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+arXiv:2301.02639v1  [math.RA]  6 Jan 2023
+Filtered skew derivations on simple artinian rings
+Adam Jones, William Woods
+January 9, 2023
+Abstract
+Given a complete, positively filtered ring (R, f) and a compatible skew derivation (σ, δ),
+we may construct its skew power series ring R[[x; σ, δ]]. Due to topological obstructions,
+even if δ is an inner σ-derivation, in general we cannot “untwist” it, i.e. reparametrise to
+find a filtered isomorphism R[[x; σ, δ]] ∼= R[[x′; σ]], as might be expected from the theory
+of skew polynomial rings; similarly when σ is an inner automorphism. We find general
+conditions under which it is possible to untwist the multiplication data, and use this to
+analyse the structure of R[[x; σ, δ]] in the simplest case when R is a matrix ring over a
+(noncommutative) noetherian discrete valuation ring.
+Contents
+1
+Introduction
+2
+1.1
+Maximal orders in semisimple artinian rings . . . . . . . . . . . . . . . . . . . .
+2
+1.2
+Untwisting skew derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+1.3
+Ideal contraction and simplicity . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+1.4
+Uniform dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+2
+Preliminaries
+6
+2.1
+Filtered rings and discrete valuation rings
+. . . . . . . . . . . . . . . . . . . . .
+6
+2.2
+Skew derivations on semisimple artinian rings
+. . . . . . . . . . . . . . . . . . .
+6
+2.3
+Compatible filtrations and skew power series rings . . . . . . . . . . . . . . . . .
+7
+2.4
+Discrete valuation rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+3
+Reparametrising filtered skew derivations
+9
+3.1
+Conditions for identical filtrations . . . . . . . . . . . . . . . . . . . . . . . . . .
+9
+4
+Skew derivations on semisimple artinian rings
+11
+4.1
+Reducing to orbits
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+11
+4.2
+Untwisting inner automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . .
+13
+4.3
+Untwisting inner σ-derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+14
+5
+Applications
+15
+5.1
+Polynomial elements
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+15
+5.2
+Uniform dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+16
+1
+
+1
+Introduction
+Let R be a ring and (σ, δ) a skew derivation on R: that is, σ is an automorphism of R, and δ
+is a left σ-derivation on R, i.e. a linear map R → R satisfying δ(ab) = δ(a)b + σ(a)δ(b) for all
+a, b ∈ R. Then we may define the skew polynomial ring R[x; σ, δ] as the unique ring which is
+equal to R[x] as a left R-module and whose multiplication is given by xr = σ(r)x + δ(r) for all
+r ∈ R.
+In the theory of skew polynomial rings (see §1.2 below), it is well known that, if σ is an
+inner automorphism of R, then there exists some x′ ∈ R[x; σ, δ] such that R[x; σ, δ] = R[x′; δ′]
+for some derivation δ′; and likewise, if δ is an inner σ-derivation of R, then there exists some
+x′ ∈ R[x; σ, δ] such that R[x; σ, δ] = R[x′; σ]. This is a crucial and frequently used simplification
+in the theory: see e.g. [7, §2.3(iii), Theorem 14.9] or [6, Proposition 3.10].
+We are primarily interested in skew power series rings, where there are many extra topological
+difficulties to deal with.
+Firstly: given an arbitrary ring R and an arbitrary skew derivation (σ, δ) on R, it is in general
+not true that there exists a well-defined multiplication of the above form on the left module
+R[[x]] without imposing some kind of convergence condition on the multiplication data. Our
+primary motivation comes from studying the completed group algebras of certain finite-rank
+pro-p groups (these completed group algebras are also known as Iwasawa algebras), where the
+following notions are appropriate. If (R, v) is a complete, N-filtered ring, and (σ, δ) is compatible
+with v (in the sense of Definition 2.4 below), then we can define the skew power series ring
+R[[x; σ, δ]] =
+��
+n≥0
+rnxn : rn ∈ R
+�
+,
+which is also a complete N-filtered ring. (If v has values in Z∪{∞} rather than N ∪{∞}, then
+we can define an appropriate notion of bounded skew power series ring – see [10] for details –
+but we do not deal with such rings in this paper.)
+Secondly: suppose that there exists t ∈ R such that δ(r) = tr − σ(r)t, an inner σ-derivation.
+Then, if we reparametrise the skew polynomial ring by changing our variable from x to x′ = x+t,
+we find that R[x; σ, δ] = R[x′; σ]: in passing from x to x′, we will say that we have untwisted δ
+from R[x; σ, δ]. This is beneficial as rings of the form R[x′; σ] (with zero derivation) are typically
+much easier to understand. (There is a similar procedure by which inner automorphisms σ can
+be untwisted.) However, under a reparametrisation like this, it is generally not true that we will
+have R[[x]] = R[[x′]] even as left R-modules (see Example 3.1), meaning that this simplification
+is not always available.
+1.1
+Maximal orders in semisimple artinian rings
+The main result of this paper uses this notion of untwisting to analyse the structure of skew
+power series rings. To state this result, we need to impose further conditions on the base ring,
+since general filtered rings are too pathological for us to be able to say much of consequence.
+The rings of interest, which we denote by O, will typically be specific maximal orders in certain
+complete, filtered semisimple artinian rings Q. These are often well-behaved enough that inner
+parts of the multiplication data (σ, δ) can always be untwisted from O[[x; σ, δ]], making a study
+of these skew power series rings tractable using our methods.
+Rings of this form are highly abundant, since beginning with a sufficiently nice filtered ring R,
+it is possible to produce such a ring O which is closely related to R due to [1, §3, Theorem C
+2
+
+and proof]. For instance, the authors of the present paper proved the results of [10] by relating
+skew power series rings over R to skew power series rings over Q(O).
+In short, we will take Q to be a semisimple artinian ring throughout, and we will assume that
+it is complete with respect to a filtration vQ. Naturally, by the Artin-Wedderburn theorem, Q
+is isomorphic to a finite direct product of full matrix rings over division rings F1, · · · , Fd, and
+we will usually construct our maximal order O in Q by simply taking maximal orders in the Fi.
+More specifically, we will assume that the data (Q, O, vQ) satisfies some or all of the following
+hypotheses:
+Hypotheses.
+(H1) We can realise Q as a product Q = A1 × · · · × Ad, where the rings A1, . . . , Ad form the
+minimal non-zero ideals of Q, O = O1 × · · · × Od for some maximal order Oi in Ai. and
+for each i = 1, . . . , d, we are given
+• a complete discrete valuation ring Di (defined as in §2.4 below),
+• its Goldie ring of quotients Fi (with its induced filtration vFi: see §2.4),
+• the full matrix rings Mn(Di) ⊆ Mn(Fi) (with the matrix filtration Mn(vFi): see
+Definition 2.1.2),
+• filtered ring isomorphisms ιi : Ai → Mn(Fi) such that Oi = ι−1
+i (Mn(Di)).
+In this context, we will write D and F for the products of the Di and Fi respectively, and
+they will be given their respective product filtrations (see Definition 2.1.1), which we will
+sometimes denote vD and vF.
+It will also sometimes be convenient to identify Mn(F) = Mn(F1) × · · · × Mn(Fd). We
+will write ι : Q → Mn(F) for the induced filtered isomorphism, and we assume that
+vQ = Mn(vF) ◦ ι.
+(H2) The skew derivation (σ, δ) is compatible with vQ (defined as in §2.3 below).
+(H3) The automorphism σ permutes the minimal nonzero ideals A1, . . . , Ad of Q transitively.
+It will additionally be convenient to name the following hypothesis:
+(S) In the context of (H1), d = 1: that is, Q is simple artinian, F is a division ring, D is a
+complete discrete valuation ring, etc.
+Remarks.
+• Hypotheses (H1) + (S) are the context of [1, §3, particularly 3.14]: our Q is there called
+Q(B), and it can be realised as a simple quotient of the artinian ring called �Q.
+• It follows from (H1) that vF is the J(D)-adic filtration on F, and hence vQ is the J(O)-adic
+filtration on Q.
+• Hypothesis (H2) is a crucial hypothesis when working with filtered skew power series rings:
+many natural and important examples of filtered skew power series rings satisfy some
+kind of compatibility criterion (see e.g. [10,13,19], [20, §§2.4–2.5]), and this compatibility
+criterion ensures that the ring multiplication is well defined (see §1.2). Note that, in
+particular, together with (H1) it implies that σ preserves O.
+• Hypothesis (H3) is a mild simplification. In fact, our results can also deal with the more
+general case where Q ∼= �d
+i=1 Mni(Fi) (note that the ni may be different!) and σ has
+multiple orbits, by applying the techniques of §4.1 to reduce easily to a case satisfying
+(H3).
+Assuming these hypotheses, our main result allows us to realise the skew power series ring
+O[[x; σ, δ]] in a form that allows us to reduce to the study of skew-power series rings over the
+division rings Di, a much less daunting task:
+3
+
+Theorem A. If (Q, O, vQ) satisfy hypotheses (H1-3), with F, D, vF, ι defined as in the state-
+ments of the hypotheses, then there exists a skew derivation (τ, θ) on F, compatible with vF,
+and an isomorphism of filtered rings ϕ : O[[x; σ, δ]] → Mn(D[[y; τ, θ]]) extending ι|O, where y
+is the image of ax − t for some a ∈ O×, t ∈ J(O).
+Note that this statement makes sense, because if (τ, θ) is compatible with vF, which is the
+J(D)-adic filtration, then it follows that τ and θ preserve D, and hence (τ, θ) restricts to a
+compatible skew-derivation of D.
+It also follows from Theorem A that the Krull dimension of O[[x; σ, δ]] is equal to the Krull
+dimension of D[[y; τ, θ]], which is 2, by similar methods to those of [20, §3.1 and Theorem 3.3].
+1.2
+Untwisting skew derivations
+In order to prove our main result, we first find general conditions under which inner parts of
+the multiplication data (σ, δ) may be untwisted from R[[x; σ, δ]]:
+Notation. Given a ring R and an invertible element a ∈ R×, we will write ca for the inner
+automorphism of R defined by ca(r) = ara−1 for all r ∈ R. Also, given an element t ∈ R, we
+will write dσ,t for the inner σ-derivation of R defined by dσ,t(r) = tr − σ(r)t for all r ∈ R.
+Let R be a ring and (σ, δ) a skew derivation on R, and fix a ∈ R× and t ∈ R. In the study
+of skew polynomial rings, it is often useful to reparametrise the ring R[x; σ, δ], i.e. replace the
+variable x with a new, more convenient variable y ∈ R[x; σ, δ], usually taken to be y = ax or
+y = x − t. It is easy to see that R[ax] = R[x − t] = R[x] as R-modules, and a calculation of
+the multiplication data shows that
+(ax)r = a(σ(r)x + δ(r)) = (aσ(r)a−1)(ax) + aδ(r)
+and
+(x − t)r = σ(r)x + δ(r) − tr = σ(r)(x − t) + δ(r) − (tr − σ(r)t),
+from which we can conclude that
+• R[x; σ, δ] = R[ax; caσ, aδ],
+• R[x; σ, δ] = R[x − t; σ, δ − dσ,t]
+as rings. This reparametrisation is a powerful tool in the study of skew polynomial rings, as it
+effectively implies that inner automorphisms and σ-derivations can be “untwisted” to become
+trivial.
+In the case of filtered skew power series rings R[[x; σ, δ]], we can no longer reparametrise arbi-
+trarily due to the topology: that is, given a prospective new variable y ∈ R[[x; σ, δ]], it is no
+longer clear when R[[x]] = R[[y]] as modules. Our second main result gives clear and broadly
+applicable sufficient conditions.
+Theorem B. Let (R, v) be a complete filtered ring, and suppose that (σ, δ) is a compatible
+skew derivation on R. Fix a ∈ R× and t ∈ R.
+(i) If v(a) = v(a−1) = 0, then Rb[[x; σ, δ]] = Rb[[ax; caσ, caδ]] as filtered rings.
+(ii) If v(t) ≥ 1, then Rb[[x; σ, δ]] = Rb[[x − t; σ, δ + dσ,t]] as filtered rings.
+(By the phrase “as filtered rings” here, we mean that they are equal as rings, and that the
+standard filtrations as defined in (2.1) are equal: i.e. in the notation of (2.1), we have fv,x = fv,ax
+in part (i) and fv,x = fv,x−t in part (ii).)
+4
+
+This is proved at the end of §3.
+1.3
+Ideal contraction and simplicity
+In §5 we will prove some results that follow as consequences from our main theorems. The first
+of these addresses the following question: when is R[[x; σ, δ]] a simple ring?
+Let R ⊆ S be rings. Then we will say that an ideal I ✁ S is R-disjoint if I ∩ R = 0.
+Let R be a simple ring and (σ, δ) a skew derivation on R. It is often useful to ask when R[x; σ, δ]
+is also a simple ring: see e.g. [5, §3] or [18]. This is clearly equivalent to the statement that
+R[x; σ, δ] has no nonzero R-disjoint ideals. However, the ideal generated by x is a nonzero
+R-disjoint ideal in the case when δ = 0, and similarly – by untwisting – there exist nonzero
+R-disjoint ideals more generally when δ is inner. This suggests that inner derivations have a
+role to play in the simplicity of R[x; σ, δ].
+The correct generalisation of “inner” is as follows. The following are equivalent [12, Theorem
+2.6, Corollary 2.7]:
+(i) R[x; σ, δ] has nonzero R-disjoint ideals,
+(ii) δ is a quasi-algebraic σ-derivation, i.e. there exists an endomorphism θ of R, an inner
+θ-derivation D of R, and elements 0 ̸= an, an−1, . . . , a1, b ∈ R (for some n ≥ 1) such that
+anδn(r) + an−1δn−1(r) + · · · + a1δ(r) = bD(r)
+for all r ∈ R. (In fact, n and θ can be chosen so that θ = σn [12, §2].)
+There are also equivalent conditions phrased in the language of invariant and semi-invariant
+polynomials. Many further such results, and references to the historical literature on these
+matters, are given in [11,12]. See [9, Theorem 3.4] or [4, §3] for examples of the usefulness of
+conditions involving R-disjoint ideals.
+Theorem C. If we assume that the data (Q, O, vQ) satisfies Hypotheses (H1–3) + (S), then
+O[[x; σ, δ]] has no nonzero O[x; σ, δ]-disjoint ideals. It follows that if Q[x; σ, δ] is a simple ring,
+then Q ⊗O O[[x; σ, δ]] is a simple ring.
+1.4
+Uniform dimension
+Recall that the uniform dimension (also called Goldie dimension or Goldie rank) of a right
+R-module M is defined as follows. We set udim(MR) = n if and only if there are uniform
+submodules U1, . . . , Un ≤ M, pairwise intersecting in zero, such that U1 ⊕ · · · ⊕ Un ≤ M is an
+essential submodule [17, 2.2.9]. If R is a ring, we write r.udim(R) := udim(RR) for its (right)
+uniform dimension.
+Uniform dimension is preserved under skew polynomial extensions in many cases of interest.
+For instance, Goodearl and Letzter showed that r.udim(R[x; σ, δ]) = r.udim(R) if R is a prime
+noetherian ring [7, Lemma 1.2], and Matczuk [16] showed that this equality holds in an even
+broader range of cases, including the case where R is semiprime right Goldie.
+More recently, the study of uniform dimension under skew power series extensions was initi-
+ated by Letzter and Wang in the paper [14]. Let S = R[[x; σ]] be a pure automorphic skew
+power series extension (i.e. δ = 0): then, if R is semiprime right noetherian, we have that
+r.udim(S) = r.udim(R) by [14, Theorem 2.8].
+5
+
+Of course, if (R, v) is a complete positively filtered ring, (σ, δ) is a compatible skew deriva-
+tion on R, and δ happens to be an inner σ-derivation of the form described in Theorem
+A(ii), say δ = dσ,t for some t ∈ R satisfying v(t) ≥ 1, then it follows from Theorem A that
+R[[x; σ, δ]] = R[[y; σ]] after setting y = x + t. This puts us immediately into the context of [14],
+allowing us to conclude that r.udim(R[[x; σ, δ]]) = r.udim(R) in this context too.
+Now assume Hypotheses (H1–3) + (S) and their notation: in particular, recall that O ∼= Mn(D).
+We prove the following result only under these rather stringent restrictions, but this is (to our
+knowledge) the first such result for skew power series extensions with nontrivial derivations, and
+unlike the previous paragraph, it covers the case of some outer σ-derivations using methods
+unlike those of [14]. We hope that, combined with the localisation process for filtered rings
+outlined in [1, §3 and Theorem C], this will spark further research for more general filtered
+skew power series rings. In §5.2, we prove:
+Theorem D. If we assume that the data (Q, O, vQ) satisfies Hypotheses (H1-3) + (S), then
+r.udim(O[[x; σ, δ]]) = r.udim(O).
+2
+Preliminaries
+2.1
+Filtered rings and discrete valuation rings
+Our conventions for filtrations (which, in this paper, are always separated Z-filtrations) are as
+follows.
+A (ring) filtration on a ring R is a function f : R → Z∪{∞} satisfying the following properties
+for all r, s ∈ R:
+(i) f(1) = 0,
+(ii) f(r + s) ≥ min{f(r), f(s)},
+(iii) f(rs) ≥ f(r) + f(s),
+(iv) f(r) = ∞ if and only if r = 0.
+We will say that (R, f) is a filtered ring for short. If f takes values in N ∪ {∞}, we will say
+that (R, f) is N-filtered or positively filtered.
+Definition 2.1.
+1. If (A, fA) and (B, fB) are filtered rings, the product filtration f := fA × fB on R = A × B
+is given by f(a, b) = min{fA(a), fB(b)}.
+2. If (A, f) is a filtered ring and n ≥ 2 is an integer, the matrix filtration g := Mn(f) on
+Mn(A) is given by g(� aijeij) = mini,j{f(aij)}, where {eij}1≤i,j≤n is the standard set of
+matrix units of Mn(A).
+2.2
+Skew derivations on semisimple artinian rings
+Let R be a ring. The pair (σ, δ) is called a skew derivation on R if σ ∈ Aut(R) and δ is a (left)
+σ-derivation of R, which means that δ is a linear map satisfying δ(rs) = δ(r)s + σ(r)δ(s) for
+all r, s ∈ R.
+Here are some basic properties:
+6
+
+Suppose that Q is a semisimple artinian ring (without topology), say Q = �d
+i=1 Ai as a product
+of two-sided ideals, where each Ai ∼= Mni(Fi) as rings for some positive integers ni and division
+rings Fi. Suppose that (σ, δ) is a skew derivation on Q. We list some well-known facts.
+Properties 2.2.
+1. There exists a permutation ρ of the indices {1, . . . , d} such that σ(Ai) = Aρ(i) and
+δ(Ai) ⊆ Ai + Aρ(i). Hence, if S is an orbit of ρ, then setting B := �
+i∈S Ai and σ′ = σ|B,
+δ′ = δ|B, we get that (σ′, δ′) is a skew derivation of B. [3, 1.1–1.3]
+2. Suppose that ρ permutes {1, . . . , d} transitively. Then n1 = · · · = nd (= n, say), so that
+there exists an isomorphism ι : Q → Mn(F1 × · · · × Fd). Writing F := F1 × · · · × Fd,
+we can then write σ as η ◦ Mn(τ)ι, where η is an inner automorphism of Q and τ is an
+automorphism of F. [3, 2.1–2.4] (Here, and elsewhere, Mn(τ)ι means ι−1Mn(τ)ι.)
+3. Suppose further that η is trivial, so σ = Mn(τ)ι. Then δ = ε + Mn(θ)ι, where ε is an
+inner σ-derivation of Q and θ is a τ-derivation of F. [3, 2.5]
+Remark 2.3. In fact, in the context of Property 2.2.3, if d > 1 then something stronger holds:
+θ can be taken to be the zero map, so that δ itself is an inner σ-derivation [3, 1.4]. However, in
+the context of filtered rings, we will allow θ to be nonzero, as this extra flexibility is crucial for
+ensuring that the decomposition δ = ε + Mn(θ)ι behaves well with respect to the filtration.
+2.3
+Compatible filtrations and skew power series rings
+Definition 2.4. Let (R, v) be a filtered ring and (σ, δ) a skew derivation on R. We will say
+that (σ, δ) is (weakly) compatible with v if v(σ(r)) = v(r) and v(δ(r)) > v(r) for all 0 ̸= r ∈ R.
+Remark 2.5. This is more general than the notion of “compatibility” used by the authors in [10],
+which could be called strong compatibility.
+Definition 2.6. Let (R, v) be a complete, positively filtered ring, i.e. R is a ring admitting a
+separated discrete filtration v : R → N ∪ {∞} with respect to which R is complete.
+The set R[[x]] :=
+�
+n≥0
+Rxn, whose elements are formal sums r0 + r1x + r2x2 + . . . over arbitrary
+ri ∈ R, is a left R-module. This is a complete filtered R-module with standard filtration
+f := fv,x : R[[x]] → N ∪ {∞},
+given by
+f
+��
+i≥0
+rixi
+�
+= inf
+i≥0{v(ri) + i},
+(2.1)
+which is separated and discrete. Note that R[x] is dense in R[[x]].
+A skew derivation (σ, δ) on R makes R[x] into a ring, with multiplication determined uniquely
+by the rule xr = σ(r)x + δ(r). We write this ring as R[x; σ, δ].
+If (σ, δ) is compatible with v, then it induces a well-defined associative multiplication on R[[x]]
+in the same way: see [10, Proposition 1.17] (cf. [19, Lemma 2.1] or [13, §3.4]), and we denote this
+ring by R[[x; σ, δ]]. The function f on R[[x; σ, δ]] as defined above is a positive ring filtration,
+and R[[x; σ, δ]] is complete with respect to f.
+7
+
+2.4
+Discrete valuation rings
+Definition 2.7. Following Ardakov [1], we will say that a discrete valuation ring is a noetherian
+domain D with the property that, for every nonzero x ∈ Q(D) (the division ring of quotients),
+we have either x ∈ D or x−1 ∈ D.
+We begin by showing that D has properties very similar to those of commutative discrete
+valuation rings.
+Lemma 2.8. Let D be a discrete valuation ring.
+(i) D is a local ring.
+(ii) All right (resp. left) ideals of D are principal.
+(iii) The lattice of right (resp. left) ideals of D is totally ordered.
+(iv) All right (resp. left) ideals of D are two-sided.
+Proof. In the language of [15], D is a noetherian total subring of the skew field Q(D), and
+statements (i–iii) follow from [15, Proposition 1.2.15].
+The proof of (iv) below is adapted from [2, Lemma 1]. We give the proof for right ideals; the
+proof for left ideals is of course similar.
+Suppose there exist right ideals of D that are not two-sided, and let J be the maximal such
+right ideal. By (ii), J = aD for some a ∈ D: then, for some r ∈ D, we have ra =: b ̸∈ aD by
+assumption. By (iii), this implies aD ⊊ bD, and so a = bs for some s ∈ D. Combining these
+two equations, we can see that b = rbs.
+Now, by the maximality of J, we have that Db ⊆ DbD = bD, so that rb = bt for some t ∈ D.
+In particular, b = bts, and so b(1 − ts) = 0. But b cannot be zero, so as D is a domain, we
+must have ts = 1, and hence (as noetherian rings are Dedekind-finite) st = 1. It follows that
+at = b, contradicting the assumption that b ̸∈ aD.
+Proposition 2.9. Let D be a discrete valuation ring.
+(i) J(D) = πD for some normal element π.
+(ii) Every nonzero ideal of D has the form πnD for some n ∈ N.
+Proof. (i) is an immediate consequence of Lemma 2.8.
+To show (ii): note that Lemma 2.8(iv) also implies that D is an FBN ring [17, 6.4.7], and so
+�∞
+n=1 πnD = 0 by [8, Theorem 9.13]. We now argue exactly as in the commutative case: indeed,
+a nonzero ideal aD must satisfy πn+1D ⊊ aD ⊆ πnD for some n by Lemma 2.8(iii), from which
+it follows that a = πnu for some u ∈ D \ πD, which must be a unit by Lemma 2.8(i).
+An element π as in the above proposition will be called a uniformiser of D.
+In the following, D will continue to denote a complete discrete valuation ring, and we will also
+set F = Q(D), O = Mn(D) and Q = Mn(F). Also write vF for the induced J(D)-adic filtration
+on F, and suppose that (τ, θ) is a skew derivation on F compatible with vF; likewise write vQ
+for the J(O)-adic filtration on Q, and suppose that (σ, δ) is a skew derivation on Q compatible
+with vQ. This puts us essentially in the situation of Hypotheses (H1–3) + (S). The following is
+now routine to check.
+Corollary 2.10.
+8
+
+(i) Let S be the multiplicatively closed set in D generated by π. Then F = S−1D = DS−1.
+Moreover, π is normal in D[[y; τ, θ]], and F⊗DD[[y; τ, θ]] = S−1D[[y; τ, θ]] = D[[y; τ, θ]]S−1.
+(ii) Let S be the multiplicatively closed set in O generated by π (where we identify D with
+its diagonal embedding in O). Then Q = S−1O = OS−1. Moreover, π is normal in
+O[[x; σ, δ]], and Q ⊗O O[[x; σ, δ]] = S−1O[[x; σ, δ]] = O[[x; σ, δ]]S−1.
+3
+Reparametrising filtered skew derivations
+Throughout this section, let (R, v) be a complete, positively filtered ring. Suppose also that
+R admits a skew derivation (σ, δ) which is compatible with v, and take y ∈ R[x] such that
+R[x] = R[y] (an equality of left R-modules).
+Given an element �m
+i=0 riyi ∈ R[y], we may define the function
+fv,y :
+m
+�
+i=0
+riyi �→ inf
+i≥0{v(ri) + i}.
+Note that fv,x is the standard ring filtration defined in (2.1). In contrast, for arbitrary elements
+y ∈ R[x], the function fv,y will not always be a ring filtration, and even when it is, it will
+generally not be equivalent to fv,x.
+Example 3.1. Take y := x + 1 ∈ Zp[x], and v the p-adic valuation on Zp. Then for all n, we
+have fv,x((x+1)n) = 0 but fv,y((x+1)n) = n. In particular, Zp[x] = Zp[y] but Zp[[x]] ̸= Zp[[y]].
+In this section, we identify two families of elements y ∈ R[x] for which fv,x and fv,y are equal
+as functions.
+3.1
+Conditions for identical filtrations
+With notation as above, we first show that fv,x and fv,y are equal when y = x − t for some
+t ∈ R satisfying v(t) ≥ 1.
+Write (x − t)n = xn + βn,1xn−1 + · · · + βn,n−1x + βn,n for all n.
+Lemma 3.2. For all n, i, we have v(βn,i) ≥ i.
+Proof. Firstly, note that (x − t)βn,ixn−i = σ(βn,i)xn+1−i + (δ(βn,i) − tβn,i)xn−i, so (writing
+βn,0 := 1 for ease of notation) we may calculate (x − t)n+1 as
+(x − t)
+� n
+�
+i=0
+βn,ixn−i
+�
+=
+n
+�
+i=0
+σ(βn,i)xn+1−i +
+n
+�
+j=0
+(δ(βn,j) − t(βn,j))xn−j
+= xn+1 +
+n
+�
+i=1
+(σ(βn,i) + δ(βn,i−1) − tβn,i−1)xn+1−i + (δ(βn,n) − tβn,n),
+by setting j = i + 1 in the second sum. That is,
+
+
+
+
+
+βn+1,0 = 1,
+βn+1,i = σ(βn,i) + δ(βn,i−1) − tβn,i−1
+(1 ≤ i ≤ n),
+βn+1,n+1 = δ(βn,n) − tβn,n,
+from which the claim follows by induction on n.
+9
+
+Lemma 3.3. Let p(x) ∈ R[x] be a polynomial, and t ∈ R such that v(t) ≥ 1.
+Then
+fv,x(p(x − t)) ≥ fv,x(p(x)).
+Proof. Write p(x) = r0 + r1x + · · · + rmxm. Then
+p(x − t) =
+m
+�
+n=0
+rn(x − t)n =
+m
+�
+n=0
+rn
+� n
+�
+i=0
+βn,ixn−i
+�
+setting βn,0 := 1
+=
+m
+�
+j=0
+� m
+�
+n=j
+rnβn,n−j
+�
+xj
+where j := n − i,
+and so
+fv,x(p(x − t)) = fv,x
+� m
+�
+j=0
+� m
+�
+n=j
+rnβn,n−j
+�
+xj
+�
+=
+inf
+0≤j≤m
+�
+v
+� m
+�
+n=j
+rnβn,n−j
+�
++ j
+�
+≥
+inf
+0≤j≤m
+inf
+j≤n≤m {v(rn) + v(βn,n−j) + j}
+as v is a filtration
+≥
+inf
+0≤j≤m
+inf
+j≤n≤m {v(rn) + n}
+by Lemma 3.2
+=
+inf
+0≤n≤m {v(rn) + n}
+= fv,x
+� m
+�
+n=0
+rnxn
+�
+= fv,x(p(x)).
+Proposition 3.4. Let t ∈ R such that v(t) ≥ 1. Then fv,x = fv,x−t.
+Proof. Take an arbitrary element p(x) ∈ R[x]. Then
+fv,x(p(x)) ≤ fv,x(p(x + t))
+applying Lemma 3.3 to − t
+= fv,x−t(p(x))
+changing variables x �→ x − t throughout
+≤ fv,x−t(p(x − t))
+applying Lemma 3.3 to t
+= fv,x(p(x))
+changing variables x �→ x + t throughout,
+from which we can conclude that fv,x(p(x)) = fv,x−t(p(x)).
+Next, we show that fv,x and fv,y are equal when y = ax where a ∈ R× and v(a) = v(a−1) = 0.
+Write (ax)n = γn,0xn + γn,1xn−1 + · · · + γn,n−1x + γn,n.
+Lemma 3.5. v(γn,i) ≥ i.
+Proof. As in Lemma 3.2, calculating (ax)n+1 = ax(γn,0xn +γn,1xn−1 +· · ·+γn,n−1x+γn,n) gives
+
+
+
+
+
+γn+1,0 = aσ(γn,0),
+γn+1,i = aσ(γn,i) + aδ(γn,i−1)
+(1 ≤ i ≤ n),
+γn+1,n+1 = aδ(γn,n),
+and we may perform induction as in Lemma 3.2.
+10
+
+Lemma 3.6. fv,x(p(ax)) ≥ fv,x(p(x)).
+Proof.
+p(ax) =
+m
+�
+n=0
+rn(ax)n =
+m
+�
+n=0
+rn
+� n
+�
+i=0
+γn,ixn−i
+�
+=
+m
+�
+j=0
+� m
+�
+n=j
+rnγn,n−j
+�
+xj
+where j := n − i,
+and so
+fv,x(p(ax)) = fv,x
+� m
+�
+j=0
+� m
+�
+n=j
+rnγn,n−j
+�
+xj
+�
+,
+and the proof now proceeds exactly as in the proof of Lemma 3.3.
+Proposition 3.7. fv,x = fv,ax.
+Proof. Take an arbitrary element p(x) ∈ R[x]. Then
+fv,x(p(x)) ≤ fv,x(p(a−1x))
+applying Lemma 3.6 to a−1
+= fv,ax(p(x))
+changing variables x �→ ax throughout
+≤ fv,ax(p(ax))
+applying Lemma 3.6 to a
+= fv,x(p(x))
+changing variables x �→ a−1x throughout,
+from which we can conclude that fv,x(p(x)) = fv,ax(p(x)).
+Finally, we fix any y ∈ R[x] such that R[x] = R[y] and fv,x = fv,y. Denote this common
+filtration by f. It follows that:
+Theorem 3.8. R[[x]] = R[[y]] as filtered left R-modules.
+Proof of Theorem B. In case (i), set y = ax, so that f := fv,x = fv,y by Proposition 3.7; in
+case (ii), set y = x − t, and use Proposition 3.4. In both cases, R[x] = R[y]. Now Theorem 3.8
+implies that R[[y]] and R[[x]] can be identified as filtered modules, and the multiplication data
+has already been calculated in §1.2, so the conclusion follows.
+4
+Skew derivations on semisimple artinian rings
+4.1
+Reducing to orbits
+For now, we do not assume any of the Hypotheses (H1–3), and we let (R, v) be an arbitrary
+filtered ring such that R admits a decomposition R ∼= B × C (as unfiltered rings).
+We will abuse notation and write R = B × C (as unfiltered rings). We will also write B (resp.
+C) for the ideal B × 0 (resp. 0 × C) of R, so that there are inclusion maps jB : B → R and
+jC : C → R and projection maps πB : R → B and πC : R → C.
+Suppose further that the filtration on R is complete and positive, and that R admits a skew
+derivation (σ, δ) which restricts to skew derivations on B and C. (That is, setting σB = πBσjB
+and δB = πBδjB, we have that (σB, δB) is a skew derivation on B, and likewise for C.)
+11
+
+Write vB and vC for the restrictions of v to B and C respectively. In general, even if (σ, δ) is
+compatible with v, it may not be true that (σB, δB) is compatible with vB, and so we must
+restrict to the case in which the decomposition R ∼= B × C and the filtration v interact nicely.
+The following lemma is an immediate consequence of the definition of the product filtration,
+as in Definition 2.1.
+Lemma 4.1. If v = vB × vC, then (σB, δB) is compatible with vB, and (σC, ��C) is compatible
+with vC.
+Hence, under the assumption v = vB × vC, we may define the filtered rings
+• B[[xB; σB, δB]], with filtration fB, satisfying fB|B = vB and fB(xB) = 1,
+• C[[xC; σC, δC]], with filtration fC, satisfying fC|C = vC and fC(xC) = 1,
+as in (2.1).
+Proposition 4.2. Suppose that v = vB × vC. Then there is an isomorphism of filtered rings
+ϕ : R[[x; σ, δ]] → B[[xB; σB, δB]] × C[[xC; σC, δC]].
+Proof. It is straightforward to check that the maps
+ϕ : R[[x; σ, δ]] → B[[xB; σB, δB]] × C[[xC; σC, δC]]
+�
+i≥0
+rixi �→
+��
+i≥0
+πB(ri)xi
+B,
+�
+i≥0
+πC(ri)xi
+C
+�
+and
+θ : B[[xB; σB, δB]] × C[[xC; σC, δC]] → R[[x; σ, δ]]
+��
+i≥0
+bixi
+B,
+�
+i≥0
+cixi
+C
+�
+�→
+�
+i≥0
+(jB(bi), jC(ci))xi
+are the mutually inverse filtered isomorphisms as required.
+Let Q′ be an arbitrary semisimple artinian filtered ring admitting a skew derivation (σ, δ), and
+write the minimal nonzero ideals of Q′ as A1, . . . , Ae, so that Q′ = �e
+i=1 Ai. In the case where
+the Ai fall into several σ-orbits, write ρ for the permutation of the indexing set {1, . . . , e}
+induced on the set {A1, . . . , Ae} by σ as in Property 2.2.1. Let S be a union of (some) orbits of
+ρ and S′ = {1, . . . , e}\S, and suppose (to avoid trivial cases) that both S and S′ are nonempty.
+Now set B′ = �
+i∈S Ai and C′ = �
+i∈S′ Ai: it follows from Property 2.2.1 that (σB′, δB′) restricts
+to a skew derivation on B′, and likewise for C′.
+Assume now that each Ai ∼= Mni(Fi), where ni ≥ 1 is some positive integer and Fi is the Goldie
+ring of quotients of a complete discrete valuation ring Di. Set Oi to be the preimage in Ai of
+Mni(Di), so that O′ = O1 × · · · × Oe is a maximal order in Q′. Moreover, if j = ρ(i) then
+Aj = σ(Ai) so Mni(Fi) ∼= Mnj(Fj), which implies that ni = nj and Fi ∼= Fj.
+Suppose that all of these rings are given their natural filtrations: that is,
+• each Di retains its discrete valuation, and Fi inherits the J(Di)-adic valuation (see §2.4),
+• Mni(Di) and Mni(Fi) are given the corresponding matrix filtrations (see Definition 2.1.2),
+• Oi and Ai inherit their filtrations from Mni(Di) and Mni(Fi) under the above isomor-
+phisms, and
+12
+
+• O′ and Q′ are given the product filtrations (see Definition 2.1.1) from the Oi and Ai
+respectively.
+Then O′, B′ and C′ as defined above will satisfy the hypotheses of Proposition 4.2. Moreover, if
+S is taken to be a single orbit of ρ with |S| = d, then (after renumbering so that S = {1, . . . , d}
+and writing n = n1 = · · · = nd) the ring O := O′ ∩ B′ as defined above, its Goldie ring of
+quotients Q := B′, etc. will satisfy Hypotheses (H1–3).
+4.2
+Untwisting inner automorphisms
+In this subsection, we assume the full force of Hypotheses (H1–3) and adopt their notation.
+Without loss of generality, reordering the Ai if necessary, we will set σ(Ai) = Ai+1 for 1 ≤ i ≤ d−1
+and σ(Ad) = A1.
+We may now invoke Property 2.2.2. In particular, there is a decomposition σ = η ◦ Mn(τ)ι,
+where η is an inner automorphism of Q, say η = ca for some a ∈ Q×, and τ is an automorphism
+of F.
+Lemma 4.3. Both η and Mn(τ)ι preserve O.
+Proof. Since η is an inner automorphism of Q, it will preserve each Ai. But σ(Ai) = Ai+1
+(with indices interpreted modulo d), so Mn(τ)ι must send Ai to Ai+1, and hence τ sends
+Fi to Fi+1. However, since σ and η are continuous, it follows that τ is continuous, and so
+Di+1 = Oi+1 ∩ Fi+1 = τ(Di), hence Mn(τ)ι preserves O. Now it follows that η = σ ◦ Mn(τ −1)ι
+preserves O.
+Our aim in this subsection is to “untwist” η by making a change of variables x �→ x′, i.e. find
+an element x′ ∈ O[[x; σ, δ]] such that
+O[[x; σ, δ]] = O[[x′; Mn(τ)ι, δ′]].
+By Theorem B(i), it would suffice if v(a) = v(a−1) = 0: this would imply that a, a−1 ∈ O, and
+we could then set x′ = a−1x, giving δ′ = a−1δ as in §1.2.
+Of course, in general, a will not necessarily have this property: for instance, if O is a complete
+discrete valuation ring with central uniformiser π, then ca = cπra for all r ∈ Z, and v(πra) will
+usually not be zero. Surprisingly, this naive obstruction is the only kind of obstruction that
+occurs.
+Proposition 4.4. There exist an element b ∈ Q×, an inner automorphism η′ = cb of Q, and
+an automorphism τ ′ of F such that σ = η′ ◦ Mn(τ ′)ι and v(b) = v(b−1) = 0.
+Proof. Suppose a = (a1, . . . , ad) ∈ Q×: then, by Lemma 4.3, we have aiOia−1
+i
+= Oi for each
+i. Let ki = v(ai), i.e. ai ∈ J(Oi)ki \ J(Oi)ki+1. So, if πj is a uniformiser of Dj, Ij is the
+identity matrix of Mn(Dj), and ˜πj := ι−1(πjIj), then we have aj = bj˜π
+kj
+j for some bj ∈ Oj with
+v(bj) = 0. Set b = (b1, . . . , bd), so that v(b) = 0.
+By Corollary 2.10(ii), ˜πj is normal in Oj, so the right ideal bjOj is in fact a two-sided ideal.
+Moreover, by Proposition 2.9(ii) and Morita equivalence, as the ideal bjOj contains the element
+bj of value 0, it must be equal to Oj. Hence bj is a unit in Oj, and we have v(bj) = v(b−1
+j ) = 0.
+Now set η′(r) = brb−1 for all r ∈ Q and τ ′(s) = Πτ(s)Π−1, where Π = (πk1
+1 I1, . . . , πkd
+d Id). The
+claim now follows from a short calculation.
+13
+
+4.3
+Untwisting inner σ-derivations
+In this subsection, let (R, f) be an arbitrary Z-filtered ring admitting a compatible skew deriva-
+tion (σ, δ), and write the f-level sets of R as FnR for n ∈ Z. We will also suppose that
+• R = Mn(A) for some ring A,
+• f = Mn(g) for some filtration g on A,
+• σ = Mn(τ) for some automorphism τ of A, and
+• δ = Mn(θ) + ε, where θ is a τ-derivation of A and ε is an inner σ-derivation of R, say
+ε = dσ,u for some u ∈ R.
+(Compare Property 2.2.3.) We will write the standard set of matrix units in R as {eij}1≤i,j≤n.
+Our aim in this subsection is to “untwist” ε by making a change of variables x �→ x′, i.e. find
+an element x′ ∈ R[[x; Mn(τ), δ]] such that
+R≥0[[x; Mn(τ), δ]] = R≥0[[x′; Mn(τ), Mn(θ)]]
+where of course, R≥0 is the positively filtered subring of R. As before, we would be done by
+Theorem B(ii) if we had f(u) ≥ 1. This is also unreasonable to expect, albeit this time for
+slightly less obvious reasons: for instance, if A is the division ring of fractions of a complete
+discrete valuation ring with uniformiser π, then Mn(θ)+dMn(τ),u = Mn(θ+dτ,πr)+dMn(τ),u−πrI
+for all r ∈ Z, and v(u − πrI) can be less than 1. However, again, under mild conditions this is
+the only obstruction that occurs.
+Proposition 4.5. In the above setup, there exist an element u′ ∈ R, a τ-derivation θ′ of A,
+and an inner σ-derivation ε′ = inn(u′) of R such that δ = Mn(θ′) + ε′ and f(u′) ≥ 1.
+Proof. Write u = �
+i,j uijeij for some coefficients uij ∈ A. Let 1 ≤ p, q ≤ n be arbitrary, and
+consider the matrix unit epq ∈ R. By assumption, δ(epq) ∈ F1R, and so
+ε(epq) ≡ −Mn(θ)(epq)
+mod F1R.
+(4.1)
+But we can calculate the left-hand side of this congruence (4.1) explicitly as
+ε(epq) =
+�
+i,j
+(uijeijepq − epquijeij) =
+�
+i
+uipeiq −
+�
+j
+uqjepj,
+and the right-hand side of (4.1) is just −θ(1A)epq, which is zero. So we may rewrite (4.1) as
+�
+i uipeiq − �
+j uqjepj ≡ 0 mod F1R, and equate corresponding entries, to get
+
+
+
+
+
+uip ∈ F1A
+i ̸= p
+uqj ∈ F1A
+j ̸= q
+upp − uqq ∈ F1A,
+and so, as p and q were arbitrary, we get u ≡ u111R mod F1R.
+Now setting u′ := u − u111R, and defining θ′(a) := θ(a) + u11a − τ(a)u11 and ε′ := dMn(τ),u′, we
+are done.
+Upshot: in this case, using Theorem B(i) we can pass to the case when δ = Mn(θ).
+Proof of Theorem A. Firstly, Proposition 4.4 shows that there exist some τ ∈ Aut(F) and some
+b ∈ Q× satisfying v(b) = v(b−1) = 0 such that σ = cb ◦ Mn(τ)ι, and both of these preserve O
+14
+
+by the same argument as in Lemma 4.3. So by Theorem B(i), we may set x′ = b−1x to get
+O[[x; σ, δ]] = O[[x′; Mn(τ)ι, δ′]] for the Mn(τ)ι-derivation δ′ := b−1δ.
+Secondly, Proposition 4.5 shows that there exist some τ-derivation θ of F and some u ∈ Q
+satisfying v(u) ≥ 1 such that δ′ = Mn(θ)ι + dMn(τ)ι,u.
+So by Theorem B(ii), we may set
+x′′ = x′ − u to get O[[x′; Mn(τ)ι, δ′]] = O[[x′′; Mn(τ)ι, Mn(θ)ι]].
+Finally, the maps
+O[[x′′; Mn(τ)ι, Mn(θ)ι]] → Mn(D)[[y; Mn(τ), Mn(θ)]]
+�
+i≥0
+qi(x′′)i �→
+�
+i≥0
+ι(qi)yi
+and
+Mn(D)[[y; Mn(τ), Mn(θ)]] → Mn(D[[z; τ, θ]])
+�
+i≥0
+�
+n
+�
+j,k=1
+cijkejk
+�
+yi �→
+n
+�
+j,k=1
+��
+i≥0
+cijkzi
+�
+ejk
+can now be checked to be filtered ring isomorphisms, and vO = Mn(vD) ◦ ι.
+5
+Applications
+Throughout this section, we assume our data (Q, O, vQ) satisfies Hypotheses (H1–3) + (S).
+5.1
+Polynomial elements
+Definition 5.1. A polynomial element of R[[x]] (resp. R[[x; σ, δ]]) is an element of R[x] (resp.
+R[x; σ, δ]).
+We first recall the following important generalisation of the Weierstrass preparation theorem
+for skew power series rings, essentially due to Venjakob.
+Theorem 5.2. Every nonzero right ideal of D[[y; τ, θ]] contains a nonzero polynomial element.
+Proof. Let v be the J(D)-adic filtration and π a uniformiser for D. Then every nonzero ele-
+ment r ∈ D[[y; τ, θ]] can be written as r = (s0 + s1y + s2y2 + . . . )πm, where all si ∈ D and
+infi≥0{v(si)} = 0, by Corollary 2.10(i). Hence s = s0 + s1y + s2y2 + . . . satisfies the hypotheses
+of [19, Theorem 3.1], and so a right-hand version of [19, Corollary 3.2] tells us that s can be
+expressed uniquely as s = Pu, where u ∈ D[[y; τ, θ]]× and P ∈ D[y; τ, θ].
+In particular, if A is a nonzero right ideal of D[[y; τ, θ]], then given any nonzero r ∈ A, we can
+write it as r = Puπm as above. Since π is normal in D[[y; τ, θ]] by Corollary 2.10(i), this is just
+r = Pπmu′ for some unit u′ ∈ D[[y; τ, θ]], and hence the polynomial element Pπm is also an
+element of A.
+Remark. Corollary 2.10(i) also implies a similar result for F ⊗D D[[y; τ, θ]].
+Corollary 5.3. Every nonzero (two-sided) ideal of O[[x; σ, δ]] contains a nonzero polynomial
+element.
+15
+
+Proof. Theorem A tells us that there exists an isomorphism ϕ : O[[x; σ, δ]] → Mn(D[[y; τ, θ]])
+extending ι, such that y = ϕ(ax − t). In particular, if P ∈ D[[y; τ, θ]] is polynomial in the
+variable y, then ϕ−1(PI) (where I is the identity matrix) is polynomial in the variable x. The
+result now follows from Theorem 5.2 and by Morita equivalence.
+Proof of Theorem C. The first statement is simply restating Corollary 5.3.
+For the second statement, take a nonzero ideal I of Q ⊗O O[[x; σ, δ]]. Then J := I ∩ O[[x; σ, δ]]
+is clearly an ideal of O[[x; σ, δ]]. By Corollary 2.10(ii), multiplying any nonzero element of I by
+an appropriate power of the regular element π will give an element of J, so J ̸= 0.
+Hence, by the first statement, we know that J ∩ O[x; σ, δ] ̸= 0, and so I ∩ Q[x; σ, δ] ̸= 0. But
+as we are assuming that Q[x; σ, δ] is simple, it must follow that I ∩ Q[x; σ, δ] = Q[x; σ, δ]. In
+particular, 1 ∈ I, and so I = Q ⊗O O[[x; σ, δ]].
+5.2
+Uniform dimension
+This subsection continues the work of [14] on uniform dimensions (Goldie ranks) of skew power
+series rings. As we have already remarked in the Introduction, Theorem B sometimes allows
+us to reduce directly to the results of [14] when the derivation is inner. However, in the special
+case of Hypotheses (H1–3) + (S), we can now prove similar results about skew power series
+rings over O for arbitrary derivations.
+Proof of Theorem D. By Theorem A, we know that O[[x; σ, δ]] ∼= Mn(D[[y; τ, θ]]) for some ap-
+propriate skew derivation (τ, θ), and hence that r.udim(O[[x; σ, δ]]) = n(r.udim(D[[y; τ, θ]])) [17,
+Example 2.11(iii)]. In the same way, r.udim(O) = n(r.udim(D)), and of course r.udim(D) = 1
+as D is a noetherian integral domain [17, Example 2.11(i)].
+It remains to show that r.udim(D[[y; τ, θ]]) = 1. So let U be a uniform right ideal of D[[y; τ, θ]],
+and let I be an arbitrary nonzero right ideal.
+By Theorem 5.2, U ∩ D[y; τ, θ] ̸= 0 and
+I ∩ D[y; τ, θ] ̸= 0, and so as D[y; τ, θ] is a prime ring [17, Theorem 1.2.9(iii)], their inter-
+section (U ∩ I) ∩ D[y; τ, θ] is also nonzero. In particular, this shows that U is an essential right
+ideal of D[[y; τ, θ]], and so r.udim(D[[y; τ, θ]]) = 1.
+References
+[1] K. Ardakov.
+Prime ideals in nilpotent Iwasawa algebras.
+Inventiones mathematicae,
+190(2):439–503, 2012.
+[2] H.-H. Brungs. Generalized discrete valuation rings. Canad. J. Math., 21:1404–1408, 1969.
+[3] G. Cauchon and J.C. Robson. Endomorphisms, derivations, and polynomial rings. Journal
+of Algebra, 53:227–238, 1978.
+[4] E. Cisneros, M. Ferrero, and M. I. Gonz´alez. Prime ideals of skew polynomial rings and
+skew Laurent polynomial rings. Math. J. Okoyama Univ., 32:61–72, 1990.
+[5] John Cozzens and Carl Faith. Simple Noetherian rings. Cambridge University Press, 2008.
+[6] K. R. Goodearl. Prime ideals in skew polynomial rings and quantized Weyl algebras. J.
+Alg., 150:324–377, 1992.
+[7] K. R. Goodearl and E. S. Letzter. Prime factor algebras of the coordinate ring of quantum
+matrices. Proc. Amer. Math. Soc., 121(4):1017–1025, 1994.
+16
+
+[8] K. R. Goodearl and R. B. Warfield, Jr. An Introduction to Noncommutative Noetherian
+Rings. Cambridge University Press, 2004.
+[9] Ronald S. Irving. Prime ideals of Ore extensions over commutative rings, II. J. Algebra,
+58:399–423, 1979.
+[10] Adam Jones and William Woods. Skew power series rings over a prime base ring (preprint).
+https://arxiv.org/abs/2112.10242.
+[11] T. Y. Lam, K. H. Leung, A. Leroy, and J. Matczuk. Invariant and semi-invariant poly-
+nomial rings. In L. Rowen, editor, Ring Theory, pages 247–261. Weizmann Science Press,
+1989.
+[12] Andr´e Leroy and Jerzy Matczuk. The extended centroid and X-inner automorphisms of
+Ore extensions. Journal of Algebra, 145:143–177, 1992.
+[13] Edward S. Letzter. Prime ideals of noetherian skew power series rings. Israel J. Math.,
+192:67–81, 2012.
+[14] Edward S. Letzter and Linhong Wang. Goldie ranks of skew power series rings of auto-
+morphic type. arXiv:0812.2010v3, 2011.
+[15] Hidetoshi Marubayashi and Freddy van Oystaeyen. Prime Divisors and Noncommutative
+Valuation Theory. Lecture Notes in Mathematics, 2059. Springer, 2012.
+[16] Jerzy Matczuk. Goldie rank of Ore extensions. Comm. Alg., 23(4):1455–1471, 1995.
+[17] J.C. McConnell and J.C. Robson. Noncommutative Noetherian Rings. American Mathe-
+matical Society, 2001.
+[18] Johan ¨Oinert, Johan Richter, and Sergei D. Silvestrov. Maximal commutative subrings
+and simplicity of Ore extensions. J. Algebra Appl., 12(4), 2013.
+[19] Otmar Venjakob. A noncommutative Weierstrass preparation theorem and applications
+to Iwasawa theory. J. Reine Angew. Math., 559:153–191, 2003.
+[20] William Woods.
+Dimension theory in iterated local skew power series rings.
+Algebr.
+Represent. Theory. Published online: https://doi.org/10.1007/s10468-022-10144-3,
+2022.
+17
+

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+arXiv:2301.05059v1  [cs.DC]  12 Jan 2023
+Distributed Self-Stabilizing MIS with Few States and Weak
+Communication
+George Giakkoupis
+Inria, Rennes, France
+george.giakkoupis@inria.fr
+Isabella Ziccardi
+Bocconi University, Milan, Italy
+isabella.ziccardi@unibocconi.it
+Abstract
+We study a simple random process that computes a maximal independent set (MIS) on a
+general n-vertex graph. Each vertex has a binary state, black or white, where black indicates
+inclusion into the MIS. The vertex states are arbitrary initially, and are updated in parallel:
+In each round, every vertex whose state is “inconsistent” with its neighbors’, i.e., it is black
+and has a black neighbor, or it is white and all neighbors are white, changes its state with
+probability 1/2. The process stabilizes with probability 1 on any graph, and the resulting set of
+black vertices is an MIS. It is also easy to see that the expected stabilization time is O(log n)
+on certain graph families, such as cliques and trees. However, analyzing the process on graphs
+beyond these simple cases seems challenging.
+Our main result is that the process stabilizes in poly(log n) rounds w.h.p. on Gn,p random
+graphs, for 0 ≤ p ≤ poly(log n) · n−1/2 and p ≥ 1/ poly(log n). Further, an extension of this
+process, with larger but still constant vertex state space, stabilizes in poly(log n) rounds on Gn,p
+w.h.p., for all 1 ≤ p ≤ 1. We conjecture that this improved bound holds for the original process
+as well. In fact, we believe that the original process stabilizes in poly(log n) rounds on any given
+n-vertex graph w.h.p. Both processes readily translate into distributed/parallel MIS algorithms,
+which are self-stabilizing, use constant space (and constant random bits per round), and assume
+restricted communication as in the beeping or the synchronous stone age models. To the best of
+our knowledge, no previously known MIS algorithm is self-stabilizing, uses constant space and
+constant randomness, and stabilizes in poly(log n) rounds in general or random graphs.
+1
+Introduction
+Finding a maximal independent set (MIS) is a fundamental problem in parallel and distributed
+computing. Given a graph G = (V, E), the objective is to identify a set of vertices S ⊆ V such
+that no two vertices u, v ∈ S are adjacent to each other (independence property), and no vertex
+u ∈ V \S can be added to S without violating independence (maximality property). The significance
+of the problem in parallel computing was first recognised in the early 80s [32, 8], due to its various
+applications in symmetry breaking [24], and it has been studied extensively every since (see [7] for
+a review of work until 2015, and [4, 17] for state of the art results).
+In this paper we explore simple distributed random processes on graphs that find an MIS
+starting from arbitrary initial states of the vertices. These processes immediately translate into self-
+stabilizing [10, 11] synchronous distributed algorithms for network systems with severely restricted
+computation and communication capabilities, such as wireless sensor networks. The processes we
+consider are also relevant to certain biological cellular networks. For example, it is known that a
+biological process occurring during the development of the nervous system of a fly is equivalent to
+computing an MIS [2, 23].
+1
+
+The main random process we consider, which we call the 2-state MIS process, is as follows.
+Each vertex has a binary state, black or white, where black indicates inclusion into the MIS. The
+vertex states are arbitrary initially and are updated in synchronous rounds. In each round, every
+vertex u whose state violates the independence or maximality properties, i.e., u is black and has
+a black neighbor, or it is white and has no black neighbor, changes its state to the opposite state
+with probability 1/2. It is easy to see that the state of a vertex stabilizes as soon as it is black
+and has no black neighbors, or it is white and has a stabilized black neighbor; and when all vertices
+have stabilized, the set of black vertices is an MIS. It is also immediate that, on any graph G, the
+process stabilizes eventually with probability 1 (due to the randomization) .1
+The 2-state MIS process can be viewed as a natural parallelization (with the addition of ran-
+domness) of a simple self-stabilizing sequential deterministic algorithm, proposed in [28, 20], where
+in each step a single node updates its state (from black to white, if the node has a black neighbor,
+and from white to black if it has no black neighbors). [28] also observed that by randomizing the
+transitions of the sequential algorithm we obtain an algorithm that stabilizes with probability 1 on
+a general adversarial scheduler model, which includes the synchronous model. A similar observa-
+tion follows from a general transformation framework proposed in [31]. The sequential algorithm is
+know to stabilize after each process has taken at most 2 steps (regardless of the scheduling order).
+However, analyzing the stabilization time of the parallel process seems a much more challenging
+problem, and has not been studied until now.
+The 2-state MIS process directly translates into a self-stabilizing MIS algorithm for the harsh
+beeping communication model [9]. In that model, in every synchronous round, each node either
+listens or beeps, and a listening node can only differentiate between none of its neighbors’ beeping,
+or at least one beeping. In our case, we can let black nodes beep in each round, while white nodes
+listen. Black nodes must be able to detect collisions (otherwise they cannot tell if they have a
+black neighbor), thus we assume the beeping model version with sender collision detection (a.k.a.
+full-duplex model) [1, 16].
+We also propose a simple variant of the 2-state MIS process, called the 3-state MIS process,
+which has an additional state and does not require collision detection (see Definition 5).
+This
+variant is suitable for the synchronous stone age model [13, 12]. The synchronous stone age model
+can be viewed as an extension of the beeping model over a constant number of channels (without
+collision detection): each node beeps in at most one channel and listens to the other channels.
+Overall, the algorithms obtained from the 2-state and 3-state MIS processes have several at-
+tractive properties: they use a constant number of states (2 or 3) and one random bit per round,
+they do not require node IDs or any global graph information (such as the number of vertices n
+or the maximum degree ∆), assume very week communication (the beeping or stone age models),
+they are self-stabilizing, and are extremely simple. We will prove that, on some families of graphs,
+these algorithms are also fast, i.e., they stabilize (from an arbitrary initial state) in a number of
+rounds that is poly-logarithmic in n, w.h.p.2 Moreover, despite that we were not able to prove such
+as strong result here, we believe that these algorithms are fast in all graphs.
+Several self-stabilizing distributed MIS algorithms have been proposed in the literature, but as
+far as we know, none possesses all the above properties. Known self-stabilizing MIS algorithms for
+the beeping model require (approximate) knowledge of n, use space that is a super-constant function
+of n, and require a super-constant number of random bits [1, 23, 16]. In the stone age model, an
+MIS algorithm proposed in [13] has similar properties as our algorithms (and is provably fast for all
+1We could have defined the process so that the transition from white to black (when the white vertex has no black
+neighbors) occurs with probability 1, but we opted for a randomized transition because it simplifies our analysis.
+2In this paper, we do not analyze the 3-state MIS process, but we expect that it behaves similarly (or better than)
+the 2-state MIS process.
+2
+
+graphs) but is not self-stabilizing; while a self-stabilizing algorithm for the model proposed recently
+in [12] is fast only on graphs whose diameter is bounded by a known constant D. Other randomized
+self-stabilizing MIS algorithms required super constant state and communication [30].
+Finally,
+known deterministic self-stabilizing MIS algorithms require distinct node IDS, super constant state
+and communication, and are in general much slower than the randomized algorithms, stabilizing
+in time linear in n or in the maximum degree ∆ [22, 18, 29, 5].
+1.1
+Our Contribution
+We first analyze the stabilization time of the 2-state MIS process on complete graphs and on graphs
+with bounded arboricity.3 We also provide an upper bound in terms of the maximum degree for
+general graphs. The proof of these results is mostly straightforward.
+Theorem 1. The stabilization time of the 2-state MIS process on n-vertex graph G is
+• O(log n) in expectation and O(log2 n) w.h.p., if G is the complete graph Kn.
+• O(log n) w.h.p., if G has bounded arboricity.
+• at most O(∆ log n) w.h.p., if the maximum degree of G is ∆.
+A main technical contribution of the paper is the analysis of the 2-state MIS process on Erd˝os-
+R´enyi Gn,p random graphs. We show a poly-logarithmic upper bound for Gn,p random graphs when
+the average degree np is at most poly(log n) · √n. The same bound is easily obtained also when
+the average degree is at least n/ poly(log n).
+Theorem 2. The stabilization time of the 2-state MIS process on a Gn,p random graph, such that
+0 ≤ p ≤ poly(log n) · n−1/2 or p ≥ 1/ poly(log n), is at most poly(log n) w.h.p.
+Our proof techniques do not yield a poly-logarithmic upper bound for the 2-state MIS process
+on Gn,p for the complete range of p. Our second technical contribution is an extension of the 2-state
+MIS process that provably stabilizes in poly-logarithmic time w.h.p. on Gn,p for all 0 ≤ p ≤ 1. The
+extended process uses a phase clock sub-process proposed in [12]. Interestingly, unlike [12], we do
+not use the phase clock for synchronization, but rather as a local non-synchronized counter (see
+Section 1.2 for a more detailed discussion).
+Theorem 3. There is an extension of the 2-state process, with 18 states, such that the stabilization
+time of the process on a Gn,p random graph, for any 0 ≤ p ≤ 1, is at most poly(log n) w.h.p.
+We believe that the bound of Theorem 3 holds for the 2-state MIS process, as well. In fact,
+we conjecture that the stabilization time of the 2-state MIS process is poly(log n) w.h.p. on any
+given n-vertex graph. We also conjecture that the same is true for the 3-state MIS process. For the
+2-state process, the best general upper bound we can hope for is O(log2 n), as the process requires
+Θ(log2 n) rounds to stabilize on the complete graph Kn w.h.p.4 For the 3-state process, we have
+no example of a graph where the stabilization time is larger than O(log n).
+3The arboricity of a graph is the minimum number of forests into which we can partition its edges.
+4It also requires Θ(log2 n) rounds in expectation to stabilize on a graph consisting of √n disjoint cliques K√n.
+3
+
+1.2
+Analysis Overview and Techniques
+Below we give an overview of the analysis of the 2-state MIS process and its extension, on Gn,p
+random graphs.
+To avoid having to deal simultaneously with the randomness of the graph and the arbitrary
+initialization of vertex states, we deal with graph randomness first. We define a family of good
+graphs, containing those graphs that satisfy all structural properties that we will need for the
+analysis, e.g., bounds on the average degree of any induced subgraph, and bounds on the number
+of common neighbors of any two vertices (see Definition 17). We then show that a Gn,p random
+graph is good w.h.p., and assume an arbitrary good graph in the analysis.
+The analysis proceeds by showing that starting from any vertex states, the process makes
+sufficient progress after O(log n) rounds, where progress is measured by the expected number of
+vertices that stabilize.
+In the 2-state MIS process, we call a vertex active if it is black and has a black neighbor, or it is
+white and has no black neighbors. Thus, active vertices change their state to a uniformly random
+state in the next step. A vertex is k-active if it is active and has at most k active neighbors.
+An elementary property of the 2-state MIS process is that if a vertex is k-active, then it becomes
+stabilized black in O(log k) rounds with probability Ω(1/k).
+We also use an extension of this
+property to sets of active vertices.5 These two properties, combined with structural properties of
+good graphs, suffice to show the desired expected progress in the case in which the number of
+non-stabilized vertices or the number of active vertices is large enough.
+The more difficult case is when the number of non-stabilized vertices is relatively small, namely
+O(p−1 log2 n), and a smaller than 1/ poly(log n) fraction of them are active. One may expect this
+to be an easy case, since the induced subgraph on a random subset of O(p−1 log2 n) vertices has
+maximum degree ∆ = O(log2 n) w.h.p. (and Theorem 1 gives an O(log3 n) bound for that ∆).
+However, the above bound on ∆ does not apply to an induced subgraph on an arbitrary subset of
+O(p−1 log2 n) vertices. Nevertheless, it is true that the average degree is O(log2 n), thus a constant
+fraction of vertices have degree O(log2 n).
+Let u be one such vertex, i.e., of degree d = O(log2 n) in the induced subgraph of non-stabilized
+vertices. To prevent u from becoming active (and thus d-active) or becoming stabilized, in each
+round at least one neighbor of u must be non-stabilized black. We show that, roughly speaking,
+if a vertex v has probability b of being non-stabilized black at some point during an interval of r
+rounds (ignoring the first few rounds, e.g., if v is black initially) then v has probability poly(b/r) of
+becoming θ-active in that interval. For the purposes of the analysis, it suffices to set r = O(log log n).
+Then θ is, roughly, bounded by the maximum number of common neighbors two nodes may have,
+thus θ ≤ poly(log n) if p ≤ poly(log n) · n−1/2 (see Section 4.1 for the relevant lemmas).
+If each of the d neighbors of u has probability less than 1/(2d) of becoming non-stabilized
+black in the next r rounds, then u has a constant probability of becoming active (or stabilize).
+On the other hand, if there is some neighbor v that has probability b ≥ 1/(2d) of becoming non-
+stabilized black in the next r rounds, we saw above that v becomes θ-active with probability at
+least poly(b/r) = 1/ poly(log n). We conclude that, with probability 1/ poly(log n), u is poly(log n)-
+active or has some poly(log n)-active neighbor at some point in the next r = O(log log n) rounds.
+It follows that u stabilizes with probability 1/ poly(log n) in the next O(log n) rounds.6
+When p > poly(log n)·n−1/2, the last case of the analysis above does not give a poly-logarithmic
+bound. A way to overcome this problem is to control how often a vertex can change its state from
+white to black. We extend the 2-state MIS process by incorporating such a control mechanism.
+5Similar properties are commonly used in the analysis of distributed MIS algorithms in the literature.
+6We suspect that a refinement of this argument may be useful for a broader class of graphs.
+4
+
+We call the new process the 3-color MIS process. It consists of two sub-processes running in
+parallel: The first is similar to the 2-state MIS process with the addition of a third color, grey; a
+black vertex now becomes gray instead of white, a gray vertex becomes white after a while, and
+other vertices treat gray vertices as white. The transition from gray to white is controlled by the
+second sub-process, called the logarithmic switch.
+In the logarithmic switch, each vertex has an on/off binary variable, and a gray vertex changes
+to white if the switch variable of the vertex is on. We would like that the logarithmic switch satisfy
+two basic properties: (i) a vertex switches from off to on every Θ(log n) rounds; and (ii) it switches
+from on to off every O(1) rounds.7 However, we do not know how to implement property (i) using
+constant states. We observe that it suffices if property (i) is satisfied only when p > poly(log n) ·
+n−1/2; for smaller p, a weaker property suffices: (i′) a vertex switches from off to on after at most
+O(log n) rounds. It is not immediately obvious how to implement this distinction, because we want
+the process to work for all 0 ≤ p ≤ 1 without knowing p (or anything else about the graph topology).
+We achieve that as follows.
+We exploit the fact that if p > poly(log n) · n−1/2 then the graph has constant diameter (in
+fact diameter 2). The logarithmic switch process we devise is similar to the phase clock process
+RandPhase proposed in [12]. RandPhase assumes that an upper bound D on the graph diameter
+is available to the process and uses D + 3 states. The core mechanism of the logarithmic switch is
+identical to that of RandPhase for D = 3 (not 2!), but the underlying graph may have arbitrary
+(and unknown) diameter. The logarithmic switch includes also a mapping of the states to the on/off
+values of the switch. Unlike RandPhase which is used for sychronization (it achieve synchronous
+phases of length D + Θ(log n)), the purpose of the logarithmic switch is not synchronization, as it
+is not required that the switch variables of different vertices change simultaneously.
+Roadmap.
+The rest of the paper is organized as follows. Section 2 contains the definition and
+some basic properties of the 2-state and 3-state MIS processes.
+Section 3 provides a proof of
+Theorem 1. Section 4 proves Theorem 2. Section 5 defines the 3-color MIS process and proves
+Theorem 3. And Appendix B reviews related work.
+Notation.
+Let G = (V, E) be a graph on n vertices. For each vertex u ∈ V , N(u) = {v: (u, v) ∈
+E} is the set of neighbors of u, and N +(u) = N(u) ∪ {u}. Similarly, for a set of vertices S ⊆ V ,
+we define N(S) = �
+u∈S N(u) \ S and N +(S) = �
+u∈S N +(u) = N(S) ∪ S. For two (not necessarily
+disjoint) sets S, T ⊆ V , we let E(S, T) = {(u, v) ∈ E : u ∈ S, v ∈ T} be the set of edges with one
+endpoint in S and the other in T. We also define E(S) = E(S, S). By G[S] we denote the induced
+subgraph of G on S ⊆ V , i.e., G[S] = (S, E(S)).
+2
+The 2-State and 3-State MIS Processes
+We define two self-stabilizing distributed graph processes that compute a maximal independent set
+when applied on any given graph.
+Definition 4 (2-State MIS Process). In the 2-state MIS process on graph G = (V, E), each vertex
+u ∈ V has a binary state from the set {black, white}, and all states are updated in parallel
+7The reason why a logarithmic switch suffices, rather than a ‘double-logarithmic’ switch is that, in the induced
+subgraph on O(p−1 log2 n) vertices consider in the last case of the analysis of the 2-state MIS process, a constant
+fraction of vertices have at most O(log n) neighbors of degree Ω(log3 n).
+5
+
+rounds. The initial state c0(u) of vertex u can be arbitrary, and in each round t = 1, 2, . . . , u’s
+state is updated from ct−1(u) to ct(u) according to the following rule.
+let NC t(u) = {ct−1(v): v ∈ N(u)}
+if
+�
+ct−1(u) = black and NC t(u) ∋ black
+�
+or
+�
+ct−1(u) = white and NC t(u) ̸∋ black
+�
+then
+let ct(u) be a uniformly random state from {black, white}
+else set ct(u) = ct−1(u)
+We say that vertex u is black or white if its state is black or white, respectively. We say that
+u is active if it is black and has some black neighbor, or it is white and has no black neighbors.
+We say that vertex u is stable, if either it is black and has no black neighbors, or it is white and
+has a neighbor that is black and stable. It is immediate from the update rule that once a vertex
+becomes stable, it remains stable thereafter, and its state no longer changes. The stabilization
+time of vertex u is the earliest round at the end of which u is stable. The stabilization time
+of the process is the earliest round at the end of which all vertices are stable. It is easy to verify
+that after the stabilization time of the process, the set of black vertices is an MIS of G.
+We let Bt = {u ∈ V : ct(u) = black} be the set of black vertices at the end of round t ≥ 0, and
+let Wt = V \ Bt be the set of white vertices. We let
+At = {u ∈ Bt : N(u) ∩ Bt ̸= ∅} ∪ {u ∈ Wt : N(u) ∩ Bt = ∅}
+denote the set of active vertices at the end of round t. We let It = {u ∈ Bt : N(u) ∩ Bt = ∅} be the
+set of stable black vertices at the and of round t (note that It is an independent set and is a subset
+of the final MIS). Finally, we let Vt = V \ N +(It) be the set of vertices that are not stable at the
+end of round t.
+Definition 5 (3-State MIS Process). In the 3-state MIS process on G = (V, E), each vertex u ∈ V
+has a state from set {black1, black0, white}, and the states are updated in parallel rounds. The
+initial state c0(u) of u is arbitrary, and in each round t ≥ 1, u’s state is updated as follows.
+let NC t(u) = {ct−1(v): v ∈ N(u)}
+if ct−1(u) = black1 or
+�
+ct−1(u) = black0 and NC t(u) ̸∋ black1
+�
+or
+�
+ct−1(u) = white
+and NC t(u) = {white}
+�
+then
+let ct(u) be a uniformly random state from {black1, black0}
+else if ct−1(u) = black0 then
+set ct(u) = white
+else set ct(u) = ct−1(u)
+In the 3-state MIS process, we say that a vertex u is black when its state is black1 or black0.
+Then the stable vertices and the stabilization times are defined as before. Note that the state of a
+stable black vertex alternates perpetually between states black1 and black0.
+In this paper we focus on the 2-state MIS process, but we expect that all our upper bound
+results should carry over to the 3-state MIS process.
+2.1
+Basic Properties of the 2-State MIS Process
+We show some elementary properties of the 2-state MIS process. In the analysis, it will be conve-
+nient to assume that at the beginning of each round t ≥ 1, we flip for each vertex u an independent
+coin φt(u) such that P[φt(u) = black] = P[φt(u) = white] = 1/2. Then if u must update its state
+to a random state in that round, i.e., if u ∈ At−1, we set ct(u) = φt(u); while if u /∈ At−1, then
+φt(u) is not used by the algorithm.
+The lemmas below apply for any graph G = (V, E), and the probabilistic statements assume
+that we know the states of vertices at the end of round t (i.e., Bt or Wt is given). The first lemma
+6
+
+says than an active vertex u with k active neighbors has probability Ω(1/k) to become stable black
+in the next O(log k) rounds.
+Lemma 6. If u ∈ At and |N(u) ∩ At| = k ≥ 1, then the probability that u ∈ It+log(k+1) is at least
+(2ek)−1.
+Proof. Let r = ⌈log(k + 1)⌉. The probability that u ∈ It+r is lower bounded by the probability
+that φt+1(v) = · · · = φt+r(v) = black holds for v = u and does not hold for any v ∈ N(u) ∩ At,
+which is
+(1/2)r · (1 − (1/2)r)k ≥ (1/2)r · e−k/(2r−1) ≥ (1/2k) · (1/e).
+(1)
+For the first inequality we used the fact (1 − 1/n)n−1 ≥ e−1, and for the second we used that
+log(k + 1) ≤ r ≤ log(k) + 1.
+The next statement is a generalization of Lemma 6 to multiple active vertices u1, . . . , uℓ. We
+will apply this result to the set of active neighbors of a vertex u, to lower bound the probability
+that u is stable after a logarithmic number of rounds (because a neighbors becomes stable black).
+The proof can be found in Appendix A.1.
+Lemma 7. Suppose that u1, . . . , uℓ ∈ At, and |N(ui) ∩ At| = ki > 0, for each 1 ≤ i ≤ ℓ. Then the
+probability that {u1, . . . , uℓ} ∩ It+log(maxi ki+1) ̸= ∅ is at least (1/5) · min
+�
+1, �
+i(2ki)−1�
+.
+3
+Simple Bounds for the 2-State MIS Process
+We show some simple bounds on the stabilization time of the 2-state MIS process on certain graph
+families, namely, the complete graph and trees (or more generally, graphs of bounded arboricity).
+We also show a basic upper bound in terms of the maximum degree on a general graph.
+Theorem 8. The stabilization time of the 2-state MIS process on the complete graph Kn = (V, E)
+is O(log n) in expectation and O(log2 n) w.h.p. More concretely, for any k > 0, the stabilization
+time is at least k · log n with probability 2−Θ(k).
+Proof. We call round t critical if |Bt| ≤ 1, and we call it stable if |Bt| = 1. Let pa be the probability
+that the next critical round is stable, given that |At| = a ≥ 2. Note that in graph Kn, At = Bt if
+|Bt| > 1, At = ∅ if |Bt| = 1, and At = V if Bt = ∅; thus |At| ̸= 1. We argue that for any a ≥ 2,
+2/3 ≤ pa ≤ 17/21.
+The lower bound follows from the observation that, for any i ≥ 2 and j ≥ 1, the conditional
+probability that round j is stable, given that it is critical and that |Aj−1| = i, is
+(i
+1)2−i
+(i
+1)2−i+2−i =
+i
+i+1 ≥
+2/3, since i ≥ 2. For the upper bound we observe that, for any i ≥ 3 and j ≥ 1, the conditional
+probability of |Bj| ∈ {2, 0}, given that |Bj| ≤ 2 and that |Aj−1| = i, is
+(i
+2)2−i+2−i
+(i
+2)2−i+(i
+1)2−i+2−i = i2−i+2
+i2+i+2 ≥
+4/7. Also, p2 = 2/3 < 17/21. Then, for any a ≥ 3, we have 1 − pa ≥ (4/7) · (1 − p2), which implies
+pa ≤ 17/21.
+Next, consider the number of rounds r from a non-stable critical round (when all nodes are
+white) until the next critical round. The probability that r > k is lower and upper bounded by
+1 − e−n2−k ≤ 1 − (1 − 2−k)n ≤ n2−k.
+7
+
+Combining the above we obtain that (i) from any given non-stable round, the probability that
+a stable round is reached in at most k = log n + 1 rounds is at least 2/3 − n2−k ≥ 1/6; (ii) from
+any given non-stable critical round, the probability that the next critical round is non-stable and is
+reached in more than k = log n − 2 rounds is at least 1 − 17/24 − e−n2−k > 1/6; and (iii) assuming
+round t = 0 is not critical, the probability that the first critical round is non-stable is at least
+1 − 17/24. These statements, together, imply that the stabilization time is at least k log n with
+probability 2−Θ(k). And from that, the expectation and high-probability bounds follow.
+Remark 9. From Theorem 8, it is immediate that the expected stabilization time of the 2-state MIS
+process is Θ(log2 n) on a graph G that is the disjoint union of √n cliques K√n. The same bound
+holds also w.h.p.
+Remark 10. A similar analysis as for Theorem 8 gives an upper bound of O(log n) on the stabi-
+lization time of the 3-state MIS process on Kn, both in expectation and w.h.p. The reason is that
+once Bt ̸= ∅ then Bt′ ̸= ∅ for all t′ ≥ t (thus the next critical round is stable).
+Theorem 11. The stabilization time of the 2-state MIS process on any graph G = (V, E) of bounded
+arboricity (e.g., G is a tree) is O(log n) w.h.p.
+Proof. Recall that the arboricity λ of G is the minimum number of forests into which its edges can
+be partitioned, and is equal up to a factor of 2 to the maximum average degree in any subgraph [26].
+Suppose that the average degree of any subgraph of G is at most d ≤ 2λ. Let St be the subset of
+Vt consisting of of all vertices u ∈ Vt with |N(u) ∩ Vt| ≤ d. Then |St| ≥ |Vt|/(d + 1). If u ∈ St \ At
+and |N(u) ∩ Vt| = du, the probability that N(u) ⊆ Wt+1 is 2−du ≥ 2−d. Thus, for each u ∈ St,
+the probability that u ∈ At ∪ At+1 is at least 2−d. And if u ∈ At ∪ At+1, Lemma 6 gives that
+u ∈ It+log(d+1)+1 with probability at least (2ed)−1. It follows
+E
+�
+|Vt+log(d+1)+1|
+�� |Vt|
+�
+≤ |Vt| − (2ed)−1 · 2−d · |Vt|/(d − 1) ≤ (1 − ǫ) · |Vt|,
+for some constant ǫ = ǫ(d). Let r = log(d + 1) + 1. Applying the above inequality iteratively,
+we obtain E[|Vrt|] ≤ (1 − ǫ)rn ≤ e−ǫrn. Thus for t = 3ǫ−1 ln n, E[|Vrt|] ≤ n−2, and by Markov’s
+inequality, P[|Vrt| ≥ 1] ≤ n−2, which implies the lemma.
+Theorem 12. The stabilization time of the 2-state MIS process on any graph G = (V, E) of
+maximum degree ∆ is at most O(∆ log n) w.h.p.
+Proof. We observe that if u ∈ Vt then N +(u)∩At ̸= ∅. Let u ∈ V0, and let (v1, t1), (v2, t2), (v3, t3), . . .
+be a random sequence of vertex-round pairs defined as follows: Let t0 = 0. For each i ≥ 1, if
+u ∈ Vti−1, then vi is an arbitrary vertex from the set N(u)∩Ati−1, and ti = min{j > ti−1 : vi /∈ Aj};
+while if u /∈ Vti−1, then (vi, ti) = (u, ti−1).
+We focus on the first r = 6e∆ log n elements of the sequence above. We bound the probability
+that u ∈ Vtr. For each 1 ≤ i ≤ r, the conditional probability that vi ∈ Iti+1 (and thus u /∈ Vti+1),
+given vi and Bti, is at least 1/(2e∆), from Lemma 6. It follows that
+P[u ∈ Vtr] ≤ (1 − 1/(2e∆))r ≤ e−r/(2e∆) = n−3.
+Next, we bound the value of tr. For each 1 ≤ i ≤ r and t ≥ ti−1, if vi ∈ At then the conditional
+probability that vi /∈ At+1, given (vi, ti) and Bt, is exactly 1/2 (in all cases). It follows that the
+probability of tr > 4r is upper bound by the probability that a sequence of 4r fair coin tosses
+contains fewer than r heads. Thus, by a Chernoff bound,
+P[tr > 4r] ≤ e−(1/2)22r/2 = e−e∆ log n < n−3.
+Combining the above results, we obtain that P[u /∈ V4r] ≥ P[{u /∈ Vtr} ∩ {tr ≤ 4r}] ≥ 1 − 2n−3.
+(Recall that r = 6e∆ log n.) Finally, a union bound over all u ∈ V competes the proof.
+8
+
+4
+The 2-State MIS Process on Random Graphs
+We first show some additional properties of the 2-state MIS process, which hold for any graph but
+are useful only when adjacent vertices do not have many common neighbors. Then we show some
+structural properties of Gn,p random graphs. Finally, we use these properties to show a poly(log n)
+upper bound on the stabilization time of the 2-state MIS process on Gn,p random graphs.
+4.1
+Refined Properties of the 2-State MIS Process
+We call a vertex k-active if it is active and has at most k active neighbors. Let
+Ak
+t = {u ∈ At : |N(u) ∩ At| ≤ k}
+be the set of k-active vertices at the end of round t. From Lemma 6, a k-active vertex has probability
+at least Ω(1/k) to become stable black in the next O(log k) rounds. It is thus desirable to have
+k-active vertices for small values k.
+In this section we establish lower bounds on the probability that a given vertex u becomes
+k-active at some point in the next r rounds, as a function of the probability that u is active (but
+has possibly more than k active neighbors) at a point in a certain subinterval of those r rounds.
+The next key lemma is the base of all the other results in the section. It lower bounds the
+probability q of a white vertex u, which is non-active and non-stable, to become k-active after a
+single round. The lower bound is expressed in terms of the probability p that u is active white after
+two rounds. The value of k depends on the number of active neighbors of u, and, crucially, on the
+number of their common neighbors with u.
+Lemma 13. Suppose that u ∈ Vt \ At,8 and let θ = |N(u) ∩ N +(At ∩ N(u))| be the number of u’s
+neighbors that are active or adjacent to an active neighbor of u at the end of round t. Let p be the
+probability that u ∈ At+2 ∩ Wt+2, and q the probability that u ∈ Ak
+t+1 where k = θ + ⌈log(1/p)⌉.
+Then q ≥ pα, where α = 1/log(4/3) ≤ 2.41.
+Proof. Let D = N(u) ∩ At. In round t + 1, each v ∈ D updates its state to a random state, while
+each v ∈ N(u) \ D remains white. Let Z = N(u) ∩ At+1 \ N +(D) be the set of active neighbors of
+u at the end of round t + 1 that are at distance at least two away from set D. Clearly, Z does not
+depend on the random choices of vertices v ∈ D in round t + 1.
+We have that u ∈ At+1 if and only if all v ∈ D update their state to white in round t + 1, i.e.,
+φt+1(v) = white.9 Also |N(u) ∩ At+1| ≤ |N(u) ∩ N +(D)| + |Z| = θ + |Z|. It follows
+q ≥ (1/2)d · P[|Z| ≤ λ],
+where d = |D| and λ = ⌈log(1/p)⌉.
+We have that u ∈ At+2 ∩ Wt+2 only if φt+1(v) or φt+2(v) = white for every v ∈ D, and
+φt+2(v) = white for every v ∈ Z. It follows that
+p ≤ (3/4)d ·
+�
+i≥0
+P[|Z| = i]/2i.
+(2)
+Let ε = P[|Z| ≤ λ]. Then
+p ≤ (3/4)d ·
+�
+ε + (1 − ε)/2λ+1�
+≤ (3/4)d · (ε + (1 − ε) · p/2) .
+8Note that u ∈ Vt \ At implies u ∈ Wt ∩ Wt+1.
+9Recall the discussion about coin flips φt(v) at the beginning of Section 2.1.
+9
+
+This implies that p ≤ ε + (1 − ε)· p/2, thus p ≤ 2ε/(1 + ε), and substituting that above yields
+p ≤ (3/4)d · (ε + (1 − ε) · ε/(1 + ε)) = (3/4)d ·
+2ε
+1 + ε.
+Finally, since (3/4)dα = (1/2)d, and for all x ∈ [0, 1],
+�
+2x
+1+x
+�α
+≤
+�
+2x
+1+x
+�2
+= x ·
+4x
+(1+x)2 ≤ x,
+pα ≤ (3/4)dα ·
+� 2ε
+1 + ε
+�α
+≤ (1/2)d · ε ≤ q.
+Next, we use the above Lemma 13 to prove a similar result over a sequence of r rounds. For
+any vertex u ∈ V and i ≥ 1, let
+θu(i) = max{|N(u) ∩ N +(S)|: S ⊆ N(u), |S| ≤ i}.
+(3)
+Lemma 14. Suppose that u ∈ Vt \ At and let d = |N(u) ∩ At|. Let pr be the probability that
+u ∈ At+1 ∪ · · · ∪ At+r, and let qr be the probability that u ∈ Ak
+t+1 ∪ · · · ∪ Ak
+t+r−1, where
+k = θu
+�
+α log
+�
+4r
+pr−2−d
+��
++
+�
+log
+�
+4r
+pr−2−d
+��
+,
+and α = 1/log(4/3). Then, for any r ≥ 2, qr ≥ r1−α ·
+�
+pr−2−d
+2
+�α
+.
+Proof. For i ≥ 0, let di = |N(u) ∩ At+i|, and define the following events: Wi is the event that
+u ∈ Wt+i; Ai is the event that u ∈ At+i; Ak
+i is the event that u ∈ Ak
+t+i; and Hi = ¯
+A0 ∩ ¯
+A1 ∩· · ·∩ ¯
+Ai.
+Let also Xi be the event that the states of the vertices at the end of round t + i are such that the
+conditional probability of Ai+2 ∩ Wi+2 is at least pr−p1
+4r
+. Let r ≥ 2 and λ = ⌊α log
+�
+4r
+pr−p1
+�
+⌋. Then
+pr =
+�
+1≤i≤r
+P[Ai ∩ Hi−1]
+= p1 +
+�
+2≤i≤r
+P[Ai ∩ Hi−1]
+= p1 +
+�
+2≤i≤r
+P[Ai ∩ Wi ∩ Hi−1]
+(since Ai ∩ Hi−1 implies Wi)
+≤ p1 +
+�
+2≤i≤r
+P[Ai ∩ Wi ∩ Hi−2]
+(since Hi−1 implies Hi−2)
+≤ p1 +
+�
+2≤i≤r
+P[Ai ∩ Wi ∩ Hi−2 ∩ {di−2 ≤ λ} ∩ Xi−2]
++
+�
+2≤i≤r
+P[Ai ∩ Wi ∩ Hi−2 ∩ {di−2 > λ}] +
+�
+2≤i≤r
+P[Ai ∩ Wi ∩ ¯
+Xi−2].
+Each of the last two sums above is at most pr−p1
+4
+, because for each non-zero sum term, we have
+P[Ai ∩ Wi ∩ Hi−2 ∩ {di−2 > λ}] ≤ P[Ai ∩ Wi | Hi−2, di−2 > λ] ≤
+�3
+4
+�λ+1
+≤ pr − p1
+4r
+,
+similarly to (2), and P[Ai ∩ Wi ∩ ¯
+Xi−2] ≤ P[Ai ∩ Wi | ¯
+Xi−2] ≤ pr−p1
+4r
+. Applying these above gives
+pr − p1
+2
+≤
+�
+2≤i≤r
+P[Ai ∩ Wi ∩ Hi−2 ∩ {di−2 ≤ λ} ∩ Xi−2]
+=
+�
+2≤i≤r
+P[Ai ∩ Wi | Hi−2, di−2 ≤ λ, Xi−2] · P[Hi−2 ∩ {di−2 ≤ λ} ∩ Xi−2].
+10
+
+Next we lower bound qr. We have
+qr ≥
+�
+1≤i≤r−1
+P[Ak
+i ∩ Hi−1]
+=
+�
+2≤i≤r
+P[Ak
+i−1 ∩ Hi−2]
+≥
+�
+2≤i≤r
+P[Ak
+i−1 ∩ Hi−2 ∩ {di−2 ≤ λ} ∩ Xi−2]
+=
+�
+2≤i≤r
+P[Ak
+i−1 | Hi−2, di−2 ≤ λ, Xi−2] · P[Hi−2 ∩ {di−2 ≤ λ} ∩ Xi−2].
+From Lemma 13, applied for round t + i − 2, using p ≥ pr−p1
+4
+and θ ≤ θu(λ), and observing that
+p1 = 2−d, we obtain
+P[Ak
+i−1 | Hi−2, di−2 ≤ λ, Xi−2] ≥ (P[Ai ∩ Wi | Hi−2, di−2 ≤ λ, Xi−2])α .
+We substitute this to the previous equation above, and then use Jensen’s inequality to complete
+the proof: Let ν = �
+2≤i≤r P[Hi−2 ∩ {di−2 ≤ λ} ∩ Xi−2] ≤ r.
+qr ≥
+�
+2≤i≤r
+�
+P[Ak
+i | Hi−2, di−2 ≤ λ, Xi−2]
+�α
+· P[Hi−2 ∩ {di−2 ≤ λ} ∩ Xi−2]
+≥ ν ·
+
+ �
+2≤i≤r
+P[Ak
+i | Hi−2, di−2 ≤ λ, Xi−2] · P[Hi−2 ∩ {di−2 ≤ λ} ∩ Xi−2]/ν
+
+
+α
+≥ ν ·
+�pr − p1
+2ν
+�α
+≥ r ·
+�pr − 2−d
+2r
+�α
+.
+Lemma 14 assumes that vertex u is initially not active. The next lemma shows a similar result
+for the case where u is active initially. In this case, in place of the probability pr that u becomes
+active at some point in the interval {t+1, . . . , t+r}, we use the probability br that u becomes black
+at some point of a subinterval {t + ℓ, . . . , t + r}. The proof proceeds by considering the first round
+after t when either u has at most k black neighbors, or u is white. If the first condition holds, then
+u has probability 1/2 of being black, and thus of being k-active. If only the second condition holds
+then we are in the case of Lemma 14. The proof can be found in Appendix A.2.
+Lemma 15. Suppose that u ∈ At. Let ℓ ≥ 2 and r ≥ ℓ + 2, let br be the probability that u ∈
+Bt+ℓ ∪· · ·∪Bt+r, and suppose that br ≥ 1/2ℓ−2. Let qr be the probability that u ∈ Ak
+t ∪· · ·∪Ak
+t+r−1,
+where
+k = θu
+�
+α log (32r/br)
+�
++ log (32r/br) + log(1/br) + 3.
+Then qr ≥ r1−α · (br/16)α, where α = 1/log(4/3).
+In the last lemma of this section, we consider the case in which Lemma 14 does not give a large
+enough lower bound for qr, even though pr is large, because the difference pr − 2−d is small. We
+proceed by essentially reducing this case to the case of Lemma 15, after a single round. The proof
+is in Appendix A.3.
+Lemma 16. Suppose that u ∈ Vt \ At, and let d = |N(u) ∩ At|. Let ℓ ≥ 5 and r ≥ ℓ + 2, let pr be
+the probability that u ∈ At+1 ∪ · · · ∪ At+r−1, let br be the probability that u ∈ Bt+ℓ ∪ · · · ∪ Bt+r, and
+11
+
+suppose that br ≥ 1/2ℓ−4 and br ≥ 2(pr − 2−d). Let qr be the probability that u ∈ Ak
+t ∪ · · · ∪ Ak
+t+r−1,
+where
+k = θu
+�
+α log (128r/br)
+�
++ log (128r/br) + log(4/br) + 3.
+Then qr ≥ r1−α · (br/64)α, where α = 1/log(4/3).
+4.2
+Structural Properties of Gn,p and Good Graphs
+We describe some structural properties that a graph must possess in order for the analysis given in
+the following sections to carry through. A graph satisfying these properties is called a good graph.
+Then we show that a random Gn,p graph is a good graph w.h.p.
+Definition 17 (Good Graphs). Let n be a positive integer and 0 < p < 1. A graph G = (V, E)
+with n vertices is (n, p)-good if it satisfies all the following properties:
+(P1) For any set S ⊆ V , the average degree of induced subgraph G[S] is at most max{8p|S|, 4 ln n}.
+(P2) For any set S ⊆ V of size |S| ≥ 40 ln(n)/p,
+|{u ∈ V \ S : |N(u) ∩ S| < p|S|/2}| ≤ |S|/2.
+(P3) For any three disjoint sets S, T, I ⊆ V such that |S| ≥ 2|T| and (S ∪ T) ∩ N(I) = ∅,
+|N(T) \ N +(S ∪ I)| ≤ |N(S) \ N +(I)| + 8 ln2(n)/p.
+(P4) For any two disjoint sets S, T ⊆ V such that |S| ≥ |T| and |T| ≤ ln(n)/p, |E(S, T)| ≤ 6|S| ln n.
+(P5) No two vertices u, v ∈ V have more than max{6np2, 4 ln n} common neighbors.
+(P6) If p ≥ 2(ln(n)/n)1/2 then diam(G) ≤ 2.
+Lemma 18. A random graph G = (V, E) drawn from Gn,p is (n, p)-good with probability 1−O(n−2).
+The proof of Lemma 18 can be found in Appendix A.4.
+4.3
+Analysis of the 2-State MIS Process on Gn,p
+In this section, we prove the following bound on the stabilization time of the 2-state MIS process
+on a random Gn,p graph.
+Theorem 19. The stabilization time of the 2-state MIS process on a random graph drawn from
+Gn,p, where p = O(
+�
+log(n)/n) or p = Ω(1/ log2.5 n), is O(log5.5 n) with probability 1 − O(n−2).
+The theorem follows by combining Lemma 18 and the next lemma, which analyzes the 2-state
+MIS process on a good graph.
+Lemma 20. The stabilization time of the 2-state MIS process on any (n, p)-good graph G = (V, E),
+where p = O(
+�
+log(n)/n) or p = Ω(1/ log2.5 n), is O(log5.5 n) with probability 1 − O(n−2).
+It is straightforward to extend the above statements so that p ≤ poly(log n) · n−1/2 or p ≥
+1/ poly(log n), for any desired poly(log n) term, by adjusting the exponent of log n in the stabiliza-
+tion time bound.
+12
+
+4.3.1
+Proof of Lemma 20
+We show that starting from any vector of vertex states, the process makes sufficient progress after
+poly(log n) rounds, where progress is measured by the expected number of vertices that become
+stable. All lemmas below assume G = (V, E) is an arbitrary (n, p)-good graph, and the probabilistic
+statements assume we know the states of the vertices at the end of round t.
+The first lemma
+considers the case in which the number of active vertices is large, namely, |At| = Ω(log(n)/p).
+Lemma 21. If |At| ≥ 80 ln(n)/p then there is a constant ǫ > 0 such that E[|Vt+log n|] ≤ (1 − ǫ)·|Vt|.
+Proof. From property (P1) in Definition 17 of good graphs, the average degree of the induced
+subgraph G[At] is at most k = max{8p|At|, 4 ln n} = 8p|At|.
+Let S be a subset of At consisting of the |At|/2 vertices u ∈ At with the smallest degree in
+G[At], i.e., for any two vertices u ∈ S and u′ ∈ At \ S, |N(u) ∩ At| ≤ |N(u′) ∩ At|. It follows that
+for all u ∈ S, |N(u) ∩ At| ≤ 2k; thus S ⊆ A2k
+t . Let R = {u ∈ V \ S : |N(u) ∩ S| < p|S|/2}. Since
+|S| = |At|/2 ≥ 40 ln(n)/p, property (P2) in Definition 17 yields |R| ≤ |S|/2. Then the number of
+vertices u ∈ Vt with |N(u) ∩ S| ≥ p|S|/2 is at least
+|Vt \ (S ∪ R)| ≥ |Vt| − (|S| + |R|) ≥ |Vt| − 3|S|/2 = |Vt| − 3|At|/4 ≥ |Vt|/4.
+Since each of those vertices u has at least p|S|/2 neighbors in S ⊆ A2k
+t , Lemma 7 gives that the
+probability at least one neighbor of u is stable black (and thus u is also stable) at the end of round
+t + log n is at least
+(1/5) · min
+�
+1, (p|S|/2) · (4k)−1�
+= (1/5) · min
+�
+1, (p|At|/4) · (32p|At|)−1�
+= 1/640.
+Then the expected number of vertices that are not stable at the end of round t + log n is
+E[|Vt+log n|] ≤ |Vt| − (|Vt|/4) · 1/640 ≤ |Vt| − |Vt|/2560.
+The next lemma considers the case in which the number of vertices that are not stable is large,
+namely |Vt| = Ω(ln2(n)/p), and |At| = O(ln(n)/p).
+Lemma 22. If |Vt| ≥ 10 ln2(n)/p and |At| ≤ 80 ln(n)/p then there is a constant ǫ > 0 such that
+E[|Vt+log n|] ≤ (1 − ǫ/ ln n) · |Vt|.
+Proof. From property (P1) in Definition 17, the average degree of graph G[At] is at most
+k = max{8p|At|, 4 ln n} ≤ 640 ln n.
+Let S be a subset of At consisting of the 2|At|/3 vertices u ∈ At with the smallest degree in G[At],
+and let T = At \ S. Then for all u ∈ S, |N(u) ∩ At| ≤ 3k; thus S ⊆ A3k
+t .
+The set Vt consist of (i) all the active vertices, u ∈ At = S ∪ T, and (ii) all the non-active
+vertices that are not in N +(It) (these vertices are white and have at least one active neighbor). We
+can thus partition Vt into the four distinct sets: S, N(S)\N(It), T \N(S), and N(T)\N +(S ∪It).
+For the sizes of these sets, we have |T \ N(S)| ≤ |T| < |S| and, by property (P3) in Definition 17,
+|N(T) \ N +(S ∪ It)| ≤ |N(S) \ N(It)| + 8 ln2(n)/p.
+Using these two inequalities, the fact that the sizes of the four sets above sum to |Vt|, and the
+assumption |Vt| ≥ 10 ln2(n)/p, we obtain
+|S| + |N(S) \ N(It)| ≥ (|Vt| − 8 ln2(n)/p)/2 ≥ |Vt|/10.
+13
+
+Therefore, at least |Vt|/10 vertices u ∈ Vt are in S or adjacent to a vertex from S. From Lemma 6,
+each u ∈ S ⊆ A3k
+t
+is stable black (and all its neighbors are stable white) at the end of round
+t + log n, with probability at least 1/(6ek). It follows that
+E[|Vt+log n|] ≤ |Vt| − (|Vt|/10) · 1/(6ek) ≤ |Vt| − |Vt|/(1.1 · 105 ln n).
+In the next lemma we analyze the remaining case, in which |Vt| = O(ln2(n)/p) and |At| =
+O(ln(n)/p). In fact, the lemma does not require a bound on |At|. Unlike the previous lemmas,
+however, it requires that p = O(
+�
+log(n)/n).
+Lemma 23. If |Vt| ≤ 10 ln2(n)/p and p ≤ c
+�
+log(n)/n, for some constant c > 0, then there is a
+constant ǫ = ǫ(c) > 0 such that E[|Vt+2 log n|] ≤
+�
+1 − ǫ/ ln3.5 n
+�
+· |Vt|.10
+Proof. From property (P1) in Definition 17, the average degree of graph G[Vt] is at most
+k = max{8p|Vt|, 4 ln n} ≤ 80 ln2 n.
+Let T be a subset of Vt consisting of the min{ln(n)/p, |Vt|/2} vertices u ∈ Vt with the largest degree
+in G[Vt], and let S = Vt \ T. Then |S| ≥ |T|, and for all u ∈ S, |N(u) ∩ Vt| is at most
+d = k|Vt|/|T| ≤ k · max{p|Vt|/ ln n, 2} ≤ 800 ln3 n.
+From property (P4) in Definition 17, the number of edges between S and T is |E(S, T)| ≤ 6|S| ln n.
+Let R = {u ∈ S : |N(u)∩T| ≤ 12 ln n}. Then |R| ≥ |S|/2 ≥ |Vt|/4. We will show for some constant
+ǫ′ = ǫ′(c) that
+P[u /∈ Vt+2 log n] ≥ ǫ′ ln−α−1 n · (ln ln n)−α,
+for all u ∈ R.
+(4)
+It follows that E[|Vt+2 log n|] ≤ |Vt| − (|Vt|/4) · ǫ′ ln−α−1 n · (ln ln n)−α. Since α = 1/log(4/3) ≤ 2.41,
+the above implies the lemma. To complete the proof it remains to show (4).
+Let u ∈ R. We partition the neighbors of u in G[Vt] into sets N(u) ∩ S and N(u) ∩ T, and let
+x = P[N(u) ∩ S ∩ A ̸= ∅]
+and
+y = P[N(u) ∩ T ∩ B ̸= ∅],
+where A = At ∪ · · · ∪ At+r−2, B = Bt+r−2 ∪ Bt+r−1 ∪ Bt+r, and r = log(48 ln n) + 6. We distinguish
+the following three cases: x + y ≤ 1/2, x ≥ 1/4, and y ≥ 1/4.
+Case x + y ≤ 1/2: With probability at least 1 − (x + y) ≥ 1/2, we have N(u) ∩ S ∩ A = ∅
+and N(u) ∩ T ∩ B = ∅. If N(u) ∩ S ∩ A = ∅ then N(u) ∩ S ⊆ Wt+r−2 (it is easy to see that
+N(u) ∩ S ∩ It+r−2 = ∅). Similarly, if N(u) ∩ T ∩ B = ∅, it is immediate that N(u) ∩ T ⊆ Wt+r−2.
+Thus, with probability at least 1/2, we have that N(u) ⊆ Wt+r−2. If N(u) ��� Wt+r−2, then either
+u ∈ At+r−2 ∩ Wt−r−2 or u ∈ It+r−2. Therefore, with probability at least 1/2, either u /∈ Vt+r−2
+or u ∈ At+r−2. If u ∈ At+r−2, then u ∈ Ad
+t+r−2 since |N(u) ∩ Vt| ≤ d, and from Lemma 6, the
+probability that u ∈ It+r−2+log n is at least (2ed)−1. Combining the last two statements yields that
+the probability of u /∈ Vt+r−2+log n is at least (1/2) · (2ed)−1 ≥ (8700 ln3 n)−1, which implies (4).
+Case x ≥ 1/4: With probability at least 1/4, there is a pair v, j such that v ∈ N(u) ∩ S,
+0 ≤ j ≤ r − 2, and v ∈ At+j. And if v ∈ At+j then v ∈ Ad
+t+j since |N(v) ∩ Vt| ≤ d, and from
+Lemma 6, the probability that v ∈ It+j+log n is at least (2ed)−1. We conclude that the probability
+that u ∈ N +(It+r−2+log n) is at least (1/4) · (2ed)−1 ≥ (17400 ln3 n)−1, which implies (4).
+Case y ≥ 1/4: There exists some v∗ ∈ N(u) ∩ T such that
+P[v∗ ∈ B] ≥ y/|N(u) ∩ T| ≥ (4 · 12 ln n)−1 = (48 ln n)−1.
+10If c is super constant, then the proof gives E[|Vt+2 log n|] ≤
+�
+1 − ǫ/(c2 ln3.5 n)
+�
+· |Vt|.
+14
+
+If v∗ ∈ At then we can apply Lemma 15, for ℓ = r − 2 ≥ log(48 ln n) + 2 and br ≥ (48 ln n)−1, to
+obtain that v∗ ∈ Aλ
+t ∪ · · · ∪ Aλ
+t+r−1 with probability at least q = r1−α · (16 · 48 ln n)−α, where
+λ = θv∗�
+α log (32r · 48 ln n)
+�
++ log (32r · 48 ln n) + log(48 ln n) + 3.
+Suppose now that v∗ ∈ Vt \ At, and let
+p∗ = P[v∗ ∈ At+1 ∪ · · · ∪ At+r−1] − 2−|N(v∗)∩At|.
+If p∗ ≥ P[v∗ ∈ B]/2 ≥ (96 ln n)−1, then we can apply Lemma 14 (using r − 1 in place of r), to
+obtain that v∗ ∈ Aλ′
+t ∪· · ·∪Aλ′
+t+r−2 with probability at least q′ = r1−α ·(p∗/2)α ≥ r1−α·(192 ln n)−α,
+where
+λ′ = θv∗�
+α log (4r · 96 ln n)
+�
++ log (4r · 96 ln n) .
+If p∗ < P[v∗ ∈ B]/2, then we can apply Lemma 16, for ℓ = r − 2 ≥ log(48 ln n) + 4 and br =
+P[v∗ ∈ B] ≥ (48 ln n)−1, to obtain that that v∗ ∈ Aλ′′
+t
+∪ · · · ∪ Aλ′′
+t+r−1 with probability at least
+q′′ = r1−α · (64 · 48 ln n)−α, where
+λ′′ = θv∗�
+α log (128r · 48 ln n)
+�
++ log (128r · 48 ln n) + log(4 · 48 ln n) + 3.
+In all the settings above, we have q, q′, q′′ ≥ ε ln−α n · (ln ln n)1−α and λ, λ′, λ′′ ≤ β ln n · ln ln n, for
+some constants ε, β > 0, where the bound on λ, λ′, λ′′ holds because property (P5) and assumption
+p ≤ c
+�
+log(n)/n imply that for any v ∈ V , θv(i) ≤ i · (6c2 + 4) log n (recall the definition of θv
+from (3)). Therefore, the probability that v∗ is (β · ln n · ln ln n)-active at the end of some round
+in {t, . . . , t + r − 1} is at least ε · ln−α n · (ln ln n)1−α, and from Lemma 6, the probability that
+v∗ ∈ It+r−1+log n is at least ε · ln−α n · (ln ln n)1−α · (2eβ · ln n · ln ln n)−1. Thus with at least that
+probability we have u /∈ Vt+r−1+log n. This completes the proof of (4).
+Putting the Pieces Together.
+First, suppose that p ≤ c
+�
+log(n)/n for some constant c > 0.
+From Lemmas 21 to 23, E[|Vt+2 log n|] ≤
+�
+1 − ǫ/ ln3.5 n
+�
+· E[|Vt|], for any t ≥ 0. Iteratively applying
+this inequality, we obtain that for any i ≥ 0,
+E[|V2i log n|] ≤
+�
+1 − ǫ/ ln3.5 n
+�i · n.
+Substituting i = 3 ln4.5 n/ǫ yields E
+�
+|V(6/ǫ) log n·ln4.5 n|
+�
+≤ n−2, and by Markov’s inequality, it follows
+P[|V(6/ǫ) log n·ln4.5 n| ≥ 1] ≤ n−2.
+If p ≥ ε/ ln2.5 n for some constant ε > 0, then we use Lemmas 21 and 22 as above to obtain that
+P[|Vt| ≥ 10 ln2(n)/p] ≤ n−2 for some t = O(log3 n). We also observe that if |Vt| < 10 ln2(n)/p then
+the maximum degree of graph (V, E(Vt)) is ∆ < |Vt| ≤ 10 ln2(n)/p ≤ 10 ln4.5(n)/ε, and Theorem 12
+yields a bound of O(∆ log n) = O(log5.5 n). Combining the two completes the proof of Lemma 20.
+Remark 24. Some of the logarithmic factors can be shaved off with a more careful analysis. For
+example, using a “pipelining” argument, one could improve the bound on halving |Vt| obtained
+from Lemma 23, from O(log n · ln3.5 n) to O(log n + ln3.5 n), thus saving one logarithmic factor.
+5
+Logarithmic Switch and the 3-Color MIS Process
+We present an extension of the 2-state MIS process, called 3-color MIS process, which uses one
+additional color, grey, and includes also a sub-process, called logarithmic switch, which runs in
+parallel to the main process. Then we analyze the 3-color MIS process on Gn,p random graphs.
+15
+
+5.1
+The Logarithmic Switch Process
+We first introduce an abstract logarithmic switch process, by specifying its properties. Then we
+describe an actual randomized graph process that satisfies these properties with high probability
+and in a self-stabilizing manner, using 6 states per vertex.
+Definition 25 (Logarithmic Switch Process). An (a, b)-logarithmic switch process on G = (V, E)
+generates for each vertex u ∈ V a binary sequence σ0(u), σ1(u), . . . , where σt(u) ∈ {on, off} for
+each t ≥ 0, such that the following properties hold for all u ∈ V .
+(S1) Every run of consecutive off values in sequence σ0(u), σ1(u), . . . has length at most a ln n.
+(S2) If diam(G) ≤ 2 then every run of consecutive off values in sequence σt(u), σt+1(u), . . . has
+length at least a
+6 ln n, where t = min{i ≥ a
+6 ln n: σi(u) = on}.
+(S3) If diam(G) ≤ 2 then every run of consecutive on values in sequence σt(u), σt+1(u), . . . has
+length at most b, where t is some constant independent of n.
+Definition 26 (Randomized Logarithmic Switch). In the randomized logarithmic switch process on
+G = (V, E), each vertex u ∈ V has a state, called level, that takes on values in the set {0, 1, . . . , 5}.
+The initial value level0(u) of u can be arbitrary, and in each round t ≥ 1 the level of u is updated
+according to the following rule, which uses a global parameter 0 < ζ < 1.
+if level t−1(u) = 5 then
+choose a random bit bt(u) such that P[bt(u) = 0] = ζ
+end
+if (level t−1(u) = 5 and bt(u) = 1) or level t−1(u) = 0 then
+set level t(u) = 5
+else set level t(u) = max{level t−1(v): v ∈ N +(u)} − 1
+Finally, we define the following mapping of the levels to the binary on/off values of Definition 25.
+For each u ∈ V and t ≥ 0,
+σt(u) =
+�
+on
+if levelt(u) ≤ 2
+off
+if levelt(u) ≥ 3.
+Lemma 27. For any graph G = (V, E), the randomized logarithmic switch process with parameter
+0 < ζ ≤ 1/2 satisfies properties (S1) to (S3) for a = 4/ζ and b = 3, with probability 1 − O(n−2),
+during the first n rounds.
+Proof. Let u ∈ V , and let Sv ⊆ V be the set of vertices at distance at most 2 from u. If u has level
+at least 3 in all rounds t, . . . , t+a ln n, then no vertex v ∈ Su has level 0 in rounds t+2, . . . , t+a ln n;
+and at least one vertex v ∈ Su must be at level 5 in all rounds t + 2, . . . , t + a ln n − 2. It follows
+that the probability there is some u ∈ V and t ≤ n such that u has level at least 3 in all rounds
+t, . . . , t + a ln n is at most
+n2(1 − ζ)a ln n−4 ≤ n2−aζ/(1 − ζ)4 ≤ 16 · n−2,
+when aζ = 4. Thus, property (S1) holds with probability at least 1 − O(n−2).
+Next we assume diam(G) ≤ 2. The rest of the proof is similar to that in [12]. Observe that
+there must be a vertex v and a round t∗ ≤ 5 such that levelt∗(u) = 5. And from the end of round
+t∗ + 2, all vertices “synchronize” in the sense that once a vertex reaches level 2 in a round, all
+vertices reach level 2 in that round, then the they all reach level 1 in the next round, then level
+0, and then 5. It follows that property (S3) holds for b = 3, starting from round t∗ + 2 ≤ 7. The
+property holds with probability 1, and for all rounds after round t∗ + 2, not just for the first n.
+16
+
+As mentioned above, after vertices have synchronized, all n vertices move from level 0 to level
+5 simultaneously, each time. When that happens, the number of rounds until there are no vertices
+left at level 5 is greater than a ln n − 6 with probability at most
+n(1 − ζ)a ln n−6 ≤ 64 · n−3,
+as before; and is smaller than r = a
+6 ln n with probability at most
+(1 − (1 − ζ)r)n ≤ e−n(1−ζ)r ≤ e−n4−ζr = e−n4−(aζ/6) ln n ≤ e−n0.07 = O(n−3).
+Combining the above, using a union bound, we obtain that property (S2) holds with probability
+1 − O(n−2).
+5.2
+The 3-Color MIS Process
+We now define the 3-color MIS process, which is an extensions of the 2-state MIS process.
+Definition 28 (3-Color MIS Process). The process consists of two (sub-)processes that run in
+parallel on G = (V, E). The first is an (a, 3)-logarithmic switch process, where a = 512, which gen-
+erates a value σt(u) ∈ {on, off} for each vertex u ∈ V in each round t ≥ 0. The second is a variant
+of the 2-state MIS process, where each vertex u ∈ V has a state ct(u) ∈ {black, white, gray}, c0(u)
+can be arbitrary, and in each round t ≥ 1, u’s state is updated as follows.
+let NC t(u) = {ct−1(v): v ∈ N(u)}
+if ct−1(u) = black and NC t(u) ∋ black then
+let ct(u) be a uniformly random state from {black, gray}
+else if ct−1(u) = white and NC t(u) ̸∋ black then
+let ct(u) be a uniformly random state from {black, white}
+else if ct−1(u) = gray and σt−1(u) = on then
+set ct(u) = white
+else set ct(u) = ct−1(u)
+There are precisely two differences in the update rule above compared to that for the 2-state
+MIS process: a black vertex with a black neighbor changes to gray with probability 1/2, rather
+than to white; and a gray vertex changes to white if its switch value is on. Note that a gray vertex
+is treated similarly to a non-active white vertex.
+A vertex is stable, if it is black and has no black neighbors, or it is not black and has a neighbor
+that is stable black. Other than that, the remaining definitions and notations are the same as in
+the 2-state MIS process, namely, of active vertices, stabilization times, Bt, Wt, At, Ak
+t , It, and
+Vt. We also let Γt = V \ (Bt ∪ Wt) denote the set of gray vertices at the end of round t.
+The definition of the 3-color MIS process above assumes an arbitrary logarithmic switch process.
+We can use the randomized logarithmic switch from Definition 26, which uses 6 states per vertex,
+to obtain a 3-color MIS process that uses 6 · 3 = 18 states in total. The probability parameter of
+the randomized switch is ζ = 4/a = 27, thus at most 7 random bits are required per round for each
+vertex (plus one more for each active vertex).
+We note that Lemmas 6 and 7 and their proofs carry over to the 3-color MIS process, without
+changes. We will use also the two simple lemmas below that are specific to the 3-color MIS process.
+Recall that a = 512 is a parameter of the logarithmic switch.
+Lemma 29. If t ≥ a ln n and u ∈ Γt then u ∈ At−a ln n ∪ · · · ∪ At−1.
+17
+
+Proof. By property (S1) in Definition 25 of the logarithmic switch, a vertex is gray for at most
+a ln n consecutive rounds. Also if a vertex becomes gray in round j > 0, it must be active black at
+the end of round j − 1. Combining these two facts implies the lemma.
+Lemma 30. If diam(G) ≤ 2, u ∈ V , t ≥ a
+6 ln n, and t′ = t + a
+6 ln n, then the expected number of
+times u is active black between rounds t and t′ is E [|{j : u ∈ Bj ∩ Vj} ∩ {t, . . . , t′}| | Bt, Wt] ≤ 4.
+Proof. From properties (S2) and (S3) in Definition 25, it is easy to see that sequence ct(u), . . . , ct′(u)
+contains at most two runs of consecutive black states. Moreover, the expected length of the prefix
+of each black run until u becomes stable black or the run finishes (and u becomes gray) is 2. It
+follows that u is non-stable black in at most 4 rounds in expectation.
+Lemmas 13 and 14 hold also for the 3-color MIS process, when u ∈ Vt \ (At ∪ Γt) and thus
+u ∈ Wt.11 The next simple lemma will be used together with Lemma 14.
+Lemma 31. Let t ≥ 0, u ∈ V , and d > 0. Let t′ ≥ t be the first round when either u is white and
+has at least d black neighbors, or u is stable. The expected number of rounds t < j < t′ at which u
+is black and has at least d black neighbors is at most 3.
+Proof. The lemma is obtained using the observations that: each time u’s state changes from white
+to black, it is equally likely that it remained white; and, when u is active black, it becomes gray in
+the next step with probability 1/2.
+5.3
+Analysis of the 3-Color MIS Process on Gn,p
+We show that the stabilization time of the 3-color MIS process on Gn,p random graphs is poly(log n),
+for the complete range of values of p.
+Theorem 32. The stabilization time of the 3-color MIS process on a random graph drawn from
+Gn,p is O(log6 n) with probability 1 − O(n−2).
+As before, it suffices to show that the above bound holds for good graphs, and apply Lemma 18.
+Lemma 33. The stabilization time of the 3-color MIS process on any (n, p)-good graph G = (V, E)
+is O(log6 n) with probability 1 − O(n−2).
+5.3.1
+Proof of Lemma 33
+The proof strategy is similar to Lemma 20’s: From any vector of vertex states at the end of round
+t, we show that the process makes sufficient progress in expectation in poly(log n) rounds. The
+main difference is that now we show that this is also true even in the case of |Vt| = O(log2(n)/p)
+when diam(G) ≤ 2, which corresponds to the case of p = Ω(
+�
+log(n)/n), by property (P6) in
+Definition 17. This is precisely the case that we could not handle in the analysis of the 2 state MIS
+process. The relevant lemma is Lemma 36.
+We first observe that Lemma 21, which considers that case of |At| = Ω(log(n)/p), carries over
+to the 3-color MIS process, without any changes in the proof.
+Next we consider the case where |At| = O(ln(n)/p), |Vt| = Ω(ln2(n)/p), and |Γt| = O(ln2(n)/p).
+The following lemma is very similar to Lemma 22, except that it requires also a bound on |Γt|.
+Recall that a = 512 is a parameter of the logarithmic switch.
+11The proofs require just minor modifications, mostly replacing some occurrences of “white” by “not black” or
+“gray”.
+18
+
+Lemma 34. If |Vt| ≥ 82a ln2(n)/p, |At| ≤ 80 ln(n)/p, and |Γt| ≤ 80a ln2(n)/p, then there is a
+constant ǫ > 0 such that E[|Vt+log n|] ≤ (1 − ǫ/ ln n) · |Vt|.
+Proof. The proof is very similar to Lemma 22’s. As before, from property (P1) in Definition 17,
+the average degree of G[At] is at most k = max{8p|At|, 4 ln n} ≤ 640 ln n. We let S be a subset of
+At consisting of the 2|At|/3 vertices u ∈ At with the smallest degree in G[At], and let T = At \ S.
+Then for all u ∈ S, |N(u) ∩ At| ≤ 3k, thus S ⊆ A3k
+t .
+The set Vt consist of (i) all active vertices, u ∈ At = S ∪T, (ii) all non-active non-stable vertices
+that have some active neighbor, and (iii) all non-active non-stable vertices have no active neighbors
+(these vertices are gray). We can thus partition Vt into the five distinct sets: S, N(S) \ N(It),
+T \ N(S), N(T) \ N +(S ∪ It), and Vt \ N +(T ∪ Sf) ⊆ Γt. We have that |Vt \ N +(T ∪ S)| ≤ |Γt| ≤
+80a ln2(n)/p, |T \ N(S)| ≤ |T| < |S|, and, by property (P3) in Definition 17,
+|N(T) \ N +(S ∪ It)| ≤ |N(S) \ N(It)| + 8 ln2(n)/p.
+Using these three inequalities, the fact that the sizes of the five sets above sum to |Vt|, the assump-
+tion |Vt| ≥ 82a ln2(n)/p, and that a ≥ 8, we obtain
+|S| + |N(S) \ N(It)| ≥ (|Vt| − (80a + 8) ln2(n)/p)/2 ≥ (|Vt| − 81a ln2(n)/p)/2 ≥ |Vt|/82a.
+Therefore, at least |Vt|/82a vertices u ∈ Vt are in S or adjacent to S. And, from Lemma 6, each
+u ∈ S ⊆ A3k
+t
+is in It+log n, with probability at least 1/(6ek). It follows
+E[|Vt+log n|] ≤ |Vt| − (|Vt|/82a) · 1/(6ek) ≤ |Vt| − |Vt|/(1.1 · 82a · 104 ln n).
+Next we assume |At| = O(ln(n)/p) and |Vt| = Ω(ln2(n)/p), as in the previous lemma, but now
+|Γt| = Ω(ln2(n)/p). We reduce this case to the previous cases using Lemma 29.
+Lemma 35. If |Vt| ≥ 83a ln2(n)/p, |At| ≤ 80 ln(n)/p, and |Γt| > 80a ln2(n)/p, then there is a
+constant ǫ > 0 such that E[|Vt+a ln n+log n|] ≤ (1 − ǫ/ ln n) · |Vt|.
+Proof. Let τ = min{j ≥ t: |Vj| ≤ 82a ln2(n)/p or |Aj| ≥ 80 ln(n)/p or |Γj| ≤ 80a ln2(n)/p}. We
+have τ ≤ t + a ln n, because if |Γt+a ln n| > 80a ln2(n)/p, then Lemma 29 implies there is some
+j ∈ {t, . . . , t + a ln n − 1} such that |Aj| ≥ |Γt+a ln n|/(a ln n) ≥ 80 ln(n)/p. We distinguish three
+cases depending on which condition in the definition of τ is satisfied first. If |Vτ| ≤ 82a ln2(n)/p,
+then
+|Vt+a ln n| ≤ |Vτ| ≤ 82a ln2(n)/p ≤ (1 − 1/83) · |Vt|.
+If |Aτ| ≥ 80 ln(n)/p, then Lemma 21 yields E[|Vt+a ln n+log n|] ≤ E[|Vt+τ+log n|] ≤ (1 − ǫ) · |Vt|.
+Last, if |Γτ| ≤ 80a ln2(n)/p and the other two conditions do not hold, then Lemma 34 gives
+E[|Vt+a ln n+log n|] ≤ (1 − ǫ/ ln n) · |Vt|.
+The next two lemmas deal with the case of |Vt| = O(ln2(n)/p). The first one assumes diam(G) ≤
+2, and thus covers the case of p = Ω(
+�
+log(n)/n), by property (P6) in Definition 17; while the second
+lemma assumes p = O(
+�
+log(n)/n) and is similar to Lemma 23.
+Lemma 36. For any t ≥ a
+6 ln n, if |Vt| ≤ 83a ln2(n)/p and diam(G) ≤ 2 then there is a constant
+ǫ > 0 such that E[|Vt+ 7
+6 a log n+log n|] ≤
+�
+1 − ǫ/ ln3 n
+�
+· |Vt|.
+Proof. From property (P1) in Definition 17, the average degree of induced subgraph G[Vt] is at most
+k = max{8p|Vt|, 4 ln n} ≤ 664a ln2 n. Let T be a subset of Vt consisting of the min{ln(n)/p, |Vt|/2}
+19
+
+vertices u ∈ Vt with the largest degree in G[Vt], and let S = Vt \ T. Then |S| ≥ |T|, and all u ∈ S,
+|N(u) ∩ Vt| is at most
+d = k|Vt|/|T| ≤ k · max{p|Vt|/ ln n, 2} ≤ 55112 ln3 n.
+From property (P4) in Definition 17, |E(S, T)| ≤ 6|S| ln n. Let R = {u ∈ S : |N(u) ∩ T| ≤ 12 ln n}.
+Then |R| ≥ |S|/2 ≥ |Vt|/4. We will show that, for some constant ǫ′ > 0,
+P[u /∈ Vt+ 7
+6 a ln n+log n] ≥ ǫ′ ln−3 n,
+for all u ∈ R.
+(5)
+From this, it follows that E[|Vt+ 7
+6 a ln n+log n|] ≤ |Vt| − (|Vt|/4) · ǫ′ ln−3 n. To complete the proof of
+the lemma it remains to prove (5).
+Let u ∈ R, and suppose that u /∈ Γt (we deal with the case u ∈ Γt at the end). From Lemma 30,
+the expected value of �
+t≤j≤t+ a
+6 ln n |(N(u) ∩ T) ∩ (Bj ∩ Vj)|, that is, the total number of times that
+vertices v ∈ N(u)∩T are active black between rounds t and a
+6 ln n, is at most 4·|N(u)∩T| ≤ 4·12 ln n.
+Then, by Markov’s inequality, that number is at most 5 · 12 ln n with probability at least 1/5. And
+since a
+6 > 5 · 12, it follows that, with probability at least 1/5, there is some j ∈ {t, . . . , t + a
+6 ln n}
+such that (N(u) ∩ T) ∩ (Bj ∩ Vj) = ∅.
+Next we claim that, if (N(u) ∩ T) ∩ (Bj ∩ Vj) = ∅ for some j ≥ t, then (i) u /∈ Vj, or (ii) u ∈ Aj′
+for some t ≤ j′ < j, or (iii) (N +(u) ∩ S) ∩ Aj ̸= ∅. Indeed, suppose that (i) and (ii) do not
+hold, i.e., u ∈ Vj and u /∈ At ∪ · · · ∪ Aj−1.
+From u ∈ Vj, it follows N +(u) ∩ Ij = ∅.
+From
+u /∈ At ∪ · · · ∪ Aj−1 and the assumption u /∈ Γt, it follows u ∈ Wj. Then, if (N(u) ∩ S) ∩ Bj ̸= ∅,
+each vertex v ∈ (N(u) ∩ S) ∩ Bj is in Aj; while if (N(u) ∩ S) ∩ Bj = ∅, then N(u) ∩ Bj = ∅ and
+u ∈ Aj. Therefore (iii) holds.
+From the above, it follows that with probability at least 1/5, there is some t ≤ j ≤ t + a
+6 ln n
+such that u /∈ Vj or (N +(u) ∩ S) ∩ Aj ̸= ∅. And if v ∈ (N +(u) ∩ S) ∩ Aj, then v ∈ Ad
+j, and from
+Lemma 6, the probability that v ∈ Ij+log n is at least 1/(6ed). We conclude that
+P[u /∈ Vt+ a
+6 ln n+log n] ≥ (1/5) · 1/(6ed) ≥ (4.5 · 106 ln3 n)−1,
+which implies (5).
+Finally, if u /∈ Γt, we consider the first round j > t such that u /∈ Γj. From property (S1),
+j ≤ t + a ln n. Then we apply the result for the previous case to complete the proof of (5).
+Lemma 37. If |Vt| ≤ 83a ln2(n)/p and p ≤ c
+�
+log(n)/n for some constant c > 0, then there is a
+constant ǫ = ǫ(c) > 0 such that E[|Vt+log1.1 n|] ≤
+�
+1 − ǫ/ ln3.9 n
+�
+· |Vt|.
+Proof Sketch. We de���ne the set S, T, R and the degree thresholds k, d as in the proof of Lemma 36,
+and we show
+P[u /∈ Vt+log1.1 n] ≥ ǫ′ ln−3.9 n,
+for all u ∈ R,
+(6)
+which implies the lemma. Next we prove (6).
+Let u ∈ R, and suppose that u /∈ Γt (we deal with case u ∈ Γt at the end). For each v ∈ N(u)∩T,
+let tv ≥ t be the first round when either v is white and has at least ℓ = ln n black neighbors, or is
+stable; and let xv be the number of rounds t ≤ j ≤ min{tv, t+r} at which v is black and has at least
+ℓ black neighbors, where r = 12 ln n · ln2 ln n. From Lemma 31, the probability that xv ≤ ln2 ln n
+for all v is at least 1 − |N(u) ∩ T| · e−Ω(ln2 ln n) = 1 − e−ω(ln ln n). For each v ∈ N(u) ∩ T let pv be
+the conditional probability that v ∈ Btv+1 ∪ · · · ∪ Bt+r, given Bt, Wt.
+If �
+v∈N(u)∩T pv ≤ 1/2, then with probability at least 1/2 − e−ω(ln ln n) > 1/3, the total number
+of rounds in which at least one v ∈ N(u) ∩ T is black and has at least ℓ black neighbors is at
+20
+
+most |N(u) ∩ T| · ln2 ln n ≤ 12 ln n · ln2 ln n ≤ r, thus there is some j ∈ {t, . . . , t + r} such that no
+v ∈ N(u) ∩ T is black and has at least ℓ black neighbors. Then we can infer that with probability
+at least 1/3 some vertex in N +(u) is stable black or is d-active at some round in {t, . . . , t + r}, in
+the same way as in the proof of Lemma 36, and then obtain (6) using Lemma 6.
+If �
+v∈N(u)∩T pv > 1/2, then there is some v∗ ∈ N(u) ∩ T such that pv∗ ≥ (2|N(u) ∩ T|)−1 ≥
+(24 ln n)−1. We can then apply Lemma 14 to v∗ at round tv∗ to show that the probability vertex
+v∗ is z-active at some round in {t, . . . , t + r}, where z = θu
+�
+α log
+4r
+pv∗−2−ℓ
+�
++ log
+4r
+pv∗−2−ℓ = O(log n ·
+log log n), is at least r1−α·
+�
+pv∗−2−ℓ
+2
+�α
+= Ω(ln3.9 n), as α ≤ 2.41 Again we obtain (6) using Lemma 6.
+Finally, as before, if u /∈ Γt, we consider the first round j > t such that u /∈ Γj, and apply the
+result for the previous case to complete the proof of (6).
+We can now conclude the proof of Lemma 33, as we did for Lemma 20. From Lemmas 21 and 34
+to 37, we have that for any t ≥ a
+6 ln n, E[|Vt+log1.1 n|] ≤
+�
+1 − ǫ/ ln3.9 n
+�
+· E[|Vt|]. Iteratively applying
+this inequality, and using by Markov’s inequality, we obtain as before P[|Vc′ ln6 n| ≥ 1] ≤ n−2, for a
+large enough constant c′ > 0. This completes the proof of Lemma 33.
+APPENDIX
+A
+Omitted Proofs
+A.1
+Proof of Lemma 7
+We assume k1 ≤ k2 ≤ · · · ≤ kℓ. For 1 ≤ i ≤ ℓ, let ri = ⌈log(ki + 1)⌉, let Ei be the event that
+φt+1(ui) = · · · = φt+ri(ui) = black, and let E = �
+i Ei. Then
+P[E] = 1 −
+�
+i
+�
+1 − 1
+2ri
+�
+≥ 1 −
+�
+i
+�
+1 − 1
+2ki
+�
+≥
+�
+1 − e− �
+i
+1
+2ki
+�
+≥
+�
+1 − e−1�
+· min
+�
+1,
+�
+i
+1
+2ki
+�
+.
+Suppose that E occurs and let j be the smallest index such that Ej occurs, i.e., ¯E1 ∩ · · · ∩ ¯Ej−1 ∩ Ej
+occurs. If gj = |N(uj) ∩ {u1, . . . , uj−1}|, then the probability that none of the kj vertices v ∈
+N(uj) ∩ At satisfies φt+1(v) = · · · = φt+rj(v) = black is
+(1 − 2−rj)kj−gj ≥ (1 − 2−rj)kj ≥ e−1,
+similarly to (1). Combining this with the previous inequality we obtain that the probability that
+ui ∈ It+ri for at least one vertex ui ∈ {u1, . . . , uℓ} is at least e−1 ·
+�
+1 − e−1�
+· min
+�
+1, �
+i
+1
+2ki
+�
+≥
+1
+5 · min
+�
+1, �
+i
+1
+2ki
+�
+.
+A.2
+Proof of Lemma 15
+Let B be the event u ∈ Bt+ℓ ∪ · · · ∪ Bt+r; then P[B] = br. Let
+τ = min{j > t: u ∈ Wj or |N(u) ∩ Bj| ≤ k}
+be the first round j > t at the end of which u is white or has at most k black neighbors. We
+have P[τ > t + ℓ] ≤ 2ℓ ≤ br/4, since τ > t + ℓ implies φt+1(u) = · · · = φt+ℓ(u) = black. Thus
+P[τ ≤ t + ℓ] ≥ 1 − br/4. Let
+x = P[|N(u) ∩ Bτ| ≤ k | τ ≤ t + ℓ].
+21
+
+We distinguish two cases, x ≥ br/4 and x ≤ br/4.
+First suppose that x ≥ br/4. For any given j > t,
+P[u ∈ Aj | τ = j, |N(u) ∩ Bτ| ≤ k] = 1/2.
+The reason is that u ∈ Bj−1 and |N(u) ∩ Bj−1| > k > 0 if τ = j > t + 1, and u ∈ At = Aj−1 if
+τ = j = t+1. In either case u ∈ Aj−1, thus the state of u at the end of round j is chosen uniformly
+at random, independently of the remaining choices in round j.
+In particular, u is black with
+probability 1/2 when 0 < |N(u)∩Bj| ≤ k, and is white with probability 1/2 when |N(u)∩Bj| = 0.
+It follows that
+P[{u ∈ Aτ} ∩ {|N(u) ∩ Bτ| ≤ k} ∩ {τ ≤ t + ℓ}] ≥ (1/2) · x · (1 − br/4) ≥ 3br/32.
+Since the event on the left side implies that u is k-active at the end of round τ ≤ t + ℓ < t + r, and
+3br/32 is greater than the desired lower bound for qr, the lemma holds in this case.
+Suppose now that x ≤ br/4. Then
+P[B ∩ {|N(u) ∩ Bτ| > k} ∩ {τ ≤ t + ℓ}] ≥ P[B] − P[τ > t + ℓ] − x ≥ br − br/4 − br/4 = br/2.
+If τ ≤ t+ℓ and |N(u)∩Bτ| > k (and thus u ∈ Wτ by τ’s definition), we define the following events:
+Ak is the event that u ∈ Ak
+τ+1 ∪ · · · ∪ Ak
+t+r−1; A is the event that u ∈ Aτ+1 ∪ · · · ∪ At+r−1; and X is
+the event that the states of vertices at the end of round τ are such that the conditional probability
+of A, given these states and τ, is at least br/4.
+If τ ≤ t + ℓ and |N(u) ∩ Bτ| > k, then event B implies A, because vertex u, which is non-active
+white at the end of round τ, cannot become black before becoming active first. Thus, from the last
+inequality above, it follows
+P[A ∩ {|N(u) ∩ Bτ| > k} ∩ {τ ≤ t + ℓ}] ≥ br/2.
+Also
+P[A ∩ X ∩ {|N(u) ∩ Bτ| > k} ∩ {τ ≤ t + ℓ}] ≥ br/2 − br/4 = br/4.
+We can now apply Lemma 14, starting from round τ ≤ t + ℓ, using d > k ≥ log(1/br) + 3 and
+pr ≥ br/4, to obtain
+P[Ak ∩ X ∩ {|N(u) ∩ Bτ| > k} ∩ {τ ≤ t + ℓ}] ≥ r1−α ·
+�br/4 − 2k
+2
+�α
+≥ r1−α ·
+�br/4 − br/8
+2
+�α
+.
+It follows that qr = P[Ak] ≥ r1−α ·
+�
+br/4−br/8
+2
+�α
+, which concludes the proof of this case.
+A.3
+Proof of Lemma 16
+Proof. We have P[u ∈ At+1] = 2−d and P[u ∈ (At+2 ∪ · · · ∪ At+r−1) \ At+1] = pr − 2−d. We also
+note that if u /∈ At+1 then u ∈ Wt+1, and u may become black in a subsequent round only after it
+becomes active. It follows that
+P[{u ∈ (Bt+ℓ ∪ · · · ∪ Bt+r) ∩ At+1] = br − P[u ∈ (Bt+ℓ ∪ · · · ∪ Bt+r) \ At+1]
+≥ br − P[u ∈ (At+2 ∪ · · · ∪ At+r−1) \ At+1]
+≥ br − (pr − 2−d)
+≥ br/2.
+22
+
+Let X be the event that the states of vertices at the end of round t+1 are such that the conditional
+probability of u ∈ Bt+ℓ ∪ · · · ∪ Bt+r is at least br/4. Then
+P[{u ∈ (Bt+ℓ ∪ · · · ∪ Bt+r) ∩ At+1} ∩ X] ≥ br/2 − P[{u ∈ (Bt+ℓ ∪ · · · ∪ Bt+r) ∩ At+1} ∩ ¯
+X ]
+≥ br/4.
+We can now apply Lemma 15, starting from round t + 1 and using br/4 in place of br, to obtain
+P[{u ∈ (Ak
+t+1 ∪ · · · ∪ Ak
+t+r−1) ∩ At+1} ∩ X] ≥ r1−α · (br/64)α .
+This implies the lemma.
+A.4
+Proof of Lemma 18
+The proof of consists of a series of lemmas. In all these lemmas, the graph G = (V, E) considered
+is a random graph drawn from Gn,p.
+Property (P1) holds trivially for sets S of size k ≤ 4 ln n. The next lemma (applied for all
+k > 4 ln n, and then combining the results using a union bound) shows that G satisfies the property
+for all larger sets, with probability at least 1 − n−Ω(log n).
+Lemma 38. Let G = (V, E) be a random graph drawn from Gn,p, and let k ≥ 1. With probability
+at least 1 − n−k, all subgraphs of G on k vertices have at most max{4pk2, 2k ln n} edges.
+Proof. The probability there is a subgraph with k vertices and at least r = max{2k ln n, 4pk2}
+edges is at most
+�n
+k
+�
+·
+�k2/2
+r
+�
+· pr ≤ nk ·
+�ek2
+2r
+�r
+· pr = ek ln n−r ln
+2r
+epk2 ≤ ek ln n−2k ln n·ln 8pk2
+epk2 ≤ n−k.
+The next lemma shows that G satisfies property (P2) with probability 1 − n−Ω(log n/p)
+Lemma 39. Let G = (V, E) be a random graph drawn from Gn,p, and let k ≥ 40 ln(n)/p. With
+probability at least 1 − n−k, every set S ⊆ V of size |S| = k satisfies
+|{u ∈ V : |N(u) ∩ S)| < pk/2}| ≤ k/2.
+Proof. For any set S of size k, and any vertex u ∈ V \ S, the expected number of neighbors of u in
+S is pk. By a Chernoff bound, the probability that u has fewer than pk/2 neighbors in S is at most
+e−pk/8. Then the probability there is some set S of size k such that at least k/2 vertices u ∈ V \ S
+have fewer than pk/2 neighbors in S, is at most
+�n
+k
+�
+·
+�n − k
+k/2
+�
+· e−(k/2)·pk/8 ≤ nk · nk/2 · e−pk2/16 = e(3/2)k ln n−pk2/16 ≤ n−k.
+Lemma 40. Let G = (V, E) be a random graph drawn from Gn,p, and let k = 3 ln(n)/p. With
+probability at least 1 − n−k, every set S ⊆ V of size |S| ≥ k satisfies |V \ N +(S)| ≤ k.
+Proof. The probability there is a set S of size k with |V \ N +(S)| ≥ k is at most
+�n
+k
+�
+·
+�n − k
+k
+�
+· (1 − p)k2 ≤ nk · nk · e−pk2 = e2k ln n−pk2 = n−k.
+The next lemma shows that G satisfies property (P3) with probability 1 − n−Ω(log n/p).
+23
+
+Lemma 41. Let G = (V, E) be a random graph drawn from Gn,p. With probability at least 1 −
+n− ln(n)/p, every triplet of disjoint sets S, T, I ⊆ V , such that |S| ≥ 2|T| and (S ∪ T) ∩ N(I) = ∅,
+satisfies
+|N(T) \ N +(S ∪ I)| − |N(S) \ N +(I))| ≤ 8 ln2(n)/p.
+(7)
+Proof. From Lemma 40, with probability at least 1−n−3 ln(n)/p, all sets S, I ⊆ V such that |S∪I| ≥
+3 ln(n)/p satisfy |V \ N +(S ∪ I)| ≤ 3 ln(n)/p, and thus
+|N(T) \ N +(S ∪ I)| ≤ |V \ N +(S ∪ I)| ≤ 3 ln(n)/p,
+which implies (7).
+Next we assume that |S ∪ I| ≤ 3 ln(n)/p. Since |S| ≥ 2|S|, we have |S ∪ T ∪ I| ≤ 4.5 ln(n)/p,
+thus there are at most n4.5 ln(n)/p different triplets S, T, I. Choose one such triplet S, T, I, before
+revealing the edges of G. Then reveal the edges incident to vertices u ∈ I; this determines N(I).
+Let U = V \ (S ∪ T ∪ N +(I)).
+The two sets on the left side of (7) can then be expressed as
+N(T) \ N +(S ∪ I) = U ∩ N(T) \ N(S), and N(S) \ N +(I) = U ∩ N(S). For every u ∈ U, the
+probability that u ∈ N(T) \ N(S) is
+p1 = P[u ∈ N(T) \ N(S)] =
+�
+1 − (1 − p)|T|�
+· (1 − p)|S|,
+and the probability that u ∈ N(S) is
+p2 = P[u ∈ N(S)] = 1 − (1 − p)|S|.
+Letting ε = (1 − p)|T| and using that |S| ≥ 2|T|, we obtain p1 ≤ (1 − ε) · ε2 and p2 ≥ 1 − ε2. Thus
+p2
+p1
+≥
+1 − ε2
+(1 − ε) · ε2 = 1 + ε
+ε2
+≥ 2.
+It follows that, by considering all vertices u ∈ U one after the other, and revealing all edges incident
+to each u at the moment u is considered, we can analyze the difference
+D = |U ∩ N(T) \ N(S)| − |U ∩ N(S)| = |N(T) \ N +(S ∪ I)| − |N(S) \ N +(I)|
+as a biased random walk on the integers starting at 0, and moving to the right with probability
+p1 and to the left with probability p2.
+The probability that the (infinite) random walk every
+reaches value i ≥ 1 is know to be (p1/p2)i ≤ 2−i. Thus, P[D ≥ i] ≤ 2−i. And the probability that
+D ≥ 8 ln2(n)/p for at least one possible triplet S, T, I is then at most
+n4.5 ln(n)/p · 2−8 ln2(n)/p ≤ n−ln n/p.
+Property (P4) holds trivially for sets S of size k ≤ 6 ln n, since |S| ≥ |T|. The next lemma
+(applied for all k > 6 ln n) shows that G satisfies the property with probability at least 1−n−Ω(log n)
+for all larger sets.
+Lemma 42. Let G = (V, E) be a random graph drawn from Gn,p, and let k ≥ 1. With probability
+at least 1 − n−2k, every pair of disjoint sets S, T ⊆ V , such that |S| = k ≥ |T| and |T| ≤ ln(n)/p,
+satisfies |E(S, T)| ≤ 6k ln n.
+Proof. For any given pair S, T, the expected value of |E(S, T)| is p · |S| · |T| ≤ k ln n, and by a
+Chernoff bound, the probability that |E(S, T)| ≥ 6k ln n is at most 2−6k ln n. Then the probability
+there is at least one pair S, T such that |E(S, T)| ≥ 6k ln n is at most
+n|S| · n|T| · 2−6k ln n ≤ n2k · 2−6k ln n ≤ n−2k.
+24
+
+Our last lemma implies that properties (P5) and (P6) hold with probability 1 − O(n−2).
+Lemma 43. In a random graph G drawn from Gn,p, the probability that no two vertices have k
+common neighbors is at least 1 − n2 · (ep2n/k)k. And the probability that diam(G) ≤ 2 is at least
+1 − n2 · e−p2(n−1).
+Proof. The probability there is a pair of vertices that have at least k common neighbors is at most
+�n
+2
+�
+·
+�n−2
+k
+�
+· p2k ≤ n2 ·
+�
+ep2n
+k
+�k
+. And the probability there is a pair of vertices with no common
+neighbors and no adjacent to each other is
+�n
+2
+�
+· (1 − p) · (1 − p2)n−2 ≤ n2 · e−p2(n−1).
+B
+Other Related Work
+In 1985, Luby [24] proposed a simple distributed randomized algorithm that finds an MIS in
+time O(log n) w.h.p. Simultaneously, Alon et al. [3] proposed a similar algorithm with the same
+performance. Both algorithms work with O(log n)-bit messages and need access to O(log n) random
+bits at each round.
+Due to various applications in radio sensor networks, restricted distributed models of communi-
+cation were introduced, in which the MIS problem has been widely studied. In the beeping model,
+introduced by Cornejo and Kuhn [9], nodes have no knowledge of the local or global structure of
+the network, do not have access to synchronized clocks and the communication among nodes relies
+completely on carrier sensing (as described in the introduction). Afek et al. [1] show that in the
+version of the beeping model where nodes are initially asleep and are woken up by an adversary,
+it is not possible to locally converge to an MIS in sub-polynomial time. Therefore, they consider
+various relaxations on the model, providing algorithms converging to an MIS in a polylogarithmic
+number of rounds. In detail, if the nodes know an upper bound on the size of the network, or if
+the beeping nodes are awakened by the neighbor’s beep, the MIS can be found in time O(log3 n)
+w.h.p. If the nodes have synchronous clocks, an MIS can be found in time O(log2 n) w.h.p. We
+remark that the authors provide a self-stabilizing algorithm just in the first setting, i.e. when an
+upper bound on the size of the network is known by the nodes and that. In all algorithms, the
+nodes have super-constnt state and have access to a super-constant number of random bits.
+In the version of the beeping model with synchronized clocks, collision detection, and simul-
+taneous wakeup, Afek et al. [2] had earlier shown that the MIS problem is solved by a biological
+process in time O(log2 n) w.h.p. [1] showed that this bound is also achievable without knowledge of
+an upper bound on the size of the network. Jeavons et al. [23] improved these results, showing that
+an MIS can be found in time O(log n) w.h.p. An improved analyisis of the local complexity of this
+algorithm was provided by Ghaffari [16]. In the same version of the beeping model without collision
+detection, Holzer and Lynch in [21], proposed a variant of the algorithm of [15], and showed that it
+converges locally in time O((log ∆ + log 1/ε) · log 1/ε) with probability at least 1 − ε on a network
+with maximum degree ∆. All these algorithms require super constant space and random bits per
+round.
+Emek et al. [13] introduced the stone age model, inspired by biological cellular networks or
+networks of microprocessor devices. In this model, the nodes can communicate by transmitting
+messages belonging to a finite communication alphabet.
+The nodes communicate in an asyn-
+chronous environment, where the pattern is decided by an adversary, and they have no knowledge
+about the size of the network. In the stone age model, the MIS problem was considered by [13, 12].
+In [13], is provided an algorithm that compute a MIS in O(log2 n) rounds. However, it assumes
+that all the nodes have the same initial state, and therefore is not self-stabilizing. In [12], they
+25
+
+provided a self-stabilizing algorithm that stabilizes in time O((D + log n) log n) w.h.p., and the
+possible number of states of each node is O(D), where D is the diameter of the graph.
+In [25], the authors introduced a randomized distributed algorithm that finds an MIS in time
+O(log n) w.h.p. In particular, the algorithm is an adaptation of Luby’s algorithm so that messages
+of just 1 bit are used. They consider an anonymous network, but in their setting, the vertices can
+distinguish between their neighbors, and each vertex needs a number of states that depends on n
+and the node degree.
+MIS algorithm has also received a lot of attention from the Self-Stabilization community. For
+a survey of those algorithms see [19].
+We first cite here the self-stabilizing algorithm for non-anonymous networks, i.e. where vertices
+have IDs. In [18], the authors provide a simple deterministic distributed algorithm that stabilizes on
+an MIS in O(n) time and O(n2) moves (i.e. total number of state changed), in a synchronous model.
+In [22], the authors proposed a deterministic two-state algorithm that works under distributed
+scheduler (an adversary that, at each time, selects arbitrarily a set of processes to execute). Both
+algorithms stabilize in time O(n2). In [29], Turau introduces a 3-state self-stabilizing algorithm that
+stabilizes in O(n) moves, under a distributed scheduler. A breakthrough was achieved by Barenboim
+et al. [5], who proposed a self-stabilizing algorithm for the MIS and other related problems, in the
+synchronous model. They prove that the algorithm stabilizes after O(∆ + log∗ n) rounds.
+Assuming anonymous networks Shukla et al. [28] proposed two deterministic two-state self-
+stabilizing algorithms, that work under a centralized scheduler (an adversary that selects one process
+to execute at each round) and stabilizes in O(n) rounds. In [30], Turau introduced a synchronous
+randomized self-stabilizing algorithm for MIS that stabilizes w.h.p. in O(log n) rounds w.h.p. The
+possible states of the nodes are O(log n).
+Next, we briefly summarize the best known upper bounds to compute an MIS in the distributed
+LOCAL model on arbitrary graphs. Barenboim et al. [6] proved that an MIS can be computed
+with a distributed deterministic algorithm in O(∆ + log∗ n) rounds and Ghaffari et al. [17] provide
+an upper bound of O(log5 n). Regarding distributed randomized algorithms, Ghaffari [15] provides
+an upper bound of O(log ∆) + 2O(√log log n) w.h.p., which, thanks to [27, 17], was improved to
+O(log ∆ + log5 log n) w.h.p. See also [14].
+The current best-known lower bound for finding an MIS is proved by Balliu et al. [4], who
+show that computing an MIS in the LOCAL model requires Ω(min{∆, log n/ log log n}) rounds
+deterministically, and Ω(min{∆, log log n/ log log log n}) rounds with a randomized algorithm.
+References
+[1] Yehuda Afek, Noga Alon, Ziv Bar-Joseph, Alejandro Cornejo, Bernhard Haeupler, and Fabian Kuhn.
+Beeping a maximal independent set. Distributed Comput., 26(4):195–208, 2013.
+[2] Yehuda Afek, Noga Alon, Omer Barad, Eran Hornstein, Naama Barkai, and Ziv Bar-Joseph. A biological
+solution to a fundamental distributed computing problem. Science, 331(6014):183—185, January 2011.
+[3] Noga Alon, L´aszl´o Babai, and Alon Itai.
+A fast and simple randomized parallel algorithm for the
+maximal independent set problem. J. Algorithms, 7(4):567–583, 1986.
+[4] Alkida Balliu, Sebastian Brandt, Juho Hirvonen, Dennis Olivetti, Mika¨el Rabie, and Jukka Suomela.
+Lower bounds for maximal matchings and maximal independent sets. J. ACM, 68(5):39:1–39:30, 2021.
+[5] Leonid Barenboim, Michael Elkin, and Uri Goldenberg. Locally-Iterative distributed (∆ + 1)-coloring
+and applications. J. ACM, 69(1):5:1–5:26, 2022.
+[6] Leonid Barenboim, Michael Elkin, and Fabian Kuhn. Distributed (∆+1)-coloring in linear (in ∆) time.
+SIAM J. Comput., 43(1):72–95, 2014.
+26
+
+[7] Leonid Barenboim, Michael Elkin, Seth Pettie, and Johannes Schneider. The locality of distributed
+symmetry breaking. J. ACM, 63(3):20:1–20:45, 2016.
+[8] Stephen A. Cook. An overview of computational complexity. Commun. ACM, 26(6):400–408, 1983.
+[9] Alejandro Cornejo and Fabian Kuhn. Deploying wireless networks with beeps. In Proc. Distributed
+Computing, 24th International Symposium, DISC, pages 148–162, 2010.
+[10] Edsger W. Dijkstra. Self-stabilizing systems in spite of distributed control. Commun. ACM, 17(11):643–
+644, 1974.
+[11] Shlomi Dolev. Self-Stabilization. MIT Press, 2000.
+[12] Yuval Emek and Eyal Keren. A thin self-stabilizing asynchronous unison algorithm with applications to
+fault tolerant biological networks. In Proc. ACM Symposium on Principles of Distributed Computing,
+PODC, pages 93–102, 2021.
+[13] Yuval Emek and Roger Wattenhofer. Stone age distributed computing. In Proc. ACM Symposium on
+Principles of Distributed Computing, PODC, pages 137–146, 2013.
+[14] Salwa Faour, Mohsen Ghaffari, Christoph Grunau, Fabian Kuhn, and V´aclav Rozhon. Local distributed
+rounding: Generalized to mis, matching, set cover, and beyond. CoRR, abs/2209.11651, 2022.
+[15] Mohsen Ghaffari. An improved distributed algorithm for maximal independent set. In Proc. Twenty-
+Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 270–277, 2016.
+[16] Mohsen Ghaffari. Distributed MIS via all-to-all communication. In Proc. ACM Symposium on Principles
+of Distributed Computing, PODC, pages 141–149, 2017.
+[17] Mohsen Ghaffari, Christoph Grunau, and V´aclav Rozhon. Improved deterministic network decomposi-
+tion. In Proc. ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 2904–2923, 2021.
+[18] Wayne Goddard, Stephen T. Hedetniemi, David Pokrass Jacobs, and Pradip K. Srimani. Self-stabilizing
+protocols for maximal matching and maximal independent sets for ad hoc networks. In Proc. 17th
+International Parallel and Distributed Processing Symposium (IPDPS 2003), page 162, 2003.
+[19] Nabil Guellati and Hamamache Kheddouci. A survey on self-stabilizing algorithms for independence,
+domination, coloring, and matching in graphs. J. Parallel Distributed Comput., 70(4):406–415, 2010.
+[20] S.M. Hedetniemi, S.T. Hedetniemi, D.P. Jacobs, and P.K. Srimani.
+Self-stabilizing algorithms for
+minimal dominating sets and maximal independent sets. Computers & Mathematics with Applications,
+46(5):805–811, 2003.
+[21] Stephan Holzer and Nancy A. Lynch. Beeping a maximal independent set fast. CoRR, abs/1704.07133,
+2017.
+[22] Michiyo Ikeda, Sayaka Kamei, and Hirotsugu Kakugawa. A space-optimal self-stabilizing algorithm for
+the maximal independent set problem. In Proc. 3rd International Conference on Parallel and Distributed
+Computing, Applications and Technologies, PDCAT, pages 70–74, 2002.
+[23] Peter Jeavons, Alex Scott, and Lei Xu. Feedback from nature: Simple randomised distributed algorithms
+for maximal independent set selection and greedy colouring. Distributed Comput., 29(5):377–393, 2016.
+[24] Michael Luby. A simple parallel algorithm for the maximal independent set problem. SIAM J. Comput.,
+15(4):1036–1053, 1986.
+[25] Yves M´etivier, John Michael Robson, Nasser Saheb-Djahromi, and Akka Zemmari. An optimal bit
+complexity randomized distributed MIS algorithm. Distributed Comput., 23(5-6):331–340, 2011.
+[26] C. St.J. A. Nash-Williams. Decomposition of finite graphs into forests. J. Lond. Math. Soc., s1-39(1):12–
+12, 1964.
+[27] V´aclav Rozhon and Mohsen Ghaffari. Polylogarithmic-time deterministic network decomposition and
+distributed derandomization. In Proc. 52nd Annual ACM SIGACT Symposium on Theory of Comput-
+ing, STOC, pages 350–363, 2020.
+27
+
+[28] Sandeep K. Shukla, Daniel J. Rosenkrantz, and Sekharipuram S. Ravi. Observations on self-stabilizing
+graph algorithms for anonymous networks. In Proc. 2nd Workshop on Self-Stabilizing Systems, SSS,
+1995.
+[29] Volker Turau. Linear self-stabilizing algorithms for the independent and dominating set problems using
+an unfair distributed scheduler. Inf. Process. Lett., 103(3):88–93, 2007.
+[30] Volker Turau.
+Making randomized algorithms self-stabilizing.
+In Proc. Structural Information and
+Communication Complexity - 26th International Colloquium, SIROCCO, pages 309–324, 2019.
+[31] Volker Turau and Christoph Weyer. Randomized self-stabilizing algorithms for wireless sensor net-
+works. In Proc. Self-Organizing Systems, First International Workshop, IWSOS, and Third Interna-
+tional Workshop on New Trends in Network Architectures and Services, EuroNGI, pages 74–89, 2006.
+[32] Leslie G. Valiant. Parallel computation. In Proc. 7th IBM Symposium on Mathematical Foundations of
+Computer Science, 1982.
+28
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+ARiADNE: A Reinforcement learning approach using Attention-based
+Deep Networks for Exploration
+Yuhong Cao1, Tianxiang Hou1, Yizhuo Wang1, Xian Yi1, Guillaume Sartoretti1†
+Abstract— In autonomous robot exploration tasks, a mobile
+robot needs to actively explore and map an unknown envi-
+ronment as fast as possible. Since the environment is being
+revealed during exploration, the robot needs to frequently
+re-plan its path online, as new information is acquired by
+onboard sensors and used to update its partial map. While
+state-of-the-art exploration planners are frontier- and sampling-
+based, encouraged by the recent development in deep reinforce-
+ment learning (DRL), we propose ARiADNE, an attention-
+based neural approach to obtain real-time, non-myopic path
+planning for autonomous exploration. ARiADNE is able to
+learn dependencies at multiple spatial scales between areas of
+the agent’s partial map, and implicitly predict potential gains
+associated with exploring those areas. This allows the agent
+to sequence movement actions that balance the natural trade-
+off between exploitation/refinement of the map in known areas
+and exploration of new areas. We experimentally demonstrate
+that our method outperforms both learning and non-learning
+state-of-the-art baselines in terms of average trajectory length
+to complete exploration in hundreds of simplified 2D indoor
+scenarios. We further validate our approach in high-fidelity
+Robot Operating System (ROS) simulations, where we consider
+a real sensor model and a realistic low-level motion controller,
+toward deployment on real robots.
+I. INTRODUCTION
+Autonomous robot exploration (ARE) refers to the task,
+in which a robot needs to autonomously explore and map an
+unknown environment as efficiently and quickly as possible.
+The robot is usually equipped with a sensor (e.g., LiDAR
+or camera) to obtain measurements of its surroundings and
+build/update a partial map of the environment. In practice,
+the (high-dimensional) collected sensor data (e.g., point
+cloud) is usually converted into a (simplified) occupancy grid
+map or Octomap [1] that can be used for further planning [2],
+[3], [4]. Such a task is also known as active SLAM [2].
+Although a robot can always slowly but accurately construct
+a map by carefully methodically covering the entire environ-
+ment, the objective of ARE is to plan the shortest path to
+complete exploration, where small noises/errors in the final
+map are tolerable. In consequence, the main challenge of
+ARE is to plan a non-myopic path that balances the trade-
+off between exploiting surroundings (i.e., refining the map in
+already-explored areas) and exploring new (usually, further
+away) areas, most importantly with only partial knowledge of
+the environment. Such an exploration path is usually planned
+† Corresponding author, to whom correspondence should be addressed.
+1
+Authors are with the Department of Mechanical Engineering,
+College of Design and Engineering, National University of Singapore.
+caoyuhong@u.nus.edu, htx24@foxmail.com,
+{wy98,yxian11}@u.nus.edu, mpegas@nus.edu.sg
+Fig. 1: Illustration of autonomous robot exploration. A
+wheeled robot is building a 3D Octomap using an onboard
+LiDAR. The purple ball indicates the next viewpoint output
+by our planner, which is tracked by the robot using a low-
+level motion controller (yellow primitives).
+incrementally online, as the partial map is updated using new
+measurements along the way.
+For example, conventional frontier-based methods [3], [5],
+[6], [7] generate multiple candidate paths, each covering
+a frontier (i.e., the boundary between explored free area
+and unexplored area), and greedily select the path with
+maximum gain, usually defined as a combination of utility
+(i.e., number of observable frontiers along the path) and
+cost (i.e., path length). However, an essential problem of
+these methods is that such myopic frontier selection does
+not guarantee optimality in the long term. A more recent
+approach [4] reasons about the whole path to cover all
+current frontiers, thus guaranteeing (near-)optimal paths in
+the current partial map. However, since the environment is
+only partially known, a previous optimal path often quickly
+becomes sub-optimal as more of the environment is revealed,
+or even worse, results in redundant movements (e.g., missing
+an unexplored shortcut between two rooms which were
+previously not known to be connected). Based on experi-
+ences from conventional exploration planners, we note that
+non-myopicity comes at two different levels in autonomous
+exploration. The first level, spatial non-myopicity, requires
+the planner to reason about the current partial map to balance
+the exploration-exploitation trade-off, while temporal non-
+myopicity requires the planner to estimate the future influence
+of current decisions (e.g., predict the changes in the partial
+map that may stem from given path planning decisions).
+To achieve non-myopicity at these two levels, we propose
+a deep reinforcement learning (DRL) based approach for
+ARE, named ARiADNE, which relies on two attention-based
+neural networks. These networks allow the agent to reason
+arXiv:2301.11575v1  [cs.RO]  27 Jan 2023
+
+about dependencies of different areas in the partial map at
+different spatial scales, thus allowing the agent to sequence
+spatially non-myopic decisions efficiently without the need
+to optimize a long path. Furthermore, our critic network
+implicitly provides the robot with the ability to estimate
+potential areas that might be found by learning the state-
+action value, which furthermore helps make decisions bene-
+ficial to the long-term efficiency, thus addressing temporally
+non-myopicity. Specifically, we first formulate autonomous
+exploration as a sequential decision-making problem on a
+collision-free graph that covers the known traversable area,
+where one of the nodes is the robot’s current position.
+We then use our attention-based neural network to select
+one neighboring node of the current robot position as the
+next viewpoint for the robot. In this work, we focus on
+training and testing our approach in indoor environments
+based on 2D occupancy grid maps, ranging from simple
+scenarios (single room) to relatively complex ones (multiple
+rooms with complicated corridors). There, we experimentally
+demonstrate that our approach outperforms state-of-the-art
+conventional methods on average. We also validate our
+approach in high-fidelity ROS (Robot Operating System)
+simulations, where we consider a real sensor model and a
+motion controller, showing the generalizability of our model
+to realistic environments.
+II. RELATED WORK
+Frontier-based vs sampling-based approaches The
+first
+frontier-based method was proposed by Yamauchi [5], in
+which the robot is constantly driven towards the nearest
+frontier. In more advanced frontier-based methods, selections
+of which frontier to visit are evaluated by a gain function,
+which considers the effect of both utility and cost [6],
+[8], [9]. On the other hand, the past few years have
+seen a number of sampling-based methods be developed,
+based on Rapidly-exploring Random Trees (RRT) [3],
+Rapidly-exploring Random Graphs [10], and Probabilistic
+Random Maps (PRM) [11]. Sampling-based methods only
+need to compute the gain of sampled paths, which avoids
+the complexity of identifying and evaluating all frontiers.
+Recent works demonstrated that frontier-based methods are
+more suitable when frontiers are sparse (e.g., 2D indoor
+scenarios), while sampling-based methods perform better
+with dense frontiers (e.g., 3D outdoor scenarios) [7], [11].
+Intuitively, Frontier-based methods become inefficient when
+there are too many frontiers to evaluate, while sampling-
+based methods underperform when informative paths are
+hard to sample.
+Planning for long-term objectives Both frontier-based and
+sampling-based methods mostly rely on greedy strategies
+to plan short-term paths. Since the robot only has access
+to a partial map of the environment, such paths inevitably
+lead to myopic performance in the long term. Recently, Cao
+et al. [4] proposed TARE to optimize the full exploration
+path and mitigate myopicity. Utilizing the full knowledge
+of the current partial map, TARE was shown to significantly
+outperform state-of-the-art sampling-based planners in large-
+scale 3D scenarios. Similar methods were also considered
+to approach the informative path planning [12] problem,
+which considers information gathering usually in obstacle-
+free environments (in fact, works on autonomous exploration
+and informative path planning mutually promote each other,
+e.g., [4] and [12], [13] and [3]).
+Learning-based exploration Niroui et al. [14] first com-
+bined frontier-based method with deep reinforcement learn-
+ing to adaptively tune the parameter of the gain function
+for frontier selections and improve performance. Schmid et
+al. [15] proposed to learn the underlying gain distribution
+based on the partial map by supervised learning, thus help-
+ing sampling-based methods more efficiently find next-best-
+views and reduce computation. While the above works focus
+on improving conventional methods using machine learning,
+some other works [16], [17], [18], [19] directly applied deep
+reinforcement learning to select a viewpoint to visit, often
+relying on convolutional neural networks (CNNs). However,
+although [16], [18] argue that DRL-based methods naturally
+optimize long-term objectives, it seems that [16], [17], [18],
+[19] are only able to reach performance slightly better than
+the nearest-frontier method so far.
+III. PROBLEM FORMULATION
+We consider a bounded and unknown environment E
+represented by a x × y 2D occupancy grid map, whose
+partial (occupancy grid) map is denoted as P. The partial
+map consists of unknown area Pu (i.e., unexplored area) and
+known area Pk (i.e., explored area), such that Pu ∪ Pk = P.
+The known area Pk is further classified into free area Pf
+(i.e., traversable area for the robot) and occupied area Po
+(i.e., obstacles) such that Pf ∪ Po = Pk. At the beginning
+of exploration, the environment is fully unknown so the
+partial map P = Pu. Then, during exploration, the unknown
+area in the sensor range ds (the sensor we use is a 360-
+degree LiDAR) is classified into either free area or occupied
+area according to sensor measurements. The objective of
+autonomous exploration is to find the shortest collision-free
+robot trajectory ψ∗ to complete exploration:
+ψ∗ = argmin
+ψ∈Ψ
+C(ψ), s.t. Pk = Pg,
+(1)
+where C : ψ −→ R+ maps a trajectory to its length and Pg
+denotes the ground truth of the environment. Although the
+ground truth is not accessible in real-world deployments, it
+is known and can be utilized to evaluate the performance
+of planners in testing. In practice, most works consider the
+closure of occupied areas as Pk = Pg [5], [4], [18], [19].
+IV. METHOD
+In this section, we cast ARE as an RL problem, and in-
+troduce our attention-based policy and critic neural networks
+as well as details of our training.
+A. Exploration as an RL Problem
+Sequential Decision-making Problem Since the free area
+is updated based on the robot’s movements, online planning
+for ARE is a sequential decision-making problem in nature.
+
+Following our previous work [20] for informative path plan-
+ning, we consider the robot trajectory ψ as a sequence of
+viewpoints ψ = (ψ0, ψ1, ...), ψi ∈ Pf. At each decision
+step t, we first uniformly distribute candidate viewpoints
+Vt = {v0, v1, ...}, ∀ vi = (xi, yi) ∈ Pf in the current free
+area Pf, similar to [4]. Then, to find collision-free paths
+between viewpoints, we connect each viewpoint with its k
+nearest neighbors through a straight line and remove edges
+that collide with occupied or unknown areas. In doing so,
+we build a collision-free graph Gt = (Vt, Et), with Vt a
+set of uniformly distributed nodes (i.e., viewpoints) over the
+free area, and Et a set of traversable edges. We finally let
+the robot select one neighboring node of its current position
+ψt as the next viewpoint. Since the decision will be taken
+upon arriving at the last selected viewpoint, the trajectory is
+a sequence of waypoints such that ψi ∈ V .
+Observation The observation of the agent is ot = (G′
+t, ψt),
+where G′
+t = (V ′
+t , Et) is the augmented graph based on the
+current collision-free graph Gt, while ψt is the robot current
+position. Note that G′
+t shares the same edge set Et as Gt.
+In addition to the node coordinates (i.e., vi = (xi, yi)),
+The properties of each node v′
+i in the augmented graph
+further include a binary signal bi, which indicates if the node
+has been visited by the agent already, and the associated
+utility ui, such that v′
+i = (xi, yi, ui, bi). We experimentally
+found that the binary signal helps improve the learning by
+allowing the robot to be aware of its previous movements.
+The utility ui represents the number of observable frontiers
+at node vi [4]. We consider observable frontiers as frontiers
+within light of sight of the node (i.e., lines between the node
+and observable frontiers are collision-free and their length
+is smaller than the sensor range). The utility ui at node
+vi is computed as ui = |Fo,i|, ∀fj ∈ Fo,i, ||fj − vi|| ≤
+ds, L(vi, fj) ∩ (P − Pf) = ∅, where Fo,i denotes the
+observable frontiers set at node vi, ds denotes the sensor
+range and L(vi, fj) the line between node vi and frontier
+fj. In practice, we scale the node coordinates and utility to
+[0, 1] before feeding the observation into the neural network.
+Action At each decision step t, given the agent’s observation
+ot, our attention-based neural network outputs a stochastic
+policy to select a node out of all neighboring nodes as the
+next viewpoint to visit. The policy is denoted as πθ(at|ot) =
+πθ(ψt+1 = vi, (ψt, vi) ∈ Et | ot), where θ represents the set
+of weights of the neural network. The robot moves to the
+next viewpoint in a straight line, and updates its partial map
+based on data collected along the way.
+Reward To encourage efficient exploration, after taking
+each movement action at, the robot receives a reward
+composed of three parts. The first part ro = |Fo,ψt+1| is
+the number of observed frontiers at the new viewpoint.
+The second part rc = −C(ψt, ψt+1) is a punishment on
+the distance between the previous and new viewpoints. A
+fixed finishing reward rf =
+� 20,
+Pk = Pg
+0,
+otherwise,
+is given
+at the end of the episode, if and only if the exploration
+task was completed. The total reward reads: rt(ot, at) =
+a · ro + b · rc + rf, where a and b are scaling parameters (in
+Fig. 2: Example decision step in the middle of an
+exploration task in our approach, showing the unknown
+area (grey cells), free area (white cells), occupied area (black
+cells), frontiers (red cells), executed trajectory (blue line),
+graph edges (tan lines), candidate viewpoints (small dots,
+whose color represents their utility), robot current position
+(purple disk), and robot starting position (light blue disk).
+practice a = 1/50, b = 1/64).
+B. Policy Network
+The policy ψθ is output by our attention-based neural
+network, which is composed of an encoder and a decoder
+(shown in Fig. 3). We first rely on the encoder to extract
+salient features from the current partial map, specifically
+by learning dependencies between nodes in the associated
+augmented graph G′. Based on these features as well as the
+current robot position, the decoder then outputs the policy
+over neighboring nodes, which can be used to decide which
+one to visit next. Note that, while policy-based RL agents
+often have a fixed action space, our decoder is inspired by the
+Pointer Network [22] to allow the action space to depend on
+the number of neighboring nodes input in the network. This
+allows our network to naturally adapt to our collision-free
+graph, where nodes have arbitrary numbers of neighbors.
+Attention Layer We use the attention layer [21] as the
+fundamental building block in our model. The input of such
+an attention layer is composed of a query vector hq and a
+key-and-value vector hk,v. The output of this layer, h′
+i, is the
+weighted sum of the value vector, where weights depend on
+the similarity between key and query:
+qi = W Qhq
+i , ki = W Khk,v
+i
+, vi = W V hk,v
+i
+,
+uij = qT
+i · kj
+√
+d
+, wij =
+euij
+�n
+j=1 euij , h′
+i =
+n
+�
+j=1
+wijvj,
+(2)
+where W Q, W K, W V ∈ Rd×d are all learnable matrices.
+Updated features are then passed through a feed-forward
+sublayer, following [21].
+Encoder In the encoder, we first linearly embed the node
+inputs V ′ into d-dimensional node features hn, where hn
+i =
+W lv′
+i +bl. We then calculate an edge mask M where mij =
+� 0, (vi, vj) ∈ Et
+1, (vi, vj) /∈ Et . The node features are then passed to
+
+Node Features
+Enhanced Node Features
+Enhanced Current Node Features
+Neighboring  Features
+Encoder
+Partial Map
+Policy
+Action
+Decoder
+Filter Neighboring Feature
+Augmented Graph
+Construct Enhanced 
+Current Node Feature
+Fig. 3: Attention-based policy network. Note that neighboring relationships in the augmented graph (tan) are also used as
+the mask [21] in attention layers in the encoder.
+multiple (6 in practice) stacked attention layers, where hq =
+hk,v = hn, each attention layer taking the output of the
+previous one as input. An edge mask is applied to allow each
+node access to its neighboring node features only, by setting
+wij = 0, ∀(i, j), mij = 1. Despite attention being restricted
+to neighboring nodes in each layer, nodes can still obtain
+non-neighboring node features by aggregating node features
+multiple times through this stacked attention structure. We
+empirically found that such structure is more suitable than
+graph transformers [23] (like in our previous work [20]) to
+learn path finding in maps with cluttered obstacles. We term
+the output of the encoder, ˆhe, the enhanced node features,
+since each of these updated node features ˆhn
+i contains the
+dependencies of v′
+i with other nodes.
+Decoder We use the decoder to output a policy based on
+enhanced node features ˆhe and the current robot position ψt.
+Denoting the current robot position as node vc = ψt, we first
+select the current node features hc and neighboring features
+hnb, ∀ˆhnb
+i , (vc, vi) ∈ Et from the corresponding enhanced
+node features. We then pass the current node features and
+enhanced node features to an attention layer, where hq =
+hc, hk,v = ˆhn, concatenate its output with hc, and project it
+back to a d-dimensional feature vector. We term this vector
+the enhanced current node features ˆhc. After that, we pass
+the enhanced current node features and neighboring features
+to a pointer layer [22], an attention layer directly outputting
+the attention weights w as the output with hq = ˆhc, hk,v =
+hnb. We finally take the output of this pointer layer as the
+robot’s policy, i.e., πθ(at | ot) = wi.
+C. Critic Network
+We train the policy network using the soft actor critic
+(SAC) algorithm [24], [25] (see details below), where a critic
+network is trained to predict state-action values. Since state-
+action values approximate long-term returns (the accumu-
+lated sum of rewards), we believe that they also implicitly
+predict potential gains (i.e., potential areas that might be
+found), which further helps the robot sequence non-myopic
+decisions. In practice, we train a critic network to approx-
+imate soft state-action values Qφ(ot, at), where φ denotes
+the set of weights of the critic network. The structure of
+the critic network is nearly the same as the policy network,
+except that there is no pointer layer at the end of the decoder.
+Instead, we directly concatenate the enhanced current node
+features and neighboring features, then project them to soft
+state-action values.
+D. Training
+Soft Actor-critic SAC aims to learn a policy that maximizes
+return while keeping its entropy as high as possible:
+π∗ = argmax E(ot,at)[
+T
+�
+t=0
+γt(rt + αH(π(.|ot)))],
+(3)
+where π∗ is the optimal policy, T the number of decision
+steps, γ the discount factor, and α the temperature parameter
+that tunes the importance of the entropy term versus the
+return. In SAC, the soft state value is calculated as: V (ot) =
+Eat[Q(ot, at)] − αlog(π(at|ot)).
+The critic loss is calculated as: JQ(φ) = Eot[ 1
+2(Qφ(ot, at)−
+(rt + γEot+1[V (ot+1)]))2].
+The
+policy
+loss
+loss
+is
+calculated
+as:
+Jπ(θ)
+=
+E(ot,at)[αlog(πθ(at|ot)) − Qφ(ot, at)].
+The temperature parameter is auto-tuned during the train-
+ing and the temperature loss is calculated as: J(α) =
+Eat[−α(logπt(at|ot) + H)],
+where H denotes the target entropy. In practice, we use
+double target networks for the critic network training, as
+in [24], [25].
+Training Details We utilize the same environments pro-
+vided in [18] for training, which are generated by a random
+dungeon generator. Each environment is a 640 × 480 grid
+map, while the sensor range ds = 80. To build the collision-
+free graph, 900 points are uniformly distributed to cover
+the whole environment, with all points in the known free
+area considered as candidate viewpoints V . We check the
+k = 20 nearest neighbor of each viewpoint, and connect
+them if such an edge is collision-free, to form the edge
+set E. We consider the exploration task to be completed
+once more than 99% of the ground truth has been explored
+(|Pk|/|Pg| > 0.99). During training, we set the max episode
+length to 128 decision steps, the discount factor to γ = 1,
+the batch size to 256, and the episode buffer size to 10, 000.
+Training starts after the episode buffer collects more than
+2000 steps data. The target entropy is set to 0.01 · log(k).
+Each training step contains 1 iteration and happens after 1
+episode finishes. We use the Adam optimizer with a learning
+
+TABLE I: Comparison with baseline ARE planners (100 scenarios for each test set). We report the average and standard
+deviation of the trajectory length to complete exploration (lower is better). For utility-based methods [6], the numbers 1, 10,
+25 represent the value of λ, which is used to tune exploitation and exploration.
+Nearest
+Utility 1
+Utility 10
+Utility 25
+NBVP
+TARE Local
+CNN
+ARiADNE
+easy
+772(±253)
+736(±266)
+732(±256)
+764(±258)
+745(±268)
+692(±228)
+779(±281)
+663(±257)
+medium
+1248(±295)
+1266(±311)
+1179(±300)
+1227(±307)
+1217(±271)
+1170(±275)
+1169(±319)
+1130(±334)
+complex
+1669(±332)
+1873(±457)
+1662(±347)
+1711(±352)
+1744(±366)
+1646(±312)
+1647(±422)
+1599(±363)
+random
+1354(±410)
+1423(±466)
+1268(±396)
+1315(±413)
+1323(±371)
+1266(±388)
+1323(±428)
+1204(±378)
+(a) simple
+(b) medium
+(c) complex
+Fig. 4: Examples scenarios from each different test set.
+rate of 10−5 for both policy and critic networks. The target
+critic network updates every 256 training steps. Our model is
+trained on a workstation equipped with a i9-10980XE CPU
+and an NVIDIA GeForce RTX 3090 GPU. We train our
+model utilizing Ray, a distributed framework for machine
+learning [26], to parallelize and accelerate data collection
+(32 instances in practice). The training needs around 24h to
+converge. We will release our full code upon acceptance.
+V. EXPERIMENT
+A. Comparison Analysis
+Most previous works often only conduct experiments in
+a few scenarios (often less than 10). However, we note
+that the performance of exploration planners exhibits high
+variance in different scenarios. Therefore, we believe a
+convincing comparison should be based on evaluation in
+a large number of testing environments. Although building
+so many testing environments is tricky and time-consuming
+even in ROS, hundreds of simplified scenarios, like the ones
+we used for training, can be generated easily. Therefore, we
+conduct comparison analyses on a fixed set of simplified
+environments, which were never seen by our trained model.
+Testing environments are divided in four sets (100 scenarios
+each), named random, easy, medium, and complex. Easy
+scenarios only contain one room, and complex scenarios
+contain multiple rooms with complicated corridors, while the
+complexity of medium scenarios lies in-between. Random
+scenarios contain a mix of easy, medium, and complex
+scenarios (but no repeated scenario from these test sets).
+We compare ARiADNE with state-of-the-art conventional
+planners, including Nearest Frontier [5], Utility-based Fron-
+tier [6], NBVP [3], and TARE Local [4]. Nearest Frontier
+always drives the robot towards the nearest frontier, while
+Utility-based Frontier evaluates the gain of each frontier
+gi = ui · e−λ·C(ψi) and drives the robot to the frontier with
+the highest gain, where ui is the utility of frontier i, ψi the
+shortest path from the robot’s current position to frontier i,
+and λ a tunable parameter used to balance exploration and
+exploitation. The same function is also used in NBVP to
+evaluate sampled trajectories. We tried a series of values of λ
+for Utility-based Frontier and NBVP, and found that λ = 10
+(a) Trajectory Analysis
+(b) ARiADNE (1618)
+(c) TARE Local (1703)
+(d) NBVP (1922)
+(e) Utility 10 (1793)
+Fig. 5: Visual comparison of our method and baselines
+in an example scenario.
+generally performs best (see Table I). Finally, TARE Local
+refers to the local planner of TARE [4], which explicitly
+plans a full trajectory to cover all frontiers (we do not
+use TARE’s global planner, since its local planning horizon
+already fits our testing environments). NBVP and TARE run
+300 and 10 iterations for each decision step respectively
+(15 and 1 in default [3], [4]), to make their decisions as
+optimal as possible. In our tests, we adopt our collision-
+free graph as the trajectory space for all baselines except
+NBVP (we found RRTs mostly generate poor zig-zag paths
+due to symmetries in our uniform graph), to alleviate the
+randomness of sampling and ensure a fair comparison. We
+further compare against a CNN-based DRL planner [18].
+Since this CNN-based planner has a fixed observation range,
+it only has a partial observation of the (partial) map, and
+relies on a frontier-based method for exploration when there
+is no nearby frontier in its field-of-view.
+We report the average and variance of the total trajectory
+length to complete exploration in Table I. Our results indicate
+that ARiADNE outperforms all baselines on average, in all
+test sets. We do not report the planning time of baseline
+methods in Table I, since we focused on implementing fun-
+damental inner workings of the baselines, without perfectly
+optimizing their computing time. In addition, we observed
+that the utility/gain computation generally takes 90% of the
+planning time for conventional methods in practice, while
+its computing time is determined by the resolution of the
+map. Therefore, computing times vary greatly in different
+exploration scenarios. Despite this, we note that our method
+can be used in real-time. Under our exploration setting, our
+
+Fig. 6: Attention weights visualization of the critic net-
+work decoder. The query vector is the node at the current
+robot’s position (purple) and the keys vector are nodes in the
+augmented graph (blue). Note how the different heads of the
+decoder learn to focus on either local or global dependencies
+of areas in the partial map.
+method takes 0.7s for the observation generation on average
+(utility computation and graph building) and less than 0.02s
+for the neural network inference on a i9-10980XE CPU.
+As discussed in the related work section, the best-tuned
+frontier-based method (Utility 10) performs well in 2D ex-
+ploration tasks (better than NBVP). Despite this, since these
+frontier-based methods are myopic, they are outperformed by
+TARE Local, which plans near-optimal long-term (full) tra-
+jectories on the current partial map. While it only constructs
+paths one viewpoint at a time, our learning-based method
+can not only reason about the whole partial map to construct
+efficient, non-myopic exploration trajectories, while learning
+to predict the potential long-term gain of decisions. We
+believe such an advantage results in the improvement of our
+method over conventional baselines (5% better than TARE
+Local in our random scenarios). Fig. 5 shows an example
+where ARiADNE plans a more efficient trajectory, while
+conventional methods suffer from redundant movements.
+However, it should be noted that considering long-term paths
+and predicting potential gains do not strictly guarantee better
+performance in every scenario (e.g., predictions could be
+wrong). In fact, ARiADNE plans the shortest path for 33
+scenarios in our random tests, while TARE Local, NBVP,
+Utility 10 perform best in 23, 21, 23 scenarios respectively.
+Finally, ARiADNE also outperforms the CNN-based plan-
+ner. We believe that our main advantage stems from the
+attention-based neural network, which efficiently learns fea-
+tures at different scales (as shown in Fig. 6, different heads
+of the decoder learn to focus on either local or global
+dependencies), while CNNs naturally only focus on local
+dependencies. Therefore, our model can better learn depen-
+dencies between different areas to reason about the entire
+partial map and avoid myopic decisions.
+B. Experimental validation
+We validate ARiADNE in a simulation environment for
+exploration provided by [27]. It contains fundamental mod-
+ules (e.g., state estimation and motion control), which allow
+us to consider a real sensor model and a low-level motion
+(a) Ground truth
+(b) Constructed Octomap
+(c) Constructed occupancy grid map
+Fig. 7: Validation of our method in simulation. Note that
+ignoring the small left-down corner is actually a wise deci-
+sion since the objective is to explore 99% of the environment.
+controller. The validation is conducted in a realistic indoor
+environment (approximately 70m × 40m) with long and
+narrow corridors connected with tables, colums, and lobby
+areas (see Fig. 7(a)). We use a wheeled robot equipped with
+a 3D Velodyne Lidar with a 130m sensor range. We convert
+collected data into an Octomap (see Fig. 7(b)) and then
+project it to a occupancy grid map for exploration planning
+(see Fig. 7(c)). The resolution of the grid map is 0.2m.
+We re-plan the path every 0.2s. Although the sensor model
+(i.e., sensor range and sensing frequency) of the robot is
+drastically different from the one used in training, our trained
+model still makes efficient decisions to avoid redundant
+movements for exploration (see the colored trajectory in
+Fig. 7(c), highlighting the generalizability of our approach.
+VI. CONCLUSION
+In this work, we propose ARiADNE, a reinforcement
+learning approach that relies on attention-based deep neural
+network for autonomous exploration. Our approach allows
+the robot to efficiently learn dependencies between different
+areas in its partial map and implicitly predict potential gains,
+thus allowing it to sequence non-myopic movement decisions
+in partially-known environments. In our tests, ARiADNE
+exhibits improvement over state-of-the-art frontier-based,
+sampling-based, and CNN-based exploration planners, in
+terms of average trajectory length to complete exploration.
+We also validate our approach in a high-fidelity ROS simula-
+tion, where we consider a real sensor model and a low-level
+motion controller, towards deployments on real robots.
+Future work will focus on extending our approach to
+autonomous exploration of 3D environments, where frontiers
+are much denser than in 2D. Second, although in this work
+we uniformly distribute nodes to construct a graph, we be-
+lieve a sparser graph containing more informative viewpoints
+may improve performance. Finally, we are also interested in
+explicitly predicting the potential gain during exploration to
+further boost planning performance.
+ACKNOWLEDGMENTS
+This work was supported by Temasek Laboratories
+(TL@NUS) under grant TL/FS/2022/01.
+
+L
+LREFERENCES
+[1] A. Hornung, K. M. Wurm, M. Bennewitz, C. Stachniss, and
+W. Burgard, “OctoMap: An efficient probabilistic 3D mapping
+framework based on octrees,” Autonomous Robots, 2013, software
+available at https://octomap.github.io. [Online]. Available: https:
+//octomap.github.io
+[2] J. A. Placed, J. Strader, H. Carrillo, N. Atanasov, V. Indelman,
+L. Carlone, and J. A. Castellanos, “A survey on active simultaneous
+localization and mapping: State of the art and new frontiers,” arXiv
+preprint arXiv:2207.00254, 2022.
+[3] A. Bircher, M. Kamel, K. Alexis, H. Oleynikova, and R. Siegwart,
+“Receding horizon” next-best-view” planner for 3d exploration,” in
+2016 IEEE international conference on robotics and automation
+(ICRA).
+IEEE, 2016, pp. 1462–1468.
+[4] C. Cao, H. Zhu, H. Choset, and J. Zhang, “Tare: A hierarchical
+framework for efficiently exploring complex 3d environments.” in
+Robotics: Science and Systems, 2021.
+[5] B. Yamauchi, “A frontier-based approach for autonomous exploration,”
+in Proceedings 1997 IEEE International Symposium on Computational
+Intelligence in Robotics and Automation CIRA’97.’Towards New Com-
+putational Principles for Robotics and Automation’.
+IEEE, 1997, pp.
+146–151.
+[6] H. H. Gonz´alez-Banos and J.-C. Latombe, “Navigation strategies for
+exploring indoor environments,” The International Journal of Robotics
+Research, vol. 21, no. 10-11, pp. 829–848, 2002.
+[7] M. Selin, M. Tiger, D. Duberg, F. Heintz, and P. Jensfelt, “Efficient
+autonomous exploration planning of large-scale 3-d environments,”
+IEEE Robotics and Automation Letters, vol. 4, no. 2, pp. 1699–1706,
+2019.
+[8] D. Holz, N. Basilico, F. Amigoni, and S. Behnke, “Evaluating the
+efficiency of frontier-based exploration strategies,” in ISR 2010 (41st
+International Symposium on Robotics) and ROBOTIK 2010 (6th Ger-
+man Conference on Robotics).
+VDE, 2010, pp. 1–8.
+[9] M. Kulich, J. Faigl, and L. Pˇreuˇcil, “On distance utility in the
+exploration task,” in 2011 IEEE International Conference on Robotics
+and Automation.
+IEEE, 2011, pp. 4455–4460.
+[10] T. Dang, M. Tranzatto, S. Khattak, F. Mascarich, K. Alexis, and
+M. Hutter, “Graph-based subterranean exploration path planning using
+aerial and legged robots,” Journal of Field Robotics, vol. 37, no. 8,
+pp. 1363–1388, 2020.
+[11] Z. Xu, D. Deng, and K. Shimada, “Autonomous uav exploration
+of dynamic environments via incremental sampling and probabilistic
+roadmap,” IEEE Robotics and Automation Letters, vol. 6, no. 2, pp.
+2729–2736, 2021.
+[12] S. Arora and S. Scherer, “Randomized algorithm for informative
+path planning with budget constraints,” in 2017 IEEE International
+Conference on Robotics and Automation (ICRA).
+IEEE, 2017, pp.
+4997–5004.
+[13] G. A. Hollinger and G. S. Sukhatme, “Sampling-based robotic infor-
+mation gathering algorithms,” The International Journal of Robotics
+Research, vol. 33, no. 9, pp. 1271–1287, 2014.
+[14] F. Niroui, K. Zhang, Z. Kashino, and G. Nejat, “Deep reinforcement
+learning robot for search and rescue applications: Exploration in
+unknown cluttered environments,” IEEE Robotics and Automation
+Letters, vol. 4, no. 2, pp. 610–617, 2019.
+[15] L. Schmid, C. Ni, Y. Zhong, R. Siegwart, and O. Andersson, “Fast
+and compute-efficient sampling-based local exploration planning via
+distribution learning,” arXiv preprint arXiv:2202.13715, 2022.
+[16] D. Zhu, T. Li, D. Ho, C. Wang, and M. Q.-H. Meng, “Deep
+reinforcement learning supervised autonomous exploration in office
+environments,” in 2018 IEEE international conference on robotics and
+automation (ICRA).
+IEEE, 2018, pp. 7548–7555.
+[17] H. Li, Q. Zhang, and D. Zhao, “Deep reinforcement learning-based
+automatic exploration for navigation in unknown environment,” IEEE
+transactions on neural networks and learning systems, vol. 31, no. 6,
+pp. 2064–2076, 2019.
+[18] F. Chen, S. Bai, T. Shan, and B. Englot, “Self-learning exploration
+and mapping for mobile robots via deep reinforcement learning,” in
+Aiaa scitech 2019 forum, 2019, p. 0396.
+[19] F. Chen, J. D. Martin, Y. Huang, J. Wang, and B. Englot, “Autonomous
+exploration under uncertainty via deep reinforcement learning on
+graphs,” in 2020 IEEE/RSJ International Conference on Intelligent
+Robots and Systems (IROS).
+IEEE, 2020, pp. 6140–6147.
+[20] Y. Cao, Y. Wang, A. Vashisth, H. Fan, and G. A. Sartoretti,
+“CAtNIPP: Context-Aware Attention-based Network for Informative
+Path Planning,” in Accepted to the 6th Annual Conference on Robot
+Learning, 2022. [Online]. Available: https://openreview.net/forum?id=
+cAIIbdNAeNa
+[21] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N.
+Gomez, Ł. Kaiser, and I. Polosukhin, “Attention is all you need,”
+Advances in neural information processing systems, vol. 30, 2017.
+[22] O. Vinyals, M. Fortunato, and N. Jaitly, “Pointer networks,” Advances
+in neural information processing systems, vol. 28, 2015.
+[23] V. P. Dwivedi and X. Bresson, “A generalization of transformer
+networks to graphs,” arXiv preprint arXiv:2012.09699, 2020.
+[24] P. Christodoulou, “Soft actor-critic for discrete action settings,” arXiv
+preprint arXiv:1910.07207, 2019.
+[25] T. Haarnoja, A. Zhou, K. Hartikainen, G. Tucker, S. Ha, J. Tan,
+V. Kumar, H. Zhu, A. Gupta, P. Abbeel, et al., “Soft actor-critic
+algorithms and applications,” arXiv preprint arXiv:1812.05905, 2018.
+[26] P. Moritz, R. Nishihara, S. Wang, A. Tumanov, R. Liaw, E. Liang,
+M. Elibol, Z. Yang, W. Paul, M. I. Jordan, et al., “Ray: A distributed
+framework for emerging ai applications,” in Proceedings of OSDI,
+2018, pp. 561–577.
+[27] C. Cao, H. Zhu, F. Yang, Y. Xia, H. Choset, J. Oh, and J. Zhang,
+“Autonomous exploration development environment and the planning
+algorithms,” in 2022 International Conference on Robotics and Au-
+tomation (ICRA).
+IEEE, 2022, pp. 8921–8928.
+

