\\circ $ has to be ignored as it corresponds to observations taken 3 months earlier, which is why this data point is not connected to the others.",
"The decrease in colour is accompanied by a decrease in absolute terms of Stokes $P^{\\prime }_Q$ ; however, this decrease in polarisation is expected with the change in phase angle.",
"Since the colour maps for this night are influenced by different seeing conditions we are unable to tell if there is a jet or if an outburst has occurred around the time of observation.",
"Very little can be said about 74P; the lack of quasi-simultaneous colour and polarisation measurements means we cannot draw any firm conclusions.",
"There are two uncharacteristic dips in $P^{\\prime }_Q$ between 5-7$^\\circ $ and 9-11$^\\circ $ .",
"The first is due to the observations being two months apart and are not a good comparison.",
"However the second dip is a little more interesting as the observations are taken a week apart and clearly something unusual occurred in the coma that has caused an increase of $\\sim 0.25\\,\\%$ in absolute terms.",
"This could be caused by smaller particles being emitted from the nucleus or the sublimation of ice [24]; however without colour information we cannot understand which.",
"Comparing the $Af\\rho $ measurements for these three comets at these large heliocentric distances 74P appears to be the most active comet, closely followed by 152P, with 67P showing the least activity.",
"The large difference in the amount of activity shown by 74P and 152P compared to 67P could suggest that 74P and 152P are relative newcomers to the inner solar system and still have a large reservoir of volatiles present on the surface of their nuclei.",
"Figure: P Q ' P^{\\prime }_Q, P U ' P^{\\prime }_U, and colour changes as a function of phase angle for comets 74P and 152P.Part of this work was supported by the COST Action MP1104 “Polarization as a tool to study the Solar System and beyond”.",
"Silvia Protopapa gratefully thanks NASA's SSO Planetary Astronomy Program (grant $\\#$ NNX15AD99G) for funding that supported this work.",
"Karri Muinonen's research is supported, in part, by the Academy of Finland grant No.",
"1257966."
]
] | 1606.05192 |
[
[
"The non-convex Burer-Monteiro approach works on smooth semidefinite\n programs"
],
[
"Abstract Semidefinite programs (SDPs) can be solved in polynomial time by interior point methods, but scalability can be an issue.",
"To address this shortcoming, over a decade ago, Burer and Monteiro proposed to solve SDPs with few equality constraints via rank-restricted, non-convex surrogates.",
"Remarkably, for some applications, local optimization methods seem to converge to global optima of these non-convex surrogates reliably.",
"Although some theory supports this empirical success, a complete explanation of it remains an open question.",
"In this paper, we consider a class of SDPs which includes applications such as max-cut, community detection in the stochastic block model, robust PCA, phase retrieval and synchronization of rotations.",
"We show that the low-rank Burer--Monteiro formulation of SDPs in that class almost never has any spurious local optima."
],
[
"Introduction",
"We consider semidefinite programs (SDPs) of the form $f^* = \\min _{X\\in {\\mathbb {S}^{n\\times n}}} \\left\\langle {C},{X}\\right\\rangle \\quad \\textrm { subject to } \\quad \\mathcal {A}(X) = b, \\ X \\succeq 0,$ where $\\left\\langle {C},{X}\\right\\rangle = \\mathrm {Tr}(C^\\top \\!",
"X)$ , $C \\in {\\mathbb {S}^{n\\times n}}$ is the symmetric cost matrix, $\\mathcal {A}\\colon {\\mathbb {S}^{n\\times n}}\\rightarrow {\\mathbb {R}^m}$ is a linear operator capturing $m$ equality constraints with right hand side $b\\in {\\mathbb {R}^m}$ and the variable $X$ is symmetric, positive semidefinite.",
"Interior point methods solve (REF ) in polynomial time [26].",
"In practice however, for $n$ beyond a few thousands, such algorithms run out of memory (and time), prompting research for alternative solvers.",
"If (REF ) has a compact search space, then it admits a global optimum of rank at most $r$ , where $\\frac{r(r+1)}{2} \\le m$ [27], [9].",
"Thus, if one restricts the search space of (REF ) to matrices of rank at most $p$ with $\\frac{p(p+1)}{2} \\ge m$ , then the globally optimal value remains unchanged.",
"This restriction is easily enforced by factorizing $X = YY^\\top \\!",
"$ where $Y$ has size $n\\times p$ , yielding an equivalent quadratically constrained quadratic program: $q^* = \\min _{Y\\in {\\mathbb {R}^{n\\times p}}} \\left\\langle {CY},{Y}\\right\\rangle \\quad \\textrm { subject to } \\quad \\mathcal {A}(YY^\\top \\! )",
"= b.$ In general, (REF ) is non-convex, making it a priori unclear how to solve it globally.",
"Still, the benefits are that it is lower dimensional than (REF ) and has no conic constraint.",
"This has motivated [15], [16] to try and solve (REF ) using local optimization methods, with surprisingly good results.",
"They developed theory in support of this observation (details below).",
"About their results, [16] write (mutatis mutandis): “How large must we take $p$ so that the local minima of (REF ) are guaranteed to map to global minima of (REF )?",
"Our theorem asserts that we need onlyThe condition on $p$ and $m$ is slightly, but inconsequentially, different in [16].",
"$\\frac{p(p+1)}{2} > m$ (with the important caveat that positive-dimensional faces of (REF ) which are `flat' with respect to the objective function can harbor non-global local minima).” The caveat—the existence or non-existence of non-global local optima, or their potentially adverse effect for local optimization algorithms—was not further discussed.",
"In this paper, assuming $\\frac{p(p+1)}{2} > m$ , we show that if the search space of (REF ) is compact and if the search space of (REF ) is a regularly defined smooth manifold, then, for almost all cost matrices $C$ , if $Y$ satisfies first- and second-order necessary optimality conditions for (REF ), then $Y$ is a global optimum of (REF ) and, since $\\frac{p(p+1)}{2} \\ge m$ , $X = YY^\\top \\!",
"$ is a global optimum of (REF ).",
"In other words, first- and second-order necessary optimality conditions for (REF ) are also sufficient for global optimality—an unusual theoretical guarantee in non-convex optimization.",
"Notice that this is a statement about the optimization problem itself, not about specific algorithms.",
"Interestingly, known algorithms for optimization on manifolds converge to second-order critical points,Second-order critical points satisfy first- and second-order necessary optimality conditions.",
"regardless of initialization [14].",
"For the specified class of SDPs, our result improves on those of [16] in two important ways.",
"Firstly, for almost all $C$ , we formally exclude the existence of spurious local optima.Before Prop.",
"2.3 in [16], the authors write: “The change of variables $X = YY^\\top \\!",
"$ does not introduce any extraneous local minima.” This is sometimes misunderstood to mean (REF ) does not have spurious local optima, when it actually means that the local optima of (REF ) are in exact correspondence with the local optima of “(REF ) with the extra constraint $\\operatorname{rank}(X) \\le p$,” which is also non-convex and thus also liable to having local optima.",
"Unfortunately, this misinterpretation has led to some confusion in the literature.",
"Secondly, we only require the computation of second-order critical points of (REF ) rather than local optima (which is hard in general [33]).",
"Below, we make a statement about computational complexity, and we illustrate the practical efficiency of the proposed methods through numerical experiments.",
"SDPs which satisfy the compactness and smoothness assumptions occur in a number of applications including Max-Cut, robust PCA, $\\mathbb {Z}_2$ -synchronization, community detection, cut-norm approximation, phase synchronization, phase retrieval, synchronization of rotations and the trust-region subproblem—see Section for references."
],
[
"A simple example: the Max-Cut problem",
"Given an undirected graph, Max-Cut is the NP-hard problem of clustering the $n$ nodes of this graph in two classes, $+1$ and $-1$ , such that as many edges as possible join nodes of different signs.",
"If $C$ is the adjacency matrix of the graph, Max-Cut is expressed as $\\max _{x\\in {\\mathbb {R}^n}} \\frac{1}{4}\\sum _{i,j=1}^n C_{ij}(1-x_ix_j) \\quad \\textrm { s.t. }",
"\\quad x_1^2 = \\cdots = x_n^2 = 1.$ Introducing the positive semidefinite matrix $X = xx^\\top \\!",
"$ , both the cost and the constraints may be expressed linearly in terms of $X$ .",
"Ignoring that $X$ has rank 1 yields the well-known convex relaxation in the form of a semidefinite program (up to an affine transformation of the cost): $\\min _{X\\in {\\mathbb {S}^{n\\times n}}} \\left\\langle {C},{X}\\right\\rangle \\quad \\textrm { s.t. }",
"\\quad \\mathrm {diag}(X) = \\mathbf {1}, \\ X \\succeq 0.$ If a solution $X$ of this SDP has rank 1, then $X=xx^\\top \\!",
"$ for some $x$ which is then an optimal cut.",
"In the general case of higher rank $X$ , [19] exhibited the celebrated rounding scheme to produce approximately optimal cuts (within a ratio of .878) from $X$ .",
"The corresponding Burer–Monteiro non-convex problem with rank bounded by $p$ is: $\\min _{Y\\in {\\mathbb {R}^{n\\times p}}} \\left\\langle {CY},{Y}\\right\\rangle \\quad \\textrm { s.t. }",
"\\quad \\mathrm {diag}(YY^\\top \\! )",
"= \\mathbf {1}.$ The constraint $\\mathrm {diag}(YY^\\top \\!",
")=\\mathbf {1}$ requires each row of $Y$ to have unit norm; that is: $Y$ is a point on the Cartesian product of $n$ unit spheres in ${\\mathbb {R}^{p}}$ , which is a smooth manifold.",
"Furthermore, all $X$ feasible for the SDP have identical trace equal to $n$ , so that the search space of the SDP is compact.",
"Thus, our results stated below apply: For $p = \\left\\lceil \\sqrt{2n}\\,\\right\\rceil $ , for almost all $C$ , even though (REF ) is non-convex, any local optimum $Y$ is a global optimum (and so is $X=YY^\\top \\!",
"$ ), and all saddle points have an escape (the Hessian has a negative eigenvalue).",
"We note that, for $p>n/2$ , the same holds for all $C$ [12].",
"${\\mathbb {S}^{n\\times n}}$ is the set of real, symmetric matrices of size $n$ .",
"A symmetric matrix $X$ is positive semidefinite ($X \\succeq 0$ ) if and only if $u^\\top \\!",
"X u \\ge 0$ for all $u\\in {\\mathbb {R}^n}$ .",
"For matrices $A, B$ , the standard Euclidean inner product is $\\left\\langle {A},{B}\\right\\rangle = \\mathrm {Tr}(A^\\top \\!",
"B)$ .",
"The associated (Frobenius) norm is $\\Vert A\\Vert = \\sqrt{\\left\\langle {A},{A}\\right\\rangle }$ .",
"$\\operatorname{Id}$ is the identity operator and $I_n$ is the identity matrix of size $n$ ."
],
[
"Main results",
"Our main result establishes conditions under which first- and second-order necessary optimality conditions for (REF ) are sufficient for global optimality.",
"Under those conditions, it is a fortiori true that global optima of (REF ) map to global optima of (REF ), so that local optimization methods on (REF ) can be used to solve the higher-dimensional, cone-constrained (REF ).",
"We now specify the necessary optimality conditions of (REF ).",
"Under the assumptions of our main result below (Theorem REF ), the search space $\\mathcal {M}& = \\mathcal {M}_p = \\lbrace Y \\in {\\mathbb {R}^{n\\times p}}: \\mathcal {A}(YY^\\top \\! )",
"= b \\rbrace $ is a smooth and compact manifold of dimension $np - m$ .",
"As such, it can be linearized at each point $Y\\in \\mathcal {M}$ by a tangent space, differentiating the constraints [3]: $Y\\mathcal {M}& = \\lbrace \\dot{Y} \\in {\\mathbb {R}^{n\\times p}}: \\mathcal {A}(\\dot{Y} Y^\\top \\!",
"+ Y \\dot{Y}^\\top \\! )",
"= 0 \\rbrace .$ Endowing the tangent spaces of $\\mathcal {M}$ with the (restricted) Euclidean metric $\\left\\langle {A},{B}\\right\\rangle = \\mathrm {Tr}(A^\\top \\!",
"B)$ turns $\\mathcal {M}$ into a Riemannian submanifold of ${\\mathbb {R}^{n\\times p}}$ .",
"In general, second-order optimality conditions can be intricate to handle [29].",
"Fortunately, here, the smoothness of both the search space (REF ) and the cost function $f(Y) & = \\left\\langle {CY},{Y}\\right\\rangle $ make for straightforward conditions.",
"In spirit, they coincide with the well-known conditions for unconstrained optimization.",
"As further detailed in Appendix , the Riemannian gradient $\\mathrm {grad}f(Y)$ is the orthogonal projection of the classical gradient of $f$ to the tangent space $Y\\mathcal {M}$ .",
"The Riemannian Hessian of $f$ at $Y$ is a similarly restricted version of the classical Hessian of $f$ to the tangent space.",
"Definition 1 A (first-order) critical point for (REF ) is a point $Y \\in \\mathcal {M}$ such that $\\mathrm {grad}f(Y) & = 0,$ where $\\mathrm {grad}f(Y) \\in Y\\mathcal {M}$ is the Riemannian gradient at $Y$ of $f$ restricted to $\\mathcal {M}$ .",
"A second-order critical point for (REF ) is a critical point $Y$ such that $\\mathrm {Hess}f(Y) \\succeq 0,$ where $\\mathrm {Hess}f(Y) \\colon Y\\mathcal {M}\\rightarrow Y\\mathcal {M}$ is the Riemannian Hessian at $Y$ of $f$ restricted to $\\mathcal {M}$ (a symmetric linear operator).",
"Proposition 1 All local (and global) optima of (REF ) are second-order critical points.",
"See [37].",
"We can now state our main result.",
"In the theorem statement below, “for almost all $C$ ” means potentially troublesome cost matrices form at most a (Lebesgue) zero-measure subset of ${\\mathbb {S}^{n\\times n}}$ , in the same way that almost all square matrices are invertible.",
"In particular, given any matrix $C\\in {\\mathbb {S}^{n\\times n}}$ , perturbing $C$ to $C+\\sigma W$ where $W$ is a Wigner random matrix results in an acceptable cost matrix with probability 1, for arbitrarily small $\\sigma > 0$ .",
"Theorem 2 Given constraints $\\mathcal {A}\\colon {\\mathbb {S}^{n\\times n}}\\rightarrow {\\mathbb {R}^m}$ , $b\\in {\\mathbb {R}^m}$ and $p$ satisfying $\\frac{p(p+1)}{2} > m$ , if (i) the search space of (REF ) is compact; and (ii) the search space of (REF ) is a regularly-defined smooth manifold, in the sense that $A_1Y, \\ldots , A_mY$ are linearly independent in ${\\mathbb {R}^{n\\times p}}$ for all $Y \\in \\mathcal {M}$ (see Appendix ), then for almost all cost matrices $C\\in {\\mathbb {S}^{n\\times n}}$ , any second-order critical point of (REF ) is globally optimal.",
"Under these conditions, if $Y$ is globally optimal for (REF ), then the matrix $X = YY^\\top \\!",
"$ is globally optimal for (REF ).",
"The assumptions are discussed in the next section.",
"The proof—see Appendix —follows directly from the combination of two intermediate results: If $Y$ is rank deficient and second-order critical for (REF ), then it is globally optimal and $X=YY^\\top \\!",
"$ is optimal for (REF ); and If $\\frac{p(p+1)}{2} > m$ , then, for almost all $C$ , every first-order critical $Y$ is rank-deficient.",
"The first step holds in a more general context, as previously established by [15], [16].",
"The second step is new and crucial, as it allows to formally exclude the existence of spurious local optima, generically in $C$ , thus resolving the caveat mentioned in the introduction.",
"The smooth structure of (REF ) naturally suggests using Riemannian optimization to solve it [3], which is something that was already proposed by [22] in the same context.",
"Importantly, known algorithms converge to second-order critical points regardless of initialization.",
"We state here a recent computational result to that effect.",
"Proposition 3 Under the numbered assumptions of Theorem REF , the Riemannian trust-region method (RTR) [2] initialized with any $Y_0\\in \\mathcal {M}$ returns in $\\mathcal {O}(1/\\varepsilon _g^2\\varepsilon _H^{} + 1/\\varepsilon _H^3)$ iterations a point $Y\\in \\mathcal {M}$ such that $f(Y) & \\le f(Y_0), & \\Vert \\mathrm {grad}f(Y)\\Vert & \\le \\varepsilon _g, & \\textrm { and } & & \\mathrm {Hess}f(Y) \\succeq -\\varepsilon _H \\operatorname{Id}.$ Apply the main results of [14] using that $f$ has locally Lipschitz continuous gradient and Hessian in ${\\mathbb {R}^{n\\times p}}$ and $\\mathcal {M}$ is a compact submanifold of ${\\mathbb {R}^{n\\times p}}$ .",
"Essentially, each iteration of RTR requires evaluation of one cost and one gradient, a bounded number of Hessian-vector applications, and one projection from ${\\mathbb {R}^{n\\times p}}$ to $\\mathcal {M}$ .",
"In many important cases, this projection amounts to Gram–Schmidt orthogonalization of small blocks of $Y$ —see Section .",
"Proposition REF bounds worst-case iteration counts for arbitrary initialization.",
"In practice, a good initialization point may be available, making the local convergence rate of RTR more informative.",
"For RTR, one may expect superlinear or even quadratic local convergence rates near isolated local minimizers [2].",
"While minimizers are not isolated in our case [22], experiments show a characteristically superlinear local convergence rate in practice [12].",
"This means high accuracy solutions can be achieved, as demonstrated in Appendix .",
"Thus, under the conditions of Theorem REF , generically in $C$ , RTR converges to global optima.",
"In practice, the algorithm returns after a finite number of steps, and only approximate second-order criticality is guaranteed.",
"Hence, it is interesting to bound the optimality gap in terms of the approximation quality.",
"Unfortunately, we do not establish such a result for small $p$ .",
"Instead, we give an a posteriori computable optimality gap bound which holds for all $p$ and for all $C$ .",
"In the following statement, the dependence of $\\mathcal {M}$ on $p$ is explicit, as $\\mathcal {M}_p$ .",
"The proof is in Appendix .",
"Theorem 4 Let $R < \\infty $ be the maximal trace of any $X$ feasible for (REF ).",
"For any $p$ such that $\\mathcal {M}_p$ and $\\mathcal {M}_{p+1}$ are smooth manifolds (even if $\\frac{p(p+1)}{2} \\le m$ ) and for any $Y\\in \\mathcal {M}_p$ , form $\\tilde{Y} = \\left[ Y | 0_{n\\times 1} \\right]$ in $\\mathcal {M}_{p+1}$ .",
"The optimality gap at $Y$ is bounded as $0 \\le 2(f(Y)-f^*) \\le \\sqrt{R} \\Vert \\mathrm {grad}f(Y)\\Vert - R \\lambda _\\mathrm {min}(\\mathrm {Hess}f(\\tilde{Y})).$ If all feasible $X$ have the same trace $R$ and there exists a positive definite feasible $X$ , then the bound simplifies to $0 \\le 2(f(Y)-f^*) \\le - R \\lambda _\\mathrm {min}(\\mathrm {Hess}f(\\tilde{Y}))$ so that $\\Vert \\mathrm {grad}f(Y)\\Vert $ needs not be controlled explicitly.",
"If $p>n$ , the bounds hold with $\\tilde{Y} = Y$ .",
"In particular, for $p=n+1$ , the bound can be controlled a priori: approximate second-order critical points are approximately optimal, for any $C$ .With $p=n+1$ , problem (REF ) is no longer lower dimensional than (REF ), but retains the advantage of not involving a positive semidefiniteness constraint.",
"Corollary 5 Under the assumptions of Theorem REF , if $p=n+1$ and $Y\\in \\mathcal {M}$ satisfies both $\\Vert \\mathrm {grad}f(Y)\\Vert \\le \\varepsilon _g$ and $\\mathrm {Hess}f(Y) \\succeq -\\varepsilon _H \\operatorname{Id}$ , then $Y$ is approximately optimal in the sense that $0 \\le 2(f(Y)-f^*) \\le \\sqrt{R} \\varepsilon _g + R \\varepsilon _H.$ Under the same condition as in Theorem REF , the bound can be simplified to $R \\varepsilon _H$ .",
"This works well with Proposition REF .",
"For any $p$ , equation (REF ) also implies the following: $\\lambda _\\mathrm {min}(\\mathrm {Hess}f(\\tilde{Y})) \\le - \\frac{2(f(Y)-f^*) - \\sqrt{R}\\Vert \\mathrm {grad}f(Y)\\Vert }{R}.$ That is, for any $p$ and any $C$ , an approximate critical point $Y$ in $\\mathcal {M}_p$ which is far from optimal maps to a comfortably-escapable approximate saddle point $\\tilde{Y}$ in $\\mathcal {M}_{p+1}$ .",
"This suggests an algorithm as follows.",
"For a starting value of $p$ such that $\\mathcal {M}_p$ is a manifold, use RTR to compute an approximate second-order critical point $Y$ .",
"Then, form $\\tilde{Y}$ in $\\mathcal {M}_{p+1}$ and test the left-most eigenvalue of $\\mathrm {Hess}f(\\tilde{Y})$ .It may be more practical to test $\\lambda _\\mathrm {min}(S)$ (REF ) rather than $\\lambda _\\mathrm {min}(\\mathrm {Hess}f)$ .",
"Lemma REF relates the two.",
"See [22] to construct escape tangent vectors from $S$ .",
"If it is close enough to zero, this provides a good bound on the optimality gap.",
"If not, use an (approximate) eigenvector associated to $\\lambda _\\mathrm {min}(\\mathrm {Hess}f(\\tilde{Y}))$ to escape the approximate saddle point and apply RTR from that new point in $\\mathcal {M}_{p+1}$ ; iterate.",
"In the worst-case scenario, $p$ grows to $n+1$ , at which point all approximate second-order critical points are approximate optima.",
"Theorem REF suggests $p = \\left\\lceil \\sqrt{2m} \\, \\right\\rceil $ should suffice for $C$ bounded away from a zero-measure set.",
"Such an algorithm already features with less theory in [22] and [12]; in the latter, it is called the Riemannian staircase, for it lifts (REF ) floor by floor."
],
[
"Related work",
"Low-rank approaches to solve SDPs have featured in a number of recent research papers.",
"We highlight just two which illustrate different classes of SDPs of interest.",
"[30] tackle SDPs with linear cost and linear constraints (both equalities and inequalities) via low-rank factorizations, assuming the matrices appearing in the cost and constraints are positive semidefinite.",
"They propose a non-trivial initial guess to partially overcome non-convexity with great empirical results, but do not provide optimality guarantees.",
"[10] on the other hand consider the minimization of a convex cost function over positive semidefinite matrices, without constraints.",
"Such problems could be obtained from generic SDPs by penalizing the constraints in a Lagrangian way.",
"Here too, non-convexity is partially overcome via non-trivial initialization, with global optimality guarantees under some conditions.",
"Also of interest are recent results about the harmlessness of non-convexity in low-rank matrix completion [18], [11].",
"Similarly to the present work, the authors there show there is no need for special initialization despite non-convexity."
],
[
"Discussion of the assumptions",
"Our main result, Theorem REF , comes with geometric assumptions on the search spaces of both (REF ) and (REF ) which we now discuss.",
"Examples of SDPs which fit the assumptions of Theorem REF are featured in the next section.",
"The assumption that the search space of (REF ), $\\mathcal {C}= \\lbrace X \\in {\\mathbb {S}^{n\\times n}}: \\mathcal {A}(X) = b, X \\succeq 0 \\rbrace ,$ is compact works in pair with the assumption $\\frac{p(p+1)}{2} > m$ as follows.",
"For (REF ) to reveal the global optima of (REF ), it is necessary that (REF ) admits a solution of rank at most $p$ .",
"One way to ensure this is via the Pataki–Barvinok theorems [27], [9], which state that all extreme points of $\\mathcal {C}$ have rank $r$ bounded as $\\frac{r(r+1)}{2} \\le m$ .",
"Extreme points are faces of dimension zero (such as vertices for a cube).",
"When optimizing a linear cost function $\\left\\langle {C},{X}\\right\\rangle $ over a compact convex set $\\mathcal {C}$ , at least one extreme point is a global optimum [28]—this is not true in general if $\\mathcal {C}$ is not compact.",
"Thus, under the assumptions of Theorem REF , there is a point $Y\\in \\mathcal {M}$ such that $X = YY^\\top \\!",
"$ is an optimal extreme point of (REF ); then, of course, $Y$ itself is optimal for (REF ).",
"In general, the Pataki–Barvinok bound is tight, in that there exist extreme points of rank up to that upper-bound (rounded down)—see for example [23] for the Max-Cut SDP and [12] for the Orthogonal-Cut SDP.",
"Let $C$ (the cost matrix) be the negative of such an extreme point.",
"Then, the unique optimum of (REF ) is that extreme point, showing that $\\frac{p(p+1)}{2} \\ge m$ is necessary for (REF ) and (REF ) to be equivalent for all $C$ .",
"We further require a strict inequality because our proof relies on properties of rank deficient $Y$ 's in $\\mathcal {M}$ .",
"The assumption that $\\mathcal {M}$ (eq.",
"(REF )) is a regularly-defined smooth manifold works in pair with the ambition that the result should hold for (almost) all cost matrices $C$ .",
"The starting point is that, for a given non-convex smooth optimization problem—even a quadratically constrained quadratic program—computing local optima is hard in general [33].",
"Thus, we wish to restrict our attention to efficiently computable points, such as points which satisfy first- and second-order KKT conditions for (REF )—see [15] and [29].",
"This only makes sense if global optima satisfy the latter, that is, if KKT conditions are necessary for optimality.",
"A global optimum $Y$ necessarily satisfies KKT conditions if constraint qualifications (CQs) hold at $Y$ [29].",
"The standard CQs for equality constrained programs are Robinson's conditions or metric regularity (they are here equivalent).",
"They read as follows, assuming $\\mathcal {A}(YY^\\top \\!",
")_i = \\left\\langle {A_i},{YY^\\top \\!",
"}\\right\\rangle $ for some matrices $A_1, \\ldots , A_m \\in {\\mathbb {S}^{n\\times n}}$ : $\\textrm {CQs hold at } Y \\textrm { if } A_1Y, \\ldots , A_mY \\textrm { are linearly independent in } {\\mathbb {R}^{n\\times p}}.$ Considering almost all $C$ , global optima could, a priori, be almost anywhere in $\\mathcal {M}$ .",
"To simplify, we require CQs to hold at all $Y$ 's in $\\mathcal {M}$ rather than only at the (unknown) global optima.",
"Under this condition, the constraints are independent at each point and ensure $\\mathcal {M}$ is a smooth embedded submanifold of ${\\mathbb {R}^{n\\times p}}$ of codimension $m$ [3].",
"Indeed, tangent vectors $\\dot{Y} \\in Y\\mathcal {M}$ (REF ) are exactly those vectors that satisfy $\\langle {A_i Y},{\\dot{Y}}\\rangle = 0$ : under CQs, the $A_iY$ 's form a basis of the normal space to the manifold at $Y$ .",
"Finally, we note that Theorem REF only applies for almost all $C$ , rather than all $C$ .",
"To justify this restriction, if indeed it is justified, one should exhibit a matrix $C$ that leads to suboptimal second-order critical points while other assumptions are satisfied.",
"We do not have such an example.",
"We do observe that (REF ) on cycles of certain even lengths has a unique solution of rank 1, while the corresponding (REF ) with $p=2$ has suboptimal local optima (strictly, if we quotient out symmetries).",
"This at least suggests it is not enough, for generic $C$ , to set $p$ just larger than the rank of the solutions of the SDP.",
"(For those same examples, at $p=3$ , we consistently observe convergence to global optima.)"
],
[
"Examples of smooth SDPs",
"The canonical examples of SDPs which satisfy the assumptions in Theorem REF are those where the diagonal blocks of $X$ or their traces are fixed.",
"We note that the algorithms and the theory continue to hold for complex matrices, where the set of Hermitian matrices of size $n$ is treated as a real vector space of dimension $n^2$ (instead of $\\frac{n(n+1)}{2}$ in the real case) with inner product $\\left\\langle {H_1},{H_2}\\right\\rangle = \\Re \\left\\lbrace \\mathrm {Tr}(H_1^*H_2^{}) \\right\\rbrace $ , so that occurrences of $\\frac{p(p+1)}{2}$ are replaced by $p^2$ .",
"Certain concrete examples of SDPs include: $\\min _X \\left\\langle {C},{X}\\right\\rangle & \\textrm { s.t. }",
"\\mathrm {Tr}(X) = 1, X\\succeq 0; \\\\\\min _X \\left\\langle {C},{X}\\right\\rangle & \\textrm { s.t. }",
"\\mathrm {diag}(X) = \\mathbf {1}, X\\succeq 0; \\\\\\min _X \\left\\langle {C},{X}\\right\\rangle & \\textrm { s.t. }",
"X_{ii} = I_d, X\\succeq 0.",
"$ Their rank-constrained counterparts read as follows (matrix norms are Frobenius norms): $\\min _{Y\\colon n\\times p} \\left\\langle {CY},{Y}\\right\\rangle & \\textrm { s.t. }",
"\\Vert Y\\Vert = 1; \\\\\\min _{Y\\colon n\\times p} \\left\\langle {CY},{Y}\\right\\rangle & \\textrm { s.t. }",
"Y^\\top \\!",
"= \\begin{bmatrix} y_1 & \\cdots & y_n \\end{bmatrix} \\textrm { and } \\Vert y_i\\Vert = 1 \\textrm { for all } i; \\\\\\min _{Y\\colon qd\\times p} \\left\\langle {CY},{Y}\\right\\rangle & \\textrm { s.t. }",
"Y^\\top \\!",
"= \\begin{bmatrix} Y_1 & \\cdots & Y_{q} \\end{bmatrix} \\textrm { and } Y_i^\\top \\!",
"Y_i^{} = I_d \\textrm { for all } i.",
"$ The first example has only one constraint: the SDP always admits an optimal rank 1 solution, corresponding to an eigenvector associated to the left-most eigenvalue of $C$ .",
"This generalizes to the trust-region subproblem as well.",
"For the second example, in the real case, $p=1$ forces $y_i = \\pm 1$ , allowing to capture combinatorial problems such as Max-Cut [19], $\\mathbb {Z}_2$ -synchronization [21] and community detection in the stochastic block model [1], [6].",
"The same SDP is central in a formulation of robust PCA [25] and is used to approximate the cut-norm of a matrix [4].",
"Theorem REF states that for almost all $C$ , $p = \\left\\lceil \\sqrt{2n}\\, \\right\\rceil $ is sufficient.",
"In the complex case, $p=1$ forces $|y_i| = 1$ , allowing to capture problems where phases must be recovered; in particular, phase synchronization [8], [31] and phase retrieval via Phase-Cut [34].",
"For almost all $C$ , it is then sufficient to set $p = \\left\\lfloor \\sqrt{n}+1 \\right\\rfloor $ .",
"In the third example, $Y$ of size $n\\times p$ is divided in $q$ slices of size $d\\times p$ , with $p \\ge d$ .",
"Each slice has orthonormal rows.",
"For $p=d$ , the slices are orthogonal (or unitary) matrices, allowing to capture Orthogonal-Cut [7] and the related problems of synchronization of rotations [35] and permutations.",
"Synchronization of rotations is an important step in simultaneous localization and mapping, for example.",
"Here, it is sufficient for almost all $C$ to let $p = \\left\\lceil \\sqrt{d(d+1)q} \\, \\right\\rceil $ .",
"SDPs with constraints that are combinations of the above examples can also have the smoothness property; the right-hand sides 1 and $I_d$ can be replaced by any positive definite right-hand sides by a change of variables.",
"Another simple rule to check is if the constraint matrices $A_1, \\ldots , A_m \\in {\\mathbb {S}^{n\\times n}}$ such that $\\mathcal {A}(X)_i = \\left\\langle {A_i},{X}\\right\\rangle $ satisfy $A_iA_j = 0$ for all $i \\ne j$ (note that this is stronger than requiring $\\left\\langle {A_i},{A_j}\\right\\rangle = 0$ ), see [22]."
],
[
"Conclusions",
"The Burer–Monteiro approach consists in replacing optimization of a linear function $\\left\\langle {C},{X}\\right\\rangle $ over the convex set $\\lbrace X\\succeq 0 : \\mathcal {A}(X) = b\\rbrace $ with optimization of the quadratic function $\\left\\langle {CY},{Y}\\right\\rangle $ over the non-convex set $\\lbrace Y\\in {\\mathbb {R}^{n\\times p}}: \\mathcal {A}(YY^\\top \\! )",
"= b\\rbrace $ .",
"It was previously known that, if the convex set is compact and $p$ satisfies $\\frac{p(p+1)}{2} \\ge m$ where $m$ is the number of constraints, then these two problems have the same global optimum.",
"It was also known from [16] that spurious local optima $Y$ , if they exist, must map to special faces of the compact convex set, but without statement as to the prevalence of such faces or the risk they pose for local optimization methods.",
"In this paper we showed that, if the set of $X$ 's is compact and the set of $Y$ 's is a regularly-defined smooth manifold, and if $\\frac{p(p+1)}{2} > m$ , then for almost all $C$ , the non-convexity of the problem in $Y$ is benign, in that all $Y$ 's which satisfy second-order necessary optimality conditions are in fact globally optimal.",
"We further reference the Riemannian trust-region method [2] to solve the problem in $Y$ , as it was recently guaranteed to converge from any starting point to a point which satisfies second-order optimality conditions, with global convergence rates [14].",
"In addition, for $p=n+1$ , we guarantee that approximate satisfaction of second-order conditions implies approximate global optimality.",
"We note that the $1/\\varepsilon ^3$ convergence rate in our results may be pessimistic.",
"Indeed, the numerical experiments clearly show that high accuracy solutions can be computed fast using optimization on manifolds, at least for certain applications.",
"Addressing a broader class of SDPs, such as those with inequality constraints or equality constraints that may violate our smoothness assumptions, could perhaps be handled by penalizing those constraints in the objective in an augmented Lagrangian fashion.",
"We also note that, algorithmically, the Riemannian trust-region method we use applies just as well to nonlinear costs in the SDP.",
"We believe that extending the theory presented here to broader classes of problems is a good direction for future work."
],
[
"Acknowledgment",
"VV was partially supported by the Office of Naval Research.",
"ASB was supported by NSF Grant DMS-1317308.",
"Part of this work was done while ASB was with the Department of Mathematics at the Massachusetts Institute of Technology.",
"We thank Wotao Yin and Michel Goemans for helpful discussions."