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@@ -0,0 +1,1644 @@
+arXiv:2301.11770v1  [math.RA]  27 Jan 2023
+CONSTRUCTION OF SOME NON-ASSOCIATIVE ALGEBRAS
+FROM ASSOCIATIVE ALGEBRAS WITH A ENDOMORPHISM
+OPERATOR, DIFFERENTIAL OPERATOR OR LEFT
+AVERAGING OPERATOR.
+WILSON ARLEY MARTINEZ, SAMIN INGRITH CERON
+Abstract. In this paper, we introduce the concepts of a Endomorphism Op-
+erator, Left Averaging Operator, Differential Operator and Rota-Baxter Op-
+erator and we construct examples of these linear maps on associative algebras
+with a left identity, skew-idempotent or idempotent element. Its maps on asso-
+ciative algebra, induces a non-associative algebra structure such as Lie algebra,
+Pre-Lie algebra, Jordan algebra, Flexible Algebra or (left) Leibniz algebra. We
+consider that the construction of non-associative algebras from associative al-
+gebras with Linear Operators as the main results of this work. In this paper
+we give a example of non-associative algebras on subspaces of square matrices
+M(3 × 3, R).
+Introduction
+Linear operators can be defined on different algebraic structures, the weell-known
+operators are the endomorphism operator and differential operator [15, 24, 16]. By
+the 1970’s, new identities for operators have emerged from studies in combinatorics,
+probability and analysis. Gian-Carlo Rota was most interested in the following
+operators:
+Endomorphism operator
+R(x · y) = R(x) · R(y),
+Differential operator
+R(x · y) = R(x) · y + x · R(y),
+Rota-Baxter operator of weight λ
+where λ is a fixed constant,
+R(x) · R(y) = R(x · R(y) + R(x) · y + λx · y),
+Average operator
+R(x) · R(y) = R(x · R(y)),
+Inverse average operator
+R(x) · R(y) = R(R(x) · y),
+Reynolds operator
+R(x) · R(y) = R(x · R(y) + R(x) · y − R(x) · R(y)).
+Received by the editors January 27, 2023 and, in revised form, January, 2023.
+2020 Mathematics Subject Classification. 17A15;17A32;17A20;17B40;47C05.
+Key words and phrases. Associative Algebras, Lie algebra, Pre-Lie algebra, Jordan algebra,
+Flexible Algebra, (left) Leibniz algebra, Rota-Baxter Operator, Endomorphism operator, Differ-
+ential operator.
+This work was completed with the support of the Universidad del Cauca.
+The author was also supported by the research group “Estructuras Algebraicas, Divulgaci´on
+Matem´atica y Teor´ıas Asociadas. @DiTa”.
+1
+
+2
+WILSON ARLEY MARTINEZ, SAMIN INGRITH CERON
+An endomorphism is a homomorphism from an algebraic structure into itself.
+Let A be a non-unital, associative algebra, α an algebra endomorphism on A and
+define ∗ : A × A −→ A by a ∗ b = α(ab) for all a, b ∈ A.
+Then (A, ∗, α) is a
+hom-associative algebra, see [30].
+The study of averaging operators from an algebraic point of view was started
+by Kamp´e de F´eriet [11] and continued and elaborated by Birkhoff [5]. Averaging
+operators has connection with developments in the theory of turbulence [6, 5], and
+was closely related to the probability theory.
+In his Ph. D. thesis in 2000 [7], Weili Cao studied averaging operators in the
+general context and the algebraic definition. He studied the naturally induced Lie
+algebra structures from averaging operators: Let R : A → A be an Averaging
+Operator on an algebra A, its map permit us to define a Lie bracket operation on
+A, by [x, y] = x · R(y) − y · R(x), ∀x, y ∈ A, see [7, 21].
+Let R : A → A be a Differential Operator on a commutative associative algebra
+A, its map induces a new Lie algebra structure called Witt type Lie algebras [29],
+defined by the bracket [x, y] = R(x)·y−x·R(y), ∀x, y ∈ A. Commutative associative
+algebras with this type of linear maps, permit us to present examples of Lie algebras.
+Rota-Baxter operators were introduced by the mathematician Glenn E. Bax-
+ter [3], in the study of differential equations applied to probability theory, and
+mainly its importance by the works of G.-C. Rota in combinatorics [4, 22, 23].
+A Rota-Baxter algebra, is an associative algebra equipped with a Rota-Baxter
+operator. Recently, noncommutative Rota-Baxter algebras have appeared in a wide
+range of areas in pure mathematics, for example the works of Loday and Ronco on
+dendriform dialgebras and trialgebras, see [18, 17] and too in applied mathematics,
+see [8].
+The following result provides a way to construct a pre-Lie algebra structure from
+Rota-Baxter operator relation on Lie Algebras or pre-Lie algebras. We find that if
+R : A → A es a Rota Baxter-Operator on an Lie Algebra (A, [, ]), its map induces
+a pre-Lie algebra structure, defined by x ∗ y = [R(x), y], ∀x, y ∈ A, see [2].
+In the case of the Rota-Baxter relation on pre-Lie algebras, it is known that
+if (A, ·) is a pre-Lie algebra and R be a Rota-Baxter operator on A. Then R is
+still a Rota-Baxter operator on (A, ∗), and the product given by x ∗ y = [R(x), y]
+= R(x) · y − y · R(x), x, y ∈ A, defines a new pre-Lie algebra (A, ∗) see [27] .
+An element u is said to be skew-idempotent with respect to a product · in
+the algebra if: u · u = −u, and an element u is a right identity if: x · u = x
+for all element x in the algebra. Associative algebras with a left identity, skew-
+idempotent or idempotent element, permit us to build examples of linear maps
+as Endomorphism Operator, Left Averaging Operator, Differential Operator and
+Rota-Baxter Operator. Associative algebras with this type of linear maps, permit
+us to present constructions of non-associative algebras.
+1. Construction of Lie algebra from Associative Algebras with a
+Endomorphism Operator
+In this section, we present in the Proposition 1.1.3 a Lie algebra structure given
+by the following bracket [x, y] = x · R(y) − y · R(x) where R is a endomorphism
+
+CONSTRUCTION OF SOME NON-ASSOCIATIVE ALGEBRAS
+3
+operator, and R is defined from a right identity on a subalgebra A: Let (A, ·) be an
+algebra, then an element u of H, A ⊆ H, is called a right identity on A if x · u = x
+for all x in A.
+1.1. Introduction. In the Definition 1.2.1 we define a left and a right identity for
+an associative algebra A and in the Proposition 1.2.3 we give a proof of the build
+of an endomorphism operator with certain properties on A useding a right identity,
+inspired by the well known result of X. Xu, [29], where establish the existence of
+Lie Algebras from an associative algebra A with an Differential Operator on the
+space A, we establish a new connection between an endomorphism operator with
+a construction of Lie algebra structures on A . We start by briefly introducing the
+definition of a Lie Algebra, a left and a right identity for A.
+Definition 1.1.1. A Lie algebra over a field F is a vector space g over F equipped
+with bilinear operation [, ] : g × g → g, called the commutator or (Lie) bracket
+which satisfies the following identities:
+[x, y]
+=
+−[y, x]
+(Antisymmetry)
+(1.1.1)
+[x, [y, z]] + [z, [x, y]] + [y, [z, x]]
+=
+0
+(Jacobi identity)
+(1.1.2)
+Remark 1.1.2. It is well known that any associative algebra becomes a Lie algebra
+with the Lie bracket given by the commutator: [x, y] = x · y − y · x. Also that the
+dimension of a Lie algebra g is its dimension as a vector space over F and Ado’s
+theorem states that every finite dimensional Lie algebra g over a field F can be
+viewed as a Lie algebra of square matrices with the commutator as bracket.
+Proposition 1.1.3. Let A be an associative algebra and let R : A → A a linear
+map such that R2(x) = R(x) and R(x) · R(y) = R(x · y) for all x, y ∈ A. Then we
+can define a Lie algebra structures on A given by
+(1.1.3)
+[x, y] = x · R(y) − y · R(x) (respectively [x, y] = R(x) · y − R(y) · x)
+Proof. Let x, y, z ∈ A; then, [x, y] = −[y, x] and
+[x, [y, z]] = x · R([y, z]) − [y, z] · R(x)
+= x · R(y · R(z) − z · R(y)) − (y · R(z) − z · R(y)) · R(x)
+= x · (R(y) · R(z) − R(z) · R(y)) − (y · R(z)) · R(x) + (z · R(y)) · R(x)
+= x · (R(y) · R(z)) − x · (R(z) · R(y)) − (y · R(z)) · R(x) + (z · R(y)) · R(x)
+[y, [z, x]] = y · R([z, x]) − [z, x] · R(y)
+= y · R(z · R(x) − x · R(z)) − (z · R(x) − x · R(z)) · R(y)
+= y · (R(z) · R(x) − R(x) · R(z)) − (z · R(x)) · R(y) + (x · R(z)) · R(y)
+= y · (R(z) · R(x)) − y · (R(x) · R(z)) − (z · R(x)) · R(y) + (x · R(z)) · R(y)
+[z, [x, y]] = z · R([x, y]) − [x, y] · R(z)
+= z · R(x · R(y) − y · R(x)) − (x · R(y) − y · R(x)) · R(z)
+= z · (R(x) · R(y) − R(y) · R(x)) − (x · R(y)) · R(z) + (y · R(x)) · R(z)
+= z · (R(x) · R(y)) − z · (R(y) · R(x)) − (x · R(y)) · R(z) + (y · R(x)) · R(z)
+Thus, [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0. Therefore, (A, [ , ]) is a Lie algebra.
+□
+
+4
+WILSON ARLEY MARTINEZ, SAMIN INGRITH CERON
+1.2. Examples of Lie algebras from the endomorphism operator. In this
+subsection we present the relationship between an endomorphism operator and a
+right identity of an associative algebra. In one direction, we show that an associa-
+tive algebra A with a right identity gives an endomorphism operator with certain
+properties on the associative algebra A that allow us to give examples of lie algebras
+from the endomorphism.
+Definition 1.2.1. Let A be an algebra and H a set containing A. An element
+u ∈ H is called a left identity for A if u · x = x for all x ∈ A. Similarly, u ∈ H is
+a right identity for A if x · u = x for all x ∈ A. An element u ∈ A which is both a
+left and a right identity for A is an identity element.
+Remark 1.2.2. Given an operation (function) · : H × A → A, an element u of H is
+called a right identity for · if x · u = x for every element x of A. That is, the map
+A → A given by x · u is the identity function on A.
+The following proposition and its examples introduce the idea of all the main
+results of this paper.
+Proposition 1.2.3. Let ( A, · ) be an associative algebra and H a set containing
+A, and suppose that there exists u ∈ H such that u · x ∈ A and x · u = x for all
+x ∈ A. Then the linear map R : A −→ A defined by R(x) = u · x satisfies
+(1.2.1)
+R(x) · R(y) = R(x · y) for all x, y ∈ A.
+Furthemore, R2(x) = R(x) if u2 = u, or R2(x) = x if u2 = 1.
+Proof. Let x, y ∈ A, then R(x · y) = u · (x · y). On the other hand we have,
+R(x)·R(y) = (u·x)·(u·y) = u·((x·u)·y) = u·(x·y). Therefore R(x)·R(y) = R(x·y)
+for all x, y ∈ A. Now, if u2 = u, then R2(x) = u · (u · x) = (u2 · x) = u · x = R(x).
+Therefore R2(x) = R(x) for all x ∈ A.
+□
+Example 1.2.4. We consider the subalgebra
+A =
+
+
+
+
+
+y
+y
+0
+n
+n
+0
+r
+r
+0
+
+ : r, n, y ∈ R
+
+
+
+under the usual matrix multiplication.
+The element u =
+
+
+a
+b
+c
+1 − a
+1 − b
+−c
+e
+f
+g
+
+ satisfies x · u = x and u · x ∈ A for all
+x ∈ A.
+Then the linear map R : A −→ A defined by
+R
+
+
+
+
+y
+y
+0
+n
+n
+0
+r
+r
+0
+
+
+
+ =
+
+
+a
+b
+c
+1 − a
+1 − b
+−c
+e
+f
+g
+
+ ·
+
+
+y
+y
+0
+n
+n
+0
+r
+r
+0
+
+
+=
+
+
+ay + bn + cr
+ay + bn + cr
+0
+(1 − a)y + (1 − b)n − cr
+(1 − a)y + (1 − b)n − cr
+0
+ey + fn + gr
+ey + fn + gr
+0
+
+
+satisfies R(x) · R(y) = R(x · y) for all x, y ∈ A.
+
+CONSTRUCTION OF SOME NON-ASSOCIATIVE ALGEBRAS
+5
+Example 1.2.5. We consider the subalgebra
+A =
+
+
+
+
+
+x
+y
+y
+w
+k
+k
+m
+n
+n
+
+ : x, w, m, y, k, n ∈ R
+
+
+
+under the usual matrix multiplication.
+The element u =
+
+
+1
+0
+0
+0
+0
+1
+0
+1
+0
+
+ satisfies u2 = I,
+x · u = x and u · x ∈ A for all
+x ∈ A.
+Then the linear map R : A −→ A defined by
+R
+
+
+
+
+x
+y
+y
+w
+k
+k
+m
+n
+n
+
+
+
+ =
+
+
+x
+y
+y
+m
+n
+n
+w
+k
+k
+
+
+satisfies R2(x) = x and R(x) · R(y) = R(x · y) for all x, y ∈ A.
+Example 1.2.6. We consider the subalgebra
+A =
+
+
+
+
+
+y
+y
+0
+n
+n
+0
+r
+r
+0
+
+ : y, n, r ∈ R
+
+
+
+under the usual matrix multiplication.
+The element u =
+
+
+1
+b
+b
+0
+1 − b
+−b
+0
+b − 1
+b
+
+ satisfies u2 = u,
+x · u = x and u · x ∈ A
+for all x ∈ A.
+Then the linear map R : A −→ A defined by
+R
+
+
+
+
+y
+y
+0
+n
+n
+0
+r
+r
+0
+
+
+
+ =
+
+
+1
+b
+b
+0
+1 − b
+−b
+0
+b − 1
+b
+
+ ·
+
+
+y
+y
+0
+n
+n
+0
+r
+r
+0
+
+
+=
+
+
+y + bn + br
+y + bn + br
+0
+n − bn − br
+n − bn − br
+0
+nb − n + br
+nb − n + br
+0
+
+
+satisfies R2(x) = R(x) and R(x) · R(y) = R(x · y) for all x, y ∈ A. Therefore, we
+can define a Lie algebra structures on A given by [x, y] = x · R(y) − y · R(x).
+2. Construction of Jordan algebra from Commutative Associative
+Algebras with a Endomorphism Operator
+Jordan algebras were introduced in the early 1930’s by a physicist, P.Jordan,
+in an attempt to generalize the formalism of quantum mechanics. Little appears
+to have resulted in this direction, but unanticipated relationships between these
+algebras and Lie groups and the foundations of geometry have been discovered.
+Definition 2.0.1. A (non-commutative) Jordan algebra is a vector space J over
+a field F of characteristic ̸= 2 with a binary operation ◦ satisfying for x, y ∈ J the
+following identity:
+(2.0.1)
+(x ◦ y) ◦ x = x ◦ (y ◦ x)
+and
+((x ◦ x) ◦ y) ◦ x = (x ◦ x) ◦ (y ◦ x).
+
+6
+WILSON ARLEY MARTINEZ, SAMIN INGRITH CERON
+Remark 2.0.2. Given an associative algebra (A, ·) we can modify the product to
+obtain a commutative algebra A+ as follows: To construct A+ we define x ∗ y =
+x · y + y · x. In this new algebra the Jordan identity is satisfied
+(2.0.2)
+((x ∗ x) ∗ y) ∗ x = (x ∗ x) ∗ (y ∗ x).
+Example 2.0.3. We consider the subalgebra
+A+ =
+
+
+
+
+
+y
+y
+0
+n
+n
+0
+r
+r
+0
+
+ : y, n, r ∈ R
+
+
+
+under the product x ∗ y = x · y + y · x is a commutative algebra (non-associative),
+where · is the usual matrix multiplication.
+The element u =
+
+
+1
+b
+b
+0
+1 − b
+−b
+0
+b − 1
+b
+
+ satisfies u2 = u,
+x · u = x and u · x ∈ A+
+for all x ∈ A+.
+Then the linear map R : A+ −→ A+ defined by
+R
+
+
+
+
+y
+y
+0
+n
+n
+0
+r
+r
+0
+
+
+
+ =
+
+
+1
+b
+b
+0
+1 − b
+−b
+0
+b − 1
+b
+
+ ·
+
+
+y
+y
+0
+n
+n
+0
+r
+r
+0
+
+
+=
+
+
+y + bn + br
+y + bn + br
+0
+n − bn − br
+n − bn − br
+0
+nb − n + br
+nb − n + br
+0
+
+
+satisfies R2(x) = R(x) and R(x)∗R(y) = R(x∗y) for all x, y ∈ A+. Therefore the
+Jordan identity is satisfied on A+, with the product given by x ◦ y = R(x) ∗ R(y).
+Proposition 2.0.4. Suppose (A, ·) is a Jordan algebra, and R : A → A is a linear
+map, such that
+(2.0.3)
+R2(x) = R(x) y R(x) · R(y) = R(x · y) for all x, y ∈ A.
+Then we can define a new (non-commutative) Jordan algebra structures on A given
+by
+x ◦ y = R(x) · y
+(respectively x ◦ y = x · R(y), x ◦ y = R(x) · R(y)).
+Proof. Let A be a Jordan algebra and R : A → A is a linear map such that
+R2(x) = R(x) y R(x) · R(y) = R(x · y) for all x, y ∈ A. So
+(x ◦ x) ◦ (y ◦ x) = R(R(x) · x) ◦ (R(y) · x)
+= (R(x) · R(x)) · (R(y) · x)
+= ((R(x) · R(x)) · R(y)) · x
+= ((R2(x) · R2(x)) · R(y)) · x
+= R((R(x) · R(x)) · y) · x
+= R((R2(x) · R(x)) · y) · x
+= R(R(R(x) · x) · y) · x
+= ((x ◦ x) ◦ y) ◦ x.
+
+CONSTRUCTION OF SOME NON-ASSOCIATIVE ALGEBRAS
+7
+This means that (x ◦ x) ◦ (y ◦ x) = ((x ◦ x) ◦ y) ◦ x for all x, y ∈ A. Since
+(x ◦ y) ◦ x = R(R(x) · y) · x = (R(x) · R(y)) · x = R(x) · (R(y) · x) = x ◦ (y ◦ x).
+So (x ◦ y) ◦ x = x ◦ (y ◦ x) for all x, y ∈ A. Therefore (A, ◦) is a Jordan algebra.
+□
+Example 2.0.5. The element u =
+
+
+1
+−1
+1
+1
+−1
+1
+1
+−1
+1
+
+ satisfy: u2 = u and x · u = x for
+all x ∈ A. The algebra of matrices
+A =
+
+
+
+
+
+x
+−x
+x
+w
+−w
+w
+p
+−p
+p
+
+ : x, w, p ∈ R
+
+
+ ,
+is a associative algebra under the usual matrix multiplication. A is a commutative
+algebra (non-associative) under the product x∗y = x·y +y ·x , where · is the usual
+matrix multiplication. Then the linear map R : A −→ A defined by
+R
+
+
+
+
+x
+−x
+x
+w
+−w
+w
+p
+−p
+p
+
+
+
+ =
+
+
+1
+−1
+1
+1
+−1
+1
+1
+−1
+1
+
+ ·
+
+
+x
+−x
+x
+w
+−w
+w
+p
+−p
+p
+
+
+=(x − w + p)
+
+
+1
+−1
+1
+1
+−1
+1
+1
+−1
+1
+
+
+satisfies R2(x) = R(x) and R(x) ∗ R(y) = R(x ∗ y) for all x, y ∈ A. Therefore, we
+can define a Jordan algebra structures on A given by x ◦ y = R(x) ∗ R(y), and from
+it we can define a new Jordan algebra structures on A given by x ◦2 y = R(x) ◦ y.
+3. Construction of (left) Leibniz algebra from Associative Algebras
+with a Endomorphism Operator.
+Leibniz algebras were first introduced by J.-L. Loday in [14] as a non-antisymmetric
+version of Lie algebras, and many results of Lie algebras have been extended to
+Leibniz algebras. Leibniz algebras play a significant role in different areas of math-
+ematics and physics.
+Definition 3.0.1. A (left) Leibniz algebra L is a vector space equipped with a
+bilinear map
+[ , ] : L × L → L
+satisfying the (left) Leibniz identity
+(3.0.1)
+[x, [y, z]] = [[x, y], z] + [y, [x, z]]for all x, y, z ∈ L.
+Remark 3.0.2. We may pass from the right to the left Leibniz algebra by considering
+a new multiplication x ◦ y = [y, x]. For a Leibniz algebra L, we define left multi-
+plication la : L → L by an element a on an element b by la(b) = [a, b]. Similarly,
+right multiplication by an element a on an element b is defined by ra(b) = [b, a].
+An algebra L over F is a left Leibniz algebra if for every x ∈ L the corresponding
+operator lx of left multiplication is a derivation of L, i.e. the mapping lx satisfies
+
+8
+WILSON ARLEY MARTINEZ, SAMIN INGRITH CERON
+lx(a · b) = lx(a) · b + a · lx(b), lx ∈ Der(L). Thus, left multiplication is a derivation
+in a left Leibniz algebra while right multiplication is not necessarily a derivation.
+Proposition 3.0.3. Let A be an associative algebra over a field F and let
+R : A → A
+be an endomorphism of A such that R2 = R. Define the binary operation [ , ] on
+A by the following rule: [a, b] = R(a) · b − b · R(a) for all elements a, b ∈ A, Then,
+with respect to the operations + and [ , ], A becomes a Leibniz algebra.
+Proof. See [25].
+□
+Lemma 3.0.4. Let A be a associative algebra and let R : A → A be a linear map.
+Suppose that R2(x) = R(x) y R(x) · R(y) = R(x · y)
+for all x, y ∈ A. Then there
+exists a Leibniz structures on A given by
+(3.0.2)
+[x, y] = R(x) · y − R(y) · R(x)
+for all x, y ∈ A.
+Proof. Let x, y, z ∈ A, then we have
+[[x, y], z] = R([x, y]) · z − R(z) · R([x, y])
+= R(R(x) · y − R(y) · R(x)) · z − R(z) · R(R(x) · y − R(y) · R(x))
+= (R(x) · R(y) − R(y) · R(x)) · z − R(z) · (R(x) · R(y) − R(y) · R(x))
+[y, [x, z]] = R(y) · [x, z] − R([x, z]) · R(y)
+= R(y) · (R(x) · z − R(z) · R(x)) − R(R(x) · z − R(z) · R(x)) · R(y)
+= R(y) · (R(x) · z − R(z) · R(x)) − (R(x) · R(z) − R(z) · R(x)) · R(y)
+Then
+[[x, y], z] + [y, [x, z]] = (R(x) · R(y) − R(y) · R(x)) · z − R(z) · (R(x) · R(y) − R(y) · R(x))
++ R(y) · (R(x) · z − R(z) · R(x)) − (R(x) · R(z) − R(z) · R(x)) · R(y)
+so
+[[x, y], z] + [y, [x, z]] = (R(x) · R(y)) · z − (R(x) · R(z)) · R(y) − R(y) · (R(z) · R(x))
++ R(z) · (R(y) · R(x))
+On the other hand, we have
+[x, [y, z]] = R(x) · [y, z] − R([y, z]) · R(x)
+= R(x) · (R(y) · z − R(z) · R(y)) − R(R(y) · z − R(z) · R(y)) · R(x)
+= R(x) · (R(y) · z − R(z) · R(y)) − (R(y) · R(z) − R(z) · R(y)) · R(x)
+Note that R2(x) = R(x) y R(x) · R(y) = R(x · y) for all x, y ∈ A, implies
+[x, [y, z]] = [[x, y], z] + [y, [x, z]] for all x, y, z ∈ A.
+□
+Example 3.0.5. The element u =
+
+
+−1
+1
+1
+−1
+1
+1
+−1
+1
+1
+
+ satisfy: u2 = u and x · u = x for
+all x ∈ A. The algebra of matrices
+A =
+
+
+
+
+
+−y
+y
+y
+−m
+m
+m
+−t
+t
+t
+
+ : y, m, t ∈ R
+
+
+ ,
+
+CONSTRUCTION OF SOME NON-ASSOCIATIVE ALGEBRAS
+9
+is a associative algebra under the usual matrix multiplication. Then the linear map
+R : A −→ A defined by
+R
+
+
+
+
+−y
+y
+y
+−m
+m
+m
+−t
+t
+t
+
+
+
+ =
+
+
+−1
+1
+1
+−1
+1
+1
+−1
+1
+1
+
+ ·
+
+
+−y
+y
+y
+−m
+m
+m
+−t
+t
+t
+
+
+= (−y + m + t)
+
+
+−1
+1
+1
+−1
+1
+1
+−1
+1
+1
+
+
+satisfies R2(x) = R(x) and R(x) · R(y) = R(x · y) for all x, y ∈ A. Therefore, we
+can define a Leibniz structures structures on A given by
+[x, y] = R(x) · y − R(y) · R(x)
+4. Construction of Pre-Lie algebra from Commutative Associative
+Algebras with a Endomorphism Operator
+In this section we present in the Propositions 4.0.3 a construction of Pre-Lie
+algebra structure given by x◦y = R(x)·R(y)−y ·R(x) where R is a endomorphism
+operator:
+Definition 4.0.1. An algebra A over F with a bilinear product ◦ which satisfies
+the following identity:
+(4.0.1)
+(x ◦ y) ◦ z − x ◦ (y ◦ z) = (y ◦ x) ◦ z − y ◦ (x ◦ z) for all x, y, z ∈ A
+is called a Left Pre-Lie algebra.
+Remark 4.0.2. There is a construction of pre-Lie algebras using a commutative
+associative algebra (A, ·) with a derivation D on A, the new product a∗b = a·D(b),
+∀a, b ∈ A makes (A, ∗) become a Novikov algebra, Novikov algebra is a pre-Lie
+algebra satisfying an additional identity: (xy)z = (xz)y, ∀x, y, z ∈ A. There are
+some generalizations of the previous result for a commutative associative algebra
+(A, ·). If D is a derivation on A , then the new product x ∗a y = x · D(y) + a · x · y,
+∀x, y ∈ A makes (A, ∗a) become a Novikov algebra for a fixed element a ∈ F or
+a ∈ A ( [10, 28]). In the case of an associative algebra (A, ·) with a Rota-Baxter
+relation of weight 1, it is known that if x ∗ y = R(x) · y − y · R(x) − x · y, ∀x, y ∈ A,
+then the product ∗ defines a pre-Lie algebra on A. ( [9, 13] )
+Proposition 4.0.3. Let A be a commutative associative algebra and let R : A → A
+a linear map such that R2(x) = R(x) and R(x) · R(y) = R(x · y) for all x, y ∈ A.
+Then we can define a Pre-Lie algebra structures on A given by
+(4.0.2) x ◦ y = R(x) · R(y) − y · R(x) (respectively x ◦ y = R(x) · y − R(y) · R(x)).
+
+10
+WILSON ARLEY MARTINEZ, SAMIN INGRITH CERON
+Proof. Let x, y, z ∈ A, then we have
+(x ◦ y) ◦ z = (R(x) · R(y) − y · R(x)) ◦ z
+= R(R(x) · R(y) − y · R(x)) · R(z) − z · R(R(x) · R(y) − y · R(x))
+= (R(x) · R(y) − R(y) · R(x)) · R(z) − z · (R(x) · R(y) − R(y) · R(x))
+= 0
+x ◦ (y ◦ z) = x ◦ (R(y) · R(z) − z · R(y))
+= R(x) · R(R(y) · R(z) − z · R(y)) − (R(y) · R(z) − z · R(y)) · R(x)
+= R(x) · (R(y) · R(z) − R(z) · R(y)) − (R(y) · R(z) − z · R(y)) · R(x)
+= −(R(y) · R(z) − z · R(y)) · R(x)
+On the other hand, we have
+(y ◦ x) ◦ z = (R(y) · R(x) − x · R(y)) ◦ z
+= R(R(y) · R(x) − x · R(y)) · R(z) − z · R(R(y) · R(x) − x · R(y))
+= (R(y) · R(x) − R(x) · R(y)) · R(z) − z · (R(y) · R(x) − R(x) · R(y))
+= 0
+y ◦ (x ◦ z) = y ◦ (R(x) · R(z) − z · R(x))
+= R(y) · R(R(x) · R(z) − z · R(x)) − (R(x) · R(z) − z · R(x)) · R(y)
+= R(y) · (R(x) · R(z) − R(z) · R(x)) − (R(x) · R(z) − z · R(x)) · R(y)
+= −(R(x) · R(z) − z · R(x)) · R(y)
+Therefore, (x ◦ y) ◦ z − x ◦ (y ◦ z) = (y ◦ x) ◦ z − y ◦ (x ◦ z).
+□
+4.1. Examples of Pre-Lie algebras from commutative associative algebras
+with the endomorphism operator. In this subsection we present in the example
+4.1.1 a commutative associative algebra A with an unusual matrix multiplication,
+that allow us to give an example of Pre-lie algebras from a endomorphism operator
+with certain properties on the algebra A.
+Example 4.1.1. Let A be the vector space of all 2 × 2 matrices over R.
+A =
+��
+m
+x
+n
+y
+�
+: m, n, x, y ∈ R
+�
+.
+A is a commutative associative algebra with the unusual matrix multiplication
+defined by:
+(4.1.1)
+�
+m
+x
+n
+y
+�
+∗
+�
+p
+z
+r
+w
+�
+=
+�
+mp
+xz
+nr
+yw
+�
+.
+The element I =
+� 1
+1
+1
+1
+�
+is the multiplicative identity.
+Example 4.1.2. The element u =
+� 1
+0
+0
+0
+�
+with the usual matrix multiplication
+satisfy: u2 = u, u · x ∈ A and x · u = x for all x ∈ A. The algebra of matrices
+A =
+�� m
+0
+n
+0
+�
+: m, n ∈ R
+�
+,
+
+CONSTRUCTION OF SOME NON-ASSOCIATIVE ALGEBRAS
+11
+is a commutative associative under the unusual matrix multiplication defined above
+in 4.1.1. Then the linear map R : A −→ A defined by
+R
+�� m
+0
+n
+0
+��
+=
+� 1
+0
+0
+0
+�
+·
+� m
+0
+n
+0
+�
+=
+�
+m
+0
+0
+0
+�
+satisfies R2(x) = R(x) and R(x) ∗ R(y) = R(x ∗ y) for all x, y ∈ A. Therefore, we
+can define a Pre-Lie algebra structures on A given by
+x ◦ y = R(x) ∗ R(y) − y ∗ R(x)
+5. Construction of Pre-Lie algebra from Commutative Associative
+Algebras with a Differential Operator
+We now present in the Propositions 5.0.1 a construction of Pre-Lie algebra struc-
+ture given by x◦y = R(x)·y where R is a differential operator, and we also introduce
+the notion of left zero divisor on A: Let ( H, · ) be a set H with a binary operation
+· on it, then an element u of H is called a left zero divisor on A ⊆ H if x · u = 0 for
+all x in A.
+Proposition 5.0.1. Let A be a commutative associative algebra and let R : A → A
+a linear map such that R2(x) = α. x and R(x)·y+x·R(y) = R(x·y) for all x, y ∈ A.
+Then we can define a Pre-Lie algebra structures on A given by x ◦ y = R(x) · y.
+Proof. Let x, y, z ∈ A, then
+(x ◦ y) ◦ z − x ◦ (y ◦ z) = (R(x) · y) ◦ z − x ◦ (R(y) · z)
+= R(R(x) · y) · z − R(x) · (R(y) · z)
+= (R2(x) · y + R(x) · R(y)) · z − R(x) · (R(y) · z)
+= (R2(x) · y) · z
+= ((α. x) · y) · z.
+On the other hand, we have
+(y ◦ x) ◦ z − y ◦ (x ◦ z) = (R(y) · x) ◦ z − y ◦ (R(x) · z)
+= R(R(y) · x) · z − R(y) · (R(x) · z)
+= (R2(y) · x + R(y) · R(x)) · z − R(y) · (R(x) · z)
+= (R2(y) · x) · z
+= ((α. y) · x) · z.
+Therefore, (x ◦ y) ◦ z − x ◦ (y ◦ z) = (y ◦ x) ◦ z − y ◦ (x ◦ z)
+□
+5.1. Examples of Pre-Lie algebras from commutative associative algebras
+with a Differential Operator.
+Proposition 5.1.1. Let ( A, · ) be an associative subalgebra of an algebra H, and
+suppose that there exists u ∈ H such that u · x ∈ A and x · u = 0 for all x ∈ A .
+Then the linear map R : A −→ A defined by R(x) = u · x satisfies
+(5.1.1)
+R(x) · y + x · R(y) = R(x · y) for all x, y ∈ A.
+
+12
+WILSON ARLEY MARTINEZ, SAMIN INGRITH CERON
+Proof. Let x, y ∈ A, then
+R(x) · y + x · R(y) = (u · x) · y + x · (u · y)
+= u · (x · y) + (x · u) · y
+= u · (x · y) + (0 · y) = u · (x · y).
+On the other hand, R(x · y) = u · (x · y). Therefore R(x) · y + x · R(y) = R(x · y) for
+all x, y ∈ A.
+□
+Example 5.1.2. We consider the subalgebra
+A =
+
+
+
+
+
+y
+y
+0
+n
+n
+0
+r
+r
+0
+
+ : r, n, y ∈ R
+
+
+
+under the usual matrix multiplication. The element u =
+
+
+a
+b
+c
+−a
+−b
+−c
+e
+f
+g
+
+ satis-
+fies x · u = 0 and u · x ∈ A for all x ∈ A.
+Then the linear map R : A −→ A defined by
+R
+
+
+
+
+y
+y
+0
+n
+n
+0
+r
+r
+0
+
+
+
+ =
+
+
+a
+b
+c
+−a
+−b
+−c
+e
+f
+g
+
+ ·
+
+
+y
+y
+0
+n
+n
+0
+r
+r
+0
+
+
+=
+
+
+ay + bn + cr
+ay + bn + cr
+0
+−ay − bn − cr
+−ay − bn − cr
+0
+ey + fn + gr
+ey + fn + gr
+0
+
+
+satisfies R(x) · y + x · R(y) = R(x · y) for all x, y ∈ A.
+Example 5.1.3. The algebra of matrices
+A =
+
+
+α
+
+
+an
+ap
+aq
+bn
+bp
+bq
+n
+p
+q
+
+ : α ∈ R
+
+
+
+where a = −βb and b, n, p, q ∈ R is a commutative associative algebra.
+The element u =
+
+
+a
+βa
+λβa
+b
+βb
+λβb
+1
+β
+λβ
+
+ where λ, β ∈ R, satisfies
+u2 = (λβ)u,
+x · u = 0 (⇔ an + bp + q = 0) and u · x ∈ A for all x ∈ A.
+Then the linear map R : A −→ A defined by
+R
+
+
+
+
+an
+ap
+aq
+bn
+bp
+bq
+n
+p
+q
+
+
+
+ =
+
+
+a
+βa
+λβa
+b
+βb
+λβb
+1
+β
+λβ
+
+ ·
+
+
+an
+ap
+aq
+bn
+bp
+bq
+n
+p
+q
+
+
+= (λβ)
+
+
+an
+ap
+aq
+bn
+bp
+bq
+n
+p
+q
+
+
+satisfies R2(x) = (λβ)R(x) and R(x) · y + x · R(y) = R(x · y) for all x, y ∈ A.
+Therefore, we can define a Pre-Lie algebra structures on A given by x◦y = R(x)·y.
+
+CONSTRUCTION OF SOME NON-ASSOCIATIVE ALGEBRAS
+13
+6. Construction of flexible algebra from Associative Algebras with
+a left averaging operator.
+Definition 6.0.1. A flexible algebra is a vector space J over a field F of charac-
+teristic ̸= 2 with a binary operation ◦ satisfying for x, y ∈ J the following identity:
+(6.0.1)
+(x ◦ y) ◦ x = x ◦ (y ◦ x).
+Remark 6.0.2. The flexible algebras was initiated by Albert ([1]) and investigated
+by the authors Myung, Okubo, Laufer, Tomber and Santilli, see for example ([20]).
+Proposition 6.0.3. Suppose (A, ·) is a flexible algebra, and R : A → A is a linear
+map, such that
+(6.0.2)
+R(x) · R(y) = R(R(x) · y) = R(x · y) for all x, y ∈ A.
+Then we can define a new flexible algebra structures on A given by
+x ◦ y = R(x) · y.
+Proof. Let A be a flexible algebra and R : A → A is a linear map such that
+R(x) · R(y) = R(R(x) · y) = R(x · y) for all x, y ∈ A. So (x ◦ y) ◦ x = (R(x) · y) ◦ x =
+R(R(x) · y) · x = (R(x) · R(y)) · x. Since x ◦ (y ◦ x) = x ◦ (R(y) · x) = R(x) · (R(y) · x)
+and (A, ·) is a flexible algebra, then we have (x ◦ y) ◦ x = x ◦ (y ◦ x) for all x, y ∈ A.
+Therefore (A, ◦) is a flexible algebra.
+□
+6.1. Examples of Flexible algebras from associative algebras with a left
+averaging operator.
+Proposition 6.1.1. Let ( A, · ) be an associative subalgebra of an algebra H. Sup-
+pose H has the following property:
+There exists u ∈ H, such that u · x ∈ A and x · u = u · x for all
+x ∈ A.
+Then the linear map R : A −→ A defined by R(a) = u ·x satisfies the left averaging
+identity
+(6.1.1)
+R(a) · R(b) = R(R(a) · b) for all a, b ∈ A.
+Furthemore, if u2 = u. Then
+(6.1.2)
+R(a) · R(b) = R(R(a) · b) = R(a · b) for all a, b ∈ A.
+Proof. Let x, y ∈ A, by the hypothesis there exists u ∈ H such that u · x ∈ A and
+x · u = u · x for all x ∈ A, then
+R(x) · R(y) = (u · x) · (u · y)
+= u · ((x · u) · y)
+= u · ((u · x) · y)
+= R(R(x) · y).
+If u2 = u, then R(x) · R(y) = u · ((u · x) · y) = u2 · (x · y) = u · (x · y) = R(x · y).
+Therefore R(x) · R(y) = R(R(x) · y) = R(x · y) for all x, y ∈ A.
+□
+Example 6.1.2. We consider the subalgebra
+A =
+
+
+
+
+
+y
+n
+0
+0
+y
+0
+0
+0
+r
+
+ : r, n, y ∈ R
+
+
+
+
+14
+WILSON ARLEY MARTINEZ, SAMIN INGRITH CERON
+under the usual matrix multiplication.
+The element u =
+
+
+1
+0
+0
+0
+1
+0
+0
+0
+0
+
+ satisfies u2 = u, x · u = u · x and u · x ∈ A for
+all x ∈ A. Then the linear map R : A −→ A defined by
+R
+
+
+
+
+y
+n
+0
+0
+y
+0
+0
+0
+r
+
+
+
+ =
+
+
+1
+0
+0
+0
+1
+0
+0
+0
+0
+
+ ·
+
+
+y
+n
+0
+0
+y
+0
+0
+0
+r
+
+ =
+
+
+y
+n
+0
+0
+y
+0
+0
+0
+0
+
+
+satisfies R(x) · R(y) = R(R(x) · y) = R(x · y) for all x, y ∈ A. We have A is a
+Lie algebra with the product [x, y] = x · R(y) − y · R(x) (see Proposition 1.1.3),
+Therefore (A, [, ]) is a flexive algebra.
+7. Construction of Rota-Baxter Operator
+In this section we present in the Propositions 7.0.5 constructions of Rota-Baxter
+Operators of weight λ = 1 and λ = 0 from associative algebra with an element
+u skew-idempotent or nilpotent of index 2 respectively, and we also introduce the
+notion of Rota-Baxter Operator of weight (λ, β).
+We recall from the Introduction:
+Definition 7.0.1. Let (A, ·) be an associative algebra. A linear map R : A → A
+is called a Rota-Baxter operator of weight λ on A if R satisfies
+(7.0.1)
+R(x) · R(y) = R (R(x) · y + x · R(y) + λ x · y) ,
+for all x, y ∈ A. A Rota-Baxter algebra (also known as a Baxter algebra) is an
+associative algebra A with a Rota-Baxter operator.
+Remark 7.0.2. One importance of the Rota-Baxter Algebra is its close relationship
+with other algebraic structures. For example pre-Lie algebras come naturally from
+a Rota Baxter-Operator on an Lie Algebras. [12], [19] .
+Definition 7.0.3. An elemet u ̸= 0 of an algebra A is called nilpotent if un = 0
+for some integer n . The least such integer is called the index of u.
+Definition 7.0.4. An elemet u ̸= 0 of an algebra A is said to be skew-idempotent
+with respect to a product · in the algebra A if: u · u = −u.
+Proposition 7.0.5. Let ( A, · ) be an associative algebra and suppose that there
+exists u ∈ A such that u2 = −u and u · x ∈ A for all x ∈ A. Then the linear map
+R : A −→ A defined by R(x) = u · x satisfies
+(7.0.2)
+R ( R(x) · y + x · R(y) + x · y ) = R(x) · R(y) for all x, y ∈ A.
+Furthemore, if u2 = 0, then R is a Rota-Baxter operator of weight zero on A.
+Proof. Let x, y ∈ A, then we have R(x) · R(y) = (u · x) · (u · y). On the other hand,
+R(R(x) · y + x · R(y) + x · y) = R((u · x) · y + x · (u · y) + x · y)
+= u · ((u · x) · y + x · (u · y) + x · y)
+= u2 · (x · y) + (u · x) · (u · y) + u · (x · y)
+= (u2 + u) · (x · y) + (u · x) · (u · y).
+
+CONSTRUCTION OF SOME NON-ASSOCIATIVE ALGEBRAS
+15
+Therefore R(R(x)·y +x·R(y)+x·y) = R(x)·R(y) for all x, y ∈ A. Now, if u2 = 0,
+then
+R(R(x) · y + x · R(y)) = (u · x) · (u · y).
+Therefore R(R(x) · y + x · R(y)) = R(x) · R(y) for all x, y ∈ A.
+□
+Example 7.0.6. The element u =
+� xy
+−x2
+y2
+−xy
+�
+satisfies u2 = 0 and u · x ∈ A for
+all x in the algebra of matrices
+A =
+��
+0
+a
+0
+b
+�
+: a, b ∈ R
+�
+considered as a subalgebra of B = M2×2 under the usual matrix multiplication. If
+we define the map R : A −→ A by
+R(
+�
+0
+a
+0
+b
+�
+) =
+�
+xy
+−x2
+y2
+−xy
+� �
+0
+a
+0
+b
+�
+=
+�
+0
+xya − x2b
+0
+y2a − xyb
+�
+,
+then R satisfies R(R(x) · y + y · R(x)) = R(x) · R(y); for all x, y ∈ A.
+Example 7.0.7. The element u =
+�
+x
+y
+−x2−x
+y
+−x − 1
+�
+, y ̸= 0 is an skew-idempotent
+in the algebra of matrices under the usual matrix multiplication, that is u2 = −u.
+We observe that u · x ∈ A for all x ∈ A. If we define the map R : A −→ A by
+R(
+�
+0
+a
+0
+b
+�
+) =
+�
+x
+y
+−x2−x
+y
+−x − 1
+� �
+0
+a
+0
+b
+�
+=
+�
+0
+xa + yb
+0
+( −x2−x
+y
+)a − (x + 1)b
+�
+then R satisfies R(x) · R(y) = R(R(x) · y + x · R(y) + x · y); for all x, y ∈ A.
+Definition 7.0.8. Let (A, ·) be an associative algebra. A linear map R : A → A
+is called a Rota-Baxter operador of weight (λ, β) on A if R satisfies
+(7.0.3)
+R(x) · R(y) = R
+�
+R(x) · y + x · R(y) + λ x · y
+�
++ β x · y, for all x, y ∈ A.
+Remark 7.0.9. A Rota-Baxter operador of weight (λ, β) for associative algebras
+allows to build examples of Dyckm-algebras [26], the main result of this section is
+the construction of Rota-Baxter operador of weight (λ, β) on associative algebras.
+Proposition 7.0.10. Let ( A, · ) be an associative algebra and suppose that there
+exists u ∈ A such that u2 = −λu − β1A and u · x ∈ A for all x ∈ A. Then the
+linear map R : A −→ A defined by R(x) = u · x satisfies
+(7.0.4)
+R ( R(x) · y + x · R(y) + λx · y ) + βx · y = R(x) · R(y) for all x, y ∈ A.
+Proof. Let x, y ∈ A, then we have
+R(R(x) · y + x · R(y) + λx · y) + βx · y = R((u · x) · y + x · (u · y) + λx · y) + βx · y
+= u · ((u · x) · y + x · (u · y) + λx · y) + βx · y
+= u2 · (x · y) + (u · x) · (u · y) + λu · (x · y) + βx · y
+= (u2 + λu + β1A) · (x · y) + (u · x) · (u · y)
+= (u · x) · (u · y)
+On the other hand, R(x) · R(y) = (u · x) · (u · y). Therefore
+R(R(x) · y + x · R(y) + λx · y) + βx · y = R(x) · R(y) for all x, y ∈ A
+
+16
+WILSON ARLEY MARTINEZ, SAMIN INGRITH CERON
+□
+Example 7.0.11. The element u =
+�
+x
+y
+−x2−λx−β
+y
+−x − λ
+�
+, where y ̸= 0 satisfy:
+u2 = −λu − β1A and u · x ∈ A for all x ∈ A. If we define the map R : A −→ A by
+R(
+� 0
+a
+0
+b
+�
+) =
+�
+x
+y
+−x2−λx−β
+y
+−x − λ
+� � 0
+a
+0
+b
+�
+=
+�
+0
+xa + yb
+0
+( −x2−λx−β
+y
+)a − (x + λ)b
+�
+then R satisfies R(x)·R(y) = R(R(x)·y + x·R(y)+ λx·y)+ βx·y; for all x, y ∈ A.
+Acknowledgment. We thank to Universidad del Cauca for the support to our
+research group “Estructuras Algebraicas, Divulgaci´on Matem´atica y Teor´ıas Aso-
+ciadas. @DiTa” under the research project with ID 5773, entitled ”Aplicaciones de
+Estructuras Algebraicas”. We also thank to the anonymous referee for their help-
+ful comments. This work is dedicated to my daughters especially to Clara Isabel
+Martinez Ceron (January 17, 2017).
+References
+1. A.A. ALBERT, Power associative rings, Trans. Amer. math. Soc. 64 (1948), 552–597.
+2. Huihui An and Chengming Bai, From rota–baxter algebras to pre-lie algebras, Journal of
+Physics A: Mathematical and Theoretical 41 (2008), no. 01.
+3. G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pacific
+J. Math. 10 (1960), 731–742.
+4.
+, Baxter algebras and combinatorial identities I, Bull. Amer. Math. Soc. 5 (1969),
+325–329.
+5. G. BIRKHOFF, “moyennes des fonctions born´ees”, Colloque d’alg`ebre et de th´eorie des
+nombres 24, pp. 143-153, Paris, 1949.
+6.
+, Averaging operators, Symposium in Lattice Theory, AMS 63 (1960).
+7. Weili Cao, An algebraic study of averaging operators, arXiv:1401.7389v1 [math.RA] (2014).
+8. A. Connes and D. Kreimer, Renormalization in quantum field theory and the riemann hilbert
+problem. i. the hopf algebra structure of graphs and the main theorem, Comm. Math. Phys.
+210 (2000), no. 1. 249–273.
+9. K. Ebrahimi-Fard, Loday-type algebras and the rota-baxter relation, Lett. Math. Phys. 61
+(2002), 139–147.
+10. V.T. Filipov, A class of simple nonassociative algebras, Mat. Zametki 45 (1989), 101–105.
+11. J.KAMP´E DE F´ERIET, Sur un probl´eme d’alg´ebre abstraite pos´e par la d´efinition de la
+moyenne dans la th´eorie de la turbulence, Ann. Soc. Sci. Bruxelles 63 (1949), 156–172.
+12. V.V. Sokolov I.Z. Golubchik, Generalized operator yang-baxter equations, integrable odes and
+nonassociative algebras, J. Nonlinear Math. Phys. 7 (2000), no. 02, 184–197.
+13. V.V. Sokolov I.Z. Golubschik, Generalized operator yang-baxter equations, integrable odes and
+nonassociative algebras, J. Nonlinear Math. Phys. 7 (2000), 184–197.
+14. Loday J.-L., “une version non commutative des algebres de lie: les algebres de leibniz”, Les
+rencontres physiciens-math´ematiciens de Strasbourg 44 (1993), 127–151.
+15. E. Kolchin, Differential algebraic groups, Academic Press, Inc., Orlando, FL, 1985. 2.
+16. Ronghua Zhang Li Guo, William Y. Sit, Differential type operators and gr¨obner-shirshov
+bases, Journal of Symbolic Computation 52 (2013), 97–123.
+17. J.-L. Loday and M. Ronco, Trialgebras and families of polytopes,, Preprint, math.AT/
+0205043, mai 2002.
+18. J.L. Loday, Dialgebras, in Dialgebras and related operads, Lecture Notes in Math., 1763
+(2001), 7–66.(preprint 2001, arXiv:math.QA/0102053).
+
+CONSTRUCTION OF SOME NON-ASSOCIATIVE ALGEBRAS
+17
+19. A. Medina, Flat left-invariant connections adapted to the automorphism structure of a lie
+group, J. Differential Geometry 16 (1981), no. 03, 445–474.
+20. H. C. MYUNG, Lie algebras and flexible lie-admissible algebras, Hadronic Press INC,
+Hadronic Press Monographs in Mathematics, 1, Massachusetts, 1982.
+21. Nguyen-Huu-Bong, Some apparent connection between baxter and averaging operators, J.
+Math. Anal. Appl. 56 (1976), no. 02, 330–345.
+22. G. Rota, Baxter operators, an introduction, in: “Gian-Carlo Rota on Combinatorics, Intro-
+ductory papers and commentaries”, Joseph P.S. Kung, Editor, Birkh¨auser, Boston, 1995.
+23. G.-C. Rota and D. Smith, Fluctuation theory and Baxter algebras, Istituto Nazionale di Alta
+Matematica, IX, 179 (1972), Reprinted in: “Gian–Carlo Rota on
+Combinatorics : Intro-
+ductory papers and commentaries”, J.P.S. Kung Ed., Contemp. Mathematicians, Birkh¨auser
+Boston, Boston, MA, 1995.
+24. M. van der Put and M. Singer, Galois theory of linear differential equations, Grundlehren der
+mathematischen Wissenschaften, 328, Springer, 2003. 2.
+25. Aleksandr A. Pypka Vladimir V. Kirichenko, Leonid A. Kurdachenko and Igor Ya Subbotin.,
+Some aspects of leibniz algebra theory, Algebra and Discrete Mathematics 24 (2017), no. 1,
+1–33.
+26. E. G. Reyes W. A. Martinez and M. Ronco, Generalizing dendriform algebras:
+Dyckm-
+algebras, rotam-algebras, and rota–baxter operators, International Journal of Geometric Meth-
+ods in Modern Physics. 18 (2021), no. 11.
+27. Dongping HOU Xiuxian LI and Chengming BAI, Rota-baxter operators on pre-lie algebras,
+Journal of Nonlinear Mathematical Physics 14 (2007), no. 2, 269–289.
+28. X. Xu, On simple novikov algebras and their irreducible modules, J. Algebra 185 (1996),
+905–934.
+29.
+, New generalized simple lie algebras of cartan type over a field with characteristic
+zero, J. Algebra 224 (2000), 23–58.
+30. D. Yau, Hom-algebras and homology, J. Lie Theory 19 (2009), no. 02, 409–421.
+Martinez, W.A.; Departmento de Matem´aticas, Universidad del Cauca , Popay´an ,
+Colombia
+Email address: wamartinez@unicauca.edu.co
+Ceron, S.I.; Departmento de Matem´aticas, Universidad del Cauca , Popay´an , Colom-
+bia
+Email address: sicbravo@gmail.com
+

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+arXiv:2301.13610v1  [gr-qc]  30 Jan 2023
+Non-Singular Bouncing Model in Energy
+Momentum Squared Gravity
+Z. Yousaf1 ∗, M. Z. Bhatti1 †, H. Aman1 ‡, P.K. Sahoo2 §
+1Department of Mathematics, University of the Punjab,
+Quaid-i-Azam Campus, Lahore-54590, Pakistan
+2Department of Mathematics,
+Birla Institute of Technology and Science-Pilani,
+Hyderabad Campus, Hyderabad-500078, India.
+Abstract
+This work is concerned to study the bouncing nature of the universe for an isotropic
+configuration of fluid Tαβ and Friedmann-Lemaˆıtre-Robertson-Walker metric scheme.
+This work is carried out under the novel f(G, TαβT αβ) gravitation by assuming a spe-
+cific model i.e, f(G, T 2) = G + αG2 + 2λT 2 with α and λ are constants, serving as free
+parameters. The terms G and T 2 served as an Gauss-Bonnet invariant and square of
+the energy-momentum trace term as an inclusion in the gravitational action respec-
+tively, and is proportional to T 2 = TαβT αβ. A specific functional form of the Hubble
+parameter is taken to provide the evolution of cosmographic parameters. A well known
+equation of state parameter, ω(t) = − k log(t+ǫ)
+t
+− 1 is used to represent the dynamical
+behavior of energy density, matter pressure and energy conditions. A detailed graph-
+ical analysis is also provided to review the bounce. Furthermore, all free parameters
+are set in a way, to make the supposed Hubble parameter act as the bouncing solution
+and ensure the viability of energy conditions. Conclusively, all necessary conditions for
+a bouncing model are checked.
+Keywords: cosmography; Hubble Parameter; f(G, TαβT αβ).
+PACS: 98.80.-k; 04.20.Cv; 04.50.Kd.
+∗zeeshan.math@pu.edu.pk
+†mzaeem.math@pu.edu.pk
+‡huzaifaaman971@gmail.com
+§pksahoo@hyderabad.bits-pilani.ac.in
+1
+
+1
+Introduction
+According to the big bang hypothesis, the whole universe was created by a single explosion,
+with all matter in the cosmos as an infinite speck [1, 2].
+This hypothesis works well in
+order to study the beginning, but lack to define different cosmological problems.
+These
+problems include the horizon problem, the flatness problem, the singularity problem, etc. In
+order to resolve these big cosmic challenges, different cosmic theories have been developed
+in literature [3–5]. The bouncing hypothesis is one of the major independent theories that
+came up with the answers related to the starting of the universe and should be enough to
+resolve the major cosmic problem of singularity.
+The bouncing cosmology works on the
+scheme of an oscillatory universe, i.e, a universe that came into being from the pre-existing
+universe without undergoing the singularity [6–8]. This whole transition of the universe not
+only explains the big-bang cosmology but also reduces one of the major issues.
+For the
+bouncing, the universe moves into the contraction phase as a matter-dominated the era of
+the universe. After the contraction, the universe starts to expand in a nonsingular manner
+for which gravity dominates the matter [9,10]. Also, density perturbations can be produced
+during the bounce era. This idea of the origination of the universe is highly accepted and
+appreciated in literature.
+General relativity (GR) was presented by Einstein and it was thought to be one of the
+best theories to explain different cosmological issues. It explains the gravity under the fabric
+of space-time. However, to understand gravity much more effectively and to provide the
+answers to the effect of gravity, dark energy, and accelerated expansion of the universe under
+the addition of different scalar fields, different attempts have been made in past to modify
+GR.
+These modifications change the geometric or matter or both parts of the Einstein
+field equations accordingly. These could help to discuss the effects of couplings of matter
+and curvature terms on the above-described items. Roshan and Shojai [11] presented the
+nonlinear form of matter term i.e, T 2 = TαβT αβ, naming it f(R, T ∈). They further indicated
+that the use of nonlinear terms may provide the prevention of early time singularities. Since
+the functional form of curvature terms has helped to introduce new gravitational theories, so
+it was considered to be effective to modify the generic action integral of GR as corrections.
+These modifications give light to the f(G) theory, for which the term G is defined as G =
+RξζαβRξζαβ − 4RξζRξζ + R2. Nojiri and Odintsov [12] introduced this f(G) theory for the
+first time in their work. They tested solar systems for this formalism and reported the phase
+change of acceleration to deceleration for the achievement of phantom line, which cooperated
+to study dark energy.
+Odintsov and Oikonomou [13] considered R + f(G) form of the
+gravitational theory to provide their contribution to the study of gravitational baryogenesis.
+Their work included the higher-order derivatives of Gauss-Bonnet terms that work in order to
+produce the baryon asymmetry. Sharif and Ikram [14] gives rise to a new theory by following
+2
+
+the footsteps of Harko. They coupled the matter part T with the geometric part of the f(G)
+theory, making it f(G, T ) cosmology. They investigated the validity of their theory with the
+help of energy conditions. Later on, Bhatti et al. [15] worked on the f(G, T ) theory to carry
+out the investigation of some physically feasible features of compact star formation. They
+inferred that the compactness of a star model grows at the core whereas the energy conditions
+remain constant. Yousaf and his mates [16] inspired by [17], have recently developed a novel
+f(G, T 2) to present the complexity of structural scalars from the use of Herrera’s method
+of splitting scalars. They considered the exponential coupling of Gauss-Bonnet terms as a
+functional form as f(G, T 2) = αGn(βGm + 1) + ηT 2, to explore the validity of their solutions
+for the Darmois and Israel conditions. They also worked on the non-static complex structures
+under the same theory to describe the effects of an electromagnetic field. They used specific
+model configuration i.e, f(G, T 2) = k1Gm(k2Gn + 1) + λT 2, in their work.
+Bouncing cosmology has gained much reputation over the past few years, because of
+its independent hypothetical nature from different standard comic problems.
+Guth [18]
+during 1980′s, had put forward his inflationary theory to tackle early and late time cosmic
+evolutionary problems. He remained successful in solving some related problems, but the
+answer to the initial singularity is still under concern. One of the best hypotheses to answer
+the singularity problem is the bouncing nature of the universe. The nature of the bouncing
+universe allows a certain universe model to transit from a pre-big crunch (contracted) phase
+into a new big bang (expanded) phase with the exclusion of singularity during the whole event
+[19]. Steinhardt and Ijjas [20] are considered to be the pioneers of the bouncing hypothesis.
+They devised a wedge diagram for a smooth bouncing method to explore the consequences
+of some cosmological problems. Sahoo et al. [21] worked on the non-singular bouncing by
+assuming the specific coupling of R and T as f(R, T ) = R + χRT , for 0 < χ < π
+4. They
+allowed such a parametric approach for the Hubble parameter to provide no singularity
+during the bounce.
+They used quintom and phantom scalar field configurations for the
+bouncing paradigm. Bamba and his collaborators [22] inspected the singularity-free concept
+of bounce by considering an exponential form of scale factor a(t) = σ exp(λt) + τ exp(λt)
+under the effect of f(G) gravity. They checked the stability of their assumed solution under
+the restricted parametric scheme.
+Yousaf et al. [23, 24] explored the bouncing universe with a specific functional form of
+Hubble parameter by taking exponential f(G, T ) form. Different cosmic models are under
+consideration for the scale factor in order to determine the value of expansion and contraction
+at the current cosmic phase and also to predict the current phase equation of state. These
+models predicted different results in the literature. However, cosmography provided us a
+benefit in processing cosmological data for explaining the universal kinematics without the
+involvement of the gravity model and hence provided that the cosmography can be employed
+with the Taylor expansions as an alternative scheme. Also, the cosmographic analysis for
+3
+
+the FLRW universe, is helpful in such a way that it can put aside the effect of the dy-
+namical field equations [25]. Gruber et al. [26] studied an alternative approach to describe
+cosmography by extending the conventional methodology. They resulted from numerical
+values of the cosmographic parameters by applying the Pad´e approximations. The testing
+of the ΛCDM model had been conveyed by Busti et al. [27] with the use of cosmographical
+analysis. Capozziello et al. provided cosmography as a non-predictive phenomenon when
+the redshift parameter becomes z ≈ 1. They used the pad´e approximations for the fifth
+order and resulted the divergence of data at the higher levels of the approximations. Lobo
+et al. [28] evaluated the dynamics of the redshift drift. They used the expanding FLRW
+universe to produce a general matter and low redshift model with the use of different vari-
+ables. However, the cross-correlation of large-scale quasars can be used and translated with
+the CMB and BAO scale data to produce the best for Hubble parameter H(z) and angular
+diametric distance SA. Also, the cosmic chronometers approach can be done to predict the
+model independent H(z) measurements which have been extensively used for cosmological
+applications [29–31]. The low redshift data set with the inclusion of the megamasers and
+chronometers had been presented by Krishnan and others [32]. They result that the Hubble
+constant H0, showed descending behavior with the redshift and having non-zero slop when
+fitted on the line by statistical means.
+Font et al. [33] studied correlation technique for
+quasars by using Lya absorption and produced the best line of fit for Planck’s data. They
+generated different results on the measurements of the Hubble parameter and the angular
+distance. One important thing is to develop such a cosmic Hubble parameter that comes
+from early to late span in such a way that it changes from a low to a high value. The Gaus-
+sian method helped to predict but provided a non-transitional behavior for both Λ and ω
+epochs. The null energy condition also proved to be an important restriction for the cut-off
+model, when compared with Hubble parameter data [34]. King et al. [35] studied the future
+approximations of the redshift by the inclusion of dark energy. They tested the equation of
+state by the linear parametrization technique. Hu et al. [34] reported different values of the
+Hubble constant by the Gaussian method. Their research produced an effective reduction
+in the Hubble crisis and proposed the non-transitional behavior of the Hubble constant.
+Different dark energy models respective to holography and agegraphy had been conducted
+by Zhang et al. [36]. They produced different energy conditions for different red shift values
+and resulted in an effective role of energy conditions for different cosmic ages.
+In this article, we implemented a functional form of the Hubble parameter that evolves
+periodically with cosmic time t and investigate the bouncing nature of the universe in
+f(G, TαβT αβ) gravity using a flat FLRW peacetime. This analysis of the bouncing uni-
+verse involves one of the most important forms of EoS parameter proposed in the litera-
+ture [37–39]. The outline is given as: Sect.2 provides a brief introduction to f(G, TαβT αβ)
+gravity with the necessary formalism of FLRW metric and modified field equations. Sect.3
+4
+
+builds the Hubble parameter as a bouncing solution for the produced field equations. The
+cosmographic parameters are also evaluated in this section. We provide the mathematical
+expressions of energy density and matter pressure for the assumed EoS parameter form in
+Sect.4. The energy conditions are also formulated in the same fashion. Detailed graphical
+profiles of energy conditions are represented in the same section to discuss the evolution
+of the universe under the influence of restricted free parameters. Finally, the concluding
+remarks are made in Sect.5.
+2
+f(G, TαβT αβ) Formalism
+The modified action for the f(G, TαβT αβ) gravity theory is defined as [16]
+Af(G,TαβT αβ) =
+√−g
+2κ2
+��
+d4x[f(G, TαβT αβ) + R] +
+�
+d4xLm
+�
+,
+(1)
+where R and G symbolize the Ricci scalar and the Gauss-Bonnet scalar terms, respectively
+and are provided as
+R ≡ gαβRαβ,
+G ≡ RξζαβRξζαβ − 4RξζRξζ + R2,
+(2)
+and κ2 = 8πG (G be the gravitational constant) and Lm = −p. Also, the term g implies the
+trace of the metric tensor gαβ with Tαβ, Rξζαβ and Rαβ indicate the stress energy-momentum
+tensor, the Riemannian tensor, and the Ricci tensor respectively. The expression for Tαβ is
+given as
+Tαβ =
+−2
+√−g
+δ(√−gLm)
+δgαβ
+.
+(3)
+Equation (3) yields the following expression, due the dependency of the matter Lagrangian
+Lm on gαβ components
+Tαβ = gαβLm − 2∂Lm
+∂gαβ .
+(4)
+Now, by taking the variation of Eq.(1) with respect to the term gαβ, we get the following
+field equations for the f(G, TαβT αβ) theory as
+Rαβ − 1
+2Rgαβ = T eff
+αβ ,
+(5)
+where the term T eff
+αβ takes the following form
+T eff
+αβ
+=
+κ2Tαβ − ΘαβfT 2(G, T 2) + 1
+2gαβf(G, T 2) − (2RRαβ − 4Rε
+αRεβ − 4RαεβηRεη
+5
+
++2Rεηδ
+α Rβεηδ)fG(G, T 2) − (2R∇2gαβ − 2R∇α∇β − 4Rαβ∇2 − 4gαβRεη∇ε∇η
++4Rε
+α∇β∇ε + 4∇ε∇αRε
+β + 4Rαεβη∇ε∇η)fG(G, T 2),
+(6)
+where,
+Θαβ ≡ δ(TµνT µν)
+δgαβ
+= 2T ξ
+α Tβξ − T Tαβ − 4T µν ∂2Lm
+∂gαβgµν − 2Lm(Tαβ − 1
+2T gαβ)
+(7)
+T 2 = TαβT αβ,
+∇2 = ∇α∇α
+(8)
+The terms fG(G, T 2) and fT 2(G, T 2) used above are defined as
+fG(G, T 2) ≡ df(G, T 2)
+dG
+,
+and fT 2(G, T 2) ≡ df(G, T 2)
+dT 2
+.
+(9)
+The trace of the above-defined field equations is produced as
+T − ΘfT 2(G, T 2) + 2GfG(G, T 2) − 2R∇2fG(G, T 2) + 4Rαβ∇α∇βfG(G, T 2) = 0.
+(10)
+Equation (10) shows the non-conversed situation of the stress energy-momentum tensor.
+Also, the properties of GR can be recovered for f(G, T 2) = 0. Similarly if we put f(G, T 2) =
+f(G), we get the properties of f(G) gravity.
+Now, as we are concerned to study the bouncing nature of the universe, so we consider
+the fluid distribution to be perfect throughout the cosmic evolution. For this, we take
+Tαβ = (ρ + p)VαVβ − pgαβ,
+(11)
+here, the four-vector velocity is defined by V β with
+V β = (1, 0, 0, 0),
+V βVβ = 1 , V β∇ζVζ = 0.
+(12)
+In addition, ρ defines the energy density part and p defines the pressure part of the stress
+energy-momentum tensor. Also the geometric background considered to be in a FLRW
+space time [40], so it implies
+ds2 = dt2 − a2(t)Σidx2
+i ,
+i = 1, 2, 3.
+(13)
+The metric component a(t) symbolizes the scale factor, that contributes to the Hubble
+parameter as H =
+˙a(t)
+a(t).
+Using Eq.(13) and Eq.(7) in Eq.(5), we get the following field
+equations
+6
+� ˙a
+a
+�2
+− 24
+� ˙a
+a
+�3
+˙fG + 24
+�¨a
+a
+� � ˙a
+a
+�2
+fG − f − 2(ρ2 + 3p2 + 4ρp)fT 2 = 2ρκ2,
+(14)
+6
+
+− 2
+�
+2¨a
+a +
+� ˙a
+a
+�2�
++ 16
+�¨a˙a
+a2
+�
+˙fG + 8
+� ˙a
+a
+�2
+¨fG − 24
+�¨a
+a
+� � ˙a
+a
+�2
+fG + f = 2pκ2.
+(15)
+To draw the conclusions on the field equations, we just need some functional form of f(G, T 2).
+As, there are many functional forms regarding the interaction of matter with the curvature
+terms, in order to deal with the issues of cosmic evolution. Various coupling models can be
+used to evaluate the formations of both energy density and matter pressure, like one can
+take f(G, T 2) = G + 2f(T 2) that may help to provide an analysis about ΛCDM epoch.
+However, the other choice is f(G, T 2) = f1(G) + f2(T 2) that may be worked as a correction
+to f(G) gravity theory because of f2(T 2).
+Similar forms have been explored in [41, 42]
+and provided some distinct results due to the direct minimal curvature matter coupling.
+Also, f(G, T 2) = f1(G) + f2(G)f3(T 2) can be taken because of an explicit non-minimally
+coupling nature between geometric parameters and matter variables [43]. So, we considered
+the following form to produce the validating results.
+f(G, T 2) = f1(G) + f2(T 2).
+(16)
+To produce a bouncing universe, we need some functional forms of f1 and f2 that not only
+describe the accelerating expansion of the universe but also explain inflation to a great
+extent. For this, the higher power curvature terms perform well to eliminate such issues.
+Elizalde [44] introduced the power forms of the curvature scalar as ηRn (n ≥ 1) and produced
+the cosmological dynamics, so we consider the specific form of the f1 as the quadratic power
+model, so
+f1(G) = G + αG2.
+(17)
+Also, we take χ2 as
+f2(T 2) = 2λT 2.
+(18)
+So, by using Eq.s (17) and (18) in the field equations, we get
+6H2 − 48αH3G ˙G + αG2 = 2κ2ρ + 6λρ2 + 18λp2 + 16λρp,
+(19)
+and
+−2(2 ˙H + 3H2) + 32( ˙H + H2)αG ˙G + 16αH2( ˙G2 + G ¨G) − αG2 = 2κ2p − 2λρ2 − 6λp2.
+(20)
+In order to reduce the complexity of the Eq.(19) and Eq.(20), we utilize p = ωρ, as the EoS
+used in [37–39]. So we get the relations,
+(3λ + 9λω2 + 8λω)ρ2 + κ2ρ − (3H2 − 24αH3G ˙G + α
+2 G2) = 0
+(21)
+7
+
+and
+(− λ
+ω2 − 3λ)p2 + κ2p + ((2 ˙H + 3H2) − 16( ˙H + H2)αG ˙G − 8αH2( ˙G2 + G ¨G) + α
+2 G2) = 0. (22)
+where, G = 24H2( ˙H + H2). Yousaf and his collaborators checked the stability of cosmic
+models in various modified gravity theories [45–48].
+3
+Hubble Parameter and Cosmography
+This section mainly focuses on describing the evolutionary behavior of these above-described
+dynamical terms. Hence, we consider a trigonometric form of the H(t) which feasibly provides
+a bounce solution [44,49], as follows
+H(t) = ζ sin(φt)h(t).
+(23)
+This parameterized form of H(t) includes ζ and φ, which are considered to be constants
+here. The choice of h(t) depends on the periodic values of the function sin(φt), so the form
+of h(t) can be chosen as periodic, that cooperates with the non-vanishing values of the above
+trigonometric function. This artificial approach of choosing such an ansatz can be considered
+as a numerical analysis of making the bouncing solution. One interesting feature is possessed
+by the term ζ, which can work well as a phase changer for the value of H(t). We consider
+h(t) as
+h(t) = exp(ϕt),
+(24)
+where ϕ acts as a constant. Finally, we have
+H(t) = ζ sin(φt) exp(ϕt).
+(25)
+This functional form of the Hubble parameter is helpful to study cosmic evolutionary expan-
+sion and contraction. This form of the Hubble parameter gives us the bounce at t = 313,
+depending upon the values of ϕ = 0.001 and φ = 0.01 provided in Fig.1.
+We have re-
+stricted the values of H(t) in the positive era of time. The basic scale factor form for this
+parameterized Hubble parameter becomes
+a(t) = exp
+�ζ exp(ϕt)(ϕ sin(φt) − φ sin(φt))
+ϕ2 + φ2
+�
+.
+(26)
+Similarly, the set of dynamical parameters that are derived from the Taylor series expansion
+of the scale factor is termed as cosmographic factors. These factors helped to obtain the
+8
+
+cosmological concordance with the assumptions of the universal homogeneity and isotropy
+on large cosmic scales [27,50]. These include deceleration, jerk and snap parameters. These
+factors allow us to check the compatibility of the scale factor and the Hubble parameter.
+The negative value of the deceleration parameter q describes the accelerated expansion of
+the universe. Similarly, jerk j and snap s determine the expansion rate of the toy universe
+model. The mathematical interpretation for these cosmography elements are defined as
+q = − 1
+H2
+1
+a
+d2a
+dt2 = −1 − 1
+ζ (e−ϕt csc(φt)(φ cot(φt) + ϕ)),
+(27)
+j = 1
+H3
+1
+a
+d3a
+dt3
+=
+1 + 1
+ζ2(e−2ϕt csc(φt)(3ζeϕt(φ cot(φt) + ϕ)
++ csc(φt)(2ϕφ cot(φt) + ϕ2 − φ2))),
+(28)
+and
+s = 1
+H4
+1
+a
+d4a
+dt4 = −
+1
+3ζ(3ζeϕt + 2 csc(φt)(φ cot(φt) + ϕ))(2e−ϕt csc(φt)(csc(φt)
+(3ζϕeϕt sin(φt) + ϕ2 − φ2) + φ cot(φt)(3ζeϕt + 2ϕ csc(φt)))).
+(29)
+Fig.1 shows the progression of the Hubble (left panel) and scale parameters (right panel)
+along the positive time axis. Similarly, the development of jerk (left panel) and snap factors
+(right panel) are provided in the fig.2. The evolution of the deceleration parameter towards
+the negative value i.e, q → −1, before the bouncing point, provided in fig.6, shows the
+accelerating universe.
+4
+Energy Conditions under the EoS Parameter
+For a specific cosmology model, energy conditions play an important role to make its val-
+idation for the restricted free parameters. These energy conditions help to maintain the
+specifications of the certain cosmic model [51–55]. Similarly, these energy conditions also
+work for the bouncing cosmology and provide a reasonable approach to validate the proce-
+dure for our toy bouncing model. These conditions are described as
+• Dominant energy condition (DEC)⇔ ρ ≥ 0 , ρ ± p ≥ 0.
+• Strong energy condition (SEC)⇔ ρ + 3p ≥ 0 , ρ + p ≥ 0.
+• Weak energy condition (WEC)⇔ ρ ≥ 0 , ρ + p ≥ 0.
+9
+
+H � 0
+Bouncing 
+H �
+H � 0
+300
+305
+310
+315
+320
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+t
+a
+a�t� � 0
+0
+50
+100
+150
+200
+250
+300
+350
+�0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+t
+a
+Figure 1: The illustrations of Hubble parameter and scale factor with fixed values of ϕ =
+0.001 and φ = 0.01.
+280
+290
+300
+310
+320
+330
+340
+�4
+�2
+0
+2
+4
+t
+j
+300
+310
+320
+330
+340
+�1.0
+�0.5
+0.0
+0.5
+1.0
+t
+s
+0
+50
+100
+150
+200
+250
+300
+350
+�1.0
+�0.5
+0.0
+0.5
+1.0
+t
+s
+Figure 2: The illustration of jerk and snap factors with fixed values of ϕ = 0.001 and
+φ = 0.01.
+10
+
+6F
+-=1
+2=0.8
+■=0.6
+M5=0.4
+■2=0.2
+0
+100
+200
+300
+400
+500H
+量5=0.8
+拉2=0.6
+2=0.4
+=0.2
+0
+100
+200
+300
+400
+500
+600
+t• Null energy condition (N EC)⇔ ρ + p ≥ 0.
+• Trace energy condition (T EC)⇔ ρ − 3p ≥ 0.
+The positivity of DEC, SEC and WEC passes on the validity and necessity of the bouncing
+concept. However, the violation of N EC has a major role. This violation is different in the
+GR context. Universal bouncing scenario is one of those ideas that provides a chance to
+discuss the singularity-free universal beginning. Many proposals in the literature suggested
+avoiding this singularity through quantum aspects, but these don’t have such reliability to fit
+in the gravitational theory. So, at this point gravitational theories allow a specific mechanism
+to check the validity of the bounce model and as well its own. Null energy condition is one
+such tool to help achieve the task. Also, it has been proved that in the context of GR, the
+violation of N EC is extremely difficult to be achieved for local-field models. So, effective
+field theories provide a chance to recognize the violation of the N EC and to allow a non-
+singular bounce [56–59].
+One such effective field is f(G, TαβT αβ) theory that provides a
+chance to study the quadratic nature of the energy terms i.e, energy density and matter
+pressure [16, 60].
+However, it also allows getting a non-singular bounce for the assumed
+gravity model form. For an excellent bouncing model, the value of H(t) turns out to be
+˙H = −4Gρπ(1 + ω) > 0 for the formulation of GR. However, if the N EC gets violated,
+we have the surety to get a bouncing scenario. To provide the mathematical formulation
+of the energy conditions, we consider Eqs.
+(21) and (22).
+Also, the EoS parameter in
+the negative regime provides the present cosmic evolution [61–63] and becomes favorable in
+the bouncing context with ω(t) ≈ −1. However, bouncing cosmology provides the possible
+geodesic evolution of the universe by avoiding the singularity along with the resolution of the
+horizon problem, flatness problem, entropy problem and many more [5]. For the modified
+gravity, EoS parameter enables us to study the universal dynamics. In this study, we used
+EoS parameter [44] to obtain the possible chance of obtaining a bounce solution in f(G, T 2)
+as
+ω(t) = −k log(t + ǫ)
+t
+− 1,
+(30)
+here k is assumed to be a constant. This particular form of the EoS parameters allows us to
+study the contracting and expanding behavior without involving the Hubble parameter as
+well as the scale factor. Elizalde et al. [44] produces cosmological dynamics by considering
+R2 gravity and logarithmic trace terms. They checked the effects of the λ parameter in the
+gravity model f(R, T ) = R+λR2+2β ln(T ) along with the bouncing solution depending on
+the two EOS parameters. Our work first described the choice of Hubble parameter and its
+effects on the dynamical field equations and then involves the EOS parameter. We only took
+one of the ω(t) value, because this state factor after the bouncing point remains negative and
+11
+
+becomes ω(t) ≈ −1. Also, the current cosmic expansion and Λ − CDM can be verified by
+this state factor. However, the dynamic properties are greatly affected under the influence
+of this EoS parameter form. Hence, the general forms of the Eqs.(21) and (22), under the
+influence of Eq.30, are presented as
+ρ
+=
+−
+1
+2λ(9ω2 + 8ω + 3)(κ2 + (κ4 − 12ζ2λ(9ω2 + 8ω + 3)e2ϕt sin2(φt)(2304αζ7e7ϕt sin4(φt)
+(sin(φt)(ζeϕt sin(φt) + ϕ) + φ cos(φt))(4ζϕeϕt sin3(φt) + 2ζφeϕt sin(2φt) sin(φt)
+−
+(φ2 − 3ϕ2) sin2(φt) + 2φ2 cos2(φt) + 3ϕφ sin(2φt)) − 96αζ4e4ϕt sin2(φt)
+(sin(φt)(ζeϕt sin(φt) + ϕ) + φ cos(φt))2 − 1))
+1
+2
+(31)
+p
+=
+1
+2(3λω2 + λ)(κ2ω2 + (κ4ω4 + 4ζω2(3λω2 + λ)eϕt(18432αζ9e9tϕ(2ϕ(2ϕ + 1) − φ2)
+sin10(tφt) + 4608αζ8e8tϕ(ϕ2(25ϕ + 22) − (13ϕ + 2)φ2) sin9(φt) + 9216αζ7φ3e7ϕt
+sin5(φt) cos3(φt)(7ζeϕt sin(φt)10ϕ + 8) + 288αζ7e7ϕt(144ϕ4 + 320ϕ3 − 16(9ϕ + 4)
+ϕφ2 − 1) sin8(φt) + 576αζ6φ4e6ϕt sin4(2φt)(ζeϕt + 2 csc(φt)) + 576αζ6ϕe6ϕt
+(48ϕ3 − 16ϕφ2 − 1) sin7(φt) − 96αζ5ϕ(3ϕ − 8)e5ϕt sin6(φt) + 192αζ4e4ϕt(3ϕ2 − φ2)
+sin5(φt) + 2 sin(φt)(5760αζ6ϕφ3e6ϕt sin3(2φt) + 48αζ4φ2e4ϕt sin2(2φt) − ϕ) + 288α
+ζ5φ2e5ϕt sin4(φt) cos2(φt)(192ζ4e4ϕt sin4(φt) + 32ζ3(31ϕ + 10)e3ϕt sin3(φt)
++16ζ2e2ϕt(ϕ(45ϕ + 56) − 7φ2) sin2(φt) + 32ζeϕt(17ϕ2 − φ2) sin(φt) − 1) − 2φ cos(φt)
+(−18432αζ9(4ϕ + 1)e9ϕt sin9(φt) − 2304αζ8e8ϕt(ϕ(75ϕ + 44) − 11φ2) sin8(φt) − 9216αζ7
+e7ϕt(3ϕ2(3ϕ + 5) − (4ϕ + 1)φ2) sin7(φt) − 288αζ6e6ϕt(192ϕ3 − 32ϕφ2 − 1)
+sin6(φt) + 96αζ5(3ϕ − 4)e5ϕt sin5(φt) − 576αζ4ϕe4ϕt sin4(φt) + 1)
+−3ζeϕt sin2(φt)))
+1
+2
+(32)
+Now, the profiles of energy density and pressure under the presence of Eq.(30), are provided
+in fig.3. The plots indicate that the energy density suffers a positive behavior for the assumed
+values of free parameters. Similarly, the negative behavior for the pressure term indicates
+that the universe is in the accelerated expansion phase. However, the positive density proves a
+strong validation for the verification of the energy conditions. Also, one can get the positive
+and alternate trends of the both terms for different time periods due to the oscillatory
+behavior of the assumed Hubble parameter. We restrict our work for the positive density
+and negative pressure behavior to ascertain the energy conditions. The evolutionary profiles
+of the energy conditions are provided in the figs. 4 and 5. The N EC plot shows the violation
+with in the bouncing regime and confirms the major verification for the universe to attain
+12
+
+Ζ
+t
+Ρ
+Ζ
+t
+p
+Figure 3: The illustration of energy density and matter pressure with fixed values of α =
+0.005, k = 0.5, ϕ = 0.001, ǫ = 0.001, φ = 0.01, κ = 1 and λ = −0.005.
+a bounce with in the framework of FLRW spacetime. The violated WEC and SEC are
+given in the left plots of the figs. 3 and 4. The violated SEC also maintains the recent
+observations for the accelerating universe [52]. One important energy condition i.e, T EC has
+also been given in this recent study. The positive profiles for the DEC and T EC are given
+in the fig.5. The evolution of these energy conditions is strictly dependent on the values
+of the free parameters used in this study. However, one can get another configuration of
+these physical factors by implementing the different free parameters. The evolution of EoS
+parameter is provided in fig.6 to encounter the negative value i.e, ω(t) ≈ −1, for the current
+expansion phase of the universe.
+5
+Discussions
+This work involves the study of bouncing cosmology for an isotropic configuration of fluid
+Tαβ and FLRW metric. We comprehend this work under f(G, TαβT αβ) theory of gravitation
+by assuming a specific model i.e, f(G, T 2) = G + αG2 + 2λT 2 with α and λ are constants,
+serving as free parameters. This is the first-ever attempt to cover bouncing cosmology in
+the f(G, TαβT αβ) theory. By the consideration of a specific functional form of the Hubble
+parameter, we discuss the evolution of cosmographic parameters.
+The assumption of a
+well-known equation of state (EoS) parameter, ω(t) = −k log(t+ǫ)
+t
+− 1, is used as a direct
+implementation to represent the dynamical behavior of energy density, matter pressure, and
+energy conditions. The free parameters are restricted to the special values provided in each
+13
+
+-51
+52
+1.0
+53
+54
+54
+100
+200
+300
+20.048
+46
+44
+1.0
+42
+20.5
+40
+100
+38
+200
+300
+0.0Ρ � p
+t
+Ζ
+Ρ � 3 p
+t
+Ζ
+Figure 4: The illustration of N EC and SEC with fixed values of α = 0.005, k = 0.5,
+ϕ = 0.001, ǫ = 0.001, φ = 0.01, κ = 1 and λ = −0.005.
+Ρ � p
+Ζ
+t
+Ρ � 3 p
+t
+Ζ
+Figure 5: The illustration of DEC and T EC with fixed values of α = 0.005, k = 0.5,
+ϕ = 0.001, ǫ = 0.001, φ = 0.01, κ = 1 and λ = −0.005.
+14
+
+204
+202
+305
+1.0
+200
+2005
+195
+198
+70.5
+100
+196
+200
+300
+0.098
+96-
+96
+94
+100
+200
+300
+30.0105
+-110
+110
+1.0
+115
+120
+120
+100
+125
+200
+300
+0.01.0
+10
+0.5
+-15
+100
+200
+00
+0.00
+50
+100
+150
+200
+250
+300
+350
+�2.0
+�1.5
+�1.0
+�0.5
+0.0
+t
+Ω
+290
+300
+310
+320
+330
+�10
+�5
+0
+5
+10
+t
+q
+0
+50
+100
+150
+200
+250
+300
+350
+�10
+�5
+0
+5
+10
+t
+q
+Figure 6: The illustration of EoS and deceleration parameters with fixed values of k = 0.5
+and ǫ = 0.001.
+graph plot and are used for H(t) to act as the bouncing solution. The viability of energy
+conditions is studied with the help of a graphical approach. Following are the concluding
+remarks for this present work.
+• The Hubble parameter H(t) used in this study is considered to have a trigonomet-
+ric functional form. The evolutionary behavior of different cosmographic factors is
+described under the same form of H(t). This parameterized form of H(t) depends
+on the periodic values of the function sin(φt) and h(t). We considered this h(t) as
+a nonvanishing function for the periodic values of sin(φt). A perfect bouncing model
+allows the Hubble parameter to show the contraction phase i.e, H < 0, and when the
+universe expands it becomes H > 0. During this expansion and contraction phase,
+there is the point in between, at where H(t) becomes zero. So, in order to produce
+such a scenario, we have arranged the constants (φ and ϕ) in the Hubble parameter
+(H(t) = ζ sin(φt) exp(ϕt)) to some specific values and notice the bounce at t = 313.
+However, t = 313 is significant in such a way that all the energy conditions necessary
+for the bounce, get satisfied accordingly till t = 313, depending on the values of φ
+and ϕ. One can also produce other values of t for bounce by restricting other values
+of φ and ϕ. The plot of H(t) is given in fig.1. The Hubble parameter gives us the
+bounce at t = 313 which is the future singularity in the scale factor, see fig.1. The
+mathematical forms of deceleration, jerk, and snap are evaluated with the same H(t).
+The deceleration parameter tends to have a negative trend i.e, q(t) approaches −1,
+which can be seen in fig.6. Similarly, the trends of jerk and snaps are given in fig.2
+with j(t) approaches to 1 and s(t) approaches to 0. All these values show a deflection
+15
+
+at the bouncing point, that fits in for the bouncing universe.
+• We ensure the configuration of the bouncing cosmology by studying energy conditions.
+These energy conditions are provided in terms of energy density and matter pressure
+derived from the modified field equations. We assumed a specific EoS parameter in the
+form ω(t) = −k log(t+ǫ)
+t
+− 1. This EoS parameter helped to maintain the positive and
+negative growth of energy density and matter pressure for the limited bouncing time
+period. The profiles of ρ and p are provided in the fig.3. However, the mathematical
+expression for these terms is evaluated in Eqs.(31) and (32).
+• Under the restricted values of the free parameters, α = 0.005, k = 0.5, ϕ = 0.001,
+ǫ = 0.001, φ = 0.01, κ = 1 and λ = −0.005, we get the violation of the N EC and SEC.
+The violated N EC derives the bouncing nature of the universe. However, the violated
+SEC and WEC provide the phase of cosmic expansion. suitable with the observational
+data. The left plots of figs.3 and 4 shows the violated SEC and WEC. Similarly, the
+positive behavior of DEC and T EC assure that the assumed model configuration is
+valid. Figure 5 represents the illustration of DEC and T EC. Also, the evolution of EoS
+can be seen in fig.6, showing that ω(t) → −1. This value of ω(t) favors the current
+accelerated expansion phase of the universe [61–63].
+• The above discussion provides that the bouncing evolution of the universe, studied in
+the framework of f(G, T 2) = G + αG2 + 2λT 2 and agrees with the recent astronomical
+observations [64, 65] i.e, all the energy conditions are fully satisfied, a great negative
+pressure behavior had been observed and provided help to study the late time acceler-
+ated universe [44]. However, this study can be used in the future for different models
+of the scale factors and Hubble parameters.
+• We finally conclude that the bouncing evolution of the universe can be studied effec-
+tively with the oscillating nature of the scale factor under the flat FLRW regime.
+References
+[1] C. J. Hogan, The little book of the big bang: A cosmic primer. Springer Science &
+Business Media, 1998.
+[2] A. H. Guth, “Eternal inflation,” Ann. N. Y. Acad. Sci., vol. 950, no. 1, pp. 66–82, 2001.
+16
+
+[3] T. Padmanabhan and T. R. Seshadri, “Does inflation solve the horizon problem?,”
+Class. Quantum Gravity, vol. 5, no. 1, p. 221, 1988.
+[4] J. Earman and J. Mosterin, “A critical look at inflationary cosmology,” Philos. Sci.,
+vol. 66, no. 1, pp. 1–49, 1999.
+[5] A. Ijjas and P. J. Steinhardt, “Bouncing cosmology made simple,” Class. Quantum
+Grav., vol. 35, no. 13, p. 135004, 2018.
+[6] E. Alesci, G. Botta, F. Cianfrani, and S. Liberati, “Cosmological singularity resolution
+from quantum gravity: The emergent-bouncing universe,” Phys. Rev. D, vol. 96, no. 4,
+p. 046008, 2017.
+[7] P. Das, S. Pan, S. Ghosh, and P. Pal, “Cosmological time crystal: Cyclic universe with
+a small cosmological constant in a toy model approach,” Phys. Rev. D, vol. 98, no. 2,
+p. 024004, 2018.
+[8] J. Mielczarek, M. Kamionka, A. Kurek, and M. Szyd�lowski, “Observational hints on the
+big bounce,” J. Cosmol. Astropart. Phys., vol. 2010, no. 07, p. 004, 2010.
+[9] Y.-F. Cai, R. Brandenberger, and P. Peter, “Anisotropy in a non-singular bounce,”
+Class. Quantum Gravity, vol. 30, no. 7, p. 075019, 2013.
+[10] Y.-F. Cai and X. Zhang, “Evolution of metric perturbations in a model of nonsingular
+inflationary cosmology,” J. Cosmol. Astropart., vol. 2009, no. 06, p. 003, 2009.
+[11] M. Roshan and F. Shojai, “Energy-momentum squared gravity,” Phys. Rev. D, vol. 94,
+no. 4, p. 044002, 2016.
+[12] S. Nojiri and S. D. Odintsov, “Modified Gauss–Bonnet theory as gravitational alterna-
+tive for dark energy,” Phys. Lett. B, vol. 631, no. 1-2, pp. 1–6, 2005.
+[13] A. V. Astashenok, S. D. Odintsov, and V. K. Oikonomou, “Modified gauss–bonnet
+gravity with the lagrange multiplier constraint as mimetic theory,” Class. Quantum
+Gravity, vol. 32, no. 18, p. 185007, 2015.
+[14] M. Sharif and A. Ikram, “Energy conditions in f (G, T) gravity,” Eur. Phys. J. C .,
+vol. 76, no. 11, pp. 1–13, 2016.
+[15] M. Z. Bhatti, M. Y. Khlopov, Z. Yousaf, and S. Khan, “Electromagnetic field and
+complexity of relativistic fluids in f (G) gravity,” Mon. Not. Roy. Astron. Soc., vol. 506,
+pp. 4543–4560, 2021.
+17
+
+[16] Z. Yousaf, M. Z. Bhatti, S. Khan, and P. K. Sahoo, “f (G, TαβTαβ) theory and complex
+cosmological structures,” Phys. Dark Universe, p. 101015, 2022.
+[17] N. Katırcı and M. Kavuk, “f(R, TµνT µν) gravity and cardassian-like expansion as one
+of its consequences,” Eur. Phys. J. Plus, vol. 129, no. 8, pp. 1–12, 2014.
+[18] A. H. Guth, “Eternal inflation and its implications,” J. Phys. A, vol. 40, no. 25, p. 6811,
+2007.
+[19] P. J. Steinhardt and N. Turok, “Cosmic evolution in a cyclic universe,” Phys. Rev. D,
+vol. 65, p. 126003, 2002.
+[20] A. Ijjas and P. J. Steinhardt, “Fully stable cosmological solutions with a non-singular
+classical bounce,” Phys. Lett. B, vol. 764, pp. 289–294, 2017.
+[21] S. Bhattacharjee and P. K. Sahoo, “Comprehensive analysis of a non-singular bounce
+in f (R, T) gravitation,” Phys. Dark Universe, vol. 28, p. 100537, 2020.
+[22] K. Bamba, A. N. Makarenko, A. N. Myagky, and S. D. Odintsov, “Bouncing cosmology
+in modified gauss–bonnet gravity,” Phys. Lett. B, vol. 732, pp. 349–355, 2014.
+[23] Z. Yousaf, M. Z. Bhatti, and H. Aman, “Cosmic bounce with α(e−βG−1)+2λ T model,”
+Phys. Scr., vol. 97, no. 5, p. 055306, 2022.
+[24] Z. Yousaf, M. Z. Bhatti, and H. Aman, “The bouncing cosmic behavior with logarithmic
+law f(G, T) model,” Chin. J. Phys., vol. 79, pp. 275–286, 2022.
+[25] M. Visser, “Jerk, snap and the cosmological equation of state,” Class. Quantum Gravity,
+vol. 21, no. 11, p. 2603, 2004.
+[26] C. Gruber and O. Luongo, “Cosmographic analysis of the equation of state of the
+universe through pad´e approximations,” Phys. Rev. D, vol. 89, no. 10, p. 103506, 2014.
+[27] V. C. Busti, A. de la Cruz-Dombriz, P. K. Dunsby, and D. Saez-Gomez, “Is cosmography
+a useful tool for testing cosmology?,” Phys. Rev. D, vol. 92, no. 12, p. 123512, 2015.
+[28] F. S. N. Lobo, J. P. Mimoso, J. Santiago, and M. Visser, “Dynamical analysis of the
+redshift drift in f l r w universes,” 2022.
+[29] Moresco et al., “A 6% measurement of the hubble parameter at z 0.45: direct evidence
+of the epoch of cosmic re-acceleration,” J. Cosmol. Astropart. Phys., vol. 2016, no. 05,
+p. 014, 2016.
+18
+
+[30] J. P. Hu, F. Y. Wang, and Z. G. Dai, “Measuring cosmological parameters with
+a luminosity-time correlation of gamma-ray bursts,” Mon. Not. Roy. Astron. Soc.,
+vol. 507, no. 1, pp. 730–742, 2021.
+[31] F. Y. Wang, J. P. Hu, G. Q. Zhang, and Z. G. Dai, “Standardized long gamma-ray
+bursts as a cosmic distance indicator,” Astrophys. J., vol. 924, no. 2, p. 97, 2022.
+[32] Krishnan et al., “Is there an early universe solution to hubble tension?,” Phys. Rev. D,
+vol. 102, no. 10, p. 103525, 2020.
+[33] Font-Riberan et al., “Quasar-lyman α forest cross-correlation from boss dr11: Baryon
+acoustic oscillations,” J. Cosmol. Astropart. Phys., vol. 2014, no. 05, p. 027, 2014.
+[34] J.-P. Hu and F.-Y. Wang, “Revealing the late-time transition of ho: relieve the hubble
+crisis,” Mon. Not. Royal Astron. Soc., vol. 517, no. 1, pp. 576–581, 2022.
+[35] A. L. King, T. M. Davis, K. D. Denney, M. Vestergaard, and D. Watson, “High-redshift
+standard candles: predicted cosmological constraints,” Mon. Not. Roy. Astron. Soc.,
+vol. 441, no. 4, pp. 3454–3476, 2014.
+[36] M.-J. Zhang, C. Ma, Z.-S. Zhang, Z.-X. Zhai, and T.-J. Zhang, “Cosmological con-
+straints on holographic dark energy models under the energy conditions,” Phys. Rev.
+D, vol. 88, no. 6, p. 063534, 2013.
+[37] E. Babichev, V. Dokuchaev, and Y. Eroshenko, “Dark energy cosmology with general-
+ized linear equation of state,” Class. Quantum Grav., vol. 22, no. 1, p. 143, 2004.
+[38] J. Haro and E. Elizalde, “Gravitational particle production in bouncing cosmologies,”
+J. Cosmol. Astropart. Phys., vol. 2015, no. 10, p. 028, 2015.
+[39] A. P. Bacalhau, N. Pinto-Neto, and S. D. P. Vitenti, “Consistent scalar and tensor
+perturbation power spectra in single fluid matter bounce with dark energy era,” Phys.
+Rev. D, vol. 97, no. 8, p. 083517, 2018.
+[40] F. Melia, “The Friedmann–Lemaˆıtre–Robertson–Walker metric,” Mod. Phys. Lett. A,
+vol. 37, no. 03, p. 2250016, 2022.
+[41] Z. Yousaf, K. Bamba, and M. Z. Bhatti, “Causes of irregular energy density in f(R, T)
+gravity,” Phys. Rev. D, vol. 93, p. 124048, 2016.
+[42] M. F. Shamir, “Bouncing universe in f (G, T) gravity,” Phys. Dark Universe, vol. 32,
+p. 100794, 2021.
+19
+
+[43] S. Nojiri, S. D. Odintsov, and V. K. Oikonomou, “Modified gravity theories on a nut-
+shell: inflation, bounce and late-time evolution,” Phys. Rep., vol. 692, pp. 1–104, 2017.
+[44] E. Elizalde, N. Godani, and G. C. Samanta, “Cosmological dynamics in R2 gravity with
+logarithmic trace term,” Phys. Dark Universe, vol. 30, p. 100618, 2020.
+[45] M. Sharif and Z. Yousaf, “Instability of meridional axial system in f(R) gravity,” Eur.
+Phys. J. C, vol. 75, p. 194, 2015.
+[46] M. Z. Bhatti, Z. Yousaf, and M. Ilyas, “Existence of wormhole solutions and energy
+conditions in f (R,T) gravity,” J. Astrophys. Astron., vol. 39, p. 69, 2018.
+[47] Z. Yousaf, “Definition of complexity factor for self-gravitating systems in Palatini f(R)
+gravity,” Phys. Scr., vol. 95, p. 075307, 2020.
+[48] M.
+M.
+M.
+Nasir,
+M.
+Z.
+Bhatti,
+and
+Z.
+Yousaf,
+“Influence
+of
+EMSG
+on
+complex systems:
+Spherically symmetric,
+static case,”
+Int. J. Mod. Phys. D,
+p. 10.1142/S0218271823500098.
+[49] M. F. Shamir, “Bouncing cosmology in gravity with logarithmic trace term,” Adv. As-
+tron., vol. 2021, p. 8852581, 2021.
+[50] J. P. Hu and F. Y. Wang, “High-redshift cosmography: Application and comparison
+with different methods,” Astron. Astrophys., vol. 661, p. A71, 2022.
+[51] S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, vol. 1.
+Cambridge university press, 1973.
+[52] M. Visser, “Energy conditions in the epoch of galaxy formation,” Science, vol. 276,
+no. 5309, pp. 88–90, 1997.
+[53] S. Nojiri and S. D. Odintsov, “Effective equation of state and energy conditions in
+phantom/tachyon inflationary cosmology perturbed by quantum effects,” Phys. Lett.
+B, vol. 571, no. 1-2, pp. 1–10, 2003.
+[54] O. Bertolami and M. C. Sequeira, “Energy conditions and stability in f (R) theories of
+gravity with nonminimal coupling to matter,” Phys. Rev. D, vol. 79, no. 10, p. 104010,
+2009.
+[55] L. Balart and E. C. Vagenas, “Regular black hole metrics and the weak energy condi-
+tion,” Phys. Lett. B, vol. 730, pp. 14–17, 2014.
+20
+
+[56] Larson et al., “Seven-year wilkinson microwave anisotropy probe (WMAP*) obser-
+vations: power spectra and WMAP-derived parameters,” Astrophys. J., Suppl. Ser,
+vol. 192, no. 2, p. 16, 2011.
+[57] R. R. Caldwell, “A phantom menace?
+cosmological consequences of a dark energy
+component with super-negative equation of state,” Phys. Lett. B, vol. 545, no. 1-2,
+pp. 23–29, 2002.
+[58] U. Alam, V. Sahni, T. Deep Saini, and A. A. Starobinsky, “Is there supernova evidence
+for dark energy metamorphosis?,” Mon. Not. Roy. Astron. Soc., vol. 354, no. 1, pp. 275–
+291, 2004.
+[59] V. K. Onemli and R. P. Woodard, “Quantum effects can render w < −1 on cosmological
+scales,” Phys. Rev. D, vol. 70, no. 10, p. 107301, 2004.
+[60] Z. Yousaf, M. Z. Bhatti, and S. Khan, “Non-static charged complex structures in
+f(G, TαβT αβ) gravity,” Eur. Phys. J. Plus, vol. 137, no. 3, pp. 1–19, 2022.
+[61] J. Hogan, “Unseen universe: Welcome to the dark side,” Nature, vol. 448, no. 7151,
+pp. 240–246, 2007.
+[62] Corasaniti et al., “Foundations of observing dark energy dynamics with the wilkinson
+microwave anisotropy probe,” Phys. Rev. D, vol. 70, no. 8, p. 083006, 2004.
+[63] J. Weller and A. M. Lewis, “Large-scale cosmic microwave background anisotropies and
+dark energy,” Mon. Not. R. Astron. Soc, vol. 346, no. 3, pp. 987–993, 2003.
+[64] S. Carloni, P. K. Dunsby, S. Capozziello, and A. Troisi, “Cosmological dynamics of rn
+gravity,” Class. Quantum Gravity, vol. 22, no. 22, p. 4839, 2005.
+[65] S. Fay, R. Tavakol, and S. Tsujikawa, “f(R) gravity theories in Palatini formalism:
+Cosmological dynamics and observational constraints,” Phys. Rev. D, vol. 75, no. 6,
+p. 063509, 2007.
+21
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+arXiv:2301.04428v1  [math.QA]  11 Jan 2023
+THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF
+THE JORDAN PLANE
+K. A. BROWN AND J. T. STAFFORD
+Abstract. The Hopf algebra D which is the subject of this paper can be
+viewed as a Drinfeld double of the bosonisation of the Jordan plane. Its prime
+and primitive spectra are completely determined. As a corollary of this anal-
+ysis it is shown that D satisfies the Dixmier-Moeglin Equivalence, leading to
+the formulation of a conjecture on the validity of this equivalence for pointed
+Noetherian Hopf algebras.
+1. Introduction
+1.1. Throughout, k will denote an algebraically closed field of characteristic 0.
+The Hopf k-algebra D of the title was defined and some initial properties were
+derived in [2], with further results in [1, 3]. Our focus here is on the prime and
+primitive spectra of D, which we completely determine. The Hopf algebra D is a
+pointed affine noetherian domain of Gelfand-Kirillov dimension 6 whose definition
+is recalled in §2.1. It is a beautiful algebra with a number of striking properties
+which make it worthy of study from several perspectives, three of which we briefly
+outline in §§1.3-1.5. First we summarise our results and explain where they are
+located.
+1.2. Results. It was proved in [1, Theorem 4.10], and explained in detail here
+in Lemma 2.1 and Theorem 2.2(iv), that the centre Z(D) of D is generated by
+elements z, ω and θ with zθ = ω2. Thus Maxspec(Z(D)) has one singular point,
+namely m0 := ⟨z, ω, θ⟩. There is one other distinctive maximal ideal of the centre,
+namely
+m+ := Z(D) ∩ D+ = ⟨z − 16, ω + 16, θ − 16⟩,
+where D+ is the augmentation ideal of D. Finally, let K denote the kernel of the
+Hopf algebra surjection π : D −→ U(sl(2, k)), mentioned in §1.3, so K is a Hopf
+ideal which is described in Theorem 2.2(i),(ii).
+The main results of this paper are given by the following theorems and give a
+complete description of the prime and primitive ideals of D.
+Theorem 1.1. Retain the above notation. The primitive ideals of D are:
+(I) the maximal ideals mD for m ∈ Maxspec(Z(D)), with m ̸= m0, m+;
+(II) the primitive ideals containing m+D, namely m+D itself together with
+P := π−1(Privspec(U(sl(2, k))));
+2010 Mathematics Subject Classification. Primary 16T05, 16D25; Secondary 16T20, 16S40,
+17B37.
+Both authors are partially supported by Leverhulme Emeritus Fellowships, respectively EM-
+2017-081 and EM-2019-015. The first author thanks Nicolas Andruskiewitsch, Ivan Angiono and
+Hector Pena Pollastri for helpful comments and for sharing early versions of their work.
+1
+
+2
+K. A. BROWN AND J. T. STAFFORD
+(III) the unique prime ideal P0 containing m0D, which has m0D = (P0)2 ⊊ P0.
+Theorem 1.2. In the above notation, the non-primitive prime ideals of D are:
+(A) {0}, K;
+(B) the principal prime ideals pD for every height one prime p of Z(D) except
+p1 = ⟨z, ω⟩ and p2 = ⟨θ, ω⟩;
+(C) height one primes P1, P2, with piD = P 2
+i ⊊ Pi for i = 1, 2, with each Pi
+generated by a normal (but not central) element. Moreover, P1 + P2 = P0.
+Theorem 1.3. Retain the above notation.
+(a) Every non-primitive prime is completely prime.
+Every primitive ideal is
+completely prime, except the co-Artinian maximal ideals (other than the
+counit), which form a subset of P.
+(b) Every prime ideal P, apart from the co-Artinian maximal ideals and (pos-
+sibly) P0, has D/P birationally equivalent to a Weyl algebra An(K), where
+1 ≤ n ≤ 2 and K is a field of transcendence degree at most 2 over k.
+(c) D satisfies the Dixmier-Moeglin Equivalence.
+Theorem 1.1 is proved in Section 4, see in particular Subsection 4.5, while Theo-
+rem 1.2 is proved in Theorem 5.3. Finally, Theorem 1.3 is proved in Subsection 5.2.
+Some questions and a conjecture (Conjecture 5.5) are scattered through the paper.
+1.2. Hopf algebras in duality. The full Drinfeld double D(H) = H ⊲⊳ H◦ of an
+infinite dimensional Hopf algebra H may often be unwieldy due to H having “too
+many” finite dimensional representations and thus leading to an unmanageably
+large finite dual H◦. This has generated significant recent interest in constructing
+doubles D(H) := H ⊲⊳ H′ where H′ is some suitable Hopf subalgebra of H◦; see
+for example [9, 20] and the papers listed in §1.1. Much is at present unclear: for
+example, what is an appropriate definition of a “suitable” Hopf subalgebra H′; and
+does a suitable algebra H′ always exist? The double D of the Jordan plane is a test
+case for these and other questions. In particular, some of the desirable properties
+exhibited by D may form a paradigm for what one might aim for in defining doubles
+in more general settings.
+1.3. Representation theory. A striking feature of the representation theory of
+D is the fact, proved in [1, Theorem 3.11] and recalled here in Theorem 2.2(iii), that
+U(sl(2, k)) is a quotient Hopf algebra of D and the finite dimensional irreducible D-
+modules are precisely the finite dimensional irreducible U(sl(2, k))-modules. This
+immediately suggests a plethora of questions, some of which are addressed in [3],
+where Verma D-modules are introduced. But many others remain untouched: for
+instance, what can be said about the category of locally finite dimensional D-
+modules, and for each primitive ideal P of D can one find a canonical irreducible
+module (hopefully a factor of a Verma module) whose annihilator equals P? The
+first step in these questions is to classify the primitive ideals of D, as we do here.
+1.4.
+Dixmier-Moeglin equivalence.
+The validity of the Dixmier-Moeglin
+Equivalence for an algebra R yields simultaneous representation-theoretic, alge-
+braic and topological characterisations of the primitive ideals amongst the prime
+ideals of R. This equivalence is a feature of some but not all Hopf k-algebras; see,
+for example, [5,6] for discussions of when it holds for a noetherian (Hopf) algebra.
+As we prove in Theorem 1.2, D satisfies the equivalence. See §5.2 for the details,
+
+THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE
+3
+where we also give a rather ambitious Conjecture 5.5, proposing a general result
+encompassing all affine Noetherian pointed Hopf C-algebras.
+Notation. Throughout, all vector spaces and all unadorned tensor products are
+understood to be over the base field k. We denote the comultiplication of a Hopf
+algebra H by ∆ and its augmentation ideal by H+. The Gelfand-Kirillov, or GK
+dimension of an object X is denoted by GKdim(X), while the global (homological)
+dimension, respectively injective dimension of a ring R is denoted by gldim(R),
+respectively injdim(R). For precision, we specify that in the Ore extension T =
+S[v; σ, ∂], multiplication is defined by
+(1.1)
+vs = sσv + ∂(s)
+for s ∈ S.
+It follows that ∂ is a σ-derivation in the sense that ∂(ab) = aσ∂(b) + ∂(a)b. This
+follows the conventions of, for example, [17, p.34].
+2. Preliminaries
+2.1. Definitions and notation. The following definitions and notation from [1])
+will remain in play throughout the paper. First, the Jordan plane is
+J := k⟨x, y : [y, x] = − 1
+2x2⟩,
+with bosonization
+H := J#C∞ = J#⟨g±1⟩, where gxg−1 = x, gyg−1 = y + x.
+Then the Drinfeld double of J is defined to be D := H⟨u, v, ζ⟩, with additional
+relations as follows:
+[u, v] = 1
+2u2; [ζ, v] = −v; [ζ, u] = −u; [u, y] = 1 − g;
+[v, x] = 1 − g + xu; [v, y] = yu − gζ; [v, g] = gu; [ζ, y] = y; [ζ, x] = x;
+[x, u] = [x, g] = [u, g] = [ζ, g] = 0.
+The coalgebra structure, which will mostly not concern us here, is determined for
+H by specifying that g is grouplike and x and y are (g, 1)−primitive; and then
+extended to D by setting u and ζ to be primitive and ∆(v) = v ⊗ 1 + 1 ⊗ v + ζ ⊗ u.
+Observe that K := k⟨u, v, ζ⟩ is a Hopf subalgebra of D and in fact, as one can
+see from the PBW theorem for D as described in [2, Proposition 2.3(ii)], D =
+J ⊗k K as vector spaces. As noted in [2, Lemma 2.2] there is a non-degenerate
+skew pairing between J and K which yields the multiplication relations between
+these subalgebras as in [14].
+2.2. Initial results. We gather together in Theorem 2.2 some of the main results
+of [2] and [1]. We must first define some elements of D, as follows. Set
+(2.1)
+q := ux + 2(1 + g),
+and s := xv + uy + (−1
+2ux + g − 1)ζ − 2(g + 1).
+The following lemma is partly explicit, partly implicit, in [1, §4]. Given a k-algebra
+automorphism σ of a k-algebra H, we say that the element h of H is σ-normal if
+ha = σ(a)h for all a ∈ H.
+Lemma 2.1. Keep the above notation.
+
+4
+K. A. BROWN AND J. T. STAFFORD
+(i) q and s are both σ-normal, where σ is the automorphism of D defined by
+σ(y) = y + 1
+2x, σ(v) = v − 1
+2u,
+with σ acting as the identity on the other generators of D.
+(ii) σ2 equals conjugation by g on D; that is, σ2(h) = ghg−1 for all h ∈ D.
+(iii) The elements z := q2g−1, θ := s2g−1 and ω := qsg−1 are in the centre
+Z(D) of D.
+Proof. (i) and (ii) are easy checks, and (iii) is immediate from (i) and (ii).
+□
+Theorem 2.2. Retain the notation introduced above.
+(i) [2, Proposition 2.7(i)] O(G) := k⟨x, u, g±1⟩ is a normal commutative Hopf
+subalgebra of D, with G = ((k, +) × (k, +)) ⋊ (k∗, ×).
+(ii) [2, Proposition 2.7(ii)] DO(G)+ is a Hopf ideal of D, with an isomorphism
+of Hopf algebras
+(2.2)
+D/DO(G)+ ∼= U(sl2(k)).
+(iii) [1, Theorem 3.11] The finite dimensional irreducible D-modules are the finite
+dimensional irreducible U(sl2(k))-modules given by the epimorphism (2.2).
+(iv) [1, Theorem 4.10] With the notation from Lemma 2.1, the centre of D is
+(2.3)
+Z(D) = k⟨z, ω, θ : zθ = ω2⟩,
+(v) [1, Remark 2.2] D is pointed.
+□
+We’ll need the following labelling of the maximal ideals of Z(D). Note that here
+there are two maximal ideals of Z(D) which require particular attention.
+Notation 2.3. (i) By Theorem 2.2((iv), Maxspec(Z(D)) consists of
+{m(α,γ) := ⟨z −α2γ−1, ω −α, θ −γ⟩ : α ∈ k, γ ∈ k∗} ˙∪ {mβ := ⟨z −β, ω, θ⟩ : β ∈ k}.
+Note that m(α,γ) can be simplified to m(α,γ) = ⟨ω − α, θ − γ⟩, while mβ = ⟨z − β, ω⟩
+when β ̸= 0.
+(ii) It is easy to calculate using the definition of the counit that
+D+ ∩ Z(D) = O(G)+D ∩ Z(D) = m(−16,16).
+We thus denote m(−16,16) by m+.
+(iii) It is clear that the singular locus of Z(D) is {m0}.
+3. Ring-theoretic preparations
+In this section we assemble some properties needed in the analysis of the primitive
+spectrum of D. The proofs are most easily approached by viewing D as an iterated
+Hopf Ore extension starting not from the base field k but from the commutative
+normal Hopf subalgebra O(G) = k⟨x, u, g±1⟩ of Theorem 2.2(i). More precisely:
+Proposition 3.1. D is an iterated Ore extension
+(3.1)
+D = O(G)[y; δ1][ζ; δ2][v; τ, δ3],
+where the derivations δ1 and δ2, the automorphism τ and the τ-derivation δ3 can
+be read off from the defining relations of D given in Subsection 2.1.
+In particular, D is a noetherian domain.
+
+THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE
+5
+Proof. Use the proof of [2, Proposition 1.6] to show that D has basis
+{gaxbucydζevf : a ∈ Z, b, . . . , f ∈ N}.
+Then the form of the relations (2.1) combined with [12, Theorem 1, p.438] show
+that it is indeed an Ore extension.
+□
+Although it will be not needed in this paper, the description (3.1) even describes
+D as an Iterated Hopf Ore Extension (IHOE), in the sense that each extension in
+that formula is itself a Hopf algebra. It also shows that, by setting
+deg x = deg u = deg g = deg g−1 = 0;
+deg y = deg ζ = deg v = 1,
+one obtains a filtration F on D with associated graded algebra
+(3.2)
+grF D = O(G)[y, ζ, v].
+So grF D is a commutative polynomial algebra in 6 variables with one variable
+inverted.
+3.1. Homological properties. In this subsection we note that D has certain use-
+ful homological properties, and we begin with the relevant definitions. A ring A
+is called Auslander Gorenstein if it has finite injective dimension and satisfies the
+Gorenstein condition: if p < q are non-negative integers and M is a finitely gener-
+ated A-module, then Extp
+A(N, A) = 0 for every submodule N of Extq
+A(M, A). The
+ring A is Auslander regular if it is Auslander Gorenstein of finite global dimension.
+Set jA(M) = min{r : Extr
+A(M, A) ̸= 0} for the homological grade of M. Then an
+Auslander Gorenstein ring A of finite GK dimension is called GK-Cohen-Macaulay
+(or just CM), provided that jA(M)+ GKdim(M) = GKdim(A) holds for each such
+M. Obviously affine commutative regular rings are both Auslander regular and
+CM.
+Proposition 3.2.
+(i) D is Auslander regular and CM.
+(ii) D is AS regular in the sense of, say, [19].
+(iii) GKdim(D) = 6 = gldim(D).
+(iv) GK dimension is an exact function on finitely generated D-modules.
+Proof. (i) By [7, Remark, p.157] the filtration F is Zariskian and so the result
+follows from [7, Theorems 3.8, 3.9 and Remark, p. 165].
+(ii) This is immediate from (i) and [11, Lemma 6.1].
+(iii) By [22, Corollary 1.4], GKdim(D) = GKdim(grF(D)) = 6.
+By Proposition 3.1 and [21, Theorem 7.5.3(i)], we have gldim(D) ≤ 6.
+By
+[1, Theorem 3.11] D has a finite dimensional module, say M and the CM condition
+implies that M has homological dimension ≥ 6. Hence gldim(D) = 6.
+(iv) Since jD is exact on finitely generated D-modules by [19, Theorem 2.3], this
+follows from the CM condition.
+□
+
+6
+K. A. BROWN AND J. T. STAFFORD
+3.2. Key lemma. The following lemma will be crucial in our analysis of the prim-
+itive spectrum of D. In its proof, given an ideal B of a noetherian ring S, we denote
+by
+√
+B the ideal of S such that
+√
+B/B is the nilradical of S/B.
+Lemma 3.3. Let M be a finitely generated (right or left) D-module such that either
+Annk[x](M) ̸= 0 or Annk[u](M) ̸= 0. Then
+(i) there exists r ≥ 1 such that
+(m+D)r ⊆ (O(G)+D)r ⊆ AnnD(M);
+(ii) GKdim(M) ≤ 3.
+Proof. (i) Let I := AnnD(M), an ideal of D. Assume that I ∩ k[x] ̸= 0, the proof
+in the other case being exactly similar, but with k⟨u, v⟩ replacing J. One easily
+confirms that every non-zero prime ideal of the Jordan plane J = k⟨x, y⟩ contains x.
+Therefore, since I ∩k[x] ̸= 0, there exists N ≥ 1 such that xN ∈ I ∩k[x] ⊆ I ∩O(G).
+Since O(G) is commutative,
+(3.3)
+x ∈
+�
+(I ∩ O(G)).
+Since I is an ideal of D, [v, I] ⊆ I; moreover, from the defining relations of D and
+the fact that O(G) = k⟨x, u, g±1⟩, [v, O(G)] ⊆ O(G). Therefore
+(3.4)
+[v, I ∩ O(G)] ⊆ I ∩ O(G).
+Since k has characteristic 0 it follows from (3.4) and [17, Lemma 3.20] that
+(3.5)
+[v,
+�
+(I ∩ O(G))] ⊆
+�
+(I ∩ O(G)).
+By (3.3) and (3.5)
+[v, x] = 1 − g + xu ∈
+�
+(I ∩ O(G)),
+so that (1 − g) ∈
+�
+(I ∩ O(G)). Then
+[v, g − 1] = [v, g] = gu ∈
+�
+(I ∩ O(G)),
+so that u ∈
+�
+(I ∩ O(G)). Since O(G)+ is generated by x, u and g − 1 we deduce
+that O(G)+D ⊆
+√
+I, proving (i).
+(ii) By (i) M is a finitely generated D/(O(G)+D)r-module for some r ≥ 1. Since
+D/O(G)+D ∼= U(sl(2, k) by Theorem 2.2(ii), and so has GK dimension 3, (ii)
+follows from this and Proposition 3.2(iv).
+□
+3.3. Ore localisations of D. To help in the analysis of its primitive spectrum we
+need four Ore localisations of D. The first of these is described in [1, Theorem 4.8],
+and the others are similar. These sets are described as follows:
+Definition 3.4. Label the following four subsets of D:
+A := {qi : i ≥ 0} ˙∪ {xj : j ≥ 0},
+B := {si : i ≥ 0} ˙∪ {xj : j ≥ 0},
+C := {qi : i ≥ 0} ˙∪ {uj : j ≥ 0},
+D := {si : i ≥ 0} ˙∪ {uj : j ≥ 0}.
+Lemma 3.5. (1) The elements x and u act ad-locally-nilpotently on D. Conse-
+quently, {xi : i ≥ 0} and {ui : i ≥ 0} are Ore sets in D.
+(2) For each Ω ∈ {A, B, C, D} the set Ω is an Ore set of regular elements of D,
+and we write the corresponding localisation as D(Ω).
+
+THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE
+7
+Proof. (1) For x this is proved in [1, Lemma 4.3(i)]. The claim for u is a similar
+easy consequence of the defining relations of D.
+(2) Localising at the powers of q is the same as localising at the powers of q2 or
+even at the powers z = q2g−1, since g is a unit. Thus, for Ω = A or Ω = C and
+appealing to Lemma 2.1(iii), we can replace q by the central element z. Similarly
+in the other two cases we can replace s by the central element θ. Thus in each
+case we wish to localise at one central and one locally ad-nilpotent element in the
+domain D. Thus it is indeed an Ore set of regular elements.
+□
+Thus each of the four rings D(Ω) is a subalgebra of the quotient division algebra
+Q(D) of D that contains D. As we next show, each of these rings is a localisation
+of the second Weyl algebra over a commutative ring.
+Notation 3.6. (i) In D(A), set pA := −2q−1x−1y,
+qA := q,
+tA := qx−1, and
+ηA := −xq−1ζ.
+(ii) In D(B), set pB := −2s−1x−1y, qB := s, tB := sx−1. ηB := −xs−1ζ.
+(iii) In D(C), set pC := 2q−1u−1v, qC := q, tC := q−1u−1, ηC := −uqζ.
+(iv) In D(D), set pD := 2s−1u−1v, qD := s, tD := s−1u−1, ηD := usζ.
+We further set zΩ := z for Ω = A and Ω = C but zΩ := θ when Ω = B, D.
+The motivation behind the above definitions becomes clear from the following
+lemma. For Ω = A, this was obtained in the proof of [1, Theorem 4.8]. The claims
+regarding the other elements can be checked by a similar direct calculation.
+Lemma 3.7. Let Ω ∈ {A, B, C, D}. Then we have the following relations in Q(D):
+[pΩ, qΩ] = 1 = [ηΩ, tΩ], with all other brackets being zero;
+□
+When Ω = A the following result is given in [1, Theorem 4.8], although we give
+a proof that works for all 4 cases simultaneously.
+Theorem 3.8. For each Ω ∈ {A, B, C, D}, the localisation D(Ω) is a localised Weyl
+algebra over its centre. More precisely:
+D(Ω) = A(Ω)
+2
+(k) ⊗ S(Ω),
+where A(Ω)
+2
+(k) denotes the localisation of the second Weyl algebra over k with gen-
+erators pΩ, q±1
+Ω , ηΩ, t±1
+Ω , while S(Ω) is the commutative ring S(Ω) = k[z±1
+Ω , ω].
+Proof. The generators z, ω and θ of Z(D) are given in Lemma 2.1(iii), from which
+it follows that the subalgebra S(Ω) of Q(D) is contained in the centre Z(D(Ω)).
+Therefore we can consider the subalgebra
+(3.6)
+E(Ω) := S(Ω)⟨pΩ, qΩ, tΩ, ηΩ⟩ ⊆ D(Ω).
+We claim that the inclusion (3.6) is an equality. In order to prove this, check that
+given generators of DΩ are contained in E(Ω). Thus, for example, when Ω = A, one
+shows that {q−1, x±1, y, ζ, g±1, u, v} ⊂ E(A), and similarly in the other cases. Thus,
+E(Ω) = D(Ω), as claimed. As noted in the proof of Lemma 3.5 the localisation of
+D at Ω involves inverting one central and one ad-nilpotent element of D. Thus,
+by Proposition 3.2(iii) and [18, Lemma 4.7], GKdim(D(Ω) = GKdim(D) = 6. We
+conclude that GKdim(E(Ω)) = GKdim(D) = 6.
+On the other hand, by Lemma 3.7 E(Ω) is a factor of the ring
+V(Ω) := S(Ω) ⊗k A(Ω)
+2
+(k),
+
+8
+K. A. BROWN AND J. T. STAFFORD
+which is also a domain of GK-dimension 6. So if E(Ω) were a proper factor of V(Ω),
+then [21, Corollary 8.3.6] would imply that GKdim(E(Ω)) < 6, giving a contradic-
+tion.
+So the only possibility is that E(Ω) ∼= V(Ω) = S(Ω) ⊗k A(Ω)
+2
+(k), as required.
+□
+4. The primitive spectrum of D
+In this section we describe the primitive spectrum of D. This splits naturally
+into several cases:
+• the primitive ideals not containing m+ or m0; these are the generic ones;
+• the ideal m+D, which is also primitive;
+• the ideal m0D, for which √m0D is a unique prime ideal P0;
+• finally, P0 is also maximal.
+The details are given in the next four subsections with the results being combined
+in Subsection 4.5.
+In this section and in Section 5.1 we will without further reference use of the
+yoga for prime ideals of Noetherian rings under Ore localisation as described in,
+for example, [17, Theorems 10.18 and 10.20]. We use Notation 2.3 to describe the
+maximal ideals of Z(D) and Definition 3.4 to define Ore sets in D.
+4.1. The generic minimal primitives. We begin by looking at the generic case.
+Theorem 4.1. Let m be a maximal ideal of Z(D) with m ̸= m+ and m ̸= m0. Then
+the following are true.
+(i) mD is a completely prime maximal ideal of D.
+(ii) The localisation of D/mD at the powers of (the image of) either x or u is
+isomorphic to a localised Weyl algebra A(Ω)
+2
+(k), where Ω ∈ {A, B, C, D}.
+(iii) GKdim(D/mD) = 4.
+(iv) mD is generated by a central regular sequence of length 2.
+(v) D/mD is CM and is Auslander Gorenstein with injdim(D/mD) < 4.
+Proof. (i), (ii) By Notation 2.3, m = ⟨z − α, ω − β, θ − γ⟩ with α, β, γ ∈ k and
+αγ = β2. Moreover, thanks to the hypothesis on m, either (a) α ̸= 0 or (b) γ ̸= 0.
+Assume (a). We prove (ii) for the localisation at the powers of x. (The arguments
+for powers of u are exactly similar, but using the Ore sets C and D rather than A
+and B.) Using the notation of §3.3 and applying Theorem 3.8, we see that mD(A)
+is a maximal ideal of D(A). Observe that, since A := {qi, xj : i, j ≥ 0} and
+z = q2g−1 ≡ α ̸= 0 mod(mD),
+A(A)
+2
+(k) is isomorphic to the localisation of D/mD at the powers of x. Define
+Pm := mD(A) ∩ D,
+so that Pm is a completely prime ideal of D with mD ⊆ Pm. By definition of Pm,
+(4.1)
+mD(A) = PmD(A).
+We claim that in fact
+(4.2)
+Pm = mD.
+
+THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE
+9
+Since D is (left) noetherian there exist e1, . . . , et ∈ Pm such that Pm = mD +
+�t
+i=1 Dei. By (4.1), for each i = 1, . . . , t there exist fi ∈ mD and si ∈ Z≥0 such
+that
+(4.3)
+ei = fix−si.
+Define s := max{si : 1 ≤ i ≤ t} ∈ Z≥0, and
+I := {τ ∈ D : Pmτ ⊆ mD}.
+Thus I is an ideal of D containing mD and, by (4.3), xs ∈ I. If s = 0 then I = D;
+otherwise we see from Lemma 3.3 that (m+)r ⊂ I for some r ≥ 1. Since also m ⊆ I
+and m ̸= m+ by hypothesis, it follows that I = D, and (4.2) is proved.
+In case (a) it remains only to prove that Pm is a maximal ideal of D. Suppose
+then that J is an ideal of D with Pm ⊊ J. Then JD(A) = D(A) by the maximality
+of the ideal PmD(A) of D(A). Again using the fact that q + mD is a unit of D/mD
+we see that xs ∈ J for some s ≥ 1. Then, as before, Lemma 3.3 implies that J = D.
+Suppose that (b) holds rather than (a). Then the element s is a unit mod mD,
+so we use the same argument as for (a), but working with D(C) rather than D(A).
+(iii) By (ii) and [18, Example 3.7 and Theorem 4.9] the localisation of D/mD at
+the powers of x has GK dimension 4. Since ad(x) acts nilpotently on D/mD by
+Lemma 3.5, it follows from [18, Theorem 4.9] that GKdim(D/mD) = 4.
+(iv) Again we assume (a) that z−α ∈ m for α ∈ k\{0}, the proof in case (b) being
+similar. We can begin a regular central sequence in mD with z − α. Since D is CM
+of GK-dimension 6 by Proposition 3.2(i, iii), it follows from [16, Theorem 7.2(b)]
+that D/(z − α)D is CM of GK-dimension 5. Moreover, by [19, Remark 2.4] the
+CM property ensures that D/(z − α)D is GK-homogeneous; that is, it contains no
+non-zero ideal with GK-dimension strictly less than 5. Since Z(D)/(z − α)Z(D) is
+a polynomial algebra we can choose y ∈ m such that m = ⟨z − α, y⟩. If y + (z − α)D
+is a zero divisor in D/(z − α)D we obtain a non-zero ideal of D/(z − α)D killed
+by mD, contradicting the GK-homogeneity of D/(z − α)D in view of (iii). Thus
+{z − α, y} is a regular central sequence in mD.
+(v) Since D is CM by Proposition 3.2(i), R = D/mD is CM with GKdim(R) = 4
+by (iv) and two applications of [16, Theorem 7.2(b)]. The Auslander Gorenstein
+property is given by (iv) and [19, §3.4, Remark (3)]. As R is simple it cannot have a
+finite dimensional module. Hence injdim(R) < 4 follows from the next lemma.
+□
+The following observation is well-known.
+Lemma 4.2. Let R be a noetherian, Auslander Gorenstein, CM ring and write
+GKdim(R) = m. Then injdim(R) ≤ m. Moreover injdim(R) = m ⇐⇒ R has a
+finite dimensional representation.
+Proof. Let n = injdim(R) and pick a finitely generated R-module M such that
+Extn
+R(M, R) ̸= 0. By the Auslander condition and the spectral sequence [19, The-
+orem 2.2] j(Enn(M)) = n for Enn = Extn(Extn(M, R), R). By the CM property
+GKdim(Enn(M)) = m − n and the result follows easily.
+□
+
+10
+K. A. BROWN AND J. T. STAFFORD
+4.2. Non-generic minimal primitives (I) - m+. The next case to consider is mD
+for m = m+, as we do here. Recall from Notation 2.3(ii) that m+ = D+ ∩ Z(D) =
+⟨ω + 16, θ − 16⟩.
+We start with a subsidiary result, which works for any field k of characteristic
+zero.
+Theorem 4.3. D is a Jacobson ring that satisfies the Nullstellensatz, in other
+words:
+(i) every prime ideal of D is an intersection of primitive ideals;
+(ii) for every simple D-module M, EndD(M) is algebraic over k. In particular,
+every primitive ideal of D contains a maximal ideal of Z(D).
+Proof. By (3.2), D has a filtration F such that the associated graded ring grF(D) is
+a commutative affine ring. Hence by [22, Corollary 1.7] there is a second filtration
+G by finite dimensional k-subspaces of D such that grF(D) is also a commutative
+and affine ring. The result now follows from [4, Theorem 0.4].
+□
+Theorem 4.4.
+(i) m+D is a completely prime, primitive ideal of D.
+(ii) The localisation of D/m+D at the powers of x or the powers of u is a
+localisation of the Weyl algebra A2(k) at powers of a generator.
+(iii) m+D is generated by a regular central sequence of length 2.
+(iv) D/m+D is Auslander Gorenstein and CM with
+GKdim(D/m+D) = 4 = injdim(D/m+D).
+(v) Every prime ideal P of D which strictly contains m+D satisfies
+O(G)+D ⊆ P,
+so the space of such primes P is homeomorphic to Spec(U(sl(2, k)).
+Proof. Recall that qA = q. Since q2 ≡ 16g ̸≡ 0 mod(m+D), Theorem 3.8 implies
+that m+D(A) is a maximal ideal of D(A), with D/m+D(A) ∼= A(A)
+2
+(k). Therefore,
+defining P+ := m+D(A) ∩ D, we deduce that P+ is a completely prime ideal of D
+with
+(4.4)
+m+D ⊆ P+.
+We will eventually show that (4.4) is an equality.
+As in the proofs of Theorem 4.1(i),(ii), let I be the right annihilator in D of
+P+/m+D. Then I contains m+D and a power of x, and hence, by Lemma 3.3,
+(4.5)
+(O(G)+D)r ⊆ I
+for some r ∈ Z≥1,
+In particular, GKdim(D/I) ≤ 3 by Lemma 3.3(ii).
+Therefore, by [18, Proposi-
+tion 5.1(d)]
+(4.6)
+GKdim(P+/m+D) ≤ 3.
+Recall that GKdim(A(A)
+2
+(k)) = 4 by [18, Example 7.3 and Theorem 4.9], so that
+(4.7)
+GKdim(D/P+) = 4
+by [18, Theorem 4.9]. Thus, from (4.6), (4.7) and Proposition 3.2(iv) it follows that
+(4.8)
+GKdim(D/m+D) = 4.
+
+THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE
+11
+By Proposition 3.2, D is CM and Auslander regular, with gldim(D) = 6 =
+GKdim(D). It therefore follows from the CM property of D together with (4.8)
+that
+(4.9)
+jD(D/m+D) = 6 − 4 = 2.
+From (4.9) and [8, Proposition 3.6] we deduce that the maximum length of a reg-
+ular sequence of elements of m+ on D is precisely 2; in particular any choice of
+a generating pair of elements of m+, for example, {z − 16, ω + 16}, is a regular
+sequence on D. Therefore, by two applications of [16, Theorem 7.2(b)],
+(4.10)
+D/m+D is CM of GK-dimension 4.
+Similarly, two applications of [19, §3.4, Remark (3)] show that D/m+D is Auslander
+Gorenstein. By Lemma 3.3(i) and Theorem 2.2(ii), U(sl(2, k)) ∼= D/DO(G)+ is a
+factor of D/m+D and so D/m+D has a non-zero finite dimensional module, M.
+Thus, by Lemma 4.2, injdim(D/m+D) = 4.
+By [19, Remark 2.4], again, the CM property for D/m+D implies that D/m+D
+is GK-homogeneous. Therefore we may conclude from (4.6) that P+ does indeed
+equal m+D.
+This proves (i) - (iv), with the exception of showing that m+D is
+primitive.
+(v) Let Q be a prime ideal of D with m+D ⊊ Q. As already noted, q is congruent
+to a unit mod m+D. Then QD(A) = D(A) by (ii), so Q must contain a power of x.
+Hence, by Lemma 3.3, O(G)+D ⊆ Q, as required.
+Finally, to see that m+D is primitive note that (v) shows that it is locally closed.
+Hence it is primitive by Theorem 4.3(i).
+□
+4.3. Non-generic minimal primitives (II) - m0. In this subsection we begin
+our study of the ideal m0D.
+Recall the definition of q, s from (2.1) and, from
+Notation 2.3(iii), that m0 := ⟨q2g−1, qsg−1, s2g−1⟩ is the unique singular point of
+Maxspec(Z(D)). Clearly the right ideal
+(4.11)
+P0 := qD + sD
+is a two-sided ideal of D since q and s are normal in D by Lemma 2.1. Moreover,
+m0D = P 2
+0 ⊂ P0 ⊆
+�
+m0D.
+As part of the next proposition we see that P0 is completely prime, so the second
+inclusion above is an equality.
+In fact P0 is a maximal ideal, but this is more
+difficult to prove, and is delayed until §4.4.
+Proposition 4.5. Retain the above notation, and set T := D/P0.
+(i) T is a localisation of a 4-step iterated Ore extension of k, namely
+T =
+�
+(k[u, x]⟨(ux + 2)−1⟩)[y; ∂1]
+�
+[v; σ, ∂2],
+where u and x commute,
+∂1(u) = − 1
+2ux − 2,
+∂1(x) = − 1
+2x2,
+∂2(u) = − 1
+2u2,
+∂2(x) = 3
+2ux + 2,
+∂2(y) = 3
+2uy − 2,
+and σ(y) = y + 1
+2x, with σ(x) = x and σ(u) = u.
+(ii) {q, s} forms a regular normal sequence of generators of P0.
+(iii) gldim(T ) ≤ 4 = GKdim(T ).
+
+12
+K. A. BROWN AND J. T. STAFFORD
+(iv) T is CM and is an Auslander regular domain.
+Proof. Throughout the proof we abuse notation by simply denoting the image in
+T of an element ω of D by ω when no confusion seems likely.
+(i),(ii) Since q := ux + 2(1 + g) and q ≡ 0 mod(P0), we can write
+(4.12)
+g ≡ − 1
+2(ux + 2) mod(P0),
+so that
+(4.13)
+ux + 2 is a unit in T.
+Using (4.12) we find that, mod(P0),
+s := xv + uy + (− 1
+2ux + g − 1)ζ − 2(g + 1) ≡ xv + uy + 2gζ − 2g − 2,
+so that, since s ∈ P0,
+(4.14)
+ζ ≡ − 1
+2g−1(ux + xv + uy)
+mod(P0)
+It follows from (4.12), (4.13) and (4.14) that
+(4.15)
+T = k⟨u, x, (ux + 2)−1, y, v⟩.
+The relations for D given in §2.1 immediately imply the following relations for the
+generators for T listed in (4.15)
+[u, x] = 0,
+[y, x] = − 1
+2x2,
+[v, x] = 3
+2ux + 2,
+[y, u] = − 1
+2ux − 2,
+[v, u] = − 1
+2u2,
+[v, y] = 3
+2uy + 1
+2xv ��� 2.
+Clearly the iterated Ore extension of k[u, x]⟨(ux + 2)−1⟩ defined in (i), which we
+temporarily label �T, satisfies precisely these relations, so there is an algebra epi-
+morphism Φ from �T onto T .
+We next show that Φ is an isomorphism, which we do by computing GKdim(T ).
+First note that GKdim( �T) = 4 by [18, Theorem 12.3.1], since it is a PBW extension
+in 2 variables of k[u, x]⟨(ux + 2)−1⟩. Thus, certainly GKdim(T ) ≤ 4. On the other
+hand D is CM of GK-dimension 6 by Proposition 3.2(i, iii). Hence, because q is
+a regular normal element of D by Lemma 2.1, D/qD is CM of GK-dimension 5
+by [16, Theorem 7.2(b) and its proof]. Moreover D/qD is GK-homogeneous by
+[19, Remark 2.4]. Since GKdim(T ) ≤ 4, this ensures that
+(4.16)
+s cannot be a zero-divisor mod qD.
+Since P0 := qD + sD, a second application of [16, Theorem 7.2(b) and its proof]
+yields GKdim(T ) = 4 and also shows that
+(4.17)
+T is CM.
+Since �T is a domain, the equality GKdim( �T) = 4 = GKdim(T ), combined with
+[18, Proposition 3.15], shows that �T = T . Thus (i) is proved, with (ii) also following
+thanks to (4.16).
+(iii) By (i), T is a 2-step iterated Ore extension of k[u.x]⟨(ux + 2)−1⟩, and so two
+applications of [21, Theorem 7.5.3(i)] gives gldim(T ) ≤ 4.
+(iv) That T is a domain is clear from (i), while the CM property was proved in
+(4.17). The Auslander Gorenstein property holds for D by Proposition 3.2(i). Thus,
+by (ii) and two applications of [16, Theorem 7.2(a)], T is also Auslander Gorenstein
+and it is then Auslander regular by (iii).
+□
+
+THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE
+13
+We remark that, by Lemma 4.2 and Theorem 2.2(iii) it follows that gldim(T ) < 4.
+We do not know the exact value of gldim(T ).
+4.4. Maximality of P0. Let T := D/P0 as in Proposition 5.3. Define also the
+following subalgebras of T :
+R := k⟨u, x, (ux + 2)−1⟩, and S := R[y; ∂1],
+so that T = S[v; σ, ∂2]. It is important to note that, by the formulæ in Proposi-
+tion 4.5, R is preserved by the σ-derivation ∂2. Moreover, since σ|R is the identity,
+∂2 actually restricts to a derivation on R.
+It is much easier to determine when an Ore extension is simple if the ring is a
+differential operator ring, in the sense that the defining automorphism is actually
+the identity. Thus we will reduce to that case. The idea follows from Lemma 2.1
+which shows that σ2 is given by the inner automorphism τg in the sense that
+σ(s) = τg(s) = gsg−1 for suitable g ∈ S. We will therefore extend R, S and T by
+√g and show that σ is then inner, and so can be removed. The details are given in
+the next few results, culminating in Proposition 4.9.
+Notation 4.6. In the algebraic closure of R, set h = (ux + 2)− 1
+2 . Write �R =
+R⟨h⟩ = k⟨u, x, h, h−1⟩. We extend the ∂i to derivations on �R by the usual rules for
+fractional powers:
+∂(h) = (−1
+2)(ux + 2)−1h∂(ux + 2),
+for ∂ = ∂1, ∂2. Set �S = �R[y; ∂1]. Finally, we can extend σ to �R and �S by setting
+σ(h) = h. Then both σ and ∂2 are naturally defined on �S as an automorphism,
+respectively σ-derivation and so �T = �S[v; σ, ∂2] is a well-defined Ore extension of
+�S.
+The following observation will prove useful.
+Lemma 4.7. �S is a free left and right S module on basis {1, h}. Similarly, �T is a
+free left and right T module on basis {1, h}.
+Proof. As h2 = (ux + 2)−1 ∈ R, the construction of �R ensures that �R is a free left
+and right R-module on basis {1, h}. We can then write
+�S =
+∞
+�
+i=0
+�Ryi =
+�
+Ryi ⊕
+�
+Rhyi =
+�
+Ryi ⊕
+�
+Ryih.
+Collecting terms shows that �S = S ⊕ Sh. As S is a domain this is necessarily a
+direct sum of free modules. The same argument works for �T.
+□
+Lemma 4.8. On �S, σ is the inner automorphism τh−1; thus σ(f) = h−1fh for
+f ∈ �S.
+Proof. Since �R is a commutative ring on which σ is the identity, the lemma holds
+trivially on �R. It therefore just remains to check that the automorphisms agree on
+y. To prove this, we rewrite h−1yh as follows.
+
+14
+K. A. BROWN AND J. T. STAFFORD
+h−1yh = (ux + 2)
+1
+2 y(ux + 2)− 1
+2
+= (ux + 2)
+1
+2 (ux + 2)− 1
+2 y + (ux + 2)
+1
+2 · ∂1
+�
+(ux + 2)− 1
+2 �
+= y + (ux + 2)
+1
+2 (− 1
+2)(ux + 2)− 3
+2 · ∂1((ux + 2))
+= y −
+1
+2(ux + 2)−1�
+(− 1
+2ux − 2)x − u( 1
+2x2)
+�
+= y −
+1
+2(ux + 2)−1�
+−(ux + 2)x
+�
+= y + 1
+2x = σ(y);
+as required.
+□
+Proposition 4.9. Set α = hv. Then �T is the Ore extension �T = �S[α; �∂2] where �∂2
+is the derivation of �S defined by �∂2(s) = h∂2(s) for s ∈ �S; thus
+�∂2(u) = − 1
+2hu2,
+�∂2(x) = h( 3
+2ux + 2)
+and
+�∂2(y) = h(( 3
+2uy − 2).
+As such, �T is a noetherian domain.
+Proof. This is a formal computation. Indeed, for s ∈ �S, Lemma 4.8 implies that
+σ(s) = h−1sh. Equivalently,
+(4.18)
+αs = hvs = hσ(s)v + h∂2(s)
+= hh−1shv + h∂2(s) = sα + h∂2(s).
+Therefore, since �T = �S[v; σ, ∂2] = � �Svi, we see that �T = � �Sαi. Since �T is a
+domain, combining this with (4.18) and [12, Theorem 1, p.438] gives the desired
+conclusion.
+□
+Our next aim will be to show that the ring �T is a simple domain, after which it
+is easy to prove the same conclusion for T . We start with some preparatory results.
+Lemma 4.10. If there exists a non-zero (∂1, �∂2)-invariant ideal I in �R, then there
+exists a non-zero (∂1, �∂2)-invariant prime ideal P in �R.
+Proof. Using [17, Lemma 3.18(b)] twice, clearly I �S is a proper non-zero ideal of �S
+and then I �T is a proper nonzero ideal of �T. Pick a prime ideal Q ⊇ I �T. Then, by
+[17, Lemmata 3.18 and 3.21], twice, Q1 = Q ∩ �S is a �∂2-invariant prime ideal of �S
+and hence Q2 = Q1 ∩ �R is a ∂1-invariant prime ideal of �R. However, since �R and
+Q1 are both �∂2-invariant, so is Q2. Thus, P = Q2 is the desired prime ideal.
+□
+Proposition 4.11. There is no proper, non-zero (∂1, �∂2)-invariant ideal I in �R.
+Proof. Suppose that there exists such an ideal I. By Lemma 4.10 we can and will
+assume that I is a prime ideal. Suppose, first, that (xu + λ) ∈ I, for some λ ∈ k.
+Then
+I ∋ ∂1(xu + λ) = (− 1
+2ux − 2)x − ( 1
+2x2)u = −(ux + 2)x
+and
+I ∋ �∂2(xu + λ) = h
+�
+− 1
+2u2x + ( 3
+2ux + 2)u
+�
+= h
+�
+ux2 + 2u
+�
+= h(ux + 2)u.
+
+THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE
+15
+As (xu + 2)−1 = h2 ∈ �R, clearly λ ̸= 2 and so the two equations imply that x ∈ I,
+respectively u ∈ I. Thus, I = x �R + u �R. But, now I ∋ ∂1(u) = − 1
+2ux − 2 and so
+I = �R, a contradiction. We conclude that
+(4.19)
+I ∩ C = ∅
+for C = {(xu + λ) : λ ∈ k∗}
+Since I is a prime ideal it follows that C ⊆ C(I) and hence that IC is a proper prime
+ideal of the localisation �RC.
+Next, if IC ∋ f = f(u) for some f(u) ∈ k[u], then IC ∋ ∂1(f) = − 1
+2(ux + 4) df
+du.
+Hence df
+du ∈ IC. By induction on deg f, this implies that IC = �RC, a contradiction.
+Thus IC ∩ k[u]∗ = ∅ and so we can further localise at S = k[u]∗ and conclude that
+ICS is a proper prime ideal of �RCS. Now consider �RCS. We have �R = k⟨u, x, h, h−1⟩
+and h−2 = (ux + 2) whence x = u−1(h−2 − 2). Thus �RCS = �RSC is a localisation
+of k(u)[h, h−1].
+The advantage of working in �RCS is that we can simplify our derivation �∂2. On
+�R and �RCS write ∂u =
+∂
+∂u and ∂x =
+∂
+∂x. Then, as derivations on either ring,
+∂1 = −( 1
+2xu + 2)∂u −
+1
+2x2∂x
+while
+�∂2 = − 1
+2hu2∂u + h( 3
+2ux + 2)∂x.
+We now set µ := −hu2(ux + 4)−1 and take
+�∂′
+2 := �∂2 + µ∂1 =
+�
+− 1
+2hu2 + µ(− 1
+2ux − 2)
+�
+∂u +
+�
+h( 3
+2ux + 2) − 1
+2x2µ
+�
+∂x.
+This element µ has been chosen so that the coefficient of ∂u in �∂′
+2 is
+− 1
+2(ux + 4)−1�
+hu2(ux + 4) − (ux + 4)hu2�
+= 0.
+Therefore,
+�∂′
+2 =
+�
+h( 3
+2ux + 2) +
+1
+2hx2u2(ux + 4)−1�
+∂x
+= (ux + 4)−1h
+�
+( 3
+2ux + 2)(ux + 4) + 1
+2x2u2�
+∂x
+= (ux + 4)−1h
+�
+2u2x2 + 8ux + 8
+�
+∂x
+= β∂x
+for
+β := 2(xu + 4)−1(ux + 2)2h.
+Since ICS is invariant under both ∂1 and �∂2, it is also invariant under �∂′
+2. Since
+β is a unit in �RCS, it follows that
+(4.20)
+ICS is also invariant under β−1 �∂′
+2 = ∂x.
+Thus, by (4.20) and the expression given above for ∂1, ICS is invariant under
+( 1
+2ux + 2)∂u, and therefore under ∂u since 1
+2ux + 2 is a unit. So ICS is invariant
+under ∂u and ∂x. Since �RCS is a localisation of k[u, x] this forces ICS = �RCS, giving
+the required contradiction.
+□
+In order to pass between T and �T we need:
+Lemma 4.12. If �T is a simple ring then so is T .
+
+16
+K. A. BROWN AND J. T. STAFFORD
+Proof. Suppose that T has a proper ideal J. Then X = �T/J �T is a (T, �T)-bimodule.
+Moreover, by Lemma 4.7 �T is a finitely generated left T -module and so X is a
+finitely generated left T -module; say X = �r
+i=1 T xi. Then, as �T is an Ore domain,
+ann �
+T (X) = �
+i ann �
+T (xi) ̸= 0. Since �T is a simple ring this implies that ann �
+T (X) =
+�T and hence that X = 0. In other words, J �T = �T.
+On the other hand, by Lemma 4.7, �T = T + T h is a free left T -module and so
+J �T = J ⊕ Jh ̸= �T. This contradiction proves the lemma.
+□
+We now put everything together and prove the main result of this subsection.
+Theorem 4.13. T is a simple ring.
+Proof. By Lemma 4.12 it suffices to prove that �T is simple. By [21, Theorem 1.8.4]
+applied to �T = �S[α; �∂2], we need to prove
+(a) �∂2 is not an inner derivation on �S, and
+(b) �S has no proper �∂2-invariant ideals.
+Now, as ∂1(x) = − 1
+2x2, the right ideal x�S is a proper two-sided ideal of �S. As
+such, it is preserved by any inner derivation of �S. But �∂2(x) = h( 3
+2ux + 2) ̸∈ x�S,
+this means �∂2 cannot be an inner derivation of �S and so (a) holds.
+Suppose that �S has a proper �∂2-invariant ideal I. Then, by [17, Lemma 3.18],
+K = I ∩ �R is a ∂1-invariant ideal of �R, while by [17, Lemma 3.19], K ̸= 0. Since
+both I and �R are both �∂2-invariant, so is K. In other words, K is a proper (∂1, �∂2)-
+invariant ideal of �R. This contradicts Proposition 4.11. Thus (b) holds and so
+[21, Theorem 1.8.4] implies that �T is simple.
+□
+Remark 4.14. We end the subsection by noting that �T is obviously birational to
+the Weyl algebra A2. We do not know if the same is true for T itself.
+4.5. The shape of the primitive spectrum of D. In this subsection we combine
+the earlier results of this section to prove Theorem 1.1. By Theorem 4.3, every
+primitive ideal P of D contains a maximal ideal of Z(D). Thus Privspec(D) is the
+disjoint union
+(4.21)
+Privspec(D) =
+˙�
+m∈Maxspec(Z(D))V(m)
+where V(m) = {P ∈ Privspec(D) : m ⊆ P}. There are thus 3 cases, corresponding
+to §§4.1, 4.2 and 4.3.
+(I) V(m), where m ∈ Maxspec(Z(D)) with m ̸= m+ and m ̸= m0. By Theorem 4.1,
+V(m) = {mD} is a single generic maximal ideal of D. Moreover D/mD is bira-
+tionally equivalent to the second Weyl algebra, with other properties as listed in
+that theorem.
+(II) V(m+). By Theorem 4.4, this consists of m+D, together with
+V(O(G)+D) := {P ∈ Privspec(D) : O(G)+D ⊂ P},
+which is homeomorphic to Privspec(U(sl(2, k))) by Theorem 2.2(ii).
+Recall that Privspec(U(sl(2, k))) is composed of the co-Artinian maximal ideals
+{Mn : n ∈ Z≥1}, where Mn = Ann(Vn), Vn being the n-dimensional irreducible
+
+THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE
+17
+U(sl2(k))−module, together with the minimal primitives of U(sl(2, k)); that is, the
+ideals (Ω − λ)U(sl(2, k)) : λ ∈ k}, where Ω is the Casimir element. Each Mn
+contains one such minimal primitive and each minimal primitive is contained in
+at most one Mn; the remaining minimal primitives are also maximal. Note that
+O(G)+D is prime but not primitive since D/O(G)+D ∼= U(sl(2, k)) and this domain
+satisfies the Nullstellensatz and has non-trivial centre k[Ω].
+(III) V(m0). This is the singleton {P0 = qD + sD = √m0}, by Proposition 4.5 and
+Theorem 4.13.
+5. Prime ideals and the Dixmier-Moeglin equivalence
+In this section we prove Theorem 1.2 from the introduction, which describes the
+prime ideals of D, and we discuss the Dixmier-Moeglin equivalence for D.
+5.1. The prime spectrum of D. We need the following lemmas for the proof of
+the main result, Theorem 5.3.
+Lemma 5.1. Let P be a nonzero prime ideal of D. Then P ∩ Z(D) ̸= {0}.
+Proof. If xi ∈ P for some i ≥ 0 then O(G)+D ⊆ P by Lemma 3.3 applied with
+M = D/P, and therefore m+ = O(G)+D ∩ Z(D) ⊆ P, proving the lemma for P.
+So we may assume that {xi : i ≥ 0} ∩ P = ∅. Similarly, we may assume that
+{qj : j ≥ 0} ∩ P = ∅, since otherwise 0 ̸= qng−2n ∈ P ∩ Z(D) for some n ≥ 0 and
+again the result follows for P.
+Hence, using Notation 3.6 and Theorem 3.8, PD(A) survives as a non-zero proper
+ideal of D(A) = D⟨q−1, x−1⟩ = A(A)
+2
+(k) ⊗k S(A), where A(A)
+2
+(k) is a localised Weyl
+algebra and S(A) = k[z±1, ω]. In particular,
+(5.1)
+PD(A) = (PD(A) ∩ S(A))D(A).
+By [17, Theorem 10.20] and the discussion in the first paragraph of this proof,
+P = PD(A) ∩ D, and therefore
+(5.2)
+P ∩ Z(D) = PD(A) ∩ Z(D) = (PD(A) ∩ S(A)) ∩ Z(D).
+Since the Z(D)-module S(A)/Z(D) is {zi}-torsion, that is {qig−2i}-torsion, it fol-
+lows from (5.1) and (5.2) that P ∩ Z(D) ̸= {0} as required.
+□
+Note that, since k is algebraically closed of characteristic 0, the defining relation
+zθ = ω2 of Z(D) can be rewritten using a linear change of variables as the quadratic
+form X2 + Y 2 = Z2. Thus a proof of the next result can be found at [15, p.51 and
+Proposition 11.4].
+Lemma 5.2. All height one primes of Z(D) are principal except p1 := ⟨z, ω⟩ and
+p2 := ⟨θ, ω⟩.
+□
+Here is the main result of this section, using in (ii) the notation of Lemma 5.2.
+This proves Theorem 1.2 from the introduction.
+Theorem 5.3. Let P be a prime but not primitive ideal of D.
+(i) There are the following three possibilities for P.
+(a) P = {0}.
+(b) P = O(G)+D.
+
+18
+K. A. BROWN AND J. T. STAFFORD
+(c) P has height one and is minimal over (P ∩ Z(D))D for a height one
+prime ideal P ∩ Z(D) of Z(D).
+(ii) In case (c), if P ∩ Z(D) = pi for i = 1, resp. i = 2, then P = qD, resp.
+P = sD. The remaining primes in case (c) are precisely the set
+{P : P = fD},
+as f ranges through the equivalence classes of irreducible elements of Z(D)
+other than the associates of z, ω, θ.
+Proof. Note first that {0} is completely prime by Proposition 3.1, and is not prim-
+itive, because D satisfies the Nullstellensatz by Theorem 4.3 and Z(D) ̸= k. This
+covers case (a).
+Let P be a non-zero prime but not primitive ideal of D. By Lemma 5.1,
+{0} ̸= p := P ∩ Z(D).
+If p = m+ then Theorem 4.4 together with the discussion at §4.5(II) shows that
+the only possibility is P = O(G)+D, which is completely prime but is again not
+primitive thanks to the Nullstellensatz, since Z(U(sl(2, k))) ̸= k. This is case (b).
+If p = m0 then P = P0, which is maximal by Theorem 4.13, so this case can’t
+happen. Similarly, p is any maximal ideal of Z(D) apart from m+ or m0, then P =
+pD is a maximal ideal of D by Theorem 4.1(i), which again gives a contradiction.
+So we are left with the case when p has height one. Assume first that p = fZ(D)
+is principal. Then, by Lemma 5.2, z = q2g−1 /∈ P, and {xi : i ≥ 0} ∩ P = ∅ by
+Lemma 3.2 Therefore, using Notation 3.6 and Theorem 3.8
+pD(A) = (P ∩ S(A))D(A) = PD(A).
+We claim that P = pD.
+To see this, note that pD = fD is principal, so that
+D/pD is CM of GK-dimension 5, by [16, Theorem 7.2(b) and its proof], and GK-
+homogeneous by [19, §3.4, Remark (3)]. Now P/pD it is killed by pD and by a
+power of q or a power of x, and so has GK-dimension less than 5, respectively by
+Theorem 4.1(iii) and 4.4(iv) or by Lemma 3.2. This forces P/pD = {0} and so
+P = pD, as claimed.
+Suppose finally that p = p1 or p = p2. In the first case, since q is a normal
+element of D by Lemma 2.1, q ∈ √pD. Thus
+(5.3)
+qD ⊆ P.
+We claim that (5.3) is an equality. To see this, note that s /∈ P, since otherwise
+P ∩Z(D) = m0, which is ruled out by hypothesis. Moreover {xj : j ≥ 0}∩P = ∅ by
+Lemma 3.2. So we can localise at the Ore set B = {sixj : i, j ≥ 0} of Definition 3.4
+and pass to the localised Weyl algebra D(B) = A(B)
+2
+(k) ⊗ S(B) of Theorem 3.8.
+However, PD(B) and qD(B) have the same intersection with the centre S(B), namely
+ωθ−1S(B) = p1S(B). Therefore PD(B) = qD(B) since the ideals of D(B) are centrally
+generated. Therefore P/qD is B-torsion, so, if it is not zero, it contains a nonzero
+element which is either killed by q and by s, or by q and x. As in the previous
+paragraph D/qD is GK-homogeneous of GK-dimension 5, and so has no such non-
+zero torsion submodule, proving that (5.3) is an equality.
+If p = p2 then the argument to show that P = sD is similar, but using the Ore
+set A; it is left to the reader.
+□
+
+THE PRIME SPECTRUM OF THE DRINFELD DOUBLE OF THE JORDAN PLANE
+19
+5.2. The Dixmier-Moeglin equivalence. The following gives evidence in favour
+of [6, Conjecture 1.3], which proposes that an affine noetherian Hopf C-algebra of
+finite GK dimension should satisfy the Dixmier-Moeglin equivalence. See [5,10] for
+definitions and background.
+Corollary 5.4. D satisfies the Dixmier-Moeglin equivalence.
+Proof. We check first using the description of the primitive spectrum in §4.5 that
+every primitive ideal is locally closed. For classes (I) and (III) this is clear since
+all these primitive ideals are maximal. The primitive ideals in (II) are homeomor-
+phic to the primitive spectrum of U(sl(2, k)), and the latter algebra satisfies the
+equivalence by [23]. Thus, by [10, Lemma II.7.15], it only remains to show that
+every rational prime ideal P is primitive, where P is rational if the centre of the
+Goldie quotient algebra of D/P is k. The non-primitive prime ideals are listed in
+Theorem 5.3 and it is easy to check case by case that none of them is rational.
+□
+Corollary 5.4 proves Theorem 1.3(c). With one exception, parts (a) and (b) of
+that theorem are proved in the results of the last two sections that describe the
+prime ideals of D. The exception is the claim that all the completely prime factors
+of D (with the possible exception of D/P0, as noted in Remark 4.14) are birationally
+equivalent to Weyl algebras. For the primitive ideals P strictly containing O(G)+D
+this follows from [13, Remarque 7.1]. For the other prime ideals, this is clear from
+the description of the prime ideals in the last two sections.
+Based on little more than the known results and counterexamples for group
+algebras and enveloping algebras, the theorem [6] for the cocommutative case, the
+recent work of Sierra and Walton on the noetherian property for enveloping algebras
+[25], together with the above result and other isolated examples, we are tempted
+to propose the following conjecture as a strengthening in the pointed setting of [6,
+Conjecture 1.3], as much in the hope of stimulating the discovery of counterexamples
+as in expectation of a positive result.
+Conjecture 5.5. Let H be an affine noetherian pointed Hopf C-algebra. Then the
+following are equivalent:
+(1) GKdim(H) is finite.
+(2) H satisfies the Dixmier-Moeglin Equivalence.
+(3) The group G(H) of grouplikes of H is nilpotent-by-finite.
+Thanks to a famous result of Roseblade [24] for group algebras, the implication
+(2) =⇒ (3) fails when k is a finite field.
+References
+[1] N. Andruskiewitsch, F. Dumas, and H. M. Pena Pollastri, On the double of the Jordan plane,
+Ark. Mat. 60 (2022), 213-229.
+[2] N. Andruskiewitsch and H. M. Pena Pollastri, On the restricted Jordan plane in odd charac-
+teristic, J. Algebra Appln. 20 (2021), no. 2140012.
+[3]
+, On the finite-dimensional representations of the double of the Jordan plane,
+arXiv2211.01581 (2022).
+[4] M. Artin, L. W. Small, and J. J. Zhang, Generic flatness for strongly noetherian rings, J.
+Algebra 221 (1999), 579-610.
+[5] J. Bell, On the importance of being primitive, Rev. Colombiana Mat. 53 (2019), 87-112.
+[6] J. Bell and W. H. Leung, The Dixmier-Moeglin equivalence for cocommutative Hopf Algebras
+of finite Gelfand-Kirillov Dimension, Alg. Rep. Theory 17 (2014), 1843-1852.
+
+20
+K. A. BROWN AND J. T. STAFFORD
+[7] J.-E. Bjork, The Auslander condition on noetherian rings, Seminaire Malliavin, Lecture Notes
+in Math. 1404 (1989), 137-173.
+[8] K. A. Brown, Unruffled extensions and flatness over central subalgebras, J. Algebra 284
+(2005), 771-800.
+[9] K. A. Brown, M. Couto, and A. Jahn, The finite dual of commutative-by-finite Hopf algebras,
+Glasgow Math. J. (2022), 1-28.
+[10] K. A. Brown and K. R. Goodearl, Lectures on Algebraic Quantum Groups, Advanced Courses
+in Math. CRM Barcelona, Birkhauser, 2002.
+[11] K. A. Brown and J. J. Zhang, Dualising complexes and twisted Hochschild (co)homology for
+noetherian Hopf algebras, J. Algebra 320 (2008), 1814-1850.
+[12] P. M. Cohn, Algebra, Vol. II, Wiley, 1977.
+[13] J. Dixmier, Quotients simples de l’alg`ebre enveloppante de sl2, J. Algebra 24 (1973), 551-564.
+[14] Y. Doi and M. Takeuchi, Multiplication alteration by two-cocycles - the quantum version,
+Comm. in Algebra 22 (1994), 5715-5732.
+[15] R. M. Fossum, The Divisor Class Group of a Krull Domain, Ergebnisse der Mathematik und
+ihrer Grenzgebiete, vol. 74, Springer, 1973.
+[16] K. R. Goodearl and T. L. Lenagan, Primitive ideals in quantum SL3 and GL3, Contemp.
+Math. 562 (2012), 115-140.
+[17] K. R. Goodearl and R. W. Warfield, An Introduction to Noncommutative Noetherian rings,
+Second edition, London Math. Soc. Student Texts, vol. 61, Cambridge University Press, 2004.
+[18] G. R. Krause and T. H. Lenagan, Growth of Algebras and Gelfand-Kirillov Dimension, Re-
+vised Edition, Graduate Studies in Math., vol. 22, Amer. Math. Soc., 2000.
+[19] T. Levasseur, Some properties of non-commutative regular graded rings, Glasgow Math. J.
+34 (1992), 277-300.
+[20] K. Li and G. Liu, The finite duals of affine prime regular Hopf algebras of GK-dimension
+one, arXiv2103.00495 (2021).
+[21] J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Revised edition, Grad-
+uate Studies in Mathematics, vol. 30, Amer. Math. Soc., Providence, RI, 2001.
+[22] J. C. McConnell and J. T. Stafford, Gelfand-Kirillov dimension and associated graded mod-
+ules, J. Algebra 125 (1989), 197-214.
+[23] C. Moeglin, Id´eaux primitifs d´es alg`ebres enveloppantes, J. Math. Pures Appl. 59 (1980),
+265-336.
+[24] J. E. Roseblade, Group rings of polycyclic groups, J. Pure Appl. Algebra 3 (1973), 307-328.
+[25] S. Sierra and C. Walton, The universal enveloping algebra of the Witt algebra is not noether-
+ian, Adv. Math. 262 (2014), 239-260.
+School of Mathematics and Statistics, University of Glasgow, Glasgow G12 8QQ,
+Scotland
+Email address: ken.brown@glasgow.ac.uk
+School of Mathematics, The University of Manchester, Manchester M13 9PL, Eng-
+land
+Email address: Toby.Stafford@manchester.ac.uk
+

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+ESTIMATE DEFORMATION CAPACITY OF
+NON-DUCTILE RC SHEAR WALLS USING
+EXPLAINABLE BOOSTING MACHINE
+Zeynep Tuna Deger (1*), Gulsen Taskin Kaya (1), John W. Wallace (2)
+(1) Istanbul Technical University, (2) University of California, Los Angeles
+(*) corresponding author
+{zeynep.tuna@itu.edu.tr, gulsen.taskin@itu.edu.tr, wallacej@ucla.edu}
+Abstract
+Machine learning is becoming increasingly prevalent for tackling challenges in earthquake
+engineering and providing fairly reliable and accurate predictions. However, it is mostly
+unclear how decisions are made because machine learning models are generally highly
+sophisticated, resulting in opaque black-box models. Machine learning models that are
+naturally interpretable and provide their own decision explanation, rather than using an
+explanatory, are more accurate in determining what the model actually computes. With
+this motivation, this study aims to develop a fully explainable machine learning model
+to predict the deformation capacity of non-ductile reinforced concrete shear walls based
+on experimental data collected worldwide. The proposed Explainable Boosting Machines
+(EBM)-based model is an interpretable, robust, naturally explainable glass-box model, yet
+provides high accuracy comparable to its black-box counterparts. The model enables the
+user to observe the relationship between the wall properties and the deformation capacity
+by quantifying the individual contribution of each wall property as well as the correlations
+among them. The mean coefficient of determination R2 and the mean ratio of predicted
+to actual value based on the test dataset are 0.92 and 1.05, respectively. The proposed
+predictive model stands out with its overall consistency with scientific knowledge, practicality,
+and interpretability without sacrificing high accuracy.
+Keywords: Explainable boosting machine, glass-box model, feature selection, general additive model, reinforced
+concrete shear wall, deformation capacity, interpretability
+1
+Introduction
+Shear walls are typically utilized as the primary elements to resist lateral loads in reinforced concrete buildings.
+Towards capacity design assumptions, shear walls are designed to exhibit ductile behavior by providing
+adequate reinforcement and proper detailing. However, experimental studies have shown that walls with an
+aspect ratio smaller than 1.5 (i.e., squat walls) and those with poor reinforcement and detailing, despite
+their higher aspect ratio, end up showing brittle failure (e.g., diagonal tension, web crushing) [1, 2, 3]. Such
+walls are often observed in buildings not designed according to modern seismic codes and are prone to severe
+damage [4, 5]. As the performance-based design and assessment approach has gained importance concordant
+arXiv:2301.04652v1  [cs.LG]  11 Jan 2023
+
+Deger ZT, Kaya GT, Wallace JW. ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC
+SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint.
+with hazard mitigation efforts, there has been an increasing need and demand for reliable models to predict
+structural behavior under seismic actions. This objective is particularly important for walls that exhibit
+shear behavior as the nonlinear deformation capacity of such walls is assumed to be zero, potentially leading
+to technical and economical over-conservation. More realistic solutions can be achieved if their behavior is
+accurately estimated and considered in seismic performance evaluation.
+The prediction of structural behavior has been achieved through the use of predictive equations or models that
+are developed based on available experimental data. Recently, machine learning (ML) methods have gained
+significant attention structural/earthquake engineering field and have demonstrated promising results despite
+the scarcity of data (compared to much larger data available in fields such as computer vision and image
+processing). Black-box models, with their high complexity and nonlinearity, often represent the input-output
+relationship better than the interpretable models in classification and regression applications. However, they
+are not necessarily consistent with true physical behavior. There are examples of misleading conclusions of
+black-box models in scientific and engineering applications [6, 7, 8]. Therefore, despite the high accuracy
+they achieve, black-box models are not completely accepted in earthquake engineering society. To leverage
+the advantages of the developments in artificial intelligence without ignoring the physical behavior, there has
+been recent research efforts that incorporates black-box machine learning methods with physical knowledge
+[8, 9, 10, 11]. This study takes this issue a step further and integrates an explainable machine learning
+approach (versus black-box) with existing physics-based understanding of seismic behavior to estimate the
+deformation capacity of non-ductile shear walls.
+2
+Literature Review
+Research efforts in the literature to estimate wall deformation capacity have produced empirical models, some
+recently adopted by building codes [12]; however, they are relatively limited compared to other behavior
+features such as shear strength or failure mode. Earlier models were mainly developed using a limited number
+of experimental results [13, 14] or were trained using a single dataset; that is, they were not trained and tested
+based on unmixed data [12, 15]. Over time, as machine learning is embraced in the earthquake engineering
+field [16, 17, 18, 19, 20, 11] and new experiments are conducted, more advanced models have been developed.
+Yet, two main issues are encountered: (i) Some models used simple approaches such as linear regression
+for the sake of interpretability [21] and sacrificed overall accuracy (or had large dispersion). One might
+think that accurate models that predict relatively complicated behavior attributes can only be achieved
+by increasing model complexity; however, literature studies have shown that this may cause problems with
+the structure and distribution of the data [22, 23]. More importantly, urging the model to develop complex
+relationships to achieve higher performance typically leads to black-box models where internal mechanisms
+include highly nonlinear, complex relations. (ii) Such black-box models achieve high overall accuracy at
+the cost of explainability [18]. Researchers that acknowledge the significance of interpretability employed
+model-agnostic, local or global explanation methods (e.g., SHapley Additive exPlanation, Local Interpretable
+Model-agnostic Explanations) to interpret the decision mechanism of their models [11]. Such algorithms are
+not fully verified [24, 25]; besides, they are approximate approaches. Moreover, despite their broadening
+use and high accuracy, the black-box models are not entirely accepted in the earthquake engineering society
+as their internal relations are opaque and, in some cases, not entirely reliable [26]. Therefore, it is critical
+to understand how the model makes the decision/estimation to (i) verify that the model is physically
+meaningful, (ii) develop confidence in the predictive model, and (iii) broaden existing scientific knowledge
+with new insights.This study addresses this need and fills this important research gap by using domain-specific
+knowledge to evaluate and validate the decisions made by ML methods. Unlike the existing ML-based
+predictive models ([11, 18]), the proposed model aims particularly at the deformation capacity of non-ductile
+shear walls and is naturally transparent and interpretable.
+Concerns regarding the trustworthiness and transparency of the black-box models motivated the development
+of a relatively new research area known as explainable artificial intelligence (XAI) [27, 28]. The XAI aims to
+provide a set of machine learning (ML) techniques for building more comprehensible and understandable
+models while maintaining a high level of learning performance. The strategies used in XAI are divided
+2
+
+Deger ZT, Kaya GT, Wallace JW. ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC
+SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint.
+into two main categories: explaining existing black-box models (post-hoc explainability) and generating
+glass-box (transparent) models. In the former, interpretability is confined to the usage of certain so-called
+explanatory algorithms that are employed to explain a black-box model, while in the latter, a predictive
+model is fully comprehensible and interpretable by humans. A model should have certain qualities to be
+considered a transparent model such as decomposability, algorithmic transparency, and simulatability [29]. The
+decomposability relates to the ability to explain each model component in terms of the inputs’ contributions
+or correlations, whereas simulatability refers to the number of parameters (input) in the model representation
+(the less is the more understandable). The algorithmically transparent models enable a clear comprehension
+of the model’ behavior for predicting any given output from its input data. Therefore, transparent models
+are highly needed approaches in fields where decisions are critical, but their performances are typically very
+low. Machine learning models that can maintain the tradeoff between performance and explainability, i.e.,
+converging to the performance of black-box models while still providing explainability, would significantly
+address the demands in earthquake engineering society. In this context, explainable boosting machine (EBM)
+[30], a recently developed method belonging to the family of Generalized Additive Models (GAMs) [31], is a
+highly accurate and transparent ML method delivering an explicit and fully explainable predictive model.
+The EBM has been utilized in the literature to solve a variety of problems, including detecting common flaws
+in data [32], diagnosing COVID-19 using blood test variables [33], predicting diseases such as Alzheimer [34],
+or Parkinson [35], and has shown to outperform black-box models with the additional benefit of being an
+inherently explainable predictive model.
+In this study, the EBM is used for the first time in the earthquake engineering field to construct an EBM-
+based predictive model for estimating deformation capacity on non-ductile RC shear walls. The inputs of
+the predictive model are designated as the shear wall design properties (e.g., wall geometry, reinforcing
+ratio), whereas the output is one of the constitutive components of the nonlinear wall behavior, that is, the
+deformation capacity. The main contributions of this research are highlighted as follows:
+• A fully transparent and interpretable predictive model is developed to estimate the deformation
+capacity of RC shear walls that are failed in pure shear or shear-flexure interaction.
+• The proposed model meets all desired properties, i.e., decomposability, algorithmic transparency,
+and simulatability, without compromising high performance.
+• This study integrates novel computational methods (i.e., EBM) and domain-based knowledge to
+formalize complex engineering knowledge. The proposed model’s overall consistency with a physics-
+based understanding of seismic behavior is verified.
+3
+The RC Shear Wall Database
+The experimental data used in this research is a sub-assembly of the wall test database utilized in Deger
+and Taskin (2022) [19] with 30 additional data [36, 37]. As the main focus is to estimate the deformation
+capacity of walls governed by shear or shear-flexure interaction, walls that did not show so-called shear failure
+indications are excluded from the database, resulting in 286 specimens of use for this research. All specimens
+were tested under quasi-static cyclic loading, whereas none was retrofitted and re-tested. The database
+consists of wall design parameters, which are herein designated as the input variables of the machine learning
+problem, namely: wall geometry (tw, lw, hw), shear span ratio (M/V lw), concrete compressive strength
+(fc), longitudinal and transverse reinforcing ratios at web (fyl, fyt), longitudinal and transverse reinforcing
+ratios at boundary elements (fybl, fysh), axial load ratio (P/Agfc), shear demand (or strength) at the section
+(Vmax), cross-section type (rectangular, barbell-shape, or flanged), curvature type (single or double). It is
+noted that single curvature and double curvature correspond to the end conditions of the specimen, i.e.,
+cantilever and fixed-fixed, respectively. Distributions of the input variables are presented in Fig.1 along with
+their box plots (shown in blue).
+The output variable of the ML problem, the deformation capacity, is taken directly as the reported ultimate
+displacement prior to its failure if the specimen is tested until failure. Otherwise, it is assumed as the
+displacement corresponding to 0.8Vmax as suggested by Park, 1989. It is noted that failure displacement
+3
+
+Deger ZT, Kaya GT, Wallace JW. ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC
+SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint.
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+MVlw
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+hw/lw
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+P/Agfc
+0
+1
+2
+3
+4
+5
+6
+rol
+0.5
+1.0
+1.5
+2.0
+2.5
+rot
+0
+2
+4
+6
+8
+10
+rosh
+0
+5
+10
+15
+20
+25
+robl
+20
+40
+60
+80
+100
+120
+140
+fc
+50
+100
+150
+200
+250
+300
+tw
+0
+500
+1000
+1500
+2000
+2500
+Vmax
+Figure 1: Distribution of the input variables in the database.
+was taken as the total wall top displacement and was not separated into shear and flexural deformation
+components.
+4
+Explainable Boosting Machines
+Explainable Boosting Machines (EBM) is a state-of-the-art machine learning technique designed as accurately
+as random forests and boosted trees while also being simple to understand [30, 38]. The EBM delivers a
+complete explainable learning model that belongs to the family of Generalized Additive Models (GAMs) [39]:
+g(f(x1, . . . , xn)) = f0 + f1(x1) + f2(x2) + . . . , fn(xn)
+(1)
+where f0 is an intercept, and each fj is called a shape function, representing the individual effect of the
+xj-th variable on the model output, f(x1, . . . , xn). The g is utilized as a link function, adapting the model to
+different settings, e.g., identity function for regression and logistic function for classification. The intercept, f0,
+is calculated as the mean response of all the outputs. Because the shape functions are trained independently
+for each input variable, the model is additive (decomposable), allowing to separately analyze the effect of
+each input variable on the model output. The EBM is designed to improve the performance of the standard
+GAM while maintaining its interpretability.
+Generalized Additive Models are more comprehensible than black-box models, but the analytical form of the
+shape functions is typically unknown, making it unsuitable for machine learning purposes. Although other
+analytical functions, such as splines or orthogonal polynomials, can be offered for defining shape functions,
+they are frequently less accurate when representing a nonlinear model [40]. The EBM uses shallow trees to
+construct the shape functions; therefore, it easily captures the nonlinearity of the data. Each input variable
+(xi) is modeled with ensemble trees such as bagging and gradient boosting. As a result, rather than employing
+the spline method, which is prevalent in traditional GAMs, the function associated with each input variable
+or interaction is produced from a vast set of shallow trees.
+The EBM offers both local and global explanations of the learning model as each variable importance is
+estimated as the average absolute value of the predicted score. Moreover, each shape function can be visualized
+(algorithmically transparent); therefore, it is possible to observe the effects of the particular feature at certain
+intervals. In the inference phase, all the terms in Eq.1 are added up and passed through the link function to
+g to compute the final prediction as shown in Fig. 2. In other words, individual predictions are generated
+using the shape functions, fi, which act as a lookup table.
+To demonstrate which feature had the largest impact on the individual forecast, the contributions can be
+sorted and shown using the principles of additivity and modularity.
+The EBM’s performance can be improved by including pairwise effects between variables in the model
+representation. For better performance, additional interactions can be incorporated; however, this may result
+in a more complex model with lower generalization performance due to the increased number of model
+parameters to be trained. The pairwise interactions are included in GA2Ms [41], which is a second-order
+4
+
+Deger ZT, Kaya GT, Wallace JW. ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC
+SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint.
+Inference Phase
+Intercept +
++
++
++
+E B M  P r e d i c t i o n  M o d e l  
+f0
++
+f1(x1)
+f2(x2)
+f3(x3)
++
++
++
+f12(x1, x5)
+x1
+x2
+x3
+x1
+x5
+score
+score
+score
+score
+Figure 2: Inference phase of EBM.
+additive model:
+g(E[y])) = f0 +
+n
+�
+i=1
+fi(xi) +
+K
+�
+i=1
+fk(xk1, xk2)
+(2)
+where K pairs of features (k1, k2) are chosen greedily (FAST algorithm) [42]. The pairwise interaction
+fij(xi, xj) could be rendered as a heatmap on the two-dimensional xi, xj - plane, still providing high
+intelligibility. Even though adding more interactions does not affect the model’s explainability, the final
+prediction model may be less comprehensible due to a large number of interactions (less simulatability).
+5
+Development of the Predictive Model
+5.1
+Overall Performance of EBM
+To assess whether the method compromises accuracy for the sake of interpretability, the performance of the
+EBM model is compared to three state-of-the-art black-box machine learning models, namely: XGBoost [43],
+Gradient Boost [44], Random Forest [45], and two glass-box models, namely Ridge Linear Regression [46],
+Decision Tree [47]. All the implementations are carried out in a Python environment. For all ML models, the
+entire database, including all twelve input variables (ten variables from Fig.1 and two binary coded variables
+for curvature type and cross-section type), is randomly split into training and test datasets with a ratio of
+90% and 10%, respectively.
+Tunning of the hyperparameters, such as learning rate, number of leaves, number of interactions, and so
+on, typically affects the performance of the corresponding regression model. For hyperparameter tuning, a
+10-fold cross-validation technique (Fig. 3) is used, wherein a subset of the data is kept as validation data,
+and the model’s performance is evaluated using various hyperparameter settings on the validation set. This
+method prevents the tuning from overfitting the training dataset.
+Training Phase
+Split Training dataset into k-folds
+Validation
+Training
+Validation
+Training
+Validation
+Training
+Validation
+Training
+Validation
+Training
+Training
+Training
+Training
+Hyperparameter tuning
+Model Evalution on validation data
+R^2, RE, PA
+Figure 3: Illustration of k-fold cross-validation technique, where k is set to 5.
+For performance evaluations, the following three metrics are used over “unseen” (i.e., not used in the training
+process) test data sets of ten random train-test data splittings: coefficient of determination (R2), relative
+5
+
+60
+40
+Score
+20
+0
+2
+m60
+40
+Score
+20
+0
+-20
+50
+100
+150
+200
+25060
+40
+Score
+20
+-20
+0
+0.1
+0.2
+0.3
+0.4
+0.5300
+60
+250
+40
+200
+20
+150
+0
+100
+-20
+50
+2
+4
+MVIWDeger ZT, Kaya GT, Wallace JW. ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC
+SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint.
+Table 1: Mean performance scores based on the test datasets over ten random splittings.
+Black-box
+Glass-box
+EBM
+XGBoost
+GB
+RF
+RLR
+DT
+R2
+0.83
+0.80
+0.79
+0.83
+0.41
+0.67
+RE (%)
+0.41
+0.40
+0.32
+0.28
+0.67
+0.47
+PA
+1.21
+1.17
+1.20
+1.15
+2.0
+1.9
+error (RE), and prediction accuracy (PA), as given in Eqs. 3, 4, and 5, respectively.
+R2
+=
+1 −
+�
+i(yi − ˆyi)2
+�
+i(yi − ¯y)2
+(3)
+RE
+=
+�
+i | ˆyi − yi|
+�
+i |yi|
+| × 100%
+(4)
+PA
+=
+�
+i
+yi
+ˆyi
+(5)
+where yi, ¯y, ˆyi, and m refer to the actual output, the mean value of yis, predicted output of corresponding
+regression model, and a number of samples in the test dataset, respectively.
+Mean performance scores of the ML models are summarized in Table 1, along with their dispersion demon-
+strated in box plots in Fig. 4. The results indicate that EBM achieves comparable performance with its
+black-box counterparts, with a correlation of determination of R2 = 0.83, a relative error of 0.41%, and a
+PA = 1.21. As seen in Fig. 4, the low R2, RE, and PA deviations of EBM imply that reliable predictions can
+be achieved regardless of the selected train-test splitting and verify the model’s robustness. Mean prediction
+accuracy (PA) shows around 20% of overestimation for EBM and the black-box methods, suggesting that
+some input variables are potentially noisy. Compared to transparent models, the EBM outperforms both
+the Decision Tree (DT) and Ridge Linear Regression (RLR) across all three metrics, indicating that it is far
+superior to the traditional glass-box approaches.
+The most remarkable advantage of the EBM method over the others is that it provides full explainability
+without sacrificing accuracy. Unlike other methods, EBM enables the user to understand how the prediction
+is made and which parameters are essential in the decision-making process. Therefore, the EBM method is
+selected as the baseline algorithm for the rest of the analysis to propose a prediction model for estimating the
+deformation capacity based on the following criteria: developing a model with fewer input variables (high
+simulatability), achieving high accuracy, and ensuring physical consistency.
+5.2
+The Proposed EBM-based Predictive Model
+The importance of the wall properties in predicting the deformation capacity is evaluated based on additive
+term contributions visualized in Fig.5. Results reveal that tw and M/V lw (or hw/lw) have the greatest
+impact on individual predictions. This is consistent with the mechanics of the behavior as walls with smaller
+thickness are shown to be more susceptible to lateral stiffness degradation due to concrete spalling, leading
+to a failure caused by lateral instabilities or out-of-plane buckling [48, 49]. The shear span ratio (or aspect
+ratio), on the other hand, both have a significant impact on deformation capacity as the higher the shear
+span ratio gets, the slender the wall is, and the higher deformations it typically can reach prior to its failure.
+The least important wall parameters, on the other hand, are identified as curvature type, cross-section type,
+and concrete compressive strength.
+Another critical aspect considered in this study is to develop the predictive model with as few input variables
+as possible. With that, the computational workload is aimed to be reduced, and a more practical and
+interpretable model is proposed for potential users. To achieve this, knowledge-based-selected combinations
+6
+
+Deger ZT, Kaya GT, Wallace JW. ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC
+SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint.
+Figure 4: Comparison of performance scores of ML methods for test samples based on ten random train
+splittings.
+Figure 5: EBM Global interpretation for twelve features included
+of four-to-five features are exhaustively evaluated to reach performance scores as high as when twelve features
+are included.
+EBM can achieve similar performance scores using four features: M/V lw, P/Agfc, tw, and Vmax. Including
+additional features (e.g., ρl, ρbl, ρsh) deemed impactful by EBM as well as experimental results [50, 51]
+has only a modest effect on the overall performance. The performances of other methods are close to their
+benchmark model (including twelve features), whereas the glass-box methods are affected by the reduction of
+input size and show much lower performances. The mean R2 drops to 0.33 for the Ridge Linear Regression,
+imparting that the input-output relation is not linear.
+The proposed EBM-based predictive model is selected to achieve the highest R2 with a prediction accuracy
+as close to 1.0 as possible. The correlation plots are presented in Fig. 6 for training and test data sets, where
+7
+
+Deger ZT, Kaya GT, Wallace JW. ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC
+SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint.
+scattered data are concentrated along the y = x line, demonstrating that the proposed model can make
+accurate predictions. It should be noted that the distribution of the residuals is concentrated around zero.
+(a)
+(b)
+(a)
+(b)
+Training Dataset
+Test Dataset
+Prediction
+Prediction
+Actual
+Actual
+Figure 6: Correlations of the model outputs with the actual values for (a) training and (b) test datasets
+As discussed above, the proposed model is an additive model in which each relevant feature is designated a
+quantitative term contribution. The EBM allows the user to explore the contribution of each feature to the
+model by plotting their shape functions (Fig.7a-d). As discussed above, the EBM method employs multiple
+decision-tree learning models; therefore, inclines and declines are undertaken with jump-looking piece-wise
+constant functions (versus smooth curves). The values, called scores, are read from these functions, and those
+from heat maps (Fig.7e-f) representing pairwise interactions (i.e., between two features) are summed up to
+calculate the prediction. The gray zones along the shape functions designate error bars that indicate the
+model’s uncertainty and data sensitivity. This typically occurs in cases of sparsity or the presence of outliers
+within the associated region.
+The shape functions in Fig.7 also indicate their correlations with the output in a graphical representation. For
+example, nonlinear patterns that can not be observed in linear approaches can be easily interpreted [52], which
+provides new insights to broaden existing experimental-based knowledge. For example, the shear demand
+Vmax (Fig.7d) reduces ductility; thus deformation capacity, as demonstrated by experimental results [53] and
+suggested by ASCE 41-17 acceptance criteria. Yet, a highly nonlinear pattern is observed when relevant
+experimental data are gathered [11, 21]. This nonlinearity can be observed in the shape function suggested
+by the proposed method. Other input variables (tw, P/Agfc, M/V lw), on the other hand, demonstrate an
+almost-linear trend. The interpretation of EBM for these variables is consistent with experimental results
+8
+
+Deger ZT, Kaya GT, Wallace JW. ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC
+SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint.
+���
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+,
+,
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+Figure 7: EBM shape functions (a-d) and pairwise interaction plots (e-f) for the proposed model. Note that
+the intercept f0= 35.528.
+in the literature, such that M/V lw (Fig.7a) and tw (Fig.7b) have a positive impact, as discussed above,
+whereas P/Agfc (Fig.7c) has an adverse influence [54, 55]. The reason for M/V lw and tw (Fig.7b) suggesting
+an inverse effect up to a certain point (M/V lw ≈ 1.2, tw ≈ 60 cm, P/Agfc ≈ 0.08) is because the model
+has an intercept value (f0, Eq.2) and specimens with smaller deformation capacities (f0 less than 35.528)
+are predicted adding up negative values. The unexpected jumps in tw are likely because there is an abrupt
+accumulation of data at tw = 100 mm and tw = 200 mm (64 and 44 specimens, respectively), which probably
+causes difficulty in decision making.
+It is noted that the EBM method offers controllability over the structure of the model proposed by, for
+instance, modifying the number of pairwise interactions. This allows the method to suggest more than one
+9
+
+Deger ZT, Kaya GT, Wallace JW. ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC
+SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint.
+−30
+−20
+−10
+0
+10
+20
+30
+Vmax (421.58)
+MVlw x tw
+MVlw x P/Agfc
+P/Agfc (0.00)
+tw (200.00)
+MVlw (2.08)
+Intercept
+Predicted (81.9) | Actual (81.8)
+−30
+−20
+−10
+0
+10
+20
+30
+MVlw x tw
+Vmax (445.60)
+MVlw x P/Agfc
+tw (130.00)
+P/Agfc (0.50)
+MVlw (2.00)
+Intercept
+Predicted (58.7) | Actual (67.2)
+x
+x
+x
+x
+(a)
+(b)
+Figure 8: Variable contribution estimates for (a) well-predicted, (b) averagely-predicted samples.
+model for the same input-output configuration for a particular train-test dataset. Reducing the number
+of interactions brings simplicity to the model; however, it typically loses accuracy as EBM relies on its
+automatically-determined interactions in the decision-making process. Given this trade-off, the number of
+interactions is set to two for the proposed model.
+5.3
+Sample-Based Explanation
+This section presents the prediction of deformation capacity for two example specimens using the proposed
+EBM-based predicted model. One specimen is predicted with excellent accuracy (almost zero error; Fig.8a),
+whereas the other is predicted with around 15% error (Fig.8b).
+Variable contribution estimates for each specimen are presented such that the intercept is constant and shown
+in gray, the additive terms with positive impact are marked in orange, and additive terms decreasing the output
+are shown in blue. Each contribution estimate is extracted from the shape functions and two-dimensional
+heat maps (Fig.7) based on the input values of a specific specimen. Overall, the model is consistent with
+physical knowledge, except Vmax has an unexpected positive impact on the output for the relatively worse
+prediction (Fig.8b). This is an excellent advantage of EBM; that is, the user can prudently understand
+how the prediction is made for a new sample and develop confidence in the predictive model (versus blind
+acceptation in black-box models).
+5.4
+Comparisons with Current Code Provisions
+ASCE 41-17 and ACI 369-17 [56] provide recommended deformation capacities for nonlinear modeling purposes,
+where shear walls are classified into the following two categories based on their aspect ratio: shear-controlled
+(hw/lw > 1.5) and flexure-controlled (hw/lw > 3.0). The deformation capacity of shear-controlled walls is
+identified as drift ratio such that ∆u/hw = 1.0 if the wall axial load level is greater than 0.5 and ∆u/hw = 2.0
+otherwise.
+Deformation capacity predictions based on the proposed EBM model are compared to ASCE 41-17/ACI
+369-17 provisions in Fig. 9. Predicted-to-actual ratios are 1.06 ± 0.49 and 6.42 ± 3.17 for EBM-based
+model and code predictions, respectively. The results imply that traditional approaches may lead to the
+overestimation of deformation capacities and cause unsafe assessments.
+10
+
+Deger ZT, Kaya GT, Wallace JW. ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC
+SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint.
+0
+5
+10
+15
+20
+25
+30
+Specimen number
+10 0
+10 1
+10 2
+10 3
+10 4
+10 5
+Ultimate displacement (mm)
+Test (actual)
+EBM-based model prediction
+ASCE 41/ACI 369 prediction
+Figure 9: Comparisons of EBM-based model predictions with code provisions.
+6
+Conclusions
+A fully transparent predictive model is developed to estimate the deformation capacity of reinforced concrete
+shear walls that are failed in pure shear or shear-flexure interaction. To achieve this, a state-of-the-art machine
+learning method, Explainable Boosting Machines (EBM), designed as accurately as random forests and
+boosted trees, is utilized. The EBM provides an additive model such that each relevant feature is designated
+a quantitative term contribution. The input-output configuration of the model is designated as the shear wall
+design properties (e.g., wall geometry, axial load ratio) and ultimate wall displacement, respectively. The
+conclusions derived from this study are summarized as follows:
+• The importance of the wall properties in predicting the deformation capacity is evaluated based
+on additive term contributions. tw and M/V lw (or hw/lw) have the greatest effect on individual
+predictions, whereas the least relevant ones are identified as curvature type, cross-section type, and
+concrete compressive strength.
+• Compared to three black-box models (XGBoost, Gradient Boost, Random Forest), the EBM achieves
+similar or better performance in terms of correlation of determination (R2), relative error (RE), and
+prediction accuracy (PA; the ratio of predicted to the actual value). The EBM achieves a mean
+R2 of 0.83 and a mean RE of 0.41% using twelve input variables based on ten random train-test
+splittings.
+• Compared to two glass-box methods (Decision Tree (DT) and Ridge Linear Regression (RLR)), the
+EBM outperforms both methods across all three metrics.
+• The dispersion of performance metrics of EBM is small, implying that the model is robust and the
+performance is relatively less data-dependent.
+• Compared to the developed model when all the available features are used, the EBM achieves
+competitive performance scores using only four input variables: M/V lw, P/Agfc, tw, and Vmax.
+Using these four features, the proposed EBM-based model achieves R2 of 0.92 and PA of 1.05 based
+on the test dataset. Using fewer variables ensures that the model is less simulatable, more practical,
+more comprehensible, and reduces the computational cost.
+• It is important to note that the decision-making process developed by the proposed EBM-based model
+has overall consistency with scientific knowledge despite several exceptions detected in sample-based
+inferences. This is an excellent advantage of the proposed model; that is, the user can assess and
+evaluate the prediction process before developing confidence in the result (versus blindly accepting as
+in black-box models).
+11
+
+Deger ZT, Kaya GT, Wallace JW. ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC
+SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint.
+• This model delivers exact intelligibility, i.e., there is no need to use local explanation methods (e.g.,
+SHAP, LIME) to interpret the learning model, which obviates the uncertainties associated with their
+approximations.
+The proposed EBM-based model is valuable in that it is simultaneously accurate, explainable, and consistent
+with scientific knowledge. The EBM’s ability to provide interpretable and transparent results would allow
+engineers to better understand the factors that affect the deformation capacity of non-ductile RC shear walls
+and make informed design decisions. The use of the EBM to estimate deformation capacity would improve the
+reliability and efficiency of structural analysis and design processes, leading to safer and more cost-effective
+buildings.
+12
+
+Deger ZT, Kaya GT, Wallace JW. ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC
+SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint.
+References
+[1] ASCE-41. ASCE Standard, ASCE/SEI, 41-17, Seismic Evaluation and Retrofit of Existing Buildings.
+American Society of Civil Engineers, 2017.
+[2] Leonardo M Massone and John W Wallace. Load-deformation responses of slender reinforced concrete
+walls. Structural Journal, 101(1):103–113, 2004.
+[3] Chadchart Sittipunt, Sharon L Wood, Panitan Lukkunaprasit, and Pichai Pattararattanakul. Cyclic
+behavior of reinforced concrete structural walls with diagonal web reinforcement. Structural Journal,
+98(4):554–562, 2001.
+[4] John W Wallace, Leonardo M Massone, Patricio Bonelli, Jeff Dragovich, René Lagos, Carl Lüders, and
+Jack Moehle. Damage and implications for seismic design of rc structural wall buildings. Earthquake
+Spectra, 28(1_suppl1):281–299, 2012.
+[5] C Arnold, B Bolt, D Dreger, E Elsesser, R Eisner, W Holmes, G McGavin, and C Theodoropoulos.
+Fema 454: Design for earthquakes: A manual for architects, 2006.
+[6] David Lazer, Ryan Kennedy, Gary King, and Alessandro Vespignani. The parable of google flu: traps in
+big data analysis. Science, 343(6176):1203–1205, 2014.
+[7] John Douglas and Hideo Aochi. A survey of techniques for predicting earthquake ground motions for
+engineering purposes. Surveys in geophysics, 29(3):187–220, 2008.
+[8] Anuj Karpatne, Ramakrishnan Kannan, and Vipin Kumar. Knowledge Guided Machine Learning:
+Accelerating Discovery using Scientific Knowledge and Data. CRC Press, 2022.
+[9] Huan Luo and Stephanie German Paal. Artificial intelligence-enhanced seismic response prediction of
+reinforced concrete frames. Advanced Engineering Informatics, 52:101568, 2022.
+[10] Siyang Zhou, Shanglin Liu, Yilan Kang, Jie Cai, Haimei Xie, and Qian Zhang. Physics-based machine
+learning method and the application to energy consumption prediction in tunneling construction.
+Advanced Engineering Informatics, 53:101642, 2022.
+[11] Muneera A Aladsani, Henry Burton, Saman A Abdullah, and John W Wallace. Explainable machine
+learning model for predicting drift capacity of reinforced concrete walls. ACI Structural Journal, 119(3),
+2022.
+[12] Saman A Abdullah and John W Wallace. Drift capacity of reinforced concrete structural walls with
+special boundary elements. ACI Structural Journal, 116(1):183, 2019.
+[13] T Paulay, MJN Priestley, and AJ Synge. Ductility in earthquake resisting squat shearwalls. In Journal
+Proceedings, volume 79, pages 257–269, 1982.
+[14] İlker Kazaz, Polat Gülkan, and Ahmet Yakut. Deformation limits for structural walls with confined
+boundaries. Earthquake Spectra, 28(3):1019–1046, 2012.
+[15] Sofia Grammatikou, Dionysis Biskinis, and Michael N Fardis. Strength, deformation capacity and failure
+modes of rc walls under cyclic loading. Bulletin of earthquake engineering, 13(11):3277–3300, 2015.
+[16] Sujith Mangalathu, Hansol Jang, Seong-Hoon Hwang, and Jong-Su Jeon. Data-driven machine-learning-
+based seismic failure mode identification of reinforced concrete shear walls. Engineering Structures,
+208:110331, 2020.
+[17] De-Cheng Feng, Wen-Jie Wang, Sujith Mangalathu, and Ertugrul Taciroglu. Interpretable xgboost-shap
+machine-learning model for shear strength prediction of squat rc walls. Journal of Structural Engineering,
+147(11):04021173, 2021.
+[18] Haoyou Zhang, Xiaowei Cheng, Yi Li, and Xiuli Du.
+Prediction of failure modes, strength, and
+deformation capacity of rc shear walls through machine learning. Journal of Building Engineering,
+50:104145, 2022.
+[19] Zeynep Tuna Deger and Gulsen Taskin. A novel gpr-based prediction model for cyclic backbone curves
+of reinforced concrete shear walls. Engineering Structures, 255:113874, 2022.
+13
+
+Deger ZT, Kaya GT, Wallace JW. ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC
+SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint.
+[20] Zeynep Tuna Deger and Gulsen Taskin Kaya. Glass-box model representation of seismic failure mode
+prediction for conventional reinforced concrete shear walls. Neural Computing and Applications, pages
+1–13, 2022.
+[21] Zeynep Tuna Deger and Cagri Basdogan. Empirical expressions for deformation capacity of reinforced
+concrete structural walls. ACI Structural Journal, 116(6), 2019.
+[22] Justin M Johnson and Taghi M Khoshgoftaar. Survey on deep learning with class imbalance. Journal of
+Big Data, 6(1):1–54, 2019.
+[23] Bhavya Kailkhura, Brian Gallagher, Sookyung Kim, Anna Hiszpanski, and T Han.
+Reliable and
+explainable machine-learning methods for accelerated material discovery. npj Computational Materials,
+5(1):1–9, 2019.
+[24] I Elizabeth Kumar, Suresh Venkatasubramanian, Carlos Scheidegger, and Sorelle Friedler. Problems
+with shapley-value-based explanations as feature importance measures. In International Conference on
+Machine Learning, pages 5491–5500. PMLR, 2020.
+[25] Cynthia Rudin. Stop explaining black box machine learning models for high stakes decisions and use
+interpretable models instead. Nature Machine Intelligence, 1(5):206–215, 2019.
+[26] Christoph Molnar. Interpretable machine learning. Lulu. com, 2020.
+[27] Amina Adadi and Mohammed Berrada. Peeking inside the black-box: a survey on explainable artificial
+intelligence (xai). IEEE access, 6:52138–52160, 2018.
+[28] Zachary C Lipton. The mythos of model interpretability: In machine learning, the concept of inter-
+pretability is both important and slippery. Queue, 16(3):31–57, 2018.
+[29] Alejandro Barredo Arrieta, Natalia Díaz-Rodríguez, Javier Del Ser, Adrien Bennetot, Siham Tabik,
+Alberto Barbado, Salvador García, Sergio Gil-López, Daniel Molina, Richard Benjamins, et al. Explainable
+artificial intelligence (xai): Concepts, taxonomies, opportunities and challenges toward responsible ai.
+Information fusion, 58:82–115, 2020.
+[30] Harsha Nori, Samuel Jenkins, Paul Koch, and Rich Caruana. Interpretml: A unified framework for
+machine learning interpretability. arXiv preprint arXiv:1909.09223, 2019.
+[31] Trevor Hastie, Robert Tibshirani, and Jerome Friedman. Additive models, trees, and related methods.
+In The Elements of Statistical Learning, pages 295–336. Springer, 2009.
+[32] Zhi Chen, Sarah Tan, Harsha Nori, Kori Inkpen, Yin Lou, and Rich Caruana. Using explainable boosting
+machines (ebms) to detect common flaws in data. In Joint European Conference on Machine Learning
+and Knowledge Discovery in Databases, pages 534–551. Springer, 2021.
+[33] Lucas M Thimoteo, Marley M Vellasco, Jorge Amaral, Karla Figueiredo, Cátia Lie Yokoyama, and Erito
+Marques. Explainable artificial intelligence for covid-19 diagnosis through blood test variables. Journal
+of Control, Automation and Electrical Systems, 33(2):625–644, 2022.
+[34] Alessia Sarica, Andrea Quattrone, and Aldo Quattrone. Explainable boosting machine for predicting
+alzheimer’s disease from mri hippocampal subfields. In International Conference on Brain Informatics,
+pages 341–350. Springer, 2021.
+[35] Alessia Sarica, Andrea Quattrone, and Aldo Quattrone. Explainable machine learning with pairwise
+interactions for the classification of parkinson’s disease and swedd from clinical and imaging features.
+Brain Imaging and Behavior, pages 1–11, 2022.
+[36] R Tokunaga and T Nakachi. Experimental study on edge confinement of reinforced concrete core walls.
+In Fifteenth World Conference on Earthquake Engineering, Lisbon, pages 1–5, 2012.
+[37] Masaya Hirosawa. Past experimental results on reinforced concrete shear walls and analysis on them.
+Kenchiku Kenkyu Shiryo, 6:33–34, 1975.
+[38] Harsha Nori, Rich Caruana, Zhiqi Bu, Judy Hanwen Shen, and Janardhan Kulkarni. Accuracy, inter-
+pretability, and differential privacy via explainable boosting. In International Conference on Machine
+Learning, pages 8227–8237. PMLR, 2021.
+14
+
+Deger ZT, Kaya GT, Wallace JW. ESTIMATE DEFORMATION CAPACITY OF NON-DUCTILE RC
+SHEAR WALLS USING EXPLAINABLE BOOSTING MACHINE,Preprint.
+[39] Trevor Hastie and Robert Tibshirani. Generalized additive models: some applications. Journal of the
+American Statistical Association, 82(398):371–386, 1987.
+[40] Mathew W McLean, Giles Hooker, Ana-Maria Staicu, Fabian Scheipl, and David Ruppert. Functional
+generalized additive models. Journal of Computational and Graphical Statistics, 23(1):249–269, 2014.
+[41] Yin Lou, Rich Caruana, and Johannes Gehrke. Intelligible models for classification and regression. In
+Proceedings of the 18th ACM SIGKDD international conference on Knowledge discovery and data mining,
+pages 150–158, 2012.
+[42] Simon N Wood. Fast stable direct fitting and smoothness selection for generalized additive models.
+Journal of the Royal Statistical Society: Series B (Statistical Methodology), 70(3):495–518, 2008.
+[43] Tianqi Chen and Carlos Guestrin. Xgboost: A scalable tree boosting system. In Proceedings of the 22nd
+acm sigkdd international conference on knowledge discovery and data mining, pages 785–794, 2016.
+[44] Jerome H Friedman. Greedy function approximation: a gradient boosting machine. Annals of statistics,
+pages 1189–1232, 2001.
+[45] Leo Breiman. Random forests. Machine learning, 45(1):5–32, 2001.
+[46] Trevor Hastie, Robert Tibshirani, Jerome H Friedman, and Jerome H Friedman. The elements of
+statistical learning: data mining, inference, and prediction, volume 2. Springer, 2009.
+[47] Leo Breiman, Jerome H Friedman, Richard A Olshen, and Charles J Stone. Classification and regression
+trees. Routledge, 2017.
+[48] Jose Miguel Vallenas, Vitelmo Victorio Bertero, and Egor Paul Popov. Hysteric behavior of reinforced
+concrete structural walls. NASA STI/Recon Technical Report N, 80:27533, 1979.
+[49] RG Oesterle, AE Fiorato, LS Johal, JE Carpenter, HG Russell, and WG Corley. Earthquake resistant
+structural walls-tests of isolated walls. Research and Development Construction Technology Laboratories,
+Portland Cement Association, 1976.
+[50] AA Tasnimi. Strength and deformation of mid-rise shear walls under load reversal. Engineering Structures,
+22(4):311–322, 2000.
+[51] MA Hube, A Marihuén, Juan Carlos de la Llera, and Bozidar Stojadinovic. Seismic behavior of slender
+reinforced concrete walls. Engineering Structures, 80:377–388, 2014.
+[52] Patrick Zschech, Sven Weinzierl, Nico Hambauer, Sandra Zilker, and Mathias Kraus. Gam (e) changer
+or not? an evaluation of interpretable machine learning models based on additive model constraints.
+arXiv preprint arXiv:2204.09123, 2022.
+[53] WG Corley, AE Fiorato, and RG Oesterle. Structural walls. Special Publication, 72:77–132, 1981.
+[54] Ioannis D Lefas, Michael D Kotsovos, and Nicholas N Ambraseys. Behavior of reinforced concrete
+structural walls: strength, deformation characteristics, and failure mechanism. Structural Journal,
+87(1):23–31, 1990.
+[55] Firooz Emamy Farvashany, Stephen J Foster, and B Vijaya Rangan. Strength and deformation of
+high-strength concrete shearwalls. ACI structural journal, 105(1):21, 2008.
+[56] ACI Committee 369. Standard Requirements for Seismic Evaluation and Retrofit of Existing Concrete
+Buildings (ACI 369-17) and Commentary. ACI (American Concrete Institute), Farmington Hills, MI,
+2017.
+15
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+Approximate Bilevel Difference Convex Programming for Bayesian Risk
+Markov Decision Processes
+Yifan Lin 1 Enlu Zhou 1
+Abstract
+We consider infinite-horizon Markov Decision
+Processes where parameters, such as transition
+probabilities, are unknown and estimated from
+data.
+The popular distributionally robust ap-
+proach to addressing the parameter uncertainty
+can sometimes be overly conservative. In this
+paper, we utilize the recently proposed formu-
+lation, Bayesian risk Markov Decision Process
+(BR-MDP), to address parameter (or epistemic)
+uncertainty in MDPs (Lin et al., 2022). To solve
+the infinite-horizon BR-MDP with a class of con-
+vex risk measures, we propose a computationally
+efficient approach of approximate bilevel differ-
+ence convex programming (ABDCP). The opti-
+mization is performed offline and produces the
+optimal policy that is represented as a finite state
+controller with desirable performance guarantees.
+We also demonstrate the empirical performance
+of the infinite-horizon BR-MDP formulation and
+proposed algorithms.
+1. Introduction
+In a Markov decision process (MDP), an agent must make
+decisions in a sequence while facing uncertainty. In this
+situation, some parameters of the MDP, such as the transi-
+tion probabilities and costs, may be unknown and must be
+estimated from available data. The problem then becomes
+how to determine the best course of action, given the limited
+or possibly absent data, in order to minimize the expected
+total cost and optimize the decision-making process under
+these uncertain parameters.
+An alternative approach to addressing the epistemic uncer-
+tainty in MDP is through the use of distributionally robust
+MDPs (DR-MDPs, (Xu & Mannor, 2010)). This method
+considers the unknown parameters as random variables and
+assumes that their distributions belong to an ambiguity set
+1H. Milton Stewart School of Industrial and Systems Engineer-
+ing, Georgia Institute of Technology, Atlanta, GA, USA. Corre-
+spondence to: Enlu Zhou <enlu.zhou@isye.gatech.edu>.
+Preliminary work.
+determined by the available data. The optimal policy is
+then found by minimizing the expected total cost using the
+most adversarial distribution within this ambiguity set. How-
+ever, these distributionally robust approaches may lead to
+overly conservative solutions that do not perform well in
+scenarios that are more likely to occur than the worst case.
+Additionally, the DR-MDP framework does not explicitly
+incorporate the dynamics of the problem, as the distribu-
+tion of the unknown parameters does not depend on the
+data process, and is therefore not time consistent, as noted
+in (Shapiro, 2021). In light of these limitations, (Lin et al.,
+2022) proposes a Bayesian risk MDP (BR-MDP) framework
+to address epistemic uncertainty in MDPs. This approach
+stems from the static stochastic optimization literature (Wu
+et al., 2018; Zhou & Xie, 2015) and involves using a nested
+risk functional based on the Bayesian posterior distributions,
+which are updated using all available data at each stage in
+the process. However, the alpha-function approximation
+algorithm proposed in (Lin et al., 2022) only applies to
+finite-horizon MDPs and provides an upper bound on the
+exact value, without any theoretical guarantee on the gap.
+In this paper, we reformulate the considered problem as a
+bilevel difference convex programming (DCP) such that we
+can employ the powerful optimization methods for DCP to
+solve infinite-horizon BR-MDP. Since the space of poste-
+rior distributions (beliefs) is uncountably infinite, we ap-
+proximate the bilevel DCP by considering only a subset of
+posterior distributions. Although the DCP is approximate,
+we show that its solution is a lower bound on the optimal
+exact value function. Using the representation of a finite
+state controller of the resulting policy, we further show an
+upper bound on the optimal exact value function. We then
+develop an iterative approach to reduce the gap between
+upper and lower bounds by incrementally generating new
+sets of posterior distributions, and show the convergence of
+the proposed algorithm.
+To summarize, the contributions of this paper are two folds.
+First, we analyze the infinite-horizon MDP with epistemic
+uncertainty under the framework of BR-MDP via a Bayesian
+perspective and show the existence and uniqueness of sta-
+tionary optimal policy. Second, we propose an approxi-
+mate difference convex programming algorithm to solve
+the proposed formulation, and show the convergence of the
+arXiv:2301.11415v1  [eess.SY]  26 Jan 2023
+
+Approximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes
+proposed algorithm.
+The rest of the paper is organized as follows. We conduct
+literature review and introduce the BR-MDP framework
+in Section 2. We show the existence and uniqueness of a
+stationary optimal policy to the infinite-horizon BR-MDP
+in Section 3.1. We provide a bilevel DCP solution to the
+infinite-horizon BR-MDP in Section 3.2. A computation-
+ally efficient approximate DCP algorithm is then shown in
+Section 3.3. We verify the theoretical results and demon-
+strate the performance of our algorithms via numerical ex-
+periments in Section 4. Finally, we conclude the paper in
+Section 5.
+2. Background
+2.1. Related Literature
+If the data used to estimate the true but unknown underlying
+MDP are not sufficient, the estimated MDP may signifi-
+cantly differ from the true MDP, leading to poor policy per-
+formance. This discrepancy (between the estimated MDP
+and the true MDP) can be seen tightly linked to the epistemic
+uncertainty about the model. There have been numerous
+approaches that address epistemic uncertainty in MDPs,
+with robust MDP (Nilim & Ghaoui, 2004; Iyengar, 2005;
+Delage & Mannor, 2010; Wiesemann et al., 2013; Petrik &
+Russel, 2019) being one of the most widely used methods.
+In robust MDPs, the optimal decisions are made based on
+their performance under the most unfavorable conditions
+within a known set of possible parameter values, known as
+the ambiguity set.
+In consideration of the overly conservativeness in the robust
+MDP approach, risk-averse approach has been proposed to
+address the epistemic uncertainty. Risk-averse approach is
+originally proposed to address the aleatoric uncertainty that
+is due to the inherent stochasticity of the underlying MDP
+(Howard & Matheson, 1972; Ruszczy´nski, 2010; Petrik &
+Subramanian, 2012; Osogami, 2012). It replaces the risk-
+neutral expectation by some general risk measures, such as
+conditional value-at-risk (CVaR, (Rockafellar & Uryasev,
+2000)). However, most of the existing approaches assume
+the agent has access to the true underlying MDP, and op-
+timize some risk measures such as CVaR in that single
+MDP (Chow & Ghavamzadeh, 2014; Tamar et al., 2015a;b;
+Sharma et al., 2019). In this paper, we consider the offline
+planning problem in MDPs, where we only have access to
+a prior belief distribution over MDPs that is constructed
+by the offline data. It should be noted that offline planning
+problem has also been considered in (Duff, 2002), where the
+author proposes a Bayes-adaptive MDP (BA-MDP) formu-
+lation with an augmented state composed of the underlying
+MDP state and the posterior distribution of the unknown
+parameters. When the agent is equipped with the learned
+optimal policy and placed in a real environment, it behaves
+as if it is adapting to its surroundings. Mostly close to the
+problem setting in this work are (Rigter et al., 2021; Lin
+et al., 2022). (Rigter et al., 2021) optimizes a CVaR risk
+functional over the total cost and simultaneously addresses
+both epistemic and aleatoric uncertainty, while (Lin et al.,
+2022) considers a nested risk functional to ensure the time
+consistency of the obtained optimal policy.
+While there are many works proposing different models and
+frameworks to address the epistemic uncertainty, developing
+computationally efficient solutions is also of great interest.
+In robust MDPs, with some mild conditions on the ambigu-
+ity set such as rectangularity, the proposed formulation can
+be solved by a second-order cone program when the horizon
+is finite, or policy iteration when the horizon is infinite (Man-
+nor & Xu, 2019). In BA-MDP and its variants, (Rigter et al.,
+2021) proposes an approximate algorithm based on Monte
+Carlo tree search and Bayesian optimization. (Lin et al.,
+2022) develops an α-function approximation algorithm us-
+ing the convexity of the CVaR risk measure. However, the
+aforementioned works consider a finite-horizon MDP and
+do not generalize well to the infinite-horizon setting.
+Compared to standard MDPs, our considered problem has
+two distinct features that make it difficult to apply value
+iteration, policy iteration, or linear programming (Puterman,
+2014). First is the resulting continuous-state MDP due to the
+augmented belief state. We note that this continuous-state
+MDP is similar to a belief-MDP, which is the equivalent way
+to represent a partially observable MDP (POMDP) by treat-
+ing the posterior distribution of the hidden state as a belief
+state. Second is the risk measure taken with respect to the
+unknown parameters in the MDPs. In this work, we propose
+an optimization-based method to solve the infinite-horizon
+BR-MDPs. It has been empirically shown in (Alagoz et al.,
+2015) that linear programming can efficiently solve a signif-
+icant number of MDPs in comparison to standard dynamic
+programming methods, such as value iteration and policy
+iteration. Furthermore, linear programming requires less
+memory and can handle MDPs with a larger number of
+states and still achieve optimality. Works that are most re-
+lated to our proposed optimization-based approach include
+(Poupart et al., 2015) who proposes an approximate lin-
+ear programming algorithm for the risk-neutral constrained
+POMDPs, and (Ahmadi et al., 2021) who proposes a differ-
+ence convex programming (DCP) for the constrained risk-
+averse MDPs. Our approach for infinite-horizon BR-MDP
+significantly differs from the above approaches in two as-
+pects. First, compared to the linear programming approach
+for risk-neutral POMDPs in (Poupart et al., 2015), we use
+bilevel DCP, due to the additional risk measure that is used
+for mitigating the epistemic uncertainty. Our considered risk
+measure brings additional challenge to exactly evaluating
+the policy, whereas policy evaluation can be easily solved
+
+Approximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes
+by a system of linear equations in (Poupart et al., 2015).
+Second, compared to the DCP for the risk-averse MDP with
+aleatoric uncertainty in (Ahmadi et al., 2021), the resulting
+continuous-state MDP in our problem has an infinite number
+of constraints, and thus requires appropriate approximation
+to make the problem computationally feasible.
+2.2. Preliminary: Bayesian Risk MDPs
+Consider
+an
+infinite-horizon
+MDP
+defined
+as
+(S, A, P, C, γ), where S is the state space, A is the
+action space, P is the transition probability with P(s′|s, a)
+denoting the probability of transitioning to state s′ from
+state s when action a is taken, C is the cost function with
+C(s, a, s′) denoting the cost when action a is taken and
+state transitions from s to s′, 0 ≤ γ < 1 is the discount
+factor. We assume the state space and action space are finite.
+A Markovian deterministic policy π is a function mapping
+from S to A. Given an initial state s, the goal is to find an
+optimal policy that minimizes the expected discounted total
+cost:
+min
+π Eπ,P,C ��∞
+t=1 γt−1C (st, at, st+1) |s1 = s
+�
+,
+where Eπ,P,C is the expectation with policy π when the
+transition probability is P and the cost is C. In practice, P
+and C are often unknown and estimated from data.
+BR-MDP is a recently proposed framework that deals with
+the epistemic uncertainty in MDPs (Lin et al., 2022). It is
+assumed that the state transition is specified by the state
+equation s′ = g(s, a, ξ′) with a known transition function
+g, which involves state s ∈ S ⊆ Rs, action a ∈ A ⊆ Ra,
+and randomness ξ ∈ Ξ ⊆ Rk, where s, a, k are the di-
+mensions of the state, action, and randomness, respec-
+tively. The state equation together with the distribution of ξ
+uniquely determines the transition probability of the MDP,
+i.e., P(s′ ∈ S′|s, a) = P({ξ ∈ Ξ : g(s, a, ξ) ∈ S′}|s, a),
+where S′ is a measurable set in S. The cost is assumed to
+be a function of state s, action a, and randomness ξ, i.e.,
+C(s, a, ξ). The distribution of ξ, denoted by f(·; θc), is
+assumed to belong to a parametric family {f(·; θ)|θ ∈ Θ},
+where Θ ⊆ Rd is the parameter space, d is the dimension
+of the parameter θ, and θc ∈ Θ is the true but unknown
+parameter value. Many problems meet the requirement of
+having a parametric assumption. For example, it is com-
+monly assumed that the demand of customers follows a
+Poisson distribution with an unknown arrival rate in inven-
+tory control.
+We begin by assuming a prior distribution, denoted by µ,
+over the parameter space Θ. This prior accounts for the
+uncertainty of the parameter estimate that comes from an
+initial set of data, and it can also take expert opinions into
+consideration. Then, given an observed realization of the
+data process, we update the posterior distribution µ accord-
+ing to the Bayes’ rule. Let the policy be a sequence of
+mappings from state s and posterior µ to the action space,
+i.e., π = {π : S × M → A}, where M is the space of
+posterior distributions. Note that this representation im-
+plies the policy is stationary. Now we present the BR-MDP
+formulation below.
+min
+π
+ρµ1Eθ1
+�
+C1(s1, a1, ξ1) + · · ·
++ γt−1ρµtEθt
+�
+Ct(st, at, ξt) + · · ·
+�
+|s1 = s, µ1 = µ
+�
+(1)
+s.t. st+1 = g(st, at, ξt), t = 1, 2, · · · ;
+(2)
+µt+1(θ) =
+µt(θ)f (ξt; θ)
+�
+Θ µt(θ)f (ξt; θ) dθ , t = 1, 2, · · · ,
+(3)
+where ρ is a risk measure, at = π(st, µt), θt is a random
+vector following distribution µt, Eθt denotes the expectation
+with respect to ξt ∼ f(·; θt) conditional on θt, and ρµt de-
+notes a risk functional with respect to θt ∼ µt. Equation (2)
+is the transition of the state st, and without loss of generality
+we assume the initial state s1 takes a deterministic value s.
+Equation (3) is the updating of the posterior µt.
+2.3. Preliminary: Risk Measure
+Let (Ω, F, P) be a probability space and Z be a linear space
+of F-measurable functions Z : Ω → R. A risk measure is
+a function ρ : Z → R which assigns to a random variable
+Z a real number representing its risk. It is said that risk
+measure ρ is convex if it possesses the properties of con-
+vexity, monotonicity, and translation invariance (F¨ollmer &
+Schied, 2002). In this paper we consider a class of convex
+risk measures which can be represented in the following
+parametric form: ρµ(Z) := infφ∈Φ Eµ[Ψ(Z, φ)],, where
+Φ ⊂ Rm and Ψ : R × Φ → R is a real-valued func-
+tion. There is a large class of risk measures which can
+be represented in the parametric form. For example, con-
+ditional value-at-risk (CVaR), defined as CVaRα(X) =
+minφ∈R
+�
+φ +
+1
+1−αE [(X − φ)+]
+�
+where (·)+ stands for
+max(0, ·), is widely used (Rigter et al., 2021; Chow et al.,
+2015). Another example is risk measures constructed from
+φ-divergence ambiguity sets (see Example 3 in (Guigues
+et al., 2021)). We refer the readers to (Shapiro et al., 2021)
+for a comprehensive discussion.
+3. Algorithm and Theoretical Analysis
+3.1. Bellman Equation and Optimality
+We can write the value function under policy π of BR-MDP
+in the following recursive forms.
+V π(s, µ) = ρµEθ
+�
+C(s, a, ξ) + γV π(s′, µ′)
+�
+s.t. s′ = g(s, a, ξ), a = π(s, µ);
+µ′(θ) =
+µ(θ)f (ξ; θ)
+�
+Θ µ(θ)f (ξ; θ) dθ.
+We refer the readers to (Lin et al., 2022) for a discussion
+
+Approximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes
+on the preference of dynamic risk measure over static risk
+measure in consideration of time consistency and derivation
+of the Bellman equation. For simplicity we only consider
+deterministic policies, but all the analysis below can be ex-
+tended to stochastic policies. As a consequence of Theorem
+5.5.3b in (Puterman, 2014), it is sufficient to consider the
+Markovian policy. The optimal value function is then de-
+noted as V ∗(s, µ) = minπ∈ΠMD V π(s, µ), where ΠMD is
+the set of Markovian deterministic policies. In the following,
+we derive the intermediate results to show V ∗ is the unique
+optimal value function to the infinite-horizon BR-MDP.
+Definition 3.1 (Bellman Operator). Let B(S, M) be the
+space of real-valued bounded measurable functions on (S ×
+M). For any bounded value function V ∈ B(S, M), define
+an operator T : B(s, µ) → B(s, µ) as:
+(T V )(s, µ) = min
+a∈A ρµ [Eθ [C(s, a, ξ) + γV (s′, µ′)]] .
+Also let T π : B(s, µ) → B(s, µ), where
+(T πV )(s, µ) = ρµ [Eθ [C(s, π(s, µ), ξ) + γV (s′, µ′)]] .
+The next two lemmas show the above Bellman operators are
+monotonic and contraction mappings. Proofs can be found
+in the appendix.
+Lemma 3.2 (Monotonicity). The operators T π and T are
+monotonic, in the sense that V ≤ V ′ implies T πV ≤ T πV ′
+and T V ≤ T V ′.
+Lemma 3.3 (Contraction Mapping). The operators T π and
+T are γ contraction for || · ||∞ norm. That is, for any two
+bounded value functions V, V ′ ∈ B(S, M), we have
+||T πV − T πV ′||∞ ≤ γ||V − V ′||∞.
+The following proposition shows that sub-solutions and
+super-solutions of the optimality equations V = T V pro-
+vide lower and upper bounds on V ∗. As a result, when
+a solution is obtained, both bounds are satisfied, meaning
+that the solution must be equivalent to V ∗. Additionally,
+this outcome serves as an important algorithmic tool for
+optimization-based methods.
+Proposition 3.4. For any V ∈ B(S, M), (i) if V ≥ T V ,
+then V ≥ V ∗; (ii) if V ≤ T V , then V ≤ V ∗.
+According to Proposition 3.4, we have V ∗ = T V ∗. By
+Banach fixed-point theorem, V ∗ is the unique optimal value
+function to the infinite horizon BR-MDP. We also have that
+the value V of a stationary policy π is the unique bounded
+solution of the equation V = T πV . Similar analysis shows
+the existence and uniqueness of the optimal stationary policy
+π∗ that satisfies V ∗ = T π∗V ∗.
+Applying the operator T on any initial value function V , we
+have the value iteration algorithm for the infinite-horizon
+BR-MDP problem. The following corollary of convergence
+rate is similar to the standard with the contraction property.
+Corollary 3.5. For any initial bounded value function
+V , the convergence rate is shown to be ||(T kV )(s, µ) −
+V ∗(s, µ)||∞ ≤ γk||V (s, µ) − V ∗(s, µ)||∞.
+3.2. Bilevel Difference Convex Programming
+The main challenge of executing the value iteration algo-
+rithm (and similarly policy iteration algorithm) lies in the
+continuous augmented state. In this work, we propose an
+optimization-based method to solve the infinite-horizon BR-
+MDPs. According to Proposition 3.4, the infinite-horizon
+BR-MDP can be solved as follows:
+max
+V
+�
+s∈S,µ∈M
+α(s, µ)V (s, µ)
+s.t. V (s, µ) ≤ ρµEθ[C(s, a, ξ) + γV (s′, µ′)]
+∀a ∈ A, s ∈ S, µ ∈ M,
+where we choose α(s, µ) to be positive scalars which satisfy
+�
+s∈S,µ∈M α(s, µ) = 1. For the considered class of convex
+risk measures, we can rewrite the above formulation as a
+bilevel difference convex program:
+min
+V
+−
+�
+s∈S,µ∈M
+α(s, µ)V (s, µ)
+(4)
+s.t.V (s, µ) − min
+φ Eµ[Ψ(Eθ[C(s, a, ξ) + γV (s′, µ′)], φ)] ≤ 0
+∀a ∈ A, s ∈ S, µ ∈ M.
+Since Ψ(z, φ) is convex in (z, φ), it remains to be con-
+vex in z after taking the minimum over φ. Thus, (4) is
+a bilevel difference convex program (see (Horst & Thoai,
+1999) for the definition of DCP). It should be noted that
+(Ahmadi et al., 2021) shows that the minimum over φ can
+be absorbed into the overall minimum problem, and φ is
+treated as a single variable. However, it is clear that the
+minimum is achieved at different φ for different augmented
+state (s, µ), thus turning (4) into a bilevel optimization prob-
+lem. When the lower-level problem is convex and satisfies
+certain regularity conditions, we can use the Karush-Kuhn-
+Tucker (KKT) conditions to reformulate the lower-level
+optimization problem, which allows us to transform the
+original bilevel optimization problem into a single-level
+(constrained) optimization problem.
+After being reduced to a single-level DCP problem, (4)
+can be solved by the convex-concave procedure (see (Lipp
+& Boyd, 2016) for such procedure), wherein the concave
+terms are replaced by a convex upper bound. We employ
+the method of disciplined convex-concave programming
+(DCCP, (Shen et al., 2016), with Python package available
+at https://github.com/cvxgrp/dccp), which converts a DCP
+problem into a disciplined convex program and subsequently
+into an equivalent cone program. However, one problem
+remains to be solved: the number of constraints in (4) is
+
+Approximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes
+infinite, due to the continuous belief state. To tackle this
+problem, we take a similar approach as (Poupart et al., 2015).
+The main idea is to start with a finite posterior set (belief
+space) ˆ
+M, and then problem (4) can be solved efficiently
+by DCCP, where the posterior distribution (belief point) not
+in the set ˆ
+M is replaced by some convex combination of
+the points in ˆ
+M. We then iteratively add to the posterior
+set new posterior distributions that are reachable from the
+current one and re-solve (4). We formally introduce the
+approximate bilevel DCP algorithm in the next section.
+3.3. Approximate Bilevel Difference Convex
+Programming
+Let ˆ
+M be the current posterior set. Let µsas′ be the one-step
+posterior distribution with observed randomness ξ indicated
+by state transition s′ = g(s, a, ξ) and current posterior µ.
+Initially the posterior set is constructed from corner (de-
+generate) points. In case the parameter space Θ is finite,
+the corner points are (1, 0, · · · , 0), (0, 1, 0, · · · ), · · · , and
+(0, · · · , 0, 1). In case the parameter space is continuous, it
+is impossible to express one-step posterior distribution (i.e.,
+µsas′) as a convex combination of those degenerate points.
+Therefore, we assume the parameter space is finite, which is
+practical in many real-world problems. It can also be viewed
+as a discrete approximation of a continuous parameter set,
+and the discretization can be chosen of any precision.
+To interpolate all µsas′ that can be reached from some µi ∈
+ˆ
+M in one step, we use some convex combination of points
+µi in ˆ
+M. Let w(µi, µsas′) be the weight wi associated with
+µi when interpolating µsas′. We can use this interpolation
+weight to define an approximate transition probability for
+posterior as:
+˜P(µ′|s, a, µ, θ) =
+�
+s′∈S
+P(s′|s, a, θ)w(µ′, µsas′).
+A sanity check that
+˜P(µ′|s, a, µ, θ) is indeed a tran-
+sition probability:
+�
+µ′∈ ˆ
+M ˜P(µ′|s, a, µ, θ)
+=
+1 and
+˜P(µ′|s, a, µ, θ) ≥ 0. We choose the convex combination
+that minimizes the weighted Euclidean norm of the differ-
+ence between µ and each µi by solving the following linear
+program:
+min
+w
+�
+i
+wi||µi − µsas′||2
+(5)
+s.t.
+�
+i
+wiµi(θ) = µsas′(θ), ∀θ ∈ Θ
+�
+i
+wi = 1, wi ≥ 0, ∀i.
+With the approximation in the constraint in (4), we obtain
+the following approximate bilevel DCP algorithm for a
+given posterior set. For ease of notation, we denote by
+C(s, a, θ) = Eθ[C(s, a, ξ)] the average cost at state s when
+action a is taken, under the parameter value θ.
+Algorithm 1 Approximate Bilevel DCP
+input: posterior set ˆ
+M
+output: policy ˆπ∗, value function ˆV ∗
+1. Solve the following approximate bilevel DCP:
+min
+V
+−
+�
+s∈S,µ∈M
+α(s, µ)V (s, µ)
+(6)
+s.t. V (s, µ) ≤ min
+φ
+�
+θ∈Θ
+µ(θ)[Ψ(γ
+�
+µ′∈ ˆ
+M,s′∈S
+P(s′|s, a, θ)
+w(µ′, µsas′)V (s′, µ′) + C(s, a, θ), φ)], ∀a ∈ A, s ∈ S, µ ∈ ˆ
+M
+where w(µ′, µsas′) is obtained by solving (5).
+2. Obtain the policy
+ˆπ∗(s, µ) = arg min
+a∈A
+ρµEθ[C(s, a, ξ) + γ ˆV ∗(s′, µ′)].
+Since the policy returned by Algorithm 1 is based on an ap-
+proximate transition probability, there is a need to evaluate
+the obtained policy. Next we show the approximate value
+function obtained by Algorithm 1 is a lower bound on the
+exact optimal value function V ∗.
+Theorem 3.6. The approximate optimal value function ˆV ∗
+found by running Algorithm 1 is a lower bound on the exact
+optimal value function V ∗.
+We also develop an upper bound on the exact optimal value
+function, using the obtained policy from Algorithm 1. The
+obtained policy is a finite state controller (see (Hansen,
+2013) for the definition of finite state controller). Let N be
+the set of nodes in the controller such that we associate a
+node ns,µ to each (s, µ) pair. The action chosen in node
+ns,µ is determined by the policy ˆπ∗(a|s, µ). For a given
+parameter θ, the transition probability to the next node
+is P(ns′,µ′|ns,µ, a) = w(µ′, µsas′)P(s′|s, a). The value
+function of the finite state controller can be computed by
+ˆV ˆπ∗(ns,µ) = min
+φ
+�
+θ∈Θ
+µ(θ)[Ψ(c(s, a, θ) + γ
+�
+ns′,µ′∈N
+w(µ′, µsas′)P(s′|s, a, θ) ˆV ˆπ∗(ns′,µ′), φ)].
+Similar to (Ahmadi et al., 2021), the value function can be
+solved efficiently by DCP. It is also known from (Hansen,
+2013) that the value function obtained by the finite state
+controller ˆV ˆπ∗ serves as an upper bound for the optimal
+value function.
+Note that the inequality ˆV ∗ ≤ V ∗ ≤ ˆV ˆπ∗ provides infor-
+mation about how well the optimal value function V ∗ is
+
+Approximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes
+approximated. As the posterior set ˆ
+M gets closer to the true
+one, the gap between the approximate value function and
+the optimal value function gets smaller.
+Next we incrementally add new posterior distributions to
+the posterior set ˆ
+M. Different methods can be employed to
+produce new posterior distributions that are added to the set
+ˆ
+M at each iteration. We take a similar approach as (Poupart
+et al., 2015), which is based on envelope techniques. It
+considers the posterior distributions that can be reached in
+one step from any posterior distribution in ˆ
+M by executing
+the policy ˆπ∗. As the number of posterior distributions to
+be added might be excessive, we can prioritize them by
+including the n reachable posterior distributions with the
+largest weighted Euclidean distance to the posterior distri-
+butions in ˆ
+M, as determined by the interpolation outlined
+in (5). Note that the point-based value iteration approach
+in (Pineau et al., 2003) shares the similar idea, that is, to
+include new posterior distribution that improves the worst-
+case density as rapidly as possible, where density is defined
+as the maximum distance from any posterior distribution to
+ˆ
+M. We summarize the new posterior set generation in the
+following algorithm.
+Algorithm 2 New Posterior Set Generation
+input: policy ˆπ∗, posterior set ˆ
+M, n
+output: newly added posterior set ˆ
+M′
+for each (s, µ) ∈ (S × ˆ
+M) and s′ ∈ S do
+µ′(θ) ∝ µ(θ)f(ξ|θ), where s′ = g(s, ˆπ∗(a|s, µ), ξ)
+distµ′ ←− distance of µ′ to ˆ
+M � ˆ
+M′
+if distµ′ > 0 (i.e., µ′ not in ˆ
+M � ˆ
+M′) then
+ˆ
+M′ ←− ˆ
+M′ �{µ′}
+end if
+if | ˆ
+M′| > n (to reduce the size of ˆ
+M′) then
+for each µ′ ∈ ˆ
+M′ do
+distµ′ ←− distance of µ′ to ˆ
+M � ˆ
+M′\{µ′}
+ˆ
+M′ ←− ˆ
+M′\{arg minµ′∈ ˆ
+M′ distµ′}
+end for
+end if
+end for
+Algorithm 3 Approximate Bilevel DCP for Infinite-horizon
+BR-MDPs
+input: threshold ϵ, n, initial augmented state (s1, µ1)
+output: policy ˆπ∗
+initialization: ˆ
+M ←− {degenerate beliefs} �{µ1}
+repeat
+obtain (ˆπ∗, ˆV ∗) by running Algorithm 1
+evaluate policy ˆπ∗ by solving a DCP and obtain ˆV ˆπ∗
+ˆ
+M ←− ˆ
+M � ˆ
+M′ generated by Algorithm 2
+until ˆV ˆπ∗(s1, µ1) − ˆV ∗(s1, µ1) ≤ ϵ
+Combining Algorithm 1 and Algorithm 2, we now present
+the full algorithm below (ABDCP), which iteratively add
+to the new posterior set and solve a bilevel difference con-
+vex program at each iteration. We are now ready to show
+Algorithm 3 converges to a near-optimal policy.
+Theorem 3.7. Algorithm 3 converges to a near-optimal
+policy ˆπ∗, i.e., V ˆπ∗(s1, µ1) − V ∗(s1, µ1) ≤ ϵ, where ϵ is
+the desired threshold.
+4. Numerical Experiments
+We illustrate the performance of the infinite-horizon BR-
+MDP formulation with different choices of risk measures
+and the proposed approximate bilevel DCP algorithm with
+two offline planning problems. Code for the experiments is
+included in the supplementary material. All algorithms are
+implemented in Python and run on a 1.4 GHz Intel Core i5
+processor with 8 GB memory. Implementing details can be
+found in the appendix.
+• Path Planning
+In the offline path planning prob-
+lem, an autonomous car (agent) navigates a two-
+dimensional terrain map represented by a 10 by 10 grid
+along roads to the destination. The agent chooses from
+four actions {up, down, left, right}. There are four
+types of roads: {highway, main road, street, lane}. The
+traffic time ξT
+i in each type of road is assumed to be
+independent and follows exponential distribution with
+different rate, denoted by θT
+i , i = 1, · · · , 4, where T
+stands for traffic time. The parameter value is assumed
+to be within the finite set. ξA
+i ∈ {0, 1}, i = 1, · · · , 4
+denotes whether there is car accident in each type of
+road, where A stands for accident. The probability
+of car accident happening in each type of road is also
+assumed to be independent, denoted by θA
+i . The param-
+eter value is assumed to be within the finite set. When
+there is an car accident, the agent receives a constant
+cost TA and makes no transition. Otherwise, the agent
+transitions to the next road depending on the action it
+takes and receives the cost, which is the traffic time for
+traversing that type of road. The agent stops when it
+reaches the destination. The discount factor γ = 0.95.
+The agent is given a historical dataset H0 containing
+past traffic times and car accident logs.
+• Multi-item Inventory Control
+In the offline multi-
+item inventory control problem, the warehouse man-
+ager decides how much to replenish from the set
+{0, 1, · · · , Si − si} for each item i ∈ [K] at each time
+stage, where Si is the storage capacity for item i, si is
+the current inventory level for item i. The customer de-
+mand is a random vector ξ = (ξ1, · · · , ξK) with each
+ξi following a Poisson distribution with parameter θi.
+The state transition is given by st+1 = max(st + at −
+ξt, 0), the cost function is given by C(st, at, ξt) =
+h · max(st + at − ξt, 0) + p · max(ξt − st − at, 0),
+
+Approximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes
+where h is the holding cost and p is the penalty cost.
+The discount factor γ = 0.95. The warehouse manager
+is given a historical dataset consisting of past customer
+demands.
+We adapt two methods to our offline planning problem and
+evaluate their performances.
+The first method (CALP)
+comes from (Poupart et al., 2015) with a risk-neutral
+POMDP formulation.
+The second method (DR-MDP)
+comes from (Xu & Mannor, 2010) with a distributionally
+robust MDP formulation. Note that the BPO approach from
+(Lee et al., 2018) solves a risk-neutral BA-MDP formula-
+tion, where two separate encoders for the physical state and
+belief state are designed to deal with the continuous latent
+parameter space. It could have been a good benchmark if
+its encoder design were made available. Apart from the two
+benchmarks, we also compare with the nominal approach
+(MLE), where a maximal likelihood estimator for the param-
+eter is computed from the given dataset and then a policy is
+obtained by solving the MDP with the plugged-in parameter
+value. In our proposed algorithm (ABDCP) for the infinite-
+horizon BR-MDP formulation, we consider two particular
+risk measures, namely expectation and CVaR with different
+risk levels α. It should be noted that when the considered
+risk measure is expectation, our algorithm can be modified
+and reduced to CALP. Similar observation is verified in
+(Poupart et al., 2006), where the BA-MDP formulation is
+transformed into a POMDP formulation.
+For each of the considered algorithms, we obtain the cor-
+responding optimal policy with the same dataset. It should
+be noted that the calculations are carried out offline. The
+obtained policy is then applied for risk-averse path planning
+and evaluated on the true system, i.e., MDP with the true
+parameter. This is referred to as one replication, and we
+repeat the experiments for 200 replications on different in-
+dependent datasets. Results for the path planning problem
+can be found in Table 1 and Table 2, with different data
+size N = 10 and N = 1000. Results for the multi-item
+inventory control problem can be found in Table 3 and Ta-
+ble 4, with different data size N = 10 and N = 1000.
+The columns report the running time, expected performance
+(cost), and the CVaR performance (cost) of our proposed al-
+gorithm and benchmarks over the 200 replications. ABDCP-
+EXP stands for our proposed algorithm ABDCP with ex-
+pectation as the risk measure. ABDCP-CVaR stands for our
+proposed algorithm ABDCP with CVaR as the risk measure.
+We also show the histogram of the actual performance over
+200 replications for our proposed algorithm and the nominal
+benchmark on the path planning problem in Figure 1. We
+summarize the main observations for the path planning prob-
+lem. Similar observations can be made for the multi-item
+inventory control problem. We include more observations
+in the appendix.
+BR-MDP hedges against epistemic uncertainty: in each
+replication, data points are randomly sampled from the true
+distribution. While facing the epistemic uncertainty, BR-
+MDP formulation optimizes over a dynamic risk measure
+that provides robustness. Table 1 shows that our proposed
+ABDCP algorithm is the most robust in the sense of balanc-
+ing the mean and variability of the actual cost. The CVaR
+cost of our proposed algorithm is also lower than the other
+benchmarks, showing that it avoids large costs. In contrast,
+the nominal approach performs badly when the data size is
+small, e.g. N = 5, indicating that it is not robust against the
+epistemic uncertainty and suffers from the scarcity of data.
+On the other hand, DR-MDP is overly conservative, even
+though it has the smallest variability. This conservativeness
+comes from two aspects. First, it always chooses to optimize
+over the worst-case scenario, which rarely happens in the
+true system. Second, the static worst-case risk measure pre-
+vents it from adapting to the data realizations, which is one
+of the motivations for the dynamic risk measure considered
+in the BR-MDP formulation.
+Convergence of ABDCP: the running time for a single
+replication on the path planning problem using our pro-
+posed ABDCP algorithm is affordable, and the proposed
+algorithm solves the infinite-horizon BR-MDP in finite time.
+In contrast, the infinite-horizon BR-MDP is intractable with
+standard value iteration or policy iteration.
+5. Conclusions
+In this paper, we consider the offline planning problem
+in MDPs with epistemic uncertainty, where we only have
+access to a prior belief distribution over MDPs that is con-
+structed by the offline data. We consider the infinite-horizon
+BR-MDP that produces a time-consistent formulation and
+provides the robustness against epistemic uncertainty. We
+develop an efficient optimization-based approximation algo-
+rithm that converges to the optimal policy. Our experiment
+results demonstrate the efficiency of the proposed approxi-
+mate algorithm, and show the robustness and the adaptivity
+to future data realization of the infinite-horizon BR-MDP
+formulation.
+One of the future directions is to study the sample complex-
+ity of the proposed algorithm. In its current form, we show
+the convergence of the proposed algorithm without analysis
+of the convergence rate. Another interesting direction is to
+utilize function approximation to improve the scalability of
+the proposed approach to more complex domains. Separate
+encoders for the physical state and belief state have been
+proposed in (Lee et al., 2018) to reduce the dimension of
+the considered BA-MDP formulation, and adaptation from
+the risk-neutral BA-MDP formulation to our risk averse BR-
+MDP formulation with the designed policy network could
+be interesting.
+
+Approximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes
+Approach
+time (sec)
+expected cost
+CVaR (α = 0.95) cost
+CVaR (α = 0.8) cost
+ABDCP-EXP (CALP)
+969.13(0.18)
+70.06(0.51)
+85.72
+82.06
+ABDCP-CVaR (α = 0.95)
+2639.38(0.22)
+67.51(0.24)
+75.67
+73.72
+ABDCP-CVaR (α = 0.8)
+2545.74(0.24)
+66.02(0.38)
+79.97
+75.50
+DR-MDP
+62.34(0.11)
+79.43(0.15)
+81.64
+80.60
+Nominal
+61.44(0.08)
+82.59(0.59)
+94.10
+92.46
+Table 1. Results for path planning problem. Running time for each replication, expected cost, and CVaR cost at different risk levels α are
+reported for different algorithms. Standard errors are reported in parentheses. Number of data points is N = 10.
+Approach
+time (sec)
+expected cost
+CVaR (α = 0.95) cost
+CVaR (α = 0.8) cost
+ABDCP-EXP (CALP)
+967.25(0.17)
+64.15(0.05)
+66.34
+65.97
+ABDCP-CVaR (α = 0.95)
+2642.26(0.21)
+65.18(0.03)
+66.14
+65.76
+ABDCP-CVaR (α = 0.8)
+2643.48(0.25)
+65.17(0.04)
+66.26
+65.84
+DR-MDP
+63.15(0.09)
+65.22(0.03)
+66.43
+66.01
+Nominal
+62.47(0.08)
+64.31(0.12)
+67.55
+65.59
+Table 2. Results for path planning problem. Running time for each replication, expected cost, and CVaR cost at different risk levels α are
+reported for different algorithms. Standard errors are reported in parentheses. Number of data points is N = 1000.
+Approach
+time (sec)
+expected cost
+CVaR (α = 0.95) cost
+CVaR (α = 0.8) cost
+ABDCP-EXP (CALP)
+1374.59(0.24)
+3478.92(15.03)
+4363.56
+4025.23
+ABDCP-CVaR (α = 0.95)
+4109.21(0.35)
+3072.84(10.24)
+3651.75
+3472.44
+ABDCP-CVaR (α = 0.8)
+4087.14(0.32)
+2831.12(12.37)
+3782.04
+3517.30
+DR-MDP
+140.76(0.13)
+3963.56(9.21)
+4424.69
+4231.12
+Nominal
+139.89(0.08)
+3987.50(18.39)
+4974.57
+4611.16
+Table 3. Results for multi-item inventory control problem. Running time for each replication, expected cost, and CVaR cost at different
+risk levels α are reported for different algorithms. Standard errors are reported in parentheses. Number of data points is N = 10.
+Approach
+time (sec)
+expected cost
+CVaR (α = 0.95) cost
+CVaR (α = 0.8) cost
+ABDCP-EXP (CALP)
+1377.27(0.22)
+1806.45(0.57)
+1825.06
+1823.71
+ABDCP-CVaR (α = 0.95)
+4114.93(0.34)
+1819.92(0.16)
+1823.63
+1821.70
+ABDCP-CVaR (α = 0.8)
+4002.62(0.33)
+1817.21(0.19)
+1824.97
+1822.39
+DR-MDP
+138.26(0.11)
+1826.03(0.12)
+1828.80
+1827.62
+Nominal
+136.38(0.10)
+1802.34(1.28)
+1836.04
+1828.99
+Table 4. Results for multi-item inventory control problem. Running time for each replication, expected cost, and CVaR cost at different
+risk levels α are reported for different algorithms. Standard errors are reported in parentheses. Number of data points is N = 1000.
+(a) ABDCP-EXP
+(b) ABDCP-CVaR(α = 0.95)
+(c) ABDCP-CVaR(α = 0.8)
+(d) Nominal
+Figure 1. Histogram of the actual performance over 200 replications for different algorithms. Number of data points is set to N = 10.
+
+25
+Mean: 70.06
+CVaR: 85.72
+requency
+20
+15
+10
+5
+60
+7075
+8
+85
+costMean: 67.51
+CVaR: 75.67
+50
+30
+20
+10
+62.5
+每.067.570.072.575.077.5
+costMean: 66.02
+CVaR: 79.97
+40
+10
+60
+70
+75
+80
+cost.597
+OT'
+25
+: 82.
+CVaR: 94.
+-ue
+uanbr
+15
+10
+5
+70
+80
+90
+costApproximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes
+References
+Ahmadi, M., Rosolia, U., Ingham, M. D., Murray, R. M.,
+and Ames, A. D. Constrained risk-averse Markov deci-
+sion processes. In Proceedings of the AAAI Conference
+on Artificial Intelligence, volume 35, pp. 11718–11725,
+2021.
+Alagoz, O., Ayvaci, M. U., and Linderoth, J. T. Optimally
+solving markov decision processes with total expected
+discounted reward function: Linear programming revis-
+ited. Computers & Industrial Engineering, 87:311–316,
+2015.
+Chow, Y. and Ghavamzadeh, M. Algorithms for CVaR
+optimization in MDPs.
+In Ghahramani, Z., Welling,
+M., Cortes, C., Lawrence, N. D., and Weinberger, K. Q.
+(eds.), Advances in Neural Information Processing Sys-
+tems, 2014.
+Chow, Y., Tamar, A., Mannor, S., and Pavone, M. Risk-
+sensitive and robust decision-making: a CVaR optimiza-
+tion approach. In Cortes, C., Lee, D. D., Sugiyama, M.,
+and Garnett, R. (eds.), Advances in Neural Information
+Processing Systems, 2015.
+Delage, E. and Mannor, S.
+Percentile optimization for
+Markov decision processes with parameter uncertainty.
+Operations research, 58(1):203–213, 2010.
+Duff, M. O. Optimal Learning: Computational procedures
+for Bayes-adaptive Markov decision processes. Ph.D.
+diss., University of Massachusetts Amherst, 2002.
+F¨ollmer, H. and Schied, A. Convex measures of risk and
+trading constraints. Finance and stochastics, 6(4):429–
+447, 2002.
+Guigues, V., Shapiro, A., and Cheng, Y. Risk-averse stochas-
+tic optimal control: an efficiently computable statistical
+upper bound. arXiv preprint arXiv:2112.09757, 2021.
+Hansen, E. A. Solving POMDPs by searching in policy
+space. arXiv preprint arXiv:1301.7380, 2013.
+Hauskrecht, M. Value-function approximations for partially
+observable Markov decision processes. Journal of artifi-
+cial intelligence research, 13:33–94, 2000.
+Horst, R. and Thoai, N. V. DC programming: overview.
+Journal of Optimization Theory and Applications, 103(1):
+1–43, 1999.
+Howard, R. A. and Matheson, J. E. Risk-sensitive Markov
+decision processes. Management science, 18(7):356–369,
+1972.
+Iyengar, G. N. Robust dynamic programming. Mathematics
+of Operations Research, 30(2):257–280, 2005.
+Lee, G., Hou, B., Mandalika, A., Lee, J., Choudhury, S., and
+Srinivasa, S. S. Bayesian policy optimization for model
+uncertainty. arXiv preprint arXiv:1810.01014, 2018.
+Lin, Y., Ren, Y., and Zhou, E. Bayesian risk Markov de-
+cision processes. In Advances in Neural Information
+Processing Systems, 2022.
+Lipp, T. and Boyd, S.
+Variations and extension of the
+convex–concave procedure. Optimization and Engineer-
+ing, 17(2):263–287, 2016.
+Mannor, S. and Xu, H. Data-driven methods for Markov
+decision problems with parameter uncertainty. In Oper-
+ations Research & Management Science in the Age of
+Analytics, pp. 101–129. INFORMS, 2019.
+Nilim, A. and Ghaoui, L. Robustness in Markov decision
+problems with uncertain transition matrices. In Thrun, S.,
+Saul, L., and Sch¨olkopf, B. (eds.), Advances in Neural
+Information Processing Systems, 2004.
+Osogami, T. Robustness and risk-sensitivity in Markov
+decision processes. In Pereira, F., Burges, C. J. C., Bottou,
+L., and Weinberger, K. Q. (eds.), Advances in Neural
+Information Processing Systems, 2012.
+Petrik, M. and Russel, R. H. Beyond confidence regions:
+Tight Bayesian ambiguity sets for robust mdps. In Wal-
+lach, H., Larochelle, H., Beygelzimer, A., d'Alch´e-Buc,
+F., Fox, E., and Garnett, R. (eds.), Advances in Neural
+Information Processing Systems, volume 32, 2019.
+Petrik, M. and Subramanian, D. An approximate solution
+method for large risk-averse Markov decision processes.
+In de Freitas, N. and Murphy, K. (eds.), Proceedings of
+the Twenty-Eighth Conference on Uncertainty in Artificial
+Intelligence, pp. 805–814, 2012.
+Pineau, J., Gordon, G., Thrun, S., et al. Point-based value
+iteration: An anytime algorithm for POMDPs. In IJCAI,
+volume 3, pp. 1025–1032, 2003.
+Poupart, P., Vlassis, N., Hoey, J., and Regan, K. An ana-
+lytic solution to discrete Bayesian reinforcement learning.
+In Proceedings of the 23rd International Conference on
+Machine Learning, pp. 697–704, 2006.
+Poupart, P., Malhotra, A., Pei, P., Kim, K.-E., Goh, B., and
+Bowling, M. Approximate linear programming for con-
+strained partially observable Markov decision processes.
+In Proceedings of the AAAI Conference on Artificial In-
+telligence, volume 29, 2015.
+Puterman, M. L.
+Markov decision processes: discrete
+stochastic dynamic programming. John Wiley & Sons,
+2014.
+
+Approximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes
+Rigter, M., Lacerda, B., and Hawes, N. Risk-averse bayes-
+adaptive reinforcement learning. In Ranzato, M., Beygelz-
+imer, A., Dauphin, Y., Liang, P., and Vaughan, J. W. (eds.),
+Advances in Neural Information Processing Systems, pp.
+1142–1154, 2021.
+Rockafellar, R. T. and Uryasev, S. Optimization of condi-
+tional value-at-risk. Journal of Risk, 2:21–41, 2000.
+Ruszczy´nski, A. Risk-averse dynamic programming for
+Markov decision processes. Mathematical programming,
+125(2):235–261, 2010.
+Shapiro, A. Tutorial on risk neutral, distributionally robust
+and risk averse multistage stochastic programming. Eu-
+ropean Journal of Operational Research, 288(1):1–13,
+2021.
+Shapiro, A., Dentcheva, D., and Ruszczynski, A. Lectures
+on stochastic programming: modeling and theory. SIAM,
+2021.
+Sharma, A., Harrison, J., Tsao, M., and Pavone, M. Robust
+and adaptive planning under model uncertainty. In Ak-
+shat Kumar, Sylvie Thi´ebaux, P. V. and Yeoh, W. (eds.),
+Proceedings of the 29th International Conference on Au-
+tomated Planning and Scheduling, pp. 410–418, 2019.
+Shen, X., Diamond, S., Gu, Y., and Boyd, S. Disciplined
+convex-concave programming. In 2016 IEEE 55th Con-
+ference on Decision and Control (CDC), pp. 1009–1014.
+IEEE, 2016.
+Tamar, A., Chow, Y., Ghavamzadeh, M., and Mannor, S.
+Policy gradient for coherent risk measures. In Cortes,
+C., Lee, D. D., Sugiyama, M., and Garnett, R. (eds.), Ad-
+vances in Neural Information Processing Systems, 2015a.
+Tamar, A., Glassner, Y., and Mannor, S. Optimizing the
+CVaR via sampling. In Twenty-Ninth AAAI Conference
+on Artificial Intelligence, 2015b.
+Wiesemann, W., Kuhn, D., and Rustem, B. Robust Markov
+decision processes. Mathematics of Operations Research,
+38(1):153–183, 2013.
+Wu, D., Zhu, H., and Zhou, E. A Bayesian risk approach
+to data-driven stochastic optimization: Formulations and
+asymptotics. SIAM Journal on Optimization, 28(2):1588–
+1612, 2018.
+Xu, H. and Mannor, S. Distributionally robust Markov
+decision processes. In Lafferty, J., Williams, C., Shawe-
+Taylor, J., Zemel, R., and Culotta, A. (eds.), Advances in
+Neural Information Processing Systems, 2010.
+Zhou, E. and Xie, W. Simulation optimization when facing
+input uncertainty. In Yilmaz, L., Chan, W. K. V., Moon,
+I., Roeder, T. M. K., Macal, C., and Rossetti, M. D. (eds.),
+Proceedings of the 2015 Winter Simulation Conference,
+pp. 3714–3724, 2015.
+
+Approximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes
+A. Technical Proof
+Proof of Lemma 3.2. Note that
+(T πV )(s, µ) = ρµ1Eθ1[C(s1, a1.ξ1) + γρµ2Eθ2[C(s2, a2, ξ2) + · · · + γV (sk, µk)]]
+for all positive integer k. As the terminal value function V (sk, µk) ≤ V ′(sk, µk) for all sk ∈ S, µk ∈ M, and using the
+monotonicity of the convex risk measure ρ(X) ≤ ρ(Y ) if X(ω) ≤ Y (ω), ∀ω ∈ Ω, we have (T πV )(s, µ) ≤ (T πV ′)(s, µ).
+Same analysis works for operator T .
+Proof of Lemma 3.3. Let Cmax = maxs∈S,µ∈M |V (s, µ) − V ′(s, µ)|, we have
+V (s, µ) − Cmax ≤ V ′(s, µ) ≤ V (s, µ) + Cmax
+(7)
+Applying T π on inequality (7), we have
+(T πV )(s, µ) − γCmax ≤ (T πV ′)(s, µ) ≤ (T πV )(s, µ) + γCmax,
+which is justified by the translation invariance of the convex risk measure. Then we have
+max
+s∈S,µ∈M |(T πV )(s, µ) − (T πV ′)(s, µ)| ≤ γ
+max
+s∈S,µ∈M |V (s, µ) − V ′(s, µ)|,
+i.e., ||T πV − T πV ′||∞ ≤ γ||V − V ′||∞. Same analysis works for operator T .
+Proof of Proposition 3.4. (i) Let π be the policy for which
+V ≥ T πV.
+(8)
+Note that such policy exists as one can choose π that yields low current cost. Applying operator T πV to both sides of
+inequality (8) and using Lemma 3.2, we have V ≥ (T π)tV , t = 1, 2, · · · . Note that the right hand side of the above
+inequality represents the cost of a finite horizon problem with stationary policy π and with final value function V . Also note
+that
+(T π)tV = ρµ1Eθ1[C(s1, a1, ξ1) + · · · + γV (st+1, µt+1)]
+≥ ρµ1Eθ1[C(s1, a1, ξ1) + · · · + γρµtEθt[C(st, at, ξt)]].
+Let t → ∞, we get V ≥ V π, where V π is the value function under policy π. Since V ∗ = minπ V π, we have V ≥ V ∗.
+(ii) Consider an arbitrary policy π and a finite horizon problem with terminal cost V (st+1, µt+1). We have under the policy
+π,
+ρµ1Eθ1[C(s1, a1, ξ1) + · · · + γV (st+1, µt+1)] = ρµ1Eθ1[C(s1, a1, ξ1) + · · · + γρµtEθt[C(st, at, ξt) + γV (st+1, µt+1)]].
+Note that ρµtEθt[C(st, at, ξt) + γV (st+1, µt+1)] ≥ T V (st, µt) ≥ V (st, µt). Therefore, we have
+ρµ1Eθ1[C(s1, a1, ξ1) + · · · + γV (st+1, µt+1)] ≥ ρµ1Eθ1[C(s1, a1, ξ1) + · · · + ρµt−1Eθt−1[C(st−1, at−1, ξt−1) + γV (st, µt)]].
+Continuing, we have
+ρµ1Eθ1[C(s1, a1, ξ1) + · · · + γV (st+1, µt+1)] ≥ V (s, µ).
+Let Cmax be an upper bound on |V (s, µ)|, ∀s ∈ S, µ ∈ M, we have
+ρµ1Eθ1[C(s1, a1, ξ1) + · · · + γρµtEθt[C(st, at, ξt)]] ≥ V (s, µ) − Cmaxγt.
+Passing to the limit t → ∞, we have for any policy π, V π(s, µ) ≥ V (s, µ). Therefore, the infimum over all policy π is
+bounded from below by V (s, µ), i.e., V ∗ ≥ V .
+
+Approximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes
+Proof of Theorem 3.6. We first show V π(s, µ) is concave in µ for any policy π. Note that
+V π(s, µ) = min
+φ Eµ[Ψ(Eθ[C(s, a, ξ) + γV π(s′, µ′)], φ)].
+For 0 ≤ t ≤ 1 and any µ1, µ2 ∈ M,
+V π(s, tµ1 + (1 − t)µ2)
+= min
+φ Etµ1+(1−t)µ2[Ψ(Eθ[C(s, a, ξ) + γV π(s′, µ′)], φ)]
+= min
+φ {Eµ1[Ψ(Eθ[C(s, a, ξ) + γV π(s′, µ′)], φ)] + (1 − t)Eµ2[Ψ(Eθ[C(s, a, ξ) + γV π(s′, µ′)], φ)]}
+≥t min
+φ1 {Eµ1[Ψ(Eθ[C(s, a, ξ) + γV π(s′, µ′)], φ1)]} + (1 − t) min
+φ2 {Eµ2[Ψ(Eθ[C(s, a, ξ) + γV π(s′, µ′)], φ2)]}
+=tV π(s, µ1) + (1 − t)V π(s, µ2).
+The same analysis works for the optimal value function V ∗. Consider running Algorithm 1 with posterior set ˆ
+M and the
+entire posterior set M. Now applying Jensen’s inequality and by Theorem 12 in (Hauskrecht, 2000), we have ˆV ∗ ≤ V ∗. Note
+that originally in (Hauskrecht, 2000), the proof is based on the fact that value function in partially observable Markov decision
+process is convex in belief and the linear programming formulation has constraint V (b) ≥ R(s, a) + γ �
+b′ P(b′|b, a)V (b′),
+where R is the reward function. Since V is convex and by linear interpolation, applying Jensen’s inequality to the right
+hand side of the constraint leads to ˆV (b) greater than V (b). Now we are in an opposite direction, by Jensen’s inequality and
+concavity of V , we have ˆV ∗ ≤ V ∗.
+Proof of Theorem 3.7. First we show Algorithm 3 terminates in finite time. Suppose not, i.e., ˆV ˆπ∗(s1, µ1)− ˆV ∗(s1, µ1) > ϵ.
+As the number of iterations increases,
+ˆ
+M will contain an increasing number of reachable posterior distributions, since
+Algorithm 3 is guaranteed to generate new reachable posterior distributions unless the current approximate optimal policy ˆπ∗
+is evaluated accurately. As the number of iterations goes to infinity, ˆ
+M will eventually contain enough posterior distributions
+to accurately evaluate all policies ˆπ∗ that Algorithm 3 produces infinitely often. Since Algorithm 3 terminates as soon as
+Algorithm 1 produces a policy that is evaluated accurately, we reach a contradiction.
+Nest we show the algorithm converges to V ∗. Suppose that the algorithm terminates, but it converges to a suboptimal policy
+˜π. By Theorem 3.6, we know that ˆV ∗ ≤ V ∗, since V ∗ ≤ ˆV ˜π and the algorithm terminates when ˆV ˜π − ˆV ∗ ≤ ϵ. Then we
+have ˆV ˜π − V ∗ ≤ ϵ, which reaches a contradiction. Thus the algorithm must converge to V ∗.
+B. Implementation Details
+B.1. Offline Path Planning
+Figure 2. Path planning terrain
+map. Colors indicate the road
+types as follows–blue: high-
+way, red: main road, orange:
+street, green: lane.
+An autonomous car (agent) navigates a two-dimensional terrain map represented by a
+10 by 10 grid along roads to the destination, as shown in Figure 2. The agent chooses
+from four actions {up, down, left, right}, as long it remains on the road. There are
+four types of roads: {highway, main road, street, lane}. The traffic time ξT
+i in each
+type of road is assumed to be independent and follows exponential distribution with dif-
+ferent rate, denoted by θT
+i , i = 1, · · · , 4, where T stands for traffic time. Specifically,
+the true rates are θT
+1 = 1, θT
+2 = 0.5, θT
+3 = 0.2, and θT
+4 = 0.1 but unknown to the
+agent. We view the parameter as a random variable, whose value is assumed to be within
+the following finite set {0.05, 0.1, 0.2, 0.25, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 1.5, 2, 2.5}.
+ξA
+i
+∈ {0, 1}, i = 1, · · · , 4 denotes whether there is car accident in each type of road,
+where A stands for accident. The probability of car accident happening in each type of road
+is also assumed to be independent, denoted by θA
+i . Specifically, the true probabilities are
+θA
+1 = 0.3, θA
+1 = 0.2, θA
+1 = 0.1 and θA
+1 = 0.05 but unknown to the agent. We view the
+parameter as a random variable, whose value is assumed to be within the following finite set
+{0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5}. When there is an car accident, the agent
+receives a constant cost TA = 10 and makes no transition. Otherwise, the agent transitions
+to the next road depending on the action it takes and receives the cost, which is the traffic
+time for traversing that type of road. The agent stops when it reaches the destination. The
+
+origin
+destinationApproximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes
+discount factor γ = 0.95. The agent is given a historical dataset H0 of size N containing past traffic times and car accident
+logs, and uses the given dataset to construct the prior for the transition rate and probability of car accident. Other parameters
+are as follows: number of points to be added at each iteration n = 20, threshold ϵ = 0.1.
+B.2. Multi-item Inventory Control
+The warehouse manager (agent) decides how much to replenish from the set {0, 1, · · · , Si − si} for each item i ∈ [K]
+at each time stage, where K = 5 is the number of different items, Si = 100 is the storage capacity for each item i, si
+is the current inventory level for each item i. The customer demand is a random vector ξ = (ξ1, · · · , ξK) with each ξi
+following a Poisson distribution with parameter θi. The true parameter is θ1 = 10, θ2 = 15, θ3 = 20, θ4 = 25, and
+θ5 = 30 but unknown to the agent. We view the parameter as a random variable, whose value is assumed to be within
+the following finite set {5, 6, 7, · · · , 33, 34, 35}. The state transition is given by st+1 = max(st + at − ξt, 0), where at
+is the amount of inventory to be replenished. Inventory level is not allowed to drop below zero (no backlog). When the
+customer demand is higher than the supply, there is a penalty cost p for each unit of unsatisfied demand. When the customer
+is lower than the supply, there is a holding cost h for each unit of overstock. In particular, for different items, p1 = 4,
+p2 = 5, p3 = 6, p4 = 7, p5 = 8, h1 = 2, h2 = 3, h3 = 4, h4 = 5, h5 = 6. The cost function at each stage is then given by
+C(st, at, ξt) = hT · max(st + at − ξt, 0) + pT · max(ξt − st − at, 0). The discount factor γ = 0.95. The agent starts with
+0 inventory and is given a historical dataset H0 of size N containing past customer demands for different items, and uses the
+given dataset to construct the prior for the rate parameter. Other parameters are as follows: number of points to be added at
+each iteration n = 20, threshold ϵ = 0.1.
+B.3. DR-MDP Details
+The DR-MDP approach, or Distributionally Robust Markov Decision Process, is a method for decision making under
+uncertainty where the ambiguity set, or the set of possible distributions for the uncertain parameters, is constructed using
+prior knowledge about the probabilistic information. However, this prior knowledge is not always readily available from a
+given data set, making the construction of the ambiguity set difficult in some cases.
+We note that the Bayesian Risk Optimization (BRO) approach has a distributionally robust optimization (DRO) interpretation.
+In particular, for a static stochastic optimization problem, it has been shown in (Wu et al., 2018) that the BRO formulation
+with the risk functional taken as Value-at-Risk (VaR) with a confidence level of 100% is equivalent to a DRO formulation with
+the ambiguity set constructed for the uncertain parameter, θ. This means that BRO and DRO can be used interchangeably,
+depending on the problem at hand and the level of uncertainty and prior knowledge about the parameters. Therefore, for a
+given problem when prior knowledge about the probabilistic information is not readily available, we adapt DR-MDP to
+our considered problem as follows: we use samples of the uncertain parameter, θ, drawn from the posterior distribution
+computed from a given data set. This allows us to construct an ambiguity set for θ using the available data, instead of relying
+on prior knowledge. Once we have samples of θ, we can obtain the optimal policy that minimizes the total expected cost
+under the most adversarial θ among the samples.
+B.4. Bilevel Optimization
+We show the bilevel DCP can be reduced to a single-level DCP. Specifically, we show this transformation for the exact
+bilevel DCP in (4), and the same technique can be applied to the approximate algorithm.
+Consider a general bilevel optimization problem:
+min
+xu,xl F(xu, xl)
+(9)
+s.t. xl ∈ arg min
+xl
+{f(xu, xl) : g(xu, xl) ≤ 0}
+G(xu, xl) ≤ 0
+where xu is the upper-level variable, xl is the lower-level variable, G denotes the upper-level constraints, g denotes the
+lower-level constraints, F denotes the upper-level objective function, f denotes the lower-level objective function. The
+Karush-Kuhn-Tucker (KKT) conditions are a set of necessary and sufficient conditions for a solution to be optimal in a
+convex optimization problem. When the lower-level problem in a bilevel optimization problem is convex and sufficiently
+regular, the KKT conditions can be used to reformulate the problem as a single-level constrained optimization problem,
+
+Approximate Bilevel Difference Convex Programming for Bayesian Risk Markov Decision Processes
+which is typically easier to solve. The general bilevel optimization problem 9 can then be reduced to the following
+single-level optimization:
+min
+xu,xl F(xu, xl)
+s.t. G(xu, xl) ≤ 0
+∇xlL(xu, xl, λ) = 0
+g(xu, xl) ≤ 0
+λg(xu, xl) = 0
+λ ≥ 0
+where L(xu, xl, λ) = f(xu, xl) + λg(xu, xl) is the Lagrangian function. In the bilevel DCP (4), V is the upper-level
+variable and φ is the lower-level variable. The constraint in (4) can be rewritten as:
+φ ∈ arg min
+φ∈Φ
+Eµ[Ψ(Eθ[C(s, a, ξ) + γV (s′, µ′)], φ)]
+V (s, µ) − Eµ[Ψ(Eθ[C(s, a, ξ) + γV (s′, µ′)], φ)] ≤ 0.
+Since the lower-level problem is convex, we can reformulate the bilevel DCP (4) as a single-level DCP problem.
+min
+V
+−
+�
+s∈S,µ∈M
+α(s, µ)V (s, µ)
+s.t.V (s, µ) − Eµ[Ψ(Eθ[C(s, a, ξ) + γV (s′, µ′)], φ)] ≤ 0
+∇φEµ[Ψ(Eθ[C(s, a, ξ) + γV (s′, µ′)], φ)] = 0
+∀a ∈ A, s ∈ S, µ ∈ M.
+B.5. Additional Observations
+We show additional observations for the path planning problem on Table 1 and Table 2.
+Larger data size reduces epistemic uncertainty: when there are more data, the posterior distribution used in BR-MDP
+formulation and the MLE estimator used in the nominal approach converges to the true parameter, which reduces to solving
+an MDP with known transition probability and cost function. Therefore, the optimal policies and the actual costs tend to be
+the same.
+Effect of risk measures: although both risk measures (expectation and CVaR) result in time-consistent optimal policy
+for each considered formulation, they provide different levels of robustness. Even though the expectation case is faster to
+compute, it provides the least robustness, especially when the data size is small. For the CVaR risk measure, different risk
+level α also affects the robustness. As α increases, the agent is more risk-averse, and the CVaR cost is smaller since it avoids
+more severe costs, as is shown in Figure 1(b) and Figure 1(c). But this comes with a price: its expected cost is higher. It is
+intuitive: even though the agent avoids severe costs, it also forfeits a chance to traverse a path that is likely to have less
+traffic, even though the likelihood is small. This is shown as a right-shift of the actual performance distribution from Figure
+1(c) and Figure 1(b).
+

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+Serenity: Library Based Python Code Analysis for
+Code Completion and Automated Machine Learning
+Wenting Zhao
+Department of Computer Science
+Cornell University
+wzhao@cs.cornell.edu
+Ibrahim Abdelaziz
+Thomas J. Watson Research Center
+IBM Research
+Ibrahim.abdelaziz1@ibm.com
+Julian Dolby
+Thomas J. Watson Research Center
+IBM Research
+dolby@us.ibm.com
+Kavitha Srinivas
+Thomas J. Watson Research Center
+IBM Research
+Kavitha.Srinivas@ibm.com
+Mossad Helali
+Department of Computer Science
+Concordia University
+mossad.helali@concordia.ca
+Essam Mansour
+Department of Computer Science
+Concordia University
+essam.mansour@concordia.ca
+Abstract
+Dynamically typed languages such as Python have become
+very popular1. Among other strengths, Python’s dynamic na-
+ture and its straightforward linking to native code have made
+it the de-facto language for many research areas such as Ar-
+tificial Intelligence. This flexibility, however, makes static
+analysis very hard. While creating a sound, or a soundy,
+analysis for Python remains an open problem, we present in
+this work Serenity, a framework for static analysis of Python
+that turns out to be sufficient for some tasks. The Serenity
+framework exploits two basic mechanisms: (a) reliance on
+dynamic dispatch at the core of language translation, and (b)
+extreme abstraction of libraries, to generate an abstraction
+of the code. We demonstrate the efficiency and usefulness of
+Serenity’s analysis in two applications: code completion and
+automated machine learning. In these two applications, we
+demonstrate that such analysis has a strong signal, and can
+be leveraged to establish state-of-the-art performance, com-
+parable to neural models and dynamic analysis respectively.
+Keywords: Python, Static Analysis, Code completion, Au-
+toML
+1
+Introduction
+Static analysis of Python is hard, due in part to features often
+regarded as strengths: its dynamic nature and its straightfor-
+ward linking to native code. Python is dynamically typed,
+so the aid static types provide to analysis of e.g. Java is not
+available. Python has a dynamic object structure; methods
+can be freely assigned and modified complicating resolving
+calls. Even basic constructs such as method calls and ob-
+ject creations can be ambiguous in the basic syntax. Beyond
+the language itself, many of the rich collection of Python li-
+braries, especially the math-heavy libraries used in machine
+learning, are implemented in native code, which makes analy-
+sis require cross-language support. For these reasons among
+1https://www.techrepublic.com/article/programming-languages-pythons-
+growth-is-absolutely-explosive-says-anaconda-ceo-and-not-slowing-
+down/
+others, to our knowledge, there is a lack of widely-used anal-
+ysis frameworks for Python, despite the value such analysis
+would have, for instance, for tools.
+However, while creating a sound, or a soundy, analysis for
+Python remains an open problem, we demonstrate Serenity2,
+a framework that turns out to be sufficient for some tasks.
+Beyond a relatively direct translation of Python Abstract
+Syntax Tree (AST) into a Control Flow Graph (CFG), Serenity
+exploits two basic mechanisms:
+1. Reliance on dynamic dispatch at the core of language
+translation. It is not possible, always, even to tell whether
+a construct is an object creation or a function call, and
+this is just one example. Our approach to such situ-
+ations is to turn them into dynamic dispatches over
+types representing constituent constructs. We detail
+how many subtleties of Python can be modeled in this
+way.
+2. Extreme abstraction of libraries. User code often makes
+heavy use of APIs to create and operate upon domain
+objects, such as arrays in numpy, but these objects are
+often fairly opaque to the user code. As such, we find
+it often suffices to treat libraries by just tracking the
+objects they create and methods called upon them. This
+is not, nor is it designed to be, soundy, let alone sound.
+We show however that this enables useful modeling
+of user code.
+We first discuss how Python is modeled and how the library
+abstraction still provides a useful analysis of user code. We
+then demonstrate that this analysis is useful where we fo-
+cus on two applications that depend on the outputs of such
+analysis:
+Code Completion is a core functionality expected in
+all IDEs, where the goal is to suggest methods and
+functions to call given prior code. We show how our
+dataflow analysis allows us to focus on relevant code
+at a point of completion, which when combined with
+2With apologies to Reinhold Niebuhr, "give us courage to model what must
+be modeled, serenity to accept what cannot be modeled, and the insight to
+know the one from the other."
+1
+arXiv:2301.05108v1  [cs.PL]  5 Jan 2023
+
+, ,
+Wenting Zhao, Ibrahim Abdelaziz, Julian Dolby, Kavitha Srinivas, Mossad Helali, and Essam Mansour
+local program context prior to the function call pro-
+duces much better code completion performance than
+the context alone.
+Automated Machine Learning which takes a given dataset
+in the form of a structured table, and creates an effec-
+tive machine learning pipeline to learn to predict some
+columns based on other columns. Prior approaches
+have been based on dynamic analysis, and we show
+static analysis does just as well. Static analysis is more
+practical, as actually running these pipelines is an ardu-
+ous and expensive task; and one can mine large open
+repositories to populate such databases using analysis.
+In the rest of the paper, we first describe Serenity’s tech-
+niques for modeling Python based on a running example
+(Sections 2 and 3). We then validate our analysis with two
+applications in code completion (Section 4) and automated
+machine learning (AutoML) (Section 5). We finally survey
+related work in Section 6 and conclude in Section 7.
+2
+A Running Example
+Figure 1 shows the running example for this paper, a snip-
+pet of Python code adapted from the multi_class_svm.py
+script of ETH150K [35] benchmark. Our modification (lines
+1 to 10) is to add some debugging options to illustrate com-
+plexities in Python. The variable fitit is assigned one of
+two functions depending on debug_level: either a closure
+or a function with an added assertion. The code loads a digits
+dataset (line 20), reads specific fields of the dataset (line 21),
+manipulates the dataset (line 22), splits the data into train
+and test splits (line 23), and finally creates X_train_bias
+(line 26. The code then creates machine learning models
+FrankWolfeSSVM (line 42) and LinearSVC (line 72). The code
+then calls them: a fitit call on a LinearSVC (line 74) takes
+the model (line 26), and fitit is called on FrankWolfeSVC
+with X_train_bias (line 79). Note that the call to the con-
+structor FrankWolfeSSVM is on line 42, and the fitit call
+on the object is on line 79, reflecting a property of most code
+- it is non-local.
+3
+Analysis
+3.1
+Background: call graph framework
+Grove et al. [18] provide a framework for expressing call
+graph algorithms for object-oriented languages. It encapsu-
+lates the bulk of the algorithm, parameterizing the algorithms
+with functions that determine how to add context sensitivity.
+The details of the framework are beyond the scope of this
+paper, but we depend on two details:
+• First, we will rely later on something called the Proce-
+dure Key Selection Function (PKS), which is essentially
+a way to specify when called functions should be ana-
+lyzed in a context-sensitive manner.
+• Second, the framework distinguishes between function
+call sites and object creation sites, which, as we shall
+1
+debug_level = 3
+2
+3
+if debug_level > 5:
+4
+def fd(model, test, train):
+5
+assert len(test) < 500
+6
+model.fit(train, test)
+7
+fitit = fd
+8
+9
+else:
+10
+fitit = lambda model, test, train:
+model.fit(train, test)
+↩→
+11
+15
+from sklearn.svm import LinearSVC
+16
+17
+from pystruct.models import MultiClassClf
+18
+from pystruct.learners import (NSlackSSVM,
+OneSlackSSVM, SubgradientSSVM,
+FrankWolfeSSVM)
+↩→
+↩→
+19
+20
+digits = load_digits()
+21
+X, y = digits.data, digits.target
+22
+X = X / 16.
+23
+X_train, X_test, y_train, y_test =
+train_test_split(X, y)
+↩→
+24
+25
+# we add a constant 1 feature for the bias
+26
+X_train_bias = np.hstack([X_train,
+np.ones((X_train.shape[0], 1))])
+↩→
+41
+42
+fw_bc_svm = FrankWolfeSSVM(model, C=.1,
+max_iter=50)
+↩→
+71
+72
+libsvm =
+LinearSVC(multi_class='crammer_singer',
+C=.1)
+↩→
+↩→
+73
+start = time()
+74
+fitit(libsvm, X_train, y_train)
+75
+time_libsvm = time() - start
+76
+print("Score with sklearn and libsvm: %f
+(took %f seconds)" %
+(libsvm.score(X_test, y_test),
+time_libsvm))
+↩→
+↩→
+↩→
+77
+78
+start = time()
+79
+fitit(fw_bc_svm, X_train_bias, y_train)
+Figure 1. A Running example
+see in Figure 3, is not possible in general in Python.
+Hence, we combine the two sets into a single one.
+The framework paper defines relevant program features
+at the top of page 694, which we excerpt here in Figure 2.
+We need two minor changes:
+InstVariables is taken to be the set of strings possibly
+used as field names, rather than a set of declared field
+names, which it is in the original framework. While
+2
+
+Serenity: Library Based Python Code Analysis for Code Completion and Automated Machine Learning
+, ,
+field names can be defined in Python, this is entirely
+optional so we ignore such definitions.
+NewSites becomes the same as the set CallSites to effect
+the second item above; that is, there is one set that
+combines all possible call sites and creation sites. This
+represents the fact that every site can potentially see
+both classes and functions.
+Class all class declarations in the program
+InstVariable all instance variable declara-
+tions of the program
+Procedure all procedure declarations of the
+program
+Variable all variable names used in the pro-
+gram
+CallSite all call sites in the program
+NewSite all new sites in the program
+LoadSite all loads of instance variables in
+the program
+StoreSite all stores to instance variables in
+the program
+Figure 2. Program features from [18]
+3.2
+Language modeling
+Figure 3 illustrates the kind of dynamism with which anal-
+ysis of Python must contend, in this case 5 different options
+for the meaning of X() on line 45 based on the value supplied
+as sys.argv[1] and sometimes sys.argv[2]:
+class1 class X (line 4) defines an ordinary class named
+X, of which line 45 creates an instance.
+class2 class X (line 11) defines a class named X that
+redefines the new operator, so line 45 just returns 0.
+def1 def X (line 16) defines a function named X, and
+calling it at line 45 returns 1.
+def2 X = lambda. . . (line 20) creates a closure and
+assigns it to X; calling the closure at line 45 returns 2.
+import The module X (line 23) overrides default module
+behavior to become callable and return 3 at line 45.
+method static X (line 32) is assigned the static method
+s of class X (line 28) which returns 5 at line 45.
+method instance X (line 41) gets a bound instance method
+(i.e. a closure over y) i of class X (list 35), returning 4
+at line 45).
+Note that all of these definitions of X can flow to the same call
+at line 45, so there is literally no syntactic distinction between
+different kinds of allocations, calls, and even modules in some
+cases. And class and function names are all first class. Thus
+analysis must handle these basic operations in a dynamic
+manner, unlike e.g. Java, where calls, allocations and imports
+have clear syntactic distinctions. Note further that even basic
+method calls require closures to handle line 41.
+1
+import sys
+2
+3
+if sys.argv[1] == "class1" or sys.argv[1] == "inst":
+4
+class X:
+5
+pass
+6
+7
+if sys.argv[1] == "inst":
+8
+X = X()
+9
+10
+elif sys.argv[1] == "class2":
+11
+class X:
+12
+def __new__(*args):
+13
+return 0
+14
+15
+elif sys.argv[1] == "def1":
+16
+def X():
+17
+return 1
+18
+19
+elif sys.argv[1] == "def2":
+20
+X = lambda: 2
+21
+22
+elif sys.argv[1] == "import":
+23
+import X
+24
+25
+elif sys.argv[1] == "method":
+26
+27
+if sys.argv[2] == "static":
+28
+class X:
+29
+def s():
+30
+return 5
+31
+32
+X = X.s
+33
+34
+elif sys.argv[2] == "instance":
+35
+class X:
+36
+v = 4
+37
+def i(self):
+38
+return self.v
+39
+40
+y = X()
+41
+X = y.i
+42
+43
+44
+print(str(X))
+45
+print(str(X()))
+Figure 3. Dynamic code examples
+The X() at line 45 is a call on X, and this allows us to
+use standard dynamic dispatch to model all of this behavior,
+using synthetic "methods" where needed to handle language
+semantics. We will use a similar "dispatch" at field accesses
+to handle the difference between class and instance fields,
+which again can only be known from the object accessed. We
+shall make use of these indirections to define our framework
+model in Section 3.3.
+3
+
+, ,
+Wenting Zhao, Ibrahim Abdelaziz, Julian Dolby, Kavitha Srinivas, Mossad Helali, and Essam Mansour
+We adopt the terminology of Grove et al. [18] to present
+our work as extensions to standard object-oriented call graph
+construction. To fit our dynamic Python context, we make
+a few changes to the core definitions of that work. These
+changes reflect that Python does not require that fields be
+declared in order to be used, and it makes no syntactic dis-
+tinction between calls and allocations. Furthermore, as is
+standard for representing first-class entities in an object-
+oriented framework, we have one class for each first-class
+entity. As Figure 3 shows, classes, functions, methods and
+modules are all first-class, so our set of classes for analysis
+includes the following:
+𝐶𝑐𝑙𝑎𝑠𝑠 a class representing program class C
+𝐶𝑖𝑛𝑠𝑡 a class representing instances of class C
+𝑀𝑖𝑛𝑠𝑡 a class representing instances of module M
+𝐷𝑖𝑛𝑠𝑡 a class representing instances of function D
+𝑆𝑖𝑛𝑠𝑡 a class representing the instance of script S
+Now most of the irregularities of Python calls and creations
+are handled by treating every call site as a CallSite for each
+receiver type ∗𝑖𝑛𝑠𝑡 and as a NewSite for every receiver type
+∗𝑐𝑙𝑎𝑠𝑠. The site on line 45 in Figure 3 would have some types
+handled by each mechanism. Fields are also handled seam-
+lessly: on line 32, X is a 𝐶𝑐𝑙𝑎𝑠𝑠, and on line 41 X is a 𝐶𝑖𝑛𝑠𝑡, so
+static and instance state are handled by making static fields
+be instance fields of the class object.
+Call graph construction starts with a root stub that creates
+an instance of the main script 𝑆𝑖𝑛𝑠𝑡 and calls it.
+3.3
+Framework modeling
+In many situations, it is difficult or impossible to find actual
+code for Python imports: there is no fixed relationship be-
+tween names in import statements and locations of actually
+source code. Even if there were, the structure of Python li-
+braries is such that large amounts of the code is native and
+hence a Python analysis framework is not applicable. Even
+if it were possible to find Python code, many libraries are
+large enough to make precise analysis challenging. In our
+case, we are interested in the behavior of application code
+rather than library internals, so we minimize these issues by
+largely not analyzing framework code.
+Our model, called Turtles3, abstracts Python frameworks
+to capture how the framework interacts with user code and
+to ignore all of its internal details. Specifically, we model
+four aspects, all using the indirections of Section 3.2:
+1. We model import statements as returning a new frame-
+work, denoted by the name of the imported module.
+The framework is an opaque object with no function-
+ality beyond implementing the model.
+2. Calls to framework functions and methods typically
+return something, which is then possibly used by the
+user code. We model every call to the framework as
+3from "turtles all the way down". This phrase is of unknown origin, see
+https://en.wikipedia.org/wiki/Turtles_all_the_way_down
+returning a new object from it; this model is transitive,
+so calls on those objects return further new objects
+from the framework. We label these objects with the
+path by which they are accessed.
+3. Accesses to fields of framework objects have little
+meaning in our model since we do not model the frame-
+work state at all. However, user code typically expects
+that a field access return something, so we model all
+such field accesses as returning the container object.
+4. Arguments to turtle methods are mostly ignored, since
+we do not model what the framework does to them;
+however, sometimes functions are passed as parame-
+ters, and we assume that the framework might call it.
+Since we do not model internal framework state, the
+model invokes callbacks from where they are passed
+as arguments.
+The framework of Grove et al. [18] provides the customiza-
+tion support needed to implement this model. We start by
+introducing a new type of class, 𝑇𝑝𝑎𝑡ℎ, that represents a tur-
+tle, i.e. an opaque model object. Item 1 is implemented by
+modeling import M statements as a call to a synthetic import
+procedure with M as its argument. This call is modeled as
+returning a 𝑇𝑀. Item 2 is implemented as a Procedure Key
+Selection Function (PKS) which takes the receiver of a type
+𝑇𝑝𝑎𝑡ℎ and the name 𝑛 of the called procedure and returns a
+new turtle of 𝑇𝑝𝑎𝑡ℎ.𝑛. Item 3 is implemented by simply re-
+turning self when reading any field of any 𝑇𝑝𝑎𝑡ℎ type. Item 4
+is implemented as a PKS that generates calls for every argu-
+ment that is of a function type (this is not illustrated in our
+example).
+3.4
+Inheritance from Turtles
+One wrinkle in our data is that application classes often
+inherit from turtle classes, meaning that method calls on
+self should logically be turtle methods when the method
+read is never assigned. That is, if a read of self is to a field
+or method that is never assigned and the class inherits from
+a turtle, the read should return a new turtle object to capture
+unknown superclass behavior. However, this is tricky to do
+because, since methods and fields can be assigned anywhere
+in the code, it is not in general possible to know if one will not
+be assigned until analysis terminates. What we need to do
+is record such reads and, when analysis terminates, process
+them as turtle reads and restart analysis. This restarting itself
+may need to be repeated, since reading one turtle could make
+more code reachable.
+3.5
+Analysis of running example
+When this analysis is applied to the running example (Fig-
+ure 1), the result is the dataflow graph shown in Figure 4. To
+illustrate our framework model, observe the import call of
+LinearSVC on line 15; as an import, this returns an object of
+type 𝐿𝑖𝑛𝑒𝑎𝑟𝑆𝑉𝐶𝑖𝑛𝑠𝑡, that is, an instance of the module. When
+4
+
+Serenity: Library Based Python Code Analysis for Code Completion and Automated Machine Learning
+, ,
+load_digits
+digits.data
+digits.target
+X
+y
+X/16
+train_test_split
+X_train
+X_test
+y_train
+y_test
+hstack
+fit (fd)
+fit (fd)
+LinearSVC
+FrankWolfeSSVM
+Invocations
+arg 0, flow
+arg > 0
+Reads
+fit (lambda)
+fit (lambda)
+Figure 4. Dataflow graph for the running example
+this is called (line 72), it returns a turtle of type 𝑇𝐿𝑖𝑛𝑒𝑎𝑟𝑆𝑉𝐶,
+illustrated by the green node labeled LinearSVC. When fit
+is called on this object in the fitit functions (line 74),
+item 2 means it returns a derived turtle of type𝑇𝐿𝑖𝑛𝑒𝑎𝑟𝑆𝑉𝐶.𝑓 𝑖𝑡,
+shown as a green node labeled fit. Since fit is called on
+LinearSVC, a black data flow edge connects them. On the
+other hand, the other non-self arguments to fit are shown
+with red arrows. Other turtle functions are shown similarly:
+load_digits, train_test_split, hstack, FrankWolfeSSVM.
+Note that analysis has no idea what these functions do, just
+that they pass data. Note that fitit is a variable holding one
+of two first-class functions, and it is called for both of the ML
+models created. To get the precise results shown in Figure 4
+requires analysis infrastructure that handles first-class func-
+tions and also does context-sensitive analysis. In particular,
+the model objects and the data flow to both the normal and
+debugging functions assigned to fit, since both potentially
+flow to fitit. In the figure, the nodes are distinguished with
+labels of the function in which they occur.
+Other nodes in Figure 4 represent local dataflow. The top-
+most two blue nodes represent reads of the data and target
+fields of data, so they have edges from the load_digits call
+and edges to their respective variables X and y. X is scaled
+by 16, shown by the nodes labeled X/16. X/16 and y then
+flow to train_test_split with red edges since they are
+arguments.
+This graph focuses on data flow, which captures patterns
+of how the various turtle APIs are used across programs.
+This allows us to learn patterns that enable our applications.
+3.6
+Implementation
+Our analysis is implemented using WALA and its support
+for both Python 2 and Python 3 using the Jython system.
+WALA is built to be extensible, and we used several features
+to ease our implementation work.
+The main extension is for handling turtles. For item 1, we
+override the model function that handles import to return a
+synthetic object with a turtle type named for the given mod-
+ule. For item 2, we override the selection of called methods
+for turtle classes so that any call goes to a synthetic method
+that creates and returns a turtle with the appropriate ex-
+tended turtle name. For item 2, this synthetic method mostly
+ignores its arguments, except generating a call to each one to
+handle callbacks. For item 3, we override the code handling
+field reads to simply return the container if it is of turtle
+type.
+The other configuration is to add aggressive context sen-
+sitivity for all turtle types. Since the synthetic methods are
+trivial anyway, it is cheap to ensure that every call site is
+analyzed separately.
+4
+Code Completion Application
+The core research question we ask is how useful Serenity’s
+analysis is and whether it can help other applications, de-
+spite the challenges in modeling dynamic languages such as
+Python accurately. As a first application, we examine a code
+completion use case, which we cover below in detail. By code
+completion, we refer to the problem where, when given a
+snippet of a program, the problem is to predict a function
+call, analogous to what an IDE does for method suggestions.
+We do not refer to code generation given natural language
+descriptions of code requirements, as in the Codex model
+that powers GitHub Co-Pilot [8] or even models that gener-
+ate entire functions in a generative style based on function
+signatures or snippets of code, such as CodeT5 [41]. Our
+observation is that for code completion, the analysis require-
+ment is that the methods be callable from a specific type, and
+so analysis for code completion is focused on detecting the
+types of objects. For languages such as Python, type infer-
+ence is hard, but our hypothesis is that code completion can
+benefit significantly from the data flow analysis that Serenity
+produces, simply because data flow can provide a focused
+context for code completion.
+Recently, there have been a plethora of neural models of
+code such as [13], [19], [22], [41] trained with the objective
+of either predicting randomly masked tokens in code, or
+predicting the very next token, which one might assume is
+consistent with the task of code completion. Our research
+question is whether one can leverage the extensive training
+of these models on millions of programs to perform code
+completion. Specifically, we asked whether data flow analy-
+sis provided by Serenity can improve code completion when
+combined with these neural models. If data flow analysis
+does provide any signal from Serenity, it should improve per-
+formance on code completion task even with the extensive
+training these language models already had. We therefore
+5
+
+, ,
+Wenting Zhao, Ibrahim Abdelaziz, Julian Dolby, Kavitha Srinivas, Mossad Helali, and Essam Mansour
+1
+print("Score with pystruct subgradient
+ssvm: %f (took %f seconds)" %
+(np.mean(y_pred == y_test),
+time_subgradient_svm))
+↩→
+↩→
+↩→
+2
+3
+# the standard one-vs-rest multi-class
+4
+# would probably be as good and faster
+5
+# but solving a different model
+6
+libsvm =
+LinearSVC(multi_class='crammer_singer',
+C=.1)
+↩→
+↩→
+7
+start = time()
+8
+libsvm.fit(X_train, y_train)
+9
+time_libsvm = time() - start
+10
+print("Score with sklearn and libsvm: %f
+(took %f seconds)" %
+(libsvm.score(X_test, y_test),
+time_libsvm))
+↩→
+↩→
+↩→
+11
+12
+13
+start = time()
+14
+fw_bc_svm.?
+Figure 5. Code snippet used for prediction
+1
+from sklearn.cross_validation import
+train_test_split
+↩→
+2
+from pystruct.models import MultiClassClf
+3
+from pystruct.learners import (NSlackSSVM,
+OneSlackSSVM,
+↩→
+4
+digits = load_digits()
+5
+digits.data
+6
+digits.target
+7
+X = X / 16.
+8
+train_test_split(X, y)
+9
+X_train_bias = np.hstack([X_train,
+np.ones((X_train.shape[0], 1))])
+↩→
+10
+model =
+MultiClassClf(n_features=X_train_bias.shape[1],
+n_classes=10)
+↩→
+↩→
+11
+fw_bc_svm = FrankWolfeSSVM(model, C=.1,
+max_iter=50)
+↩→
+12
+fw_bc_svm.?
+Figure 6. Code snippet corresponding to a slice from the
+analysis graph
+modeled code completion as a fine tuning task, and varied
+the training inputs of fine tuning to be one of the three
+conditions shown below:
+• All code as text prior to the function call
+• A slice of the code restricted to source expressions that
+are relevant to a function call in data flow
+• Both code as text, as well as the slice, separated by a
+token to distinguish the two inputs.
+For all text code prior to a function call, there are limits on
+how many tokens modern language models can fit. That is,
+when the code goes beyond the limit, truncation is needed
+in order for the models to run. A widely-used truncation
+strategy is to only keep 𝑛 tokens prior to the prediction point,
+where 𝑛 is the maximum sequence length, which can lead to
+fairly local information, as shown in Figure 5 for our running
+example shown in Figure 1. The key prediction in Figure 5
+is to predict what method will be called on fw_bc_svm, but
+notice that the construction of fw_bc_svm is out of the scope
+of the truncation4.
+For obtaining the slice restricted purely to dataflow, given
+a program and its corresponding dataflow graph, to predict
+the function call, we start at a node that we would like to
+predict, reverse all edges coming into the node, and find
+all reachable nodes. Each node in the reachability set corre-
+sponds to a source expression in the original program, and
+we only include the expressions that are not sub-expressions
+of any other expressions as features. Then, we order these
+expressions according to their positions in the source files,
+and add in variable names from the analysis artifacts so the
+code looks more or less like real code that the language mod-
+els have been trained on. Figure 6 shows an example of such
+a dataflow based slice looks for the code in Figure 1. Here we
+start the fw_bc_svm.fit call in Figure 4, reverse all edges
+coming into the node, and perform a reachability analysis,
+to gather the slice, adding variable names such as digits =
+load_digits(). In this example, dataflow analysis does give
+important information useful for predicting the function call,
+because the slice brings in non-local but relevant code such
+as the definition of fw_bc_svm into the scope of text that
+can be fed to a neural model.
+4.1
+Dataset
+We used the popular benchmark of ETH150K [35], which
+comes with 100K programs used for training, and the re-
+maining 50K used as a testing set. ETH150K was analyzed
+using Serenity, and 147,288 of 150,000 files were successfully
+analyzed. For the analyzed files, we parsed each file with a
+Python AST parser, and gathered all function calls. For each
+function call identified by the AST, we examined whether
+we could find the function in the analysis output, and if it
+was found in the output, we checked if the source location of
+the call matched that in the AST. Our observation has been
+that the Jython source mappings can be wrong sometimes,
+so we used both metrics to measure the completeness of the
+analysis. The analysis found 58.77% of function calls in the
+AST with matching source locations, and 67.36% of function
+calls when the requirements to match source was relaxed.
+Manual inspection on a few cases where source locations did
+not match indicated that the problem was indeed mapping
+4In this example, truncation was set to 1024 tokens, as per the requirements
+of one of the CuBERT models [22]
+6
+
+Serenity: Library Based Python Code Analysis for Code Completion and Automated Machine Learning
+, ,
+1
+def f_Hp(self, pars, p, inpt, target):
+2
+eps = 1E-6
+3
+deriv = self.fprime(pars, inpt,
+target)
+↩→
+4
+offseted = self.fprime(pars + p *
+eps, inpt, target)
+↩→
+5
+return (offseted - deriv) / eps
+Figure 7. Example of code where a leaf node is an expression
+being incorrect in Jython. Further investigation revealed that
+many of the missing calls are instances of Python primitives
+that Serenity does not model and treats as no-ops, such as
+repr and FutureWarning. A small fraction was found to
+be genuinely dead code, especially when Python files were
+integral parts of a larger application, as they often are in
+ETH150K.
+To generate the slices, we started with leaf nodes, and
+restricted ourselves to cases where the nodes had at least a
+depth of 1 when the edges were reversed. We note that in a
+majority of cases, leaf nodes were actually expressions, as
+shown in the example code in Figure 7. We ignored these
+in creating our dataset because we were focused on a prob-
+lem that cannot be solved by a pure lexical analysis of code.
+When we restricted ourselves to nodes that were potentially
+function calls rather than expressions, we generated slices
+from 65.35% of the programs where there existed at least one
+slice where the leaf node was likely a function call. For the
+train and test sets of programs, we generated 334,415 slices
+and 162,847 slices respectively by iterating over all the leaf
+nodes in dataflow graphs. Once we consider leaf nodes as
+nodes for our prediction, there were a total of over 65K labels
+that were generated across train and test sets for code com-
+pletion. Figure 8 plots the cumulative frequency distribution
+of labels against the number of labels. As shown in the Figure,
+the distribution of labels follows the usual power law, but we
+note that the most popular label appeared across train and
+test only 1.7% of the time, and the top 10 labels cumulatively
+appeared only 11.0% of the time. In other words, this is a
+difficult classification problem5. We note that this method
+of declaring code completion is more realistic compared to
+other means for code completion (such as measuring next
+token prediction), in the sense that this is often the case that
+IDEs focus on.
+4.2
+Language model selection
+To decide on the best neural model to use as a basis for our
+code completion experiments, we tested a number of code
+related language models including CodeBERT [13], Graph-
+CodeBERT [19], CuBERT [22] and CodeT5 [41]. CodeBERT[13]
+5We will make the datasets and code for all the work reported in this section
+available as open source.
+ 0
+ 0.1
+ 0.2
+ 0.3
+ 0.4
+ 0.5
+ 0.6
+ 0.7
+ 0.8
+ 0.9
+ 1
+ 1
+ 4
+ 16
+ 64
+ 256
+ 1024
+ 4096
+ 16384
+Cumulative Frequency
+No. of Labels
+Cumulative frequency of labels
+Figure 8. Distribution of labels for the classification task
+is a bimodal model trained on datasets with natural lan-
+guage (NL) -programming language (PL) pairs (e.g. docu-
+mentation/code pairs) across six programming languages
+(Python, Java, JavaScript, PHP, Ruby, and Go). Similarly,
+GraphCodeBERT uses NL-PL pairs for pretraining a code
+language model, but based on local data flow graphs ex-
+tracted from Abstract Syntax Trees. CuBERT [22] is another
+BERT-based model fine-tuned on multiple classification tasks
+such as checking the presence of certain bugs and predicting
+exception types. CuBERT is trained only on Python code,
+and furthermore uses language level tokens as inputs to the
+model. CodeT5 [41] is an encoder-decoder model based on
+T5 architecture [33] with code-specific knowledge trained
+to distinguish which tokens are identifiers and recover them
+when they are masked out. CodeT5 is fine-tuned using multi-
+ple CodeXGLUE benchmarks including understanding tasks
+such as code defect detection and clone detection, and gen-
+eration tasks like code summarization and translation.
+Figure 9 shows the performance of these different models
+on the code completion task with no fine tuning for the top-1
+and top-5 cases. We modeled code completion as a mask pre-
+diction task, with the function call to be predicted being the
+masked token. As shown in the Figure 9, the best performing
+model was CuBERT on this task, which is not surprising
+because it was the only model trained exclusively on Python
+and used language level tokens unlike the other models. We
+note that the performance of CodeT5 was surprisingly poor,
+but we think this may in part be due to the fact that it is
+trained on NL-PL pairs and it is strictly a generative model,
+7
+
+, ,
+Wenting Zhao, Ibrahim Abdelaziz, Julian Dolby, Kavitha Srinivas, Mossad Helali, and Essam Mansour
+21
+19
+13
+2.7
+33
+27
+23
+3.1
+0
+5
+10
+15
+20
+25
+30
+35
+40
+CuBERT
+CodeBERT
+GraphCodeBert
+CodeT5
+Accuracy
+Complete (Top1)
+Complete (Top5)
+Figure 9. Accuracy on top-5 and top-1 test data for base lan-
+guage models. Performance on CuBERT for slices is based on
+150,739 and 137,987 examples for complete and slice, respec-
+tively because of tokenization issues. For all other systems
+the number of testing examples was 167,816.
+for which we needed to specify a length of generation. It
+also needed the most tuning in terms of specifying different
+search strategies for final token prediction, so it is possible
+that we did not choose the optimal search strategy for it. For
+our purposes though, we chose CuBERT as a base, primarily
+because we expected to benefit most from fine tuning. We
+point out that given our label distribution, for the language
+model to provide even 21% performance on complete for
+top-1 and 33% for top-5 is quite good.
+We turn now to the problem of fine tuning CuBERT with
+training inputs to see if analysis does in fact improve code
+completion as we defined it. Note that CuBERT’s pretraining
+was performed by feeding the model the logical lines of 5
+million programs - so at the minimal, some fine tuning for
+the code context where the function call is to be predicted
+is needed. As stated earlier, we contrasted three different
+training conditions:
+• complete: where we gave the model text starting from
+the call, backwards, as shown in Figure 5
+• slice: where we used a backwards slice as shown in
+Figure 6
+• combined where the text from complete and slice
+were concatenated as input to the model using a sepa-
+rator token.
+The test was on complete text, or combined. We chose
+these conditions because we observed from examples that
+for the problem of code generation, data flow is not sufficient
+by itself. Figure 10 shows such an example. In this code, lines
+5 and 15 contain the clue needed to make the prediction
+of id, but they are unrelated to the receiver for which the
+call is being made on line 22. Yet, the local pattern of code
+has the same variable names, and the same set of calls are
+repeated across functions, suggesting that id may be a good
+candidate label. By contrast, the corresponding slice con-
+tains minimal information as shown in Figure 11, since the
+1
+response.json.return_value = dict(response,
+total_count=3, limit=0, offset=0)
+↩→
+2
+projects =
+self.redmine.project.all()
+↩→
+3
+self.assertEqual(projects.limit, 0)
+4
+self.assertEqual(projects.offset,
+0)
+↩→
+5
+self.assertEqual(projects[0].id, 1)
+6
+self.assertEqual(projects[1].id, 2)
+7
+self.assertEqual(projects[2].id, 3)
+8
+9
+def test_offset_limit(self):
+10
+response_with_limit_offset =
+{'total_count': 2, 'limit': 3,
+'offset': 1, 'projects':
+response['projects'][1:3]}
+↩→
+↩→
+↩→
+11
+self.response.json.return_value =
+response_with_limit_offset
+↩→
+12
+projects =
+self.redmine.project.all()[1:3]
+↩→
+13
+self.assertEqual(projects.limit, 3)
+14
+self.assertEqual(projects.offset,
+1)
+↩→
+15
+self.assertEqual(projects[0].id, 2)
+16
+self.assertEqual(projects[1].id, 3)
+17
+18
+def test_offset_limit_mimic(self):
+19
+projects =
+self.redmine.project.all()[1:3]
+↩→
+20
+self.assertEqual(projects.limit, 3)
+21
+self.assertEqual(projects.offset,
+1)
+↩→
+22
+self.assertEqual(projects[0].?
+Figure 10. Code snippet where local text can help prediction
+1
+from tests import unittest, mock, Redmine,
+URL
+↩→
+2
+Redmine(self.url)
+3
+projects = self.redmine.project.all()[1:3]
+4
+self.assertEqual(projects[0].?
+Figure 11. Code where data flow lacks sufficient context
+receiver projects[0] was defined just within the function
+test_offset_limit_mimic.
+We note however that sometimes the slice can help even
+when the truncation does not cut off key information for
+prediction. Figure 12 shows one such example. The predicted
+function is partial is imported in line 3, but the actual call
+is on line 28. On the other hand, in Figure 13, the import is
+the only call prior to the line, so the slice can make relevant
+information proximal, such that the neural model can pay
+greater attention to proximal elements of the code.
+8
+
+Serenity: Library Based Python Code Analysis for Code Completion and Automated Machine Learning
+, ,
+1
+import sys
+2
+import logging
+3
+from functools import partial
+4
+from datetime import datetime
+5
+from abc import ABCMeta, abstractmethod
+6
+import json
+7
+from _config import AttrDict
+8
+9
+__all__ = ['multikey_getter_gen',
+'unescape_json', 'LogParser',
+'JSONParser', 'LogLine',
+↩→
+↩→
+10
+'AccessLog', 'CommonLogFormat',
+'uWSGIParser']
+↩→
+11
+12
+def multikey_getter_gen(parser, keys,
+is_indices=False, delimiter="\t"):
+↩→
+13
+"""Generator meta-function to return a
+function
+↩→
+14
+parsing a logline and returning
+multiple keys (tab-delimited)"""
+↩→
+15
+if is_indices:
+16
+keys = map(int, keys)
+17
+18
+def multikey_getter(line, parser,
+keyset):
+↩→
+19
+data = parser(line.strip())
+20
+return
+delimiter.join((unicode(data[k])
+for k in keyset))
+↩→
+↩→
+21
+22
+def multiindex_getter(line, parser,
+keyset):
+↩→
+23
+data = parser(line.strip())
+24
+return delimiter.join((unicode(
+data.by_index( idx-1,
+raw=True)) for idx in keys))
+↩→
+↩→
+25
+26
+if is_indices is True:
+27
+# Field indices
+28
+return ?
+Figure 12. Example of where complete text may have text
+relevant to the prediction, but distant from call site
+1
+partial = #!/usr/bin/env python #
+2
+from functools import partial
+3
+keys = map(int, keys)
+4
+return ?
+Figure 13. Example of where dataflow is very focused
+4.3
+Model details
+We use the CuBERT model released by [22]6, which has 24
+layers with 16 attention heads and 1024 hidden units and
+6The CuBERT model can be accessed at github.com/google-research/google-
+research/tree/master/cubert
+was pretrained on 4M unique Python files on Github. At
+fine-tuning, we set the batch size to 10 and trained the model
+using 8 Tesla V100 with 32GB memory. The learning rate
+is 5e-5, and we gradually warmed up the learning rate for
+the first 300 gradient updates, which are the default val-
+ues provided by the HuggingFace library [43]. The training
+stops after 20 epochs, or ends after the evaluation accuracy
+hasn’t improved for three epochs. For the complete and
+slice models we used the 512 tokens model, and when we
+used combined, we used the 1024 tokens model such that
+the exact same tokens present in complete and slice could
+be used together along with the separator.
+We apply CuBERT’s tokenization to Python programs in
+ETH150K where the Python programs are first tokenized us-
+ing the standard Python tokenizer (the tokenize module)7,8.
+Then we further break down the program tokens into 49,558
+subwords using subword tokenization [39], as performed by
+the cuBERT tokenizer.
+4.4
+Results of fine tuning
+Figure 14 shows the accuracy in predicting the function call
+exactly across the different training and test conditions. As
+shown in the Figure 14, training on slice was at 47% ac-
+curacy when tested on the complete text (slice-complete),
+which is significantly above the 21% of top-1 baseline from
+cuBERT. Training on complete however was much better
+at 62% on the same text (complete-complete), which is
+not surprising given that inspection of examples (e.g., Fig-
+ure 10) show that complete often contains the expressions
+in slice when the dataflow is local, and furthermore, ben-
+efits from repetition in coding patterns that might hint at
+labels in the absence of any real connection. The key ques-
+tion is whether slices provide any benefit over and above
+what benefit is gained from complete. Training on combined
+suggests that slices do provide a strong signal, with a 65%
+accuracy on complete text (combined-complete), and 69%
+accuracy on the combined text (combined-combined). We
+also compared top-5 performance across conditions to allow
+comparison to the baseline language models - not surpris-
+ingly this result improved accuracy across all conditions,
+with the combined-combined condition showing the best
+performance at 78%. The results show that data flow analysis
+can significantly augment code completion performance.
+4.5
+How do slices help code completion?
+We conducted an analysis of how slices might help code
+completion performance; i.e., to understand if slices help
+the model complete code better for rare labels compared
+7github.com/python/cpython/blob/main/Lib/tokenize.py
+8We note that tokenize only outputs tokens for the code snippet that is
+free of any syntax errors; otherwise, it returns either IndentationError or
+TokenError. To predict function calls we often feed code that is incomplete,
+thus syntactically incorrect; therefore, we had to modify the original module
+so that it always returns what has been already tokenized thus far.
+9
+
+, ,
+Wenting Zhao, Ibrahim Abdelaziz, Julian Dolby, Kavitha Srinivas, Mossad Helali, and Essam Mansour
+21
+47
+62
+65
+69
+33
+66
+74
+75
+78
+0
+10
+20
+30
+40
+50
+60
+70
+80
+CuBERT baseline
+slice-complete
+complete-complete combined-complete combined-combined
+Accuracy
+Top-1
+Top-5
+Figure 14. Results of fine tuning on the different "training -
+testing" conditions; i.e., training on {slice, complete or com-
+bined} and tested on {complete or combined}
+ 0
+ 0.1
+ 0.2
+ 0.3
+ 0.4
+ 0.5
+ 0.6
+ 0.7
+ 0.8
+ 0.9
+ 1
+ 1
+ 4
+ 16
+ 64
+ 256
+ 1024
+Proportion of labels output correctly
+Label Counts
+Combined-Complete
+Complete-Complete
+Combined-Combined
+Figure 15. Accuracy on labels with different counts
+to more common ones, since statistical approaches likely
+work better for common labels, but less well for rare ones.
+Figure 15 shows the performance of the different models; the
+presence of slices at training and test enhance code comple-
+tion performance for rare labels more than common labels,
+although the advantage does seem to be present for common
+ones as well. The combined-combined model was 15% ac-
+curate on labels with count 1, of about 18,000 labels, and that
+number rapidly approaches 40% for labels with count 3. As
+labels become more frequent, the differences between the
+sklearn.datasets
+sklearn.load_digits
+sklearn.load_digits.data
+sklearn.load_digits.target
+sklearn.train_test_split
+sklearn.LinearSVC
+sklearn.svm
+sklearn.cross_validation
+Figure 16. KGpip’s training graph for our running example
+after filtering out as input to the AutoML system.
+models gets more noisy but the combined-combined case
+still holds an advantage.
+4.6
+Comparison to existing work
+In this space, comparisons are tricky because there is no com-
+mon or standard benchmark and because the exact problem
+varies. We chose ETH150K, which is at least a well-known
+code repository, but work that e.g. relies on its own sample
+of GitHub makes results incomparable. The exact problem
+varies too, with some tools, like us, predicting function calls,
+others predicting only method calls and still others predict
+the next token for all tokens. There is a real need for a bench-
+mark in this space; as part of helping build such a benchmark,
+we will release our own slice and complete dataset to the
+community. [25] reports overall accuracy numbers around
+0.7, which is almost the same as ours, but that paper is pre-
+dicting the next token across all token types, so could benefit
+from the fact that some predictions (e.g. ’)’ followed by ’:’ in
+def) follow from the grammar. Pythia [37] uses a neural net-
+work rather than a language model, but reports comparable
+accuracy numbers for top-1. Their top-5 number is higher,
+but their predictions seem to be for method calls, rather than
+all functions as we do, for which the receiver may provide
+context to aid prediction.
+These approaches may be complementary, too. Our work
+showed that adding slice data to local context greatly aided
+the accuracy of our models. Other approaches also rely on
+mostly local information, and could potentially benefit from
+slices as well. We plan to investigate this further in our future
+work.
+5
+Automated ML Pipelines Application
+The problem of automated machine learning pipelines (Au-
+toML) focuses on automatically building pipelines by per-
+forming a search over valid data transformations and learn-
+ers, along with hyper-parameter optimization for each learner.
+Our research question is whether we can perform learner and
+transformation selection based on mining large repositories
+10
+
+Serenity: Library Based Python Code Analysis for Code Completion and Automated Machine Learning
+, ,
+of abstracted ML python scripts obtained statically by Seren-
+ity. Unlike dynamic analysis, Serenity’s static analysis of ML
+pipelines has the advantage of scaling to millions of scripts
+due to its low cost. Specifically, our question is whether the
+extracted semantics of a set of pipelines by Serenity can help
+in predicting a new pipeline when combined with neural
+graph models.
+We developed an AutoML system, called KGpip, described
+in detail in a separate work [20], which builds a database
+of datasets and their corresponding historically used ML
+pipelines using Serenity analysis. KGpip formulates the au-
+tomation of a ML pipeline as a graph generation problem. It
+is based on two hypotheses that a neural graph generator
+will: i) capture more succinctly multiple pipelines seen in
+practice for a given dataset, and ii) capture statistical similar-
+ities between different pipelines more effectively. In KGpip,
+we filter out Serenity’s analysis to remove non-ML related
+components such as calls to libraries other than target ML
+libraries (Sklearn, XGBoost, and LGBM), and nodes indicat-
+ing location of calls within a pipeline script, among others.
+Figure 16 shows the filtered version of the graph for our
+running example. We also show in Figure 17 an overview of
+how KGpip works at training and inference phases. Using
+code analysis graphs obtained from Serenity and dataset em-
+beddings, KGpip trains a graph generation model optimized
+to output a ML pipeline as close as possible to the target
+pipelines of the training data. At inference time, KGpip iden-
+tifies the closest dataset to the input dataset and uses its
+embedding as input to the graph generation model which in
+turn outputs a set of possible ML pipelines. These pipelines
+are then validated and fed to a hyper-parameter optimizer to
+get the best pipeline that results in the highest performance
+on the input dataset.
+KGpip is designed to work with AutoML systems, such
+as AutoSklearn [15] and FLAML [40], to utilize their hyper-
+parameter optimizers. With a collection of 2000 ML python
+scripts, we trained a graph generation neural network that
+learns to generate a ML pipeline graph for a given dataset.
+We conducted a comprehensive evaluation using 77 datasets
+from different benchmarks, such as AutoML and Penn Ma-
+chine Learning Benchmark (PMLB), and different ML portals,
+such as Kaggle and OpenML. Table 1 shows the overall KG-
+pip performance which significantly improves the selection
+of data transformation and learning algorithms of state-of-
+the-art AutoML systems, namely, Auto-Sklearn and FLAML.
+We note that AutoSklearn consults a database of pipelines
+and datasets, and picks pipelines to start the search based
+on a nearest neighbors to an unseen dataset, except that Au-
+toSklearn’s dataset consists of effective pipelines based on
+actual execution. We also compared KGpip to AL [6], which
+uses dynamic code analysis on existing machine learning
+pipelines to select optimized pipelines. AL was unable to
+process 60 datasets because it ran out of time in searching
+for pipelines. On a smaller set of 17/77 datasets on which
+Average Performance
+T-Test
+AutoSklearn
+0.71 (0.24)
+-
+KGpip + AutoSklearn
+0.77 (0.22)
+0.0002
+FLAML
+0.71 (0.27)
+-
+KGpip + FLAML
+0.77 (0.20)
+0.0132
+Table 1. Performance (average and stdev) of KGpip com-
+pared to FLAML and AutoSklearn. Both variations of KGpip
+show significant improvements compared to existing sys-
+tems, both with 2-tailed T-test 𝑝 < 0.05
+AL was able to work, AL achieved an average performance
+of 0.36 compared to 0.745, 0.705, 0.79 and 0.765 by FLAML,
+Auto-Sklearn, KGpip + FLAML, and KGpip + AutoSklearn,
+respectively. This comparison with both AL and AutoSklearn
+clearly illustrates the value provided by Serenity, compared
+even to approaches that rely on dynamic runtime analysis.
+6
+Related Work
+Static Analysis for Python: Static analysis of Python has
+attracted considerable interest lately, and there have been
+a range of approaches. Type inference has been a focus of
+much work, some using techniques such as abstract inter-
+pretation and, more recently, there has been work using
+machine learning. Likely the best-known work is MyPy [1]
+and Pytype [2]. MyPy focuses on checking and inferring
+types that conform to PEP 484 [38], which defines a syntax
+for Python types. MyPy focuses on inference within a single
+function, since types are expressed at function boundaries
+in PEP 484. Pytype does type inference, and it can handle
+cases where a variable has different types at different points.
+It also does relatively little interprocedural analysis.
+Moat et al. [28] present an abstract interpretation for type
+inference of Python that models a variety of domains to
+compute more accurate information, and it makes use of
+the recency abstraction for aliasing. However, it is currently
+limited in its support for interprocedural analysis, which is
+enabled by inlining. Fritz and Hage [16] present a dataflow
+analysis for type inference that provides a range of tradeoffs
+for cost and precision. It does handle features like first-class
+functions.
+Machine learning is also used for type inference of Python.
+TypeWriter [31] trained a neural model using a corpus of
+code, with labeled data derived from user annotations. Type-
+Writer considers comments in code as inputs to the neural
+model, unlike program analysis based type inference. While
+such systems are certainly performing analysis, their ap-
+proach and mechanism are quite different from ours.
+There have been other analyses of Python, often for spe-
+cial purposes. Ariadne [11] makes use of WALA, as we do,
+but focuses on inferring the shapes of tensors in machine
+11
+
+, ,
+Wenting Zhao, Ibrahim Abdelaziz, Julian Dolby, Kavitha Srinivas, Mossad Helali, and Essam Mansour
+Figure 17. An overview of KGpip training and inference workflows
+learning programs. Unlike our approach to libraries, Ariadne
+models ML libraries as needed for tensor-related operations.
+Code completion: Code completion has been a prominent
+area of research towards achieving better productivity when
+working within an IDE. One of the main challenges in this
+domain is about code representation where the vast majority
+of work has used either tokens or abstract syntax trees to
+represent code (we refer the reader to [3] for a detailed survey
+of this area). [25] represents Python and JavaScript codes
+as ASTs and use a pointer network for better predicting
+Out of Vocabulary words in code completion. Pythia [37] is
+another approach that uses ASTs with an LSTM model for
+code completion.
+A number of approaches also tried to leverage representa-
+tions based on data and control flow [4, 7, 14]. On JavaScript,
+[21] utilized a program dependence graph to detect code du-
+plication. [4] use AST based representation augmented with
+local data and control flows for predicting variable names
+and variables misuse. [14] combines token based represen-
+tations of code with edges based on object uses, and AST
+nodes to predict the documentation of a method. To perform
+code completion over Java API calls, [29, 30] used a mostly
+intraprocedural analysis for mining graphs augmented with
+control and data flow.
+With the rise of pre-trained language models such as BERT
+[10] and GPT [5, 32], many recent approaches [13, 19, 22,
+26, 41] started to leverage the already existing rich language
+understanding in these models and fine tune it for various
+code understanding tasks such as code summarization, trans-
+lation, completion, bug detection, etc. CodeBERT[13] is a
+BERT based model trained on pairs of natural language and
+programming language samples across six programming
+languages. GraphCodeBERT [19] uses BERT as well, but
+represents the code using data flow graphs based on ASTs.
+The dataflow in GraphCodeBERT is completely local and
+not interprocedural, as in Serenity. For instance as an exam-
+ple, it adds edges from all variables used in an expression
+to their definitions. CuBERT [22] and CodeT5 [41] are an-
+other two models based on BERT and T5 [33] architectures,
+respectively.
+Unlike our approach, all these methods represented code
+either as a sequence of tokens [13, 22], ASTs [25, 27, 37, 41],
+or data flows derived from ASTs [19].
+AutoML approaches: Several AutoML frameworks have
+been proposed recently [6, 12, 15, 40]. In most AutoML sys-
+tems, learner and pre-processing selection is driven by a
+database of actual executions of pipelines and data. For in-
+stance, [15, 36] compute a database of dataset meta-features
+such as number of rows and columns, while [6] mines a
+repository of run-time information of inputs to the learners
+and preprocessors via dynamic code analysis of public ML
+pipelines available e.g. on Kaggle. The predicted learners and
+preprocessors are based on a similarity measure between the
+target dataset and stored features. In KGpip we utilized dense
+vector embeddings derived from raw contents of datasets to
+measure this similarity and graph neural networks to select
+the learners/preprocessors and generate the pipeline.
+Some existing systems such as TPOT [24] or Recipe [9]
+use evolutionary algorithms for pipeline generation. Others
+approach it as a probabilistic matrix factorization [17], an
+AI planning problem when combined with a user specified
+grammar [23, 42], or a bayesian optimization problem com-
+bined with Monte Carlo Tree Search [34]. None of these
+approaches however use analysis to build up their database.
+7
+Conclusion
+In this paper, we introduced Serenity; a framework for Python
+code static analysis. Serenity relies on two mechanisms (a)
+dynamic dispatching at the core of language translation,
+and (b) extreme abstraction of libraries. To demonstrate the
+12
+
+E
+CSV
+Deep graph
+jupyter
+generator
+model
+CsV
+ML Pipelines
+Hyperparameter
+Skeleton
+Optimization
+Auto-
+Sklearn
+FLAMISerenity: Library Based Python Code Analysis for Code Completion and Automated Machine Learning
+, ,
+utility of Serenity’s analysis, we used it in two important
+code-related applications: code generation and automated
+machine learning. Serenity’s analysis showed very promis-
+ing performance in both applications, allowing us in some
+cases to outperform approaches based on dynamic analysis,
+and perform competitively for code completion. We also im-
+plemented Serenity as an open-source implementation based
+on WALA, a popular framework for program analysis.
+References
+[1] [n.d.]. MyPy. http://mypy-lang.org. Accessed: 2021-11-18.
+[2] [n.d.]. Pytype. https://github.com/google/pytype. Accessed: 2021-11-
+18.
+[3] Miltiadis Allamanis, Earl T Barr, Premkumar Devanbu, and Charles
+Sutton. 2018. A survey of machine learning for big code and natural-
+ness. ACM Computing Surveys (CSUR) 51, 4 (2018), 1–37.
+[4] Miltiadis Allamanis, Marc Brockschmidt, and Mahmoud Khademi.
+2018. Learning to Represent Programs with Graphs. In ICLR. OpenRe-
+view.net.
+[5] Tom B Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared
+Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish
+Sastry, Amanda Askell, et al. 2020. Language models are few-shot
+learners. arXiv preprint arXiv:2005.14165 (2020).
+[6] José P. Cambronero and Martin C. Rinard. 2019. AL: Autogenerating
+Supervised Learning Programs. In Proceedings of the ACM on Program-
+ming Languages, Vol. 3. https://doi.org/10.1145/3360601
+[7] Kwonsoo Chae, Hakjoo Oh, Kihong Heo, and Hongseok Yang. 2017.
+Automatically Generating Features for Learning Program Analysis
+Heuristics for C-like Languages. Proc. ACM Program. Lang. 1, OOPSLA,
+Article 101 (Oct. 2017), 25 pages.
+[8] Mark Chen, Jerry Tworek, Heewoo Jun, Qiming Yuan, Henrique Ponde
+de Oliveira Pinto, Jared Kaplan, Harri Edwards, Yuri Burda, Nicholas
+Joseph, Greg Brockman, Alex Ray, Raul Puri, Gretchen Krueger,
+Michael Petrov, Heidy Khlaaf, Girish Sastry, Pamela Mishkin, Brooke
+Chan, Scott Gray, Nick Ryder, Mikhail Pavlov, Alethea Power, Lukasz
+Kaiser, Mohammad Bavarian, Clemens Winter, Philippe Tillet, Fe-
+lipe Petroski Such, Dave Cummings, Matthias Plappert, Fotios
+Chantzis, Elizabeth Barnes, Ariel Herbert-Voss, William Hebgen Guss,
+Alex Nichol, Alex Paino, Nikolas Tezak, Jie Tang, Igor Babuschkin,
+Suchir Balaji, Shantanu Jain, William Saunders, Christopher Hesse, An-
+drew N. Carr, Jan Leike, Josh Achiam, Vedant Misra, Evan Morikawa,
+Alec Radford, Matthew Knight, Miles Brundage, Mira Murati, Katie
+Mayer, Peter Welinder, Bob McGrew, Dario Amodei, Sam McCandlish,
+Ilya Sutskever, and Wojciech Zaremba. 2021. Evaluating Large Lan-
+guage Models Trained on Code. (2021). arXiv:2107.03374 [cs.LG]
+[9] Alex G. C. de Sá, Walter José G. S. Pinto, Luiz Otavio V. B. Oliveira, and
+Gisele L. Pappa. 2017. RECIPE: A Grammar-Based Framework for Au-
+tomatically Evolving Classification Pipelines. In Genetic Programming,
+James McDermott, Mauro Castelli, Lukas Sekanina, Evert Haasdijk,
+and Pablo García-Sánchez (Eds.). 246–261. https://doi.org/10.1007/978-
+3-319-55696-3_16
+[10] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova.
+2018. Bert: Pre-training of deep bidirectional transformers for language
+understanding. arXiv preprint arXiv:1810.04805 (2018).
+[11] Julian Dolby, Avraham Shinnar, Allison Allain, and Jenna Reinen. 2018.
+Ariadne: Analysis for Machine Learning Programs. In Proceedings of
+the 2nd ACM SIGPLAN International Workshop on Machine Learning
+and Programming Languages (Philadelphia, PA, USA) (MAPL 2018).
+Association for Computing Machinery, New York, NY, USA, 1–10.
+https://doi.org/10.1145/3211346.3211349
+[12] Iddo Drori, Lu Liu, Yi Nian, Sharath C Koorathota, Jung-Shian Li,
+Antonio Khalil Moretti, Juliana Freire, and Madeleine Udell. 2019.
+AutoML using Metadata Language Embeddings. ArXiv (2019). https:
+//arxiv.org/abs/1910.03698
+[13] Zhangyin Feng, Daya Guo, Duyu Tang, Nan Duan, Xiaocheng Feng,
+Ming Gong, Linjun Shou, Bing Qin, Ting Liu, Daxin Jiang, and Ming
+Zhou. 2020. CodeBERT: A Pre-Trained Model for Programming and
+Natural Languages. CoRR abs/2002.08155 (2020). arXiv:2002.08155
+https://arxiv.org/abs/2002.08155
+[14] Patrick Fernandes, Miltiadis Allamanis, and Marc Brockschmidt. 2018.
+Structured Neural Summarization. CoRR abs/1811.01824 (2018).
+[15] Matthias Feurer, Aaron Klein, Katharina Eggensperger, Jost Springen-
+berg, Manuel Blum, and Frank Hutter. 2015. Efficient and Robust
+Automated Machine Learning. In Proceedings of the International Con-
+ference on Neural Information Processing Systems (NeurIPS). 2962–2970.
+https://dl.acm.org/doi/10.5555/2969442.2969547
+[16] Levin Fritz and Jurriaan Hage. 2017. Cost versus Precision for Approxi-
+mate Typing for Python. In Proceedings of the 2017 ACM SIGPLAN Work-
+shop on Partial Evaluation and Program Manipulation (Paris, France)
+(PEPM 2017). Association for Computing Machinery, New York, NY,
+USA, 89–98. https://doi.org/10.1145/3018882.3018888
+[17] Nicolo Fusi, Rishit Sheth, and Melih Elibol. 2018. Probabilistic Ma-
+trix Factorization for Automated Machine Learning. In Proceedings of
+the International Conference on Neural Information Processing Systems
+(NeurIPS). 3352–3361. https://dl.acm.org/doi/10.5555/3327144.3327254
+[18] David Grove and Craig Chambers. 2001. A Framework for Call Graph
+Construction Algorithms. ACM Trans. Program. Lang. Syst. 23, 6 (nov
+2001), 685–746. https://doi.org/10.1145/506315.506316
+[19] Daya Guo, Shuo Ren, Shuai Lu, Zhangyin Feng, Duyu Tang, Shujie Liu,
+Long Zhou, Nan Duan, Alexey Svyatkovskiy, Shengyu Fu, Michele
+Tufano, Shao Kun Deng, Colin B. Clement, Dawn Drain, Neel Sundare-
+san, Jian Yin, Daxin Jiang, and Ming Zhou. 2021. GraphCodeBERT:
+Pre-training Code Representations with Data Flow. In 9th International
+Conference on Learning Representations, ICLR 2021, Virtual Event, Aus-
+tria, May 3-7, 2021. OpenReview.net. https://openreview.net/forum?
+id=jLoC4ez43PZ
+[20] Mossad Helali, Essam Mansour, Ibrahim Abdelaziz, Julian Dolby, and
+Kavitha Srinivas. 2022. A Scalable AutoML Approach Based on Graph
+Neural Networks. Proc. VLDB Endow. 15, 11 (sep 2022), 2428–2436.
+https://doi.org/10.14778/3551793.3551804
+[21] Chun-Hung Hsiao, Michael J. Cafarella, and Satish Narayanasamy.
+2014. Reducing MapReduce Abstraction Costs for Text-centric Appli-
+cations. In 43rd International Conference on Parallel Processing, ICPP
+2014, Minneapolis, MN, USA, September 9-12, 2014. 40–49.
+https:
+//doi.org/10.1109/ICPP.2014.13
+[22] Aditya Kanade, Petros Maniatis, Gogul Balakrishnan, and Kensen Shi.
+2020. Learning and evaluating contextual embedding of source code.
+In Proceedings of the 37th International Conference on Machine Learning,
+ICML 2020, 12-18 July 2020 (Proceedings of Machine Learning Research).
+PMLR.
+[23] Michael Katz, Parikshit Ram, Shirin Sohrabi, and Octavian Udrea.
+2020. Exploring Context-Free Languages via Planning: The Case
+for Automating Machine Learning. In Proceedings of the International
+Conference on Automated Planning and Scheduling (ICAPS). 403–411.
+https://ojs.aaai.org//index.php/ICAPS/article/view/6686
+[24] Trang T Le, Weixuan Fu, and Jason H Moore. 2020. Scaling tree-based
+automated machine learning to biomedical big data with a feature set
+selector. Bioinformatics 36, 1 (2020), 250–256. https://doi.org/10.1093/
+bioinformatics/btz470
+[25] Jian Li, Yue Wang, Michael R Lyu, and Irwin King. 2017. Code com-
+pletion with neural attention and pointer networks. arXiv preprint
+arXiv:1711.09573 (2017).
+[26] Shuai Lu, Daya Guo, Shuo Ren, Junjie Huang, Alexey Svyatkovskiy,
+Ambrosio Blanco, Colin Clement, Dawn Drain, Daxin Jiang, Duyu
+Tang, et al. 2021.
+CodeXGLUE: A Machine Learning Benchmark
+Dataset for Code Understanding and Generation.
+arXiv preprint
+13
+
+, ,
+Wenting Zhao, Ibrahim Abdelaziz, Julian Dolby, Kavitha Srinivas, Mossad Helali, and Essam Mansour
+arXiv:2102.04664 (2021).
+[27] Chris Maddison and Daniel Tarlow. 2014. Structured generative models
+of natural source code. In International Conference on Machine Learning.
+PMLR, 649–657.
+[28] R. Monat, A. Ouadjaout, and A. Miné. 2020. Static type analysis by
+abstract interpretation of Python programs. In Proc. of the 34th Euro-
+pean Conference on Object-Oriented Programming (ECOOP’20) (virtual
+conference) (Leibniz International Proceedings in Informatics (LIPIcs),
+Vol. 166). Dagstuhl Publishing, 17:1–17:29.
+https://doi.org/10.4230/
+LIPIcs.ECOOP.2020.17
+http://www-apr.lip6.fr/~mine/publi/article-
+monat-al-ecoop20.pdf.
+[29] Anh Tuan Nguyen and Tien N. Nguyen. 2015. Graph-based Statistical
+Language Model for Code. In Proceedings of the 37th International
+Conference on Software Engineering - Volume 1 (Florence, Italy) (ICSE
+’15). IEEE Press, Piscataway, NJ, USA, 858–868.
+http://dl.acm.org/
+citation.cfm?id=2818754.2818858
+[30] Tung Thanh Nguyen, Hoan Anh Nguyen, Nam H. Pham, Jafar M. Al-
+Kofahi, and Tien N. Nguyen. 2009. Graph-based Mining of Multiple
+Object Usage Patterns. In Proceedings of the the 7th Joint Meeting of
+the European Software Engineering Conference and the ACM SIGSOFT
+Symposium on The Foundations of Software Engineering (Amsterdam,
+The Netherlands) (ESEC/FSE ’09). ACM, New York, NY, USA, 383–392.
+https://doi.org/10.1145/1595696.1595767
+[31] Michael Pradel, Georgios Gousios, Jason Liu, and Satish Chandra.
+2020. Typewriter: Neural type prediction with search-based validation.
+In Proceedings of the 28th ACM Joint Meeting on European Software
+Engineering Conference and Symposium on the Foundations of Software
+Engineering. 209–220.
+[32] Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei,
+Ilya Sutskever, et al. 2019. Language models are unsupervised multitask
+learners. OpenAI blog 1, 8 (2019), 9.
+[33] Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan
+Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J Liu. 2019.
+Exploring the limits of transfer learning with a unified text-to-text
+transformer. arXiv preprint arXiv:1910.10683 (2019).
+[34] Herilalaina Rakotoarison, Marc Schoenauer, and Michèle Sebag. 2019.
+Automated Machine Learning with Monte-Carlo Tree Search. In Pro-
+ceedings of the International Joint Conference on Artificial Intelligence
+(IJCAI). 3296–3303. https://doi.org/10.24963/ijcai.2019/457
+[35] Veselin Raychev, Pavol Bielik, and Martin Vechev. 2016. Probabilistic
+Model for Code with Decision Trees. SIGPLAN Not. 51, 10 (oct 2016),
+731–747. https://doi.org/10.1145/3022671.2984041
+[36] Matthias Reif, Faisal Shafait, and Andreas Dengel. 2012. Meta-learning
+for evolutionary parameter optimization of classifiers. Machine Learn-
+ing 87, 3 (2012), 357–380. https://doi.org/10.1007/s10994-012-5286-7
+[37] Alexey Svyatkovskiy, Ying Zhao, Shengyu Fu, and Neel Sundaresan.
+2019. Pythia: AI-assisted code completion system. In Proceedings of the
+25th ACM SIGKDD International Conference on Knowledge Discovery &
+Data Mining. 2727–2735.
+[38] Guido van Rossum, Jukka Lehtosalo, and Lukasz Langa. 2014. PEP 484
+– Type Hints. https://www.python.org/dev/peps/pep-0484/
+[39] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion
+Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. 2017. At-
+tention is all you need. In Advances in neural information processing
+systems. 5998–6008.
+[40] Chi Wang, Qingyun Wu, Markus Weimer, and Erkang Zhu.
+2021.
+FLAML: A Fast and Lightweight AutoML Library. In
+Proceedings of Machine Learning and Systems (MLSys), Vol. 3.
+434–447.
+https://proceedings.mlsys.org/paper/2021/file/
+92cc227532d17e56e07902b254dfad10-Paper.pdf
+[41] Yue Wang, Weishi Wang, Shafiq Joty, and Steven C. H. Hoi. 2021.
+CodeT5: Identifier-aware Unified Pre-trained Encoder-Decoder Mod-
+els for Code Understanding and Generation. In Proceedings of the
+2021 Conference on Empirical Methods in Natural Language Processing,
+EMNLP 2021.
+[42] Marcel Wever, Felix Mohr, and Eyke Hüllermeier. 2018. ML-Plan for
+Unlimited-Length Machine Learning Pipelines. In AutoML Workshop
+at the International Conference on Machine Learning (ICML). https:
+//ris.uni-paderborn.de/download/3852/3853/38.pdf
+[43] Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond,
+Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Rémi Louf,
+Morgan Funtowicz, Joe Davison, Sam Shleifer, Patrick von Platen,
+Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, Teven Le Scao, Syl-
+vain Gugger, Mariama Drame, Quentin Lhoest, and Alexander M. Rush.
+2020. Transformers: State-of-the-Art Natural Language Processing.
+In Proceedings of the 2020 Conference on Empirical Methods in Natural
+Language Processing: System Demonstrations. Association for Computa-
+tional Linguistics, Online, 38–45. https://www.aclweb.org/anthology/
+2020.emnlp-demos.6
+14
+