],
[
"Proofs and additional lemmas",
"We start by working out explicit formulas for the Riemannian gradient and Hessian which appear in Definition REF .",
"Let $\\mathrm {Proj}_Y \\colon {\\mathbb {R}^{n\\times p}}\\rightarrow Y\\mathcal {M}$ be the orthogonal projector to the tangent space at $Y$ (eq.",
"(REF )), and let $\\nabla f(Y) & = 2CY, & \\nabla ^2 f(Y)[\\dot{Y}] & = 2C\\dot{Y}$ be the (Euclidean) gradient and Hessian of the cost function (REF ).",
"The Riemannian gradient and Hessian of $f$ on $\\mathcal {M}$ are related to these as follows [3]: $\\mathrm {grad}f(Y) & = \\mathrm {Proj}_Y \\nabla f(Y), \\\\ \\forall \\dot{Y} \\in Y\\mathcal {M}, \\quad \\mathrm {Hess}f(Y)[\\dot{Y}] & = \\mathrm {Proj}_Y ( Y \\mapsto \\mathrm {grad}f(Y) (Y)[\\dot{Y}].$ Let us focus on the gradient first.",
"Since $\\mathrm {grad}f(Y)$ is a tangent vector at $Y$ (REF ),For non-symmetric $B\\in {\\mathbb {R}^{n\\times n}}$ , note that $\\mathcal {A}(B) = \\mathcal {A}\\big ( \\frac{B+B^\\top \\!",
"}{2} \\big )$ .",
"$\\mathcal {A}(\\mathrm {grad}f(Y)Y^\\top \\! )",
"& = 0,$ and since it is the orthogonal projection of $\\nabla f(Y)$ to the tangent space, there exists $\\mu \\in {\\mathbb {R}^m}$ such that $\\mathrm {grad}f(Y) + 2 \\mathcal {A}^*(\\mu )Y & = \\nabla f(Y) = 2CY,$ where $\\mathcal {A}^* \\colon {\\mathbb {R}^m}\\rightarrow {\\mathbb {S}^{n\\times n}}$ is the adjoint of $\\mathcal {A}$ .",
"Indeed, considering symmetric matrices $A_1, \\ldots , A_m$ such that $\\mathcal {A}(X)_i = \\left\\langle {A_i},{X}\\right\\rangle $ , matrices $\\mathcal {A}^*(\\mu )Y = \\mu _1 A_1Y + \\cdots + \\mu _m A_mY$ span the normal space to the manifold at $Y$ .",
"Right-multiply (REF ) with $Y^\\top \\!",
"$ and apply $\\mathcal {A}$ to obtain $\\mathcal {A}\\left(\\mathcal {A}^*(\\mu )YY^\\top \\!",
"\\right) & = \\mathcal {A}(CYY^\\top \\!",
").$ Under the assumption that the $A_iY$ 's are linearly independent, $\\mu $ is the unique solution to this linear system—for KKT points, these are the Lagrange multipliers.",
"Furthermore, contrary to classical KKT conditions, $\\mu $ is defined for all feasible $Y$ (not only for KKT points) and can be found by solving (REF ).For the Max-Cut SDP for example, $\\mathcal {A}= \\mathrm {diag}$ and $\\mu = \\mathrm {diag}(CYY^\\top \\!",
")$ .",
"This $\\mu $ is a well-defined, differentiable function of $Y$ .Eq.",
"(REF ) is equivalent to $G\\mu = \\mathcal {A}(CYY^\\top \\!",
")$ , where $G_{ij} = \\left\\langle {A_iY},{A_jY}\\right\\rangle $ .",
"For all $Y\\in \\mathcal {M}$ , $G$ is invertible since $A_1Y, \\ldots , A_mY$ are linearly independent.",
"Hence, $\\mu = G^{-1} \\mathcal {A}(CYY^\\top \\!",
")$ is differentiable in $Y$ at $Y\\in \\mathcal {M}$ .",
"Using this definition of $\\mu $ , let $S & = S(Y) = S(YY^\\top \\! )",
"= C - \\mathcal {A}^*(\\mu ).$ First-order critical points then satisfy (using (REF )): $\\frac{1}{2}\\mathrm {grad}f(Y) = SY = 0.$ We note in passing that $\\mu (Y)$ is feasible for the dual of (REF ) exactly when $S(Y) \\succeq 0$ : $d^* = \\max _{\\mu \\in {\\mathbb {R}^m}} b^\\top \\!",
"\\mu \\textrm { subject to } C - \\mathcal {A}^*(\\mu ) \\succeq 0,$ which illustrates the importance of $S$ as a dual certificate for (REF ).",
"Now let us turn to the Hessian of $f$ .",
"Equation (REF ) requires computation of the differential of $\\mathrm {grad}f(Y)$ , which is $( Y \\mapsto \\mathrm {grad}f(Y) \\big )(Y)[\\dot{Y}] & = ( Y \\mapsto 2SY \\big )(Y)[\\dot{Y}] = 2S \\dot{Y} + 2\\dot{S}Y,$ where $\\dot{S} \\triangleq S̥(Y)[\\dot{Y}]$ is a symmetric matrix.",
"Because of eq.",
"(REF ), $\\dot{S} = \\mathcal {A}^*(\\nu )$ for some $\\nu \\in {\\mathbb {R}^m}$ .",
"Hence, for any tangent vector $\\dot{Z}\\in Y\\mathcal {M}$ (REF ), we have $\\langle {\\dot{Z}},{\\dot{S} Y}\\rangle = \\langle {\\dot{Z} Y^\\top \\!",
"},{\\mathcal {A}^*(\\nu )}\\rangle = \\langle {\\mathcal {A}(\\dot{Z} Y^\\top \\!",
")},{\\nu }\\rangle = 0$ : $\\dot{S} Y$ is orthogonal to the tangent space at $Y$ .",
"Using (REF ), we find that $\\frac{1}{2}\\mathrm {Hess}f(Y)[\\dot{Y}] = \\mathrm {Proj}_Y S\\dot{Y}.$ The second-order condition for $Y$ is that $\\mathrm {Hess}f(Y)$ be positive semidefinite on $Y\\mathcal {M}$ .",
"Using that $\\mathrm {Proj}_Y$ is a self-adjoint operator, it follows that this condition is equivalent to: $\\forall \\dot{Y} \\in Y\\mathcal {M}, \\quad \\frac{1}{2} \\langle {\\dot{Y}},{\\mathrm {Hess}f(Y)[\\dot{Y}]}\\rangle = \\langle {\\dot{Y}},{S\\dot{Y}}\\rangle \\ge 0.$ We now show that rank-deficient second-order critical points are globally optimal.",
"We obtain this result as a corollary to a more informative statement about optimality gap at approximately second-order critical points (assuming exact rank deficiency).",
"The lemmas also show how $S$ can be used to control the optimality gap at approximate critical points without requiring rank deficiency.",
"This is valid for any $p$ and any $C$ .",
"Lemma 6 For any $Y$ on the manifold $\\mathcal {M}$ , if $\\Vert \\mathrm {grad}f(Y)\\Vert \\le \\varepsilon _g$ and $S(Y) \\succeq -\\frac{\\varepsilon _H}{2} I_n$ , then the optimality gap at $Y$ with respect to (REF ) is bounded as $0 \\le 2(f(Y) - f^*) \\le \\varepsilon _H R + \\varepsilon _g \\sqrt{R},$ where $R = \\max _{X\\in \\mathcal {C}} \\mathrm {Tr}(X) < \\infty $ measures the size of the compact set $\\mathcal {C}$ (REF ).",
"If $I_n\\in \\mathrm {im}(\\mathcal {A}^*)$ , the right hand side of (REF ) simplifies to $\\varepsilon _H R$ .",
"This holds in particular if all $X\\in \\mathcal {C}$ have same trace and $\\mathcal {C}$ has a relative interior point (Slater condition).",
"By assumption on $S(Y)$ (eq.",
"(REF )), $\\forall \\tilde{X}\\in \\mathcal {C}, \\quad -\\frac{\\varepsilon _H}{2} \\mathrm {Tr}(\\tilde{X}) \\le \\langle {S(Y)},{\\tilde{X}}\\rangle & = \\langle {C},{\\tilde{X}}\\rangle - \\langle {\\mathcal {A}^*(\\mu (Y))},{\\tilde{X}}\\rangle = \\langle {C},{\\tilde{X}}\\rangle - \\langle {\\mu (Y)},{b}\\rangle .$ This holds in particular for $\\tilde{X}$ optimal for (REF ).",
"Thus, we may set $\\langle {C},{\\tilde{X}}\\rangle = f^*$ ; and certainly, $\\mathrm {Tr}(\\tilde{X}) \\le R$ .",
"Furthermore, $\\langle {\\mu (Y)},{b}\\rangle = \\langle {\\mu (Y)},{\\mathcal {A}(YY^\\top \\!",
")}\\rangle = \\langle {C-S(Y)},{YY^\\top \\!",
"}\\rangle = f(Y) - \\langle {S(Y)Y},{Y}\\rangle .$ Combining the typeset equations and using $\\mathrm {grad}f(Y) = 2S(Y)Y$ , we find $0 \\le 2(f(Y) - f^*) \\le \\varepsilon _H R + \\langle {\\mathrm {grad}f(Y)},{Y}\\rangle .$ In general, we do not assume $I_n \\in \\mathrm {im}(\\mathcal {A}^*)$ and we get the result by Cauchy–Schwarz on (REF ) and $\\Vert Y\\Vert = \\sqrt{\\mathrm {Tr}(YY^\\top \\! )}",
"\\le \\sqrt{R}$ : $0 \\le 2(f(Y)-f^*) \\le \\varepsilon _H R + \\varepsilon _g \\sqrt{R}.$ But if $I_n \\in \\mathrm {im}(\\mathcal {A}^*)$ , then we show that $Y$ is a normal vector at $Y$ , so that it is orthogonal to $\\mathrm {grad}f(Y)$ .",
"Formally: there exists $\\nu \\in {\\mathbb {R}^m}$ such that $I_n = \\mathcal {A}^*(\\nu )$ , and $\\langle {\\mathrm {grad}f(Y)},{Y}\\rangle & = \\langle {\\mathrm {grad}f(Y) Y^\\top \\!",
"},{I_n}\\rangle = \\langle {\\mathcal {A}(\\mathrm {grad}f(Y) Y^\\top \\!",
")},{\\nu }\\rangle = 0,$ since $\\mathrm {grad}f(Y) \\in Y\\mathcal {M}$ (REF ).",
"This indeed allows to simplify (REF ).",
"To conclude, we show that if $\\mathcal {C}$ has a relative interior point $X^{\\prime }$ (that is, $\\mathcal {A}(X^{\\prime }) = b$ and $X^{\\prime } \\succ 0$ ) and if $\\mathrm {Tr}(X)$ is a constant for all $X$ in $\\mathcal {C}$ , then $I_n \\in \\mathrm {im}(\\mathcal {A}^*)$ .",
"Indeed, ${\\mathbb {S}^{n\\times n}}= \\mathrm {im}(\\mathcal {A}^*) \\oplus \\ker \\mathcal {A}$ , so there exist $\\nu \\in {\\mathbb {R}^m}$ and $M \\in \\ker \\mathcal {A}$ such that $I_n = \\mathcal {A}^*(\\nu ) + M$ .",
"Thus, for all $X$ in $\\mathcal {C}$ , $0 = \\mathrm {Tr}(X - X^{\\prime }) = \\left\\langle {\\mathcal {A}^*(\\nu )+M},{X-X^{\\prime }}\\right\\rangle = \\left\\langle {M},{X-X^{\\prime }}\\right\\rangle .$ This implies that $M$ is orthogonal to all $X-X^{\\prime }$ .",
"These span $\\ker \\mathcal {A}$ since $X^{\\prime }$ is interior.",
"Indeed, for any $H\\in \\ker \\mathcal {A}$ , since $X^{\\prime }\\succ 0$ , there exists $\\varepsilon > 0$ such that $X \\triangleq X^{\\prime }+\\varepsilon H \\succeq 0$ and $\\mathcal {A}(X) = b$ , so that $X \\in \\mathcal {C}$ .",
"Hence, $M \\in \\ker \\mathcal {A}$ is orthogonal to $\\ker \\mathcal {A}$ .",
"Consequently, $M = 0$ and $I_n = \\mathcal {A}^*(\\nu )$ .",
"Lemma 7 If $Y\\in \\mathcal {M}$ is column rank deficient and $\\mathrm {Hess}f(Y) \\succeq -\\varepsilon _H \\operatorname{Id}$ , then $S(Y) \\succeq -\\frac{\\varepsilon _H}{2} I_n$ .",
"By assumption, there exists $z\\in {\\mathbb {R}^{p}}$ , $\\Vert z\\Vert = 1$ such that $Yz = 0$ .",
"Thus, for any $x\\in {\\mathbb {R}^n}$ , we can form $\\dot{Y} = xz^\\top \\!",
"$ : it is a tangent vector since $Y\\dot{Y}^\\top \\!",
"= 0$ (REF ).",
"Then, condition (REF ) combined with the assumption on $\\mathrm {Hess}f(Y)$ tells us $-\\varepsilon _H \\Vert x\\Vert ^2 \\le \\langle {\\dot{Y}},{\\mathrm {Hess}f(Y)[\\dot{Y}]}\\rangle = 2\\langle {\\dot{Y}},{S\\dot{Y}}\\rangle = 2\\langle {xz^\\top \\!",
"z x^\\top \\!",
"},{S}\\rangle = 2 x^\\top \\!",
"S x.$ This holds for all $x\\in {\\mathbb {R}^n}$ , hence $S \\succeq -\\frac{\\varepsilon _H}{2} I_n$ as required.",
"Corollary 8 If $Y\\in \\mathcal {M}_p$ is a column rank-deficient second-order critical point for (REF ), then it is optimal for (REF ) and $X = YY^\\top \\!",
"$ is optimal for (REF ).",
"In particular, for $p>n$ , all second-order critical points are optimal.",
"The first part of this corollary also appears as [15], where the statement is made about local optima rather than second-order critical points.",
"At this point, we can already give a short proof of Theorem REF .",
"[Proof of Theorem REF ] Since $\\tilde{Y} \\tilde{Y}^\\top \\!",
"= YY^\\top \\!",
"$ , $S(\\tilde{Y}) = S(Y)$ ; in particular, $f(\\tilde{Y}) = f(Y)$ and $\\Vert \\mathrm {grad}f(\\tilde{Y})\\Vert = \\Vert \\mathrm {grad}f(Y)\\Vert $ .",
"Since $\\tilde{Y}$ has deficient column rank, apply Lemmas REF and REF .",
"For $p>n$ , there is no need to form $\\tilde{Y}$ as $Y$ necessarily has deficient column rank.",
"Based on Corollary REF , to establish Theorem REF it is sufficient to show that, for almost all $C$ , all second-order critical points are rank deficient already for small $p$ .",
"We show that in fact this is true even for first-order critical points.",
"The argument is by dimensionality counting on ${\\mathbb {S}^{n\\times n}}$ : the set of all possible cost matrices $C$ .",
"Lemma 9 Under the assumptions of Theorem REF , for almost all $C$ , all critical points of (REF ) are rank deficient.",
"Let $Y$ be a critical point for (REF ).",
"By the first-order condition $S(Y)Y=0$ (REF ) and the definition of $S(Y) = C - \\mathcal {A}^*(\\mu (Y))$ (REF ), there exists $\\mu \\in {\\mathbb {R}^m}$ such that $\\operatorname{rank}Y \\le \\operatorname{null}(C-\\mathcal {A}^*(\\mu )) \\le \\max _{\\nu \\in {\\mathbb {R}^m}} \\operatorname{null}(C-\\mathcal {A}^*(\\nu )),$ where $\\operatorname{null}$ denotes the nullity (dimension of the kernel).",
"This first step in the proof is inspired by [36].",
"If the right hand side evaluates to $\\ell $ , then there exists $\\nu $ such that $M = C-\\mathcal {A}^*(\\nu )$ and $\\operatorname{null}(M) = \\ell $ .",
"Writing $C = M + \\mathcal {A}^*(\\nu )$ , we find that $C & \\in \\mathcal {N}_\\ell + \\mathrm {im}(\\mathcal {A}^*)$ where the $+$ is a set-sum and $\\mathcal {N}_\\ell $ denotes the set of symmetric matrices of size $n$ with nullity $\\ell $ .",
"This set has dimension $\\dim \\mathcal {N}_\\ell = \\frac{n(n+1)}{2} - \\frac{\\ell (\\ell +1)}{2},$ whereas $\\dim \\mathrm {im}(\\mathcal {A}^*) = \\operatorname{rank}(\\mathcal {A}^*) \\le m$ .",
"Assume the right hand side of (REF ) evaluates to $p$ or more.",
"Then, a fortiori, $C \\in \\bigcup _{\\ell = p,\\ldots ,n} \\mathcal {N}_\\ell + \\mathrm {im}(\\mathcal {A}^*).$ The set on the right hand side contains all “bad” $C$ 's, that is, those for which (REF ) offers no information about the rank of $Y$ .",
"The dimension of that set is bounded as follows, using that the dimension of a finite union is at most the maximal dimension, and the dimension of a finite sum of sets is at most the sum of the set dimensions: $\\dim \\left( \\bigcup _{\\ell = p,\\ldots ,n} \\mathcal {N}_\\ell + \\mathrm {im}(\\mathcal {A}^*) \\right) \\le \\dim \\left( \\mathcal {N}_p + \\mathrm {im}(\\mathcal {A}^*) \\right) \\le \\frac{n(n+1)}{2} - \\frac{p(p+1)}{2} + m.$ Since $C\\in {\\mathbb {S}^{n\\times n}}$ lives in a space of dimension $\\frac{n(n+1)}{2}$ , almost no $C$ verifies (REF ) if $\\frac{n(n+1)}{2} - \\frac{p(p+1)}{2} + m < \\frac{n(n+1)}{2}.$ Hence, if $\\frac{p(p+1)}{2} > m$ , then, for almost all $C$ , critical points verify $\\operatorname{rank}(Y) < p$ .",
"Theorem REF follows as a corollary of Corollary REF and Lemma REF ."
],
[
"Numerical experiments",
"As an example, we run five different solvers on (REF ) with a collection of graphs used in [15], [16] known as the Gset.Downloaded from: http://web.stanford.edu/~yyye/yyye/Gset/ on June 6, 2016.",
"The solvers are as follows, all run in Matlab.",
"The first three are based on a low-rank factorization while the last two are interior point methods (IPM).",
"Manopt runs the Riemannian Trust-Region method on (REF ), via the Manopt toolbox [13], with $p = \\left\\lceil \\frac{\\sqrt{8n+1}}{2} \\right\\rceil $ and random initialization.",
"The number of inner iterations allowed to solve the trust-region subproblem is 500.",
"The solver returns when $\\frac{1}{2} \\Vert \\mathrm {grad}f(Y)\\Vert = \\Vert SY\\Vert \\le 10^{-6}$ .",
"Code is in Matlab.",
"Manopt+ runs the same algorithm as above, but with $p$ increasing from 2 to $\\left\\lceil \\frac{\\sqrt{8n+1}}{2} \\right\\rceil $ in 5 steps.",
"The point $Y$ computed at a lower $p$ is appended with columns of i.i.d.",
"random Gaussian variables with standard deviation $10^{-5}$ and mean 0, then rows are normalized to produce $Y_+$ : the initial point for the next value of $p$ .",
"The randomization allows to escape near-saddle points (in practice).",
"Code is in Matlab.",
"SDPLR runs the original Burer–Monteiro algorithm implemented by its authors [15].",
"Code is in C interfaced through C-mex.",
"HRVW runs an IPM whose implementation is tailored to (REF ), implemented by its authors [20].",
"Code is in Matlab.",
"CVX runs SDPT3 [32] on (REF ) via CVX [17].",
"Code is in C interfaced through C-mex.",
"After the solvers return, we project their answers to the feasible set.",
"Manopt and SDPLR return a matrix $Y$ : it is sufficient to normalize each row to ensure $X = YY^\\top \\!",
"$ is feasible for (REF ) (for Manopt, this step is not necessary).",
"HRVW and CVX return a symmetric matrix $X$ .",
"We compute its Cholesky factorization $X = RR^\\top \\!",
"$ —if $X$ is not positive semidefinite, we first project using an eigenvalue decomposition.",
"Then, each row of $R$ is normalized so that $X = RR^\\top \\!",
"$ is feasible for (REF ).",
"Computation time required for these projections is not included in the timings.",
"We report three metrics for each graph and each solver.",
"Cut bound: a bound on the maximal cut value (lower is better).",
"If $C$ is the adjacency matrix of the graph and $D$ is the degree matrix, then $L = D-C$ is the Laplacian and $\\max _{X} \\frac{1}{4}\\left\\langle {L},{X}\\right\\rangle $ s.t.",
"$\\mathrm {diag}(X) = \\mathbf {1}, X \\succeq 0$ is a bound on the maximal cut.",
"Using Lemma REF applied to (REF ), a candidate optimizer $X$ yields a bound $\\frac{1}{4}\\left\\langle {L},{X}\\right\\rangle - \\frac{n}{4}\\lambda _\\mathrm {min}(S)$ .",
"$\\lambda _\\mathrm {min}(S)$ : by Lemma REF , this is a measure of optimality for $X$ (feasible), where $S = C - \\mathrm {diag}(\\mathrm {diag}(CX))$ .",
"It is nonpositive and must be as close to 0 as possible.",
"We compute it using bisection and the Cholesky factorization to ensure accuracy.",
"Time: computation time in seconds for the solver to runMatlab R2015a on $2 \\times 6$ cores processors with hyperthreading, Intel(R) Xeon(R) CPU E5-2640 @ 2.50GHz, 256Gb RAM, Springdale Linux 6.",
"(this excludes time taken to project the solution to the feasible set and to compute the reported metrics.)",
"Based on the results reported in Table , we make the following main observations: (i) the Manopt approach (optimization on manifolds, also advocated in [22]) consistently reaches high accuracy solutions, being often orders of magnitude more accurate than other methods, as judged from $\\lambda _\\mathrm {min}(S)$ ; (ii) incremental rank solvers (Manopt+ and SDPLR) are often the fastest solvers for large instances; and (iii) the tailored IPM HRVW is faster and typically more accurate than the IPM called by CVX (which is generic software).",
"The latter point hints that one must be careful in dismissing IPMs based on experiments using generic software, although it remains clear from Table that IPMs scale poorly compared to the low-rank factorization methods tested here.",
"In particular, CVX runs into memory trouble for the larger problem instances reported.On Graph 77, running CVX leads to Matlab error “Number of elements exceeds maximum flint $2^{53}-1$ .” To save time, we did not run CVX on the largest graphs."
],
[
"Numerical experiments: results",
"lllllll Results of the experiments described in Section .",
"Graph Metric Manopt Manopt+ SDPLR HRVW CVX Graph Metric Manopt Manopt+ SDPLR HRVW CVX Graph 1 Cut bound 12083.2 12083.2 12083.2 12083.2 12083.2 800 nodes $\\lambda _\\mathrm {min}(S)$ $-3 \\cdot 10^{-11}$ $-2 \\cdot 10^{-11}$ $-9 \\cdot 10^{-6}$ $-2 \\cdot 10^{-5}$ $-3 \\cdot 10^{-6}$ 19176 edges Time [s] 2.1 3.2 6.6 1.9 35.0 Graph 2 Cut bound 12089.4 12089.4 12089.4 12089.4 12089.4 800 nodes $\\lambda _\\mathrm {min}(S)$ $-2 \\cdot 10^{-10}$ $-8 \\cdot 10^{-12}$ $-5 \\cdot 10^{-6}$ $-3 \\cdot 10^{-5}$ $-7 \\cdot 10^{-7}$ 19176 edges Time [s] 1.6 3.1 7.8 2.0 33.7 Graph 3 Cut bound 12084.3 12084.3 12085.5 12084.3 12084.3 800 nodes $\\lambda _\\mathrm {min}(S)$ $-3 \\cdot 10^{-11}$ $-1 \\cdot 10^{-11}$ $-6 \\cdot 10^{-3}$ $-4 \\cdot 10^{-5}$ $-2 \\cdot 10^{-6}$ 19176 edges Time [s] 2.1 4.5 9.8 2.0 34.0 Graph 4 Cut bound 12111.5 12111.5 12111.5 12111.5 12111.5 800 nodes $\\lambda _\\mathrm {min}(S)$ $-2 \\cdot 10^{-11}$ $-2 \\cdot 10^{-10}$ $-1 \\cdot 10^{-5}$ $-3 \\cdot 10^{-5}$ $-6 \\cdot 10^{-6}$ 19176 edges Time [s] 1.8 3.2 10.6 2.2 33.7 Graph 5 Cut bound 12099.9 12099.9 12099.9 12099.9 12099.9 800 nodes $\\lambda _\\mathrm {min}(S)$ $-3 \\cdot 10^{-12}$ $-8 \\cdot 10^{-12}$ $-1 \\cdot 10^{-5}$ $-3 \\cdot 10^{-5}$ $-1 \\cdot 10^{-6}$ 19176 edges Time [s] 1.5 2.5 6.7 2.2 33.7 Graph 6 Cut bound 2656.2 2656.2 2660.8 2656.2 2656.2 800 nodes $\\lambda _\\mathrm {min}(S)$ $-4 \\cdot 10^{-12}$ $-8 \\cdot 10^{-12}$ $-2 \\cdot 10^{-2}$ $-7 \\cdot 10^{-6}$ $-9 \\cdot 10^{-6}$ 19176 edges Time [s] 1.4 2.6 5.5 2.4 34.1 Graph 7 Cut bound 2489.3 2489.3 2489.3 2489.3 2489.3 800 nodes $\\lambda _\\mathrm {min}(S)$ $-2 \\cdot 10^{-11}$ $-2 \\cdot 10^{-11}$ $-1 \\cdot 10^{-5}$ $-9 \\cdot 10^{-6}$ $-4 \\cdot 10^{-7}$ 19176 edges Time [s] 6.4 2.6 5.9 2.0 35.7 Graph 8 Cut bound 2506.9 2506.9 2506.9 2506.9 2506.9 800 nodes $\\lambda _\\mathrm {min}(S)$ $-5 \\cdot 10^{-12}$ $-9 \\cdot 10^{-12}$ $-4 \\cdot 10^{-5}$ $-1 \\cdot 10^{-5}$ $-1 \\cdot 10^{-6}$ 19176 edges Time [s] 1.2 1.8 10.6 2.2 34.0 Graph 9 Cut bound 2528.7 2528.7 2528.7 2528.7 2528.7 800 nodes $\\lambda _\\mathrm {min}(S)$ $-1 \\cdot 10^{-9}$ $-8 \\cdot 10^{-12}$ $-8 \\cdot 10^{-6}$ $-1 \\cdot 10^{-5}$ $-1 \\cdot 10^{-6}$ 19176 edges Time [s] 0.9 1.8 5.7 2.4 34.8 Graph 10 Cut bound 2485.1 2485.1 2485.1 2485.1 2485.1 800 nodes $\\lambda _\\mathrm {min}(S)$ $-5 \\cdot 10^{-11}$ $-8 \\cdot 10^{-12}$ $-6 \\cdot 10^{-6}$ $-8 \\cdot 10^{-6}$ $-2 \\cdot 10^{-6}$ 19176 edges Time [s] 1.2 1.6 5.3 2.1 33.9 Graph 11 Cut bound 629.2 629.2 629.2 629.2 629.2 800 nodes $\\lambda _\\mathrm {min}(S)$ $-3 \\cdot 10^{-9}$ $-7 \\cdot 10^{-12}$ $-5 \\cdot 10^{-6}$ $-1 \\cdot 10^{-6}$ $-4 \\cdot 10^{-8}$ 1600 edges Time [s] 13.6 13.6 3.9 2.0 31.5 Graph 12 Cut bound 623.9 623.9 623.9 623.9 623.9 800 nodes $\\lambda _\\mathrm {min}(S)$ $-1 \\cdot 10^{-10}$ $-4 \\cdot 10^{-12}$ $-3 \\cdot 10^{-6}$ $-3 \\cdot 10^{-6}$ $-9 \\cdot 10^{-8}$ 1600 edges Time [s] 8.8 7.3 1.9 2.0 31.7 Graph 13 Cut bound 647.1 647.1 647.1 647.1 647.1 800 nodes $\\lambda _\\mathrm {min}(S)$ $-1 \\cdot 10^{-9}$ $-2 \\cdot 10^{-12}$ $-2 \\cdot 10^{-6}$ $-2 \\cdot 10^{-6}$ $-1 \\cdot 10^{-7}$ 1600 edges Time [s] 6.9 6.7 1.3 2.2 31.4 Graph 14 Cut bound 3191.6 3191.6 3191.6 3191.6 3191.6 800 nodes $\\lambda _\\mathrm {min}(S)$ $-1 \\cdot 10^{-10}$ $-3 \\cdot 10^{-12}$ $-3 \\cdot 10^{-5}$ $-3 \\cdot 10^{-5}$ $-1 \\cdot 10^{-6}$ 4694 edges Time [s] 1.5 5.3 4.4 2.5 34.1 Graph 15 Cut bound 3171.6 3171.6 3171.6 3171.6 3171.6 800 nodes $\\lambda _\\mathrm {min}(S)$ $-1 \\cdot 10^{-10}$ $-5 \\cdot 10^{-12}$ $-6 \\cdot 10^{-6}$ $-5 \\cdot 10^{-6}$ $-3 \\cdot 10^{-7}$ 4661 edges Time [s] 3.4 6.5 5.4 3.2 34.6 Graph 16 Cut bound 3175.0 3175.0 3175.1 3175.0 3175.0 800 nodes $\\lambda _\\mathrm {min}(S)$ $-9 \\cdot 10^{-12}$ $-2 \\cdot 10^{-12}$ $-6 \\cdot 10^{-4}$ $-1 \\cdot 10^{-5}$ $-6 \\cdot 10^{-7}$ 4672 edges Time [s] 6.6 6.2 3.8 3.1 34.8 Graph 17 Cut bound 3171.3 3171.3 3171.5 3171.3 3171.3 800 nodes $\\lambda _\\mathrm {min}(S)$ $-5 \\cdot 10^{-12}$ $-2 \\cdot 10^{-12}$ $-1 \\cdot 10^{-3}$ $-1 \\cdot 10^{-5}$ $-1 \\cdot 10^{-7}$ 4667 edges Time [s] 6.1 6.3 3.5 2.9 34.5 Graph 18 Cut bound 1166.0 1166.0 1166.0 1166.0 1166.0 800 nodes $\\lambda _\\mathrm {min}(S)$ $-4 \\cdot 10^{-12}$ $-3 \\cdot 10^{-12}$ $-3 \\cdot 10^{-6}$ $-4 \\cdot 10^{-6}$ $-1 \\cdot 10^{-6}$ 4694 edges Time [s] 1.8 2.9 4.2 3.2 35.1 Graph 19 Cut bound 1082.0 1082.0 1082.0 1082.0 1082.0 800 nodes $\\lambda _\\mathrm {min}(S)$ $-4 \\cdot 10^{-10}$ $-4 \\cdot 10^{-12}$ $-4 \\cdot 10^{-6}$ $-3 \\cdot 10^{-6}$ $-8 \\cdot 10^{-7}$ 4661 edges Time [s] 1.9 2.8 4.3 3.4 34.5 Graph 20 Cut bound 1111.4 1111.4 1112.1 1111.4 1111.4 800 nodes $\\lambda _\\mathrm {min}(S)$ $-2 \\cdot 10^{-12}$ $-3 \\cdot 10^{-12}$ $-3 \\cdot 10^{-3}$ $-4 \\cdot 10^{-6}$ $-2 \\cdot 10^{-6}$ 4672 edges Time [s] 2.8 3.7 2.9 3.6 34.1 Graph 21 Cut bound 1104.3 1104.3 1104.3 1104.3 1104.3 800 nodes $\\lambda _\\mathrm {min}(S)$ $-2 \\cdot 10^{-11}$ $-6 \\cdot 10^{-12}$ $-4 \\cdot 10^{-6}$ $-2 \\cdot 10^{-6}$ $-6 \\cdot 10^{-6}$ 4667 edges Time [s] 2.7 4.3 3.5 3.7 34.1 Graph 22 Cut bound 14135.9 14135.9 14136.0 14135.9 14137.2 2000 nodes $\\lambda _\\mathrm {min}(S)$ $-8 \\cdot 10^{-12}$ $-8 \\cdot 10^{-12}$ $-3 \\cdot 10^{-5}$ $-3 \\cdot 10^{-5}$ $-2 \\cdot 10^{-3}$ 19990 edges Time [s] 5.5 4.9 22.5 25.7 177.7 Graph 23 Cut bound 14142.1 14142.1 14142.1 14142.1 14143.5 2000 nodes $\\lambda _\\mathrm {min}(S)$ $-2 \\cdot 10^{-11}$ $-3 \\cdot 10^{-11}$ $-8 \\cdot 10^{-6}$ $-3 \\cdot 10^{-5}$ $-3 \\cdot 10^{-3}$ 19990 edges Time [s] 7.0 9.1 16.3 23.8 182.8 Graph 24 Cut bound 14140.9 14140.9 14140.9 14140.9 14142.1 2000 nodes $\\lambda _\\mathrm {min}(S)$ $-1 \\cdot 10^{-11}$ $-7 \\cdot 10^{-12}$ $-1 \\cdot 10^{-5}$ $-2 \\cdot 10^{-5}$ $-2 \\cdot 10^{-3}$ 19990 edges Time [s] 4.5 5.7 24.3 24.8 173.3 Graph 25 Cut bound 14144.2 14144.2 14148.8 14144.2 14145.8 2000 nodes $\\lambda _\\mathrm {min}(S)$ $-1 \\cdot 10^{-9}$ $-9 \\cdot 10^{-12}$ $-9 \\cdot 10^{-3}$ $-9 \\cdot 10^{-6}$ $-3 \\cdot 10^{-3}$ 19990 edges Time [s] 4.8 18.1 16.7 23.8 175.0 Graph 26 Cut bound 14132.9 14132.9 14132.9 14132.9 14134.2 2000 nodes $\\lambda _\\mathrm {min}(S)$ $-7 \\cdot 10^{-12}$ $-1 \\cdot 10^{-11}$ $-4 \\cdot 10^{-6}$ $-2 \\cdot 10^{-5}$ $-3 \\cdot 10^{-3}$ 19990 edges Time [s] 6.8 6.5 14.4 23.1 177.6 Graph 27 Cut bound 4141.7 4141.7 4145.0 4141.7 4143.1 2000 nodes $\\lambda _\\mathrm {min}(S)$ $-1 \\cdot 10^{-11}$ $-7 \\cdot 10^{-12}$ $-7 \\cdot 10^{-3}$ $-9 \\cdot 10^{-6}$ $-3 \\cdot 10^{-3}$ 19990 edges Time [s] 3.7 4.4 10.8 23.5 175.9 Graph 28 Cut bound 4100.8 4100.8 4100.8 4100.8 4102.2 2000 nodes $\\lambda _\\mathrm {min}(S)$ $-2 \\cdot 10^{-9}$ $-6 \\cdot 10^{-12}$ $-3 \\cdot 10^{-5}$ $-7 \\cdot 10^{-6}$ $-3 \\cdot 10^{-3}$ 19990 edges Time [s] 3.0 8.0 19.6 26.5 176.8 Graph 29 Cut bound 4208.9 4208.9 4208.9 4208.9 4210.0 2000 nodes $\\lambda _\\mathrm {min}(S)$ $-2 \\cdot 10^{-11}$ $-2 \\cdot 10^{-11}$ $-5 \\cdot 10^{-6}$ $-2 \\cdot 10^{-6}$ $-2 \\cdot 10^{-3}$ 19990 edges Time [s] 12.2 8.3 17.7 24.5 180.6 Graph 30 Cut bound 4215.4 4215.4 4215.4 4215.4 4216.6 2000 nodes $\\lambda _\\mathrm {min}(S)$ $-7 \\cdot 10^{-11}$ $-6 \\cdot 10^{-12}$ $-5 \\cdot 10^{-6}$ $-6 \\cdot 10^{-6}$ $-2 \\cdot 10^{-3}$ 19990 edges Time [s] 19.8 10.5 11.6 25.2 176.7 Graph 31 Cut bound 4116.7 4116.7 4119.1 4116.7 4118.0 2000 nodes $\\lambda _\\mathrm {min}(S)$ $-2 \\cdot 10^{-11}$ $-5 \\cdot 10^{-12}$ $-5 \\cdot 10^{-3}$ $-7 \\cdot 10^{-6}$ $-3 \\cdot 10^{-3}$ 19990 edges Time [s] 4.1 8.9 16.2 26.2 170.6 Graph 32 Cut bound 1567.6 1567.6 1567.6 1567.6 1567.8 2000 nodes $\\lambda _\\mathrm {min}(S)$ $-2 \\cdot 10^{-10}$ $-8 \\cdot 10^{-12}$ $-1 \\cdot 10^{-6}$ $-1 \\cdot 10^{-6}$ $-3 \\cdot 10^{-4}$ 4000 edges Time [s] 45.6 25.4 13.9 21.7 142.6 Graph 33 Cut bound 1544.3 1544.3 1544.3 1544.3 1544.4 2000 nodes $\\lambda _\\mathrm {min}(S)$ $-7 \\cdot 10^{-10}$ $-5 \\cdot 10^{-12}$ $-1 \\cdot 10^{-6}$ $-9 \\cdot 10^{-7}$ $-1 \\cdot 10^{-4}$ 4000 edges Time [s] 31.2 17.3 9.9 23.0 141.2 Graph 34 Cut bound 1546.7 1546.7 1546.7 1546.7 1546.8 2000 nodes $\\lambda _\\mathrm {min}(S)$ $-1 \\cdot 10^{-9}$ $-5 \\cdot 10^{-12}$ $-2 \\cdot 10^{-6}$ $-1 \\cdot 10^{-6}$ $-2 \\cdot 10^{-4}$ 4000 edges Time [s] 31.3 22.0 7.7 23.6 143.9 Graph 35 Cut bound 8014.7 8014.7 8014.7 8014.7 8015.3 2000 nodes $\\lambda _\\mathrm {min}(S)$ $-1 \\cdot 10^{-9}$ $-4 \\cdot 10^{-11}$ $-5 \\cdot 10^{-6}$ $-9 \\cdot 10^{-6}$ $-1 \\cdot 10^{-3}$ 11778 edges Time [s] 19.4 17.4 26.0 34.5 187.7 Graph 36 Cut bound 8006.0 8006.0 8006.0 8006.0 8006.6 2000 nodes $\\lambda _\\mathrm {min}(S)$ $-9 \\cdot 10^{-10}$ $-3 \\cdot 10^{-11}$ $-1 \\cdot 10^{-5}$ $-2 \\cdot 10^{-5}$ $-1 \\cdot 10^{-3}$ 11766 edges Time [s] 12.0 36.9 41.1 37.0 193.3 Graph 37 Cut bound 8018.6 8018.6 8019.4 8018.6 8019.5 2000 nodes $\\lambda _\\mathrm {min}(S)$ $-2 \\cdot 10^{-10}$ $-1 \\cdot 10^{-11}$ $-1 \\cdot 10^{-3}$ $-1 \\cdot 10^{-5}$ $-2 \\cdot 10^{-3}$ 11785 edges Time [s] 11.2 15.4 38.4 35.2 191.1 Graph 38 Cut bound 8015.0 8015.0 8015.0 8015.0 8015.5 2000 nodes $\\lambda _\\mathrm {min}(S)$ $-1 \\cdot 10^{-10}$ $-1 \\cdot 10^{-11}$ $-2 \\cdot 10^{-5}$ $-1 \\cdot 10^{-5}$ $-1 \\cdot 10^{-3}$ 11779 edges Time [s] 13.1 14.2 44.7 37.5 193.0 Graph 39 Cut bound 2877.6 2877.6 2877.8 2877.6 2878.4 2000 nodes $\\lambda _\\mathrm {min}(S)$ $-4 \\cdot 10^{-9}$ $-7 \\cdot 10^{-12}$ $-3 \\cdot 10^{-4}$ $-4 \\cdot 10^{-6}$ $-2 \\cdot 10^{-3}$ 11778 edges Time [s] 16.9 12.2 31.9 39.3 195.8 Graph 40 Cut bound 2864.8 2864.8 2866.2 2864.8 2865.6 2000 nodes $\\lambda _\\mathrm {min}(S)$ $-1 \\cdot 10^{-11}$ $-2 \\cdot 10^{-11}$ $-3 \\cdot 10^{-3}$ $-3 \\cdot 10^{-6}$ $-2 \\cdot 10^{-3}$ 11766 edges Time [s] 9.2 9.4 40.8 40.9 189.0 Graph 41 Cut bound 2865.2 2865.2 2868.1 2865.2 2865.8 2000 nodes $\\lambda _\\mathrm {min}(S)$ $-4 \\cdot 10^{-10}$ $-1 \\cdot 10^{-11}$ $-6 \\cdot 10^{-3}$ $-4 \\cdot 10^{-6}$ $-1 \\cdot 10^{-3}$ 11785 edges Time [s] 5.3 8.6 30.8 40.9 189.8 Graph 42 Cut bound 2946.3 2946.3 2948.3 2946.3 2947.0 2000 nodes $\\lambda _\\mathrm {min}(S)$ $-9 \\cdot 10^{-12}$ $-7 \\cdot 10^{-12}$ $-4 \\cdot 10^{-3}$ $-6 \\cdot 10^{-6}$ $-1 \\cdot 10^{-3}$ 11779 edges Time [s] 7.9 8.1 32.9 41.8 188.4 Graph 43 Cut bound 7032.2 7032.2 7032.2 7032.2 7033.2 1000 nodes $\\lambda _\\mathrm {min}(S)$ $-3 \\cdot 10^{-12}$ $-4 \\cdot 10^{-12}$ $-6 \\cdot 10^{-6}$ $-2 \\cdot 10^{-5}$ $-4 \\cdot 10^{-3}$ 9990 edges Time [s] 1.9 2.3 3.6 3.8 36.4 Graph 44 Cut bound 7027.9 7027.9 7029.2 7027.9 7029.4 1000 nodes $\\lambda _\\mathrm {min}(S)$ $-1 \\cdot 10^{-8}$ $-3 \\cdot 10^{-12}$ $-5 \\cdot 10^{-3}$ $-2 \\cdot 10^{-5}$ $-6 \\cdot 10^{-3}$ 9990 edges Time [s] 2.9 3.9 3.7 3.6 38.0 Graph 45 Cut bound 7024.8 7024.8 7024.8 7024.8 7025.9 1000 nodes $\\lambda _\\mathrm {min}(S)$ $-1 \\cdot 10^{-9}$ $-5 \\cdot 10^{-12}$ $-2 \\cdot 10^{-5}$ $-8 \\cdot 10^{-6}$ $-5 \\cdot 10^{-3}$ 9990 edges Time [s] 1.3 6.1 4.9 3.5 37.4 Graph 46 Cut bound 7029.9 7029.9 7029.9 7029.9 7030.8 1000 nodes $\\lambda _\\mathrm {min}(S)$ $-2 \\cdot 10^{-10}$ $-3 \\cdot 10^{-12}$ $-2 \\cdot 10^{-5}$ $-1 \\cdot 10^{-5}$ $-4 \\cdot 10^{-3}$ 9990 edges Time [s] 12.9 2.3 3.1 3.7 38.3 Graph 47 Cut bound 7036.7 7036.7 7036.7 7036.7 7037.8 1000 nodes $\\lambda _\\mathrm {min}(S)$ $-8 \\cdot 10^{-10}$ $-9 \\cdot 10^{-12}$ $-1 \\cdot 10^{-5}$ $-1 \\cdot 10^{-5}$ $-5 \\cdot 10^{-3}$ 9990 edges Time [s] 10.4 4.1 8.2 3.8 39.2 Graph 48 Cut bound 6000.0 6000.0 6000.0 6000.0 6000.0 3000 nodes $\\lambda _\\mathrm {min}(S)$ $4 \\cdot 10^{-16}$ $3 \\cdot 10^{-16}$ $-6 \\cdot 10^{-10}$ $-3 \\cdot 10^{-6}$ $5 \\cdot 10^{-18}$ 6000 edges Time [s] 2.8 4.3 3.5 47.7 307.3 Graph 49 Cut bound 6000.0 6000.0 6000.0 6000.0 6000.0 3000 nodes $\\lambda _\\mathrm {min}(S)$ $4 \\cdot 10^{-16}$ $4 \\cdot 10^{-16}$ $-1 \\cdot 10^{-9}$ $-3 \\cdot 10^{-6}$ $-4 \\cdot 10^{-16}$ 6000 edges Time [s] 3.9 5.1 4.9 46.1 299.7 Graph 50 Cut bound 5988.2 5988.2 5988.2 5988.2 5988.2 3000 nodes $\\lambda _\\mathrm {min}(S)$ $-2 \\cdot 10^{-12}$ $-1 \\cdot 10^{-14}$ $-1 \\cdot 10^{-7}$ $-3 \\cdot 10^{-6}$ $2 \\cdot 10^{-16}$ 6000 edges Time [s] 6.0 5.0 5.4 45.7 318.4 Graph 51 Cut bound 4006.3 4006.3 4006.3 4006.3 4006.9 1000 nodes $\\lambda _\\mathrm {min}(S)$ $-2 \\cdot 10^{-9}$ $-4 \\cdot 10^{-12}$ $-8 \\cdot 10^{-6}$ $-1 \\cdot 10^{-5}$ $-3 \\cdot 10^{-3}$ 5909 edges Time [s] 5.8 7.8 10.7 5.4 41.4 Graph 52 Cut bound 4009.6 4009.6 4010.0 4009.6 4010.2 1000 nodes $\\lambda _\\mathrm {min}(S)$ $-4 \\cdot 10^{-12}$ $-9 \\cdot 10^{-12}$ $-1 \\cdot 10^{-3}$ $-5 \\cdot 10^{-6}$ $-2 \\cdot 10^{-3}$ 5916 edges Time [s] 6.4 8.8 6.5 5.2 39.6 Graph 53 Cut bound 4009.7 4009.7 4009.7 4009.7 4010.5 1000 nodes $\\lambda _\\mathrm {min}(S)$ $-1 \\cdot 10^{-10}$ $-1 \\cdot 10^{-11}$ $-6 \\cdot 10^{-6}$ $-1 \\cdot 10^{-5}$ $-3 \\cdot 10^{-3}$ 5914 edges Time [s] 4.2 8.5 8.3 5.0 39.1 Graph 54 Cut bound 4006.2 4006.2 4006.2 4006.2 4006.9 1000 nodes $\\lambda _\\mathrm {min}(S)$ $-2 \\cdot 10^{-10}$ $-3 \\cdot 10^{-12}$ $-3 \\cdot 10^{-5}$ $-5 \\cdot 10^{-6}$ $-3 \\cdot 10^{-3}$ 5916 edges Time [s] 2.9 6.6 6.1 4.8 39.1 Graph 55 Cut bound 11039.5 11039.5 11039.5 11039.5 11039.7 5000 nodes $\\lambda _\\mathrm {min}(S)$ $-2 \\cdot 10^{-12}$ $-3 \\cdot 10^{-12}$ $-5 \\cdot 10^{-6}$ $-6 \\cdot 10^{-6}$ $-2 \\cdot 10^{-4}$ 12498 edges Time [s] 26.6 20.6 22.2 411.4 1588.0 Graph 56 Cut bound 4760.0 4760.0 4760.0 4760.0 4760.3 5000 nodes $\\lambda _\\mathrm {min}(S)$ $-7 \\cdot 10^{-12}$ $-2 \\cdot 10^{-12}$ $-1 \\cdot 10^{-5}$ $-2 \\cdot 10^{-6}$ $-3 \\cdot 10^{-4}$ 12498 edges Time [s] 20.1 16.3 32.9 475.9 1550.1 Graph 57 Cut bound 3885.5 3885.5 3885.5 3885.5 3885.7 5000 nodes $\\lambda _\\mathrm {min}(S)$ $-1 \\cdot 10^{-9}$ $-8 \\cdot 10^{-12}$ $-2 \\cdot 10^{-6}$ $-2 \\cdot 10^{-6}$ $-1 \\cdot 10^{-4}$ 10000 edges Time [s] 218.0 78.8 38.3 269.8 1012.4 Graph 58 Cut bound 20136.2 20136.2 20138.1 20136.2 20136.7 5000 nodes $\\lambda _\\mathrm {min}(S)$ $-3 \\cdot 10^{-9}$ $-5 \\cdot 10^{-11}$ $-2 \\cdot 10^{-3}$ $-7 \\cdot 10^{-6}$ $-4 \\cdot 10^{-4}$ 29570 edges Time [s] 55.4 44.0 321.5 497.9 1865.7 Graph 59 Cut bound 7312.3 7312.3 7315.0 7312.3 7313.0 5000 nodes $\\lambda _\\mathrm {min}(S)$ $-7 \\cdot 10^{-12}$ $-3 \\cdot 10^{-11}$ $-2 \\cdot 10^{-3}$ $-4 \\cdot 10^{-6}$ $-5 \\cdot 10^{-4}$ 29570 edges Time [s] 51.3 35.6 353.1 511.3 1869.0 Graph 60 Cut bound 15222.3 15222.3 15222.3 15222.3 15222.6 7000 nodes $\\lambda _\\mathrm {min}(S)$ $-3 \\cdot 10^{-11}$ $-4 \\cdot 10^{-12}$ $-2 \\cdot 10^{-5}$ $-2 \\cdot 10^{-6}$ $-2 \\cdot 10^{-4}$ 17148 edges Time [s] 58.6 30.9 63.6 1326.9 3581.9 Graph 61 Cut bound 6828.1 6828.1 6828.2 6828.1 6828.4 7000 nodes $\\lambda _\\mathrm {min}(S)$ $-2 \\cdot 10^{-11}$ $-4 \\cdot 10^{-12}$ $-7 \\cdot 10^{-5}$ $-2 \\cdot 10^{-6}$ $-2 \\cdot 10^{-4}$ 17148 edges Time [s] 113.4 40.2 55.8 1263.3 3795.6 Graph 62 Cut bound 5430.9 5430.9 5430.9 5430.9 5431.1 7000 nodes $\\lambda _\\mathrm {min}(S)$ $-1 \\cdot 10^{-9}$ $-6 \\cdot 10^{-11}$ $-9 \\cdot 10^{-7}$ $-2 \\cdot 10^{-6}$ $-1 \\cdot 10^{-4}$ 14000 edges Time [s] 813.8 242.8 110.8 862.4 2124.3 Graph 63 Cut bound 28244.4 28244.4 28245.9 28244.4 28245.0 7000 nodes $\\lambda _\\mathrm {min}(S)$ $-7 \\cdot 10^{-9}$ $-8 \\cdot 10^{-9}$ $-8 \\cdot 10^{-4}$ $-9 \\cdot 10^{-6}$ $-3 \\cdot 10^{-4}$ 41459 edges Time [s] 238.9 97.6 663.0 1454.7 4583.9 Graph 64 Cut bound 10465.9 10465.9 10466.6 10465.9 10466.6 7000 nodes $\\lambda _\\mathrm {min}(S)$ $-3 \\cdot 10^{-9}$ $-2 \\cdot 10^{-11}$ $-4 \\cdot 10^{-4}$ $-5 \\cdot 10^{-6}$ $-4 \\cdot 10^{-4}$ 41459 edges Time [s] 140.4 109.5 1014.8 1609.4 4439.8 Graph 65 Cut bound 6205.5 6205.5 6205.5 6205.5 6205.7 8000 nodes $\\lambda _\\mathrm {min}(S)$ $-1 \\cdot 10^{-9}$ $-1 \\cdot 10^{-11}$ $-6 \\cdot 10^{-7}$ $-1 \\cdot 10^{-6}$ $-1 \\cdot 10^{-4}$ 16000 edges Time [s] 567.2 168.5 154.4 1075.2 2861.5 Graph 66 Cut bound 7077.2 7077.2 7077.2 7077.2 7077.4 9000 nodes $\\lambda _\\mathrm {min}(S)$ $-2 \\cdot 10^{-9}$ $-5 \\cdot 10^{-11}$ $-2 \\cdot 10^{-7}$ $-9 \\cdot 10^{-7}$ $-6 \\cdot 10^{-5}$ 18000 edges Time [s] 762.6 215.3 218.1 1525.7 3915.7 Graph 67 Cut bound 7744.4 7744.4 7744.4 7744.4 - 10000 nodes $\\lambda _\\mathrm {min}(S)$ $-1 \\cdot 10^{-9}$ $-3 \\cdot 10^{-11}$ $-3 \\cdot 10^{-7}$ $-1 \\cdot 10^{-6}$ - 20000 edges Time [s] 816.4 339.0 267.3 2005.4 - Graph 70 Cut bound 9861.5 9861.5 9861.5 9861.5 - 10000 nodes $\\lambda _\\mathrm {min}(S)$ $-2 \\cdot 10^{-10}$ $-6 \\cdot 10^{-13}$ $-2 \\cdot 10^{-6}$ $-2 \\cdot 10^{-6}$ - 9999 edges Time [s] 143.3 82.9 102.2 3167.3 - Graph 72 Cut bound 7808.5 7808.5 7808.5 7808.5 - 10000 nodes $\\lambda _\\mathrm {min}(S)$ $-6 \\cdot 10^{-10}$ $-8 \\cdot 10^{-12}$ $-8 \\cdot 10^{-7}$ $-1 \\cdot 10^{-6}$ - 20000 edges Time [s] 720.8 262.6 199.0 1902.7 - Graph 77 Cut bound 11045.7 11045.7 11045.7 11045.7 - 14000 nodes $\\lambda _\\mathrm {min}(S)$ $-8 \\cdot 10^{-10}$ $-4 \\cdot 10^{-11}$ $-7 \\cdot 10^{-7}$ $-1 \\cdot 10^{-6}$ - 28000 edges Time [s] 1578.5 513.0 515.1 5249.1 - Graph 81 Cut bound 15656.2 15656.2 15656.2 15656.2 - 20000 nodes $\\lambda _\\mathrm {min}(S)$ $-5 \\cdot 10^{-10}$ $-6 \\cdot 10^{-11}$ $-1 \\cdot 10^{-6}$ $-3 \\cdot 10^{-6}$ - 40000 edges Time [s] 4152.8 1539.7 1035.6 16576.6 -"
],
[
"Regularity assumption",