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+arXiv:2301.05549v1  [quant-ph]  12 Jan 2023
+1
+On the explainability of quantum neural networks
+based on variational quantum circuits
+Ammar Daskin
+Abstract—Ridge functions are used to describe and study the
+lower bound of the approximation done by the neural networks
+which can be written as a linear combination of activation
+functions. If the activation functions are also ridge functions,
+these networks are called explainable neural networks.
+In this paper, we first show that quantum neural networks
+which are based on variational quantum circuits can be written
+as a linear combination of ridge functions. Consequently, we
+show that the interpretability and explainability of such quantum
+neural networks can be directly considered and studied as an
+approximation with the linear combination of ridge functions.
+Index Terms—Quantum neural networks, explainability, inter-
+pretability, ridge functions
+I. INTRODUCTION
+Neural networks have applications almost in every field
+of the science ranging from health to banking. The ability
+to interpret the result of a model and explain the learning
+behavior may be deemed important especially in critical in-
+dustries such as medicine and health care[1–3]. Limitations
+of the approximation rates of the classical neural networks
+can be understood better by using linear combination of ridge
+functions as an approximation to neural networks.
+The power and the limitations of quantum neural networks
+are yet to be fully understood. In this paper, we show that
+quantum neural networks can be written as a sum of ridge
+functions. Therefore, the math and methodologies that are used
+to understand classical neural networks can be used to study
+quantum ones.
+A. Approximation with ridge functions
+For a random variable y, if we have the observations
+y1, . . . , yn at points x1, . . . , xn, a standard regression model
+can be described by yi = ˆyi + ri, where ˆyi defines the
+dependence of yi on xi When xi are univariate real values,
+the assumption is that the dependence is smooth. This leads
+the following estimation for the regression model[4]:
+E(y | x) = f(x),
+(1)
+where f is a smoothing function. In linear smoothing, ˆy =
+(ˆy1, . . . , ˆyn)T can be written in the form of matrix vector
+transformation: ˆy = Sy, where S is the smoother matrix
+that does not depend on y. When there are more than one
+predictors, the estimating regression surface is hard because
+of the curse of dimensionality (the data sparseness in high
+A.
+Daskin
+was
+with
+the
+Department
+of
+Computer
+Engineering,
+Istanbul
+Medeniyet
+University,
+Istanbul,
+Turkey
+e-mail:
+(see
+https://sites.google.com/view/adaskin).
+dimensions)[5]. The general approach is to use the one-
+dimensional smoother as the building block in an additive
+model [4]. Given predictors xijs for each yi outcome, i.e.
+{yi, xi1, . . . , xip}, the additive model can be described as:
+E(yi|xi1, . . . , xip) = α +
+p
+�
+j=1
+fj(xij) + ǫ.
+(2)
+where ǫ is the inherent error, α is a constant parameter, and fjs
+represent unspecified smooth functions. Fitting can be done
+by using the backfitting algorithm [5, 6] which is in matrix
+form equivalent to Gauss-Seidel method in numerical linear
+algebra[7].
+Generalizing this model leads to the projection pursuit
+model proposed in [5] where the regression surface is pre-
+dicted by a linear combination of ridge functions as in the
+following form:
+f(x) =
+K
+�
+k=1
+fk(wk · x).
+(3)
+Here, wks represent weight vectors (projection indices) and
+fks are ridge functions [8–11]: Any multivariate function
+fk : Rd → R. The vector wk is called the direction and fk
+gives a constant on certain hyper-planes whose normal vectors
+are parallel to this direction. Ridge functions are used in
+approximation theory, partial differential equations, and neural
+networks. For instance, a feed forward neural network can be
+defined as [8]:
+�
+k
+γkσ (wk · x + bk) ,
+(4)
+where bk, αk, and wk represents parameters that describe the
+neural network. σ is a univariate function (activation function).
+The degree of approximation by σ functions can be bounded
+by the degree of approximation by ridge functions (we refer
+the reader to Ref.[8] for the properties and other uses of the
+ridge functions). The lower bound of the approximation of the
+neural networks can be also studied through the relations of
+the activation functions with ridge functions (e.g., [12–14]).
+If σ is chosen as a ridge function these networks are recently
+called explainable neural networks [15]: e.g. an example
+architecture which has three important structures, a projection
+layer, sub-network, and a combination layer is described to
+learn the following:
+f(x) = µ +
+K
+�
+k=1
+γkfk(wk · x).
+(5)
+Here, µ and γks are shift and scaling parameters, respectively.
+In comparison to standard neural networks, the learning in
+
+2
+this model can be understood by the ”explainable” features:
+linear projections and uni-variate functions (in other words, the
+mechanisms used to learn the model can be clearly explained
+by studying the constitutes that are ridge functions.).
+II. EXPLAINABILITY OF QUANTUM NEURAL NETWORKS
+Quantum neural networks[16–19] are generally based on
+variational quantum circuits[20] and can be described by
+⟨x| W(θ) ˆOW(θ) |x⟩ ,
+(6)
+where ˆO represents the measurement operator, |x⟩ is the input
+vector formatted as a quantum state and W(θ) is a unitary
+matrix generated by the quantum gates with the angle values
+defined by θ. Here, by abuse of the notation, we can consider
+ˆO as a selector set on the parts of W(θ) |x⟩: e.g., to obtain
+the measurement output of the first qubit in |0⟩ state, in vector
+forms, we select the first half of the output and combine their
+squared absolute values. Then, the output quantum state of the
+quantum circuit applied to |x⟩ can be rewritten as:
+�
+i∈ ˆ
+O
+|
+�
+wi|x
+�
+|2 =
+�
+i∈ ˆ
+O
+fi(
+�
+wi|x
+�
+),
+(7)
+where ⟨wi| represents a row of the unitary matrix W(θ).
+In variational quantum circuits, generally any change of
+the vector element of θ may affect multiple row of W(θ).
+This can affect the studies that try to understand the quantum
+neural network model. Therefore, to make any fi independent
+from each other, we can use the linear combination of unitary
+matrices[21, 22]. By following Ref.[22], we can write the rows
+of any matrix W(θ) as the first rows of matrices and combine
+them on a block diagonal matrix:
+V =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+⟨w1|
+•
+...
+•
+
+
+
+
+
+
+N×N
+...
+
+
+
+
+
+
+⟨wN|
+•
+...
+•
+
+
+
+
+
+
+N×N
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+N 2×N 2
+(8)
+where N is the dimension of W(θ). Using the direct sum,
+we can write V =
+N
+�
+i
+Vi with Vi representing the unitary
+matrix for wi. Note that any N-dimensional vector can be
+formed with O(N) quantum gates as the leading row of a
+unitary matrix by using its Schmidt decomposition. Therefore,
+the construction V for a generic W(θ) requires at most O(N 2)
+gates (See Ref.[22] for complexity analysis of the circuit.).
+The equivalent quantum state to the output of W(θ) |x⟩
+can be generated as a part of the outcome of the following
+transformation:
+|ψ⟩ = V
+
+
+
+
+|x⟩
+...
+|x⟩
+
+
+
+
+N 2×1
+.
+(9)
+In a simplified form let |ψi⟩ represents the ith element of |ψ⟩
+with 0 ≥ i < N 2, we can define a new selector operator that
+selects every |ψi⟩, where i mod N = 0. That means we can
+still use the definition similar to Eq.(7):
+N 2−1
+�
+i,i mod N=0
+|
+�
+wi|x
+�
+|2
+(10)
+Note that by writing a quantum operator in this way, we simply
+make the weight vectors independent from each other. That
+means the impact of an angle-change in one quantum gate
+can be limited to affect only one vector. Therefore, this model
+is at least as much powerful as using a single unitary matrix
+where a change may affect multiple rows. In addition, studying
+the approximations in this model may be easier since we can
+easily see how many independent weight vectors are needed
+and how each weight vector affect the result and training.
+A. Explainability of networks that are represented by expo-
+nentials
+Ridge functions are also used to approximate the integral
+forms of functions[23]: One of the most commonly used one
+is given by the following Fourier form:
+Ψwk := eix·wk,
+(11)
+Here, the complex plane can be rewritten using the only real
+valued functions, therefore Ψwk can be considered as a ridge
+function for each wk. Therefore, quantum neural networks
+such as [24] can be also explained as an approximation through
+the linear combination of ridge functions.
+This can be also used to understand quantum circuits that
+are defined through the Ising type Hamiltonian[25] or machine
+learning tasks that are solved through the adiabatic quantum
+computation [26, 27]
+III. DISCUSSION AND CONCLUSION
+For a given set of weight vectors, in literature there are many
+works on the conditions to approximate a given function with
+unknown ridge functions. For instance[28], consider C(X) is
+the space of continuous functions on X, then for a given
+{w1, w2}, for an appropriate choice of some continuous
+functions h1 and h2, one can write g1(h1(x)) + g2(h2(x)) =
+C(X) if and only if the lengths of h1 − h2 paths in X are
+bounded by some positive integer. From the Kolmogorov-
+Arnold representation theorem [29], we know that any multi-
+variate continuous function can be approximated as a sum of
+univariate functions. By this theorem, it can be also explained
+how the hidden layers in the classical neural networks help
+in approximations[29–31]. However, a continuous function
+
+3
+may not be represented exactly by an approximation with
+ridge functions [10, 29]. Therefore, the approximation rates
+in quantum neural networks that reduces to Eq.(7) and (10)
+can be well understood. On the other hand, this indicates that
+the limitations of the approximations with the ridge functions
+may be also limitations for these types of quantum neural
+networks.
+In this brief paper, we show that quantum neural networks
+can be understood as a linear combination of ridge functions,
+which is used to understand the interpretability and explain-
+ability of the classical neural networks. In particular, Eq.(7)
+and Eq.(10) can be used to describe quantum neural network
+models. In addition, since it can be written in the form of a
+linear combination of ridge functions, the approximation errors
+and upper and lower bounds on the errors of quantum neural
+networks can be studied through this formulation.
+In quantum neural networks which can be reduced to
+Eq.(7) and (10), the approximation is done by using N ridge
+functions. Here, N is in general exponential in the number of
+qubits and can be considered a very large number. For general
+function approximations, using these equations may be helpful
+to make a decision on the size of N and required number of
+operations and qubits before any application. However, the
+problems solved by the neural networks are in general not
+easy to define by functions, therefore it is not easy to decide
+how many functions (and quantum operations and qubits) are
+required to make the model more trainable.
+REFERENCES
+[1] E. Tjoa and C. Guan, “A survey on explainable artificial
+intelligence (xai): Toward medical xai,” IEEE transac-
+tions on neural networks and learning systems, vol. 32,
+no. 11, pp. 4793–4813, 2020.
+[2] P. Linardatos, V. Papastefanopoulos, and S. Kotsiantis,
+“Explainable ai: A review of machine learning inter-
+pretability methods,” Entropy, vol. 23, no. 1, p. 18, 2020.
+[3] L. H. Gilpin, D. Bau, B. Z. Yuan, A. Bajwa, M. Specter,
+and L. Kagal, “Explaining explanations: An overview of
+interpretability of machine learning,” in 2018 IEEE 5th
+International Conference on data science and advanced
+analytics (DSAA), pp. 80–89, IEEE, 2018.
+[4] A. Buja, T. Hastie, and R. Tibshirani, “Linear smoothers
+and additive models,” The Annals of Statistics, vol. 17,
+no. 2, pp. 453–510, 1989.
+[5] J. H. Friedman, M. Jacobson, and W. Stuetzle, “PROJEC-
+TION PURSUIT REGRESSION,” J. Am. Statist. Assoc.,
+vol. 76, p. 817, 1981.
+[6] L. Breiman and J. H. Friedman, “Estimating optimal
+transformations for multiple regression and correlation,”
+Journal of the American Statistical Association, vol. 80,
+no. 391, pp. 580–598, 1985.
+[7] G. H. Golub and C. F. Van Loan, Matrix computations.
+JHU press, 2013.
+[8] A. Pinkus, Ridge Functions. Cambridge Tracts in Math-
+ematics, Cambridge University Press, 2015.
+[9] E. J. Candes, Ridgelets: theory and applications. Stan-
+ford University, 1998.
+[10] V. Maiorov, “On best approximation by ridge functions,”
+Journal of Approximation Theory, vol. 99, no. 1, pp. 68–
+94, 1999.
+[11] P. P. Petrushev, “Approximation by ridge functions and
+neural networks,” SIAM Journal on Mathematical Anal-
+ysis, vol. 30, no. 1, pp. 155–189, 1998.
+[12] V. Maiorov and A. Pinkus, “Lower bounds for ap-
+proximation by mlp neural networks,” Neurocomputing,
+vol. 25, no. 1-3, pp. 81–91, 1999.
+[13] V. E. Ismailov, “Approximation by neural networks with
+weights varying on a finite set of directions,” Journal of
+Mathematical Analysis and Applications, vol. 389, no. 1,
+pp. 72–83, 2012.
+[14] J. M. Klusowski and A. R. Barron, “Minimax lower
+bounds for ridge combinations including neural nets,”
+in 2017 IEEE International Symposium on Information
+Theory (ISIT), pp. 1376–1380, IEEE, 2017.
+[15] J. Vaughan, A. Sudjianto, E. Brahimi, J. Chen, and V. N.
+Nair, “Explainable neural networks based on additive
+index models,” stat, vol. 1050, p. 5, 2018.
+[16] M. Schuld, I. Sinayskiy, and F. Petruccione, “The quest
+for a quantum neural network,” Quantum Information
+Processing, vol. 13, no. 11, pp. 2567–2586, 2014.
+[17] Y. Kwak, W. J. Yun, S. Jung, and J. Kim, “Quantum
+neural networks: Concepts, applications, and challenges,”
+in 2021 Twelfth International Conference on Ubiquitous
+and Future Networks (ICUFN), pp. 413–416, IEEE,
+2021.
+[18] Z.-A. Jia, B. Yi, R. Zhai, Y.-C. Wu, G.-C. Guo, and
+G.-P. Guo, “Quantum neural network states: A brief
+review of methods and applications,” Advanced Quantum
+Technologies, vol. 2, no. 7-8, p. 1800077, 2019.
+[19] F. V. Massoli, L. Vadicamo, G. Amato, and F. Falchi,
+“A leap among quantum computing and quantum neural
+networks: A survey,” ACM Computing Surveys, vol. 55,
+no. 5, pp. 1–37, 2022.
+[20] A. Kandala, A. Mezzacapo, K. Temme, M. Takita,
+M. Brink, J. M. Chow, and J. M. Gambetta, “Hardware-
+efficient
+variational quantum eigensolver for
+small
+molecules and quantum magnets,” Nature, vol. 549,
+no. 7671, pp. 242–246, 2017.
+[21] A. M. Childs and N. Wiebe, “Hamiltonian simulation us-
+ing linear combinations of unitary operations,” Quantum
+Information & Computation, vol. 12, no. 11-12, pp. 901–
+924, 2012.
+[22] A. Daskin, A. Grama, G. Kollias, and S. Kais, “Universal
+programmable quantum circuit schemes to emulate an
+operator,” The Journal of chemical physics, vol. 137,
+no. 23, p. 234112, 2012.
+[23] A. Pinkus, Integral Representations, p. 141–151. Cam-
+bridge Tracts in Mathematics, Cambridge University
+Press, 2015.
+[24] A. Daskin, “A simple quantum neural net with a pe-
+riodic activation function,” in 2018 IEEE International
+Conference on Systems, Man, and Cybernetics (SMC),
+pp. 2887–2891, IEEE, 2018.
+[25] R. Xia, T. Bian, and S. Kais, “Electronic structure
+calculations and the ising hamiltonian,” The Journal of
+
+4
+Physical Chemistry B, vol. 122, no. 13, pp. 3384–3395,
+2017.
+[26] E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser,
+“Quantum computation by adiabatic evolution,” arXiv
+preprint quant-ph/0001106, 2000.
+[27] T. Albash and D. A. Lidar, “Adiabatic quantum com-
+putation,” Reviews of Modern Physics, vol. 90, no. 1,
+p. 015002, 2018.
+[28] V. E. Ismailov, “Representation of multivariate functions
+by sums of ridge functions,” Journal of mathematical
+analysis and applications, vol. 331, no. 1, pp. 184–190,
+2007.
+[29] J. Schmidt-Hieber, “The kolmogorov–arnold represen-
+tation theorem revisited,” Neural networks, vol. 137,
+pp. 119–126, 2021.
+[30] R. Hecht-Nielsen, “Kolmogorov’s mapping neural net-
+work existence theorem,” in Proceedings of the interna-
+tional conference on Neural Networks, vol. 3, pp. 11–14,
+IEEE Press New York, NY, USA, 1987.
+[31] V. K˚urkov´a, “Kolmogorov’s theorem and multilayer neu-
+ral networks,” Neural networks, vol. 5, no. 3, pp. 501–
+506, 1992.
+

FNE5T4oBgHgl3EQfVQ8Q/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf,len=321
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='05549v1  [quant-ph]  12 Jan 2023  1  On the explainability of quantum neural networks  based on variational quantum circuits  Ammar Daskin  Abstract—Ridge functions are used to describe and study the  lower bound of the approximation done by the neural networks  which can be written as a linear combination of activation  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' If the activation functions are also ridge functions,  these networks are called explainable neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  In this paper, we first show that quantum neural networks  which are based on variational quantum circuits can be written  as a linear combination of ridge functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Consequently, we  show that the interpretability and explainability of such quantum  neural networks can be directly considered and studied as an  approximation with the linear combination of ridge functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  Index Terms—Quantum neural networks, explainability, inter-  pretability, ridge functions  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' INTRODUCTION  Neural networks have applications almost in every field  of the science ranging from health to banking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' The ability  to interpret the result of a model and explain the learning  behavior may be deemed important especially in critical in-  dustries such as medicine and health care[1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Limitations  of the approximation rates of the classical neural networks  can be understood better by using linear combination of ridge  functions as an approximation to neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  The power and the limitations of quantum neural networks  are yet to be fully understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' In this paper, we show that  quantum neural networks can be written as a sum of ridge  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Therefore, the math and methodologies that are used  to understand classical neural networks can be used to study  quantum ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Approximation with ridge functions  For a random variable y, if we have the observations  y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' , yn at points x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' , xn, a standard regression model  can be described by yi = ˆyi + ri, where ˆyi defines the  dependence of yi on xi When xi are univariate real values,  the assumption is that the dependence is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' This leads  the following estimation for the regression model[4]:  E(y | x) = f(x),  (1)  where f is a smoothing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' In linear smoothing, ˆy =  (ˆy1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' , ˆyn)T can be written in the form of matrix vector  transformation: ˆy = Sy, where S is the smoother matrix  that does not depend on y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' When there are more than one  predictors, the estimating regression surface is hard because  of the curse of dimensionality (the data sparseness in high  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  Daskin  was  with  the  Department  of  Computer  Engineering,  Istanbul  Medeniyet  University,  Istanbul,  Turkey  e-mail:  (see  https://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='com/view/adaskin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  dimensions)[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' The general approach is to use the one-  dimensional smoother as the building block in an additive  model [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Given predictors xijs for each yi outcome, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  {yi, xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' , xip}, the additive model can be described as:  E(yi|xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' , xip) = α +  p  �  j=1  fj(xij) + ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  (2)  where ǫ is the inherent error, α is a constant parameter, and fjs  represent unspecified smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Fitting can be done  by using the backfitting algorithm [5, 6] which is in matrix  form equivalent to Gauss-Seidel method in numerical linear  algebra[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  Generalizing this model leads to the projection pursuit  model proposed in [5] where the regression surface is pre-  dicted by a linear combination of ridge functions as in the  following form:  f(x) =  K  �  k=1  fk(wk · x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  (3)  Here, wks represent weight vectors (projection indices) and  fks are ridge functions [8–11]: Any multivariate function  fk : Rd → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' The vector wk is called the direction and fk  gives a constant on certain hyper-planes whose normal vectors  are parallel to this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Ridge functions are used in  approximation theory, partial differential equations, and neural  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' For instance, a feed forward neural network can be  defined as [8]:  �  k  γkσ (wk · x + bk) ,  (4)  where bk, αk, and wk represents parameters that describe the  neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' σ is a univariate function (activation function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  The degree of approximation by σ functions can be bounded  by the degree of approximation by ridge functions (we refer  the reader to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' [8] for the properties and other uses of the  ridge functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' The lower bound of the approximation of the  neural networks can be also studied through the relations of  the activation functions with ridge functions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=', [12–14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  If σ is chosen as a ridge function these networks are recently  called explainable neural networks [15]: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' an example  architecture which has three important structures, a projection  layer, sub-network, and a combination layer is described to  learn the following:  f(x) = µ +  K  �  k=1  γkfk(wk · x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  (5)  Here, µ and γks are shift and scaling parameters, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  In comparison to standard neural networks, the learning in  2  this model can be understood by the ”explainable” features:  linear projections and uni-variate functions (in other words, the  mechanisms used to learn the model can be clearly explained  by studying the constitutes that are ridge functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' EXPLAINABILITY OF QUANTUM NEURAL NETWORKS  Quantum neural networks[16–19] are generally based on  variational quantum circuits[20] and can be described by  ⟨x| W(θ) ˆOW(θ) |x⟩ ,  (6)  where ˆO represents the measurement operator, |x⟩ is the input  vector formatted as a quantum state and W(θ) is a unitary  matrix generated by the quantum gates with the angle values  defined by θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Here, by abuse of the notation, we can consider  ˆO as a selector set on the parts of W(θ) |x⟩: e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=', to obtain  the measurement output of the first qubit in |0⟩ state, in vector  forms, we select the first half of the output and combine their  squared absolute values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Then, the output quantum state of the  quantum circuit applied to |x⟩ can be rewritten as:  �  i∈ ˆ  O  |  �  wi|x  �  |2 =  �  i∈ ˆ  O  fi(  �  wi|x  �  ),  (7)  where ⟨wi| represents a row of the unitary matrix W(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  In variational quantum circuits, generally any change of  the vector element of θ may affect multiple row of W(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  This can affect the studies that try to understand the quantum  neural network model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Therefore, to make any fi independent  from each other, we can use the linear combination of unitary  matrices[21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' By following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' [22], we can write the rows  of any matrix W(θ) as the first rows of matrices and combine  them on a block diagonal matrix:  V =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  ⟨w1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  N×N  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  ⟨wN| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  N×N  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  N 2×N 2  (8)  where N is the dimension of W(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Using the direct sum,  we can write V =  N  �  i  Vi with Vi representing the unitary  matrix for wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Note that any N-dimensional vector can be  formed with O(N) quantum gates as the leading row of a  unitary matrix by using its Schmidt decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Therefore,  the construction V for a generic W(θ) requires at most O(N 2)  gates (See Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' [22] for complexity analysis of the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  The equivalent quantum state to the output of W(θ) |x⟩  can be generated as a part of the outcome of the following  transformation:  |ψ⟩ = V  \uf8eb  \uf8ec  \uf8ec  \uf8ed  |x⟩  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  |x⟩  \uf8f6  \uf8f7  \uf8f7  \uf8f8  N 2×1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  (9)  In a simplified form let |ψi⟩ represents the ith element of |ψ⟩  with 0 ≥ i < N 2, we can define a new selector operator that  selects every |ψi⟩, where i mod N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' That means we can  still use the definition similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' (7):  N 2−1  �  i,i mod N=0  |  �  wi|x  �  |2  (10)  Note that by writing a quantum operator in this way, we simply  make the weight vectors independent from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' That  means the impact of an angle-change in one quantum gate  can be limited to affect only one vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Therefore, this model  is at least as much powerful as using a single unitary matrix  where a change may affect multiple rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' In addition, studying  the approximations in this model may be easier since we can  easily see how many independent weight vectors are needed  and how each weight vector affect the result and training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Explainability of networks that are represented by expo-  nentials  Ridge functions are also used to approximate the integral  forms of functions[23]: One of the most commonly used one  is given by the following Fourier form:  Ψwk := eix·wk,  (11)  Here, the complex plane can be rewritten using the only real  valued functions, therefore Ψwk can be considered as a ridge  function for each wk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Therefore, quantum neural networks  such as [24] can be also explained as an approximation through  the linear combination of ridge functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  This can be also used to understand quantum circuits that  are defined through the Ising type Hamiltonian[25] or machine  learning tasks that are solved through the adiabatic quantum  computation [26, 27]  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' DISCUSSION AND CONCLUSION  For a given set of weight vectors, in literature there are many  works on the conditions to approximate a given function with  unknown ridge functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' For instance[28], consider C(X) is  the space of continuous functions on X, then for a given  {w1, w2}, for an appropriate choice of some continuous  functions h1 and h2, one can write g1(h1(x)) + g2(h2(x)) =  C(X) if and only if the lengths of h1 − h2 paths in X are  bounded by some positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' From the Kolmogorov-  Arnold representation theorem [29], we know that any multi-  variate continuous function can be approximated as a sum of  univariate functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' By this theorem, it can be also explained  how the hidden layers in the classical neural networks help  in approximations[29–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' However, a continuous function  3  may not be represented exactly by an approximation with  ridge functions [10, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Therefore, the approximation rates  in quantum neural networks that reduces to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' (7) and (10)  can be well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' On the other hand, this indicates that  the limitations of the approximations with the ridge functions  may be also limitations for these types of quantum neural  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  In this brief paper, we show that quantum neural networks  can be understood as a linear combination of ridge functions,  which is used to understand the interpretability and explain-  ability of the classical neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' In particular, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' (7)  and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' (10) can be used to describe quantum neural network  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' In addition, since it can be written in the form of a  linear combination of ridge functions, the approximation errors  and upper and lower bounds on the errors of quantum neural  networks can be studied through this formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  In quantum neural networks which can be reduced to  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' (7) and (10), the approximation is done by using N ridge  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Here, N is in general exponential in the number of  qubits and can be considered a very large number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' For general  function approximations, using these equations may be helpful  to make a decision on the size of N and required number of  operations and qubits before any application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' However, the  problems solved by the neural networks are in general not  easy to define by functions, therefore it is not easy to decide  how many functions (and quantum operations and qubits) are  required to make the model more trainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  REFERENCES  [1] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Tjoa and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Guan, “A survey on explainable artificial  intelligence (xai): Toward medical xai,” IEEE transac-  tions on neural networks and learning systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 32,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 4793–4813, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [2] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Linardatos, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Papastefanopoulos, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Kotsiantis,  “Explainable ai: A review of machine learning inter-  pretability methods,” Entropy, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 23, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 18, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [3] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Gilpin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Bau, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Yuan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Bajwa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Specter,  and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Kagal, “Explaining explanations: An overview of  interpretability of machine learning,” in 2018 IEEE 5th  International Conference on data science and advanced  analytics (DSAA), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 80–89, IEEE, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Buja, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Hastie, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Tibshirani, “Linear smoothers  and additive models,” The Annals of Statistics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 17,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 453–510, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Friedman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Jacobson, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Stuetzle, “PROJEC-  TION PURSUIT REGRESSION,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=',  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 76, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 817, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [6] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Breiman and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Friedman, “Estimating optimal  transformations for multiple regression and correlation,”  Journal of the American Statistical Association, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 80,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 391, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 580–598, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [7] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Golub and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Van Loan, Matrix computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  JHU press, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Pinkus, Ridge Functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Cambridge Tracts in Math-  ematics, Cambridge University Press, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [9] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Candes, Ridgelets: theory and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Stan-  ford University, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [10] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Maiorov, “On best approximation by ridge functions,”  Journal of Approximation Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 99, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 68–  94, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [11] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Petrushev, “Approximation by ridge functions and  neural networks,” SIAM Journal on Mathematical Anal-  ysis, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 30, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 155–189, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [12] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Maiorov and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Pinkus, “Lower bounds for ap-  proximation by mlp neural networks,” Neurocomputing,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 25, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 1-3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 81–91, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [13] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Ismailov, “Approximation by neural networks with  weights varying on a finite set of directions,” Journal of  Mathematical Analysis and Applications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 389, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 1,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 72–83, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [14] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Klusowski and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Barron, “Minimax lower  bounds for ridge combinations including neural nets,”  in 2017 IEEE International Symposium on Information  Theory (ISIT), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 1376–1380, IEEE, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [15] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Vaughan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Sudjianto, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Brahimi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Chen, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  Nair, “Explainable neural networks based on additive  index models,” stat, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 1050, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 5, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [16] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Schuld, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Sinayskiy, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Petruccione, “The quest  for a quantum neural network,” Quantum Information  Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 13, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 2567–2586, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [17] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Kwak, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Yun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Jung, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Kim, “Quantum  neural networks: Concepts, applications, and challenges,”  in 2021 Twelfth International Conference on Ubiquitous  and Future Networks (ICUFN), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 413–416, IEEE,  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [18] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Jia, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Yi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Zhai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Wu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Guo, and  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Guo, “Quantum neural network states: A brief  review of methods and applications,” Advanced Quantum  Technologies, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 2, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 7-8, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 1800077, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [19] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Massoli, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Vadicamo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Amato, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Falchi,  “A leap among quantum computing and quantum neural  networks: A survey,” ACM Computing Surveys, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 55,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 1–37, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [20] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Kandala, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Mezzacapo, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Temme, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Takita,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Brink, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Chow, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Gambetta, “Hardware-  efficient  variational quantum eigensolver for  small  molecules and quantum magnets,” Nature, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 549,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 7671, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 242–246, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Childs and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Wiebe, “Hamiltonian simulation us-  ing linear combinations of unitary operations,” Quantum  Information & Computation, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 12, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 11-12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 901–  924, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Daskin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Grama, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Kollias, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Kais, “Universal  programmable quantum circuit schemes to emulate an  operator,” The Journal of chemical physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 137,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 23, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 234112, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [23] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Pinkus, Integral Representations, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 141–151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Cam-  bridge Tracts in Mathematics, Cambridge University  Press, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [24] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Daskin, “A simple quantum neural net with a pe-  riodic activation function,” in 2018 IEEE International  Conference on Systems, Man, and Cybernetics (SMC),  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 2887–2891, IEEE, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [25] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Xia, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Bian, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Kais, “Electronic structure  calculations and the ising hamiltonian,” The Journal of  4  Physical Chemistry B, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 122, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 13, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 3384–3395,  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [26] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Farhi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Goldstone, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Gutmann, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Sipser,  “Quantum computation by adiabatic evolution,” arXiv  preprint quant-ph/0001106, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [27] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Albash and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Lidar, “Adiabatic quantum com-  putation,” Reviews of Modern Physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 90, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 1,  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 015002, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [28] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Ismailov, “Representation of multivariate functions  by sums of ridge functions,” Journal of mathematical  analysis and applications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 331, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 184–190,  2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [29] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Schmidt-Hieber, “The kolmogorov–arnold represen-  tation theorem revisited,” Neural networks, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 137,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 119–126, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [30] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' Hecht-Nielsen, “Kolmogorov’s mapping neural net-  work existence theorem,” in Proceedings of the interna-  tional conference on Neural Networks, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 11–14,  IEEE Press New York, NY, USA, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content='  [31] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' K˚urkov´a, “Kolmogorov’s theorem and multilayer neu-  ral networks,” Neural networks, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 5, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}
+page_content=' 501–  506, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE5T4oBgHgl3EQfVQ8Q/content/2301.05549v1.pdf'}

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+arXiv:2301.02899v1  [math.AG]  7 Jan 2023
+Burnside rings and volume forms with logarithmic
+poles
+Antoine Chambert-Loir
+Université Paris Cité, Institut de Mathématiques de Jussieu-Paris Rive Gauche, F-75013,
+Paris, France
+E-mail: antoine.chambert-loir@u-paris.fr
+Maxim Kontsevich
+Institut des Hautes Études Scientifiques, 35 route de Chartres, 91440 Bures-sur-Yvette,
+France
+E-mail: maxim@ihes.fr
+Yuri Tschinkel
+Courant Institute, NYU, 251 Mercer St. New York, NY 10012, USA
+Simons Foundation, 160 5th Av., New York, NY 10010, USA
+E-mail: tschinkel@cims.nyu.edu
+Abstract. — We develop a theory of Burnside rings in the context of birational equivalences
+of algebraic varieties equipped with logarithmic volume forms.
+We introduce a residue
+homomorphism and construct an additive invariant of birational morphisms. We also define
+a specialization homomorphism.
+Résumé. —
+Nous proposons une théorie d’anneaux de Burnside dans le contexte de la
+géométrie birationnelle des variétés algébriques munies d’une forme volume à pôles logarith-
+miques. Nous introduisons un homomorphisme « résidu », construisons un invariant additif
+des morphismes birationnels. Nous définissons aussi un homomorphisme de spécialisation.
+2000 Mathematics Subject Classification. — 14E08, 14E07, 14D06.
+Key words and phrases. — Birational geometry, Burnside rings, logarithmic volume
+forms, specialization.
+Contents
+1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+2
+2. Logarithmic differential forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+3. Burnside rings for logarithmic forms. .. . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+4. Residues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+9
+5. A complex of Burnside rings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
+6. Algebraic structure of Burn(k) after localization at 2. . . . . . . . . . . 17
+7. Birational morphisms preserving volume forms. . . . . . . . . . . . . . . . . . 20
+8. Specialization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
+References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
+
+2
+ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL
+1. Introduction
+The study of birationality of algebraic varieties is a classical and well studied
+subject, with many open problems. In some cases, it is interesting to study birational
+maps preserving additional structure, for example group actions, symplectic forms,
+or volume forms. Such a study is already implicit in many questions of birational
+geometry, eg, in the notion of crepant resolution of singularities.
+In this paper, we consider the case of varieties endowed with volume forms with
+logarithmic poles and develop a formalism of Burnside rings along the lines of their
+counterpart introduced by Kontsevich & Tschinkel (2019) to establish the spe-
+cialization of rationality, and its equivariant version by Kresch & Tschinkel
+(2022b).
+Let k be a field of characteristic zero. For each integer n, we define
+Burnn(k)
+as the free abelian group on birational equivalence classses of pairs (X, ω) consisting
+of an integral smooth proper k-variety X of dimension n equipped with an n-form ω
+with at most logarithmic poles.
+The graded abelian group
+Burn(k) =
+�
+n∈N
+Burnn(k)
+carries a ring structure, induced by taking products of varieties, decomposed into ir-
+reducible components, and equipped with the external product of the volume forms.
+In section 4, we define morphisms of abelian groups
+∂ : Burnn(k) → Burnn−1(k).
+When X is smooth and the polar divisor of ω has strict normal crossings, the image
+of the class [X, ω] is given by the following formula. Let (Dα)α∈A be the family
+of irreducible components of the polar divisor of ω. For each subset A of A , the
+intersection DA = �
+α∈A Dα is a union of integral smooth varieties of codimension |A|;
+taking iterated residues, we may equip it with a volume form with logarithmic
+poles ωA. Then
+∂([X, ω]) =
+�
+∅̸=A⊆A
+(−1)|A|−1[DA, ωA] · T|A|−1,
+where
+T = [P1, dt/t].
+In particular, the existence of the map ∂ relies on the birational invariance of this
+expression, see theorem 4.7.
+This construction is reminiscent of the boundary map in polar homology
+(Khesin & Rosly, 2003; Khesin et al, 2004; Gorchinskiy & Rosly, 2015).
+However, apart from the obvious difference that we only record birational types of
+strata, rather than the strata themselves, our formula takes into account strata of
+all codimensions, rather than those of codimension one.
+
+BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES
+3
+The map ∂ is additive. Furthermore, we prove in theorem 4.10 that
+∂(a · b) = εn · ∂(a) · b + a · ∂(b) − T · ∂(a) · ∂(b),
+when a ∈ Burnm(k) and b ∈ Burnn(k). Here ε is the class of the point Spec(k)
+equipped with the volume form equal to −1.
+In theorem 5.1, we show that
+∂ ◦ ∂ = 0.
+These formulas may look complicated. However, as we explain in §6, they simplify
+significantly after inverting 2.
+Inspired by the constructions of Lin et al (2020); Lin & Shinder (2022);
+Kresch & Tschinkel (2022a), we define in §7 a homomorphism
+c: Bir(X, ω) → Burnn−1(k),
+from the group of birational automorphisms of the pair (X, ω), where X is an n-
+dimensional integral proper smooth variety equipped with a logarithmic volume
+form ω. As in the above references, our map c is defined at the groupoid level of
+birational maps preserving logarithmic volume forms.
+When the birational isomorphism ϕ: (X, ω) ��� (Y, η) is described by a diagram
+W
+X
+Y
+←
+→
+p
+←
+→
+q
+←
+→
+ϕ
+of smooth proper integral k-varieties, with birational morphisms p and q, the two
+logarithmic volume forms p∗ω and q∗η on W are equal, and the element c(ϕ) ∈
+Burnn−1(k) is given by
+c(ϕ) =
+�
+E∈Exc(q)
+[E, p∗ωE] −
+�
+D∈Exc(p)
+[D, q∗ηD]
+where Exc(p) is the set of irreducible components of the exceptional divisor of p,
+and where, for each such component D, the logarithmic volume form p∗ωD on D is
+obtained by taking the residue of p∗ω along D (and similarly for q).
+Finally, consider a discrete valuation ring with residue field k and field of frac-
+tions K and let t be a uniformizing element. In this context, we define a specialization
+map
+ρt : Burnn(K) → Burnn(k).
+The image of the class [X, ω] involves the combinatorics of a good model (X , ω)
+over the valuation ring, and a certain subcomplex of the Clemens complex of the
+special fiber. In the particular case where X is smooth, the polar divisor of ω is
+a relative divisor with normal crossings, and denoting by ωo the restriction of ω to
+the special fiber Xo, one has
+ρt([X, ω]) = [Xo, ωo].
+Note that the existence of such a specialization map implies, as in theorem 1
+of Kontsevich & Tschinkel (2019), or as in (Nicaise & Shinder, 2019), that
+
+4
+ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL
+birational equivalence of varieties with logarithmic volume forms is preserved under
+“good specializations”.
+In the geometric case, where the valuation is the local ring of a curve C at point o,
+the construction of the specialization map can be viewed as a restriction to the
+special fiber of a normalization of a global residue map ∂ that takes place on a proper
+model whose special fiber is a divisor with normal crossings. The normalization
+procedure extracts a subcomplex of the Clemens complex of the special fiber. A
+similar situation appeared in the study of Tamagawa measures on analytic manifolds
+(Chambert-Loir & Tschinkel, 2010).
+Related constructions emerged from the seminal work of Kontsevich & Soibelman
+(2006) inspired by mirror symmetry, and its subsequent developments, eg, by
+Mustaţă & Nicaise (2015); Nicaise & Xu (2016); Boucksom & Jonsson
+(2017); Jonsson & Nicaise (2020).
+Our constructions use essentially only formal properties of the residue maps. Con-
+sequently, one can envision analogous theories for logarithmic forms of smaller de-
+gree, Milnor K-theory, or even for the cycle modules of Rost (1996).
+Acknowledgments. — The third author was partially supported by NSF grant
+2000099.
+2. Logarithmic differential forms
+2.1. Kähler differentials. — Let k be a field of characteristic zero and let K be
+a finitely generated extension of k; let n be its transcendence degree. The space of
+Kähler differentials ΩK/k is the K-vector space generated by symbols da, for a ∈ K,
+subject to the relations:
+(1) For a ∈ k, one has da = 0;
+(2) For a, b ∈ K, one has d(a + b) = da + db and d(ab) = adb + b(da).
+For any integer m ⩾ 0, we may consider its mth exterior power Ωm
+K/k, which is
+a K-vector space of dimension
+�n
+m
+�
+; in particular, it vanishes if m > n, Ω1
+K/k has
+dimension n, and Ωn
+K/k has dimension one. One has Ω0
+K/k = k, canonically.
+Elements of Ωn
+K/k, for n = tr. degk(K), are also called volume forms.
+For a ∈ K×, we also write dlog a = da/a ∈ ΩK/k.
+2.2. Models. — Let m be an integer and let ω ∈ Ωm
+K/k.
+A model of K is an
+integral k-scheme X together with a k-isomorphism K ≃ k(X); we say that this
+model is proper, resp. smooth if X is proper, resp. smooth over k. Given such a
+model, ω induces a meromorphic global section ωX of Ωm
+X/k. The polar ideal of ωX
+is the subsheaf of OX whose local sections are the a ∈ OX such that aωX is induced
+by a regular m-form. Let D be the zero-locus of this ideal. Its complement U is
+the largest open subscheme of X such that ωX is induced by a regular m-form on U.
+If X is smooth, then ωX is locally free, hence the scheme D is an effective divisor
+(Hartogs’s principle); we call it the polar divisor of ω on X.
+
+BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES
+5
+2.3. Logarithmic forms. — By Hironaka’s theorem on embedded resolution of
+singularities, there exist smooth projective models (X, ωX) of (K, ω) such that the
+polar divisor D of ωX has normal crossings.
+Following (Deligne, 1970, chap. 2, §3), we then say that ωX has at most loga-
+rithmic poles, or that ω has at most logarithmic poles on X, if both ωX and dωX
+have at most simple poles along D.
+The following lemma implies that this condition is essentially independent of the
+choice of X such that the polar divisor of ωX has normal crossings.
+Lemma 2.4. — Let g : X′ → X be a morphism of smooth k-varieties, let D be a
+divisor with normal crossings in X and let D′ be a divisor with normal crossings
+in X′ such that D′ = g−1(D). Let ω be a regular m-form on X
+D and let ω′ = g∗ω.
+(1) If ω has at most logarithmic poles along D, then ω′ has at most logarithmic
+poles along D′.
+(2) The converse holds if g is proper and surjective.
+Proof. — The first assertion is (Deligne, 1970, chap. II, prop. 3.2, (iv)). Let us
+prove the second one.
+Consider the generic point η of X and a point η′ ∈ X′
+D′ which is algebraic
+over k(η). The Zariski closure X′
+1 of η′ is proper and generically finite over X, and
+D′
+1 = D′ ∩ X′
+1 is a divisor. There is a proper modification h: X′
+2 → X′
+1 such that
+D′
+2 = h−1(D′
+1) has normal crossings. By the first part, the form h∗ω′|X′
+1 has at most
+logarithmic poles along D′
+2. Replacing g by g◦h, we may assume that g is generically
+finite.
+Since the sheaf of forms with at most logarithmic poles along D is locally free and
+X is smooth, we can delete from X a subset of codimension at least 2. Thus, we
+may assume that g is flat, D is smooth and irreducible, and g is étale outside of D.
+It suffices to argue étale locally at the generic point of D. By the local description
+of ramified morphisms, there are étale local coordinates (z1, . . . , zn) on X such that
+Dred = V(z1), local coordinates (z′
+1, . . . , z′
+n) on X′ such that g∗z1 = (z′
+1)e, g∗z2 = z′
+2,
+etc., where e is the ramification index of g along D. Let d be the order of the pole
+of ω along D; write ω = α/zd
+1 + β ∧ dz1/zd
+1, where α, β are regular forms which do
+not involve dz1. Then
+ω′ = g∗ω = g∗α/(z′
+1)de + e g∗β ∧ dz′
+1/(z′
+1)1+(d−1)e.
+Assume, by contradiction, that d ⩾ 2, so that de ⩾ 2 and 1 + (d − 1)e ⩾ 2. Since ω′
+has at most logarithmic poles along D, we get g∗α = 0 and g∗β = 0. This implies
+that both α and β are multiples of z1, contradicting the hypothesis that d was the
+order of the pole of ω along D. Therefore, d ⩽ 1. This concludes the proof.
+2.5. — We say that an m-form ω ∈ Ωm
+K/k is logarithmic if for all proper smooth
+models X of K such that the polar divisor of ωX has normal crossings, the meromor-
+phic differential form ωX has at most logarithmic poles. By resolution of singularities
+
+6
+ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL
+(Hironaka, 1964), two models are dominated by a third one, hence lemma 2.4, im-
+plies that it suffices that this condition is satisfied on some proper smooth model
+for which the polar divisor of ωX has normal crossings.
+Analogously, if X is a reduced k-variety, then we say that a meromorphic m-form ω
+on X is logarithmic “everywhere” if for all proper birational models (X′, ω′) of (X, ω),
+the meromorphic m-form ω′ on X′ has at most logarithmic poles. It suffices that
+this holds on one such model.
+3. Burnside rings for logarithmic forms
+3.1. Burnside rings. — Let k be a field of characteristic zero and n an integer
+such that n ⩾ 0. Kontsevich & Tschinkel (2019) defined the Burnside group
+Burnn(k) as the free abelian group on isomorphism classes of finitely generated
+extensions of k of transcendence degree n.
+Any integral k-variety X of dimension n has a class [X] in Burnn(k). This gives rise
+to alternative useful presentations of Burnn(k), for example involving only classes
+of integral projective smooth varieties.
+The group
+Burn(k) =
+�
+n≥0
+Burnn(k)
+carries a natural commutative ring structure, with multiplication defined by taking
+products of (smooth projective) k-varieties:
+[X] · [X′] = [X × X′].
+3.2. Definition of a Burnside group for volume forms. — Let k be a field
+of characteristic zero and let n be an integer ⩾ 0. We define Burnn(k) to be the
+free abelian group on isomorphisms classes of pairs (K, ω), where
+– K is a finitely generated extension of k of transcendence degree n and
+– ω ∈ Ωn
+K/k is a logarithmic volume form.
+We write
+[K, ω] ∈ Burnn(k)
+for the class of a pair (K, ω).
+Remark 3.3. — This definition has obvious more geometric formulations.
+For
+example, we can take for generators equivalence classes of pairs (X, ω), where
+– X is a smooth integral k-scheme of dimension n, and
+– ω a regular volume form on X which is logarithmic “everywhere”,
+modulo the smallest equivalence relation that identifies (X, ω) and (X′, ω′) if there
+exists an open immersion f : X′ → X such that ω′ = f ∗ω.
+Alternatively, we can assume that X is proper, smooth and integral, the form ω is
+a logarithmic volume form on X, and consider the smallest equivalence relation that
+identifies (X, ω) and (X′, ω′) if there exists a proper birational morphism f : X′ → X
+such that ω′ = f ∗ω. By the weak factorization theorem of (Abramovich et al,
+
+BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES
+7
+2002), this equivalence relation is generated by such morphisms f which are blowing-
+ups along smooth centers in good position with respect to the polar divisor of X.
+In both contexts, if X is an n-dimensional k-variety and ω is a meromorphic n-form
+on X which is logarithmic “everywhere”, then we define [X, ω] to be the sum, over
+all irreducible components Y of X which have dimension n, of the classes [Y, ω|Y].
+Example 3.4. — Finitely generated extensions of k of transcendence degree 0 are
+finite extensions of k. Let K be such an extension. Since k has characteristic zero,
+one has Ω1
+K/k = 0. However, Ω0
+K/k, which is its 0th exterior power, is canonically
+isomorphic to K. Consequently, Burn0(k) is the free abelian group on isomorphism
+classes of pairs (K, λ), where K is a finite extension of k and λ ∈ K.
+We will let 1 = [Spec(k), 1] and ε = [Spec(k), −1].
+Example 3.5. — Let K = k(t). The differential form dt/t is a logarithmic volume
+form; indeed X = P1
+k is a model of K and this form has poles of order 1 at 0 and ∞,
+and no other poles. We write T for the class of (k(t), dt/t).
+Note that the k-isomorphism of K that maps t to 1/t maps dt/t to its opposite;
+consequently, we also have T = [k(t), −dt/t] = ε · T.
+In the context of birational geometry in presence of logarithmic volume forms,
+“rational varieties” would have class in Tn, and similarly for stable birationality.
+3.6. Multiplicative structure. — We view the direct sum
+Burn(k) =
+�
+n∈N
+Burnn(k)
+as a graded abelian group. It is endowed with a multiplication such that
+[X, ω] · [X′, ω′] = [X × X′, ω ∧ ω′]
+when X, X′ are proper, smooth and integral and ω, resp. ω′ are logarithmic volume
+forms on X, resp. X′, and Y ranges over the set of irreducible components of X×X′.
+Let s: X′ × X → X × X′ be the isomorphism exchanging the two factors. One has
+s∗(ω ∧ ω′) = (−1)nn′ω′ ∧ ω,
+if n = dim(X), n′ = dim(X′), ω is a volume form on X and ω′ is a volume form
+on X′. Consequently,
+a · b = εnn′ · b · a
+for a ∈ Burnn(k) and b ∈ Burnn′(k). In particular, classes in Burnn(k), for even n,
+are central in Burn(k).
+We remark that the element T ∈ Burn1(k) is central as well. Let indeed a ∈
+Burnn(k). If n is even, then a · T = T · a. Otherwise, we have a · T = ε · T · a, but
+we have seen in example 3.5 that T = ε · T. As a consequence, a · T = T · a.
+However, the ring Burn(k) is not commutative. Indeed, consider curves X, X′
+without automorphisms and no nonconstant morphism between them. Then the
+switch is the only isomorphism from X′ × X to X × X′. Take nonzero logarithmic
+
+8
+ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL
+1-forms ω, ω′ on X, X′ respectively. The classes [X × X′, ω ∧ ω′] and [X′ × X, ω′ ∧ ω]
+are then distinct.
+3.7. Functoriality. — Let k′ be an extension of k. Then there is a natural ring
+homomorphism
+Burn(k) → Burn(k′)
+described as follows. Let (X, ω) be an integral k-variety of dimension n equiped with
+a logarithmic q-form. Let X′ = X ⊗k k′ be its base change to k′, and let ω′ be the
+volume form on X′ deduced from ω by base change. Then the class of (X, ω) maps to
+the sum of classes (Y, ω′|Y), where Y runs the (finite) set of irreducible components
+of X′.
+If k′ is a finite extension of k, we also have a trace map
+Trk′/k : Burn(k′) → Burn(k)
+obtained by averaging over a set of representatives of automorphisms of the Galois
+closure of k′ over k modulo those preserving k′.
+3.8. Relation with the classical Burnside group. — Forgetting the form ω
+gives a ring morphism
+π: Burn(k) → Burn(k).
+On the other hand, if K is a finitely generated extension of k of transcendence
+degree n, we can endow it with the zero n-form. The resulting map
+̟: Burn(k) → Burn(k)
+identifies Burn(k) with an ideal of Burn(k). One has π ◦ ̟ = id.
+3.9. Variations on the theme. — The construction of the Burnside ring Burn(k)
+admits several natural variants that are relevant in more specific contexts. Some of
+them will be used in later sections.
+3.9.1. A relative ring. — Let n be an integer.
+For any k-scheme S, we define
+Burnn(S/k) as the free abelian group on triples (X, ω, u) where X is an integral
+smooth n-dimensional k-scheme, ω ∈ Ωn
+X/k is a regular volume form which is loga-
+rithmic “everywhere”, and u: X → S is a morphism, modulo the smallest equivalence
+relation that identifies (X, ω, u) and (X′, ω′, u′) if there exists an open immersion
+f : X′ → X such that ω′ = f ∗ω and u′ = u ◦ f.
+Let h: S → T be a morphism of k-schemes. It induces a morphism of abelian
+groups
+h∗ : Burnn(S/k) → Burnn(T/k)
+such that h∗([X, ω, u]) = [X, ω, h ◦ u] for any triple (X, ω, u) as above.
+3.9.2. Pluriforms. — One can replace volume forms with volume r-pluriforms, that
+is, elements of (Ωn
+K/k)⊗r, for some given integer r. The corresponding logarithmic
+condition requires that the pluriform has poles of order at most r on an adequate
+model. Note that when r is even, the obtained ring is commmutative.
+
+BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES
+9
+3.9.3. Forms up to scalars. — In the construction, we may wish to identify (K, ω)
+and (K′, ω′) if there exists λ ∈ k×, resp. λ ∈ {±1}, and a k-isomorphism f : K → K′
+such that f ∗ω′ = λω. These variants also give rise to a commutative ring.
+3.9.4. Group actions. — Let G be a profinite group scheme over k. One can also
+consider pairs (K, ω), where the field K is endowed with an action of G leaving the
+form ω invariant. The obtained ring will be denoted by BurnG(k).
+4. Residues
+4.1. Residue of a volume form. — Let X be an equidimensional smooth k-
+variety of dimension n.
+Let D be a smooth divisor on X. We denote by Ωm
+X/k(log D) the sheaf of m-forms
+on X with logarithmic poles along D, locally of the form η ∧d log f +η′, where η and
+η′ are regular and f is a local equation of D. The residue map is the homomorphism
+of OX-modules
+ρD : Ωm
+X/k(log D) → Ωm−1
+D/k ,
+characterized by the relation
+ρD(η ∧ d log f + η′) = η|D
+for every local sections η ∈ Ωm−1
+X/k and η′ ∈ Ωm
+X/k, and any local generator f of the
+ideal of D.
+If ω is a logarithmic m-form on X, there is an open subset U of X such that
+U ∩ D ̸= ∅ and such that ω|U belongs to Ωm
+X/k(log D). Its residue ρD(ω|U) is then a
+meromorphic section of Ωm−1
+D/k .
+Lemma 4.2. — Let ω be a logarithmic differential form of degree m on X. Then
+ρD(ω) is a logarithmic (m − 1)-form on D.
+Proof. — We may assume that the sum of D and of the polar divisor of ω has strict
+normal crossings. The assertion is then evident in local coordinates.
+4.3. Blowing-ups and normal bundles. — Let Y be a smooth closed sub-
+scheme of X. The blow-up BlY(X) of X along Y is a smooth k-variety. The blowing-
+up morphism bY : BlY(X) → X is an isomorphism over the complement of Y. If Y
+is nowhere dense and nonempty, then EY = b−1
+Y (Y) is a smooth divisor in BlY(X).
+In general, EY = b−1
+Y (Y) identifies, as an Y-scheme, with the projectivization of
+the normal bundle NY(X) of Y in X.
+Let W be a closed smooth subscheme of X. Assume that W and Y are transversal.
+Then the Zariski closure of b−1
+Y (W
+(Y ∩ W)) is called the strict transform of W
+in BlY(X). It identifies with BlY∩W(W).
+Let now ω be a logarithmic m-form on X.
+Then the form b∗
+Yω on BlY(X) is
+logarithmic; assuming that Y is nonempty and nowhere dense, we can consider the
+residue ρY(ω) of b∗
+Yω along EY. It is a logarithmic (n − 1)-form on P(NY(X )).
+
+10
+ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL
+Definition 4.4. — Let X be an irreducible proper smooth k-variety, let n be its
+dimension and let ω ∈ Ωn
+X/k be logarithmic volume form whose polar divisor D has
+strict normal crossings. Let (Dα)α∈A be the family of its irreducible components; for
+A ⊆ A , we let DA = �
+α∈A Dα. We then define an element ρ(X, ω) in Burnn−1(X/k)
+by the formula:
+ρ(X, ω) =
+�
+∅̸=A⊆A
+(−1)|A|−1ρDA(X, ω).
+(In this formula and all similar ones below, it is always implicit that the terms
+where DA = ∅ are omitted.)
+4.5. Iterated residues. — We retain the notation of definition 4.4
+Fix a logarithmic volume form ω on X and a nonempty subset A of A such
+that DA ̸= ∅. It will be useful to compute inductively the logarithmic volume form
+ρDA(ω) that appears in definition 4.4.
+Let bA : ˜X → X be the blowing-up of X along DA and let E be its exceptional
+divisor.
+When A = {α} has a single element, DA is the divisor Dα, the blowing-up mor-
+phism bA is an isomorphism and the exceptional divisor identifies with DA. Then
+ρDA(X, ω) = [Dα, ρDα(ω), jα],
+where jα is the immersion of Dα into X.
+This construction can be pursued in higher codimension, using iterated residues.
+Fix a total order on A . There is a unique, strictly increasing sequence (α1, . . . , αm)
+in A such that A = {α1, . . . , αm}. Given the chosen order on A , we may apply the
+iterated residues construction and obtain a logarithmic form of degree n − m
+ρDA(ω) = ρDα1 ◦ · · · ◦ ρDαm(ω).
+On a nonempty open subset U of X that meets DA, we may write
+ω = η ∧ dlog(fα1) ∧ . . . dlog(fαm),
+for a regular form η, and then one has ρDA(ω) = η|U∩DA.
+Denote by bA the blowing-up of X along DA and by EA its exceptional divisor;
+recall that EA identifies with the projectivized normal bundle NDA(X) of DA in X.
+Using local equations for the divisors Dα, for α ∈ A, we trivialize NDA(X) on a dense
+open subscheme of DA; this gives a birational isomorphism of EA with DA × Pm−1
+(with m = |A|), and a local computation gives the formula
+ρDA(X, ω) = [DA, ρDA(ω)] · Tm−1
+in Burnn−1(DA/k).
+When m ⩾ 2, the definition of ρDA actually depends on the chosen order of A , but
+only up to a sign, so that the class [DA, ρDA(ω)] is well defined up to multiplication
+by the class ε ∈ Burn0(k). On the other hand, it is multiplied by Tm−1 and we
+have observed that ε · T = T.
+
+BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES
+11
+Proposition 4.6. — Let (X, ω), D, and (Dα)α∈A be as in definition 4.4. Let Y be
+a strict irreducible subvariety of X; let AY be the set of all α ∈ A such that Y ̸⊆ Dα;
+we assume that �
+α∈AY Dα meets Y transversally.
+Let g : X′ → X be the blowing-up of X along Y and let ω′ = g∗ω; it is a logarithmic
+form, its polar divisor has strict normal crossings, and we have
+g∗ρ(X′, ω′) = ρ(X, ω)
+in Burn(X/k).
+Proof. — Let E = g−1(Y) be the exceptional divisor; for each α ∈ A , let D′
+α be the
+strict transform of Dα. The blow-up X′ is smooth; the divisor E + �
+α∈A D′
+α has
+strict normal crossings and contains the polar divisor of ω′.
+Let B be the set of all β ∈ A such that Y ⊆ Dβ, so that DB is the minimal
+stratum containing Y.
+We now split the discussion into two cases.
+(1) Assume that dim(Y) < dim(DB). Since g is ramified along E, its Jacobian
+vanishes along E. Since ω has poles of order at most one, the form ω′ = g∗ω is
+regular at the generic point of E. Consequently, the polar divisor of ω′ does not
+contain E and we have to compare
+�
+∅̸=A⊆A
+(−1)|A|−1ρD′
+A(ω′)
+with
+�
+∅̸=A⊆A
+(−1)|A|−1ρDA(ω).
+Since g is a local isomorphism around the generic points of Dα, for α ∈ A , we
+see that the polar divisor of ω′ is equal to �
+α∈A D′
+α. For every nonempty subset A
+of A , one has
+g∗ρDA(X′, ω′) = ρDA(X, ω)
+for every nonempty subset A of A , which implies the desired formula in this case.
+(2) Assume that dim(Y) = dim(DB). In this case, Y is an irreducible component
+of DB. Since D∅ = X and Y ̸= X, we have B ̸= ∅. We have to compare the expression
+�
+∅̸=A⊆A
+(−1)|A|−1ρD′
+A(ω′) +
+�
+A⊆A
+(−1)|A|ρE∩D′
+A(ω′)
+with
+�
+∅̸=A⊆A
+(−1)|A|−1ρDA(ω).
+The argument takes place in a neighborhood of Y, which allows us to assume that
+Y = DB.
+Let A be a nonempty subset of A . One has D′
+A = ∅ whenever B ⊆ A, and the
+corresponding terms are absent from the second expression. On the other hand,
+if B ̸⊆ A, the morphism g identifies D′
+A with the blow-up of DA along DA ∩ Y =
+DA∪B. In particular, g induces a birational isomorphism from D′
+A to DA, so that
+
+12
+ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL
+g∗ρD′
+A(X′, ω′) = ρDA(X, ω). Moreover, E ∩ D′
+A is the projectivized normal bundle
+PNDA∪B(DA), and
+g∗ρE∩D′
+A(X′, ω′) = ρDA∪B(X, ω).
+Similarly, one has
+g∗ρE(X′, ω′) = ρDB(X, ω).
+This gives a formula of the form
+g∗ρ(X′, ω′) =
+�
+∅̸=A⊆A
+B̸⊆A
+(−1)|A|−1ρDA(X, ω) +
+�
+A⊆A
+B̸⊆A
+(−1)|A|ρDA∪B(X, ω)
+=
+�
+∅̸=A⊆A
+n′
+AρDA(X, ω),
+where
+n′
+A =
+
+
+
+(−1)|A|−1
+if B ̸⊆ A,
+�
+C⊆A
+B̸⊆C
+C∪B=A
+(−1)|C|
+if B ⊆ A.
+It suffices to prove that n′
+A = nA for any nonempty subset A of A . This is obvious
+when B ̸⊆ A, so let us assume that B ⊆ A. In the sum that defines n′
+A, we write
+C = (C
+B)∪C′, where C′ = C∩B is a subset of B; the condition C∪B = A means
+C
+B = A
+B; the condition B ̸⊆ C means C′ ̸= B. Consequently, we have
+n′
+A = (−1)|A B| �
+C′⊆B
+C′̸=B
+(−1)|C′|
+= (−1)|A B|
+� �
+C′⊆B
+(−1)|C′| − (−1)|B|
+�
+= (−1)|A B| �
+(1 − 1)|B| − (−1)|B|�
+= (−1)|A|−1,
+since |B| ⩾ 1. This concludes the proof of the proposition.
+Theorem 4.7. — Let (X, ω) be as in definition 4.4. If X is proper, then the image
+of ρ(X, ω) in Burnn−1(k) only depends on the class [X, ω] ∈ Burnn(k). It gives rise
+to a morphism of abelian groups
+∂n : Burnn(k) → Burnn−1(k).
+Proof. — By the definition of Burnn(k) involving pairs (X, ω) where X is proper,
+it suffices to consider two pairs (X, ω) and (X′, ω′) as in definition 4.4 which are
+related by a proper birational morphism g : X′ → X such that g∗ω = ω′. By the
+weak factorization theorem of Abramovich et al (2002), in order to prove the
+theorem, we may assume that g is a blowing-up of X along a smooth subvariety
+which is transversal to the polar divisor of ω. In this case, proposition 4.6 asserts
+
+BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES
+13
+that g∗��(X′, ω′) = ρ(X, ω) in Burn(X/k). In particular, the images in Burn(k) of
+ρ(X′, ω′) and ρ(X, ω) are equal.
+Example 4.8. — The meromorphic differential form dt/t on P1
+k has residues 1
+and −1 at 0 and ∞ respectively. By construction, we thus have
+∂1(T) = [Spec(k), 1] + [Spec(k), −1] = 1 + ε.
+Let n be an integer such that n ⩾ 2 and let us compute ∂n(Tn). We view Tn as
+the class of Pn, with homogeneous coordinates [1 : x1 : . . . : xn], and with the toric
+differential form
+ωn = (dx1/x1) ∧ . . . (dxn/xn).
+Its divisor is the sum of the toric hyperplanes D0, . . . , Dn. Each of these hyperplanes
+identifies with Pn−1, and ρDj(ωn) is (−1)n−jωn−1. Let A = {0, . . . , n}. If A = A ,
+then DA = ∅. Otherwise, we see by induction that DA is isomorphic to Pn−|A| and
+ρDA(ωn) identifies with ±ωn−|A|, so that
+[DA, ρDA(ωn)] · T|A|−1 = [Gm
+n−1, ±ωn−1] = Tn−1,
+since n − 1 ⩾ 1. Then,
+∂n(Tn) =
+�
+∅̸=A⊆A
+(−1)|A|−1[DA, ρDA(ωn)] · T|A|−1
+=
+�
+∅̸=A⊊A
+(−1)|A|−1Tn−1.
+Now,
+�
+∅̸=A⊊A
+(−1)|A|−1 = 1 − (1 − 1)n+1 + (−1)n+1 =
+�
+2
+if n is odd;
+0
+if n is even.
+We get ∂n(Tn) = 2Tn−1 if n is odd and ∂n(Tn) = 0 if n is even. (Remind that
+n ⩾ 2.) Since T = ε · T, the following formula unifies the various cases: for n ⩾ 1,
+we have
+∂n(Tn) = (1 + (−1)n−1ε) · Tn−1.
+Proposition 4.9. — For every class b ∈ Burnn(k), we have
+∂n+1(b · T) = −∂n(b) · T + b · ∂1T.
+Proof. — We may assume that b = [X, ω], where X is a proper integral smooth
+variety of dimension n, and ω is a logarithmic volume form on X whose polar divisor
+has strict normal crossings. Let (Dα)α∈A be the family of its irreducible components.
+We view b·T as the class of [X×P1, ω ∧dt/t]. The polar divisor of ω ∧dt/t is equal
+to
+�
+α∈A
+Dα × P1 + X × {0} + X × {∞}.
+
+14
+ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL
+It has strict normal crossings, and its strata are of the form DA × P1, for nonempty
+A ⊆ A , or DA × {0}, or DA × {∞}, for A ⊆ A . This decomposes ∂n+1(b × T) as
+the sum of three terms.
+The first one is
+�
+∅̸=A⊆A
+[DA × P1, ρDA×P1(ω ∧ dt/t)] · T|A|−1.
+For any nonempty subset A of A , one has
+ρDA×P1(ω ∧ dt/t) = ±ρDA(ω) ∧ dt/t,
+so that
+[DA × P1, ρDA×P1(ω ∧ dt/t)] · T|A|−1 = [DA, ρDA(ω)] · T · T|A|−1.
+Consequently, the first term equals ∂n(b) × T.
+Write D0 = X × {0} and D∞ = X × {∞}, and identify both divisors to X. For a
+subset A of A , we have
+ρDA∪{0}(ω ∧ dt/t) = ρDA ◦ ρD0(ω ∧ dt/t) = ρDA(ω).
+Consequently, the second term is equal to
+�
+A⊆A
+(−1)|A|[DA, ρDA(ω)] · T|A| = [X, ω] − ∂n(b) · T.
+Similarly, the third term is equal to
+[X, −ω] − ∂n(b) · T.
+Summing up these three terms, we get
+∂n+1(b × T) = −∂n(b) · T + [X, ω] + [X, −ω].
+We now recall that ∂1(T) = [Spec(k), 1] + [Spec(k), −1], so that
+[X, ω] + [X, −ω] = [X, ω] · ∂1(T) = b · ∂1(T).
+This concludes the proof.
+Theorem 4.10. — Let a ∈ Burnm(k) and b ∈ Burnn(k); we have
+∂m+n(a · b) = εn · ∂m(a) · b + a · ∂n(b) − T · ∂m(a) · ∂n(b)
+in Burnm+n−1(k).
+Proof. — It suffices to treat the case where a and b are classes of proper integral
+smooth varieties (X, ω), (Y, η), endowed with meromorphic volume forms whose
+polar divisors have strict normal crossings and no multiplicities. Let (Dα)α∈A be
+the irreducible components of the polar divisor of ω, let (Eβ)β∈B be the irreducible
+components of the polar divisor of η. Then [X, ω]·[Y, η] is the class of [X×Y, ω ∧η];
+the polar divisor of ω ∧ η is equal to
+�
+α∈A
+Dα × Y +
+�
+β∈B
+X × Eβ.
+
+BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES
+15
+We fix a total order on the disjoint union of A and B such that the elements of A
+are smaller than those of B. For any subsets A, B of A and B, observe that we
+have
+ρDA∪B(ω ∧ η) = ±ρDA(ω) ∧ ρEB(η),
+where ρDA has to be understood as the identity when A is empty, and similarly
+for ρEB. The sign is 1 when A = ∅; when B = ∅, it is equal to (−1)|A|n; we won’t
+need to use its explicit value in the other cases. Then we can write ∂([X, ω] · [Y, η])
+as
+�
+A⊆A
+B⊆B
+A∪B̸=∅
+(−1)|A|+|B|−1[DA × EB, ±ρA(ω) ∧ ρEB(η)] · T|A∪B|−1
+and we split it into the sum of three terms, according to which B = ∅, or A = ∅, or
+none of them is empty. The first two terms are respectively equal to
+�
+∅̸=A⊆A
+(−1)|A|−1[DA × Y, (−1)n|A|ρDA(ω) ∧ η] · T|A|−1 = ∂([X, (−1)nω]) · [Y, η]
+and
+�
+∅̸=B⊆B
+(−1)|B|−1[X × EB, ω ∧ ρEB(η)] · T|B|−1 = [X, ω] · ∂([Y, η]),
+since T belongs to the center of Burn(k). As for the third one, we obtain
+−
+�
+∅̸=B⊆B
+(−1)|B|−1
+�
+∅̸=A⊆A
+(−1)|A|−1[DA, ρDA(ω)] · [EB, ρEB(η)] · T|A|+|B|−2
+which equals
+−∂([X, ω]) · ∂([Y, η]) · T.
+Finally, we get
+���m+n(a · b) = ∂m+n([X, ω] · [Y, η])
+= ∂m([X, (−1)nω) · [Y, η] + [X, ω] · ∂n([Y, η])
+− T · ∂m([X, ω]) · ∂n([Y, η])
+= εn · ∂m(a) · b + a · ∂n(b) − T · ∂m(a) · ∂n(b)
+as was to be shown.
+In particular, using the computation of example 4.8, we obtain the following
+generalization of proposition 4.9.
+Corollary 4.11. — For any a ∈ Burnm(k) and any integer n, we have
+∂m+n(a · Tn) =
+�
+∂m(a) · Tn
+if n is even;
+−∂m(a) · Tn + a · ∂n(Tn)
+if n is odd.
+Remark 4.12. — For the variant of Burn(k) where we consider forms up to sign,
+the formula of theorem 4.10 simplifies to
+∂m+n(a · b) = ∂m(a) · b + a · ∂n(b) − T · ∂m(a) · ∂n(b).
+
+16
+ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL
+5. A complex of Burnside rings
+Theorem 5.1. — For any integer n ⩾ 2, we have
+∂n−1 ◦ ∂n = 0.
+In other words, the residue morphisms of Burnside groups give rise to a complex
+· · · → Burnn(k) → Burnn−1(k) → · · · → Burn1(k) → Burn0(k)
+Proof. — It suffices to prove the following result:
+Let (X, ω) be an integral
+proper smooth variety of dimension n equipped with a meromorphic volume
+form ω whose polar divisor has strict normal crossings and no multiplicities; then
+∂n−1(∂n([X, ω])) = 0.
+Let (Dα)α∈A be the family of irreducible components of the polar divisor of ω
+in X. By definition, one has
+∂n([X, ω]) =
+�
+∅̸=A⊆A
+(−1)|A|−1ρDA(X, ω).
+Fix a total order on A . Let (α1, . . . , αm) be a strictly increasing sequence in A
+and let A = {α1, . . . , αm}. We have seen in §4.5 that ρDA(X, ω) can be defined via
+iterated residue maps:
+ρDA([X, ω]) = [DA, ρDα1 ◦ · · · ◦ ρDαm(ω)] · T|A|−1 = [DA, ωA] · T|A|−1
+where we wrote ωA for the composition ρDα1 ◦ · · · ◦ ρDαm(ω). When |A| is odd, we
+have
+∂(ρDA([X, ω])) = ∂([DA, ωA]) · T|A|−1,
+while when |A| is even, we have
+∂(ρDA([X, ω])) = −∂([DA, ωA]) · Ta−1 + [DA, ωA] · ∂(T|A|−1).
+Consequently, we have
+∂ ◦ ∂([X, ω]) =
+�
+∅̸=A⊆A
+∂([DA, ωA]) · T|A|−1 −
+�
+∅̸=A⊆A
+|A| even
+[DA, ωA] · ∂(T|A|−1).
+The polar divisor of the form ωA on DA is equal to �
+β̸∈A Dβ ∩ DA, so that, by
+definition (and computation of ∂ via iterated residues),
+∂([DA, ωA]) =
+�
+∅̸=B⊆∁A
+(−1)|B|−1[DAB, ωA∪B] · T|B|−1.
+Also, when A is nonempty and of even cardinality, ∂(T|A|−1) = 2T|A|−2. When we
+put these two formulas into the antepenultimate one and collect the various terms,
+we obtain
+∂ ◦ ∂([X, ω]) =
+�
+C⊆A
+2⩽|C|
+nC[DC, ωC] · T|C|−2,
+
+BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES
+17
+where
+nC = −
+�
+∅̸=A,B
+A∪B=C,A∩B=∅
+(−1)|B| − 2δ|C| is even.
+In the first sum, the terms A = ∅ or B = ∅ are omitted, while if we put them in, we
+obtain
+�
+A∪B=C
+A∩B=∅
+(−1)|B| =
+|C|
+�
+b=0
+�|C|
+b
+�
+(−1)b = (1 − 1)|C| = 0
+since |C| ⩾ 1. Consequently,
+nC = 1 + (−1)|C| − 2δ|C| is even = 0.
+This concludes the proof.
+6. Algebraic structure of Burn(k) after localization at 2
+In this section, we study the algebraic structure of the Burnside ring Burn(k),
+endowed with its elements ε, T and the operator ∂.
+6.1. — By construction, Burn(k) = �
+n⩾0 Burnn(k) is an associative unital Z⩾0-
+graded ring, ε ∈ Burn0(k), T ∈ Burn1(k) and ∂ is a homogeneous additive map
+of degree −1. They satisfy the following relations, for homogeneous elements a, b ∈
+Burn(k):
+b · a = ε|a||b| · a · b
+(§3.6);
+(1)
+ε2 = 1
+(example 3.4);
+(2)
+T = ε · T
+(example 3.5);
+(3)
+∂(T) = 1 + ε
+(example 4.8);
+(4)
+∂(a · b) = ε|b| · ∂(a) · b + a · ∂(b) − T · ∂(a) · ∂(b)
+(theorem 4.10);
+(5)
+∂(∂(a)) = 0
+(theorem 5.1).
+(6)
+By (1), the element ε is central, and by (2), we may view Burn(k) as an algebra over
+Z[ε]/(ε2 − 1). After inverting 2, the algebra Burn(k) splits into two components
+Burnε=1(k) and Burnε=−1(k), one over which ε = 1, and the other over which
+ε = −1.
+In the rest of this section, we implicitly assume that 2 is inverted, without changing
+the notation.
+
+18
+ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL
+6.2. Sector ε = −1. — Here, we have T = −T, hence T = 0 since 2 is invertible.
+As a consequence, after replacing ∂ with ∂′ : a �→ (−1)1+|a|∂(a), one gets from (5)
+the usual graded Leibniz rule
+∂′(a · b) = ∂′(a) · b + (−1)|a|a · ∂′(b)
+and therefore Burnε=−1(k) is a classical differential graded (super-)commutative
+algebra, similar to, eg, the de Rham complex.
+6.3. Sector ε = 1. — The algebra Burnε=1 is now commutative (and not graded
+commutative). This reflects the intuition in our constructions that they speak about
+volume forms (as opposed to top-degree differential forms) for which we have com-
+mutativity (as reflected by the change of order of integration in multiple integrals).
+Lemma 6.4. — The map F: a �→ a − T �� ∂(a) is a ring endomorphism of
+Burnε=1(k), and F2 = id. Moreover, one has F ◦ ∂ = ∂ = −∂ ◦ F.
+Proof. — This map is additive. One has F(1) = 1 − T · ∂(1) = 1. Let us show
+multiplicativity. Indeed, for a, b ∈ Burnε=1(k), one has
+F(a) · F(b) = (a − T · ∂(a)) · (b − T · ∂(b))
+= a · b − T · ∂(a) · b − T · a · ∂(b) + T2 · ∂(a) · ∂(b)
+= a · b − T · (∂(a) · b + a · ∂(b) − T · ∂(a) · ∂(b))
+= a · b − T · ∂(a · b)
+(using (5))
+= F(a · b).
+Since ∂2 = 0, one has
+F(∂(a)) = ∂(a) − T · ∂(∂(a)) = ∂(a).
+On the other hand,
+∂(F(a)) = ∂(a − T · ∂(a))
+= ∂(a) − ∂(T · ∂(a))
+= ∂(a) − ∂(T) · ∂(a) − T · ∂(∂(a)) + T · ∂(T) · ∂(∂(a))
+= −∂(a)
+using that ∂(T) = 2 and ∂2 = 0.
+Consequently, for a ∈ Burnε=1(k), we have
+F2(a) = F(a) − T · ∂(F(a)) = a − T · ∂(a) + T · ∂(a) = a
+since ∂ ◦ F = −∂.
+
+BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES
+19
+6.5. — To simplify the notation, write B = Burnε=1(k). Since F2 = id and 2 is
+invertible, the algebra B splits as a direct sum
+B = B+ ⊕ B−,
+such that F acts as id on B+ and as − id on B−. Moreover, B+ is a subalgebra.
+Since the operator ∂ anticommutes with F, it induces maps
+∂± : B+ → B−,
+∂∓ : B− → B+.
+Note that
+F(T) = T − T · ∂(T) = −T,
+so that T ∈ B−. Consequently, the multiplication by T map induces two maps
+t± : B+ → B−,
+t∓ : B− → B+.
+Lemma 6.6. — The map ∂ vanishes on B+. Equivalently, ∂± = 0.
+The maps 1
+2∂∓ and t± are inverses the one of the other.
+Proof. — For a ∈ B+, one has ∂(a) = −∂(F(a)) = −∂(a), since ∂ ◦ F = −∂, hence
+∂(a) = 0.
+On the other hand, for a ∈ B+, one has
+∂(T · a) = 2 · a + T · ∂(a) − 2T · ∂(a) = 2a − T · ∂(a) = a + F(a) = 2a
+while for a ∈ B−, we have
+T · ∂(a) = a − F(a) = 2a.
+This concludes the proof of the lemma.
+In particular, we see that the cohomology of the differential ∂ vanishes in the
+sector Burnε=1(k) = B.
+6.7. — It follows from the lemma that we have a ring isomorphism
+B = B+[t](t2 − T2),
+from which we see that all the algebraic structure of B+ (namely δ, T, F) can be
+canonically reconstructed from a unital commutative associative Z⩾0-graded ring B+
+endowed with an element in degree +2 (namely, the element T2).
+Remark 6.8. — The situation clarifies even more if we invert the class T. Then
+we can write ∂(a) = (a − F(a))/T, and all relations happen to follow from the fact
+that F is an involution such that F(T) = −1. Indeed,
+∂2(a) = ∂(a) − F(∂(a))
+T
+= 1
+T
+�a − ∂(a)
+T
+− F(a − ∂(a)
+T
+�
+= 0
+
+20
+ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL
+explains that ∂2 = 0. Moreover, for a, b ∈ B, we have
+∂(a · b) = a · b − F(a · b)
+T
+= a · b − F(a) · F(b)
+T
+= a − F(a)
+T
+· b + a · b − F(b)
+T
+− T · a − F(a)
+T
+· b − F(b)
+T
+= ∂(a) · b + a · ∂(b) − T · ∂(a) · ∂(b).
+7. Birational morphisms preserving volume forms
+7.1. — Let (X, ωX) be a smooth integral k-variety of dimension n equipped with
+a meromorphic form with poles of order at most one on X and let f : Y → X be a
+proper birational morphism.
+Let E be an exceptional divisor in Y, that is, such that dim(f(E)) < dim(E).
+Then the Jacobian of p vanishes along E. Since ω has poles of order at most one
+on X, the meromorphic form f ∗ω on Y is regular at the generic point of E; its
+restriction to E is a meromorphic form with poles of order at most one and we may
+consider the class [E, f ∗ωX|E] in Burnn−1(k).
+We define c(f; X, ω) to be the sum of all such classes [E, f ∗ω|E] in the free abelian
+group Burnn−1(k).
+Lemma 7.2. — Let g : Z → Y be a proper birational morphism of smooth integral
+varieties of dimension n. Then g ◦ f is a proper birational morphism and one has
+c(g ◦ f; X, ω) = c(g; Y, f ∗ω) + c(f; X, ω)
+in Burnn−1(k).
+Proof. — An integral divisor F in Z is exceptional for g ◦ f if and only if one of the
+two mutually excluding situations happen:
+– The divisor F is exceptional for g;
+– Or g(F) is a divisor in Y which is exceptional for f.
+Moreover, any divisor E in Y which is exceptional for f appears once and only as
+a divisor of the form g(F). In the first case, F contributes to c(g ◦ f; X, ω) by a
+term [F, (g ◦ f)∗ω|F], and it contributes to c(g; Y, f ∗ω) by precisely the same term.
+In the second case, F contributes to c(g ◦ f; X, ω) by a term [F, g∗(f ∗ω|E)], while E
+contributes to c(f; X, ω) by the term [E, f ∗ω|E]. Since g is birational around the
+generic point of E, they coincide, and this concludes the proof.
+7.3. — Let (X, ωX) and (Y, ωY) be proper smooth k-varieties equipped with loga-
+rithmic volume forms and let
+ϕ: (X, ωX) ��� (Y, ωY)
+
+BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES
+21
+be a birational map preserving the volume forms. By definition, this means that
+there exists a diagram
+W
+X
+Y
+←
+→
+p
+←
+→
+q
+←
+→
+ϕ
+of integral k-varieties such that that p and q are proper and birational, and such
+that p∗ω = q∗ω′ on W. In this situation, we may assume that W is smooth.
+Lemma 7.4. — With this notation, the element
+c(ϕ) = c(q) − c(p) ∈ Burnn−1(k)
+only depends on the birational map ϕ, and not on the choice of the triple (W, p, q).
+Proof. — Consider two possible diagrams X
+p←− V
+q−→ Y and X
+r←− W
+s−→ Y describ-
+ing ϕ. Considering for example a resolution of singularities U of V ×X W, we can fit
+these two diagrams in a common commutative diagram of the following form:
+U
+V
+W
+X
+Y
+←
+→
+u
+←
+→
+v
+←
+→
+p
+←
+→
+q
+←
+→
+r
+←
+→
+s
+←
+→
+ϕ
+The equalities p∗ωX = q∗ωY and r∗ωX = s∗ωY imply that
+(p ◦ u)∗ωX = u∗p∗ωX = u∗q∗ωY = (q ◦ u)∗ωY = (s ◦ v)∗ωY.
+By lemma 7.2, we then have
+c(p) − c(q) = c(p ◦ u) − c(q ◦ u) = c(r ◦ v) − c(s ◦ v) = c(r) − c(s).
+This concludes the proof.
+Theorem 7.5. — If ψ: (Y, ωY) ��� (Z, ωZ) is another birational map preserving
+volume forms, then one has
+c(ψ ◦ ϕ) = c(ψ) + c(ψ).
+Proof. — Consider two diagrams X
+p←− V
+q−→ Y and Y
+r←− W
+s−→ Y describing ϕ
+and ϕ. Considering for example a resolution of singularities U of V ×Y W, we can
+
+22
+ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL
+fit these two diagrams in a common commutative diagram of the following form:
+U
+V
+W
+X
+Y
+Z
+←
+→
+u
+←
+→
+v
+←
+→
+p
+←
+→
+q
+←
+→
+r
+←
+→
+s
+←
+→
+ϕ
+←
+→
+ψ
+and the diagram X
+p◦u
+←−− U
+s◦v
+−−→ describes the birational map ψ◦ϕ. Since q◦u = r◦v,
+we then have
+c(ψ ◦ ϕ) = c(p ◦ u) − c(s ◦ v)
+= c(p ◦ u) − c(q ◦ u) + c(r ◦ v) − c(s ◦ v)
+= c(p) − c(q) + +c(r) − c(s)
+= c(ϕ) + c(ψ),
+as was to be shown.
+Corollary 7.6. — Let Bir(X, ω) be the set of birational automorphisms of X pre-
+serving ω. The map c induces a homomorphism of abelian groups
+Bir(X, ω) → Burnn−1(k).
+Its kernel contains the group of automorphisms of X that preserve ω.
+8. Specialization
+Let K be the field of fractions of a discrete valuation ring R with residue field k.
+Fix a uniformizer t ∈ R.
+In this context, Kontsevich & Tschinkel (2019) have defined two (distinct)
+specialization morphisms
+ρt : Burnn(K) → Burnn(k),
+relating the Burnside groups of K and k (see 3.1), one of which is a ring homo-
+morphism.
+(The latter homomorphism actually depends on the choice of t, see
+example 6.2 of (Kresch & Tschinkel, 2022b).)
+The goal of this section is to define a similar homomorphism
+ρt : Burn(K) → Burn(k)
+for varieties with logarithmic volume forms.
+
+BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES
+23
+8.1. — Let X be an integral proper scheme over R, of relative dimension n, whose
+special fiber ∆ is a divisor with strict normal crossings.
+Let (∆α)α∈A be the family of irreducible components of the special fiber ∆; for
+α ∈ A , let eα be the multiplicity of ∆α in ∆. For every nonempty subset A of A ,
+let ∆A be the intersection of all divisors ∆α, for α ∈ A and eA be the greatest
+common divisor of the eα, for α ∈ A; let also ∆◦
+A be the complement ∆A
+�
+α̸∈A ∆α.
+The first specialization morphism of (Kontsevich & Tschinkel, 2019) is de-
+fined by
+(8.2)
+ρt([XK]) =
+�
+∅̸=A⊆A
+(−1)|A|−1[∆A]L|A|−1,
+where L ∈ Burn(k) is the class of the affine line.
+Although this map is not multiplicative, it proved sufficient for many applications
+to rationality problems.
+To ensure multiplicativity, a more delicate construction was necessary, valued in
+the Burnside ring
+Burn�µ(k)
+of varieties endowed with an action of the profinite group �µ, limit of finite groups of
+roots of unity.
+Fix a nonempty subset A of A . We identify the normal bundle of ∆A in X as a
+direct sum of line bundles:
+N∆A(X ) ≃
+�
+α∈A
+N∆α(X )|∆A.
+Let us consider its open subscheme N ◦
+∆A(X ) obtained by restricting to ∆◦
+A and
+taking out all “coordinate” hyperplanes. This furnishes a morphism
+νA : N ◦
+∆A(X ) →
+�
+α∈A
+N∆α(X )⊗eα|∆◦
+A.
+Since the uniformizer t has divisor − �
+α∈A eα∆α on X , it trivializes the line bundle
+on the target of νA. We set ∆′
+A = ν−1
+A (t). By construction, the projection ∆′
+A → ∆A
+is a torsor with group µeA.
+With this notation, the correct, multiplicative, specialization map of (Kontsevich & Tschinkel,
+2019) is given by the formula
+�ρt(X) =
+�
+∅̸=A⊆A
+(−1)|A|−1[∆′
+A]L|A|−1
+in Burn�µ(k).
+Remark 8.3. — The relation between the two specialization morphisms is as fol-
+lows. Fix a nonempty subset A of A . The group Gm acts diagonally on N ◦
+∆A(X )
+(the factors of index α /∈ A don’t act), and this induces an action of the finite group
+of roots of unity of order eA on ∆′
+A, hence an action of �µ, so that �ρt(X) naturally
+lives in the equivariant Burnside ring Burn�µ(k). Moreover, taking the �µ-invariants
+
+24
+ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL
+of ∆′
+A, we get ∆◦
+A, so that the specialization map ρt is the composition of �ρt with
+the map
+Burn�µ(k) → Burn(k)
+obtained by taking �µ-invariants.
+Taking invariants does not commute with taking products, in general, so that ρt
+is not multiplicative.
+8.4. — Let us explain how to define analogous specialization homomorphisms in
+our context of Burnside groups with volume forms.
+For simplicity, we only consider the case where K has transcendence degree 1
+over k, in which case the idea can be explained geometrically as follows. We assume
+that there exists an smooth integral curve C together with a k-point o ∈ C(k) such
+that K = k(C) and R = OC,o. We fix a local parameter t ∈ R such that V(t) = o.
+Let us consider a pair (X, ω) consisting of an integral proper K-variety X of di-
+mension n and a logarithmic n-form ω on X.
+8.5. — Consider a regular flat proper model X is of X over C, let ∆ = (Xo)red
+be its reduced special fiber, and consider a divisor D with relative normal cross-
+ings on X .
+We assume that the divisor D + ∆ has normal crossings.
+In this
+situation, Deligne (1970, §3.3.2) says that a meromorphic relative differential m-
+form on X /C is logarithmic with respect to D + ∆ if it is (locally) the image of
+a logarithmic m-form ˜ω in Ωm
+X /k with poles D + ∆ under the natural morphism
+Ωm
+X /k → Ωm
+X /C.
+Consider a logarithmic relative n-form ω on X /C. We consider an associated
+volume form ω′ on X , defined locally by
+ω′ = ˜ω ∧ dt/t,
+where ˜ω is any local lift of ω. This form ω′ is logarithmic and we can compute its
+“residue along ∆” as in §4, only taking into account the strata of the polar divisor
+of ω′ which are contained in the special fiber ∆.
+There exists a subset Ao of A and a subset Bo of B such that the polar divisor
+of ω′ is given by
+�
+α∈Ao
+∆α +
+�
+β∈Bo
+Dβ.
+We thus set
+ρt(X , ω) =
+�
+∅̸=A⊆Ao
+B⊆Bo
+(−1)|A|+|B|−1ρ∆A∩DB(X , ω).
+This is an element of Burnn(Xo/k).
+Proposition 8.6. — Let Y be an irreducible closed subscheme of X which is
+transverse to D + ∆ and let g : X ′ → X be the blowing-up of X along Y . The
+form g∗ω on X ′ is logarithmic and we have
+g∗ρt(X ′, g∗ω) = ρt(X , ω)
+
+BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES
+25
+in Burnn(Xo/k).
+Proof. — With the notation of §4, the difference
+ρ(X , ˜ω) − ρt(X , ω)
+is exactly the part of ρ(X , ˜ω) which lies over the complement of the special fiber Xo
+in X . We have seen in theorem 4.7 that
+g∗ρ(X ′, ˜ω′) = ρ(X , ω),
+and a similar formula holds over X
+Xo. This implies the proposition.
+8.7. — Starting from a smooth proper K-variety X and a logarithmic volume
+form ω on X, we can define a model X /C, with D and ∆ as above, but the form ω
+will not necessarily extend to a logarithmic relative form with respect to D +∆, nor
+does the volume form ˜ω on X . However, this can be achieved by multiplying ω by
+a suitable power of the uniformizing element.
+Let us write the polar divisor of ˜ω on X as
+divX (˜ω) = D + ∆ =
+�
+α∈A
+dα∆α +
+�
+β∈B
+dβDβ.
+With this notation, the condition for ˜ω to be logarithmic on X is just that
+dα ⩾ −1,
+dβ ⩾ −1.
+In particular, while the conditions at the horizontal components follow from their
+counterparts on the generic fiber, those for the vertical components are not auto-
+matic. On the other hand, for any κ ∈ Z, the form tκ˜ω is logarithmic if and only
+if
+κeα + dα ⩾ −1
+for all α ∈ A , that is, if and only if κ ⩾ κ(ω), where κ(ω) is defined by
+κ(ω) = inf
+α∈A
+1 − dα
+eα
+.
+Since the rational number κ(ω) is defined in terms of logarithmic forms, it only
+depends on the class of (X, ω) in Burnn(K), and not on the actual model which is
+chosen to compute it.
+8.8. — We assume for the moment that κ(ω) ∈ Z. This holds in particular if the
+special fiber Xo is reduced. Let then Ao be the subset of A consisting of all α such
+that
+κ(ω)eα + dα = −1,
+and let Bo be the subset of B consisting of all β such that dβ = −1. The polar
+divisor of tκ˜ω is equal to
+�
+α∈Ao
+∆α +
+�
+β∈Bo
+Dβ,
+
+26
+ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL
+and we set
+ρt(X , ω) = ρt(X , tκ(ω)ω)
+in Burnn(Xo/k).
+In the particular case where D is empty, the strata of the Clemens complex of
+the special fiber that actually appear in the definition of this class are those defined
+by Kontsevich & Soibelman (2006), more precisely, by the adjustment provided
+by Mustaţă & Nicaise (2015).
+8.9. — In the general case, the rational number κ(ω) is not an integer. Let us
+consider the finite ramified extension Kd = K(t1/d) of K, whose ramification index d
+is a multiple of the denominator of κ(ω), but which induces an isomorphism on
+the residue field. Geometrically, this furnishes a morphism π: Cd → C which is
+ramified at the point o, together with a lift of o in Cd(k) (still denoted by o), and a
+distinguished uniformizing element t1/d.
+We consider the extension of (X, ω) to Kd and introduce a model (Xd, ωd) as
+above, over Cd. Now, the corresponding κ-parameter is integral, so that any choice
+of a uniformizing element t1/d in Rd induces a class ρt1/d(Xd, ωd) in Burn(k). In
+fact, we can assume that the scheme Xd carries an action of the group scheme µd of
+dth roots of unity induced by its action on Spec(Rd), leaving the logarithmic form ωd
+invariant. In other words, we obtain a class in the group Burn�µ(k).
+Combinining these classes, we obtain the desired group homomorphism
+�ρt : Burn(K) → Burn�µ(k).
+In fact, as explained in (Nicaise, 2013, §2.3), especially proposition 2.3.2, one
+can compute the normalisation of X ⊗ Rd in terms of the given model X . This
+gives an explicit decomposition of �ρt(X, ω) as a sum
+�
+∅̸=A⊆Ao
+(−1)|A|−1[D′
+A, ν∗
+AωA] · T|A|−1,
+where νA : D′
+A → DA is the µdA-torsor introduced in §8.1 for the definition of the
+classical specialization map.
+Remark 8.10. — In the case of specialization of rationality, it has proved fruitful
+to consider models with singularities on the special fiber, mild enough so that the
+special fiber computes the specialization of the birational type of the generic fiber.
+This is in particular the case for rational double points.
+A parallel study can be developped in the context of varieties with logarithmic
+forms.
+Following (Kontsevich & Tschinkel, 2019) and keeping track of the various
+logarithmic volume forms on the strata, we have:
+Theorem 8.11. — The morphism �ρt is a ring homomorphism.
+
+BURNSIDE RINGS AND VOLUME FORMS WITH LOGARITHMIC POLES
+27
+References
+D. Abramovich, K. Karu, K. Matsuki & J. Włodarczyk (2002), “Torifica-
+tion and factorization of birational maps”. Journal of the American Mathematical
+Society, 15 (3), pp. 531–572 (electronic).
+S. Boucksom & M. Jonsson (2017), “Tropical and non-Archimedean limits of
+degenerating families of volume forms”. Journal de l’École polytechnique - Math-
+ématiques, 4, pp. 87–139.
+A. Chambert-Loir & Yu. Tschinkel (2010), “Igusa integrals and volume asymp-
+totics in analytic and adelic geometry”. Confluentes Mathematici, 2, pp. 351–429.
+P. Deligne (1970), Equations différentielles à points singuliers réguliers, Lect.
+Notes Math. 163, Springer, Cham.
+S. Gorchinskiy & A. Rosly (2015), “A Polar Complex for Locally Free Sheaves”.
+International Mathematics Research Notices, 2015 (10), pp. 2784–2829.
+H. Hironaka (1964), “Resolution of singularities of an algebraic variety over a
+field of characteristic zero. I, II”. Annals of Mathematics. Second Series, 79, pp.
+109–203, 205–326.
+M. Jonsson & J. Nicaise (2020), “Convergence of p-adic pluricanonical measures
+to Lebesgue measures on skeleta in Berkovich spaces”. Journal de l’École poly-
+technique — Mathématiques, 7, pp. 287–336.
+B. Khesin & A. Rosly (2003), “Polar Homology”. Canadian Journal of Mathe-
+matics, 55 (5), pp. 1100–1120.
+B. Khesin, A. Rosly & R. Thomas (2004), “A polar de Rham theorem”. Topology,
+43 (5), pp. 1231–1246.
+M. Kontsevich & Y. Soibelman (2006), “Affine structures and non-Archimedean
+analytic spaces”. The Unity of Mathematics, Progr. Math. 244, pp. 321–385,
+Birkhäuser Boston, Boston, MA.
+M. Kontsevich & Y. Tschinkel (2019), “Specialization of birational types”.
+Inventiones mathematicae, 217 (2), pp. 415–432.
+A. Kresch & Y. Tschinkel (2022a), “Burnside groups and orbifold invariants of
+birational maps”. arXiv:2208.05835.
+A. Kresch & Y. Tschinkel (2022b), “Equivariant birational types and Burnside
+volume”. Annali Scuola Normale Superiore - Classe Di Scienze, 23 (2), pp. 1013–
+1052.
+H.-Y. Lin & E. Shinder (2022), “Motivic invariants of birational maps”.
+arXiv:2207.07389.
+H.-Y. Lin, E. Shinder & S. Zimmermann (2020), “Factorization centers in di-
+mension two and the Grothendieck ring of varieties”. arXiv:2012.04806.
+M. Mustaţă & J. Nicaise (2015), “Weight functions on non-Archimedean analytic
+spaces and the Kontsevich–Soibelman skeleton”. Algebraic Geometry, 2 (3), pp.
+365–404.
+J. Nicaise (2013), “Geometric criteria for tame ramification”. Mathematische
+Zeitschrift, 273 (3-4), pp. 839–868.
+
+28
+ANTOINE CHAMBERT-LOIR, MAXIM KONTSEVICH & YURI TSCHINKEL
+J. Nicaise & E. Shinder (2019), “The motivic nearby fiber and degeneration of
+stable rationality”. Inventiones mathematicae, 217 (2), pp. 377–413.
+J. Nicaise & C. Xu (2016), “The essential skeleton of a degeneration of algebraic
+varieties”. American Journal of Mathematics, 138 (6), pp. 1645–1667.
+M. Rost (1996), “Chow Groups with Coefficients”. Documenta Mathematica,
+1 (16), pp. 319–393.
+

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+SIMPLE LYAPUNOV SPECTRUM FOR LINEAR HOMOGENEOUS
+DIFFERENTIAL EQUATIONS WITH Lp PARAMETERS
+DINIS AMARO, MÁRIO BESSA, AND HELDER VILARINHO
+Abstract. In the present paper we prove that densely, with respect to an Lp-like
+topology, the Lyapunov exponents associated to linear continuous-time cocycles
+Φ : R× M → GL(2, R) induced by second order linear homogeneous differential
+equations ¨x + α(ϕt(ω))˙x + β(ϕt(ω))x = 0 are almost everywhere distinct. The
+coefficients α, β evolve along the ϕt-orbit for ω ∈ M and ϕt : M → M is an
+ergodic flow defined on a probability space. We also obtain the corresponding
+version for the frictionless equation ¨x + β(ϕt(ω))x = 0 and for a Schrödinger
+equation ¨x + (E − Q(ϕt(ω)))x = 0, inducing a cocycle Φ : R × M → SL(2, R).
+Keywords: Linear cocycles; Linear differential systems; Multiplicative ergodic
+theorem; Lyapunov exponents; second order linear homogeneous differential equa-
+tions.
+2010 Mathematics Subject Classification: Primary: 34D08, 37H15, Secondary:
+34A30, 37A20.
+1. Introduction
+1.1. Non-autonomous linear differential equations. The behaviour of the Lya-
+punov exponents which are determined by the asymptotic growth of the expression
+log ∥Φt
+A∥1/t where Φt
+A is a matricial solution of the autonomous differential equa-
+tion ˙U(t) = A · U(t) and A is a square matrix of the same order as U(t), is a simple
+exercise of linear algebra. Standard linear algebraic computations allows us to de-
+termine the Lyapunov spectrum which is defined by the Lyapunov exponents and
+its eigendirections. The dynamics of a perturbed system like ˙U(t) = B·U(t), where
+B is a perturbation of A, is a problem that is well understood (see e.g. [26]). A much
+more complicated and interesting situation was considered in the pioneering works
+of Lyapunov and intended to consider the non-autonomous case ˙U(t) = A(t) · U(t),
+where A is a matrix depending continuously on t. Not only the asymptotic de-
+meanor of log ∥Φt
+A∥1/t as well as its stability proves to be a substantially more
+difficult issue. A standard way of looking to non-autonomous linear differential
+equations is to consider the language of linear cocycles (see §2.1 for full details)
+where being non-autonomous is captured by a labelling through an orbit of a given
+flow ϕt on a certain phase space.
+1.2. The quest for positive Lyapunov exponents. A positive (or negative) Lya-
+punov exponent gives us the average exponential rate of divergence (or conver-
+gence) of two neighboring trajectories whereas zero exponents give us the ab-
+sence of any kind of exponential behavior.
+Pesin’s theory guarantee a strong
+stable/unstable manifold theory in the presence of non-zero Lyapunov exponents.
+These geometric tools underlie much of the central results in today’s dynamical
+Date: January 13, 2023.
+1
+arXiv:2301.04909v1  [math.DS]  12 Jan 2023
+
+systems. Consequently, there is no doubt that detecting non-zero Lyapunov expo-
+nents is an important question in dynamics an issue dating back to the late six-
+tiees and the work of Millionshchikov [28]. It the early eightees Cornelis and
+Wojtkowski [11], and Ledrappier [25] obtained criteria for the positivity of the
+Lyapunov exponents and in the nineties Knill [30] and Nerurkar [29] proved that
+non-zero Lyapunov exponents are a C0-dense phenomena for certain cocycles. In
+the late nineties Arnold and Cong [7] proved the Lp-denseness of positive Lya-
+punov exponents and their strategy was widespread in [13] by two of the authors.
+Using Moser-type methods based on the concept of rotation number allowed Fab-
+bri and Johnson to obtain abundance of positive Lyapunov exponents for linear
+differential systems evolving on SL(2, R) and based on a translation on the torus
+(see [20, 21, 22] and also the work with Zampogni [23]). Clearly, finding a po-
+sitive Lyapunov exponent in SL(2, R) immediately enable us to obtain a negative
+Lyapunov exponent and thus the simplicity of the Lyapunov spectrum (i.e. all
+Lyapunov exponents are different). Several results on the positivity of Lyapunov
+exponents established in the last ten years or so bring up different new approaches
+[17, 13, 34, 19, 14, 35]. As a paradigmatic example we recall [10] where Avila
+obtained abundance of simple spectrum, on a quite large scope of topologies and
+on the two dimensional case.
+1.3. Asymptotic behaviour of second order linear homogeneous differential
+equations from Lyapunov’s viewpoint. It has been known for almost two cen-
+turies that there are serious constraints when we try to apply analytic methods to
+integrate most functions. Indeed, Liouville theory (see e.g [32]) explicitly describes
+what kind of problems can arise when solving differential equations. The qualita-
+tive theory of differential equations created by Poincaré and Lyapunov turn out to
+be a clever approach to deal with this setback. Here we intend to analyze the as-
+ymptotic behavior of the solutions of second order homogeneous linear differential
+equations of the form
+¨x(t) + α(ϕt(ω))˙x(t) + β(ϕt(ω))x(t) = 0,
+(1)
+with coefficients α and β displaying Lp regularity, varying in time along the orbits
+of a flow ϕt and allowing an Lp-small perturbation on the parameters. Namely, we
+will describe its Lyapunov spectrum taking into account the possibility of making
+a Lp-type perturbation on its coefficients. Instead of deal with a single equation
+we will consider infinite equations simultaneously as explained now: we consider
+a time-continuous cocycle based on an ergodic flow ϕt : M → M with respect to a
+probability measure in M and with a dynamics on the fiber defined by a linear flow
+Φt
+A which is solution of the linear variational equation ˙U(ω, t) = A(ϕt(ω)) · U(ω, t)
+with generator
+A:
+M
+−→
+R2×2
+ω
+�−→
+�
+0
+1
+−β(ω)
+−α(ω)
+�
+(2)
+Differential equations like (1) appear in large scale in physics, engineering, com-
+plex biological systems and numerous applications of mathematics. The quintes-
+sential example is the simple damped pendulum free from external forces where α
+and β are functions depending on ω ∈ M evolving along a flow ϕt : M → M for
+t ∈ R. When α and β are first integrals (i.e. functions that are constant along the
+orbits of the flow ϕt) related with ϕt, then (1) can be solved by simple algorithms of
+2
+
+an elementary course on differential equations. When the parameters vary in time,
+explicit solutions could be hard to get. This is the case when the frictional force α
+and the frequency of the oscillator β change over time which, we must admit, is the
+most plausible to happen in nature. Notice that generators like A in (2) generate a
+particular class of solutions. Clearly, when α � 0 the solutions evolve on a sub-
+class of the general linear group GL(2, R) and when α = 0 the solutions evolve on
+a subclass of the special linear group SL(2, R). Therefore, a specific study should
+be made taking into consideration that perturbations must belong to our class and
+not to the wider class of generators of cocycles evolving in GL(2, R) or even in
+SL(2, R). Questions related to this particular class were treated in several works
+like e.g. [8, 9, 12, 24, 27, 3].
+Fixing position and momentum (x(0), ˙x(0)) we intend to study the asymptotic
+behavior when t → ∞ of the pair (x(t), ˙x(t)) namely asymptotic exponential growth
+rate given by the Lyapunov exponent. In the present work and broadly speaking we
+intend to answer the following question:
+Is it possible to perturb the coefficients α and β, in an Lp-topology,
+in order to obtain two distinct Lyapunov exponents?
+Of course that, when considering the autonomous case in (2), say α and β not
+depending on ω previous question is easily answered. Indeed, consider Aβ in (2)
+with α = 0, then A0 has a solution with trivial Lyapunov spectrum (a single Lya-
+punov exponent equal to 0) but any Aβ with small β � 0 will produce a solution
+with simple Lyapunov spectrum (two Lyapunov exponents equal to ± √β). The
+difficulty increases significantly when we consider the non-autonomous case.
+The precise concepts that allow an adequate formalisation to express the above
+question will be presented in Theorem 1 and Corollaries 1 and 2.
+2. Definitions and statement of the results
+2.1. Linear cocycles. In this section we present some definitions that will be use-
+ful in the sequel. Let (M, M, µ) be a probability space and let ϕ: R × M → M be a
+metric dynamical system (or flow) in the sense that is a measurable map and
+(1) ϕt : M → M given by ϕt(ω) = ϕ(t, ω) preserves the measure µ for all t ∈ R;
+(2) ϕ0 = IdM and ϕt+s = ϕt ◦ ϕs for all t, s ∈ R.
+Unless stated otherwise we will consider along the text that the flow is ergodic
+in the usual sense that there exist no invariant sets except zero measure sets and
+their complements. Let B(X) be the Borel σ-algebra of a topological space X.
+A (continuous-time) linear random dynamical system (RDS) on (R2, B(R2)), or a
+(continuous-time) linear cocycle, over ϕ is a (B(R) ⊗ M/B(GL(2, R))-measurable
+map
+Φ : R × M → GL(2, R)
+such that the mappings Φ(t, ω) forms a cocycle over ϕ, i.e.,
+(1) Φ(0, ω) = Id for all ω ∈ M;
+(2) Φ(t + s, ω) = Φ(t, ϕs(ω)) ◦ Φ(s, ω), for all s, t ∈ R and ω ∈ M,
+and t �→ Φ(t, ω) is continuous for all ω ∈ M. We recall that having ω �→ Φ(t, ω)
+measurable for each t ∈ R and t �→ Φ(t, ω) continuous for all ω ∈ M implies that
+Φ is measurable in the product measure space. These objects are also called linear
+differential systems (LDS) in the literature.
+3
+
+2.2. Kinetic linear cocycles. We begin by considering as motivation the non-
+autonomous linear differential equation which describes a motion of the damped
+harmonic oscillator as the simple pendulum along the path (ϕt(ω))t∈R, with ω ∈ M
+described by the flow ϕ. Let K ⊂ R2×2 be the set of matrices 2 × 2 of type
+�0
+1
+b
+a
+�
+(3)
+with a, b ∈ R. Denote by G the set of measurable applications A : M → R2×2
+and by K ⊂ G the set of kinetic measurable applications A : M → K. As usual
+we identify two applications on G that coincide on a µ full measure subset of M.
+Consider measurable maps α: M → R and β: M → R. Take the differential
+equation given in (1). Considering y(t) = ˙x(t) we may rewrite (1) as the following
+vectorial first order linear system
+˙X = A(ϕt(ω)) · X,
+(4)
+where X = X(t) = (x(t), y(t))T = (x(t), ˙x(t))T and A ∈ K is given by (2). For all
+1 ≤ p < ∞ we define
+Gp =
+�
+A ∈ G:
+�
+M
+∥A∥pdµ < ∞
+�
+,
+where ∥ · ∥ denotes de standard Euclidean matrix norm. It is clear that for all
+1 ≤ p < q < ∞ we have Gq ⊂ Gp. It follows from [5, Thm. 2.2.2] (see also
+Lemma 2.2.5 and Example 2.2.8 in this reference) that if A ∈ G1 then it generates
+a unique (up to indistinguishability) linear RDS ΦA satisfying
+ΦA(t, ω) = Id +
+� t
+0
+A(ϕs(ω)) · ΦA(s, ω) ds.
+(5)
+The solution ΦA(t, ω) defined in (5) is called the Carathéodory solution or weak
+solution. Given an initial condition X(0) = v ∈ R2, we say that t �→ ΦA(t, ω)v
+solves or is a solution of (4), or that (4) generates ΦA(t, ω). Note that ΦA(0, ω)v = v
+for all ω ∈ M and v ∈ R2. If the solution (5) is differentiable in time (i.e. with
+respect to t) and satisfies for all t
+d
+dtΦA(t, ω)v = A(ϕt(ω)) · ΦA(t, ω)v
+and
+ΦA(0, ω)v = v,
+(6)
+then it is called a classical solution of (4). Of course that t �→ ΦA(t, ω)v is con-
+tinuous for all ω and v. Due to (6) we call A : M → K a (kinetic) ‘infinitesimal
+generator’ of ΦA. Sometimes, due to the relation between A and ΦA, we refer
+to both A and ΦA as a kinetic linear cocyle/RDS/LDS. If (4) has initial condition
+X(0) = v then ΦA(0, ω)v = v and X(t) = ΦA(t, ω)v.
+Let K0 ⊂ K stand for the traceless kinetic cocycles derived from matrices as
+in (3) but with a = 0. For 1 ≤ p < ∞ set K p = K ∩ Gp and K p
+0 = K0 ∩ Gp ⊂ K p.
+2.3. The Lp topology. We begin by defining an Lp-like topology generated by a
+metric that compares the infinitesimal generators on G. Given 1 ≤ p < ∞ and
+A, B ∈ G we set
+ˆσp(A, B) :=
+�����������
+��
+M
+∥A(ω) − B(ω)∥p dµ(ω)
+� 1
+p
+,
+∞ if the above integral does not exists,
+4
+
+and define
+σp(A, B) :=
+�������
+ˆσp(A,B)
+1+ ˆσp(A,B),
+if ˆσp(A, B) < ∞
+1,
+if ˆσp(A, B) = ∞ .
+Clearly, σp is a distance in G. It can be understood has a version of the Lp-distance.
+Next topological content results were mainly proved in [3]. The remaining state-
+ments follow straightforwardly.
+Proposition 2.1. Consider 1 ≤ p < ∞. Then:
+(i) σp(A, B) ≤ σq(A, B) for all 1 ≤ p ≤ q < ∞ and all A, B ∈ G.
+(ii) If A ∈ G1 then sup0≤t≤1 log+ ∥ΦA(t, ω)±1∥ ∈ L1(µ).
+(iii) If A ∈ Gp then for any B ∈ G satisfying σp(A, B) < p we have B ∈ Gp.
+(iv) The sets (K p, σp) and (K p
+0 , σp) are closed, for all 1 ≤ p < ∞.
+(v) For all 1 ≤ p < ∞, (K p, σp) and (K p
+0 , σp) are complete metric spaces
+and, therefore Baire spaces.
+Next results are elementary in measure theory nevertheless we will use it often.
+They capture the whole idea of making huge perturbations on the uniform norm
+but small perturbations in the σp-distance as long the support is small in measure.
+Lemma 2.2. Let 1 ≤ p < ∞. Given A ∈ Gp and ϵ > 0 there exists δ > 0 such that
+if F ∈ M and µ(F ) < δ, then
+�
+F ∥A(ω)∥p dµ(ω) < ϵ.
+Proof. The proof is made by contradiction. Suppose that exists ϵ > 0 and Fn ∈ M,
+for each n ∈ N, such that µ(Fn) < 1
+2n and
+�
+Fn
+∥A(ω)∥p dµ(ω) ≥ ϵ.
+(7)
+Letting F = lim supn Fn, by the Borel-Cantelli lemma µ(F ) = 0, and so
+�
+F
+∥A(ω)∥p dµ(ω) = 0.
+(8)
+The following leads to a contradiction:
+ϵ
+(7)
+≤
+lim sup
+�
+Fn
+∥A(ω)∥p dµ(ω) = lim sup
+�
+∥A(ω)∥pχFn(ω) dµ(ω)
+⋆≤
+�
+lim sup ∥A(ω)∥pχFn(ω) dµ(ω) =
+�
+∥A(ω)∥pχF (ω) dµ(ω)
+=
+�
+F
+∥A(ω)∥p dµ(ω)
+(8)= 0,
+where in ⋆ we used the reverse Fatou lemma.
+□
+Corollary 2.3. Let 1 ≤ p < ∞, A ∈ Gp and ϵ > 0 be given. Consider B ∈ Gp such
+that A(ω) � B(ω) if and only if ω ∈ F for some F ∈ M (that is, B only differs from
+A in F ). Then there exists δ > 0 such that if µ(F ) < δ we have σp(A, B) < ϵ.
+Proof. Is is enough to prove that ˆσp(A, B) < ϵ. For that, apply Lemma 2.2 for
+(A − B) ∈ Gp and ϵ p.
+□
+5
+
+2.4. Statement of Theorem 1 and a tour on its proof. Let 1 ≤ p < ∞ and
+A ∈ K p. Since K p ⊂ K1 ⊂ G1, from Proposition 2.1 the cocycle ΦA satisfies the
+following integrability condition
+sup
+0≤t≤1
+log+ ∥ΦA(t, ω)±1∥ ∈ L1(µ).
+(9)
+Hence, under condition (9) Oseledets theorem (see e.g. [31, 5]) guarantees that
+for µ almost every ω ∈ M, there exists a ΦA-invariant splitting, called Oseledets
+splitting, of the fiber R2
+ω = E1
+ω ⊕ E2
+ω and real numbers λ1(A, ω) ≥ λ2(A, ω), called
+Lyapunov exponents, such that:
+λ(A, ω, vi) := lim
+t→±∞
+1
+t log ∥ΦA(t, ω)vi∥ = λi(A, ω),
+for any vi ∈ Ei
+ω \ {⃗0} and i = 1, 2. If the flow ϕt is ergodic, then the Lyapunov
+exponents (and the dimensions of the associated subbundles) are constant µ almost
+everywhere, and we refer to them as λ1(A) and λ2(A), with λ1(A) ≥ λ2(A). We say
+that A (or ΦA) has one-point Lyapunov spectrum or trivial Lyapunov spectrum if
+for µ a.e. ω ∈ M, λ1(A, ω) = λ2(A, ω). Otherwise we say A (or ΦA) has simple
+Lyapunov spectrum. For details on these results see [5] (in particular, Example
+3.4.15).
+We are now in conditions to state our main result that establishes the existence
+of a σp-dense subset of K p displaying simple spectrum:
+Theorem 1. Let ϕt : M → M be ergodic. For any 1 ≤ p < ∞, A ∈ K p and ϵ > 0,
+there exists B ∈ K p exhibiting simple Lyapunov spectrum satisfying σp(A, B) < ϵ.
+This result shows in particular that the σp-generic subset of K p in which the
+trivial spectrum prevails, obtained in [3], can not contain σp-open sets. The stra-
+tegy to prove that for each kinetic cocycle satisfying the integrability condition
+there is another kinetic cocycle, arbitrarily close with a simple spectrum, borrow
+some ideas of [7, 13] where the authors obtained a similar result for the discrete
+time case and for more general cocycles. However, the context of continuous-time
+cocycles and the restriction to a very particular family of cocycles, such as the one
+we are considering in this paper, bring several difficulties that have no similarities
+in previous works. We have to face the situation that kinetic cocycles are rigid1
+and to obtain the desired perturbation we will make a step-by-step perturbation
+algorithm that we now describe:
+(1) We begin by coding ϕ by a special flow to avoid overlaps and then consider
+a thin time-1 flowbox VR concatenated to an also thin time-1 flowbox VS ,
+so that o VR ∪ VS will be a time-2 flowbox;
+(2) We cut the original dynamics in VR (respectively VS ) and paste a simple
+constant traceless infinitesimal generator R2π, whose solution basically ro-
+tates an angle 2πη in time-η. Outside VR ∪ VS we keep the same dynamic
+of A. By simple we mean that we can easily obtain the identity by just
+doing a time-1 iteration. Call A0 this new cocycle;
+1 The pertubative arguments in [7, 13] were easier to make because since dim SL(2, R) = 3 three
+degrees of freedom were available. In our kinetic scenario we have to perform the same perturbations
+but with only a single degree of freedom.
+6
+
+(3) Since VR ∪ VS is a thin flowbox, A0 will be arbitrarily σp-near A. If A0
+has simple spectrum we are over, otherwise we prove Theorem 1 for A0
+instead of A;
+(4) Inside VR we cut the dynamics of A0 and paste a tailor-made rotation R
+such that for each ω entering in VR we rotate in time-1 a vector vω into a
+fixed special direction given by v = (1, 1). The vector vω will be used to
+forcefully create an Oseledets direction so we can calculate the Lyapunov
+exponents. Here we rotate any angle by a small σp-perturbation since by
+(1) VR is thin. A key observation is that the trace keeps unchanged, and
+that is the main motivation to the previous placement of R2π on VR. Call
+B0 this new cocycle. If B0 has simple spectrum we are over, otherwise we
+prove Theorem 1 for B0 instead of A0;
+(5) Inside VS we cut the dynamics of B0 and paste a constant infinitesimal
+generator S which stretch the vector v in time-1 by a known magnitude e.
+No problem arises with the (eventually large) size of the uniform norm of
+the perturbation because the σp-distance is small due to the thickness of
+VS . Again the trace keeps unchanged. Call B this new cocycle;
+(6) Now we use ergodicity and compute the Lyapunov exponents of points
+who will inevitably have to return to VR ∪ VS infinitely many times;
+(7) The stretch S is a perturbation that is concerned with providing an expan-
+sion along an invariant direction. As it is difficult to find different kinetic
+cocycles which keep the same invariant directions here it becomes clear
+why we have chosen back there the identity after time 1 (more precisely a
+rotation by 2π) given by R2π;
+(8) Finally, the concern to keep the trace constant in (4) and (5) will bear fruit
+since if a perturbation increases a Lyapunov exponent and simultaneously
+the sum of the two Lyapunov exponents of the original cocycle and the
+perturbed one remains the same, then only one thing could have happened:
+the perturbed cocycle cannot have trivial spectrum but instead must display
+a Lyapunov exponent smaller than the Lyapunov exponent of the original
+cocycle.
+The following table summarises the step-by-step construction from the linear
+differential systems A to B:
+Table 1. Step-by-step description of the several perturbations
+Cocycle
+M \ (VR ∪ VS )
+VR
+VS
+A
+A
+A
+A
+A0
+A
+R2π
+R2π
+B0
+A
+R
+R2π
+B
+A
+R
+S
+We use an approach slightly different from the previous works [7, 13, 6, 18].
+Moreover, to avoid overlapping in the perturbations, we will encode the base flow
+through a special flow in a Kakutani Castle (as in [2, 33]). On the other hand, to
+estimate the proximity of the perturbed cocycle to the original one, we also use a
+control over the measure of VR ∪VS that support the two perturbations taking into
+account Corollary 2.3.
+7
+
+It should be noted that, in addition to the difficulties inherent in the context
+of continuous-time cocycles, performing these perturbations (rotation and stretch)
+are not trivial, as we do not have the usual mechanisms like those that exist in
+the context in cocycles that evolve in GL(2, R) or SL(2, R), or, more generally,
+cocycles that satisfy the accessibility condition (also recognized as twisting) and
+saddle-conservative (also known as pinching), which allow the realization of these
+processes in a less demanding way, as, for example, in [4, 7, 15, 16, 13].
+As our perturbations are all traceless we get from Theorem 1 that conservative
+kinetic cocycles have non-zero Lyapunov exponents σp-densely.
+Corollary 1. Let ϕt : M → M be ergodic. For any 1 ≤ p < ∞, A ∈ K p
+0 and
+ϵ > 0, there exists B ∈ K p
+0 exhibiting non-zero Lyapunov exponents satisfying
+σp(A, B) < ϵ.
+Finally, we present Corollary 1 with a somewhat different look, namely by con-
+sidering the one-dimensional Schrödinger operator on L2(R) and with an Lp poten-
+tial Q: M → R given by:
+Hω :
+L2(R)
+−→
+L2(R)
+φ
+�−→
+�
+− d2
+dt2 + Q(ϕt(ω))
+�
+φ
+(10)
+In particular we like to describe the Lyapunov spectrum of the time-independent
+Schrödinger equation
+Hωφ = Eφ,
+(11)
+where E ∈ R is a given energy. Putting together (10) and (11) we deduce a kinetic
+cocycle as in (2) but with α(ω) = 0 and β(ω) = E − Q(ω) for all ω ∈ M. We fix the
+energy E and focus on the LDS
+AE :
+M
+−→
+R2×2
+ω
+�−→
+�
+0
+1
+−E + Q(ω)
+0
+�
+(12)
+called one-dimensional Schrödinger LDS with potential Q. As a direct conse-
+quence of Corollary 1 we have:
+Corollary 2. Let ϕt : M → M be ergodic. Given 1 ≤ p < ∞, ϵ > 0 and a one-
+dimensional Schrödinger LDS with a fixed energy E as in (12) and with potential
+Q, there exists ˜Q such that the one-dimensional Schrödinger LDS with the same
+energy E and potential ˜Q exhibits non-zero Lyapunov exponents and ∥ ˜Q−Q∥Lp < ϵ.
+3. On the perturbations
+3.1. Special flows. Consider a measure space Σ, a map T : Σ → Σ, a T -invariant
+probability measure ˜µ defined in Σ and a roof function h: Σ → R+ satisfying
+h(ω) ≥ H > 0, for some H > 0 and all ω ∈ Σ, and
+�
+Σ h(ω)d˜µ(ω) < ∞. Define the
+space Mh ⊆ Σ × R+ by
+Mh = �(ω, t) ∈ Σ × R+ : 0 ≤ t ≤ h(ω)�
+with the identification between the pairs (ω, h(ω)) and (T (ω), 0). The semiflow
+defined on Mh by S s(ω, r) = (T n(ω), r + s − �n−1
+i=0 h(T i(ω))), where n ∈ N is
+uniquely defined by
+n−1
+�
+i=0
+h(T i(ω)) ≤ r + s <
+n
+�
+i=0
+h(T i(ω))
+8
+
+is called a suspension semiflow. If T is invertible then (S t)t is a flow. Furthermore,
+if ℓ denotes the one dimensional Lebesgue measure the measure µ = (˜µ×ℓ)/
+�
+h d˜µ
+defined on Mh by
+�
+g dµ =
+1
+�
+h d˜µ
+� �� h(ω)
+0
+g(ω, t)dt
+�
+d˜µ(ω),
+∀g ∈ C0(Mh)
+is a probability measure and it is invariant by the suspension semiflow (S t)t. Flows
+with such representation are called special flows (or flows built under a function)
+and are denoted by (ϕt, Σ, T , h). It is well-known (see [1, Theorem 2]) that any
+ergodic flow is isomorphic to a special flow. Along this work we assume that the
+base flow is a special flow (ϕt, Σ, T , h) and, without any loss of generality, that
+H > 2. To avoid overloading the notation we write M instead of Mh.
+3.2. Perturbations supported in time-τ flowboxes. Take A ∈ G and a non-
+periodic orbit ω ∈ M.
+We will consider a perturbation B = Bω,τ of A only
+along a segment of the orbit of ω with extremes ω and ϕτ(ω) for τ > 0. Let
+P ∈ G be given and define B: M → R2×2 such that B( ˆω) = A( ˆω) for all ˆω outside
+ϕ[0,τ](ω) = {ϕs(ω) : s ∈ [0, τ]} and B( ˆω) = P( ˆω) otherwise. The map B is called a
+(local) perturbation of A by P supported on ϕ[0,τ](ω). Given Σ0 ⊂ Σ and 0 ≤ a < b
+we define the set
+ϕ[a,b](Σ0) =
+�
+ϕt(ω): ω ∈ Σ0, t ∈ [a, b]
+�
+.
+Given A ∈ G1, P ∈ G, Σ0 ⊂ Σ and a > 0, we may extend the local perturbations
+of A by P to be supported on the flowbox ϕ[a,b](Σ0), with 0 ≤ a < b < H, in the
+following way: for ω ∈ ϕ[a,b](Σ0) we project ω in ˜ω ∈ ϕa(Σ0) i.e. ω = ϕr( ˜ω), for
+some 0 ≤ r ≤ b−a, and let B ˜ω,b−a be (local) perturbation of A by P = P ˜ω supported
+on ϕ[0,b−a]( ˜ω) and define
+B(ω) :=
+� A(ω),
+if ω � ϕ[a,b](Σ0)
+B ˜ω,b−a(ω),
+if ω ∈ ϕ[a,b](Σ0) .
+To distinguish the situations we refer for B(ω) as a global perturbation of A by
+P supported in ϕ[a,b](Σ0), where we always suppose that P(ω) = P ˜ω(ω) for all
+ω ∈ ϕ[a,b](Σ0).
+3.3. Rotating and Stretching. Next two results provide local and global argu-
+ments to rotate over prescribed directions under a small σp-perturbation. This will
+be used to generate a suitable invariant direction. The first one allows us to perform
+a uniform bounded kinetic perturbation in a local segment of orbit which rotates
+a given vector. The second one thickens Lemma 3.1 by broaden the rotation in a
+single orbit to rotations in a flowbox.
+Lemma 3.1. Given ω ∈ M, u, v ∈ R2 \ {0}, A ∈ K p, there is γ � 0, and a
+perturbation Bω,1 ∈ K p of A supported on ϕ[0,1](ω) such that:
+(i) ∥Bω,1( ˆω)∥ ≤ 4π2 for all ˆω on ϕ[0,1](ω), and
+(ii) ΦBω,1(1, ω)u = γ v.
+Proof. Let θ = ∡(Ru, Rv) ∈ ]0, 2π] measured clockwise. Set a constant infinitesi-
+mal generator R: M → R2×2 given by
+R(ω) = Rθ(ω) =
+� 0
+1
+−θ2
+0
+�
+.
+(13)
+9
+
+We consider the perturbation B = Bω,1 ∈ K p of A by R supported on ϕ[0,1](ω).
+The infinitesimal generator in (13) generates a linear differential system with fun-
+damental classical solution (6) given, for all ω ∈ M and t ∈ R by the ‘clockwise
+elliptical rotation’ defined by:
+ΦR(t, ω) =
+�
+cos(θt)
+θ−1 sin(θt)
+−θ sin(θt)
+cos(θt)
+�
+,
+(14)
+and such that ΦB(1, ω)u = ΦR(1, ω)u = γv, for some γ � 0 fulfilling (ii).
+□
+From Corollary 2.3 it follows that we may extend the local perturbation Bω,1
+given by the rotation Rθ(ω) as in Lemma 3.1, to a global perturbation, tuned for
+each orbit segment, to obtain a new generator that is σp-close to the original, once
+we have a smaller measure of the flowbox were the perturbation takes place. This
+is pointed in the next basic measure theoretic result which is an immediate conse-
+quence of Corollary 2.3.
+Lemma 3.2 (Global). For all 1 ≤ p < ∞, A ∈ Gp, a > 0 and ϵ > 0, there exists
+a measurable set Σ0 ⊂ Σ with ˜µ(Σ0) > 0 such that for any global perturbation
+B ∈ Gp of A supported in the flowbox ϕ[a,a+1](Σ0), with ∥B(ϕt(ω))∥ ≤ 4π2 for all
+ω ∈ Σ0 and t ∈ [a, a + 1], we have that σp(A, B) < ϵ.
+Let us fix a suitable constant and traceless infinitesimal generator
+S =
+�0
+1
+1
+0
+�
+.
+(15)
+As S has simple expression we integrate it obtaining:
+ΦS (t, ω) = eS t =
+�cosh t
+sinh t
+sinh t
+cosh t
+�
+(16)
+We notice that (16) has eigenvalues σS
+1 = et and σS
+2 = e−t with associated eigen-
+vectors vS
+1 = (1, 1) and vS
+2 = (−1, 1), respectively. Observe that ES
+1 = R · vS
+1 is a
+unstable direction and ES
+2 = R · vS
+2 is a stable direction.
+Next trivial remark will be of utmost importance in the sequel because it com-
+bines three main ingredients: invariance of certain 1-dimensional directions, some
+expansiveness along this direction and all this done in traceless kinetic infinitesi-
+mal generators.
+Remark 3.1 (Invariance and stretch). Considering θ = 2π in (14), say R2π, we get
+e · vS
+1 = e · ΦR2π(1, ω) vS
+1 = ΦS (1, ω) vS
+1 .
+(17)
+4. Proof of Theorem 1
+Let A ∈ K p, 1 ≤ p < ∞ and ϵ > 0 be given. We assume that ΦA has a single
+Lyapunov exponent λ(A). The sequence of perturbations are summarized in Table
+1.
+10
+
+4.1. Defining A0 (picking out good coordinates): Let Σ0 ⊂ Σ be as in Lemma 3.2.
+For r > 0 we assume that we have flowboxes defined by VR := ϕ[0,1](Br) and
+VS := ϕ[1,2](Br), where Br ∈ Σ0 is such that 0 < ˜µ(Br) ≤ r. Consider A0 ∈ K1
+defined as:
+A0(ω) :=
+� A(ω),
+if ω � VR ∪ VS
+R2π,
+if ω ∈ VR ∪ VS
+.
+By Corollary 2.3 if r is sufficiently small when compared with ϵ we get
+σp(A, A0) < ϵ
+3.
+(18)
+If ΦA0 has simple spectrum we are over. Otherwise, we prove the theorem for A0
+instead of A.
+4.2. Defining B0 (rotating on VR): Set
+k(ω) = inf
+t≥0
+�
+t: ϕ−t(ω) ∈ ϕ1(Br)
+�
+.
+We will define the a random vector field g(ω). We start with the normalized image
+under the cocycle associated with ΦA0 of the vector v =
+vS
+1
+∥vS
+1 ∥ =
+� √
+2
+2 ,
+√
+2
+2
+�
+:
+g(ω) :=
+���������
+v,
+if
+ω ∈ ϕ1(Br)
+ΦA0(k(ω),ϕ−k(ω)(ω))v
+∥ΦA0(k(ω),ϕ−k(ω)(ω))v∥,
+if
+ω � (VR \ Br)
+and set from now on E(ω) = span {g(ω)}.
+Let B0 be a perturbation of A0 supported in the flowbox VR as in Lemma 3.2
+such that for all ω ∈ Br we have ΦB0(1, ω)g(ω) = κv for some κ ∈ R, that is:
+B0(ω) :=
+� R(ω),
+if ω ∈ VR
+A0(ω),
+otherwise
+.
+Observe that the rotation must be tuned for each ω0 ∈ Br, in the sense that for
+ω = ϕt(ω0) ∈ VR, with 0 ≤ t ≤ 1, we set R(ω) = Rθ(ω0) with θ = ∡(g(ω0), v). In
+particular, for all ω0 ∈ Br we have Φ(1, ω0)g(ω) = κv, for some κ ∈ R. Moreover,
+A0 and B0 have the same trace. Indeed, A0 = B0 outside VR and in VR we have
+B0 = R and A0 = R2π, which are both traceless (see (13)). Therefore, by Liouville’s
+formula for all ω and t ≥ 0
+det ΦB0(t, ω) = det ΦA0(t, ω).
+(19)
+For ω ∈ VR \ Br define
+g(ω) = ΦB0(k(ω), ϕ−k(ω)(ω))v
+∥ΦB0(k(ω), ϕ−k(ω)(ω))v∥.
+(20)
+Notice that for ω ∈ Br, since ΦB0(1, ω)Rg(ω) = Rv we get
+ΦB0(1, ω)Rg(ω)(ω) = Rg(ϕ1(ω)).
+(21)
+Let ˜ω ∈ ϕ1(Br) and τ > 0 be such that ϕt( ˜ω) � VR for all t ∈]0, τ[. Then, for all
+t ∈ [0, τ] we have the ΦB0-invariance of g:
+ΦB0(t, ˜ω)Rg( ˜ω) = ΦB0(t, ˜ω)Rv = ΦA0(t, ˜ω)Rv = Rg(ϕt( ˜ω)).
+(22)
+11
+
+If ϕt( ˜ω) ∈ VR for some t ∈]0, τ[ then considering s > 0 such that ϕs(ω) ∈ Br we
+get:
+ΦB0(t, ˜ω)Rg( ˜ω)
+=
+ΦB0(t − s, ϕs( ˜ω))ΦA0(s, ˜ω)Rv
+=
+ΦB0(t − s, ϕs( ˜ω))Rg(ϕs( ˜ω))
+(20)
+=
+Rg(ϕt( ˜ω)).
+Finally, (21), (26) and last equality gives that the vector field g is ΦB0-invariant.
+Again by Corollary 2.3 if r is sufficiently small we get
+σp(A0, B0) < ϵ
+3.
+(23)
+If ΦB0 has simple spectrum we are over. Otherwise, we prove the theorem for
+B0 instead of A0.
+4.3. Defining B (stretching on VS ): We define
+B(ω) :=
+� B0(ω),
+if ω � VS
+S,
+if ω ∈ VS
+.
+Observe that B and B0 have the same trace. Indeed, B = B0 outside VS and in
+VS we have B0 = R2π which are both traceless (see (13) and (15)). Therefore, by
+Liouville’s formula and (19) for all ω and t ≥ 0
+det ΦB(t, ω) = det ΦB0(t, ω) = det ΦA0(t, ω).
+(24)
+From Corollary 2.3, once more, if r is sufficiently small we get
+σp(B0, B) < ϵ
+3.
+(25)
+Notice that the invariance of the direction E(ω) under ΦB fails when ϕt(ω) enters
+VS . However, for ˜ω ∈ ϕ1(Br) we have by (17) and (26)
+ΦB(1, ˜ω)Rg( ˜ω) = ΦS (1, ˜ω)Rv = Rv = RΦR2π(1, ˜ω)v = ΦA0(1, ˜ω)Rv = Rg(ϕ1( ˜ω))
+and so
+ΦS (1, ˜ω)E( ˜ω) = E(ϕ1( ˜ω)),
+(26)
+which will be enough for our purposes; see Figure 1.
+Figure 1. The traceless perturbation scheme with the invariant di-
+rections and the stretch effect.
+Let λ1(B) ≥ λ2(B) be the Lyapunov exponents of ΦB. We assume that ΦB0 has
+one-point spectrum, say λ1(B0) = λ2(B0) = λ(B0), because otherwise the theorem
+12
+
+dB(T, W)g()
+ΦB(1,
+B(1, T())
+P Bo
+3is proved. Let λ(B0) be the single Lyapunov exponent of ΦB0. Hence we have
+λ(B0) = λ(B0, ω, vS
+1 ) for a.e. ω. By the Oseledets theorem we have
+2λ(B0) =
+�
+log
+���det(ΦB0(1, ω))
+���dµ
+(27)
+and
+λ1(B) + λ2(B) =
+�
+log
+���det(ΦB(1, ω))
+���dµ.
+(28)
+The two previous equalities together with (24) allows us to conclude that
+2λ(B0) = λ1(B) + λ2(B)
+(29)
+and so, if we show that λ1(B) > λ(B0) then we get λ1(B) > λ2(B) and Theorem 1
+is proved. Recall that the random vector field g is invariant by ΦB0 but in what ΦB
+concerns, the invariance fails as the base dynamics enters VS . However, by (26)
+the invariance is recovered in the moment the base dynamics is leaving VS .
+For ω ∈ M let us consider the real map b0(·, ω) for all t ∈ R in such a way that
+b0(t, ω)g(ϕt(ω)) = ΦB0(t, ω)g(ω).
+(30)
+Claim 4.1. The map b0(t, ω) forms a cocycle over ϕt.
+Indeed, since ΦB0(0, ω) = Id for all ω ∈ M we have b0(0, ω) = 1 and for all s, t,
+evaluating b0(t + s, ω) at g(ϕt+s(ω)), we have
+b0(t + s, ω)g(ϕt+s(ω))
+(30)
+=
+ΦB0(t + s, ω)g(ω)
+=
+ΦB0(t, ϕs(ω)) · ΦB0(s, ω)g(ω)
+(30)
+=
+ΦB0(t, ϕs(ω)) · b0(s, ω)g(ϕs(ω))
+=
+b0(s, ω) ΦB0(t, ϕs(ω))g(ϕs(ω))
+=
+b0(t, ϕs(ω))b0(s, ω)g(ϕt+s(ω)),
+and so b0(t + s, ω) = b0(t, ϕs(ω))b0(s, ω).
+Since the random vector field g is not completely invariant by ΦB we consider
+two distinct situations. Set ϕ{1,2}(Br) = ϕ1(Br) ∪ ϕ2(Br). For ω ∈ M and τ ≥ 0 such
+that ϕt(ω) � VS \ ϕ{1,2}(Br), for all 0 ≤ t ≤ τ, we consider the real map b(·, ω) for
+all t ∈ [0, τ] in such a way that
+b(t, ω)g(ϕt(ω)) = ΦB(t, ω)g(ω)
+(31)
+and, for all ω ∈ ϕ1(Br), we set b(1, ω) ∈ R in such a way that
+ΦB(1, ω)g(ω) = b(1, ω)g(ϕ1(ω)).
+(32)
+If ϕt(ω) � VS \ ϕ{1,2}(Br), for all 0 ≤ t ≤ τ, we have B(ϕt(ω)) = B0(ϕt(ω)) and
+b(t, ω)g(ϕt(ω)) = ΦB(t, ω)g(ω) = ΦB0(t, ω)g(ω) = b0(t, ω)g(ϕt(ω)).
+(33)
+In particular this holds between the output of VS to the next input in VS .
+Claim 4.2. If ϕt(ω), ϕs(ω) � VS \ ϕ{1,2}(Br), b(t, ω) forms a cocycle over ϕt in the
+sense that b(t + s, ω) = b(t, ϕs(ω))b(s, ω).
+13
+
+The proof follows similarly to Claim 4.1 taking also into account (32).
+Pick ω in a full measure subset of points that visits infinitely often Br and for
+which the conclusion of Birkhoff’s Ergodic theorem holds. Without loss of gene-
+rality we may assume that ω � Vr ∪ VS . For t ≥ 0 set
+Jt(ω) = #
+�
+j ∈ N: j ≤ t, ϕ j(ω) ∈ ϕ2(Br)
+�
+.
+Recall that
+λ(B, ω, g(ω))
+=
+lim
+t→∞
+1
+t log ∥ΦB(t, ω)g(ω)∥,
+and we may split the previous orbit in the limit by considering the time for ϕt(ω)
+to enter VS , the time-1 moment crossing the flowbox VS , where we use (17), and,
+again, the time it takes to return to VS and so on. For simplicity, let us define
+recursively
+s0 = s0(ω) = min{t: ϕt(ω) ∈ ϕ1(Br)},
+ℓ0 = ℓ0(ω) = s0 + 1,
+sn = s0(ϕℓn−1(ω)) and ℓn = sn + 1, for n ≥ 1,
+∆n = sn − ℓn−1, for n ≥ 1,
+˜ωn = ϕsn(ω) ∈ ϕ1(Br) and ˆωn = ϕℓn(ω) ∈ ϕ2(Br), for n ≥ 1.
+Now, in one hand, since B0 has one-point spectrum, for µ-a.e. ω,
+λ(B0, ω)
+=
+λ(B0, ω, g(ω))
+=
+lim
+t→∞
+1
+t log ∥ΦB0(t, ω)g(ω)∥
+(30)
+=
+lim
+t→∞
+1
+t log |b0(t, ω)|.
+(34)
+On the other hand, by Remark 3.1 and (32) we have for ˜ω ∈ ϕ1(Br) that
+b(1, ˜ω)g(ϕ1( ˜ω)) = ΦB(1, ˜ω)g( ˜ω)
+(17)
+= e·ΦB0(1, ˜ω)g( ˜ω) = e·b0(1, ˜ω)g(ϕ1( ˜ω)). (35)
+Without loss of generality, we can consider the following limits over the un-
+bounded set {t ≥ 0: ϕt(ω) ∈ ϕ1(Br)}. From Birkhoff’s Ergodic theorem we have
+λ(B, ω, g(ω))
+=
+lim
+t→∞
+1
+t log ∥ΦB(t, ω)g(ω)∥
+(31)+(32)
+=
+lim
+t→∞
+1
+t
+���������log |b(s0, ω)| +
+Jt(ω)−1
+�
+j=0
+log |b(∆j+1, ˆωsj)b(1, ˜ωsj)|
+���������
+(33)+(35)
+=
+lim
+t→∞
+1
+t
+���������log |b0(s0, ω)| +
+Jt(ω)−1
+�
+j=0
+log |b0(∆j+1, ˆωsj) · e · b0(1, ˜ωsj)|
+���������
+Claim 4.1
+=
+lim
+t→∞
+1
+t log |b0(t, ω)| + lim
+t→∞
+Jt(ω)
+t
+(30)
+=
+lim
+t→∞
+1
+t log ∥ΦB0(t, ω)g(ω)∥ + lim
+t→∞
+1
+t
+� t
+0
+1VS (ϕt(ω)) dt
+=
+λ(B0, ω, g(ω)) + µ(VS ),
+which implies λ1(B, ω) > λ(B0, ω), hence λ1(B) > λ(B0). From (29), we get
+λ1(B) > λ(B0) > λ2(B) so that B has simple spectrum. Moreover, by (18), (23) and
+(25) we have σp(A, B) < ϵ and Theorem 1 is now proved. □
+14
+
+Clearly when considering the set K1
+0 on Corollary 1 the equalities (27) and (28)
+become 2λ(B0) = λ1(B) + λ2(B) = 0. Hence the conclusion this time will be that
+λ1(B) > 0 for B ∈ K1
+0 arbitrarily σp-close to A and also λ2(B) = −λ1(B) < 0.
+Acknowledgements: The authors were partially supported by FCT - ‘Fundação
+para a Ciência e a Tecnologia’, through Centro de Matemática e Aplicações (CMA-
+UBI), Universidade da Beira Interior, project UIDB/MAT/00212/2020. MB was
+partially supported by the Project ‘Means and Extremes in Dynamical Systems’
+(PTDC/MAT-PUR/4048/2021). MB also like to thank CMUP for providing the
+necessary conditions in which this work was developed.
+References
+[1] W. Ambrose, Representation of ergodic flows, Annals of Mathematics 42 (1941), 3, 723–739.
+[2] W. Ambrose, S. Kakutani, Structure and continuity of measure preserving transformations,
+Duke Math. J., 9: (1942), 25–42.
+[3] D. Amaro, M. Bessa, H. Vilarinho Genericity of trivial Lyapunov spectrum for Lp-cocycles
+derived from second order linear homogeneous differential equations (Submitted).
+[4] A. Arbieto, J. Bochi, Lp-generic cocycles have one-point Lyapunov spectrum, Stochastics and
+Dynamics 3 (2003) 73–81. Corrigendum. ibid, 3 (2003) 419–420.
+[5] L. Arnold, Random Dynamical Systems, Springer Verlag, 1998.
+[6] L. Arnold, N. Cong, Linear cocycles with simple Lyapunov spectrum are dense in L∞, Ergod.
+Th. & Dynam. Sys., 19, (1999) 1389–1404.
+[7] L. Arnold, N. Cong, On the simplicity of the Lyapunov spectrum of products of random matri-
+ces, Ergod. Th. & Dynam. Sys. 17 (1997) 1005–1025.
+[8] L. Arnold, H. Crauel, J.-P. Eckmann, editors Lyapunov Exponents. Proceedings, Oberwolfach
+1990, volume 1486 of Springer Lecture Notes in Math. Springer-Verlag, Berlin Heidelberg New
+York, 1991.
+[9] L. Arnold, V. Wihstutz, editors, Lyapunov Exponents. Proceedings, Bremen 1984, volume 1186
+of Springer Lecture Notes in Mathematics. SpringerVerlag, Berlin Heidelberg New York, 1986.
+[10] A. Avila, Density of positive Lyapunov exponents for S L(2, R)-cocycles, J. Amer. Math. Soc.
+24 (4) (2011) 999–1014.
+[11] E. Cornelis, M. Wojtkowski, A criterion for the positivity of the Liapunov characteristic expo-
+nent, Ergod. Theory & Dyn. Syst. 4 (1984) 527–539.
+[12] M. Bessa, Perturbations of Mathieu equations with parametric excitation of large period, Ad-
+vances in Dynamical Systems and Applications, 7, 1, (2012) 17–30.
+[13] M. Bessa, H. Vilarinho, Fine properties of Lp-cocycles which allows abundance of simple and
+trivial spectrum. Journal of Differential Equations, 256, 7 (2014) 2337–2367.
+[14] M. Bessa, J. Bochi, M. Cambrainha, C. Matheus, P. Varandas, D. Xu, Positivity of the Top
+Lyapunov Exponent for Cocycles on Semisimple Lie Groups over Hyperbolic Bases, Bull Braz
+Math Soc, New Series (2018) 49:73–87.
+[15] J. Bochi, Genericity of zero Lyapunov exponents, Ergod. Th. & Dynam. Sys. 22 (2002) 1667–
+1696.
+[16] J. Bochi, M.Viana, The Lyapunov exponents of generic volume-preserving and symplectic maps,
+Ann. of Math. 161 (3) (2005) 1423–1485.
+[17] Bonatti, C., Gómez-Mont, X., Viana, M., Généricité d’exposants de Lyapunov non-nuls pour
+des produits déterministes de matrices. Ann. Inst. H. Poincaré Anal. Non Linéaire 20, (2003)
+579–624.
+[18] N. D. Cong, A generic bounded linear cocycle has simple Lyapunov spectrum, Ergod. Th. &
+Dynam. Sys. (2005),25, 1775-1797.
+[19] Duarte, P., Klein, S., Positive Lyapunov exponents for higher dimensional quasiperiodic cocy-
+cles. Commun. Math. Phys. 332(1), (2014) 189–219.
+15
+
+[20] R. Fabbri, Genericity of hyperbolicity in linear differential systems of dimension two, (Italian)
+Boll. Unione Mat. Ital., Sez. A, Mat. Soc. Cult. 8 (1) Suppl. (1998) 109–111.
+[21] R. Fabbri, R. Johnson, Genericity of exponential dichotomy for two-dimensional differential
+systems, Ann. Mat. Pura Appl. IV. Ser. 178 (2000) 175–193.
+[22] R. Fabbri, R. Johnson, On the Lyapounov exponent of certain SL(2, R)-valued cocycles, Differ.
+Equ. Dyn. Syst. 7 (3) (1999) 349–370.
+[23] R. Fabbri, R. Johnson, L. Zampogni, On the Lyapunov exponent of certain SL(2, R)-valued
+cocycles II, Differ. Equ. Dyn. Syst. 18 (1-2) (2010) 135–161.
+[24] X. Feng, K. Loparo, Almost sure instability of the random harmonic oscillator, SIAM J. Appl.
+Math. 50, 3, (1990) 744–759.
+[25] Ledrappier, F.: Positivity of the exponent for stationary sequences of matrices. In: Arnold, L.,
+Wihstutz, V. (eds.) Lyapunov Exponents (Bremen, 1984). Lecture Notes in Mathematics, vol.
+1886, pp. 56–73, Springer, New York (1986)
+[26] T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer, 1980.
+[27] A. Leizarowitz, On the Lyapunov exponent of a harmonic oscillator driven by a finite-state
+Markov process, SIAM J. Appl. Math., 49, 2, (1989) 404–419.
+[28] V. M. Millionshchikov, Systems with integral separateness which are dense in the set of all
+linear systems of differential equations, Differential Equations 5 (1969) 850–852.
+[29] M. Nerurkar, Positive exponents for a dense set of continuous cocycles which arise as solutions
+to strongly accessible linear differential systems, Contemp. Math. Ser. AMS 215 (1998) 265–
+278.
+[30] O. Knill, Positive Lyapunov exponents for a dense set of bounded measurable SL(2, R) cocycles,
+Ergodic Theory Dynam. Systems 12 (2) (1992) 319–331.
+[31] V. Oseledets, A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynami-
+cal systems, Transl. Moscow Math. Soc. 19 (1968) 197-231.
+[32] R. H. Risch, The problem of integration in finite terms, Trans. Amer. Math. Soc. 139 (1969),
+167–189.
+[33] D. Rudolph, A Two-Valued Step Coding for Ergodic Flows, Math. Z. 150 (1976) 201–220.
+[34] M. Viana, Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov expo-
+nents, Ann. of Math. 167 (2) (2008) 643–680.
+[35] D. Xu, Density of positive Lyapunov exponents for symplectic cocycles, J. Eur. Math. Soc., 21,
+10, (2019), 3143–3190.
+Centro de Matem´atica e Aplica¸c˜oes (CMA-UBI), Universidade da Beira Interior, Rua Marquˆes
+d’Ávila e Bolama, 6201-001, Covilh˜a, Portugal.
+Email address: dinis.amaro@ubi.pt
+Email address: bessa@ubi.pt
+Email address: helder@ubi.pt
+16
+

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+STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER
+UNCERTAINTY: A TENSOR TRAIN APPROACH
+HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA
+Abstract. We propose an algorithm to solve optimization problems constrained by partial
+(ordinary) differential equations under uncertainty, with almost sure constraints on the state
+variable.
+To alleviate the computational burden of high-dimensional random variables,
+we approximate all random fields by the tensor-train decomposition. To enable efficient
+tensor-train approximation of the state constraints, the latter are handled using the Moreau-
+Yosida penalty, with an additional smoothing of the positive part (plus/ReLU) function by
+a softplus function. We derive theoretical bounds on the constraint violation in terms of
+the Moreau-Yosida regularization parameter and smoothing width of the softplus function.
+This result also proposes a practical recipe for selecting these two parameters. When the
+optimization problem is strongly convex, we establish strong convergence of the regularized
+solution to the optimal control. We develop a second order Newton type method with a
+fast matrix-free action of the approximate Hessian to solve the smoothed Moreau-Yosida
+problem. This algorithm is tested on benchmark elliptic problems with random coefficients,
+optimization problems constrained by random elliptic variational inequalities, and a real-
+world epidemiological model with 20 random variables. These examples demonstrate mild
+(at most polynomial) scaling with respect to the dimension and regularization parameters.
+1. Introduction
+Over last two decades optimization problems constrained by physical laws, such as partial
+(ordinary) differential equations (PDEs/ODEs), have emerged as a prominent research area.
+This is fueled by many applications in science and engineering, such as controlling pathogen
+propagation in built environment [26, 25], shape and topology optimization [36, 28], optimal
+strategies to predict shutdowns due to pandemics [11]. The optimization variables consist of
+state (y) and control/design (u). However, often due to noisy measurements and ambiguous
+models due to incomplete physics, the underlying physical laws contain uncertainty. This
+has led to significant theoretical and algorithmic developments in the area of optimization
+problems constrained by physical laws under uncertainty. See for instance [23, 4, 14, 3] and
+the references therein. These papers focus on problems with control constraints.
+The literature on state-constrained optimization problems under uncertainty is scarce.
+For instance, [12, 17] use probability constraints, and [15, 13, 16] consider almost surely
+Date: 20 January 2023.
+2020 Mathematics Subject Classification.
+49J55, 93E20, 49K20, 49K45, 90C15, 65D15, 15A69, 15A23 .
+Key words and phrases. almost surely constraints, state constraints, risk neutral, tensor train, reduced
+space, preconditioner, variational inequality.
+HA is partially supported by NSF grant DMS-2110263 and the AirForce Office of Scientific Research under
+Award NO: FA9550-22-1-0248. SD is thankful for the support from Engineering and Physical Sciences Re-
+search Council (EPSRC) New Investigator Award EP/T031255/1 and New Horizons grant EP/V04771X/1.
+1
+arXiv:2301.08684v1  [math.OC]  20 Jan 2023
+
+2
+HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA
+type constraints.
+It is well-known that even in the deterministic setting, the state con-
+strained problems are highly challenging. One of the fundamental difficulties is that the
+state constraints are imposed in the sense of continuous functions. As a result, the Lagrange
+multipliers corresponding to those constraints are Radon measures that exhibit low regu-
+larity [6]. The situation is much more delicate in the stochastic setting. We refer to the
+aforementioned references for a detailed discussion on this topic. Motivated by the deter-
+ministic setting, [13] introduces a Moreau-Yosida based approximation scheme to solve the
+state-constrained optimization problems when the PDE constraints are given by an elliptic
+equation with random coefficients. Further extensions of this work are considered in [1, 16].
+However, all of these papers approximate expectations of random fields by Monte-Carlo-type
+methods, which may converge slowly.
+In [3], we introduced an algorithm (TTRISK) based on the tensor train (TT) decom-
+position [30] to solve risk-averse optimization problems with control constraints, and the
+conditional value-at-risk (CVaR) [32] risk measure. We demonstrated that the extra com-
+putational cost due to the uncertainty can scale proportionally to error−0.5 when the TT
+approximation is used, in contrast to a error−2 scaling of Monte Carlo quadratures.
+In
+the current paper, we continue this program and develop a TT based algorithm for state-
+constrained optimization problems.
+For simplicity of presentation, we only consider the
+risk-neutral setting, i.e., the objective function is given by the expected value of a quantity
+of interest. Similarly to [13, 16], we tackle the state constrains using Moreau-Yosida based
+relaxation with a softplus smoothing. The main contributions of this paper are listed next:
+(i) We consider an ε-softplus regularization of the positive part function (·)+ = max{·, 0}
+and derive a probabilistic estimate of state constraint violation in terms of Moreau-
+Yosida regularization parameter γ and ε. In particular, we show that selecting ε ∝ γ−1/2
+ensures the convergence of the constraint violation with a rate γ−1/2. This result is
+motivated by [13, Prop. 2]. Notice that the ε-smoothing is carried out because the
+irregular function (·)+ may lack an efficient TT decomposition.
+(ii) When the optimization problem is strongly convex, we establish strong convergence
+of the regularized solution to the optimal control. Our final results can be seen as
+generalizations of the results in deterministic setting.
+(iii) We derive a second order Newton type method to solve the regularized problem with
+a fast matrix-free action of the approximate Hessian.
+(iv) We test the proposed method on elliptic equations in one and two physical dimensions
+and random coefficients, as well as an ODE example (motivated by a realistic applica-
+tion) with 20 random variables, and show that the algorithm is free from the curse of
+dimensionality.
+(v) The proposed approach has been also successfully applied to an example where the
+PDE constraint is given by an elliptic variational inequality.
+Outline: The remainder of the paper is organized as follows. In Section 2, we provide a
+rigorous mathematical formulation of the problem under consideration. Section 3 is devoted
+to the Moreau-Yosida approximation, derivation of the second order Newton method and
+approximation error estimates due to the Moreau-Yosida approximation. In Section 4, we
+provide a brief description of the TT format. This is followed by practical aspects of Moreau-
+Yosida approximations in Section 5. Finally, in Section 6, we provide a series of numerical
+
+STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY
+3
+experiments. At first, we consider an optimization problem with an elliptic PDE in one
+spatial dimension as constraints. This is followed by a two-dimensional case. After these
+benchmarks, an optimization problem with an elliptic variational inequality as constraint is
+considered in Section 6.3. The numerical experiments conclude with a realistic ODE example
+for designing optimal lockdown strategies in Section 6.4.
+2. Problem Formulation
+Let (Ω, F, P) denote a complete probability space, where Ω represents the sample space,
+F is the Borel σ-algebra of events on the power set of Ω, and P : Ω → [0, 1] is an appropriate
+probability measure. We denote by E[·] the expectation with respect to P. Let U be a real
+deterministic reflexive Banach space of optimization variables (control or design) defined on
+an open, bounded and connected set D ⊂ Rn with Lipschitz boundary. We denote by ∥ · ∥U
+the norm on U, and the duality pairing between U and U ∗ as ⟨·, ·⟩U∗,U. Let Y = L2(Ω, F, P; ˆY)
+and Z = L2(Ω, F, P; ˆZ) be Bochner spaces of random fields, based on deterministic Banach
+spaces ˆY �→ L2(D) �→ ˆY∗ and ˆZ, with corresponding norms and duality pairings
+∥y∥2
+Y = E[∥y(ω)∥2
+ˆY],
+⟨y, v⟩Y∗,Y = E
+�
+⟨y(ω), v(ω)⟩ ˆY∗, ˆY
+�
+,
+and similarly for Z. Let Uad ⊆ U be a closed convex nonempty subset and let c : Y×Uad×Ω →
+Z denote, e.g., a partial differential operator, then consider the equality constraint
+c(y, u; ω) = 0,
+in Z,
+a.s. ω ∈ Ω,
+where a.s. indicates “almost surely” with respect to the probability measure P.
+In this paper, we consider the optimization problems of the form
+min
+y,u R[J(y, u; ω)]
+(2.1)
+s.t
+c(y, u; ω) = 0,
+in Z,
+a.s. ω ∈ Ω,
+(2.2)
+where R represents the risk measure and R[J(y, u; ω)] is a deterministic cost function. More
+precisely, we will focus on the so-called risk-neutral formulation; that is, R is simply the
+expectation, denoted by E. We are particularly interested in the case in which the state
+variable y is constrained by a random variable:
+y ≤ ymax(ω)
+a.s.,
+(2.3)
+where we assume that ymax ∈ Y.
+In what follows, we discuss the Moreau-Yosida approximation for (2.1)-(2.3) and derive a
+Newton type method. Throughout the paper, without explicitly stating, we will make use
+of the following assumption.
+Assumption 2.1 (unique forward solution). There exists an injective operator S(ω) : Uad →
+Y (maybe nonlinear) such that c(S(ω)u, u; ω) = 0 ∀u ∈ Uad a.s.
+This allows us to define the reduced-space cost function
+j(u) := R[J(S(ω)u, u; ω)].
+(2.4)
+
+4
+HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA
+The resulting reduced optimization problem is given by
+min
+u∈Uad j(u)
+s.t
+y ≤ ymax(ω)
+a.s.
+(2.5)
+3. Smoothed Moreau-Yosida approximation
+Solving (2.5) with state constraints involve computation of the indicator function of an
+active set and/or Lagrange multiplier as a random field that is nonnegative on a compli-
+cated high-dimensional domain.
+This may be difficult for many function approximation
+methods, especially for tensor decompositions that are considered in this paper. We tackle
+this difficulty by first turning the constrained optimization problem (2.5) into an uncon-
+strained optimization problem with the Moreau-Yosida penalty, and further by smoothing
+the indicator function in the penalty term.
+The classical Moreau-Yosida problem reads, with γ ≥ 0 denoting the regularization pa-
+rameter,
+min
+u∈Uad jγ(u),
+where
+jγ(u) := j(u) + γ
+2E
+���(Su − ymax(ξ))+
+��2
+L2(D)
+�
+,
+(3.1)
+where the so-called positive part or ReLU function (·)+ reads (s)+ = s if s ≥ 0 and 0, oth-
+erwise. Here, we have removed the need to optimize the Lagrange multiplier (corresponding
+to the inequality constraints) over the nonnegative cone, but the function approximation of
+a nonsmooth high-dimensional random field (Su − ymax(ξ))+ (and derivatives thereof) may
+be still inefficient.
+For this reason, we replace the ReLU function in the penalty term by a smoothed version.
+In this paper, we use the softplus function
+gε(s) = ε · log(1 + exp(s/ε)) ∈ C∞(R),
+g0(s) = lim
+ε→0 gε(x) = (s)+,
+(3.2)
+although other (e.g. piecewise polynomial) functions are also possible [24, 1]. Now, the cost
+function becomes
+jγ,ε(u) := j(u) + γ
+2E
+���gε(Su − ymax)
+��2
+L2(D)
+�
+.
+(3.3)
+3.1. Discretization and Derivatives of the Cost. In practice, the operator S involves
+the solution of a differential equation, which needs to be discretized (using e.g.
+Finite
+Element methods and/or time integration schemes). For a given mesh parameter h > 0, we
+introduce the discretized (maybe nonlinear) operator Sh(ω) : Uad → Rny, where ny is the
+total number of degrees of freedom in the discrete solution. We denote the induced Bochner
+space Yh ∼= L2
+h(Ω, D) := L2(Ω, F, P; Rny). The L2-norm can be written as an expectation of
+a vector quadratic form,
+∥y∥2
+L2
+h(Ω,D) = E
+�
+y(ω)⊤My(ω)
+�
+,
+∀y ∈ L2
+h(Ω, D),
+where M = M⊤ > 0 ∈ Rny×ny is a mass matrix. The discretized problem cost is denoted
+by jh(u) ≈ j(u), and the discretized constraint is yh
+max ∈ Yh. Now, the semi-discretized
+Moreau-Yosida cost function (3.3) becomes
+jγ,ε,h(u) := jh(u) + γ
+2E
+�
+∥gε(Shu − yh
+max)∥2
+M
+�
+.
+(3.4)
+
+STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY
+5
+To derive a Newton type method, we compute the expressions of gradient and Hessian:
+∇ujγ,ε,h = ∇ujh + γE
+�
+S∗
+h · diag(g′
+ε(Shu − yh
+max)) · Mgε(Shu − yh
+max)
+�
+,
+(3.5)
+∇uujγ,ε,h = ∇2
+uujh + γE
+�
+S∗
+h · diag(g′
+ε)Mdiag(g′
+ε) · S′
+h
+�
+(3.6)
++ γE
+�
+S∗
+h · (tendiag(g′′
+ε) ×3 (Mgε)) · S′
+h
+�
+(3.7)
++ γE
+�
+∇uS∗
+h ×3 (diag(g′
+ε(Shu − yh
+max)) · Mgε(Shu − yh
+max))
+�
+,
+(3.8)
+where tendiag(·) is producing a 3-dimensional tensor out of vector by putting the vector
+elements along the diagonal, and zero elements otherwise, and ×3 is the tensor-vector con-
+traction product over the 3d mode of the tensor. If Sh is a nonlinear operator, S′
+h = ∇uSh(u)
+denotes the gradient of an image of u, and S∗
+h is the adjoint of S′
+h.
+3.2. Matrix-free Fixed Point Gauss-Newton Hessian. The exact assembly of all terms
+of the Hessian (3.6)–(3.8) can be too computationally expensive, since this involves dense
+tensor-valued random fields (such as ∇uS∗
+h). To simplify the computations, we can firstly
+omit the terms (3.7) and (3.8) which contain order-3 tensors. Secondly, we can replace the
+exact expectation by a fixed-point evaluation. Rewriting (2.1) using Assumption 2.1 we can
+define J(u; ω) = J(S(ω)u, u; ω) and its discretized version Jh(u; ω) = J(Sh(ω)u, u; ω). The
+Hessian of jh can then be written as
+∇2
+uujh = E
+�
+∇2
+uuJh(u; ω)
+�
+.
+For practical computations, it is convenient to parametrize all random fields with inde-
+pendent identically distributed (i.i.d.) random variables with a known probability density
+function. Those variables can then be sampled independently, and an expectation can be
+computed simply by quadrature. Therefore, we will use the following assumption.
+Assumption 3.1 (finite noise). There exists a d-dimensional random vector ξ(ω) ∈ Rd
+with a product probability density function π(ξ) = π(ξ1) · · · π(ξd), such that any random field
+y ∈ Y can be expressed as a function of ξ, y(ω) = y(ξ(ω)) a.s., and
+E[y] =
+ˆ
+Rd y(ξ)π(ξ)dξ.
+In particular, the vector ξ can often be derived from a parametrization of the forward
+solution operator Sh(ω) = Sh(ξ(ω)), and/or the constraint yh
+max(ω) = yh
+max(ξ(ω)).
+Example 3.2. Let y = S(ν(ω))u be the resolution of an elliptic PDE
+−∇(κ(x; ν(ω))∇y) = u,
+where the diffusivity
+κ(x; ν(ω)) = κ0(x) +
+p
+�
+k=1
+ψk(x)νk(ω)
+and the constraint
+ymax(x; η(ω)) = y0(x) +
+q
+�
+k=1
+φk(x)ηk(ω)
+
+6
+HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA
+are given by Karhunen-Loeve expansions (see e.g., [27]), where ν and η are independent
+random variables. Then, we can define ξ = (ν1, . . . , νp, η1, . . . , ηq).
+Now we can replace ∇2
+uujh = E[∇2
+uuJh(u; ξ)] by
+˜∇2
+uujh = ∇2
+uuJh(u; E[ξ]).
+This is exact if ∇2
+uuJh is linear in ξ, but we can take it as an approximation in the general case
+too. Now to apply ˜∇2
+uujh to a vector we just need to apply one deterministic ∇2
+uuJh(u; E[ξ]),
+which involves solving one forward, one adjoint, and two linear sensitivity (of state and
+adjoint) deterministic problems in the most general setting [4, Ch. 1, Algo. 2].
+Similarly we approximate the second term in (3.6) by
+γS∗
+h(ξ∗)MS′
+h(ξ∗),
+where
+ξ∗ = E
+�
+ξ · 1⊤g′
+ε(Shu − yh
+max(ξ))
+�
+E
+�
+1⊤g′
+ε(Shu − yh
+max(ξ))
+�
+is the mean of the random variable with respect to the probability density πg′ ∝ π ·
+(1⊤g′
+ε(Shu − yh
+max)), and 1 ∈ Rny is the constant vector, averaging the spatial components.
+Note that 1⊤g′
+ε(Shu − yh
+max) is a nonnegative function bounded by ny, so π1⊤g′
+ε(Shu − yh
+max)
+is nonnegative and normalizable, and πg′ is indeed a probability density.
+Finally, we obtain a deterministic approximate Hessian
+˜H = ∇2
+uuJh(u; E[ξ]) + γS∗
+h(ξ∗)MS′
+h(ξ∗),
+(3.9)
+which can be applied to a vector by solving 2 forward, 2 adjoint, and 2 sensitivity problems.
+3.3. Probability of the Constraint Violation. In the rest of this section, we prove certain
+properties about the quality of the solution of the smoothed problem (3.3) with respect to
+the constraint, and the exact solution of (2.1)–(2.3). This needs a few properties of the
+softplus smoothing function.
+Lemma 3.3. For any ε ≥ 0, the softplus function (3.2) satifies: gε(s) ≥ (s)+ for any s ∈ R,
+g′
+ε(s) ≥ 0.5 for s ≥ 0, and g′
+ε(s) ≤ 0.5 for s ≤ 0.
+Proof. Using the monotonicity of the logarithm,
+gε(s) = ε log
+�
+1 + exp(s/ε)
+�
+≥
+�
+ε log
+�
+exp(s/ε)
+�
+= s = (s)+,
+s ≥ 0,
+0 = (s)+,
+s < 0.
+The remaining inequalities follow simply from the monotonicity of the sigmoid function
+g′
+ε(s) = 1/(1 + exp(−s/ε)) and that g′
+ε(0) = 0.5.
+□
+Theorem 3.4. Let uγ,ε be a minimizer of (3.3), and assume that j(u) ≥ 0 for any u ∈ Uad.
+Then for any δ > 0, we have
+P
+�
+∥(S(ω)uγ,ε − ymax(ω))+∥2
+L2(D) > δ
+�
+≤ C1 + C2γε2
+γδ
+,
+where C1 = 2j(u∗), C2 = log2 2 · ∥1∥2
+L2(D), and u∗ is a minimizer of (2.1)–(2.3).
+Remark 3.5. This motivates the condition ε ≲ 1/√γ to overcome the effect of smoothing.
+
+STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY
+7
+Proof. Using Markov’s inequality, we obtain
+P
+�
+∥(Suγ,ε − ymax(ω))+∥2
+L2(D) > δ
+�
+≤
+E
+���(Suγ,ε − ymax(ω))+
+��2
+L2(D)
+�
+δ
+≤
+E
+���gε(Suγ,ε − ymax(ω))
+��2
+L2(D)
+�
+δ
+,
+where in the second inequality we used Lemma 3.3. Since uγ,ε minimizes (3.3), it holds
+j(uγ,ε) + γ
+2E[∥gε(Suγ,ε − ymax(ω))∥2
+L2(D)] ≤ j(u∗) + γ
+2E[∥gε(Su∗ − ymax(ω))∥2
+L2(D)]
+for any u∗ ∈ Uad such as the minimizer of (2.1) constrained to (2.3). Dividing by γ/2 and
+neglecting j(uγ,ε) ≥ 0, we get
+E[∥gε(Suγ,ε − ymax(ω))∥2
+L2(D)] ≤ C1
+γ + E[∥gε(Su∗ − ymax(ω))∥2
+L2(D)].
+For the latter term, (2.3) implies Su∗ − ymax(ω) ≤ 0 a.s., and due to monotonicity of gε,
+gε(Su∗ − ymax(ω)) ≤ gε(0) = ε · log 2
+a.s.
+Taking this upper bound out of the expectation and norm, we obtain
+E[∥gε(Suγ,ε−ymax(ω))∥2
+L2(D)] ≤ C1
+γ +ε2·log2 2·E[∥1∥2
+L2(D)] = C1
+γ +ε2·log2 2·∥1∥2
+L2(D), (3.10)
+and the estimate on probability follows by the Markov’s inequality.
+□
+3.4. Strong Convergence with Strongly Convex Cost. To prove the strong conver-
+gence of the minimizer of (3.3) to the minimizer of (2.1)–(2.3) we need further assumptions
+on the cost and smoothing functions.
+Assumption 3.6 (Bounded derivative of the cost). There exists L < ∞ such that
+∥j′(u)∥U∗ ≤ L
+∀u ∈ Uad.
+Assumption 3.7 (α-strong convexity of the cost). There exists α > 0 such that
+⟨j′(u) − j′(v), u − v⟩U∗,U ≥ α∥u − v∥2
+U,
+∀u, v ∈ Uad.
+Assumption 3.8 (Smoothing function). The smoothing function gε possesses the following
+properties
+g′
+ε(s) ≥ 0.5,
+gε(s) ≥ s,
+for
+s ≥ 0,
+g′
+ε(s) ≤ 0.5,
+for
+s ≤ 0,
+(3.11)
+and either:
+gε(s)s ≥ −ηmax(ε),
+for
+s ≤ 0,
+(3.12)
+or, for any random field y(ω) ∈ Y such that y(ω) ≤ 0 a.s.,
+⟨y, gε(y)⟩Y∗,Y ≥ −ηint(ε),
+(3.13)
+where ηmax(ε), ηint(ε) ≥ 0, ∀ε > 0, ηmax(ε), ηint(ε) → 0 as ε → 0.
+Notice that all the conditions in (3.11) are satisfied by the softplus function (3.2) (see
+Lemma 3.3). We only need to check (3.12) or alternatively (3.13).
+
+8
+HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA
+Conjecture 3.9. Our numerical experiments demonstrate that for the softplus function
+(3.2) it holds ηmax(ε) = O(ε2) and ηint(ε) = O(ε3), although we are only able to prove the
+latter estimate under specific conditions (Lemma 3.11 and Theorem 3.12).
+Now we are able to prove the strong convergence of the smoothed optimal control.
+Theorem 3.10. Under Assumptions 2.1 and 3.6–3.8, linear operator S, and ε = εγ depen-
+dent on γ in such a way that
+γ min{ηmax(εγ), ηint(εγ)} → 0,
+as
+γ → ∞,
+and ⟨f, f⟩Y∗,Y = ∥f∥2
+L2(Ω,D) for any f ∈ Y, the minimizer uγ of (3.3) converges to the
+solution u∗ of the exact problem (2.1)–(2.3),
+α∥uγ − u∗∥2
+U + γ
+2∥(Suγ − ymax)+∥2
+L2(Ω,D) → 0,
+γ → ∞.
+Proof. The optimality condition for the smoothed problem, ⟨∇ujγ,ε(uγ), v − uγ⟩U∗,U ≥ 0,
+∀v ∈ Uad, can be expanded by introducing an auxiliary variable λγ to match the gradient of
+the Moreau-Yosida term:
+⟨j′(uγ) + S∗λγ, v − uγ⟩U∗,U ≥ 0,
+(3.14)
+γg′
+ε(Suγ − ymax)gε(Suγ − ymax) = λγ.
+(3.15)
+In turn, the KKT conditions for the original problem read
+⟨j′(u∗) + S∗λ∗, v − u∗⟩U∗,U ≥ 0
+∀v ∈ Uad
+(3.16)
+λ∗ ≥ 0
+Su∗ − ymax ≤ 0
+⟨λ∗, Su∗ − ymax⟩Y∗,Y = 0.
+(3.17)
+Adding (3.16) with v = uγ to (3.14) with v = u∗, and casting S∗ onto another side of the
+duality pairing, we get
+0 ≥ ⟨j′(uγ) + S∗λγ − j′(u∗) − S∗λ∗, uγ − u∗⟩U∗,U
+= ⟨j′(uγ) − j′(u∗), uγ − u∗⟩U∗,U + ⟨λγ, Suγ − Su∗⟩Y∗,Y + ⟨j′(u∗), uγ − u∗⟩U∗,U.
+(3.18)
+Due to the strong convexity, (3.18), and Assumption 3.6 we arrive at
+α∥uγ − u∗∥2
+U + ⟨λγ, Suγ − Su∗⟩Y∗,Y ≤ ⟨j′(u∗), u∗ − uγ⟩U∗,U ≤ ∥j′(u∗)∥U∗∥u∗ − uγ∥U. (3.19)
+The second term on the left hand side can be bounded as follows.
+Using the fact that
+ymax − Su∗ ≥ 0 a.s. and the definition of λγ, we obtain that
+⟨λγ, Suγ − Su∗⟩Y∗,Y = ⟨λγ, (Suγ − ymax) + (ymax − Su∗)⟩Y∗,Y
+≥ ⟨λγ, Suγ − ymax⟩Y∗,Y
+= γ⟨g′
+ε(Suγ − ymax)gε(Suγ − ymax), Suγ − ymax⟩Y∗,Y
+= γ⟨g′
+ε(Suγ − ymax)(Suγ − ymax), gε(Suγ − ymax)⟩Y∗,Y
+= γ⟨g′
+ε(Suγ − ymax)(Suγ − ymax)+, gε(Suγ − ymax)⟩Y∗,Y
++ γ⟨g′
+ε(Suγ − ymax)(Suγ − ymax)−, gε(Suγ − ymax)⟩Y∗,Y,
+(3.20)
+
+STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY
+9
+where we have split Suγ − ymax into positive and negative parts, with (s)− = min(s, 0)
+denoting the negative part. Next using Assumption 3.8 in (3.20), we readily obtain that
+⟨λγ, Suγ − Su∗⟩Y∗,Y ≥ γ⟨0.5(Suγ − ymax)+, (Suγ − ymax)+⟩Y∗,Y
++ γ⟨0.5(Suγ − ymax)−, gε(Suγ − ymax)⟩Y∗,Y
+(3.21)
+≥ γ
+�
+0.5∥(Suγ − ymax)+∥2
+L2(Ω,D) − 0.5ηint(ε)
+�
+.
+(3.22)
+Alternatively, we can bound (3.21) using (3.12) to arrive at
+⟨λγ, Suγ − Su∗⟩Y∗,Y ≥ γ
+�
+0.5∥(Suγ − ymax)+∥2
+L2(Ω,D) − 0.5ηmax(ε)∥1∥2
+L2(Ω,D)
+�
+.
+In either case, (3.19) implies that uγ is bounded in Uad. Therefore, there exists a weakly
+converging subsequence uγ ⇀ ˆu in U as γ → ∞. Since, Uad is closed convex, therefore
+ˆu ∈ Uad. If ε = εγ → 0 as γ → ∞, Assumption 3.8 (for both ηmax and ηint) implies that
+0.5γ∥(Suγ −ymax)+∥2
+L2(Ω,D) is bounded, which means ∥(Suγ −ymax)+∥2
+L2(Ω,D) → 0 as γ → ∞.
+Since S is injective and linear, ∥(Suγ − ymax)+∥2
+L2(Ω,D) is continuous and convex, hence [38,
+Theorem 2.12]:
+0 = lim inf
+γ→∞ ∥(Suγ − ymax)+∥2
+L2(Ω,D) ≥ ∥(Sˆu − ymax)+∥2
+L2(Ω,D).
+Since D is a connected domain of positive measure, this yields |(Sˆu − ymax)+| = 0, that is,
+Sˆu ≤ ymax a.s. Adding again (3.16) and (3.14) and using strong convexity of j, but keeping
+both λγ and λ∗, we get
+α∥uγ − u∗∥2
+U ≤ ⟨λ∗ − λγ, Suγ − Su∗⟩Y∗,Y
+(3.23)
+≤ ⟨λ∗, (Suγ − ymax) + (ymax − Su∗)⟩Y∗,Y
+(3.24)
+− γ
+2∥(Suγ − ymax)+∥2
+L2(Ω,D) + γ
+2 min{∥1∥2
+L2(Ω,D)ηmax(εγ), ηint(εγ)},
+(3.25)
+where we used (3.22) with the negative sign. If γηmax(εγ) → 0 or γηint(εγ) → 0, then
+0 ≤ lim
+γ→∞[α∥uγ − u∗∥2
+U] ≤ lim
+γ→∞⟨λ∗, Suγ − ymax⟩Y∗,Y = ⟨ λ∗
+����
+≥0
+, Sˆu − ymax
+�
+��
+�
+≤0
+⟩Y∗,Y ≤ 0
+(3.26)
+due to (3.17), so uγ → u∗, thereby completing the proof of the theorem.
+□
+Lemma 3.11. For the softplus function (3.2) it holds for any ε ≥ 0:
+ˆ 0
+−∞
+sgε(s)ds ≥ −ε3.
+Proof. The proof uses elementary calculus and is given in Appendix A.
+□
+In order to search for a rate of convergence, we establish the following result:
+Theorem 3.12. Suppose Assumptions 2.1, 3.1 and 3.6–3.8 hold, ˆY is a space of scalar
+functions, the operator S is linear, and |∂(Su − ymax)/∂ξ1| ≥ c > 0 a.s. ∀u ∈ Uad. Suppose
+that ⟨f, g⟩ ˆY∗, ˆY =
+´
+D f(x)g(x)dx ∀f, g ∈ ˆY, and maxξ1∈R π(ξ1) = P < ∞. Let ε = ε0/√γ
+
+10
+HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA
+with any ε0 > 0. Then the minimizer uγ of (3.3) converges to the solution u∗ of the exact
+problem (2.1)–(2.3), and
+∥uγ − u∗∥2
+U ≤ Cε3
+0γ−1/2 + 1
+α⟨λ∗, Suγ − ymax⟩Y∗,Y → 0,
+γ → ∞,
+where C > 0 is independent of γ and ε0.
+Remark 3.13. For the classical Moreau-Yosida penalty with ε0 = 0, we recover existing
+convergence estimates [21, 2] that depend only on ⟨λ∗, Suγ − ymax⟩Y∗,Y. This term converges
+to 0 as shown in (3.26), but the rate of this convergence can be estimated only if bounds on
+∥λ∗∥L2(Ω,D) or ∥Suγ −ymax∥Y can be established from other sources, such as the discretization
+of Y [21, Theorem 3.7].
+Proof. We aim at refining the estimate (3.22). Specifically, we need to lower-bound ⟨(Suγ −
+ymax)−, gε(Suγ − ymax)⟩Y∗,Y, where (y)− = min(y, 0).
+For brevity, let f(x, ξ) = Suγ −
+ymax(x, ξ). Using the particular form of duality pairing and Assumption 3.1, we can write
+⟨(Suγ − ymax)−, gε(Suγ − ymax)⟩Y∗,Y =
+ˆ
+Rd
+ˆ
+D
+(f)−gε(f)dxπ(ξ1) · · · π(ξd)dξ
+=
+ˆ
+D
+ˆ
+f(x,ξ)≤0
+fgε(f)π(ξ1) · · · π(ξd)dξdx.
+(3.27)
+Introduce a change of variables
+�
+����
+ξ1
+ξ2...
+ξd
+�
+���� →
+�
+����
+f(x, ξ)
+ξ2...
+ξd
+�
+����
+with the Jacobian
+J :=
+���������
+det
+�
+����
+∂f
+∂ξ1
+∂f
+∂ξ2
+· · ·
+∂f
+∂ξd
+0
+1
+· · ·
+0
+...
+0
+· · ·
+0
+1
+�
+����
+���������
+=
+����
+∂f
+∂ξ1
+���� ≥ c > 0.
+Now we can express (3.27) using univariate integration,
+⟨(Suγ − ymax)−, gε(Suγ − ymax)⟩Y∗,Y =
+ˆ
+D
+ˆ 0
+min f
+ˆ
+Rd−1 fgε(f)J−1π(ξ1(f)) · · · π(ξd)dξ2 · · · dξddfdx
+≥
+ˆ
+D
+ˆ 0
+−∞
+fgε(f)1
+cPdfdx
+≥ −|D|P 1
+cε3,
+where in the second line we used that the expression under the integral is nonpositive, and
+´
+π(x2)dx2 = · · · =
+´
+π(xd)dxd = 1, and in the third line we used Lemma 3.11.
+
+STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY
+11
+Now we can replace (3.22) as follows:
+⟨λγ, Suγ − Su∗⟩Y∗,Y ≥ γ
+�
+0.5∥(Suγ − ymax)+∥2
+L2(Ω,D) − 0.5|D|P 1
+cε3
+�
+.
+Proceeding as in Theorem 3.10, we replace (3.25) by
+α∥uγ − u∗∥2
+U ≤ ⟨λ∗, Suγ − ymax⟩Y∗,Y + γ
+2|D|P 1
+cε3.
+Setting ε = ε0/√γ, we obtain that
+∥uγ − u∗∥2
+U ≤ 1
+α⟨λ∗, Suγ − ymax⟩Y∗,Y + |D|P
+2cα
+� �� �
+C
+ε3
+0
+γ1/2.
+Thus the proof is complete.
+□
+Remark 3.14. This theorem can be generalized to vector-valued functions straightforwardly.
+Indeed, if fi(x, ξ) denotes the ith component of a vector function, the duality pairing (3.27)
+reads
+⟨(f)−, gε(f)⟩Y∗,Y =
+ˆ
+Rd
+ˆ
+D
+�
+i
+(fi)−gε(fi)dxπ(ξ)dξ =
+�
+i
+ˆ
+D
+ˆ
+fi(x,ξ)≤0
+figε(fi)π(ξ)dξdx,
+and ξ1 can be changed to fi for each term of the sum over i.
+The assumption of a lower bound of the Jacobian is practical.
+The Karhunen-Loeve
+expansion as in Example 3.2 is normally derived as the eigenvalue expansion of the covariance
+function of e.g.
+κ.
+By the Perron-Frobenius theorem, ψ1(x) = ∂κ/∂ξ1 > 0.
+Further,
+∂y/∂κ ̸= 0 due to ellipticity. Hence ∂(Su)/∂ξ1 ̸= 0 whenever either u or boundary conditions
+or source term are nonzero. The remaining assumptions of Thm. 3.12 are also reasonable for
+practical solutions of regularized optimization problems. A convenient observation is that
+ε = ε0/√γ is the sufficient condition on the law of decay of the smoothing parameter for
+both Theorems 3.4 and 3.12.
+4. Tensor-Train decomposition
+Throughout this section, we use Assumption 3.1. Recall that the bottleneck is the com-
+putation of the expectation in e.g. gradient (3.5). While it may be possible to use a Monte
+Carlo quadrature, its convergence is usually slow, which may make estimates of small values
+of the gradient near the optimum particularly inaccurate. In this section, we describe the
+Tensor-Train (TT) decomposition as a function approximation technique that allows fast
+computation of the expectation. The original TT decomposition [30] was proposed for ten-
+sors (such as tensors of expansion coefficients), and the functional TT (FTT) decomposition
+[5, 19] has extended this idea to multivariate functions.
+Let us introduce a basis {ℓi(ξk)}
+nξ
+i=1 in each random variable ξk, k = 1, . . . , d, and a
+quadrature with nodes Z = {zj} and weights {wj} which is exact on this basis,
+E[ℓi] =
+nξ
+�
+j=1
+wjℓi(zj).
+
+12
+HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA
+For example, we can take Lagrange interpolation polynomials built upon a Gaussian quadra-
+ture, or orthogonal polynomials up to degree nξ − 1 together with the roots of the degree-nξ
+polynomial, or Fourier modes and the rectangular quadrature with the number of nodes
+corresponding to the highest frequency. Then we can approximate any random field y ∈ Y
+in the tensor product basis,
+y(ξ) ≈
+nξ
+�
+i1=1
+· · ·
+nξ
+�
+id=1
+Yi1,...,idℓi1(ξ1) · · · ℓid(ξd).
+Note that the expansion coefficients Y form a tensor of nd
+ξ entries, which is impossible to
+store directly if d is large. The TT decomposition aims to factorize this tensor further to a
+product of tensors of manageable size.
+Definition 4.1. A tensor Y ∈ Rnξ×···×nξ is said to be approximated by the TT decomposition
+with a relative approximation error ϵ if there exist 3-dimensional tensors Y(k) ∈ Rrk−1×nξ×rk,
+k = 1, . . . , d, such that
+˜Yi1,...,id :=
+r0,...,rd
+�
+s0,...,sd=1
+Y(1)
+s0,i1,s1Y(2)
+s1,i2,s2 · · · Y(d)
+sd−1,id,sd,
+(4.1)
+and ∥Y− ˜Y∥F = ϵ∥Y∥F. The factors Y(k) are called TT cores, and the ranges of summation
+indices r0, . . . , rd ∈ N are called TT ranks. Note that without loss of generality we can let
+r0 = rd = 1.
+Plugging in the basis and redistributing the summations we obtain the FTT approximation
+˜y(ξ) :=
+r0,...,rd
+�
+s0,...,sd=1
+y(1)
+s0,s1(ξ1)y(2)
+s1,s2(ξ2) · · · y(d)
+sd−1,sd(ξd),
+where
+y(k)
+sk−1,sk(ξk) =
+nξ
+�
+i=1
+Y(k)
+sk−1,i,skℓi(ξk),
+k = 1, . . . , d.
+Smooth [35], weakly correlated [33] or certainly structured [20] functions have been shown
+to induce rapidly converging TT approximations.
+Given the TT decomposition, its expectation can be computed by first integrating each
+TT core, and then multiplying the TT cores one by one. Let
+V(k)
+sk−1,sk =
+nξ
+�
+j=1
+wjy(k)
+sk−1,sk(zj) =
+nξ
+�
+i,j=1
+wjLi,jY(k)
+sk−1,i,sk,
+where
+Li,j = ℓi(zj).
+(4.2)
+Now we multiply the matrices V(k) ∈ Rrk−1×rk in order:
+E[˜y] =
+���
+V(1)V(2)�
+V(3)
+�
+· · · V(d)
+�
+.
+(4.3)
+Note that each step in (4.3) is a product of 1 × rk−1 vector by rk−1 × rk matrix. In turn, the
+univariate quadrature (4.2) requires n2
+ξrk−1rk floating point operations if the Vandermonde
+matrix L is dense, and nξrk−1rk if it’s sparse, for example, if Lagrange polynomials are used.
+
+STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY
+13
+Introducing r := maxk rk, we conclude that the expectation of a TT decomposition can be
+computed with a complexity O(dr2) which is linear in the dimension.
+To compute a TT approximation, we employ the TT-Cross algorithm [31]. We start with
+an empirical risk minimization problem
+min
+Y(1),...,Y(d)
+N
+�
+j=1
+�
+˜y(ξj) − y(ξj)
+�2
+,
+where Ξ = {ξj} is a certain set of samples. To avoid minimization over all Y(1), . . . , Y(d)
+simultaneously (which is non-convex), we switch to an alternating direction approach: iterate
+over k = 1, . . . , d, solving in each step
+min
+Y(k)
+N
+�
+j=1
+�
+˜y(ξj) − y(ξj)
+�2
+.
+(4.4)
+This problem can be solved by linear normal equations. Indeed, introduce a matrix Y̸=k ∈
+RN×(rk−1nξrk) with elements
+(Y̸=k)j,t =
+�
+s0,...,sk−2
+y(1)
+s0,s1(ξj
+1) · · · y(k−1)
+sk−2,sk−1(ξj
+k−1)ℓi(ξj
+k)
+�
+sk+1,...,sd
+y(k+1)
+sk,sk+1(ξj
+k+1) · · · y(d)
+sd−1,sd(ξj
+d),
+where t = (sk−1 − 1)nξrk + (i − 1)rk + sk, and a vector y(k) ∈ Rrk−1nξrk with elements
+y(k)
+t
+= Y(k)
+sk−1,i,sk. Now ˜y(Ξ) = Y̸=ky(k), and (4.4) is minimized by
+y(k) = (Y⊤
+̸=kY̸=k)−1(Y⊤
+̸=ky(Ξ)).
+(4.5)
+To both select “good” sample set Ξ and simplify the assembly of Y̸=k, we restrict the set
+to have the Cartesian form
+Ξ = Ξ<k × Z × Ξ>k,
+where Ξ<k = {(ξ1, . . . , ξk−1)}, Ξ>k = {(ξk+1, . . . , ξd)} with nestedness conditions
+(ξ1, . . . , ξk−1, ξk) ∈ Ξ<k+1 ⇒ (ξ1, . . . , ξk−1) ∈ Ξ<k,
+(ξk, ξk+1, . . . , ξd) ∈ Ξ>k−1 ⇒ (ξk+1, . . . , ξd) ∈ Ξ>k.
+This makes
+Y̸=k = Y<k ⊗ L ⊗ Y>k,
+where
+(Y<k)j,s =
+�
+s0,...,sk−2
+y(1)
+s0,s1(ξj
+1) · · · y(k−1)
+sk−2,s(ξj
+k−1),
+(ξj
+1, . . . , ξj
+k−1) ∈ Ξ<k,
+(Y>k)j,s =
+�
+sk+1,...,sd
+y(k+1)
+s,sk+1(ξj
+k+1) · · · y(d)
+sd−1,sd(ξj
+d),
+(ξj
+k+1, . . . , ξj
+d) ∈ Ξ>k.
+Moreover, Y<k+1 and Y>k−1 are submatrices of
+Y≤k :=
+�
+��
+Y<ky(k)(z1)
+...
+Y<ky(k)(znξ)
+�
+��
+and
+Y≥k :=
+�
+y(k)(z1)Y>k
+· · ·
+y(k)(znξ)Y>k
+�
+,
+(4.6)
+
+14
+HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA
+respectively. This allows us to build the sampling sets by selecting rk rows of Y≤k (resp.
+columns of Y≥k) by the maximum volume principle [18], which needs only O(nξr3) floating
+point operations per single matrix Y≤k or Y≥k. The rk indices of e.g. rows of Y≤k con-
+stituting the maximum volume submatrix Y<k are also indices of the rk tuples in Ξ<k × Z
+constituting the next “left” set Ξ<k+1. The “right” set Ξ>k−1 is constructed analogously.
+This closes the recursion and allows us to carry out the alternating iteration in either di-
+rection, k = 1, . . . , d or k = d, . . . , 1. By this construction, the cardinality of Ξ<k+1 and
+Ξ>k−1 is rk. Hence, the cardinality of Ξ is rk−1nξrk, and one full iteration of the TT-Cross
+algorithm needs O(dnξr2) samples of y.
+One drawback of the “naive” TT-Cross algorithm outlines above is that the TT ranks are
+fixed. To adapt them to a desired error tolerance, several modifications have been proposed:
+merge ξk, ξk+1 into one variable, optimize the corresponding larger TT core, and separate it
+into two actual TT cores using truncated singular value decomposition (SVD) [34] or matrix
+adaptive cross approximation [8]; oversample Ξ<k or Ξ>k with random or error-targeting
+points [10]; oversample the selection of submatrices from (4.6) by using the rectangular
+maximum volume principle [29].
+However, in this paper we can pursue a somewhat more natural regression approach [7]. We
+will always need to approximate a vector function, where different components correspond
+to different degrees of freedom of an ODE or a PDE solution, or different components of
+a gradient. Since the procedure to evaluate y is now taking two arguments (ξ and, say,
+m = 1, . . . , M indexing extra degrees of freedom), we can replace the normal equations (4.5)
+by
+y(k)(m) = (Y⊤
+̸=kY̸=k)−1(Y⊤
+̸=ky(Ξ, m)),
+which can be reshaped into a 4-dimensional tensor ˆY(k) ∈ Rrk−1×nξ×rk×M with elements
+ˆY(k)
+sk−1,i,sk,m = y(k)
+t (m). To compute the usual 3-dimensional TT core, we can use a simple
+Principal Component Analysis (PCA), which selects ˆr slices Y(k)
+sk−1,i,1, . . . , Y(k)
+sk−1,i,ˆr with the
+minimal ˆr such that
+min
+W
+�
+sk−1,i,sk,m
+�
+�
+ˆr
+�
+s=1
+Y(k)
+sk−1,i,sWs,sk,m − ˆY(k)
+sk−1,i,sk,m
+�
+�
+2
+≤ tol2 · ∥ ˆY(k)∥2
+F.
+Note that this problem is solved easily by the truncated SVD, where the new TT rank ˆr
+can be chosen anywhere between 1 and min{rk−1nξ, rkM} to satisfy the error tolerance tol.
+After replacing rk with ˆr, the TT-Cross iteration k = 1, . . . , d can proceed as previously.
+In the last step (k = d), the PCA step is omitted, and we obtain the so-called block TT
+decomposition [9], which in the functional form reads
+˜y(ξ, m) =
+�
+s0,...,sd
+y(1)
+s0,s1(ξ1) · · · y(d−1)
+sd−2,sd−1(ξd−1)ˆy(d)
+sd−1,sd(ξd, m).
+The “backward” iteration k = d, . . . , 1 can be generalized similarly.
+
+STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY
+15
+5. Practical computation of the smoothed Moreau-Yosida optimization
+To compute the gradient of the cost function (3.5), we need to approximate the function
+under the expectation,
+Gε,h
+u (ξ) := Sh(ξ)∗ · diag(g′
+ε(Sh(ξ)u − yh
+max(ξ))) · Mgε(Sh(ξ)u − yh
+max(ξ)),
+(5.1)
+using the TT-Cross, followed by taking the expectation of the TT decomposition1 This can be
+performed in two ways. To begin with, we can apply the TT-Cross algorithm to approximate
+directly Gε,h
+u (ξ). For each sample ξj ∈ Ξ, one needs to solve one forward problem to compute
+Sh(ξj)u, and one adjoint problem to apply Sh(ξj)∗ to the rest of the function. Recall that
+the TT-Cross needs O(dnξr2) samples, hence O(dnξr2) solutions of the forward, adjoint
+and sensitivity problems. However, the maximal TT rank r of the softplus and sigmoid
+functions typically grows proportional to 1/ε. When the solution of the forward and adjoint
+problem is expensive (for example, in the PDE-constrained optimization), this may result in
+an excessive computational complexity.
+Alternatively, we can first compute TT approximations ˜y(ξ) ≈ Sh(ξ)u and ˜Sh(ξ)∗ ≈
+Sh(ξ)∗, followed by TT approximations ˜gε(ξ) :≈ gε(˜y(ξ) − yh
+max(ξ)), ˜g′
+ε(ξ) :≈ g′
+ε(˜y(ξ) −
+yh
+max(ξ)), and finally ˜Gε,h
+u (ξ) ≈ ˜Sh(ξ)∗diag(˜g′
+ε(ξ))˜gε(ξ) using the approximate solution ˜y(ξ),
+which does not require the solution of the PDE anymore. The bottleneck now is the ap-
+proximation of the matrix-valued function Sh(ξ)∗ ∈ Rnu×ny. If both ny and nu are large (for
+example, in a case of a distributed control), the computation of Sh(ξ)∗ for each sample of ξ
+requires assembling this large dense matrix, equivalent to the solution of the adjoint prob-
+lem with nu right hand sides. Nevertheless, the tensor approximation of Sh(ξ)∗ converges
+usually much faster (e.g. exponentially) compared to the approximation of Gε,h
+u (ξ) directly,
+hence the TT approximation of Sh(ξ)∗ may need much smaller TT ranks compared to the
+TT approximation of Gε,h
+u (ξ). In turn, the TT-Cross applied to Sh(ξ)∗ requires much fewer
+solutions of the forward problem. For a moderate nu this makes it faster to precompute ˜y(ξ)
+and ˜Sh(ξ)∗. The entire pseudocode of the smoothed Moreau-Yosida optimization is listed in
+Algorithm 1.
+6. Numerical examples
+We start with γ0 = 1 and double γℓ+1 = 2γℓ in the course of the Newton iterations until
+a desired value of γ∗ is reached. According to Theorem 3.4, we choose εℓ = 0.5/√γℓ. The
+iteration is stopped when γL has reached the maximal desired value γ∗, and the step size has
+become smaller than δmin = 10−3. We always take a zero control as the initial guess u0, and
+θ = 10−4. All computations are carried out in MATLAB 2020b on a Intel Xeon E5-2640 v4
+CPU, using TT-Toolbox (https://github.com/oseledets/TT-Toolbox).
+6.1. One-dimensional Elliptic PDE. We consider an elliptic PDE example from [22, 13].
+Here, a misfit functional
+j(u) = 1
+2E
+�
+∥y(u, ω, x) − yd(x)∥2
+L2(D)
+�
++ α
+2 ∥u(x)∥2
+L2(D)
+1Note that Gε,h
+u (ξ) is a vector function with M being the number of degrees of freedom in the discretized
+u.
+
+16
+HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA
+Algorithm 1 Inexact projected Newton optimization with smoothed a.s. constraints
+Require: Procedures to compute Shu, jh(u), ∇ujh(u), constraint yh
+max, initial and maximal
+Moreau-Yosida parameters γ0, γ∗, initial smoothing parameter ε0, initial control u0, ap-
+proximation and stopping tolerance tol, maximal number of iterations L, Armijo tuning
+parameter θ ∈ (0, 1), minimal step size δmin ∈ (0, 1).
+Ensure: Optimized control uγ∗,h.
+1: Set iteration number ℓ = 0, step size δ = 1, u−1 = u0.
+2: while ℓ < L and δ > δmin and ∥uℓ − uℓ−1∥U > tol · ∥uℓ∥U or ℓ = 0 or γℓ < γ∗ do
+3:
+Set ε = ε0/√γℓ.
+4:
+Approximate ˜Gε,h
+uℓ (ξ) ≈ Gε,h
+uℓ (ξ) as shown in (5.1) using TT-Cross up to tolerance tol.
+5:
+Approximate ˜g′
+ε(ξ) ≈ g′
+ε(Sh(ξ)uℓ − yh
+max(ξ)) using TT-Cross up to tolerance tol.
+6:
+Compute the gradient ∇ujγℓ,ε,h = ∇ujh(uℓ) + γℓE[ ˜Gε,h
+uℓ (ξ)]
+7:
+Compute the anchor point ξ∗ = E[ξ · 1⊤˜g′
+ε(ξ)]/E[1⊤˜g′
+ε(ξ)].
+8:
+Compute the Newton direction v = − ˜H−1∇ujγℓ,ε,h using (3.9).
+9:
+Set step size δ = 1.
+10:
+while jh(PUad(uℓ + δv)) > jh(uℓ) + δθ⟨v, ∇ujγℓ,ε,h⟩U∗,U and δ > δmin do
+11:
+Set δ = δ/2.
+12:
+end while
+13:
+Set uℓ+1 = PUad(uℓ + δv).
+14:
+Set γℓ+1 = min{2γℓ, γ∗}.
+15:
+Set ℓ = ℓ + 1.
+16: end while
+17: return uγ∗,h = uℓ.
+is optimized subject to the stochastic PDE constraint2
+ν(ω)∆y(u, ω, x) = g(ω, x) + u(x),
+(ω, x) ∈ Ω × D,
+ν(ω) = 10ξ1(ω)−2,
+g(ω, x) = ξ2(ω)
+100 ,
+y|x=0 = −1 − ξ3(ω)
+1000 ,
+y|x=1 = −2 + ξ4(ω)
+1000
+(6.1)
+where D = (0, 1), and ξ(ω) = (ξ1(ω), . . . , ξ4(ω)) ∼ U(−1, 1)4 is uniformly distributed. We
+take the desired state yd(x) = − sin(50x/π) and the regularization parameter α = 10−2.
+Moreover, we add the constraints
+y(u, ω, x) ≤ ymax ≡ 0
+a.s.,
+and
+− 0.75 ≤ u(x) ≤ 0.75
+a.e.
+We discretize (6.1) in the spatial coordinate x using linear finite elements on a uniform
+grid with ny interior points, and in each random variable ξi(ω) using nξ Gauss-Legendre
+quadrature nodes on (−1, 1). Note that we exclude the boundary points x = 0 and x = 1
+due to the Dirichlet boundary conditions. This spatial discretization is used for both y and
+u.
+2Note that [22, 13] considered the constraint y ≥ 0, so here we reverse the sign of y to make the constraint
+in the form (2.3).
+
+STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY
+17
+Firstly, we study precomputation of the surrogate solution ˜y(ξ) and adjoint operator
+˜S∗
+h(ξ).
+We fix ny = 63, nξ = 65, the TT approximation tolerance 10−7 and the final
+Moreau-Yosida regularization parameter γ∗ = 1000.
+The direct computation of the TT
+approximation of (5.1) requires 995 seconds of the CPU time due to the maximal TT rank
+of 87. In contrast, ˜S∗
+h has the maximal TT rank of 8, and the computation of ˜S∗
+h requires
+only 64 seconds despite a larger ny × ny TT core carrying the spatial variables. Using the
+surrogates ˜y and ˜S∗
+h, the remaining computation of ∇ujγ,ε,h can be completed in less than 15
+seconds. The relative difference between the two approximations of ∇ujγ,ε,h is below the TT
+approximation tolerance. This shows that the surrogate forward solution can significantly
+speed up Algorithm 1 without degrading its convergence, so we use it in all remaining
+experiments in this subsection.
+0
+0.2
+0.4
+0.6
+0.8
+1
+−0.6
+−0.4
+−0.2
+0
+0.2
+0.4
+0.6
+x
+u
+γ∗ = 3000
+γ∗ = 1000
+γ∗ = 300
+γ∗ = 100
+0
+0.2
+0.4
+0.6
+0.8
+1
+−1.6
+−1.4
+−1.2
+−1
+−0.8
+−0.6
+−0.4
+−0.2
+0
+ymax = 0
+x
+y
+Figure 1. Left: control signals uγ∗(x) for different γ∗. Right: mean (solid
+lines) and 95% confidence interval (shaded area, for γ∗ = 3000 only) of the
+state y(uγ∗, ω, x).
+In Figure 1 we show the solutions (control and state) for varying final Moreau-Yosida
+penalty parameter γ∗, fixing ny = 63, nξ = 129 and the TT approximation tolerance of
+10−6. We see that the solution converges with increasing γ∗, and larger γ∗ yields a smaller
+probability of the constraint violation, albeit at a larger misfit cost j(u), as shown in Figure 2.
+In particular, γ∗ > 300 gives a solution with less than 1% of the constraint violation, such
+that the empirical 95% confidence interval computed using 1000 samples of the converged
+field y(uγ∗) (see Fig. 1, right) is entirely within the constraint.
+Finally, we study the convergence in the approximation parameters more systematically
+in Figure 3.
+In each plot we fix two out of three parameters: the final Moreau-Yosida
+penalty γ∗, the number of discretization points in the random variables nξ, and the number
+
+18
+HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA
+102
+103
+104
+10−3
+10−2
+γ∗
+102
+103
+104
+0.35
+0.36
+0.37
+0.38
+0.39
+γ∗
+Figure 2. Left: probability of the constraint violation, P(y(uγ∗, ω, x) > 0).
+Right: total final cost j(uγ∗).
+102
+102.5
+103
+10−1.5
+10−1
+10−0.5
+γ∗
+relative error
+u
+y
+γ−0.75
+∗
+γ−0.5
+∗
+20
+40
+10−5
+10−4
+10−3
+10−2
+nξ
+relative error
+u
+y
+31
+65
+127
+255
+10−3
+10−2
+10−1
+ny
+relative error
+u
+y
+n−1.5
+y
+Figure 3. Relative L2-norm difference from y and u to the reference solutions
+with γ∗ = 104 with fixed nξ = 257, ny = 63 (left), nξ = 129 with fixed γ∗ = 100,
+ny = 63 (middle) and ny = 511 with fixed γ∗ = 100, nξ = 25 (right).
+of discretization points in space ny. In addition, we fix the TT approximation threshold
+to 10−8 to reduce its influence. We observe a convergence in line with the γ−1/2
+∗
+rate of
+Theorem 3.4, exponential in nξ (which is often the case for a polynomial approximation of
+smooth functions [37]) until the tensor approximation error is hit, and between first and
+second order in ny, which seems to be an interplay of the discretization consistency of the
+linear finite elements (second order) and box constraints (first order).
+
+STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY
+19
+6.2. Two-dimensional elliptic PDE. Now consider a two-dimensional extension of the
+previous problem,
+ν(ω)∆y(u, ω, x) = g(ω, x) + u(x),
+(ω, x) ∈ Ω × D,
+(6.2)
+y|x1=0 = b1(ω)(1 − x2) + b2(ω)x2,
+y|x2=1 = b2(ω)(1 − x1) + b3(ω)x1
+(6.3)
+y|x1=1 = b4(ω)(1 − x2) + b3(ω)x2,
+y|x2=0 = b1(ω)(1 − x1) + b4(ω)x1,
+(6.4)
+ν(ω) = 10ξ1(ω)−2,
+g(ω, x) = ξ2(ω)
+100 ,
+(6.5)
+b1(ω) = −1 − ξ3(ω)
+1000 ,
+b2(ω) = −2 + ξ4(ω)
+1000
+,
+(6.6)
+b3(ω) = −1 − ξ5(ω)
+1000 ,
+b4(ω) = −2 + ξ6(ω)
+1000
+,
+(6.7)
+where D = (0, 1)2, and ξ(ω) = (ξ1(ω), . . . , ξ6(ω)) ∼ U(−1, 1)6 is uniformly distributed. We
+optimize the regularized misfit functional
+j(u) = 1
+2E
+�
+∥y(u, ω, x) − yd(x)∥2
+L2(D)
+�
++ α
+2 ∥u(x)∥2
+L2(D)
+with the desired state yd(x) = − sin(50x1/π) cos(50x2/π) and the regularization parameter
+α = 10−2, subject to constraints
+y(u, ω, x) ≤ ymax ≡ 0
+a.s.,
+and
+− 0.75 ≤ u(x) ≤ 0.75
+a.e.
+We smooth the almost sure constraint by the Moreau-Yosida method with the ultimate
+penalty parameter γ∗ = 102.
+We discretize both y and u in (6.2) using bilinear finite elements on a ny × ny rectangular
+grid. For the two-dimensional problem, the operator ˜S∗
+h is a dense matrix of size n2
+y × n2
+y,
+which we are unable to precompute. Therefore, we use the TT-Cross to approximate Gε,h
+u (ξ)
+directly.
+In Figure 4 we show the optimal control, mean and standard deviation of the solution for
+ny = 63 and nξ = 17. We see that the mean solution reflects the desired state subject to the
+constraints. The final cost j(uγ∗) is about 0.222634, and the probability of the constraint
+violation is 0.0139223. The Newton method took L = 37 iterations to converge, the maximal
+TT rank of ˜y(ξ) was 10 which was the same in all iterations, the maximal rank of g′
+ε(˜y−yh
+max)
+was 300, attained at the iteration after reaching γ∗ (iteration 9), and the maximal rank of
+˜Gε,h
+u (ξ) was 56 (in the final iterations). The computation took about a day of CPU time.
+However, these TT ranks are comparable to those in the one-dimensional example. This
+shows that the proposed technique can be also applied to a high-dimensional physical space,
+including complex domains and non-uniform grids, since the TT structure is independent of
+the spatial discretization.
+6.3. Variational inequality constraints. In this section we minimize the regularized mis-
+fit
+j(u) = 1
+2E[∥y(u, ω, x) − yd(x)∥2
+L2(D)] + 1
+2∥u(x)∥2
+L2(D)
+(6.8)
+
+20
+HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+-0.9
+-0.8
+-0.7
+-0.6
+-0.5
+-0.4
+-0.3
+-0.2
+-0.1
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+0.05
+0.1
+0.15
+0.2
+0.25
+Figure 4. Left: control signal uγ∗(x). Middle: mean E[y(uγ∗, ω, x)]. Right:
+standard deviation
+�
+E[(y(uγ∗, ω, x) − E[y(uγ∗, ω, x)])2].
+subject to a random elliptic variational inequality (VI) constraint,
+y(u, ω, x) ≤ 0 :
+⟨A(ω)y(u, ω, x)−f(ω, x)−B(ω, x)u, y(u, ω, x)−v⟩ ≤ 0,
+∀v : v ≤ 0. (6.9)
+We use Example 5.1 from [1] (with the reversed sign of y), where D = (0, 1)2, A = −∆,
+B = Id, and deterministic functions constructing the desired state:
+ˆy(x) =
+�
+160(x3
+1 − x2
+1 + 0.25x1)(x3
+2 − x2
+2 + 0.25x2)
+in (0, 0.5)2,
+0,
+otherwise,
+ˆζ(x) = max(0, −2|x1 − 0.8| − 2|x1x2 − 0.3| + 0.5),
+yd(x) = −ˆy − ˆζ + ∆ˆy.
+In contrast, the right hand side depends on the random variables,
+f(ξ(ω), x) = ∆ˆy + ˆy + ˆζ + b(ξ(ω), x),
+b(ξ(ω), x) =
+��d
+i=1
+√λiφi(x)ξi(ω),
+in (0, 0.5) × (0, 1),
+0,
+otherwise.
+The Karhunen-Loeve expansion in b(ξ, x) is an affine-uniform random field, with ξi(ω) ∼
+U(−1, 1), φi(x) = 2 cos(πjx2) cos(πkx1) and λi =
+1
+100 exp(− π
+4(j2 + k2)), where the pairs
+(j, k), j, k = 1, 2, . . . , are permuted such that λ1 ≥ λ2 ≥ · · · .
+The VI (6.9) is replaced by the penalized problem
+Ay + 1
+εgε(y) = f(ξ, x) + Bu,
+(6.10)
+so we minimize (6.8) with y(u, ξ, x) plugged in from (6.10). The latter equation is solved
+via the Newton method, initialized with y = 0 as the initial guess, and stopped when the
+relative difference between two consecutive iterations of y falls below 10−12. The problem is
+discretized in x via the piecewise bilinear finite elements on a uniform ny × ny grid with cell
+size h = 1/(ny + 1). The homogeneous Dirichlet boundary conditions y = 0 on ∂D allow us
+to store only interior grid points. This gives us a discrete problem of minimizing
+jh(u) = 1
+2E[∥y(u, ξ) − yd∥2
+Mh] + 1
+2∥u∥2
+Mh
+(6.11)
+
+STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY
+21
+subject to
+Ahy + 1
+εgε(y) = f(ξ) + u,
+(6.12)
+where Ah, Mh ∈ Rn2
+y×n2
+y are the stiffness and mass matrices, respectively.
+The state part of the cost
+jy(u, ξ) = 1
+2∥y(u, ξ) − yd∥2
+Mh
+and its gradient
+∇ujy(u, ξ) = S∗
+h(ξ)Mh(y(u, ξ) − yd)
+are approximated by the TT-Cross (as functions of ξ), which allows one to compute the ex-
+pectation of ˜jy(u, ξ) ≈ jy(u, ξ) and ∇u˜jy(u, ξ) ≈ ∇ujy(u, ξ) easily. The forward model (6.12)
+is solved at each evaluation of ξ in the TT-Cross. However, to avoid excessive computations,
+the Hessian of (6.11) is approximated by that anchored at the mean point ξ = 0:
+∇uujh(u) ≈ ˜H := S∗
+h(0)MhS′
+h(0) + Mh.
+The Newton system ˜H−1∇ujh is solved iteratively by using the CG method, since the matrix-
+vector product with ˜H requires the solution of only one forward and one adjoint problem,
+S∗
+h · v = S′
+h · v =
+�
+Ah + diag
+�1
+εg′
+ε(y)
+��−1
+v,
+∀v ∈ Rn2
+y.
+(6.13)
+In Table 1 we vary the dimension of the random variable d, the number of quadrature
+points in each random variable nξ, and the approximation tolerance in the TT-Cross (tol).
+The spatial grid size is fixed to ny = 31, which is comparable with the resolution in [1],
+and the smoothing parameter ε = 10−6. As a reference solution u∗, we take the control
+computed with d = 20, nξ = 5 and tol = 10−4. We see that the control and the cost can be
+approximated quite accurately even with a very low order of the polynomial approximation
+in ξ. It also seems unnecessary to keep 20 terms in the Karhunen-Loeve expansion.
+The computation complexity is dominated by the solutions of the forward and adjoint
+problems. The article [1] reports a “# PDE solves” in a path-following stochastic variance
+reduced gradient method solving (6.8)–(6.9). We believe this indicates the number of the
+complete solutions of the PDE (6.12). However, each solution of (6.12) to the increment
+tolerance 10−12 requires 23–25 Newton iterations, each of which requires the linear system
+solution of the form (6.13), Moreover, the anchored outer Hessian ˜H requires two extra linear
+solves. Therefore, in Table 1, we show both the number of PDE solutions till convergence,
+Npde, and the number of all linear system solutions Nlin, occurred during the optimization
+of (6.11) till the relative increment of u falls below the TT-Cross tolerance. In addition,
+we report the maximal TT ranks of the state cost gradient and the state itself. Note that
+assembly of the full state is not needed during the optimization of (6.11) – only certain
+samples of y(u, ξ) are needed in the TT-Cross approximation of ∇ujh. To save the computing
+time, the TT tensor of the entire state is computed only after the optimization of u has
+converged.
+In Figure 5 we show the mean optimized forward state and the control.
+The results
+coincide qualitatively with those in [1]. If we consider the computational cost necessary to
+
+22
+HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA
+Table 1. Cost, error in the control, number of solutions of n2
+y × n2
+y linear
+system as in (6.13), number of complete forward PDE solutions (6.12), and
+the TT ranks of the cost gradient and forward solution.
+d
+nξ
+tol
+jh(u)
+∥u−u∗∥Mh
+∥u∗∥Mh
+Nlin
+Npde
+r(∇u˜jy)
+r(˜y)
+10
+5
+10−4
+1.261333069
+1.1473e-06
+1070007
+44584
+85
+316
+20
+3
+10−3
+1.261333069
+2.9012e-05
+46312
+1976
+7
+29
+20
+3
+10−4
+1.261333069
+4.2713e-06
+433134
+18153
+56
+183
+20
+5
+10−4
+1.261333069
+—
+1840467
+76243
+102
+402
+Figure 5. Left: mean optimised state E[−y] with d = 20, nξ = 3 and tol =
+10−3. Middle: variance of the optimized state E[(y −E[y])2]. Right: optimised
+control u.
+compute the optimal control only, we can notice that Npde is significantly lower than the
+291808 PDE solves in the stochastic variance reduced gradient method of [1].
+6.4. SEIR ODE model. Now consider a slightly simplified version of the epidemiological
+ODE model used for the propagation of COVID-19 in the UK using the data from March-
+May 2020 [11].
+This is a compartmental differential equation model with the following
+compartments.
+• Susceptible (S).
+• Exposed (E), but not yet infectious.
+• Infected SubClinical type 1 (ISC1): may require hospitalization in the future.
+• Infected SubClinical type 2 (ISC2): will recover without hospitalization.
+• Infected Clinical type 1 (IC1): individuals in the hospital who may decease.
+• Infected Clinical type 2 (IC2): individuals in the hospital who will recover.
+• Recovered (R) and immune to reinfections.
+• Deceased (D).
+In turn, each of these compartments are split into 5 further sub-compartments corresponding
+to age bands: 0-19, 20-39, 40-59, 60-79 and 80+. The number of individuals in each com-
+partment is denoted by the name of the compartment and age band index, For example, Si
+denotes the number of susceptible individuals in the ith age band (i = 1, . . . , 5), Ei denotes
+the number of exposed individuals in the ith age band, and so on. Variables corresponding
+
+0.06
+0.04
+0.02
+0
+0.2
+0.4
+0.5
+0.6
+0.810-7
+.5
+0.5
+O
+0
+0.5
+0.5
+1.50.06
+0.04
+0.02
+0
+0.2
+0.4
+0.5
+0.6
+0.8STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY
+23
+to different age bands but same compartment are collected into vectors, S = (S1, . . . , S5),
+E = (E1, . . . , E5) and so on.
+Some of the variables introduced above are coupled to others only one way, and can be
+removed from the actual simulations.
+First, when the number of infected individuals is
+small compared to the population size (which is typically the case in the early stages of the
+epidemic), the relative variation of S is small. Hence, S can be taken constant instead of
+solving an ODE on it. Similarly, none of the variables depend on R and D, so they can be
+excluded from a coupled system of ODEs too, and computed separately after the solution of
+the ODEs. With these considerations in mind, the forward model reads as follows:
+d
+dt
+�
+�����
+E
+ISC1
+ISC2
+IC1
+IC2
+�
+�����
+−
+�
+�����
+−κI
+Au
+Au
+0
+0
+κ · diag(ρ)
+−ηCI
+0
+0
+0
+κ · diag(1 − ρ)
+0
+−ηRI
+0
+0
+0
+ηC · diag(ρ′)
+0
+−νI
+0
+0
+ηC · diag(1 − ρ′)
+0
+0
+−ηR,CI
+�
+�����
+�
+�����
+E
+ISC1
+ISC2
+IC1
+IC2
+�
+�����
+= 0.
+(6.14)
+Here I ∈ R5×5 is the identity matrix and diag(·) produces a diagonal matrix from a vec-
+tor. The control is defined in terms of the intensity of lockdown measures, and affects the
+susceptible-infected interaction matrix Au = χ · diag(S) · Cu · diag( 1
+N ), where
+Cu = diag(chome)Chome + diag(cwork
+u
+)Cwork + diag(cschool
+u
+)Cschool + diag(cother
+u
+)Cother (6.15)
+is the matrix of contact intensities between the age compartments. The total contact inten-
+sity is a sum of pre-pandemic contact intensity matrices in the four setting Chome, Cwork, Cschool
+and Cother, multiplied by the reduction factors chome, cwork
+u
+, cschool
+u
+and cother
+u
+due to the lock-
+down measures.
+Since home contacts cannot be controlled, chome = (1, . . . , 1), but the
+remaining factors vary proportionally to the lockdown control applied from day 17 onwards,
+cµ
+u(t) =
+�
+�
+�
+(1, 1, 1, 1, 1)⊤,
+t < 17,
+(c123(1 − uµ(t)), c123(1 − uµ(t)), c123(1 − uµ(t)), c4, c5)⊤,
+17 ≤ t ≤ 90,
+(c123(1 − uµ(90)), c123(1 − uµ(90)), c123(1 − uµ(90)), c4, c5)⊤,
+t > 90,
+(6.16)
+where µ ∈ {work, school, other}, uµ are the intensities of lockdown measures applied to each
+setting µ, and c123, c4, c5 are the initial contact intensities in the corresponding age groups.
+Note that the control will be optimized only on the time interval [17, 90]. Before day 17
+the contact intensities are not reduced (no lockdown). From day 90 onwards we continue
+applying the last value of the control.
+In addition, the model depends on the following parameters:
+• χ: probability of S–ISC interactions.
+• κ = 1/dL: average rate of an Exposed individual becoming SubClinical. It is inversely
+proportional to the average number of days dL an individual stays in the Exposed
+state.
+• ηC = 1/dC: average rate of a SubClinical individual becoming Clinical. Similarly, dC
+is the average time spent in the SubClinical state.
+• ηR = 1/dR: rate of recovery from ISC2.
+• ηR,C = 1/dR,C: rate of recovery from IC2.
+
+24
+HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA
+• ν = 1/dD: rate of decease in the IC1 state.
+• ρ = (ρ1, . . . , ρ5)⊤ ∈ R5: correction coefficients of the Exposed → SubClinical 1 tran-
+sition rate for different age bands.
+• ρ′ = (ρ′
+1, . . . , ρ′
+5)⊤ ∈ R5: correction coefficients of the SubClinical → Clinical 1 tran-
+sition.
+• N = (N1, . . . , N5)⊤ ∈ R5: total number of individuals in each age group.
+• N 0: total number of infected individuals on day 0.
+• N in = (0.1, 0.4, 0.35, 0.1, 0.05)⊤N 0: age partition of the initial number of infected
+individuals.
+The ODE (6.14) is initialized by setting
+E(0) = N in
+3 ,
+ISC1(0) = 2
+3diag(ρ)N in,
+ISC2(0) = 2
+3diag(1−ρ)N in,
+IC1(0) = IC2(0) = 0.
+The population sizes S = N are taken from the Office for National Statistics, mid 2018
+estimate.
+However, none of the model parameters above are known beforehand. In [11], those were
+treated as random variables, and their distributions were estimated from observed numbers
+of infections and hospitalizations during the first 90 days using Approximate Bayesian Com-
+putation (ABC). In general, these variables are correlated through the posterior distribution,
+sampling from which is a daunting problem. Here, we replace the joint ABC posterior dis-
+tribution by independent uniform distributions with a scaled posterior standard deviation
+centered around the posterior mean:
+χ ∼ U(0.13 − 0.03σ, 0.13 + 0.03σ),
+dL ∼ U(1.57 − 0.42σ, 1.57 + 0.42σ),
+(6.17)
+dC ∼ U(2.12 − 0.80σ, 2.12 + 0.80σ),
+dR ∼ U(1.54 − 0.40σ, 1.54 + 0.40σ),
+dR,C ∼ U(12.08 − 1.51σ, 12.08 + 1.51σ),
+dD ∼ U(5.54 − 2.19σ, 5.54 + 2.19σ),
+ρ1 ∼ U(0.06 − 0.03σ, 0.06 + 0.03σ),
+ρ2 ∼ U(0.05 − 0.03σ, 0.05 + 0.03σ),
+ρ3 ∼ U(0.08 − 0.04σ, 0.08 + 0.04σ),
+ρ4 ∼ U(0.54 − 0.22σ, 0.54 + 0.22σ),
+ρ5 ∼ U(0.79 − 0.14σ, 0.79 + 0.14σ),
+ρ′
+1 ∼ U(0.26 − 0.23σ, 0.26 + 0.23σ),
+ρ′
+2 ∼ U(0.28 − 0.25σ, 0.28 + 0.25σ),
+ρ′
+3 ∼ U(0.33 − 0.27σ, 0.33 + 0.27σ),
+ρ′
+4 ∼ U(0.26 − 0.11σ, 0.26 + 0.11σ),
+ρ′
+5 ∼ U(0.80 − 0.13σ, 0.80 + 0.13σ),
+N 0 ∼ U(276 − 133σ, 276 + 133σ),
+c123 ∼ U(0.63 − 0.21σ, 0.63 + 0.21σ),
+c4 ∼ U(0.57 − 0.23σ, 0.57 + 0.23σ),
+c5 ∼ U(0.71 − 0.23σ, 0.71 + 0.23σ).
+Here, σ is the standard deviation scaling parameter, taken to be 0.03 in our experiment.
+This distribution behaves qualitatively similar to the posterior distribution in the vicinity
+of the posterior mean.
+It provides sufficient randomness to benchmark the constrained
+optimization method, while admitting independent sampling and gridding, needed for the
+TT approximations. That is, (6.17) form a random vector
+ξ = (χ, dL, dC, dR, dR,C, dD, ρ1, ρ2, ρ3, ρ4, ρ5, ρ′
+1, ρ′
+2, ρ′
+3, ρ′
+4, ρ′
+5, N 0, c123, c4, c5)
+of d = 20 independent random variables, the state vector is
+y(ξ, t) = (E1, . . . , E5, ISC1
+1
+, . . . , ISC1
+5
+, ISC2
+1
+, . . . , ISC2
+5
+, IC1
+1 , . . . , IC1
+5 , IC2
+1 , . . . , IC2
+5 ),
+
+STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY
+25
+and the ODE (6.14) constitutes the forward problem.
+For the inverse problem, we use the total number of deceased patients as the cost function.
+The rate of decease is proportional to the number of Clinical type 1 individuals, so the total
+number of deceased individuals can be computed as
+D(ξ, t) = ν
+ˆ t
+0
+IC1(ξ, s)ds.
+(6.18)
+To regularize the problem, we add also the norm of the control u(t) = (uwork(t), uschool(t), uother(t)).
+Thus, the total cost function reads
+j(u) = 1
+2E[D(ξ, T)] + α
+2
+ˆ 90
+17
+∥u(t)∥2
+2dt,
+(6.19)
+where T = 100 is the final simulation time, and α is the regularization parameter, which we
+set to 100 in our experiment. Note that the norm of the control is taken only over the time
+interval [17, 90] where the control varies.
+We introduce the following constraints. Firstly, we limit the control components to the
+intervals uwork ∈ [0, 0.69], uschool ∈ [0, 0.9] and uother ∈ [0, 0.59]. Next, we constrain the
+R number at the end of the variable control interval, R(ξ, 90) ≤ 1. In our model, the R
+number can be computed as R(ξ, t) = λmax(K), where
+K = −
+�
+�����
+0
+Au
+Au
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+�
+�����
+�
+�����
+−κI
+0
+0
+0
+0
+κ · diag(ρ)
+−ηCI
+0
+0
+0
+κ · diag(1 − ρ)
+0
+−ηRI
+0
+0
+0
+ηC · diag(ρ′)
+0
+−νI
+0
+0
+ηC · diag(1 − ρ′)
+0
+0
+−ηR,CI
+�
+�����
+−1
+,
+and λmax denotes the maximal in modulus eigenvalue. Recall that R < 1 implies that the
+epidemic decays, while R > 1 corresponds to an expanding epidemic. The full smoothed
+Moreau-Yosida cost function becomes
+jγ,ε(u) = 1
+2E[D(ξ, T)] + α
+2
+ˆ 90
+17
+∥u(t)∥2
+2dt + γ
+2E
+���gε(R(ξ, 90) − 1)
+��2�
+.
+(6.20)
+Since the control is applied nonlinearly in the model, computation of derivatives of the cost
+function (6.20) is complicated. Thus, instead of the Newton method, we use the projected
+gradient descent method, where the gradient of (6.20) is calculated using finite differencing
+with anisotropic step sizes 10−6 �� max(|u|, 0.1). The ODE (6.14) is solved using an implicit
+Euler method with a time step 0.1.
+In this experiment, we use a fixed Moreau-Yosida
+parameter γ = 5 · 105 in all iterations, and the smoothing width is chosen as ε = 50/√γ.
+The iteration is stopped when the cost value does not decrease in two consecutive iterations.
+Each random variable (6.17) is discretized with n = 3 Gauss-Legendre quadrature nodes, and
+the TT approximations are carried out with a relative error tolerance of 10−2. The control
+u(t) is discretized using 7 Gauss-Legendre nodes on [17, 90] with a Lagrangian interpolation
+in between.
+In Figure 6, we compare optimizations without constraining R(ξ, 90) (left), and with the
+a.s. constraint (right) as described above. We plot the time evolution of the mean and
+confidence interval of the total number of hospitalized individuals, IC(t) = IC1(t) + IC2(t).
+
+26
+HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA
+0
+20
+40
+60
+80
+100
+0
+5
+10
+15
+20
+25
+t (days)
+hospitalizations (thousands)
+0
+20
+40
+60
+80
+100
+0
+5
+10
+15
+20
+25
+t (days)
+hospitalizations (thousands)
+0
+20
+40
+60
+80
+100
+0
+0.2
+0.4
+0.6
+0.8
+1
+t (days)
+u
+work
+school
+other
+0
+20
+40
+60
+80
+100
+0
+0.2
+0.4
+0.6
+0.8
+1
+t (days)
+u
+work
+school
+other
+Figure 6. Top: optimized IC = IC1 + IC2, mean (blue circles) and 95%
+confidence interval (shaded area). Bottom: optimized control signals. Left:
+unconstrained optimization, Right: optimization constrained with R(ξ, 90) ≤
+1 a.s. approximated with γ = 5 · 105. Black dashed lines indicate the end of
+the optimization time horizon t = 90.
+The unconstrained scenario is a finite horizon optimization problem, which drives the control
+to near zero values at the end of the controllable time interval, t = 90, due to the zero terminal
+condition on the adjoint state. Naturally, this leads to infection growing again for t > 90,
+since we extrapolate these small values of the control from t = 90 onwards.
+In contrast, if we constrain the R number at the end of the optimization interval to be
+below 1 almost surely, this drives the control to higher values again. If we extrapolate these
+
+STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY
+27
+control values beyond the optimization window, the epidemic continues decaying, albeit with
+a slightly larger uncertainty. This indicates that almost sure constraints can suggest a more
+resilient control in risk-critical applications.
+Appendix A. Proof of Lemma 3.11
+Introduce a new variable t = exp(s/ε), then
+ˆ 0
+−∞
+s log(1 + exp(s/ε))ds =
+ˆ 1
+0
+ε log(t) log(1 + t)
+t/ε
+dt
+= ε2
+ˆ 1
+0
+log(t) log(t + 1)d log(t)
+= ε2
+2 (log(t))2 log(t + 1)
+��1
+0 − ε2
+2
+ˆ 1
+0
+(log(t))2d log(t + 1).
+The first term is zero at t = 1, and at t = 0 we can use that 0 ≤ log(t + 1) ≤ t for 0 ≤ t < 1
+and limt→0(log(t))2 log(t + 1) ≤ limt→0(log(t))2t = 0. For the second term, we proceed as
+follows,
+ˆ 0
+−∞
+s log(1 + exp(s/ε))ds = −ε2
+2
+ˆ 1
+0
+(log(t))2
+t + 1
+dt
+≥ −ε2
+2
+ˆ 1
+0
+(log(t))2dt
+= −ε2
+2 t(log(t))2��1
+0
+�
+��
+�
+0
++ε2
+ˆ 1
+0
+log(t)dt
+= ε2 t log t|1
+0 − ε2
+ˆ 1
+0
+dt = −ε2.
+The proof is completed by recalling that sgε(s) = ε · s log(1 + exp(s/ε)).
+References
+[1] A. Alphonse, C. Geiersbach, M. Hinterm¨uller, and T. M. Surowiec. Risk-averse optimal control of
+random elliptic variational inequalities. arXiv preprint 2210.03425, 2022.
+[2] H. Antil, T.S. Brown, D. Verma, and M. Warma. Optimal control of fractional PDEs with state and
+control constraints. Pure Appl. Funct. Anal., 7(5):1533–1560, 2022.
+[3] H. Antil, S. Dolgov, and A. Onwunta. Ttrisk: Tensor train decomposition algorithm for risk averse
+optimization. Numerical Linear Algebra with Applications, n/a(n/a):e2481.
+[4] H. Antil, D.P. Kouri, M.-D. Lacasse, and D. Ridzal, editors. Frontiers in PDE-constrained optimization,
+volume 163 of The IMA Volumes in Mathematics and its Applications. Springer, New York, 2018. Papers
+based on the workshop held at the Institute for Mathematics and its Applications, Minneapolis, MN,
+June 6–10, 2016.
+[5] D. Bigoni, A. P. Engsig-Karup, and Y. M. Marzouk. Spectral tensor-train decomposition. SIAM J. Sci.
+Comput., 38(4):A2405–A2439, 2016.
+[6] E. Casas. Control of an elliptic problem with pointwise state constraints. SIAM J. Control Optim.,
+24(6):1309–1318, 1986.
+
+28
+HARBIR ANTIL, SERGEY DOLGOV, AND AKWUM ONWUNTA
+[7] S. Dolgov, B. N. Khoromskij, A. Litvinenko, and H. G. Matthies. Polynomial Chaos Expansion of
+random coefficients and the solution of stochastic partial differential equations in the Tensor Train
+format. SIAM J. Uncertainty Quantification, 3(1):1109–1135, 2015.
+[8] S. Dolgov and D. Savostyanov. Parallel cross interpolation for high–precision calculation of high–
+dimensional integrals. Comput. Phys. Commun., 246:106869, 2020.
+[9] S. V. Dolgov, B. N. Khoromskij, I. V. Oseledets, and D. V. Savostyanov. Computation of extreme
+eigenvalues in higher dimensions using block tensor train format. Comput. Phys. Commun., 185(4):1207–
+1216, 2014.
+[10] S. V. Dolgov and D. V. Savostyanov. Alternating minimal energy methods for linear systems in higher
+dimensions. SIAM Journal on Scientific Computing, 36(5):A2248–A2271, 2014.
+[11] R. Dutta, S. N. Gomes, D. Kalise, and L. Pacchiardi. Using mobility data in the design of optimal
+lockdown strategies for the COVID-19 pandemic. PLoS Comput. Biol., 17(8):1–25, 2021.
+[12] M. H. Farshbaf-Shaker, R. Henrion, and D. H¨omberg. Properties of chance constraints in infinite di-
+mensions with an application to PDE constrained optimization. Set-Valued Var. Anal., 26(4):821–841,
+2018.
+[13] D.B. Gahururu, M. Hinterm¨uller, and T.M. Surowiec. Risk-neutral pde-constrained generalized nash
+equilibrium problems. Mathematical Programming, 2022.
+[14] S. Garreis, T. M. Surowiec, and M. Ulbrich. An interior-point approach for solving risk-averse PDE-
+constrained optimization problems with coherent risk measures. SIAM J. Optim., 31(1):1–29, 2021.
+[15] C. Geiersbach and W. Wollner. Optimality conditions for convex stochastic optimization problems in
+Banach spaces with almost sure state constraints. SIAM J. Optim., 31(4):2455–2480, 2021.
+[16] Caroline Geiersbach and Michael Hinterm¨uller. Optimality Conditions and Moreau–Yosida Regular-
+ization for Almost Sure State Constraints. ESAIM Control Optim. Calc. Var., 28:Paper No. 80, 36,
+2022.
+[17] A. Geletu, A. Hoffmann, P. Schmidt, and P. Li. Chance constrained optimization of elliptic PDE systems
+with a smoothing convex approximation. ESAIM Control Optim. Calc. Var., 26:Paper No. 70, 28, 2020.
+[18] S. A. Goreinov, I. V. Oseledets, D. V. Savostyanov, E. E. Tyrtyshnikov, and N. L. Zamarashkin. How
+to find a good submatrix. In V. Olshevsky and E. Tyrtyshnikov, editors, Matrix Methods: Theory,
+Algorithms, Applications, pages 247–256. World Scientific, Hackensack, NY, 2010.
+[19] A. Gorodetsky, S. Karaman, and Y. Marzouk. A continuous analogue of the tensor-train decomposition.
+Comput. Methods Appl. Mech. Engrg., 347:59–84, 2019.
+[20] W. Hackbusch and B. N. Khoromskij. Low-rank Kronecker-product approximation to multi-dimensional
+nonlocal operators. I. Separable approximation of multi-variate functions. Computing, 76(3-4):177–202,
+2006.
+[21] M. Hinterm¨uller and M. Hinze. Moreau-Yosida regularization in state constrained elliptic control prob-
+lems: Error estimates and parameter adjustment. SIAM Journal on Numerical Analysis, 47(3):1666–
+1683, 2009.
+[22] M. Hoffhues, W. R¨omisch, and T. M. Surowiec. On quantitative stability in infinite-dimensional opti-
+mization under uncertainty. Optimization Letters, 15(8):2733–2756, 2021.
+[23] D. P. Kouri and T. M. Surowiec. Risk-averse PDE-constrained optimization using the conditional value-
+at-risk. SIAM J. Optim., 26(1):365–396, 2016.
+[24] K. Kunisch and D. Wachsmuth. Sufficient optimality conditions and semi-smooth Newton methods for
+optimal control of stationary variational inequalities. ESAIM Control Optim. Calc. Var., 18(2):520–547,
+2012.
+[25] R. L¨ohner, H. Antil, S. Idelsohn, and E. O˜nate. Detailed simulation of viral propagation in the built
+environment. Comput. Mech., 66(5):1093–1107, 2020.
+[26] R. L¨ohner, H. Antil, A. Srinivasan, S Idelsohn, and E. O˜nate. High-fidelity simulation of pathogen
+propagation, transmission and mitigation in the built environment. Archives of Computational Methods
+in Engineering, pages 1–26, 2021.
+[27] G. J. Lord, C. E. Powell, and T. Shardlow. An Introduction to Computational Stochastic PDEs. West
+Nyack: Cambridge University Press, 2014.
+
+STATE-CONSTRAINED OPTIMIZATION PROBLEMS UNDER UNCERTAINTY
+29
+[28] K. Maute. Topology optimization under uncertainty. In Topology optimization in structural and contin-
+uum mechanics, pages 457–471. Springer, 2014.
+[29] A. Yu. Mikhalev and I. V. Oseledets. Rectangular maximum–volume submatrices and their applications.
+Linear Algebra Appl., 538:187–211, 2018.
+[30] I. V. Oseledets. Tensor train decomposition. SIAM J. Sci. Comp., 33(5):2295 – 2317, 2011.
+[31] I. V. Oseledets and E. E. Tyrtyshnikov. TT-cross approximation for multidimensional arrays. Linear
+Algebra Appl., 432(1):70–88, 2010.
+[32] R. T. Rockafellar and S. Uryasev. Optimization of conditional value-at-risk. The Journal of Risk, 2:21
+– 41, 2000.
+[33] P. B. Rohrbach, S. Dolgov, L. Grasedyck, and R. Scheichl. Rank bounds for approximating Gaussian
+densities in the Tensor-Train format. SIAM/ASA Journal on Uncertainty Quantification, 10(3):1191–
+1224, 2022.
+[34] D. V. Savostyanov and I. V. Oseledets. Fast adaptive interpolation of multi-dimensional arrays in tensor
+train format. In Proceedings of 7th International Workshop on Multidimensional Systems (nDS). IEEE,
+2011.
+[35] R. Schneider and A. Uschmajew. Approximation rates for the hierarchical tensor format in periodic
+Sobolev spaces. J. Complexity, 2013.
+[36] J. Soko�lowski and J. P. Zol´esio. Introduction to shape optimization, volume 16 of Springer Series in
+Computational Mathematics. Springer-Verlag, Berlin, 1992. Shape sensitivity analysis.
+[37] L. N. Trefethen. Spectral methods in MATLAB. SIAM, Philadelphia, 2000.
+[38] F. Tr¨oltzsch. Optimal Control of Partial Differential Equations: Theory, Methods and Applications.
+American Mathematical Society, 2010.
+Harbir Antil, The Center for Mathematics and Artificial Intelligence (CMAI) and De-
+partment of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA.
+Email address: hantil@gmu.edu
+Sergey Dolgov, Department of Mathematical Sciences, University of Bath, Bath, BA2
+7AY, UK.
+Email address: s.dolgov@bath.ac.uk
+Akwum Onwunta, Department of Industrial and Systems Engineering, Lehigh University,
+Bethlehem, PA 18015, USA.
+Email address: ako221@lehigh.edu
+

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+A prototype tank for the SWGO detector
+Sofia Grusovin,∗ Giovanni Consolati,𝑎, 𝑓 Alessandro de Angelis,𝑏,𝑒 Cornelia Arcaro,𝑒
+Francesca Bisconti,𝑐,𝑔 Andrea Chiavassa,𝑐,𝑔 Michele Doro,𝑏,𝑒 Fausto Guarino,𝑑,ℎ
+Mosè Mariotti𝑏,𝑒 and Elisa Prandini𝑏,𝑒
+𝑎Department of Aerospace Science and Technology, Politecnico di Milano,
+Via LaMasa 34, 20156 Milano, Italy
+𝑏Dipartimento di Fisica e Astronomia G. Galilei, Università degli Studi di Padova,
+Via F. Marzolo 8, 35131 Padova, Italy
+𝑐Dipartimento di Fisica, Università degli Studi di Torino,
+Via Pietro Giuria 1, 10125 Torino, Italy
+𝑑Dipartimento di Fisica "Ettore Pancini", Università degli Studi di Napoli Federico II,
+Strada Comunale Cinthia, 80126 Napoli, Italy
+𝑒INFN Padova,
+Via Francesco Marzolo 8, 35131 Padova, Italy
+𝑓 INFN Milano,
+Via Celoria 16, 20133 Milano, Italy
+𝑔INFN Torino,
+Via Pietro Giuria 1, 10125 Torino, Italy
+ℎINFN Napoli,
+Strada Comunale Cinthia, 80126 Napoli, Italy
+E-mail: sofia.grusovin@protonmail.com, giovanni.consolati@polimi.it,
+alessandro.deangelis@unipd.it, cornelia.arcaro@pd.infn.it,
+fbiscont@to.infn.it, achiavas@to.infn.it, michele.doro@unipd.it,
+guarino@na.infn.it, mose.mariotti@unipd.it, elisa.prandini@unipd.it
+The Southern Wide-field Gamma-ray Observatory (SWGO) is an international collaboration work-
+ing on realizing a next-generation observatory located in the Southern hemisphere, which offers
+a privileged view of our galactic center. We are working on the construction of a prototype water
+Cherenkov detector at Politecnico di Milano using a flexible testing facility for several candidate
+light sensors and configurations.
+A structure able to hold different types of detectors in multiple configurations has been designed,
+built and tested in Politecnico’s labs. Furthermore, an analytical study of muons and electrons
+showers has been carried out using the SWGO observatory simulation software to examine the
+correlation between the detection capabilities of the prototype tank and its water level.
+*** 27th European Cosmic Ray Symposium - ECRS ***
+*** 25-29 July 2022 ***
+*** Nijmegen, the Netherlands ***
+∗Speaker
+© Copyright owned by the author(s) under the terms of the Creative Commons
+Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).
+https://pos.sissa.it/
+arXiv:2301.02449v1  [astro-ph.IM]  6 Jan 2023
+
+A prototype tank for the SWGO detector
+Sofia Grusovin
+1.
+Introduction
+The Southern Wide-field Gamma ray Observatory (SWGO) is an international collaboration
+with the objective of realizing a new-generation observatory for Very-High-Energy (VHE) gamma
+rays. The observatory will be located in South America at a latitude between -30° and -10°, at
+an altitude of at least 4.4 km. Some similar facilities already exist, such as HAWC in Mexico[10]
+and LHAASO in China[9], but SWGO will be the first observatory of its kind in the Southern
+hemisphere, which has a privileged view of the galactic center region[1, 4, 8] (fig. 1b).
+(a) Qualitative representation of SWGO’s
+layout.[1]
+(b) FoV of SWGO and HAWC.[1]
+Figure 1: SWGO design.
+The observatory will be based on thousends of Water Cherenkov units (fig. 1a). In the design of
+such units, there are many variables to assess, such as the detector’s geometry, the types of sensors
+to be used and their configuration, and the materials; therefore some test facilities are needed[5].
+2.
+The prototype tank
+The Italian partners of the SWGO collaboration (Istituto Nazionale di Fisica Nucleare, Politec-
+nico di Milano, Università degli Studi di Torino, Università degli Studi di Padova and Università
+degli Studi di Napoli Federico II) are working on the realization of a test facility to be used for
+SWGO. It will consist of a cylindrical tank (3.32 m diameter and 3.12 m height) with an external
+structure made of galvanized steel and an internal liner in AQUATEX® PVC which could be then
+changed for different tests (fig. 2). The prototype tank is installed at Politecnico di Milano, and
+its objective is to act as a test facility for different types of sensors and sensors’ configurations,
+different types of liners, different tank configurations and anything else that could be useful to the
+collaboration.
+Multi-PMT modules and light traps for example are being studied at the universities of Padova
+and Napoli respectively and the prototype tank will be used to test them. Multi-PMT modules are
+made by 3” PMTs, like KM3 Hyper-Kamiokade (fig. 4a), which has been considered as inspiration
+although a MoU is still being finalized. Light traps on the other hand use Wavelength Shiftters
+(WLS), which are materials that absorb light and re-emit it at a lower energy in a different random
+direction, so the light ends up trapped inside because of internal reflection (fig. 4b). This means
+that a small sensor can be used to capture the light by using a large WLS surface[2, 3].
+2
+
+PARTICLEDETECTORARRAY
+NottoscafeTHEFERMI BUBBLES
+SWGO
+ImageCredit:NASA
+Invisible to HAWC
+Image Credit:HAWCCollab
+(Preliminary)
+HAWC
+Invisible to HAWC
+&SWGO
+THE GALACTIC CENTRE
+Image Credit: SARAOA prototype tank for the SWGO detector
+Sofia Grusovin
+(a) B6 building in Politecnico di Milano’s Bovisa campus: site of the test
+installation and first candidate for the tank’s location.
+(b) The prototype tank inside
+B6 labs after test installation.
+Figure 2: Test installation of the Prototype Tank.
+(a) Hyper-K prototype considered as inspi-
+ration.
+(b) WLS light traps: concept and tests.
+Figure 3: Experimental sensors design.
+2.1 Main requirements
+The first main requirement to be considered for the tank’s installation site is that the floor must
+resist the tank’s pressure due to the large volume of water contained. An analysis on detection
+capabilities as a function of the water level has therefore been done first, in order to choose the
+appropriate water level (see section 3). The analysis’s results could also be useful in the future for
+the studies on tank geometry.
+Since the prototype tank is a test facility, it will be of main importance to be able to change
+the sensors’ configurations frequently, so an appropriate structure was designed to hold the sensors.
+During the design process, another main requirement that had to be taken into account was the need
+of maintaining high water purity inside the tank, which posed limits on the materials’ choices.
+3
+
+photon
+detectorA prototype tank for the SWGO detector
+Sofia Grusovin
+3.
+Study on particle detection as a function of the water level
+A preliminary study has been conducted on two showers of muons with the energies of 1 GeV
+and 10 GeV: muons in fact are the only particles that will most likely be detected at Milano’s altitude.
+The shower’s interaction with the tank has been studied with 5 different water levels: 1.65 m, 2.00
+m, 2.35 m, 2.70 m and 3.05 m (fig. 4).
+(a) Prototype tank with 2.0 m water level.
+(b) Prototype tank with 2.7 m water level.
+Figure 4: Visual output of the simulation of a muon (green) entering the tank and producing photons (red)
+in two different scenarios.
+An additional study has been carried out on electron showers with the energy of 1 GeV to
+verify the model, since a different behavior was expected from the tank’s interaction with muons
+and electrons. Muons are more penetrating particles with respect to electrons: the latter lose energy
+faster, thus producing Cherenkov light only during a brief segment in the upper part of the detector;
+muons on the other hand usually cross the whole tank. A lower number of PE detected by the PMTs
+can be expected for electrons with a higher water level since they are produced only in the upper
+part of the tank and many of them would therefore be absorbed by the water before reaching the
+photosensors.
+Both studies have been carried out using the HAWCSim framework, which makes use of
+Geant4[6] to simulate the interaction of the particle with the tank itself and the water, includ-
+ing the production of Cherenkov photons that can be detected by the Photo Multiplier Tubes (PMTs)
+installed inside the tank (fig. 4).
+The tank’s dimensions and materials used for the simulations were the ones reported in section 2.
+The sensors’ configuration chosen (which will be the prototype tank’s reference configuration) was
+made by:
+• 1 × 10” (253 mm) PMT situated in the center of the tank
+• 4 × 5” (128 mm) PMT equidistant from the center in square configuration
+In HAWCSim, three models of PMT from the Hamamatsu company were available at the time
+of this analysis: 8” R5912 PMT, 10” R7081HQE PMT and 3” R12199 PMT. To simulate the four
+5” PMTs, the 8” R5912 PMTs were used and then scaled to 5” during the analysis phase[5].
+4
+
+A prototype tank for the SWGO detector
+Sofia Grusovin
+3.1 Particles generation
+12000 particles (for each case: 1 GeV muons, 10 GeV muons and 1 GeV electrons) have been
+generated in a disk of fixed radius over the tank (r = 1.78 m, 10 cm larger than the tank’s radius; h
+= 3.22 m, 10 cm larger than the tank’s height). Their azimuth angle 𝜙 has been randomly selected
+in the range 𝜃 - 360° and zenith angle 𝜃 extracted from the distribution cos(𝜃)2. In each scenario,
+the first 10000 particles entering the tank have been analyzed. Since they had random direction in
+fact, not all the particles would enter the tank, therefore generating 12000 particles assured having
+at least 10000 that could be used for the analysis[7].
+3.2 Results
+As it can be seen from fig. 5b, the detection efficiency increases linearly with the water
+level and the standard deviation of the first photon time decreases with the water level (fig. 5c),
+meaning that the detector becomes more sensitive with more water. The number of photoelectrons
+detected is important to notice because here the difference between muons and electrons can be
+clearly appreciated (fig. 5a). The number of PE detected decreasing with higher levels of water for
+electrons was the expected behavior, in contrast with muons, for which that number increases, so
+the model was considered to be reliable.
+(a) Number of PE detected.
+(b) Detection efficiency.
+(c) Standard deviation of the first photon time.
+Figure 5: Plots of results for muons (1 GeV and 10 GeV) and electrons (1 GeV).
+4.
+Design of the PMT holder
+The material chosen for the PMT holder has been stainless steel AISI 304, which had good
+mechanical properties and was compatible with purified water.
+5
+
+Number of PE detected by PMT 1 (1o"
+140
+120
+Number of PE detected
+100
+80
+60
+40
+Npe 1GeV PMT 1 mu-
+20
+Npe 10GeV PMT 1 mu-
+Npe 1GeV PMT 1 e-
+0.
+1.65
+2.00
+2.35
+2.70
+3.05
+Water levelDetection efficiency of PMT 1 (1o"
+1
+0.9
+0.8
+Detection efficiency
+0.7
+0.6
+0.5
+0.4
+0.3
+Eff. 1GeV PMT 1 mu-
+0.2
+Eff. 10GeV PMT 1 mu-
+0.1
+Eff. 1GeV PMT 1 e-
+0
+1.65
+2.00
+2.35
+2.70
+3.05
+Water levelSD of the first photon time on PMT 1 (10")
+3.5
+ Standard deviation
+2.5
+1.5
+std dev.f.time1GeV PMT1
+1
+mu-
+std deV. f.time 10GeV PMT 1
+0.5
+mu-
+std deV. f.time 1GeV PMT 1 e-
+0
+1.65
+2.00
+2.35
+2.70
+3.05
+Water levelA prototype tank for the SWGO detector
+Sofia Grusovin
+The first idea was to design a structure that could hold as many sensors configurations as
+possible. A CAD model of such structure had been realized (fig. 6), and it was a good compromise
+between weight, robustness and flexibility requirements. This design however had to be adapted to
+parts available on the market and it was not easy to find similar pieces. In addition to that, the PMTs
+of the reference configuration were nearly ready to be tested, while the other sensors that have to be
+tested here are still in development.
+VII - 19 sensors (1 + 6 + 6 + 6)
+VI - 13 sensors (1 + 6 + 6)
+V - 5 sensors (1 + 4)
+IV - 10 sensors (1 + 3 + 6)
+I - 4 sensors (1 + 3)
+IIa and IIb - 4 sensors compact (1 + 3)
+III - 7 sensors (1 + 6)
+(a) Initial structure scheme.
+(b) Initial structure cad model.
+Figure 6: First design of the PMT holder.
+It has therefore been decided to design a simpler structure to be used for the reference configu-
+ration, designing it directly with parts that were known to be available on the market, with the idea
+of disassembling it and reuse the parts for the bigger structure when it will be needed (fig. 7 and
+fig. 8).
+A loading analysis has been performed in solidworks applying the mass of the reference
+configuration PMTs and it has been shown that the structure could easily sustain the weight (fig. 9).
+To properly hold the PMTs, the same C-profiles used for the main structure have been used since
+the materials were already available and they were able to guarantee good stability. An additional
+gasket has to be placed between the C-profiles and the PMT in order to protect the sensor.
+Figure 7: 3D model of the cross holder.
+6
+
+A prototype tank for the SWGO detector
+Sofia Grusovin
+(a) Part a: flat bar.
+(b) Part b: C-profile.
+(c) Part c: L-profile.
+Figure 8: Parts composing the cross holder: part a is the base piece bought from the supplier (1000 mm x
+50 mm x 1 mm perforated flat bar) from which part b and part c can be obtained.
+(a) Von Mises stress with static maximum load.
+(b) Displacement with static maximum load.
+Figure 9: Cross holder: results of the solidworks static analysis.
+The parts have been manufactured in the workshop of the Department of Aerospace Science
+and Technology (Politecnico di Milano) by cutting and bending the flat bars. The C-profiles and
+the L-profiles could be obtained by cutting and folding the original perforated flat bars.
+When the structure was complete, a placement test with the five reference configuration PMTs
+has been done. The appropriate rubber for the gaskets, compatible with purified water, had not been
+ordered yet, so foam rubber has been used. The structure was not deformed by the PMTs’ weight
+and the sensors were held stably (fig. 10).
+5.
+Conclusions
+After considering the data coming from the study on particle detection as a function of the
+water level, it has been decided to maximize detection performances allowing the possibility to fill
+the tank completely.
+Since the PMT holder is ready to be used, the first test with the reference configuration will
+start soon, followed by tests with the novel sensors designs from Napoli end Padova described in
+section 2.
+7
+
+von Mises (N/m^2)
+7,581e+06
+6,824 +06
+6,066e +06
+5,308e +06
+4,550t+06
+3,793 +06
+3,015 +06
+2,277e +06
+1,519e +06
+7.,617e+05
+3,943± -03URES (mm)
+3,077e-02
+2,769e-02
+2,462±-02
+2,154e-02
+1,846e-02
+1,539e-02
+1,231e-02
+9,231e-03
+6,154e-03
+3,077e-03
+1,000e-30A prototype tank for the SWGO detector
+Sofia Grusovin
+Figure 10: PMT holder: holding test.
+References
+[1] INFN and SWGO collaboration, INFN SWGO letter of intent, 2020.
+[2] J.E. Ward, J. Cortina and D. Guberman, Light-Trap: a SiPM upgrade for VHE astronomy and
+beyond, Journal of Instrumentation 11 (2016).
+[3] M. Mariotti et al., Optimized wavelength shifters light traps with SiPM photo sensors for
+SWGO, Internal SWGO paper.
+[4] J. Hinton, The Southern Wide-field Gamma-ray Observatory: status and srospects, Proceed-
+ings of Science 395 ICRC2021 (2022) 023.
+[5] F. Bisconti and A. Chiavassa, Study of water Cherenkov detector designs for the SWGO
+experiment, Proceedings of Science 395 ICRC2021 (2022) 895.
+[6] S. Agostinelli et al., GEANT4 — a simulation toolkit, Nuclear Instruments and Methods in
+Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equip-
+ment 506 (2003) 250.
+[7] F. Bisconti and A. Chiavassa, Study of water Cherenkov detector design for ground-based
+gamma-ray experiments, arXiv e-prints (2022).
+[8] A. Albert et al., Science case for a wide field-of-view very-high-energy gamma-ray observatory
+in the Southern hemisphere, arXiv e-prints (2019).
+[9] F. A. Aharonian, LHAASO: A PeVatrons explorer, Science China Physics, Mechanics &
+Astronomy volume 64 (2021) 109531.
+[10] G. Sinnis, The HAWC TeV gamma-ray observatory, Il nuovo cimento C 34 (2021) 65.
+8
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+page_content=' 80126 Napoli,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' Italy  𝑒INFN Padova,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  Via Francesco Marzolo 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' 35131 Padova,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' Italy  𝑓 INFN Milano,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  Via Celoria 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' 20133 Milano,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' Italy  𝑔INFN Torino,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  Via Pietro Giuria 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' 10125 Torino,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' Italy  ℎINFN Napoli,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  Strada Comunale Cinthia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' 80126 Napoli,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' Italy  E-mail: sofia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='grusovin@protonmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='com, giovanni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='consolati@polimi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='it,  alessandro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='deangelis@unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='it, cornelia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='arcaro@pd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='it,  fbiscont@to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='it, achiavas@to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='it, michele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='doro@unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='it,  guarino@na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='it, mose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='mariotti@unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='it, elisa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='prandini@unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='it  The Southern Wide-field Gamma-ray Observatory (SWGO) is an international collaboration work-  ing on realizing a next-generation observatory located in the Southern hemisphere, which offers  a privileged view of our galactic center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' We are working on the construction of a prototype water  Cherenkov detector at Politecnico di Milano using a flexible testing facility for several candidate  light sensors and configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  A structure able to hold different types of detectors in multiple configurations has been designed,  built and tested in Politecnico’s labs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' Furthermore, an analytical study of muons and electrons  showers has been carried out using the SWGO observatory simulation software to examine the  correlation between the detection capabilities of the prototype tank and its water level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  *** 27th European Cosmic Ray Symposium - ECRS ***  *** 25-29 July 2022 ***  *** Nijmegen, the Netherlands ***  ∗Speaker  © Copyright owned by the author(s) under the terms of the Creative Commons  Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='it/  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='02449v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='IM]  6 Jan 2023  A prototype tank for the SWGO detector  Sofia Grusovin  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  Introduction  The Southern Wide-field Gamma ray Observatory (SWGO) is an international collaboration  with the objective of realizing a new-generation observatory for Very-High-Energy (VHE) gamma  rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' The observatory will be located in South America at a latitude between -30° and -10°, at  an altitude of at least 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='4 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' Some similar facilities already exist, such as HAWC in Mexico[10]  and LHAASO in China[9], but SWGO will be the first observatory of its kind in the Southern  hemisphere, which has a privileged view of the galactic center region[1, 4, 8] (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  (a) Qualitative representation of SWGO’s  layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' [1]  (b) FoV of SWGO and HAWC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' [1]  Figure 1: SWGO design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  The observatory will be based on thousends of Water Cherenkov units (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' In the design of  such units, there are many variables to assess, such as the detector’s geometry, the types of sensors  to be used and their configuration, and the materials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' therefore some test facilities are needed[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  The prototype tank  The Italian partners of the SWGO collaboration (Istituto Nazionale di Fisica Nucleare, Politec-  nico di Milano, Università degli Studi di Torino, Università degli Studi di Padova and Università  degli Studi di Napoli Federico II) are working on the realization of a test facility to be used for  SWGO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' It will consist of a cylindrical tank (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='32 m diameter and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='12 m height) with an external  structure made of galvanized steel and an internal liner in AQUATEX® PVC which could be then  changed for different tests (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' The prototype tank is installed at Politecnico di Milano, and  its objective is to act as a test facility for different types of sensors and sensors’ configurations,  different types of liners, different tank configurations and anything else that could be useful to the  collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  Multi-PMT modules and light traps for example are being studied at the universities of Padova  and Napoli respectively and the prototype tank will be used to test them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' Multi-PMT modules are  made by 3” PMTs, like KM3 Hyper-Kamiokade (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' 4a), which has been considered as inspiration  although a MoU is still being finalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' Light traps on the other hand use Wavelength Shiftters  (WLS), which are materials that absorb light and re-emit it at a lower energy in a different random  direction, so the light ends up trapped inside because of internal reflection (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' This means  that a small sensor can be used to capture the light by using a large WLS surface[2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  2  PARTICLEDETECTORARRAY  NottoscafeTHEFERMI BUBBLES  SWGO  ImageCredit:NASA  Invisible to HAWC  Image Credit:HAWCCollab  (Preliminary)  HAWC  Invisible to HAWC  &SWGO  THE GALACTIC CENTRE  Image Credit: SARAOA prototype tank for the SWGO detector  Sofia Grusovin  (a) B6 building in Politecnico di Milano’s Bovisa campus: site of the test  installation and first candidate for the tank’s location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  (b) The prototype tank inside  B6 labs after test installation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  Figure 2: Test installation of the Prototype Tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  (a) Hyper-K prototype considered as inspi-  ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  (b) WLS light traps: concept and tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  Figure 3: Experimental sensors design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='1 Main requirements  The first main requirement to be considered for the tank’s installation site is that the floor must  resist the tank’s pressure due to the large volume of water contained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' An analysis on detection  capabilities as a function of the water level has therefore been done first, in order to choose the  appropriate water level (see section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' The analysis’s results could also be useful in the future for  the studies on tank geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  Since the prototype tank is a test facility, it will be of main importance to be able to change  the sensors’ configurations frequently, so an appropriate structure was designed to hold the sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  During the design process, another main requirement that had to be taken into account was the need  of maintaining high water purity inside the tank, which posed limits on the materials’ choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  3  photon  detectorA prototype tank for the SWGO detector  Sofia Grusovin  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  Study on particle detection as a function of the water level  A preliminary study has been conducted on two showers of muons with the energies of 1 GeV  and 10 GeV: muons in fact are the only particles that will most likely be detected at Milano’s altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  The shower’s interaction with the tank has been studied with 5 different water levels: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='65 m, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='00  m, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='35 m, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='70 m and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='05 m (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  (a) Prototype tank with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='0 m water level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  (b) Prototype tank with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='7 m water level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  Figure 4: Visual output of the simulation of a muon (green) entering the tank and producing photons (red)  in two different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  An additional study has been carried out on electron showers with the energy of 1 GeV to  verify the model, since a different behavior was expected from the tank’s interaction with muons  and electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' Muons are more penetrating particles with respect to electrons: the latter lose energy  faster, thus producing Cherenkov light only during a brief segment in the upper part of the detector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  muons on the other hand usually cross the whole tank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' A lower number of PE detected by the PMTs  can be expected for electrons with a higher water level since they are produced only in the upper  part of the tank and many of them would therefore be absorbed by the water before reaching the  photosensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  Both studies have been carried out using the HAWCSim framework, which makes use of  Geant4[6] to simulate the interaction of the particle with the tank itself and the water, includ-  ing the production of Cherenkov photons that can be detected by the Photo Multiplier Tubes (PMTs)  installed inside the tank (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  The tank’s dimensions and materials used for the simulations were the ones reported in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  The sensors’ configuration chosen (which will be the prototype tank’s reference configuration) was  made by:  1 × 10” (253 mm) PMT situated in the center of the tank  4 × 5” (128 mm) PMT equidistant from the center in square configuration  In HAWCSim, three models of PMT from the Hamamatsu company were available at the time  of this analysis: 8” R5912 PMT, 10” R7081HQE PMT and 3” R12199 PMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' To simulate the four  5” PMTs, the 8” R5912 PMTs were used and then scaled to 5” during the analysis phase[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  4  A prototype tank for the SWGO detector  Sofia Grusovin  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='1 Particles generation  12000 particles (for each case: 1 GeV muons, 10 GeV muons and 1 GeV electrons) have been  generated in a disk of fixed radius over the tank (r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='78 m, 10 cm larger than the tank’s radius;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' h  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='22 m, 10 cm larger than the tank’s height).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' Their azimuth angle 𝜙 has been randomly selected  in the range 𝜃 - 360° and zenith angle 𝜃 extracted from the distribution cos(𝜃)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' In each scenario,  the first 10000 particles entering the tank have been analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' Since they had random direction in  fact, not all the particles would enter the tank, therefore generating 12000 particles assured having  at least 10000 that could be used for the analysis[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='2 Results  As it can be seen from fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' 5b, the detection efficiency increases linearly with the water  level and the standard deviation of the first photon time decreases with the water level (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' 5c),  meaning that the detector becomes more sensitive with more water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' The number of photoelectrons  detected is important to notice because here the difference between muons and electrons can be  clearly appreciated (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' The number of PE detected decreasing with higher levels of water for  electrons was the expected behavior, in contrast with muons, for which that number increases, so  the model was considered to be reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  (a) Number of PE detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  (b) Detection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  (c) Standard deviation of the first photon time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  Figure 5: Plots of results for muons (1 GeV and 10 GeV) and electrons (1 GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  Design of the PMT holder  The material chosen for the PMT holder has been stainless steel AISI 304, which had good  mechanical properties and was compatible with purified water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  5  Number of PE detected by PMT 1 (1o"  140  120  Number of PE detected  100  80  60  40  Npe 1GeV PMT 1 mu-  20  Npe 10GeV PMT 1 mu-  Npe 1GeV PMT 1 e-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='65  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='35  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='70  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='05  Water levelDetection efficiency of PMT 1 (1o"  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='8  Detection efficiency  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='3  Eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' 1GeV PMT 1 mu-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='2  Eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' 10GeV PMT 1 mu-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='1  Eff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' 1GeV PMT 1 e-  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='65  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='35  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='70  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='05  Water levelSD of the first photon time on PMT 1 (10")  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='5  Standard deviation  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='5  std dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='time1GeV PMT1  1  mu-  std deV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='time 10GeV PMT 1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='5  mu-  std deV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='time 1GeV PMT 1 e-  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='65  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='35  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='70  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='05  Water levelA prototype tank for the SWGO detector  Sofia Grusovin  The first idea was to design a structure that could hold as many sensors configurations as  possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' A CAD model of such structure had been realized (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' 6), and it was a good compromise  between weight, robustness and flexibility requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' This design however had to be adapted to  parts available on the market and it was not easy to find similar pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' In addition to that, the PMTs  of the reference configuration were nearly ready to be tested, while the other sensors that have to be  tested here are still in development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  VII - 19 sensors (1 + 6 + 6 + 6)  VI - 13 sensors (1 + 6 + 6)  V - 5 sensors (1 + 4)  IV - 10 sensors (1 + 3 + 6)  I - 4 sensors (1 + 3)  IIa and IIb - 4 sensors compact (1 + 3)  III - 7 sensors (1 + 6)  (a) Initial structure scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  (b) Initial structure cad model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  Figure 6: First design of the PMT holder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  It has therefore been decided to design a simpler structure to be used for the reference configu-  ration, designing it directly with parts that were known to be available on the market, with the idea  of disassembling it and reuse the parts for the bigger structure when it will be needed (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' 7 and  fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  A loading analysis has been performed in solidworks applying the mass of the reference  configuration PMTs and it has been shown that the structure could easily sustain the weight (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  To properly hold the PMTs, the same C-profiles used for the main structure have been used since  the materials were already available and they were able to guarantee good stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' An additional  gasket has to be placed between the C-profiles and the PMT in order to protect the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  Figure 7: 3D model of the cross holder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  6  A prototype tank for the SWGO detector  Sofia Grusovin  (a) Part a: flat bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  (b) Part b: C-profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  (c) Part c: L-profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  Figure 8: Parts composing the cross holder: part a is the base piece bought from the supplier (1000 mm x  50 mm x 1 mm perforated flat bar) from which part b and part c can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  (a) Von Mises stress with static maximum load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  (b) Displacement with static maximum load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  Figure 9: Cross holder: results of the solidworks static analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  The parts have been manufactured in the workshop of the Department of Aerospace Science  and Technology (Politecnico di Milano) by cutting and bending the flat bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' The C-profiles and  the L-profiles could be obtained by cutting and folding the original perforated flat bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  When the structure was complete, a placement test with the five reference configuration PMTs  has been done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' The appropriate rubber for the gaskets, compatible with purified water, had not been  ordered yet, so foam rubber has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' The structure was not deformed by the PMTs’ weight  and the sensors were held stably (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  Conclusions  After considering the data coming from the study on particle detection as a function of the  water level, it has been decided to maximize detection performances allowing the possibility to fill  the tank completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  Since the PMT holder is ready to be used, the first test with the reference configuration will  start soon, followed by tests with the novel sensors designs from Napoli end Padova described in  section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  7  von Mises (N/m^2)  7,581e+06  6,824 +06  6,066e +06  5,308e +06  4,550t+06  3,793 +06  3,015 +06  2,277e +06  1,519e +06  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=',617e+05  3,943± -03URES (mm)  3,077e-02  2,769e-02  2,462±-02  2,154e-02  1,846e-02  1,539e-02  1,231e-02  9,231e-03  6,154e-03  3,077e-03  1,000e-30A prototype tank for the SWGO detector  Sofia Grusovin  Figure 10: PMT holder: holding test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  References  [1] INFN and SWGO collaboration, INFN SWGO letter of intent, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' Ward, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' Cortina and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' Guberman, Light-Trap: a SiPM upgrade for VHE astronomy and  beyond, Journal of Instrumentation 11 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' Mariotti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=', Optimized wavelength shifters light traps with SiPM photo sensors for  SWGO, Internal SWGO paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' Hinton, The Southern Wide-field Gamma-ray Observatory: status and srospects, Proceed-  ings of Science 395 ICRC2021 (2022) 023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  [5] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' Bisconti and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' Chiavassa, Study of water Cherenkov detector designs for the SWGO  experiment, Proceedings of Science 395 ICRC2021 (2022) 895.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=', GEANT4 — a simulation toolkit, Nuclear Instruments and Methods in  Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equip-  ment 506 (2003) 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  [7] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' Bisconti and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' Chiavassa, Study of water Cherenkov detector design for ground-based  gamma-ray experiments, arXiv e-prints (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' Albert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=', Science case for a wide field-of-view very-high-energy gamma-ray observatory  in the Southern hemisphere, arXiv e-prints (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  [9] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' Aharonian, LHAASO: A PeVatrons explorer, Science China Physics, Mechanics &  Astronomy volume 64 (2021) 109531.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  [10] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content=' Sinnis, The HAWC TeV gamma-ray observatory, Il nuovo cimento C 34 (2021) 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}
+page_content='  8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNE0T4oBgHgl3EQfiwE-/content/2301.02449v1.pdf'}