"Originally, Theorems REF and REF had the assumption that the search space of the factorized problem, $\\mathcal {M}& = \\lbrace Y \\in {\\mathbb {R}^{n\\times p}}: \\mathcal {A}(YY^\\top \\! )",
"= b \\rbrace ,$ is a manifold.",
"From this assumption, we stated incorrectly that the tangent space at $Y$ of $\\mathcal {M}$ , denoted by $Y\\mathcal {M}$ , is given by (REF ): $Y\\mathcal {M}& = \\lbrace \\dot{Y} \\in {\\mathbb {R}^{n\\times p}}: \\mathcal {A}(\\dot{Y} Y^\\top \\!",
"+ Y \\dot{Y}^\\top \\! )",
"= 0 \\rbrace .$ This identity is used in a number of places of the proofs.",
"In general, $\\mathcal {M}$ being an embedded submanifold of ${\\mathbb {R}^{n\\times p}}$ only implies the left hand side is included in the right hand side.If $\\dot{Y} \\in Y\\mathcal {M}$ , by definition, there exists a smooth curve $\\gamma \\colon {\\mathbb {R}}\\rightarrow \\mathcal {M}$ such that $\\gamma (0) = Y$ and $\\gamma ^{\\prime }(0) = \\dot{Y}$ .",
"Since $\\gamma (t) \\in \\mathcal {M}$ for all $t$ , we have $\\mathcal {A}(\\gamma (t)\\gamma (t)^\\top \\! )",
"= b$ for all $t$ .",
"Differentiating on both sides with respect to $t$ and evaluating at 0 gives $\\mathcal {A}(\\dot{Y}Y^\\top \\!",
"+ Y\\dot{Y}^\\top \\! )",
"= 0$ .",
"Below, we give an example where $\\mathcal {M}$ is a manifold yet the two sets are not equal.",
"In order to restore equality, we strengthened the assumption, requiring constraint qualifications to hold at all feasible points (see (REF )): $\\forall Y \\in \\mathcal {M}, \\quad A_1Y, \\ldots , A_mY \\textrm { are linearly independent in } {\\mathbb {R}^{n\\times p}},$ where $A_i$ , $i = 1, \\ldots , m$ , are the symmetric constraint matrices such that $\\mathcal {A}(X)_i = \\left\\langle {A_i},{X}\\right\\rangle $ .",
"This ensures the map $\\Phi (Y) = \\mathcal {A}(YY^\\top \\! )",
"- b$ is full rank on $\\mathcal {M}= \\Phi ^{-1}(0)$ , from which it follows by a standard result in differential geometry (see for example [24]) that $\\mathcal {M}$ is a smooth embedded submanifold of ${\\mathbb {R}^{n\\times p}}$ of dimension $np - m$ .",
"Then, the left hand side of (REF ) has dimension $np - m$ , and it is included in the right hand side, which itself is a linear space of dimension $np - m$ , so that they are equal.",
"We now describe an SDP such that $\\mathcal {M}$ is indeed a manifold, yet (REF ) does not hold.",
"Consider $n = 2, m = 2$ , $b = (1, 1)^\\top \\!",
"$ and $A_1 & = \\begin{pmatrix}1 & 0 \\\\ 0 & 1\\end{pmatrix}, & A_2 & = \\begin{pmatrix}1 & 0 \\\\ 0 & \\frac{1}{4}\\end{pmatrix}.$ The search space of the SDP, $\\mathcal {C}= \\lbrace X = X^\\top \\!",
"\\in {\\mathbb {R}}^{n\\times n} : \\mathcal {A}(X) = b, X \\succeq 0\\rbrace = \\left\\lbrace \\begin{pmatrix}1 & 0 \\\\ 0 & 0\\end{pmatrix} \\right\\rbrace ,$ is degenerate but it is compact.",
"Furthermore, the set $\\mathcal {M}$ is a smooth manifold for $p = 1$ : $\\mathcal {M}_{p=1} & = \\left\\lbrace Y = \\begin{pmatrix}y_1 \\\\ y_2\\end{pmatrix} \\in {\\mathbb {R}}^2 : y_1^2 + y_2^2 = 1 \\textrm { and } y_1^2 + \\frac{1}{4}y_2^2 = 1 \\right\\rbrace = \\lbrace (1, 0)^\\top \\!",
", (-1, 0)^\\top \\!",
"\\rbrace .$ The dimension of the manifold is 0, so that $Y\\mathcal {M}= \\lbrace 0\\rbrace $ for all $Y\\in \\mathcal {M}$ .",
"Consider now the right hand side of (REF ), $K_Y & = \\lbrace \\dot{Y} \\in {\\mathbb {R}^{n\\times p}}: \\mathcal {A}(\\dot{Y} Y^\\top \\!",
"+ Y \\dot{Y}^\\top \\! )",
"= 0 \\rbrace = \\lbrace \\dot{Y} \\in {\\mathbb {R}^{n\\times p}}: \\langle {A_1Y},{\\dot{Y}}\\rangle = \\langle {A_2Y},{\\dot{Y}}\\rangle = 0 \\rbrace .$ These are the vectors orthogonal to $A_1Y, A_2Y$ .",
"For $Y = (\\pm 1, 0)^\\top \\!",
"$ , we get $A_1Y = A_2Y = (\\pm 1, 0)^\\top \\!",
"$ : they are colinear, so $K_Y$ has dimension 1 at all $Y \\in \\mathcal {M}$ : $Y\\mathcal {M}\\ne K_Y$ .",
"Similarly, at $p = 2$ , the set $\\mathcal {M}$ becomes a circle embedded in ${\\mathbb {R}}^4$ : $\\mathcal {M}_{p=2} & = \\left\\lbrace Y = \\begin{pmatrix}y_1 & y_2 \\\\ y_3 & y_4\\end{pmatrix} \\in {\\mathbb {R}}^{2\\times 2} : y_1^2 + y_2^2 + y_3^2 + y_4^2 = 1 \\textrm { and } y_1^2 + y_2^2 + \\frac{1}{4}(y_3^2 + y_4^2) = 1 \\right\\rbrace \\\\& = \\left\\lbrace Y = \\begin{pmatrix}y_1 & y_2 \\\\ y_3 & y_4\\end{pmatrix} \\in {\\mathbb {R}}^{2\\times 2} : y_1^2 + y_2^2 = 1 \\textrm { and } y_3 = y_4 = 0 \\right\\rbrace .$ This manifold has dimension 1 (and so do all its tangent spaces).",
"Yet, $K_Y$ has dimension 3 for all $Y \\in \\mathcal {M}$ .",
"Indeed, we can parameterize $\\mathcal {M}_{p=2}$ as the matrices $\\begin{pmatrix}\\cos \\theta & \\sin \\theta \\\\ 0 & 0\\end{pmatrix}$ for all $\\theta \\in {\\mathbb {R}}$ .",
"It is easy to verify that $A_1Y = A_2Y \\ne 0$ for all $Y\\in \\mathcal {M}_{p=2}$ , so that the codimension of $K_Y$ is 1, here too in disagreement with $Y\\mathcal {M}$ .",
"Notice also that in this example we have $\\frac{p(p+1)}{2} > m$ ."
]
] | 1606.04970 |
[
[
"Notes on heat engines and negative temperatures"
],
[
"Abstract We show that a Carnot cycle operating between a positive canonical-temperature bath and a negative canonical-temperature bath has efficiency equal to unity.",
"It follows that a negative canonical-temperature cannot be identified with an absolute temperature.",
"We illustrate this with a spin in a varying magnetic field."
],
[
"Notes on heat engines and negative temperatures Michele Campisi michele.campisi@sns.it NEST, Scuola Normale Superiore & Istituto Nanoscienze-CNR, I-56126 Pisa, Italy We show that a Carnot cycle operating between a positive canonical-temperature bath and a negative canonical-temperature bath has efficiency equal to unity.",
"It follows that a negative canonical-temperature cannot be identified with an absolute temperature.",
"We illustrate this with a spin in a varying magnetic field.",
"In the context of the issue of negative temperatures it is often claimed that negative temperatures imply Carnot efficiencies larger than one [1], [2].",
"Such claims are based on the use of the Carnot engine formula for the efficiency $\\eta = 1-T_C/T_H$ with a positive $T_C$ and a negative $T_H$ .",
"This procedure is erroneous for a simple reason: In deriving Eq.",
"(REF ) one uses the assumption that $T_C$ and $T_H$ have the same sign.",
"Hence the formula cannot be used with two temperatures of opposite signs.",
"We recall the derivation of Eq.",
"(REF ) for convenience.",
"I should emphasise here that this is a basic thermodynamic derivation that does not refer to any specific statistical ensemble.",
"Consider a Carnot cycle.",
"Consider first the isothermal expansion at $T_H$ .",
"The heat entering the system is $Q_H= T_H \\Delta S_H$ .",
"The adiabatic expansion has $ Q= 0$ , hence $\\Delta S=0$ .",
"With the isothermal compression, the system must go back to the very initial entropy, hence $\\Delta S_C = -\\Delta S_H = Q_C/T_C$ .",
"Therefore $Q_C = - Q_H T_C/T_H$ .",
"Imagine $Q_H>0$ .",
"If $T_C$ and $T_H$ have equal sign, then $Q_C<0$ .",
"So the heat intake $Q_\\text{in}$ is given by $Q_\\text{in}=Q_H$ .",
"Using the first law of thermodynamics (total work output $W$ is given by the total heat balance $Q_\\text{in}+Q_\\text{out}=W$ ) and the efficiency definition $\\eta = W/Q_\\text{in}$ Eq.",
"(REF ) immediately follows.",
"Imagine now instead that $T_C$ and $T_H$ have opposite signs, then heat enters the system in both isotherms [3], hence $Q_\\text{in}=Q_H+Q_C=W$ , and the correct formula is now $\\eta =1 \\quad [\\text{sign}(T_C) \\ne \\text{sign}(T_H)]$ Mis-identification of $\\eta $ with $W/Q_H$ , as in Ref.",
"[1], [2] would lead to the absurd conclusion that $\\eta >1$ .",
"The very concept on an absolute scale of temperatures is constructed upon Eq.",
"(REF ) [4].",
"Since Eq.",
"(REF ) only holds under the provision that both quantities $T_H, T_C$ have same sign (conventionally positive), there apparently is no room for interpreting any negative quantity, e.g.",
"a negative canonical-temperature (see below), as an absolute temperature.",
"We consider a canonical Carnot cycle where two isothermal transformations are alternated by two adiabatic transformations.",
"By “canonical cycle”, we mean that the system is in a canonical Gibbs state at all times during the cycle.",
"The isothermal transformations take place while the system stays in contact with a thermal bath characterised by the given inverse canonical-temperature (c-temperature) $\\beta $ .",
"A thermal bath at inverse c-temperature $\\beta $ is a physical system with the property of leading a system of interest to the state $e^{-\\beta H(\\lambda )}/Z(\\lambda ,\\beta )$ when the two are allowed to interact for a sufficiently long time.",
"$ H(\\lambda )$ is the Hamiltonian of the system of interest, which depends on the work-parameter $\\lambda $ .",
"We leave aside the question if a thermal bath with negative $\\beta $ can exist or can be engineered.",
"Figure: A canonical Carnot cycle of a spin 1/21/2 between baths of opposite canonical temperature sign.",
"The cycle can be decomposed in two cycles each of unit efficiency.",
"The overall efficiency is therefore η=1\\eta =1.A Carnot cycle between opposite c-temperature baths is illustrated in Fig.",
"REF for the case of a single spin $1/2$ with Hamiltonian $H(B) = -B \\sigma ^z/2$ , with $\\sigma _z$ a Pauli operator.",
"The magnetic field $B$ plays the role of work parameter.",
"The system is prepared at c-temperature $T_H<0$ and $B_1>0$ , its state is $\\rho _1\\propto e^{-B_1 \\sigma ^z/k_B T_H}$ .",
"It undergoes then an isothermal transformation of the magnetic field to $B_2>B_1$ at c-temperature $T_H$ , thus reaching the state $\\rho _2\\propto e^{-B_2 \\sigma ^z/k_B T_H}$ .",
"Contact with the thermal reservoir is now removed and the magnetic field is brought to $B_3<0$ .",
"Independent of the speed of the magnetic field reversal, no jumps between the two spin states occur because the spin Hamiltonian commutes with itself at all times.",
"Accordingly the state $\\rho $ remains unvaried and the new positive c-temperature $T_C=T_H B_3/B_2>0$ is reached: $\\rho _3=\\rho _2=\\propto e^{-B_3 \\sigma ^z/k_B T_C}$ .",
"This reversal does not suffer the problems of passage through null c-temperature mentioned in [3], [5], [2].",
"The spin is now brought into contact with the bath at c-temperature $T_C$ and the magnetic field is isothermally switched to $B_4=B_1 T_C/T_H<0$ , so that the state $\\rho _4\\propto e^{-B_4 \\sigma ^z/k_B T_H} = e^{-B_1 \\sigma ^z/k_B T_C}=\\rho _1$ is reached.",
"A switch of the magnetic field back to $B_1$ after thermal contact is removed, closes the Carnot cycle.",
"Using the quantum mechanical formula $U = \\mbox{Tr}\\rho H$ it is straightforward to calculate the internal energy of each state and accordingly the heat exchanged with each bath.",
"One finds, at variance with the ordinary Carnot engine which withdraws energy from the hot bath only, that this engine withdraws heat from both baths $Q_H>0, Q_C>0$ .",
"Hence the heat intake is $Q_\\text{in}= Q_H+ Q_C=W$ and the efficiency is one: $\\eta = 1$ .",
"This result can also be obtained by doing no math by noting that the Cycle is composed of two sub-Carnot-cycles, one operating between c-temperatures 0 and $T_C$ (with efficiency $\\eta =1-0/T_C =1$ ), and the other between $T_H$ and 0 (also with efficiency $1- 0/T_H=1$ [3]).",
"The overall efficiency is therefore 1.",
"This research was supported by a Marie Curie Intra European Fellowship within the 7th European Community Framework Programme through the project NeQuFlux grant n. 623085 and by the COST action MP1209 “Thermodynamics in the quantum regime”."
]
] | 1606.05244 |
[
[
"Mass assembly and morphological transformations since $z\\sim3$ from\n CANDELS"
],
[
"Abstract [abridged] We quantify the evolution of the stellar mass functions of star-forming and quiescent galaxies as a function of morphology from $z\\sim 3$ to the present.",
"Our sample consists of ~50,000 galaxies in the CANDELS fields ($\\sim880$ $arcmin^2$), which we divide into four main morphological types, i.e.",
"pure bulge dominated systems, pure spiral disk dominated, intermediate 2-component bulge+disk systems and irregular disturbed galaxies.",
"Our main results are: Star-formation: At $z\\sim 2$, 80\\% of the stellar mass density of star-forming galaxies is in irregular systems.",
"However, by $z\\sim 0.5$, irregular objects only dominate at stellar masses below $10^9M\\odot$.",
"A majority of the star-forming irregulars present at $z\\sim 2$ undergo a gradual transformation from disturbed to normal spiral disk morphologies by $z\\sim 1$ without significant interruption to their star-formation.",
"Rejuvenation after a quenching event does not seem to be common except perhaps for the most massive objects.",
"Quenching: We confirm that galaxies reaching a stellar mass of $M_*\\sim10^{10.8}M_\\odot$ ($M^*$) tend to quench.",
"Also, quenching implies the presence of a bulge: the abundance of massive red disks is negligible at all redshifts over 2~dex in stellar mass.",
"However the dominant quenching mechanism evolves.",
"At $z>2$, the SMF of quiescent galaxies above $M^*$ is dominated by compact spheroids.",
"Quenching at this early epoch destroys the disk and produces a compact remnant unless the star-forming progenitors at even higher redshifts are significantly more dense.",
"At $1<z<2$, the majority of newly quenched galaxies are disks with a significant central bulge.",
"This suggests that mass-quenching at this epoch starts from the inner parts and preserves the disk.",
"At $z<1$, the high mass end of the passive SMF is globally in place and the evolution mostly happens at stellar masses below $10^{10}M_\\odot$."
],
[
"Introduction",
"Lying at the centers of dark matter potential wells, galaxies are the building blocks of our universe.",
"How they assemble their mass and acquire their morphology are two central open questions today.",
"The answer requires a complete understanding of the complex baryonic physics which dominate at these scales.",
"At first order, however, a galaxy is a system that transforms gas into stars.",
"The life of a galaxy is therefore a balance between processes that trigger star formation by accelerating gas cooling and others which tend to prevent star formation by expelling or heating gas (e.g. [63]).",
"Stellar mass functions (SMFs) are a key first-order observable which allow one to statistically trace back the formation of stars in the universe.",
"Comparison with predicted SMFs constrains the mechanisms which trigger, enhance or inhibit star formation.",
"Deep NIR surveys over large areas undertaken in the last years probe the evolution of the stellar-mass functions from $z\\sim 4$ (e.g.",
"[93], [55], [81]).",
"They have shown that some form of feedback, to avoid the over-formation of stars both at the high mass and low mass ends, is necessary.",
"Another key result is that the abundance of passive galaxies steadily increases from $z\\sim 4$ to $z\\sim 0$ .",
"The quenching of star formation is therefore a key process in the evolution of baryons.",
"It causes a bimodal color distribution at least from $z\\sim 3$ (e.g.",
"[125]) and is probably the main explanation for the decrease of the star-formation rate density in the universe (e.g. [64]).",
"What makes a galaxy quench is still an open and extensively debated question.",
"The evolution of the SMFs of passive and star-forming galaxies suggests that stellar mass (or more generally halo mass) is a fundamental property tightly linked to the star-formation activity.",
"The $z\\sim 0$ SMF has a knee ($M^{*}$ ) around $\\sim 10^{10.7}M_\\odot $ , and this mass scale seems to be independent of redshift.",
"Galaxies tend to quench when they reach that characteristic mass (e.g.",
"[91], [55], [80]).",
"This mass-quenching process primarily takes place at $z>1$ because, at later times, there are more passive than star-forming galaxies with this mass, so mass-quenching becomes less relevant.",
"Therefore, at late times, most of the quenching activity happens below $\\sim 10^{10.7}M_\\odot $ , and this tends to flatten the low mass end of the passive galaxy SMF (e.g. [80]).",
"Since most of these galaxies are satellites, this quenching is generally referred to as environmental quenching.",
"Even though this empirical description of quenching has been extremely successful in explaining the global trends, the actual physical mechanisms behind quenching are still largely unconstrained.",
"Stellar mass functions alone do not provide information on how the formation of stars affects galaxy structure.",
"It is however well established that star formation activity is strongly correlated with morphology.",
"Galaxies which live on the main sequence of star-formation tend to have a disk-like morphology with low Sersic indices, while passive galaxies tend to have early-type morphologies and Sersic indices larger than 2 (e.g. [130]).",
"Whether this is a cause or a consequence is not yet known [62].",
"Several studies claim that the observed relation between structure and star-formation is in fact a consequence of very dissipative quenching processes.",
"A large amount of gas would be driven into the central parts of the galaxies producing a central burst of star-formation and therefore a bulge with high central stellar mass density (e.g.",
"[4], [2]).",
"However, recent evidence suggests that the dominant quenching mechanism at intermediate stellar masses might be simply a shutting off of the gas supply through strangulation (e.g.",
"[90]) without significant morphological transformations even at very high redshifts (e.g.",
"[39]).",
"The observed correlation between central stellar mass density and star-formation rate could be mostly explained by the fading of the disk after the strangulation event (e.g.",
"[27]).",
"This would also explain the relative large abundance of fast-rotating passive galaxies in the local universe (see [24] for a review).",
"Properly quantifying how the joint distribution of morphology and mass evolves might shed new light on which are the main quenching processes.",
"It also provides a new element of comparison with recent numerical and empirical simulations which now predict morphologies and structure (e.g [124]).",
"However, there is currently no benchmark measurement of this type.",
"Large surveys such as SDSS [11] and more recently GAMA [73] have enabled a good quantification of the morphological dependence of the SMF at low redshift (the larger volume of the SDSS means it is able to probe rarer higher masses than GAMA).",
"Pushing to higher redshift requires better angular resolution over large areas.",
"As a result, there are very few complete studies of the morphological dependence of the SMF at high redshift.",
"[23] made a first attempt but stopped at $z\\sim 0.8$ .",
"Most of the studies at higher redshift are based on smaller subsets of objects (e.g.",
"[32], [21], [76]) or broad morphologies [74].",
"Future space based wide surveys, such as EUCLID and WFIRST-AFTA, will clearly be a major step forward.",
"In the meanwhile, the largest area observed by the Hubble Space Telescope both in the optical and infrared is the CANDELS survey [47], [58].",
"Even though it does not reach the same coverage as ground based surveys such as UltraVista [70], it probes at high angular resolution the rest-frame optical morphologies of galaxies from $z\\sim 3$ with a similar depth to the deepest NIR ground-based surveys.",
"In this sense, it is currently the best available dataset for establishing robust constraints on the abundance of different morphologies in the early universe.",
"This is the main purpose of the present work.",
"In [52], we used new deep-learning techniques to estimate the morphologies of all galaxies with $H<24.5$ in the five CANDELS fields with unprecedented accuracyThe catalog is available at http://rainbowx.fis.ucm.es/Rainbow_navigator_public/.",
"We now use these morphologies, together with robust stellar mass estimates from extensive multi-band imaging, to study the evolution of the SMFs of quiescent and star-forming galaxies of different morphologies from $z\\sim 3$ , for the first time.",
"We then discuss the implications for the dominant quenching processes and morphological transformations.",
"The data on the mass functions are made public so that they can be directly compared with the predictions of different models.",
"The paper is organized as follows.",
"In § , we describe the dataset used as well as the main physical parameters we measure (morphologies, structural parameters, stellar masses etc.).",
"In § we describe the methodology used to derive the stellar mass functions.",
"§ discusses their evolution.",
"Finally in § , we discuss the implications for the star formation histories of the different morphologies and the evolution of the quenching mechanisms at different cosmic epochs.",
"Throughout the paper, we assume a flat cosmology with $\\Omega _M=0.3$ , $\\Omega _\\Lambda =0.7$ and $H_0= 70$ $km.s^{-1}.Mpc^{-1}$ and we use magnitudes in the AB system.",
"All stellar masses were scaled to a [29] IMF.",
"Galaxies in the 5 CANDELS fields (UDS, COSMOS, EGS, GOODS-S, GOODS-N) are selected in the F160W by applying a magnitude cut F160W$<$ 24.5 mag (AB).",
"The total area is $\\sim 880$ $arcmin^2$ .",
"We use the CANDELS public photometric catalogs for UDS [43] and GOODS-S [48] and soon-to-be published CANDELS catalogs for COSMOS, EGS (Stefanon et al.",
"2016) and GOODS-N (Barro et al.",
"2016).",
"The magnitude cut is required to ensure the availability of morphologies [52] a key quantity for the analysis presented in this work.",
"The stellar mass completeness resulting from this magnitude cut is extensively discussed in section REF given its importance to derive reliable stellar mass functions."
],
[
"Structural properties",
"We use the publicly available 2D single Sersic fits from [116] to estimate basic structural parameters (radii, Sersic indeces, axis ratios).",
"The fist were done using galfit [92] on the three NIR images (F105W,F125W,F160W).",
"The expected uncertainty on the main parameters is less than $20\\%$ for the magnitude cut applied in this work as widely discussed in [116], [115]."
],
[
"Morphological classification",
"We use the deep-learning morphology catalog described in [52].",
"In brief, the ConvNets-based algorithm is trained with visual morphologies available in GOODS-S and then applied to the remaining 4 fields.",
"Following the CANDELS classification scheme, we assign 5 numbers to each galaxy: $f_{sph}$ , $f_{disk}$ , $f_{irr}$ , $f_{PS}$ , $f_{Unc}$ .",
"These measure the frequency with which hypothetical classifiers would have flagged the galaxy as having a spheroid, a disk, presenting an irregularity, being compact (or a point source), and being unclassifiable/unclear.",
"For a given image, ConvNets are able to predict the various $f_{type}$ values with negligible bias on average, scatter of $\\sim 10\\%-15\\%$ , and fewer than $1\\%$ mis-classifications [52].",
"In what follows, we primarily use the H band (F160W) since our sample is dominated by galaxies at $z>1$ , where NIR filters probe the optical rest-frame.",
"For $z<1$ galaxies, we also explored the I band filters (814W, 850LP) but because the classes we define below are quite broad, the classifications do not change significantly (also see Kartaltepe et al.",
"2015).",
"In addition, as we show below, at low redshifts our classifications match those in the SDSS rather well: morphological k-corrections do not have a big impact on our results.",
"In this work we distinguish 4 main morphological types defined as follows: spheroids [SPH]: $f_{sph}>2/3$ AND $f_{disk}<2/3$ AND $f_{irr}<0.1$ late-type disks [DISK]: $f_{sph}<2/3$ AND $f_{disk}>2/3$ AND $f_{irr}<0.1$ early-type disks [DISKSPH]: $f_{sph}>2/3$ AND $f_{disk}>2/3$ AND $f_{irr}<0.1$ irregulars [IRR]: $f_{sph}<2/3$ AND $f_{irr}>0.1$ The thresholds above are somewhat arbitrary but have been calibrated through visual inspection first to make sure that they indeed result in distinct morphological classes (see also [56]).",
"In appendix we show some randomly selected postage stamps of the different morphological classes in the COSMOS/CANDELS field sorted by stellar mass and redshift.",
"Slight changes to the thresholds used to define these classes do not affect our main results.",
"The SPH class contains bulge dominated galaxies with little or no disk: it should be close to the classical Elliptical classification used in the local universe.",
"The DISK class is made of galaxies in which the disk component dominates over the bulge (typically Sb-c galaxies).",
"Between both classes, lies the DISKSPH class in which there is no clear dominant component: it should include typical S0 galaxies and early-type spirals (Sa).",
"We also distinguish galaxies with clear asymmetry in their light profiles.",
"This category should capture the variety of irregular systems usually observed in the high redshift universe (e..g clumpy, chain, tadpole etc.).",
"This irregular class might contain a wide variety of galaxies with different physical properties since the classification is based on the irregularity of the light profile.",
"Notice that IRR is defined with no condition on $f_{disk}$ ; therefore, this class can include many late-type disks at low redshifts (i.e.",
"Sds).",
"We have verified that the different classes have distinct structural properties.",
"Spheroids are more compact, rounder ($b/a\\sim 0.8$ ) and have larger Sersic indices ($n\\sim 4-5$ ) than all other morphologies at all stellar masses and at all redshifts.",
"On the other extreme, disks are larger, more elongated ($b/a\\sim 0.5$ ) and have Sersic indices close to 1, as expected.",
"Disk+sperhoids systems lie somewhat in between: they have Sersic indices $\\sim 2$ , but are less compact than the spheroids and have similar axis ratios to disks (in agreement with the visual classification; also see Huertas-Company et al.",
"2015a).",
"Although a detailed analysis of the structural properties of the different morphologies will be presented elsewhere, appendix shows that the different morphologies also have different stellar mass bulge-to-total mass ratios."
],
[
"Stellar masses and completeness",
"SED fitting is used to estimate photometric redshifts and stellar masses used in this work.",
"The detailed methodology is described in [130], [129] and [4], [3].",
"Therefore only the main points are discussed here.",
"Photometric redshifts are the result of combining different codes to improve the individual performance.",
"The technique is fully described in [35].",
"Based on the best available redshifts (spectroscopic or photometric) we then estimate stellar mass-to-light ratios from the PEGASE01 models [41].",
"For these, we assume solar metallicity, exponentially declining star formation histories, a [26] extinction law and a Salpeter IMF (1955).",
"The $M_*/L$ values are then converted to a Chabrier IMF by applying a constant 0.22 dex shift.",
"The stellar mass is estimated by multiplying the $M_*/L$ value by the Sersic-based $L$ (from galfit 2D fits - see section REF ).",
"See also Bernardi et al.",
"2013 and 2016 for extensive discussion of the systematics associated with all these choices.",
"The stellar mass completeness of the sample is estimated following the methodology of [82] and [55] separately for star-forming and quiescent galaxies.",
"We first compute the lowest stellar mass ($M_{lim}^*$ ) which could be observed for each galaxy of magnitude $H$ given the applied magnitude cut ($H<24.5$ ): $log(M^*_{lim})=log(M_*)+0.4(H-24.5)$ .",
"We then estimate the completeness as the 90th percentile of the distribution of $M_{lim}$ , i.e.",
"the stellar mass for which $90\\%$ of the galaxies have lower limiting stellar masses.",
"By adopting this threshold, we make sure that at most $10\\%$ of the low mass galaxies are lost in each redshift bin.",
"Figure REF shows the distribution of galaxies in our sample in the mass-redshift plane and the adopted stellar mass completeness as a function of redshift for passive and all galaxies.",
"The sample is roughly complete for galaxies above $10^{10}$ solar masses at $z\\sim 3$ and goes down to $10^9$ at $z\\sim 0.5$ (see also table REF ).",
"As a sanity check, we use an alternative estimate of the stellar mass completeness by taking advantage of the fact that the CANDELS data are significantly deeper than the H-band selected sample used here ($H<24.5$ ).",
"We therefore compute in bins of redshift, the stellar mass at which 90% of the galaxies in the full CANDELS catalog are also included in our bright selection.",
"The resulting stellar mass completeness is over-plotted in figure REF .",
"It agrees reasonably well with the one estimated independently using the methodology by [82].",
"The largest differences are observed at the high mass-end.",
"It can be a consequence of low statistics in these stellar mass bins.",
"In the following we will adopt therefore the first estimate, keeping in mind however that at high redshift we might under-estimate the completeness.",
"Figure: Stellar mass as a function of redshift for all galaxies (left panel) and quiescent galaxies (right panel) for our H<24.5H<24.5 selected sample.",
"Blue points show the minimum stellar mass which can be observed for a given galaxy computed as explained in the main text.",
"The red line shows the 90th percentile of the distribution of M lim M_{lim} which is the adopted mass completeness in this work (see text for details).",
"The orange line shows the mass completeness estimated using the full depth CANDELS catalog (see text for details)"
],
[
"Quiescent / star-forming separation",
"Rest-frame magnitudes (U, V and J) are computed based on the best-fit redshifts and stellar templates (see section REF ) and are then used to separate the passive and star-forming populations as widely used in the previous literature [126].",
"This color-color separation has the advantage of properly distinguishing galaxies reddened by dust from real passive galaxies with old stellar populations."
],
[
"Estimation of morphological stellar mass functions",
"We use the $V_{max}$ estimator [99] to derive the stellar mass functions in this work.",
"It has the advantage of being very simple but can easily diverge when the incompleteness becomes too important.",
"For this reason, we restrict our analysis to stellar masses above the thresholds derived in section REF and quoted in table REF .",
"Recent works have shown that above the completeness limits, very consistent results are obtained with maximum-likelihood methods (e.g. [55]).",
"For simplicity, we restrict our analysis to one single estimator throughout this work."
],
[
"Uncertainties",
"We consider 3 sources of errors which contribute to the uncertainties on the SMFs.",
"Namely Poisson errors ($\\sigma _P$ ), cosmic variance ($\\sigma _{CV}$ ) and errors associated with the estimation of stellar masses and photometric redshifts ($\\sigma _{T}$ ).",
"Poisson errors reflect exclusively statistical uncertainties due to the limited number of galaxies in each bin.",
"They are proportional to the the square root of the number of objects.",
"Cosmic variance errors are related to the fact that we observe a small area in the sky so our measurements can be affected by statistical fluctuations in the number of galaxies due to the underlying large scale density fluctuations.",
"Cosmic variance can be computed from the galaxy bias and the dark matter cosmic variance assuming a CDM model.",
"We use the tool of [78] to estimate the fractional error in density given the size of the CANDELS fields and also their spatial distribution.",
"Finally, uncertainties in stellar mass and redshifts do have an impact on the density of galaxies.",
"Stellar masses are obtained through SED fitting assuming a photometric redshift (spectroscopic redshifts are available for a minority of sources).",
"There are therefore systematic (e.g.",
"template errors, IMF assumptions, SFHs) and statistical errors associated with this methodology.",
"To estimate this uncertainty, we take advantage of the various measurements of stellar masses and redshifts existing in CANDELS.",
"E.g., the 3D-HST team has computed an independent set of photometric redshifts and derived stellar masses using the FAST and EAZY codes [107].",
"They used BC03 models and a Chabrier IMF.",
"A comparison of the two should provide an estimate of the errors induced in the SMFs due to errors in redshifts and stellar masses.",
"We therefore generated a set of 50 catalogs by randomly combining stellar masses and photometric redshifts from the CANDELS and 3D-HST catalogs and recomputed the SMFs for each of them.",
"We then measured the scatter in the final 50 SMFs in bins of redshift and stellar mass.",
"This scatter combines the statistical error associated with estimating $M_*$ from fitting noisy photometry to a given set of templates, with the systematic error associated with the fact that the templates used have built-in assumptions about the star formation history (bursty or not?",
"dusty or not?",
"etc.).",
"We note however that this approach certainly under estimates the errors.",
"There is in fact a large overlap in assumptions made, notably the exponentially declining tau models and Calzetti reddening law.",
"Additionally, although the photometric extractions by the 3D-HST and CANDELS teams were done independently, the actual data on which the photometry is based are nearly identical.",
"This is clearly not ideal.",
"Nevertheless, we lump these together and add in quadrature to the other two terms.",
"Hence, to each bin we assign an uncertainty $\\sigma =\\sqrt{\\sigma _{P}^2+\\sigma _{CV}^2+\\sigma _T^2}$ .",
"Figure REF shows the different fractional errors on the number density of galaxies as a function of stellar mass and redshift for the total sample.",
"Cosmic variance dominates the error budget for stellar masses below $\\sim 10^{11}$ .",
"It is in fact always greater than $10\\%$ while Poisson and template fitting errors are generally below $\\sim 5\\%$ .",
"At larger stellar masses, the small number statistics generate an increase in the Poisson and template errors, which can exceed $50\\%$ at the very highest masses.",
"In the morphology divided samples, the number of objects is obviously reduced and therefore the statistical errors dominate over comic variance effects at all stellar masses.",
"Similar trends are observed when the objects are separated into star-forming and quiescent samples.",
"Figure: Fractional errors on the number densities of galaxies as a function of stellar mass and redshift for the total sample used in this work.",
"The left, middle and right-hand columns show poisson errors, template fitting-related errors and the effects of cosmic variance."
],
[
"Schechter function fits",
"The non-parametric Vmax estimator is fitted with a Schechter or double Schechter model, depending on the sample.",
"Given that our sample is not large, especially when it is divided into different morphological types, we preferentially use a single Schechter fit for $z>0.8$ .",
"Only in the lower redshift bins, where the SMFs reach lower stellar masses and an upturn is observed, do we adopt a double Schechter as done in previous works (Pozzetti et al.",
"2010, Ilbert et al.",
"2013, Muzzin et al.",
"2013).",
"In all cases, we only fit data points above the completeness limit to avoid biases related to the fact that the $1/V_{max}$ estimator tends to underestimate the number densities beyond the completeness limits."
],
[
"Evolution of the morphological stellar mass functions",
"In this section, we discuss the evolution of the SMFs for different morphologies."
],
[
"Full sample evolution",
"Figure REF shows the stellar mass functions for each of the 4 morphological types defined previously; the different panels show results for 7 redshift bins.",
"Figure REF shows the same information in a different format: each panel shows the evolution of the SMF for a fixed morphological type.",
"The redshift bin sizes were determined by a trade-off between number of objects and lookback time as seen in tables REF and REF .",
"The functions are only plotted above the mass completeness limit derived in section REF .",
"Best-fit parameters are shown in table REF .",
"The global mass functions, i.e., not subdivided by morphology, (black region in figure REF ) are also shown and compared with recent measurements in the UltraVista survey by Ilbert et al.",
"(2013) and Muzzin et al.",
"(2013).",
"There is good agreement despite the significantly smaller volume probed by the CANDELS fields.",
"This suggests that our completeness limits are well-estimated.",
"Volume effects are mostly visible in the lowest redshift bin, where the CANDELS SMFs show a lack of very massive galaxies.",
"Before we consider CANDELS in more detail, it is worth remarking on the cyan curve (same in each panel), which shows the SMF in the SDSS from [9].",
"The large volume of the SDSS means Poisson errors are negligible, so the shaded region encompasses the systematic differences between different $M_*/L$ estimates.",
"At low $z$ , the UltraVista measurements are in good agreement with the SDSS; moreover, they lie below it at higher redshifts, as one might expect.",
"In contrast, Figure 5 of Ilbert et al.",
"(2013) shows that their $z\\sim 0.5$ SMF lies above the $z\\sim 0.1$ SDSS SMF, which does not make physical sense.",
"This is because their Figure 5 used the SDSS estimate of Moustakas et al.",
"(2013).",
"Bernardi et al.",
"(2016) discuss why their estimate is to be preferred; note that their work was not motivated by this problem, so the fact that evolution makes better physical sense when their SMF is used as the low redshift benchmark provides additional support for their analysis.",
"Very briefly, the main reason is that Moustakas et al.",
"(2013) used SDSS Model magnitudes, which underestimate the total luminosity of bright galaxies (Bernardi et al.",
"2010, 2013; D'Souza et al.",
"2015; see especially Figures 2 and 3 in Bernardi et al.",
"2016 and related discussion).",
"This accounts for about half the difference from Bernardi et al.",
"; the remainder is due to M*/L.",
"Section 4.3 of Bernardi et al.",
"(2016) discusses this in more detail (see, e.g., their Figures 14-16).",
"We refer the reader to Bernardi et al.",
"(2016) for a more complete discussion.",
"If the evolution is driven by star-formation (no mergers), then figure REF shows that the stellar mass of galaxies below $M^*$ increases by more than 1 dex in the redshift range $0.5<z<3$ .",
"More massive galaxies increase their stellar mass by less than 0.5 dex.",
"Therefore, we confirm previous reports of a mass-dependent evolution for the global population.",
"In addition, Table REF shows that $Log(M^*/M_\\odot )\\sim 10.85\\pm 0.1$ is approximately independent of redshift.",
"Figure: Stellar mass functions for 4 morphological types in different redshift bins as labelled.",
"Red, blue, orange and green shaded regions in the top panels show the number densities of spheroids, disks, disk+spheroids and irregular/clumpy systems respectively.",
"The bottom panels show the fractions of each morphological type with the same color code.",
"The black regions show the global stellar mass functions.",
"The pink triangles and brown squares are the measurements by and respectively in the UltraVista survey.",
"The points are only plotted when their redshift bins are the same than the ones used in this work.",
"We also show for reference in all panels the SMF for all SDSS galaxies (cyan shaded region) from Bernardi et al.",
"(2016).Figure: Evolution of the stellar mass functions at fixed morphology.",
"Same as figure but binned by morphological type.",
"Each color shows a different redshift bin.",
"We also over-plot the local SMFs from Bernardi et al.",
"(2016) for the total sample and the ones divided by morphology from Moffet et al.",
"(2016) (best fit Schechter functions).Table: Best-fit parameters with single and double schechter functions to the stellar mass functions of the four morphological types defined in this work.",
"The parameters of the double Schechter are set to -99 whenever a single Schechter was used.",
"Values of -99 are also used when the fit did not converge.The key new ingredient of the present work is the evolution at fixed morphology.",
"Morphological evolution, as well as the mass dependence of the dominant morphology, are both clearly observed.",
"At $0.2<z<0.5$ , the population of $\\sim M^*$ galaxies ($10<log(M_*/M_\\odot )<11$ ) is essentially uniformly distributed between disks with low bulge fractions, spheroids with large B/Ts and intermediate objects with 2 components meaning that here is no clear dominant morphology at this mass scale.",
"Above $log(M_*/M_\\odot )=11$ ), objects with a clear bulge component tend to dominate the population.",
"Below $10^{10}M_\\odot $ , the population is basically dominated by objects with small bulges or without.",
"Irregular objects only start dominating the population at $log(M_*/M_\\odot ) < 9$ .",
"This morphological distribution remains globally unchanged from $z\\sim 1$ .",
"Our low mass SMFs match the local SMFs recently derived in the GAMA survey (Moffet et al.",
"2016) quite well, as shown in figure REF .",
"We over-estimate the abundance of irregulars at $log(M_*/M_\\odot )>10$ compared to them.",
"This is probably a consequence of our definition of irregulars based on the asymmetry of the light profile.",
"It has however little impact on the other morphologies since their abundance is still very low at the high mass end.",
"The agreement with their work confirms the robustness of our automated classifications.",
"Above $z\\sim 1$ irregular objects start dominating even at higher-masses.",
"At $z>2$ , the morphological mix changes radically: there are basically only 2 types of galaxies at these redshifts – irregulars account for 70% of the objects, and bulge dominated galaxies (spheroids) for the remaining $30\\%$ (based on the extrapolations of the Schechter fits).",
"This has a number of interesting implications.",
"First, at these early epochs, the majority of disks are irregular (probably a signature of unstable disks as probed by recent IFU surveys, e.g. [128]).",
"Note that this is not a morphological k-correction effect, since we are probing the optical rest-frame band at this epoch.",
"Second, symmetric disks and bulge+disks systems only begin to appear between $z\\sim 2$ and $z\\sim 1$ ; objects classified as DISK+SPH account for fewer than 5% of the objects at $z>2$ .",
"This is also observed in the top right panel of figure REF .",
"Disks and disk+spheroid mass functions experience the most dramatic evolution.",
"One might worry that the apparent disappearance of $z\\ge 2$ disks is due to surface brightness dimming.",
"This is unlikely though for several reasons.",
"Extensive simulations (e.g.",
"Kartaltepe et al.",
"2015, van der Wel et al.",
"2014) have shown that disks should be detectable at the depth of the CANDELS survey for the magnitude selection used in this work.",
"Also, these are fairly massive galaxies so there is not much room for the presence of a massive disk.",
"In fact, preliminary results of figure REF show a clear correlation between the morphological classification and the stellar mass bulge-to-disk ratio which would have been erased if surface brightness dimming was an issue.",
"These global trends are captured in the top left panel of figure which summarizes the evolution of the stellar mass density (integrated over all galaxies with $log(M_*/M_\\odot )\\ge 8$ ).",
"Since this lower limit lies below the completeness limit, the result relies on extrapolating the best Schechter fits to low masses.",
"We first observe the previously reported 2-speed growth of the mass density on the full sample (black line) in good agreement with previous measurements.",
"From $z\\sim 4$ to 1, the total mass density increases by a factor of $\\sim 6 $ .",
"From $z\\sim 1$ onwards the growth flattens: $\\rho _*$ at $z=0$ is larger by only a factor of $\\sim 2$ .",
"As we discuss in the following sections, this is a consequence of both the decrease in the specific star-formation rate below $z\\sim 1$ (e.g.",
"[97]) and of quenching at large stellar masses.",
"Regarding the morphological evolution above $10^8$ solar masses (resulting from the extrapolation of the best Schechter fits), the key observed trends observed in figure are At $z>2$ , more than 70% of the stellar mass density is in irregular galaxies (see also [32]).",
"The stellar mass density in irregulars decreases over time from $Log(\\rho _*/M_\\odot Mpc^{-3})\\sim 7.7$ at $z\\sim 1.5$ to $\\sim 7.1$ at $z\\sim 0.3$ .",
"This is clear evidence of morphological transformations as we will discuss in the following sections.",
"At $z>2$ , 30% of the stellar mass density is in compact spheroids with large B/T.",
"This suggests that bulge growth at this epoch destroys disks.",
"The emergence of regular disks (S0a-Sbc) happens between $z\\sim 2$ and $z\\sim 1$ .",
"In this period, the stellar density in both pure disks and bulge+disk systems increases by a factor of $\\sim 30$ .",
"Below $z\\sim 1$ , the stellar mass density is equally distributed among disks, spheroids and mixed systems.",
"Figure: NO_CAPTION"
],
[
"Evolution of the star-forming population",
"Figure REF shows the evolution of the stellar mass functions of star-forming galaxies as a function of morphological type.",
"To guide the eye, the cyan region (same in all panels) shows the $z\\sim 0.1$ SDSS determination as a reference.",
"This curve was obtained by following the analysis of Bernardi et al.",
"(2016), but selecting the subset of objects for which the log of the specific star formation rate determined by the MPA-JHU (e.g.",
"Kauffmann et al.",
"2003) group is greater than $-11$ dex.",
"Table REF summarizes the best fit Schechter function parameters for our CANDELS analysis.",
"In agreement with previous work, the SMF of all star-forming galaxies increases steeply at the low-mass end, and evolves very little at the high-mass end.",
"This is a consequence of quenching: when star-forming galaxies exceed a critical mass, they quench and so are removed from the SF sample (e.g.",
"Ilbert et al.",
"2013, Peng et al.",
"2010).",
"Figure: Stellar-mass functions of star-forming galaxies divided in 4 morphological types and in different redshift bins as labelled.",
"Red, blue, orange and green shaded regions in the top panels show the number densities of spheroids, disks, disk+spheroids and irregular/clumpy systems respectively.",
"The black regions show the global mass functions.",
"The pink triangles are the values measured by Ilbert et al.",
"(2013) and the brown squares the values of Muzzin et al.",
"(2013) in the same redshift bins.",
"The bottom panels show the fractions of each morphological type with the same color code.",
"The cyan shaded region shows the SMF for the SDSS star-forming galaxies from Bernardi et al.",
"(2016) (log(SSFR)>-11log(SSFR)>-11).Our new results show that the morphological mix of star-forming galaxies also experiences a pronounced evolution.",
"At $0.2<z<0.5$ , the typical morphology of a star-forming galaxy differs significantly from that in the full sample.",
"Purely bulge dominated systems (spheroids) account for $\\le 5\\%$ of the objects at all stellar masses.",
"Star-forming galaxies at low redshifts are therefore dominated by regular systems with no pronounced asymmetries and with low bulge fractions (i.e.",
"disks) over 2 decades in stellar mass ($9<log(M_*/M_\\odot )<11$ ).",
"Irregular disks start to dominate only at very low-masses ($log(M_*/M_\\odot )\\le 9$ ).",
"Bulge+disk systems are also a minority, but account for $\\sim 40\\%$ of the population at stellar masses greater than $log(M_*/M_\\odot )\\sim 10.7$ .",
"The presence of the bulge component is therefore tightly linked to the star-formation activity of the galaxy as widely documented in the recent literature (e.g. [129]).",
"As observed for the full sample, this morphological mix seems to have remained rather stable since $z\\sim 1$ .",
"At higher redshifts, the relative abundance of irregulars and normal disks is inverted: disturbed systems become the dominant morphological class of star-forming galaxies.",
"The relative abundance steadily increases from $z\\sim 1$ to $z\\sim 2$ .",
"At $z>2$ , irregular systems are almost 100% of the star-forming population.",
"While we confirm a population of star-forming spheroids at $z>2$ (e.g.",
"[119]), they account for only $\\sim 5-10\\%$ of the star-forming population at these redshifts.",
"This strongly suggests that bulge formation at these early epochs requires rapid consumption of gas and therefore the quenching of star-formation.",
"These trends are summarized in the middle panel of figure which shows the evolution of the integrated stellar mass density of star-forming galaxies with $M_*/M_\\odot >10^8$ .",
"The main observed features are: Stars are formed in systems with a disk.",
"The abundance of star-forming spheroids is below $10\\%$ .",
"There is a transition of the galaxy morphology which hosts star-formation.",
"At $z<1-1.5$ most of the stars in star-forming systems are in regular disks with low ($\\sim 50\\%$ ) and intermediate ($\\sim 20\\%$ ) B/T while at $z>1.5$ they are predominantly in irregular systems ($>80\\%$ ).",
"The stellar mass density in irregular galaxies decreases with redshift (by a factor of $\\sim 3$ ); therefore, irregulars are being transformed into other morphologies.",
"Table: Best-fit parameters for a single Schechter function to the star-forming and quiescent SMFs of the four morphological types defined in this work.",
"A=all.",
"S=spheroids, D=disks, DS=disks+spheroids and I=irregulars.",
"Quiescent galaxies at z>2.5z>2.5 are not fitted because there are too few values above completeness.",
"Quiescent irregulars are also very few.",
"Although the fit works, the mass function is not always well described by a Schechter function.",
"To emphasize this we have set the error on M * M^* to 99.9."
],
[
"Evolution of the quiescent population",
"Figure REF shows the evolution of the SMFs of quiescent galaxies as a function of morphological type.",
"We also show in all panels the SMF of quiescent galaxies ($log(SSFR)<-11$ ) in the SDSS based on the recent measurements of Bernardi et al.",
"(2016).",
"The quiescent SMF, summed over all morphological types, agrees with the one measured in Ilbert et al.",
"(2013).",
"There are some discrepancies, especially in the $0.5<z<0.8$ bin, which can be a consequence of both cosmic variance and of a difference in the colors used to select passive galaxies.",
"In any case, the broad evolution trends remain the same.",
"Quiescent galaxies first appear at the high-mass end.",
"The quiescent SMF increases rapidly at the high-mass end up to $z\\sim 1$ .",
"From $z\\sim 1$ to the present, the low mass population of passive galaxies starts to emerge.",
"Figure: Stellar mass functions of quiescent galaxies divided in four morphological types and in different redshift bins as labelled.",
"Red, blue, orange and green shaded regions in the top panels show the number densities of spheroids, disks, disk+spheroids and irregular/clumpy systems respectively.",
"The black regions show the global mass function.",
"Pink filled triangles and brown squares show the recent SMFs by Ilbert et al.",
"(2013) and Muzzin et al.",
"(2013) in the UltraVista survey respectively.",
"The bottom panels show the fractions of each morphological type with the same color code.",
"The cyan shaded region shows the SMF for the SDSS quiescent galaxies from Bernardi et al.",
"(2016) (log(SSFR)<-11log(SSFR)<-11).The morphological dissection which our analysis allows provides additional information on how quenching mechanisms affect the morphologies of galaxies.",
"At low redshift, the morphologies of passive galaxies are dominated by two types, pure spheroids and disk+bulge systems.",
"The fraction of quenched late-type spirals is almost negligible ($~\\sim 5\\%$ ), in agreement with measurements in the local universe [69].",
"Only below $10^9M_\\odot $ do red disk systems seem to be more abundant.",
"The quenching mechanisms above $10^{10}M_\\odot $ are therefore linked to the presence of a bulge.",
"At $z>2$ , the population of quiescent galaxies is dominated by pure spheroids while the abundance of passive disk+bulge systems (intermediate B/T) is less than $5\\%$ at $z\\sim 2$ .",
"The fraction of disk+bulge systems grows at lower $z$ .",
"This suggests that most of the newly quenched galaxies between $z\\sim 2$ and $z\\sim 0$ have a disk component.",
"These trends are also captured in figure (which is integrated down to $10^8M_\\odot $ ).",
"To summarize, the main observed trends are: $\\sim 80-90\\%$ of the stellar mass density of quiescent galaxies is in galaxies with bulges (spheroids and bulge+disk systems) from z>2.",
"Stars in dead galaxies are therefore almost exclusively in systems with a bulge component.",
"The relative distribution between the two types changes with time.",
"At $z>2$ , almost all stars are in spheroids while at $z<1$ stars are equally distributed in disky passive galaxies and spheroids.",
"The stellar mass density in disky passive galaxies increases therefore much faster than in spheroids (a factor of $\\sim 40$ compared to a factor 4 for spheroids).",
"The majority of newly quenched galaxies between $z\\sim 2$ and $z\\sim 0.5$ preserve a disk component.",
"This constrains the dominant quenching mechanisms as discussed in section REF ."
],
[
"Discussion",
"The evolution of stellar mass functions can be used to indirectly infer the star formation histories of the different morphologies.",
"The evolution also allows an estimate of when different morphologies emerge.",
"We now discuss the implications of our results for morphological transformations and quenching processes at different stellar mass scales from $z\\sim 3$ to the present."
],
[
"Inferred star-formation histories at fixed morphology",
"As is well known, the stellar mass density is the integral of the star-formation rate density corrected for the amount of mass loss: $\\rho _*(t)=\\int _{0}^{t}SFRD(t^{^{\\prime }})(1-0.05ln(1+(t-t^{^{\\prime }})/0.3))dt^{^{\\prime }}$ SFRD stands here for star-formation rate density.",
"The previous equation assumes a parametrization of the return fraction provided by [31] with a Chabrier IMF.",
"Several works have already done this exercise and compared the inferred SFRD evolution with the one obtained from direct measurements, finding different results.",
"[127] first reported a discrepancy of $\\sim 0.6$ dex between inferred and direct measurement of the SFHs.",
"Ilbert et al.",
"(2013) revisited this issue with more recent measurements.",
"They found a reasonable agreement with direct measurements from the data compilation of [7], especially at $z<2$ , reducing the previous tensions.",
"We repeat those efforts here, but add morphological information.",
"This enables the first estimates of the formation of stars in different morphologies.",
"We first assume that the SFRD evolution can be parametrized with a Lilly-Madau law as done by [7]: $SFRD(z)=\\frac{C}{10^{A(z-z_0)}+10^{B(z-z_0)}}$ Then we fit, for each morphological type, an SFRD following the parametrization of equation REF using equation REF and the measured of stellar mass densities reported in figure .",
"The results are shown in figure REF .",
"The global inferred SFRD evolution agrees reasonably well with the one derived by Ilbert et al.",
"(2013) using the same methodology but on a completely different dataset and with different assumptions on stellar masses.",
"This suggests that the method is robust.",
"With $A$ fixed to $-1$ , we find best fit parameters of: $C=0.11\\pm 0.02$ , $B=0.21\\pm 0.04$ and $z_0=0.98\\pm 0.13$ .",
"This confirms a peak of star formation activity at $z\\sim 2$ .",
"Our measurements also agree reasonably well with the most recent compilation of different direct measurements performed by [64], especially at low redshifts.",
"At $z>2$ , direct measurements estimate a star-formation rate density that is $\\sim 1.25$ times larger than our inferred values.",
"The interpretation of the SFHs at fixed morphologies is more complex since galaxies can transform their morphologies over time.",
"Hence the SFRD we infer cannot be directly interpreted as the star-formation activity of a single morphological type.",
"Rather, it captures the combined effect of stars formed in-situ in a given morphology and of new stars which were formed in another morphological type and then merged or transformed.",
"For morphologies with a very low quiescent fraction, it is reasonable to assume that the SFRD will be dominated by in-situ star-formation.",
"However, for morphologies which are mostly quiescent, the inferred history is most probably driven by morphological transformations and mergers.",
"To help in the interpretation of the evolution of the SFRD, figure REF shows the evolution of the quiescent fractions of the different morphological types.",
"As we noted before, it appears that irregulars and disks with low B/T fractions are predominantly star-forming.",
"The quiescent fraction in this population is below $10\\%$ .",
"In contrast, the quiescent fraction in (massive) spheroids exceeds $\\sim 70\\%$ at all redshifts.",
"Of course, this is based on the assumption that the SFHs of individual morphologies can indeed be properly parametrized by equation REF .",
"Nevertheless, the analysis of the SFRDs reveals some interesting first-order trends.",
"There is a clear transition in the dominant morphology hosting star-formation.",
"At $z>2$ , star-formation mostly takes place in irregular systems.",
"The fact that the best fit exceeds the global SFRD is clearly an effect of the over simplification of our model.",
"The SFH of irregulars peaks indeed at $z\\sim 2.5-3$ and sharply decreases thereafter.",
"But this does not mean that they stop forming stars: their quiescent fraction is $\\le 5\\%$ at all redshifts.",
"Rather, they must transform into other morphologies.",
"At $z\\sim 1.5$ , stars are indeed formed in normal symmetric disks both with and without bulge.",
"Also interesting is that, at high redshift, the star formation rate density in spheroids is significantly larger than in disks.",
"Again, this does not mean that spheroids are forming more stars than disks, since stars which formed in another morphology and then transformed into spheroids would be credited to spheroids by our parametrization.",
"Since most of spheroids are quiescent, it is likely that the SFH in spheroids is actually dominated by the contribution from transformations.",
"The SFH suggests that the formation of spheroids was most efficient at $z>2$ .",
"Figure: Inferred star-formation histories for different morphological types.",
"The black solid line shows the global sample and the different colors show the star-formation histories of different morphologies.",
"The dotted line is the SFRD inferred by on UltraVista following the same methodology.",
"The black dashed-dotted line is the most recent compilation of direct measurements by ."
],
[
"Constraints on morphological transformations of star-forming galaxies: rejuvenation or continuous star-formation?",
"The SFHs suggest a transition in the morphology where most stars are forming.",
"A more complete understanding requires accounting for the impact of mergers and morphological transformations.",
"Figure REF shows the redshift evolution of the SMF of the two morphologies which dominate the star-forming population, i.e.",
"disks (left) and irregulars (right).",
"In the panel on the left, the abundances increase monotically with time; the opposite is true in the panel on the right.",
"Since the vast majority of irregulars are star-forming, the decrease in the panel on the right indicates that massive irregulars disappear from the irregular class, so they must begin contributing to another morphological class.",
"What do these objects become?",
"If they continue forming stars, then it is plausible that they transform into symmetric disks.",
"This would be consistent with the low quiescent fractions ($<10\\%$ ) of both (the irregular and the disk) populations.",
"It would also be qualitively consistent with the obvious increase with time of the number of normal star-forming disks (left hand panel of figure REF ).",
"I.e., the measured evolution of the disk SMF must result from the combined effects of morphological transformations and in-situ star-formation.",
"At $z>1.5$ , where irregulars still dominate the population of star-forming galaxies, the evolution in the panel on the right is probably dominated by transformations.",
"Below $z\\sim 1$ , the reservoir of massive irregular galaxies is exhausted, so from this point on, the evolution becomes dominated by genuine star-formation within disks, rather than transformations from irregulars.",
"This does not actually constrain the individual detailed channels.",
"It does not mean that all star-forming galaxies move straight from an irregular to a pure disk.",
"Only general statistical trends are reflected in the SMF.",
"The individual paths followed by galaxies are necessarily more complex and diverse.",
"As a matter of fact, star-formation in a galaxy is not necessarily continuous and galaxies can experience several morphological transformations during their lives.",
"A galaxy can easily destroy a disk, quench and then rejuvenate by rebuilding a disk and going back to the disk SF population (e.g.",
"[50], [54]).",
"Also a galaxy might appear as an irregular if seen in a merger phase and then go-back to the disk population.",
"Although this last possibility does not seem to be very common since the evolution of the high mass end of the spheroid mass function does not significantly evolve.",
"[112] recently looked at the individual paths followed by massive galaxies in the EAGLE simulation.",
"They found however that the fraction of rejuvenated disks represents less than $2\\%$ of the star-forming population, suggesting that most of the SF galaxies should keep star-forming and experience a gradual morphological transformation as suggested by the global evolution of the SMFs.",
"Separating rejuvenated disks from disks with continuous star-formation in our sample requires an accurate age determination of our disk population.",
"While this is not currently possible with the available data (broad or medium band photometry), we can try to place some constraints.",
"As shown in [112] most of the rejuvenated galaxies go through a compact quenched phase which lasts $\\sim 4$ Gyrs, after which they rebuild a disk.",
"Since rejuvenated objects also build a bulge, once they accrete the disk, an important fraction of them should end up as a star-forming 2-component system in our classification system.",
"Thus, to first-order, star-forming bulge+disks systems are potential good candidates for being rejuvenated objects.",
"The fraction of these systems among the star-forming population is $\\sim 5\\%$ which suggests rejuvenation is not common.",
"Only above $\\sim 10^{11}M_\\odot $ , and at low redshifts, does the fraction increase to $30-40\\%$ .",
"Major mergers should play an important role at these mass scales [91], [24].",
"So these massive systems might result from mergers followed by the accretion of a disk as suggested in numerical simulations [54].",
"Another possible explanation for this population of star-forming bulge+disks systems is that they are in fact transiting in the other direction.",
"i.e.",
"they are in the process of inside-out quenching (e.g [111]).",
"The fact that they are a small fraction of the population would then suggest that the transitioning phase is short, in line with expectations (e.g [112]).",
"We discuss this in the following section.",
"At the the low-mass end, the irregular SMFs show little evidence of evolution.",
"This suggests an equilibrium between the arrival of new systems and galaxy growth followed by morphological transformations.",
"This is expected in a CDM scenario where the halo mass function also evolves little at low masses (e.g.",
"[15]).",
"The growth of haloes is indeed compensated by the emergence of new smaller systems.",
"The assumption that these new small haloes will be populated by irregular galaxies initially is in qualitative agreement with the observed mild evolution of the abundance of low mass irregulars.",
"Figure: Evolution of the quiescent fraction at fixed morphology.",
"Different colors show different redshift bins as labeled.Figure: Evolution of the stellar mass function of star-forming galaxies at fixed morphological type.",
"The two dominant morphologies of star-forming galaxies, disks (left) and irregulars (right), are shown."
],
[
"Constraints on quenching processes",
"The morphological evolution of quiescent and star-forming galaxies contrains the quenching mechanisms which operate at different epochs."
],
[
"Quenching at $M^*$",
"The analysis of the evolution of the global SMFs suggests that galaxies reaching masses close to $M^*$ ($\\sim 10^{10.8}M\\odot $ ) tend to quench and populate the quiescent stellar mass function (e.g.",
"Peng et al.",
"2010, Ilbert et al.",
"2013).",
"This is now known as mass quenching (i.e.",
"Peng et al.",
"2010).",
"However, the physical cause of this decrease in star-formation is still unclear.",
"It may be a consequence of halo heating, which prevents the inflow of gas that is required to feed further star formation.",
"On the other hand, quenching may be due to more violent processes such as AGN feedback, or some kind of violent disk instabilities (VDI), which are expected to have a stronger impact on the morphology.",
"At low redshift, by studying the metallicity of local quiescent galaxies Peng et al.",
"(2015) suggested that strangulation might be a dominant process.",
"At high redshift, more violent mechanisms such as VDI might be needed to compactify objects (e.g Barro et al.",
"2013, 2015).",
"In section REF , we showed that the quiescent population at $z>2$ is dominated by spheroids, and that a population of disky passive galaxies emerges at later epochs.",
"In figure REF , we show the evolution of the dominant morphologies of quiescent galaxies, i.e.",
"spheroids and disk+spheroids.",
"The plot shows that the SMFs evolve differently: Especially around $10.5<log(M_*)<11$ , where mass quenching is expected to be the dominant process, the spheroidal SMF seems to increase with time by a smaller amount than that of disk+spheroids.",
"We quantify this effect in figure REF : in this mass range, the number density of pure spheroids evolves little from $z\\sim 2$ to the present, whereas that of quiescent galaxies with disks increases $\\sim 100\\times $ .",
"As discussed in section REF , this is unlikely an observational bias due to cosmic dimming.",
"Figure: Evolution of the stellar mass function of quiescent spheroids (left) and quiescent disk+spheroids (right).One can take a step forward in the interpretation by making the simplistic assumption that pure spheroids are created after some kind of violent dissipative process that destroys the disk and rapidly quenches star-formation.",
"Pure SPHs are indeed compact, round and dense (Huertas-Company et al.",
"2015a).",
"Gas-rich major mergers or violent disk instabilities followed by some AGN feedback (e.g.",
"Ceverino et al., Bongiorno et al.",
"2016) are possible processes.",
"In contrast, disky passive galaxies can be assumed to be predominantly a consequence of a more gradual mechanism related to the lack of available fresh gas (e.g.",
"strangulation, Peng et al.",
"2015) or morphological quenching [68].",
"Then, the evolution observed in figure REF can be interpreted as a signature of a transition in the dominant quenching mechanism.",
"At $2<z<3$ violent processes such as mergers and VDIs seem to be rather common channel for quenching since the number of spheroids increases in this period by a factor of $\\sim 10$ .",
"At lower redshift though, VDIs appear to be less common in light of the weak evolution of the abundance of spheroids and the passive evolution of their stellar populations.",
"At $z<2$ , the majority of newly quenched $\\sim M^*$ galaxies preserves a passive disk component.",
"Therefore the most common mass quenching path could be more related to some kind of strangulation that provokes an aging of the stellar populations without significantly altering the morphology.",
"This agrees with Peng et al.",
"(2015).",
"It also means that the population of $\\sim M^*$ star-forming bulge+disk systems are probably in the process of quenching from the inside-out (e.g.",
"[111]): Although they have already built a bulge, the star-formation has not yet ceased.",
"Since this phase is expected to last at most $\\sim 2$ Gyrs this would explain why these objects are so uncommon.",
"The previous discussion starts from an assumption that the two populations of passive galaxies formed in different processes.",
"This does not need to be true.",
"The fact that galaxies which quenched at later epochs appear larger and less dense can also simply be a consequence of the fact that their star-forming progenitors are also larger given the observed size increase in star-forming galaxies (e.g.",
"[115], [104]).",
"If so, this would not imply a change in the dominant quenching mechanism.",
"The recent simulations of [39] suggest that the formation of massive quiescent galaxies at very high redshift is also predominantly a consequence of a low amount of gas accretion.",
"Size measurements of star-forming galaxies at $z>3$ [95], [104] also show that the typical effective radii of star-forming galaxies at these redshifts are $1-2kpc$ , consistent with the sizes of spheroids at $z<2-3$ .",
"This would imply that the amount of required compaction might be less.",
"Figure: Number density evolution of quiescent galaxies with 10.5<log(M * /M ⊙ )<1110.5<log(M_*/M_\\odot )<11 divided by morphology.",
"Red regions show pure spheroids and brown are early-type disks.",
"The last redshift bin is not shown due to incompleteness."
],
[
"Quenching in sub-$M^*$ galaxies",
"At $z\\sim 1.5$ , the SMF of quiescent galaxies with stellar masses close to $M_*$ is mostly in place as seen in the bottom panel of figure REF and also in previous works analyzing the global mass functions.",
"This means that most of the newly quenched galaxies at $z<1$ have lower stellar masses, i.e.",
"$log(M_*/M_\\odot )<10$ .",
"They are expected to bepredominantly satellite galaxies and the quenching process for these systems is generally known as environmental quenching.",
"The effect of environmental quenching is clearly observed in the low mass end of the quiescent mass function which begins to turn upward at $z<1$ .",
"The analysis of the morphological SMFs derived in this work also reveals interesting properties of the mechanisms of environmental quenching.",
"In fact, at these mass scales, the shape of the SMF of passive bulge+disks systems and spheroids is significantly different.",
"While the abundance of disky systems clearly decreases at the low mass end (Fig.",
"REF ) the one of spheroids tends to increase and mimic the upturn of the global quiescent SMF.",
"Therefore, the low mass end of the quiescent SMF is in fact predominantly populated by spheroids which are significantly more abundant than disk+bulge systems.",
"This means that the environmental quenching process happening at these mass scales will in general destroy the disk and keep only the central component so that the morphology appears like a roundish bulge dominated system.",
"Mechanisms like ram-pressure stripping could indeed remove the disk as a satellite galaxy enters a massive halo.",
"Also relevant is that the abundance of red spirals, i.e.",
"passive disk galaxies with no bulge also increases at $z<1$ (see figures and REF ).",
"One possible formation mechanism is strangulation as they enter a massive halo as satellites.",
"These two mechanisms seem to coexist at the low mass-end.",
"We emphasize that this is only a first-order interpretation.",
"A proper quantification of the environments of these low-mass galaxies needs to be undertaken in order to conclude on the effect of environment on quenching.",
"This is beyond the scope of this work and will be explored in the near future."
],
[
"Summary and conclusions",
"We have studied the evolution of the stellar mass functions of quiescent and star-forming galaxies in the redshift range $0.2<z<3$ at fixed morphological type covering an area of $\\sim 880$ $arcmin^2$ .",
"Our sample consists of $\\sim 50,000$ galaxies with $H<24.5$ .",
"The stellar mass completeness goes from $log(M_*/M_\\odot )\\sim 8$ at $z\\sim 0.2$ to $log(M_*/M_\\odot )\\sim 10.3$ at $z\\sim 3$ .",
"Galaxies are divided into four main morphological classes based on a deep-learning classification, i.e pure bulge dominated spheroids, pure disks, intermediate 2-component systems and irregular or disturbed objects.",
"Each morphology has clearly differentiated structural properties.",
"Our main conclusions are summarized below: Our global SMFs agree with recent measurements from large NIR ground-based surveys.",
"Volume effects are only seen in the lowest redshift bins.",
"We find mass-dependent evolution of the global and star-forming stellar mass function: the low mass end evolves faster than the high mass end in agreement with previous work.",
"This is a consequence of mass-quenching being efficient for galaxies which reach a typical stellar mass of $log(M_*/M_\\odot )\\sim 10.8$ .",
"The stellar mass density of quiescent galaxies with $log(M_*/M_\\odot )>8$ increases by a factor of 5 between $z\\sim 3$ and $z\\sim 1$ .",
"At $z<1$ the passive SMF flattens at the low mass-end; this is usually interpreted a signature of environment quenching on satellite galaxies.",
"The inclusion of statistical morphological information brings additional insight.",
"See also figure REF .",
"At $z>2$ , the morphological distribution of massive galaxies is bimodal: spheroids and irregulars.",
"All star-forming galaxies are irregulars.",
"Taking into account recent dynamical studies of star-forming objects at $z>1$ , this might be a signature of unstable and turbulent disks.",
"The quiescent galaxies are pure compact spheroids with no clear evidence of a disk component.",
"At these redshifts, the high mass end of the passive population is building-up rapidly.",
"The morphological distribution suggests therefore a violent quenching mechanism as main channel to quench galaxies at $z>2$ .",
"Strong dissipative processes such as very-gas rich mergers or violent disk instabilities are known to rapidly bring a large amount of gas into the central parts of the galaxy, leading to a massive, compact and dense remnant as observed.",
"Alternatively, they might be the result of quenching of small star-forming systems at higher redshifts.",
"Between $1<z<2$ the majority of normal disks observed in the local universe emerge.",
"The SMF of normal star-forming spiral disks evolves rapidly during this time.",
"The evolution is a combination of in-situ star-formation and morphological transformations from irregular disks.",
"At $z\\sim 1.5$ , star-formation occurs primarily in normal spiral disks.",
"To lowest order, this morphological transition does not seem to interrupt star-formation.",
"Rejuvenation does not play an important role, although this has to be confirmed with a careful age analysis of the stellar population of late-type disks.",
"The morphological mix of quiescent galaxies also evolves significantly between $1<z<2$ .",
"Most of the newly massive quenched galaxies in this redshift range have a disk component but with a larger bulge than the star-forming ones.",
"The number density increases $100\\times $ while that of quiescent spheroidals stays roughly constant.",
"The efficiency with which spheroids form decreases and the dominant quenching process does not destroy the disk.",
"This suggests a transition in the main quenching mechanism.",
"Strangulation and/or morphological quenching are possible explanations.",
"At $z<1$ , there is little evolution of the morphological mix above $10^{10.8}M_\\odot $ .",
"At the highest masses, the abundance of bulgeless systems decreases; nearly $100\\%$ of the population has a significant bulge; star-forming objects with a large bulge represent $\\sim 40\\%$ of the population.",
"Galaxies with masses $\\sim 10^{10.8}M_\\odot $ are equally likely to be spheroids as symmetric late and early-type spirals.",
"Most ($95\\%$ ) passive galaxies are spheroids or early-type spiral/S0 galaxies while most (90%) star-forming galaxies are late-type spirals.",
"Below $\\sim 10^{10}M_\\odot $ , irregular objects dominate the star-forming population.",
"Quenching mostly happens at this low mass-end: it creates a population of low-mass bulge dominated systems and leads to an increase in the fraction of red spirals.",
"This suggests both ram-pressure and strangulation as the main quenching mechanisms.",
"Figure: Cartoon summarizing the main statistical trends reported in this work.",
"The green, blue, orange and red curves are the real measured SMFs of irregulars, disks, early-type disks and spheroids respectively.",
"The arrows indicate quenching.",
"Stamps are typical morphologies emerging at a given epoch.",
"Blue frames for star-forming and red for quiescent."
],
[
"Acknowledgements",
"Thanks to R. Sheth for comments on an early draft.",
"Thanks to the anonymous referee for a constructive and quick report."
],
[
"Morphologies",
"Figures REF to REF show postage stamps of a random subset set of galaxies in each of the 4 main morphological classes used in this work, over a range of different stellar masses and redshifts.",
"Figure: Postage stamps of galaxies classified as spheroids (∼Es\\sim Es) sorted by increasing stellar mass (vertical direction) and redshift (horizontal direction).Figure: Postage stamps of galaxies classified as disk+spheroids (∼\\sim S0s and early-type spirals) sorted by increasing stellar mass (vertical direction) and redshift (horizontal direction).Figure: Postage stamps of galaxies classified as disks (∼\\sim late-type spirals) sorted by increasing stellar mass (vertical direction) and redshift (horizontal direction).Figure: Postage stamps of galaxies classified as irregulars sorted by increasing stellar mass (vertical direction) and redshift (horizontal direction)."
],
[
"Bulge-to-total ratios of different morphologies",
"As an additional sanity check and given that many models use the stellar-mass disk to bulge ratio as a proxy for morphology, figure REF shows the stellar mass bulge-to-total (B/Ts) ratios for a subsample of galaxies from our dataset.",
"Bulge fractions are obtained by fitting a 2-component Sersic+exponential model on 7 HST filters (from near UV to NIR) simultaneously using Megamorph [51].",
"Sizes of both components and Sersic indices of the bulges are allowed to change with wavelength following a polynomial of order 2.",
"We then fit the 7 point SEDs of bulges and disks separately with BC03 templates and estimate the stellar masses of the two components separately.",
"While a detailed discussion of the procedure is beyond the scope of this paper (details are provided in Dimauro et al., in preparation), here we simply want to highlight the fact that the morphologies estimated independently with deep-learning do match the expected distribution of B/Ts reasonably well.",
"I.e., DISKs and IRRs tend to have bulge fractions smaller than 0.2 whereas SPHs have B/T greater than $\\sim 0.6$ .",
"DISKSPHs have a broader distribution of B/T values.",
"Note, however, that bulge/disk decompositions do not capture the irregularities in the light profile which are an important elements in this work.",
"Figure: Stellar mass bulge-to-total ratios for the different morphologies.",
"The visual morphologies defined in this work are compared for some galaxies to the distribution of stellar mass bulge-to-total ratios.",
"The expected trends are observed."