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+MedKLIP: Medical Knowledge Enhanced Language-Image Pre-Training
+Chaoyi Wu1,2,
+Xiaoman Zhang1,2,
+Ya Zhang1,2,
+Yanfeng Wang1,2,
+Weidi Xie1,2
+1Cooperative Medianet Innovation Center, Shanghai Jiao Tong University
+2Shanghai AI Laboratory
+{wtzxxxwcy02, xm99sjtu, ya zhang, wangyanfeng, weidi}@sjtu.edu.cn
+https://chaoyi-wu.github.io/MedKLIP/
+Abstract
+In this paper, we consider the problem of enhancing
+self-supervised visual-language pre-training (VLP) with
+medical-specific knowledge, by exploiting the paired image-
+text reports from the radiological daily practice. In par-
+ticular, we make the following contributions: First, un-
+like existing works that directly process the raw reports, we
+adopt a novel report filter to extract the medical entities,
+avoiding unnecessary complexity from language grammar
+and enhancing the supervision signals; Second, we pro-
+pose a novel entity embedding module by querying an ex-
+ternal knowledge description base, to exploit the rich con-
+text of additional information that the medical domain af-
+fords, and implicitly build relationships between entities in
+the language embedding space; Third, we propose a novel
+Transformer-based fusion model for spatially aligning the
+entity description with visual signals at the image patch
+level only with self-supervised learning, thus enabling the
+ability for spatial grounding; Fourth, we conduct thorough
+experiments to validate the effectiveness of our proposed
+architecture, and benchmark on numerous public bench-
+marks e.g., ChestX-ray14, RSNA Pneumonia, SIIM-ACR
+Pneumothorax, COVIDx CXR-2, COVID Rural, and Ede-
+maSeverity. In both zero-shot and fine-tuning settings, our
+model has demonstrated strong performance compared with
+the former methods on disease classification and grounding.
+1. Introduction
+With the rapid development of deep learning, numerous
+works have been proposed to facilitate computer-aided di-
+agnosis in the medical field [16, 17, 39, 48]. Despite the
+tremendous progress, these models are normally trained to
+recognize or segment the structures that fall into a certain
+closed set of anatomical or disease categories, whenever a
+new disease comes to be of interest, a costly procedure for
+data annotation, model re-training, and ethics proof will be
+required, fundamentally limiting its practical values. As an
+a.
+b.
+Pre-train Model
+Which one?
+ 
+ 
+R_2
+R_2
+R_1
+--
+R_1
+--
+…
+--
+R_N
+--
+R_N
+--
+Report Base
+R_2
+R_1
+--
+…
+--
+R_N
+--
+Report Base
+Increased right lower lobe opacity, 
+concerning for infection. No evidence of 
+pneumothorax.
+R_2: Final Report 
+Increased right lower lobe opacity, 
+concerning for infection. No evidence of 
+pneumothorax.
+R_2: Final Report 
+Supervision
+Pre-train Model
+Which one?
+ 
+ 
+R_2
+R_1
+--
+…
+--
+R_N
+--
+Report Base
+Increased right lower lobe opacity, 
+concerning for infection. No evidence of 
+pneumothorax.
+R_2: Final Report 
+Supervision
+Pre-train Model
+Knowledge 
+Description Base
+Entity Descriptions
+Entities
+ Position? Exist?  
+Entity
+Position
+Exist
+Opacity
+Right lower lobe
+TRUE
+Pneumothorax
+Unspecified
+FALSE
+ 
+ 
+Report Filter
+Final Report
+Increased right lower lobe opacity, 
+concerning for infection. No evidence of 
+pneumothorax.
+Final Report
+Increased right lower lobe opacity, 
+concerning for infection. No evidence of 
+pneumothorax.
+Supervision
+Report Base
+Pre-train Model
+Knowledge 
+Description Base
+Entity Descriptions
+Entities
+ Position? Exist?  
+Entity
+Position
+Exist
+Opacity
+Right lower lobe
+TRUE
+Pneumothorax
+Unspecified
+FALSE
+ 
+ 
+Report Filter
+Final Report
+Increased right lower lobe opacity, 
+concerning for infection. No evidence of 
+pneumothorax.
+Supervision
+Report Base
+Figure 1. Our method mainly considers combining medical knowl-
+edge with VLP. a. shows the standard VLP flowchart which uses
+text-image retrieval as a proxy task. b. is our MedKLIP flowchart.
+We adopt a report filter module to decompose raw reports at entity
+level and further use knowledge descriptions to explain entities.
+Our model can realize zero-shot classification and grounding.
+alternative, recent research considers to train the model by
+exploiting a large number of multi-modal medical data, that
+is generated from daily clinical routine, for example, the
+most common example is the dataset of X-ray images with
+paired radiological reports [15,25,28].
+This paper focuses on self-supervised vision-language
+representation learning in the medical domain, with the goal
+of zero-shot disease diagnosis (classification) and ground-
+ing. Undoubtedly, such tasks have also been widely inves-
+tigated in the computer vision community, with significant
+progress made in the past years, for example, CLIP [43],
+ALBEF [30], BLIP [29], etc. However, to achieve such a
+goal in the medical domain, different challenges must be
+resolved, which requires research efforts from the commu-
+nity: First, data availability, training Foundation Models in
+arXiv:2301.02228v1  [eess.IV]  5 Jan 2023
+
+computer vision normally require over millions of image-
+text pairs at ease, while in the medical domain, only a few
+hundred thousand pairs are available [28]. The limited data
+amount challenges language models to understand the re-
+ports in free form [7]. Second, the problem considered in
+medical diagnosis is naturally fine-grained, that requires
+distinguishing between fine appearance details to under-
+stand the disease, as a consequence, domain knowledge is
+essential; Third, robustness is crucial, it is, therefore, prefer-
+able to have explainability, where diagnosis results come
+along with the visual grounding, to help radiologists to un-
+derstand the system, and build trust between humans and
+machines.
+Unlike existing work in medical VLP (Vision-Language
+Pre-training) [7, 22, 40, 56] that na¨ıvely matches raw re-
+ports with image scans, we propose a novel knowledge-
+enhanced visual-language model that takes medical prior
+into consideration and enables us to address the aforemen-
+tioned challenges explicitly, as shown in Fig. 1: First, we
+propose a report filter to extract useful medical entities,
+and simplify each report into sets of triplets, denoted as
+{entity, position, exist}. Consequently, decomposing re-
+ports into entities leads to an effective representation of the
+reports with minimal information loss, enriching supervi-
+sion signals at the detailed entity level; Second, we map
+these entities into fine-grained descriptions by querying a
+well-defined medical knowledge base, and compute the text
+embedding for these descriptions, to implicitly build rela-
+tionships between entities; Third, we adopt a transformer-
+based architecture for aligning the image patches with en-
+tity descriptions, that simultaneously infer the likelihood of
+certain diseases and the visual evidence in the form of a
+spatial heatmap, i.e., providing grounding for explainability
+purpose.
+We train the model on the most widely-used medi-
+cal image-report dataset MIMIC-CXR [28] and rigorously
+evaluate on numerous public benchmarks, e.g., ChestX-
+ray14 [50], RSNA Pneumonia [44], SIIM-ACR Pneumoth-
+orax [1], COVIDx CXR-2 [41], COVID Rural [12, 47],
+and EdemaSeverity [8]. We get a state-of-the-art zero-shot
+classification and grounding performance on different dis-
+eases, spanning different image distributions, with further
+fine-tuning, our model still exceeds previous models signif-
+icantly.
+2. Related Work
+Vision-Language Pre-training Models. Vision-Language
+Pre-training (VLP) models have achieved great success in
+natural scenarios. Generally, there are two typical structures
+for VLP models. One is two-stream methods [5,27,30]. The
+other is single-stream methods [10, 32]. These impressive
+results promote VLP methods in medical. Different from
+natural data, medical VLP suffers from a serious Lack of
+Data (224k [28] vs 400M [5]). Most medical VLP meth-
+ods follow the two-stream methods [8, 22, 40, 56]. Con-
+VIRT [56] first proposed to use contrastive learning as a
+proxy task in medical. LoVT and GLoRIA then focus on
+improving the local alignment performance [8,22]. BioViL
+notices the language pattern in reports is different from
+other natural texts and re-designs the language model used
+for medical VLP [7]. These works have greatly contributed
+to the development of this aspect. However, they still view
+medical texts and images as common natural data and use a
+classical pipeline to handle them, instead of leveraging the
+rich prior knowledge in medical.
+Medical Named-Entity-Recognition Models.
+Various
+natural language processing (NLP) approaches have been
+proposed to extract useful information from radiology re-
+ports [25,37,42,45]. These early methods considered only
+the disease, causing they lose a lot of information. Some
+Analysis tools [4,6] have also been developed to extract key
+clinical concepts and their attributes from biomedical text.
+Further state-of-the-art works [26, 51] are proposed to ex-
+tract relationship between different entities without demand
+of pre-defined close disease set, retaining most of useful in-
+formation with high accuracy. This progress inspires us a
+lot and provide a new perspective for VLP. However, how
+to take advantage of this Named-Entity-Recognition (NER)
+models have not been discussed sufficiently in VLP field.
+Medical Knowledge Enhanced Models. Leveraging ex-
+ternal medical knowledge to enhance deep learning models
+is not a new topic [52]. Depending the approaches of us-
+ing medical knowledge, They can be classified into model-
+based and input-based two types. In model-based types, the
+authors may imitate the radiological practice to design the
+model [20, 23, 31, 49] or change the model structure based
+on diagnostic patterns [11, 18, 38]. In input-based types,
+the knowledge is viewed as an extra input to calculate fea-
+tures [46, 53, 54] or to guide the final loss [9, 24]. Multi-
+task learning and attention mechanism [33,34,36] are often
+adopted to realize medical knowledge combination. How-
+ever, most of these works are task-specific [52] because
+medical knowledge lacks an effective shared representation
+space across various diseases while we demonstrate that
+leveraging medical entity descriptions with text encoding
+has the potential to provide such a space.
+3. Method
+In this section, we start by describing our considered
+problem scenario in Sec. 3.1, followed by the details of
+our proposed Transformer-based architecture in Sec. 3.2,
+including, the visual encoder, knowledge-enhanced text en-
+coder, and the fusion module for aligning the visual and lan-
+guage signals. In Sec. 3.3, we describe the training proce-
+dure for our proposed model with the paired image-reports
+
+Text Encoder
+Text Encoder
+Visual 
+Encoder
+Visual 
+Encoder
+Entity Descriptions：
+Pneumonia is an  condition of the lung …
+Pneumothorax is an abnormal collection of air …
+Opacity is defined as an area of hazy opacification …
+Fracture is a break of bone especially in rib bone…
+… 
+Fusion Module
+Fusion Module
+Contrastive Loss
+CE Loss
+…
+=
++
+Chest 1 view, 8/21/2011
+History: 50 years male   Comparison: None
+Impression: Increased right lower lobe opacity, concerning 
+for infection. No evidence of pneumothorax.
+Final Report
+Chest 1 view, 8/21/2011
+History: 50 years male   Comparison: None
+Impression: Increased right lower lobe opacity, concerning 
+for infection. No evidence of pneumothorax.
+Final Report
+CE Head
+CE Head
+Contrastive Head
+Contrastive Head
+Report Filter
+…
+…
+Entity
+Position
+Exist
+Opacity
+Right lower lobe
+TRUE
+Pneumothorax
+Unspecified
+FALSE
++ Negative Locations
+Report Filter
+Report
+Location Embeddings 
+for Contrastive
+0/1 Labels 
+for CE
+Text Filter
+Text Filter
+It is located at [location]
+Text Encoder
+Text Encoder
+It is located at [location]
+Text Encoder
+Main Framework
+Figure 2. The whole framework of our method. The figure mainly contains the fourth module: Visual Encoder, Knowledge-enhanced
+Language Encoding, Fusion Module. Knowledge-enhanced Language Encoding contains Text Encoder and Report Filter. Report Filter
+extracts entities from the raw reports and Text Encoder further embeds them. Visual Encoder is used to encoder the input of visual
+modalities and Fusion Module is used for cross-modality interaction. The details of Report Filter can be found in the right sub-figure. A
+report is first filtered by a pre-trained filter and viewed as a set of triplets. The “Position” part is mixed with some negative positions for
+contrastive loss and the “Exist” part is used for CE loss.
+sourced from the daily routine X-ray scans.
+3.1. Problem Scenario
+Assuming we are given a training set with N samples,
+i.e., Dtrain = {(X1, T1), . . . , (XN, TN)}, where Xi, Ti re-
+fer to the X-ray image and its corresponding medical report
+generated in the daily routine scans, respectively, our goal
+is to train a visual-language model that enables us to diag-
+nose the existence of certain diseases and localize the vi-
+sual evidence spatially. Specifically, at inference time, we
+can freely ask the system to identify the likelihood of the
+patient getting a certain disease (may or may not be seen
+during training), with its visual description for the disease
+of interest:
+ˆsi, ˆmi = Φfusion(Φvisual(Xi), Φtextual([description])),
+(1)
+where Xi ∈ RH×W ×3 refers to an image sample from the
+test set, with H, W denoting height and width respectively.
+ˆsi ∈ [0, 1] refers to the inferred likelihood of the patient
+having a certain disease indicated by the input description,
+and ˆmi ∈ RH×W ×1 denotes a predicted spatial heatmap,
+with high activation on pixels that potentially provide the
+visual indication for such disease. In the following section,
+we will detail the individual components of our architecture,
+namely, the visual encoder, text encoder, and fusion mod-
+ule, and training them with the available training set (Dtrain).
+3.2. Architecture
+In this section, we detail our proposed framework, con-
+sisting of three main components, namely, visual encoding,
+knowledge-enhanced language encoding, and fusion mod-
+ule, as shown in Fig. 2. Note that, we hereon only consider
+single sampled image-reports pair (Xi, Ti), and ignore the
+subscript in notations for simplicity.
+3.2.1. Visual Encoding
+Given an X-ray image scan X ∈ RH×W ×3, we can com-
+pute the features with a visual backbone:
+V = Φvisual(X) ∈ Rh×w×d,
+(2)
+h, w, d refer to the height, width, and feature dimension
+of the output feature map, in our case, we adopt a stan-
+dard ResNet-50 as the visual backbone, and take the out-
+put from the 4th residual block. Note that, we make the
+such an architectural choice for a fair comparison with ex-
+isting work [7,22,40,56], while other visual backbones, e.g.,
+ViT [14], can equally be applied.
+3.2.2. Knowledge-enhanced Language Encoding
+The goal of this module is to extract useful information
+from the text report, by incorporating medical domain
+knowledge. In particular, we design two stages, namely re-
+port filtering, and entity encoding.
+Report Filtering. To start with, we propose to condense
+the report and transform it into a set of entity triplets, i.e.,
+removing the unnecessary complexity from language gram-
+mar, as shown in Figure 2 (right). In detail, we use a pre-
+trained text filter [26, 55] to extract valuable information
+from the report, for example, the medical entities and their
+corresponding positions on the image.
+Specifically, given a report T with a set of sentences,
+T = {s1, s2, ..., sM}, the filter independently operates on
+each of the sentences, and construct a number of triplets
+from the report, with the extracted entity (most are dis-
+eases), spatial position, and a label indicating the existence
+of the disease:
+Φfilter(sj) = {entityn, positionn, existn}, n ∈ [0, tj],
+(3)
+
+where tj represents the total number of entities contained
+in one sentence, with n = 0 indicating the special case that
+there is no valid entity. Note that, the position refers to the
+spatial position of the entity lying in an image, it is not to
+be confused with the positional embedding in Transformer.
+Entity Encoding. Here, we replace the entities by querying
+detailed visual descriptions from a medical-purpose knowl-
+edge base, for example, “Pneumonia” → “It is a condition
+of the lung primarily affecting the small air sacs known as
+alveolar. It may present with opacities and pleural effusion
+and it can increase the diagnostic accuracy of lung consoli-
+dation”. Note that, such descriptions for the medical termi-
+nologies can easily be sourced from either existing educa-
+tional textbooks or online resources12. Despite its simplic-
+ity, converting the entities into descriptions is crucial for
+more reliable and open-vocabulary diagnosis, as it further
+decomposes the medical entities into visual attributes that
+are shared by different diseases, encouraging the model to
+capture a deep understanding of the visual evidence.
+To encode the entity, we use the ClinicalBERT [3] as a
+pre-trained text encoder, to first compute the embedding for
+the entity description and position, and then adopt a linear
+MLP to flexibly project the embedding to desired dims:
+e = Φtextual([description]) ∈ Rd,
+(4)
+p = Φtextual(“it is located at [position]”) ∈ Rd′.
+(5)
+Each triplet has now been converted into {e, p, l}, l ∈ {0, 1}
+denotes the existence of the entity.
+Discussion We have made two major differences compared
+to the existing visual-language models in computer vision,
+First, the information in medical reports is often more con-
+densed, normally describing the existence of abnormality
+and their positions in the image, thus, adopting the filter
+operation can avoid unnecessary complexity from grammar,
+while still retaining most of the useful information in re-
+ports.
+Second, entities tend to be medical terminologies
+that are only understandable to audiences with a medical
+background, enrich the encoding by visual descriptions can
+significantly help the model to capture a deep understand-
+ing of the visual evidence for diseases, specifically, for seen
+diseases, such shared visual attributes enable to build the
+implicit relationship, while for unseen diseases, their visual
+evidence may have already been well understood by pro-
+cessing the descriptions of the seen ones, as they tend to be
+shared among diseases.
+1Wikipedia https://en.wikipedia.org/wiki/Wiki
+2UMLS [6] https://www.nlm.nih.gov/research/umls/
+index.html
+3.2.3. Fusion Module
+After extracting all the entities and their corresponding po-
+sitions from the reports, we select the top |Q| most com-
+monly appearing entities in reports, and compute the textual
+embeddings for their corresponding descriptions, denoted
+as an entity set Q = {e1, e2, ..., e|Q|}, and top |P| posi-
+tion embeddings as a position set P = {p1, p2, ..., p|P |}.
+For a certain image, its computed visual representation and
+the entity set will be passed into a fusion module for align-
+ment, consisting of multiple Transformer Decoder layers.
+We treat the entity set Q as Query, and the image features
+V as Key and Value into the Transformer decoders, the out-
+puts from the fusion module are further fed into two linear
+MLP layers, one is used for inferring the existence of the
+entity, and the other generates an embedding to indicate the
+entity’s spatial position:
+{ˆs, ˆp, ˆm} = Φfusion(V, Q),
+(6)
+where ˆs ∈ R|Q| represents the existence prediction for each
+entity query, and ˆp ∈ R|Q|×d′ represents the prediction em-
+bedding of spatial position for the entities. ˆm denotes the
+average of the cross-attention maps sourced from Trans-
+former Decoder layers. The training procedure will be de-
+tailed in Sec. 3.3.
+Discussion.
+In contrast to the existing approaches [56]
+that aligns the reports with the entire image, our adopted
+Transformer decoder enables to compute correspondences
+between text and image at the patch level. Consequently,
+the image features V are more suitable for downstream seg-
+mentation tasks and the average of the cross-attention map
+in each layers can be used directly for zero-shot grounding.
+3.3. Training
+To train the proposed model, we leverage the corre-
+sponding triplets, for entities that are not mentioned in the
+considered report, we simply ignore them while computing
+loss. For simplicity, the following formulations are based
+on the assumption that the considered query do have corre-
+sponding triplets. In specific, for the existence prediction
+ˆs, we use binary cross-entropy with the existence label, de-
+noted as Lcls; to supervise the position prediction for each
+entity query, we adopt contrastive learning, randomly sam-
+ple M position embeddings from the position set:
+Lloc = − 1
+|Q|
+|Q|
+�
+k=1
+e⟨ˆpk,pk⟩
+e⟨ˆpk,pk⟩ + �M
+u=1 e⟨ˆpk,PI(k,u)⟩ ,
+(7)
+where ⟨·, ·⟩ represents the inner product of two vectors
+and I(·, ·) is a random index sampling function. P is un-
+normalized in calculation.
+The final loss is the sum of the two:
+Ltotal = α1Lloc + α2Lcls,
+(8)
+
+where α1, α2 refer to two hyper-parameters controlling the
+ratio of the two losses, and we set them to be 1.0 by default.
+4. Experiment
+In this section, we start by introducing the dataset used
+for experiments, e.g., pre-training, and various downstream
+datasets. Then we describe the implementation details and
+the considered baselines.
+4.1. Pre-training Dataset
+MIMIC-CXR v2 [19, 28] consists of over 227k studies of
+paired image-report data, they are sourced from 65,379 pa-
+tients at different scanning. Each study can have one or two
+images (different scan views), totaling 377,110 images.
+4.2. Datasets for Downstream Tasks
+ChestX-ray14 [50] contains 112,120 frontal-view X-ray
+images of 30,805 unique patients, collected from the year
+of 1992 to 2015 by NIH(National Institutes of Health), with
+labels of 14 common diseases provided. We split the dataset
+into 0.8/0.1/0.1 for train/valid/test.
+RSNA Pneumonia [44] contains more than 260k frontal-
+view chest X-rays with corresponding pneumonia opac-
+ity masks collected by RSNA (Radiological Society of
+North America). Commonly, it is treated as a classifica-
+tion tasks [7, 22]. We split the dataset into 0.6/0.2/0.2 for
+train/valid/test.
+SIIM-ACR Pneumothorax [1] contains more than 12k
+frontal-view chest X-rays with pneumothorax masks col-
+lected by SIIM-ACR (Society for Imaging Informatics in
+Medicine and American College of Radiology). Similarly
+to RSNA Pneumonia dataset, it can be both used as clas-
+sification and segmentation tasks. We split the dataset into
+0.6/0.2/0.2 for train/valid/test.
+COVIDx CXR-2 [41] and COVID Rural [12, 47] aim to
+evaluate on diagnosing COVID-19. COVIDx CXR-3 con-
+tains 29,986 images from 16,648 patients with COVID-19
+classification labels. We use it as a classification dataset and
+split it into 0.7/0.2/0.1 for train/valid/test. Additionally,
+we use COVID Rural dataset for COVID-19 segmentation.
+It contains more than 200 chest X-rays with segmentation
+masks, and we split it into 0.6/0.2/0.2 for train/valid/test.
+Edema Severity [8] contains 6,524 examples from MIMIC-
+CXR with pulmonary edema severity labels (0 to 3, increas-
+ing severity) extracted from the radiology reports. Of these,
+141 radiologists were examined by radiologists, and con-
+sensus was reached on severity level. It can be seen as a
+typical fine-grained classification task. We split the dataset
+into 0.6/0.2/0.2 for train/valid/test.
+4.3. Implementation Details
+This section details the implementation for architecture,
+pre-training, zero-shot inference and fine-tuning procedure.
+Model architecture. As input to the model, images are re-
+sized into 224 × 224 × 3. We use the first four layers of
+ResNet50 [21] as our visual backbone (Φvisual), and adopt
+a MLP layer to transform the output feature dimension into
+d = 256. As a result, the output feature maps from vi-
+sual encoder is V ∈ R14×14×256. On the report side, we
+extract the entities with a pre-trained text filter, as described
+in [26], and compute the entity and position embedding with
+a pre-trained ClinicalBERT [2], its default embedding dim
+is d′ = 768. We obtain |Q| = 75 entities and |P| = 51 po-
+sitions that most frequently appear in the reports, following
+[55]. We sample M = 7 negative positions for each en-
+tity to calculate contrastive loss for training entity position
+training. In the fusion module, We adopt 4 Transformer De-
+coder layers with 4 heads in each layer.
+Pre-training. At this stage, both the filtering operation and
+language encoding use pre-trained networks, while the vi-
+sual encoder and fusion module are trained end-to-end on
+the image-text pairs. We use AdamW [35] optimizer with
+lr = 1 × 10−4 and lrwarm up = 1 × 10−5. We train on a
+GeForce RTX 3090 GPU with batch size 32 for 60 epochs.
+The first 5 epochs are set for warming up.
+Inference. At inference time, given a test image, we aim
+to infer the existence of certain entity / disease, and ground
+their visual evidence. For those entities that have appeared
+at training time, we simply adopt the corresponding ele-
+ments from the entity query set, while for those unseen
+ones, we replace the entity with a brief description, and
+treat that as an added query to the model. The existence
+output can be directly applied for classification and the av-
+erage cross-attention of different layers in the transformer-
+based fusion module between specific query and the visual
+features are used for grounding.
+Fine-tuning. For the downstream tasks, with large amount
+of training data, we can fine-tune the model end-to-
+end, with our pre-trained visual backbone as initializa-
+tion. Specifically, for image classification task, we adopt
+ResNet50 [21] and initialize its first four layers with our
+pre-trained visual encoder. For image segmentation task,
+we use ResUNet [13] as backbone and initialize its encoder
+with our pre-trained image encoder.
+4.4. Baselines
+We compare with various existing state-of-the-art med-
+ical image-text pre-train methods, namely, ConVIRT [56],
+GLoRIA [22] and BioViL [7]. Since ConVIRT and GLo-
+RIA are pre-trained on an in-house dataset, we re-train their
+models on MIMIC-CXR dataset for fair comparison. For
+BioViL, we use the officially released models by the au-
+
+Dataset
+RSNA Pneumonia
+SIIM-ACR Pneumothorax
+ChestX-ray14
+Methods
+AUC↑
+F1↑
+ACC↑
+AUC↑
+F1↑
+ACC↑
+AUC↑
+F1↑
+ACC↑
+ConVIRT [56]
+0.8042
+0.5842
+0.7611
+0.6431
+0.4329
+0.5700
+0.6101
+0.1628
+0.7102
+GLoRIA [22]
+0.7145
+0.4901
+0.7129
+0.5342
+0.3823
+0.4047
+0.6610
+0.1732
+0.7700
+BioViL [7]
+0.8280
+0.5833
+0.7669
+0.7079
+0.4855
+0.6909
+0.6912
+0.1931
+0.7916
+Ours
+0.8694
+0.6342
+0.8002
+0.8924
+0.6833
+0.8428
+0.7676
+0.2525
+0.8619
+Table 1. Comparison with other state-of-the-art methods on zero-shot classification task. AUC, F1 and ACC scores are reported. For
+ChestX-ray14, the metrics all refer to the macro average on the 14 diseases.
+Prompt Type
+Direct Covid-19
+Covid-19 Description
+Methods
+AUC↑
+F1↑
+ACC↑
+AUC↑
+F1↑
+ACC↑
+ConVIRT [56]
+0.6159
+0.7057
+0.6113
+0.5208
+0.6902
+0.5266
+GLoRIA [22]
+0.6319
+0.6938
+0.5710
+0.6659
+0.7007
+0.6083
+BioViL [7]
+0.6137
+0.6958
+0.5461
+0.5382
+0.6910
+0.5375
+Ours
+0.6561
+0.7066
+0.5917
+0.7396
+0.7670
+0.7006
+Table 2.
+Comparison with other state-of-the-art
+methods on zero-shot Covid-19 classification task.
+AUC, F1 and ACC scores are reported.
+“Direct
+covid-19” refers to directly use “Covid-19” to con-
+struct the prompt sentence while “Covid-19 Descrip-
+tion” refers to replace the name “Covid-19” with its
+medical description.
+thors. For zero-shot setting, we use the prompt as men-
+tioned by BioViL [7]. For fine-tuning, we all use ResNet50
+as the visual encoder as described in Sec. 4.3.
+4.5. Metrics
+AUC refers to the area under the receiver operating charac-
+teristic (ROC) curve, that is commonly used for detection
+and binary classification tasks.
+F1 and ACC are used as supplementary metrics for classi-
+fication tasks. Specifically, F1 comprehensively measures
+the recall and precision of the model, and ACC is the short
+of Accuracy. The final binary prediction threshold is chosen
+to maximise the F1 score. The ACC score is also calculated
+under this threshold.
+Pointing Game is used for evaluating the grounding perfor-
+mance. In specific, we extract the region with max response
+in the output heat-map, for one instance, if the region hit
+the ground-truth mask, it is considered a positive prediction,
+otherwise negative. Finally, accuracy can be calculated as
+the pointing game score.
+Dice and IOU are commonly used for segmentation tasks.
+For zero-shot segmentation, we search the segmentation
+threshold with 0.01 interval for all methods, and report the
+maximal Dice score for each model.
+Precision and Recall refer to the detection Precision and
+Recall.
+For medical, it is important that lesions are de-
+tected even without fine segmentation.
+Additionally, in
+some hard cases, especially for the zero-shot setting, Dice
+and IOU may be too strict to reflect the performance differ-
+ence. Precision and recall scores can compensate for these.
+We choose the IOU threshold as 0.1 to calculate the scores.
+5. Results
+In this section, we will report the experimental results. In
+general, we split the results into two parts: zero-shot setting
+and fine-tuning setting. In the zero-shot case (Sec. 5.1), we
+carry out the ablation study and compare it with the other
+SOTA image-text pre-train methods. We mainly consider
+classification and segmentation tasks; In the fine-tuning
+case (Sec. 5.2), we evaluate the model’s transferability by
+fine-tuning the model with 1%, 10%, and 100% data por-
+tion. Additionally, we also add a disease grading down-
+stream task, which can be seen as a fine-grade classification
+task, showing that our pre-trained model can be transferred
+to the downstream tasks at ease.
+5.1. Zero-shot
+In this section, we compare our method with the other
+state-of-the-art methods under zero-shot setting, classifica-
+tion, and grounding. Due to the space limitation, we include
+the entire ablation study in the supplementary material, re-
+ferring to it for more details and analysis, and all compar-
+isons here are made using our best model with position con-
+trastive loss and entity description encoder.
+5.1.1. Classification
+Seen Diseases. As shown in Tab. 1, we compare with ex-
+isting methods on three widely-used datasets, demonstrat-
+ing consistent performance improvement. Specifically, on
+pneumonia and pneumothorax datasets, despite the images
+being collected by different clinics with different diseases,
+our model improves the AUC score from 0.83 to 0.87 on
+RSNA pneumonia dataset and from 0.71 to 0.89 on SIIM-
+ACR pneumothorax dataset, as shown in Tab. 1.
+This
+shows that our method can better deal with the multi-center
+and multi-disease data distribution in medical. While on
+ChestX-ray14 dataset, we improve the average AUC scores
+from 0.69 to 0.77, we refer the reader to supplementary ma-
+terial for a more detailed comparison of 14 diseases.
+Unseen Diseases. In particular, we use covid-19 to evaluate
+the systems. Note that, our considered setting is different
+
+Methods
+Pointing Game↑
+Recall↑
+Precision↑
+IoU↑
+Dice↑
+GLoRIA [22]
+0.7607
+0.8330
+0.1621
+0.2182
+0.3468
+BioViL [7]
+0.8342
+0.8521
+0.5034
+0.3029
+0.4386
+Ours
+0.8721
+0.8661
+0.6420
+0.3172
+0.4649
+(a) Zero-shot grounding on Pneumonia
+Methods
+Pointing Game↑
+Recall↑
+Precision↑
+GLoRIA [22]
+0.0651
+0.2377
+0.0585
+BioViL [7]
+0.0252
+0.1963
+0.1429
+Ours
+0.1975
+0.3562
+0.1940
+(b) Zero-shot grounding on Pneumothorax
+Table 3. Comparison with other state-of-the-art methods on zero-shot region grounding tasks. (a) shows the results on RSNA Pneumonia
+dataset. (b) shows the results on SIIM-ACR Pneumothorax dataset. The pneumothorax region tends to be thin and narrow and much more
+challenging for grounding, we thus only consider pointing game, recall, and precision. Our method can achieve better performance on
+different metrics, especially on the pointing game score. ConVIRT as the basic method proposed earliest can not realize this function.
+Prompt Type
+Direct covid-19
+Covid-19 Description
+Methods
+Pointing Game↑
+Recall↑
+Precision↑
+IoU↑
+Dice↑
+Pointing Game↑
+AR↑
+AP↑
+IoU↑
+Dice↑
+GLoRIA [22]
+0.0364
+0.2906
+0.1073
+0.0645
+0.1141
+0.2727
+0.2821
+0.1336
+0.0596
+0.1075
+BioViL [7]
+0.4000
+0.2564
+0.2703
+0.1198
+0.1967
+0.1818
+0.2393
+0.1637
+0.0861
+0.1427
+Ours
+0.1818
+0.1880
+0.1497
+0.0747
+0.1289
+0.5818
+0.5214
+0.4959
+0.1373
+0.2278
+Table 4. Comparison with other state-of-the-art methods on zero-shot covid-19 opacity region grounding task. “Direct covid-19” refers
+to directly use “Covid-19” to construct the prompt sentence while “Covid-19 Description” refers to replace the name “Covid-19“ with its
+medical description. Our method can achieve better performance on different metrics.
+from existing approaches, where all entities have been ex-
+posed to the model at training time, and prediction can be
+made by a retrieval-type approach, i.e., compute the simi-
+larity between the image and the entity embedding by en-
+coding the disease name with a language encoder [7], while
+we are considering a stricter setting for openset classifica-
+tion. Covid-19 is a new disease that only appeared in 2019,
+MIMIC-CXR reports collected in the year 2015 do not in-
+clude any entity of covid-19, thus it requires the system to
+have the ability to diagnose truly unseen diseases.
+As shown in Tab. 2, existing approaches that only rely
+on disease name struggles to make the correct diagnosis.
+While with our proposed approach by introducing medical
+knowledge, i.e., using entity descriptions, our methods can
+understand the complex medical entity descriptions unseen
+in the training set, and significantly boost the performance
+0.66 to 0.74 on AUC and from 0.59 to 0.70 on ACC.
+5.1.2. Grounding
+In addition to the plain diagnosis, explainability can be
+equally critical in healthcare, improving the reliability and
+trustiness of the machine learning systems. Here, we con-
+sider providing explainability by grounding the abnormal-
+ity in the prediction and compare against the existing ap-
+proaches.
+Similarly, we split the diseases into seen and
+unseen ones, depending on whether their names have ap-
+peared in the medical reports. Specifically, “Pneumonia”
+and “Pneumothorax” are viewed as seen, and “Covid-19” is
+viewed as unseen. Due to the space limitation, we refer the
+reader to supplementary material for visualization results.
+Seen Diseases.
+We show the results for grounding on
+RSNA Pneumonia opacity and SIIM-ACR Pneumothorax
+collapse in Tab. 3. As shown in Tab. 3a, our proposed model
+surpasses existing approaches on all metrics, for example,
+we improve the pointing game score from 0.83 to 0.87, the
+detection Recall from 0.85 to 0.87, the detection precision
+from 0.50 to 0.64, the IOU from 0.30 to 0.32 and the Dice
+from 0.44 to 0.46. While on SIIM-ACR dataset (Tab. 3b),
+the pneumothorax region tends to be thin and narrow, local-
+izing it can often be more challenging than that of opacity
+grounding [7], we thus only consider pointing game, recall,
+and precision. Similarly, our method can achieve signifi-
+cantly better performance than prior approaches.
+Unseen Diseases. We also conduct the zero-shot ground-
+ing experiment on the unseen disease, namely, Covid-19, as
+shown in Tab. 4. Our model has shown consistent improve-
+ments in all metrics, e.g., boosting the pointing game score
+from 0.40 to 0.58. One observation to be noticed is that, re-
+sults in Tab. 4 are mostly consistent with those in Tab. 2, i.e.,
+better classification results tend to lead to better grounding.
+Overall, our model with knowledge-enhanced language en-
+coding has facilitated the visual encoder to learn underlying
+evidence relating to the diseases, therefore, leading to more
+interpretable representations than prior approaches.
+5.2. Fine-tuning
+In this section, we consider the fine-tuning scenario, with
+the pre-trained model as initialization, and trained end-to-
+end on the downstream tasks. We test on three different
+tasks, namely, classification, segmentation, and grading. In
+classification and segmentation, the test splits and metrics
+are the same as in the “zero-shot” section. Grading is a new
+task we introduce in fine-tuning setting, which can be seen
+as a fine-grained classification task.
+
+Dataset
+Pneumonia
+Pneumothorax
+Covid-19
+ChestX-ray14
+Data Portion
+1%
+10%
+100%
+1%
+10%
+100%
+1%
+10%
+100%
+1%
+10%
+100%
+Scratch
+0.7107
+0.8150
+0.8626
+0.4347
+0.6120
+0.6571
+0.7861
+0.9162
+0.9554
+0.6005
+0.7365
+0.7924
+ConVIRT [56]
+0.8398
+0.8562
+0.8761
+0.7134
+0.7826
+0.9004
+0.8675
+0.9541
+0.9726
+0.6615
+0.7658
+0.8128
+GLoRIA [22]
+0.8599
+0.8666
+0.8846
+0.7439
+0.8538
+0.9014
+0.9065
+0.9381
+0.9728
+0.6710
+0.7642
+0.8184
+BioViL [7]
+0.8233
+0.8538
+0.8836
+0.6948
+0.7775
+0.8689
+0.8989
+0.9529
+0.9729
+0.6952
+0.7527
+0.8245
+Ours
+0.8731
+0.8799
+0.8931
+0.8527
+0.9071
+0.9188
+0.9224
+0.9657
+0.9729
+0.7721
+0.7894
+0.8323
+Table 5. Comparison of AUC scores with other state-of-the-art methods on fine-tuning classification task. The macro average of AUC
+scores on 14 diseases are reported for ChestX-ray14 dataset.
+Diseases
+Pneumonia
+Pneumothorax
+Covid-19
+Data Portion
+1%
+10%
+100%
+1%
+10%
+100%
+1%
+10%
+100%
+Scratch
+0.4347
+0.6047
+0.7068
+0.2133
+0.3323
+0.7447
+0.1481
+0.2367
+0.3228
+ConVIRT [56]
+0.5706
+0.6491
+0.7201
+0.5406
+0.6121
+0.7352
+0.1995
+0.2724
+0.3737
+GLoRIA [22]
+0.6555
+0.6907
+0.7328
+0.5673
+0.5778
+0.7694
+0.1889
+0.2809
+0.3869
+BioViL [7]
+0.6824
+0.7038
+0.7249
+0.6267
+0.6998
+0.7849
+0.2113
+0.3239
+0.4162
+Ours
+0.7064
+0.7162
+0.7579
+0.6659
+0.7210
+0.7937
+0.2445
+0.3539
+0.4399
+Table 6. Comparison of Dice scores with other state-of-the-art methods on fine-tuning segmentation tasks. Three diseases are reported,
+and for each disease, three data portions, 1%, 10%, 100% are adopted to show the performance change under different data amounts.
+Methods
+0
+1
+2
+3
+AVG
+AUC↑
+F1↑
+ACC↑
+AUC↑
+F1↑
+ACC↑
+AUC↑
+F1↑
+ACC↑
+AUC↑
+F1↑
+ACC↑
+AUC↑
+F1↑
+ACC↑
+Scratch
+0.7631
+0.7036
+0.6738
+0.5383
+0.3593
+0.3223
+0.6692
+0.4328
+0.7012
+0.8420
+0.5694
+0.8770
+0.7031
+0.5163
+0.6436
+ConVIRT [56]
+0.8453
+0.7769
+0.7793
+0.6099
+0.3938
+0.4629
+0.7202
+0.4843
+0.6445
+0.9047
+0.6154
+0.8809
+0.7700
+0.5676
+0.6919
+GLoRIA [22]
+0.8304
+0.7577
+0.7520
+0.6208
+0.3991
+0.4922
+0.7339
+0.4958
+0.7037
+0.9246
+0.6667
+0.9102
+0.7774
+0.5798
+0.7145
+BioViL [7]
+0.8034
+0.7378
+0.7148
+0.6035
+0.3912
+0.4570
+0.6860
+0.4497
+0.6777
+0.9229
+0.6500
+0.9160
+0.7540
+0.5572
+0.6914
+Ours
+0.8502
+0.7646
+0.7539
+0.6641
+0.4140
+0.5392
+0.7605
+0.5266
+0.7031
+0.8845
+0.6250
+0.9160
+0.7898
+0.5826
+0.7280
+Table 7. Comparison with other state-of-the-art methods on fine-tuning edema severity grading multi-class classification task. AUC score
+is reported in the Table. “0,1,2,3” in the table represents the severity level and final macro average scores are reported.
+5.2.1. Classification
+We experiment on four different datasets, using 1%, 10%,
+100% of the data for fine-tuning, that is consistent with the
+existing work [7,22,56]. As shown in Tab. 5, our model has
+demonstrated substantial improvements in the AUC scores
+over the existing approaches across all datasets, reflecting
+that our pre-trained representation is of higher quality than
+existing models. We refer the readers to the supplementary
+material for more detailed comparison results.
+5.2.2. Segmentation
+In Tab. 6, we conduct fine-tuning experiments on three dif-
+ferent diseases for segmentation. We pick 1%, 10%, 100%
+of the data for fine-tuning. For all three different diseases
+with different image distributions, our methods surpass the
+existing state-of-the-art methods by a large margin, espe-
+cially under the low-data regime.
+5.2.3. Grading
+Besides diagnosis, grading the disease severity level also
+plays an important role. Here, we adopt our pre-trained fea-
+tures and train them for the multi-class classification task,
+with 0 to 3 representing different severity levels. As shown
+in Tab. 7, for each level, the AUC, F1, and ACC scores are
+calculated as one class against all other ones, for example,
+0 vs {1, 2, 3}. Final macro average scores of four levels are
+computed. On the majority of severity levels, our method
+can achieve the best results.
+6. Conclusion
+In this paper, we introduce a novel medical knowledge
+enhanced VLP model. First, we propose a report filter to
+extract useful medical entities with more useful supervi-
+sion signals, simplifying complex raw reports with mini-
+mal information loss. Second, we translate entities into de-
+tailed medical descriptions and embed them with a text en-
+coder enabling the network to understand complex medical
+expert-level knowledge. Finally, a transformer-based struc-
+ture is proposed to do local region alignment. In experi-
+ments, We evaluate our method on different datasets under
+various settings. Our method shows strong zero-shot clas-
+sification and grounding abilities, even facing unseen dis-
+eases. Besides, in fine-tuning setting, our method still out-
+performs state-of-the-art methods significantly, showing the
+superiority of our method.
+
+References
+[1] Society for imaging informatics in medicine: Siim-
+acr pneumothorax segmentation. https://www.
+kaggle.com/c/siim-acr-pneumothorax-
+segmentation. 2019. 2, 5
+[2] Emily Alsentzer, John Murphy, William Boag, Wei-
+Hung Weng, Di Jin, Tristan Naumann, and Matthew
+McDermott. Publicly available clinical BERT embed-
+dings. In Proceedings of the 2nd Clinical Natural Lan-
+guage Processing Workshop, pages 72–78, Minneapo-
+lis, Minnesota, USA, June 2019. Association for Com-
+putational Linguistics. 5
+[3] Emily Alsentzer, John R Murphy, Willie Boag, Wei-
+Hung Weng, Di Jin, Tristan Naumann, WA Redmond,
+and Matthew BA McDermott. Publicly available clin-
+ical bert embeddings.
+NAACL HLT 2019, page 72,
+2019. 4
+[4] Alan R Aronson and Franc¸ois-Michel Lang.
+An
+overview of metamap: historical perspective and re-
+cent advances. Journal of the American Medical In-
+formatics Association, 17(3):229–236, 2010. 2
+[5] Federico
+Bianchi,
+Giuseppe
+Attanasio,
+Raphael
+Pisoni, Silvia Terragni, Gabriele Sarti, and Sri Lak-
+shmi. Contrastive language-image pre-training for the
+italian language.
+arXiv preprint arXiv:2108.08688,
+2021. 2
+[6] Olivier Bodenreider.
+The unified medical lan-
+guage system (umls): integrating biomedical termi-
+nology.
+Nucleic acids research, 32(suppl 1):D267–
+D270, 2004. 2, 4
+[7] Benedikt Boecking, Naoto Usuyama, Shruthi Ban-
+nur, Daniel C Castro, Anton Schwaighofer, Stephanie
+Hyland, Maria Wetscherek, Tristan Naumann, Aditya
+Nori, Javier Alvarez-Valle, et al. Making the most of
+text semantics to improve biomedical vision–language
+processing.
+In European conference on computer
+vision, pages 1–21, 2022.
+Official Implementa-
+tion: https://github.com/microsoft/hi-
+ml/tree/main/hi-ml-multimodal. 2, 3, 5,
+6, 7, 8, 17
+[8] Geeticka Chauhan, Ruizhi Liao, William Wells, Jacob
+Andreas, Xin Wang, Seth Berkowitz, Steven Horng,
+Peter Szolovits, and Polina Golland. Joint modeling of
+chest radiographs and radiology reports for pulmonary
+edema assessment.
+In International Conference on
+Medical Image Computing and Computer-Assisted In-
+tervention, pages 529–539. Springer, 2020. 2, 5
+[9] Sihong Chen, Jing Qin, Xing Ji, Baiying Lei, Tianfu
+Wang, Dong Ni, and Jie-Zhi Cheng. Automatic scor-
+ing of multiple semantic attributes with multi-task
+feature leverage: a study on pulmonary nodules in
+ct images.
+IEEE transactions on medical imaging,
+36(3):802–814, 2016. 2
+[10] Yen-Chun Chen, Linjie Li, Licheng Yu, Ahmed
+El Kholy, Faisal Ahmed, Zhe Gan, Yu Cheng, and
+Jingjing Liu. Uniter: Universal image-text represen-
+tation learning. In European conference on computer
+vision, pages 104–120. Springer, 2020. 2
+[11] Hui Cui, Yiyue Xu, Wanlong Li, Linlin Wang, and
+Henry Duh. Collaborative learning of cross-channel
+clinical attention for radiotherapy-related esophageal
+fistula prediction from ct. In International Conference
+on Medical Image Computing and Computer-Assisted
+Intervention, pages 212–220. Springer, 2020. 2
+[12] Shivang Desai, Ahmad Baghal, Thidathip Wong-
+surawat, Piroon Jenjaroenpun, Thomas Powell, Shay-
+maa Al-Shukri, Kim Gates, Phillip Farmer, Michael
+Rutherford, Geri Blake, et al. Chest imaging repre-
+senting a covid-19 positive rural us population. Scien-
+tific data, 7(1):1–6, 2020. 2, 5
+[13] Foivos I Diakogiannis, Franc¸ois Waldner, Peter Cac-
+cetta, and Chen Wu.
+Resunet-a:
+A deep learn-
+ing framework for semantic segmentation of remotely
+sensed data. ISPRS Journal of Photogrammetry and
+Remote Sensing, 162:94–114, 2020. 5
+[14] Alexey
+Dosovitskiy,
+Lucas
+Beyer,
+Alexander
+Kolesnikov,
+Dirk
+Weissenborn,
+Xiaohua
+Zhai,
+Thomas Unterthiner, Mostafa Dehghani, Matthias
+Minderer, Georg Heigold, Sylvain Gelly, et al.
+An
+image is worth 16x16 words: Transformers for image
+recognition at scale. In International Conference on
+Learning Representations, 2020. 3
+[15] Jared A Dunnmon, Alexander J Ratner, Khaled
+Saab, Nishith Khandwala, Matthew Markert, Hersh
+Sagreiya, Roger Goldman, Christopher Lee-Messer,
+Matthew P Lungren, Daniel L Rubin, et al. Cross-
+modal data programming enables rapid medical ma-
+chine learning. Patterns, 1(2):100019, 2020. 1
+[16] Bradley J Erickson, Panagiotis Korfiatis, Zeynettin
+Akkus, and Timothy L Kline. Machine learning for
+medical imaging. Radiographics, 37(2):505, 2017. 1
+[17] Andre Esteva, Brett Kuprel, Roberto A Novoa, Justin
+Ko, Susan M Swetter, Helen M Blau, and Sebas-
+tian Thrun.
+Dermatologist-level classification of
+skin cancer with deep neural networks.
+nature,
+542(7639):115–118, 2017. 1
+[18] Leyuan Fang, Chong Wang, Shutao Li, Hossein Rab-
+bani, Xiangdong Chen, and Zhimin Liu.
+Attention
+to lesion: Lesion-aware convolutional neural network
+for retinal optical coherence tomography image clas-
+sification.
+IEEE transactions on medical imaging,
+38(8):1959–1970, 2019. 2
+
+[19] Ary L Goldberger, Luis AN Amaral, Leon Glass,
+Jeffrey M Hausdorff, Plamen Ch Ivanov, Roger G
+Mark, Joseph E Mietus, George B Moody, Chung-
+Kang Peng, and H Eugene Stanley. Physiobank, phys-
+iotoolkit, and physionet: components of a new re-
+search resource for complex physiologic signals. cir-
+culation, 101(23):e215–e220, 2000. 5
+[20] Ivan Gonzalez-Diaz.
+Dermaknet: Incorporating the
+knowledge of dermatologists to convolutional neural
+networks for skin lesion diagnosis.
+IEEE journal
+of biomedical and health informatics, 23(2):547–559,
+2018. 2
+[21] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian
+Sun.
+Deep residual learning for image recognition.
+In Proceedings of the IEEE conference on computer
+vision and pattern recognition, pages 770–778, 2016.
+5
+[22] Shih-Cheng Huang, Liyue Shen, Matthew P Lungren,
+and Serena Yeung. Gloria: A multimodal global-local
+representation learning framework for label-efficient
+medical image recognition.
+In Proceedings of the
+IEEE/CVF International Conference on Computer Vi-
+sion, pages 3942–3951, 2021. Official Implementa-
+tion: https://github.com/marshuang80/
+gloria. 2, 3, 5, 6, 7, 8, 17
+[23] Xin Huang, Yu Fang, Mingming Lu, Fengqi Yan, Jun
+Yang, and Yilu Xu. Dual-ray net: automatic diagno-
+sis of thoracic diseases using frontal and lateral chest
+x-rays. Journal of Medical Imaging and Health Infor-
+matics, 10(2):348–355, 2020. 2
+[24] Sarfaraz Hussein, Kunlin Cao, Qi Song, and Ulas
+Bagci.
+Risk stratification of lung nodules using 3d
+cnn-based multi-task learning. In International con-
+ference on information processing in medical imaging,
+pages 249–260. Springer, 2017. 2
+[25] Jeremy Irvin, Pranav Rajpurkar, Michael Ko, Yifan
+Yu, Silviana Ciurea-Ilcus, Chris Chute, Henrik Mark-
+lund, Behzad Haghgoo, Robyn Ball, Katie Shpan-
+skaya, et al.
+Chexpert: A large chest radiograph
+dataset with uncertainty labels and expert comparison.
+In Proceedings of the AAAI conference on artificial in-
+telligence, volume 33, pages 590–597, 2019. 1, 2
+[26] Saahil Jain, Ashwin Agrawal, Adriel Saporta, Steven
+Truong, Tan Bui, Pierre Chambon, Yuhao Zhang,
+Matthew P Lungren, Andrew Y Ng, Curtis Langlotz,
+et al. Radgraph: Extracting clinical entities and re-
+lations from radiology reports.
+In Thirty-fifth Con-
+ference on Neural Information Processing Systems
+Datasets and Benchmarks Track (Round 1), 2021. 2,
+3, 5
+[27] Chao Jia, Yinfei Yang, Ye Xia, Yi-Ting Chen, Zarana
+Parekh, Hieu Pham, Quoc Le, Yun-Hsuan Sung, Zhen
+Li, and Tom Duerig. Scaling up visual and vision-
+language representation learning with noisy text su-
+pervision.
+In International Conference on Machine
+Learning, pages 4904–4916. PMLR, 2021. 2
+[28] AEWP Johnson, Tom Pollard, Roger Mark, Seth
+Berkowitz, and Steven Horng. Mimic-cxr database.
+PhysioNet10, 13026:C2JT1Q, 2019. 1, 2, 5
+[29] Junnan Li, Dongxu Li, Caiming Xiong, and Steven
+Hoi. Blip: Bootstrapping language-image pre-training
+for unified vision-language understanding and gener-
+ation. arXiv preprint arXiv:2201.12086, 2022. 1
+[30] Junnan Li, Ramprasaath Selvaraju, Akhilesh Gotmare,
+Shafiq Joty, Caiming Xiong, and Steven Chu Hong
+Hoi. Align before fuse: Vision and language represen-
+tation learning with momentum distillation. Advances
+in neural information processing systems, 34:9694–
+9705, 2021. 1, 2
+[31] Liu Li, Mai Xu, Xiaofei Wang, Lai Jiang, and Han-
+ruo Liu. Attention based glaucoma detection: a large-
+scale database and cnn model. In Proceedings of the
+IEEE/CVF conference on computer vision and pattern
+recognition, pages 10571–10580, 2019. 2
+[32] Liunian Harold Li, Mark Yatskar, Da Yin, Cho-Jui
+Hsieh, and Kai-Wei Chang. Visualbert: A simple and
+performant baseline for vision and language. arXiv
+preprint arXiv:1908.03557, 2019. 2
+[33] Xiaomeng Li, Xiaowei Hu, Lequan Yu, Lei Zhu, Chi-
+Wing Fu, and Pheng-Ann Heng. Canet: cross-disease
+attention network for joint diabetic retinopathy and di-
+abetic macular edema grading. IEEE transactions on
+medical imaging, 39(5):1483–1493, 2019. 2
+[34] Qing Liao, Ye Ding, Zoe L Jiang, Xuan Wang,
+Chunkai Zhang, and Qian Zhang.
+Multi-task deep
+convolutional neural network for cancer diagnosis.
+Neurocomputing, 348:66–73, 2019. 2
+[35] Ilya Loshchilov and Frank Hutter. Decoupled weight
+decay regularization. In International Conference on
+Learning Representations, 2018. 5
+[36] Gabriel Maicas, Andrew P Bradley, Jacinto C Nasci-
+mento, Ian Reid, and Gustavo Carneiro. Training med-
+ical image analysis systems like radiologists. In In-
+ternational Conference on Medical Image Computing
+and Computer-Assisted Intervention, pages 546–554.
+Springer, 2018. 2
+[37] Matthew BA McDermott, Tzu Ming Harry Hsu, Wei-
+Hung Weng, Marzyeh Ghassemi, and Peter Szolovits.
+Chexpert++: Approximating the chexpert labeler for
+speed, differentiability, and probabilistic output.
+In
+Machine Learning for Healthcare Conference, pages
+913–927. PMLR, 2020. 2
+
+[38] Masahiro
+Mitsuhara,
+Hiroshi
+Fukui,
+Yusuke
+Sakashita,
+Takanori
+Ogata,
+Tsubasa
+Hirakawa,
+Takayoshi Yamashita, and Hironobu Fujiyoshi. Em-
+bedding human knowledge into deep neural network
+via attention map. arXiv preprint arXiv:1905.03540,
+2019. 2
+[39] Yasuhide Miura, Yuhao Zhang, Emily Tsai, Curtis
+Langlotz, and Dan Jurafsky. Improving factual com-
+pleteness and consistency of image-to-text radiology
+report generation. In Proceedings of the 2021 Confer-
+ence of the North American Chapter of the Associa-
+tion for Computational Linguistics: Human Language
+Technologies, pages 5288–5304, 2021. 1
+[40] Philip M¨uller, Georgios Kaissis, Congyu Zou, and
+Daniel R¨uckert. Joint learning of localized representa-
+tions from medical images and reports. arXiv preprint
+arXiv:2112.02889, 2021. 2, 3
+[41] Maya Pavlova, Naomi Terhljan, Audrey G Chung,
+Andy Zhao, Siddharth Surana, Hossein Aboutalebi,
+Hayden Gunraj, Ali Sabri, Amer Alaref, and Alexan-
+der Wong.
+Covid-net cxr-2:
+An enhanced deep
+convolutional neural network design for detection of
+covid-19 cases from chest x-ray images. Frontiers in
+Medicine, 9, 2022. 2, 5
+[42] Yifan Peng, Xiaosong Wang, Le Lu, Mohammadhadi
+Bagheri, Ronald Summers, and Zhiyong Lu.
+Neg-
+bio: a high-performance tool for negation and un-
+certainty detection in radiology reports. AMIA Sum-
+mits on Translational Science Proceedings, 2018:188,
+2018. 2
+[43] Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya
+Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sas-
+try, Amanda Askell, Pamela Mishkin, Jack Clark,
+et al. Learning transferable visual models from natural
+language supervision. In International Conference on
+Machine Learning, pages 8748–8763. PMLR, 2021. 1
+[44] George Shih, Carol C Wu, Safwan S Halabi, Marc D
+Kohli, Luciano M Prevedello, Tessa S Cook, Arjun
+Sharma, Judith K Amorosa, Veronica Arteaga, Maya
+Galperin-Aizenberg, et al. Augmenting the national
+institutes of health chest radiograph dataset with ex-
+pert annotations of possible pneumonia. Radiology.
+Artificial intelligence, 1(1), 2019. 2, 5
+[45] Akshay Smit, Saahil Jain, Pranav Rajpurkar, Anuj Pa-
+reek, Andrew Y Ng, and Matthew Lungren.
+Com-
+bining automatic labelers and expert annotations for
+accurate radiology report labeling using bert. In Pro-
+ceedings of the 2020 Conference on Empirical Meth-
+ods in Natural Language Processing (EMNLP), pages
+1500–1519, 2020. 2
+[46] Jiaxing Tan, Yumei Huo, Zhengrong Liang, and Li-
+hong Li. Expert knowledge-infused deep learning for
+automatic lung nodule detection. Journal of X-ray Sci-
+ence and Technology, 27(1):17–35, 2019. 2
+[47] Haiming Tang, Nanfei Sun, and Yi Li. Deep learn-
+ing segmentation model for automated detection of the
+opacity regions in the chest x-rays of the covid-19 pos-
+itive patients and the application for disease severity.
+medRxiv preprint, 2020. 2, 5
+[48] Joseph J Titano, Marcus Badgeley, Javin Schefflein,
+Margaret Pain, Andres Su, Michael Cai, Nathaniel
+Swinburne, John Zech, Jun Kim, Joshua Bederson,
+et al. Automated deep-neural-network surveillance of
+cranial images for acute neurologic events.
+Nature
+medicine, 24(9):1337–1341, 2018. 1
+[49] Kun Wang, Xiaohong Zhang, Sheng Huang, Feiyu
+Chen, Xiangbo Zhang, and Luwen Huangfu. Learn-
+ing to recognize thoracic disease in chest x-rays with
+knowledge-guided deep zoom neural networks. IEEE
+Access, 8:159790–159805, 2020. 2
+[50] Xiaosong Wang, Yifan Peng, Le Lu, Zhiyong Lu,
+Mohammadhadi Bagheri, and Ronald M Summers.
+Chestx-ray8: Hospital-scale chest x-ray database and
+benchmarks on weakly-supervised classification and
+localization of common thorax diseases. In Proceed-
+ings of the IEEE conference on computer vision and
+pattern recognition, pages 2097–2106, 2017. 2, 5
+[51] Joy T Wu, Nkechinyere Nneka Agu, Ismini Lourent-
+zou,
+Arjun
+Sharma,
+Joseph
+Alexander
+Paguio,
+Jasper Seth Yao, Edward Christopher Dee, William G
+Mitchell, Satyananda Kashyap, Andrea Giovannini,
+et al. Chest imagenome dataset for clinical reason-
+ing. In Thirty-fifth Conference on Neural Information
+Processing Systems Datasets and Benchmarks Track
+(Round 2), 2021. 2
+[52] Xiaozheng Xie, Jianwei Niu, Xuefeng Liu, Zhengsu
+Chen, Shaojie Tang, and Shui Yu. A survey on in-
+corporating domain knowledge into deep learning for
+medical image analysis.
+Medical Image Analysis,
+69:101985, 2021. 2
+[53] Yutong Xie, Yong Xia, Jianpeng Zhang, Yang Song,
+Dagan Feng, Michael Fulham, and Weidong Cai.
+Knowledge-based collaborative deep learning for
+benign-malignant lung nodule classification on chest
+ct. IEEE transactions on medical imaging, 38(4):991–
+1004, 2018. 2
+[54] Wenkai Yang, Juanjuan Zhao, Yan Qiang, Xiaotang
+Yang, Yunyun Dong, Qianqian Du, Guohua Shi, and
+Muhammad Bilal Zia.
+Dscgans: Integrate domain
+knowledge in training dual-path semi-supervised con-
+ditional generative adversarial networks and s3vm for
+ultrasonography thyroid nodules classification. In In-
+ternational conference on medical image computing
+
+and computer-assisted intervention, pages 558–566.
+Springer, 2019. 2
+[55] Ke Yu, Shantanu Ghosh, Zhexiong Liu, Christopher
+Deible, and Kayhan Batmanghelich. Anatomy-guided
+weakly-supervised abnormality localization in chest
+x-rays. In International Conference on Medical Im-
+age Computing and Computer-Assisted Intervention,
+pages 658–668. Springer, 2022. 3, 5, 14, 15
+[56] Yuhao Zhang, Hang Jiang, Yasuhide Miura, Christo-
+pher D Manning, and Curtis P Langlotz. Contrastive
+learning of medical visual representations from paired
+images and text. In Machine Learning for Healthcare,
+2022. Highest Starred Implementation: https://
+github.com/edreisMD/ConVIRT-pytorch.
+2, 3, 4, 5, 6, 8, 17
+
+Supplementary
+A. The Entity Description Base and Position Set
+14
+B. Details of Fusion Module
+16
+C. Ablation Study
+16
+D. Detailed results on ChestX-ray14
+17
+E. Visualization Results
+18
+
+A. The Entity Description Base and Position Set
+Tab. 8 shows the descriptions we used to translate different entities. We have kept 75 entities in query set Q, following [55].
+“Tail abnorm obs” entity represents some tailed entities and “exluded obs” represents some entities useless for diagnosis.
+The last “covid-19” description row is only referred to for inference since it does not appear in pre-train reports.
+Table 8. The Entity description used for translate single entity name. The description can be easily found from the open website.
+Entity
+Description
+normal
+It means the absence of diseases and infirmity, indicating the structure is normal.
+clear
+The lungs are clear and normal. No evidence for other diseases on lung.
+sharp
+This means that an anatomical structure s boundary or edge is clear and normal, meaning it is free of diseases.
+sharply
+“Sharply seen means that an anatomical structure is clearly visible.
+unremarkable
+This represents some anatomical structures are normal, usually modifying cardiac and mediastinal silhouettes.
+intact
+The bonny structure is complete and normal, meaning no fractures.
+stable
+The modified anatomical structures are normal and stable. No evidence for diseases.
+free
+It usually refers to free air and is associate with pneumothorax, atelectasis, pneumoperitoneum and emphysema.
+effusion
+A pleural effusion is accumulation of excessive fluid in the pleural space, the potential space that surrounds each lung. A pleural
+effusion infiltrates the space between the visceral pleura and the parietal pleura.
+opacity
+It is defined as an area of hazy opacification due to air displacement by fluid, airway collapse, fibrosis, or a neoplastic process. It is
+causes include infections, interstitial lung disease, and pulmonary edema.
+pneumothorax
+A pneumothorax is an abnormal collection of air in the pleural space between the lung and the chest wall. It may be caused by
+pneumonia or fibrosis and other diseases.
+edema
+Pulmonary edema, also known as pulmonary congestion, is excessive liquid accumulation in the tissue and air spaces of the lungs. It
+will show fluid in the alveolar walls.
+atelectasis
+It is the collapse or closure of a lung resulting in reduced or absent gas exchange. Findings can include lung opacification and loss of
+lung volume.
+tube
+It is a surgical drain that is inserted through the chest wall and into the pleural space or the mediastinum to remove undesired substances
+such as air.
+consolidation
+It is a region of normally compressible lung tissue that has filled with liquid instead of air. Consolidation must be present to diagnose
+pneumonia: the signs of lobar pneumonia are characteristic and clinically referred to as consolidation.
+process
+Acute process means there is abnormality in the anotomy structure.
+abnormality
+It means the exist of diseases and infirmity, indicating the structure is abnormal.
+enlarge
+It usually modifies cardiac silhouette and heart. Cardiomegaly is a medical condition in which the heart is enlarged.
+tip
+It refers to the top head of the tube.
+low
+The presence of low lung volumes may be a sign of a restrictive lung condition such as pulmonary fibrosis or sarcoidosis.
+pneumonia
+Pneumonia is an inflammatory condition of the lung primarily small air sacs known as alveoli. Pneumonia may present with opacities.
+Complications such as pleural effusion may also be found increasing the diagnostic accuracy of lung consolidation and pleural effusion
+line
+It refers to venous access line ot PICC lines.
+congestion
+Pulmonary congestion is defined as accumulation of fluid in the lungs, resulting in impaired gas exchange and arterial hypoxemia.
+catheter
+catheter is a tube placed in the body to drain and collect urine from the bladder
+cardiomegaly
+Cardiomegaly (sometimes megacardia or megalocardia) is a medical condition in which the heart is enlarged.
+fracture
+Fracture is a break in a rib bone.
+air
+It refers to the free air or gas in pleural space, indicating pneumothorax. Air displacement by fluid may lead to opacity.
+tortuous
+The Aorta is slightly tortuous. Sometimes it may refer to varicose veins
+lead
+It refers to the leading head of the tube.
+disease
+It means the exist of diseases and abnormalty, indicating the structure is abnormal.
+calcification
+Pulmonary calcification is a common asymptomatic finding. Pulmonary calcifications are caused mainly by two mechanisms: the
+dystrophic form and the metastatic form
+prominence
+It means the exist of some observation.
+device
+It refer to some equipments like picc tub, valve catheter, pacemaker hardware, arthroplastmarker icd defib, device support equipment
+and mediport
+engorgement
+Pulmonary vascular engorgement means obstruction of the normal flux of blood within the blood vessel network of the lung resulting
+in engorgement of pulmonary vessels
+picc
+A peripherally inserted central catheter (PICC), also called a PICC line, is a long, thin tube that s inserted through a vein in your arm
+and passed through to the larger veins near your heart.
+clip
+Surgical clips or vascular clips usually represent the one kind of medical equipments.
+elevation
+If tissues or anatomical structures are elevated, they are raised up higher than the normal location.
+expand
+It means the lungs are normally expanded and clear, indicating the absence of pneumothorax.
+nodule
+A lung nodule or pulmonary nodule is a relatively small focal density in the lung. it may be confused with the projection of a structure
+of the chest wall or skin, such as a nipple, a healing rib fracture or lung cancer.
+wire
+Sternotomis wires means the center line of the chest.
+fluid
+It refers to the water of liquid in the lung and it may indicate edema and other diseases.
+degenerative
+Degenerative disease is the result of a continuous process based on degenerative cell changes
+pacemaker
+pacemaker device usually represents the one kind of medical equipments.
+thicken
+Pleural thickening is an increase in the bulkiness of one or both of the pulmonary pleurae. It may cause by pulmonary Infection,
+empyema, tuberculosis or lung cancer.
+
+Entity
+Description
+marking
+It represents interstitial markings or bronchovascular markings
+scar
+A scar (or scar tissue) is an area of fibrous tissue that replaces normal tissues after an injury.
+hyperinflate
+Hyperinflated lungs are larger-than-normal lungs as a result of trapped air.
+blunt
+Blunting of the costophrenic angles is usually caused by a pleural effusion, as already discussed. Other causes of costophrenic angle
+blunting include lung disease in the region of the costophrenic angle, and lung hyperexpansion.
+loss
+The etiology of lung volume loss can be listed as follow: airway obstruction or compression, obesity, scoliosis, restrictive diseases
+such as pulmonary fibrosis and interstitial lung disease, tuberculosis.
+widen
+The mediastinum is not widened or enlarged.
+collapse
+Collapse lung refers to pneumothorax or atelectasis.
+density
+The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass per unit volume.
+emphysema
+Emphysema, or pulmonary emphysema, is a lower respiratory tract disease, characterized by air-filled spaces (pneumatosis) in the
+lungs, that can vary in size and may be very large.
+aerate
+Aeration (also called aerification or aeriation) is the process by which air is circulated through, mixed with or dissolved in a liquid or
+other substances that act as a fluid (such as soil).
+mass
+A lung mass is an abnormal growth or area in the lungs and it can also view as lung cancer.
+crowd
+Crowding of the bronchovascular structures is an important direct sign of volume loss. The atelectatic lung enhances densely after
+contrast administration because of closeness of the pulmonary arteries and arterioles within the collapsed lobe.
+infiltrate
+A pulmonary infiltrate is a substance denser than air, such as pus, blood, or protein, which lingers within the parenchyma of the lungs.
+Pulmonary infiltrates are associated with pneumonia, tuberculosis and sarcoidosis.
+obscure
+Some anatomy structures are not clear and is difficult to understand or see.
+deformity
+It means some body parts are abnormal or unjuried.
+hernia
+Lung hernia (Sibson hernia) is a protrusion of lung outside of thoracic wall. the hernia is noted after chest trauma, thoracic surgery or
+certain pulmonary diseases.
+drainage
+Tube drainage represents the one kind of medical equipment.
+distention
+Distension generally refers to an enlargement, dilation, or ballooning effect. It may refer to: Abdominal distension.
+shift
+The mediastinal shift is the deviation of the mediastinal structures towards one side of the chest cavity, usually seen on chest radiograph.
+It indicates a severe asymmetry of intrathoracic pressures.
+stent
+Tracheal stent represents the one kind of medical equipments
+pressure
+Pulmonary venous pressure is intermediate between mean PAP and LAP over all physiologic pressures
+lesion
+Lung nodules, pulmonary nodules, white spots, lesions—these terms all describe the same phenomenon: an abnormality in the lungs.
+finding
+Some observation on body parts, usually indicating abnormalty.
+borderline
+Borderline size of the cardiac silhouette means the cardiac silhouette is not enlarged and normal.
+hardware
+It represents the one kind of medical equipments.
+dilation
+The state of being larger or more open than normal
+chf
+Heart failure — sometimes known as congestive heart failure — occurs when the heart muscle doesn’t pump blood as well as it should.
+When this happens, blood often backs up and fluid can build up in the lungs, causing shortness of breath.
+redistribution
+If the pulmonary edema is due to heart failure or fluid overload, you may also see cardiomegaly and distension of the pulmonary veins,
+particularly in the upper lung fields.
+aspiration
+Aspiration pneumonia occurs when food or liquid is breathed into the airways or lungs, instead of being swallowed.
+tail abnorm obs
+Some very rare diseases.
+excluded obs
+Some meaningless observations.
+covid-19
+It is a contagious disease caused by a virus. Ground-glass opacities, consolidation, thickening, pleural effusions commonly appear in
+infection.
+Additionally, we keep 51 positive positions, following [55], to form the position set P, as {trachea, left hilar, right hilar,
+hilar unspec, left pleural, right pleural, pleural unspec, heart size, heart border, left diaphragm, right diaphragm, di-
+aphragm unspec, retrocardiac, lower left lobe, upper left lobe, lower right lobe middle right lobe, upper right lobe,
+left lower lung,
+left mid lung,
+left upper lung left apical lung,
+left lung unspec,
+right lower lung,
+right mid lung,
+right upper lung right apical lung, right lung unspec, lung apices, lung bases, left costophrenic right costophrenic,
+costophrenic unspec, cardiophrenic sulcus, mediastinal, spine clavicle, rib, stomach, right atrium, right ventricle, aorta,
+svc, interstitium, parenchymal, cavoatrial junction, cardiopulmonary, pulmonary, lung volumes, unspecified, other}.
+“Other” is used to reprepresnt some tailed positions.
+
+B. Details of Fusion Module
+In the transformer-based fusion module, the queries are first passed through a self-attention layer and then followed by a
+multi-head attention layer between the modified queries and image features. In each head, the image features are processed
+by a linear key head and a linear value head as key embeddings and value embeddings independently. The value is weighted-
+added based on the attention map, which is calculated by the soft-max dot product of the keys and queries. Finally, a feed-
+forward network gathers the vector of different heads resulting in the output of this layer. The output vector of the former
+layer is considered as the entity query vector of the next layer. In formulation, if denoting Φi
+fusion(·, ·) : RN×D2 ×RP 2×D2 �→
+RN×D2 as the i-th layer, the procedure is expressed as:
+Φi
+fusion(ri−1, V) = ri, r0 = Q.
+(9)
+C. Ablation Study
+Our final method mainly contains three key parts, transformer-based fusion module, position location contrastive
+loss (PosCL), and entity description encoder (DE). We gradually remove the modules to analyze their effectiveness.
+“w/o (DE)” refers to removing DE module and “w/o (PosCL+DE)” refers to only maintaining the fusion module with basic
+CE loss. Tab. 9 and Tab. 10 shows the quantitative results.
+Transformer Decoder. The lines about “w/o (PosCL + DE)” in tables demonstrate the performance of the basic model
+modified only by base CE loss. This model can exceed many former methods. This indicates the complex syntax will hurt
+the network to capture the useful entities significantly and our filtering operation combined with medical NER can greatly
+relieve the problem.
+Position Contrastive Loss. The PosCL can significantly help the network to ground the abnormalities. As shown in the
+results by adding PosCL the classification results can be further improved, e.g., from 0.75 to 0.76 on AUC in ChestX-ray14
+dataset. Besides classification, location contrastive loss can bring more gain in grounding. These results show position is
+another vital element in reports especially for grounding tasks. Our filtered triplets can conclude and clean the reports with
+little information loss and make the network learn the report information more straightforward.
+Entity Description Encoder. By adding entity descriptions, we want to realize two goals. First, in addition to just learning
+from the image-report data, the network can actively learn the relationship between different entities based on the entity
+descriptions. As shown in tables, adding descriptions in most scenarios can help the network better understand the entity and
+bring gain to the final metric scores. Second,the description encoder enables our model to handle openset new diseases.
+Since the entity list is a close set during pre-training, our method will be only able to handle the seen diseases without
+DE, while, with a description encoder, our method can handle unseen diseases and understand complex medical disease
+knowledge.
+Dataset
+RSNA Pneumonia
+SIIM-ACR Pneumothorax
+ChestX-ray14
+Methods
+AUC↑
+F1↑
+ACC↑
+AUC↑
+F1↑
+ACC↑
+AUC↑
+F1↑
+ACC↑
+w/o (PosCL + DE)
+0.8532
+0.6079
+0.7669
+0.8768
+0.6672
+0.8187
+0.7502
+0.2374
+0.8541
+w/o (DE)
+0.8537
+0.6241
+0.8146
+0.9017
+0.7008
+0.8584
+0.7621
+0.2452
+0.8606
+Ours
+0.8694
+0.6342
+0.8002
+0.8924
+0.6833
+0.8428
+0.7676
+0.2525
+0.8619
+Table 9. Ablation study on zero-shot classification task. AUC, F1 and ACC scores are reported. For ChestX-ray 14, the metrics all refer to
+the macro average on the 14 diseases.
+Methods
+Pointing Game↑
+Recall↑
+Precision↑
+IoU↑
+Dice↑
+w/o (PosCL + DE)
+0.7979
+0.8961
+0.4036
+0.2783
+0.4230
+w/o (DE)
+0.8424
+0.8226
+0.6520
+0.3118
+0.4610
+Ours
+0.8721
+0.8661
+0.6420
+0.3172
+0.4649
+(a) Zero-shot grounding on Pneumonia
+Methods
+Pointing Game↑
+Recall↑
+Precision↑
+w/o (PosCL + DE)
+0.1786
+0.3151
+0.1336
+w/o (DE)
+0.2080
+0.3178
+0.1711
+Ours
+0.1975
+0.3562
+0.1940
+(b) Zero-shot grounding on Pneumothorax
+Table 10. Ablation study on zero-shot grounding tasks. (a) shows the results on RSNA Pneumonia dataset. (b) shows the results on
+SIIM-ACR Pneumothorax dataset.
+
+D. Detailed results on ChestX-ray14
+We further show the detailed performance of 14 different diseases on ChestX-ray14 dataset. Tab. 11 shows the results on
+the zero-shot setting. Our method can exceed the former methods for most diseases. The radar Fig. 3Y shows more visually
+how our model compares with other solutions under the zero-shot setting. Our method can exceed the former methods for
+most diseases. Under 100% fine-tuning settings, we achieved similarly excellent results as shown in Tab. 12.
+Methods
+Ate.
+Car.
+Eff.
+Inf.
+Mas.
+Nod.
+Pna.
+Pnx.
+Con.
+Ede.
+Emp.
+Fib.
+Thi.
+Her.
+AVG
+ConVIRT [56]
+0.4533 0.4601 0.7262 0.6238 0.6790 0.6322 0.6097 0.6698 0.6855 0.7699 0.4701 0.5293 0.6098 0.6220 0.6101
+GLoRIA [22]
+0.6680 0.7647 0.7975 0.6159 0.6722 0.5293 0.6755 0.4785 0.7306 0.8212 0.6033 0.5104 0.6721 0.7144 0.6610
+BioViL [7]
+0.5026 0.6328 0.7914 0.5791 0.7029 0.6126 0.6866 0.7516 0.7455 0.8533 0.7136 0.6751 0.6560 0.7692 0.6909
+w/o (PosCL + DE) 0.7131 0.8100 0.8635 0.6361 0.7776 0.6740 0.6903 0.8124 0.7915 0.8869 0.7480 0.6780 0.6429 0.7784 0.7502
+w/o (DE)
+0.7420 0.8270 0.8663 0.6336 0.7867 0.6974 0.7238 0.8310 0.8037 0.8887 0.7865 0.6715 0.5414 0.8691 0.7621
+ours
+0.7506 0.8299 0.8636 0.6280 0.7885 0.6947 0.7236 0.8361 0.8079 0.8888 0.7950 0.6511 0.5783 0.9097 0.7676
+Table 11. Comparison with other state-of-the-art methods on zero-shot ChestX-ray 14 diseases classification task. For each disease,
+AUC score is reported and the macro average AUC score is also reported. We use the first three letters to represent one disease but for
+“pneumonia” and “pneumothorax” we use the first two and the last letters.
+Methods
+Ate.
+Car.
+Eff.
+Inf.
+Mas.
+Nod.
+Pna.
+Pnx.
+Con.
+Ede.
+Emp.
+Fib.
+Thi.
+Her.
+AVG
+Scratch
+0.7835 0.8116 0.8563 0.6537 0.7788 0.6912 0.7004 0.8561 0.8090 0.8869 0.8564 0.7534 0.7454 0.9106 0.7924
+ConVIRT [56] 0.8012 0.8360 0.8511 0.6613 0.8004 0.7490 0.6998 0.8666 0.8079 0.9023 0.9014 0.7933 0.7468 0.9627 0.8128
+GLoRIA [22]
+0.8263 0.8326 0.8596 0.6641 0.8179 0.7348 0.7104 0.8452 0.8129 0.8977 0.9310 0.7886 0.7608 0.9750 0.8184
+BioViL [7]
+0.8185 0.8543 0.8607 0.6660 0.8302 0.7633 0.7090 0.8595 0.8287 0.9031 0.9251 0.7912 0.7638 0.9696 0.8245
+ours
+0.8291 0.8594 0.8719 0.6565 0.8382 0.7647 0.7378 0.8807 0.8275 0.9083 0.9224 0.7977 0.7784 0.9796 0.8323
+Table 12. Comparison with other state-of-the-art methods on fine-tuning ChestX-ray 14 diseases classification task. For each disease,
+AUC score is reported and the macro average AUC score is also reported. We use the first three letters to represent one disease but for
+“pneumonia” and “pneumothorax” we use the first two and the last letters.
+Figure 3. The radar figure of our method and other
+methods of ChestX-ray14 14 diseases. AUC scores
+are reported and, as shown, our method exceeds the
+previous state-of-the-art on most diseases.
+
+E. Visualization Results
+Fig. 4 shows visualization results of our model on zero-shot grounding task. As shown in figure, the ground truth of
+“Pneumonia” is given by bounding box and generally related to a large area region. Thus the metrics on this are higher than
+other two datasets. Our network captures its regions very well. For “Pneumothorax”, its abnormality pattern is different from
+other diseases, which aim to capturing the collapsed part of the lung, rendering darker areas on the images rather than brighter
+opacity. Its ground-truth masks are generally thin and narrow while our network can still highlight its location. For “Covid-
+19”, its image textual was similar to “Pneumonia”, but since this is a totally new disease, grounding its regions is much more
+challenging. It requires the model to build relationships between them based on their complex definition and symptoms. The
+visualization results suggest that our model successfully achieve this, supporting that, for other unseen diseases, our model
+can also understand their complex descriptions.
+(a) Pneumonia
+(b) Pneumothorax
+(c) Covid-19
+GT
+Prediction
+GT
+Prediction
+GT
+Prediction
+GT
+Prediction
+GT
+Prediction
+GT
+Prediction
+(a) Pneumonia
+(b) Pneumothorax
+(c) Covid-19
+GT
+Prediction
+GT
+Prediction
+GT
+Prediction
+Figure 4. The visualization of zero-shot grounding results of our method. Each column represents the results on one disease and the left
+in it is the ground-truth and right is the heatmap predication of our model. The brighter the red on the figure, the more likely the model
+considering this region to be associated with abnormalities.
+
+(A)

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+A REALIZATION OF POSET ASSOCIAHEDRA
+ANDREW SACK
+Abstract. Given any connected poset P, we give a simple realization of Galashin’s poset
+associahedron A (P) as a convex polytope in RP . The realization is inspired by the de-
+scription of A (P) as a compactification of the configuration space of order-preserving
+maps P → R. In addition, we give an analogous realization for Galashin’s affine poset
+cyclohedra.
+1. Introduction
+Given a finite connected poset P, the poset associahedron A (P) is a simple, convex
+polytope of dimension |P| − 2 introduced by Galashin [6]. Poset associahedra arise as a
+natural generalization of Stasheff’s associahedra [7, 11, 15, 16], and were originally discovered
+by considering compactifications of the configuration space of order-preserving maps P → R.
+These compactifications are generalizations of the Axelrod–Singer compactification of the
+configuration space of points on a line [1, 8, 13]. Galashin constructed poset associahedra
+by performing stellar subdivisions on the polar dual of Stanley’s order polytope [14], but did
+not provide an explicit realization. Various poset associahedra and cyclohedra have already
+been studied including permutohedra, associahedra, operahedra [9], type B permutohedra [5],
+and cyclohedra [2].
+Poset associahedra bear resemblance to graph associahedra, where the face lattice of each
+is described by a tubing criterion. However, neither class is a subset of the other. When
+Carr and Devadoss introduced graph associahedra in [3], they distinguish between bracketings
+and tubings of a path, where the idea of bracketings does not naturally extend to any simple
+graph.
+In the case of poset associahedra, the idea of bracketings does extend to every
+connected poset.
+Galashin [6] also introduces affine posets, and analagous affine order polytopes and affine
+poset cyclohedra. In this paper, we provide a simple realization of poset associahedra and
+affine poset cyclohedra as an intersection of half spaces, inspired by the compactification
+description and by a similar realization of graph associahedra due to Devadoss [4]. In inde-
+pendent work [10], Mantovani, Padrol, and Pilaud found a realization of poset associahedra
+as sections of graph associahedra. The authors of [10] also generalize from posets to oriented
+building sets (which combine a building set with an oriented matroid).
+Date: January 30, 2023.
+Key
+words
+and
+phrases. Poset,
+associahedron,
+cyclohedron,
+realization,
+configuration
+space,
+compactification.
+This material is based upon work supported by the National Science Foundation Graduate Research
+Fellowship Program under Grant No. DGE-2034835. Any opinions, findings, and conclusions or recommen-
+dations expressed in this material are those of the author(s) and do not necessarily reflect the views of the
+National Science Foundation.
+1
+arXiv:2301.11449v1  [math.CO]  26 Jan 2023
+
+1
+2
+3
+4
+5
+1
+2
+3
+4
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+1
+2
+3
+4
+5
+Examples
+Non-examples
+Figure 1. Examples and non-examples of proper tubings.
+2. Background
+2.1. Poset Associahedra. We start by defining the poset associahedron.
+Definition 2.1. Let (P, ⪯) be a finite poset. We make the following definitions:
+• A subset τ ⊆ P is connected if it is connected as an induced subgraph of the Hasse
+diagram of P.
+• τ ⊆ P is convex if whenever a, c ∈ τ and b ∈ P such that a ⪯ b ⪯ c, then b ∈ τ.
+• A tube of P is a connected, convex subset τ ⊆ P such that 2 ≤ |τ|.
+• A tube τ is proper if |τ| ≤ |P| − 1.
+• Two tubes σ, τ ⊆ P are nested if σ ⊆ τ or τ ⊆ σ. Tubes σ and τ are disjoint if
+τ ∩ σ = ∅.
+• For disjoint tubes σ, τ we say σ ≺ τ if there exists a ∈ σ, b ∈ τ such that a ≺ b.
+• A proper tubing T of P is a set of proper tubes of P such that any pair of tubes
+is nested or disjoint and the relation ≺ extends to a partial order on T. That is,
+whenever τ1, . . . , τk ∈ T with τ1 ≺ · · · ≺ τk then τk ̸≺ τ1. This is referred to as the
+acyclic tubing condition.
+• A proper tubing T is maximal if it is maximal by inclusion on the set of all proper
+tubings.
+Figure 1 shows examples and non-examples of proper tubings.
+Definition 2.2. For a finite poset P, the poset associahedron A (P) is a simple, convex
+polytope of dimension |P| − 2 whose face lattice is isomorphic to the set of proper tubings
+ordered by reverse inclusion. That is, if FT is the face corresponding to T, then FS ⊂ FT if
+one can make S from T by adding tubes. Vertices of A (P) correspond to maximal tubings
+of P.
+We realize poset associahedra as an intersection of half-spaces. Let P be a finite poset
+and let n = |P|. We work in the ambient space RP
+Σ=0, the space of real-valued functions on
+P that sum to 0. For a subset τ ⊆ P, define a linear function ατ on RP
+Σ=0 by
+ατ(p) :=
+�
+i≺·j
+i,j∈τ
+pj − pi.
+Here the sum is taken over all covering relations contained in τ. We define the half-space hτ
+and the hyperplane Hτ by
+hτ := {p ∈ RP
+Σ=0 | ατ(p) ≥ n2|τ|}
+and
+Hτ := {p ∈ RP
+Σ=0 | ατ(p) = n2|τ|}.
+The following is our main result in the finite case:
+2
+
+-4
+-3
+-2
+-1
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+...
+...
+-4
+-3
+-2
+-1
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+...
+...
+˜P
+A maximal tubing of ˜P
+Figure 2.
+An affine poset of order 4 and a maximal tubing
+Theorem 2.3. If P is a finite, connected poset, the intersection of HP with hτ for all proper
+tubes τ gives a realization of A (P).
+2.2. Affine Poset Cyclohedra. Now we describe affine poset cyclohedra.
+Definition 2.4. An affine poset of order n ≥ 1 is a poset ˜P = (Z, ⪯) such that:
+(1) For all i ∈ Z, i ⪯ i + n;
+(2) ˜P is n-periodic: For all i, j ∈ Z, i ⪯ j ⇔ i + n ⪯ j + n;
+(3) ˜P is strongly connected: for all i, j ∈ Z, there exists k ∈ Z such that i ⪯ j + kn.
+The order of ˜P is denoted | ˜P| := n.
+Following Galashin [6], we give analagous versions of Definition 2.1. We give them only
+where they differ from the finite case.
+Definition 2.5. Let ˜P = (Z, ⪯) be an affine poset.
+• A tube of ˜P is a connected, convex subset τ ⊆ P such that 2 ≤ |τ| and either τ = ˜P
+or τ has at most one element in each residue class modulo n.
+• A collection of tubes T is n-periodic is for all τ ∈ T, k ∈ Z, τ + kn ∈ T.
+• A proper tubing T of ˜P is an n-periodic set of proper tubes of ˜P that satisfies the
+acyclic tubing condition and such that any pair of tubes is nested or disjoint.
+Figure 2 gives an example of an affine poset of order 4 and a maximal tubing of that poset.
+Definition 2.6. For an affine poset ˜P, the affine poset cyclohedron C ( ˜P) is a simple, convex
+polytope of dimension | ˜P| − 1 whose face lattice is isomorphic to the set of proper tubings
+ordered by reverse inclusion. Vertices of C ( ˜P) correspond to maximal tubings of ˜P.
+We also realize affine poset cyclohedra as an intersection of half-spaces. Let ˜P be an affine
+poset and let n = | ˜P|. Fix some constant c ∈ R+. We define the space of affine maps R ˜P as
+the set of bi-infinite sequences ˜x = (˜xi)i∈Z such that ˜xi = ˜xi+n + c for all i ∈ Z. Let C ⊂ R ˜P
+be the subspace consisting of all constant maps. We work in the ambient space R ˜P/C where
+the constant c in the definition of affine maps is given by c = n2(n+1).
+3
+
+a
+b
+c
+d
+e
+→
+ab
+c
+de
+→
+ab
+cde
+→
+abcde
+a
+b
+c
+d
+e
+f
+→
+ab
+c
+d
+e
+f
+→
+abc
+d
+e
+f
+→
+abcd
+e
+f
+→
+abcde
+f
+→
+abcdef
+Figure 3.
+Multiplication of a word and of a generalized word
+For a finite subset τ ⊆ P, define a linear function ατ on R ˜P/C by
+ατ(˜x) :=
+�
+i≺·j
+i,j∈τ
+˜xj − ˜xi.
+Again, the sum is taken over all covering relations contained in τ. We define the half-space
+hτ and the hyperplane Hτ by
+hτ := {p ∈ R
+˜P/C | ατ(p) ≥ n2|τ|}
+and
+Hτ := {p ∈ R
+˜P/C | ατ(p) = n2|τ|}.
+Remark 2.7. Observe that for any tube τ and k ∈ Z, hτ = hτ+kn.
+The following is our main result in the affine case:
+Theorem 2.8. If ˜P is an affine poset, the intersection of hτ for all proper tubes τ gives a
+realization of C ( ˜P).
+2.3. An interpretation of tubings. When P is a chain, A (P) recovers the classical as-
+sociahedron.
+There is a simple interpretation of proper tubings that explains all of the
+conditions above in terms of generalized words.
+We can understand the classical associahedron as follows: Let P = ({1, ..., n}, ≤) be a
+chain. We can think of the chain as a word we want to multiply together with the rule that
+two elements can be multiplied if they are connected by an edge. A maximal tubing of P
+is a way of disambiguating the order in which one performs the multiplication. If a pair of
+adjacent elements x and y have a pair of brackets around them, they contract along the edge
+connecting them and replace x and y by their product.
+Similarly, we can understand the Hasse diagram of an arbitrary poset P as a generalized
+word we would like to multiply together. Again, we are allowed to multiply two elements
+if they are connected by an edge, but when multiplying elements, we contract along the
+edge connecting them and then take the transitive reduction of the resulting directed graph.
+That is, we identify the two elements and take the resulting quotient poset. A maximal
+4
+
+tubing is again a way of disambiguating the order of the multiplication. See Figure 3 for
+an illustration of this multiplication. This perspective is discussed in relation to operahedra
+in [9, Section 2.1] when the Hasse diagram of P is a rooted tree.
+3. Configuration spaces and compactifications
+We turn our attention to the relationship between poset associahedra and configuration
+spaces. For a poset P, the order cone
+L (P) := {p ∈ RP
+Σ=0 | pi ≤ pj for all i ⪯ j}
+is the set of order preserving maps P → R whose values sum to 0.
+Fix a constant c ∈ R+. The order polytope, first defined by Stanley [14] and extended by
+Galashin [6], is the (|P| − 2)-dimensional polytope
+O(P) := {p ∈ L (P) | αP(p) = c}.
+Remark 3.1. When P is bounded, that is, has a unique maximum ˆ1 and minimum ˆ0,
+this construction is projectively equivalent to Stanley’s order polytope where we replace the
+conditions of the coordinates summing to 0 and αP(p) = c with the conditions pˆ0 = 0 and
+pˆ1 = 1, see [6, Remark 2.5].
+Galashin [6] obtains the poset associahedra by an alternative compactification of O◦(P),
+the interior of O(P). We describe this compactification informally, as it serves as motivation
+for the realization in Theorem 2.3.
+A point is on the boundary of O(P) when any of the inequalities in the order cone achieve
+equality. The faces of of O(P) are in bijection with proper tubings of P such that all tubes
+are disjoint. Let T be such a tubing. If p is in the face corresponding to T and τ ∈ T then
+pi = pj for i, j ∈ τ.
+We can think of the point p in the face corresponding to T as being “what happens in
+O(P)” when for each τ ∈ T, the coordinates are infinitesimally close. However, by taking
+all coordinates in τ to be equal, we lose information about their relative ordering. In A (P),
+we still think of the coordinates in τ as being infinitesimally close, but we are still inter-
+ested in their configuration. Upon zooming in, this is parameterized by the order polytope
+of the subposet (τ, ⪯). We iterate this process, allowing points in τ to be infinitesimally
+closer, and so on. We illustrate this in Figure 4. This idea is a common explanation of the
+Axelrod–Singer compactification of O◦(P) when P is a chain, see [1, 8, 13].
+The idea of the realization in Theorem 2.3 is to replace the notions of infinitesimally
+close and infinitesimally closer with being exponentially close and exponentially closer. For
+p ∈ L (P), ατ acts a measure of how close the coordinates of p|τ are. We can make this
+precise with the following definition and lemma.
+Definition 3.2. For S ⊆ P and p ∈ RP, define the diameter of p relative to S by
+diamS(p) = max
+i,j∈S |pi − pj|.
+That is, diamS(p) is the diameter of {pi : i ∈ S} as a subset of R.
+Lemma 3.3. Let τ ⊆ P be a tube and let p ∈ L (P). Then
+diamτ(p) ≤ ατ(p) ≤ n2
+4 diamτ(p).
+5
+
+1
+2
+3
+4
+5
+6
+123
+456
+1
+2
+3
+4
+5
+6
+1
+2
+3
+5
+4
+6
+Tubing in O(P)
+Point in O(P)
+Tubing in A (P)
+Point in A (P)
+Figure 4.
+A vertex in O(P) vs. A (P).
+Proof. By the triangle inequality and as τ is connected, diamτ(p) ≤ ατ(p). For the other
+inequality,
+ατ(p) =
+�
+i≺·j
+i,j∈τ
+pj − pi
+≤
+�
+i≺·j
+i,j∈τ
+diamτ(p)
+≤ 1
+4n2 diamτ(p)
+The inequality in the last line comes from the fact that there are at most n2
+4 covering
+relations in P, which follows from Mantel’s Theorem and the fact that Hasse diagrams are
+triangle-free.
+□
+In particular, for p ∈ L (P), if p ∈ Hτ, then {pi | i ∈ τ} is clustered tightly together
+compared to any tube containing τ. If p ∈ hτ, then {pi | i ∈ τ} is spread far apart compared
+to any tube contained in τ.
+4. Realizing poset associahedra
+We are now prepared to prove Theorem 2.3. Define
+A (P) :=
+�
+σ⊂P
+hσ ∩ HP
+where the intersection is over all tubes of P. Note that A (P) ⊆ L (P) as if i ≺· j is a
+covering relation, then for p ∈ h{i,j}, pi ≤ pj.
+Theorem 2.3 follows as a result of three lemmas:
+Lemma 4.1. If T is a maximal tubing, then
+vT :=
+�
+τ∈T∪{P}
+Hτ
+is a point.
+6
+
+Lemma 4.2. If T is a collection of tubes that do not form a proper tubing, then
+�
+τ∈T
+Hτ ∩ A (P) = ∅.
+Lemma 4.3. If T is a maximal tubing and τ /∈ T is a proper tube, then ατ(vT) > n2|τ|. That
+is, vT lies in the interior of hτ.
+Lemma 4.1 follows from a standard induction argument.
+Proof of Lemma 4.2. If T is not a collection of tubes that do proper tubing, then at least
+one of the following two cases holds:
+(1) There is a pair of non-nested and non-disjoint tubes τ1, τ2 in T.
+(2) There is a sequence of disjoint tubes τ1, ..., τk such that τ1 ≺ · · · ≺ τk ≺ τ1.
+The idea of the proof is as follows: For S ⊆ P, define the convex hull of S as
+conv(σ) := {b ∈ P | ∃a, c ∈ S : a ≤ b ≤ c}.
+Observe that if p ∈ L (P), then diamS(p) ≤ diamconv(S)(p). Take σ = conv(τ1 ∪ · · · ∪ τk).
+One can show that σ is a tube, so Lemma 3.3 tells us that for each τi, diamτi(p) is very
+small compared to n2|σ|. As the tubes either intersect or are cyclic, one can show this forces
+diamσ(p) to also be small, so ασ(p) < n2|σ|.
+More concretely, suppose that
+p ∈
+�
+Hτi ∩ L (P).
+Note that for all i, |σ| > |τi| + 1 and diamτi(p) ≤ n2(|σ|−1). In case (1), let a, b ∈ σ. There
+exists some x ∈ τ1 ∩ τ2, so
+|pa − pb| ≤ |pa − px| + |px − pb|
+≤ diamτ1(p) + diamτ2(p)
+≤ 2n2(|σ|−1)
+< n2(|σ|).
+Hence diamσ(p) < n2|σ|, so by Lemma 3.3, p /∈ hσ.
+Now we move to case (2). Suppose there is a sequence of disjoint tubes τ1, ..., τk such that
+for each i there exists xi, yi ∈ ��i where xi ≺ yi+1 where we take the indices mod k. Then:
+pyi − diamτi(p) ≤ pxi
+pxi ≤ pyi+1
+pyi+1 ≤ pxi+1 + diamτi+1
+Furthermore, since τi and τi+1 are disjoint, |τi| ≤ |σ|−2 and diamτi ≤ n2(|σ|−2). Combining
+these we get
+pyi ≤ pyi+1 + 2n2(|σ|−2).
+Then we have:
+py1 ≤ pyi + 2in2(|σ|−2)
+and
+pyi + 2in2(|σ|−2) ≤ py1 + 2(k + 1)n2(|σ|−2).
+7
+
+4
+5
+2
+1
+3
+6
+7
+8
+9
+τ
+4
+5
+2
+1
+3
+6
+7
+8
+9
+τ
+σ
+A
+B
+Maximal Tubing T and tube τ
+σ, A, and B labelled
+Figure 5. An example illustrating the proof of Lemma 4.3.
+A
+B
+σ
+τ
+Figure 6.
+If diamA(p) and diamB(p) are small and diamσ(p) is large, then
+diamτ(p) is large.
+These yield
+py1 − pyi ≤ 2in2(|σ|−2)
+and
+pyi − py1 ≤ 2(k − i + 1)n2(|σ|−2).
+As i, k − i + 1 ≤ k ≤ n
+2, we have |py1 − pyi| ≤ n2(|σ|−1). Finally, if zi ∈ τi, zj ∈ τj, then
+|pzi − pzj| ≤ |pzi − pyi| + |pyi − py1| + |py1 − pyj| + |pyj − pzj|
+≤ 4n2(|σ|−1)
+< n2|σ|.
+Hence diamσ(p) < n2|σ|, and by Lemma 3.3, p /∈ hσ.
+□
+Proof of Lemma 4.3. Let T be a maximal tubing of P and let τ /∈ T be a tube. Define the
+convex hull of τ relative to T by
+convT(τ) := min{σ ∈ T | τ ⊂ σ}.
+Let σ = convT(τ). T partitions σ into a lower set A and an upper set B where A and B
+are either tubes or singletons. Furthermore, A and B both intersect τ. See Figure 5 for an
+example illustrating this.
+The idea of the proof is as follows: Let p = vT. By Lemma 3.3, diamA(p) and diamB(p)
+are both very small compared to diamσ(p). Then for any a ∈ A, b ∈ B, |pa − pb| must be
+large. As τ intersects both A and B, diamτ(p) must be large and hence p ∈ hτ. See Figure 6
+for an illustration of this. More precisely, we show that for any i ∈ A, j ∈ B, pj −pi > (n2)|τ|,
+which implies that p lies in the interior of hτ.
+8
+
+Observe that:�
+x≺·y
+py − px =
+�
+x≺·y
+x,y∈A
+(py − px)
+�
+��
+�
+≤(n2)|σ|−1
+< 1
+8 (n2)|σ|
++
+�
+x≺·y
+x,y∈B
+(py − px)
+�
+��
+�
+≤(n2)|σ|−1
+< 1
+8 (n2)|σ|
++
+�
+x≺·y
+x∈A,y∈B
+(py − px).
+Fix i ∈ A and j ∈ B. By Lemma 3.3, for any x ∈ A, y ∈ B,
+py − px ≤ pj − pi + diamA(p) + diamB(p)
+≤ pj + pi + 2n2(|σ|−1).
+Again, noting that the number of covering relations in σ is at most n2
+4 we obtain:
+�
+x≺·σy
+x∈A,y∈B
+(py − px) ≤
+�
+x≺·σy
+x∈A,y∈B
+(pj − pi + 2(n2)|σ|−1)
+≤ n2
+4
+�
+pj − pi + 2(n2)|σ|−1�
+= n2
+4 (pj − pi) + 1
+2(n2)|σ|.
+Combining all of this we get:
+�
+x≺·σy
+py − px = (n2)|σ|
+< 1
+8(n2)|σ| + 1
+8(n2)|σ| + 1
+2(n2)|σ| + n2
+4 (pj − pi)
+≤ 3
+4(n2)|σ| + n2
+4 (pj − pi)
+Then (n2)|σ|−1 < (pj − pi) and as |τ| ≤ |σ| − 1, p is in the interior of hτ.
+□
+Remark 4.4. A similar approach for realizing graph associahedra is taken by Devadoss [4].
+One difference is that Devadoss realizes graph associahedra by cutting off slices of a simplex
+whereas we cut off slices of an order polytope.
+5. Realizing affine poset cyclohedra
+The proofs in the affine case are nearly identical to the finite case with some additional
+technical components. The similarity comes from the fact that Lemma 3.3 still applies. We
+highlight where the proofs are different. Let ˜P be an affine poset of order n.
+Define
+C ( ˜P) :=
+�
+σ⊂P
+hσ
+and
+L ( ˜P) := {p ∈ R
+˜P/C | pi ≤ pj for all i ⪯ j}.
+where the intersection is over all tubes of ˜P. Note that C ( ˜P) ⊆ L ( ˜P) as if i ≺· j is a
+covering relation, then for p ∈ h{i,j}, pi ≤ pj. Theorem 2.8 follows as a result of 3 lemmas:
+9
+
+Lemma 5.1. If T is a maximal tubing, then
+vT :=
+�
+τ∈T
+Hτ
+is a point.
+Lemma 5.2. If T is a collection of tubes that do not form a proper tubing, then
+�
+τ∈T
+Hτ ∩ C ( ˜P) = ∅.
+Lemma 5.3. If T is a maximal tubing and τ /∈ T is a proper tube, then ατ(vT) > n2|τ|. That
+is, vT lies in the interior of hτ.
+Proof of Lemma 5.1. Let T be a maximal tubing and take any σ ∈ T such that |τ| = n.
+Then restricting to ˜P|σ, Lemma 4.1 implies that
+�
+τ∈T
+τ⊆σ
+Hτ
+is a point. However, as T is n-periodic,
+�
+τ∈T
+τ⊆σ
+Hτ =
+�
+τ∈T
+Hτ.
+□
+Proof of Lemma 5.2. By Remark 2.7, we can assume T is n-periodic. The proof is almost
+identical to the proof of Lemma 4.2. Define
+L ( ˜P) := {p ∈ R
+˜P/C | pi ≤ pj for all i ⪯ j}.
+and note that
+L ( ˜P) ⊆ R
+˜P/C
+�
+i,j∈ ˜P
+i≺·j
+h{i,j}.
+Let
+p ∈
+�
+Hτi ∩ L ( ˜P).
+We again break into two cases:
+(1) There is a pair of non-nested and non-disjoint tubes τ1, τ2 in T.
+(2) All tubes in T are pairwise nested or disjoint and there is a sequence of disjoint tubes
+τ1, ..., τk such that τ1 ≺ · · · ≺ τk ≺ τ1.
+The only difference in the proof occurs in case (1). Here, it is possible that there exists
+x ∈ τ1 ∪τ2 such that x+n ∈ τ1 ∪τ2 as well. In this case, the proof of Lemma 4.2 still implies
+that diamτ1∪τ2(p) ≤ diamτ1(p) + diamτ2(p) ≤ 2n2n. However, |px+n − px| = n2(n+1).
+□
+Proof of Lemma 5.3. Let T be a maximal tubing and τ /∈ T be a proper tube. Let p = vT.
+We claim that ατ(p) > n2|τ|.
+The only difference from the proof of Lemma 4.3 is that τ may not be contained by any
+tube in τ so convT(τ) may not be well-defined. In this case, there exists A ∈ T such that
+10
+
+ˆ0
+a
+b
+c
+ˆ1
+P
+ε = 1
+3
+ε = 1
+9
+ε =
+1
+27
+Figure 7.
+O(P) as a limit of A (P)
+|A| = n, A ∩ τ ̸= ∅, and (A + n) ∩ τ ̸= ∅. Here, (A + n) acts the same as B in the finite
+case, except the argument is much simpler.
+Let i ∈ A ∩ τ, j ∈ (A + n) ∩ τ. Observe that diamA(p), diam(A+n)(p) ≤ n2n and that
+i + n ∈ (A + n). Then
+|pj − pi| ≥ (pj − n2n) − pi
+≥ pi+n − pi
+= n2(n+1).
+Hence diamτ(p) > n2|τ| and by Lemma 3.3, ατ(p) > n2|τ|.
+□
+6. Remarks and Questions
+Remark 6.1. Let (P, ⪯) be a bounded poset. In Remark 3.1, we discuss how O(P) can be
+realized as the set of all p ∈ RP such that pˆ0 = 0, pˆ1 = 1, and pi ≤ pj whenever i ⪯ j. We
+can similarly realize A (P) as follows: Fix 0 < ε <
+1
+n2.
+For a proper tube τ ⊂ P, let
+h′
+τ = {p ∈ RP | ατ(p) < εn−|τ|}.
+Then A (P) is realized as the intersection over all h′
+τ with the hyperplanes
+{pˆ0 = 0} and {pˆ1 = 1}.
+Letting ε → 0, we obtain O(P) as a limit of A (P) as shown in Figure 7.
+Remark 6.2. The key piece to the realizations in Theorems 2.3 and 2.8 is the linear form
+ατ, where ατ acts as an approximation of diamτ. In particular, let τ be a tube and let
+p ∈ L (P). Then:
+• ατ(p) ≥ 0.
+• ατ(p) = 0 ⇔ p|τ is constant.
+• If σ ⊆ τ is a tube, then ασ(p) ≤ ατ(p).
+However, there are many other options for choice of ατ that could fill this role. Some other
+options include:
+11
+
+(1) Sum over all pairs i ≺ j in τ.
+ατ(p) =
+�
+i≺j
+i,j∈τ
+pj − pi.
+(2) Let A and B be the set of minima and maxima of the restriction P|τ respectively.
+ατ(p) =
+�
+i≺j
+i∈A,j∈B
+pj − pi.
+(3) Fix a spanning tree T in the Hasse diagram of τ. Let E = {(i, j) | i ≺·T j} be the set
+of edges in T.
+ατ(p) =
+�
+(i,j)∈E
+pj − pi.
+An advantage of this option is that we would have
+diamτ(p) ≤ ατ(p) ≤ (n − 1) diamτ(p).
+A similar realization can be obtained for each choice of of ατ.
+Question 6.3. Recall that for a simple d-dimensional polytope P, the f-vector and h-vector
+of P are given by (f0, . . . , fd) and (h0, . . . , hd) where fi is the number of i-dimensional faces
+and
+d
+�
+i=0
+fiti =
+d
+�
+i=0
+hi(t + 1)i.
+Postnikov, Reiner, and Williams [12] found a statistic on maximal tubings of graph associa-
+hedra of chordal graphs where
+�
+T
+tstat(T) =
+�
+hiti.
+In particular, they define a map T �→ wT from maximal tubings of a graph on n vertices
+to the set of permutations Sn such that stat(T) = des(wT), the number of descents of wT.
+It would be interesting to find a similar statistic on maximal tubings of poset associahedra.
+For a simple polytope P, one can orient the edges of P according to a generic linear form
+and take stat(v) = outdegree(v) [17, §8.2]. It may be possible to use our realization to find
+the desired statistic.
+Acknowledgements
+The author is grateful to Pavel Galashin for his many helpful comments and suggestions
+and to Stefan Forcey for fruitful conversations.
+References
+[1]
+Scott Axelrod and Isadore M Singer. “Chern-Simons perturbation theory. II”. In: Jour-
+nal of Differential Geometry 39.1 (1994), pp. 173–213.
+[2]
+Raoul Bott and Clifford Taubes. “On the self-linking of knots”. In: Journal of Mathe-
+matical Physics 35.10 (1994), pp. 5247–5287.
+[3]
+Michael Carr and Satyan L Devadoss. “Coxeter complexes and graph-associahedra”.
+In: Topology and its Applications 153.12 (2006), pp. 2155–2168.
+12
+
+[4]
+Satyan L Devadoss. “A realization of graph associahedra”. In: Discrete Mathematics
+309.1 (2009), pp. 271–276.
+[5]
+Sergey Fomin and Nathan Reading. “Root systems and generalized associahedra”. In:
+arXiv preprint math/0505518 (2005).
+[6]
+Pavel Galashin. “Poset associahedra”. In: arXiv preprint arXiv:2110.07257 (2021).
+[7]
+Mark Haiman. “Constructing the associahedron”. In: Unpublished manuscript, MIT
+(1984).
+[8]
+Pascal Lambrechts, Victor Turchin, and Ismar Voli´c. “Associahedron, cyclohedron and
+permutohedron as compactifications of configuration spaces”. In: Bulletin of the Belgian
+Mathematical Society-Simon Stevin 17.2 (2010), pp. 303–332.
+[9]
+Guillaume Laplante-Anfossi. “The diagonal of the operahedra”. In: Advances in Math-
+ematics 405 (2022), p. 108494.
+[10]
+Chiara Mantovani, Arnau Padrol, and Vincent Pilaud. “Acyclonestohedra: when ori-
+ented matroids meet nestohedra”. in prep.
+[11]
+Kyle Petersen. Eulerian Numbers. Oct. 2015. isbn: 978-1-4939-3090-6. doi: 10.1007/
+978-1-4939-3091-3.
+[12]
+Alex Postnikov, Victor Reiner, and Lauren Williams. “Faces of Generalized Permuto-
+hedra”. In: Documenta Mathematica 13 (2008), pp. 207–273.
+[13]
+Dev P Sinha. “Manifold-theoretic compactifications of configuration spaces”. In: Selecta
+Mathematica 10.3 (2004), pp. 391–428.
+[14]
+Richard P Stanley. “Two poset polytopes”. In: Discrete & Computational Geometry
+1.1 (1986), pp. 9–23.
+[15]
+Jim Stasheff. “From Operads to ‘Physically’ Inspired Theories”. In: (Sept. 1996).
+[16]
+Dov Tamari. “Mono¨ıdes pr´eordonn´es et chaˆınes de Malcev”. In: Bulletin de la Soci´et´e
+math´ematique de France 82 (1954), pp. 53–96.
+[17]
+G¨unter M Ziegler. Lectures on polytopes. Vol. 152. Springer Science & Business Media,
+2012.
+Department of Mathematics, University of California, Los Angeles, CA 90095, USA
+Email address: andrewsack@math.ucla.edu
+13
+

NdFJT4oBgHgl3EQfGyyu/content/tmp_files/load_file.txt

@@ -0,0 +1,438 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf,len=437
+page_content='A REALIZATION OF POSET ASSOCIAHEDRA  ANDREW SACK  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Given any connected poset P, we give a simple realization of Galashin’s poset  associahedron A (P) as a convex polytope in RP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' The realization is inspired by the de-  scription of A (P) as a compactification of the configuration space of order-preserving  maps P → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' In addition, we give an analogous realization for Galashin’s affine poset  cyclohedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Introduction  Given a finite connected poset P, the poset associahedron A (P) is a simple, convex  polytope of dimension |P| − 2 introduced by Galashin [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Poset associahedra arise as a  natural generalization of Stasheff’s associahedra [7, 11, 15, 16], and were originally discovered  by considering compactifications of the configuration space of order-preserving maps P → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  These compactifications are generalizations of the Axelrod–Singer compactification of the  configuration space of points on a line [1, 8, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Galashin constructed poset associahedra  by performing stellar subdivisions on the polar dual of Stanley’s order polytope [14], but did  not provide an explicit realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Various poset associahedra and cyclohedra have already  been studied including permutohedra, associahedra, operahedra [9], type B permutohedra [5],  and cyclohedra [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Poset associahedra bear resemblance to graph associahedra, where the face lattice of each  is described by a tubing criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' However, neither class is a subset of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' When  Carr and Devadoss introduced graph associahedra in [3], they distinguish between bracketings  and tubings of a path, where the idea of bracketings does not naturally extend to any simple  graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  In the case of poset associahedra, the idea of bracketings does extend to every  connected poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Galashin [6] also introduces affine posets, and analagous affine order polytopes and affine  poset cyclohedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' In this paper, we provide a simple realization of poset associahedra and  affine poset cyclohedra as an intersection of half spaces, inspired by the compactification  description and by a similar realization of graph associahedra due to Devadoss [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' In inde-  pendent work [10], Mantovani, Padrol, and Pilaud found a realization of poset associahedra  as sections of graph associahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' The authors of [10] also generalize from posets to oriented  building sets (which combine a building set with an oriented matroid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Date: January 30, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Key  words  and  phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Poset,  associahedron,  cyclohedron,  realization,  configuration  space,  compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  This material is based upon work supported by the National Science Foundation Graduate Research  Fellowship Program under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' DGE-2034835.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Any opinions, findings, and conclusions or recommen-  dations expressed in this material are those of the author(s) and do not necessarily reflect the views of the  National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='11449v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='CO]  26 Jan 2023  1  2  3  4  5  1  2  3  4  1  2  3  4  5  1  2  3  4  5  1  2  3  4  1  2  3  4  5  Examples  Non-examples  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Examples and non-examples of proper tubings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Background  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Poset Associahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' We start by defining the poset associahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Let (P, ⪯) be a finite poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' We make the following definitions:  A subset τ ⊆ P is connected if it is connected as an induced subgraph of the Hasse  diagram of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  τ ⊆ P is convex if whenever a, c ∈ τ and b ∈ P such that a ⪯ b ⪯ c, then b ∈ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  A tube of P is a connected, convex subset τ ⊆ P such that 2 ≤ |τ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  A tube τ is proper if |τ| ≤ |P| − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Two tubes σ, τ ⊆ P are nested if σ ⊆ τ or τ ⊆ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Tubes σ and τ are disjoint if  τ ∩ σ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  For disjoint tubes σ, τ we say σ ≺ τ if there exists a ∈ σ, b ∈ τ such that a ≺ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  A proper tubing T of P is a set of proper tubes of P such that any pair of tubes  is nested or disjoint and the relation ≺ extends to a partial order on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' That is,  whenever τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' , τk ∈ T with τ1 ≺ · · · ≺ τk then τk ̸≺ τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' This is referred to as the  acyclic tubing condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  A proper tubing T is maximal if it is maximal by inclusion on the set of all proper  tubings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Figure 1 shows examples and non-examples of proper tubings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' For a finite poset P, the poset associahedron A (P) is a simple, convex  polytope of dimension |P| − 2 whose face lattice is isomorphic to the set of proper tubings  ordered by reverse inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' That is, if FT is the face corresponding to T, then FS ⊂ FT if  one can make S from T by adding tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Vertices of A (P) correspond to maximal tubings  of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  We realize poset associahedra as an intersection of half-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Let P be a finite poset  and let n = |P|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' We work in the ambient space RP  Σ=0, the space of real-valued functions on  P that sum to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' For a subset τ ⊆ P, define a linear function ατ on RP  Σ=0 by  ατ(p) :=  �  i≺·j  i,j∈τ  pj − pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Here the sum is taken over all covering relations contained in τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' We define the half-space hτ  and the hyperplane Hτ by  hτ := {p ∈ RP  Σ=0 | ατ(p) ≥ n2|τ|}  and  Hτ := {p ∈ RP  Σ=0 | ατ(p) = n2|τ|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  The following is our main result in the finite case:  2  4  3  2  1  0  1  2  3  4  5  6  7  8  9  10  11  12  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  4  3  2  1  0  1  2  3  4  5  6  7  8  9  10  11  12  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  ˜P  A maximal tubing of ˜P  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  An affine poset of order 4 and a maximal tubing  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' If P is a finite, connected poset, the intersection of HP with hτ for all proper  tubes τ gives a realization of A (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Affine Poset Cyclohedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Now we describe affine poset cyclohedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' An affine poset of order n ≥ 1 is a poset ˜P = (Z, ⪯) such that:  (1) For all i ∈ Z, i ⪯ i + n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  (2) ˜P is n-periodic: For all i, j ∈ Z, i ⪯ j ⇔ i + n ⪯ j + n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  (3) ˜P is strongly connected: for all i, j ∈ Z, there exists k ∈ Z such that i ⪯ j + kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  The order of ˜P is denoted | ˜P| := n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Following Galashin [6], we give analagous versions of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' We give them only  where they differ from the finite case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Let ˜P = (Z, ⪯) be an affine poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  A tube of ˜P is a connected, convex subset τ ⊆ P such that 2 ≤ |τ| and either τ = ˜P  or τ has at most one element in each residue class modulo n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  A collection of tubes T is n-periodic is for all τ ∈ T, k ∈ Z, τ + kn ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  A proper tubing T of ˜P is an n-periodic set of proper tubes of ˜P that satisfies the  acyclic tubing condition and such that any pair of tubes is nested or disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Figure 2 gives an example of an affine poset of order 4 and a maximal tubing of that poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' For an affine poset ˜P, the affine poset cyclohedron C ( ˜P) is a simple, convex  polytope of dimension | ˜P| − 1 whose face lattice is isomorphic to the set of proper tubings  ordered by reverse inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Vertices of C ( ˜P) correspond to maximal tubings of ˜P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  We also realize affine poset cyclohedra as an intersection of half-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Let ˜P be an affine  poset and let n = | ˜P|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Fix some constant c ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' We define the space of affine maps R ˜P as  the set of bi-infinite sequences ˜x = (˜xi)i∈Z such that ˜xi = ˜xi+n + c for all i ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Let C ⊂ R ˜P  be the subspace consisting of all constant maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' We work in the ambient space R ˜P/C where  the constant c in the definition of affine maps is given by c = n2(n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  3  a  b  c  d  e  →  ab  c  de  →  ab  cde  →  abcde  a  b  c  d  e  f  →  ab  c  d  e  f  →  abc  d  e  f  →  abcd  e  f  →  abcde  f  →  abcdef  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Multiplication of a word and of a generalized word  For a finite subset τ ⊆ P, define a linear function ατ on R ˜P/C by  ατ(˜x) :=  �  i≺·j  i,j∈τ  ˜xj − ˜xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Again, the sum is taken over all covering relations contained in τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' We define the half-space  hτ and the hyperplane Hτ by  hτ := {p ∈ R  ˜P/C | ατ(p) ≥ n2|τ|}  and  Hτ := {p ∈ R  ˜P/C | ατ(p) = n2|τ|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Observe that for any tube τ and k ∈ Z, hτ = hτ+kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  The following is our main result in the affine case:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' If ˜P is an affine poset, the intersection of hτ for all proper tubes τ gives a  realization of C ( ˜P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' An interpretation of tubings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' When P is a chain, A (P) recovers the classical as-  sociahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  There is a simple interpretation of proper tubings that explains all of the  conditions above in terms of generalized words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  We can understand the classical associahedron as follows: Let P = ({1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=', n}, ≤) be a  chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' We can think of the chain as a word we want to multiply together with the rule that  two elements can be multiplied if they are connected by an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' A maximal tubing of P  is a way of disambiguating the order in which one performs the multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' If a pair of  adjacent elements x and y have a pair of brackets around them, they contract along the edge  connecting them and replace x and y by their product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Similarly, we can understand the Hasse diagram of an arbitrary poset P as a generalized  word we would like to multiply together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Again, we are allowed to multiply two elements  if they are connected by an edge, but when multiplying elements, we contract along the  edge connecting them and then take the transitive reduction of the resulting directed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  That is, we identify the two elements and take the resulting quotient poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' A maximal  4  tubing is again a way of disambiguating the order of the multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' See Figure 3 for  an illustration of this multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' This perspective is discussed in relation to operahedra  in [9, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='1] when the Hasse diagram of P is a rooted tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Configuration spaces and compactifications  We turn our attention to the relationship between poset associahedra and configuration  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' For a poset P, the order cone  L (P) := {p ∈ RP  Σ=0 | pi ≤ pj for all i ⪯ j}  is the set of order preserving maps P → R whose values sum to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Fix a constant c ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' The order polytope, first defined by Stanley [14] and extended by  Galashin [6], is the (|P| − 2)-dimensional polytope  O(P) := {p ∈ L (P) | αP(p) = c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' When P is bounded, that is, has a unique maximum ˆ1 and minimum ˆ0,  this construction is projectively equivalent to Stanley’s order polytope where we replace the  conditions of the coordinates summing to 0 and αP(p) = c with the conditions pˆ0 = 0 and  pˆ1 = 1, see [6, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Galashin [6] obtains the poset associahedra by an alternative compactification of O◦(P),  the interior of O(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' We describe this compactification informally, as it serves as motivation  for the realization in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  A point is on the boundary of O(P) when any of the inequalities in the order cone achieve  equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' The faces of of O(P) are in bijection with proper tubings of P such that all tubes  are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Let T be such a tubing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' If p is in the face corresponding to T and τ ∈ T then  pi = pj for i, j ∈ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  We can think of the point p in the face corresponding to T as being “what happens in  O(P)” when for each τ ∈ T, the coordinates are infinitesimally close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' However, by taking  all coordinates in τ to be equal, we lose information about their relative ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' In A (P),  we still think of the coordinates in τ as being infinitesimally close, but we are still inter-  ested in their configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Upon zooming in, this is parameterized by the order polytope  of the subposet (τ, ⪯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' We iterate this process, allowing points in τ to be infinitesimally  closer, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' We illustrate this in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' This idea is a common explanation of the  Axelrod–Singer compactification of O◦(P) when P is a chain, see [1, 8, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  The idea of the realization in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='3 is to replace the notions of infinitesimally  close and infinitesimally closer with being exponentially close and exponentially closer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' For  p ∈ L (P), ατ acts a measure of how close the coordinates of p|τ are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' We can make this  precise with the following definition and lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' For S ⊆ P and p ∈ RP, define the diameter of p relative to S by  diamS(p) = max  i,j∈S |pi − pj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  That is, diamS(p) is the diameter of {pi : i ∈ S} as a subset of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Let τ ⊆ P be a tube and let p ∈ L (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Then  diamτ(p) ≤ ατ(p) ≤ n2  4 diamτ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  5  1  2  3  4  5  6  123  456  1  2  3  4  5  6  1  2  3  5  4  6  Tubing in O(P)  Point in O(P)  Tubing in A (P)  Point in A (P)  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  A vertex in O(P) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' A (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' By the triangle inequality and as τ is connected, diamτ(p) ≤ ατ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' For the other  inequality,  ατ(p) =  �  i≺·j  i,j∈τ  pj − pi  ≤  �  i≺·j  i,j∈τ  diamτ(p)  ≤ 1  4n2 diamτ(p)  The inequality in the last line comes from the fact that there are at most n2  4 covering  relations in P, which follows from Mantel’s Theorem and the fact that Hasse diagrams are  triangle-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  □  In particular, for p ∈ L (P), if p ∈ Hτ, then {pi | i ∈ τ} is clustered tightly together  compared to any tube containing τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' If p ∈ hτ, then {pi | i ∈ τ} is spread far apart compared  to any tube contained in τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Realizing poset associahedra  We are now prepared to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Define  A (P) :=  �  σ⊂P  hσ ∩ HP  where the intersection is over all tubes of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Note that A (P) ⊆ L (P) as if i ≺· j is a  covering relation, then for p ∈ h{i,j}, pi ≤ pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='3 follows as a result of three lemmas:  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' If T is a maximal tubing, then  vT :=  �  τ∈T∪{P}  Hτ  is a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  6  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' If T is a collection of tubes that do not form a proper tubing, then  �  τ∈T  Hτ ∩ A (P) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' If T is a maximal tubing and τ /∈ T is a proper tube, then ατ(vT) > n2|τ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' That  is, vT lies in the interior of hτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='1 follows from a standard induction argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' If T is not a collection of tubes that do proper tubing, then at least  one of the following two cases holds:  (1) There is a pair of non-nested and non-disjoint tubes τ1, τ2 in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  (2) There is a sequence of disjoint tubes τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=', τk such that τ1 ≺ · · · ≺ τk ≺ τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  The idea of the proof is as follows: For S ⊆ P, define the convex hull of S as  conv(σ) := {b ∈ P | ∃a, c ∈ S : a ≤ b ≤ c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Observe that if p ∈ L (P), then diamS(p) ≤ diamconv(S)(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Take σ = conv(τ1 ∪ · · · ∪ τk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  One can show that σ is a tube, so Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='3 tells us that for each τi, diamτi(p) is very  small compared to n2|σ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' As the tubes either intersect or are cyclic, one can show this forces  diamσ(p) to also be small, so ασ(p) < n2|σ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  More concretely, suppose that  p ∈  �  Hτi ∩ L (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Note that for all i, |σ| > |τi| + 1 and diamτi(p) ≤ n2(|σ|−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' In case (1), let a, b ∈ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' There  exists some x ∈ τ1 ∩ τ2, so  |pa − pb| ≤ |pa − px| + |px − pb|  ≤ diamτ1(p) + diamτ2(p)  ≤ 2n2(|σ|−1)  < n2(|σ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Hence diamσ(p) < n2|σ|, so by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='3, p /∈ hσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Now we move to case (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Suppose there is a sequence of disjoint tubes τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=', τk such that  for each i there exists xi, yi ∈ τi where xi ≺ yi+1 where we take the indices mod k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Then:  pyi − diamτi(p) ≤ pxi  pxi ≤ pyi+1  pyi+1 ≤ pxi+1 + diamτi+1  Furthermore, since τi and τi+1 are disjoint, |τi| ≤ |σ|−2 and diamτi ≤ n2(|σ|−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Combining  these we get  pyi ≤ pyi+1 + 2n2(|σ|−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Then we have:  py1 ≤ pyi + 2in2(|σ|−2)  and  pyi + 2in2(|σ|−2) ≤ py1 + 2(k + 1)n2(|σ|−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  7  4  5  2  1  3  6  7  8  9  τ  4  5  2  1  3  6  7  8  9  τ  σ  A  B  Maximal Tubing T and tube τ  σ, A, and B labelled  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' An example illustrating the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  A  B  σ  τ  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  If diamA(p) and diamB(p) are small and diamσ(p) is large, then  diamτ(p) is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  These yield  py1 − pyi ≤ 2in2(|σ|−2)  and  pyi − py1 ≤ 2(k − i + 1)n2(|σ|���2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  As i, k − i + 1 ≤ k ≤ n  2, we have |py1 − pyi| ≤ n2(|σ|−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Finally, if zi ∈ τi, zj ∈ τj, then  |pzi − pzj| ≤ |pzi − pyi| + |pyi − py1| + |py1 − pyj| + |pyj − pzj|  ≤ 4n2(|σ|−1)  < n2|σ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Hence diamσ(p) < n2|σ|, and by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='3, p /∈ hσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  □  Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Let T be a maximal tubing of P and let τ /∈ T be a tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Define the  convex hull of τ relative to T by  convT(τ) := min{σ ∈ T | τ ⊂ σ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Let σ = convT(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' T partitions σ into a lower set A and an upper set B where A and B  are either tubes or singletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Furthermore, A and B both intersect τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' See Figure 5 for an  example illustrating this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  The idea of the proof is as follows: Let p = vT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='3, diamA(p) and diamB(p)  are both very small compared to diamσ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Then for any a ∈ A, b ∈ B, |pa − pb| must be  large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' As τ intersects both A and B, diamτ(p) must be large and hence p ∈ hτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' See Figure 6  for an illustration of this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' More precisely, we show that for any i ∈ A, j ∈ B, pj −pi > (n2)|τ|,  which implies that p lies in the interior of hτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  8  Observe that:�  x≺·y  py − px =  �  x≺·y  x,y∈A  (py − px)  �  ��  �  ≤(n2)|σ|−1  < 1  8 (n2)|σ|  +  �  x≺·y  x,y∈B  (py − px)  �  ��  �  ≤(n2)|σ|−1  < 1  8 (n2)|σ|  +  �  x≺·y  x∈A,y∈B  (py − px).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Fix i ∈ A and j ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='3, for any x ∈ A, y ∈ B,  py − px ≤ pj − pi + diamA(p) + diamB(p)  ≤ pj + pi + 2n2(|σ|−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Again, noting that the number of covering relations in σ is at most n2  4 we obtain:  �  x≺·σy  x∈A,y∈B  (py − px) ≤  �  x≺·σy  x∈A,y∈B  (pj − pi + 2(n2)|σ|−1)  ≤ n2  4  �  pj − pi + 2(n2)|σ|−1�  = n2  4 (pj − pi) + 1  2(n2)|σ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Combining all of this we get:  �  x≺·σy  py − px = (n2)|σ|  < 1  8(n2)|σ| + 1  8(n2)|σ| + 1  2(n2)|σ| + n2  4 (pj − pi)  ≤ 3  4(n2)|σ| + n2  4 (pj − pi)  Then (n2)|σ|−1 < (pj − pi) and as |τ| ≤ |σ| − 1, p is in the interior of hτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  □  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' A similar approach for realizing graph associahedra is taken by Devadoss [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  One difference is that Devadoss realizes graph associahedra by cutting off slices of a simplex  whereas we cut off slices of an order polytope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Realizing affine poset cyclohedra  The proofs in the affine case are nearly identical to the finite case with some additional  technical components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' The similarity comes from the fact that Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='3 still applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' We  highlight where the proofs are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Let ˜P be an affine poset of order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Define  C ( ˜P) :=  �  σ⊂P  hσ  and  L ( ˜P) := {p ∈ R  ˜P/C | pi ≤ pj for all i ⪯ j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  where the intersection is over all tubes of ˜P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Note that C ( ˜P) ⊆ L ( ˜P) as if i ≺· j is a  covering relation, then for p ∈ h{i,j}, pi ≤ pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='8 follows as a result of 3 lemmas:  9  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' If T is a maximal tubing, then  vT :=  �  τ∈T  Hτ  is a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' If T is a collection of tubes that do not form a proper tubing, then  �  τ∈T  Hτ ∩ C ( ˜P) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' If T is a maximal tubing and τ /∈ T is a proper tube, then ατ(vT) > n2|τ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' That  is, vT lies in the interior of hτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Let T be a maximal tubing and take any σ ∈ T such that |τ| = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Then restricting to ˜P|σ, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='1 implies that  �  τ∈T  τ⊆σ  Hτ  is a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' However, as T is n-periodic,  �  τ∈T  τ⊆σ  Hτ =  �  τ∈T  Hτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  □  Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' By Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='7, we can assume T is n-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' The proof is almost  identical to the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Define  L ( ˜P) := {p ∈ R  ˜P/C | pi ≤ pj for all i ⪯ j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  and note that  L ( ˜P) ⊆ R  ˜P/C  �  i,j∈ ˜P  i≺·j  h{i,j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Let  p ∈  �  Hτi ∩ L ( ˜P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  We again break into two cases:  (1) There is a pair of non-nested and non-disjoint tubes τ1, τ2 in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  (2) All tubes in T are pairwise nested or disjoint and there is a sequence of disjoint tubes  τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=', τk such that τ1 ≺ · · · ≺ τk ≺ τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  The only difference in the proof occurs in case (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Here, it is possible that there exists  x ∈ τ1 ∪τ2 such that x+n ∈ τ1 ∪τ2 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' In this case, the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='2 still implies  that diamτ1∪τ2(p) ≤ diamτ1(p) + diamτ2(p) ≤ 2n2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' However, |px+n − px| = n2(n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  □  Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Let T be a maximal tubing and τ /∈ T be a proper tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Let p = vT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  We claim that ατ(p) > n2|τ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  The only difference from the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='3 is that τ may not be contained by any  tube in τ so convT(τ) may not be well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' In this case, there exists A ∈ T such that  10  ˆ0  a  b  c  ˆ1  P  ε = 1  3  ε = 1  9  ε =  1  27  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  O(P) as a limit of A (P)  |A| = n, A ∩ τ ̸= ∅, and (A + n) ∩ τ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Here, (A + n) acts the same as B in the finite  case, except the argument is much simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Let i ∈ A ∩ τ, j ∈ (A + n) ∩ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Observe that diamA(p), diam(A+n)(p) ≤ n2n and that  i + n ∈ (A + n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Then  |pj − pi| ≥ (pj − n2n) − pi  ≥ pi+n − pi  = n2(n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Hence diamτ(p) > n2|τ| and by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='3, ατ(p) > n2|τ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Remarks and Questions  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Let (P, ⪯) be a bounded poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' In Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='1, we discuss how O(P) can be  realized as the set of all p ∈ RP such that pˆ0 = 0, pˆ1 = 1, and pi ≤ pj whenever i ⪯ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' We  can similarly realize A (P) as follows: Fix 0 < ε <  1  n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  For a proper tube τ ⊂ P, let  h′  τ = {p ∈ RP | ατ(p) < εn−|τ|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Then A (P) is realized as the intersection over all h′  τ with the hyperplanes  {pˆ0 = 0} and {pˆ1 = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Letting ε → 0, we obtain O(P) as a limit of A (P) as shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' The key piece to the realizations in Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='8 is the linear form  ατ, where ατ acts as an approximation of diamτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' In particular, let τ be a tube and let  p ∈ L (P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Then:  ατ(p) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  ατ(p) = 0 ⇔ p|τ is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  If σ ⊆ τ is a tube, then ασ(p) ≤ ατ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  However, there are many other options for choice of ατ that could fill this role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Some other  options include:  11  (1) Sum over all pairs i ≺ j in τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  ατ(p) =  �  i≺j  i,j∈τ  pj − pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  (2) Let A and B be the set of minima and maxima of the restriction P|τ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  ατ(p) =  �  i≺j  i∈A,j∈B  pj − pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  (3) Fix a spanning tree T in the Hasse diagram of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Let E = {(i, j) | i ≺·T j} be the set  of edges in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  ατ(p) =  �  (i,j)∈E  pj − pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  An advantage of this option is that we would have  diamτ(p) ≤ ατ(p) ≤ (n − 1) diamτ(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  A similar realization can be obtained for each choice of of ατ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Recall that for a simple d-dimensional polytope P, the f-vector and h-vector  of P are given by (f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' , fd) and (h0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' , hd) where fi is the number of i-dimensional faces  and  d  �  i=0  fiti =  d  �  i=0  hi(t + 1)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Postnikov, Reiner, and Williams [12] found a statistic on maximal tubings of graph associa-  hedra of chordal graphs where  �  T  tstat(T) =  �  hiti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  In particular, they define a map T �→ wT from maximal tubings of a graph on n vertices  to the set of permutations Sn such that stat(T) = des(wT), the number of descents of wT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  It would be interesting to find a similar statistic on maximal tubings of poset associahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  For a simple polytope P, one can orient the edges of P according to a generic linear form  and take stat(v) = outdegree(v) [17, §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' It may be possible to use our realization to find  the desired statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Acknowledgements  The author is grateful to Pavel Galashin for his many helpful comments and suggestions  and to Stefan Forcey for fruitful conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  References  [1]  Scott Axelrod and Isadore M Singer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' “Chern-Simons perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' II”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' In: Jour-  nal of Differential Geometry 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='1 (1994), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' 173–213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  [2]  Raoul Bott and Clifford Taubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' “On the self-linking of knots”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' In: Journal of Mathe-  matical Physics 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='10 (1994), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' 5247–5287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  [3]  Michael Carr and Satyan L Devadoss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' “Coxeter complexes and graph-associahedra”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  In: Topology and its Applications 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='12 (2006), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' 2155–2168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  12  [4]  Satyan L Devadoss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' “A realization of graph associahedra”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' In: Discrete Mathematics  309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='1 (2009), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' 271–276.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  [5]  Sergey Fomin and Nathan Reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' “Root systems and generalized associahedra”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' In:  arXiv preprint math/0505518 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  [6]  Pavel Galashin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' “Poset associahedra”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' In: arXiv preprint arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='07257 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  [7]  Mark Haiman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' “Constructing the associahedron”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' In: Unpublished manuscript, MIT  (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  [8]  Pascal Lambrechts, Victor Turchin, and Ismar Voli´c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' “Associahedron, cyclohedron and  permutohedron as compactifications of configuration spaces”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' In: Bulletin of the Belgian  Mathematical Society-Simon Stevin 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='2 (2010), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' 303–332.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  [9]  Guillaume Laplante-Anfossi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' “The diagonal of the operahedra”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' In: Advances in Math-  ematics 405 (2022), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' 108494.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  [10]  Chiara Mantovani, Arnau Padrol, and Vincent Pilaud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' “Acyclonestohedra: when ori-  ented matroids meet nestohedra”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  [11]  Kyle Petersen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Eulerian Numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' isbn: 978-1-4939-3090-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='1007/  978-1-4939-3091-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  [12]  Alex Postnikov, Victor Reiner, and Lauren Williams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' “Faces of Generalized Permuto-  hedra”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' In: Documenta Mathematica 13 (2008), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' 207–273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  [13]  Dev P Sinha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' “Manifold-theoretic compactifications of configuration spaces”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' In: Selecta  Mathematica 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='3 (2004), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' 391–428.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  [14]  Richard P Stanley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' “Two poset polytopes”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' In: Discrete & Computational Geometry  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='1 (1986), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' 9–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  [15]  Jim Stasheff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' “From Operads to ‘Physically’ Inspired Theories”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' In: (Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  [16]  Dov Tamari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' “Mono¨ıdes pr´eordonn´es et chaˆınes de Malcev”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' In: Bulletin de la Soci´et´e  math´ematique de France 82 (1954), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' 53–96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  [17]  G¨unter M Ziegler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Lectures on polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content=' Springer Science & Business Media,  2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='  Department of Mathematics, University of California, Los Angeles, CA 90095, USA  Email address: andrewsack@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='ucla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}
+page_content='edu  13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdFJT4oBgHgl3EQfGyyu/content/2301.11449v1.pdf'}

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+arXiv:2301.03824v1  [astro-ph.EP]  10 Jan 2023
+Draft version January 11, 2023
+Typeset using LATEX preprint style in AASTeX63
+Photosynthetic Fluorescence from Earth-Like Planets around Sun-Like and Cool Stars
+Yu Komatsu,1, 2 Yasunori Hori,1, 2 Masayuki Kuzuhara,1, 2 Makiko Kosugi,1, 2, 3
+Kenji Takizawa,1, 3 Norio Narita,4, 1, 5 Masashi Omiya,1, 2 Eunchul Kim,3
+Nobuhiko Kusakabe,1, 2 Victoria Meadows,6, 7 and Motohide Tamura1, 2, 8
+1Astrobiology Center, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan.
+2National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan.
+3National Institute for Basic Biology, 38 Nishigonaka, Myodaiji, Okazaki, Aichi 444-8585, Japan.
+4Komaba Institute for Science, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan.
+5Instituto de Astrof´ısica de Canarias (IAC), 38205 La Laguna, Tenerife, Spain
+6Department of Astronomy and Astrobiology Program, University of Washington, Box 351580, Seattle, Washington
+98195, USA.
+7NASA Nexus for Exoplanet System Science, Virtual Planetary Laboratory Team, Box 351580, University of
+Washington, Seattle, Washington 98195, USA.
+8Department of Astronomy, Graduate School of Science, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo
+113-0033, Japan.
+(Received August 15; Revised November 12; Accepted November 15, 2022)
+Submitted to ApJ
+ABSTRACT
+Remote sensing of the Earth has demonstrated that photosynthesis is traceable as the
+vegetation red edge (VRE), which is the steep rise in the reflection spectrum of vegeta-
+tion, and as solar-induced fluorescence. This study examined the detectability of bio-
+logical fluorescence from two types of photosynthetic pigments, chlorophylls (Chls) and
+bacteriochlorophylls (BChls), on Earth-like planets with oxygen-rich/poor and anoxic
+atmospheres around the Sun and M dwarfs. Atmospheric absorption, such as H2O, CH4,
+O2, and O3, and the VRE obscure the fluorescence emissions from Chls and BChls. We
+found that BChl-based fluorescence for wavelengths of 1000–1100 nm, assuming the
+spectrum of BChl b-bearing purple bacteria, could provide a suitable biosignature but
+only in the absence of the water cloud coverage or other strong absorbers near 1000 nm.
+The Chl fluorescence is weaker for several reasons, e.g., spectral blending with the
+VRE. The apparent reflectance excess is greatly increased in both Chl and BChl cases
+around TRAPPIST-1 due to fluorescence and stellar absorption lines. This could be a
+promising feature for detecting the fluorescence around ultracool red dwarfs by follow-
+up ground-based observations with high spectral resolution; however, it requires a long
+time around Sun-like stars, even for a LUVOIR-like space mission. Moreover, the simul-
+taneous detection of fluorescence and VRE is key to identifying traces of photosynthesis
+Corresponding author: Yu Komatsu
+yu.komatsu@nao.ac.jp
+
+2
+Komatsu et al.
+because absorption, reflectance, and fluorescence are physically connected. For further
+validation of fluorescence detection, the nonlinear response of biological fluorescence as
+a function of light intensity could be considered.
+Keywords: astrobiology, planets and satellites: atmospheres, planets and satellites: sur-
+faces, planets and satellites: terrestrial planets
+1. INTRODUCTION
+The ultimate goal of characterizing rocky planets is to identify potential biosignatures, spec-
+tral fingerprints of atmospheric gases, and surface features produced by biological activities
+(Des Marais et al. 2002; Schwieterman et al. 2018; Meadows et al. 2018). The simultaneous identifi-
+cation of oxygen, ozone, and methane on rocky habitable planets shows promise as a way to detect
+Earth-like life. Oxygenic photosynthesis produces a unique feature in the reflection spectrum on a
+planetary surface, called the vegetation red edge (VRE), as well as biosignature gases (Kiang et al.
+2007a). The VRE is the steep difference in the reflection spectrum of the surface vegetation around
+700 nm due to chlorophyll (Chl) absorption in the visible region and the large reflectance by cell
+structures in the near-infrared (NIR) region (Gates et al. 1965; Jacquemoud & Baret 1990). Remote
+sensing of the Earth and Earthshine observations provide spectral indices involved in the VRE, such
+as the NDVI, which is a normalized difference in the reflection spectrum of the Earth between the
+visible and NIR wavelength regions. The Moderate Resolution Imaging Spectroradiometer (MODIS)
+onboard NASA’s Terra satellite at 16-day intervals at 500 m and 1 km resolutions shows that the
+NDVI varies from
+0.05 to nearly 0.9, whose upper limit is obtained at a dense forest site during
+the peak growing season (Huete et al. 2002). Whereas remote sensing observes local areas on Earth,
+Earthshine observations provide disk-averaged spectra of the Earth, leading to fruitful insights into
+exoplanet applications. The apparent reflectance change in the Earth’s disk-averaged spectrum due
+to surface vegetation is less than 2% (Monta˜n´es-Rodr´ıguez et al. 2006). The NDVI calculated from
+the Earthshine observations varies up to ∼0.10, depending on different views of the Earth, and is
+reduced by cloud coverage (Tinetti et al. 2006). The application of NDVI to disk-averaged spectra
+assuming Earth-like exoplanets requires caution because remote sensing observes only local areas on
+the Earth to map vegetation. For instance, Livengood et al. (2011) found that additional spectral
+bands to NDVI are required to distinguish between the Earth vegetation and the Moon surface.
+The VRE signals from exoplanets around stars other than a Sun-like star are challenging to predict
+due to the complexity of photosynthetic mechanisms in different light environments. However, the
+VRE on exoplanets may still be recognizable as an anomalous time-varying due to seasonal variability
+of the vegetation, and step-function-like spectroscopic feature at wavelengths different from those on
+the Earth (Seager et al. 2005). Tinetti et al. (2006) proposed that if a three-photon photosynthetic
+scheme were working on exoplanets around M dwarfs, where there was little or no visible light, then
+the red edge of vegetation could also be shifted into the NIR. However, according to Takizawa et al.
+(2017), even around M dwarfs, the evolution of photosynthesis in water may drive a preference for
+using visible light rather than NIR, even after organisms colonize land surfaces. Moreover, the light
+absorption properties of land vegetation could be optimized after long-term adaptive evolution de-
+pending on stellar irradiations as estimated by Lehmer et al. (2021). Anoxygenic photosynthesis as
+performed by organisms such as purple bacteria, is thought to precede the emergence of oxygenic
+
+Photosynthetic fluorescence on Exoplanets
+3
+photosynthesis, whose global effect was characterized by the great oxidation event (∼2.3 billion years
+ago (Ga)). Sanrom´a et al. (2013) discussed the detectability of light reflected from purple bacteria
+with bacteriochlorophyll (BChl) as a photosynthetic pigment. They showed that purple bacteria
+exhibit detectable features, and their VRE peak is redder than higher plants, assuming an Earth-
+like planet before the rise of oxygen.
+In a comprehensive study of different pigment reflectivity,
+Schwieterman et al. (2015) showed that both nonphotosynthetic pigments and photosynthetic pig-
+ments affect the disk-averaged spectra. Furthermore, as for false positive detection, the reflectance
+features of some minerals on the Earth are similar to the VRE ones (Seager et al. 2005; Schwieterman
+2018). Thus, extracting the VRE signal from reflected light should require knowledge of the surface
+environment on an exoplanet and high-resolution spectroscopic observations.
+Fluorescence is another photosynthesis-related phenomenon that could also be a remote-sensing
+biosignature. Fluorescence is one of the de-excitation processes of photosynthetic pigments from the
+excited states to the ground state, along with intersystem crossing and inner conversion. Photosyn-
+thetic organisms on the Earth use Chls or BChls as light-absorbing pigments and electron donors/ac-
+ceptors in the primary reactions of photosynthesis. The photon energy captured by Chls/BChls is
+mainly transferred to the reaction center (RC), which is the pigment-protein complex at the center of
+the photosystem used for photochemical reactions. A part of photon energy is, however, dissipated
+as heat or emitted as fluorescence from light-harvesting antenna systems, which are pigment-protein
+complexes surrounding RC that capture light energy and deliver the energy to the RC. Excess photon
+energy is preferentially removed as heat dissipation, rather than fluorescence. As a result, fluores-
+cence yield tends to be a smaller percentage of the excess energy and fluctuates with the degree
+of the excitation energy transfer (EET) between Chls, and heat dissipation. The fluorescence yield
+of photosynthetic organisms is estimated to be ∼5%, whereas that of free Chls/BChls in organic
+solvents is ∼ 30% (Grimm et al. 2006).
+Plants and other oxygenic phototrophs use two different photosystems in sequence, that is, pho-
+tosystem II (PSII) and photosystem I (PSI). The energy level of the RC of PSII is higher, being
+equivalent to 680 nm, than that of PSI. In general, Chl fluorescence is mainly emitted from PSII
+because the excess light energy in PSI is immediately dissipated as heat. Therefore, the fluorescence
+spectrum of a cell has a peak at 680 nm, and the distribution of fluorescence emission extends to
+wavelengths up to 780 nm. Note that fluorescence emissions at 680 nm under highly concentrated
+Chls conditions, such as a leaf structure, decrease due to reabsorption by peripheral Chls with a
+red-absorption band. Conversely, the six BChls (BChl a, b, c, d, e, and g) used in non-oxygenic
+photosynthetic bacteria, such as purple bacteria, green sulfur and nonsulfur bacteria and heliobac-
+teria (Kiang et al. 2007b), mainly absorb far-red light in vivo. The BChl b in purple bacteria has
+the longest wavelength absorbance (1010 nm) and fluorescence (1050 nm) emissions. However, the
+detailed characteristics of fluorescence from BChls, such as fluorescence yield and its variation in
+light environments, remain poorly understood.
+In contrast to the VRE which tracks the vegetation mass in the remote sensing of the Earth, fluo-
+rescence can be used as an indicator of active photosynthesis. The fluorescence signal emitted from
+the global ground vegetation, which is called solar-induced fluorescence (SIF), can be detected by
+remote sensing from satellites as excess light seen in the absorption of Fraunhofer lines in sunlight
+reflected from the Earth, which is the apparent increase in the reflectance spectrum due to fluores-
+cence (Maier et al. 2004). The observation of SIF is fundamentally challenging because the small SIF
+
+4
+Komatsu et al.
+signal is overwhelmed by large background signals in the reflected sunlight. Then, high-resolution
+spectroscopy utilizes specific wavelengths with large solar absorption, which means the low intensity
+of reflected light. The SIF is observed as the in-filling effect at these wavelengths. This methodology
+works because a large contrast is ensured between the Sun and the reflected light from the Earth at
+specific wavelengths. Thus, SIF has been observed in absorption bands by the Fourier high-dispersion
+spectrometers onboard many environmental satellites (e.g., GOSAT (Hamazaki et al. 2005; Lee et al.
+2013), GOME-2 (Callies et al. 2000), and GOSAT-2 (Nakajima et al. 2012)), which produce the time-
+series SIF map of Earth (Frankenberg et al. 2014; Sun et al. 2018). We can extract information on
+the ground vegetation and atmospheric/surface environment, especially the gross primary production
+(GPP), from the changes in the fluorescence map by calibrating the remote observations with the
+results of local ground observations (Sun et al. 2018). Such as the SIF in Earth observations, the
+detection of photosynthetic fluorescence in a planet around stars will investigate the surface envi-
+ronment and vegetation conditions on exoplanets. High-resolution spectroscopy would be inevitable
+for the exofluorescence detection, and the contrast between a planet and its host star should be high
+enough at specific wavelengths. Biofluorescence, similar to that shown by coral reefs on Earth, has
+been suggested as a new potential biosignature for exoplanets experiencing strong UV radiation from
+F stars (O’Malley-James & Kaltenegger 2018) and M stars (O’Malley-James & Kaltenegger 2019).
+It might work if the fluorescence were emitted very efficiently according to gained photons in their
+habitats. As mentioned above, photosynthetic pigments are a potential emitter of biofluorescence.
+However, the yield and detectability of photosynthetic fluorescence on the surface of exoplanets have
+not yet been examined.
+Finding surface biosignatures on Earth-like exoplanets, including the potential detectability of
+biofluorescence, would be one of the important goals of future astronomy and may become possible
+with future space missions such as the Large UV/Optical/IR Surveyor (LUVOIR) or the Habit-
+able Exoplanet Observatory (HabEx), and next-generation extremely large ground-based telescopes
+(TMT, ELT, and GMT) observing in reflected light. Thus, it is important to quantitatively evaluate
+the detectability of any potential surface biosignature using expected specifications of specific future
+missions.
+This study made the first attempt to investigate the detectability of photosynthetic fluorescence on
+Earth-like exoplanets. The remainder of this paper is structured as follows: Section 2 describes the
+surface vegetation model for an Earth-like planet in the habitable zone and fluorescence emissions
+based on the photoresponse of photosynthetic organisms. Section 3 shows the expected fluorescence
+emissions in the reflected light spectra on an Earth-like planet around an M dwarf or the Sun. In Sec-
+tion 4, we discuss the physiological conditions of photosynthesis that enhance fluorescence emissions
+and its unique features for future detection, including false-positive signals and seasonal changes.
+Additionally, we present the detectability of biofluorescence by a future space-based telescope assum-
+ing the LUVOIR telescope parameters, and the key spectral feature possibly useful for the detection
+by follow-up observations with high-dispersion spectroscopy. In the last section, we summarize our
+paper.
+2. MATERIALS AND METHODS
+We assume that the radiation from a planetary surface is the sum of the reflected light on the
+surface and the fluorescence emission from photosynthesis. The outgoing flux on the surface and at
+
+Photosynthetic fluorescence on Exoplanets
+5
+Figure 1.
+(a) Incident radiation and (b) photon flux density at the top of atmosphere (TOA) of an
+Earth-like planet around the Sun, GJ667C, and TRAPPIST-1. The spectral data of the Sun, GJ667C, and
+TRAPPIST-1 were obtained from Meftah et al. (2018), France et al. (2016), and Lincowski et al. (2018),
+respectively.
+the top of atmosphere (TOA) is given by:
+F ↑
+surface(λ)=F ↓
+surface(λ)R(λ) + Ffluor.(λ),
+(1)
+F ↑
+TOA(λ)=F ↑
+surface(λ)T(λ),
+(2)
+where λ is the wavelength, T(λ) is the atmospheric transmittance (see Section 2.1), R(λ) is the
+surface reflectance of a planet (see Section 2.3), F ↑
+surface(λ) is the upward flux from a planetary
+surface, F ↑
+TOA(λ) is the reflected flux at the TOA, and Ffluor.(λ) is the net fluorescence emission from
+photosynthesis. F ↓
+surface(λ) = F ↓
+TOA(λ)T(λ) is the downward flux from the planetary atmosphere to
+the surface, where F ↓
+TOA is the incident flux from a host star at the TOA (see Figure 1 and Section
+2.1 below).
+We neglect the effects of thermal emission in all the cases and Rayleigh scattering
+in most cases, as both processes contribute little radiation to our spectral region of interest (600-
+1000 nm). The transmittance T(λ) in the atmosphere of an Earth-like planet through geological
+evolution was obtained from Rugheimer & Kaltenegger (2018), which was calculated by a 1D coupled
+radiative/convective-photochemical model for a planetary atmosphere (see also Pavlov & Kasting
+2002; Kasting & Ackerman 1986; Segura et al. 2005).
+2.1. Stellar Radiation
+Two nearby M dwarfs, GJ667 C and TRAPPIST-1, have candidate planets in a habitable zone (HZ).
+We considered fluorescence emissions from photosynthesis on an Earth-like planet in an HZ around
+GJ667 C, TRAPPIST-1, and the Sun. We extracted the incident stellar flux from high-resolution
+spectral data for the Sun (Meftah et al. 2018), GJ667 C (France et al. 2016), and TRAPPIST-1
+(Lincowski et al. 2018). The incident flux F ↓
+TOA received by an Earth-like planet around GJ667 C,
+and TRAPPIST-1 is scaled by the current location of GJ667C c, and TRAPPIST-1e. GJ667 C, and
+TRAPPIST-1 are modeled as M1V and M8V stars. Figure 1 shows the incident flux received by an
+Earth-like planet at the TOA around the Sun, GJ667 C, and TRAPPIST-1.
+2.2. Fluorescence from Photosynthesis
+
+le3
+tons/s/m2/μm)
+lel
+Sun
+(un/zw/m)
+2.0
+GJ667C
+TRAPPIST-1
+0.8
+1.5
+(mmol-pho
+0.6
+Density
+1.0
+0.4
+Flux
+0.5
+Photon Flux
+0.2
+0.0
+0.0
+0
+250
+500
+750
+10001250
+15001750
+2000
+0
+250
+500
+750
+10001250150017502000
+Wavelength(nm)
+Wavelength[nm]6
+Komatsu et al.
+Fluorescence emissions from a planetary surface F
+′
+fluor. are expressed as:
+F
+′
+fluor.(λ) = scvπF std
+fluor. × f(λ),
+(3)
+where cv is the surface coverage of vegetation (see Section 2.3), s is the scaling factor from the stan-
+dard observed fluorescence emission reflecting the photosynthetic activity, and F std
+fluor. is the standard
+fluorescence intensity from vegetation based on field measurements. The spectral shape of fluores-
+cence emissions from a photosynthetic organism at wavelength λ is defined by f(λ). In this study,
+F std
+fluor. = 1.0 (W m−2 µm−1 sr−1) (Du et al. 2019; Yao et al. 2021) and s = 0, 1, 5, and 10. The net
+fluorescence intensity Ffluor. is calculated by considering acquired photons at the habitat using F
+′
+fluor.
+in Equation (3) as:
+Ffluor.(λ) = χ
+χ0
+F
+′
+fluor.(λ),
+(4)
+χ ≡
+�
+n(λ)σ(λ)dλ,
+(5)
+χ0 ≡
+�
+nsun,ref.(λ)σchls(λ)dλ,
+(6)
+where χ is the light absorption efficiency, n(λ) is the photon flux density at the planetary surface, and
+σ(λ) is the absorption coefficient of a photosynthetic pigment. χ0 represents the standard absorption
+efficiency on Earth. The subscript chls on σ(λ) represents chlorophylls (see Chl:abs in Figure 2).
+nsun,ref.(λ) is the photon flux density on the surface of the Earth from the reference solar spectral
+irradiance at an air mass of 1.5 (National Renewable Energy Laboratory), which corresponds to a
+typical irradiance for Earth vegetation.
+We considered an incident flux from a star under two sky conditions, a clear sky and 60% cloud
+cover, to estimate the reflectance at the TOA and χ on the ground, in accordance with the setup
+for the simulation. We assumed the clear sky condition, if not specified, and the cloud condition
+appeared only in Section 4.3.1 (Figure 12). In the cloudy condition, 60% of the radiation is reflected
+in three kinds of clouds, and 40% of the radiation reaches the ground. For the clouds, we assumed that
+40% are low water clouds, 40% are high water clouds, and 20% are high ice clouds (Gao & Kaufman
+2003) at 1, 6, and 12 km altitude, respectively. To model Earth-like conditions, the effect of Rayleigh
+scattering in a planetary atmosphere was also considered in the cloudy condition using a previously
+described empirical approach by Bucholtz (1995) (see Appendix A for more details).
+Equation (4) indicates that F
+′
+fluor. is linearly scaled to the number of incoming photons that are
+absorbed by chlorophylls at the planetary surface.
+In other words, chlorophylls can emit strong
+fluorescence if the spectral shapes of n(λ) and σ(λ) match well.
+Note that n(λ) is exactly the
+same as F ↓
+surface(λ), and its unit is shown in Figure 1(b) (F ↓
+TOA in the figure). This treatment in
+Equations (3) and (4) can be applied to the relationship between the incoming photons and the
+photons emitted as fluorescence on an Earth-like planet around various stars other than the Sun.
+Figure 2 shows the normalized spectra of fluorescence f(λ) and absorption coefficient σ(λ) for Chls
+and BChls. The peak wavelength of f(λ) is red-shifted from that of σ(λ), which is called the Stokes
+shift (Lakowicz 2006). There are two absorption bands in the σ(λ) of chlorophylls: the B band
+(known as the Soret band) in the short-wavelength region and the Q band in the long-wavelength
+region. The primary fluorescence emission is derived from the Q absorption band. In this study,
+
+Photosynthetic fluorescence on Exoplanets
+7
+Figure 2. Fluorescence (f(λ): solid curves) and photoabsorption (σ(λ): dashed curves) spectra for chloro-
+phylls (Chl: black) and bacteriochlorophylls (BChl: red). The absorption coefficient of chlorophylls in units
+of cm2 µg−1 was obtained from Feret et al. (2008). The fluorescence spectrum is expressed by the Gaussian
+functions given in Frankenberg et al. (2012) and Guanter et al. (2010). For bacteriochlorophylls, f(λ) and
+σ(λ) adopt those of the LH1–RC complex of a bacteriochlorophyll b containing purple photosynthetic bac-
+teria (Magdaong et al. 2016). The nondimensional absorption spectrum for BChl is normalized at the peak
+value in the longest absorption band, the Q band, of the Chl. Two fluorescence spectra are normalized at
+their peak values.
+we modeled f(λ) for Chls as the superposition of two Gaussian distributions (Frankenberg et al.
+2012; Guanter et al. 2010) with means of 680 nm (PSII) and 740 nm (PSI and PSII). σ(λ) for Chl
+uses the model vegetation with chlorophylls (σchls(λ)) (Feret et al. 2008). We obtained f(λ) and
+σ(λ) for BChls from the spectral data for the LH1–RC complex, the supramolecular complex of
+the light-harvesting core antenna (LH1), and the RC in a bacteriochlorophyll b containing purple
+photosynthetic bacteria (see Figure 3 in Magdaong et al. 2016)). Note that we used only σ(λ) in
+the Q band for calculating χ and χ0 because free Chls and BChl–protein complexes in each solution
+affect each spectrum in the B band to different degrees.
+2.3. Surface Vegetation
+To determine the detectability of vegetation fluorescence, we use two leaf models for our exper-
+iments: one which assumes the reflectance spectrum and fluorescence of standard chlorophyll and
+another that uses a scaled version of the spectrum of bacteriochlorophyll. The reflectance of a planet
+is expressed as R(λ) = �
+i ciri(λ), where i denotes the surface type, ci is the fraction of the surface
+coverage of type i, and ri the reflectance of type i. We obtained the reflection spectra for various
+surface types including vegetation, ocean, and coast from the USGS Digital Spectral Library and
+the ASTER Spectral Library (Baldridge et al. 2009). The detailed compositions used in this paper
+are summarized in Table 1. The reflectance of the surface vegetation rv is estimated from radiation
+transfer calculations for a modeled leaf (Jacquemoud & Baret 1990; Feret et al. 2008), using σ(λ)
+over all the wavelengths shown in Figure 2. Figure 3 shows the reflectance of a Chl-based leaf (“stan-
+
+Absorption coefficient (cm2/μg)
+Normalized fluorescence
+0.16
+1.0
+0.14
+0.8
+0.12
+0.10
+0.6
+0.08
+0.4
+0.06
+0.04
+0.2
+0.02
+0.00
+0.0
+400
+600
+800
+1000
+1200
+Wavelength (nm)
+Chl: abs
+Chl: fl
+BChl: abs
+BChl: fl8
+Komatsu et al.
+Figure 3. The reflectance of vegetation estimated from radiation transfer calculations for two leaf models:
+Chl (“standard”) and BChl (“hypothetical”)
+(Jacquemoud & Baret 1990; Feret et al. 2008).
+The light
+absorption spectrum for Chls and BChls uses σ(λ) in Figure 2.
+dard”) and a BChl-based leaf (“hypothetical”). In the latter case, we assumed the vegetation on a
+different planet has a photosynthetic pigment whose optical property is the same as BChl exhibiting
+the VRE in the longer wavelength region as shown in Figure 3. As the input to the radiative transfer
+calculations, we used the absorption spectra of Chl (Feret et al. 2008) and BChl (Magdaong et al.
+2016). The unitless absorption spectrum for BChl is normalized at the peak in that for Chl, unlike
+the calculations of χ and χ0 in Section 2.2. As shown in Figure 3, both Chl- and BChl-based leaves
+show a large reflectance (i.e., the VRE) in the wavelength ranges around 700–750 and 1000–1100 nm.
+The green bump around 500 nm is observed in the reflectance for Chl, and larger and broader bumps
+are observed from ∼500 to 950 nm for BChls by the larger difference in the wavelength between the
+B and Q bands than observed for Chl. Like many kinds of photosynthetic organisms, the organisms
+with BChl could have acquired accessory pigments such as carotenoids (Cars) that absorb photons
+with wavelengths between the B and Q bands of Chl (Cogdell 1978). The effective light absorption by
+accessory pigments can suppress the increase in reflectance. With or without accessory pigments, the
+bump for BChl does not affect fluorescence emissions in the wavelength (see Figures 7, 8, and 10).
+The low reflectance from ∼500 to 700 nm (1000 nm), due to the light absorption by Chls (BChls),
+affects the reflectance of the planet. The degree of reduction in the overall planetary reflectance
+varies depending on the surface coverage by vegetation.
+3. RESULTS
+We considered three fluorescence cases on an Earth-like planet at different stages of atmospheric
+evolution around the Sun, GJ667 C, and TRAPPIST-1 for different surface biosignatures: Earth-like
+(Chl) vegetation, hypothetical BChl-based vegetation, and biological fluorescence without any surface
+vegetation. Our models for the surface compositions, vegetation, fluorescence types, and atmospheric
+compositions, i.e., transmittance, are summarized in Table 1. Mod-earth corresponds to the surface
+condition for the Modern Earth, leading to a lesser contribution of fluorescence emissions than in the
+other two cases. The veg-only models are considered optimistic conditions for fluorescence emissions
+where vegetation covers the whole planetary surface. The veg-land models, with 70 % ocean, 2%
+coast, and 28% land covered with the vegetation, lie between the mod-earth and veg-only models. As
+mentioned in Section 2, we considered two leaf models for land vegetation: Chl-based vegetation and
+
+0.5
+Reflectance
+0.4
+0.3
+0.2
+Chl
+0.1
+BChl
+500
+1000
+1500
+2000
+Wavelength (nm)Photosynthetic fluorescence on Exoplanets
+9
+BChl-based vegetation. For the atmospheric compositions of an Earth-like planet, we adopted the
+Modern Earth model at 0.0 Ga (oxygen-rich atmosphere), the Paleoproterozoic Earth model at 2.0 Ga
+(oxygen-poor atmosphere), and the Archean Earth model at 3.9 Ga (anoxic atmosphere) (see Table 1
+in Rugheimer & Kaltenegger 2018). As an extreme case, we assumed the presence of photosynthetic
+bacteria with BChl spread over the land and ocean on an Archean-Earth-like planet with no surface
+vegetation. We assumed a clear sky for all atmospheric conditions in Section 3.
+Model name
+Surface compositions
+Surface vegetation
+Fluorescence type
+T(λ)
+cv
+veg-only 0C
+Chl surf.
+Chl fluor.
+0.0 Ga
+veg-only 2C
+100% vegetation
+2.0 Ga
+1.00
+veg-only 0B
+BChl surf.
+BChl fluor.
+0.0 Ga
+veg-only 2B
+2.0 Ga
+veg-land 0C
+Chl surf.
+Chl fluor.
+0.0 Ga
+veg-land 2C
+70% ocean, 2% coast
+2.0 Ga
+0.28
+veg-land 0B
+and 28% vegetation
+BChl surf.
+BChl fluor.
+0.0 Ga
+veg-land 2B
+2.0 Ga
+mod-earth 0C
+Chl surf.
+Chl fluor.
+0.0 Ga
+mod-earth 2C
+70% ocean, 2% coast
+2.0 Ga
+0.168
+mod-earth 0B
+and 28 % mixed land
+BChl surf.
+BChl fluor.
+0.0 Ga
+mod-earth 2B
+(incl. 16.8% vegetation)
+2.0 Ga
+anoxic B
+70% ocean, 2% coast and
+-
+BChl fluor.
+3.9 Ga
+0.72
+28% mixed land at 3.9 Ga
+Table 1. Surface composition, vegetation, its fluorescence types, and atmospheric transmittance (T(λ))
+for all the cases in this paper. Mixed land is composed of 60% vegetation (16.8% in total), 15% snow, 9%
+granite, 9% basalt, and 7% sand (Baldridge et al. 2009); mixed land at 3.9 Ga means the land model of the
+Archean Earth at 3.9 Ga, which is composed of 35% basalt, 40% granite, 15% snow, and 10% sand. Chl
+surf. and BChl surf. correspond to reflection spectra of Chl and BChl in Figure 3, respectively. The spectral
+shapes of fluorescence emissions f(λ) for Chl fluor. and BChl fluor. correspond to the fluorescence spectra
+of Chl and BChl in Figure 2, respectively; their intensities Fflour. are scaled in Equations (3) and (4). cv is
+given by the relationship between the surface coverage of vegetation and the fluorescence emission. s={0,
+1.0, 5.0, 10.0}. We obtained T(λ) at 0.0, 2.0, and 3.9 Ga from Rugheimer & Kaltenegger (2018).
+3.1. Case-1: Planets with Earth-Like Vegetation
+In case-1, Earth-like vegetation (Chl) emits fluorescence on the surface of an Earth-like planet. The
+fluorescence emissions from chlorophyll are visible at the wavelengths from 650 to 800 nm, as shown
+in Figure 2. To determine the contribution of fluorescence from planets, the reflectance is defined
+as F ↑
+TOA(λ)/F ↓
+TOA(λ) and calculated. Figures 4 and 5 show the reflectance of an Earth-like planet
+with the Modern Earth’s atmosphere (0.0 Ga) and an oxygen-poor atmosphere (2.0 Ga), respectively.
+The O2, O3, CH4, and H2O absorption features in the atmosphere are imprinted in the reflectivity
+in the visible–NIR wavelengths from 600–800 nm. The oxygen-poor-atmosphere models show less
+conspicuous patterns in the reflectance profile in the 700 to 750 nm wavelength region. The reflec-
+tivity between 600 and 700 nm is nearly constant but increases with decreasing surface coverage of
+
+10
+Komatsu et al.
+vegetation. The VRE is observed as the steep rise in the reflectance from 700 to 750 nm (also see
+Figure 3), whereas the reflectance excess due to fluorescence is quite small, even in optimistic condi-
+tions (veg-only models). Note that the red curve with 1Ffluor. around the Sun in the mod-earth model
+(Figure 4), corresponding to the modern earth fluorescence, is hardly seen. Around TRAPPIST-1,
+however, sharp increase in the reflectance around 770 nm is due to the strong absorption of potassium
+in the stellar atmosphere. As a result, we observed similar features in the light reflected from an
+Earth-like planet with different atmospheric compositions around TRAPPIST-1 (see Section 4.3.2
+for further discussion).
+Figure 6 shows the reflectance excess due to fluorescence emissions on an Earth-like planet with
+the Modern Earth’s atmosphere. Atmospheric absorptions, such as H2O, O2, and O3, weaken the
+Gaussian features in the fluorescence emissions from an Earth-like planet around the Sun.
+The
+fluorescence from chlorophylls around 740 nm is less pronounced for a planet around M dwarfs than
+one around the Sun because of weaker radiation flux in the wavelength region of 700–750 nm (see
+Figure 1). In addition, a sudden increase in reflectance due to the VRE obscures the fluorescence
+emission around 740 nm (see Figures 4 and 5). As a result, the Chl fluorescence around 680 nm
+emitted from PSII on an Earth-like planet would be the most promising feature for detection (see
+Figure 2). Note that nonphotochemical quenching processes can decrease the fluorescence intensity
+around 680 nm, and the fluorescence emission is further reduced by the reabsorption of photons within
+the canopy (Porcar-Castell et al. 2021).
+3.2. Case-2: Planets with Bacteriochlorophylls-Based Vegetation
+In case-2, BChl-based vegetation, as the major photosynthetic pigment, covers the surface of a
+planet. The BChls are assumed to emit the same degree of fluorescence intensity as the Earth’s
+vegetation. As shown in Figure 2, fluorescence from BChls occurs in the wavelength range from 1000
+to 1100 nm. In contrast to case-1, fluorescence emissions with 5 and 10Ffluor. show strong features
+around 1050 nm in almost all conditions in Figures 7 and 8. Identifying the fluorescence on the Earth’s
+vegetation level (≲ Ffluor.) is still challenging even in the optimistic case, that is, (a) veg-only 0B.
+The reflectivity between 1000 and 1050 nm becomes slightly higher for mod-earth models with less
+surface vegetation coverage. As shown in Figure 9, the BChl organisms efficiently absorb photons and
+emit fluorescence with less absorption and scattering in the planetary atmosphere. The fluorescence
+emissions from BChls that we assumed are invulnerable to blending with the steep increase in the
+reflectance by the VRE. As a result, we found a more significant fluorescence contribution to the
+reflected light in case-2.
+Atmospheric properties, such as chemical compositions and cloud coverage, change the fluorescence
+profile. The water absorption is weak for wavelengths from 1000 to 1100 nm. If the major absorption
+bands of a photosynthetic pigment lie in wavelengths longer or shorter than 1000–1100 nm, the pres-
+ence of water vapor in the atmosphere complicates the detection of fluorescence emissions. A strong
+absorption due to CH4 in an oxygen-poor atmosphere also hides fluorescence near 1000 nm (see the
+GJ667C models in Figure 8). The BChl organisms bearing BChl b and their Stokes shift are ideal for
+detecting fluorescence in wavelengths longer than the characteristic wavelength of fluorescence from
+Chls. Thus, fluorescence in the wavelength range of 1000 -1100 nm could be a suitable biosignature
+for photosynthetic organisms, such as bacteriochlorophylls, on planetary surfaces unless they coexist
+with strong absorbers near 1000 nm.
+
+Photosynthetic fluorescence on Exoplanets
+11
+Figure 4. Reflectance of an Earth-like planet with the Modern Earth’s atmosphere (0.0Ga) around the
+Sun, GJ667C, and TRAPPIST-1. The three colors represent the reflected light from a planet with Ffluor.
+(s = 1: red), 5Ffluor. (s = 5: blue), and 10Ffluor. (s = 10: green), where Ffluor. is the fluorescence emission
+from chlorophylls observed on the Earth. No-fluorescence emission models are also indicated by gray lines.
+We assumed Earth-like vegetation (chlorophylls) covers the planetary surface (see Table 1 for model details).
+The reflectance is defined here as F ↑
+TOA(λ)/F ↓
+TOA(λ), where F ↑
+TOA(λ) is the light reflected from the ground
+at the top of atmosphere (TOA), and F ↓
+TOA(λ) is the flux at TOA induced by stars. For each case around
+TRAPPIST-1, the reflectance with a logarithmic scale is also shown as the inset plot.
+The VRE with a sharp rise in the reflectance is observed in the wavelength range from 1050 to
+1100 nm in case-2, as shown in Figure 3. Reflectance excess due to BChl fluorescence is 0.01–0.05
+for the Modern Earth atmosphere models (see Figure 9), whereas that due to the VRE is 0.4–0.5
+(0.1–0.15) for veg-only models (veg-land and mod-earth models). Bacteriochrolophylls’ fluorescence
+causes a slight increase in reflectance around 1000 -1100 nm compared to the VRE. Such nonprominent
+fluorescence emission with a Gaussian shape in the wavelength different from the VRE feature can
+be extracted from the reflectance profile using data processing such as principal component analysis
+(PCA). Photosynthetic organisms different from those around the Sun are expected to exhibit VRE
+and fluorescence features in different wavelengths. Thus, not only spectral features due to atmospheric
+molecules but also the simultaneous detection of the VRE and the fluorescence will help identify
+traces of photosynthesis on an exoplanet. Probably, when we found a possible signal of VRE, the
+fluorescence would be useful for further validation, because the VRE signal is stronger than the
+fluorescence one.
+3.3. Case-3: Anoxic World (without VRE)
+
+(a) veg-only 0C
+(b) veg-land oC
+(c) mod-earth 0C
+0.8
+0.25
+0.25
+10 Fflour.
+0.20
+0.20
+5 Fflour.
+1 Fflour.
+0.15
+0.15-
+0.4
+O Fflour.
+0.10
+0.10
+0.05
+0.05-
+0.0
+0.0Q
+0.0Q
+009
+650
+700
+750
+800
+600
+650
+700
+750
+800
+600
+650
+700
+750
+800
+0.8
+0.25
+0.25
+0.20
+0.20
+J667C
+0.15
+0.15
+0.4-
+0.10
+0.10
+0.2
+0.05
+0.05
+0.0
+0.0Q
+0.00
+600
+650
+700
+750
+800
+600
+650
+700
+750
+800
+009
+650
+700
+750
+800
+0.8
+0.25
+0.25
+0.2010
+0.2010
+TRAPPIST-1
+0.1510-
+0.15
+0.4
+600
+700
+800
+600
+700
+800
+600
+700
+800
+0.10
+0.10
+20.2
+0.05
+0.05
+0.0
+0.0Q
+0.0Q
+600
+650
+700
+750
+800
+600
+650
+700
+750
+800
+600
+650
+700
+750
+800
+wavelength (nm)
+wavelength (nm)
+wavelength (nm)12
+Komatsu et al.
+Figure 5. The same as Figure 4, but for an Earth-like planet with an oxygen-poor atmosphere (2.0 Ga).
+Figure 6. Reflectance excess due to chlorophyll fluorescence emissions on an Earth-like planet with the
+Modern Earth’s atmosphere.
+
+(a) veg-only 0C
+(b) veg-land 0C
+(c) mod-earth 0C
+0.10
+10 Fflour.
+0.03
+0.03
+5 Fflour.
+1 Fflour.
+0.02
+0.02
+Sun
+0.05
+0.01
+0.01
+0.00
+0.00
+0.00
+650
+700
+750
+800
+650
+700
+750
+600
+800
+650
+600
+600
+700
+750
+800
+0.10
+0.03
+0.03
+GJ667C
+0.02
+0.02
+0.05
+0.01
+0.01
+0.0Q
+0.0Q
+0.00
+600
+650
+700
+750
+800
+600
+650
+700
+750
+800
+600
+650
+700
+750
+800
+0.10-
+0.03
+0.03
+10-
+Difference in
+10-
+Reflectance
+TRAPPIST-1
+0.02
+10-5
+10-5
+600
+700
+800
+600
+700
+800
+600
+700
+800
+0.05
+0.01
+0.01
+0.0Q
+0.0Q
+0.0Q
+600
+650
+700
+750
+800
+600
+650
+700
+750
+800
+600
+650
+700
+750
+800
+wavelength (nm)
+wavelength (nm)
+wavelength (nm)(a) veg-only 2C
+(b) veg-land 2C
+(c) mod-earth 2C
+0.8
+0.25
+0.25
+10 Fflour.
+9 0.6
+0.20
+0.20
+5 Fflour.
+0.15
+0.15-
+sun
+1 Fflour.
+0.4
+O Fflour.
+0.10.
+0.10-
+0.05
+0.05-
+0.9
+0.00
+0.00
+009
+650
+700
+750
+800
+600
+650
+700
+750
+800
+600
+650
+700
+750
+800
+0.8
+0.25
+0.25
+0.20
+0.20
+J667C
+0.15
+0.15
+0.4
+0.10.
+0.10
+0.2
+0.05
+0.05.
+0.0
+0.0Q:
+0.00
+600
+650
+700
+750
+800
+600
+650
+700
+750
+800
+600
+650
+700
+750
+800
+0.8
+0.25
+0.25
+10
+100
+0.20+
+0.20{0-1
+TRAPPIST-1
+10
+0-1
+0.15↓0
+-2
+0.4
+600
+700
+800
+600
+700
+800
+600
+700
+800
+0.10
+0.10
+0.2
+0.05
+0.05
+0.0
+0.0Q:
+0.0Q
+600
+650
+700
+750
+800
+600
+650
+700
+750
+800
+600
+650
+700
+750
+800
+wavelength (nm)
+wavelength (nm)
+wavelength (nm)Photosynthetic fluorescence on Exoplanets
+13
+Figure 7. The same as Figure 4 but for the reflectance of a planet covered with bacteriochlorophyll-based
+vegetation.
+Figure 8. The same as Figure 7, but for a planet with an oxygen-poor atmosphere (2.0 Ga).
+
+(a) veg-only 2B
+(b) veg-land 2B
+(c) mod-earth 2B
+0.6
+0.20
+0.20
+10 Fflour.
+0.5
+5 Fflour.
+0.15
+0.15
+0.4
+sun
+1 Fflour.
+0.3
+0.10
+0.10
+O Fflour.
+0.2
+0.05
+0.05
+0.1
+0.0
+0.00
+0.00
+900
+950
+1000 1050 1100 1150 1200
+900
+950
+1000 1050 1100 1150 1200
+900
+950 1000 1050 1100 1150 1200
+0.6
+0.20
+0.20
+0.5
+nce
+0.15
+0.15
+J667C
+0.4
+ctal
+0.3
+0.10
+0.10
+D
+G
+0.2
+0.05
+0.05
+0.1
+0.0
+0.00
+0.00
+900
+950
+1000 1050 1100 1150 1200
+900
+950 1000 1050 1100 1150 1200
+900
+950 1000 1050 1100 1150 1200
+0.6
+0.20
+0.20
+0.5
+ince
+0.15
+0.15
+TRAPPIST-1
+0.4
+Reflectar
+0.3
+0.10
+0.10
+0.2
+0.05
+0.05
+0.1
+0.0
+0.00
+0.00
+900
+950 1000 1050 1100 1150 1200
+006
+950 1000 1050 1100 1150 1200
+006
+950 1000 1050 1100 1150 1200
+wavelength (nm)
+wavelength (nm)
+wavelength (nm)(a) veg-only OB
+(b) veg-land OB
+(c) mod-earth 0B
+0.6
+0.20
+0.20
+10 Fflour.
+0.5
+5 Fflour.
+0.15
+0.15
+0.4
+sun
+1 Fflour.
+0.3
+0.10
+0.10
+O Fflour.
+0.2
+0.05
+0.05
+0.1
+0.0
+0.00
+0.00
+900
+950 1000 1050 1100 1150 1200
+900
+950 1000 1050 1100 1150 1200
+900
+950 1000 1050 1100 1150 1200
+0.6
+0.20
+0.20
+0.5
+nce
+0.15
+0.15
+0.4
+ctal
+0.3
+0.10-
+0.10-
+D
+G
+0.2
+0.05
+0.05
+0.1
+0.0
+0.0Q
+0.00
+900
+950 1000 1050 1100 1150 1200
+900
+950 1000 1050 1100 1150 1200
+900
+950 1000 1050 1100 1150 1200
+0.6
+0.20
+0.20
+0.5
+ince
+0.15
+0.15
+TRAPPIST-1
+0.4
+Reflectar
+0.3
+0.10
+0.10
+0.2
+0.05
+0.05
+0.1
+0.0
+0.00
+0.00
+900
+950 1000 1050 1100 1150 1200
+006
+950 1000 1050 1100 1150 1200
+006
+950 1000 1050 1100 1150 1200
+wavelength (nm)
+wavelength (nm)
+wavelength (nm)14
+Komatsu et al.
+Figure 9. Reflectance excess due to bacteriochlorophyll fluorescence emissions on an Earth-like planet with
+the Modern Earth’s atmosphere.
+In case-3, an Earth-like planet has the same reduced atmosphere as the Archean Earth at 3.9 Ga.
+Anoxic bacteria with photosynthetic pigments such as bacteriochlorophylls may spread over the
+surface of a planet with a CO2-rich atmosphere. Anoxic bacteria are assumed to live in the ocean
+and coast (i.e., cv = 0.72) and emit only fluorescence whose intensity is comparable to the standard
+emission from land plants, without the distinct reflectance of a vegetation surface.
+Fluorescence
+emissions from anoxic bacteria adopt those from bacteriochlorophylls on the Earth. Figure 10 shows
+the reflectance of an Archean-Earth-like planet with BChl-based bacteria. In the reflection spectra, a
+strong water absorption appears around 950 and 1150 nm. The relatively high reflectance across the
+wavelength range is mainly from the light reflected by the land. We observe fluorescence emissions
+in the wavelength range between 1000 and 1100 nm owing to the lack of light reflected from BChl-
+bearing oceanic bacteria, including the VRE feature. Intense absorption in the stellar atmosphere
+enhances the apparent reflectance of a planet around TRAPPIST-1 (see also Figures 4 and 5, and
+Section 4.3.2).
+4. DISCUSSION
+This study demonstrated reflectance with photosynthetic fluorescence on an Earth-like planet
+around the Sun and two M dwarfs. This section reviews the biological processes of photosynthe-
+sis and then considers the future detection of biofluorescence on an exoplanet. In Section 4.1, we
+discuss the possible physiological conditions that enhance the fluorescence emissions on a planet
+based on our understanding of Chl fluorescence. In Section 4.2, we discuss the possible false positive
+or negative detection of fluorescence (Section 4.2.1), and the potential usage of the nonlinear photore-
+
+(a) veg-only OB
+(b) veg-land OB
+(c) mod-earth 0B
+0.15
+0.04
+0.04
+10 Fflour.
+nce
+5 Fflour.
+0.03
+0.03-
+0.10
+1 Fflour.
+sun
+0.02
+0.02
+0.05
+0.01
+0.01
+0.00:
+0.00
+0.00
+900
+1000
+1100
+1200
+900
+1000
+1100
+1200
+900
+1000
+1100
+1200
+0.15
+0.04
+0.04
+0.03
+0.03
+J667C
+0.02
+0.02
+G
+0.01
+0.01
+0.00
+0.00
+0.00
+900
+1000
+1100
+1200
+900
+1000
+1100
+1200
+900
+1000
+1100
+1200
+0.15
+0.04
+0.04
+Difference in
+TRAPPIST-1
+ince
+0.03
+0.03-
+0.10
+Reflectal
+0.02
+0.02-
+0.05
+R
+0.01
+0.01-
+0.00
+0.00
+0.00
+900
+1000
+1100
+1200
+900
+1000
+1100
+1200
+900
+1000
+1100
+1200
+wavelength (nm)
+wavelength (nm)
+wavelength (nm)Photosynthetic fluorescence on Exoplanets
+15
+Figure 10. The reflectance of an Earth-like planet with an anoxic atmosphere and no land vegetation
+(anoxic B).
+
+anoxic B
+0.15
+Reflectance
+0.10
+Sun
+0.05
+0.0Q
+900
+1000
+1100
+1200
+0.15
+Reflectance
+10 Fflour.
+GJ667C
+0.10
+5 Fflour.
+1 Fflour.
+0.05
+O Fflour.
+0.00
+006
+1000
+1100
+1200
+0.15
+Reflectance
+TRAPPIST-1
+0.10
+0.05
+0.00
+006
+1000
+1100
+1200
+wavelength (nm)16
+Komatsu et al.
+sponse in fluorescence yield to excitation light intensity to distinguish between biofluorescence and
+the false positive/negative signals of fluorescence (Section 4.2.2). Finally, in Section 4.3, we show the
+fluorescence detection with telescopes. We present the detectability of fluorescence from an Earth
+twin around a Sun-like star u sing the noise model for a LUVOIR-A-like mission (Section 4.3.1), and
+the remarkable enhancement in the reflectance due to the absorption lines of stars, which could be
+a promising feature for detection by high-dispersion spectroscopy, especially around ultracool stars
+(Section 4.3.2).
+4.1. Possible Physiological Conditions for Supporting Fluorescence Detection
+This study adopted the typical fluorescence spectrum of Chl-containing plants and LH1–RC purified
+from BChl b-bearing purple bacteria. The fluorescence spectrum of the LH1–RC complex suspended
+in buffer solution was measured under laboratory conditions with a low concentration of LH1–RC in
+the solution to avoid the reabsorption of fluorescence. Cells having LH1-RC in vivo would result in
+an ∼ 50 nm shift in the spectral peak wavelength toward longer wavelengths under dense conditions,
+because the reabsorption of fluorescence reduces the shorter-wavelength part of fluorescence. A red-
+shifted fluorescence spectrum should still be observable because it is located within the atmospheric
+window. For the fluorescence intensity of vascular plants on the ground, we referred to the standard
+value (Ffluor.) for the fluorescence model on exoplanets in our simulations. The possible detection of
+fluorescence emissions on exoplanets would require ≳ 5Ffluor. with BChl (see Figures 7, 8, and 10).
+There are four potential factors that increase the fluorescence yield in photosynthetic organisms from
+the biophysical viewpoint of photosynthetic studies on existing phototrophs on the Earth:
+1. Increasing Chl/BChl concentration per land area
+A high concentration of Chls and BChls enhances their fluorescence intensity. In general, the
+Chl/BChl concentration in a cell increases for capturing as many photons as possible under
+low light conditions. Fluorescence increases linearly with Chl/BChl concentration when cell
+density is low. In contrast, the fluorescence intensity reaches a saturation level in highly dense
+environments due to the reabsorption of fluorescence by cells (Du et al. 2017).
+2. Small spectral overlap between absorption and fluorescence
+The large separation between the main absorption band and its fluorescence band increases
+the fluorescence intensity of concentrated cells. In photosynthetic organisms, the excitation
+energy is transferred between Chls, and the Chl fluorescence tends to be emitted from long-
+wavelength Chls (LWC), which has the reddest absorption band in a photosystem because
+the excess excitation energy is easily trapped at the lowest energy level. A redshift in the
+peak wavelength of fluorescence and a blueshift in absorption, which can be caused by the
+modification of the vibronic interactions of pigments between surrounding proteins and solvent,
+reduce the spectral overlap between fluorescence and absorption. The fluorescence emission
+from LWCs is red-shifted to over 50 nm from that of bulk Chls in some conditions. Although
+most plants have a small amount of LWCs in PSII and the Chl fluorescence is absorbed well
+under high Chl concentrations, far-red absorbable LWC contributing to PSII has been reported
+in some eukaryote algae (Fujita & Ohki 2004; Wilhelm & Jakob 2006; Kotabov´a et al. 2014;
+Wolf et al. 2018; Kosugi et al. 2020). These algae show a significant fluorescence emission at
+far-red-light wavelengths (700–800 nm) at room temperature, and some of them decrease the
+overlap (Fujita & Ohki 2004; Kosugi et al. 2020).
+
+Photosynthetic fluorescence on Exoplanets
+17
+3. Low photosynthetic efficiency
+Photon loss in photosynthetic processes reduces the photon yield of fluorescence. Excitation
+yield in PSII has increased throughout the evolutionary processes of photosystems. For ex-
+ample, the increase in light use efficiency in oxygenic photosynthesis on Earth was achieved
+by changing the light-harvesting antenna protein from the membrane superficial phycobili-
+some in cyanobacteria to the light-harvesting Chl binding protein in eukaryotic algae. Fur-
+thermore, the subsequent modification of LHCs achieved a higher photosynthetic quantum
+yield in the evolution process. The maximum excitation yield in PSII of vascular plants is
+estimated to be ∼ 0.9, whereas that of green algae and cyanobacteria is ∼ 0.8 and ∼ 0.6,
+respectively (Schuurmans et al. 2015). Suppose phototrophs on an exoplanet are in the early
+stage of evolution. In that case, the expected fluorescence yield may be high to compensate for
+the low efficiency of photon yields in primitive photosynthesis.
+4. Suppression of heat dissipation
+Photon loss by the heat dissipation in photosynthetic pigments suppresses the photon yield of
+fluorescence. Heat dissipation occurs in the vibrational relaxation of excited pigment molecules,
+Chls, or accessory pigments such as carotenoids. Additionally, light-dependent protection mech-
+anisms to dissipate the excess light energy as heat are inherent in all the cyanobacteria, algae,
+and plants. The efficiency of heat dissipation largely depends on the molecular configuration
+and the environment of pigments binding to proteins. The energy conversion rate from light to
+heat in photosystems is crucial in estimating photosynthetic fluorescence on other planets.
+Therefore, the fluorescence yield in photosynthetic pigments should fluctuate over time due to pho-
+tosynthetic activity and heat dissipation.
+4.2. Further Identification for Confirming Photosynthetic Fluorescence
+4.2.1. Potential false positive/negative of biological fluorescence detection from exoplanets
+Photosynthetic pigments on an exoplanet may be different from those on Earth, and the wavelength
+relevant to fluorescence emission from exovegetation remains to be unknown. A possible fluorescence
+signal on other planets can be a false positive or negative detection of biological activities. Poten-
+tial main sources causing false positive/negative could be surface reflectance or fluorescence from
+minerals on exoplanets.
+Both Chl and BChl fluorescence in our study can be contaminated by
+mineral fluorescence, but it is not plausible to expect the fluorescent minerals to cover a fraction
+of a planetary surface comparable to Earth’s vegetation as far as our knowledge of the Earth’s
+environment.
+Recently, solar-induced mineral luminescence (SML) has been extracted from SIF
+data obtained by remote sensing of the Earth (K¨ohler et al. 2021). They revealed that about 10%
+of non-vegetated areas are weakly luminescent and speculated that luminescence came from some
+spots covered by carbonate with Mn2+ and was comparable to SIF (or Chl fluorescence). However,
+those areas are negligible on the planetary scale. On the other hand, mineral fluorescence could
+pollute, to an extent, fluorescence in near-infrared, which includes the BChl fluorescence. For in-
+stance, silicate (e.g., pyroxene and olivine) shows a prominent absorption around 1000 nm caused by
+Fe2+ (Bishop et al. 2019; Klima et al. 2011; Sunshine & Pieters 1998). Its fluorescence could appear
+in a slightly longer wavelength from the absorption, whose energy corresponds to the Stokes shift, like
+other near-infrared fluorescent materials (Jackson et al. 2021; Selvaggio et al. 2020). While there are
+
+18
+Komatsu et al.
+a variety of fluorescent minerals (e.g., fluorite, calcite, corundum), we do not deny the possibility that
+the unexpectedly strong mineral fluorescence could be observed on exotic planets such as a carbide
+exoplanet (Allen-Sutter et al. 2020) whose surface could be covered by diamond with lattice defects,
+e.g., due to nitrogen-vacancy center (Schirhagl et al. 2014). To understand potential fluorescence fea-
+tures from surface components of an exoplanet, e.g., rocks and minerals, characterizing atmospheric
+features is helpful. Besides, as mentioned so far, the simultaneous detection of vegetation reflectance
+(VRE) and fluorescence features could help identify photosynthesis.
+4.2.2. Nonlinear photoresponse in photosynthesis
+Photosynthetic organisms regulate metabolic processes to maximize the use of available photons
+under light conditions and emit biological fluorescence by converting light energy via photochemical
+reactions. The nonlinear response of the fluorescence yield to the excitation light intensity would
+be a clue to finding the presence of photosynthetic organisms. If a planet is in an elliptical orbit,
+the incident flux received by the planet from its host star varies with time. Fluorescence emissions
+from nonbiological processes increase with incident light intensity. In contrast, a saturation level
+of the fluorescence intensity from biological activities, such as photosynthesis, exists because the
+quantum yields of Chl fluorescence vary according to the light environment and atmospheric CO2
+concentrations.
+The quantum yields of Chl fluorescence are primarily involved in the reduction
+states of electron acceptors of photosystems for electron transports and excitation energy quenching
+by photoprotection mechanisms (see Genty et al. 1989; Krause & Weis 1991; Baker 2008). A sudden
+intense light can induce the reduction in the electron acceptors of PSII, where oxidation of water to
+generate O2 occurs as a primary step in photosynthesis. The presence of photoprotection mechanisms
+also modulates the quantum yields of Chl fluorescence. When dark- or dim-light-adapted leaves are
+suddenly irradiated with intense light, Chl fluorescence quantum yields rapidly increase by up to five
+times. Accordingly, the relationship between fluorescence yield and excitation light intensity (i.e.,
+the number of absorbed photons) provides a hint to explore the origin of fluorescence on a planet.
+4.3. Detectability of Biological Fluorescence by Future Telescopes
+4.3.1. The Earth-Sun System as an Earth Twin in a LUVOIR-A-Like Mission
+We investigated the detectability of fluorescence from an Earth twin around a Sun-like star at 10 pc
+from the Earth, assuming a LUVOIR-A-like space telescope. Figure 11 presents the simulated spectra
+of a second Earth around a Sun-like star at 10 pc with the biological fluorescence. We applied the
+noise model used in Robinson et al. (2016) and Kopparapu et al. (2021), which accounts for planet
+photons, stellar photon noise, and background noise, e.g., zodi, exozodi, read-out, and dark current
+noises with the throughput assuming the LUVOIR-A telescope. The parameters and the formalism
+used in this paper are presented in Appendix B. Figures 11(a–c) show the results of the most optimistic
+model for the fluorescence signal (veg-only 0B) from the Earth-Sun system observed from 10 pc with
+a 15 m space telescope.
+The original data are the same as those of the Sun in Figure 7(a). In
+Figure 11(a), Fp/Fs observed at the telescope for each wavelength bin is shown as solid lines, with
+the random noise as the 1σ error bars for each bin, in 9000 hours of exposure time, where Fp is
+the reflected light from the planet and Fs is the starlight. Figure 11(b) depicts a magnification of
+the spectrum in Figure 11(a). Some error bars are outside the solid line, but the spectral feature of
+fluorescence emission is recognizable for each case in the figure. Figure 11(c) shows the SNR with
+the same observation time as that in Figure 11(a). The difference between 0 and 5 Ffluor. is larger
+
+Photosynthetic fluorescence on Exoplanets
+19
+than 1σ. To detect the fluorescence with 3σ error, ∼ 50000 hours of exposure time are required,
+and with 5σ, ∼ 100000 hours, ten years, are expected (not shown in figures). Thus, fluorescence
+detection would require years for observation, even by the LUVOIR-A-like space telescope, and it
+is extremely challenging to observe one target. In less optimistic models, namely, the veg-land 0B
+model around the Sun in Figure 7(b), the detection of fluorescence signals is even more challenging,
+as shown in Figure 11(d). As discussed in Section 3.2, the fluorescence in mod-earth 0B is difficult
+to identify. Moreover, cloud coverage obscures the VRE features as well as atmospheric features
+on exoplanets (Seager et al. 2005; Tinetti et al. 2006; Kaltenegger et al. 2007). The reflectance in
+Figure 12 indicates how clouds suppress the fluorescence signal. Even in the most optimistic model,
+the fluorescence in the reflectance is significantly reduced and can hardly be observed. In the mod-
+earth model, it is impossible to identify the fluorescence signals. The only possible way to observe
+surface vegetation with significant cloud coverage, except for atmospheric gases, would be the VRE
+(∼0.1 in reflectance in the optimistic model). Thus, the existence of water clouds that are expected
+in Earth-like planets with surface water seems to be critical for fluorescence detection. However,
+around TRAPPIST-1, as the relevant argument was shown in Session 4.3.2, we found that the Chl
+fluorescence in the K I lines was insensitive to the coverage by Earth clouds, which could be an
+advantage in the Chl detection over BChl one.
+The fluorescence feature would be poorly determined with 900 hours of exposure time with 1σ
+errors, whereas the VRE feature can be identified. Even for a LUVOIR-A-like space telescope, an
+enormous observational time would be needed to identify the fluorescence in addition to the VRE
+with more confidence for detecting traces of photosynthesis. We also investigated the detectability of
+fluorescence by a space telescope with a different diameter. A 6 m space telescope is recommended for
+future space missions, according to Astro2020 Decadal Survey. With a 6-m diameter, ∼ 300,000 hours
+of observation time are required to identify fluorescence. When we adopt a 30 m space telescope with
+1σ errors, the required exposure time is reduced to ∼ 800 hours. Furthermore, one of the background
+noises, i.e., the readout noise, can be suppressed with data processing because of increasing reads in
+an exposure as implemented for H2RG infrared detectors (e.g., Brandt et al. 2017; Kuzuhara et al.
+2018). When the readout noise is assumed to be zero all over the wavelengths, the required observation
+times are reduced to ∼ 250,000, ∼ 7,000 and ∼ 500 hours with the 6-, 15-, and 30-m diameters.
+4.3.2. Apparent Enhancement in Fluorescence around Ultracool Stars and Possible Detection with
+High-Dispersion Spectroscopy
+Figure 13 shows the contribution of fluorescence around three host stars. Around TRAPPIST-1 the
+apparent enhancement in reflectance induced by fluorescence is significant compared to around the
+other two stars because TRAPPIST-1 has strong absorption features spanning the wavelengths of
+the fluorescence. Within the TRAPPIST-1 stellar absorption features, reflected light from the planet
+is reduced, allowing the fluorescence emission to become a much larger fraction of the outgoing
+flux (reflected + fluorescence) at these wavelengths. This is analogous to the methodology of SIF
+detection with remote sensing observations and the retrieval processes by determining how much the
+fluorescence influences the Fraunhofer lines (Maier et al. 2004). These spectroscopic features may be
+widely used for fluorescence detection around ultracool stars.
+Figure 13(a) shows that the reflectance is highly enhanced due to the absorption lines of K I in
+the stellar spectrum of TRAPPIST-1, which is not affected by water clouds (Figure 12). The degree
+of enhancement for each line depends on the atmospheric compositions of an Earth-like planet. Fig-
+
+20
+Komatsu et al.
+Figure 11. Simulated spectrum with the biological fluorescence on a second Earth around a Sun-like star
+at 10 pc from the Earth, assuming a LUVOIR-A-like space telescope. (a–c) The results from the veg-only
+0B model and (d) Fp/Fs with the veg-land 0B model. (a) Fp/Fs with 9000 hours of observation time. The
+solid line shows Fp/Fs and the error bar indicates the noise at each wavelength. (b) A magnification of
+Fp/Fs in (a). (c) The SNR in (a).
+ure 13(b,c) presents a spiky feature due to absorption of FeH and VO, as commonly observed around
+ultracool stars. Therefore, observing the possible fluorescence signal with high spectral resolution
+using extremely large ground telescopes would be worthwhile.
+5. CONCLUSIONS
+In this paper, we explored fluorescence from photosynthesis as a biosignature on an exoplanet for
+future observations in great detail and identified the situations in which the signal could be enhanced,
+and the regions of the spectrum where fluorescence from chlorophylls and bacteriochlorophylls could
+be most detectable for Earth-like planets around different stars. We also described how we could
+enhance the possibility to more definitively detect the action of photosynthesis. For direct imaging
+observations, however, we found that the detection of fluorescence emissions would be extremely
+challenging to observe and especially not feasible for the planned 6m space telescope. More details
+are provided as follows.
+We considered fluorescence emissions from Chl- and BChl-based vegetation in a clear-sky condition
+on an Earth-like planet around the Sun and two M dwarfs (GJ667 C and TRAPPIST-1). Chl- and
+BChl-based leaves show a VRE in wavelengths around 700–750 and 1000–1100 nm. The fluorescence
+emissions from Chls and BChls occur at wavelengths from 650 to 800 nm and 1000 to 1100 nm, cor-
+responding to the longest Q absorption band of each pigment. The two peaks of Chl fluorescence
+
+1e-9
+1e-10
+1.0
+(a)
+(b)
+0.8
+2.0
+10 Fflour.
+1.5
+5 Fflour.
+F
+0.4
+O Fflour.
+1.0
+0.2
+0.5
+0.0
+0.0
+006
+950
+1000 1050 1100 1150 1200
+980
+1000
+1020
+1040
+1060
+Wavelength [nm]
+Wavelength [nm]
+1e-10
+3.5
+(C)
+d)
+3.0
+2.5
+101
+FS 2.0
+SNR
+ 1.5
+10 Fflour.
+5 Fflour.
+1.0
+1 Fflour.
+0.5
+O Fflour.
+100.
+0.0
+900
+950
+1000 1050 1100 1150 1200
+900
+950
+1000 1050 1100 1150 1200
+Wavelength [nm]
+Wavelength[nmlPhotosynthetic fluorescence on Exoplanets
+21
+Figure 12. The effect of cloud on the reflectance with veg-only 0B and 0C models. The models are the
+same as the veg-only 0B in Figure 7 and the veg-only 0C in Figure 4 but with cloud coverage.
+at 680 and 740 nm arise from the PSII and PSI, respectively. Thus, atmospheric absorption bands,
+such as H2O, CH4, O2, and O3, and the VRE could be overlapped with the fluorescence emissions
+from Chls and BChls. Chl fluorescence emission from PSI is blended with the steep VRE feature.
+Fluorescence emitted from PSII on an Earth-like planet is the most promising feature for observation,
+but it may also be reduced by nonphotochemical quenching processes and reabsorption of photons by
+surrounding Chls. Conversely, the fluorescence emitted from BChls is not suppressed by the sharp
+increase in the reflectance due to the VRE and atmospheric absorption by, for example, water va-
+por, except for CH4 absorption around 1000 nm. Therefore, the BChl fluorescence in the wavelength
+range of 1000–1100 nm, rather than Chl fluorescence, may be a more promising biosignature from
+photosynthetic organisms on a planetary surface. In both cases of Chl- and BChl-based vegetation,
+the simultaneous detection of the VRE and fluorescence is significant for identifying photosynthetic
+activity on an exoplanet, because we do not know exactly what kind of vegetation exists in the planet
+in principal and we need more information for further validation to identify the trace of photosyn-
+thesis. If BChl-bearing photosynthetic bacteria inhabit water without any leaf or tree structures,
+the fluorescence spectrum is the only surface reflectance feature that can be used to access such
+underwater photosynthetic organisms, although the fluorescence signal would be reduced according
+to the opacity of overlying liquid water.
+Based on our understanding of photosynthesis, the intensity of fluorescence is lower in photosyn-
+thetic bacteria compared to land plants. Here, we presented four factors that enhance the fluorescence
+emission for possible detection of biological fluorescence on an exoplanet: (1) increase in Chl/BChl
+concentration per land area, (2) small overlap of absorption and fluorescence spectrum, (3) low
+
+veg-only OB
+veg-only 0C
+0.5
+1.2
+0.4
+1.0
+0.3
+0.8
+Sun
+0.6
+0.2
+0.4
+0.1
+0.2
+0.0
+0.Q:
+900
+950 1000 1050 1100 1150 1200
+720
+740
+760
+780
+800
+0.5
+1.2
+ 0.4
+1.0
+10 Fflour.
+nc
+J667C
+0.8
+5 Fflour.
+0.6
+1 Fflour.
+efl
+0.4
+O Fflour.
+R 0.1
+0.2
+0.0
+0.0
+006
+950 1000 1050 1100 1150 1200
+720
+740
+760
+780
+800
+0.5
+1.2
+0.4
+TRAPPIST-1
+1.0
+nc
+0.8
+0.6
+0.2
+efl
+0.4
+R
+0.1
+0.2
+0.0
+0.0
+900
+950 1000 1050 1100 1150 1200
+720
+740
+760
+780
+800
+wavelength (nm)
+wavelength (nm)22
+Komatsu et al.
+Figure 13. The apparent enhancement of fluorescence in reflectance due to stellar absorption around the
+three template stars: (a) veg-only 0C model (Figure 4), (b) veg-only 0B model (Figure 7), and (c) anoxic B
+model (Figure 10).
+photosynthetic efficiency, and (4) suppression of heat dissipation. This study assumed a linear pho-
+toresponse of fluorescence to excitation light intensity. If a planet is on a large elliptical orbit and
+the telescope has sufficient sensitivity to temporally resolve changes in fluorescence as a function of
+time, the nonlinear photoresponse from the biological fluorescence can be identified. Assuming a
+LUVOIR-A-like mission, an enormous duration (around 9000 hours) would be required to detect the
+BChl fluorescence emission, whose fluorescence yield is 5–10 times larger than that of vegetation on
+Earth in the optimistic cases for an Earth-Sun twin at a distance of 10 pc from the Earth. In addition,
+the cloud coverage significantly affects the detection of fluorescence as well as other spectral features
+because the cloud more strongly obscures fluorescence emissions than the VRE feature. Interestingly,
+the fluorescence in the reflectance was found to be remarkably enhanced in all three cases around
+TRAPPIST-1 because of its strong absorption in the stellar atmosphere, like the SIF detection by
+remote sensing using Fraunhofer lines. The reflectance excess due to K I absorption and VO/FeH
+absorption can be a promising feature for characterizing the fluorescence around ultracool stars in
+Chl and BChl cases. Note that Chl fluorescence in K I lines was still prominent with water clouds.
+Thus, one of the most important future works would be the mock observation assuming a 30
+m class ground-based telescope to investigate how the apparent enhancement in reflectance due
+to stellar absorption could help the fluorescence detection around ultracool stars. In addition, to
+better support the future detection of fluorescence emissions on an exoplanet, further studies are
+required from various perspectives. For example, planetary spectra for a wide range of atmospheric
+and surface conditions consistent with biological fluorescence emission should be estimated and tested
+
+(a) veg-land oC
+(b) veg-land OB
+(c) anoxic B
+0.08
+0.20
+0.7-
+10 Fflour.
+5 Fflour.
+0.06
+0.15
+0.5
+1 Fflour.
+Sun
+0.04
+0.10.
+0.3
+O Fflour.
+0.02
+0.05
+0.1
+0.9
+0.00-
+0.00.
+766
+767
+768
+769
+770
+771
+1000
+1020
+1040
+1060
+1000
+1020
+1040
+1060
+0.08
+0.20
+0.7.
+0.06-
+0.15
+J667C
+0.04-
+0.10.
+0.3
+G
+0.02
+0.05
+0.1
+0.9
+0.00
+0.00
+767
+770
+766
+768
+769
+771
+1000
+1020
+1040
+1060
+1000
+1020
+1040
+1060
+0.5
+0.08
+0.20
+ 0.4
+TRAPPIST-1
+Reflectance
+0.06-
+0.15
+0.3
+0.04-
+0.10.
+0.2
+0.02
+0.05-
+0.1
+0.9
+0.00
+0.00.
+766
+767
+768
+769
+770
+771
+1000
+1020
+1040
+1060
+1000
+1020
+1040
+1060
+wavelength (nm)
+wavelength (nm)
+wavelength (nm)Photosynthetic fluorescence on Exoplanets
+23
+using radiation transfer calculations because our studies considered still-limited conditions. Moreover,
+we need to conduct simulations on how the fluorescence is observed on an exoplanet when a global SIF
+map data from remote sensing of the Earth are applied. Also, experimental validation of prominent
+NIR fluorescence emissions is needed in some species of photosynthetic organisms and conditions.
+ACKNOWLEDGMENTS
+We would like to thank one anonymous reviewer for constructive comments to improve the paper.
+We also thank Tatsuya Miyauchi, Haruki Oshio, Yu Someya, Tomoki Kiyono, and Masanori Takeda
+for fruitful discussions at NIES on SIF detection by remote sensing, which led to the draft idea of
+this study, and Kouki Hikosaka (Tohoku University) and Hibiki Noda (NIES) for further discussions
+and for introducing SIF identification by remote sensing. The data for the LUVOIR noise model was
+helpfully provided by Geronimo Villanueva and Ravi Kopparapu (NASA/Goddard). Y.H. and N.N.
+were supported by a Grant-in-Aid for Scientific Research on Innovative Areas (JSPS KAKENHI
+grant number 18H05439).
+PyAstronomy (https://github.com/sczesla/PyAstronomy) was used in
+mock observations assuming a space telescope. In several cases, numerical data were extracted from
+figures in published papers using WebPlotDigitizer (https://automeris.io/WebPlotDigitizer/).
+APPENDIX
+A. EMPIRICAL RAYLEIGH SCATTERING
+The effect of Rayleigh scattering is implemented empirically as follows (Bucholtz 1995):
+τR(λ)=βs(λ)Ts
+Ps
+� z′
+0
+P(z)
+T(z)dz,
+(A1)
+where τR is the Rayleigh optical depth at altitude z′; T(z) and P(z) are the temperature and pressure
+at z, respectively. We adopted the T − P profile in the U.S. standard atmosphere 1976 from 0 to
+60 km to compute the Rayleigh scattering cross-section in the atmosphere of an Earth-like planet.
+The actual T − P profile in the atmosphere of an Earth-like planet around a star other than the Sun
+is quite different from that in the Earth’s atmosphere. Rayleigh scattering, however, has a negligible
+effect on the transmittance at wavelengths from 600 to 1100 nm (≈ 6 % in transmittance at 600 nm,
+reducing with increasing wavelength, and then < 1 % at 1100 nm for an Earth-like planet around the
+Sun, for instance), which is closely related to the fluorescence from Chls and BChls. Ts and Ps are
+the temperature and pressure at standard conditions on Earth, respectively (Ts = 288.15 K and Ps
+= 1013.25 mbars). The total Rayleigh volume-scattering coefficient βs is expressed as:
+βs(λ) = Aλ−B−Cλ−D/λ,
+(A2)
+where the coefficients A, B, C, and D are empirically determined (see Table 3 in Bucholtz (1995)).
+B. LUVOIR NOISE MODEL
+We implemented a noise model assuming a LUVOIR-A-like mission. The formalism and the pa-
+rameters are based on Robinson et al. (2016), but, as shown in Table 2, we updated some parameters
+
+24
+Komatsu et al.
+Parameter
+Description
+Adopted Value
+D
+Mirror Diameter
+6, 15, 30 m
+C
+Raw Contrast
+10−10
+R
+Instrumental spectral resolution
+70
+TTele
+Accounts for light lost due to contamination
+0.95
+and inefficiencies in the main collecting area
+Tread
+Read-out efficiency
+0.75
+TQE
+Raw quantum efficiency
+0.9
+fpa
+Fraction of planetary light that falls within photometric aperture
+1
+X
+Width of photometric aperture as multiple of λ/D
+0.61 arcsec
+Nez
+Number of Exozodis
+4.5
+De−
+Dark current (UVIS/NIR)
+3E-5/2E-3 e−/s
+Re−
+Read noise per pixel (UVIS/NIR)a
+0/2.5 e−
+θIWA
+Inner working angle of the coronagraph as multiple of λ/D
+3
+λ0
+Diffraction limit at the wavelength
+500 nm
+Table 2. Parameters for simulations based on a LUVOIR-A-like mission.
+aTaken from the Planetary Spectrum Generator for LUVOIR/A-VIS and A-NIR, which is maintained by
+NASA (https://psg.gsfc.nasa.gov/instrument.php).
+(with several treatments) following Kopparapu et al. (2021) for our simulations with the LUVOIR-A
+telescope.
+The total noise in the observation Ctotal is calculated by:
+Ctotal =Cp + Cs + Cb,
+(B3)
+where Cp is the number of planet photons, Cs is the stellar photon noise (leakage through the
+coronagraph), and Cb is the background noise, which is the sum of zodi Cz, exozodi Cez, dark current
+CD, and readout noise CR. The internal thermal noise is ignored because the thermal contribution is
+negligible in our wavelengths of interest. Note that the noise in Equation B3 corresponds to variance
+rather than the standard deviation. The noise count is expressed as:
+Cnoise =
+�
+Cp + Cs + 2Cb
+(B4)
+where the double Cb accounts for the on-off observation with and without the planet. The on-off
+observation corresponds to the subtraction of point spread functions of a central star. S/N for each
+wavelength λ is defined by:
+S/N = Cp
+Cnoise
+.
+(B5)
+The Fp and Fs are now defined to be the reflected light from a planet and the stellar flux acquired
+by the telescope at a wavelength (bin) λ. When observing Fp/Fs, the 1σ error at λ is given as:
+σ(λ)= Fp
+Fs
+1
+S/N.
+(B6)
+
+Photosynthetic fluorescence on Exoplanets
+25
+The end-to-end throughput for planetary fluxes is calculated as:
+Ttotal =TTeleTcorToptTreadTQE,
+(B7)
+where TTele is an account for light lost due to contamination and inefficiencies in the main collecting
+area, Tread is the read-out efficiency, and TQE is the raw quantum efficiency for the detector. The
+coronagraphic Tcor and the optical Topt throughputs are the same as in Figure 9 in Kopparapu et al.
+(2021).
+We updated the formalism on noise from zodis, exozodis, and readout as follows; In Robinson et al.
+(2016), the spectral shape of zodis (exozodis) was assumed to be equal to that of the Sun (the host
+star). Instead, we explicitly adopt the normalized reflectance on solar zodis, ˜R⊙,λ, in the model
+to better account for the zodical light in a exoplanetary system. We calculate ˜R⊙,λ by tracing the
+spectral data from observations of the zodical light (see Figure 8 in Kawara et al. (2017) and Figure
+10 in Tsumura et al. (2010)) with the normalization in the V band. Using ˜R⊙,λ, the noise from zodis
+is expressed as:
+Cz = πλ2D2
+4hcR
+F⊙,λ(1au)
+F⊙,V (1au)
+˜R⊙,λF0,V 10−Mz,V /2.5TtotalΩ∆texp,
+(B8)
+where F⊙,λ is the solar flux density at λ, F⊙,V is the solar flux density in the V band, h is the
+Planck constant, c is the speed of light, Mz,V = 23 mag arcsec−2 is the V -band zodical-light surface
+brightness, and ∆texp is the exposure time. The circular photometry aperture size is expressed as
+Ω = π(Xλ/D)2. Assuming the exozodis’s reflectance to be the same as ˜R⊙,λ, the noise from exozodis
+is written as:
+Cez = πλ2D2
+4hcR
+�1au
+r
+�2 Fs,λ(1au)
+Fs,V (1au)
+Fs,V (1au)
+F⊙,V (1au)
+˜R⊙,λF0,V Nez10−Mez,V /2.5TtotalΩ∆texp,
+(B9)
+where Fs,λ is the stellar flux density at λ, Fs,V is the stellar flux density in the V band, and r is the
+distance between the planet and the parent star. Mez,V = 22 mag arcsec−2 is the V -band exozodical
+light surface brightness. Even if the original treatment of exozodical light is adopted, our results
+do not significantly vary. We calculate the read-out noise (CR) to be CR = NpixNreadR2
+e− instead of
+CR = NpixNreadRe− in Robinson et al. (2016) to more realistically incorporate the noise propagation,
+where Npix is the number of contribution pixels, Nread is the number of reads at each observation,
+and Re− is the read noise count.
+REFERENCES
+Allen-Sutter, H., Garhart, E., Leinenweber, K.,
+et al. 2020, Planet. Sci. J., 1, 39
+Baker, N. R. 2008, Annu. Rev. Plant Biol., 59, 89
+Baldridge, A., Hook, S., Grove, C., & Rivera, G.
+2009, Remote Sensing of Environment, 113, 711,
+doi: https://doi.org/10.1016/j.rse.2008.11.007
+Bishop, J. L., Bell III, J. F., Bell, J., & Moersch,
+J. E. 2019, Remote Compositional Analysis:
+Techniques for Understanding Spectroscopy,
+Mineralogy, and Geochemistry of Planetary
+Surfaces, Vol. 24 (Cambridge University Press)
+
+26
+Komatsu et al.
+Brandt, T. D., Rizzo, M., Groff, T., et al. 2017,
+Journal of Astronomical Telescopes,
+Instruments, and Systems, 3, 048002,
+doi: 10.1117/1.JATIS.3.4.048002
+Bucholtz, A. 1995, Applied Optics, 34, 2765
+Callies, J., Corpaccioli, E., Eisinger, M., Hahne,
+A., & Lefebvre, A. 2000, ESA bulletin, 102, 28
+Cogdell, R. J. 1978, Philosophical Transactions of
+the Royal Society of London. B, Biological
+Sciences, 284, 569
+Des Marais, D. J., Harwit, M. O., Jucks, K. W.,
+et al. 2002, Astrobiology, 2, 153
+Du, S., Liu, L., Liu, X., et al. 2019, Sensors, 19,
+3009
+Du, S., Liu, L., Liu, X., & Hu, J. 2017, Remote
+Sensing, 9, 911
+Feret, J.-B., Fran¸cois, C., Asner, G. P., et al. 2008,
+Remote sensing of environment, 112, 3030
+France, K., Loyd, R. P., Youngblood, A., et al.
+2016, The Astrophysical Journal, 820, 89
+Frankenberg, C., O’Dell, C., Berry, J., et al. 2014,
+Remote Sensing of Environment, 147, 1
+Frankenberg, C., O’Dell, C., Guanter, L., &
+McDuffie, J. 2012, Atmospheric Measurement
+Techniques, 5, 2081
+Fujita, Y., & Ohki, K. 2004, Plant and cell
+physiology, 45, 392
+Gao, B.-C., & Kaufman, Y. J. 2003, Journal of
+Geophysical Research: Atmospheres, 108
+Gates, D. M., Keegan, H. J., Schleter, J. C., &
+Weidner, V. R. 1965, Applied optics, 4, 11
+Genty, B., Briantais, J.-M., & Baker, N. R. 1989,
+Biochimica et Biophysica Acta (BBA)-General
+Subjects, 990, 87
+Grimm, B., Porra, R. J., R¨udiger, W., Scheer, H.,
+et al. 2006
+Guanter, L., Alonso, L., G´omez-Chova, L., et al.
+2010, Journal of Geophysical Research:
+Atmospheres, 115
+Hamazaki, T., Kaneko, Y., Kuze, A., & Kondo, K.
+2005, in Enabling sensor and platform
+technologies for spaceborne remote sensing, Vol.
+5659, SPIE, 73–80
+Huete, A., Didan, K., Miura, T., et al. 2002,
+Remote sensing of environment, 83, 195
+Jackson, C. T., Jeong, S., Dorlhiac, G. F., &
+Landry, M. P. 2021, iScience, 24, 102156
+Jacquemoud, S., & Baret, F. 1990, Remote
+sensing of environment, 34, 75
+Kaltenegger, L., Traub, W. A., & Jucks, K. W.
+2007, The Astrophysical Journal, 658, 598
+Kasting, J. F., & Ackerman, T. P. 1986, Science,
+234, 1383
+Kawara, K., Matsuoka, Y., Sano, K., et al. 2017,
+Publications of the Astronomical Society of
+Japan, 69
+Kiang, N. Y., Segura, A., Tinetti, G., et al. 2007a,
+Astrobiology, 7, 252
+Kiang, N. Y., Siefert, J., & Blankenship, R. E.
+2007b, Astrobiology, 7, 222
+Klima, R. L., Dyar, M. D., & Pieters, C. M. 2011,
+Meteorit. Planet. Sci., 46, 379
+K¨ohler, P., Fischer, W. W., Rossman, G. R., et al.
+2021, Geophys. Res. Lett., 48
+Kopparapu, R., Arney, G., Haqq-Misra, J.,
+Lustig-Yaeger, h., & Villanueva, G. 2021, The
+Astrophysical Journal, 908, 164
+Kosugi, M., Ozawa, S.-I., Takahashi, Y., et al.
+2020, Biochimica et Biophysica Acta
+(BBA)-Bioenergetics, 1861, 148139
+Kotabov´a, E., Jareˇsov´a, J., Kaˇna, R., et al. 2014,
+Biochimica et Biophysica Acta
+(BBA)-Bioenergetics, 1837, 734
+Krause, G., & Weis, E. 1991, Annual review of
+plant biology, 42, 313
+Kuzuhara, M., Hirano, T., Kotani, T., et al. 2018,
+in Ground-based and Airborne Instrumentation
+for Astronomy VII, Vol. 10702, International
+Society for Optics and Photonics, 1070260
+Lakowicz, J. R. 2006, Principles of fluorescence
+spectroscopy (Springer)
+Lee, J.-E., Frankenberg, C., van der Tol, C., et al.
+2013, Proceedings of the Royal Society B:
+Biological Sciences, 280, 20130171
+Lehmer, O. R., Catling, D. C., Parenteau, M. N.,
+Kiang, N. Y., & Hoehler, T. M. 2021, Frontiers
+in Astronomy and Space Sciences, 8,
+doi: 10.3389/fspas.2021.689441
+Lincowski, A. P., Meadows, V. S., Crisp, D., et al.
+2018, The Astrophysical Journal, 867, 76
+Livengood, T. A., Deming, L. D., A’hearn, M. F.,
+et al. 2011, Astrobiology, 11, 907
+Magdaong, N. C. M., Niedzwiedzki, D. M.,
+Goodson, C., & Blankenship, R. E. 2016, The
+Journal of Physical Chemistry B, 120, 5159
+Maier, S. W., G¨unther, K. P., & Stellmes, M.
+2004, Digital imaging and spectral techniques:
+Applications to precision agriculture and crop
+physiology, 66, 207
+
+Photosynthetic fluorescence on Exoplanets
+27
+Meadows, V. S., Reinhard, C. T., Arney, G. N.,
+et al. 2018, Astrobiology, 18, 630
+Meftah, M., Dam´e, L., Bols´ee, D., et al. 2018,
+Astronomy & Astrophysics, 611, A1
+Monta˜n´es-Rodr´ıguez, P., Pall´e, E., Goode, P., &
+Mart´ın-Torres, F. 2006, The Astrophysical
+Journal, 651, 544
+Nakajima, M., Kuze, A., & Suto, H. 2012, in
+Sensors, Systems, and Next-Generation
+Satellites XVI, Vol. 8533, SPIE, 21–30
+O’Malley-James, J. T., & Kaltenegger, L. 2018,
+Monthly Notices of the Royal Astronomical
+Society, 481, 2487
+—. 2019, Monthly Notices of the Royal
+Astronomical Society, 488, 4530
+Pavlov, A., & Kasting, J. 2002, Astrobiology, 2, 27
+Porcar-Castell, A., Malenovsk`y, Z., Magney, T.,
+et al. 2021, Nature plants, 7, 998
+Robinson, T. D., Stapelfeldt, K. R., & Marley,
+M. S. 2016, Publications of the Astronomical
+Society of the Pacific, 128, 025003
+Rugheimer, S., & Kaltenegger, L. 2018, The
+Astrophysical Journal, 854, 19
+Sanrom´a, E., Pall´e, E., Parenteau, M., et al. 2013,
+The Astrophysical Journal, 780, 52
+Schirhagl, R., Chang, K., Loretz, M., & Degen,
+C. L. 2014, Annu. Rev. Phys. Chem., 65, 83
+Schuurmans, R. M., van Alphen, P., Schuurmans,
+J. M., Matthijs, H. C., & Hellingwerf, K. J.
+2015, PloS one, 10, e0139061
+Schwieterman, E. W. 2018, Surface and Temporal
+Biosignatures, ed. H. J. Deeg & J. A. Belmonte
+(Cham: Springer International Publishing),
+1–29, doi: 10.1007/978-3-319-30648-3 69-1
+Schwieterman, E. W., Cockell, C. S., & Meadows,
+V. S. 2015, Astrobiology, 15, 341
+Schwieterman, E. W., Kiang, N. Y., Parenteau,
+M. N., et al. 2018, Astrobiology, 18, 663
+Seager, S., Turner, E. L., Schafer, J., & Ford,
+E. B. 2005, Astrobiology, 5, 372
+Segura, A., Kasting, J. F., Meadows, V., et al.
+2005, Astrobiology, 5, 706
+Selvaggio, G., Chizhik, A., Nißler, R., et al. 2020,
+Nat. Commun., 11, 1495
+Sun, Y., Frankenberg, C., Jung, M., et al. 2018,
+Remote Sensing of Environment, 209, 808
+Sunshine, J. M., & Pieters, C. M. 1998, Journal of
+Geophysical Research: Planets, 103, 13675
+Takizawa, K., Minagawa, J., Tamura, M.,
+Kusakabe, N., & Narita, N. 2017, Scientific
+reports, 7, 1
+Tinetti, G., Meadows, V. S., Crisp, D., et al. 2006,
+Astrobiology, 6, 881
+Tsumura, K., Battle, J., Bock, J., et al. 2010, The
+Astrophysical Journal, 719, 394
+Wilhelm, C., & Jakob, T. 2006, Photosynthesis
+Research, 87, 323
+Wolf, B. M., Niedzwiedzki, D. M., Magdaong, N.
+C. M., et al. 2018, Photosynthesis research, 135,
+177
+Yao, L., Yang, D., Liu, Y., et al. 2021, A New
+Global Solar-induced Chlorophyll Fluorescence
+(SIF) Data Product from TanSat
+Measurements, Springer
+