]
] | 1606.04952 |
[
[
"Distance geometry approach for special graph coloring problems"
],
[
"Abstract One of the most important combinatorial optimization problems is graph coloring.",
"There are several variations of this problem involving additional constraints either on vertices or edges.",
"They constitute models for real applications, such as channel assignment in mobile wireless networks.",
"In this work, we consider some coloring problems involving distance constraints as weighted edges, modeling them as distance geometry problems.",
"Thus, the vertices of the graph are considered as embedded on the real line and the coloring is treated as an assignment of positive integers to the vertices, while the distances correspond to line segments, where the goal is to find a feasible intersection of them.",
"We formulate different such coloring problems and show feasibility conditions for some problems.",
"We also propose implicit enumeration methods for some of the optimization problems based on branch-and-prune methods proposed for distance geometry problems in the literature.",
"An empirical analysis was undertaken, considering equality and inequality constraints, uniform and arbitrary set of distances, and the performance of each variant of the method considering the handling and propagation of the set of distances involved."
],
[
"Introduction",
"Let $G = (V, E)$ be an undirected graph.",
"A $k$ -coloring of $G$ is an assignment of colors $\\lbrace 1, 2, \\dots , k\\rbrace $ to the vertices of $G$ so that no two adjacent vertices share the same color.",
"The chromatic number $\\chi _G$ of a graph is the minimum value of $k$ for which $G$ is $k$ -colorable.",
"The classic graph coloring problem, which consists in finding the chromatic number of a graph, is one of the most important combinatorial optimization problems and it is known to be NP-hard [13].",
"There are several versions of this classic vertex coloring problem, involving additional constraints, in both edges as vertices of the graph, with a number of practical applications as well as theoretical challenges.",
"One of the main applications of such problems consists of assigning channels to transmitters in a mobile wireless network.",
"Each transmitter is responsible for the calls made in the area it covers and the communication among devices is made through a channel consisting of a discrete slice of the electromagnetic spectrum.",
"However, the channels cannot be assigned to calls in an arbitrary way, since there is the problem of interference among devices located near each other using approximate channels.",
"There are three main types of interferences: co-channel, among calls of two transmitters using the same channels; adjacent channel, among calls of two transmitters using adjacent channels and co-site, among calls on the same cell that do not respect a minimal separation.",
"It is necessary to assign channels to the calls such that interference is avoided and the spectrum usage is minimized [3], [15], [14].",
"Thus, the channel assignment scenario is modeled as a graph coloring problem by considering each transmitter as a vertex in a undirected graph and the channels to be assigned as the colors that the vertices will receive.",
"Some more general graph coloring problems were proposed in the literature in order to take the separation among channels into account, such as the T-coloring problem, also known as the Generalized Coloring Problem (GCP) where, for each edge, the absolute difference between colors assigned to each vertex must not be in a given forbidden set [12].",
"The Bandwidth Coloring Problem, a special case of T-coloring where the absolute difference between colors assigned to each vertex must be greater or equal a certain value [20], and the coloring problem with restrictions of adjacent colors (COLRAC), where there is a restriction graph for which adjacent colors in it cannot be assigned to adjacent vertices [1].",
"The separation among channels is a type of distance constraint, so we can see the channel assignment as a type of distance geometry (DG) problem [19] since we have to place the channels in the transmitters respecting some distances imposed in the edges, as can be seen in Figure REF .",
"One method to solve DG problems is the branch-and-prune approach proposed by [16], [17], where a solution is built and if at some point a distance constraint is violated, then we stop this construction (prune) and try another option for the current solution in the search space.",
"See also: [21], [16], [9], [10], [6], [8], [7].",
"For graph theoretic concepts and terminology, see the book by [4].",
"The main contribution of this paper consists of a distance geometry approach for special cases of T-coloring problems with distance constraints, involving a study of graph classes for which some of these distance coloring problems are unfeasible, and branch-prune-and-bound algorithms, combining concepts from the branch-and-bound method and constraint propagation, for the considered problems.",
"The remainder of this paper is organized as follows.",
"Section defines the distance geometry models for some special graph coloring problems.",
"Section shows some properties regarding the structure of those distance geometry graph coloring problems, including the determination of feasibility for some graphs classes.",
"Section formulates the branch-prune-and-bound (BPB) algorithms proposed for the problems and shows properties regarding optimality results.",
"Section shows results of some experiments done with the BPB algorithms using randomly generated graphs for each proposed model.",
"Finally, Section concludes the paper and states the next steps for ongoing research.",
"Figure: Example of channel assignment with distance constraints, wherethe separation is given by the weight in each edge.",
"Theimage on the right shows the network as an undirected graph and the projection of vertices on the real number line,but considering only natural numbers."
],
[
"Distance geometry and graph colorings",
"We propose an approach in distance geometry for special vertex coloring problems with distance constraints, based on the Discretizable Molecular Distance Geometry Problem (DMDGP), which is a special case of the Molecular Distance Geometry Problem, where the set $V$ of vertices from the input graph $G$ are ordered such that the set $E$ of edges contain all cliques on quadruplets of consecutive vertices, that is, any four consecutive vertices induce a complete graph ($\\forall i \\in \\lbrace 4, \\dots , n\\rbrace \\ \\forall j, k \\in \\lbrace i-3, \\dots , i\\rbrace \\ (\\lbrace j, k\\rbrace \\in E)$ ) [16].",
"Furthermore, a strict triangular inequality holds on weights of edges between consecutive vertices in such ordering ($\\forall i \\in \\lbrace 2, \\dots , n-1\\rbrace \\ d_{i-1, i+1} < d_{i-1, i}\\ +\\ d_{i, i+1}$ ).",
"All coordinates are given in $\\mathbb {R}^3$ space.",
"The position for a point $i$ (where $i \\ge 4$ ) can be determined using the positions of the previous three points $i-1, i-2$ and $i-3$ by intersecting three spheres with radii $d_{i-3,i}, d_{i-2, i}$ and $d_{i-1, i}$ , obtaining two possible points that are checked for feasibility.",
"A similar reasoning can be used in vertex coloring problems with distance constraints, where the distances that must be respected involve the absolute difference between two values $x(i)$ and $x(j)$ (respectively, the color points attributed to $i$ and $j$ ), but for these problems the space considered is actually unidimensional.",
"The positioning of a vertex $i$ can be determined by using a neighbor $j$ that is already positioned.",
"Thus, we have a 0-sphere, consisting of a projection of a 1-sphere (a circle), which itself is a projection of a 2-sphere (the three-dimensional sphere), as shown in Figure REF .",
"The 0-sphere is a line segment with a radius $d_{i,j}$ , and feasible colorings consist of treating the intersections of these 0-spheres.",
"Figure REF exemplifies the correlations between these types of spheres and shows the example from Figure REF as the positioning of these line segments.",
"Figure: Some types of nn-spheres.",
"A (n-1n-1)-sphere is a projection of a nn-sphere on a lower dimension.Figure: Example from Figure using 0-spheres (line segments).Figure: Specific order of 0-spheres that leads to the optimal solution for Figure .In this work we focus on problems with exact distances between colors, and also in the analysis of different types of BPB algorithms and integer programming models.",
"Based on DMDGP, which is a decision problem involving equality distance constraints, the basic distance graph coloring model we consider also involves equality constraints between colors of two neighbor vertices $i$ and $j$ .",
"That is, the absolute difference between them must be exactly equal to an arbitrary weight imposed on the edge $(i, j)$ , and the solution candidate must satisfy all given constraints.",
"We can formally define as follows.",
"Given a graph $G=(V,E)$ , we define $d_{i,j}$ as a positive integer weight associated to an edge $(i,j) \\in E(G)$ .",
"In distance coloring, for each vertex $i$ , a color must be determined for it (denoted by $x(i)$ ) such that the constraints imposed on the edges between $i$ and its neighbors are satisfied.",
"A variation of the classic graph coloring problem consists in finding the minimum span of $G$ , that is, in determining that the maximum $x(i)$ , or color used, be the minimum possible.",
"Based on these preliminary definitions, we describe the following distance geometry vertex coloring problems.",
"Definition 1 Coloring Distance Geometry Problem (CDGP): Given a simple weighted undirected graph $G = (V, E)$ , where, for each $(i, j) \\in E$ , there is a weight $d_{i,j} \\in \\mathbb {N}$ , find an embedding $x: V \\rightarrow \\mathbb {N}$ (that is, an embedding of $G$ on the real number line, but considering only the natural number points) such that $|x(i) - x(j)| = d_{i,j}$ for each $(i, j) \\in E$ .",
"CDGP involves equality constraints, and thus is named as Equal Coloring Distance Geometry Problem and labeled as EQ-CDGP.",
"A solution for this problem consists of a tree, whose vertices are colored with colors that respect the equality constraints involving the weighted edges (see Figure REF ).",
"Since CDGP (or EQ-CDGP) is a decision problem, only a feasible solution is required.",
"This problem is NP-complete, as shown below.",
"Theorem 1 EQ-CDGP is NP-complete.",
"To prove that EQ-CDGP $\\in $ NP-complete, we must show that EQ-CDGP $\\in $ NP and EQ-CDGP $\\in $ NP-hard.",
"1.",
"EQ-CDGP $\\in $ NP.",
"Given, for a graph $G = (V, E)$ , an embedding $x: V \\rightarrow \\mathbb {N}$ , its feasibility can be checked by taking each edge $(i, j) \\in E$ and examining if its endpoints do not violate the corresponding distance constraint, that is, if $|x(i) - x(j)| = d_{i, j}$ .",
"If all distance constraints are valid, then $x$ is a certificate for a positive answer to the EQ-CDGP instance, meaning that a certificate for a YES answer can be verified in $O(|E|)$ time, which is linear.",
"Thus, EQ-CDGP $\\in $ NP.",
"2.",
"EQ-CDGP $\\in $ NP-hard.",
"Since EQ-CDGP is equivalent to 1-Embeddability with integer weights, which is NP-hard [22], we can use the same proof for the latter problem to show that EQ-CDGP is also NP-hard.",
"The proof is made by reducing the Partition problem, which is known to be NP-complete [11] to EQ-CDGP.",
"Consider a Partition instance, consisting of a set $I$ of $r$ integers, that is, $M = \\lbrace m_1, m_2, \\dots , m_r\\rbrace $ .",
"Let $G$ be a weighted graph $G = (V, E)$ , where $G$ is a cycle such that $|V| = |E| = r$ and, for each edge $(i, j)$ , its weight is a natural number denoted by $d_{i, j}$ .",
"This graph is constructed from $M$ by considering: $V = \\lbrace i_0, i_1, \\dots , i_{r-1}\\rbrace .$ $E = \\lbrace (i_b, i_{b+1 \\text{ mod } r})\\ |\\ 0 \\le b \\le r\\rbrace .$ $d_{i_b, i_{b+1 \\text{ mod } r}} = m_b\\ (\\forall 0 \\le b \\le r).$ Now, let $x: V \\rightarrow \\mathbb {N}$ be an embedding of the vertices on the number line.",
"If it is a valid embedding, then we can define two sets: $S_1 = \\lbrace m_b\\ |\\ x(i_b) < x(i_{b+1 \\text{ mod } r})\\rbrace $ .",
"$S_2 = \\lbrace m_b\\ |\\ x(i_b) > x(i_{b+1 \\text{ mod } r})\\rbrace $ .",
"We have that $S_1$ and $S_2$ are disjoint subsets of $M$ (that is, they form a partition of $M$ ) where the sum of all $S_1$ elements is equal to the sum of all $S_2$ elements, that is, if the cyclic graph constructed from $G$ admits an embedding on the line (which means that its solution to EQ-CDGP is YES), then $M$ has a YES solution for Partition and vice-versa.",
"This reduction can be made in $O(r)$ time, thus, EQ-CDGP $\\in $ NP-hard.",
"To illustrate the reasoning from Theorem REF , let $M$ be an instance of Partition such that $M =\\lbrace 1, 4, 5, 6, 7, 9\\rbrace $ .",
"Figure REF shows its corresponding EQ-CDGP solution.",
"Figure: Partition instances and corresponding transformations to EQ-CDGP.Since most graph coloring problems in the literature and in real world applications are optimization problems, we define an optimization version of this basic distance geometry graph coloring problem, as shown below.",
"Definition 2 Minimum Equal Coloring Distance Geometry Problem (MinEQ-CDGP): Given a simple weighted undirected graph $G = (V, E)$ , where, for each $(i, j) \\in E$ , there is a weight $d_{i,j} \\in \\mathbb {N}$ , find an embedding $x: V \\rightarrow \\mathbb {N}$ such that $|x(i) - x(j)| = d_{i,j}$ for each $(i, j) \\in E$ whose span $S$ , defined as $S = \\max _{i \\in V} x(i)$ , that is, the maximum used color, is the minimum possible.",
"Figure REF shows an example of this model and its corresponding 0-sphere visualization.",
"Figure: MinEQ-CDGP instance with solution and its 0-sphere representation.On the other hand, instead of equalities, we can consider inequalities, such that the weight $d_{i,j}$ on an edge $(i, j)$ is actually a lower bound for the distance to be respected between the color points $x(i)$ and $x(j)$ , that is, $|x(i) - x(j)| \\ge d_{ij}$ .",
"Thus, we can modify MinEQ-CDGP to deal with this scenario, which becomes the following model.",
"Definition 3 Minimum Greater than or Equal Coloring Distance Geometry Problem(MinGEQ-CDGP): Given a simple weighted undirected graph $G = (V, E)$ , where, for each $(i, j) \\in E$ , there is a weight $d_{i,j} \\in \\mathbb {N}$ , find an embedding $x: V \\rightarrow \\mathbb {N}$ such that $|x(i) - x(j)| \\ge d_{i,j}$ for each $(i, j) \\in E$ whose span ($\\max \\limits _{i \\in V} x(i)$ ) is the minimum possible.",
"MinGEQ-CDGP is equivalent to the bandwidth coloring problem (BCP) [20], which itself is equivalent to the minimum span frequency assignment problem (MS-FAP) [14], [3].",
"Figure REF In Figure REF , this model, along with its 0-sphere representation, is exemplified.",
"Figure: MinGEQ-CDGP instance with solution and its 0-sphere representation."
],
[
"Special cases",
"For the models previously stated, we can identify some specific scenarios for which additional properties can be identified.",
"The first special case is for EQ-CDGP, the decision distance coloring problem, where all distances are the same, that is, the input is a graph with uniform edge weights, as stated below.",
"Definition 4 Coloring Distance Geometry Problem with Uniform Distances (EQ-CDGP-Unif): Given a simple weighted undirected graph $G = (V, E)$ , and a nonnegative integer $\\varphi $ , find an embedding $x: V \\rightarrow \\mathbb {N}$ such that $|x(i) - x(j)| = \\varphi $ for each $(i, j) \\in E$ .",
"For the optimization version, we can also define this special case, as shown below.",
"Definition 5 Minimum Equal Coloring Distance Geometry Problem with Uniform Distances (MinEQ-CDGP-Unif): Given a simple weighted undirected graph $G = (V, E)$ , and a nonnegative integer $\\varphi $ , find an embedding $x: V \\rightarrow \\mathbb {N}$ such that $|x(i) - x(j)| = \\varphi $ for each $(i, j) \\in E$ whose span ($\\max _{i \\in V} x(i)$ ) is the minimum possible.",
"In this model, an input graph can be defined by its sets of vertices and edges and the $\\varphi $ value, instead of a set of weights for each edge.",
"A similar special case exists for MinGEQ-CDGP, as stated in the following definition.",
"Definition 6 Minimum Greater than or Equal Coloring Distance Geometry Problem with Uniform Distances (MinGEQ-CDGP-Unif): Given a simple weighted undirected graph $G = (V, E)$ , and a nonnegative integer $\\varphi $ , find an embedding $x: V \\rightarrow \\mathbb {N}$ such that $|x(i) - x(j)| \\ge \\varphi $ for each $(i, j) \\in E$ whose span ($\\max _{i \\in V} x(i)$ ) is the minimum possible.",
"When $\\varphi =1$ , MinGEQ-CDGP-Unif is equivalent to the classic graph coloring problem (Figure REF ).",
"A summary of all distance coloring models, including special cases, is given in Table REF .",
"Table: Summary of distance coloring models.Figure: Examples of instances for the special cases of distance coloring models withconstant edge weights and feasible solutions for them."
],
[
"Feasibility conditions of distance graph coloring problems",
"In this section, we discuss feasibility conditions related to our proposed EQ-CDGP problems.",
"Clearly, the problems involving inequality constraints are always feasible.",
"This is the case for the MinGEQ-CDGP and MinGEQ-CDGP problems (and the special cases with uniform distances, MinGEQ-CDGP-Unif and MinGEQ-CDGP-Unif).",
"However, this is not so for versions that involve only equality constraints, EQ-CDGP and its special case with uniform distances, the EQ-CDGP-Unif problem."
],
[
"Feasibility conditions for EQ-CDGP-Unif",
"Graphs that admit a solution for the EQ-CDGP-Unif problem are characterized by the following theorem.",
"Theorem 2 A graph $G$ has solution YES for EQ-CDGP-Unif problem if and only if $G$ is bipartite.",
"Let $G$ be a graph, input to a EQ-CDGP-Unif problem, where for each edge $v_iv_j$ of $G$ , the distance required is $d_{ij}=\\varphi $ , $\\varphi \\in \\mathbb {N}$ , constant.",
"Suppose $G$ has a YES solution for the problem such that $x: V \\rightarrow \\mathbb {N}$ is a certificate for that solution.",
"Let $x(i)$ be the color assigned to $v_i \\in V$ .",
"Choose an arbitrary path $v_1,v_2,...,v_k$ of $G$ , not necessarily simple.",
"Then $|x(i)-x(j)|=\\varphi $ , for $|i-j|=1$ .",
"The latter implies $x(i)=x(i+2)$ , $i=1,2,...,k-2$ .",
"Consequently, if the path contains the same vector $v_i$ twice, their corresponding indices are the same.",
"That is, all edges of $G$ are necessarily even, and $G$ is bipartite.",
"Conversely, if $G$ is bipartite, its vertices admit a proper coloring with two distinct colors.",
"Assign the value $x(i)$ to the vertices of the first color, and the value $\\varphi + 1$ to the second one.",
"Then $|x(i)-x(j)|=\\varphi $ , for each edge $v_iv_j$ of $G$ , and EQ-CDGP-Unif has a YES solution.",
"As an alternative way of proving that if a graph is bipartite then it has a YES solution for EQ-CDGP, observe that, since the input graph is bipartite, it is also 2-colorable (considering the classic graph coloring problem), that is, the entire graph can be colored using only two different colors, which can be determined by considering a single edge from the graph.",
"All edges $(i, j)$ have the same distance constraint, that is, $|x(i) - x(j)| = \\varphi $ , so the two colors that will be used are {1, 1+$\\varphi $ }, which form the solution for the EQ-CDGP-Unif instance.",
"In order to prove the converse statement, that is, if a graph has a YES solution for EQ-CDGP, it is bipartite, we will use a proof by contrapositive, which states that if a graph is not bipartite, then it has a NO solution for EQ-CDGP.",
"This will be done by mathematical induction on odd cycles, since a graph is not bipartite if, and only if, it contains an odd cycle.",
"Let $|V| = 2z+1$ .",
"The proof will be by induction on $z$ (the number of vertices).",
"Base case: $z=1$ .",
"We have the cycle $C_3$ , with three vertices ($V = \\lbrace 1, 2, 3\\rbrace $ ) and three edges ($\\lbrace (1,2), (1,3), (2,3)\\rbrace $ ), with $|x(i)-x(j)| = \\varphi $ for all of them.",
"Without loss of generality, let $x(1) = 1$ and $x(2) = 1+\\varphi $ .",
"Then we have that: Since $(1,3) \\in E$ and $x(1) = 1$ , then $|x(3)-1| = \\varphi $ .",
"All colors must be positive integers, so $x(3) = 1+\\varphi $ .",
"Since $(2,3) \\in E$ and $x(2) = 1+\\varphi $ , then $|x(3)-(1+\\varphi )| = \\varphi \\ \\ \\Rightarrow \\ \\ |x(3)-1-\\varphi | = \\varphi $ .",
"By this inequation, $x(3) = 1$ or $x(3) = 1+ 2\\varphi $ .",
"From this result, we have that $x(3) = 1+\\varphi $ and ($x(3) = 1$ or $x(3) = 1+ 2\\varphi $ ) at the same time, which is impossible.",
"Then $C_3$ has a NO solution for EQ-CDGP, as seen in Figure REF .",
"Induction hypothesis: The cycle $C_{2z+1}$ has a NO solution for EQ-CDGP.",
"Inductive step: By the inductive hypothesis, the cycle $C_{2z+1}$ is infeasible for EQ-CDGP.",
"If we consider the cycle $C_{2(z+1)+1} = C_{2z+3}$ , we have that the size of the cycle increases by two vertices, but it will still be an odd cycle.",
"If we add only one vertex, that is, we consider the cycle $C_{2z+1+1} = C_{2z+2}$ , we will have an even cycle.",
"Since all even cycles are bipartite, they are feasible in EQ-CDGP according to Theorem REF .",
"Now, consider that another vertex is added to $C_{2z+2}$ , becoming $C_{2z+3}$ .",
"Without loss of generality, consider that the new vertex $2z+3$ is adjacent to vertices $2z+2$ and 1, that is, we have $\\lbrace (2z+2, 2z+3), (2z+3, 1)\\rbrace \\subseteq E$ , and $x(2z+2) = 1+\\varphi $ and $x(1) = 1$ (these colors can be seen as having been assigned when we added only one vertex, generating an even cycle).",
"Then we have that: Since $(2z+2, 2z+3) \\in E$ and $x(2z+2) = 1+\\varphi $ , then $|x(2z+3)-(1+\\varphi )| = \\varphi \\ \\ \\Rightarrow \\ \\ |x(2z+3)-1-\\varphi | = \\varphi $ .",
"By this inequation, $x(2z+3) = 1$ or $x(2z+3) = 1+ 2\\varphi $ .",
"Since $(2z+3, 1) \\in E$ and $x(1) = 1$ , then $|x(2z+3)-1| = \\varphi $ .",
"All colors must be positive integers, so $x(2z+3) = 1+\\varphi $ .",
"From this result, we have that $x(2z+3) = 1+\\varphi $ and ($x(2z+3) = 1$ or $x(2z+3) = 1+ 2\\varphi $ ) at the same time, which is impossible.",
"Therefore $C_{2z+3}$ has a NO solution for EQ-CDGP, as can be seen in Figure REF .",
"Figure: C 3 C_3 graph that has a NO solution for EQ-CDGP-Const when all distances are the same.Figure: Odd cycle C 2z+3 C_{2z+3} that has a NO solution for EQ-CDGP-Const when all distances are the same.As a complementary result, graphs which have odd-length cycles as induced subgraphs will always have a NO solution for EQ-CDGP-Unif, because a graph is bipartite if, and only if, it contains no odd-length cycles.",
"Since the recognition of bipartite graphs can be done in linear time using a graph search algorithm such as depth-first search (DFS), the EQ-CDGP-Unif problem can be solved in linear time."
],
[
"Feasibility conditions for EQ-CDGP",
"Clearly, Theorem REF does not apply when the distances are arbitrarily defined.",
"Depending on the edge weights, bipartite graphs may have NO solutions for EQ-CDGP, and graphs which include odd-length cycles may have YES solutions.",
"Figure REF shows examples of instances considering each case.",
"However, this decision problem can be easily solved for trees, as shown below.",
"Figure: Examples of instances for the special cases of distance coloring models withconstant edge weights and feasible solutions for them.Theorem 3 Let $G = (V, E, d)$ , be a tree, where $\\forall (i, j) \\in E\\ \\ d_{i, j}$ is an arbitrary positive integer.",
"Then $G$ always has a YES solution for EQ-CDGP.",
"We describe a simple algorithm for assigning colors that satisfy the EQ-CDGP problem.",
"Initially unmark all vertices.",
"Choose an arbitrary vertex $v_i$ , assign any positive integer value $x(v_i)$ to $v_i$ , and mark $v_i$ .",
"Iteratively, choose an unmarked vertex $v_j$ , adjacent to some marked vertex $v_k$ .",
"Assign the value $x(v_j)=d_{jk}+x(v_j)$ and mark $v_j$ .",
"Repeat the iteration until all vertices become marked.",
"The algorithm described in Theorem REF has linear time complexity.",
"It is important to note that when this procedure is applied to a MinEQ-CDGP instance, that is, the optimization problem with equality constraints, it only guarantees that a feasible solution is found for a tree, not the optimal one."
],
[
"Algorithmic techniques and methods to solve EQ-CDGP models",
"In this section, we show some algorithmic strategies to solve our distance geometry graph coloring models, and discuss some algorithmic strategies considering the EQ-CDGP models proposed in the previous section."
],
[
"Branch-prune-and-bound methods",
"For solving the three distance geometry graph coloring models shown in Section , we developed three algorithms that combine concepts from constraint propagation and optimization techniques.",
"A branch-and-prune (BP) algorithm was proposed by [16] for the Discretizable Molecular Distance Geometry Problem (DMDGP), based on a previous version for the MDGP by [18].",
"The algorithm proceeds by enumerating possible positions for the vertices that must be located in three-dimensional space ($\\mathbb {R}^3$ ), by manipulating the set of available distances.",
"The position for a vertex $i$ , where $i \\in [4, n]$ and $n$ is the number of vertices that must be placed in $\\mathbb {R}^3$ , is determined with respect to the last three vertices that have already been positioned, following the ordering and sphere intersection cited in Section .",
"However, a distance between the currently positioned vertex and a previous one that was placed before the last three can be violated, which requires feasibility tests to guarantee that the solution being built is valid.",
"The authors applied the Direct Distance Feasibility (DDF) pruning test, where $\\forall (i, j) \\in E\\ \\ |||x(i) - x(j)|| - d_{i, j}| < \\epsilon $ , and where $\\epsilon $ is a given tolerance.",
"In this work, we adapted these concepts to study and solve our proposed distance geometry coloring models.",
"One of the first reflections that can be made is that for the distance geometry coloring models, there are no initial assumptions to be respected, and thus, there is no explicit vertex ordering to be considered, so we build the ordering by an implicit enumerating process.",
"We mix concepts from branch-and-prune for DMDGP and branch-and-bound procedures to obtain partial solutions (sequences of vertices that have already been colored) that cannot improve on the current best solution.",
"Our branch-prune-and-bound (BPB) method works as follows.",
"First, a vertex $i$ that has not been colored yet is selected as a starting point.",
"This vertex receives the color 1, which is the lowest available (since all colors are positive integers).",
"Then a neighbor $j$ of $i$ that has not been colored yet is selected.",
"A color selection algorithm is used for setting a color to $j$ and the process is repeated recursively for neighbors of $j$ that have not been colored yet.",
"When an uncolored neighbor of the current vertex cannot be found, a uncolored vertex of the graph is used.",
"Pseudocode for this general procedure is given in Algorithm REF .",
"[t] Branch-prune-and-bound general algorithm.",
"0.9 [1] graph $G$ (with set $V$ of vertices and set $E$ of edges), function $d: E \\rightarrow \\mathbb {N}$ of distances for each edge, previous vertex $i$ , current vertex $j$ to be colored, current partial coloring $x$ , best complete coloring found $x_{best}$ , upper bound $ub$ , array $pred$ of predecessors from each vertex (initially all set to -1) and enumeration tree depth $dpt$ .",
"Branch-Prune-And-Bound$G = (V, E), d, i, j, x, x_{best}, ub, pred, dpt$ each neighbor $k$ of $j$ $predec[k] = -1$ $predec[k] \\leftarrow i$ Set current vertex as predecessor of neighbors $i = -1$ $i \\leftarrow predec[j]$ If this call did not come from a neighbor, use predecessor information $colorsAvail \\leftarrow $ SelectColors$G, d, i, j, x, ub$ $colorsAvail \\ne \\emptyset $ $color \\leftarrow $ element of $colorsAvail$ $colorsAvail \\leftarrow colorsAvail - \\lbrace color\\rbrace $ $x(j) \\leftarrow color$ $\\max \\limits _{v \\in V\\ |\\ v \\text{ is colored}} x(v) \\ge ub$ Remove color from $i$ continue Discard this possible partial solution by bounding FeasibilityTest$G, d, f, x, i$ = false Remove color from $i$ Distance violation, discard partial solution by pruning $dpt = |V|$ If true, then all vertices are colored $\\max \\limits _{v \\in V} x(v)$ $<$ $\\max \\limits _{v \\in V} x_{best}(v)$ $x_{best} \\leftarrow x$ $ub \\leftarrow \\max \\limits _{v \\in V} x(v)$ $hasNeighbor \\leftarrow $ false each neighbor $k$ of $j$ $k$ is not colored $hasNeighbor \\leftarrow $ true Branch-Prune-And-Bound$G, d, f, j, k, x, x_{best}, ub, dpt+1$ $hasNeighbor =$ false each vertex $k$ of $G$ such that $predec[k] \\ne -1$ Only from vertices with predecessors $k$ is not colored Branch-Prune-And-Bound$G, d, f, -1, k, x, x_{best}, ub, dpt+1$ Remove color from $i$ $x_{best}$ We propose different strategies for selecting a color for a vertex and illustrate how the feasibility checking can be done in different levels of the procedure.",
"Each of these cases are discussed below.",
"Color selection for a vertex There are two possibilities for determining which colors a vertex can use, determined by the call to SelectColors()), which returns a set of possible colors for a vertex.",
"The first one, denoted by BPB-Prev, is based on the original BP algorithm by [16].",
"When a vertex $i$ has to be colored, the single previously colored vertex $j$ is taken into account.",
"If $j$ is an invalid vertex, which means that $i$ is not an uncolored neighbor of $j$ , then the only color that $i$ can receive is 1.",
"Otherwise, the function returns a set of cardinality at most 2, whose elements are: $x(j) + d_{i, j}$ .",
"$x(j) - d_{i, j}$ (returned only if $x(j) > d_{i, j}$ ).",
"This means that this criterion uses only information from the previous vertex to determine colors, which makes the BPB that uses it an inexact algorithm, something that the original BP for DMDGP also is [16].",
"However, to counter this in our BPB, when a vertex is colored, its neighbors are marked so that they can use the current vertex as a predecessor in case the search restarts from one of such neighbors.",
"Since we assume the input graph is connected and the algorithm essentially walks through the graph, this information helps to find the true optimal solution.",
"This procedure is done in $O(1)$ time, since only two arithmetic operations are made to determine the colors.",
"An example of this color selection possibility is given in Figure REF .",
"When using this criterion, we apply the feasibility checking at each colored vertex.",
"However, an alternative is to prune only infeasible solutions where all vertices have colors, that is, we apply the feasibility test only at the last level of the enumeration tree.",
"An example of this alternative is shown in Figure REF , where it is possible to see that this strategy makes the tree grow very large.",
"The second selection criterion is undertaken using information from all colored neighbors to determine the color for the current vertex $i$ .",
"This is done by solving a system of absolute value inequalities (or equalities, in the case of MinEQ).",
"Those inequalities arise from the distance constraints for the edges.",
"Let $i$ be the vertex that must be colored.",
"The color $x(i)$ must be the solution of a system of absolute value (in)equalities where there is one for each colored neighbor $j$ and each one is as follows: $|x(j) - x(i)|\\ \\ \\text{OP}\\ \\ d_{i, j}$ Where OP is either “=” (for MinEQ-CDGP type problems) or “$\\ge $ ” (for MinGEQ-CDGP type problems).",
"The color that will be assigned to $j$ is the smallest value that satisfies all (in)equalities.",
"We note that this procedure always returns a set of cardinality 1, that is, only one color (since only the lowest index is returned) which is also feasible for the partial solution and eventually leads to the optimal solution, although it requires more work per vertex.",
"This selection strategy runs in $O(ub)$ time, where $ub$ is an upper bound for the span, since, to solve the system, we have to mark each possible solution in the interval $[1, ub]$ and select the smallest value.",
"Figure REF shows an example of an enumeration tree using this color selection strategy.",
"Figure: Partial enumeration of solutions starting from vertex 2 for the MinEQ-CDGP instance defined by Figures and using BPB-Prev, with color selection based only on the previous vertex and feasibility checking at each partial solution.Figure: Partial enumeration of solutions starting from vertex 2 for the MinEQ-CDGP instance defined by Figures and using BPB-Prev-FeasCheckFull with color selection based only on the previous vertex and feasibility checking only when all vertices are colored.The backtracking points are indicated when the solution is pruned.Figure: Partial enumeration of solutions starting from vertex 2 for the MinEQ-CDGP instance defined by Figures and using BPB-Select, where a color is determined using a system of absolute value expressions (equalities or inequalities).Feasibility checking When building a partial solution we must verify if it is feasible when not all distances are taken into account at the same time, especially on BPB-Prev.",
"We used a similar feasibility test to the Direct Distance Feasibility (DDF) used on the BP algorithm for the DMDGP.",
"Let $i$ be the vertex that has just been colored.",
"Then we must check, for each neighbor $j$ that has already been colored, if the condition $|x(i)-x(j)| \\ge d_{i, j}$ (if $f((i, j)) = 0$ ) or $|x(i)-x(j)| = d_{i, j}$ (if $f((i, j)) = 1$ ).",
"This test can be seen as a variation of DDF setting $\\epsilon $ to zero and allowing inequalities in the test.",
"We denote this procedure as Direct Distance Coloring Feasibility (DDCF) and its pseudocode is given in Algorithm REF .",
"Direct Distance Coloring Feasibility (DDCF) check 0.9 [1] graph $G$ (with set $V$ of vertices and set $E$ of edges), problem type $t$ (MinGEQ-CDGP or MinEQ-CDGP), matrix $d$ of distances for each edge, current coloring $x$ and vertex $i$ .",
"DDCF-Check$G, d, f, x, i$ each neighbor $k$ of $i$ $k$ is colored $t$ = MinGEQ-CDGP Inequality constraint $|x(k)-x(i)| > d_{i,j}$ false Equality constraint $|x(k)-x(i)| \\ne d_{i,j}$ false true We note that when selecting a color using the first criterion (only taking into account the previously colored vertex) the feasibility check can be made at each colored vertex or only when all vertices have been colored (which will require that the function DDF-Check() is called for each vertex).",
"Each check (for only one vertex) runs in $O(|V|)$ time, and if the entire coloring is checked (that is, for all vertices), it runs in $O(|V|^2)$ time.",
"We also note that, when using the second criterion (using a system of absolute value (in)equalities), the feasibility check can be skipped, since the color that it returns is always feasible.",
"The combination of these selection criteria and the corresponding feasibility checks result in three possible BPB algorithms, which are summarized in Table REF .",
"Table: Summary of branch-prune-and-bound methods."
],
[
"Computational experiments",
"In order to analyze the behavior of the proposed distance geometry coloring problems and the branch-prune-and-bound algorithms, we made two main sets of experiments: the first one involved generating many random graphs with different numbers of vertices according to some configurations and counting how many include even or odd cycles (while the rest are trees), since some of the properties of distance geometry coloring are related to these types of graphs.",
"All algorithms used in these experiments were implemented in C language (compiled with gcc 4.8.4 using options -Ofast -march=native -mtune=native) and executed on a computer equipped with an Intel Core i7-3770 (3,4GHz), 8GB of memory and Linux Mint 17 operating system.",
"We describe each set of experiments below."
],
[
"Counting members of graph classes in random instances",
"Using Theorems REF and REF , we have information about some types of graphs which always have feasible embeddings for EQ-CDGP and EQ-CDGP-Unif.",
"Based on this, we generated a large amount of random graphs with different number of vertices and counted how many were cyclic (and based on that, how many there were for each possibility of having even or odd cycles) and how many were trees.",
"Each random graph always starts as a random spanning tree, that is, a connected undirected graph $G = (V, E)$ , where $|E| =|V|-1$ .",
"To generate this initial tree, we used a random walk algorithm proposed independently by [5] and [2].",
"The procedure works by using a set $V^*$ of the vertices outside the tree and a set $W$ of edges of the spanning tree.",
"Then, whenever the random walk reaches a vertex $j$ outside the tree, the edge $(i, j)$ is added to $E$ and $j$ is removed from $V^*$ .",
"This continues until $V^* = \\varnothing $ .",
"We note that this amounts to making a random walk in a complete graph of $|V|$ vertices and it generates trees in a uniform manner, that is, for all possible spanning trees of a given complete graph, each one has the same probability of being generated by the algorithm.",
"After the initial tree is generated, we add random new edges to it until the graph has the desired number of edges.",
"This parameter is also randomly set, sampled from interval $\\left[|V|-1, \\frac{|V| (|V|-1)}{2}\\right]$ .",
"This interval ensures that the generated graph is always connected and is, at least, a tree and, at most, a complete graph.",
"In Table REF , we outline statistics obtained from using the described procedure to generate 1,000,000 (one million) random graphs for each $|V| \\in \\lbrace 50, 100, 150, 200, 250, 300, 350, 400, 450, 500\\rbrace $ .",
"As we can observe, most of the graphs (more than 99%) generated have odd cycles, which translates into a very small set of possible EQ-CDGP-Unif instances with feasible embeddings for this configuration of random graphs.",
"By increasing the number of vertices, more possibilities for generating edges appear, but the number of possible connections which will lead to trees or graphs with even cycles is very small.",
"In fact, we can deduce that this configuration generated very few bipartite graphs.",
"For EQ-CDGP (with arbitrary distances), the space of instances with guaranteed feasible embeddings is even smaller, since only trees are certain to have them.",
"However, as shown in Section , odd and even cycles can have embeddings depending on how the edges are weighted.",
"In Figure REF , we can observe the growth of the average number of edges between all generated graphs for each number of vertices.",
"Since the number of edges in a graph is proportional to the square of the number of edges (since $\\frac{|V| (|V|-1)}{2}\\ \\in \\ O(|V|^2)$ , the curve follows a similar pattern, being a half parabola.",
"Table: Number of random graphs with even, odd or no cycles (trees) and bipartite graphs generated for each number of vertices.",
"For each size,1,000,000 graphs were generated.Figure: Growth of the average number of edges generated when the number of vertices increases.Figure: Total number of bipartite graphs generated from 1,000,000 random graphs of each number of vertices."
],
[
"Results for branch-prune-and-bound algorithms",
"In order to use some of the random graphs in experiments with the BPB algorithms, we selected four graphs of each type (with even cycle, with odd cycle and trees) for each number of vertices and randomly weighted the edges with a uniform distribution in the interval $[1, 30]$ .",
"We made two subsets of experiments: the first one involved only the EQ-CDGP and EQ-CDGP-Const models, in order to find a feasible solution for each of its instances that were generated (that is, the algorithms use the pruning procedure, but not bounding - equivalently, stopping the search as soon as the first feasible solution is found), and the second one involved the optimization models for each discussed model (MinEQ-CDGP, MinEQ-CDGP-Const, MinGEQ-CDGP and MinGEQ-CDGP-Const).",
"Tables REF , REF and REF give detailed results with 4 to 8, 9 to 16 and 18 to 20 vertices, respectively, considering each BPB algorithm applied to the decision versions.",
"We can see that BPB-Prev reaches a feasible solution faster than the other methods (that is, it solves the decision problem in less time), but it also returns the first feasible solution with a worst span than BPB-Select (noting that BPB-Prev-FeasCheckFull always returns the same span from BPB-Prev because only the pruning point is changed).",
"However, it is much slower to prove that an instance is infeasible (that is, the answer to the decision problem is NO).",
"This is explained by the fact that the enumeration tree is smaller in BPB-Select, since instead of two color possibilities for each vertex, there is only one.",
"Although the time complexity of determining the color for a vertex in BPB-Select is higher (as shown in Table REF ), this is compensated by generating a smaller tree and that the feasibility check is not explicitly needed, since it is guaranteed by the color selection algorithm.",
"We also note that BPB-Prev-FeasCheckFull has the worst CPU times for infeasible instances, because the method will keep branching the enumeration tree to find a feasible solution, but since feasibility checking is only done at the leaf nodes, the tree will tend to become the full enumeration tree.",
"In the same manner, Table REF shows the results for the BPB algorithms considering the optimization versions and applied only to feasible MinEQ-CDGP instances.",
"For almost all of these instances, BPB-Prev is the best method, BPB-Prev-FeasCheckFull shows similar performance and BPB-Select has worse CPU times.",
"The ties between BPB-Prev and BPB-Prev-FeasCheckFull are explained by the fact that although, in the latter method, the feasibility checking at the leaf nodes increases the time required to find a feasible solution, we keep using the bounding procedures at each node, which reduces the amount of work needed to find the optimal solution.",
"We also note that, for the 4th Tree instance with 20 vertices, BPB-Select does not find the true optimum for the problem.",
"This happens because the method is recursively applied only to vertices which have recorded neighbors, in the same manner as the other two BPBs, but the system of absolute value expressions returns only one color, which may not be the one for the optimal solution when applying the recursion only on some vertices.",
"On some experiments, we detected that, if we generate all vertice orders and apply the color selection of BPB-Select, on them, the optimal solution is found, but the CPU times become very high, since this procedure does not take advantage of the 0-sphere intersection characteristic.",
"The last experiments were made by applying the BPB algorithms on all instances considered for the algorithms, but by transforming them to MinGEQ-CDGP (changing the $=$ in constraints to $\\ge $ ).",
"Since they are always feasible for MinGEQ-CDGP because of its equivalence to BCP, we only have to consider optimization problems, as was already done in Section .",
"The same pattern of previous experiments occur here, with BPB-Prev being the best method of the three, however, the CPU time difference between it and BPB-Prev-FeasCheckFull becomes much more apparent here, since there are many more feasible solutions for MinGEQ-CDGP than MinEQ-CDGP.",
"Table: Results for BPB algorithms (optimization) applied to MinEQ-CDGP instances - 4 to 20 vertices.Table: Results for BPB algorithms (optimization) applied to MinGEQ-CDGP instances - 4 to 10 vertices.Table: Results for BPB algorithms (optimization) applied to MinEQ-CDGP instances - 12 to 20 vertices."