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+Hadronic observables from master-field simulations
+Marco Cè,𝑎,∗ Mattia Bruno,𝑏 John Bulava,𝑐 Anthony Francis,𝑑 Patrick Fritzsch,𝑒
+Jeremy R. Green,𝑐 Maxwell T. Hansen 𝑓 and Antonio Rago𝑔,ℎ
+𝑎Albert Einstein Center for Fundamental Physics (AEC) and Institut für Theoretische Physik, Universität
+Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
+𝑏Dipartimento di Fisica “Giuseppe Occhialini”, Università degli Studi di Milano-Bicocca and INFN -
+Sezione di Milano Bicocca, Piazza della Scienza 3, 20126 Milan, Italy
+𝑐Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, 15738 Zeuthen, Germany
+𝑑Institute of Physics, National Yang Ming Chiao Tung University, Hsinchu, Taiwan 30010
+𝑒School of Mathematics and Hamilton Mathematics Institute, Trinity College Dublin, Dublin 2, Ireland
+𝑓 Higgs Centre for Theoretical Physics, School of Physics and Astronomy, The University of Edinburgh,
+Edinburgh EH9 3FD, United Kingdom
+𝑔IMADA and CP3, University of Southern Denmark, Odense, Denmark
+ℎDepartment of Theoretical Physics, CERN, 1211 Geneva 23, Switzerland
+E-mail: marcoce@itp.unibe.ch
+Substantial progress has been made recently in the generation of master-field ensembles. This
+has to be paired with efficient techniques to compute observables on gauge field configurations
+with a large volume. Here we present the results of the computation of hadronic observables,
+including hadron masses and meson decay constants, on large-volume and master-field ensembles
+with physical volumes of up to (18 fm)4 and 𝑚 𝜋𝐿 up to 25, simulated using 𝑁f = 2 + 1 stabilized
+Wilson fermions. We obtain sub-percent determinations from single gauge configurations with the
+combined use of position-space techniques, volume averages and master-field error estimation.
+The 39th International Symposium on Lattice Field Theory,
+8th-13th August, 2022,
+Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany
+∗Speaker
+© Copyright owned by the author(s) under the terms of the Creative Commons
+Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).
+https://pos.sissa.it/
+arXiv:2301.05156v1  [hep-lat]  12 Jan 2023
+
+Hadronic observables from master-field simulations
+Marco Cè
+1.
+Introduction
+Gauge-field configurations in a lattice theory with a mass gap have the stochastic locality property,
+that is, gauge-invariant local fields at large physical separations are stochastically independent.
+The master-field paradigm introduced by Lüscher [1] proposes to use stochastic locality to obtain
+observable estimates from a single or at most a few representative gauge-field configurations on very
+large lattices, making use of the invariance under translations of the theory and of volume averages.
+As a first application of this paradigm, stochastic locality has been used to compute the
+topological susceptibility at 𝑇 > 𝑇𝑐 in master-field simulations of SU(3) Yang–Mills theory [2]. In
+a theory with fermions such as QCD, numerical simulations are performed after integrating out
+fermions exactly. Hadronic observables in QCD are expressed in terms of contractions of quark
+propagators whose locality is not manifest. Moreover, the sheer size of the lattices requires stabilising
+measures that have been studied in ref. [3]. These include a slight modification of the standard
+𝑂(𝑎)-improved lattice Dirac operator, replacing the HMC with the stochastic molecular dynamics
+(SMD) algorithm, employing quadruple-precision lattice sums and uniform-norm stopping criteria
+for the Dirac equation solver. Recent progress in master-field simulations has been presented at the
+Lattice 2021 conference [4, 5] and at this conference [6]. In these proceedings we further develop
+the position-space techniques introduced in ref. [5], by presenting an estimator for position-space
+correlators that scales efficiently with the volume.
+Estimation of observables on master fields is explained in details in ref. [1]. In summary, the
+expectation value ⟨O(𝑥)⟩ of a local field O(𝑥) is obtained averaging over translations
+⟪O(𝑥)⟫ = 1
+𝑉
+∑︁
+𝑧
+O(𝑥 + 𝑧),
+⟨O(𝑥)⟩ = ⟪O(𝑥)⟫ + 𝑂
+�
+𝑉−1/2�
+,
+(1)
+with the variance of this estimator given by
+𝜎2
+⟪O⟫(𝑥) =
+�
+[⟪O(𝑥)⟫ − ⟨O(𝑥)⟩]2�
+= 1
+𝑉
+∑︁
+𝑦
+⟨O(𝑦)O(0)⟩𝑐
+= 1
+𝑉
+������
+∑︁
+|𝑦|≤𝑅
+⟨O(𝑦)O(0)⟩𝑐 + 𝑂
+�
+e−𝑚𝑅�������
+= 1
+𝑉
+������
+∑︁
+|𝑦|≤𝑅
+⟪O(𝑦)O(0)⟫𝑐 + 𝑂
+�
+e−𝑚𝑅�
++ 𝑂
+�
+𝑉−1/2�������
+,
+(2)
+where in the second line we first used the fact that the connected correlator of the local field O(𝑥)
+decays exponentially with spacetime separation, and then we applied again translation averages.
+2.
+Position-space correlators
+In this work we focus on correlation functions in position space
+𝐶𝑃𝑃(𝑥) → 𝑐2
+𝑃
+4𝜋2
+𝑚 𝜋
+|𝑥| 𝐾1(𝑚 𝜋|𝑥|),
+(3a)
+𝐶𝐴𝑃,𝜇(𝑥) → 𝑐𝐴𝑐𝑃
+4𝜋2
+𝑥𝜇
+|𝑥|
+𝑚 𝜋
+|𝑥| 𝐾2(𝑚 𝜋|𝑥|),
+(3b)
+𝐶𝐴𝐴,𝜇𝜈(𝑥) →
+𝑐2
+𝐴
+4𝜋2
+�
+−𝛿𝜇𝜈
+1
+𝑥2 𝐾2(𝑚 𝜋|𝑥|) + 𝑥𝜇𝑥𝜈
+𝑥2
+�𝑚 𝜋
+|𝑥| 𝐾1(𝑚 𝜋|𝑥|) + 4
+𝑥2 𝐾2(𝑚 𝜋|𝑥|)
+��
+,
+(3c)
+𝐶𝑁 𝑁 (𝑥) →
+𝑐2
+𝑁
+4𝜋2
+𝑚2
+𝑁
+|𝑥|
+�
+𝐾1(𝑚𝑁 |𝑥|) + /𝑥
+|𝑥| 𝐾2(𝑚𝑁 |𝑥|)
+�
+,
+(3d)
+2
+
+Hadronic observables from master-field simulations
+Marco Cè
+Table 1: Parameters of the master-field lattices used in this study (with 𝑎 ≈ 0.094 fm and 𝑚 𝜋 ≈ 270 MeV,
+see also ref. [4]), together with information on the statistics used in the observable computation as explained
+in section 3.
+𝐿/𝑎
+𝐿 [fm]
+𝑚 𝜋𝐿
+𝑛cnfg
+𝑏/𝑎
+|𝐺|
+𝑛shift
+𝑏shift/𝑎
+𝑛point
+A
+96
+9
+12.5
+5
+48
+8
+512
+12
+4096
+B
+192
+18
+25
+2
+48
+128
+32
+24
+4096
+where the subscript indicates the two-point function of either pseudoscalar densities 𝑃 = ¯𝑢𝛾5𝑑, axial
+current 𝐴𝜇 = ¯𝑢𝛾𝜇𝛾5𝑑 or nucleon spinor 𝑁 = 𝜖𝑎𝑏𝑐(𝑢𝑇
+𝑎𝐶𝛾5𝑑𝑏)𝑢𝑐, as a function of the source-sink
+separation 𝑥.
+In eqs. (3), the asympotic behaviour for 𝑥 → ∞ of these correlators is given assuming the
+symmetries of the continuum theory in an infinite volume. From position-space correlators one
+can extract simple hadronic observables, including the masses 𝑚 𝜋 and 𝑚𝑁 and the decay constant
+𝑓𝜋 = 𝑐𝐴/𝑚 𝜋, as demonstrated in ref. [5].
+Once computed on the lattice as discussed in the following section, these correlators as a
+function of the four-dimensional source-sink separation 𝑥 include lattice discretization effects that
+break the rotational symmetry and depend on the direction of 𝑥. In this study, we limit ourselves to
+the radial correlators ˚𝐶(𝑟) introduced in ref. [5] that are averaged over 𝑆3(𝑟) = {𝑥 ∈ R4 : |𝑥| = 𝑟},
+the 3-sphere of radius 𝑟, and by construction depend only on the radial coordinate 𝑟 = |𝑥|. While
+˚𝐶𝑃𝑃(𝑟) = 𝐶𝑃𝑃(𝑥), for the 𝐴𝑃-correlator 𝐶𝐴𝑃,𝜇(𝑥) we contract the open 𝜇 index with the only
+available four-vector 𝑥𝜇 to obtain a scalar, ˚𝐶𝐴𝑃(𝑟) = 𝑥𝜇𝐶𝐴𝑃,𝜇(𝑥) → 𝑐𝐴𝑐𝑃
+4𝜋2 𝑚 𝜋𝐾2(𝑚 𝜋𝑟). In the case
+of 𝐶𝐴𝐴,𝜇𝜈 there are two ways to obtain a scalar,
+˚𝐶 (1)
+𝐴𝐴(𝑟) = 𝛿𝜇𝜈𝐶𝐴𝐴,𝜇𝜈(𝑥),
+˚𝐶 (2)
+𝐴𝐴(𝑟) = 𝑥𝜇𝑥𝜈𝐶𝐴𝐴,𝜇𝜈(𝑥),
+(4)
+and similarly for the nucleon correlator that is a spinor, with /𝑥 = 𝛾𝜇𝑥𝜇,
+˚𝐶 (1)
+𝑁 𝑁 (𝑟) ≡ tr 𝐶𝑁 𝑁 (𝑥),
+˚𝐶 (2)
+𝑁 𝑁 (𝑟) ≡ tr /𝑥𝐶𝑁 𝑁 (𝑥).
+(5)
+On the lattice, an estimator of these radial correlators is given by
+˚𝐶(𝑟) =
+1
+r4(𝑟2)
+∑︁
+|𝑥 |=𝑟
+𝐶(𝑥)
+(6)
+where r4 is defined in ref. [5].
+We note that the symmetry of 𝑆3(𝑟) is broken not only by 𝑎 ≠ 0 but also by the finite size of the
+hypercubic box and by the fact that we choose antiperiodic (instead of periodic) boundary conditions
+in one of the four dimensions for quarks. However, as we show in section 5 these boundary effects
+are not visible at the current level of precision on the master-field lattices in table 1 considered here,
+differently from what we observed on smaller volumes [5].
+3.
+Grid of point sources estimator
+The simplest way to compute the correlators introduced in section 2 numerically is to solve the
+Dirac equation on a point source, that is, a source spinor that is supported on a single lattice point,
+and subsequently perform the suitable contractions of spinor and space-time indeces. A consequence
+3
+
+Hadronic observables from master-field simulations
+Marco Cè
+of this naive strategy applied on gauge-field configurations with a large volume is that the effort for
+each correlator point source scales proportionally with the volume, which is clearly not optimal.
+Indeed, most of the resources are spent in computing the correlator at a distance from the source of
+multiple correlation lengths, which has an exponentially suppressed contribution to the physics and
+in most of the cases is completely dominated by noise.
+Instead, we would like to exploit stochastic locality to define estimators that scale efficiently
+with the volume and are suitable for master-field applications. Taking as an example the radial
+correlators ˚𝐶(𝑟) introduced in section 2, let us assume that we are interested in physics that can be
+extracted from correlators up to a maximum radial source-sink separation 𝑟max. Ref. [1] sketches a
+decomposition of the lattice in space-time domains, or blocks, that are physically large, such that all
+the lattice points within an 𝑟max distance from a source point at the centre of each block are within
+the same block. This implies a block size 𝑏 > 2𝑟max. Solving the Dirac equation in each block,
+imposing Dirichlet boundary conditions at the block boundary of the gauge field, one can decouple
+the computational cost of the estimator from the volume of the global lattice. However, this method
+introduces boundary effects that can be large for sink points close to the boundaries [1, 7], see also
+refs. [8, 9]. We leave the exploration of this direction for future work, and we focus here on a simpler
+approach that does not require a dedicated correction computation.
+We introduce a set of lattice points 𝐺 that are separated (on average) by a physical distance
+constant in the volume, such that the number of points |𝐺| ∝ 𝑉, that is, it grows proportionally with
+the volume. On these point we introduce stochastic sources that satisfy
+�
+𝜂𝑖(𝑥)𝜂†
+𝑗(𝑦)
+�
+𝜂 = 𝛿𝑖 𝑗𝛿𝑥𝑦𝐼
+for 𝑥, 𝑦 ∈ 𝐺,
+(7)
+where 𝐼 is the identity matrix in spin and colour space. By contracting at the sink with stochastic
+noise corresponding to each coordinate 𝑦 ∈ 𝐺 one obtains |𝐺| ∝ 𝑉 samples of the quark propagator,
+one for each 𝑦 ∈ 𝐺, from a single global-lattice inversion that is 𝑂(𝑉) computationally. Each sample
+has a spurious contribution of stochastic nature from source points 𝑥 ≠ 𝑦, which is suppressed
+by averaging the quark propagator over a number of sources 𝑛src and does not contribute to the
+expectation value. Mesonic two-point functions that contract two quark propagators require 𝑛src ≥ 2
+to obtain an unbiased estimator, that in the case of the pseudoscalar-density two-point function reads
+𝐶𝐺
+𝑃𝑃(𝑥; 𝑦) =
+1
+𝑛src(𝑛src − 1)
+∑︁
+𝑖≠𝑗
+Re
+�
+𝜓†
+𝑖 (𝑥 + 𝑦)𝜓 𝑗(𝑥 + 𝑦)𝜂†
+𝑗(𝑦)𝜂𝑖(𝑦)
+�
+,
+(8)
+where 𝜓𝑖(𝑥) = �
+𝑦 𝐷−1(𝑥; 𝑦)𝜂𝑖(𝑦) and the double sum over 𝑖 ≠ 𝑗 can be computed in 𝑂(𝑛src) cost.
+In this approach, since |𝐺| ∝ 𝑉, efficient scaling of the solutions of the Dirac equation is
+achieved. Moreover, 𝐺 implicitly realises a domain decomposition by labelling each lattice point
+with the closest 𝑦 ∈ 𝐺.1 Eq. (8) is in principle valid for any 𝑥 and 𝑦, but if only (𝑥, 𝑦) pairs that are
+in the same domain are considered then one can compute efficiently all the |𝐺| ∝ 𝑉 contributions
+with a single 𝑂(𝑉) pass over the whole lattice. This realises the optimal volume scaling for the
+contractions too. It also lowers the required 𝑛src since the “correct” source 𝑦 is always the closest to
+the sink 𝑥 and spurious contributions are further suppressed by the longer source-sink separation.2
+1Up to points equidistant from two or more 𝑦 ∈ 𝐺 that require additional conditions to be assigned to a domain.
+2These spurious contributions are only stochastic and do not modify the expectation value, although we note that they
+can have different quantum numbers and decay slower than the correlator being estimated.
+4
+
+Hadronic observables from master-field simulations
+Marco Cè
+𝑏
+𝑟max =
+√
+2𝑏/2
+𝑟max =
+√
+2𝑏/2
+𝑦 ∈ 𝐺
+𝑦 ∈ 𝐺
+𝑥
+Figure 1: Sketch of the estimator with a grid of point sources over a two-dimensional window of the lattice.
+The set 𝐺 of source points
+∈ 𝐺 is a regular grid with spacing 𝑏 and even point only. A mesonic two-point
+function is evaluated at sink point 𝑥 that is in the domain defined by 𝑦 ∈ 𝐺 and within a distance 𝑟max from 𝑦.
+One of the spurious contributions from the “wrong” source is shown in light grey.
+Moreover, it implies 𝑟max = min𝑥,𝑦∈𝐺 |𝑥 − 𝑦|/2, that is, the minimum of the semidistance of points
+in 𝐺. Therefore, 𝐺 has to be sparse enough for correlators at the relevant radial separations 𝑟 ≤ 𝑟max
+to be accessible.
+We study this setup on two sets of a few master fields whose parameters are given in table 1.
+The master fields in both sets are hypercubic boxes with equal extent in each dimension denoted
+by 𝐿, such that the volume is 𝑉 = 𝐿4. The 𝐿 = 192𝑎 master fields denoted by B (𝑛cnfg = 2) have
+exactly 16 times, twice in each dimension, the volume of the ones with 𝐿 = 96𝑎 in set A (𝑛cnfg = 5)
+and otherwise identical parameters, and we can thus define equivalent 𝐺s on both sets and study the
+volume scaling. We employ U(1) noise that satisfies eq. (7). The simplest choice for 𝐺 is a regular
+grid with spacing 𝑏, which matches the domain decomposition proposed in ref. [1], with 𝑏 = 48𝑎
+being a suitable choice in our case. However, the definition of 𝐺 is more flexible. In this work, we
+employ a grid with only even (or equivalently odd) points, which results in 𝑟max =
+√
+2𝑏/2 ≃ 33.94𝑎
+instead of 𝑏/2 = 24𝑎, at the cost of halving the number of points on the grid.3 The total number
+of points is thus |𝐺| = (𝐿/𝑏)4/2 that evaluates to 8 and 128 for A and B respectively. We fix
+𝑛src = 2 and with the current precision we do not observe deviations from the expected behaviour,
+especially at 𝑟 close to 𝑟max, that can be attributed to spurious contributions. Further optimisation
+such as systematically and exactly removing the closer spurious contributions, e.g. with hierarchical
+probing [11], are not explored here.
+The statistics obtained with a single source, e.g. eight points on each master field in A, is limited
+by the need of balancing the density of 𝐺 with a lower limit on the 𝑟max suitable to extract long-range
+physics. To increase the statistics we simply propose to recompute eq. (8) on 𝑛shift sources, each
+time shifting 𝐺 to have a distinct support. This is done four times for each direction in the case of A
+and twice for each direction in B. An extra factor of two is obtained by pairing each even-only 𝐺
+with the corresponding odd-only, leading to 𝑛shift = 512 and 32 for A and B respectively. Combined
+with |𝐺|, the final result is the same number of source points 𝑛point = 4096 for both volumes, on
+3This results in a doubled |𝐺|𝑟4max/𝑉 density. Indeed, it corresponds to a 𝐷4 lattice (or equivalently 𝐹4 lattice) that
+has the densest known packing of equal spheres in four dimensions [10].
+5
+
+Hadronic observables from master-field simulations
+Marco Cè
+a regular grid with spacing 𝑏shift = 12𝑎 and 24𝑎 for A and B respectively. Ignoring that on the A
+lattices source points are on average twice as close and thus potentially more correlated than on B,
+in our setup we have same statistics for each gauge field configuration for both A and B. Crucially,
+thanks to the optimal volume scaling of the stochastic grid correlator, this matching statistic has
+been obtained at an equivalent computational cost.
+4.
+Master-field errors
+The estimator in section 3 applied to the radial correlator leads to a collection of up to 4096
+correlators for each master-field configuration on a regular grid of source points with spacing
+𝑏shift = 𝐿/8. Applying stochastic locality, the expectation value
+� ˚𝐶(𝑟)
+� is given up to volume-
+suppressed corrections by the translation average
+� ˚𝐶(𝑟)
+�
+= ⟪ ˚𝐶(𝑟)⟫ + 𝑂
+�
+𝑉−1/2�
+= 1
+𝑉
+∑︁
+𝑦∈𝐺
+˚𝐶(𝑟; 𝑦) + 𝑂
+�
+𝑉−1/2�
+(9)
+where the 𝑦 in ˚𝐶(𝑟; 𝑦) denotes the source point. The error of this estimator can be estimated applying
+eq. (2) with O(𝑦) = ˚𝐶(𝑟; 𝑦)
+�
+[⟪ ˚𝐶(𝑟)⟫ −
+� ˚𝐶(𝑟)
+�
+]2�
+= 1
+𝑉
+������
+∑︁
+|𝑦|≤𝑅
+⟪ ˚𝐶(𝑟; 𝑦) ˚𝐶(𝑟; 0)⟫𝑐 + 𝑂
+�
+e−𝑚𝑅�
++ 𝑂
+�
+𝑉−1/2�������
+,
+(10)
+where again the sum over the source coordinates 𝑦 is performed over the grid of point sources.
+Finding the optimal 𝑅 to truncate the sum in the r.h.s. has a clear analogy with the well-known Γ
+method introduced by Wolff to deal with autocorrelation in Monte Carlo time and estimate an error
+with less errors [12], and leads to a generalisation of the Madras–Sokal formula for the statistical
+error of the error [13, 14]. This can be implemented in a resource efficient way by computing
+the correlation between grid points with higher-dimensional fast Fourier transforms. The optimal
+𝑅 depends on the observable. In particular, since each value of the correlator radial source-sink
+separation 𝑟 defines a distinct observable with different spacetime support, 𝑅 is a function of 𝑟.
+Alternatively, one can apply a four-dimensional binning of the point sources in the grid into
+blocks. For instance, blocks of size (24𝑎)4 bin 16 point sources on A and only one point source on
+B according to the spacing 𝑏shift in table 1, while blocks of size (48𝑎)4 bin 256 and 16 point sources
+respectively. We tested these two bin sizes and observed that this leads to a stable error estimate. In
+the following, we show results obtained in the more conservative case, that is, with blocks of size
+(48𝑎)4.
+We note that master-field error estimation can be combined with standard methods based on
+an ensemble of gauge field configurations, e.g. with a five-dimensional variant of the Γ method in
+spacetime coordinates and Monte Carlo time. Explorations in this direction can be found in ref. [15].
+5.
+Numerical results
+We computed 𝑚 𝜋, 𝑚𝑁 and 𝑓𝜋 using position-space correlators on the sets of master fields
+whose parameters are listed in table 1. The results for these hadronic observables are listed in table 2.
+We employed the technique already studied in ref. [5] to extract the pion mass 𝑚 𝜋 from the
+long-distance behaviour in eq. (3a) of the position-space correlator ˚𝐶𝑃𝑃(𝑟). In those proceedings
+6
+
+Hadronic observables from master-field simulations
+Marco Cè
+Table 2: Numerical results for hadronic observable with errors estimated à la master field.
+𝐿/𝑎
+𝑎𝑚 𝜋
+𝑎𝑚𝑁
+𝑎 𝑓 bare
+𝜋
+A
+96
+0.126 28(33)
+0.500(6)
+0.0890(3)
+B
+192
+0.126 01(19)
+0.487(8)
+0.0885(4)
+5
+10
+15
+20
+25
+30
+r/a
+0.08
+0.10
+0.12
+0.14
+0.16
+0.18
+0.20
+ameff
+covariant
+one-state fit
+two-state fit
+5
+10
+15
+20
+25
+30
+r/a
+0.08
+0.10
+0.12
+0.14
+0.16
+0.18
+0.20
+ameff
+covariant
+one-state fit
+two-state fit
+Figure 2: Effective mass of the ˚𝐶𝑃𝑃(𝑟) correlator as a function of 𝑟 for master fields in set A (left plot) and
+set B (right plot). On top of the data points with master-field errors shown in blue, we show the results of a
+one-state fit in a green band and of a two-states fit in a red band. The thickness of the bands is the statistical
+error.
+the technique was applied to correlators computed with point sources on an ensemble of gauge field
+configurations with a (6 fm)3 space volume, performing a standard error estimation. Here we have a
+larger volume that allows us to use the grid of point sources as described in section 3 and estimate
+the error à la master field, see section 4. On top of the same number of samples 𝑛point = 4096 for
+each configuration, we have 5 configurations in set A and 2 in set B. This means that we have a larger
+statistics for the 𝐿 = 96𝑎 master fields from which we expect a ≈ 1.58 reduction of the error.
+The effective mass4 of ˚𝐶𝑃𝑃(𝑟) is shown in the two plots in figure 2. For each set, two fits are
+performed: a “one-state” fit having 𝑐𝑃 and 𝑚 𝜋 as free parameters, and a “two-states” one with an
+added “excited state” term 𝑎1(𝑚1/𝑟)𝐾1(𝑚1/𝑟) with two extra free parameters 𝑎1 and 𝑚1 > 𝑚 𝜋. We
+choose appropriate values for the smaller 𝑟 of the correlator data that enter the fit, with different
+choices for one-state and two-states fits. Instead, all the data up to largest available 𝑟 = 𝑟max enter
+the fit, since we do not observe any boundary effect that constrains us otherwise. The two fits on
+each set give compatible results and the corresponding effective mass is shown in figure 2.
+From the one-state fits we obtain the results in table 2, which show a good agreement between
+the two sets. Contrary to the expectation based on 𝑛cnfg, the error is 40 % smaller on set B. A
+possible explanation for this fact is the 𝑏shift = 12𝑎 of the samples of set A, halved with respect to set
+B, which can lead to a reduced effective number of samples due to stronger correlations in space.
+Similarly, we extract 𝑚𝑁 from the two contractions in eq. (5) of the position-space nucleon
+correlator in eq. (3d) as done in ref. [5], but employing the techniques of sections 3 and 4. The results in
+table 2 are from the one-state fits to ˚𝐶 (1)
+𝑁 𝑁 (𝑟) with the free parameters 𝑐𝑁 and 𝑚𝑁 , and are compatible
+4See eq. (10) in ref. [5] for the definition of the effective mass of the radial correlator.
+7
+
+Hadronic observables from master-field simulations
+Marco Cè
+5
+10
+15
+20
+25
+30
+r/a
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+ameff
+covariant, trNN
+covariant, tr/xNN
+one-state fit, trNN
+one-state fit, tr/xNN
+two-state fit, trNN
+two-state fit, tr/xNN
+5
+10
+15
+20
+25
+30
+r/a
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+ameff
+covariant, trNN
+covariant, tr/xNN
+one-state fit, trNN
+one-state fit, tr/xNN
+two-state fit, trNN
+two-state fit, tr/xNN
+Figure 3: Effective mass of the ˚𝐶 (𝑖)
+𝑁 𝑁 (𝑟) correlators as a function of 𝑟 for master fields in set A (left plot) and
+set B (right plot), where 𝑖 = 1 corresponds to the tr 𝑁𝑁 contraction and 𝑖 = 2 to the tr /𝑥𝑁𝑁 one. On top of the
+data points with master-field errors shown in blue and orange for 𝑖 = 1 and 2 respectively, we show the results
+of a one-state fit in green and brown bands and of a two-states fit in red and purple bands. The thickness of the
+bands is the statistical error.
+with the results of two-states fits with the replacement ˚𝐶𝑁 𝑁 (𝑟) → ˚𝐶𝑁 𝑁 (𝑟)[1+𝑎1(𝑚 𝜋/𝑟)𝐾1(𝑚 𝜋𝑟)]
+where 𝑎1 is an extra free parameter and 𝑚 𝜋 is fixed. The fit to ˚𝐶 (2)
+𝑁 𝑁 (𝑟) shows similar results,
+although with a slightly larger central value that can be attributed to different discretization effects.
+The effective masses corresponding to data and fits are shown in figure 3. In the case of 𝑚𝑁 , we
+observe a larger error on set B, compatible with the lower statistics and showing no indication of
+correlation-in-space effects.
+We also extract the pion decay constant 𝑓 bare
+𝜋
+, where the bare indicates that we do not include the
+axial-current renormalization factor, from a combined fit of the four correlators ˚𝐶𝑃𝑃, ˚𝐶𝐴𝑃, ˚𝐶 (1)
+𝐴𝐴 and
+˚𝐶 (2)
+𝐴𝐴. As fit function we employ the long-distance behaviours derived from eqs. 3, which depends
+on the free parameters 𝑐𝑃, 𝑐𝐴 and 𝑚 𝜋. As shown from the plots of the ratio between data and fit
+functions in figure 4, ˚𝐶𝐴𝑃 approaches the asymptotic behaviour at a smaller value of 𝑟, followed
+by ˚𝐶𝑃𝑃 and ˚𝐶 (2)
+𝐴𝐴.
+˚𝐶 (1)
+𝐴𝐴 converges to the asymptotic behaviour at a much larger 𝑟, with the ratio
+being initially negative and changing sign around 𝑟 ≈ 14𝑎. The values of 𝑚 𝜋 obtained from these
+combined fits are consistent with the previous fits to only the ˚𝐶𝑃𝑃 correlators. The decay constant is
+then given by 𝑓 bare
+𝜋
+= 𝑐𝐴/𝑚 𝜋 and shown in table 2. Like in the case of 𝑚𝑁 , the values on set A and
+B are compatible, with a slightly larger error for set B that is consistent with the lower number of
+master field configurations.
+6.
+Conclusions
+We have shown that position-space correlators can be used to extract hadron masses and decay
+constants with short-distance and cut-off effects under control. Crucially, the statistical error can
+be estimated à la master field, obtaining an efficient scaling of the computational effort with the
+increased volume.
+In this work we studied sphere-averaged radial correlators, but potentially more information is
+encoded in correlators as function of four-dimensional coordinates. This requires understanding
+effects that break rotational symmetry at finite lattice spacing and is an interesting topic for further
+studies.
+8
+
+Hadronic observables from master-field simulations
+Marco Cè
+0.020
+0.022
+0.024
+0.026
+|cP|2
+|cP|2 fit
+˚CPP
+0.0016
+0.0018
+0.0020
+|cAc†
+P|
+|cAc†
+P| fit
+˚CAP
+10
+20
+30
+r/a
+0.00011
+0.00012
+0.00013
+0.00014
+|cA|2
+|cA|2 fit
+˚C(1)
+AA
+10
+20
+30
+r/a
+0.00011
+0.00012
+0.00013
+0.00014
+|cA|2
+|cA|2 fit
+˚C(2)
+AA
+0.020
+0.022
+0.024
+0.026
+|cP|2
+|cP|2 fit
+˚CPP
+0.0016
+0.0018
+0.0020
+|cAc†
+P|
+|cAc†
+P| fit
+˚CAP
+10
+20
+30
+r/a
+0.00011
+0.00012
+0.00013
+0.00014
+|cA|2
+|cA|2 fit
+˚C(1)
+AA
+10
+20
+30
+r/a
+0.00011
+0.00012
+0.00013
+0.00014
+|cA|2
+|cA|2 fit
+˚C(2)
+AA
+Figure 4: Plots of the ratio between correlator data and their fitted long-distance behaviours for master fields
+in set A (top row) and set B (bottom row). The amplitude in the denominator is set to one, so that the actual
+amplitude for each correlator is shown on the vertical axis. In each row, four plots are shown for ˚𝐶𝑃𝑃, ˚𝐶 (1)
+𝐴𝐴
+(left column), ˚𝐶𝐴𝑃 and ˚𝐶 (2)
+𝐴𝐴 (right column), with the correlator data with master field errors shown in blue.
+The amplitude parameters of the corresponding fit function, which are functions of 𝑐𝑃 and 𝑐𝐴, are shown in
+an orange horizontal line with a pale orange error band.
+Position-space methods find applications in computations of quantities that go beyond the
+simple hadronic quantities considered here, such as for example the hadronic vacuum polarisation
+contribution to the anomalous magnetic moment of the muon [16, 17], including the so-called
+window contribution [18]. The estimators presented here provide a straightforward path to the
+computation of this quantities in the master-field paradigm.
+Acknowledgements: The research of MB is funded through the MUR program for young researchers “Rita
+Levi Montalcini”. AF acknowledges support by the Ministry of Science and Technology Taiwan (MOST)
+under grant 111-2112-M-A49-018-MY2. JRG acknowledges support from the Simons Foundation through the
+Simons Bridge for Postdoctoral Fellowships scheme. MTH is supported by UKRI Future Leader Fellowship
+MR/T019956/1 and in part by UK STFC grant ST/P000630/1. This work was performed using the DiRAC
+Data Intensive service at Leicester, operated by the University of Leicester IT Services, which forms part
+of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS capital
+funding via STFC capital grants ST/K000373/1 and ST/R002363/1 and STFC DiRAC Operations grant
+ST/R001014/1. DiRAC is part of the National e-Infrastructure. We acknowledge PRACE for awarding
+us access to SuperMUC-NG at GCS@LRZ, Germany, where some computations were performed Many
+9
+
+Hadronic observables from master-field simulations
+Marco Cè
+simulations were performed on a dedicated HPC cluster at CERN. We gratefully acknowledge the computer
+resources and the technical support provided by these institutions.
+References
+[1] M. Lüscher, Stochastic locality and master-field simulations of very large lattices, EPJ Web
+Conf. 175 (2018) 01002 [1707.09758].
+[2] L. Giusti and M. Lüscher, Topological susceptibility at 𝑇 > 𝑇c from master-field simulations of
+the SU(3) gauge theory, Eur. Phys. J. C 79 (2019) 207 [1812.02062].
+[3] A. Francis, P. Fritzsch, M. Lüscher and A. Rago, Master-field simulations of 𝑂(𝑎)-improved
+lattice QCD: Algorithms, stability and exactness, Comput. Phys. Commun. 255 (2020) 107355
+[1911.04533].
+[4] P. Fritzsch, J. Bulava, M. Cè, A. Francis, M. Lüscher and A. Rago, Master-field simulations of
+QCD, PoS LATTICE2021 (2022) 465 [2111.11544].
+[5] M. Cè, M. Bruno, J. Bulava, A. Francis, P. Fritzsch, J.R. Green et al., Approaching the
+master-field: Hadronic observables in large volumes, PoS LATTICE2021 (2022) 383
+[2110.15375].
+[6] P. Fritzsch, Master-field simulations of QCD and the exponential clover action, PoS
+LATTICE2022 247.
+[7] M. Cè, L. Giusti and S. Schaefer, Domain decomposition, multilevel integration, and
+exponential noise reduction in lattice QCD, Phys. Rev. D 93 (2016) 094507 [1601.04587].
+[8] L. Giusti and M. Saccardi, Four-dimensional factorization of the fermion determinant in lattice
+QCD, Phys. Lett. B 829 (2022) 137103 [2203.02247].
+[9] M. Saccardi and L. Giusti, Four-dimensional domain decomposition for the factorization of the
+fermion determinant, PoS LATTICE2022 (2022) 386 [2211.06902].
+[10] J.H. Conway and N.J.A. Sloane, Sphere Packings, Lattices and Groups, Springer (1999),
+10.1007/978-1-4757-6568-7.
+[11] A. Stathopoulos, J. Laeuchli and K. Orginos, Hierarchical probing for estimating the trace of
+the matrix inverse on toroidal lattices, SIAM J. Sci. Comput. 35 (2013) S299 [1302.4018].
+[12] U. Wolff, Monte Carlo errors with less errors, Comput. Phys. Commun. 156 (2004) 143
+[hep-lat/0306017].
+[13] N. Madras and A.D. Sokal, The pivot algorithm: A highly efficient Monte Carlo method for the
+self-avoiding walk, J. Statist. Phys. 50 (1988) 109.
+[14] M. Cè, M. Bruno, J. Bulava, A. Francis, P. Fritzsch, J.R. Green et al. in preparation.
+[15] C. Lehner, The hadronic vacuum polarization (RBC/UKQCD), 2022.
+https://indico.ph.ed.ac.uk/event/112/contributions/1660/.
+[16] H.B. Meyer, Lorentz-covariant coordinate-space representation of the leading hadronic
+contribution to the anomalous magnetic moment of the muon, Eur. Phys. J. C 77 (2017) 616
+[1706.01139].
+[17] M. Cè, A. Gérardin, K. Ottnad and H.B. Meyer, The leading hadronic contribution to the
+running of the weinberg angle using covariant coordinate-space methods, PoS
+LATTICE2018 (2018) 137 [1811.08669].
+[18] E.-H. Chao, H.B. Meyer and J. Parrino, Coordinate-space calculation of the window
+observable for the hadronic vacuum polarization contribution to (𝑔 − 2)𝜇, 2211.15581.
+10
+

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+page_content=' The University of Edinburgh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  Edinburgh EH9 3FD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' United Kingdom  𝑔IMADA and CP3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' University of Southern Denmark,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Odense,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Denmark  ℎDepartment of Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' CERN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' 1211 Geneva 23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Switzerland  E-mail: marcoce@itp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='unibe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='ch  Substantial progress has been made recently in the generation of master-field ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' This  has to be paired with efficient techniques to compute observables on gauge field configurations  with a large volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Here we present the results of the computation of hadronic observables,  including hadron masses and meson decay constants, on large-volume and master-field ensembles  with physical volumes of up to (18 fm)4 and 𝑚 𝜋𝐿 up to 25, simulated using 𝑁f = 2 + 1 stabilized  Wilson fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' We obtain sub-percent determinations from single gauge configurations with the  combined use of position-space techniques, volume averages and master-field error estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  The 39th International Symposium on Lattice Field Theory,  8th-13th August, 2022,  Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany  ∗Speaker  © Copyright owned by the author(s) under the terms of the Creative Commons  Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='it/  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='05156v1  [hep-lat]  12 Jan 2023  Hadronic observables from master-field simulations  Marco Cè  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  Introduction  Gauge-field configurations in a lattice theory with a mass gap have the stochastic locality property,  that is, gauge-invariant local fields at large physical separations are stochastically independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  The master-field paradigm introduced by Lüscher [1] proposes to use stochastic locality to obtain  observable estimates from a single or at most a few representative gauge-field configurations on very  large lattices, making use of the invariance under translations of the theory and of volume averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  As a first application of this paradigm, stochastic locality has been used to compute the  topological susceptibility at 𝑇 > 𝑇𝑐 in master-field simulations of SU(3) Yang–Mills theory [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' In  a theory with fermions such as QCD, numerical simulations are performed after integrating out  fermions exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Hadronic observables in QCD are expressed in terms of contractions of quark  propagators whose locality is not manifest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Moreover, the sheer size of the lattices requires stabilising  measures that have been studied in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' These include a slight modification of the standard  𝑂(𝑎)-improved lattice Dirac operator, replacing the HMC with the stochastic molecular dynamics  (SMD) algorithm, employing quadruple-precision lattice sums and uniform-norm stopping criteria  for the Dirac equation solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Recent progress in master-field simulations has been presented at the  Lattice 2021 conference [4, 5] and at this conference [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' In these proceedings we further develop  the position-space techniques introduced in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' [5], by presenting an estimator for position-space  correlators that scales efficiently with the volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  Estimation of observables on master fields is explained in details in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' In summary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' the  expectation value ⟨O(𝑥)⟩ of a local field O(𝑥) is obtained averaging over translations  ⟪O(𝑥)⟫ = 1  𝑉  ∑︁  𝑧  O(𝑥 + 𝑧),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  ⟨O(𝑥)⟩ = ⟪O(𝑥)⟫ + 𝑂  �  𝑉−1/2�  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  (1)  with the variance of this estimator given by  𝜎2  ⟪O⟫(𝑥) =  �  [⟪O(𝑥)⟫ − ⟨O(𝑥)⟩]2�  = 1  𝑉  ∑︁  𝑦  ⟨O(𝑦)O(0)⟩𝑐  = 1  𝑉  ������  ∑︁  |𝑦|≤𝑅  ⟨O(𝑦)O(0)⟩𝑐 + 𝑂  �  e−𝑚𝑅�������  = 1  𝑉  ������  ∑︁  |𝑦|≤𝑅  ⟪O(𝑦)O(0)⟫𝑐 + 𝑂  �  e−𝑚𝑅�  + 𝑂  �  𝑉−1/2�������  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  (2)  where in the second line we first used the fact that the connected correlator of the local field O(𝑥)  decays exponentially with spacetime separation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' and then we applied again translation averages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  Position-space correlators  In this work we focus on correlation functions in position space  𝐶𝑃𝑃(𝑥) → 𝑐2  𝑃  4𝜋2  𝑚 𝜋  |𝑥| 𝐾1(𝑚 𝜋|𝑥|),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  (3a)  𝐶𝐴𝑃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='𝜇(𝑥) → 𝑐𝐴𝑐𝑃  4𝜋2  𝑥𝜇  |𝑥|  𝑚 𝜋  |𝑥| 𝐾2(𝑚 𝜋|𝑥|),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  (3b)  𝐶𝐴𝐴,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='𝜇𝜈(𝑥) →  𝑐2  𝐴  4𝜋2  �  −𝛿𝜇𝜈  1  𝑥2 𝐾2(𝑚 𝜋|𝑥|) + 𝑥𝜇𝑥𝜈  𝑥2  �𝑚 𝜋  |𝑥| 𝐾1(𝑚 𝜋|𝑥|) + 4  𝑥2 𝐾2(𝑚 𝜋|𝑥|)  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  (3c)  𝐶𝑁 𝑁 (𝑥) →  𝑐2  𝑁  4𝜋2  𝑚2  𝑁  |𝑥|  �  𝐾1(𝑚𝑁 |𝑥|) + /𝑥  |𝑥| 𝐾2(𝑚𝑁 |𝑥|)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  (3d)  2  Hadronic observables from master-field simulations  Marco Cè  Table 1: Parameters of the master-field lattices used in this study (with 𝑎 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='094 fm and 𝑚 𝜋 ≈ 270 MeV,  see also ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' [4]), together with information on the statistics used in the observable computation as explained  in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  𝐿/𝑎  𝐿 [fm]  𝑚 𝜋𝐿  𝑛cnfg  𝑏/𝑎  |𝐺|  𝑛shift  𝑏shift/𝑎  𝑛point  A  96  9  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='5  5  48  8  512  12  4096  B  192  18  25  2  48  128  32  24  4096  where the subscript indicates the two-point function of either pseudoscalar densities 𝑃 = ¯𝑢𝛾5𝑑, axial  current 𝐴𝜇 = ¯𝑢𝛾𝜇𝛾5𝑑 or nucleon spinor 𝑁 = 𝜖𝑎𝑏𝑐(𝑢𝑇  𝑎𝐶𝛾5𝑑𝑏)𝑢𝑐, as a function of the source-sink  separation 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  In eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' (3), the asympotic behaviour for 𝑥 → ∞ of these correlators is given assuming the  symmetries of the continuum theory in an infinite volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' From position-space correlators one  can extract simple hadronic observables, including the masses 𝑚 𝜋 and 𝑚𝑁 and the decay constant  𝑓𝜋 = 𝑐𝐴/𝑚 𝜋, as demonstrated in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  Once computed on the lattice as discussed in the following section, these correlators as a  function of the four-dimensional source-sink separation 𝑥 include lattice discretization effects that  break the rotational symmetry and depend on the direction of 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' In this study, we limit ourselves to  the radial correlators ˚𝐶(𝑟) introduced in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' [5] that are averaged over 𝑆3(𝑟) = {𝑥 ∈ R4 : |𝑥| = 𝑟},  the 3-sphere of radius 𝑟, and by construction depend only on the radial coordinate 𝑟 = |𝑥|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' While  ˚𝐶𝑃𝑃(𝑟) = 𝐶𝑃𝑃(𝑥), for the 𝐴𝑃-correlator 𝐶𝐴𝑃,𝜇(𝑥) we contract the open 𝜇 index with the only  available four-vector 𝑥𝜇 to obtain a scalar, ˚𝐶𝐴𝑃(𝑟) = 𝑥𝜇𝐶𝐴𝑃,𝜇(𝑥) → 𝑐𝐴𝑐𝑃  4𝜋2 𝑚 𝜋𝐾2(𝑚 𝜋𝑟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' In the case  of 𝐶𝐴𝐴,𝜇𝜈 there are two ways to obtain a scalar,  ˚𝐶 (1)  𝐴𝐴(𝑟) = 𝛿𝜇𝜈𝐶𝐴𝐴,𝜇𝜈(𝑥),  ˚𝐶 (2)  𝐴𝐴(𝑟) = 𝑥𝜇𝑥𝜈𝐶𝐴𝐴,𝜇𝜈(𝑥),  (4)  and similarly for the nucleon correlator that is a spinor, with /𝑥 = 𝛾𝜇𝑥𝜇,  ˚𝐶 (1)  𝑁 𝑁 (𝑟) ≡ tr 𝐶𝑁 𝑁 (𝑥),  ˚𝐶 (2)  𝑁 𝑁 (𝑟) ≡ tr /𝑥𝐶𝑁 𝑁 (𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  (5)  On the lattice, an estimator of these radial correlators is given by  ˚𝐶(𝑟) =  1  r4(𝑟2)  ∑︁  |𝑥 |=𝑟  𝐶(𝑥)  (6)  where r4 is defined in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  We note that the symmetry of 𝑆3(𝑟) is broken not only by 𝑎 ≠ 0 but also by the finite size of the  hypercubic box and by the fact that we choose antiperiodic (instead of periodic) boundary conditions  in one of the four dimensions for quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' However, as we show in section 5 these boundary effects  are not visible at the current level of precision on the master-field lattices in table 1 considered here,  differently from what we observed on smaller volumes [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  Grid of point sources estimator  The simplest way to compute the correlators introduced in section 2 numerically is to solve the  Dirac equation on a point source, that is, a source spinor that is supported on a single lattice point,  and subsequently perform the suitable contractions of spinor and space-time indeces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' A consequence  3  Hadronic observables from master-field simulations  Marco Cè  of this naive strategy applied on gauge-field configurations with a large volume is that the effort for  each correlator point source scales proportionally with the volume, which is clearly not optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  Indeed, most of the resources are spent in computing the correlator at a distance from the source of  multiple correlation lengths, which has an exponentially suppressed contribution to the physics and  in most of the cases is completely dominated by noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  Instead, we would like to exploit stochastic locality to define estimators that scale efficiently  with the volume and are suitable for master-field applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Taking as an example the radial  correlators ˚𝐶(𝑟) introduced in section 2, let us assume that we are interested in physics that can be  extracted from correlators up to a maximum radial source-sink separation 𝑟max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' [1] sketches a  decomposition of the lattice in space-time domains, or blocks, that are physically large, such that all  the lattice points within an 𝑟max distance from a source point at the centre of each block are within  the same block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' This implies a block size 𝑏 > 2𝑟max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Solving the Dirac equation in each block,  imposing Dirichlet boundary conditions at the block boundary of the gauge field, one can decouple  the computational cost of the estimator from the volume of the global lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' However, this method  introduces boundary effects that can be large for sink points close to the boundaries [1, 7], see also  refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' We leave the exploration of this direction for future work, and we focus here on a simpler  approach that does not require a dedicated correction computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  We introduce a set of lattice points 𝐺 that are separated (on average) by a physical distance  constant in the volume, such that the number of points |𝐺| ∝ 𝑉, that is, it grows proportionally with  the volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' On these point we introduce stochastic sources that satisfy  �  𝜂𝑖(𝑥)𝜂†  𝑗(𝑦)  �  𝜂 = 𝛿𝑖 𝑗𝛿𝑥𝑦𝐼  for 𝑥, 𝑦 ∈ 𝐺,  (7)  where 𝐼 is the identity matrix in spin and colour space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' By contracting at the sink with stochastic  noise corresponding to each coordinate 𝑦 ∈ 𝐺 one obtains |𝐺| ∝ 𝑉 samples of the quark propagator,  one for each 𝑦 ∈ 𝐺, from a single global-lattice inversion that is 𝑂(𝑉) computationally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Each sample  has a spurious contribution of stochastic nature from source points 𝑥 ≠ 𝑦, which is suppressed  by averaging the quark propagator over a number of sources 𝑛src and does not contribute to the  expectation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Mesonic two-point functions that contract two quark propagators require 𝑛src ≥ 2  to obtain an unbiased estimator, that in the case of the pseudoscalar-density two-point function reads  𝐶𝐺  𝑃𝑃(𝑥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' 𝑦) =  1  𝑛src(𝑛src − 1)  ∑︁  𝑖≠𝑗  Re  �  𝜓†  𝑖 (𝑥 + 𝑦)𝜓 𝑗(𝑥 + 𝑦)𝜂†  𝑗(𝑦)𝜂𝑖(𝑦)  �  ,  (8)  where 𝜓𝑖(𝑥) = �  𝑦 𝐷−1(𝑥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' 𝑦)𝜂𝑖(𝑦) and the double sum over 𝑖 ≠ 𝑗 can be computed in 𝑂(𝑛src) cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  In this approach, since |𝐺| ∝ 𝑉, efficient scaling of the solutions of the Dirac equation is  achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Moreover, 𝐺 implicitly realises a domain decomposition by labelling each lattice point  with the closest 𝑦 ∈ 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='1 Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' (8) is in principle valid for any 𝑥 and 𝑦, but if only (𝑥, 𝑦) pairs that are  in the same domain are considered then one can compute efficiently all the |𝐺| ∝ 𝑉 contributions  with a single 𝑂(𝑉) pass over the whole lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' This realises the optimal volume scaling for the  contractions too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' It also lowers the required 𝑛src since the “correct” source 𝑦 is always the closest to  the sink 𝑥 and spurious contributions are further suppressed by the longer source-sink separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='2  1Up to points equidistant from two or more 𝑦 ∈ 𝐺 that require additional conditions to be assigned to a domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  2These spurious contributions are only stochastic and do not modify the expectation value, although we note that they  can have different quantum numbers and decay slower than the correlator being estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  4  Hadronic observables from master-field simulations  Marco Cè  𝑏  𝑟max =  √  2𝑏/2  𝑟max =  √  2𝑏/2  𝑦 ∈ 𝐺  𝑦 ∈ 𝐺  𝑥  Figure 1: Sketch of the estimator with a grid of point sources over a two-dimensional window of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  The set 𝐺 of source points  ∈ 𝐺 is a regular grid with spacing 𝑏 and even point only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' A mesonic two-point  function is evaluated at sink point 𝑥 that is in the domain defined by 𝑦 ∈ 𝐺 and within a distance 𝑟max from 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  One of the spurious contributions from the “wrong” source is shown in light grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  Moreover, it implies 𝑟max = min𝑥,𝑦∈𝐺 |𝑥 − 𝑦|/2, that is, the minimum of the semidistance of points  in 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Therefore, 𝐺 has to be sparse enough for correlators at the relevant radial separations 𝑟 ≤ 𝑟max  to be accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  We study this setup on two sets of a few master fields whose parameters are given in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  The master fields in both sets are hypercubic boxes with equal extent in each dimension denoted  by 𝐿, such that the volume is 𝑉 = 𝐿4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' The 𝐿 = 192𝑎 master fields denoted by B (𝑛cnfg = 2) have  exactly 16 times, twice in each dimension, the volume of the ones with 𝐿 = 96𝑎 in set A (𝑛cnfg = 5)  and otherwise identical parameters, and we can thus define equivalent 𝐺s on both sets and study the  volume scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' We employ U(1) noise that satisfies eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' The simplest choice for 𝐺 is a regular  grid with spacing 𝑏, which matches the domain decomposition proposed in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' [1], with 𝑏 = 48𝑎  being a suitable choice in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' However, the definition of 𝐺 is more flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' In this work, we  employ a grid with only even (or equivalently odd) points, which results in 𝑟max =  √  2𝑏/2 ≃ 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='94𝑎  instead of 𝑏/2 = 24𝑎, at the cost of halving the number of points on the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='3 The total number  of points is thus |𝐺| = (𝐿/𝑏)4/2 that evaluates to 8 and 128 for A and B respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' We fix  𝑛src = 2 and with the current precision we do not observe deviations from the expected behaviour,  especially at 𝑟 close to 𝑟max, that can be attributed to spurious contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Further optimisation  such as systematically and exactly removing the closer spurious contributions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' with hierarchical  probing [11], are not explored here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  The statistics obtained with a single source, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' eight points on each master field in A, is limited  by the need of balancing the density of 𝐺 with a lower limit on the 𝑟max suitable to extract long-range  physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' To increase the statistics we simply propose to recompute eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' (8) on 𝑛shift sources, each  time shifting 𝐺 to have a distinct support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' This is done four times for each direction in the case of A  and twice for each direction in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' An extra factor of two is obtained by pairing each even-only 𝐺  with the corresponding odd-only, leading to 𝑛shift = 512 and 32 for A and B respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Combined  with |𝐺|, the final result is the same number of source points 𝑛point = 4096 for both volumes, on  3This results in a doubled |𝐺|𝑟4max/𝑉 density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Indeed, it corresponds to a 𝐷4 lattice (or equivalently 𝐹4 lattice) that  has the densest known packing of equal spheres in four dimensions [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  5  Hadronic observables from master-field simulations  Marco Cè  a regular grid with spacing 𝑏shift = 12𝑎 and 24𝑎 for A and B respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Ignoring that on the A  lattices source points are on average twice as close and thus potentially more correlated than on B,  in our setup we have same statistics for each gauge field configuration for both A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Crucially,  thanks to the optimal volume scaling of the stochastic grid correlator, this matching statistic has  been obtained at an equivalent computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  Master-field errors  The estimator in section 3 applied to the radial correlator leads to a collection of up to 4096  correlators for each master-field configuration on a regular grid of source points with spacing  𝑏shift = 𝐿/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Applying stochastic locality, the expectation value  � ˚𝐶(𝑟)  � is given up to volume-  suppressed corrections by the translation average  � ˚𝐶(𝑟)  �  = ⟪ ˚𝐶(𝑟)⟫ + 𝑂  �  𝑉−1/2�  = 1  𝑉  ∑︁  𝑦∈𝐺  ˚𝐶(𝑟;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' 𝑦) + 𝑂  �  𝑉−1/2�  (9)  where the 𝑦 in ˚𝐶(𝑟;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' 𝑦) denotes the source point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' The error of this estimator can be estimated applying  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' (2) with O(𝑦) = ˚𝐶(𝑟;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' 𝑦)  �  [⟪ ˚𝐶(𝑟)⟫ −  � ˚𝐶(𝑟)  �  ]2�  = 1  𝑉  ������  ∑︁  |𝑦|≤𝑅  ⟪ ˚𝐶(𝑟;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' 𝑦) ˚𝐶(𝑟;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' 0)⟫𝑐 + 𝑂  �  e−𝑚𝑅�  + 𝑂  �  𝑉−1/2�������  ,  (10)  where again the sum over the source coordinates 𝑦 is performed over the grid of point sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  Finding the optimal 𝑅 to truncate the sum in the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' has a clear analogy with the well-known Γ  method introduced by Wolff to deal with autocorrelation in Monte Carlo time and estimate an error  with less errors [12], and leads to a generalisation of the Madras–Sokal formula for the statistical  error of the error [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' This can be implemented in a resource efficient way by computing  the correlation between grid points with higher-dimensional fast Fourier transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' The optimal  𝑅 depends on the observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' In particular, since each value of the correlator radial source-sink  separation 𝑟 defines a distinct observable with different spacetime support, 𝑅 is a function of 𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  Alternatively, one can apply a four-dimensional binning of the point sources in the grid into  blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' For instance, blocks of size (24𝑎)4 bin 16 point sources on A and only one point source on  B according to the spacing 𝑏shift in table 1, while blocks of size (48𝑎)4 bin 256 and 16 point sources  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' We tested these two bin sizes and observed that this leads to a stable error estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' In  the following, we show results obtained in the more conservative case, that is, with blocks of size  (48𝑎)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  We note that master-field error estimation can be combined with standard methods based on  an ensemble of gauge field configurations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' with a five-dimensional variant of the Γ method in  spacetime coordinates and Monte Carlo time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Explorations in this direction can be found in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  Numerical results  We computed 𝑚 𝜋, 𝑚𝑁 and 𝑓𝜋 using position-space correlators on the sets of master fields  whose parameters are listed in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' The results for these hadronic observables are listed in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  We employed the technique already studied in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' [5] to extract the pion mass 𝑚 𝜋 from the  long-distance behaviour in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' (3a) of the position-space correlator ˚𝐶𝑃𝑃(𝑟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' In those proceedings  6  Hadronic observables from master-field simulations  Marco Cè  Table 2: Numerical results for hadronic observable with errors estimated à la master field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  𝐿/𝑎  𝑎𝑚 𝜋  𝑎𝑚𝑁  𝑎 𝑓 bare  𝜋  A  96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='126 28(33)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='500(6)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='0890(3)  B  192  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='126 01(19)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='487(8)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='0885(4)  5  10  15  20  25  30  r/a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='20  ameff  covariant  one-state fit  two-state fit  5  10  15  20  25  30  r/a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='20  ameff  covariant  one-state fit  two-state fit  Figure 2: Effective mass of the ˚𝐶𝑃𝑃(𝑟) correlator as a function of 𝑟 for master fields in set A (left plot) and  set B (right plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' On top of the data points with master-field errors shown in blue, we show the results of a  one-state fit in a green band and of a two-states fit in a red band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' The thickness of the bands is the statistical  error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  the technique was applied to correlators computed with point sources on an ensemble of gauge field  configurations with a (6 fm)3 space volume, performing a standard error estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Here we have a  larger volume that allows us to use the grid of point sources as described in section 3 and estimate  the error à la master field, see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' On top of the same number of samples 𝑛point = 4096 for  each configuration, we have 5 configurations in set A and 2 in set B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' This means that we have a larger  statistics for the 𝐿 = 96𝑎 master fields from which we expect a ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='58 reduction of the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  The effective mass4 of ˚𝐶𝑃𝑃(𝑟) is shown in the two plots in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' For each set, two fits are  performed: a “one-state” fit having 𝑐𝑃 and 𝑚 𝜋 as free parameters, and a “two-states” one with an  added “excited state” term 𝑎1(𝑚1/𝑟)𝐾1(𝑚1/𝑟) with two extra free parameters 𝑎1 and 𝑚1 > 𝑚 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' We  choose appropriate values for the smaller 𝑟 of the correlator data that enter the fit, with different  choices for one-state and two-states fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Instead, all the data up to largest available 𝑟 = 𝑟max enter  the fit, since we do not observe any boundary effect that constrains us otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' The two fits on  each set give compatible results and the corresponding effective mass is shown in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  From the one-state fits we obtain the results in table 2, which show a good agreement between  the two sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Contrary to the expectation based on 𝑛cnfg, the error is 40 % smaller on set B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' A  possible explanation for this fact is the 𝑏shift = 12𝑎 of the samples of set A, halved with respect to set  B, which can lead to a reduced effective number of samples due to stronger correlations in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  Similarly, we extract 𝑚𝑁 from the two contractions in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' (5) of the position-space nucleon  correlator in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' (3d) as done in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' [5], but employing the techniques of sections 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' The results in  table 2 are from the one-state fits to ˚𝐶 (1)  𝑁 𝑁 (𝑟) with the free parameters 𝑐𝑁 and 𝑚𝑁 , and are compatible  4See eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' (10) in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' [5] for the definition of the effective mass of the radial correlator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  7  Hadronic observables from master-field simulations  Marco Cè  5  10  15  20  25  30  r/a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='8  ameff  covariant, trNN  covariant, tr/xNN  one-state fit, trNN  one-state fit, tr/xNN  two-state fit, trNN  two-state fit, tr/xNN  5  10  15  20  25  30  r/a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='8  ameff  covariant, trNN  covariant, tr/xNN  one-state fit, trNN  one-state fit, tr/xNN  two-state fit, trNN  two-state fit, tr/xNN  Figure 3: Effective mass of the ˚𝐶 (𝑖)  𝑁 𝑁 (𝑟) correlators as a function of 𝑟 for master fields in set A (left plot) and  set B (right plot), where 𝑖 = 1 corresponds to the tr 𝑁𝑁 contraction and 𝑖 = 2 to the tr /𝑥𝑁𝑁 one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' On top of the  data points with master-field errors shown in blue and orange for 𝑖 = 1 and 2 respectively, we show the results  of a one-state fit in green and brown bands and of a two-states fit in red and purple bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' The thickness of the  bands is the statistical error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  with the results of two-states fits with the replacement ˚𝐶𝑁 𝑁 (𝑟) → ˚𝐶𝑁 𝑁 (𝑟)[1+𝑎1(𝑚 𝜋/𝑟)𝐾1(𝑚 𝜋𝑟)]  where 𝑎1 is an extra free parameter and 𝑚 𝜋 is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' The fit to ˚𝐶 (2)  𝑁 𝑁 (𝑟) shows similar results,  although with a slightly larger central value that can be attributed to different discretization effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  The effective masses corresponding to data and fits are shown in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' In the case of 𝑚𝑁 , we  observe a larger error on set B, compatible with the lower statistics and showing no indication of  correlation-in-space effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  We also extract the pion decay constant 𝑓 bare  𝜋  , where the bare indicates that we do not include the  axial-current renormalization factor, from a combined fit of the four correlators ˚𝐶𝑃𝑃, ˚𝐶𝐴𝑃, ˚𝐶 (1)  𝐴𝐴 and  ˚𝐶 (2)  𝐴𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' As fit function we employ the long-distance behaviours derived from eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' 3, which depends  on the free parameters 𝑐𝑃, 𝑐𝐴 and 𝑚 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' As shown from the plots of the ratio between data and fit  functions in figure 4, ˚𝐶𝐴𝑃 approaches the asymptotic behaviour at a smaller value of 𝑟, followed  by ˚𝐶𝑃𝑃 and ˚𝐶 (2)  𝐴𝐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  ˚𝐶 (1)  𝐴𝐴 converges to the asymptotic behaviour at a much larger 𝑟, with the ratio  being initially negative and changing sign around 𝑟 ≈ 14𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' The values of 𝑚 𝜋 obtained from these  combined fits are consistent with the previous fits to only the ˚𝐶𝑃𝑃 correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' The decay constant is  then given by 𝑓 bare  𝜋  = 𝑐𝐴/𝑚 𝜋 and shown in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Like in the case of 𝑚𝑁 , the values on set A and  B are compatible, with a slightly larger error for set B that is consistent with the lower number of  master field configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  Conclusions  We have shown that position-space correlators can be used to extract hadron masses and decay  constants with short-distance and cut-off effects under control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Crucially, the statistical error can  be estimated à la master field, obtaining an efficient scaling of the computational effort with the  increased volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  In this work we studied sphere-averaged radial correlators, but potentially more information is  encoded in correlators as function of four-dimensional coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' This requires understanding  effects that break rotational symmetry at finite lattice spacing and is an interesting topic for further  studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  8  Hadronic observables from master-field simulations  Marco Cè  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
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+page_content='00014  |cA|2  |cA|2 fit  ˚C(2)  AA  Figure 4: Plots of the ratio between correlator data and their fitted long-distance behaviours for master fields  in set A (top row) and set B (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' The amplitude in the denominator is set to one, so that the actual  amplitude for each correlator is shown on the vertical axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' In each row, four plots are shown for ˚𝐶𝑃𝑃, ˚𝐶 (1)  𝐴𝐴  (left column), ˚𝐶𝐴𝑃 and ˚𝐶 (2)  𝐴𝐴 (right column), with the correlator data with master field errors shown in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  The amplitude parameters of the corresponding fit function, which are functions of 𝑐𝑃 and 𝑐𝐴, are shown in  an orange horizontal line with a pale orange error band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  Position-space methods find applications in computations of quantities that go beyond the  simple hadronic quantities considered here, such as for example the hadronic vacuum polarisation  contribution to the anomalous magnetic moment of the muon [16, 17], including the so-called  window contribution [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' The estimators presented here provide a straightforward path to the  computation of this quantities in the master-field paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  Acknowledgements: The research of MB is funded through the MUR program for young researchers “Rita  Levi Montalcini”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' AF acknowledges support by the Ministry of Science and Technology Taiwan (MOST)  under grant 111-2112-M-A49-018-MY2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' JRG acknowledges support from the Simons Foundation through the  Simons Bridge for Postdoctoral Fellowships scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' MTH is supported by UKRI Future Leader Fellowship  MR/T019956/1 and in part by UK STFC grant ST/P000630/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' This work was performed using the DiRAC  Data Intensive service at Leicester, operated by the University of Leicester IT Services, which forms part  of the STFC DiRAC HPC Facility (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='dirac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' The equipment was funded by BEIS capital  funding via STFC capital grants ST/K000373/1 and ST/R002363/1 and STFC DiRAC Operations grant  ST/R001014/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' DiRAC is part of the National e-Infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' We acknowledge PRACE for awarding  us access to SuperMUC-NG at GCS@LRZ, Germany, where some computations were performed Many  9  Hadronic observables from master-field simulations  Marco Cè  simulations were performed on a dedicated HPC cluster at CERN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' We gratefully acknowledge the computer  resources and the technical support provided by these institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  References  [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Lüscher, Stochastic locality and master-field simulations of very large lattices, EPJ Web  Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' 175 (2018) 01002 [1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='09758].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  [2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Giusti and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Lüscher, Topological susceptibility at 𝑇 > 𝑇c from master-field simulations of  the SU(3) gauge theory, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' C 79 (2019) 207 [1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='02062].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Francis, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Fritzsch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Lüscher and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Rago, Master-field simulations of 𝑂(𝑎)-improved  lattice QCD: Algorithms, stability and exactness, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' 255 (2020) 107355  [1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='04533].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  [4] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Fritzsch, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Bulava, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Cè, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Francis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Lüscher and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Rago, Master-field simulations of  QCD, PoS LATTICE2021 (2022) 465 [2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='11544].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Cè, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Bruno, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Bulava, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Francis, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Fritzsch, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Green et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=', Approaching the  master-field: Hadronic observables in large volumes, PoS LATTICE2021 (2022) 383  [2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='15375].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  [6] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Fritzsch, Master-field simulations of QCD and the exponential clover action, PoS  LATTICE2022 247.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Cè, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Giusti and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Schaefer, Domain decomposition, multilevel integration, and  exponential noise reduction in lattice QCD, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' D 93 (2016) 094507 [1601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='04587].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  [8] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Giusti and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Saccardi, Four-dimensional factorization of the fermion determinant in lattice  QCD, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' B 829 (2022) 137103 [2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='02247].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Saccardi and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Giusti, Four-dimensional domain decomposition for the factorization of the  fermion determinant, PoS LATTICE2022 (2022) 386 [2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='06902].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  [10] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Conway and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Sloane, Sphere Packings, Lattices and Groups, Springer (1999),  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='1007/978-1-4757-6568-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Stathopoulos, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Laeuchli and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Orginos, Hierarchical probing for estimating the trace of  the matrix inverse on toroidal lattices, SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' 35 (2013) S299 [1302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='4018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  [12] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Wolff, Monte Carlo errors with less errors, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' 156 (2004) 143  [hep-lat/0306017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  [13] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Madras and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Sokal, The pivot algorithm: A highly efficient Monte Carlo method for the  self-avoiding walk, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' 50 (1988) 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Cè, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Bruno, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Bulava, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Francis, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Fritzsch, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Green et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' in preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  [15] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Lehner, The hadronic vacuum polarization (RBC/UKQCD), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  https://indico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='uk/event/112/contributions/1660/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  [16] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Meyer, Lorentz-covariant coordinate-space representation of the leading hadronic  contribution to the anomalous magnetic moment of the muon, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' C 77 (2017) 616  [1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='01139].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Cè, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Gérardin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Ottnad and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Meyer, The leading hadronic contribution to the  running of the weinberg angle using covariant coordinate-space methods, PoS  LATTICE2018 (2018) 137 [1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='08669].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  [18] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Chao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Meyer and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content=' Parrino, Coordinate-space calculation of the window  observable for the hadronic vacuum polarization contribution to (𝑔 − 2)𝜇, 2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='15581.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}
+page_content='  10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9E4T4oBgHgl3EQflA2r/content/2301.05156v1.pdf'}

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@@ -0,0 +1,1844 @@
+arXiv:2301.04287v1  [math.NT]  11 Jan 2023
+ON INVERTED KLOOSTERMAN SUMS OVER FINITE FIELDS
+XIN LIN AND DAQING WAN
+ABSTRACT. The classical n-variable Kloosterman sums over finite fields are well understood by
+Deligne’s theorem from complex point of view and by Sperber’s theorem from p-adic point of view.
+In this paper, we study the complex and p-adic estimates of inverted n-variable Kloosterman sums,
+addressing a question of N. Katz (1995). We shall give two complex estimates. The first one is
+elementary based on Gauss sums. The second estimate is deeper, depending on the cohomological
+results of Adolphson-Sperber, Denef-Loeser and Fu for twisted toric exponential sums. This deeper
+result assumes that the characteristic p does not divide n + 1. Combining with Dwork’s p-adic
+theory, we also determine the exact p-adic valuations for zeros and poles of the L-function associated
+to inverted n-variable Kloosterman sums in the case p ≡ 1 mod (n + 1). As we shall see, the
+inverted n-variable Kloosterman sum is more complicated than the classical n-variable Kloosterman
+sum in all aspects in the sense that our understanding is less complete, partly because the Hodge
+numbers are now mostly 2 instead of 1.
+1. INTRODUCTION
+Let Fq be the finite field of q elements with characteristic p. Let ψ : Fq → C∗ be a nontrivial
+additive character and let χ1, . . . , χm : F∗
+q → C∗ be multiplicative characters. A classical problem
+in number theory is to give a good estimate for the mixed character sum
+�
+xi∈F∗q
+χ1(f1) · · · χm(fm)ψ(f),
+(1.1)
+where f, f1, . . . , fm ∈ Fq
+�
+x±1
+1 , . . . , x±1
+n
+�
+are Laurent polynomials. Reducing m if necessary, we
+may assume that all the χi’s are non-trivial. Using the Gauss sum
+G(χ) =
+�
+x∈F∗q
+χ(x)ψ(x),
+one obtains the well known relation
+�
+xi,yj∈F∗q
+χ1(y1) · · · χm(ym)ψ (f + y1f1 + · · · + ymfm)
+= G(χ1) · · · G(χm)
+�
+xi∈F∗q
+χ1(f1) · · · χm(fm)ψ(f).
+As the Gauss sums are well-understood, the study of (1.1) is reduced to the study of the following
+type of twisted toric exponential sum
+�
+xi∈F∗q
+χ1(x1) · · · χn(xn)ψ (f(x1, · · · , xn)) ,
+(1.2)
+where some of the χi’s may be trivial. This type of twisted toric exponential sum has been studied
+extensively in the literature, most notably by Adolphson-Sperber [AS87a, AS89, AS90, AS93] via
+Dwork’s p-adic cohomology, and by Denef-Loeser [DL91] and Fu [Fu09,Fu16] via Grothendieck’s
+ℓ-adic cohomology. For the complex estimate, both approaches depend on Deligne’s theorem on
+the Weil conjectures. A sharp estimate is obtained when f is non-degenerate with respect to its
+Newton polyhedron ∆(f). For arbitrary f, the sum is still far from well understood.
+2020 Mathematics Subject Classification. 11T23, 11L05, 11L07, 11S40.
+Key words and phrases. Inverted Kloosterman sums, Exponential sums, L-function, Finite field.
+1
+
+2
+XIN LIN AND DAQING WAN
+An important example of toric exponential sums is the classical n-variable Kloosterman sum,
+where χ1 = · · · = χn = 1 and
+f(x1, · · · , xn) = x1 + · · · + xn +
+b
+x1 · · · xn
+, b ∈ F∗
+q.
+In this case, the complex weights were determined by Deligne’s well known theorem, and the p-
+adic slopes were determined by Sperber’s theorem [Spe80]. It should be noted that for twisted
+n-variable Kloosterman sum (when some of the χi’s are non-trivial), the p-adic slopes are not
+completely determined in general, except in the case n = 1 for which Adolphson-Sperber [AS87b]
+obtained the generic p-adic slopes. If we invert the above Laurent polynomial and consider the
+following rational function
+f(x1, · · · , xn) =
+1
+x1 + · · · + xn +
+b
+x1···xn
+, b ∈ F∗
+q,
+which is no longer a Laurent polynomial, we are led to the so-called inverted Kloosterman sum.
+The study of such sums goes back to N. Katz [Kat95].
+More precisely, in this paper, we study the following twisted inverted n-variable Kloosterman
+sum defined by
+Sn(χ, b) =
+�
+x1···xn+1=b, xi∈F∗q
+x1+···+xn+1̸=0
+χ1(x1) · · · χn+1(xn+1)ψ
+�
+1
+x1 + · · · + xn+1
+�
+=
+�
+x1+···+xn+
+b
+x1···xn ̸=0
+xi∈F∗q
+χ1(x1) · · · χn(xn)χn+1
+�
+b
+x1 · · · xn
+�
+ψ
+�
+1
+x1 + · · · + xn +
+b
+x1···xn
+�
+,
+where b ∈ F∗
+q and n ≥ 1. When n = 1, Katz [Kat95] obtained a sharp upper bound for S1(χ, b).
+This result along with the papers of Angel [Ang96] and Evans [Eva95] proves that finite upper
+half plane graphs are Ramanujan in characteristic 2. In [Kat95], Katz raised the question: what
+can be said for Sn(χ, b) when n ≥ 1? The aim of this paper is to study these inverted n-variable
+Kloosterman sums from both complex and p-adic point of views. As we shall see, this class of sums
+is very interesting as various new features and additional difficulties arise.
+Remark. The exact sum introduced in [Kat95] is the following related sum,
+Tn(χ, b) =
+�
+x1···xn+1=1
+x1+···+xn+1̸=0
+χ1(x1) · · · χn+1(xn+1)ψ
+�
+b
+x1 + · · · + xn+1
+�
+.
+Upon the change of variables xi → bxi, one sees that
+Tn(χ, b) = Sn(χ,
+1
+bn+1 )χ1 · · · χn+1(b).
+Thus, the two families of sums Sn(χ, b) and Tn(χ, b) are essentially equivalent. We work with
+Sn(χ, b) as it is the closer inverted analogue of the classical Kloosterman sum.
+For complex estimate of Sn(χ, b), the best one can hope for would be a square root cancellation
+in the sum Sn(χ, b), i.e.
+Sn(χ, b) = On(q
+n
+2 ).
+As we shall see, this is not true for n > 2 when χ1 = · · · = χn+1, in which case, Sn(χ, b) has the
+main term −qn−1χ1(b) whose exponent n − 1 is larger than the exponent n/2.
+We shall give two different estimates for Sn(χ, b). The first estimate is based on an elementary
+method via Gauss sums which already shows the new feature of a non-trivial main term when
+χ1 = · · · = χn+1. We obtain the following simple estimate for Sn(χ, b).
+
+ON INVERTED KLOOSTERMAN SUMS OVER FINITE FIELDS
+3
+Theorem 1.1. Notations as above. If χ1 = · · · = χn+1, we have
+����Sn(χ, b) + (q − 1)n
+q
+χ1(b)
+���� ≤ q
+n+1
+2 .
+If χi ̸= χj for some i ̸= j, we have
+|Sn(χ, b)| ≤ q
+n+1
+2 .
+For large q, the error term q(n+1)/2 is not the optimal square root cancellation yet. To obtain the
+deeper square root cancellation with error term On(qn/2), we reduce Sn(χ, b) to a certain twisted
+toric exponential sum S∗
+k(χ, f) which can be handled by the results of Adolphson-Sperber, Denef-
+Loeser and Fu. We check that the related Laurent polynomial f is non-degenerate if p does not
+divide n + 1. This gives our second estimate.
+Theorem 1.2. Notations as above. Suppose p ∤ (n + 1). If χ1 = · · · = χn+1, we have
+|Sn(χ, b) + (q − 1)n − (−1)n
+q
+χ1(b)| ≤ (2n + 1)q
+n
+2 .
+If χi ̸= χj for some i ̸= j, we have
+|Sn(χ, b)| ≤ 2(n + 1)q
+n
+2 .
+It is clear that the estimate in Theorem 1.2 is better than the estimate in Theorem 1.1 if q >
+4(n + 1)2. It is expected that Theorem 1.2 (with possibly a better constant) remains true when p
+divides n + 1. But in this singular case, the above toric sum results do not apply and one would
+need a different approach.
+For p-adic slopes, we focus on the simpler untwisted case. When χ1 = · · · = χn+1, we study
+the p-adic valuations for the reciprocal roots and poles of the generating L-function of the inverted
+n-variable Kloosterman sum. We show that the Newton polygon agrees with the Hodge polygon
+if and only if p ≡ 1 mod n + 1. As a consequence, this completely determines the p-adic slope
+sequence when p ≡ 1 mod n + 1. This is described more precisely below.
+Our approach is to reduce the generating L-function to a certain untwisted toric L-function. To
+construct the relationship between the two L-functions, we need to consider the inverted Klooster-
+man sum defined over every finite extension Fqk. Suppose χ1 = · · · = χn+1, the inverted n-variable
+Kloosterman sum over Fqk is defined by
+Sk,n(b) =
+�
+x1+···+xn+
+b
+x1···xn ̸=0
+xi∈F∗
+qk
+ψ
+�
+Trk
+�
+1
+x1 + · · · + xn +
+b
+x1···xn
+��
+,
+where b ∈ F∗
+q and Trk : Fqk → Fq is the trace map. The generating L-function of Sk,n(b) is defined
+by
+Ln(b, T) = exp
+� ∞
+�
+k=1
+Sk,n(b)T k
+k
+�
+.
+Applying some systematic results available for the related toric L-function, we obtain the complex
+and p-adic absolute values for all the reciprocal roots and poles of Ln(b, T) under given restrictions
+on p.
+Theorem 1.3. Suppose p ∤ (n + 1). The L-function is a rational function of the following form:
+Ln(b, T)(−1)n+1 = (1 − T)(n+1)
+n
+�
+j=2
+�
+1 − qj−1T
+�(n
+j)(−1)j−1 2n
+�
+i=1
+(1 − αiT).
+As complex numbers, the reciprocal roots αi satisfy |αi| = q
+n
+2 for all 1 ≤ i ≤ 2n.
+
+4
+XIN LIN AND DAQING WAN
+If p ≡ 1 mod (n + 1), viewing the αi’s as p-adic numbers, the slope sequence {vq(αi)}2n
+i=1 in
+increasing order is given by
+{0, 1, 1, 2, 2, . . . , n − 1, n − 1, n}.
+Our results show that for the inverted n-variable Kloosterman sum, if p does not divide n + 1,
+the primitive middle cohomology has dimension 2n, pure of weight n and with Hodge numbers
+{1, 2, 2, · · · , 2, 1}. This is in contrast to the classical n-variable Kloosterman sum, where the middle
+cohomology has dimension n + 1, pure of weight n and with Hodge numbers {1, 1, 1, · · · , 1, 1}.
+The larger Hodge numbers suggest that new features and additional difficulties would likely arise
+in studying inverted n-variable Kloosterman sums.
+As a corollary of the first part in Theorem 1.3, we get a slightly better bound for Sk,n(b).
+Corollary 1.4. If p ∤ (n + 1), for all integers k ≥ 1, we have
+|Sk,n(b) + (qk − 1)n − (−1)n(qk + 1)
+qk
+| ≤ 2nq
+nk
+2 .
+When k = 1, the exponential sum Sk,n(b) reduces to Sn(χ, b) with χ1 = · · · = χn+1. Explicitly,
+the estimate in Corollary 1.4 is better than the estimate in the first case of Theorem 1.2. We remark
+that the condition p ≡ 1 mod (n + 1) in the second part of Theorem 1.3 is necessary and sufficient
+for the same conclusion to hold. Thus, if p ̸≡ 1 mod (n + 1), the slope sequence will be strictly
+different, but we do not know the exact slope sequence in this case.
+The rest of this paper is organized as follows. In section 2, we review some technical methods
+including Adolphson-Sperber’s theorems and Dwork’s theory on toric exponentials sums. In section
+3, we use these methods to prove the main results.
+We end the introduction by mentioning several recent references [FW21] [Li21] [YZ22] [CL22]
+[LC22] which applied some of the related toric techniques in treating different classes of non-toric
+n variable exponential sums arising from analytic number theory.
+2. PRELIMINARIES ON TORIC EXPONENTIAL SUMS
+2.1. Rationality of the toric L-function. Let f ∈ Fq
+�
+x±1
+1 , . . . , x±1
+n
+�
+be a Laurent polynomial and
+its associated twisted toric exponential sum is defined to be
+S∗
+k(χ, f) =
+�
+xi∈F∗
+qk
+χ1(Nk(x1)) · · · χn(Nk(xn))ψ(Trk(f)),
+(2.1)
+where Trk : Fqk → Fq is the trace map, Nk : Fqk → Fq is the norm map, χ1, . . . , χn : F∗
+q → C∗ are
+multiplicative characters and ψ : Fq → C∗ is a nontrivial additive character. A classical problem in
+number theory is to estimate the absolute values of S∗
+k(χ, f).
+A well known theorem of Dwork-Bombieri-Grothendieck [Dwo60,Bom66,Gro68] says that the
+generating L-function of S∗
+k(χ, f) is a rational function:
+L∗(χ, f, T) = exp
+� ∞
+�
+k=1
+S∗
+k(χ, f)T k
+k
+�
+=
+�d1
+i=1(1 − αiT)
+�d2
+j=1(1 − βjT)
+,
+where all the reciprocal zeros and poles are non-zero algebraic integers. In particular, when all of χi
+are trivial characters, the untwisted toric exponential sum and the associated L-function are denoted
+by S∗
+k(f) and L∗(f, T) respectively.
+Through logarithmic derivatives, we have
+S∗
+k(χ, f) =
+d2
+�
+j=1
+βk
+j −
+d1
+�
+i=1
+αk
+i ,
+k ∈ Z≥1.
+(2.2)
+Thus, the estimate of S∗
+k(χ, f) is reduced to understanding all the absolute values of the reciprocal
+zeros αi and poles βj. Deligne’s theorem on Riemann hypothesis [Del80] describes the bounds for
+
+ON INVERTED KLOOSTERMAN SUMS OVER FINITE FIELDS
+5
+all the absolute values of αi and βj in general. The complex absolute values of reciprocal zeros and
+poles satisfy
+|αi| = qui/2, |βj| = qvj/2, ui ∈ Z ∩ [0, 2n], vj ∈ Z ∩ [0, 2n].
+For non-archimedean absolute values, Deligne proved that |αi|ℓ = |βj|ℓ = 1 when ℓ is a prime and
+ℓ ̸= p. For p-adic absolute values, one has
+|αi|p = q−ri, |βj|p = q−sj, ri ∈ Q ∩ [0, n], sj ∈ Q ∩ [0, n].
+The integer ui (resp. vj) is called the weight of αi (resp. βj) and the rational number ri (resp. sj)
+is called the slope of αi (resp. βj).
+In the past few decades, there has been tremendous interest in determining the weights and slopes
+of the generating L-functions. Without any further condition on the Laurent polynomial f, it is even
+hard to determine the number of reciprocal roots and poles. Most of the existing work about the
+weights and slopes relies on a suitable smoothness condition. For toric exponential sums, this
+usually means the non-degenerate condition, see below for the precise definition.
+Let
+(2.3)
+f(x1, . . . xn) =
+J
+�
+j=1
+ajxVj
+be a Laurent polynomial with aj ∈ F∗
+q and Vj = (v1j, . . . , vnj) ∈ Zn (1 ≤ j ≤ J). The Newton
+polyhedron of f, ∆(f), is defined to be the convex closure in Rn generated by the origin and the
+lattice points Vj (1 ≤ j ≤ J). For δ ⊂ ∆(f), let the Laurent polynomial
+f δ =
+�
+Vj∈δ
+ajxVj
+be the restriction of f to δ.
+Definition 2.1 (non-degenerate). A Laurent polynomial f is called non-degenerate if for each closed
+face δ of ∆(f) of arbitrary dimension which doesn’t contain the origin, the partial derivatives
+�∂f δ
+∂x1
+, . . . , ∂f δ
+∂xn
+�
+have no common zeros with x1 . . . xn ̸= 0 over the algebraic closure of Fq.
+When f is non-degenerate, Adolphson-Sperber [AS89] proved that the untwisted toric L-function
+L∗(f, T)(−1)n−1 is a polynomial and improved the bound for weight.
+Theorem 2.2 ( [AS89]). For any non-degenerate f ∈ Fq[x±1
+1 , . . . , x±1
+n ], the associated L-function
+L∗(f, T)(−1)n−1 is a polynomial of degree n! Vol(∆(f)). Namely,
+L∗(f, T)(−1)n−1 =
+n! Vol(∆(f))
+�
+i=1
+(1 − αiT), αi ̸= 0.
+For any multiplicative characters χi (nontrivial or trivial), the degree and weights of the twisted
+toric L-function are studied and completed by Adolphson-Sperber [AS91, AS93], [DL91]and Fu
+[Fu09] under the non-degeneracy assumption. These results lead to the following bound for the
+twisted toric exponential sum.
+Theorem 2.3 ( [DL91] [AS93] [Fu09]). Let f ∈ Fq
+�
+x±1
+1 , . . . , x±1
+n
+�
+be a Laurent polynomial with
+∆ = ∆(f). If f is non-degenerate, one has
+|S∗
+k(χ1, . . . , χn, f)| ≤ n! Vol(∆)q
+nk
+2 .
+For the slopes of the L-function, the situation is somewhat simpler in the untwisted case, oth-
+erwise, even the description of Adolphson-Sperber’s “Hodge lower bound" is a little cumbersome.
+Thus, the definitions and theories discussed in the following subsections focus on the untwisted
+L-function.
+
+6
+XIN LIN AND DAQING WAN
+2.2. Newton polygon and Hodge polygon. To determine the q-adic slopes of its reciprocal roots,
+we introduce the q-adic Newton polygon.
+Definition 2.4 (Newton polygon). Let L(T) = �n
+i=0 aiT i ∈ 1+TQp[T], where Qp is the algebraic
+closure of Qp. The q-adic Newton polygon of L(T) is defined to be the lower convex closure of the
+set of points {(k, ordq(ak)) |k = 0, 1, . . . , n} in R2.
+Lemma 2.5 ( [Kob84]). Notations as above. Let L(T) = (1−α1T) . . . (1−αnT) be the factoriza-
+tion of L(T) in terms of reciprocal roots αi ∈ Qp. Let λi = ordq αi. If λ is the slope of the q-adic
+Newton polygon of L(T) with horizontal length l, then precisely l of the λi are equal to λ.
+The q-adic Newton polygon of L∗(f, T)(−1)n−1 is denoted as NP(f). Lemma 2.5 relates NP(f)
+to the q-adic valuation of reciprocal roots of toric L-functions. The definition of NP(f) relies on
+the coefficients of L-function, which makes it hard to compute directly. When f is non-degenerate,
+Adolphson and Sperber proved that L∗(f, T)(−1)n−1 is a polynomial and NP(f) has a topological
+lower bound called Hodge polygon, which is easier to determine. Thus, we shall compute Hodge
+polygon and consider when the Newton polygon coincides with this lower bound.
+Let ∆ be an n-dimensional integral polytope containing the origin in Rn. For u ∈ Rn, the
+weight function w(u) represents the smallest non-negative real number c such that u ∈ c∆. Denote
+w(u) = ∞ if such c doesn’t exist. Assume δ is a co-dimension 1 face of ∆ not containing the
+origin. Let D(δ) be the least common multiple of the denominators of the coefficients in the linear
+equation defining δ, normalized to have constant term 1. We define the denominator of ∆ to be the
+least common multiple of all such D(δ) given by:
+D = D(∆) = lcmδD(δ),
+where δ runs over all the co-dimension 1 faces of ∆ that don’t contain the origin. It’s easy to check
+w(Zn) ⊆
+1
+D(∆)Z≥0 ∪ {+∞}.
+For a non-negative integer k, let
+(2.4)
+W∆(k) = #
+�
+u ∈ Zn|w(u) = k
+D
+�
+be the number of lattice points in Zn with weight k/D. Its generating function is known to be a
+rational function of the following form
+∞
+�
+k=0
+W∆(k)tk/D =
+�nD
+k=0 H∆(k)tk/D
+(1 − t)n
+.
+This leads to
+Definition 2.6 (Hodge number). Let ∆ be an n-dimensional integral polytope containing the origin
+in Rn. For a non-negative integer k, the k-th Hodge number of ∆ is defined to be
+H∆(k) =
+n
+�
+i=0
+(−1)i
+�n
+i
+�
+W∆(k − iD).
+(2.5)
+It is known that
+H∆(k) = 0,
+if
+k > nD.
+Based on the Hodge numbers, we define the Hodge polygon of a given polyhedron ∆ ∈ Rn as
+follows.
+Definition 2.7 (Hodge polygon). The Hodge polygon HP(∆) of ∆ is the lower convex polygon in
+R2 with vertices (0,0) and
+Qk =
+�
+k
+�
+m=0
+H∆(m), 1
+D
+k
+�
+m=0
+mH∆(m)
+�
+,
+k = 0, 1, . . . , nD,
+
+ON INVERTED KLOOSTERMAN SUMS OVER FINITE FIELDS
+7
+where H∆(k) is the k-th Hodge number of ∆, k = 0, 1, . . . , nD.
+That is, HP(∆) is a polygon starting from origin (0,0) with a slope k/D side of horizontal length
+H∆(k) for k = 0, 1, . . . , nD. The vertex Qk is called a break point if H∆(k + 1) ̸= 0 where
+k = 1, 2, . . . , nD − 1.
+Note that the horizontal length H∆(k) is the number of lattice points of weight k/D in a certain
+fundamental domain corresponding to a basis of the p-adic cohomology space used to compute
+the L-function. By a theorem of Adolphson-Sperber, the Hodge polygon is a lower bound of the
+corresponding Newton polygon.
+Theorem 2.8 ( [AS89]). For every prime p and non-degenerate Laurent polynomial f with ∆(f) =
+∆ ⊂ Rn, we have
+NP(f) ≥ HP(∆),
+where NP(f) is the q-adic Newton polygon of L∗(f, T)(−1)n−1. Furthermore, the endpoints of
+NP(f) and NP(∆) coincide.
+Definition 2.9 (ordinary). A Laurent polynomial f is called ordinary if NP(f) = HP(∆).
+It is clear that the ordinary property of a Laurent polynomial depends on its Newton polyhedron
+∆ and on the coefficients of f(x). Applying the facial decomposition theorem [Wan93], we reduce
+the ordinary property of f to its smaller pieces which are easier to deal with.
+Theorem 2.10 (Facial decomposition theorem [Wan93]). Let f be a non-degenerate Laurent poly-
+nomial over Fq. Assume ∆ = ∆(f) is n-dimensional and δ1, . . . , δh are all the co-dimension 1
+faces of ∆ which don’t contain the origin. Let f δi denote the restriction of f to δi. Then f is
+ordinary if and only if f δi is ordinary for 1 ≤ i ≤ h.
+2.3. Boundary decomposition theorems. Before describing the boundary decomposition, we ex-
+press the L-function in terms of the Fredholm determinant of an infinite Frobenius matrix via
+Dwork’s trace formula.
+2.3.1. Dwork’s trace formula. Let Qp be the field of p-adic numbers and Ω be the completion of
+Qp. A fixed primitive p-th root of unity in Ω is denoted as ζp. Let π be the element of Qp(ζp)
+satisfies
+∞
+�
+m=0
+πpm
+pm = 0, π ≡ ζp − 1
+mod (ζp − 1)2,
+and
+ordp π =
+1
+p − 1.
+Then, π is a uniformizer of Qp(π) and thus Qp(π) = Qp(ζp). Let Ep(t) be the Artin-Hasse
+exponential series,
+Ep(t) = exp
+� ∞
+�
+m=0
+tpm
+pm
+�
+=
+∞
+�
+m=0
+λmtm ∈ Zp[[x]].
+In Dwork’s terminology, a splitting function θ(t) is defined to be
+θ(t) = Ep(πt) =
+∞
+�
+m=0
+λmπmtm.
+A Laurent polynomial f ∈ Fq[x±1
+1 , . . . , x±1
+n ] is written as
+f =
+J
+�
+j=1
+¯ajxVj,
+where Vj ∈ Zn and ¯aj ∈ F∗
+q. Let aj be the Teichmüller lifting of ¯aj in Ω satisfying aq
+j = aj. Let
+F(f, x) =
+J
+�
+j=1
+θ(ajxVj) =
+�
+r∈Zn
+Fr(f)xr.
+
+8
+XIN LIN AND DAQING WAN
+The coefficients are given by
+Fr(f) =
+�
+u
+(
+J
+�
+j=1
+λujauj
+j )πu1+···+uJ,
+r ∈ Zn,
+where the sum is over all the solutions of the following linear system
+J
+�
+j=1
+ujVj = r
+with
+uj ∈ Z≥0,
+and λm is m-th coefficient of the Artin-Hasse exponential series Ep(t).
+Assume ∆ = ∆(f). Let L(∆) = Zn ∩ C(∆) be the set of lattice points in the closed cone
+generated by origin and ∆. For a given point r ∈ Rn, define the weight function to be
+w(r) := inf
+⃗u
+
+
+
+J
+�
+j=1
+uj|
+J
+�
+j=1
+ujVj = r,
+uj ∈ R≥0
+
+
+ .
+The infinite semilinear Frobenius matrix A1(f) is the following matrix whose rows and columns
+are indexed by the lattice points in L(∆) with respect to the weights:
+A1(f) = (ar,s(f)) = (Fps−r(f)πw(r)−w(s)),
+where r, s ∈ L(∆). The infinite linear Frobenius matrix Aa(f) is defined to be
+Aa(f) = A1(f)Aτ
+1(f) · · · Aτ a−1
+1
+(f),
+where τ is the absolute Frobenius automorphism.
+Dwork’s trace formula can be expressed in terms of the matrix Aa(f) as follows, see [Wan04]
+Theorem 2.11. We have
+(2.6)
+L∗(f, T)(−1)n−1 =
+n
+�
+i=0
+det(I − TqiAa(f))(−1)i(n
+i).
+Equivalently,
+(2.7)
+det(I − TAa(f)) =
+∞
+�
+i=0
+�
+L∗(f, qiT)(−1)n−1�(n+i−1
+i
+) .
+Now it suffices to understand the determinant det(I−TAa(f)). Based on the fact that ordp Fr(f) ≥
+w(r)
+p−1 , we have the following estimate
+ordp(ar,s(f)) ≥ w(ps − r) + w(r) − w(s)
+p − 1
+≥ w(s).
+Let ξ be an element in Ω satisfying ξD = πp−1. Then A1(f) can be written in a block form,
+A1(f) =
+
+
+
+
+
+
+
+
+A00
+ξA01
+· · ·
+ξiA0i
+· · ·
+A10
+ξA11
+· · ·
+ξiA1i
+· · ·
+...
+...
+...
+...
+Ai0
+ξAi1
+· · ·
+ξiAii
+· · ·
+...
+...
+...
+...
+
+
+
+
+
+
+
+
+,
+where the block Aii is a p-adic integral W∆(i)×W∆(i) matrix. This implies that the q-adic Newton
+polygon of det(I −TA1(f)) has a natural lower bound which can be identified with the chain level
+version of the Hodge polygon.
+
+ON INVERTED KLOOSTERMAN SUMS OVER FINITE FIELDS
+9
+Definition 2.12. Let P(∆) be the polygon in R2 with vertices (0, 0) and
+Pk =
+�
+k
+�
+m=0
+W∆(m), 1
+D
+k
+�
+m=0
+mW∆(m)
+�
+,
+k = 0, 1, 2, . . .
+The chain level version of Adolphson-Sperber’s lower bound and the ordinary property are as
+follows.
+Proposition 2.13 ( [AS87a]). The q-adic Newton polygon of det(I − TAa(f)) lies above P(∆).
+Proposition 2.14 ( [Wan04]). Notations as above. Assume f is non-degenerate with ∆ = ∆(f).
+Then NP(f) = HP(∆) if and only if the q-adic Newton polygon of det(I − TAa(f)) coincides
+with its lower bound P(∆).
+2.3.2. Boundary decomposition. Let f ∈ Fq[x±1
+1 , . . . , x±1
+n ] with ∆ = ∆(f), where ∆ is an n-
+dimensional integral convex polyhedron in Rn containing the origin. Let C(∆) be the cone gener-
+ated by ∆ in Rn.
+Definition 2.15. The boundary decomposition
+B(∆) = { the interior of a closed face in C(∆) containing the origin}
+is the unique interior decomposition of C(∆) into a disjoint union of relatively open cones.
+If the origin is a vertex of ∆, then it is the unique 0-dimensional open cone in B(∆). Recall that
+A1(f) = (ar,s(f)) is the infinite semilinear Frobenius matrix whose rows and columns are indexed
+by the lattice points in L(∆). For Σ ∈ B(∆), we define A1(Σ, f) to be the submatrix of A1(f)
+with r, s ∈ Σ. Let f Σ be the restriction of f to the closure of Σ. Then A1(Σ, f Σ) denotes the
+submatrix of A1(f Σ) with r, s ∈ Σ.
+Let B(∆) = {Σ0, . . . , Σh} such that dim(Σi) ≤ dim(Σi+1), i = 0, . . . , h − 1. Define Bij =
+(ar,s(f)) with r ∈ Σi and s ∈ Σj (0 ≤ i, j ≤ h). After a permutation of basis vectors, the infinite
+semilinear Frobenius matrix can be written as
+(2.8)
+A1(f) =
+
+
+
+
+
+B00
+B01
+· · ·
+B0h
+B10
+B11
+· · ·
+B1h
+...
+...
+...
+...
+Bh0
+Bh1
+· · ·
+Bhh
+
+
+
+
+ ,
+where Bij = 0 for i > j. Then det(I −TA1(f)) = �h
+i=0 det(I −TBii) and we have the boundary
+decomposition theorem.
+Theorem 2.16 (Boundary decomposition [Wan93]). Let f ∈ Fq[x±1
+1 , . . . , x±1
+n ] with ∆ = ∆(f).
+Then we have the following factorization
+det(I − TA1(f)) =
+�
+Σ∈B(∆)
+det
+�
+I − TA1(Σ, f Σ)
+�
+.
+2.4. Diagonal local theory. In this subsection, we introduce some non-degenerate and ordinary
+criteria when the Laurent polynomial is diagonal.
+Definition 2.17. A Laurent polynomial f ∈ Fq[x±1
+1 , . . . , x±1
+n ] is called diagonal if f has exactly n
+non-constant terms and ∆(f) is an n-dimensional simplex in Rn.
+Let f be a diagonal Laurent polynomial over Fq. Write
+f(x1, x2, . . . xn) =
+n
+�
+j=1
+ajxVj,
+where aj ∈ F∗
+q and Vj = (v1j, . . . , vnj) ∈ Zn for 1 ≤ j ≤ n. Let ∆ = ∆(f). The vertex matrix of
+∆ is defined to be
+M(∆) = (V1, . . . , Vn),
+
+10
+XIN LIN AND DAQING WAN
+where the i-th column is the i-th exponent of f. Since f is diagonal, M(∆) is invertible.
+Proposition 2.18. Suppose f ∈ Fq[x±1
+1 , . . . , x±1
+n ] is diagonal with ∆ = ∆(f). Then f is non-
+degenerate if and only if p is relatively prime to det(M(∆)).
+Let S(∆) be the solution set of the following linear system
+M(∆)
+
+
+
+
+
+r1
+r2
+...
+rn
+
+
+
+
+ ≡ 0 (mod1),
+ri ∈ Q ∩ [0, 1).
+It’s easy to prove that S(∆) is an abelian group and its order is given by
+|det M(∆)| = n! Vol(∆).
+(2.9)
+Let Sp(∆) denote the prime to p part of S(∆). It is an abelian subgroup of order equal to the
+prime to p factor of det M(∆). In particular, Sp(∆) = S(∆) if p is relatively prime to det M(∆).
+By the Stickelberger theorem for Gauss sums, we have the following ordinary criterion for a non-
+degenerate Laurent polynomial [Wan04].
+Proposition 2.19. A diagonal Laurent polynomial f is ordinary at p if and only if the norm function
+|r| = r1 + · · · + rn on Sp(∆) is stable under the p-action: That is, for each r ∈ Sp(∆), we have
+|r| = |{pr}|, where {pr} is the class of pr in Sp(∆).
+3. PROOF OF THE MAIN THEOREMS
+We prove the main theorems in this section.
+3.1. Proof of Theorem 1.1. Recall that for integer n ≥ 1, the twisted inverted n-variable Kloost-
+erman sum is defined to be
+Sn(χ, b) =
+�
+x1···xn+1=b
+x1+···+xn+1̸=0
+χ1(x1) · · · χn+1(xn+1)ψ
+�
+1
+x1 + · · · + xn+1
+�
+,
+where b ∈ F∗
+q, ψ : Fq → C∗ is a nontrivial additive character and χ1, . . . , χn+1 : F∗
+q → C∗ are
+multiplicative characters. Let χ : F∗
+q → C∗ denote a multiplicative character. By the orthogonality
+of characters, we have
+Sn(χ, b) =
+1
+q(q − 1)
+�
+λ,xi∈F∗q
+�
+u∈Fq
+ψ (u (x1 + · · · + xn+1 − λ)) χ1(x1) · · · χn+1(xn+1)
+× ψ
+�1
+λ
+� �
+χ
+χ
+�x1 · · · χn+1
+b
+�
+=
+1
+q(q − 1)
+�
+λ∈F∗q
+�
+xi∈F∗q
+χ1(x1) · · · χn+1(xn+1)ψ
+� 1
+λ
+� �
+χ
+χ
+�x1 · · · χn+1
+b
+�
++
+1
+q(q − 1)
+�
+λ∈F∗q
+�
+xi∈F∗q
+�
+u∈F∗q
+ψ (u (x1 + · · · + xn+1 − λ))
+× χ1(x1) · · · χn+1(xn+1)ψ
+� 1
+λ
+� �
+χ
+χ
+�x1 · · · χn+1
+b
+�
+=S1 + S2.
+(3.1)
+Then
+S1 =
+1
+q(q − 1)
+�
+λ∈F∗q
+ψ
+�1
+λ
+� �
+χ
+χ−1(b)
+�
+xi∈F∗q
+(χχ1) (x1) · · · (χχn+1) (xn+1)
+
+ON INVERTED KLOOSTERMAN SUMS OVER FINITE FIELDS
+11
+=
+1
+q(q − 1)
+�
+λ∈F∗q
+ψ
+�1
+λ
+� �
+χ
+χ−1(b)
+n+1
+�
+i=1
+
+ �
+xi∈F∗q
+(χχi) (xi)
+
+
+=
+
+
+
+−(q − 1)n
+q
+χ1(b),
+if χ1 = · · · = χn+1,
+0,
+otherwise.
+(3.2)
+If χ is trivial, the Gauss sum G(χ) = −1. If χ is non-trivial, |G(χ)| = √q. Then
+S2 =
+1
+q(q − 1)
+�
+λ,u∈F∗q
+�
+χ
+χ−1(b)
+�
+xi∈F∗q
+(χχ1) (x1)ψ(ux1) · · · (χχn+1) (xn+1)ψ(uxn+1)
+× ψ(−uλ)ψ
+� 1
+λ
+�
+=
+1
+q(q − 1)
+�
+λ,u∈F∗q
+�
+χ
+χ−1(b)χn+1χ1 · · · χn+1(u)ψ(−uλ)ψ
+�1
+λ
+�
+G(χχ1) · · · G(χχn+1)
+=
+1
+q(q − 1)
+�
+χ
+χ−1(b)
+
+ �
+λ∈F∗q
+χn+1χ1 · · · χn+1
+�
+− 1
+λ
+�
+ψ
+� 1
+λ
+�
+ G(χn+1χ1 · · · χn+1)
+× G(χχ1) · · · G(χχn+1)
+=
+1
+q(q − 1)
+�
+χ
+χ−1(b)χn+1χ1 · · · χn+1(−1)G(χn+1χ1 · · · χn+1)G(χn+1χ1 · · · χn+1)
+× G(χχ1) · · · G(χχn+1).
+(3.3)
+Since |G(χ)| ≤ √q, it follows that |S2| ≤ q
+n+1
+2 . Combining (3.1) and (3.2), we can deduce the
+following bounds.
+����Sn(χ, b) + (q − 1)n
+q
+χ1(b)
+���� ≤ q
+n+1
+2 ,
+if χ1 = · · · = χn+1,
+and
+|Sn(χ, b)| ≤ q
+n+1
+2 ,
+if χi ̸= χj for some i ̸= j.
+This proves Theorem 1.1.
+3.2. Proof of Theorem 1.2. The twisted inverted Kloosterman sum Sn(χ, b) has the expression
+Sn(χ, b) =
+�
+x1+···+xn+
+b
+x1···xn ̸=0
+xi∈F∗q
+χ1(x1) · · · χn(xn)χn+1
+�
+b
+x1 · · · xn
+�
+× ψ
+�
+1
+x1 + · · · + xn +
+b
+x1···xn
+�
+=
+�
+z
+�
+x1+···+xn+
+b
+x1···xn
+�
+=1
+z, xi∈F∗q
+χn+1(b) (χ1χn+1)(x1) · · · (χnχn+1)(xn)ψ (z)
+= 1
+q
+�
+z, xi∈F∗q
+y∈Fq
+χn+1(b) (χ1χn+1)(x1) · · · (χnχn+1)(xn)
+× ψ
+�
+z + y
+�
+1 − z
+�
+x1 + · · · + xn +
+b
+x1 · · · xn
+���
+
+12
+XIN LIN AND DAQING WAN
+= χn+1(b)
+q
+
+ �
+z, xi∈F∗q
+(χ1χn+1)(x1) · · · (χnχn+1)(xn)ψ (z) + En(χ, b)
+
+
+=
+
+
+
+
+
+−(q − 1)n
+q
+χ1(b) + 1
+qχ1(b)En(χ, b),
+if χ1 = · · · = χn+1,
+1
+qχn+1(b)En(χ, b),
+if χi ̸= χj for some i ̸= j,
+(3.4)
+where
+En(χ, b) =
+�
+y,z,xi∈F∗q
+(χ1χn+1)(x1) · · · (χnχn+1)(xn)
+× ψ
+�
+z + y
+�
+1 − z
+�
+x1 + · · · + xn +
+b
+x1 · · · xn
+���
+.
+In order to prove Theorem 1.2, it suffices to estimate En(χ, b).
+Let f ∈ Fq[x±1
+1 , . . . , x±1
+n+2] be the Laurent polynomial defined by
+f(x1, · · · , xn+2) = xn+1
+�
+1 − xn+2
+�
+x1 + · · · + xn +
+b
+x1 · · · xn
+��
++ xn+2.
+As defined in (2.1), En(χ, b) is the twisted toric exponential sum associated to f. Let ∆ = ∆(f)
+denote the Newton polyhedron corresponding to f. Clearly, dim ∆ = n + 2 and ∆ has n + 4
+vertices in Rn+2: V0 = (0, · · · , 0)(the origin), V1 = (1, 0, · · · , 0, 1, 1), V2 = (0, 1, · · · , 0, 1, 1),
+. . . , Vn = (0, 0, · · · , 1, 1, 1), Vn+1 = (−1, · · · , −1, 1, 1), Vn+2 = (0, · · · , 0, 1, 0) and Vn+3 =
+(0, · · · , 0, 0, 1). Furthermore, ∆ has exactly 2 co-dimension 1 faces not containing the origin.
+Explicitly, they are
+δ1 : xn+1 = 1
+and
+δ2 : xn+2 = 1.
+Vertices V1, . . . , Vn+2 determine the face δ1 and vertices V1, . . . , Vn+1, Vn+3 determine the face δ2.
+Let M(δi) be the vertex matrix of δi, we have
+M(δ1) =
+
+
+
+
+
+
+
+
+
+1
+0
+· · ·
+0
+−1
+0
+0
+1
+· · ·
+0
+−1
+0
+...
+...
+...
+...
+...
+...
+0
+0
+· · ·
+1
+−1
+0
+1
+1
+· · ·
+1
+1
+1
+1
+1
+· · ·
+1
+1
+0
+
+
+
+
+
+
+
+
+
+,
+M(δ2) =
+
+
+
+
+
+
+
+
+
+1
+0
+· · ·
+0
+−1
+0
+0
+1
+· · ·
+0
+−1
+0
+...
+...
+...
+...
+...
+...
+0
+0
+· · ·
+1
+−1
+0
+1
+1
+· · ·
+1
+1
+0
+1
+1
+· · ·
+1
+1
+1
+
+
+
+
+
+
+
+
+
+.
+(3.5)
+Explicitly, each f δi is diagonal for i = 1, 2. The restriction of f to δi is defined by
+f δi =
+�
+Vj∈δi
+ajxVj.
+Proposition 3.1.
+(i). The denominator D = 1.
+(ii). f is non-degenerate if and only if p ∤ (n + 1).
+(iii). Vol(∆) = 2n + 2
+(n + 2)!.
+Proof. The denominator D = 1 can be deduced immediately from the equation of δi. Since δ1 and
+δ2 are the co-dimension 1 faces of ∆(f) not containing the origin, it suffices to prove f δ1 and f δ2
+are non-degenerate. By Proposition 2.18, f δi is non-degenerate if and only if p is relatively prime
+to det(M(δi)). By formula (3.5),
+det(M(δ1)) = −(n + 1)
+and
+det(M(δ2)) = n + 1.
+(3.6)
+This proves (ii).
+
+ON INVERTED KLOOSTERMAN SUMS OVER FINITE FIELDS
+13
+V0
+V1
+V2
+V3
+V4
+FIGURE 1. ∆ for n = 1
+Let ∆i be the polytope generated by δi and the origin. The facial decomposition of ∆ implies
+that
+Vol(∆) = Vol(∆1) + Vol(∆2).
+By formula (2.9) and (3.6), we obtain (iii).
+□
+Combining Theorem 2.3 with Proposition 3.1, if p ∤ (n + 1), we have
+|En(χ, b)| ≤ (n + 2)! Vol(∆) · q
+n+2
+2
+= 2(n + 1)q
+n+2
+2 ,
+(3.7)
+where p is the characteristic of Fq. Putting (3.4) and (3.7) together, we then obtain the following
+bounds when p ∤ (n + 1).
+|Sn(χ, b) + (q − 1)n
+q
+χ1(b)| ≤ 2(n + 1)q
+n
+2 ,
+if χ1 = · · · = χn+1,
+and
+|Sn(χ, b)| ≤ 2(n + 1)q
+n
+2 ,
+if χi ̸= χj for some i ̸= j.
+In the case χ1 = · · · = χn+1, the twisted sum En(χ, b) becomes the following untwisted toric
+exponential sum
+En(χ, b) =
+�
+y,z,xi∈F∗q
+ψ
+�
+z + y
+�
+1 − z
+�
+x1 + · · · + xn +
+b
+x1 · · · xn
+���
+.
+Since the origin is a vertex of ∆ and the polynomial inside the additive character has no constant
+term, 1 is a trivial eigenvalue of the middle dimensional cohomology. Removing this trivial eigen-
+value from the error term, one gets
+|En(χ, b) − (−1)n+2| ≤ (2n + 1)q
+n
+2 ,
+if χ1 = · · · = χn+1,
+and hence the slightly sharper estimate
+|Sn(χ, b) + (q − 1)n + (−1)n+1
+q
+χ1(b)| ≤ (2n + 1)q
+n
+2 ,
+if χ1 = · · · = χn+1.
+This proves Theorem 1.2.
+3.3. Proof of Theorem 1.3. Similar to formula (3.4), we relate the untwisted inverted Kloosterman
+sum Sk,n(b) to toric exponential sum S∗
+k(f).
+Sk,n(b) =
+�
+x1+···+xn+
+b
+x1···xn ̸=0
+xi∈F∗
+qk
+ψ
+�
+Trk
+�
+1
+x1 + · · · + xn +
+b
+x1···xn
+��
+
+14
+XIN LIN AND DAQING WAN
+=
+�
+z
+�
+x1+···+xn+
+b
+x1···xn
+�
+=1
+z, xi∈F∗
+qk
+ψ (Trk (z))
+= 1
+qk
+�
+z, xi∈F∗
+qk
+y∈Fqk
+ψ
+�
+Trk
+�
+z + y
+�
+1 − z
+�
+x1 + · · · + xn +
+b
+x1 · · · xn
+����
+= −(qk − 1)n
+qk
++ 1
+qk S∗
+k(f).
+(3.8)
+where f is the Laurent polynomial given by
+f(x1, · · · , xn+2) = xn+1
+�
+1 − xn+2
+�
+x1 + · · · + xn +
+b
+x1 · · · xn
+��
++ xn+2
+and
+S∗
+k(f) =
+�
+xi∈F∗
+qk
+ψ
+�
+Trk
+�
+xn+2 + xn+1
+�
+1 − xn+2
+�
+x1 + · · · + xn +
+b
+x1 · · · xn
+����
+.
+The L-functions associated to Sk,n(b) and S∗
+k(f) are defined as
+Ln(b, T) = exp
+� ∞
+�
+k=1
+Sk,n(b)T k
+k
+�
+and
+L∗(f, T) = exp
+� ∞
+�
+k=1
+S∗
+k(f)T k
+k
+�
+.
+It follows from formula (3.8) that
+Ln(b, T) = exp
+� ∞
+�
+k=1
+−
+�
+qk − 1
+�n · T k
+qk · k
+�
+L∗ (f, T/q)
+=
+n
+�
+i=0
+exp
+�
+(−1)n−i+1
+�n
+i
+� ∞
+�
+k=1
+�
+qi−1T
+�k
+k
+�
+L∗ (f, T/q)
+= L∗ (f, T/q)
+n
+�
+i=0
+�
+1
+1 − qi−1T
+�(−1)n−i+1(n
+i)
+.
+(3.9)
+The main purpose of this subsection is to determine the slopes and weights of Ln(b, T). Based
+on formula (3.9), it suffices to consider L∗(f, T) instead. Let ∆ = ∆(f) denote the Newton
+polyhedron corresponding to f. Some of the geometric properties about ∆ have been discussed in
+subsection 3.2. In Proposition 3.1, we proved that f is non-degenerate if and only if p ∤ (n + 1). In
+this case, the L-function L∗(f, T)(−1)n+1 is a polynomial of degree 2n+2. To determine the slopes
+of the reciprocal roots of L∗(f, T)(−1)n+1, we shall compute the Hodge polygon and consider when
+it coincides with the Newton polygon.
+Proposition 3.2. The Laurent polynomial f is ordinary if and only if p ≡ 1 mod (n + 1).
+Proof. By facial decomposition theorem, it suffices to consider f δi for i = 1, 2. Let S(δi) be the
+solution set of the following linear system
+M(δi)
+
+
+
+
+
+r1
+r2
+...
+rn+2
+
+
+
+
+ = u ∈ Zn+2,
+where rj ∈ Q ∩ [0, 1).
+(3.10)
+
+ON INVERTED KLOOSTERMAN SUMS OVER FINITE FIELDS
+15
+For i = 1 and a given point u = (x1, . . . , xn+2)T , linear system (3.10) equals to
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+x1 = r1 − rn+1,
+x2 = r2 − rn+1,
+· · ·
+xn = rn − rn+1,
+xn+1 = r1 + · · · + rn+2,
+xn+2 = r1 + · · · + rn+1,
+where rj ∈ Q ∩ [0, 1).
+(3.11)
+Note that xj ∈ Z, where 1 ≤ j ≤ n + 2. For any r = (r1, . . . , rn+2)T ∈ S(δ1), we have
+r1 = · · · = rn = rn+1 ∈
+Z
+n + 1
+and
+rn+2 = 0.
+Let Sp(δi) denote the prime to p part of S(δi). In particular, Sp(δi) = S(δi) if p ∤ det(M(δi)).
+Suppose p ∤ (n + 1), the norm function |r| and |{pr}| are given by
+|r| = (n + 1)r1
+and
+|{pr}| = (n + 1){pr1}.
+Then |r| on Sp(δ1) is stable under the p-action if and only if p ≡ 1 mod (n + 1). To see this, it
+suffices to consider the unique point r = (
+1
+n+1, · · · ,
+1
+n+1, 0) with norm 1. This condition holds for
+Sp(δ2) through a similar proof. By Proposition 2.19, we obtain Proposition 3.2.
+□
+Theorem 3.3. The n + 2 Hodge numbers of ∆ are {1, 2, 2, · · · , 2, 1}. Namely,
+H∆(0) = 1, H∆(1) = · · · = H∆(n) = 2, H∆(n + 1) = 1.
+Proof. Let ∆i be the polytope generated by the origin and δi. Let u = (x1, . . . , xn+2)T ∈ C(∆i)
+be a lattice point with the weight w(u) = k, where 0 ≤ k ≤ n + 2. For i = 1, 2, consider the linear
+system (3.10). Since f δi is diagonal, system (3.10) has a unique solution r = (r1, . . . , rn+2)T for a
+fixed point u ∈ C(∆i). In this case, the weight is given by
+w(u) = r1 + · · · + rn+2 = |r|.
+When i = 1, the linear equations (3.11) has exact one solution u = (0, . . . , 0, k, k)T . Since
+xn+2 = �n+1
+j=1 rj = k and 0 ≤ rj < 1, we get the restriction 0 ≤ k < n + 1. The Hodge number
+H∆1(k) counts the number of lattice points u of weight k/D in a fundamental domain: That is,
+H∆1(k) =
+�
+1,
+for 0 ≤ k < n + 1,
+0,
+for k ≥ n + 1.
+The generating function of H∆1(k) is
+H1(x) = 1 + x + · · · + xn.
+By formula (2.5), we get the generating function of W∆1(k) as follow.
+W1(x) =
+∞
+�
+k=0
+W∆1(k)xk =
+H1(x)
+(1 − x)n+2 = 1 − xn+1
+(1 − x)n+3 .
+Let H2(x) and W2(x) be the generating function of H∆2(k) and W∆2(k), respectively. Similarly,
+we can prove H2(x) = H1(x) and W2(x) = W1(x). The polytope ∆1
+� ∆2 is determined by
+V1, . . . , Vn+1, whose generating function is given by
+W3(x) =
+∞
+�
+k=0
+W∆1
+� ∆2(k)xk = 1 − xn+1
+(1 − x)n+2 .
+By facial decomposition, we have
+W∆(k) = W∆1(k) + W∆2(k) − W∆1
+� ∆2(k),
+(3.12)
+
+16
+XIN LIN AND DAQING WAN
+which implies
+W(x) =
+∞
+�
+k=0
+W∆(k)xk = W1(x) + W2(x) − W3(x) = 1 + 2x + · · · + 2xn + xn+1
+(1 − x)n+2
+.
+This gives the Hodge numbers of ∆ via formula (2.5), that is,
+H∆(0) = 1, H∆(1) = · · · = H∆(n) = 2, H∆(n + 1) = 1.
+□
+When f is ordinary, the slopes of L∗(f, T)(−1)n+1 can be deduced from Theorem 3.3.
+Theorem 3.4. If p ≡ 1 mod (n + 1), the slope sequence of L∗(f, T)(−1)n+1 is given by
+{0, 1, 1, 2, 2, . . . , n, n, n + 1}.
+Proof. This theorem follows from Lemma 2.5, Proposition 3.2 and Theorem 3.3.
+□
+Note that the converse of this theorem is also true, as Proposition 3.2 shows that the condition
+p ≡ 1 mod (n + 1) is a necessary and sufficient condition for f to be ordinary.
+Now we are ready to consider the weights for the reciprocal roots of L∗(f, T)(−1)n+1.
+Theorem 3.5. Suppose p ∤ (n + 1). We have
+L∗(f, T)(−1)n+1 = (1 − T)(1 − qT)
+2n
+�
+i=1
+(1 − βiT).
+For each 1 ≤ i ≤ 2n, the reciprocal root βi satisfies |βi| = q
+n+2
+2 .
+Proof. Since the origin is a vertex of ∆, we decompose the cone C(∆) via boundary decomposition
+B(∆). Let N(i) be the number of i-dimensional face Σi of C(∆), where 0 ≤ i ≤ dim∆. For
+Newton polyhedron ∆ = ∆(f), we have N(0) = 1 and N(1) = n + 3. Note that Σi is an
+open cone and Σi ∈ B(∆). Let Σi be the closure of Σi. For simplicity, we denote the Fredholm
+determinants as
+D(T) = det (I − TA1(f)) ,
+D◦
+i (T) = det
+�
+I − TA1(Σi, f Σi)
+�
+,
+Di(T) = det
+�
+I − TA1(Σi, f Σi)
+�
+.
+The unique 0-dimensional cone Σ0 is the origin and D0(T) = D◦
+0(T) = 1 − T. When i = 1, each
+f Σ1 can be normalized to x by variable substitution. That is,
+L∗(f Σ1, T) = exp
+� ∞
+�
+k=1
+−T k
+k
+�
+= 1 − T.
+By formula (2.7), we have
+D1(T) =
+∞
+�
+i=0
+�
+L∗ �
+f Σ1, qiT
+��(i
+i)
+= (1 − T) (1 − qT)
+�
+1 − q2T
+�
+· · · .
+Since the only boundary of Σ1 are Σ1 and Σ0, we get D◦
+1(T) after eliminating D◦
+0(T), i.e.,
+D◦
+1(T) = D1(T)
+D◦
+0(T) = (1 − qT)
+�
+1 − q2T
+� ∞
+�
+i=3
+�
+1 − qiT
+�
+.
+Theorem 2.16 shows that D(T) can be expressed as a product of D◦
+i (T) as follow.
+D(T) =
+n+2
+�
+i=1
+N(i)
+�
+j=1
+D◦
+i (T) = (1 − T)(1 − qT)n+3 · · · ,
+
+ON INVERTED KLOOSTERMAN SUMS OVER FINITE FIELDS
+17
+Note that L∗(f, T)(−1)n+1 is a polynomial of degree 2(n + 1) if f is non-degenerate. Combining
+formula (2.6), we obtain
+L∗(f, T)(−1)n+1 =
+D(T)D(q2T)(n+2
+2 ) · · ·
+D(qT)n+2D(q3T)(n+2
+3 ) · · ·
+= (1 − T)(1 − qT)
+2n
+�
+i=1
+(1 − βiT),
+where |βi| = q
+wi
+2 ≤ q
+n+2
+2 . That is, wi ≤ n + 2.
+If βi is a reciprocal root of L∗(f, T)(−1)n+1, the conjugate βi is a reciprocal root of the conjugate
+L-function
+L∗(f, T)
+(−1)n+1
+= L∗(−f, T)(−1)n+1.
+By Theorem 2.8, the Newton polygon and Hodge polygon coincide at the end points. Applying this
+to the product L∗(f, T)(−1)n+1L∗(f, T)
+(−1)n+1
+, we deduce that
+2(n+1)2 = 2(
+n
+�
+i=1
+2i+n+1) = ordq(1·q2 ·
+2n
+�
+i=1
+βiβi) = 2+
+2n
+�
+i=1
+wi ≤ 2+2n(n+2) = 2(n+1)2.
+It follows that the inequality must be an equality, that is, all wi = n + 2.
+□
+Formula (3.9) relates L∗(f, T) to Ln(b, T). The valuations for the reciprocal roots and poles of
+Ln(b, T) follow from Theorem 3.4 and 3.5.
+Theorem 3.6. Suppose p ∤ (n + 1). We have
+Ln(b, T)(−1)n+1 = (1 − T)(n+1)
+n
+�
+j=2
+�
+1 − qj−1T
+�(n
+j)(−1)j−1 2n
+�
+i=1
+(1 − αiT).
+For each 1 ≤ i ≤ 2n, the reciprocal root αi satisfies |αi| = q
+n
+2 . If p ≡ 1 mod (n + 1), the slope
+sequence of the αi’s is given by {0, 1, 1, 2, 2, . . . , n − 1, n − 1, n}.
+Based on weights of toric L-function, we get the following slightly more precise upper bound for
+its associated exponential sum.
+Corollary 3.7. If p ∤ (n + 1), we have
+|Sk,n(b) + (qk − 1)n − (−1)n(qk + 1)
+qk
+| ≤ 2nq
+nk
+2 .
+Proof. Theorem 3.5 implies that
+|S∗
+k(f) − (−1)n(qk + 1)| ≤ 2nq
+(n+2)k
+2
+.
+Combining formula (3.8), we get the bound for Sk,n(b).
+□
+Remark. We finish this paper with two open problems on the estimates of inverted Kloosterman
+sums. If n + 1 is divisible by p, the related Laurent polynomial f is degenerate and thus the results
+for toric exponential sums are not tenable. In this case, it is an open problem to determine the
+optimal square root cancellation for Sn(χ, b) in general. The case n = 1 with p = 2 is already
+handled in [Kat95]. The second question concerns the q-adic slope sequence. If p is not equivalent
+to 1 modulo n + 1, the Newton polygon corresponding to f is strictly above its Hodge polygon.
+Under this assumption, can one still obtain the explicit q-adic slope sequence?
+REFERENCES
+[Ang96] Jeff Angel, Finite upper half planes over finite fields, Finite Fields Appl. 2 (1996), no. 1, 62–86. MR 1371720
+[AS87a] Alan Adolphson and Steven Sperber, Newton polyhedra and the degree of the L-function associated to an
+exponential sum, Invent. Math. 88 (1987), no. 3, 555–569. MR 884800
+[AS87b]
+, Twisted Kloosterman sums and p-adic Bessel functions. II. Newton polygons and analytic continua-
+tion, Amer. J. Math. 109 (1987), no. 4, 723–764. MR 900037
+[AS89]
+, Exponential Sums and Newton Polyhedra: Cohomology and Estimates, Ann. Math. 130 (1989), no. 2,
+367–406. MR 1014928
+
+18
+XIN LIN AND DAQING WAN
+[AS90]
+, Exponential sums on (Gm)n, Invent. Math. 101 (1990), no. 1, 63–79. MR 1055711
+[AS91]
+, On twisted exponential sums, Math. Ann. 290 (1991), no. 4, 713–726. MR 1119948
+[AS93]
+, Twisted exponential sums and Newton polyhedra, J. Reine Angew. Math. 443 (1993), 151–177.
+MR 1241131
+[Bom66] Enrico Bombieri, On exponential sums in finite fields, Les Tendances Géom. En Algèbre et Théorie Des Nom-
+bres, Éditions du Centre National de la Recherche Scientifique (CNRS), Paris, 1966, pp. 37–41. MR 0204413
+[CL22]
+Chao Chen and Xin Lin, L-functions of certain exponential sums over finite fields, Math. Z. 300 (2022), no. 2,
+1851–1871. MR 4363799
+[Del80]
+Pierre Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. (1980), no. 52, 137–252.
+MR 601520
+[DL91]
+J. Denef and F. Loeser, Weights of exponential sums, intersection cohomology, and Newton polyhedra, Invent.
+Math. 106 (1991), no. 1, 275–294. MR 1128216
+[Dwo60] Bernard Dwork, On the rationality of the zeta function of an algebraic variety, Amer. J. Math. 82 (1960),
+631–648. MR 140494
+[Eva95] Ronald Evans, Spherical functions for finite upper half planes with characteristic 2, Finite Fields Appl. 1
+(1995), no. 3, 376–394. MR 1341954
+[Fu09]
+Lei Fu, Weights of twisted exponential sums, Math. Z. 262 (2009), no. 2, 449–472. MR 2504886
+[Fu16]
+, ℓ-adic GKZ hypergeometric sheaves and exponential sums, Adv. Math. 298 (2016), 51–88.
+MR 3505737
+[FW21]
+Lei Fu and Daqing Wan, On Katz’s (A, B)-exponential sums, Q. J. Math. 72 (2021), no. 3, 773–793.
+MR 4310299
+[Gro68] Alexander Grothendieck, Formule de Lefschetz et rationalité des fonctions L [see 1608788], Dix Exposés Sur
+La Cohomologie Des Schémas, Adv. Stud. Pure Math., vol. 3, North-Holland, Amsterdam, 1968, pp. 31–45.
+MR 3202554
+[Kat95]
+Nicholas M. Katz, A note on exponential sums, Finite Fields Appl. 1 (1995), no. 3, 395–398. MR 1341955
+[Kob84] Neal Koblitz, p-adic numbers, p-adic analysis, and zeta-functions, second ed., Graduate Texts in Mathematics,
+vol. 58, Springer-Verlag, New York, 1984. MR 754003
+[LC22]
+Xin Lin and Chao Chen, L-functions of certain exponential sums over finite fields II, J. Number Theory 241
+(2022), 198–220. MR 4472439
+[Li21]
+Jiyou Li, Newton polygons of L-functions associated to Deligne polynomials, Finite Fields Appl. 75 (2021),
+Paper No. 101880, 10. MR 4272552
+[Spe80]
+Steven Sperber, Congruence properties of the hyper-Kloosterman sum, Compositio Math. 40 (1980), no. 1,
+3–33. MR 558257
+[Wan93] Daqing Wan, Newton polygons of zeta functions and L functions, Ann. Math. 137 (1993), 249–293.
+MR 1207208
+[Wan04]
+, Variation of p-adic Newton polygons for L-functions of exponential sums, Asian J. Math. 8 (2004),
+no. 3, 427–472. MR 2129244
+[YZ22]
+Liping Yang and Hao Zhang, Generic Newton polygons for L-functions of (A, B)-exponential sums, Finite
+Fields Appl. 78 (2022), Paper No. 101980, 20. MR 4349885
+DEPARTMENT OF MATHEMATICS, SHANGHAI MARITIME UNIVERSITY, SHANGHAI 201306, PR CHINA.
+Email address: xlin1126@hotmail.com
+DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, IRVINE, CA 92697-3875 USA.
+Email address: dwan@math.uci.edu
+