],
[
"Concluding remarks",
"In this paper, we proposed some distance geometry models for graph coloring problems with distance constraints that can be applied to important, modern real world applications, such as in telecommunications for channel assignment in wireless networks.",
"In these proposed coloring distance geometry problems (CDGPs), the vertices of the graph are embedded on the real line and the coloring is treated as an assignment of natural numbers to the vertices, while the distances correspond to line segments, whose objective is to determine a feasible intersection of them.",
"We tackle such problems under the graph theory approach, to establish conditions of feasibility through behavior analysis of the problems in certain classes of graphs, identifying prohibited structures for which the occurrence indicates that it can not admit a valid solution, as well as identifying classes graphs that always admit valid solution.",
"We also developed exact and approximate enumeration algorithms, based on the Branch-and-Prune (BP) algorithm proposed for the Discretizable Molecular Distance Geometry Problem (DMDGP), combining concepts from constraint propagation and optimization techniques, resulting in Branch-Prune-and-Bound algorithms (BPB), that handle the set of distances in different ways in order to get feasible and optimal solutions.",
"The computational experiments involved equality and inequality constraints and both uniform and arbitrary sets of distances, where we measure the number of prunes and bounds and the CPU time needed to reach the best solution.",
"The main contribution of this paper consists of a distance geometry approach for special cases of T-coloring problems with distance constraints, involving a study of graph classes for which some of these distance coloring problems are unfeasible, and branch-prune-and-bound algorithms, combining concepts from the branch-and-bound method and constraint propagation, for the considered problems.",
"Ongoing and future works include improving the BPB formulations by domain reduction and more specific constraints; developing hybrid methods, involving integer and constraint programming; and applying heuristics, in order to solve the proposed distance coloring models more efficiently.",
"Studying problems posed to specific classes of graphs, in order to establish other characterizations of feasibility conditions for more general CDGP problems, is also a subject of research in progress.",
"Acknowledgement We would like to thank Dr. Leo Liberti, professor at LIX - Ãcole Polytechnique, France, for his valuable contribution to this work, having identified a relationship between coloring problems in graphs with additional constraints on the edges and distance geometry problems, and for discussing some of the ideas presented here."
]
] | 1606.04978 |
[
[
"Global adiabaticity and non-Gaussianity consistency condition"
],
[
"Abstract In the context of single-field inflation, the conservation of the curvature perturbation on comoving slices, $\\R_c$, on super-horizon scales is one of the assumptions necessary to derive the consistency condition between the squeezed limit of the bispectrum and the spectrum of the primordial curvature perturbation.",
"However, the conservation of $\\R_c$ holds only after the perturbation has reached the adiabatic limit where the constant mode of $\\R_c$ dominates over the other (usually decaying) mode.",
"In this case, the non-adiabatic pressure perturbation defined in the thermodynamic sense, $\\delta P_{nad}\\equiv\\delta P-c_w^2\\delta\\rho$ where $c_w^2=\\dot P/\\dot\\rho$, usually becomes also negligible on superhorizon scales.",
"Therefore one might think that the adiabatic limit is the same as thermodynamic adiabaticity.",
"This is in fact not true.",
"In other words, thermodynamic adiabaticity is not a sufficient condition for the conservation of $\\R_c$ on super-horizon scales.",
"In this paper, we consider models that satisfy $\\delta P_{nad}=0$ on all scales, which we call global adiabaticity (GA), which is guaranteed if $c_w^2=c_s^2$, where $c_s$ is the phase velocity of the propagation of the perturbation.",
"A known example is the case of ultra-slow-roll(USR) inflation in which $c_w^2=c_s^2=1$.",
"In order to generalize USR we develop a method to find the Lagrangian of GA K-inflation models from the behavior of background quantities as functions of the scale factor.",
"Applying this method we show that there indeed exists a wide class of GA models with $c_w^2=c_s^2$, which allows $\\R_c$ to grow on superhorizon scales, and hence violates the non-Gaussianity consistency condition."
],
[
"Introduction",
"A period of accelerated expansion during the early stages of the evolution of the Universe, called inflation [1], [2], [3], is able to account for several otherwise difficult to explain features of the observed Universe such the high level of isotropy of the cosmic microwave background (CMB) [4] radiation and the small value of the curvature.",
"Some of the simplest inflationary models are based on a single slowly-rolling scalar field, and they are in good agreement with observations.",
"It is commonly assumed in slow-roll models that adiabaticity in the thermodynamic sense, $\\delta P_{nad}\\equiv \\delta P-c_w^2\\delta \\rho =0$ where $c_w^2=\\dot{P}/\\dot{\\rho }$ , implies the conservation of the curvature perturbation on uniform density slices $\\zeta $ , and hence the conservation of the curvature perturbation on comoving slices ${\\mathcal {R}}_c$ , on super-horizon scales.",
"In [5] it was shown that there can be important exceptions, i.e.",
"in some cases thermodynamic adiabaticity does not necessarily imply the super-horizon conservation of ${\\mathcal {R}}_c$ and $\\zeta $ , and that they can differ from each other.",
"This can happen even for models in which $c_w^2=c_s^2$ .",
"An example is ultra-slow-roll (USR) inflation [6], [7], which has exact adiabaticity $\\delta P_{nad}=0$ on all scales.",
"In USR inflation, both ${\\mathcal {R}}_c$ and $\\zeta $ exhibit super-horizon growth but their behavior is very different from each other.",
"As has been stressed in [8], the non-freezing of ${\\mathcal {R}}_c$ has important phenomenological consequences.",
"Since the freezing of ${\\mathcal {R}}_c$ on superhorizon scales is a necessary ingredient [9] for Maldacena's consistency relation [10] to hold, models that do not conserve ${\\mathcal {R}}_c$ can actually violate that consistency condition.",
"In this paper focusing on K-inflation, i.e., Einstein-scalar models with a general kinetic term, we explore in a general way other single field models which have $c_w^2=c_s^2$ , hence satisfy $\\delta P_{nad}=0$ on all scales which we call globally adiabatic (GA), but which may not conserve ${\\mathcal {R}}_c$ .",
"We find a generalization of the USR model.",
"A different generalization without imposing the condition $c_w^2=c_s^2$ was discussed in [11], [12].",
"The method we adopt is based on establishing a general condition for the non-conservation of ${\\mathcal {R}}_c$ in terms of the dependence of the background quantities, in particular the slow-roll parameter $\\epsilon \\equiv -\\dot{H}/H^2$ and the sound velocity $c_s$ , on the scale factor $a$ .",
"We first derive the necessary condition for the comoving curvature perturbation ${\\mathcal {R}}_c$ to grow on superhorizon scales.",
"Next we determine $\\rho (a)$ and $P(a)$ by solving the continuity equation.",
"Then using the equivalence between barotropic fluids and $K$ -inflationary models which satisfy the condition $c_w^2=c_s^2$ [13], [14], we determine the corresponding Lagrangian for the equivalent scalar field model.",
"Using this method we obtain a new class of GA scalar field models which do not conserve ${\\mathcal {R}}_c$ .",
"Throughout the paper we denote the proper-time derivative by a dot ($\\dot{~}=d/dt$ ), the conformal-time derivative by a prime ($\\prime {~}=d/d\\eta =a\\,d/dt$ ) and the Hubble expansion rates in proper and conformal times by $H=\\dot{a}/a$ and ${\\cal H}=a^{\\prime }/a$ , respectively.",
"We also use the terminology “adiabaticity” for thermodynamic adiabaticity $\\delta P_{nad}=0$ throughout the paper."
],
[
"Conservation of ${\\mathcal {R}}_c$ and global adiabaticity",
"We set the perturbed metric as $ds^2&=&a^2\\Bigl [-(1+2A)d\\eta ^2+2\\partial _jB dx^jd\\eta \\nonumber \\\\&& \\qquad +\\left\\lbrace \\delta _{ij}(1+2{\\cal R})+2\\partial _i\\partial _j E\\rbrace dx^idx^j\\right\\rbrace \\Bigr ]\\,.$ In [5] it was shown that independently of the gravity theory and for generic matter the energy-momentum conservation equations imply $\\delta P_{nad}=\\left[\\left(\\frac{c_w}{c_s}\\right)^2-1\\right](\\rho +P) A_c \\,,$ where the subscript $c$ means a quantity evaluated on comoving slices defined by $\\delta T^i_0=0$ (or equivalently slices on which the scalar field is homogeneous).",
"In the case of general relativity, the additional relation $A_c=\\dot{{\\mathcal {R}}}_c/H$ gives an important relation for the time derivative of ${\\mathcal {R}}_c$ $\\delta P_{ nad}=\\left[\\left(\\frac{c_w}{c_s}\\right)^2-1\\right](\\rho +P)\\frac{\\dot{{\\mathcal {R}}_c}}{H}.",
"$ The non-adiabatic pressure perturbation is given according to its thermodynamics definition $\\delta P_{nad}\\equiv \\delta P - c_w^2 \\delta \\rho .$ This definition of $\\delta P_{nad}$ is important because it is gauge invariant and $\\delta P_{nad}=\\delta P_{ud}$ , where $\\delta P_{ud}$ is the pressure perturbation on uniform density slices.",
"It appears in the equation for the curvature perturbation on uniform density slices $\\zeta \\equiv {\\mathcal {R}}_{ud}$ obtained from the energy conservation law [15], $\\zeta ^{\\prime }=-\\frac{{\\cal H}\\delta P_{nad}}{(\\rho +P)}+\\frac{1}{3}\\mathop \\Delta ^{(3)}\\left(v-E^{\\prime }\\right)_{ud}\\,$ where $v$ is the 3-velocity potential ($v=\\delta \\phi /\\phi ^{\\prime }$ for a scalar field).",
"In general, the curvature perturbations on uniform density and comoving slices are related as $\\zeta ={\\mathcal {R}}_c+\\frac{\\delta P_{ nad}}{3(\\rho +P)(c^2_s-c^2_w)} \\,.$ A common interpretation of these equations (see for example [16], [17]) is that when $\\delta P_{nad}=0$ with $c_w^2 \\ne c_s^2$ , $\\zeta $ and ${\\mathcal {R}}_c$ are equal because of eq.",
"(REF ), and they are both conserved on super-horizon scales because of eq.",
"(REF ).",
"The equation (REF ) is the key relation to understand how ${\\mathcal {R}}_c$ depends on the non-adiabatic pressure $\\delta P_{nad}$ .",
"First of all let us note that this equation is valid on any scale.",
"The advantage of it with respect to eq.",
"(REF ) is that it does not involve gradient terms, so it allows us to directly relate $\\delta P_{\\rm nad}$ to $\\dot{{\\mathcal {R}}}_{c}$ if $c_w^2\\ne c_s^2$ , while in eq.",
"(REF ) $\\dot{\\zeta }$ depends on spatial gradients, which in the case of USR are not negligible on super-horizon scales [5].",
"This explains while in USR in which $c_w^2=c_s^2=1$ , both ${\\mathcal {R}}_c$ and $\\zeta $ are not conserved despite $\\delta P_{nad}=0$ .",
"It should be noted here that for slow-roll attractor models $c_w^2\\ne c_s^2$ in general, and ${\\mathcal {R}}_c$ is time-varying on sub-horizon scales.",
"This implies that the non-adiabatic pressure perturbation $\\delta P_{nad}$ on sub-horizon scales is not zero.",
"In other words, the attractor models are adiabatic only on super-horizon scales, and we call these models super-horizon adiabatic (SHA).",
"From eq.",
"(REF ) we can immediately deduce that in general relativity there are two possible scenarios for the non-conservation of ${\\mathcal {R}}_c$ , $\\mbox{(1)}\\quad &c_s^2=c_w^2\\,,\\quad \\delta P_{nad}=0\\,,\\nonumber \\\\\\mbox{(2)}\\quad &c_s^2\\ne c_w^2\\,,\\quad \\delta P_{nad}\\ne 0 \\, .$ The second case was studied in [11], [12].",
"Here we focus on the first case.",
"It is trivial to see that because of the gauge invariance of $\\delta P_{nad}$ the condition $c_w^2=c_s^2$ automatically implies $\\delta P_{nad}=0$ .",
"The models satisfying the condition $c_s^2-c_w^2=\\delta P_{nad}=0$ are adiabatic on any scale, and because of this we call them globally adiabatic (GA).",
"In GA models an explicit calculation can reveal the super-horizon behavior of ${\\mathcal {R}}_c$ , and $\\zeta $ , as was shown in [5] in the case of USR.",
"Below, we develop an inversion method to find a new class of models that violate the conservation of ${\\mathcal {R}}_c$ without solving the perturbations equations."
],
[
"Globally adiabatic K-essence models",
"The condition $c_w^2=c_s^2$ has been studied in the context of K-inflation [13] described by the action $S=\\frac{1}{2}\\int d^4x\\sqrt{-g}\\left[M^2_{Pl}R+2P(X,\\phi )\\right] \\,,$ and it was shown that it is satisfied by scalar field models with the Lagrangian of the form, $P(X,\\phi )=u(X g(\\phi ))=u(Y)\\,.",
"$ These models are equivalent to a barotropic perfect fluid, i.e.",
"a fluid with equation of state $P(\\rho )$ .",
"See also [18], [19], [20], [21].",
"We note again that these models are adiabatic on any scale (GA), contrary to the slow-roll attractor models, which are adiabatic only on super-horizon scales (SHA).",
"The fact that they are mutually exclusive can be readily seen by considering the hypothetical case of $\\delta P_{nad}=0$ and $c_w^2\\ne c_s^2$ .",
"In this case Eq.",
"(REF ) which is valid on any scale would mean ${\\mathcal {R}}_c$ should be frozen on all scales.",
"In contrast, the condition $c_w^2=c_s^2$ allows for the curvature perturbation to evolve both on sub-horizon and super-horizon scales.",
"In [13] it was shown that is possible to associate any barotropic perfect fluid with an equivalent K-inflation model according to $2\\int ^P\\frac{du}{F(u)}=\\log (Y) \\,, $ where $F(P)=\\rho (P)+P$ and $Y=g(\\phi )X$ with $X=-g^{\\mu \\nu }\\partial _\\mu \\phi \\partial _\\nu \\phi /2$ .",
"These models are the ones which could violate the conservation of ${\\mathcal {R}}_c$ for adiabatic perturbations, since they satisfy $c_w^2=c_s^2$ .",
"It is noted of course that the global adiabaticity is not the sufficient condition for the non-conservation of ${\\mathcal {R}}_c$ .",
"Not all GA models violate the conservation of ${\\mathcal {R}}_c$ on super-horizon scales."
],
[
"General conditions for super-horizon growth of ${\\mathcal {R}}_c$",
"From the equation for the curvature perturbation on comoving slices, $\\frac{\\partial }{\\partial t}\\left(\\frac{a^3\\epsilon }{c_s^2}\\frac{\\partial }{\\partial t}{\\mathcal {R}}_c\\right)-a\\epsilon \\Delta {\\mathcal {R}}_c=0 \\,,$ we can deduce, after re-expressing the time derivative in terms of the derivative respect to the scale factor $a$ , that on superhorizon scales there is (apart from a constant solution) a solution of the form, ${\\mathcal {R}}_c &\\propto & \\int ^a\\frac{da}{a} f(a)\\,;\\quad f(a)\\equiv {\\frac{c_s^2(a)}{Ha^3 \\epsilon (a)}} \\,, $ where we have introduced the function $f(a)$ for later convenience.",
"In conventional slow-roll inflation $c_s^2$ and $\\epsilon $ are both slowly varying, hence the integral rapidly approaches a constant, rendering ${\\mathcal {R}}_c$ conserved.",
"The time dependent part of the above solution corresponds to the decaying mode.",
"The necessary and sufficient condition for super-horizon freezing is that there exists some $\\delta >0$ for which $\\lim _{a\\rightarrow \\infty }a^\\delta f(a)=0 .",
"$ By definition of inflation, $H$ must be sufficiently slowly varying; $\\epsilon =-\\dot{H}/H^2\\ll 1$ .",
"So we may neglect the time dependence of $H$ in (REF ) at leading order, while $\\epsilon $ and $c_s^2$ may vary rapidly in time.",
"For models for which $\\epsilon \\approx a^{-n}$ and $c_s^2\\approx a^q$ we get $f \\propto a^{q+n-3} \\,,$ hence the condition for freezing is $q+n-3<0 \\,.$ If this condition is violated, i.e.",
"$q+n-3\\ge 0$ , then the solution (REF ) will grow on super-horizon scales.",
"This happens for example in USR., which corresponds to $c_s^2=1$ and $\\epsilon \\propto a^{-6}$ , i.e.",
"$q=0$ , and $n=6$ .",
"(The super-horizon growth of ${\\mathcal {R}}_c$ in USR can also be understood as a direct consequence of the non-attractor nature of USR [22].)",
"In general, we expect that $q$ would not become very large.",
"This implies $\\epsilon $ should decrease sufficiently rapidly.",
"Conversely, if $\\epsilon $ decreases sufficiently rapidly, then the growth of ${\\mathcal {R}}_c$ on superhorizon scales will follow."
],
[
"Barotropic model",
"We have shown that GA models could violate the super-horizon conservation of ${\\mathcal {R}}_c$ , so now we will look for GA K-essence models which do indeed violate it, based on the freezing condition in eq.",
"(REF ).",
"Inspired by the equivalence between barotropic fluids and GA K-essence models [13] we will first look for barotropic fluids that can give the growing curvature perturbation on superhorizon scales.",
"From the very beginning we will set $c_w^2=c_s^2$ .",
"Using the Friedmann equation we can write the slow-roll parameter $\\epsilon $ as $\\epsilon = -\\frac{\\dot{H}}{H^2}=\\frac{3}{2}\\frac{\\rho +P}{\\rho } \\,.$ In terms of the scale factor and $\\epsilon $ the energy conservation equation reads $\\frac{d\\rho }{da}+\\frac{3}{a}(\\rho +p)=\\frac{d\\rho }{da}+\\frac{2\\epsilon \\rho }{a}=0 .$ We may now define the quantity $b(a)=2\\epsilon \\rho $ .",
"It appears naturally in the continuity equation and plays a crucial role in regards to the super-horizon behavior of curvature perturbations because the function $f(a)$ can be re-written in terms of it as $f(a)\\propto \\frac{H c_s^2}{a^3 b(a)} \\,.$ Integrating the energy conservation equation we get $\\rho (a)=\\rho _0\\exp \\left[-2\\int _{a_0}^a{\\frac{\\epsilon }{a}da}\\right]=\\int {-\\frac{b(a)}{a} da} \\,.$ Using eq.",
"(REF ), we then obtain $P(a)=\\left(\\frac{2}{3}\\epsilon -1\\right)\\rho \\,.$ The sound velocity is given by $c_w^2=c_s^2=\\frac{dP}{d\\rho }&=-1+\\frac{1}{3}\\frac{db(a)}{d\\rho }\\nonumber \\\\&=-1+\\frac{1}{3}\\frac{db(a)}{da}\\Big /\\left(\\frac{d\\rho }{da}\\right)\\nonumber \\\\&=-1-\\frac{a}{3b(a)}\\frac{db(a)}{da}\\,.$ We now consider the behavior of $f(a)$ introduced in (REF ).",
"As mentioned before, we consider the case when $\\epsilon $ decreases sufficiently rapidly.",
"In this case, $\\rho =3H^2M_P^2$ approaches a constant rapidly.",
"Hence the time dependence of $\\rho $ may be neglected compared to that of other quantities that vary far more rapidly.",
"With this approximation, assuming $\\epsilon \\propto a^{-n}$ , we find $c_s^2\\approx \\frac{n-3}{3}\\,,$ which means $q\\approx 0$ , and $f(a)={\\frac{c_s^2(a)}{Ha^3 \\epsilon (a)}}\\propto a^{n-3}\\,,$ which satisfies the condition for the growth if $n>3$ , in accordance with the original anticipation.",
"In passing, it is interesting to note that the condition $n>3$ implies $c_s^2>0$ , a necessary condition to avoid the gradient instability of the perturbation.",
"Thus virtually all GA models that are free from the gradient instability exhibit superhorizon growth of the comoving curvature perturbation ${\\mathcal {R}}_c$ ."
],
[
"Scalar field model",
"Let us now find a scalar field model that corresponds to the barotropic model discussed in the previous section.",
"As a warm-up, let us consider the USR case, whose fluid interpretation has already been studied in [23].",
"In this case, we exactly have $c_s^2=1$ .",
"From eq.",
"(REF ), this implies $b/2=\\epsilon \\rho \\bigl (=3(\\rho +P)/2\\bigr )\\propto a^{-6}$ .",
"Also $c_s^2=1$ implies $\\rho =P+const.$ Inserting this into eq.",
"(REF ) gives $\\frac{2dP}{2P+const.",
"}=\\frac{dY}{Y}\\,.$ Thus up to a constant term $P$ and $Y$ are the same, $P=Y+const..$ Absorbing $g(\\phi )$ in $Y$ into the definition of the scalar field by $g^{1/2}d\\phi \\rightarrow d\\phi $ , this is indeed the Lagrangian for a minimally coupled massless scalar with a cosmological constant: $L=P(\\phi ,X)=X-V_0\\,.",
"$ This is consistent with $\\rho +P=2X\\propto \\epsilon \\rho \\propto a^{-6}$ .",
"Let us generalize the USR case.",
"As in the previous section, we consider models that have the behavior of $\\epsilon \\rho $ as $2\\epsilon \\rho =b(a)\\,,$ where $b(a)$ should decrease faster than $a^{-3}$ asymptotically at $a\\rightarrow \\infty $ but otherwise is an arbitrary function.",
"Then we have $F(P)\\equiv \\rho +P=2H^2\\epsilon =\\frac{2\\epsilon \\rho }{3}=\\frac{b(a)}{3}\\,,$ which gives $\\frac{dY}{Y}=2\\frac{dP}{F(P)}=6\\frac{dP}{2\\epsilon \\rho }=6\\frac{dP}{b(a)}\\,.$ For $dP$ , using the energy conservation law, we may rewrite it as $dP&=d\\left(-\\rho +F(P)\\right)=-d\\rho +\\frac{db(a)}{3}\\nonumber \\\\&=3\\frac{da}{a}(\\rho +P)+\\frac{db(a)}{3}=b(a)\\frac{da}{a}+\\frac{db(a)}{3}\\,.$ Therefore we have $\\frac{dY}{Y}=6\\frac{dP}{b(a)}=6\\frac{da}{a}+2\\frac{db}{b}\\,.$ Hence $Y\\propto a^6b^2\\,.",
"$ This is consistent with the USR case in which $b(a) \\propto a^{-6}$ and $Y=X \\propto a^{-6}$ .",
"This relation is quite useful since it allows to rewrite the freezing function $f(a)$ as $f(a)\\propto \\frac{H c_s^2}{\\sqrt{Y}} \\,,$ from which we can deduce that $Y(a)$ determines the super-horizon behavior of ${\\mathcal {R}}_c$ .",
"In particular, for the models we are considering in which $c_s$ is constant, we infer that super-horizon growth can happen in the limit $Y\\rightarrow 0$ .",
"For a given choice of $b(a)$ , eq.",
"(REF ) can be inverted to give the scale factor as a function of $Y$ , $a=a(Y)$ .",
"Also eq.",
"(REF ) can be integrated to give $P=P(a)$ .",
"Combining these two, one can obtain the Lagrangian for the scalar field, $L=P=P(Y)$ .",
"Note that in GA models there is a one-to-one correspondence between the scale factor and state variables such as $P(a)$ and $\\rho (a)$ , which is the reason why we can also write a barotropic equation of state $P(\\rho )=P(a(\\rho ))$ .",
"Once any of the functions $P(a),\\rho (a),b(a),\\epsilon (a),Y(a)$ is specified, all the others are specified too, as well as the equation of state $P(\\rho )$ or its scalar field equivalent Lagrangian $P(Y)$ , which is in fact the basis of the inversion method that we are developing in this paper."
],
[
"examples",
"Here we give a couple of specific K-inflation models that are globally adiabatic and violate the convervation of ${\\mathcal {R}}_c$ .",
"Given the parametric behaviour of $b\\equiv 2 \\epsilon \\rho $ , our inversion method allows us to deduce the Lagrangian."
],
[
"Ex 1: Generalized USR",
"Let us consider a specific case where $b(a)$ is a power-law function, $2\\epsilon \\rho =b(a)=c a^{-n} \\,.$ where $c$ is a constant.",
"We assume $n>3$ in order to have the growth on superhorizon scales.",
"From eq.",
"(REF ) we have $a\\propto Y^{1/(6-2n)}\\,.$ Now eq.",
"(REF ) gives $P&=\\int ^a\\left(b(a)\\frac{da}{a}+\\frac{db(a)}{3}\\right)\\nonumber \\\\&=-\\frac{c}{n}a^{-n}+\\frac{c}{3}a^{-n}+const.\\nonumber \\\\&=\\frac{n-3}{3n}b(a)+const..$ Plugging eq.",
"(REF ) into this, we finally obtain $L=P(Y)= Y^{n/(2n-6)}-V_0 \\,.",
"$ Since this may be regarded as a natural generalization of the USR case, which corresponds to the case $n=6$ , we call it the generalized USR (GUSR) model.",
"Lagrangians involving $Y^{\\alpha }$ terms have already been studied in [11], [24], [25], but those models are either not exactly globally adiabatic because of the presence of a not constant potential or they satisfy the relation $\\epsilon \\propto a^{-n}$ only approximately and during a limited time range, while for GUSR $\\epsilon \\propto a^{-n}$ is an exact relation and is valid at any time.",
"As the Lagrangian is of the type described in eqs.",
"(REF ) and (REF ) (remember that after a field transformation $Y$ can be made equal to $X$ ), we understand that this scalar field model is indeed equivalent to a barotropic fluid.",
"Hence we have $c_w^2=c_s^2$ , and therefore $\\delta P_{ nad}=0$ .",
"Indeed the second condition for super-horizon growth of ${\\mathcal {R}}_c$ given in eq.",
"(REF ) is satisfied.",
"More precisely, we note that for the GUSR model, the sound velocity is exactly constant, $c_w^2=c_s^2=\\frac{n-3}{3} \\,.$ The power spectrum of the comoving curvature perturbation can be explicitly computed for this model.",
"One finds [26] that the spectral index is a function of $n$ : $n_s-1= 6-n$ , in agreement with the scale invariant spectrum of the original ultra slow-roll inflation in which one has $n=6$ .",
"Hence, the model can be constrained by the observational value.",
"Note as well, from eq.",
"(REF ), that to have a slightly red-tilted spectrum, we need a slightly superluminal speed of sound."
],
[
"Ex 2: Lambert Inflation",
"As another example, let us consider the case when $\\epsilon $ is a power-law function, $\\epsilon (a)=\\epsilon _0{a}^{-n}\\,.$ As before, we assume $n>3$ .",
"In this case, since $d\\log \\rho /d\\log a=-2\\epsilon \\propto a^{-n}$ , we find $\\rho (a)=\\rho _0\\exp \\left[\\frac{2\\epsilon }{n}\\right] \\,.$ It is clear that $\\rho $ approaches a constant $\\rho _0$ asymptotically at $a\\rightarrow \\infty $ .",
"Inserting eq.",
"(REF ) and eq.",
"(REF ) into eq.",
"(REF ), the sound velocity is given by $c_w^2=c_s^2=-1-\\frac{1}{3}\\left(\\frac{d\\log \\epsilon }{d\\log a}+\\frac{d\\log \\rho }{d\\log a}\\right)=\\frac{n-3+2\\epsilon }{3}\\,.$ Thus $c_s^2$ is time dependent, but it rapidly approaches a constant as $\\epsilon $ decays out.",
"Also from eq.",
"(REF ) and eq.",
"(REF ), we find $b(a)=2\\epsilon \\rho =2\\epsilon \\rho _0\\exp \\left[\\frac{2\\epsilon }{n}\\right]\\,.$ Thus we have $Y\\propto a^6 b^2\\propto a^{6-2n}\\exp \\left[\\frac{4\\epsilon }{n}\\right]\\propto \\epsilon ^{(2n-6)/n}\\exp \\left[\\frac{4\\epsilon }{n}\\right] \\,,$ which implies $Y^{n/(2n-6)}\\propto \\frac{4\\epsilon }{2n-6}\\exp \\left[\\frac{4\\epsilon }{2n-6}\\right]\\,.$ To find the Lagrangian, we manipulate eq.",
"(REF ) as $dP&=b\\frac{da}{a}+\\frac{db}{3}=-\\frac{b}{n}\\frac{d\\epsilon }{\\epsilon }+\\frac{db}{3}\\nonumber \\\\&=-\\frac{2}{n}\\rho _0e^{2\\epsilon /n}d\\epsilon +\\frac{db}{3}\\,.$ Therefore, integrating this we obtain $P=\\rho _0e^{2\\epsilon /n}\\left(-1+\\frac{2\\epsilon }{3}\\right)+const..$ One can invert eq.",
"(REF ) to find $\\epsilon $ as a function of $Y$ , and then insert it into the above to obtain the Lagrangian.",
"Specifically, we introduce the Lambert function $W(x)$ defined by the inverse function of $X(z)=ze^z$ , $z=X^{-1}(ze^z)\\equiv W(ze^z)\\,.$ Setting ${Y}^{n/(2n-6)}=ze^z\\,;\\quad z=\\frac{4\\epsilon }{2n-6}\\,,$ we have $\\frac{4\\epsilon }{2n-6}=W(y)\\,;\\quad y\\equiv Y^{n/(2n-6)}\\,.$ Inserting this into eq.",
"(REF ), we finally obtain $L&=P(Y)\\nonumber \\\\&=\\rho _0\\left(\\frac{n-3}{3}W(y)-1\\right)\\exp \\left[\\frac{n-3}{n}W(y)\\right]-V_0\\,,$ where $y=y(Y)$ is given in eq.",
"(REF ).",
"Note that this model has been derived without making any approximation, and it gives exactly $\\epsilon \\propto a^{-n}$ .",
"However, as we mentioned before, in the late time limit, there is no difference between $\\epsilon \\propto a^{-n}$ and $\\rho \\epsilon \\propto a^{-n}$ .",
"Thus the two models discussed above are essentially the same at late times.",
"This can be easily checked by expanding $W(y)$ around $y=0$ , $W(y)=y-y^2+\\cdots .$ At leading order in $y=Y^{n/(2n-6)}$ , this gives $P(Y)=\\frac{n-3}{3}\\rho _0Y^{n/(2n-6)}-\\rho _0-V_0\\,.$ By absorbing the constant coefficient into $g(\\phi )$ in the definition of $Y$ , $Y=g(\\phi )X$ , and absorbing $\\rho _0$ into the constant $V_0$ , eq.",
"(REF ) reduces to $P=Y^{n/(2n-6)}-V_0\\,,$ which indeed coincides with the GUSR model, see eq.",
"(REF ).",
"Higher order terms in the expansion give an infinite class of models of the type $u(Y)&=&\\sum _i \\beta _i Y^{n_i} \\,, $ where $\\beta _i$ are appropriate coefficients.",
"Finally, note that in USR and as well in the two examples considered here, the shift symmetry in the potential ($V(\\phi )=V_0$ ) is a direct consequence of the demand $c_w^2=c_s^2$ , which in turn follows from the global adiabaticity of the model.",
"That is in line with the general statement [27], [28] that for a $k$ -essence theory to describe a fluid, one needs a shift symmetry (i.e., there is no physical clock, the model is of the non-attractor type)."
],
[
"Conclusions",
"By introducing the notion of global adiabaticity, namely, $c_w^2=c_s^2$ and $\\delta P_{nad}=\\delta P-c_w^2\\delta \\rho =0$ , where $c_w^2=\\dot{P}/\\dot{\\rho }$ and $c_s$ is the propagation (phase) speed of the perturbation, we have determined the general conditions for the non-conservation of the curvature perturbations on comoving slices ${\\mathcal {R}}_c$ on super-horizon scales.",
"We have found that globally adiabatic K-essence models can exhibit this behavior.",
"We have then developed a method to construct the Lagrangian of a K-essence globally adiabatic (GA) model by specifying the behavior of background quantities such as $\\epsilon \\rho $ where $\\epsilon $ is the slow-roll parameter, using the equivalence between barotropic fluids and GA K-essence models.",
"We have applied the method to find the equations of state of the fluids and derive the Lagrangian of the equivalent single scalar field models.",
"Interestingly, we have found that the requirement to avoid the gradient instability, ie, $c_s^2>0$ is almost identical to the condition for the non-conservation on superhorizon scales.",
"The advantage of our approach is that we have not solved any perturbation equation explicitly, since we have proceeded in the opposite way solving the inversion problem consisting of requiring certain properties to the behavior of the perturbation equation we are interested in.",
"In other words, instead of starting from a Lagrangian and then solve the perturbations equations we have determined the equation of state or equivalently the Lagrangian which admits a solution of the perturbation equation with the particular behavior we are interested in.",
"We have shown that the main difference between attractor models and GA models is that the latter are adiabatic on all scales, while attractor models are approximately adiabatic in the sense of $\\delta P_{nad}=0$ only on super-horizon scales and $c_w^2\\ne c_s^2$ .",
"The detailed study of the new models found in this paper will be done in a separate upcoming work [26] but we can already predict that they can be compatible with observational constraints on the spectral index thanks to the extra parameter $n$ which is not present in USR.",
"Furthermore they can violate the Maldacena's consistency condition and consequently produce large local shape non-Gaussianity.",
"In the future it will be interesting to apply the inversion method we have developed to other problems related to primordial curvature perturbations, or to develop a similar method for the adiabatic sound speed as function of the scale factor.",
"The work of MS was supported by MEXT KAKENHI No. 15H05888.",
"SM is funded by the Fondecyt 2015 Postdoctoral Grant 3150126.",
"This work was supported by the Dedicacion exclusica and Sostenibilidad programs at UDEA, the UDEA CODI project IN10219CE and 2015-4044, and Colciencias mobility project COSOMOLOGY AFTER BICEP."
]
] | 1606.04906 |
[
[
"Perfect Embezzlement of Entanglement"
],
[
"Abstract Van Dam and Hayden introduced a concept commonly referred to as embezzlement, where, for any entangled quantum state $\\phi$, there is an entangled catalyst state $\\psi$, from which a high fidelity approximation of $\\phi \\otimes \\psi$ can be produced using only local operations.",
"We investigate a version of this where the embezzlement is perfect (i.e., the fidelity is 1).",
"We prove that perfect embezzlement is impossible in a tensor product framework, even with infinite-dimensional Hilbert spaces and infinite entanglement entropy.",
"Then we prove that perfect embezzlement is possible in a commuting operator framework.",
"We prove this using the theory of C*-algebras and we also provide an explicit construction.",
"Next, we apply our results to analyze perfect versions of a nonlocal game introduced by Regev and Vidick.",
"Finally, we analyze the structure of perfect embezzlement protocols in the commuting operator model, showing that they require infinite-dimensional Hilbert spaces."
],
[
"Introduction",
"It is well known that an entangled quantum state cannot be produced by local operations alone.",
"Van Dam and Hayden [2] proposed a method that, in a certain sense, appears to produce additional entanglement by local operations.",
"They showed that, for any entangled state $\\phi $ and $\\epsilon > 0$ , starting with a special entangled catalyst state $\\psi $ , applying local operations, can produce a state that approximates $\\phi \\otimes \\psi $ within fidelity $1-\\epsilon $ .",
"Although the entanglement entropy of the state produced cannot exceed that of $\\psi $ , when $\\epsilon $ is small, it is difficult to distinguish between the state produced and $\\phi \\otimes \\psi $ .",
"The name embezzlement reflects the fact that the protocol “steals\" entanglement from $\\psi $ in order to produce entanglement elsewhere, but in a manner that is difficult to detect.",
"In the method of [2], fidelity $1-\\epsilon $ can be attained for any $\\epsilon > 0$ , using a catalyst $\\psi $ with entanglement entropy $O(\\log (1/\\epsilon ))$ .",
"Moreover, it is shown in [2] that the entanglement entropy of the catalyst must be $\\Omega (\\log (1/\\epsilon ))$ to attain this fidelity.",
"Thus, high fidelity embezzlement requires a large amount of entanglement to begin with.",
"We consider the question: what kinds of embezzlement are possible when the amount of entanglement in $\\psi $ is allowed to be infinite?",
"The aforementioned results do not rule out perfect (i.e., fidelity 1) embezzlement in such cases.",
"On the other hand, the catalytic states $\\psi _{\\epsilon }$ in [2] do not converge to a valid quantum vector state as $\\epsilon $ approaches 0.",
"This question provides a setting in which the consequences of notions of infinite entanglement can be explored.",
"We first show that in the tensor product framework, where catalytic states are in the tensor product of two Hilbert spaces, perfect embezzlement is impossible, even if the spaces are infinite dimensional and the entanglement entropy is infinite.",
"Next, we consider a commuting operator framework, where the notion of “local\" is formalized differently: there is one joint Hilbert space, accessible to both Alice and Bob; however, the operations that Alice performs on this space must commute with those of Bob.",
"This formalism is used in quantum field theory (see [12], [9], [11], [4], [6], [5] for more discussion about this framework and its relationship with the tensor product framework).",
"A natural adaptation of the commuting operator framework to the setting of embezzlement is the following.",
"The catalytic state $\\psi $ is in a jointly accessibe Hilbert space, that we refer to as the resource space $\\mathcal {R}$ .",
"There are also two additional Hilbert spaces: $\\mathcal {H}_A$ , accessible to Alice only; and $\\mathcal {H}_B$ , accessible to Bob only.",
"The goal of the protocol is to transform a product state to an entangled state in $\\mathcal {H}_A \\otimes \\mathcal {H}_B$ while using $\\psi $ catalytically, and using operators that are commuting in the following sense.",
"Alice can apply a unitary operator on $\\mathcal {H}_A \\otimes \\mathcal {R}$ and Bob can apply a unitary operator on $\\mathcal {H}_B \\otimes \\mathcal {R}$ ; however, $U_A \\otimes I_{\\mathcal {H}_B}$ and $I_{\\mathcal {H}_A} \\otimes U_B$ must commute on $\\mathcal {H}_A \\otimes \\mathcal {R}\\otimes \\mathcal {H}_B$ , as illustrated in Figure REF .",
"Figure: Commuting operator framework as a circuit diagram.We focus on the problem of embezzling a Bell state of the form ${\\textstyle {\\frac{1}{\\sqrt{2}}}}|0\\rangle \\otimes |0\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|1\\rangle \\otimes |1\\rangle $ (though our methodology adapts to more general states).",
"In this case, $\\mathcal {H}_A = \\mathcal {H}_B = \\mathbb {C}^{2}$ .",
"A perfect embezzlement protocol consists of a resource space $\\mathcal {R}$ , a catalytic state $\\psi \\in \\mathcal {\\mathcal {R}}$ , and commuting unitary operators $U_A$ and $U_B$ , such that $(U_A \\otimes I_{\\mathcal {H}_B})&(I_{\\mathcal {H}_A} \\otimes U_B) |0\\rangle \\otimes \\psi \\otimes |0\\rangle \\\\&= \\textstyle {\\frac{1}{\\sqrt{2}}}|0\\rangle \\otimes \\psi \\otimes |0\\rangle +\\textstyle {\\frac{1}{\\sqrt{2}}}|1\\rangle \\otimes \\psi \\otimes |1\\rangle .",
"\\nonumber $ We show that, in this commuting operator framework, a perfect embezzlement protocol exists, where the resource space is a countably infinite dimensional (i.e., separable) Hilbert space.",
"We show this in two ways: one is a simple existence proof, based on the theory of C*-algebras, which does not yield explicit unitary operations; the other is by an explicit construction.",
"Next, we consider coherent embezzlement, which was introduced in [10] (where it is referred to as $T_2$ ) and is a refinement of coherent state exchange, introduced in [8].",
"Coherent embezzlement is related to embezzlement but has the property that it is operationally testable in a sense similar to that of nonlocal games (whereas embezzlement itself does not have this property).",
"We give reductions between perfect embezzlement and perfect coherent embezzlement to prove that perfect coherent embezzlement is impossible in the tensor product framework; whereas it is possible in the commuting operator framework.",
"Finally, we prove a theorem concerning the structure of pairs of unitaries that achieve perfect embezzlement in terms of properties of their constituent operators.",
"We show that at least one of these operators must contain a non-unitary isometry, a term that we will define later.",
"Since non-unitary isometries do not exist in finite dimensions, this implies that perfect embezzlement in the commuting-operator model cannot be achieved with a finite dimensional resource space."
],
[
"Perfect embezzlement is impossible in a tensor product framework",
"In [2], it is proved that, for any protocol that embezzles within fidelity $1 - \\epsilon $ , the entanglement entropy of the catalyst must be $\\Omega (\\log (1/\\epsilon ))$ .",
"It follows that perfect embezzlement is impossible with finite-dimensional entanglement in the tensor product framework.",
"Here, we extend this impossibility result to tensor products of arbitrary Hilbert spaces (where the dimension of the spaces and entanglement entropy can be infinite).",
"In the tensor product framework, the resource space is of the form $\\mathcal {R} = \\mathcal {R}_A \\otimes \\mathcal {R}_B$ , where $\\mathcal {R}_A$ and $\\mathcal {R}_B$ are arbitrary Hilbert spaces.",
"Alice has access to $\\mathcal {H}_A \\otimes \\mathcal {R}_A$ and Bob has access to $\\mathcal {H}_B \\otimes \\mathcal {R}_B$ .",
"Alice and Bob can each apply any unitary operation to the registers that they have access to, as illustrated in Figure REF (left), where the input state is $|0\\rangle \\otimes \\psi \\otimes |0\\rangle $ , for some state $\\psi \\in \\mathcal {R}_A \\otimes \\mathcal {R}_B$ .",
"The protocol performs perfect embezzlement if its output state is $\\textstyle {\\frac{1}{\\sqrt{2}}}|0\\rangle \\otimes \\psi \\otimes |0\\rangle +\\textstyle {\\frac{1}{\\sqrt{2}}}|1\\rangle \\otimes \\psi \\otimes |1\\rangle $ .",
"We also define a potentially stronger model, that we refer to as embezzlement with ancillas, which includes the possibility of Alice and Bob employing additional registers as part of their protocol, as illustrated in Figure REF (right).",
"Figure: Circuit diagram for embezzlement (left) and embezzlement with ancillas (right) in the tensor product framework.Registers ℛ A \\mathcal {R}_A and ℛ B \\mathcal {R}_B contain a bipartite resource state that must be used catalytically.Registers ℋ A =ℋ B =ℂ 2 \\mathcal {H}_A = \\mathcal {H}_B = \\mathbb {C}^{2} are intialized to state |00〉|00\\rangle and are transformed to state 1 2|00〉+1 2|11〉{\\textstyle {\\frac{1}{\\sqrt{2}}}}|00\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|11\\rangle .Registers 𝒢 A \\mathcal {G}_A and 𝒢 B \\mathcal {G}_B are ancillas, whose initial state is unentangled, but they need not be used catalytically.The input to the circuit is of the form $\\gamma _A \\otimes |0\\rangle \\otimes \\psi \\otimes |0\\rangle \\otimes \\gamma _B$ , where $\\psi \\in \\mathcal {R}_A \\otimes \\mathcal {R}_B$ is the catalyst state, and $\\gamma _A \\in \\mathcal {G}_A$ and $\\gamma _B \\in \\mathcal {G}_B$ are the initial states of Alice and Bob's respective ancilla registers, $\\mathcal {G}_A$ and $\\mathcal {G}_B$ (which can be infinite dimensional).",
"If we express the Hilbert space as $(\\mathcal {H}_A \\otimes \\mathcal {H}_B) \\otimes (\\mathcal {R}_A \\otimes \\mathcal {R}_B) \\otimes (\\mathcal {G}_A \\otimes \\mathcal {G}_B)$ then the input state can be written as $|00\\rangle \\otimes \\psi \\otimes (\\gamma _A \\otimes \\gamma _B)$ .",
"The protocol performs perfect embezzlement if and only if the output state is of the form $\\bigl ({\\textstyle {\\frac{1}{\\sqrt{2}}}}|00\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|11\\rangle \\bigr )\\otimes \\psi \\otimes \\gamma _{AB},$ for some state $\\gamma _{AB} \\in \\mathcal {G}_A \\otimes \\mathcal {G}_B$ .",
"Theorem 2.1 Perfect embezzlement is impossible in the tensor product framework, even if Alice and Bob are allowed to use ancillas.",
"The proof is a straightforward application of the Schmidt decomposition for vectors in tensor products of arbitrary Hilbert spaces.",
"For arbitrary (not necessarily separable) Hilbert spaces $\\mathcal {H}$ and $\\mathcal {K}$ and any $\\phi \\in \\mathcal {H} \\otimes \\mathcal {K}$ , it is possible to express $\\phi = \\sum _{j = 0}^{\\infty } \\alpha _j\\, u_j\\otimes v_j,$ where $\\alpha _j \\ge 0$ , $\\sum _{j = 0}^{\\infty } |\\alpha _j|^2 = 1$ , $ \\alpha _j \\ge \\alpha _{j+1}$ , $u_0, u_1, \\dots $ are orthonormal vectors in $\\mathcal {H}$ , and $v_0, v_1, \\dots $ are orthonormal vectors in $\\mathcal {K}$ .",
"Moreover, given these conditions, the coefficients $\\alpha _j$ are unique.",
"For the convenience of the reader, we include a proof of this in Appendix .",
"Now taking a Schmidt decomposition of $\\gamma _A \\otimes |0\\rangle \\otimes \\psi \\otimes |0\\rangle \\otimes \\gamma _B$ , with respect to $\\big (\\mathcal {G}_A \\otimes \\mathcal {H}_A \\otimes \\mathcal {R}_A \\big ) \\otimes \\big (\\mathcal {G}_B \\otimes \\mathcal {H}_B \\otimes \\mathcal {R}_B \\big )$ , we obtain Schmidt coefficients $\\alpha _j$ .",
"Suppose that a perfect embezzlement protocol exists.",
"Then, since $U_A$ and $U_B$ are local unitaries, the Schmidt coefficients of the initial state $|00\\rangle \\otimes \\psi \\otimes \\gamma _A \\otimes \\gamma _B$ must be the same as those of the final state $({\\textstyle {\\frac{1}{\\sqrt{2}}}}|00\\rangle + {\\textstyle {\\frac{1}{\\sqrt{2}}}}|11\\rangle )\\otimes \\psi \\otimes \\gamma _{AB}$ .",
"But this is a contradiction, since the largest Schmidt coefficient of the input state is $\\alpha _0$ (which is nonzero) and the largest Schmidt coefficient of the output state is at most ${\\textstyle {\\frac{1}{\\sqrt{2}}}}\\alpha _0$ .",
"Therefore, there is no perfect embezzlement protocol in the tensor product framework."
],
[
"Perfect embezzlement is possible in a commuting operator framework",
"In this section we show that, since one can approximately embezzle a Bell state to any level of precision in finite dimensions (by the results of [2]), one can perfectly embezzle in infinite dimensions in the commuting operator framework.",
"Readers unfamiliar with the theory of C*-algebras might prefer to read our primer on C*-algebras in Appendix before tackling this section.",
"At the end of the section, we explain how to generalize the technique to more general entangled states.",
"We begin by showing that each commuting operator framework, where $\\mathcal {H}_A = \\mathcal {H}_B = \\mathbb {C}^2$ , yields a set of eight operators on the resource space.",
"To study the most general commuting framework, it is natural to consider the relations that any such set of operators must satisfy and look for a “universal\" model for such sets of operators.",
"We will show that the eight operators arising from a commuting operator framework are always a representation of a certain C*-algebra and that the catalyst vector yields a state on this C*-algebra.",
"We will show that the commuting operator framework together with the catalyst vector achieves perfect embezzlement of a Bell state if and only if the state on this C*-algebra induced by the catalyst vector satisfies a set of four equations.",
"In this manner the question of whether or not one can perfectly embezzle a Bell state is reduced to a question about the existence of a state on this C*-algebra that satisfies our four equations.",
"Finally, we show that perfect embezzlement of a Bell state is possible in the commuting operator framework by showing the existence of such a state.",
"The “universal\" C*-algebra that one needs was first introduced by L.G.",
"Brown [1], who referred to it as the universal C*-algebra of a non-commutative unitary for reasons that will, hopefully, be clear.",
"Our viewpoint shows that in a certain sense questions about embezzlement can be interpreted as questions about states on these particular quantum group C*-algebras.",
"We think that this perspective is new and should lead to interesting links between these two areas.",
"Let's return to the scenario of Figure REF .",
"Alice's unitary operation, $U_A: \\mathbb {C}^2 \\otimes \\mathcal {R} \\rightarrow \\mathbb {C}^2 \\otimes \\mathcal {R}$ can be represented by a $2 \\times 2$ matrix of operators on $\\mathcal {R}$ , $U_A= \\begin{pmatrix} U_{00} & U_{01} \\\\ U_{10} & U_{11} \\end{pmatrix}= \\sum _{i,j=0}^1 |i\\rangle \\langle j| \\otimes U_{ij} $ where $U_A( |j\\rangle \\otimes h) = \\sum _{i=0}^1 |i\\rangle \\otimes U_{ij} h$ .",
"In this case, $ U_A^* = \\begin{pmatrix} U_{00}^* & U_{10}^* \\\\ U_{01}^* & U_{11}^* \\end{pmatrix} $ and the fact that $U_A$ is unitary can be expressed by eight equations involving these operators that are best expressed as $U_A^*U_A = \\begin{pmatrix} I_{\\mathcal {R}} & 0\\\\ 0 & I_{\\mathcal {R}} \\end{pmatrix} = U_AU_A^*,$ where we apply the usual rules of matrix multiplication, being careful to remember that since the entries of $U_A$ are operators, not numbers, they need not commute.",
"We also recall that when $\\mathcal {R}$ is infinite dimensional, then it is necessary that both $U_A^*U_A$ and $U_AU_A^*$ be the identity to guarantee that $U_A$ is unitary.",
"Finally, in the special case that $\\dim (\\mathcal {R}) =1$ so that these entries are numbers, then we are back to the usual case of a $2 \\times 2$ complex unitary matrix.",
"Conversely, if we let $U_A$ be any $2 \\times 2$ matrix of operators on $\\mathcal {R}$ that satisfies Eq.",
"(REF ) then $U_A$ will define a unitary on $\\mathbb {C}^2 \\otimes \\mathcal {R}$ .",
"Similarly, Bob's unitary $U_B: \\mathcal {R} \\otimes \\mathbb {C}^2 \\rightarrow \\mathcal {R} \\otimes \\mathbb {C}^2$ is represented by a $2 \\times 2$ matrix of operators on $\\mathcal {R}$ , $U_B= ( V_{ij})$ , whose entries satisfy the same eight equations.",
"Finally, to have a commuting operator framework as in Figure REF , we need $(U_A \\otimes I_2)(I_2 \\otimes U_B) = (I_2 \\otimes U_B)(U_A \\otimes I_2)$ .",
"The following proposition translates this condition into equations involving the operator entries.",
"Proposition 3.1 Let $U_{ij}, V_{kl}, 0 \\le i,j,k,l \\le 1$ be operators on the Hilbert space $\\mathcal {R}$ such that $U_A=( U_{ij}),$ and $U_B= (V_{kl})$ are unitaries.",
"Then $(U_A \\otimes I_2)(I_2 \\otimes U_B)= (I_2 \\otimes U_B)(I_2 \\otimes U_A)$ if and only if $U_{ij}V_{kl} = V_{kl}U_{ij}$ and $U_{ij}^*V_{kl} = V_{kl}U_{ij}^*$ for all $i,j,k,l$ .",
"We have that $(U_A \\otimes I_2)(I_2 \\otimes U_B)( |j\\rangle \\otimes h \\otimes |l\\rangle ) = \\sum _{i,k=0}^1 |i\\rangle \\otimes U_{ij}V_{kl}h \\otimes |k\\rangle ,$ and similarly, $(I_2 \\otimes U_B)(U_A \\otimes I_2) ( |j\\rangle \\otimes h \\otimes |l\\rangle ) = \\sum _{i,k=0}^1 |i\\rangle \\otimes V_{kl}U_{ij}h \\otimes |k\\rangle .$ Thus, we see that $(U_A \\otimes I_2)(I_2 \\otimes U_B) = (I_2 \\otimes U_B)(U_A \\otimes I_2)$ is equivalent to $U_{ij}V_{kl} = V_{kl}U_{ij}$ , for all $i,j,k,l.$ However, if an invertible operator commutes with another operator, then its inverse also commutes with that operator.",
"Hence, $(U_A \\otimes I_2)^{-1} = (U_A^* \\otimes I_2)$ commutes with $(I_2 \\otimes U_B)$ and this is equivalent to $U_{ij}^*V_{kl}= V_{kl}U_{ij}^*$ , for all $i,j,k,l.$ The above equations are generally summarized by saying that the set of operators $\\lbrace U_{ij} \\rbrace $ *-commutes with the set $\\lbrace V_{kl} \\rbrace $ .",
"Thus, having a commuting operator framework is equivalent to having two unitaries $U_A=(U_{ij})$ , and $U_B=(V_{kl})$ whose entries *-commute.",
"We wish to study “universal\" properties of $2 \\times 2$ matrices of operators $(U_{ij})$ that give rise to a unitary.",
"To do this we begin with a unital *-algebra $\\mathcal {U}_2$ with generators, denoted 1 and $u_{ij}, \\, 0 \\le i,j \\le 1$ , subject to the eight equations, $\\begin{pmatrix} u_{00} & u_{01} \\\\ u_{10} & u_{11} \\end{pmatrix}\\begin{pmatrix} u_{00}^* & u_{10}^* \\\\ u_{01}^* & u_{11}^* \\end{pmatrix}=\\begin{pmatrix} u_{00}^* & u_{10}^* \\\\ u_{01}^* & u_{11}^* \\end{pmatrix}\\begin{pmatrix} u_{00} & u_{01} \\\\ u_{10} & u_{11} \\end{pmatrix}= \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix}.$ Thus, whenever there is a Hilbert space $\\mathcal {H}$ and four operators, $U_{ij}$ on that space such that the operator-matrix $U=(U_{ij})$ defines a unitary operator on $\\mathcal {H} \\otimes \\mathbb {C}^2$ , then there is a *-homomorphism, $ \\pi : \\mathcal {U}_2 \\rightarrow B(\\mathcal {H}) \\text{ with } \\pi (u_{ij}) = U_{ij}.$ For $x \\in \\mathcal {U}_2$ he sets $\\Vert x\\Vert = \\sup \\lbrace \\Vert \\pi (x)\\Vert \\colon \\pi \\text{ a *-homomorphism} \\rbrace $ , where the supremum is taken over all Hilbert spaces and all $\\pi $ 's as above.",
"This defines a norm on $\\mathcal {U}_2$ and that the completion is a C*-algebra, we shall denote $U_{\\operatorname{nc}}(2)$ .",
"The subscript $\\operatorname{nc}$ stands for “non-commuting\" and is intended to remind us that the generators $u_{ij}$ do not commute.",
"(This approach generalizes naturally to $d \\times d$ matrices of operators, for $d > 2$ , where the C*-algebra is denoted as $U_{\\operatorname{nc}}(d)$ .)",
"Note that in the commuting operator framework, the set $U_{ij}$ and the set $V_{kl}$ each gives rise to a *-homomorphism and that these two *-homomorphisms commute.",
"Thus, it is not hard to see that we have a one-to-one correspondence between commuting operator frameworks and *-homomorphisms of $U_{\\operatorname{nc}}(2) \\otimes U_{\\operatorname{nc}}(2)$ into $B(\\mathcal {R})$ .",
"Since we want to consider all commuting operator frameworks, we are lead to study $U_{\\operatorname{nc}}(2) \\otimes _{\\max } U_{\\operatorname{nc}}(2)$ .",
"The study of states on this algebra turns out to be closely related to embezzlement constructions as the following result shows.",
"Theorem 3.2 There exists a perfect embezzlement protocol in the commuting operator framework if and only if there exists a state $s: U_{\\operatorname{nc}}(2) \\otimes _{\\max } U_{\\operatorname{nc}}(2) \\rightarrow \\mathbb {C}$ such that $s(u_{00} \\otimes u_{00}) = \\frac{1}{\\sqrt{2}}$ , $s(u_{10} \\otimes u_{00}) = \\,0$ , $s(u_{00} \\otimes u_{10}) = \\,0$ , $s(u_{10} \\otimes u_{10}) = \\frac{1}{\\sqrt{2}}$ .",
"First assume that a perfect embezzlement protocol exists in a commuting operator framework.",
"Let $\\mathcal {R}$ be a Hilbert space, let $\\psi \\in \\mathcal {R}$ be a unit vector, let $U_A= \\big ( U_{ij} \\big )$ and $U_B= \\big ( V_{kl} \\big )$ be unitaries on $\\mathbb {C}^2 \\otimes \\mathcal {H}$ and $\\mathcal {H} \\otimes \\mathbb {C}^2$ , respectively, such that $U_A \\otimes I_2$ commutes with $I_2 \\otimes U_B$ and let $\\psi $ be a catalyst vector for perfect embezzlement of a Bell state, Define $\\pi : U_{\\operatorname{nc}}(2) \\otimes _{\\max } U_{\\operatorname{nc}}(2) \\rightarrow B(\\mathcal {H})$ to be the *-homomorphism defined by $\\pi (u_{ij} \\otimes 1) = U_{i,j}, \\, \\pi (1 \\otimes u_{kl}) = V_{kl}$ .",
"Since $\\psi $ is a catalyst vector, $ (U_A \\otimes I_2)(I_2 \\otimes U_B) ( |0\\rangle \\otimes \\psi \\otimes |0\\rangle ) = \\textstyle {\\frac{1}{\\sqrt{2}}}( |0\\rangle \\otimes \\psi \\otimes |0\\rangle + |1\\rangle \\otimes \\psi \\otimes |1\\rangle ).",
"$ Now define a state on $U_{\\operatorname{nc}}(2) \\otimes _{\\max } U_{\\operatorname{nc}}(2)$ by $s(x) = \\langle \\pi (x) \\psi , \\psi \\rangle $ .",
"We have that $\\big ( I_2 \\otimes U_B\\big ) \\big ( U_A \\otimes I_2 \\big ) |0\\rangle \\otimes \\psi \\otimes |0\\rangle & = \\sum _{i,j=0}^1 |i\\rangle \\otimes U_{i,0}V_{j,0} \\psi \\otimes |j\\rangle \\\\ & = \\textstyle {\\frac{1}{\\sqrt{2}}}|0\\rangle \\otimes \\psi \\otimes |0\\rangle + \\textstyle {\\frac{1}{\\sqrt{2}}}|1\\rangle \\otimes \\psi \\otimes |1\\rangle ,$ which is equivalent to $ U_{00}V_{00} \\psi = U_{10}V_{10} \\psi = \\textstyle {\\frac{1}{\\sqrt{2}}}\\psi \\text{\\ \\ and \\ \\ } U_{00} V_{10} \\psi = U_{10}V_{00} \\psi = 0 .$ From these equations, it follows that the state $s$ satisfies the four conditions.",
"Conversely, assume that $s: U_{\\operatorname{nc}}(2) \\otimes _{\\max } U_{\\operatorname{nc}}(2) \\rightarrow \\mathbb {C}$ is a state that satisfies the 4 conditions.",
"Let $\\pi _s: U_{\\operatorname{nc}}(2) \\otimes _{\\max } U_{\\operatorname{nc}}(2) \\rightarrow B(\\mathcal {H}_s)$ and $\\psi \\in \\mathcal {H}_s$ be the GNS representation of the state so that $s(x) = \\langle \\pi _s(x) \\psi , \\psi \\rangle $ .",
"If we define $U_A: \\mathbb {C}^2 \\otimes \\mathcal {H} \\rightarrow \\mathbb {C}^2 \\otimes \\mathcal {H}$ by $U_A= \\big ( \\pi (u_{ij} \\otimes 1) \\big )$ and $U_B: \\mathcal {H} \\otimes \\mathbb {C}^2 \\rightarrow \\mathcal {H} \\otimes \\mathbb {C}^2$ by $U_B = \\big ( \\pi (1 \\otimes u_{i,j}) \\big )$ , then $U_A \\otimes I_2$ commutes with $I_2 \\otimes U_B$ .",
"The operator on the direct sum of four copies of $\\mathcal {H}$ given by $ \\begin{pmatrix} U_{00}V_{00} & U_{01}V_{00} & U_{00}V_{01} & U_{01}V_{01} \\\\ U_{10}V_{00} & U_{11}V_{00} & U_{10}V_{01} & U_{11}V_{01}\\\\ U_{00} V_{10} & U_{01}V_{10} & U_{00}V_{11} & U_{01}V_{11}\\\\ U_{10}V_{10} & U_{11}V_{10} & U_{10} V_{11} & U_{11}V_{11} \\end{pmatrix} $ is unitary.",
"Hence, $ 1 &= |\\langle U_{00}V_{00} \\psi , \\psi \\rangle |^2 + |\\langle U_{10}V_{10} \\psi , \\psi \\rangle |^2 \\le \\Vert U_{00} V_{00} \\psi \\Vert ^2 + \\Vert U_{10}V_{10}\\psi \\Vert ^2 \\\\ &\\le \\Vert U_{00} V_{00} \\psi \\Vert ^2 + \\Vert U_{10}V_{10}\\psi \\Vert ^2 + \\Vert U_{00}V_{10}\\psi \\Vert ^2 + \\Vert U_{10} V_{00} \\psi \\Vert ^2 =1, $ from which it follows that $ U_{00}V_{00} \\psi = U_{10}V_{10} \\psi = \\textstyle {\\frac{1}{\\sqrt{2}}}\\psi \\text{\\ \\ and \\ \\,} U_{00} V_{10} \\psi = U_{10}V_{00} \\psi = 0 .$ Thus, $\\big ( I_2 \\otimes U_B\\big ) \\big ( U_A \\otimes I_2 \\big ) |0\\rangle \\otimes \\psi \\otimes |0\\rangle & = \\sum _{i,j=0}^1 |i\\rangle \\otimes U_{i,0}V_{j,0} \\psi \\otimes |j\\rangle \\\\ & = \\textstyle {\\frac{1}{\\sqrt{2}}}|0\\rangle \\otimes \\psi \\otimes |0\\rangle + \\textstyle {\\frac{1}{\\sqrt{2}}}|1\\rangle \\otimes \\psi \\otimes |1\\rangle $ and we have a perfect embezzlement protocol.",
"Thus, we have proven that perfect embezzlement in the commuting operator framework is equivalent to the existence of a state on $U_{\\operatorname{nc}}(2) \\otimes _{\\max } U_{\\operatorname{nc}}(2)$ that satisfies the four equations above.",
"We now prove that such a state exists.",
"Theorem 3.3 There exists a state $s: U_{\\operatorname{nc}}(2) \\otimes _{\\max } U_{\\operatorname{nc}}(2) \\rightarrow \\mathbb {C}$ that satisfies the four equations of the previous theorem and consequently perfect embezzlement is possible in the commuting operator framework.",
"By the results of [2], we have finite dimensional Hilbert spaces $H_n$ unit vectors $h_n\\in H_n$ and unitary operators $U_n, V_n$ on $H_n \\otimes \\mathbb {C}^2$ , such that $(U_n \\otimes I_2) (I_2 \\otimes V_n)(|0\\rangle \\otimes h_n \\otimes |0\\rangle ) -\\frac{1}{\\sqrt{2}}(|0\\rangle \\otimes h_n \\otimes |0\\rangle + |1\\rangle \\otimes h_n \\otimes |1\\rangle )$ has norm less than $1/n$ .",
"These operators induce *-homomorphisms, $\\pi _n: U_{\\operatorname{nc}}(2) \\otimes _{\\max } U_{\\operatorname{nc}}(2) \\rightarrow B(H_n)$ and states $s_n: U_{\\operatorname{nc}}(2) \\otimes _{\\max } U_{\\operatorname{nc}}(2) \\rightarrow \\mathbb {C}$ defined by $s_n(x) = \\langle \\pi _n(x) h_n , h_n \\rangle $ .",
"These states satisfy: $\\bigl | s_n( u_{00} \\otimes u_{00}) - \\frac{1}{\\sqrt{2}} \\bigr | < \\frac{1}{n}$ , $| s_n(u_{10} \\otimes u_{00}) | < \\frac{1}{n}$ , $| s_n( u_{00} \\otimes u_{10}) | < \\frac{1}{n}$ , $ \\bigl | s_n(u_{10} \\otimes u_{10}) - \\frac{1}{\\sqrt{2}} \\bigr | < \\frac{1}{n}$ .",
"Now by the fact that the state space of any unital C*-algebra is compact in the weak*-topology, we may take a limit point $s$ of this sequence of states.",
"Since the value of $s(u_{i,j} \\otimes u_{k,l})$ must be a limit of the values of $s_n$ on these same elements, $s$ will be a state on $U_{\\operatorname{nc}}(2) \\otimes _{\\max } U_{\\operatorname{nc}}(2)$ that satisfies the 4 conditions exactly.",
"Remark 3.4 From [2], each of the states, $s_n$ appearing in the above proof is actually a state on $U_{\\operatorname{nc}}(2) \\otimes _{\\min } U_{\\operatorname{nc}}(2)$ .",
"Hence, by taking a limit point, we obtain a state $s: U_{\\operatorname{nc}}(2) \\otimes _{\\min } U_{\\operatorname{nc}}(2) \\rightarrow \\mathbb {C}$ that satisfies the 4 equations of Theorem REF .",
"If we apply the GNS construction or any other method to represent it as $s(x) = \\langle \\pi (x) \\psi , \\psi \\rangle $ on some Hilbert space $\\mathcal {H}$ , where $\\pi : U_{\\operatorname{nc}}(2) \\otimes _{\\min } U_{\\operatorname{nc}}(2) \\rightarrow B(\\mathcal {H})$ , then the representation $\\pi $ and the Hilbert space cannot decompose as a tensor product.",
"Otherwise we would achieve perfect embezzlement in a tensor product framework.",
"Hence, we obtain an example of a state on a minimal tensor product, such that it cannot be represented using a *-homomorphism that is a spatial tensor product.",
"In fact, no state on $U_{\\operatorname{nc}}(2) \\otimes _{\\min } U_{\\operatorname{nc}}(2)$ that satisfies just those 4 equations can have a spatial tensor product representation.",
"Remark 3.5 The coefficients that appear in Theorem REF are a consequence of the fact that we are embezzling a Bell state.",
"If we wish instead for a perfect embezzlement protocol of a more general vector state, say, of the form $\\sum _{i,j=0}^{d-1} \\alpha _{ij} |i\\rangle \\otimes |j\\rangle $ , then this is equivalent to the existence of a state $s$ on $U_{\\operatorname{nc}}(d) \\otimes _{\\max } U_{\\operatorname{nc}}(d)$ satisfying $s(u_{i0} \\otimes u_{j0}) = \\alpha _{ij}$ , for $0 \\le i,j < d$ .",
"Moreover, it is shown in [2] that every vector in $\\mathbb {C}^d \\otimes \\mathbb {C}^d$ can be approximately embezzled in a finite dimensional scenario.",
"Therefore, arguing as above, there is always a state $s$ on $U_{\\operatorname{nc}}(d) \\otimes _{\\min } U_{\\operatorname{nc}}(d)$ satisfying the $d^2$ equations."
],
[
"Explicit construction of a perfect embezzlement protocol in a commuting-operator framework",
"The previous section proves the existence of a perfect embezzlement protocol, but without constructing one explicitly.",
"Some of the steps of the proof are nonconstructive.",
"In Theorem REF , an abstract state is obtained by invoking an existence theorem using the weak*-compactness of the set of all states; moreover, in Theorem REF , the Hilbert space is obtained by applying the GNS representation of the state, which is based on the completion of an abstract C*-algebra.",
"In this section, we give an explicit commuting-operator protocol for perfect embezzlement.",
"We explain the technique for Bell states, and, at the end of the section, explain how to it extends to more general entangled states."
],
[
"The resource space $\\mathcal {R}$ and shift operations on this space",
"The resource space is the Hilbert space $\\ell ^2$ , whose orthonormal basis is countably infinite.",
"In order to define the operations used in the protocol, it is useful to think of this space in terms of countably infinite tensor products of states, where all but finitely many of them are fixed.",
"First, consider the set of infinite tensor products of 2-qubit computational basis states, where all but finitely many of them are in state $|00\\rangle $ .",
"We can express these states as $|\\dots x_2 x_1 x_0\\,,\\, \\dots y_2 y_1 y_0\\rangle $ , or as $|x,y\\rangle $ (where $x, y \\in \\mathbb {N}_0$ , and $x_j$ and $y_j$ are the binary digits of $x$ and $y$ , respectively, in position $j$ ).",
"Next, consider the set of infinite tensor products of 2-qubit Bell basis states where all but finitely many of them are $\\frac{1}{\\sqrt{2}}|00\\rangle + \\frac{1}{\\sqrt{2}}|11\\rangle $ .",
"Let us denote these states as $|0.x_{-1}x_{-2}x_{-3}\\dots \\,,\\, 0.y_{-1}y_{-2}y_{-3}\\dots \\rangle $ , with the convention that the qubits in position $-j$ (i.e., the $j$th qubit pair, corresponding to bits $x_{-j}$ and $y_{-j}$ ) are in the Bell basis state $\\textstyle {\\frac{1}{\\sqrt{2}}}|0y_{-j}\\rangle + \\textstyle {\\frac{1}{\\sqrt{2}}}(-1)^{x_{-j}}|1\\overline{y_{-j}}\\rangle .$ A convenient way of denoting an orthonormal basis for the (spatial) tensor product of the Hilbert spaces generated by the two aforementioned sets is as the set of all $|\\dots x_2 x_1 x_0\\,\\mbox{\\Large .",
"}\\, x_{-1} x_{-2} \\dots \\,,\\ \\dots y_2 y_1 y_0\\,\\mbox{\\Large .",
"}\\, y_{-1} y_{-2}\\dots \\rangle ,$ with all but finitely many $x_j$ and $y_{j}$ set to 0.",
"Equivalently, each basis state can be written as $|x,y\\rangle $ , where $x$ and $y$ are dyadic rational numbersA dyadic rational number is of the form $x = a/2^b$ , where $a, b \\in \\mathbb {N}_0$ .",
"Each dyadic $x$ can be written in binary as $x = x_{\\ell } \\dots x_2 x_1 x_0 \\hspace*{0.85358pt}\\mbox{\\Large .",
"}\\hspace*{0.85358pt} x_{-1} x_{-2} \\dots x_{-r}$ .",
"Formally, for all $j \\in \\mathbb {Z}$ , bit $j$ of $x$ is defined as $x_j = \\lfloor x 2^{-j} \\rfloor \\bmod 2$ .",
"Note that $2x$ is $x$ with all the binary digits shifted left by 1..",
"Intuitively, these basis states can be thought of as two-way infinite tensor products, as illustrated in Figure REF .",
"Figure: Schematic picture of the tensor product structure of the basis states.",
"Each circle represents a qubit.",
"In positions 0,1,2,⋯0, 1, 2, \\dots the qubits are in computational basis states.",
"In positions -1,-2,⋯-1, -2, \\dots the qubits are in Bell basis states.On the left are computational basis states (with all but finitely many in state $|00\\rangle $ ).",
"On the right are Bell basis states (with all but finitely many in state $\\frac{1}{\\sqrt{2}}|00\\rangle + \\frac{1}{\\sqrt{2}}|11\\rangle )$ .",
"We now define a left shift $L$ on the Hilbert space spanned by these basis states.",
"Intuitively, $L$ shifts the two-way infinite tensor product to the left by one.",
"Formally, we define $L$ as the product of two unitaries.",
"First, define $L_1$ as the left shift of the digits of $x$ and $y$ $L_1 & |\\dots x_2 x_1 x_0 \\,.\\, x_{-1} x_{-2} \\dots \\,,\\ \\dots y_2 y_1 y_0\\,.\\, y_{-1} y_{-2} \\dots \\rangle \\\\&=|\\dots x_1 x_0 x_{-1}\\,.\\, x_{-2} y_{-3} \\dots \\,,\\ \\dots y_1 y_0 y_{-1} \\,.\\, y_{-2} y_{-3} \\dots \\rangle ,$ or, equivalently, as $L_1 |x,y\\rangle = |2x,2y\\rangle $ .",
"$L_1$ is unitary because it is a permutation of the basis states.",
"Note that $L_1$ does not implement the desired left shift $L$ because the qubits in position $-1$ are in the Bell basis whereas the qubits in position 0 are in the computational basis.",
"A basis conversion is needed when position $-1$ is shifted to position 0.",
"Define $L_2$ to perform this basis conversion in position 0 as $L_2 & |\\dots x_2 x_1 x_0 \\,.\\, x_{-1} x_{-2} \\dots \\,,\\ \\dots y_2 y_1 y_0 \\,.\\, y_{-1} y_{-2}\\dots \\rangle \\\\&={\\textstyle {\\frac{1}{\\sqrt{2}}}}|\\dots x_2 x_1 0 \\,.\\, x_{-1} x_{-2} \\dots \\,,\\ \\dots y_2 y_1 y_0 \\,.\\, y_{-1} y_{-2}\\dots \\rangle \\\\&\\ \\ + {\\textstyle {\\frac{1}{\\sqrt{2}}}}(-1)^{x_0}|\\dots x_2 x_1 1 \\,.\\, x_{-1} x_{-2} \\dots \\,,\\ \\dots y_2 y_1 \\overline{y_0} \\,.\\, y_{-1} y_{-2}\\dots \\rangle .$ This can be equivalently expressed in terms of arithmetic operations on dyadic rationals as $L_2 |x,y\\rangle ={\\textstyle {\\frac{1}{\\sqrt{2}}}}|x - x_0,y\\rangle + {\\textstyle {\\frac{1}{\\sqrt{2}}}}(-1)^{x_0}|x - x_0 + 1, y - 2y_0 + 1\\rangle .$ $L_2$ is unitary because it is a direct sum of $4 \\times 4$ unitaries.",
"Finally, define $L = L_2 L_1$ , which is unitary because $L_1$ and $L_2$ are unitary.",
"An interesting property of $L$ is that applying this operation to the state $|0.0,0.0\\rangle $ yields ${\\textstyle {\\frac{1}{\\sqrt{2}}}}|0.0,0.0\\rangle + {\\textstyle {\\frac{1}{\\sqrt{2}}}}|1.0,1.0\\rangle $ .",
"In the tensor product picture, $L$ leaves the state of all the qubits intact except for the qubits in position 0, whose state changes from $|00\\rangle $ to $\\frac{1}{\\sqrt{2}}|00\\rangle + \\frac{1}{\\sqrt{2}}|11\\rangle $ .",
"This is performing something like an embezzlement transformation (in a manner reminiscent of the imaginary “Hilbert hotel\"); however, this $L$ does not decompose into two commuting operations that have the structure illustrated in Figure REF .",
"In order to obtain such a decomposition, we need to enlarge our Hilbert space.",
"We begin with some intuition.",
"In Figure REF , assume that Alice possesses the qubits in the first row and Bob possesses the qubits in the second row.",
"When Alice's qubits are shifted to the left by one, the picture changes to that of Figure REF .",
"Figure: Schematic picture of the tensor product of basis states when Alice's qubits (top row) are shifted left by one.Pairs of circles connected by lines are in the Bell basis.Such states are orthogonal to all states of the form of Figure .This can be equivalently expressed by shifting the labels of Alice's qubits as in Figure REF .",
"Figure: An alternative schematic picture of the tensor product of Figure , where the labels of the qubits on the top row are adjusted to reflect the shift in the top row.More generally, for an arbitrary $r \\in \\mathbb {Z}$ , a left shift of the first row by $r$ is illustrated in Figure REF .",
"Figure: Schematic picture of a left shift of Alice's qubits (top row) by r∈ℤr \\in \\mathbb {Z}.With the picture of Figure REF in mind, define the Hilbert space $\\mathcal {R}$ as having orthonormal basis states of the form $|r,x,y\\rangle $ , where $x, y$ are dyadic rationals and $r \\in \\mathbb {Z}$ represents the leftward shift of Alice's qubits.",
"We can interpret $|r,x,y\\rangle $ as an encoding of the following logical state.",
"For all $j \\in \\mathbb {Z}$ , Alice's logical qubit in position $j+r$ and Bob's logical qubit in position $j$ are in the joint state ${\\left\\lbrace \\begin{array}{ll}|x_{j}y_{j}\\rangle & \\text{if $j \\ge 0$} \\\\[2mm]{\\textstyle {\\frac{1}{\\sqrt{2}}}}|0y_{j}\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}(-1)^{x_{j}}|1\\overline{y_{j}}\\rangle & \\text{if $j < 0$.}\\end{array}\\right.",
"}$ Now we define the Alice left shift $L_A$ as simply $L_A |r,x,y\\rangle = |r+1,x,y\\rangle .$ $L_A$ is obviously unitary and commutes with $L$ , since they act on different components of $|r,x,y\\rangle $ .",
"Next, define the Bob left shift $L_B$ as $L_B = L_{A}^{*}L$ (a left shift of both Alice and Bob's qubits followed by a right shift of Alice's qubits).",
"Note that $L_A$ and $L_B$ commute, since $L_A$ and $L$ commute.",
"Also, since $L$ is a left shift by both Alice and Bob and $L_{A}^{*}$ is a right shift by Alice, $L_B$ has no net effect on Alice's logical qubits."
],
[
"Swap operations between $\\mathcal {H}_A$ , {{formula:5ce0fc52-fcb6-4283-ae7b-84f3bd08f3dc}} and {{formula:3ba76a35-7dd7-4b48-878e-0858d3c7d485}}",
"Prior to defining our embezzlement protocol, we define swap operations between $\\mathcal {H}_A$ and the logical qubit of Alice in position 0 of $\\mathcal {R}$ , and between $\\mathcal {H}_B$ and the logical qubit of Bob in position 0 of $\\mathcal {R}$ .",
"The Bob swap $S_B$ is defined simply as the unitary operation that acts on $\\mathcal {H}_B \\otimes \\mathcal {R}$ as $S_B |t\\rangle \\otimes |r,\\, x,\\, \\dots y_1 y_0 \\,.\\, y_{-1} \\dots \\rangle =|y_0\\rangle \\otimes |r,\\, x,\\, \\dots y_1 t \\,.\\, y_{-1} \\dots \\rangle ,$ or, equivalently, as $S_B|t\\rangle \\otimes |r,x,y\\rangle = |y_0\\rangle \\otimes |r,x,y-y_0+t\\rangle $ .",
"$S_B$ is clearly unitary and commutes with $L_A$ since they act on different components of each basis state $|r,x,y\\rangle \\in \\mathcal {R}$ .",
"The corresponding Alice swap, acting on $\\mathcal {H}_A \\otimes \\mathcal {R}$ , is more complicated than $S_B$ .",
"First define $\\tilde{S}_A$ (a naïve Alice swap) as $\\tilde{S}_A |s\\rangle \\otimes |r,\\, \\dots x_1 x_0 \\,.\\, x_{-1} \\dots ,\\, y\\rangle =|x_0\\rangle \\otimes |r,\\, \\dots x_1 s \\,.\\, x_{-1} \\dots ,\\, y\\rangle ,$ or, equivalently, as $\\tilde{S}_A|s\\rangle \\otimes |r,x,y\\rangle = |x_0\\rangle \\otimes |r,x-x_0+s,y\\rangle $ .",
"$\\tilde{S}_A$ does not swap with Alice's logical qubit in position 0—moreover, $\\tilde{S}_A$ does not commute with $L_B$ .",
"To swap with Alice's logical qubit in position 0, it is convenient to first define the controlled-$L$, denoted as $C$ , acting on $\\mathcal {R}$ as $C |r,x,y\\rangle = L^{r}|r,x,y\\rangle ,$ which makes sense because $L^{r}$ acts only on the second and third component of $|r,x,y\\rangle $ .",
"$C$ is unitary because each $L^{r}$ is unitary and $C$ is a direct sum of all $L^{r}$ .",
"Intuitively, $C^*|r,x,y\\rangle $ is a state in which Alice's literal qubit in position 0 corresponds to Alice's logical qubit in position 0 in $|r,x,y\\rangle $ .",
"Now we define the actual Alice swap as $S_A = C \\tilde{S}_A C^{*}.$ Clearly $S_A$ is unitary and, for each $|r,x,y\\rangle \\in \\mathcal {R}$ , its effect is localized to Alice's logical qubit in position 0.",
"$S_A$ and $S_B$ commute because $S_B$ is localized to Bob's logical qubit in position 0.",
"Moreover, $S_A$ and $L_B$ commute because, for $|r,x,y\\rangle \\in \\mathcal {R}$ , $L_B$ is localized to Bob's logical qubits."
],
[
"The embezzlement protocol",
"The idea is to start with state $|0,\\,0.0,\\,0.0\\rangle $ , perform $L_A$ and $L_B$ , and then swap the two qubits in position 0 of $\\mathcal {R}$ into $\\mathcal {H}_A$ and $\\mathcal {H}_B$ .",
"Alice performs $U_A = S_A L_A$ and Bob performs $U_B = S_B L_B$ .",
"Clearly $U_A$ and $U_B$ commute.",
"Recall that $L_A L_B = L$ .",
"The state evolves as: 0. initial state $|0\\rangle \\otimes |0\\rangle \\otimes |0,\\,0.0,\\,0.0\\rangle $ 1. after $L_A L_B$ $|0\\rangle \\otimes |0\\rangle \\otimes \\bigl ({\\textstyle {\\frac{1}{\\sqrt{2}}}}|0,\\,0.0,\\,0.0\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|0,\\,1.0,\\,1.0\\rangle \\bigr )$ 2. after $S_A S_B$ $\\bigl ({\\textstyle {\\frac{1}{\\sqrt{2}}}}|0\\rangle \\otimes |0\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|1\\rangle \\otimes |1\\rangle \\bigr )\\otimes |0,\\,0.0,\\,0.0\\rangle $ This completes the protocol for perfect embezzlement in the commuting operator framework.",
"Remark 4.1 It is easy to adapt the above method to embezzle a more general entangled state, say, of the form $\\phi = \\sum _{i,j=0}^{d-1} \\alpha _{ij} |i\\rangle \\otimes |j\\rangle $ .",
"First, redefine $\\mathcal {R}$ to be in terms of basis states of the form $|r,x,y\\rangle $ , where $x$ and $y$ are $d$ -adic rational numbers (i.e., with digits in $\\mathbb {Z}_d$ ).",
"Then it suffices to set the operation $L_2$ (which is the basis change part of the left shift operation) to be any unitary operation on $\\mathbb {C}^d \\otimes \\mathbb {C}^d$ that maps $|0\\rangle \\otimes |0\\rangle $ to $\\phi $ .",
"The other parts of the protocol are essentially the same."
],
[
"Coherent embezzlement games",
"A purported protocol for embezzlement cannot be tested in the way that nonlocal games can, because Alice and Bob can perform local operations that perfectly map $|0\\rangle \\otimes |0\\rangle $ to $\\frac{1}{\\sqrt{2}}|0\\rangle \\otimes |0\\rangle +\\frac{1}{\\sqrt{2}}|1\\rangle \\otimes |1\\rangle $ using the resource of only a single (concealed) Bell state.",
"Leung, Toner and Watrous [8] proposed a coherent state exchange game that is related to embezzlement but is operationally testable.",
"In this game, Alice and Bob each receive a qutrit from a referee as input and they each return a qubit to the referee, who performs a measurement on the returned state to determine whether they win or lose.",
"There is no perfect strategy for this game using finite entanglement.",
"It is shown in [8] that, for all $\\epsilon > 0$ : there exists a strategy that succeeds with probability $1 - \\epsilon $ using $O(\\log (1/\\epsilon ))$ -entropy entanglement; moreover, to succeed with probability $1 - \\epsilon $ requires entanglement with entropy $\\Omega (\\log (1/\\epsilon ))$ .",
"Regev and Vidick [10] presented a simplification of the coherent state exchange game that has the above properties, but where the outputs are classical bits instead of qubits (the inputs are still qutrits).",
"In [10], this is called the $T_2$ game.",
"We refer to this as the coherent embezzlement game, to highlight its close relationship with embezzlement.",
"In this section, we begin by reviewing the definition of the coherent embezzlement (a.k.a.",
"$T_2$ ) game.",
"Then we show that a perfect strategy for embezzlement can be converted to a perfect strategy for coherent embezzlement—and vice versa.",
"By such reductions, we prove that there is a perfect strategy for coherent embezzlement in the commuting operator framework (Theorem REF ), but there is no such perfect strategy in the tensor product framework (Theorem REF ).",
"We now define the coherent embezzlement game [10].",
"Alice and Bob each receive two qutrits as input and they each produce a classical bit as output.",
"The input state that Alice and Bob jointly receive is either $\\phi _0$ or $\\phi _1$ , where $\\phi _0 &= {\\textstyle {\\frac{1}{\\sqrt{2}}}}|0\\rangle \\otimes |0\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}\\bigl ({\\textstyle {\\frac{1}{\\sqrt{2}}}}|1\\rangle \\otimes |1\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|2\\rangle \\otimes |2\\rangle \\bigr ) \\\\\\phi _1 &= {\\textstyle {\\frac{1}{\\sqrt{2}}}}|0\\rangle \\otimes |0\\rangle -{\\textstyle {\\frac{1}{\\sqrt{2}}}}\\bigl ({\\textstyle {\\frac{1}{\\sqrt{2}}}}|1\\rangle \\otimes |1\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|2\\rangle \\otimes |2\\rangle \\bigr ).$ Call Alice and Bob's output bits $a$ and $b$ respectively.",
"The winning condition is that: when the input is $\\phi _0$ , $a \\oplus b = 0$ ; when the input is $\\phi _1$ , $a \\oplus b = 1$ .",
"Note that the winning condition does not require Alice and Bob's resource state to be used in a catalytic manner; Alice and Bob are free to destroy this state in their strategy.",
"In this sense, the coherent embezzlement game is simpler than embezzlement, whose definition depends critically on restoring the resource state.",
"In the remainder of this section, for technical convenience, we represent qutrits as pairs of qubits using the encoding $0 \\equiv 00$ , $1 \\equiv 10$ (for Alice) or 01 (for Bob), and $2 \\equiv 11$ .",
"With this encoding, the input states that Alice and Bob jointly receive can be written as, for $c \\in \\lbrace 0,1\\rbrace $ , $\\phi _c &= {\\textstyle {\\frac{1}{\\sqrt{2}}}}|00\\rangle \\otimes |00\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}(-1)^c\\bigl ({\\textstyle {\\frac{1}{\\sqrt{2}}}}|10\\rangle \\otimes |01\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|11\\rangle \\otimes |11\\rangle \\bigr ) \\\\&= {\\textstyle {\\frac{1}{\\sqrt{2}}}}|0\\rangle |00\\rangle |0\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}(-1)^c|1\\rangle \\bigl ({\\textstyle {\\frac{1}{\\sqrt{2}}}}|00\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|11\\rangle \\bigr )|1\\rangle $ (where the first two qubits are Alice's input and the last two qubits are Bob's input).",
"Theorem 5.1 There is a perfect strategy for the coherent embezzlement game in the commuting operator framework.",
"Let $\\mathcal {R}$ , $U_A$ , and $U_B$ be as defined in the unitary embezzlement protocol of section .",
"For coherent embezzlement, the input Hilbert spaces are $\\mathcal {H}_{A_1} \\otimes \\mathcal {H}_{A_2} = \\mathcal {H}_{B_1} \\otimes \\mathcal {H}_{B_2} = \\mathbb {C}^2\\otimes \\mathbb {C}^2$ .",
"We will show that the protocol in Figure REF performs coherent embezzlement.",
"Figure: Circuit diagram of a protocol for coherent embezzlement from a protocol for embezzlement based on U A U_A, U B U_B, and ψ∈ℛ\\psi \\in \\mathcal {R}.First note that, since the controlled $U^*_A$ and $U^*_B$ perform the inverse of embezzlement when their control qubits are in state $|1\\rangle $ , they perform a mapping on $\\mathcal {H}_{A_1} \\otimes \\mathcal {H}_{A_2} \\otimes \\mathcal {H}_{B_2} \\otimes \\mathcal {H}_{B_1}$ such that $\\phi _c =&{\\textstyle {\\frac{1}{\\sqrt{2}}}}|0\\rangle |00\\rangle |0\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}(-1)^c|1\\rangle \\bigl ({\\textstyle {\\frac{1}{\\sqrt{2}}}}|00\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|11\\rangle \\bigr )|1\\rangle \\\\[1mm]&\\mapsto \\ \\ {\\textstyle {\\frac{1}{\\sqrt{2}}}}|0\\rangle |00\\rangle |0\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}(-1)^c|1\\rangle |00\\rangle |1\\rangle .$ Note that, on the Hilbert space $\\mathcal {H}_{A_1} \\otimes \\mathcal {H}_{B_1}$ , this is the pure state ${\\textstyle {\\frac{1}{\\sqrt{2}}}}|00\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}(-1)^c|11\\rangle $ .",
"Finally, since the Hadamard gates perform the mapping ${\\textstyle {\\frac{1}{\\sqrt{2}}}}|00\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}(-1)^c|11\\rangle \\mapsto {\\textstyle {\\frac{1}{\\sqrt{2}}}}|0c\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|1\\overline{c}\\rangle ,$ the result follows.",
"Theorem 5.2 There is no perfect strategy for the coherent embezzlement game in the tensor product framework (where Alice and Bob are allowed to use ancillas).",
"The idea is that a perfect strategy for coherent embezzlement can be converted into a perfect strategy for embezzlement.",
"Suppose that there is a perfect strategy for coherent embezzlement.",
"Without loss of generality, it can be assumed that this strategy is of the form depicted in Figure REF , where $\\psi \\in \\mathcal {R}_A \\otimes \\mathcal {R}_B$ , Alice and Bob each perform a local unitary operation followed by measurement of one of their qubits.",
"Figure: Circuit diagram of an arbitrary protocol for coherent embezzlement.ψ∈ℛ A ⊗ℛ B \\psi \\in \\mathcal {R}_A \\otimes \\mathcal {R}_B is the resource state.Without loss of generality, two local unitaries are performed, followed by measurements of two specific qubits.Since we are assuming that the strategy is perfect, for input state $\\phi _0$ , the output bits satisfy $a\\oplus b = 0$ , and for input state $\\phi _1$ , the output bits satisfy $a\\oplus b = 1$ .",
"Now, consider the protocol in Figure REF .",
"Figure: Modification of circuit in Figure that performs embezzlement using the resource state ψ∈ℛ A ⊗ℛ B \\psi \\in \\mathcal {R}_A \\otimes \\mathcal {R}_B.We begin by showing that, for each $c \\in \\lbrace 0,1\\rbrace $ , the effect of the first three steps of the protocol in Figure REF is to map $\\phi _c\\otimes \\psi \\otimes \\gamma _A\\otimes \\gamma _B$ to $(-1)^c \\phi _c\\otimes \\psi \\otimes \\gamma _A\\otimes \\gamma _B$ .",
"Since $a \\oplus b = c$ , After the $U_A \\otimes U_B$ operations have been performed, the state must be of the form $\\alpha |0\\rangle \\otimes |c\\rangle \\otimes \\xi _0 + \\beta |1\\rangle \\otimes |\\overline{c}\\rangle \\otimes \\xi _1,$ when expressed the state in the Hilbert space $\\mathcal {H}_{A_1} \\otimes \\mathcal {H}_{B_1} \\otimes (\\mathcal {H}_{A_2} \\otimes \\mathcal {H}_{B_2} \\otimes \\mathcal {R}_A \\otimes \\mathcal {R}_B \\otimes \\mathcal {G}_A \\otimes \\mathcal {G}_B).$ Therefore, after the two $Z$ gates, the state is $(-1)^c\\bigl (\\alpha |0\\rangle \\otimes |c\\rangle \\otimes \\xi _0 + \\beta |1\\rangle \\otimes |\\overline{c}\\rangle \\otimes \\xi _1\\bigr ),$ and this is mapped by $U^{*}_{A} \\otimes U^{*}_{B}$ to $(-1)^c \\phi _c\\otimes \\psi \\otimes \\gamma _A\\otimes \\gamma _B$ .",
"This implies that the first three steps of the modified protocol maps $|00\\rangle \\otimes |00\\rangle = {\\textstyle {\\frac{1}{\\sqrt{2}}}}\\phi _0 + {\\textstyle {\\frac{1}{\\sqrt{2}}}}\\phi _1$ on $\\mathcal {H}_{A}\\otimes \\mathcal {H}_B$ (with the other registers in state $\\psi \\otimes \\gamma _A\\otimes \\gamma _B$ ) to ${\\textstyle {\\frac{1}{\\sqrt{2}}}}|10\\rangle \\otimes |01\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|11\\rangle \\otimes |11\\rangle = {\\textstyle {\\frac{1}{\\sqrt{2}}}}\\phi _0 - {\\textstyle {\\frac{1}{\\sqrt{2}}}}\\phi _1.$ Finally, the two $X$ gates of the protocol in Figure REF map the state ${\\textstyle {\\frac{1}{\\sqrt{2}}}}|10\\rangle \\otimes |01\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|11\\rangle \\otimes |11\\rangle $ to ${\\textstyle {\\frac{1}{\\sqrt{2}}}}|00\\rangle \\otimes |00\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|01\\rangle \\otimes |10\\rangle $ .",
"Therefore, the entire protocol maps $|00\\rangle \\otimes |00\\rangle \\otimes \\psi \\otimes \\gamma _A\\otimes \\gamma _B$ to $\\bigl ({\\textstyle {\\frac{1}{\\sqrt{2}}}}|00\\rangle \\otimes |00\\rangle &+{\\textstyle {\\frac{1}{\\sqrt{2}}}}|01\\rangle \\otimes |10\\rangle \\bigr )\\otimes \\psi \\otimes \\gamma _A\\otimes \\gamma _B \\\\& =|0\\rangle ({\\textstyle {\\frac{1}{\\sqrt{2}}}}|00\\rangle +{\\textstyle {\\frac{1}{\\sqrt{2}}}}|11\\rangle )|0\\rangle \\otimes \\psi \\otimes \\gamma _A\\otimes \\gamma _B.$ This is perfect embezzlement in the registers $\\mathcal {H}_{A_2} \\otimes \\mathcal {H}_{B_2}$ .",
"Since this protocol violates Theorem REF , there cannot exist a perfect strategy for coherent embezzlement using entanglement of the form $\\psi \\in \\mathcal {R}_A \\otimes \\mathcal {R}_B$ ."
],
[
"Perfect embezzlement requires non-unitary isometries",
"In this section we obtain further information about the nature of the unitary operators that appear in perfect embezzlement protocols.",
"We show that some of the operators that occur in such protocols must contain non-unitary isometries.",
"This result justifies the need for the register shifts that appear in the explicit protocol of the earlier section.",
"Also, since non-unitary isometries only exist in infinite dimensions, this result implies that perfect embezzlement is impossible with a finite dimensional resource space.",
"We begin by reviewing the key facts about unitaries and isometries.",
"Given a Hilbert space $\\mathcal {H}$ a linear map $V: \\mathcal {H} \\rightarrow \\mathcal {H}$ is an isometry provided that $\\Vert Vh\\Vert =\\Vert h\\Vert $ for all $h \\in \\mathcal {H}$ .",
"Note that isometries are necessarily one-to-one.",
"The map $V$ is a unitary provided that it is an isometry and it is onto.",
"A simple dimension count shows that in finite dimensions every isometry is necessarily a unitary.",
"An example of an isometry that is not a unitary is the unilateral shift S. This is the operator on the Hilbert space $\\ell ^2(\\mathbb {N})$ which has an orthonormal basis $\\lbrace |j\\rangle : j \\in \\mathbb {N} \\rbrace $ , defined by $S(|j\\rangle ) = |j+1\\rangle $ .",
"It is easily seen that this operator is an isometry and the $|1\\rangle $ is orthogonal to the range of $S$ , so that $S$ is not onto.",
"Note that the kernel of $S^*$ is equal to the span of $|1\\rangle $ .",
"We call a linear map $V: \\mathcal {H} \\rightarrow \\mathcal {H}$ a non-unitary isometry provided that it is an isometry that is not onto.",
"It is not hard to show that if $V$ is a non-unitary isometry, $e_1$ is any unit vector orthogonal to the range of $V$ , and we set $e_n = V^n e_1,$ then this sequence is orthonormal and $V$ acts as a unilateral shift on this subspace.",
"So, in this sense, non-unitary isometries always contain a space on which they act like unilateral shifts.",
"Recall that a subspace $\\mathcal {M}$ is called invariant for an operator $C$ provided that $C(\\mathcal {M}) \\subseteq \\mathcal {M}$ .",
"Lemma 6.1 Let $\\mathcal {H}$ be a Hilbert space, $C: \\mathcal {H} \\rightarrow \\mathcal {H}$ be a contraction (i.e., $\\Vert Ch\\Vert \\le \\Vert h \\Vert , \\, \\forall h \\in \\mathcal {H}$ ), and $\\mathcal {M} \\subseteq \\mathcal {H}$ be a subspace that is invariant for $C$ .",
"If the restriction of $C$ to $\\mathcal {M}$ acts as a unitary $U$ on $\\mathcal {M}$ , then $\\mathcal {M}$ is also invariant for $C^*$ and the restriction of $C^*$ to $\\mathcal {M}$ is $U^*$ .",
"This is easiest to see using operator matrices.",
"Decomposing $\\mathcal {H} = \\mathcal {M} \\oplus \\mathcal {M}^{\\perp }$ we may write $C= \\begin{pmatrix} U &X \\\\Y & Z \\end{pmatrix}$ where $U: \\mathcal {M} \\rightarrow \\mathcal {M},$ $X: \\mathcal {M}^{\\perp } \\rightarrow \\mathcal {M}$ , $Y: \\mathcal {M} \\rightarrow \\mathcal {M}^{\\perp }$ , and $Z: \\mathcal {M}^{\\perp } \\rightarrow \\mathcal {M}^{\\perp }$ .",
"The fact that $\\mathcal {M}$ is invariant implies that $Y=0$ .",
"The hypothesis that the restriction of $C$ to $\\mathcal {M}$ is a unitary, means that $U$ is a unitary.",
"But $C^*= \\begin{pmatrix} U^* & X^*\\\\ 0 & Z^* \\end{pmatrix}$ is a contraction with $U^*$ a unitary.",
"This forces that for every $h \\in \\mathcal {M},$ $X^*h =0$ .",
"Thus, $X^*=0$ and $\\mathcal {M}$ is invariant for $C^*$ .",
"Note that if $U$ was a non-unitary isometry, then $U^*$ would have a kernel and we could no longer conclude that $X^*=0$ .",
"The next concept that we shall need is the polar decomposition of an operator.",
"Given an operator $A$ on a Hilbert space $\\mathcal {H}$ , we set $|A|= (A^*A)^{1/2}$ .",
"Let $\\mathcal {R}(A)$ denote the closure of the range of $A$ and let $\\mathcal {R}(|A|)$ denote the closure of the range of $|A|$ .",
"Note that for any $x \\in \\mathbb {\\mathcal {\\unknown.",
"}}H,$ we have $ \\Vert Ax\\Vert ^2 = \\langle Ax, Ax \\rangle = \\langle x, A^*Ax \\rangle =\\langle |A|x, |A|x \\rangle = \\Vert |A|x\\Vert ^2.$ From this equality, it follows that there is a well-defined linear isometry $W: \\mathcal {R}(|A|) \\rightarrow \\mathcal {R}(A)$ defined by setting $W(|A|h) = Ah$ , so that $A= W|A|.$ If we extend $W$ to a map $U: \\mathcal {H} \\rightarrow \\mathcal {H}$ by sitting $U= WQ$ where $Q$ is the orthogonal projection onto $\\mathcal {R}(|A|)$ then we still have $A= U|A|$ .",
"The map $U$ is called a partial isometry.",
"This representation $A=U|A|$ is called the polar decomposition of $A$ .",
"Lemma 6.2 Let $C$ be a contraction and let $C=U|C|$ be its polar decomposition.",
"If $h$ is a unit vector such that $||C^nh||=1, \\forall n \\in \\mathbb {N}$ , then the closed subspace $\\mathcal {M}$ generated by $\\lbrace C^nh: n \\ge 0 \\rbrace $ is an invariant subspace for $C$ and for any $v \\in \\mathcal {M},$ $\\Vert Cv\\Vert =\\Vert v\\Vert $ .",
"Clearly $\\mathcal {M}$ is invariant for $C$ .",
"Set $P = |C|$ so that $C=UP$ with $0 \\le P \\le I$ and $||Ph||=||UPh||= ||Ch||=1$ .",
"This implies that $Ph=h$ .",
"Thus, $Ch= Uh$ .",
"Since $1= ||C^2h|| = ||UPUPh|| = ||UP(Uh)|| = ||P(UH)|| $ we have that $P(Uh) = Uh$ and so $C^2h = UPUPh =UPUh=U^2h.$ Inductively, $P(U^nh) = U^nh$ so that $C^nh = U^n h$ .",
"Thus, $Cv= Uv$ for any $v \\in \\mathcal {M}$ .",
"Since $P(U^nh) = U^n h$ for all $n \\ge 0$ , we have that $\\mathcal {M} \\subseteq \\mathcal {R}(|C|)$ .",
"But $U$ acts isometrically on $\\mathcal {R}(|C|)$ and since $U$ and $C$ are equal on this subspace, $C$ acts isometrically on $\\mathcal {M}$ .",
"Theorem 6.3 Let $\\mathcal {H}_R, \\psi , U_A=(U_{ij}),$ and $U_B= (V_{kl})$ be a perfect embezzlement protocol.",
"If $\\mathcal {M}$ is the closed subspace of $\\mathcal {H}$ spanned by $\\lbrace U_{00}^{*n} \\psi : n \\ge 0 \\rbrace $ , then the restriction of $U_{00}^*$ to this invariant subspace is a non-unitary isometry.",
"Recall that the $U_{ij}$ 's must *-commute with the $V_{kl}$ 's.",
"The embezzlement relations tell us that $ U_{00}V_{00}\\psi = \\psi /\\sqrt{2}, \\ \\ U_{10}V_{00}\\psi =0, \\ \\ U_{10}V_{10}\\psi = \\psi /\\sqrt{2}, \\ \\ U_{00}V_{10}\\psi =0.$ The fact that $U_A$ and $U_B$ are unitaries implies that $ I= V^*_{00}V_{00}+ V^*_{10}V_{10}= U_{00}^*U_{00} + U_{10}^*U_{10} $ so that $U_{00} \\psi &= U_{00}(V_{00}^*V_{00} + V_{10}^*V_{10})\\psi = V_{00}^*U_{00}V_{00}\\psi = V_{00}^*\\psi /\\sqrt{2}, \\\\V_{00}\\psi &= V_{00}(U_{00}^*U_{00} +U_{10}^*U_{10})\\psi = U_{00}^*\\psi / \\sqrt{2}.$ Iterating, yields $(V_{00}^*)^n\\psi = (\\sqrt{2}U_{00})^n \\psi \\mbox{\\ \\ and \\ \\ } V_{00}^n \\psi = (U_{00}^*/\\sqrt{2})^n \\psi .$ Since $(1/\\sqrt{2})^n \\psi = (U_{00}V_{00})^n\\psi = U_{00}^nV_{00}^n \\psi =U_{00}^n(U_{00}^*/\\sqrt{2})^n \\psi $ , we have that $U_{00}^n U_{00}^{*n}\\psi =\\psi .$ Because $||U_{00}|| \\le 1$ we have that $||U_{00}^{*n} \\psi || =1$ for all $n$ .",
"Hence, by the previous lemma, $U_{00}^*$ acts isometrically on $\\mathcal {M}$ .",
"Now if $U_{00}^*$ acted unitarily, then by the earlier lemma, for vectors in this space $U_{00}$ would be the inverse and in particular would act isometrically on $\\mathcal {M}$ .",
"But then we would have that $\\Vert V_{00}^*\\psi \\Vert = \\sqrt{2} \\Vert U_{00}\\psi \\Vert = \\sqrt{2}$ .",
"This is impossible, because $U_B$ is a unitary and so $\\Vert V_{00}^*\\Vert \\le 1$ .",
"This contradiction shows that $U_{00}^*$ must be a non-unitary isometry on $\\mathcal {M}$ .",
"This yields the following fact.",
"Recall that for a perfect embezzlement protocol, we are only assuming that the operators commute, not that the resource space has a bipartite tensor structure.",
"Corollary 6.4 Perfect embezzlement is impossible in the commuting-operator framework if the resource space is finite-dimensional.",
"If $\\mathcal {H}_R$ is finite dimensional, then $\\mathcal {M}$ is also finite dimensional.",
"But every isometry on a finite dimensional space is necessarily a unitary, contradicting the fact that $U_{00}^*$ is a non-unitary isometry."
],
[
"Acknowledgements",
"We would like to thank Marius Junge, Debbie Leung, Volkher Scholz, and John Watrous for helpful discussions.",
"This research was supported in part by Canada's NSERC, a David R. Cheriton Scholarship, and a Mike and Ophelia Lazaridis Fellowship."
],
[
"The Schmidt and polar decompositions in infinite dimensions",
"In this section, for the convenience of the reader, we gather together a few useful results from operator theory that are not well known within the QIT community.",
"We are claiming no originality.",
"Definition 1.1 Let $W: H_1 \\rightarrow H_2$ .",
"Then $W$ is called an isometry if $\\Vert Wh_1\\Vert _2 = \\Vert h_1\\Vert _1$ for every $h_1 \\in H_1$ .",
"$W$ is called a coisometry iff $W^*:H_2 \\rightarrow H_1$ is an isometry.",
"$W$ is called a partial isometry if the restriction of $W$ to $ker(W)^{\\perp }$ is an isometry.",
"In this case the space $ker(W)^{\\perp }$ is called the initial space of $W$ and $ran(W)^- = ker(W^*)^{\\perp }$ is called the final space of $W$ .",
"Proposition 1.2 Let $H$ and $K$ be Hilbert spaces of arbitrary dimension, let $\\lbrace e_{\\alpha }: \\alpha \\in A \\rbrace $ and $\\lbrace f_{\\beta } : \\beta \\in B \\rbrace $ be o.n.",
"bases for $H$ and $K$ , respectively.",
"Then $\\lbrace e_{\\alpha } \\otimes f_{\\beta }: \\alpha \\in A, \\beta \\in B \\rbrace $ is an o.n.",
"basis for $H \\otimes K.$ Proposition 1.3 (The Polar Decomposition) Let $X: H_1 \\rightarrow H_2$ and let $|X|= (X^*X)^{1/2}.$ Then there is a unique partial isometry $W:H_1 \\rightarrow H_2$ with initial space $ker(X)^{\\perp }=ker(|X|)^{\\perp }= ran(|X|)^-$ and final space $ran(X)^-= ker(X^*)^{\\perp }$ such that $X= W|X|.$ To prove, one simply sets $W(|X|h) = Xh$ and shows that this is well-defined and satisfies the properties.",
"Note that $W$ is an isometry iff $ker(X) = (0)$ and is a coisometry iff $ran(X)^- = H_2.$ Proposition 1.4 Let $\\lbrace e_k \\rbrace $ be an o.n.",
"sequence in a Hilbert space, set $u_i = \\sum _k u_{i,k} e_k$ and let $U=(u_{i,j})$ .",
"Then $\\lbrace u_i \\rbrace $ is o.n.",
"iff $UU^*= I,$ i.e., $U$ is a coisometry.",
"Theorem 1.5 (The Infinite Dimensional Schmidt Decomposition) Let $H$ and $K$ be Hilbert spaces of arbitrary dimension and let $x \\in H \\otimes K$ .",
"Then there are countable orthonormal sets $ u_k \\in H$ and $v_k \\in K$ and $d_k \\ge 0$ and $d_k \\ge d_{k+1}, \\forall k$ , such that $x= \\sum _k d_k u_k \\otimes v_k,$ and so $\\Vert x\\Vert ^2= \\sum _k d_k^2.$ Moreover, if $x = \\sum _k c_k w_k \\otimes z_k$ is another such representation of $x$ , then $c_k=d_k$ for all $k$ .",
"Pick any orthonormal bases $\\lbrace e_{\\alpha } : \\alpha \\in A \\rbrace $ and $\\lbrace f_{\\beta }: \\beta \\in B \\rbrace $ .",
"By Proposition REF , we can expand $x = \\sum _{\\alpha , \\beta } z_{\\alpha , \\beta } e_{\\alpha } \\otimes f_{\\beta }$ .",
"We know that only countably many of the coefficients are non-zero, so we only need countably many $\\alpha $ 's and countably many $\\beta $ 's.",
"So we can write $x = \\sum _{i,j} z_{i,j} e_i \\otimes f_j.$ and $\\sum _{i,j} |z_{i,j}|^2 = \\Vert x\\Vert ^2$ .",
"Let $h_j = \\sum _i z_{i, j} e_i$ , so that $x = \\sum _j h_j\\otimes f_j$ , and let $H = \\text{span}\\lbrace h_j\\rbrace $ .",
"Let $\\lbrace |i\\rangle \\rbrace $ be an orthonormal basis of $H$ , and write $h_j = \\sum _{i} x_{i, j} |i\\rangle $ .",
"This gives us $x = \\sum _{i, j} x_{i, j}|i\\rangle \\otimes f_j$ .",
"Let $X = \\sum _{i, j} x_{i, j} |i\\rangle \\langle j|$ be the matrix of a map from $H$ to $H$ .",
"Note that $X$ is Hilbert-Schmidt and so compact and also has dense range because $\\text{span}\\lbrace |i\\rangle \\rbrace = \\text{span} \\lbrace h_j\\rbrace = \\text{span}\\lbrace \\sum _{i} x_{i, j} |i\\rangle \\rbrace $ .",
"By Proposition REF , performing polar decomposition on $X$ yields $X = W |X|$ where $W$ is a partial isometry, and $|X| = (X^* X)^{1/2}$ .",
"Since $|X|$ is compact and positive, it has an orthonormal basis of eigenvectors.",
"This defines a unitary $V$ such that $V |X| V^* = D$ , where $D =\\sum _{k} d_k |k\\rangle \\langle k|$ is a diagonal matrix and the $d_k$ 's are the singular values of $X$ arranged in decreasing order.",
"Conjugating $D$ by $V$ , we get $|X| = V^* D V$ .",
"Combining this with the polar decomposition, we get $X = W|X| = WV^* D V = U D V$ where $U = WV^*$ is a partial isometry.",
"Moreover, since $X$ has dense range, $U$ is a coisometry.",
"Let $U = \\sum _{i, j} u_{i,j} |i\\rangle \\langle j|$ , and $V = \\sum _{i, j} v_{i, j}|i\\rangle \\langle j|$ .",
"Then, $\\sum _{i,j} |x_{i,j}|^2 &=\\mbox{Tr}(X^*X)= \\mbox{Tr}(V^*D^*U^*UDV) \\\\&= \\mbox{Tr}(V^*D^2V) = \\mbox{Tr}(D^2) = \\sum _k d_k^2.$ Now let $u_k = \\sum _{i} u_{i, k} |i\\rangle $ and $v_k = \\sum _j v_{k, j} f_j$ .",
"Since $U$ is a coisometry, by Proposition REF , $\\lbrace u_k\\rbrace $ is an orthonormal set.",
"Now we have $x = \\sum _{i, j} x_{i, j} |i\\rangle \\otimes f_j = \\sum _{i, j, k} u_{i, k} d_k v_{k, j} |i\\rangle \\otimes f_j = \\sum _{k} d_k u_k\\otimes v_k.$ The statement about the uniqueness of the sequence $d_k$ follows from the fact that these numbers are the singular values of the Hilbert-Schmidt matrix $X$ and that any other choice of basis for representing $X$ would give rise to a matrix that is obtained from $X$ by pre and post multiplying by unitaries, which does not alter the singular values."
],
[
"A primer on C*-algebras",
"For readers unfamiliar with C*-algebras, we briefly mention the definitions and tools that we shall use.",
"For very readable general references we recommend [3] or [7].",
"Given a Hilbert space $\\mathcal {H}$ we let $B(\\mathcal {H})$ denote the set of bounded linear operators from $\\mathcal {H}$ to $\\mathcal {H}$ .",
"By a C*-algebra of operators we mean a subset $\\mathcal {A} \\subseteq B(\\mathcal {H})$ for some Hilbert space $\\mathcal {H}$ satisfying: $X, Y \\in \\mathcal {A}, \\lambda \\in \\mathbb {C} \\Rightarrow (\\lambda X+Y) \\in \\mathcal {A}$ and $XY \\in \\mathcal {A}$ , $X \\in \\mathcal {A} \\Rightarrow X^* \\in \\mathcal {A}$ , where $X^*$ denotes the adjoint of the operator $X$ (sometimes denoted by $X^{\\dag }$ in the physics literature), $\\mathcal {A}$ is closed in the operator norm, i.e., if $X_n \\in \\mathcal {A}$ , $X \\in B(\\mathcal {H})$ and $\\Vert X_n - X \\Vert \\rightarrow 0,$ then $X \\in \\mathcal {A}$ .",
"The first condition is the definition of what it means to say that $\\mathcal {A}$ is an algebra over the complex field.",
"The second condition is that $\\mathcal {A}$ be invariant under the taking of operator adjoints and the third is that it be a closed subset of $B(\\mathcal {H})$ in a certain topology.",
"C*-algebras of operators also have an abstract characterization.",
"An algebra $\\mathcal {A}$ over the complex numbers that is equipped with a norm $\\Vert \\cdot \\Vert $ that satisfies $\\Vert xy\\Vert \\le \\Vert x\\Vert \\cdot \\Vert y\\Vert $ is called a normed algebra.",
"If a normed algebra is complete, i.e., if every Cauchy sequence converges, then it is called a Banach algebra.",
"Given an algebra $\\mathcal {A}$ over the complex numbers, a *-map is a map from $\\mathcal {A}$ to $\\mathcal {A}$ , $^*: \\mathcal {A} \\rightarrow \\mathcal {A}$ satisfying $(x+y)^*= x^*+y^*$ , $(\\lambda x)^* = \\overline{\\lambda } x^*,$ $(xy)^*= y^*x^*$ and $(x^*)^*=x$ .",
"An algebra equipped with a *-map is called a *-algebra.",
"A map $\\pi : \\mathcal {A} \\rightarrow \\mathcal {B}$ between two algebras that is linear and satisfies $\\pi (xy) = \\pi (x) \\pi (y)$ is called a homomorphism.",
"If both algebras are also *-algebras and the map satisfies $\\pi (x^*) = \\pi (x)^*$ then $\\pi $ is called a *-homomorphism.",
"Finally, an (abstract) C*-algebra $\\mathcal {A}$ is a Banach *-algebra that satisfies $\\Vert x^*x\\Vert = \\Vert x\\Vert ^2, \\, \\forall x \\in \\mathcal {A}$ .",
"Note that $B(\\mathcal {H})$ is an abstract C*-algebra and so is every C*-algebra of operators.",
"The celebrated Gelfand-Naimark-Segal theorem shows that every abstract C*-algebra is in an appropriate sense a C*-algebra of operators.",
"Theorem 2.1 (Gelfand-Naimark-Segal) Let $\\mathcal {A}$ be an abstract C*-algebra.",
"Then there is a Hilbert space $\\mathcal {H}$ and a map $\\pi : \\mathcal {A} \\rightarrow B(\\mathcal {H})$ such that: $\\pi $ is a *-homomorphism, $\\Vert \\pi (x)\\Vert = \\Vert x\\Vert $ for all $x \\in \\mathcal {A}$ .",
"Moreover, if $\\mathcal {A}$ has a unit element $1 \\in \\mathcal {A}$ then, in addition, one can arrange that $\\pi (1) = I_{\\mathcal {H}}$ .",
"A map satisfying the second condition is called an isometry.",
"Clearly, an isometry is one-to-one.",
"Conversely, it is a theorem that every one-to-one *-homomorphism is an isometry.",
"A one-to-one, onto *-homomorphism is called a *-isomorphism.",
"The two conditions in the above theorem also guarantee that the range of $\\pi ,$ $\\mathcal {B}= \\pi (\\mathcal {A})$ is a C*-algebra of operators.",
"Thus, $\\pi $ is a *-isomorphism from the abstract C*-algebra onto a C*-algebra of operators.",
"A key element of the proof of the above theorem is their theorem on representations of states.",
"A state on an abstract unital C*-algebra $\\mathcal {A}$ is any linear functional, $s: \\mathcal {A} \\rightarrow \\mathbb {C}$ such that $s(1) =1$ and $s(x^*x) \\ge 0$ for every $x \\in \\mathcal {A}$ .",
"Theorem 2.2 (GNS state representation theorem) Let $s:\\mathcal {A} \\rightarrow \\mathbb {C}$ be a state on a unital C*-algebra.",
"Then there exists a Hilbert space $\\mathcal {H}_s$ , a unit vector $\\xi \\in \\mathcal {H}_s$ and a unital *-homomorphism, $\\pi _s: \\mathcal {A} \\rightarrow B(\\mathcal {H}_s)$ such that $s(x) = \\langle \\xi | \\pi _s(x) \\xi \\rangle $ for all $x \\in \\mathcal {A}$ and such that the subspace of vectors of the form $\\lbrace \\pi _s(x) \\xi : x \\in \\mathcal {A} \\rbrace $ is dense in $\\mathcal {H}_s$ .",
"Thus, the theorem says that for each abstract state there is a way to realize the C*-algebra as a C*-algebra of operators such that the state becomes a vector state."
],
[
"The State Space",
"Given a unital C*-algebra $\\mathcal {A}$ , the set of all states on $\\mathcal {A}$ , denoted $S(\\mathcal {A})$ is a convex set.",
"Moreover, it is endowed with a topology, called the weak*-topology and in this topology it is a compact set.",
"A net of states $\\lbrace s_{\\lambda } \\rbrace $ converges to a state $s$ in this topology if and only if $\\lim _{\\lambda } | s_{\\lambda }(a) - s(a)| =0$ for every $a \\in \\mathcal {A}$ ."
],
[
"Tensor Products of C*-algebras",
"Let $\\mathcal {A}$ and $\\mathcal {B}$ be two unital C*-algebras, and let $\\mathcal {A}\\otimes \\mathcal {B}$ be their algebraic tensor product.",
"Given $x= \\sum _i a_i \\otimes b_i$ and $y = \\sum _j c_j \\otimes d_j$ in $\\mathcal {A} \\otimes \\mathcal {B}$ we define their product by $ xy = \\sum _{i,j} a_i c_j \\otimes b_i \\otimes d_j,$ and a *-map by $x^* = \\sum _i a_i^* \\otimes b_i^*.$ Endowed with these two operations, $\\mathcal {A} \\otimes \\mathcal {B}$ becomes a *-algebra.",
"Note that the *-subalgebra $\\lbrace a \\otimes 1: a \\in \\mathcal {A} \\rbrace $ can be identified with $\\mathcal {A}$ and similarly, $\\lbrace 1 \\otimes b: b \\in \\mathcal {B} \\rbrace $ can be identified with $\\mathcal {B}$ .",
"Also $(a \\otimes 1)(1 \\otimes b) = a \\otimes b = (1 \\otimes b)(1 \\otimes a)$ so that these “copies\" of $\\mathcal {A}$ and $\\mathcal {B}$ commute.",
"There are two important ways to give this *-algebra a norm so that it can be completed to become a C*-algebra.",
"Given $x \\in \\mathcal {A} \\otimes \\mathcal {B}$ we set $\\Vert x\\Vert _{\\max } = \\sup \\lbrace \\Vert \\pi (x) \\Vert : \\ \\pi : \\mathcal {A} \\otimes \\mathcal {B} \\rightarrow B(\\mathcal {H}) \\text{ is a unital *-homomorphism}\\rbrace , $ where the supremum is taken over all Hilbert spaces $\\mathcal {H}$ and all unital *-homomorphisms.",
"The completion of $\\mathcal {A} \\otimes \\mathcal {B}$ in this norm is a C*-algebra denoted $\\mathcal {A} \\otimes _{\\max } \\mathcal {B}$ .",
"Alternatively, if $\\pi _1: \\mathcal {A} \\rightarrow B(\\mathcal {H}_1)$ and $\\pi _2: \\mathcal {B} \\rightarrow B(\\mathcal {H}_2)$ are unital *-homomorphisms, then setting $\\pi (a \\otimes b) = \\pi _1(a) \\otimes \\pi _2(b) \\in B(\\mathcal {H}_1 \\otimes \\mathcal {H}_2)$ and extending linearly, defines a unital *-homomorphism from $\\mathcal {A} \\otimes \\mathcal {B}$ into $B(\\mathcal {H}_1 \\otimes \\mathcal {H}_2)$ denoted by $\\pi = \\pi _1 \\otimes \\pi _2$ .",
"Given $x \\in \\mathcal {A} \\otimes \\mathcal {B}$ we set $ \\Vert x\\Vert _{\\min } = \\sup \\lbrace \\Vert \\pi _1 \\otimes \\pi _2(x) \\Vert : \\,\\, \\pi _1: \\mathcal {A} \\rightarrow B(\\mathcal {H}_1), \\, \\, \\pi _2: \\mathcal {B} \\rightarrow B(\\mathcal {H}_2) \\\\ \\text{ are unital *-homomorphisms}\\rbrace .$ The completion of $\\mathcal {A} \\otimes \\mathcal {B}$ in this norm is a C*-algebra denoted $\\mathcal {A} \\otimes _{\\min } \\mathcal {B}$ ."
]
] | 1606.05061 |
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