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+Predator Extinction arose from Chaos of the Prey: the Chaotic
+Behavior of a Homomorphic Two-Dimensional Logistic Map in the
+Form of Lotka-Volterra Equations
+Wei Shan Lee∗, Hou Fai Chan, Ka Ian Im, Kuan Ieong Chan, and U Hin Cheang
+Pui Ching Middle School Macau
+Macao Special Administrative Region, People’s Republic of China.
+Abstract
+A two-dimensional homomorphic logistic map that preserves features of Lotka-Volterra Equations
+was proposed.
+In order to examine the Lotka-Volterra chaos, in addition to ordinary iteration plots
+of population, Lyapunov exponents either calculated directly from eigenvalues of Jacobian of the 2D
+logistic mapping, or from time-series algorithms of both Rosenstein and Eckmann et al. were calculated,
+among which discrepancies were compared. Bifurcation diagrams may be divided into five categories
+depending on different topological shapes, among which flip bifurcation and Neimark-Sacker bifurcation
+were observed, the latter showing closed orbits around limit circles in the phase portrait and phase space
+diagram. Our model restored the 1D logistic map of the prey at the absence of the predator, as well as the
+normal competing behavior between two species when the initial population of the two is equal. In spite
+of the possibility for two species going into chaos simultaneously, it is also possible that with the same
+inter-species parameters as normal but with predator population 10 times more than that of the prey,
+under certain growth rate the latter becomes chaotic, and former dramatically reduces to zero, referring
+to total annihilation of the predator species. Interpreting humans as the predator and natural resources
+as the prey in the ecological system, the aforementioned conclusion may imply that not only excessive
+consumption of the natural resources, but its chaotic state triggered by overpopulation of humans may
+also backfire in a manner of total extinction on human species. Fortunately, a little chance may exist for
+survival of human race, as isolated fixed points in bifurcation diagram of the predator reveals.
+Keywords Chaos, Neimark-Sacker Bifurcation, Logistic Map, Lyapunov Exponents, Lotka-Volterra Equa-
+tions, Extinction of Species
+1
+Introduction
+Understanding interactions between human beings and natural resources plays an important role in estab-
+lishing sustainable economy and society. Relationships of these two may be studied by the prey and predator
+model after we realize that humans beings may be regarded as the predator while natural resources may be
+thought of as the prey[1]. Afterwards, researches on the prey-predator models may be implemented to this
+field instinctively[2]-[3].
+Generally speaking, there are two main approaches of studies in the literature to this prey and predator
+model. The first is to study differential equations, while the other is to study the iterations in difference
+equations, whose forms may be inspired by directly applying the forward Euler’s Scheme to acquire coun-
+terpart of the former[4]-[10]. The discrete model could be more promising than the continuous one, because
+it has more abundant dynamic characteristics in chaotic behaviors[8], whereas it would be more difficult for
+solutions to continuous models to reach chaos in low dimensional cases. Taking some examples about the first
+approach, studies[11]-[12] have performed on chaos of Lotka-Volterra differential equations with dimensions
+higher than three, and researchers[13] claimed that it is impossible to reach chaos for two species in the
+form of differential Lotka-Volterra Equations, whose general solutions were obtained in sinusoidal forms by
+Evans and Findley[14]. Additionally, based on the Lotka-Volterra model, Dunbar[15] confirmed the existence
+∗email: wslee@g.puiching.edu.mo
+1
+arXiv:2301.11669v1  [nlin.CD]  27 Jan 2023
+
+of traveling wave solutions for two reaction diffusion systems. Besides that, Das and Gupta[16] proposed
+solutions to the fractional-order time derivative Lotka-Volterra equations using an analytical approach for
+nonlinear problems known as the homotopy perturbation method (HPM).
+On the other hand, there are also several studies on the discrete difference equations.
+For instance,
+Bessoir and Wolf [17] made pioneering contributions to the application of 1D logistic equation on biological
+and ecological studies. The same equation was also used to interpret, analyze and predict data according to
+the COVID-19 by many researchers[18]. Mareno and English[19] implemented the 1D logistic equation to
+the coupled 2D logistic one, and demonstrated that for large growth rate the system underwent a Neimark-
+Sacker bifurcation. Li et al.[20] imposed an equal individual effect intensity, corresponding to equal growth
+rate in the 1D logistic map, on the two oligopolists in the homomorphic Kopel model and observed three
+different kinds of bifurcation. Furthermore, Elhadi and Sprott[21] proposed a two-dimensional mapping, one
+of which is the ordinary 1D logistic map while the other consists of a perturbation term of the former and is
+also modulated by the first. Shilnikov and Rulkov[22] studied chaos behaviors in two-dimensional difference
+equations that reproduced spike-bursting activities in biological neurons, improving further on the previous
+research based on the three-dimensional system of ODEs. In spite of applying the forward Euler’s Scheme to
+acquire the difference equation, researchers also made use of exponential forms corresponding to solutions on
+the differential equations. For example, Ishaque et al.[23] studied a three dimensional predator-prey-parasite
+model with an exponential form describing interactions among healthy or infected Tilapia fish as the prey, and
+Pelican birds as the predator. Tassaddiq et al.[24] worked on discrete-time exponential difference equation
+of Leslie-Gower predator-prey model together with a Holling type III functional response, and indicated
+the advantage on this type of discretization method. Previous study[25] suggested a heteromorphic term
+describing the decreasing effects on the predator that was only linear to the population of that species,
+contrary to the corresponding quadratic term in the prey. Hassell et al[26] applied the predator-prey model
+to insect parasitoids and anthropods, and found out that local movements of the two species may cause
+extermination of the entire ecological system with chaos, and it is difficult to maintain population stability
+for large growth rate of anthropods. Besides, researchers[27] also pointed out that human misbehavior may
+be the reason for an ecological system to go into chaotic states.
+However, there is no convincing reason for the prey and the predator to have different forms in the
+difference equations.
+Intuition in mathematical symmetry naturally came to our mind that a successful
+predator-prey difference model should resemble the symmetry structure as in Lotka-Volterra differential
+equations. Moreover, solutions to Lotka-Volterra Equations in sinusoidal forms cannot explain extinction of
+species.
+We proposed homomorphic two-dimensional logistic maps that preserve both forms of Lotka-Volterra
+Equations and the 1D logistic equation. In our model, we conjectured a quadractic form in both corresponding
+terms of the prey and the predator, treating both species on the equal stance. Structures of Bifurcation
+diagrams showed that there could be six different categories in our dynamic system. For each categories, we
+examined population iterations, phase portraits, phase space diagrams, and topological types of fixed points.
+Lyapunov exponents either calcaulted from eigenvalues of Jacobian of the 2D mapping or from time-series
+algorithms either Rosenstein[28] or Eckmann et al.[29] were also calculated. Comparisons among those results
+were also discussed.
+The advantages of our model include the following.
+First, we may be able to establish a standard
+bifurcation diagram of 1D logistic map about the prey with nonzero initial predator population, growth rates
+in both species, and predation parameters. Second, our model may also describe the normal behavior of rise
+and fall on the population of the two species when interacting with each other. Third, besides simultaneous
+chaos in both species, the main discovery in our research was that the predator may go extinct under the
+circumstance of chaos in the prey that the predator overpopulation should be blamed.
+2
+Theorems
+We first review the one-dimensional logistic equation and the two-dimensional Lotka-Volterra Equations,
+comparing the similarity and difference between the two sets of equations, which inspires us on the idea
+to establish two-dimensional logistic equations that maintain important features about both of the above
+equations.
+2
+
+2.1
+Review on the 1D logistic equation and Litka-Volterra Equations
+To begin with, the one-dimensional logistic equation may be written as
+xn+1 = µ0xn(1 − xn),
+(1)
+where µ0 denotes to the growth rate.
+On the other hand, the two-dimensional Lotka-Volterra Equations[30], [31] describe interactions between
+the prey and predator in an environment where there is sufficient food supply for the prey, whose only natural
+enemy is the predator. The formula may be written as follows:
+�
+�
+�
+�
+�
+�
+�
+dx
+dt = µ0x − µ1xy;
+dy
+dt = −ν0y + ν1xy,
+(2a)
+(2b)
+where µ0, µ1, ν0, and ν1 are all positive inter-species parameters. x denotes population of the prey while
+y, population of the predator, both being positive real numbers. µ0 and ν1 are, respectively, growth rate
+of the prey and the predator. While µ1 refers to the parameter for predation to occur upon the prey at
+the presence of predator, ν0 refers to all effects that decrease population of the predator, which may include
+disease, death, or emigration[32]. With forward Euler’s scheme, one may immediately write the difference
+version of Eq.(2) as follows
+�xn+1 − xn = µ0xn − µ1xnyn;
+yn+1 − yn = −ν0yn + ν1xnyn.
+(3a)
+(3b)
+However, there is an obvious drawback about the above simultaneous equations: it does not preserve the
+feature of 1D logistic equation, because the first term on the right hand side is either linear to xn or yn,
+whereas the right hand side in Eq.(1) is quadratic to xn.
+We now discuss our idea on establishing the two-dimensional map that resembles interaction terms of the
+prey and the predator as in Lotka-Volterra. First, we may rewrite the linear term of x on the right hand side
+of Eq.(2a) into quadratic, which looks like µ0x(1 − x), together with the homomorphic corresponding term
+in Eq.(2b) as ν0y(1 − y). Thus, the modified Lotka-Volterra Equations[33] are
+�
+�
+�
+�
+�
+�
+�
+dx
+dt = µ0x(1 − x) − µ1xy;
+dy
+dt = −ν0y(1 − y) + ν1xy,
+(4a)
+(4b)
+whose resemblance in the form of difference equations are, therefore,
+�xn+1 − xn = µ0xn(1 − xn) − µ1xnyn;
+yn+1 − yn = −ν0yn(1 − yn) + ν1xnyn.
+(5a)
+(5b)
+Even though it seems more reasonable to direct resemblance of Lotka-Volterra Equations, Eq.(5) fails to
+restore Eq.(1) when parameters other than µ0 are all set to be zero. Fortunately, we may modify this by
+dropping xn and yn terms on the left hand side of the above equations
+�xn+1 = µ0xn(1 − xn) − µ1xnyn;
+yn+1 = −ν0yn(1 − yn) + ν1xnyn,
+(6a)
+(6b)
+which is the desired form. We divided into two cases to study further about properties of Eq.(5) and Eq.(6)
+in Sec. 2.3 and Sec. 2.4. But before that, in next subsection we first discuss a very important lemma that
+allows us to study stability behaviors of fixed points.
+3
+
+2.2
+Stability of fixed points
+Suppose a mapping of two-dimensional iterations xn+1 and yn+1 are written as xn+1 = f(x, y) and yn+1 =
+g(x, y). The Jacobian is, therefore,
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+J = ∂(f, g)
+∂(x, y)
+=
+� ∂f
+∂x
+∂f
+∂y
+∂g
+∂x
+∂g
+∂y
+�
+,
+(7a)
+(7b)
+whose eigenvalus are ω0 and ω1. It is well known[6], [8] that a fixed point may be divided into the following
+four topological types based on their stability behaviors. First, it could be a sink and locally asymptotic
+stable if eigenvalues of Eq.(7) satisfy |ω0| < 1 and |ω1| < 1. Second, it could be a source and locally unstable
+if eigenvalues satisfy |ω0| > 1 and |ω1| > 1. Third, a fixed point could be a saddle if one of the absolute
+values of the eigenvalues is greater than 1 while the other is smaller than 1. At last, a fixed point could be
+non-hyperbolic if one of the absolute values of the eigenvalues is equal to 1. The stability of a non-hyperbolic
+fixed point is fragile[34], which means that its stability is easily influenced by the small nonlinear terms.
+Instead of calculating the range of eigenvalues directly, most of the time it is more convenient to work with
+the quadratic formula consisting of eigenvalues ω0 and ω1, namely, Ω(ω) = ω2 −Tr(J)ω +det(J), where Tr(J)
+and det(J) are trace and determinant of Jacobian in Eq.(7), respectively, and there could be a correspondence
+on the stability behavior around a fixed point between the roots of the quadratic formual, ω0 and ω1, which
+are also eigenvalues of Jacobian, through the following Lemma
+Lemma 1. Let Ω(ω) = ω2 − Tr(J)ω + det(J), be a quadratic formula where where Tr(J) and det(J) are trace
+and determinant of Jacobian in Eq.(7), respectively. Then
+1. If Ω(1) > 0, then
+(a) |ω0| < 1 and |ω1| < 1 and hence the fixed point is a sink if and only if Ω(−1) > 0 and det(J) < 1;
+(b) |ω0| > 1 and |ω1| > 1 and hence the fixed point is a source if and only if Ω(−1) > 0 and det(J) > 1;
+(c) One of |ω0| and |ω1| is smaller than 1 while the other greater than 1 and hence the fixed point is
+a saddle if and only if Ω(−1) < 0;
+(d) Either |ω0| or |ω1| is equal to 1 and hence the fixed point is a non-hyperbolic whenever
+i. ω0 = −1 and ω1 ̸= −1 if and only if Ω(−1) = 0 and Tr(J) ̸= 2.
+ii. ω0 and ω1 are a pair of complex conjugates and |ω0| = |ω1| = 1 if and only if |Tr(J)| < 2 and
+det(J) = 1.
+iii. ω0 = ω1 = −1 if and only if Ω(−1) = 0 and Tr(J) = 2.
+2. If Ω(1) = 0, then either |ω0| or |ω1| has to be equal to 1. Therefore the fixed point is a non-hyperbolic.
+Absolute value of the other root is greater than, equal to, or smaller than 1 if and only if, correspondingly,
+absolute value of det(J) is greater than, equal to, or smaller than 1.
+3. If Ω(1) < 0, then either |ω0| or |ω1| has to be real and greater than 1. Therefore, the fixed point is a
+saddle. Further,
+(a) the other root is smaller or equal to −1 if and only if, correspondingly, Ω(−1) < −1 or Ω(−1) = −1.
+(b) absolute value of the other root is smaller than 1 if and only if Ω(−1) > 0.
+1 makes it easier for us to study analytically the stability of fixed points.
+2.3
+Properties of Eq.(5)
+Setting up xn+1 = f(x, y) and yn+1 = g(x, y), the two-dimensional logistic equations in Eq.(5) have the
+mappings
+�f(x, y) = µ0x(1 − x) − µ1xy + x;
+g(x, y) = −ν0y(1 − y) + ν1xy + y.
+(8a)
+(8b)
+4
+
+Eq.(8) has Jacobian, as indicated in Eq.(7),
+J =
+�µ0(1 − 2x) − µ1y + 1
+−µ1x
+ν1y
+ν0(−1 + 2y) + ν1x + 1
+�
+,
+(9)
+with eigenvalues ω0 and ω1 being, respectively,
+�
+�
+�
+�
+�
+�
+�
+ω0 = −µ0x + ν0y + 1 + 1
+2(xν1 − yµ1) + 1
+2(µ0 − ν0) + ω
+2
+ω1 = −µ0x + ν0y + 1 + 1
+2(xν1 − yµ1) + 1
+2(µ0 − ν0) − ω
+2
+(10a)
+(10b)
+where
+ω =
+�
+4(ν0 + µ1
+2 )2y2 +
+��
+(4µ0 − 2ν1)µ1 + 8ν0(µ0 + ν1
+2 )
+�
+x − 4(ν0 + µ0)(ν0 + µ1
+2 )
+�
+y
+(11)
++ 4
+�
+(µ0 + ν1
+2 )x − µ0
+2 − ν0
+2
+�2� 1
+2
+Further, fixed points, at which pairs of x and y stay still irrespective of time-series iterations[34], are those
+pairs of points (x∗, y∗) such that
+�x∗ = µ0x∗(1 − x∗) − µ1x∗y∗ + x∗
+y∗ = ν0y∗(1 − y∗) − ν1x∗y∗ + y∗,
+(12a)
+(12b)
+from which four pairs of fixed points (x∗, y∗) may be derived as
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+E0 = (0, 0)
+E1 = (0, 1)
+E2 = (1, 0)
+E3 =
+� ν0(µ1 − µ0)
+−µ0ν0 + µ1ν1
+,
+µ0(ν1 − ν0)
+−µ0ν0 + µ1ν1
+�
+,
+(13a)
+(13b)
+(13c)
+(13d)
+provided that the denominator in Eq.(13d) is not zero. On the contrary, however, when µ0ν0 = µ1ν1, Eq.(5)
+has fixed points E0, E1,and E2.
+Keeping Lamma 1 in Sec. 2.2 in mind, we may be able to examine the topological type of each fixed
+point in Eq.(13) as in Theorem 1:
+Theorem 1. The topological types of fixed points in Eq.(13) are
+1. For E0 = (0, 0),
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Ω(1) = −µ0ν0
+Ω(−1) = (2 + µ0)(2 − ν0)
+det(J) = (1 + µ0)(1 − ν0)
+Tr(J) = 2 + µ0 − ν0.
+Because Ω(1) < 0, therefore, E0 is always a saddle.
+2. For E1 = (0, 1),
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Ω(1) = ν0(µ0 − µ1)
+Ω(−1) = (2 + ν0)(2 + µ0 − µ1)
+det(J) = (1 + ν0)(1 + µ0 − µ1)
+Tr(J) = 2 + µ0 − µ1 + ν0.
+In this case,
+5
+
+(a) E1 cannot be a sink;
+(b) if µ0 > µ1, then E1 is a source;
+(c) if µ0 < µ1, then E1 is a saddle.
+(d) if µ0 = µ1, then E1 is a non-hyperbole;
+3. For E2 = (1, 0),
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Ω(1) = −µ0(ν1 − ν0)
+Ω(−1) = (2 − µ0)(2 + ν1 − ν0)
+det(J) = (1 − µ0)(1 + ν1 − ν0)
+Tr(J) = 2 − µ0 + (ν1 − ν0).
+In this case,
+(a) if µ0 < 2 and ν1 < ν0 < ν1 + 1 and ν1 < ν0, or µ0 < 2 and ν0 = ν1 + 1, or µ0 < 2 and
+ν1 + 1 < ν0 < ν1 + 2, then E2 is a sink;
+(b) if µ0 > 2 and ν0 > ν1 + 2, then E2 is a source;
+(c) if µ0 > 2 and ν1 < ν0 < ν1 + 2, or µ0 < 2 and ν0 > ν1 + 2, or ν1 > ν0, then E2 is a saddle;
+(d) if ν1 = ν0, then E2 is a non-hyperbole.
+4. For E3 =
+�
+ν0(µ1−µ0)
+−µ0ν0+µ1ν1 ,
+µ0(ν1−ν0)
+−µ0ν0+µ1ν1
+�
+,
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Ω(1) = −µ0ν0(ν0 − ν1)(µ0 − µ1)
+µ0ν0 − µ1ν1
+Ω(−1) = −ν0(ν0 − ν1 + 2)µ2
+0 + ν0(µ1 + 2)(ν0 − ν1 + 2)µ0 − 4µ1ν1
+µ0ν0 − µ1ν1
+det(J) = −ν0(ν0 − ν1 + 1)µ2
+0 + ν0(µ1 + 1)(ν0 − ν1 + 1)µ0 − µ1ν1
+µ0ν0 − µ1ν1
+Tr(J) = −µ2
+0ν0 + ν0(ν0 + µ1 − ν1 + 2)µ0 − 2µ1ν1
+µ0ν0 − µ1ν1
+.
+In this case,
+(a) if µ0 < µ1 and µ2
+0ν0 < µ2
+1ν1, or µ0 > µ1 and µ2
+0ν0 > µ2
+1ν1, then it is a saddle;
+(b) if µ2
+0/µ2
+1 = ν1/ν0, we have a non-hyperbole in the interior region.
+In order to plot bifurcation diagrams, we further assume that µ1 = αµ0, ν0 = βµ0, and ν1 = γµ0, where
+α, β, and γ are parameters. In this case the original µ1, ν0, and ν1 vary with µ0. Under this circumstance,
+the nontrivial fixed points E3 becomes
+E3 =
+�β(α − 1)
+αγ − β , γ − β
+αγ − β
+�
+,
+which is independent of the growth rate parameter µ0. Eigenvalues of Jacobian at E3 in Eq.(9) is
+�
+�
+�
+�
+�
+�
+�
+�
+�
+ω0 =
+1
+2αγ − 2β
+�
+Ξ0 + αγ − β
+|αγ − β|Ξ1
+�
+ω1 =
+1
+2αγ − 2β
+�
+Ξ0 − αγ − β
+|αγ − β|Ξ1
+�
+,
+(18a)
+(18b)
+where
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Ξ0 = 2αγ − β2µ0 −
+�
+2 + (α − γ − 1)µ0
+�
+β
+Ξ1 = µ0
+�
+β
+�
+(4βγ − 4γ2 + β)α2 − (2β2 + 2βγ − 4γ2 + 2β)α + β(β − γ + 1)2
+�� 1
+2
+(19a)
+(19b)
+6
+
+2.4
+Properties of Eq.(6)
+Similar to Eq.(8), the two-dimensional logistic equations in Eq.(6) have the mappings
+�f(x, y) = µ0x(1 − x) − µ1xy;
+g(x, y) = −ν0y(1 − y) + ν1xy.
+(20a)
+(20b)
+Eq.(20) has Jacobian that is slightly different from Eq.(9)
+J =
+�µ0(1 − 2x) − µ1y
+−µ1x
+ν1y
+ν0(−1 + 2y) + ν1x
+�
+,
+(21)
+with eigenvalues ω0 and ω1 being, respectively,
+�
+�
+�
+�
+�
+�
+�
+ω0 = −µ0x + ν0y + 1
+2(xν1 − yµ1) + 1
+2(µ0 − ν0) + ω
+2
+ω1 = −µ0x + ν0y + 1
+2(xν1 − yµ1) + 1
+2(µ0 − ν0) − ω
+2
+(22a)
+(22b)
+where ω is the same as in Eq.(11). Fixed points for Eq.(6) are
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+E′
+0 = (0, 0)
+E′
+1 = (0, 1 + 1
+ν0
+)
+E′
+2 = (1 − 1
+µ0
+, 0)
+E′
+3 =
+�−µ0ν0 + µ1ν0 + µ1 + ν0
+−µ0ν0 + µ1ν1
+, −µ0ν0 + µ0ν1 − µ0 − ν1
+−µ0ν0 + µ1ν1
+�
+(23a)
+(23b)
+(23c)
+(23d)
+However, when µ0ν0 = µ1ν1, Eq.(6) has fixed points E′
+0, E′
+1, and E′
+2.
+We may also make use of Lamma 1 in Sec. 2.2 to examine the topological type of each fixed point in
+Eq.(6) as in Theorem 2:
+Theorem 2. The topological types of fixed points in Eq.(23) are
+1. For E′
+0 = (0, 0),
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Ω(1) = −(µ0 − 1)(ν0 + 1)
+Ω(−1) = −(µ0 + 1)(ν0 − 1)
+det(J) = −µ0ν0
+Tr(J) = µ0 − ν0.
+In this case,
+(a) if µ0 < 1 and ν0 < 1, then E′
+0 is a sink.
+(b) E′
+0 cannot be a source.
+(c) if µ0 > 1, then E′
+0 is a saddle.
+(d) if µ0 = 1, then E′
+0 is a non-hyperbole.
+2. For E′
+1 = (0, 1 + 1
+ν0 ),
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Ω(1) = (ν0 + 1)((µ0 − µ1 − 1)ν0 − µ1)
+ν0
+Ω(−1) = (ν0 + 3)((µ0 − µ1 + 1)ν0 − µ1)
+ν0
+det(J) = (ν0 + 2)((µ0 − µ1)ν0 − µ1)
+ν0
+Tr(J) = ν2
+0 + (µ0 − µ1 + 2)ν0 − µ1
+ν0
+.
+In this case,
+7
+
+(a) E′
+1 cannot be a sink.
+(b) if µ0 > µ1ν0+µ1+ν0
+ν0
+, then E′
+1 is a source.
+(c) if µ0 < µ1ν0+µ1+ν0
+ν0
+, then E′
+1 is a saddle.
+(d) if µ0 = µ1ν0+µ1+ν0
+ν0
+, then E′
+1 is a non-hyperbole.
+3. For E′
+2 = (1 −
+1
+µ0 , 0),
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Ω(1) = (µ0 − 1)((ν0 − ν1 + 1)µ0 + ν1)
+µ0
+Ω(−1) = (µ0 − 3)((ν0 − ν1 − 1)µ0 + ν1)
+µ0
+det(J) = (µ0 − 2)((ν0 − ν1)µ0 + ν1)
+µ0
+Tr(J) = −µ2
+0 + (ν0 − ν1 − 2)µ0 + ν1
+µ0
+.
+In this case,
+(a) if 1 < µ0 < 2 and ν1µ0−µ0−ν1
+µ0
+< ν0 < ν1µ0+µ0−ν1
+µ0
+or µ0 = 2 and ν1
+2 − 1 < ν0 < ν1
+2 + 1 with ν1 > 2,
+or 2 < µ0 < 3 and ν1µ0−µ0−ν1
+µ0
+< ν0 < ν1µ0+µ0−ν1
+µ0
+, then E′
+2 is a sink;
+(b) if 0 < µ0 < 1 and ν0 < ν1µ0−µ0−ν1
+µ0
+, or µ0 > 3 and ν0 > ν1µ0+µ0−ν1
+µ0
+, then E′
+2 is a source;
+(c) if 1 < µ0 < 3 and ν0 > ν1µ0+µ0−ν1
+µ0
+, or µ0 > 3 and ν1µ0−µ0−ν1
+µ0
+< ν0 < ν1µ0+µ0−ν1
+µ0
+, or 0 < µ0 < 1
+and ν0 > ν1µ0−µ0−ν1
+µ0
+, or 1 < µ0 and ν0 < ν1µ0−µ0−ν1
+µ0
+, then E′
+2 is a saddle;
+(d) if µ1 = 1 or ν0 = ν1µ0−µ0−ν1
+µ0
+, then E′
+2 is a non-hyperbole.
+4. For E′
+3 =
+�
+ν0(µ1−µ0)
+−µ0ν0+µ1ν1 ,
+µ0(ν1−ν0)
+−µ0ν0+µ1ν1
+�
+,
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Ω(1) = −µ0ν0(ν0 − ν1)(µ0 − µ1)
+µ0ν0 − µ1ν1
+Ω(−1) = −ν0(ν0 − ν1 + 2)µ2
+0 + ν0(µ1 + 2)(ν0 − ν1 + 2)µ0 − 4µ1ν1
+µ0ν0 − µ1ν1
+det(J) = −ν0(ν0 − ν1 + 1)µ2
+0 + ν0(µ1 + 1)(ν0 − ν1 + 1)µ0 − µ1ν1
+µ0ν0 − µ1ν1
+Tr(J) = −µ2
+0ν0 + ν0(ν0 + µ1 − ν1 + 2)µ0 − 2µ1ν1
+µ0ν0 − µ1ν1
+.
+In this case,
+(a) if µ0 < µ1 and µ2
+0ν0 < µ2
+1ν1, or µ0 > µ1 and µ2
+0ν0 > µ2
+1ν1, then it is a saddle;
+(b) if µ2
+0/µ2
+1 = ν1/ν0, we have a non-hyperbole in the interior region.
+In terms of α, β, and γ, E′
+3 in Eq.(23d) is E′
+3 =
+�
+αβµ0−βµ0+α+β
+µ0(αγ−β)
+, −βµ0+γµ0−γ−1
+µ0(αγ−β)
+�
+, at which the eigenvalues
+of Jacobian in Eq.(21) is
+�
+�
+�
+�
+�
+�
+�
+�
+�
+ω0 =
+1
+2αγ − 2β
+�
+Ξ′
+0 + αγ − β
+|αγ − β|Ξ′
+1
+�
+ω1 =
+1
+2αγ − 2β
+�
+Ξ′
+0 − αγ − β
+|αγ − β|Ξ′
+1
+�
+,
+(28a)
+(28b)
+8
+
+where
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Ξ′
+0 = (2γ − 1)α − β2µ0 −
+�
+αµ0 + (−µ0 + 1)γ − µ0 + 4
+�
+β
+Ξ′
+1 =
+�
+(µ0 − 1)
+�
+(−4βµ0 − 4) α2 + 4β (µ0 − 1) α + β2 (µ0 − 1)
+�
+γ2
++
+�
+4 (βµ0 + 1)2 α2 − 2β (µ0 − 1) (βµ0 + 1) α − 2β2µ0 (µ0 − 1) (β + 1)
+�
+γ
++
+�
+(βµ0 + 1) α − βµ0 (β + 1)
+�2
+� 1
+2
+(29a)
+(29b)
+(29c)
+Eq.(28) shows that whenever Ξ′ 2
+1
+is negative, ω0 and ω1 are complex conjugates. Unlike the previous case,
+fixed points E′
+1, E′
+2, and E′
+3 now are dependent on the growth rate µ0.
+2.5
+Lyapunov exponents
+In addition, the chaotic behavior may be better examined by introducing Lyapunov exponents, which are
+defined as, base 2 being chosen to conform to Wolf et al[35],
+�
+�
+�
+�
+�
+�
+�
+λx ≡ log2 |w0| = ln |w0|
+ln 2
+λy ≡ log2 |w1| = ln |w1|
+ln 2 .
+(30a)
+(30b)
+The positive value of λx or λy, together with the negative value of total sum of the Lyapunov exponent,
+either � λx < 0 or � λy < 0, are strong inference of chaos for the prey or the predator[36]. For comparison,
+Lyapunov exponents of time series data of prey and predator populations were also calculated by both
+algorithms of Rosenstein[28] and Eckmann et al.[29] with the package of NOnLinear measures for Dynamical
+Systems (nolds)[37]. For Rosenstein algorithm, embedding dimension for delay embedding was emb dim = 10,
+and the step size between time series data points was set to τ = 1 second. While number of data points
+(trajectory len) was set to 20 and was used for the distance trajectories between two neighboring points,
+the mean period of time series data, obtained from the fast Fourier transform, was used as the minimal
+temporal separation(min tsep) between two neighbors.
+Search of the suitable lag was terminated when
+number of potential neighbors for a vector was found to be smaller than minimal neighbors, which was set
+as min neighbors = 20. At last, the RANSAC-fitting was used for the line fitting. As for the algorithm
+proposed by Eckmann et al., the matrix dimension was set to 2, and embedding dimension was also set to
+10 as in Rosenstein algorithm. Moreover, τ = 1 s, the minimal number of neighbors (min nb) was 4, and
+min tsep = 0 were used in the algorithm.
+There are at least four disadvantages for the above algorithms, as mentioned by Escot and Galan[36]: lack
+of the ability to estimate full Lyapunov spectrum, not resilient to noise in time-series data, poor detection
+performance in nonlinearity with an adequate sample size, and no theoretical derivations for the algorithms
+about their consistency and asymptotic distributions, making it impossible to statistical inferences respect
+to chaos.
+3
+Results and Discussions
+In the present study, we focused only on drawings of equations in Sec. 2.4. We observed that there could be
+five different bifurcation diagrams with various kinds of combinations of parameters. The first category is
+Normal, referring to the normal competitive behavior on the increasing and decreasing on numbers about
+species between the prey and the predator. The second category is Standard, referring to the standard
+bifurcation diagram as shown in the well-known 1D logistic equation in the prey at the absence of the
+predator. The third category is named as Paraclete, referring to overlapping structure in the bifurcation
+diagram of the prey.
+The fourth category is Extinction, connoting to extinction of the predator when
+the prey becomes chaotic. The last category is Vorticella Strange, meaning that the bifurcation diagram
+resembles the shape of a vorticella but with more complex inner structures before the two species become
+chaotic at the same time. Categories and parameters were summarized in Table 1. initX and initY indicate
+9
+
+initial values of the prey and the predator, respectively. Special attention should be paid to the cases of
+Normal and Extinction, where the inter-species parameters are deliberately made the same but the initial
+population were different: for Normal, the two species have the same initial population whereas for Extinction,
+the predator has 10 times more population than the prey. The discrepancy on initial population in these two
+cases shows completely different evolution consequences.
+In the following figures, discussions on Lyapounov exponents, Equation, Rosenstein, Eckmann X, and
+Eckmann Y in the legend of λx and λy refer to calculations directly from Eq.(30), from time-series algorithm
+of Rosenstein, Eckmann et al of the prey, and Eckmann et al of the predator, respectively. Codes, together
+with animations on population iterations, phase portraits and phase diagrams under different growth rates,
+may be retrieved via Ref([38]).
+initX
+initY
+α
+β
+γ
+Normal
+0.200
+0.200
+1.000
+0.001
+0.500
+Standard
+0.100
+0.500
+1.000
+0.100
+0.500
+Paraclete
+0.010
+0.100
+5.000
+0.010
+0.900
+Extinction
+0.010
+0.100
+1.000
+0.001
+0.500
+Vorticella Strange
+0.100
+0.500
+0.875
+0.018
+1.000
+Table 1: Parameters used for equations in Sec. 2.4.
+3.1
+Normal
+Our dynamical system may describe normal competitiveness between two species. Under the circumstance
+of equal initial population, Figure 1a shows steadily increasing population of x at the absence of the predator
+when µ0 < 3.0. At the appearance of y after µ0 > 3.0, the prey gradually diminishes with more number of
+the predator.
+Figure 1b ensures that under this scenario there is no chaos, for the Lyapunou exponents calculated by
+every algorithm are negative. However, results are different from algorithms. First for λx, there is a trench
+around µ0 = 2 by Equation, whereas all other algorithms fail to reproduce. Rosenstein, Eckmann X, and
+Eckmann Y only reproduced shallow dip around 1.5 < µ0 < 2.5. In addition, We observe that Rosenstein
+and Eckmann X have quite similar results in the whole range of µ0, except that Rosenstein has a slightly
+higher value. Furthermore, Eckmann X and Eckmann Y have almost identical values when µ0 > 1.66, but
+Eckmann Y digresses a lot from the other three curves at low growth rate below µ0 = 1.66. As for λy, all
+algorithms show close spectrum µ0 > 1.692, with larger values for Rosenstein. The four algorithms divide
+into two groups of results below µ0 = 1.692, with Rosenstein and Eckmann X showing an increasing tail that
+is different from the other two algorithms showing curves of dropping.
+3.2
+Standard
+Figure 2a shows bifurcation diagram and Lyapunov exponents of Standard. Our model shows that even
+with nonzero initial population and nonzero inter-species relationships of α, β, and γ, we may still acquire
+flip bifurcation for 1D logistic equation[39] for the prey at the absence of the predator. A flip bifurcation
+is a counterpart in discrete dynamic system to describe the concept of periodic doubling in the continuous
+dynamic system[40].
+Figure 2b shows the Lyapunov exponents. Rosenstein algorithm did not show results in in this case,
+because singular value decomposition did not converge when doing linear least squares, meaning that positive
+or negative infinity appeared when we tried to deal with pseudo-inverse matrix. Eckmann Y cannot work,
+either, for y = 0 in the whole range. We may see that for overall trend of λx, both algorithms have λx < 0
+for µ0 < 3.0, whereas λx has both positive and negative values for µ0 > 3.5.
+It is widely accepted[41]
+that values of Lyapunov exponents occur interchangeably between positive and negative infer chaos, which
+is consistent with the shaded area in Figure 2a. Another inconsistency occurs with 3.0 < µ0 < 3.5 with
+λx > 0 for Equation but λx < 0 for Eckmann X, where x exhibits a flip bifurcation from 2-cycle into 4-cycle.
+Nevertheless, this inconsistency may not be a problem for us to distinguish chaos from happening. There is
+10
+
+(a)
+(b)
+Figure 1: Competitive behavior and Lyapunov exponents of Normal.
+11
+
+0.6
+0.5 
+0.4 
+0.2 
+0.1 
+0.0 
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5 
+4.0
+Logistic Map
+0.25
+0.20 
+0.15 -
+y
+0.10 
+0.05 
+000
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+μoLyapunov Exponents
+0
+入x
+-4
+-6 -
+Equation
+......
+Rosenstein
+Eckmann X
+Eckmann Y
+-8
+1.0 
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+-0
+-2 
+-4 -
+ g-
+-8 +
+-10 -
+Equation
+Rosenstein
+Eckmann X
+-12
+EckmannY
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+μoa break around µ0 = 2 for Eckmann X in both λx and λy, at which Equation shows a deep trench in λx.
+Also, near µ0 ≈ 1.245, Equation shows a smaller trench while Eckmann X produces a rising tail.
+Figure 3 studies the population in the course of time (iteration) at µ0 equal to 2.700 (1-cycle), 3.000
+(2-cycle where the flip bifurcation occurs) and 3.500 (4-cycle), 3.700 (at which the system goes into chaos),
+3.845 (the system going back to more stable 3-cycle), and 3.945 (where the system returns to chaos again)
+in the successive order. Zero predator population is obtained throughout course of time.
+Figure 4 studies topological types of fixed points. Plots of imaginary or real part of eigenvalues in Eq.(22)
+were evaluated at the fixed points E′
+1 in Eq.(23b) (as shown in the upper row), E′
+2 in Eq.(23c) (as shown
+in the middle row), and E′
+3 in Eq.(23d) (as shown in the bottom row). Color-bar at the right-hand side
+stands for various growth rate µ0, while circles with four different sizes in the legend represent, from the
+smallest to the biggest, topological types of sink, source, saddle, and non-hyperbole. We may see that ω0
+and ω1 at the fixed points E′
+1 were pure real numbers with absolute values greater than 1, making the fixed
+point a source for all µ0. While ω0 and ω1 at the fixed points E′
+2 were also pure real numbers, the absolute
+values vary across 1, making the fixed point E′
+2 topological types of sink, source, and saddle, with possible
+non-hyperboles occurring at either low µ0 = 1 or high µ0 = 3.75 if we apply Theorem 2.3(d).
+Figure 5 shows absolute values of eigenvalues ω0 and ω1 on fixed points E′
+1, E′
+2 and E′
+3. Topological types
+of fixed points may be double-checked more straightforwardly with the figure. The first column shows that
+E′
+1 is a source because ω0 and ω1 are always greater than 1. The second column demonstrates that E′
+2 are
+a sink when µ0 < 2.141, and when 2.141 < µ0 < 3, E′
+2 is a saddle. Non-hyperboles can also be examined
+at µ0 = 1 for E′
+2, at µ0 = 3 for E′
+2 (in both cases ω0 = 1 and ω1 ̸= 1), and at µ0 = 3.75 for E′
+2 and E′
+3 (in
+which ω0 ̸= 1 and ω1 = 1). That µ0 = 3.75 is located in chaos region, making the fixed point vulnerable to
+nonlinear terms in the dynamic system. At last, when µ0 > 3, E′
+2 is a source.
+Figure 6 shows phase portrait and phase space diagram about Standard. µ0 values are represented by the
+color bar at the right-hand side. Figure 6a refers to phase portrait, where topological types are also shown
+in the legend. An isolated initial coordinate (0.1, 0.50) marked as a source at the upper-left corner. Flip
+bifurcation occurs at (0.665, 0, 000). No limit circles were found in the ��ip bifurcation. The oblique black
+straight line, starting from (0.732, 0.000) to (0.689, 0.062), consisting of fixed points E′
+3 that is enclosed by a
+thicker yellow cloak indicates that, along the axis, E′
+3 is a saddle, with nonzero y values. This result seems
+to contradict to the previous one, saying that predator population is always zero. However, since our initial
+population is (x, y)=(0.1, 0.5), it does not lie in the above range. Therefore, the system with the chosen
+inter-species constants is not attracted by the saddle points along the oblique black line, confirming that
+with none-zero initial population of the predator and non-zero inter-species constants, the predator could
+appear, but only for a while. Afterwards, the predator dies out in the course of time, leaving the prey to be
+the only surviving species in the paradisaic. This observation may explain why the predator species may not
+survive long in some specific ecological system. Further, Figure 6b is phase space diagram for the prey. It
+is meaningless to discuss phase space diagram for the predator because there are only two points, (0.0, 0.0)
+and (0.5, −0.5), in this case.
+12
+
+(a)
+(b)
+Figure 2: Bifurcation diagram and Lyapunov exponents of Standard.
+13
+
+1.0 
+0.8 
+0.6 -
+X
+0.4
+0.2 
+0.0
+0.04
+0.02
+0.00
+0.02
+0.04
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+uoLyapunov Exponents
+-2
+-6
+Equation
+Eckmann X
+EckmannY
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+0
+-2 
+-4
+-6
+-8 -
+Equation
+Eckmann X
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+μo(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+Figure 3: Population vs. iteration of Standard.
+14
+
+1.0 
+0.8 
+0.6 
+μo = 3.945
+0.4
+0.2 
+0.0
+0
+25
+50 
+75
+100
+125
+150
+175
+200
+0.5
+0.4 
+0.3 
+n
+0.2
+0.1 
+0.0
+0
+25
+50
+75
+100
+125
+150
+175
+200
+n1.0 
+0.8
+0.6
+μo = 2.700
+0.4 
+0.2 
+0.0 
+0
+25
+50
+75
+100
+125
+150
+175
+200
+0.5
+0.4 
+0.3 
+0.1 
+0.0 
+25
+50
+75
+100
+125
+150
+175
+200
+n1.0 
+0.8 
+0.6 
+μo = 3.000
+0.4
+0.2 
+0.0
+0
+25
+50
+75
+100
+125
+150
+175
+200
+0.5
+0.4 
+0.3 
+0.2
+0.1 
+0.0 
+0
+25
+50
+75
+100
+125
+150
+175
+200
+n1.0 
+0.8 
+0.6 
+0.4 
+Ho= 3.500
+0.2 
+0.0
+0
+25
+50 
+75
+100
+125
+150
+175
+200
+0.5
+0.4 
+0.3 
+0.2
+0.1 
+0.0
+0
+25
+50
+75
+100
+125
+150
+175
+200
+n1.0 
+0.8 
+0.6 
+μo = 3.700
+0.4
+0.2 
+0.0
+0
+25
+50
+75
+100
+125
+150
+175
+200
+0.5
+0.4 
+0.3 
+0.2
+0.1 
+0.0
+0
+25
+50
+75
+100
+125
+150
+175
+200
+n1.0 
+0.8 
+0.6 
+0.4
+0.2 
+0.0
+0
+25
+50 
+75
+100
+125
+150
+175
+200
+0.5
+0.4 
+0.3 
+0.2
+0.1 
+0.0
+25
+50
+75
+100
+125
+150
+175
+200
+nFigure 4: Analysis on eigenvalues for the case of Standard. Eigenvalues described in Eq.(22) for fixed points
+E′
+1 (Eq.(23b)), E′
+2 (Eq.(23c)), and E′
+3 (Eq.(23d)) are plotted in the upper row, middle row, and lower row,
+respectively. Types of topology are indicated with circles of various sizes. Color bar stands for different µ0.
+.
+Figure 5: Absolute values of eigenvalues vs. growth rate at fixed points for Standard. Prominent coordinates
+that help us understand stability and distinguish the topological type about a fixed point are recorded as
+follows. Upper middle panel: (1.244, 0.000), and (3, 75, 1, 00). Upper-right corner panel: (3.75, 1.00). E′
+2 and
+E′
+3 are both non-hyperbolic at µ0 = 1, 3, and 3.75.
+15
+
+0.04 
+0.04
+10.4
+0.02
+0.02
+10.2
+Im(wo)
+Im(wi)
++ 00°0
+0.00
+0.02 -
+0.02
+9.8 
+0.04 -
+0.04 
+9'6
+2.10
+2.15
+2.20
+2.25
+2.30
+2.35
+2.40
+10.4
+10.2
+10.0
+9.8
+9.6
+2.10
+2.15
+2.20
+2.25
+2.30
+2.35
+2.40
+0.04 -
+0.04
+1.00 
+0.02
+0.02
+0.75
+Im(wo)
+Im(wi)
+0.00-
+0.00
+0.02 -
+0.02
+0.25
+0.04
+0.04
+0.00 
+2.0
+-1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.000.25
+0.500.75
+1.00
+1.251.50
+1.75
+2.00
+1.8 
+0.04
+0.04
+1.6 
+0.02
+0.02
+Im(wo)
+Im(wi)
+0.00 -
+0.00
+0.02 -
+0.02
+1.2 
+0.04
+0.04
+1.0 J
+2.8
+2.6
+2.4
+2.2
+2.0
+-1.8
+1.6
+1.0
+1.2 
+1.4
+1.6
+1.8
+1.6
+1.8
+2.0
+2.2
+2.4
+2.6
+2.8
+Re(wo)
+1°ml 
+μo
+Re(wi)
+Type
+source
+non-hyperbolic
+sink
+O
+saddle1.8 
+10.4
+1.0 -
+1.6
+0.8 
+10.2
+0.6
+1.4
+0.4 
+1.2 
+9.8
+0.2
+9.6
+1.0
+0.0
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5 
+4.0
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+2.40
+2.00
+2.8 -
+1.75 
+2.35
+2.6 
+1.50
+2.30 
+1.25
+2.4 
+1.00
+2.2 
+0.75
+2.20
+2.0 
+0.50
+2.15
+0.25
+1.8 
+2.10 -
+0.00
+1.6 
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+μo. Column for Ei
+μo. Column for E2
+μo. Column for E3(a)
+(b)
+Figure 6:
+Analysis on phase
+portrait and phase space dia-
+gram about Standard. Topolog-
+ical types of sink, source, sad-
+dle and non-hyperbole are repre-
+sented with different sizes from
+the smallest to the largest as
+shown in the legends.
+Oblique
+black line in Figure 6a indicates
+saddle fixed points, demonstrat-
+ing that the predator cannot sur-
+vive long in the ecological sys-
+tem. Figure 6b shows the phase
+space of x. Phase diagram of the
+predator is not shown here, be-
+cause it only contains two points
+(0.0, 0.0), and (0.5, −0.5), which
+makes the figure boring.
+16
+
+Type
+0.5 
+non-hyperbolic
+sink
+saddle
+O
+source
+0.4 
+y
+0.2 
+0.1 
+0.0 
+0.0
+0.2
+0.4 
+0.6
+0.8
+1.0
+μo
+X0.6 
+0.4 
+0.2 -
+0.0 
+0.4
+0.6
+-0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+μo3.3
+Paraclete
+Figure 7a represents two sets of overlapping bifurcation diagrams: one is the Standard, the other with
+vorticella-shaped, which starts to appear after µ0 = 2.29.
+Between 2.29 < µ0 < 2.45, shaded regions
+appearing vertically with gaps are not chaos but transient states of population. At µ0 = 3.256, the vorticella-
+shaped has bifurcation that start to be chaotic, at which we would explain later that it should be classified
+as Neimark-Sacker bifurcation, and mingles together with the flip bifurcation of Standard after µ0 = 3.46, at
+which the four-cycles occurs.
+Figure 7b shows the Lyapunov exponents for Paraclete. For the same reason in Standard, Rosenstein
+algorithm did not show results in in this case, either. We may see that for overall trend of λx, algorithms
+of both Equation and Eckmann X have λx < 0 for µ0 < 3.0, whereas λx has both positive and negative
+values for µ0 > 3.0, inferring chaos. Eckmann Y only shows a small portion, not spectrum, because y = 0 for
+µ0 < 2.29 leads to failure on producing full spectrum of both λx and λy. Therefore, where we may see only
+some segment of λx after µ0 > 3.0 that almost overlaps with the curve of Eckmann X. Inconsistency also
+occurs at low growth rate in λy spectrum, for Equation shows stern-drooping tails while Eckmann X shows
+a raising-up one.
+Figure 8 studies the population in the course of time (iteration) at µ0 equal to 2.790 (stable population),
+3.025 (2-cycle), 3.255 (at which Neimark-sacker bifurcation is on the way), 3.460 (4-cycle), 3.845 (3-cycle in
+Standard), and 3.945 (chaos region) in the successive order.
+Figure 9 studies topological types of fixed points. Plots of imaginary or real part of eigenvalues in Eq.(22)
+were evaluated at the fixed points E′
+1 in Eq.(23b) (as shown in the upper row), E′
+2 in Eq.(23c) (as shown in
+the middle row), and E′
+3 in Eq.(23d) (as shown in the bottom row). Color-bar at the right-hand side stands
+for various growth rate µ0, while circles with four different sizes in the legend represent, from the smallest
+to the biggest, topological types of sink, source, saddle, and non-hyperbole. We may see that ω0 and ω1 at
+the fixed points E′
+1 were pure real numbers with absolute values greater than 1, making the fixed point a
+source for all µ0. While ω0 and ω1 at the fixed points E′
+2 were also pure real numbers, the absolute values
+vary across 1, making the fixed point E′
+2 topological types of sink, source, and saddle, with only one possible
+non-hyperbole occurring at |ω0| = 1.
+Figure 10 shows absolute values of eigenvalues ω0 and ω1 on fixed points E′
+1, E′
+2 and E′
+3. Topological
+types of fixed points may be double-checked more straightforwardly with the figure. The first column shows
+that E′
+1 is a source because ω0 and ω1 are always greater than 1. The second column demonstrates that E′
+2
+are a sink when µ0 < 2.141, and when 2.141 < µ0 < 3, E′
+2 is a saddle. At last, when µ0 > 3, E′
+2 is a source.
+As we look closely into the third column, it shows that E′
+3 is non-hyperbolic at µ0 ≈ 2.137 for ω0 ̸= 1
+and ω1 = 1, which is vulnerable to nonlinear terms in the dynamic system. It explains why the transient
+state under that growth rate is not shown in Figure 7a. Also, bending points along curves plotted in the
+third-column figures demonstrate that transient states occur within 2.258 < µ0 < 2.475. More interestingly,
+the third column manifests that E′
+3 represents Neimark-Sacker bifurcation at µ0 = 3.25636 because of the fol-
+lowing facts: first, ω0 and ω1 are complex conjugates with modulus 1, and second, as µ0 varies across 3.25636
+from smaller to larger value, topological type of E′
+3 changes from a sink (stable) to a source (unstable)[19].
+Figure 11 shows phase portrait and phase space diagram about Paraclete. µ0 values are represented by
+the color bar at the right-hand side. Figure 11a refers to phase portrait, showing Neimark-Sacker bifurcation
+established at (0.352, 0.068) with µ0 ≈ 3.256 at the center of limit circles.
+Black straight lines indicate
+oblique axis consisting of E′
+3, including sink and source, and horizontal axis composed of E′
+2, including
+source and saddle, under different µ0. Topological types are also shown in the legend. Dots with larger µ0
+representing chaos spread outside around the limit circles. Figure 11b is phase space diagram for the prey
+centered at (0.352, 0) and Figure 11c refers to phase space diagram for the predator centered at (0, 0.068).
+Not surprisingly, the center of limit circles in Figure 11b has the same x value as that of Figure 11a. Similarly,
+same y value for the center of limit circles in Figure 11c and in Figure 11a.
+17
+
+(a)
+(b)
+Figure 7: Bifurcation diagram and Lyapunov exponents of Paraclete.
+18
+
+1.0
+0.8
+0.6
+X
+0.4 -
+0.2
+0.0
+0.150
+0.125
+0.100
+>0.075
+0.050
+0.025
+0.000
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+uo0
+-2 -
+-4
+6-
+Equation
+Eckmann X
+EckmannY
+-8
+2 
+-2
+-4 -
+-6于
+Equation
+Eckmann X
+-8 
+Eckmann Y
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+μo(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+Figure 8: Population vs. iteration of Paraclete.
+19
+
+1.0 
+0.8
+0.6
+0.4 
+0.2 
+μo = 2.790
+0.0 
+0
+25
+50
+75
+100
+125
+150
+175
+200
+0.150
+0.125
+0.100
+0.075
+0.050
+0.025
+0.000 -
+0.025
+0.050
+25
+50
+75 
+100
+125
+150
+175
+200
+n1.0
+0.8
+0.6
+0.4
+0.2 
+μo = 3.025
+0.0
+0
+25
+50 
+75
+100
+125
+150
+175
+200
+0.150
+0.125
+0.100
+0.075
+0.050
+0.025
+0.000
+0.025
+-0.050
+0
+25
+50
+75
+100
+125
+150
+175
+200
+n1.0
+0.8
+0.6
+X
+0.4
+0.2
+μo = 3,255
+0.0
+0
+25
+50 
+100
+125
+150
+175
+200
+0.150
+0.125
+0.100
+0.075
+0.050
+0.025
+0.000
+0.025
+0.050
+0
+25
+50
+75
+100
+125
+150
+175
+200
+n1.0 
+0.8
+0.6
+ux
+0.4
+0.2 
+μo = 3.460
+0.0
+0
+25
+50 
+75
+100
+125
+150
+175
+200
+0.150
+0.125
+0.100
+0.075
+0.050
+0.025
+0.000
+-0.025
+-0.050
+0
+25
+50
+75
+100
+125
+150
+175
+200
+n1.0 
+0.8
+0.6
+0.4
+0.2 
+以o -:3.845.
+0.0
+0
+25
+50 
+75
+100
+125
+150
+175
+200
+0.150
+0.125
+0.100
+0.075
+0.050
+0.025
+0.000
+-0.025
+-0.050
+0
+25
+50
+75
+100
+125
+150
+175
+200
+n1.0
+0.8
+0.6
+n
+X
+0.4
+0.2
+3.945
+0.0
+0
+25
+50 
+75
+100
+125
+150
+175
+200
+0.150
+0.125
+0.100
+0.075
+0.050
+0.025
+0.000
+-0.025
+-0.050
+0
+25
+50
+75
+100
+125
+150
+175
+200
+nFigure 9: Analysis on eigenvalues for the case of Paraclete. Legends have the same meaning described in
+Figure 4.
+Figure 10: Absolute values of eigenvalues vs. growth rate at fixed points for Paraclete. Important coordinates:
+Upper middle panel (2.14, 1.00). Upper-right corner panel (2.137, 1.000), (2.457, 0.440), and (3.256, 1.000).
+Lower-right corner panel (2.258, 0.000), (2.475, 0.427), and (3.256, 1.000).
+20
+
+0.04 
+0.04
+515.0 
+0.02
+0.02
+512.5
+Im(wo)
+Im(wi)
+0.00 
+0.00
+0.02 -
+0.02
+507.5
+0.04 -
+0.04
+505.0
+2.010
+2.015
+2.020
+2.025
+2.030
+2.035
+2.040
+-516
+-514
+-512
+510
+508
+-506
+504
+2.010
+2.015
+2.020
+2.025
+2.030
+2.035
+2.040
+3
+0.04 -
+0.04
+2.5 
+0.02
+2.0 -
+0.02
+Im(wo)
+Im(wi)
++00'0
+0.00
+0.02 -
+1.0
+0.02
+0.5 
+0.04
+0.04
+EEEELEEF
+0.0 
+2.0
+1.5
+-1.0
+0.5
+0.0
+0.5
+1.0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+0.000.250.500.751.00
+1.251.501.752.00
+2
+1.25 于
+0.00 
+1.50 
+1.00 
+0.25
+0.50 
+1.25 
+Im(wi)
+0.50
+0.75
+1.00
+0.75 
+0.25
+1.25
+0.50
+0.00 
+0.8
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Re(wi)
+[wol
+μo
+Re(wo)
+Type
+source
+non-hyperbolic
+saddle
+sink516
+2.5
+1.6
+514 
+1.4 
+2.0
+512 
+1.2 
+1.5
+1.0
+1.0 
+508 
+0.8 
+506
+0.5 
+0.6
+504
+0.0 
+0.4
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5 
+4.0
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5 
+4.0
+2.040
+2.00
+1.2 
+1.75
+2.035
+1.50
+1.0
+2.030
+1.25 
+0.8 -
+1.00 
+0.6
+0.75
+2.020
+0.4 
+0.50
+2.015
+0.2 
+0.25 
+2.010
+0.00 
+0.0
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5 
+4.0
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+μo. Column for Ei
+μo. Column for E2
+μo. Column for E3(a)
+(b)
+(c)
+Figure 11:
+Analysis on phase
+portrait
+and
+phase
+space
+diagram
+about
+Paraclete.
+(11a)Phase
+portrait
+shows
+Neimark-Sacker bifurcation es-
+tablished at (0.352, 0.068) with
+µ0 ≈ 3.256 located at the center
+of limit circles.
+Oblique axis
+and horizontal axis consisting
+of E′
+3
+and E′
+2
+with different
+µ0, respectively.
+We may see
+from the legend that topological
+types of E′
+3
+are mostly sink
+and source, while those of E′
+2
+are mostly source and saddle.
+Phase portrait and phase space
+diagram for the prey have same
+x in (11b) for the center limit
+circle. Similarly, phase portrait
+and phase space diagram for the
+predator have same y in (11c)
+for the center limit circle.
+21
+
+Type
+non-hyperbolic
+sink
+saddle
+0.14
+source
+0.12
+0.10
+0.08-
+0.06-
+0.04 -
+0.02
+0.00 
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+μo0.6 
+0.4 
+0.2 -
+0.0 
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+μo0.075
+0.050
+0.025
+0.000 -
+0.025
+0.050
+0.075
+0.100
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+0.14
+V
+μo3.4
+Extinction
+Figure 12a shows the bifurcation diagram of Extinction. There is a flip bifurcation for x around µ0 ≈ 2.99;
+however, the 2-cycle collides at µ0 ≈ 3.165 and returns back to 1-cycle. Shaded regions between 3.165 < µ0 <
+3.434 for x and 3.10 < µ0 < 3.343 for y do not refer to chaos but belong to transient states. Afterwards, the
+bifurcation diagram goes back to Normal for both x and y, and represent pleasant conditions of predictable
+values before x goes into 3-cycle at µ0 ≈ 3.828 as in Standard, where y drops to zero cliff-fallingly at
+µ0 ≈ 3.824, manifesting to extinction of the predator for the prey is around the 3-cycle state. Fortunately,
+there could be still few little chances for the predator to survive at (µ0, y) = (3.845, 0.219), (3.887, 0.226),
+and (3.897, 0.228), where we may see three isolated fixed points appear with a vertical tail of transient states.
+When the prey becomes fully chaotic, the predator population reduces back to zero dramatically again, and
+never has any further opportunity to rise back. This astonishing phenomenon could be the most profound
+finding in the study, which states that the prey in chaos generated by overpopulation of the predator would
+erase the entire predator species.
+Figure 12b shows Lyapunov exponents for Extinction that is similar to those in Figure 1b, except for the
+regions at 3.0 < µ0 < 3.18 and µ0 > 3.86, the former showing a bum by Equation, which is also the same
+region for the prey to be in 2-cycle, and the latter presenting chaos for x and extinction for y. Discrepancy
+appears for Eckmann Y that is different from the other when µ0 < 1.662, where it shows a decreasing tail
+while the other algorithms show an increasing trend. Unlike Figure 1b, whose results show that Eckmann
+X is always in between Rosenstein and Eckmann Y in the range of 1.5 < µ0 < 3.0, Figure 12b shows more
+intertwines at µ0 = 1.886 for λx, and at µ0 = 1.717 and µ0 = 1.891 for λy. Similar to Figure 1b, the four
+algorithms also divides into two groups of curves below µ0 = 1.717, with Rosenstein and Eckmann X going
+upward, together with Equation and Eckmann Y going downward.
+Figure 13 shows population iteration of Extinction. As we can see, flip bifurcation starts at µ0 = 3.000 as
+in Figure(13a), while in Figure(13b) the two fixed points collide, and after transient states (n > 200), they
+have tendency to merging into a single fixed point, as we explained earlier in Figure 12a on the characteristics
+about the shaded region between 3.165 < µ0 < 3.434 for x. We further demonstrate that at µ0 = 3.500 in
+Figure(13c), after transient(n > 175), bifurcation collapses to one. Furthermore, 3-cycle is opened in x at
+µ0 = 3.84, as indicated in Figure(13d), with extinction of y at the exactly the same moment. However, a
+sunlight of survival for y is shed on the window at µ0 = 3.845, as shown in Figure(13e), where the two species
+may still exist under predictable population. Finally, Figure 13f portends the predator extinction under
+chaos of the prey.
+Figure 14 studied topological types of fixed points.
+Plots of imaginary or real part of eigenvalues in
+Eq.(22) were evaluated also at the three fixed points E′
+1, E′
+2, and E′
+3 as in Figure 4. We may see that ω0
+at the fixed points E′
+1 is pure real numbers with absolute values greater than 1, whereas ω1 keeps the value
+−1000 for all µ0, making the fixed point a source (unstable) for all µ0. While ω0 < 1 and ω1 > 1 at the
+fixed points E′
+2, the fixed point E′
+2 has a topological type of saddle, with possible non-hyperboles occurring
+at |ω0| = 1 and |ω1| = 0.
+Stability of fixed points may also be examined in Figure 20. Figures at the first column shows that that
+E′
+1 is a source, for both eigenvalues have absolute values greater than 1. In the second column, we see that
+E′
+2 changes its stability from a sink to source when µ0 varies at 3, at which flip bifurcation occurs; meanwhile,
+E′
+3 changes from source to sink.
+Figure 16 shows phase portrait and phase diagrams for Extinction. No limit circles are found in this
+particular case.
+22
+
+(a)
+(b)
+Figure 12: Bifurcation diagram and Lyapunov exponents of Extinction.
+23
+
+Logistic Map
+1.0
+0.8 -
+0.6 
+0.4 
+0.2
+0.0
+0.20 -
+0.15 -
+0.10 -
+0.05 -
+0.00-
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+μoLyapunovExponents
+0
+-2
+-6-
+Equation
+Rosenstein
+Eckmann X
+Eckmann Y
+8
+0-
+-2-
+-4
+-6
+-8 +
+Equation
+-10 -
+Rosenstein
+Eckmann X
+EckmannY
+-12
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+μo(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+Figure 13: Population vs. iteration of Extinction.
+24
+
+1.0 -
+0.8 -
+0.6
+n
+0.4
+μo = 3.000
+0.2 
+0.0
+0
+25
+50
+75
+100
+125
+150
+175
+200
+0.30 
+0.25 -
+0.20 
+0.15
+0.10 -
+0.05
+0.00
+0.05
+0
+25
+50
+100
+125
+150
+175
+200
+n1.0 -
+0.8 -
+0.6 
+u
+0.4
+μo = 3.375
+0.2 
+0.0
+0
+25
+50
+75
+100
+125
+150
+175
+200
+0.30 
+0.25 -
+0.20 
+0.15
+0.10 -
+0.05
+0.00
+0.05
+0
+25
+50
+75
+100
+125
+150
+175
+200
+n1.0 -
+0.8 -
+0.6 
+n
+0.4
+μo = 3.500
+0.2 
+0.0
+0
+25
+50
+75
+100
+125
+150
+175
+200
+0.30 
+0.25 -
+0.20
+0.15
+0.10 -
+0.05
+0.00
+0.05
+0
+25
+50
+75
+100
+125
+150
+175
+200
+n1.0
+0.8 -
+0.6
+0.4
+μo = 3.840
+0.2 
+0.0
+0
+25
+50
+75
+100
+125
+150
+175
+200
+0.30 
+0.25 -
+0.20 -
+0.15
+0.10 -
+0.05
+0.00 -
+0.05
+0
+25
+50
+100
+125
+150
+175
+200
+n1.0 -
+0.8 -
+0.6
+0.4
+μo = 3.845
+0.2 
+0.0
+0
+25
+50
+75
+100
+125
+150
+175
+200
+0.30
+0.25 -
+0.20 
+0.15
+0.10 -
+0.05 
+0.00
+0.05
+0
+25
+50
+75
+100
+125
+150
+175
+200
+n1.0 -
+0.8 -
+0.6
+C
+0.4
+μo = 3.945
+0.2 
+0.0
+0
+25
+50
+75
+100
+125
+150
+175
+200
+0.30 
+0.25 -
+0.20 
+0.15
+0.10 -
+0.05
+0.00 -
+0.05
+0
+25 
+50
+75
+100
+125
+150
+175
+200
+nFigure 14: Analysis on eigenvalues for the case of Extinction. Legends have the same meaning described in
+Figure 4.
+Figure 15: Absolute values of eigenvalues vs. growth rate at fixed points for Extinction. From the figures at
+the first column, it is clearly shown that E′
+1 is a source. Also, at µ0 = 3 where flip bifurcation occurs, E′
+2
+changes its stability from a sink to source, whereas E′
+3 from source to sink.
+25
+
+0.04 
+0.04
+1040-
+0.02
+0.02
+1020
+Im(wo)
+Im(wi)
+0.00 
+0.00
+0.02 -
+0.02
+980 -
+0.04
+0.04
+096
+2.00102.00152.0020
+2.00252.00302.00352.0040
+1040
+1020
+-1000
+980
+-960
+2.0010 2.0015 2.0020 2.00252.0030 2.00352.0040
+3
+1.5 
+0.04
+0.04
+0.02
+0.02
+Im(wo)
+Im(wi)
+1.0 
+Iml
+0.00-
+0.00
+0.02 -
+0.02
+0.5 
+0.04
+0.04
++ 00
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+0.000.250.500.751.00
+1.251.501.75
+2.00
+0.04
+0.04
+1.5 
+0.02
+0.02
+Im(wo)
+Im(wi)
+00°0
+0.00
+0.02 -
+0.02
+0.5 
+0.04
+0.04
+1.8-1.6-1.4
+-1.2
+-1.0
+0.80.60.40.2
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2 
+1.4
+1.6
+1.8
+Re(wo)
+Re(wi)
+[wol
+μo
+Type
+source
+non-hyperbolic
+sink
+O
+saddle1.8 
+1.4
+1040
+1.6 
+1.2
+1.4 
+1020
+1.0
+1.2 
+0.8 
+1.0 
+0.6 
+0.8
+980
+0.6 
+0.4 
+0.4 
+0.2 
+960
+0.2
+0.0
+1.0
+1.5 
+2.0
+2.5
+3.0
+3.5 
+4.0
+1.0 
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+1.0
+1.5
+2.0
+2.5
+3.0 
+3.5
+4.0
+1.8 
+2.0040
+2.00
+1.6 
+1.75
+2.0035
+1.4 J
+1.50
+2.0030
+1.2 
+1.25 
+1.0
+1.00 
+0.8 
+0.75
+2.0020
+0.6 
+0.50
+2.0015
+0.4
+0.25
+0.2 
+2.0010
+0.00
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5 
+4.0
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5 
+4.0
+μo. Column for Ei
+μo. Column for E2
+μo. Column for Es(a)
+(b)
+(c)
+Figure 16:
+Analysis on phase
+portrait and phase space dia-
+gram about Extinction. No limit
+circles were found in this case.
+26
+
+Type
+0.25
+non-hyperbolic
+sink
+O
+saddle
+source
+0.15 -
+y
+0.10 -
+0.05
+0.00 
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+μo0.6 
+0.4 
+0.2 -
+0.0 
+0.4
+0.6
+-0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+μo0.04
+0.02
+0.00
+0.02
+-0.04 -
+-0.06
+-0.08 
+0.10
+0.00
+0.05
+0.10
+0.15
+0.20
+μo
+y3.5
+Vorticella Strange
+The last bifurcation type in our study is Vorticella Strange, which means that bifurcation diagram looks
+like a vorticella, only with more complicated internal structures.
+Figure 17a shows bifurcation diagram.
+When µ0 < 2.0, x grows steadily without y. After presence of the predator, population of the prey starts to
+decrease. Both species show predictable population before µ0 = 3.025, at which we classify a Neimark-Sacker
+bifurcation as in Paraclete. Transient states occur between 3.025 < µ0 < 3.200, as indicated in the shaded
+region. Later, a 6-cycle appears at µ0 = 3.24, which is also confirmed in Figure 18b. The 6-cycle becomes
+3-cycle at µ0 = 3.40, as shown in Figure 18c, followed by chaos at µ0 = 3.485, which is also demonstrated
+in Figure 18d. The system goes back to 6-cycle around µ0 = 3.540, as we may also confirm in Figure 18e.
+Afterwards, the system goes back to chaos, as shown in Figure 18f.
+Figure 17b shows Lyapunov spectrum of Voticella Strange. All four algorithms barely show positive spectra
+for both x and y before µ0 = 3.5. On the contrary, afterµ0 = 3.5, four algorithms show positive Lyapunov
+exponents, where the system falls into chaos. For λx, Eckmann Y shows a decreasing tail below µ0 < 1.5,
+inconsistent from the other algorithms. Besides Equation showing two valleys between 1.500 < µ0 < 2.304
+and 3.224 < µ0 < 3.463, the other three algorithms only provide flat spectra in the two regions.
+Figure 19 shows analysis on eigenvalues of Vorticella Strange. At the first row, we see that both eigenvalues
+have zero imaginary parts, with absolute real parts of both greater than 1 (see also first column in Figure
+20). Thus, E′
+1 is a source. Similar analysis may also be done on E′
+2, as shown in the second row in Figure
+19 as well as the second column in Figure 20. At µ0 = 2.0, it turns from a sink to a saddle, and it maintains
+as a saddle between 2.0 < µ0 < 3.0, after which it turns to a source. Finally, the third column in Figure
+20 demonstrates that Neimark-Sacker bifurcation occurs at µ0 ≈ 3.025, with coordinates (0.341, 0.377), as
+shown in Figure 21a.
+Similar bifurcation diagram was found by Hu et al.[10], and was identified as the Hopf bifurcation.
+However, the criteria for Hopf bifurcation in the two-dimensional system includes that the two eigenvalues
+are purely conjugate imaginary pair with zero real part[42]. Since Figure 19 shows that ω1 has a non-zero
+real part, our Vorticella Strange cannot be a Hopf bifurcation.
+For the sake of integrity, phase portrait and phase space diagram of Vorticell Strange are also shown in
+Figure 21. Similar detailed meaning is discussed in the Section of Paraclete.
+4
+Conclusions
+We successfully built up a discrete-time model of two-dimensional mapping that resembles features of dif-
+ferential Lotka-Volterra Equations. We studied the topological types of fixed points, and found out that
+fascinating feather-like structures were formed around limit circles pierced through by the axis of fixed points
+in the phase portraits and phase space diagrams under various growth rates with Neimark-Sacker bifurcation.
+We further divided our dynamical systems into five categories by different shapes of bifurcation diagrams:
+Normal, Standard, Paraclete, Extinction, and Vorticella Strange. In every case, we studied the stability and
+topological types of the fixed points by criteria discussed in Lamma 1. In addition to plot the population
+vs. iteration, we also calculated Lyapunov exponents both by eigenvalues of Jacobian of mapping functions,
+and by Rosenstein algorithm and Eckmann et al. algorithm. Discrepancies clearly showed that Lyapunov
+exponents calculated by time-series algorithms may be unreliable within the range of chaos and at low growth
+rates, as well as when the Lyapunov exponents change dramatically. Furthermore, our model not only re-
+gained the 1D logistic mapping of the prey under zero predator yet with non-zero initial predator population
+and with non-zero inter-species constants, but it also showed the normal competitiveness of the prey and
+the predator without chaos. The quintessence in the current research was that, besides the possibility for
+the prey and the predator to become chaotic altogether, it is also probable for the predator to go extinct at
+the chaotic state of the prey. In other words, human overpopulation would cause chaos in natural resources,
+ultimately in return erase the entire human race. Luckily, even under this difficult circumstance slim chance
+is still left upon us to continue our race under some specific growth rate, as we may see some isolated fixed
+points remain in the predator bifurcation diagram.
+Our model may inspire conjecture on other relationships between two physical quantities, because mathe-
+matically what we demonstrated was that one quantity may dramatically reduce to zero at the state of chaos
+27
+
+(a)
+(b)
+Figure 17: Bifurcation diagram and Lyapunov exponents of Vorticella Strange.
+28
+
+Logistic Map
+1.0
+0.8 
+0.6
+0.4 
+0.2 -
+0.0 
+1.0 
+0.8 -
+0.6 
+0.4 
+0.2 -
+0'0
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+μoLyapunov Exponents
+0-
+-2
+Equation
+6
+Rosenstein
+Eckmann X
+Eckmann Y
+2
+F9-
+Equation
+8 
+Rosenstein
+Eckmann X
+EckmannY
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+uo(a)
+(b) fig:
+(c)
+(d)
+(e)
+(f)
+Figure 18: Population vs. iteration of Vorticella Strange.
+29
+
+1.0 -
+μo = 3,025
+0.8 
+0.6 
+0.4 
+0.2 
+0.0
+50
+75
+100
+125
+150
+175
+200
+1.0
+0.8
+0.6 
+0.4
+..
+0.2 
+0.0
+0
+25
+50
+75
+100
+125
+150
+175
+200
+n1.0 -
+μo = 3.240
+0.8
+0.6 
+0.4
+0.2 
+0.0
+25
+50 
+75
+100
+125
+150
+175
+200
+1.0
+0.8
+0.6 
+0.4
+0.2 
+0.0
+0
+25
+50
+75
+100
+125
+150
+175
+200
+n1.0 -
+μo = 3.400
+0.8
+0.6
+0.4
+0.2 
+0.0
+25
+50 
+75
+100
+125
+150
+175
+200
+1.0
+0.8
+0.6 
+0.4
+0.2 
+0.0
+0
+25
+50
+75
+100
+125
+150
+175
+200
+n1.0 
+μo = 3.485
+0.8
+0.6 
+0.4
+0.2 
+0.0
+0
+25
+50
+75
+100
+125
+150
+175
+200
+1.0
+0.8
+0.6 
+0.4
+0.2
+0.0
+0
+25
+50
+75
+100
+125
+150
+175
+200
+n1.0 
+μo = 3.540
+0.8 
+0.6 
+0.4
+0.2 
+0.0
+0
+25
+50
+75
+100
+125
+150
+175
+200
+1.0
+0.8
+0.6 
+0.4
+0.2 
+0.0
+0
+25
+50
+75
+100
+125
+150
+175
+200
+n1.0 
+μo = 3.700
+0.8 
+0.6 
+0.4
+0.2 
+0.0
+0
+25
+50
+75
+100
+125
+150
+175
+200
+1.0
+0.8
+0.6 
+0.4
+0.2
+0.0
+25
+50
+75
+100
+125
+150
+175
+200
+nFigure 19: Analysis on eigenvalues for the case of Vorticella Strange.
+Legends have the same meaning
+described in Figure 4.
+.
+Figure 20: Absolute values of eigenvalues vs. growth rate at fixed points for Vorticella Strange. Impor-
+tant coordinates include: Upper middle panel (3.025, 1.935). Upper-right corner panel (2.299, 0.491), and
+(3.035, 1.000). Lower-left corner panel (2.069, 0.000), and (2.370, 0.490).
+30
+
+48.5
+0.04
+0.04
+48.4
+0.02
+Im(wi)
+0.02 
+Im(wo)
+0.00 -
+0.00 +
+0.02
+0.02
+48.2
+0.04
+0.04
+2.02
+2.03
+2.04
+2.05
+2.06
+2.07
+-48.50-48.45-48.40-48.35-48.30-48.2548.20-48.15
+2.02
+2.03
+2.04
+2.05
+2.06
+2.07
+3
+0.04
+0.04
+0.02
+0.02
+2
+Im(wo)
+Im(wi)
+ml
+00°0
++00'0
+0.02
+0.02
+0.04
+0.04
+0+
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+0.000.250.50°0.75
+1.00
+1.25
+1.501.75
+1.25 
+0.00
+1.50 
+1.00 
+0.25
+Im(wi)
+0.50
+1.25 
+0.50
+0.75
+0.25
+1.00
+0.75 
+0.00
+1.25
+0.50-
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4 
+1.6
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+Re(wo)
+Re(wi)
+[wol
+μo
+Type
+source
+non-hyperbolic
+saddle
+sink48.50
+1.6
+48.45
+2.5 
+1.4
+48.40
+2.0
+48.35
+1.2 
+1.5 
+1.0 
+48.25
+1.0 
+0.8
+48.20
+0.5 
+0.6
+48.15
+0.0 
+1.0
+1.5 
+2.0
+2.5
+3.0
+3.5
+4.0
+1.0
+1.5 
+2.0
+2.5 
+3.0
+3.5
+4.0
+1.0 
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+2.07
+1.4
+1.75
+1.2 
+2.06
+1.50
+1.0 
+1.25 手
+2.05
+0.8
+[wol 
+1.00 
+2.04
+0.6
+0.75 
+0.50
+0.4 
+2.03
+0.25 手
+0.2 
+2.02
+0.00
+0.0
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5 
+4.0
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+μo. Column for Ei
+μo. Column for E2
+μo. Column for E3(a)
+(b)
+(c)
+Figure 21:
+Analysis on phase
+portrait and phase space dia-
+gram about Vorticella Strange.
+(21a)Phase
+portrait
+shows
+Neimark-Sacker bifurcation es-
+tablished at (0.341, 0.377) with
+µ0 ≈ 3.025. See also caption in
+Figure 11.
+31
+
+Type
+1.0
+non-hyperbolic
+sink
+O
+saddle
+source
+0.8 -
+0.6
+y
+0.4 
+0.2 
+0.0 
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+μo
+X0.6
+0.4 -
+0.2 
+0.0 
+0.2
+0.4
+0.6
+0.8
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+μo0.4 
+0.2
+0.0 -
+0.2 
+0.4 -
+0.6 
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+μo
+yof the other. Therefore, it could be highly possible that, for example, the superconducting state, which refers
+to the zero resistance, may be achieved with chaos of some physical quantity that has relationship between
+resistance in the form of simultaneous difference equations.
+5
+Acknowledgment
+We thank Pui Ching Middle School in Macau PRC for the kindness to support this research project.
+References
+[1]
+Peter Roopnarine, Ecology and the Tragedy of the Commons, Sustainability 2013, 5(2), 749-773.
+[2]
+Ankit Kumar et al., A Computer-Based Simulation Showing Balance of the Population of Predator and
+Prey and the Effects of Human Intervention, 2021 IOP Conf. Ser.: Mater. Sci. Eng. 1031 012049
+[3]
+Cheng Sok Kin et al., Predicting Earth’s Carrying Capacity of Human Population as the Predator
+and the Natural Resources as the Prey in the Modified Lotka-Volterra Equations with Time-dependent
+Parameters, arXiv:1904.05002v2 [q-bio.PE]. Retrieved via https://doi.org/10.48550/arXiv.1904.
+05002
+[4]
+K.A. Hasan and M. F. Hama, ”Complex Dynamics Behaviors of a Discrete Prey-Predator Model with
+Beddington-DeAngelis Functional Response”, Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 45, 2179
+- 2195.
+[5]
+A George Maria Selvam et al, ”Bifurcation and Chaos Control for Prey Predator Model with Step Size
+in Discrete Time”, 2020 J. Phys.: Conf. Ser. 1543 012010.
+[6]
+Boshan Chen and Jiejie Chen, ”Bifurcation and Chaotic behavior of a discrete singular biological eco-
+nomic system.” Applied Mathematics and Computation, 219(2012) 2371-2386.
+[7]
+Yong Li et al, ”Flip and Neimark-Sacker Bifurcations of a Discrete Time Predator-Prey Model”, IEEE
+Access, Vol. 7, pp 123430-123435, 2019.
+[8]
+C. Wang and X. Li, J. Math. Ann. Appl., 422(2015) 920-939.
+[9]
+X.L. Liu and D.M. Xiau, Complex dynamic behaviors of a discrete-time predator-prey system, Chaos
+Solitons Fract., 32(2007)80-94.
+[10] Z. Hu, Z. Teng, and L. Zhang, Stability and bifurcation analysis of a discrete predator-prey model with
+nonmonotonic functional response, Nonlinear Anal. Real World Appl. 12(4)(2011)2356-2377.
+[11] Massimo Cencini, Fabio Cecconi, and Angelo Vulpiani, Chaos: From Simple Models To Complex Systems
+(Advances in Statistical Mechanics), Wspc; Illustrated edition (September 1, 2009)
+[12] J A Vano, J C Wildenberg, M B Anderson, J K Noel and J C Sprott, Chaos in low-dimensional
+Lotka–Volterra models of competition, Nonlinearity 19 (2006) 2391–2404.
+[13] Rubin H. Landau, Manuel J. P´aez, and Cristian C. Bordeianu, Computational Physics, 2015 WI-
+LEY VCH, pp. 356-357.
+[14] C.M. Evans and G.L. Findley, ”A new transformation for the Lotka-Volterra problem”, Journal of
+Mathematical Chemistry 25 (1999) 105-110.
+[15] Dunbar, S. R. (1983). Traveling wave solutions of diffusive Lotka-Volterra equations. Journal of Mathe-
+matical Biology, 17(1), 11-32.
+[16] Das, S. and Gupta, P. K. (2011). A mathematical model on fractional Lotka–Volterra equations. Journal
+of Theoretical Biology, 277(1), 1-6.
+32
+
+[17] T. Bessoir and A. Wolf, Chaos Simulations, Physics Academic Software, North Carolina State University,
+Raleigh, NC 27695-8202, USA.
+[18] Pelinovsky, E., Kurkin, A., Kurkina, O., Kokoulina, M., and Epifanova, A. (2020). Logistic equation
+and COVID-19. Chaos, Solitons & Fractals, 140, 110241.
+[19] A. Mareno, L. Q. English, ”Flip and Neimark–Sacker Bifurcations in a Coupled Logistic Map System”,
+Discrete Dynamics in Nature and Society, vol. 2020, Article ID 4103606, 14 pages, 2020. https://doi.
+org/10.1155/2020/4103606
+[20] Bo Li, Qizhi He and Ruoyu Chen,”Neimark–Sacker bifurcation and the generate cases of Kopel oligopoly
+model with different adjustment speed”, Advances in Difference Equations (2020) 2020:72, https:
+//doi.org/10.1186/s13662-020-02545-9.
+[21] Zeraoulia Elhadj and J. C. Sprott, ”The effect of modulating a parameter in the logistic map”, Chaos
+18, 023119 (2008).
+[22] Andrey L. Shilnikov and Nikolai F. Rulkov, Origin of Chaos in a Two-Dimensional Map Modeling
+Spiking-Bursting Neural Activity, International Journal of Bifurcation and Chaos, Vol. 13, No. 11(2003),
+3325-3340.
+[23] Waqas Ishaque, Qamar Din, Muhammad Taj and Muhammad Asad Iqbal,”Bifurcation and chaos control
+in a discrete-time predator–prey model with nonlinear saturated incidence rate and parasite interaction”,
+Advances in Difference Equations (2019) 2019:28, https://doi.org/10.1186/s13662-019-1973-z
+[24] Asifa Tassaddiq, Muhammad Sajjad Shabbir, Qamar Din and Humera Naaz, ”Discretization, Bifurcation
+and Control for a Class of Predator–Prey Interactions”, Fractal Fract. 2022, 6(1), 31; https://doi.or
+g/10.3390/fractalfract6010031
+[25] Abd-Elalim Elsadany, H.A. EL-Metwally, E.M. Elabbasy, and H.N. Agiza, Chaos and bifurcation of a
+nonlinear discrete prey-predator system, Computational Ecology and Software, 2012, 2(3): 169-198.
+[26] Michael P. Hassell, Hugh N. Comins & Robert M. Mayt, ”Spatial structure and chaos in insect population
+dynamics”, Nature volume 353, pages 255–258 (1991)
+[27] A. A. Berryman and J.A. Millstein, ”Are Ecological Systems Chaotic-And If Not, Why Not?”, Trends
+Ecol Evol. 1989 Jan;4(1):26-8. doi: 10.1016/0169-5347(89)90014-1.
+[28] M. T. Rosenstein, J. J. Collins, and C. J. De Luca, “A practical method for calculating largest Lyapunov
+exponents from small data sets,” Physica D: Nonlinear Phenomena, vol. 65, pp. 117–134, 1993.
+[29] J. P. Eckmann, S. O. Kamphorst, D. Ruelle, and S. Ciliberto, “Liapunov exponents from time series,”
+Physical Review A, vol. 34, no. 6, pp. 4971–4979, 1986.
+[30] A.J. Lotka, ”Contribution to the Theory of Periodic Reaction”, J. Phys, Chem. 14(3), 271–274, 1910.
+[31] V. Volterra, ”Variazioni e fluttuazioni del numero d’individui in specie animali conviventi”. Mem. Acad.
+Lincei Roma. 2: 31–113, 1926.
+[32] Lotka-Volterra Equation Wikipedia, https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra e
+quations
+[33] Immanuel M. Bonze, Lotka-Volterra Equation and Replicator Dynamics: A Two-Dimensional Classifi-
+cation, Biol. Cybern 48, 201-221, 1983. Eq 4 in our work has the same form as Eq 2 in this reference.
+[34] Steve H. Strogatz, Nonlinear Dynamics and Chaos, p.156, Addison Wesley, 2nd Ed.
+[35] Alan Wolf, Jack B. Swift, Harry L. Swinney and John A. Vastano, Determining Lyapunov Exponents
+from a Time Series, Physica 16D, 285-317, 1985.
+[36] Lorenzo Escot and Julio E. Sandubete Galan, ”A brief methodological note on chaos theory and its
+recent applications based on new computer resources”, Revista: ENERGEIA (ISSN: 1666-5732) Vol.
+VII, N´um. 1, 2020, pp 53-64.
+33
+
+[37] Sch¨olzel, Christopher. (2019, June 16). Nonlinear measures for dynamical systems (Version 0.5.2). Zen-
+odo. http://doi.org/10.5281/zenodo.3814723
+[38] Codes and animations may be retrieved via https://github.com/weishanlee/LotkaVolterraChaos
+[39] Stephen T. Thornton and Jerry B. Marrion, Classical Dynamics of Particles and Systems, 5th Ed., CH4,
+ISBN-10:0534408966
+[40] IGI Global https://www.igi-global.com/dictionary/flip-bifurcation/11262
+[41] Herbert Goldstein, Charles Poole, John Safko, Classical Mechanics, 3rd Ed, CH11, ISBN-10:0321188977.
+[42] J. Hale, and H Ko¸cak, Dynamics and Bifurcations. CH 11, Vol. 3, Berlin: Springer-Verlag (1991).
+ISBN-13: 9783540971412.
+34
+

ZNFPT4oBgHgl3EQfuTUu/content/tmp_files/2301.13155v1.pdf.txt

@@ -0,0 +1,1417 @@
+Published as a conference paper at ICLR 2023
+ADVANCING RADIOGRAPH REPRESENTATION LEARN-
+ING WITH MASKED RECORD MODELING
+Hong-Yu Zhou1,2∗
+Chenyu Lian1∗
+Liansheng Wang1
+Yizhou Yu2
+1School of Informatics, Xiamen University
+2Department of Computer Science, The University of Hong Kong
+whuzhouhongyu@gmail.com, dopaminel@foxmail.com,
+lswang@xmu.edu.cn, yizhouy@acm.org
+ABSTRACT
+Modern studies in radiograph representation learning (R2L) rely on either self-
+supervision to encode invariant semantics or associated radiology reports to in-
+corporate medical expertise, while the complementarity between them is barely
+noticed. To explore this, we formulate the self- and report-completion as two com-
+plementary objectives and present a unified framework based on masked record
+modeling (MRM). In practice, MRM reconstructs masked image patches and
+masked report tokens following a multi-task scheme to learn knowledge-enhanced
+semantic representations. With MRM pre-training, we obtain pre-trained mod-
+els that can be well transferred to various radiography tasks. Specifically, we
+find that MRM offers superior performance in label-efficient fine-tuning. For in-
+stance, MRM achieves 88.5% mean AUC on CheXpert using 1% labeled data,
+outperforming previous R2L methods with 100% labels. On NIH ChestX-ray,
+MRM outperforms the best performing counterpart by about 3% under small la-
+beling ratios. Besides, MRM surpasses self- and report-supervised pre-training in
+identifying the pneumonia type and the pneumothorax area, sometimes by large
+margins. Code and models are available at https://github.com/RL4M/
+MRM-pytorch.
+1
+INTRODUCTION
+Findings:
+[MASK] cardiac, mediastinal and
+hilar [MASK] are normal.
+[MASK] [MASK] [MASK] normal.
+lungs are [MASK] .
+…
+…
+Inputs
+Report
+Radiograph
+Paired
+Representations
+Findings:
+the cardiac, mediastinal and
+hilar contours are normal.
+pulmonary vasculature is normal.
+lungs are clear .
+Outputs
+Figure 1: Illustration.
+MRM learns trans-
+ferable radiograph representations via recon-
+structing masked records, i.e., masked radio-
+graph patches and masked reports tokens.
+Radiograph representation learning (R2L) has been
+among the core problems of medical image analysis.
+Previously, downstream radiograph analysis tasks
+counts on pre-trained models on ImageNet (Deng
+et al., 2009) or large X-ray datasets (Wang et al.,
+2017; Irvin et al., 2019; Johnson et al., 2019; Bus-
+tos et al., 2020) to alleviate the shortage of expert
+labeling. The emergence of self-supervised repre-
+sentation learning (Doersch et al., 2015; Agrawal
+et al., 2015; Wang & Gupta, 2015) provides a choice
+to conduct pre-training with negligible human inter-
+vention by exploiting self-supervision. However, the
+self-supervised paradigm ignores the introduction of
+medical expertise (e.g., anatomy), reducing its trans-
+ferability to downstream tasks with limited label in-
+formation.
+On the other hand, free-text radiology reports written
+by experienced radiologists often contain rich do-
+main knowledge. To leverage this, researchers developed automated rule-based labelers (Wang
+et al., 2017; Irvin et al., 2019) to extract structured labels from unstructured texts. Nevertheless,
+∗Work done while visiting Xiamen University. First two authors contributed equally.
+1
+arXiv:2301.13155v1  [cs.CV]  30 Jan 2023
+
+Published as a conference paper at ICLR 2023
+these labelers have several practical limitations. First, some procedures of the label extraction work-
+flow, such as rulemaking and natural language processing, still require the intensive involvement of
+experts and engineers. Besides, the developed labelers can hardly adapt to new scenarios due to the
+fixed rules and lexicons.
+Against this background, report-supervised R2L was proposed (Zhang et al., 2020) to acquire super-
+vision from radiology reports. In practice, this paradigm leverages words and sentences in free-text
+reports as supervision to guide deep neural networks to learn radiograph representations, outper-
+forming the archetypical label- and self-supervised pre-training by observable margins in various
+downstream tasks (Zhang et al., 2020; Zhou et al., 2022). The report-supervised R2L highlights
+the importance of the incorporation of domain knowledge. This differs from the self-supervised
+paradigm, which focuses on learning invariant semantic representations. Nonetheless, current stud-
+ies view the self- and report-supervised R2L as separate, discrete choices, preventing their combina-
+tions.
+Driven by this analysis, we present a unified framework based on masked record modeling (MRM),
+where the self- and report-completion tasks are modeled as two complementary objectives. Specif-
+ically, masked image reconstruction integrates semantics into pre-trained models, while masked
+report restoration facilitates the incorporation of medical expertise.
+As a result, MRM learns
+knowledge-enhanced semantic representations that generalize well. In practice, MRM masks ran-
+dom patches and tokens from the input radiograph and associated radiology report with high mask-
+ing ratios. Following a multi-task scheme, MRM asks the radiography pre-trained model to learn
+visual representations that can not only reconstruct the missing patches but also restore the missing
+tokens from the non-masked token embeddings along with mask tokens.
+With MRM pre-training, we can train radiography models on MIMIC-CXR (Johnson et al., 2019)
+with improved generalization performance. With a pre-trained ViT-B/16 model, we achieve 88.5%
+mean AUC when fine-tuned on CheXpert (Irvin et al., 2019) with only 1% labels. This outperforms
+all previous counterparts with 100% labeled data. On NIH ChestX-ray (Wang & Gupta, 2015),
+MRM surpasses the report-supervised paradigm by about 3% when the labeling ratios1 are 1% and
+10%. On pneumonia identification tasks, MRM outperforms self- and report-supervised baselines,
+sometimes by substantial margins. These observations help verify the effectiveness of MRM in
+learning more transferable radiograph representations.
+2
+RELATED WORK
+2.1
+REPORT-SUPERVISED RADIOGRAPH REPRESENTATION LEARNING
+Recently, report-supervised learning (Zhang et al., 2020; Liao et al., 2021; Huang et al., 2021; Zhou
+et al., 2022; Boecking et al., 2022) emerges as a new R2L paradigm that automatically acquires
+supervision from free-text radiology reports. Zhang et al. (2020) proposed ConVIRT to contrast the
+radiograph features with latent embeddings of sentences in radiology reports. Liao et al. (2021) and
+Huang et al. (2021) explored the alignment between local patches and words in the report. Zhou et al.
+(2022) presented a Transformer-based R2L framework that conducts autoregressive report modeling
+and study-report matching. Report-supervised R2L takes the advantage of label-supervised learning,
+which is the incorporation of domain knowledge. Compared to the self-supervised paradigm, report-
+supervised R2L lays no emphasis on learning semantically invariant representations. To address
+the discrepancy between them, we formalize self- and report-completion as two complementary
+objectives, based on which we propose to encode both semantics and medical expertise into latent
+representations following a multi-task scheme.
+2.2
+VISUAL REPRESENTATION LEARNING VIA IMAGE-LANGUAGE PRE-TRAINING
+Learning visual representations from image-language pairs has achieved tremendous success in nat-
+ural image tasks (Sariyildiz et al., 2020; Desai & Johnson, 2021; Radford et al., 2021; Mu et al.,
+2021; Li et al., 2021; Geng et al., 2022; Wang et al., 2022; Chen et al., 2022; Singh et al., 2022; Yu
+et al., 2022; Dou et al., 2022; Arici et al., 2021; Kwon et al., 2022). Similar to ConVIRT (Zhang
+1The labeling ratio X% means that X% of the training set from a fully annotated downstream dataset are
+used for supervised fine-tuning.
+2
+
+Published as a conference paper at ICLR 2023
+Tokenizer
+…… There is no focal 
+consolidation, effusion, 
+or pneumothorax.  ……. 
+No free air below the 
+right hemidiaphragm is 
+seen. ……
+Masked language modeling (MLM) loss
+GAP
+Duplication
+Low-resolution input
+…… There [MASK] no 
+[MASK] consolidation, 
+[MASK] , or [MASK]. ……. 
+No [MASK] air [MASK] 
+the [MASK] [MASK] is 
+seen. ……
+…… There is no focal
+consolidation, effusion, 
+or pneumothorax.  ……. 
+No free air below the 
+right hemidiaphragm is 
+seen. ……
+Paired
+Rand. mask
+(ratio = 50%)
+Rand. mask
+(ratio = 75%)
+Image
+encoder
++
+Report
+decoder
+Image
+decoder
+Sub-sampling
+Masked image modeling (MIM) loss
+High-resolution output
+…
+…
+Non-masked report token embed.
+Non-masked image patch embed.
+Non-masked hybrid token embed.
+Mask token embed.
+Input report
+Masked report generation
+Hybrid representations
+Masked report restoration
+Input radiograph
+Masked LR image generation
+Radiograph representations
+Masked HR image restoration
+Figure 2: OVERVIEW. During the pre-training stage, MRM requires the image encoder to provide
+radiograph representations to simultaneously support the restoration of masked radiograph patches
+and masked associated radiology report tokens. The masked language and image modeling losses
+are only calculated on image and report tokens highlighted in pink . Embed., LR, and HR stand
+for embeddings, low-resolution, and high-resolution, respectively.
+et al., 2020), image-language contrast has been widely adopted to conduct pre-training (Radford
+et al., 2021; Mu et al., 2021; Li et al., 2021; Yu et al., 2022; Dou et al., 2022; Gao et al., 2022).
+Nowadays, efforts have been made to train a unified encoder for vision and language data (Geng
+et al., 2022; Wang et al., 2022; Chen et al., 2022; Geng et al., 2022). Akin to our approach, SLIP (Mu
+et al., 2021) combines SimCLR (Chen et al., 2020) and CLIP (Radford et al., 2021) to train a vi-
+sion encoder using image-language pairs. However, SLIP only slightly outperforms SimCLR in
+fine-tuning, while requiring large batch sizes and tens of millions of image-language pairs for pre-
+training. In contrast, our MRM surpasses various self-supervised methodologies by large margins
+and can be pre-trained using only hundreds of thousands of radiograph-report pairs, enabling effec-
+tive medical visual representation learning with limited annotations and computing resources.
+3
+MASKED RECORD MODELING
+We propose MRM (i.e., Masked Record Modeling) to learn radiograph representations using record-
+level supervision. As the name implies, MRM acquires supervision signals from both radiographs
+and associated radiology reports. The motivation behind is to learn knowledge-enhanced semantic
+latent representations by reconstructing masked radiograph patches and masked radiology report
+tokens in medical records.
+Fig. 2 presents an overview of MRM. We first apply random masking to each low-resolution radio-
+graph and its associated radiology report (with different high masking ratios). Then, we forward
+the obtained non-masked image patches to the image encoder to acquire non-masked image patch
+embeddings. These embeddings serve two purposes: (i) assist non-masked report tokens to restore
+the masked report tokens; (ii) restore the high-resolution masked radiograph patches. To achieve
+the first goal, we add the globally averaged radiograph representation to each non-masked report
+token embedding and pass the resulting hybrid representations to the report decoder for masked re-
+port restoration. As for the second purpose, we conduct a novel patch restoration task to explicitly
+encode more local details into radiograph representations by reconstructing high-resolution patches
+from low-resolution inputs.
+3.1
+REPORT COMPREHENSION
+Masked report generation. In our scenario, each radiology report is associated with a radiograph.
+To convert the free-text report into tokens, we use WordPiece (Wu et al., 2016) as the default to-
+kenizer, whose vocabulary has approximately 30k tokens. After tokenization, we randomly mask
+3
+
+2CPublished as a conference paper at ICLR 2023
+a number of report tokens with [MASK]. Compared to BERT (Devlin et al., 2018) that randomly
+masks 15% tokens, we use a 50% probability of masking each token in the report. The insight
+behind the use of a higher masking ratio is that we want the model to lean more upon the image
+embeddings to finish the report-completion task.
+Hybrid representations for storing multi-modal information. We then transform non-masked
+report tokens into token embeddings using a simple lookup table2, which stores randomly initialized
+embeddings of a fixed dictionary and size. In practice, we retrieve embeddings using indices. Then,
+the global embedding of the associated radiograph is added to each non-masked token embedding.
+The resulting non-masked hybrid embeddings are supposed to include the multi-modal information
+from the radiograph and associated radiology report, which ought to be helpful for restoring the
+masked tokens.
+Masked report restoration. To reconstruct the masked tokens, we forward latent embeddings of
+both hybrid tokens and mask tokens to the report decoder (a light-weight transformer model), where
+fixed (i.e., unlearnable) positional embeddings are added to encode the position information. We
+train the report decoder using the masked language modeling objective.
+3.2
+RADIOGRAPH UNDERSTANDING
+Masked image generation with low resolution. We propose to learn radiograph representations
+by reconstructing high-resolution radiograph patches from low-resolution inputs. The motivation
+behind is to encode more local information into latent embeddings via super-resolution imaging.
+As shown in Fig. 2, we sub-sample each high-resolution radiograph by a factor of two to generate
+a low-resolution input. Following He et al. (2022), we split low-resolution radiograph into non-
+overlapping image patches, where 75% patches are randomly masked.
+Radiograph representations. We add fixed unlearnable positional embeddings to linearly trans-
+formed non-masked image patches.
+Next, we forward the resulting patch embeddings to the
+transformer-based image encoder, which produces non-masked image patch embeddings. Then, the
+global average pooling (GAP) is applied to all non-masked embeddings, whereby a global feature
+is obtained. Here, we hypothesize that the image-level information brought by the global feature
+is helpful to the restoration of masked report tokens. Based on this hypothesis, we duplicate and
+add the global feature to each non-masked report token embedding, producing the hybrid token
+embeddings that encode the multi-modal information.
+Masked image restoration with high resolution. Non-masked image and mask token represen-
+tations with added positional embeddings are passed to the image decoder for the restoration of
+masked radiograph patches. Specifically, the image decoder is required to restore a high-resolution
+(2× the input resolution) patch from each input token via a shared fully-connected (FC) layer (across
+all tokens). In practice, the proposed restoration procedure explicitly requires the learned image rep-
+resentations to include more local details that often matter a lot in medical diagnosis.
+3.3
+MULTI-TASK MODELING
+Suppose each input radiograph consists of two set IM and IN . The masked set IM={x1, . . . , xh}
+(ground truths) contains h high-resolution image patches that serve as reconstruction targets. The
+non-masked set IN ={s1, . . . , sk} comprises k low-resolution patches that are treated as model in-
+puts. Likewise, we denote the associated radiology report using the masked set RM={u1, . . . , up}
+(ground truths) and the non-masked set RN ={v1, . . . , vq} (inputs). Here, x, s, u, and v stand for
+the masked image patch, non-masked image patch, masked report token, and non-masked report
+token, respectively. For model parameters, we use ΘE, ΘD, and ΘR to denote the parameters of the
+image encoder, image decoder, and report decoder, respectively.
+For the restoration of masked report tokens, we forward hybrid representations to the report decoder
+and minimize the negative log-likelihood function. During the training stage, the objective function
+2https://pytorch.org/docs/stable/generated/torch.nn.Embedding.html.
+4
+
+Published as a conference paper at ICLR 2023
+LR (i.e., the MLM loss in Fig. 2) of the above optimization procedure can be summarized as follows:
+LR(RM, RN , IN ) = −
+p
+�
+i=1
+log P (ui | v1:q, s1:k; ΘE, ΘR) ,
+(1)
+where P stands for the conditional probability. We ignore the mask tokens for simplicity.
+Similarly, we can formalize the objective function of the high-resolution masked radiograph restora-
+tion (cf. the MIM loss in Fig. 2) as follows:
+LI(IM, IN ) = MSE (fΘD(fΘE(s1:k)), x1:h) .
+(2)
+In practice, we adopt the mean squared error (MSE) to measure the differences between the predicted
+and ground-truth image patches with high resolution, where all pixel values are normalized to [0, 1].
+The total multi-task training objective function (to be minimized) of the multi-modal restoration is
+as follows:
+L(RM, RN , IM, IN ) = LR(RM, RN , IN ) + λLI(IM, IN )
+(3)
+where λ is a hyper-parameter that controls the relative impacts of two objective functions. After
+pre-training, we can transfer the weight parameters of the image encoder (i.e., ΘE) to various down-
+stream tasks for fine-tuning.
+4
+EXPERIMENTS
+In this section, we mainly compare MRM against report- and self-supervised R2L methodologies on
+5 well-established public datasets. Average results are reported over three training runs.
+4.1
+MIMIC-CXR FOR PRE-TRAINING
+We conduct pre-training on MIMIC-CXR (Johnson et al., 2019), one of the largest X-ray datasets,
+that contains more than 370,000 radiograph images from over 220,000 patient studies. Each radio-
+graph is paired with one associated radiology report.
+4.2
+DATASETS FOR FINE-TUNING
+We validate the transferability of learned radiograph representations on X-ray based classification
+and segmentation tasks via end-to-end fine-tuning. Specifically, we evaluate the pre-trained model
+on 4 X-ray datasets in the classification tasks, which are NIH ChestX-ray (Wang et al., 2017),
+CheXpert (Irvin et al., 2019), RSNA Pneumonia (Shih et al., 2019), and COVID-19 Image Data
+Collection (Cohen et al., 2020). For the segmentation task, we fine-tune the pre-trained model on
+SIIM-ACR Pneumothorax Segmentation.3
+CheXpert introduces a multi-label classification problem on chest X-rays. We follow the official
+guideline (Irvin et al., 2019) and report the model performance on 5 selected pathologies, i.e., at-
+electasis, cardiomegaly, consolidation, edema, and pleural effusion. Considering the official test
+set of CheXpert is not available to the public, we follow ConVIRT (Zhang et al., 2020) to regard
+the official validation set as the test set. Meanwhile, we randomly sample 5,000 images from the
+official training set to build the validation set. The training/validation/test split each constitutes
+218,414/5,000/234 images of the whole dataset.
+RSNA Pneumonia defines a binary classification problem, where each chest radiograph is cat-
+egorized as either pneumonia or normal.
+We adopt the official data split, where the train-
+ing/validation/test set comprises 25,184/1,500/3,000 images, respectively.
+NIH ChestX-ray consists of about 112,120 frontal-view chest radiograph images, where a multi-
+label classification problem on 14 chest pathologies is introduced. The training/validation/test split
+each constitutes 70%/10%/20% of the whole dataset.
+COVID-19 Image Data Collection is a relatively small dataset, which involves 900 chest radio-
+graphs. We follow Zhou et al. (2022) to conduct fine-tuning on this small-scale dataset to investigate
+3https://www.kaggle.com/c/siim-acr-pneumothorax-segmentation.
+5
+
+Published as a conference paper at ICLR 2023
+Methods
+Input
+Size
+Pre-train.
+Data
+CheXpert
+RSNA Pneumonia
+SIIM
+1%
+10%
+100%
+1%
+10%
+100%
+10%
+100%
+Our MRM
+224
+MI-CXR
+88.5± 0.7
+88.5± 0.6
+88.7± 0.3
+91.3± 0.6
+92.7± 0.4
+93.3± 0.4
+73.2± 0.5
+91.4± 0.3
+CNN-based
+ConVIRT
+224
+CheXpert
+85.9
+86.8
+87.3
+77.4
+80.1
+81.3
+43.2
+59.9
+GLoRIA
+224
+CheXpert
+86.6
+87.8
+88.1
+86.1
+88.0
+88.6
+46.9
+63.4
+ConVIRT
+224
+MI-CXR
+87.0
+88.1
+88.1
+88.8
+91.5
+92.7
+-
+-
+MedKLIP†
+224
+MI-CXR
+-
+-
+-
+87.3
+88.0
+89.3
+72.1
+79.4
+BioViL
+480
+PubMed +
+MI-III/CXR
+-
+-
+-
+88.1
+88.4
+89.1
+-
+-
+Transformer-based
+GLoRIA∗
+224
+MI-CXR
+86.5± 0.8
+87.5± 0.6
+87.8± 0.5
+89.7± 0.8
+91.2± 0.5
+92.1± 0.3
+71.8± 0.7
+90.9± 0.4
+REFERS
+224
+MI-CXR
+87.2± 0.8
+88.1± 0.5
+88.2± 0.3
+89.4± 0.7
+91.6± 0.7
+92.7± 0.4
+72.1± 0.5
+89.7± 0.2
+M3AE
+224
+MI-CXR
+86.2± 0.6
+87.3± 0.6
+87.9± 0.4
+89.0± 0.5
+90.8± 0.6
+92.3± 0.3
+72.0± 0.7
+90.4± 0.3
+MGCA†
+224
+MI-CXR
+88.8
+89.1
+89.7
+89.1
+89.9
+90.8
+59.3
+64.2
+Table 1: COMPARISONS ON CHEXPERT, RSNA PNEUMONIA, AND SIIM. We report AUC scores
+of different labeling ratios when fine-tuning on CheXpert and RSNA Pneumonia. In compari-
+son, dice scores are presented on SIIM. The best results are bolded. MI- stands for the MIMIC
+dataset series. Note that ResNet-50 and ViT-B/16 are treated as the default backbones for CNN-
+and Transformer-based methods, respectively. * denotes our implementation of GLoRIA using ViT-
+B/16. Approaches with † leverage disease-level annotations for pre-training. Specifically, numbers
+of MGCA on CheXpert and RSNA Pneumonia are linear classification results.
+Labeling Ratios
+Methods
+Average
+Atelectasis
+Cardiomegaly
+Consolidation
+Edema
+Effusion
+Emphysema
+Fibrosis
+Hernia
+Infiltration
+Mass
+Nodule
+Pleural Thickening
+Pneumonia
+Pneumothorax
+1%
+Our MRM
+79.4± 0.8
+78.8
+90.3
+80.0
+86.5
+86.9
+82.0
+71.9
+90.0
+67.2
+82.3
+69.6
+72.3
+69.6
+84.0
+MedKLIP
+77.2
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+REFERS
+76.7
+77.5
+85.6
+78.6
+84.9
+85.4
+79.5
+72.3
+77.1
+67.5
+76.2
+66.5
+71.6
+69.3
+81.7
+Model Genesis
+70.3
+72.1
+67.1
+75.8
+76.1
+80.6
+72.6
+64.8
+73.5
+65.7
+65.2
+62.2
+67.6
+64.8
+76.2
+C2L
+71.1
+75.1
+67.1
+77.6
+75.1
+83.4
+71.5
+66.8
+70.0
+63.8
+70.1
+66.2
+68.1
+65.7
+74.4
+Context Restoration
+67.8
+69.1
+64.4
+73.2
+73.8
+78.1
+70.0
+62.1
+70.2
+65.2
+62.4
+59.1
+65.0
+62.2
+73.8
+TransVW
+71.3
+74.5
+68.9
+76.7
+79.8
+81.1
+67.9
+68.7
+68.2
+66.8
+66.5
+66.2
+68.5
+68.8
+75.0
+ImageNet Pre-training
+69.8
+73.3
+69.6
+76.0
+81.7
+80.5
+67.1
+64.9
+64.8
+65.8
+67.0
+62.3
+65.7
+65.0
+74.0
+10%
+Our MRM
+84.0± 0.5
+82.3
+90.9
+81.1
+89.0
+88.8
+92.2
+84.8
+94.0
+70.1
+86.6
+75.1
+78.6
+74.3
+88.4
+MedKLIP
+78.9
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+REFERS
+80.9
+80.1
+89.8
+79.5
+87.8
+87.5
+88.2
+77.2
+86.1
+69.6
+82.0
+72.8
+74.2
+72.2
+85.6
+Model Genesis
+76.0
+77.2
+72.8
+77.5
+85.7
+85.2
+81.0
+75.3
+78.0
+68.4
+73.1
+69.5
+72.2
+67.7
+80.4
+C2L
+76.6
+78.0
+75.5
+77.5
+84.1
+85.7
+81.2
+73.7
+79.5
+67.4
+77.5
+71.7
+72.0
+67.3
+81.9
+Context Restoration
+73.8
+75.5
+70.6
+77.1
+84.5
+84.2
+79.4
+73.1
+67.5
+68.1
+70.9
+66.9
+71.7
+65.2
+79.1
+TransVW
+74.4
+76.5
+70.8
+77.6
+83.0
+84.8
+79.7
+69.9
+74.7
+68.5
+72.1
+68.3
+72.4
+63.2
+79.6
+ImageNet Pre-training
+74.4
+74.2
+79.8
+75.9
+85.7
+83.2
+80.4
+72.1
+74.0
+64.1
+71.7
+65.6
+69.6
+66.2
+79.7
+100%
+Our MRM
+85.9± 0.3
+84.2
+93.0
+82.2
+91.0
+89.6
+94.3
+86.7
+94.4
+71.8
+88.2
+78.5
+81.4
+77.3
+90.2
+MedKLIP
+83.2
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+-
+REFERS
+84.7
+83.0
+92.3
+82.1
+90.2
+88.7
+91.4
+83.9
+93.3
+74.1
+85.5
+76.7
+78.5
+77.0
+89.1
+Model Genesis
+81.0
+78.8
+84.5
+79.2
+87.8
+86.6
+89.7
+81.0
+85.2
+71.1
+81.9
+73.2
+75.8
+73.0
+85.6
+C2L
+82.2
+81.1
+90.2
+81.0
+88.1
+88.0
+88.3
+80.8
+86.8
+72.0
+82.7
+74.1
+76.2
+75.3
+85.9
+Context Restoration
+78.7
+75.8
+82.9
+76.4
+86.6
+84.8
+88.2
+78.6
+83.0
+70.0
+79.6
+69.5
+73.2
+69.4
+84.0
+TransVW
+81.7
+79.8
+85.0
+80.0
+88.2
+87.1
+90.1
+81.8
+85.9
+72.3
+82.6
+74.4
+76.6
+74.0
+86.1
+ImageNet Pre-training
+80.0
+78.3
+89.3
+77.6
+87.9
+85.9
+87.4
+78.5
+88.8
+65.9
+79.9
+70.7
+74.5
+71.0
+84.7
+Table 2: COMPARISONS ON NIH CHESTX-RAY. Besides self-supervised and transfer learning
+baselines, we also present the performance of REFERS and MedKLIP (with competitive perfor-
+mance on CheXpert, RSNA Pneumonia, and SIIM) as references. AUC scores are displayed. The
+highest AUC scores in each labeling ratio are bolded.
+the effectiveness of various pre-training methodologies when the amount of annotations is limited.
+There are two tasks included. The first task requires the model to distinguish COVID-19 cases
+from non-COVID-19 pneumonia cases, where the training/validation/test set comprises 356/54/99
+radiographs, respectively. The second task is to distinguish viral pneumonia cases from bacterial
+pneumonia ones, where the training/validation/test set contains 297/43/86 cases, respectively.
+SIIM-ACR Pneumothorax Segmentation (SIIM) aims to facilitate the development of segmen-
+tation models to identify pneumothorax disease in chest radiographs. SIIM contains over 120,000
+frontal-view chest X-rays with precise manual segmentation of pneumothorax. We follow Huang
+et al. (2021) to construct the training/validation/test split, where each constitutes 70%/15%/15% of
+the whole dataset.
+6
+
+Published as a conference paper at ICLR 2023
+4.3
+BASELINES
+4.3.1
+REPORT-SUPERVISED METHODOLOGIES
+We first compare MRM against a range of pre-training approaches, which use radiology reports
+as supervision to learn radiograph representations. There are 4 report-supervised approaches in-
+volved in the baseline comparisons, which are ConVIRT (Zhang et al., 2020), GLoRIA (Huang
+et al., 2021), BioViL (Boecking et al., 2022), and REFERS (Zhou et al., 2022). Specifically, Con-
+VIRT (Zhang et al., 2020) proposed to learn medical visual representations by contrasting paired
+radiographs and sentences from radiology reports. GLoRIA (Huang et al., 2021) improved ConVIRT
+by contrasting radiograph sub-regions and words in the reports. BioViL (Boecking et al., 2022) and
+REFERS (Zhou et al., 2022) incorporated masked language modeling loss into contrastive learn-
+ing. Moreover, REFERS introduced a multi-view fusion attention to better align the representations
+of each radiograph and its associated report. In addition, MGCA (Wang et al., 2023) and Med-
+KLIP (Wu et al., 2023) were included as two recent baselines4. Apart from above baselines, we also
+include M3AE (Geng et al., 2022), a recent masked multi-modal pre-training method aside from the
+application to medical data, for comparison.
+In experiments, we fine-tune pre-trained models of MRM and other report-supervised methods on
+CheXpert (classification), RSNA Pneumonia (classification), and SIIM (segmentation).
+4.3.2
+SELF-SUPERVISED AND TRANSFER LEARNING METHODS
+Besides report-supervised approaches, we also include self-supervised and transfer learning ap-
+proaches in our comparisons. Specifically, Context Restoration (Chen et al., 2019), Model Gen-
+esis (Zhou et al., 2021), and TransVW (Haghighi et al., 2021) are based on predictive SSL, while
+C2L (Zhou et al., 2020) was developed on top of contrastive learning. In addition, MRM is also
+compared to ImageNet pre-training (Wang et al., 2017). In practice, we conduct the comparisons
+with self-supervised and transfer learning approaches on NIH ChestX-ray and COVID-19 Image
+Data Collection.
+Methods
+COVID-19 vs. Others
+Viral vs. Bacterial
+Our MRM
+85.8± 0.4
+91.5± 0.3
+REFERS
+82.1
+80.4
+Model Genesis
+76.0
+71.8
+C2L
+77.8
+73.0
+Context Restoration
+74.6
+69.8
+TransVW
+76.1
+71.5
+ImageNet Pre-training
+74.1
+70.3
+Table 3: COMPARISONS ON COVID-
+19 IMAGE DATA COLLECTION.
+The
+best results are bolded.
+Methods
+1%
+10%
+100%
+Our MRM
+79.4
+84.0
+85.9
+- Masked modeling & Super-resolution restoration
+69.9
+75.2
+80.3
+- Masked report modeling (LR)
+74.7
+81.3
+85.1
+- Masked radiograph modeling (LI)
+76.7
+82.2
+84.7
+- Super-resolution restoration
+78.8
+83.7
+85.7
++ Hybrid features for image restoration
+78.9
+83.6
+85.7
+Table 4: ABLATIONS ON NIH CHESTX-RAY.
+AUC scores of three different labeling ratios are
+reported. The best results are bolded.
+4.4
+RESULTS
+4.4.1
+COMPARISONS WITH REPORT-SUPERVISED BASELINES
+In Table 1, we present the comparative results with report-supervised methodologies on CheXpert,
+RSNA Pneumonia, and SIIM (segmentation). Specifically, we investigate the performance when
+fine-tuning with limited and full supervision. We provide reconstruction examples and segmentation
+results in the appendix.
+From Table 1, we observe no obvious performance gap between CNN- and Transformer-based
+report-supervised pre-training methods. For instance, after implementing GLoRIA on MIMIC-CXR
+with ViT-B/16, we observe performance drops and improvements on CheXpert and RSNA Pneumo-
+nia, respectively. This contrast demonstrates that replacing CNN with Transformer may not bring
+performance gains to report-supervised pre-training. Among all baselines, REFERS is the best per-
+forming approach.
+Nonetheless, MRM consistently outperforms various baselines on all three datasets under different
+labeling ratios. Specifically, MRM maintains more advantages over previous pre-training method-
+4MGCA (Wang et al., 2023) and MedKLIP (Wu et al., 2023) were released after the submission deadline of
+ICLR 2023. We added results in the camera ready for better comparisons.
+7
+
+Published as a conference paper at ICLR 2023
+Fusion
+1%
+10%
+100%
+GAP
+79.4
+84.0
+85.9
+GMP
+77.2
+83.8
+86.0
+(a) Multi-modal fusion strategies
+PI
+1%
+10%
+100%
+75%
+79.4
+84.0
+85.9
+50%
+78.8
+84.4
+86.1
+0%
+75.4
+82.0
+84.4
+(b) Masking ratios for radiographs
+λ
+1%
+10%
+100%
+1
+79.4
+84.0
+85.9
+2
+78.6
+83.5
+85.4
+0.5
+79.0
+83.7
+85.9
+(c) Loss controlling factor λ
+Table 5: Ablation studies on choices of multi-modal fusion and hyper-parameters. GAP and GMP
+stand for global average and maximum pooling, respectively. Experiments are performed on NIH
+ChestX-ray under various labeling ratios. The best results are bolded.
+ologies when fine-tuning with limited annotations, which is quite meaningful for medical image
+analysis as large amounts of specialist annotations (from radiologists or clinicians) are usually hard
+to access. It is worth noting that MRM achieves 88.5% when using only 1% labeled data on CheX-
+pert, better than previous counterparts with 100% annotations.
+We see that M3AE generally performs worse than our MRM in all labeling ratios, especially under
+extremely limited data. The underperformance of M3AE may be attributed to the fact that it requires
+a large amount of multi-modal data to learn transferable joint representations for images and texts.
+4.4.2
+COMPARISONS WITH SELF-SUPERVISED AND TRANSFER LEARNING BASELINES
+Table 2 presents the average and per-class classification results on NIH ChestX-ray. Compared to
+self-supervised learning baselines, MRM achieves large improvements in almost every chest pathol-
+ogy. On average, MRM outperforms C2L (Zhou et al., 2020) and TransVW (Haghighi et al., 2021),
+the best two self-supervised pre-training methodologies, by about 8% when the amount of available
+labeled data is extremely limited (i.e., the labeling ratio is 1%). Similarly, we also observe remark-
+able improvements when comparing MRM to ImageNet Pre-training (Wang et al., 2017). These
+phenomena demonstrate that the radiograph representations learned by MRM are more transferable
+than those from previous self-supervised and transfer learning methods.
+Compared to REFERS (i.e., the best performing report-supervised baseline in Table 1), MRM still
+provides notable improvements in most chest pathologies. Specifically, MRM is more advantageous
+when the amount of labeled data is limited. For instance, MRM surpasses REFERS by 2.7% and
+3.1% on average when the labeling ratios are 1% and 10%, respectively. These comparisons again
+verify the effectiveness of MRM over previous report-supervised pre-training counterparts.
+We also investigate the impacts of pre-training on the real-world scenario with extremely limited
+specialist supervision. In Table 3, we compare the performance of a range of pre-training method-
+ologies on two binary classification tasks, which are distinguishing COVID-19 from non-COVID-19
+pneumonia, and differentiating between viral pneumonia and bacterial pneumonia. Again, MRM
+outperforms self-supervised, transfer learning, and report-supervised pre-training methodologies by
+substantial margins. Compared to REFERS, MRM brings nearly 4% and 11% improvements to
+two tasks, respectively, further enhancing the practicability of the diagnosis system trained with
+extremely limited supervision.
+4.5
+ABLATION ANALYSIS
+Advantages over single-task pre-training paradigm. First of all, we remove the two masked mod-
+eling objectives and super-resolution restoration task, resulting in substantial performance drops.
+These results verify the necessity of using masked modeling and super-resolution restoration in
+MRM. After removing the masked report modeling objective, MRM only acquires supervision
+signals from self-supervision. Thus, the whole framework degenerates into a self-supervised pre-
+training methodology. From Table 4, we observe that removing LR leads to dramatic performance
+degradation in different labeling ratios. Moreover, we find that introducing the masked report mod-
+eling is greatly helpful to the fine-tuning performance with limited labeling resources (1% and
+10%). For instance, adding LR brings about 5-percent improvements to MRM. Similarly, remov-
+ing the masked radiograph modeling also leads to notable performance drops in all labeling ratios.
+These results demonstrate the necessity of introducing multi-task objectives in masked modeling
+pre-training.
+8
+
+Published as a conference paper at ICLR 2023
+Is super-resolution restoration helpful? As Table 4 shows, the proposed super-resolution restora-
+tion provides consistent performance gains in different labeling ratios. The underlying reason may
+be that the low to high resolution restoration process helps preserve more local information into
+latent representations, which enhances the transferable ability to downstream tasks.
+Would it be beneficial to introduce multi-modal information to image restoration? We investi-
+gate this question by adding non-masked report token embeddings to image patch embeddings and
+passing the resulting hybrid features to the image decoder. As Table 4 shows, introducing multi-
+modal information to masked image restoration does not improve the fine-tuning performance. We
+leave the exploration of the reason behind to future work.
+Ablations on multi-modal fusion and hyper-parameters. In Table 5a, we present the experimental
+results of using different strategies for radiograph-report fusion. We find that global average pooling
+(GAP) outperforms global maximum pooling (GMP) by 2.2% when access to labeled data is quite
+limited while achieving comparable results as the labeling ratio increases. We also investigate the
+impact of applying different masking ratios to input radiographs (cf. Table 5b). Specifically, we find
+that a ratio of 75% performs the best on the extremely small labeling ratio (i.e., 1%), while a ratio
+of 50% achieves slightly better results when the labeling ratio becomes larger. In MRM, we set the
+default masking ratio for radiographs to 75% because this operation leads to fewer input patches,
+accelerating the pre-training process and reducing the memory cost. The insight behind applying a
+high masking ratio (i.e., 75%) is that it addresses the heavy spatial redundancy of radiographs. By
+applying a high masking ratio, we reduce the redundancy and create a surrogate task, requiring the
+model to understand the high-level semantics holistically. We also perform ablative experiments to
+investigate the influence of λ in Eq. 3, whose results are displayed in Table 5c. We see that λ = 1
+is an optimal choice, while a smaller λ value (i.e, 0.5) performs better than a larger value (i.e., 2.0),
+indicating that the MLM objective may play a more important role than the MIM objective during
+the pre-training stage.
+4.6
+IMPLEMENTATION DETAILS.
+Our code is implemented using PyTorch 1.8.2 (Paszke et al., 2019).
+The pre-training experi-
+ments were conducted on 4 GeForce RTX 3080Ti GPUs, and the training time is about 2 days
+for 200 epochs, requiring 12GB memory from each GPU. The training batch size is 256. We use
+AdamW (Loshchilov & Hutter, 2017) as the default optimizer, where the initial learning rate is
+1.5e−4, weight decay is 0.05, β1 is 0.9, and β2 is 0.95. The MSE and cross-entropy losses are used
+for masked image and language modeling, respectively. In practice, we set λ in Eq. 3 to 1.
+For fine-tuning on SIIM, we train the segmentation network on 4 GeForce RTX 3080Ti GPUs.
+AdamW is the default optimizer, where the initial learning rate is 2e−5, weight decay is 0.05, β1 is
+0.9, and β2 is 0.999. For fine-tuning on other datasets, we train the classification network on a single
+GeForce RTX 3080Ti GPU, where the default optimizer is SGD with momentum 0.9. The training
+cost is a mix of focal and dice losses.
+For fine-tuning on CheXpert, RSNA Pneumonia, NIH ChestX-ray, and COVID-19 Image Data Col-
+lection, we adopt the cross-entropy loss. We search the best initial learning rate from 3e-2, 3e-3, and
+5e-4 to get the best performance on validation set.
+For both the pre-training and the fine-tuning of image classification task, the network is ”warmed
+up” by increasing the learning rate linearly to the set value, and then learning rate is decreased using
+the cosine decay schedule.
+5
+CONCLUSION
+We present masked record modeling (MRM) for radiograph representation learning. MRM for-
+malizes the radiograph understanding and radiology report comprehension as two complementary
+masked modeling objectives. With MRM pre-training, we achieve better results on well-established
+datasets. Specifically, MRM outperforms previous self- and report-supervised counterparts by large
+margins when the labeled data is extremely limited. These observations verify the effectiveness
+of MRM, and we hope they will enable our field to explore the use of multi-task supervision for
+learning more transferable visual representations.
+9
+
+Published as a conference paper at ICLR 2023
+REFERENCES
+Pulkit Agrawal, Joao Carreira, and Jitendra Malik. Learning to see by moving. In Proceedings of
+the IEEE International Conference on Computer Vision, pp. 37–45, 2015.
+Tarik Arici, Mehmet Saygin Seyfioglu, Tal Neiman, Yi Xu, Son Train, Trishul Chilimbi, Belinda
+Zeng, and Ismail Tutar. Mlim: Vision-and-language model pre-training with masked language
+and image modeling. arXiv preprint arXiv:2109.12178, 2021.
+Benedikt Boecking, Naoto Usuyama, Shruthi Bannur, Daniel C Castro, Anton Schwaighofer,
+Stephanie Hyland, Maria Wetscherek, Tristan Naumann, Aditya Nori, Javier Alvarez-Valle, et al.
+Making the most of text semantics to improve biomedical vision–language processing. arXiv
+preprint arXiv:2204.09817, 2022.
+Aurelia Bustos, Antonio Pertusa, Jose-Maria Salinas, and Maria de la Iglesia-Vay´a. Padchest: A
+large chest x-ray image dataset with multi-label annotated reports. Medical Image Analysis, 66:
+101797, 2020.
+Liang Chen, Paul Bentley, Kensaku Mori, Kazunari Misawa, Michitaka Fujiwara, and Daniel Rueck-
+ert. Self-supervised learning for medical image analysis using image context restoration. Medical
+Image Analysis, 58:101539, 2019.
+Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple framework for
+contrastive learning of visual representations. In International Conference on Machine Learning,
+pp. 1597–1607. PMLR, 2020.
+Xi Chen, Xiao Wang, Soravit Changpinyo, AJ Piergiovanni, Piotr Padlewski, Daniel Salz, Sebastian
+Goodman, Adam Grycner, Basil Mustafa, Lucas Beyer, et al. Pali: A jointly-scaled multilingual
+language-image model. arXiv preprint arXiv:2209.06794, 2022.
+Joseph Paul Cohen, Paul Morrison, Lan Dao, Karsten Roth, Tim Q Duong, and Marzyeh Ghas-
+semi. Covid-19 image data collection: Prospective predictions are the future. arXiv preprint
+arXiv:2006.11988, 2020.
+Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. ImageNet: A large-scale
+hierarchical image database. In Proceedings of the IEEE/CVF conference on Computer Vision
+and Pattern Recognition, pp. 248–255. Ieee, 2009.
+Karan Desai and Justin Johnson. Virtex: Learning visual representations from textual annotations.
+In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp.
+11162–11173, 2021.
+Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. Bert: Pre-training of deep
+bidirectional transformers for language understanding. North American Chapter of the Associa-
+tion for Computational Linguistics, 2018.
+Carl Doersch, Abhinav Gupta, and Alexei A Efros. Unsupervised visual representation learning by
+context prediction. In Proceedings of the IEEE International Conference on Computer Vision, pp.
+1422–1430, 2015.
+Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas
+Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al. An
+image is worth 16x16 words: Transformers for image recognition at scale.
+arXiv preprint
+arXiv:2010.11929, 2020.
+Zi-Yi Dou, Aishwarya Kamath, Zhe Gan, Pengchuan Zhang, Jianfeng Wang, Linjie Li, Zicheng Liu,
+Ce Liu, Yann LeCun, Nanyun Peng, et al. Coarse-to-fine vision-language pre-training with fusion
+in the backbone. arXiv preprint arXiv:2206.07643, 2022.
+Yuting Gao, Jinfeng Liu, Zihan Xu, Jun Zhang, Ke Li, and Chunhua Shen. Pyramidclip: Hierarchical
+feature alignment for vision-language model pretraining. arXiv preprint arXiv:2204.14095, 2022.
+Xinyang Geng, Hao Liu, Lisa Lee, Dale Schuurams, Sergey Levine, and Pieter Abbeel. Multimodal
+masked autoencoders learn transferable representations. arXiv preprint arXiv:2205.14204, 2022.
+10
+
+Published as a conference paper at ICLR 2023
+Fatemeh Haghighi, Mohammad Reza Hosseinzadeh Taher, Zongwei Zhou, Michael B Gotway, and
+Jianming Liang. Transferable visual words: Exploiting the semantics of anatomical patterns for
+self-supervised learning. IEEE Transactions on Medical Imaging, 40(10):2857–2868, 2021.
+Kaiming He, Xinlei Chen, Saining Xie, Yanghao Li, Piotr Doll´ar, and Ross Girshick. Masked au-
+toencoders are scalable vision learners. In Proceedings of the IEEE/CVF Conference on Computer
+Vision and Pattern Recognition, pp. 16000–16009, 2022.
+Shih-Cheng Huang, Liyue Shen, Matthew P Lungren, and Serena Yeung. Gloria: A multimodal
+global-local representation learning framework for label-efficient medical image recognition. In
+Proceedings of the IEEE International Conference on Computer Vision, pp. 3942–3951, 2021.
+Jeremy Irvin, Pranav Rajpurkar, Michael Ko, Yifan Yu, Silviana Ciurea-Ilcus, Chris Chute, Henrik
+Marklund, Behzad Haghgoo, Robyn Ball, Katie Shpanskaya, et al.
+Chexpert: A large chest
+radiograph dataset with uncertainty labels and expert comparison. In Proceedings of the AAAI
+Conference on Artificial Intelligence, volume 33, pp. 590–597, 2019.
+Alistair EW Johnson, Tom J Pollard, Nathaniel R Greenbaum, Matthew P Lungren, Chih-ying Deng,
+Yifan Peng, Zhiyong Lu, Roger G Mark, Seth J Berkowitz, and Steven Horng. Mimic-cxr-jpg, a
+large publicly available database of labeled chest radiographs. arXiv preprint arXiv:1901.07042,
+2019.
+Gukyeong Kwon, Zhaowei Cai, Avinash Ravichandran, Erhan Bas, Rahul Bhotika, and Stefano
+Soatto. Masked vision and language modeling for multi-modal representation learning. arXiv
+preprint arXiv:2208.02131, 2022.
+Yangguang Li, Feng Liang, Lichen Zhao, Yufeng Cui, Wanli Ouyang, Jing Shao, Fengwei Yu,
+and Junjie Yan. Supervision exists everywhere: A data efficient contrastive language-image pre-
+training paradigm. In International Conference on Learning Representations, 2021.
+Ruizhi Liao, Daniel Moyer, Miriam Cha, Keegan Quigley, Seth Berkowitz, Steven Horng, Polina
+Golland, and William M Wells. Multimodal representation learning via maximization of local
+mutual information. In International Conference on Medical Image Computing and Computer-
+Assisted Intervention, pp. 273–283. Springer, 2021.
+Ilya Loshchilov and Frank Hutter.
+Decoupled weight decay regularization.
+arXiv preprint
+arXiv:1711.05101, 2017.
+Norman Mu, Alexander Kirillov, David Wagner, and Saining Xie. Slip: Self-supervision meets
+language-image pre-training. arXiv preprint arXiv:2112.12750, 2021.
+Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor
+Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al.
+Pytorch: An imperative style,
+high-performance deep learning library. Advances in Neural Information Processing Systems, 32:
+8026–8037, 2019.
+Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal,
+Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al. Learning transferable visual
+models from natural language supervision. In International Conference on Machine Learning,
+pp. 8748–8763. PMLR, 2021.
+Mert Bulent Sariyildiz, Julien Perez, and Diane Larlus. Learning visual representations with caption
+annotations. In European Conference on Computer Vision, pp. 153–170. Springer, 2020.
+George Shih, Carol C Wu, Safwan S Halabi, Marc D Kohli, Luciano M Prevedello, Tessa S Cook,
+Arjun Sharma, Judith K Amorosa, Veronica Arteaga, Maya Galperin-Aizenberg, et al. Augment-
+ing the national institutes of health chest radiograph dataset with expert annotations of possible
+pneumonia. Radiology. Artificial intelligence, 1(1), 2019.
+Amanpreet Singh, Ronghang Hu, Vedanuj Goswami, Guillaume Couairon, Wojciech Galuba, Mar-
+cus Rohrbach, and Douwe Kiela. Flava: A foundational language and vision alignment model.
+In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp.
+15638–15650, 2022.
+11
+
+Published as a conference paper at ICLR 2023
+Fuying Wang, Yuyin Zhou, Shujun Wang, Varut Vardhanabhuti, and Lequan Yu. Multi-granularity
+cross-modal alignment for generalized medical visual representation learning. In Advances in
+Neural Information Processing Systems, 2023.
+Wenhui Wang, Hangbo Bao, Li Dong, Johan Bjorck, Zhiliang Peng, Qiang Liu, Kriti Aggarwal,
+Owais Khan Mohammed, Saksham Singhal, Subhojit Som, et al. Image as a foreign language:
+Beit pretraining for all vision and vision-language tasks. arXiv preprint arXiv:2208.10442, 2022.
+Xiaolong Wang and Abhinav Gupta. Unsupervised learning of visual representations using videos.
+In Proceedings of the IEEE International Conference on Computer Vision, pp. 2794–2802, 2015.
+Xiaosong Wang, Yifan Peng, Le Lu, Zhiyong Lu, Mohammadhadi Bagheri, and Ronald M Sum-
+mers. Chestx-ray8: Hospital-scale chest x-ray database and benchmarks on weakly-supervised
+classification and localization of common thorax diseases. In Proceedings of the IEEE/CVF con-
+ference on Computer Vision and Pattern Recognition, pp. 2097–2106, 2017.
+Chaoyi Wu, Xiaoman Zhang, Ya Zhang, Yanfeng Wang, and Weidi Xie. Medklip: Medical knowl-
+edge enhanced language-image pre-training. medRxiv, pp. 2023–01, 2023.
+Yonghui Wu, Mike Schuster, Zhifeng Chen, Quoc V Le, Mohammad Norouzi, Wolfgang Macherey,
+Maxim Krikun, Yuan Cao, Qin Gao, Klaus Macherey, et al. Google’s neural machine trans-
+lation system: Bridging the gap between human and machine translation.
+arXiv preprint
+arXiv:1609.08144, 2016.
+Jiahui Yu, Zirui Wang, Vijay Vasudevan, Legg Yeung, Mojtaba Seyedhosseini, and Yonghui
+Wu.
+Coca:
+Contrastive captioners are image-text foundation models.
+arXiv preprint
+arXiv:2205.01917, 2022.
+Yuhao Zhang, Hang Jiang, Yasuhide Miura, Christopher D Manning, and Curtis P Langlotz. Con-
+trastive learning of medical visual representations from paired images and text. arXiv preprint
+arXiv:2010.00747, 2020.
+Hong-Yu Zhou, Shuang Yu, Cheng Bian, Yifan Hu, Kai Ma, and Yefeng Zheng. Comparing to
+learn: Surpassing imagenet pretraining on radiographs by comparing image representations. In
+International Conference on Medical Image Computing and Computer-Assisted Intervention, pp.
+398–407. Springer, 2020.
+Hong-Yu Zhou, Xiaoyu Chen, Yinghao Zhang, Ruibang Luo, Liansheng Wang, and Yizhou Yu.
+Generalized radiograph representation learning via cross-supervision between images and free-
+text radiology reports. Nature Machine Intelligence, 4(1):32–40, 2022.
+Zongwei Zhou, Vatsal Sodha, Jiaxuan Pang, Michael B Gotway, and Jianming Liang.
+Models
+genesis. Medical Image Analysis, 67:101840, 2021.
+12
+
+Published as a conference paper at ICLR 2023
+A
+APPENDIX
+A.1
+DISCUSSION: DIFFERENCES FROM MASKED VISION-AND-LANGUAGE PRE-TRAINING
+BESIDES MEDICAL DATA
+To our knowledge, M3AE (Geng et al., 2022), MLIM (Arici et al., 2021), and MaskVLM (Kwon
+et al., 2022) are three recent efforts that are most closely related to ours, all of which use masked
+modeling for vision and language pre-training. In the following, we clarify the differences between
+our MRM and these approaches from two perspectives: motivation and implementation.
+Motivation. M3AE (Geng et al., 2022) and MLIM (Arici et al., 2021) aim to learn joint image-
+text representations. MaskVLM (Kwon et al., 2022) is developed for improving vision+language
+tasks, such as image-text retrieval, visual question answering, and natural language for visual rea-
+soning. These characteristics differ from our methodology, which aims to learn a representation for
+radiographs only for disease diagnosis even though both radiographs and reports are used during
+training.
+Implementation. As aforementioned, M3AE (Geng et al., 2022) and MLIM (Arici et al., 2021)
+implement unified multi-modal transformers that take imaging and textual as inputs. As a result,
+they require much more pre-training data, which is often an order of magnitude larger than ours.
+This is not practical and applicable in the medical field, where access to the data is quite limited.
+MaskVLM (Kwon et al., 2022) applies masked modeling and contrastive learning to image-text
+pairs, where a binary matching task is employed to tell whether an image and a text are aligned or
+not. Besides, MaskVLM builds cross-modality encoders to incorporate multi-modal information. In
+contrast, our MRM is much simpler. MRM uses masking modeling as the only training objective,
+and a simple GAP-addition workflow is proposed for fusing multi-modal information.
+A.2
+CONFIGURATIONS OF NETWORKS
+The image encoder a ViT-like encoder (Dosovitskiy et al., 2020) that includes a patch embedding
+layer followed by twelve transformer blocks. The architecture of the image decoder is very similar
+to that of the encoder. The decoder architecture consists of a decoder embedding layer, four trans-
+former blocks, and one fully-connected layer to predict masked patches. The numbers of attention
+heads in the image encoder and decoder are twelve and six, respectively. We add a 2D sin-cos posi-
+tional embedding to input patches. The report decoder is a light-weight transformer that includes an
+embedding layer and six transformer blocks, followed by a one-layer fully-connected predictor. The
+number of attention heads in the report decoder is six. We adopt a learnable positional embedding
+in the report decoder.
+A.3
+RECONSTRUCTION ANALYSIS
+Fig. 3 presents example results on MIMIC-CXR. We find that even with high masking ratios, MRM
+can still produce satisfactory reconstructions, though some details are missing. Specifically, the re-
+constructed reports are surprisingly close to the ground truths, which we attribute to the introduction
+of hybrid multi-modal representations. For instance, MRM can tell the position of rib fractures based
+on the radiograph input. Besides, we obtain some interesting mistakes. In the first example, MRM
+recognizes the clips over the left lung as calcifications (potentially in nipple shadow). This observa-
+tion again shows that the report reconstructions rely on the image input as the clips are masked in
+the input radiograph.
+A.4
+SEGMENTATION ANALYSIS
+In this section, we visualize the segmentation results of GLoRIA (Huang et al., 2021),
+REFERS (Zhou et al., 2022), and our MRM.
+13
+
+Published as a conference paper at ICLR 2023
+[CLS] final [MASK] [MASK] [MASK] chest [MASK] pa and [MASK] ) indication : ___f with [MASK] onset [MASK] // [MASK] for infection [MASK] : [MASK] [MASK] [MASK] 
+[MASK] [MASK] [MASK] none . [MASK] : [MASK] is [MASK] focal [MASK] [MASK] pleural [MASK] or pneumothorax [MASK] [MASK] [MASK] [MASK] that [MASK] likely 
+represent nipple shadows . [MASK] [MASK] silhouette [MASK] [MASK] . [MASK] [MASK] over the left [MASK] , potentially [MASK] [MASK] [MASK] . the [MASK] upper 
+abdomen [MASK] [MASK] . [MASK] [MASK] of [MASK] [MASK] left sixth [MASK] seventh [MASK] are [MASK] [MASK] [MASK] : no acute cardiopulmonary process . 
+[CLS] final report examination : chest ( pa and lat ) indication : ___f with new onset confusion // eval for infection technique : chest pa and lateral comparison :
+none . findings : there is no focal consolidation , pleural effusion or pneumothorax . bilateral nodular opacities that most likely represent nipple shadows . the
+cardiomediastinal silhouette is normal . calcifications project over the left lung , potentially a nipple shadow . the imaged upper abdomen is unremarkable . healed
+fractures of the posterior left sixth and seventh ribs are noted . impression : no acute cardiopulmonary process . 
+[CLS] final report examination : chest ( pa and lat ) indication : ___f with new onset ascites // eval for infection technique : chest pa and lateral comparison :
+none . findings : there is no focal consolidation , pleural effusion or pneumothorax . bilateral nodular opacities that most likely represent nipple shadows . the
+cardiomediastinal silhouette is normal . clips project over the left lung , potentially within the breast . the imaged upper abdomen is unremarkable . chronic
+deformity of the posterior left sixth and seventh ribs are noted . impression : no acute cardiopulmonary process . 
+[CLS] [MASK] report [MASK] [MASK] [MASK] ( [MASK] [MASK] lat ) [MASK] : [MASK] [MASK] [MASK] with shortness [MASK] [MASK] [MASK] : [MASK] [MASK] and [MASK] 
+comparison : [MASK] findings : [MASK] cardiac , mediastinal and hilar [MASK] are normal . [MASK] [MASK] [MASK] normal. lungs are [MASK] . [MASK] pleural [MASK] 
+[MASK] pneumothorax [MASK] [MASK] . multiple clips [MASK] again seen [MASK] [MASK] the left [MASK] . remote [MASK] - sided rib fractures are also re - [MASK] . 
+impression : [MASK] [MASK] cardiopulmonary abnormality . 
+[CLS] final report examination : chest ( pa and lat ) indication : history : ___f with shortness of breath technique : chest pa and lateral comparison : ___ findings : 
+the cardiac , mediastinal and hilar contours are normal . pulmonary vasculature is normal . lungs are clear . no pleural effusion or pneumothorax is present . 
+multiple clips are again seen projecting over the left axilla . remote left- sided rib fractures are also re - demonstrated . impression : no acute cardiopulmonary 
+abnormality .
+[CLS] final report examination : chest ( pa and lat ) indication : history : ___f with shortness of breath technique : chest pa and lateral comparison : ___ findings : 
+the cardiac , mediastinal and hilar contours are normal . pulmonary vasculature is normal . lungs are clear . no pleural effusion or pneumothorax is present . 
+multiple clips are again seen projecting over the left breast . remote left - sided rib fractures are also re - demonstrated . impression : no acute cardiopulmonary 
+abnormality .
+Inputs
+Recons.
+GT
+Inputs
+Recons.
+GT
+Figure 3: Example results on MIMIC-CXR. For each triplet, we show the masked radiograph and
+report (Inputs), our MRM reconstruction (Recons.), and the ground truth (GT). The masking ra-
+tios are 75% (radiograph) and 50% (report). Predicted and corresponding ground truth words are
+highlighted in pink and green , respectively.
+14
+
+OPublished as a conference paper at ICLR 2023
+Radiograph
+GLoRIA*
+REFERS
+Our MRM
+GT
+15
+
+ECGVPublished as a conference paper at ICLR 2023
+Radiograph
+GLoRIA*
+REFERS
+Our MRM
+GT
+Figure 4: Segmentation results of MRM, GLoRIA, and REFERS. GT stands for the ground truth
+masks. * means the GLoRIA implementation is based on ViT-B/16, the same backbone as used in
+REFERS and MRM.
+16
+
+LPORTABLE

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+Optimal control problem for Stokes system: Asymptotic
+analysis via unfolding method in a perforated domain
+Swati Garg and Bidhan Chandra Sardar∗
+Department of Mathematics
+Indian Institute of Technology Ropar
+Rupnagar-140001, Punjab, India
+swati.19maz0006@iitrpr.ac.in, swatigargmks@gmail.com
+bcsardar@iitrpr.ac.in, bcsardar31@gmail.com
+January 4, 2023
+Abstract
+This article’s subject matter is the study of the asymptotic analysis of the optimal
+control problem (OCP) constrained by the stationary Stokes equations in a periodi-
+cally perforated domain. We subject the interior region of it with distributive controls.
+The Stokes operator considered involves the oscillating coefficients for the state equa-
+tions. We characterize the optimal control and, upon employing the method of periodic
+unfolding, establish the convergence of the solutions of the considered OCP to the solu-
+tions of the limit OCP governed by stationary Stokes equations over a non-perforated
+domain. The convergence of the cost functional is also established.
+Keywords: Stokes equations, Homogenization, Optimal control, Perforated domain, Un-
+folding operator
+1
+Introduction
+In this article, we consider the optimal control problem (OCP) governed by generalized
+stationary Stokes equations in a periodically perforated domain O∗
+ε (see Section 2, on the
+domain description). The size of holes in the perforated domain is of the same order as
+that of the period, and the holes are allowed to intersect the boundary of the domain. The
+control is applied in the interior region of the domain, and we wish to study the asymptotic
+AMS subject classifications: 35B27, 35B40, 35Q93, 49J20, 76D07
+∗Corresponding author
+1
+arXiv:2301.01136v1  [math.OC]  3 Jan 2023
+
+analysis (homogenization) of an interior OCP subject to the constrained stationary Stokes
+equations with oscillating coefficients.
+One can find several works in the literature regarding the homogenization of Stokes
+equations over a perforated domain. Using the multiple-scale expansion method, the au-
+thors in [16] studied the homogenization of Stokes equations in a porous medium with the
+Dirichlet boundary condition on the boundary of the holes. They obtained the Darcy’s law
+as the limit law in the homogenized medium.
+In [9], the authors considered the Stokes
+system in a periodically perforated domain with non-homogeneous slip boundary conditions
+depending upon some parameter γ. Upon employing the Tartar’s method of oscillating test
+functions they obtained under homogenization, the limit laws, viz., Darcy’s law ( for γ < 1),
+Brinkmann’s law (for γ = 1), and Stokes’s type law (for γ > 1). In [25], the author studied
+a similar problem using the method of periodic unfolding in perforated domains by [10].
+Further, the type of behavior as seen in [9] was already observed in [12] by the authors while
+studying the homogeneous Fourier boundary conditions for the two-dimensional Stokes equa-
+tion. Likewise, in [1, 2], the author examined the Stokes equation in a perforated domain
+with holes of size much smaller than the small positive parameter ε, wherein they considered
+the boundary conditions on the holes to be of the Dirichlet type in [1] and the slip type
+in [2]. The domain geometry, more specifically, the size of the holes, determines the kind of
+limit law in these works. Also, the author in [6] employed the Γ− convergence techniques to
+get comparable results.
+A few works concern the homogenization of the OCPs governed by the elliptic systems
+over the periodically perforated domains with different kinds of boundary conditions on the
+boundary of holes (of the size of the same order as that of the period). In this regard, with
+the use of different techiniques, viz., H0− convergence in [18], two-sclae convergence in [23],
+and unfolding methods in [7,21], the homogenized OCPs were thus obtained over the non-
+perforated domains. Further, in context to the Stokes system, the authors in [22] studied
+the homogenization of the OCPs subject to the Stokes equations with Dirichlet boundary
+conditions on the boundary of holes, where the size of the holes is of the same order as that of
+the period. Here, the authors could obtain the homogenized system, pertaining only to the
+case when the set of admissible controls was unconstrained. For more literature concerning
+the homogenization of optimal control problems in perforated domains, the reader is reffered
+to [13–15,19,24] and the references therein.
+The present article introduces an interior OCP subject to the generalized stationary
+Stokes equations in a periodically perforated domain O∗
+ε. On the boundary of holes that
+do not intersect the outer boundary, the homogeneous Neumann boundary condition is
+prescribed, while on the rest part of the boundary, the homogeneous Dirichlet boundary
+condition is prescribed. The underlying objective of this article is to study the homogeniza-
+tion of this OCP. More specifically, we consider the minimization of the L2−cost functional
+(3.1), which is subject to the constrained generalized stationary Stokes equations (3.2).
+The Stokes equations are generalized in the sense that we consider a second-order elliptic
+linear differential operator in divergence form with oscillating coefficients, i.e., − div (Aε∇),
+first studied for the fixed domain in [4, Chapter 1], instead of the classical Laplacian operator.
+2
+
+Here, the action of the scalar operator − div (Aε∇) is defined in a ”diagonal” manner on any
+vector u = (u1, . . . , un), with components u1, . . . , un in the H1 Sobolev space. That is, for
+1 ≤ i ≤ n, we have (− div (Aε∇u))i = − div (Aε∇ui). The main difficulty observed during
+the homogenization was identifying the limit pressure terms appearing in the state and the
+adjoint systems, which we overcame by introducing suitable corrector functions that solved
+some cell problems. We thus obtained the limit OCP associated with the stationary Stokes
+equation in a non-perforated domain.
+The layout of this article is as follows: In the next section, we introduce the periodically
+perforated domain O∗
+ε along with the notations that will be useful in the sequel. Section
+3 is devoted to a detailed description of the considered OCP and the derivation of the
+optimality condition, followed by the characterization of the optimal control. In Section 4,
+we derive a priori estimates of the solutions to the considered OCP and its corresponding
+adjoint problem. In Section 5, we recall the definition of the method of periodic unfolding
+in perforated domains (see, [8,11]) and a few of its properties. Section 6, refers to the limit
+(homogenized) OCP. Finally, we derive the main convergence results in Section 7.
+2
+Domain description and Notation
+2.1
+Domain description
+Let {b1, ..., bn} be a basis of Rn (n ≥ 2), and W be the associated reference cell defined as
+W =
+�
+w ∈ Rn | w =
+n
+�
+i=1
+wibi, (w1, . . . , wn) ∈ (0, 1)n
+�
+.
+Let us denote O, W, and W ∗ = W\Y by an open bounded subset of Rn, a compact subset
+of W, and the perforated reference cell, respectively. It is assumed that the boundary of Y
+is Lipschitz continuous and has a finite number of connected components.
+Also, let ε > 0 be a sequence that converges to zero and set
+T =
+�
+ζ ∈ Rn | ζ =
+n
+�
+i=1
+zibi, (z1, . . . , zn) ∈ Zn
+�
+,
+Zε = {ζ ∈ T | ε(ζ + W) ⊂ O} .
+We take into account the perforated domain O∗
+ε (see Figure 1) given by O∗
+ε = O\Yε, where
+Yε = ∪ζ∈T ε(ζ + Y ). Now, let us denote �
+Oε as the interior of the largest union of ε(ζ + W)
+cells such that ε(ζ + W) ⊂ O, while Λε ⊂ O as containing the parts from ε(ζ + W) cells
+intersecting the boundary ∂O. More precisely, we write Λε = O\ �
+Oε, where
+�
+Oε = interior
+�
+∪ζ∈Zε ε(ζ + W)
+�
+.
+The associated perforated domains are defined as
+�
+O∗
+ε = ˆOε\Yε,
+ˆΛ∗
+ε = O∗
+ε\ �
+O∗
+ε.
+3
+
+Figure 1:
+The Perforated domain O∗
+ε and the reference cell W.
+Also, we denote the boundary of the perforated domain O∗
+ε as
+∂O∗
+ε = Γε
+1 ∪ Γε
+0,
+where Γε
+1 = ∂ �
+Oε ∩ ∂Yε and Γε
+0 = ∂O∗
+ε\Γε
+1,
+which means that Γε
+1 denotes the boundary of set of holes contained in �
+Oε.
+In Figure 1, �
+O∗
+ε and ˆΛ∗
+ε respectively represent the dark perforated part and the remaining
+part of the perforated domain O∗
+ε. While, Γε
+1 and Γε
+0 respectively represent the boundary
+of holes contained in �
+O∗
+ε and the boundary of holes contained in ˆΛ∗
+ε along with the outer
+boundary ∂O. In the following, we introduce a few notations that we shall use throughout
+this article.
+2.2
+Notation
+• Aε(x) = A( x
+ε) a.e. in O, for all ε > 0.
+• vε = (vε1, . . . , vεn), for any bold symbol vector function vε.
+• v = (v1, . . . , vn), for any bold symbol vector function v.
+• ηε denotes the outward normal unit vector to Γε
+1.
+• η denotes the outward normal unit vector to ∂O.
+• M t denotes the transpose of any matrix M.
+• �ψ is the zero extension of any function ψ outside O∗
+ε to the whole of O.
+• �ψ = (�
+ψ1, · · · , �
+ψn), for any vector function ψ.
+• |F| is the Lebesgue measure of the measurable set F.
+4
+
+M• Θ = |W ∗|
+|W| , the proportion of the perforated reference cell W ∗ in the reference cell W.
+• MW ∗(φ) is the mean value of φ on the perforated reference cell W ∗.
+• MW ∗(φ) = (MW ∗(φ1), · · · , MW ∗(φn)), for vector function φ.
+• {D → R}, the set of all real valued functions defined on domain D.
+• D(Ω), is the space of infinitely many times differentiable functions with compact sup-
+port in Ω, for any open set Ω ∈ Rn.
+3
+Problem description and Optimality condition
+Let us consider the following OCP associated with Stokes system:
+inf
+θε∈(L2(O∗ε))n
+�
+Jε(θε) = 1
+2
+�
+O∗ε
+|uε(θε) − ud|2 + τ
+2
+�
+O∗ε
+|θε|2
+�
+,
+(3.1)
+subject to
+�
+�
+��
+�
+�
+�
+�
+�
+�
+− div (Aε∇uε) + ∇pε
+= θε
+in O∗
+ε,
+div(uε)
+= 0
+in O∗
+ε,
+ηε · Aε∇uε − pεηε
+= 0
+on Γε
+1,
+uε
+= 0
+on Γε
+0,
+(3.2)
+where the desired state ud = (ud1, . . . , udn) is defined on the space (L2(O))n, θε is a control
+function defined on the space (L2(O∗
+ε))n and τ > 0 is a given regularization parameter. Here,
+the matrix Aε(x) = A( x
+ε), where A(x) = (aij(x))1≤i,j≤n defined on the space (L∞(O))n×n
+is assumed to obey the uniform ellipticity condition: there exist real constants m1, m2 > 0
+such that m1||λ||2 ≤ �n
+i,j=1 aij(x)λiλj ≤ m2||λ||2 for all λ ∈ Rn, which is endowed with
+an Eucledian norm denoted by || · ||. Also, we understand the action of scalar boundary
+operator ηε · Aε∇ on the vector uε|Γε
+1 in a ”diagonal” manner: (ηε · Aε∇uε)i = ηε · Aε∇uεi,
+for 1 ≤ i ≤ n.
+We introduce the function space (H1
+Γε
+0(O∗
+ε))n := {φ ∈ (H1(O∗
+ε))n | φ|Γε
+0 = 0}. This is a
+Banach space endowed with the norm
+||φ||(H1
+Γε
+0(O∗ε))n := ||∇φ||(L2(O∗ε))n×n,
+∀φ ∈ (H1
+Γε
+0(O∗
+ε))n.
+Definition 2.1. We say a pair (uε, pε) ∈ (H1
+Γε
+0(O∗
+ε))n × L2(O∗
+ε) is a weak solution to (3.2)
+if, for all φ ∈ (H1
+Γε
+0(O∗
+ε))n,
+�
+O∗ε
+Aε∇uε : ∇φ dx −
+�
+O∗ε
+pε div(φ) dx =
+�
+O∗ε
+θε · φ dx,
+(3.3)
+5
+
+and, for all w ∈ L2(O∗
+ε),
+�
+O∗ε
+div(uε) w dx = 0.
+(3.4)
+Here, (: ) and (·) represent the summation of the component-wise multiplication of the
+matrix entries and the usual scalar product of vectors, respectively.
+The existence of a
+unique weak solution (uε(θε), pε) ∈ (H1
+Γε
+0(O∗
+ε))n × L2(O∗
+ε) of the system (3.2) follows anal-
+ogous to [5, Theorem IV.7.1]. Also, for each ε > 0, there exists a unique solution to the
+problem (3.1) that can be proved along the same lines as in [20, Chapter 2, Theorem 1.2].
+We call the optimal solution to (3.1) by the triplet (uε, pε, θε), with uε, pε, and θε as optimal
+state, pressure, and control, respectively.
+Optimality Condition: The optimality condition is given by J′
+ε(θ) · (θ − θε) ≥ 0, for all
+θ ∈ (L2(O∗
+ε))n (see, [20, Chapter 2, Page 48]). One can obtain the further simplification of
+this condition as
+�
+O∗ε(vε + τ θε) · (θ − θε) ≥ 0, for all θ ∈ (L2(O∗
+ε))n (see, [20, Chapter 2]),
+where the pair (vε, qε) is the solution to the following adjoint problem:
+�
+�
+�
+�
+�
+�
+�
+�
+�
+− div
+�
+At
+ε∇vε
+�
++ ∇qε
+= uε − ud
+in O∗
+ε,
+div(vε)
+= 0
+in O∗
+ε,
+ηε · At
+ε∇vε − qεηε
+= 0
+on Γε
+1,
+vε
+= 0
+on Γε
+0.
+(3.5)
+We call vε and qε, the adjoint state and pressure, respectively. The existence of unique weak
+solution (vε, qε) to (3.5) can now be proved in a way similar to that of system (3.2).
+The following theorem characterizes the optimal control, the proof of which follows analogous
+to standard procedure laid in [20, Chapter 2, Theorem 1.4].
+Theorem 3.1. Let
+�
+uε, pε, θε
+�
+be the optimal solution of the problem (3.1) and (vε, qε)
+solves (3.5), then the optimal control is characterized by
+θε = −1
+τ vε a.e. in O∗
+ε.
+(3.6)
+Conversely, suppose that a triplet (ˇuε, ˇpε, ˇθε) ∈
+�
+H1
+Γε
+0(O∗
+ε)
+�n
+× L2(O∗
+ε) × (L2(O∗
+ε))n and a
+pair (ˇvε, ˇqε) ∈
+�
+H1
+Γε
+0(O∗
+ε)
+�n
+× L2(O∗
+ε) solves the following system:
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+− div (Aε∇ˇuε) + ∇ˇpε = −1
+τ ˇvε
+in O∗
+ε,
+− div
+�
+At
+ε∇ˇvε
+�
++ ∇ˇqε = ˇuε − ud
+in O∗
+ε,
+div(ˇuε) = 0, div(ˇvε) = 0
+in O∗
+ε,
+ηε · Aε∇ˇuε − ˇpεηε = 0
+on Γε
+1,
+ηε · At
+ε∇ˇvε − ˇqεηε = 0
+on Γε
+1,
+ˇvε = 0, ˇuε = 0
+on Γε
+0.
+6
+
+Then the triplet (ˇuε, ˇpε, − 1
+τ ˇvε) is the optimal solution of (3.1).
+4
+A priori estimates
+This section concerns the derivation of estimates for the optimal solution to the problem
+(3.1) and the associated solution to the adjoint problem (3.5). These estimates are uniform
+and independent of the parameter ε. Towards attaining this aim, we first evoke the following
+two lemmas:
+Lemma 4.1 (Lemma A.4, [3]). There exists a constant C ∈ R+, independent of ε, such
+that
+||v||L2(O∗ε)n ≤ C||∇v||(L2(O∗ε))n×n,
+∀ v ∈ (H1
+Γε
+0(O∗
+ε))n.
+Lemma 4.2 (Lemma 5.1, [12]). For each ε > 0 and qε ∈ L2(O∗
+ε), there exists gε ∈
+(H1
+Γε
+0(O∗
+ε))n and a constant C ∈ R+, independent of ε, such that
+div(gε) = qε and ||∇gε||(L2(O∗ε))n×n ≤ C(O) ||qε||L2(O∗ε).
+(4.1)
+Theorem 4.3. For each ε > 0, let
+�
+uε, pε, θε
+�
+be the optimal solution of the problem (3.1)
+and (vε, qε) solves the corresponding adjoint problem (3.5). Then, one has θε ∈ (H1
+Γε
+0(O∗
+ε))n
+and there exists a constant C ∈ R+, independent of ε such that
+��¯θε
+��
+(L2(O∗ε))n ≤ C,
+(4.2)
+∥¯uε∥(H1
+Γε
+0(O∗ε))n ≤ C,
+(4.3)
+∥¯vε∥(H1
+Γε
+0(O∗ε))n ≤ C,
+(4.4)
+∥¯pε∥L2(O∗ε) ≤ C,
+(4.5)
+∥¯qε∥L2(O∗ε) ≤ C.
+(4.6)
+Proof. Let uε(0) denotes the solution to (3.2) corresponding to θε = 0. In view of Lemma
+4.1, one can show that ∥uε(0)∥(L2(O∗ε))n ≤ 0, i.e., uε(0) = 0 in (L2(O∗
+ε))n. Using this and
+the optimality of solution (uε, pε, θε) to problem (3.1), we have
+∥uε(θ) − ud∥2
+(L2(O∗ε))n + τ∥θε∥2
+(L2(O∗ε))n ≤ ∥uε(0) − ud∥2
+(L2(O∗ε))n ≤ C,
+which gives estimate (4.2). Now, let us take uε as a test function in (3.3). Considering
+(4.2) and the uniform ellipticity condition of matrix Aε, one obtains upon applying the
+Cauchy-Schwarz inequality along with the Lemma 4.1, the following:
+m1∥∇uε∥2
+(L2(O∗ε))n×n ≤
+�
+O∗ε
+Aε∇uε : ∇uε dx ≤ C ∥θε∥(L2(O∗ε))n∥∇uε∥(L2(O∗ε))n×n,
+from which estimate (4.3) follows.
+7
+
+Owing to Lemma 4.2, for given pε ∈ L2(O∗
+ε), there exists gε ∈ (H1
+Γε
+0(O∗
+ε))n satisying div(gε) =
+pε. Corresponding to θε, taking v = gε in (3.3), we get
+∥pε∥2
+L2(O∗ε) =
+�
+O∗ε
+Aε∇uε : ∇gε dx −
+�
+O∗ε
+θε · gε dx.
+(4.7)
+In view of (4.1), (4.2) and (4.3), and the uniform ellipticity condition of the matrix Aε,
+one obtains from (4.7) upon employing the Cauchy-Schwarz inequality and Lemma 4.1, the
+following:
+∥pε∥2
+L2(O∗ε) ≤
+�
+m2∥∇uε∥(L2(O∗ε))n×n + C∥θε∥(L2(O∗ε))n
+�
+∥∇gε∥(L2(O∗ε))n×n,
+which gives the estimate (4.5). Likewise, one can easily obtain the estimates (4.4) and (4.6)
+following the above discussion. Finally, from (3.6), we obtain that θε ∈ (H1
+Γε
+0(O∗
+ε))n.
+5
+The method of periodic unfolding for perforated do-
+mains
+We evokes the definition of the periodic unfolding operator and few of its properties as
+stated in [8,11]. Given x ∈ Rn, we denote the greatest integer and the fractional parts of x
+respectively by [x]W and {x}W. That is, [x]W = �n
+j=1 kjbj be the unique integer combination
+of periods and {x}W = x − [x]W. In particular, we have for ε > 0,
+x = ε
+��x
+ε
+�
+W +
+�x
+ε
+�
+W
+�
+,
+∀ x ∈ Rn.
+Definition 5.1. The unfolding operator T ∗
+ε : {O∗
+ε → R} → {O × W ∗ → R} is defined as
+T ∗
+ε (u) (x, y) =
+�
+u
+�
+ε
+� x1
+ε
+�
+W + εy
+�
+a.e.
+for (x, y) ∈ �
+Oε × W ∗,
+0
+a.e.
+for (x, y) ∈ Λε × W ∗.
+Also, for any domain D ⊇ O∗
+ε and vector u = (u1, · · · , un) ∈ ({D → R})n, we define its
+unfolding by
+T ∗
+ε (u) := (T ∗
+ε (u1), · · · , T ∗
+ε (un)).
+Proposition 5.2. In the following there are the properties of the unfolding operator:
+(i) T ∗
+ε is linear and continuous from L2(O∗
+ε) to L2(O × W ∗).
+(ii) Let u, v ∈ L2(O∗
+ε). Then T ∗
+ε (uv) = T ∗
+ε (u) T ∗
+ε (v) .
+(iii)
+Let u ∈ L2 (O) . Then T ∗
+ε (u) → u strongly in L2 (O × W ∗) .
+(iv)
+Let u ∈ L1 (O∗
+ε) . Then
+�
+�
+O∗ε
+u(x) dx =
+�
+O∗ε
+u(x) dx −
+�
+ˆΛ∗ε
+u(x) dx =
+1
+|W ∗|
+�
+O×W ∗ T ∗
+ε (u)(x, y) dxdy.
+8
+
+(v)
+For each ε > 0, let {uε} ∈ L2 (O) and uε → u strongly in L2 (O) .
+Then T ∗
+ε (uε) → u strongly in L2 (O × W ∗) .
+(vi) Let v ∈ L2 (W ∗) be a W-periodic function and vε(x) = v
+� x
+ε
+�
+. Then,
+T ∗
+ε (vε) (x, y) =
+�
+v(y)
+a.e. for (x, y) ∈ �
+Oε × W ∗,
+0
+a.e. for (x, y) ∈ Λε × W ∗.
+(vii) Let fε ∈ L2 (O∗
+ε) be uniformly bounded. Then, there exists f ∈ L2(O × W ∗) such that
+T ∗
+ε (fε) ⇀ f weakly in L2(O × W ∗), and
+�fε ⇀
+1
+|W|
+�
+W ∗ f(·, y) dy weakly in L2(O).
+Proposition 5.3. Let O ⊂ Rn be bounded with Lipschitz boundary. Let fε ∈ H1(O∗
+ε) be
+such that fε = 0 on ∂O ∩ ∂O∗
+ε and satisfy,
+∥∇fε∥(L2(O∗ε))n ≤ C§.
+Then, there exists f ∈ H1
+0(O) and �f ∈ L2 �
+O; H1
+per (W ∗)
+�
+with MW ∗( �f) = 0, such that up to
+a subsequence,
+�
+T ∗
+ε (∇fε) ⇀ ∇f + ∇y �f
+weakly in (L2 (O × W ∗))n ,
+T ∗
+ε (fε) → f
+strongly in L2 (O; H1 (W ∗)) .
+6
+Limit optimal control problem
+This section presents the limit (homogenized) system corresponding to the problem (3.1),
+which we considered in the beginning.
+Let us consider the function space
+�
+H1
+0(O)
+�n :=
+�
+ϕ ∈ (H1(O))n | ϕ|∂O = 0
+�
+,
+which is a Hilbert space for the norm
+∥ϕ∥(H1
+0(O))n := ∥∇ϕ∥(L2(O))n×n
+∀ ϕ ∈ (H1
+0(O))n.
+We now consider the limit OCP associated with the Stokes system
+inf
+θ∈(L2(O))n
+�
+J(θ) = Θ
+2
+�
+O
+|u − ud|2 dx + τΘ
+2
+�
+O
+|θ|2 dx
+�
+,
+(6.1)
+§The symbol C represents a generic constant that is positive and independent of ε.
+9
+
+subject to
+�
+�
+�
+�
+�
+�
+�
+�
+�
+−
+n
+�
+j,α,β=1
+∂
+∂xα
+�
+bαβ
+ij
+∂uj
+∂xβ
+�
++ ∇p
+= θ
+in O,
+div (u)
+= 0
+in O,
+u
+= 0
+on ∂O,
+(6.2)
+where the tensor B = (bαβ
+ij ) = (bαβ
+ij )1≤i,j,α,β≤n is constant, elliptic, and for 1 ≤ i, j, α, β ≤ n,
+is given by
+bαβ
+ij = aαβ
+ij −
+1
+|W ∗|
+�
+W ∗ A(y)∇y
+�
+P β
+j − χβ
+j
+�
+: ∇yχα
+i dy,
+with aαβ
+ij =
+1
+|W ∗|
+�
+W ∗ A(y)∇y
+�
+P β
+j − χβ
+j
+�
+: ∇yP α
+i dy as the entries of the constant tensor A0,
+P β
+j = P β
+j (y) = (0, . . . , yj, . . . , 0) with yj at the β-th position, and for 1 ≤ j, β ≤ n, the
+correctors (χβ
+j , Πβ
+j ) ∈ (H1(W ∗))n × L2(W ∗) solves the cell problem
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+− divy
+�
+A(y)∇y(P β
+j − χβ
+j )
+�
++ ∇yΠβ
+j
+= 0
+in W ∗,
+η · A(y)∇y(P β
+j − χβ
+j ) − Πβ
+j η
+= 0
+on ∂W ∗\∂W,
+divy(P β
+j − χβ
+j )
+= 0
+in W ∗,
+(χβ
+j , Πβ
+j )
+W ∗- periodic,
+MW ∗(χβ
+j )
+= 0.
+(6.3)
+The existence of this unique pair (u, p) ∈ (H1
+0(O))n × L2(O) can be found in [4, Chapter
+1]. Further, the problem (6.1) is a standard one and there exists a unique weak solution to
+it, one can follow the arguments introduced in [20, Chapter 2, Theorem 1.2]. We call the
+triplet (u, p, θ) ∈ (H1
+0(O))n × L2(O) × (L2(O))n, the optimal solution to (6.1), with u, p,
+and θ as the optimal state, pressure, and control, respectively.
+Now, we introduce the limit adjoint system associated with (6.2): Find a pair (v, q) ∈
+(H1
+0(O))n × L2(O) which solves the system
+�
+�
+�
+�
+�
+−
+n
+�
+i,α,β=1
+∂
+∂xβ
+�
+bβα
+ji
+∂vi
+∂xα
+�
++ ∇q
+= u − ud
+in O,
+div (v)
+= 0
+in O,
+(6.4)
+where the tensor Bt = (bβα
+ji ) = (bβα
+ji )1≤i,j,α,β≤n is constant, elliptic, and for 1 ≤ i, j, α, β ≤ n,
+is given by
+bβα
+ji = aβα
+ji −
+1
+|W ∗|
+�
+W ∗ At(y)∇y
+�
+P β
+j − Hβ
+j
+�
+: ∇yHα
+i dy,
+with aβα
+ji =
+1
+|W ∗|
+�
+W ∗ At(y)∇y
+�
+P β
+j − Hβ
+j
+�
+: ∇yP α
+i dy as the entries of the constant tensor
+At
+0. Also, for 1 ≤ j, β ≤ n, the correctors (Hβ
+j , Zβ
+j ) ∈ (H1(W ∗))n × L2(W ∗) solves the cell
+10
+
+problem
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+− divy
+�
+At(y)∇y(P β
+j − Hβ
+j )
+�
++ ∇yZβ
+j
+= 0
+in W ∗,
+η · At(y)∇y(P β
+j − Hβ
+j ) − Zβ
+j η
+= 0
+on ∂W ∗\∂W,
+divy(P β
+j − Hβ
+j )
+= 0
+in W ∗,
+(Hβ
+j , Zβ
+j )
+W ∗- periodic,
+MW ∗(Hβ
+j )
+= 0.
+(6.5)
+In the following, we state a result similar to Theorem 3.1 that characterizes the optimal
+control θ in terms of the adjoint state v and the proof of which follows analogous to the
+standard procedure laid in [20, Chapter 2, Theorem 1.4].
+Theorem 6.1. Let
+�
+u, p, θ
+�
+be the optimal solution to (6.1) and (v, q) be the corresponding
+adjoint solution to (6.4), then the optimal control is characterized by
+θ = −1
+τ v a.e. in O.
+(6.6)
+Conversely, suppose that a triplet
+(ˇu, ˇp, ˇθ) ∈ (H1
+0(O))n × L2(O) × (L2(O))n and a pair
+(ˇv, ˇq) ∈ (H1
+0(O))n × L2(O), respectively, satisfy the following systems:
+�
+�
+�
+�
+�
+−
+n
+�
+j,α,β=1
+∂
+∂xα
+�
+bαβ
+ij
+∂ˇuj
+∂xβ
+�
++ ∇ˇp
+= − 1
+τ ˇv
+in O,
+div (ˇu)
+= 0
+in O,
+and
+�
+�
+�
+�
+�
+−
+n
+�
+i,α,β=1
+∂
+∂xβ
+�
+bβα
+ji
+∂ˇvi
+∂xα
+�
++ ∇ˇq
+= ˇu − ud
+in O,
+div (ˇv)
+= 0
+in O.
+Then, the triplet
+�ˇu, ˇp, − 1
+τ ˇv
+�
+is the optimal solution to (6.1).
+7
+Convergence results
+We present here the key findings on the convergence analysis of the optimal solutions to the
+problem (3.1) and its corresponding adjoint system (3.5) by using the method of periodic
+unfolding for perforated domains described in Section 5.
+Theorem 7.1. For given ε > 0, let the triplets (uε, pε, θε) and (u, p, θ), respectively, be the
+11
+
+optimal solutions of the problems (3.1) and (6.1). Then
+T ∗
+ε (Aε) → A
+strongly in (L2(O × W ∗))n×n,
+(7.1a)
+�
+θε ⇀ Θ θ
+weakly in
+�
+L2 (O)
+�n ,
+(7.1b)
+�
+uε ⇀ Θ u
+weakly in (H1
+0(O))n,
+(7.1c)
+�
+vε ⇀ Θ v
+weakly in (H1
+0(O))n,
+(7.1d)
+�pε ⇀ Θ
+n A0∇u: I + Θ p
+weakly in L2(O),
+(7.1e)
+�qε ⇀ Θ
+n At
+0∇v: I + Θ q
+weakly in L2(O),
+(7.1f)
+where A0 is a tensor as defined in Section 6, I is the n × n identity matrix, θ is characterized
+through (6.6) and the pairs (vε, qε) and (v, q) solve respectively the systems (3.5) and (6.4).
+Moreover,
+lim
+ε→0 Jε(θε) = J(θ).
+(7.2)
+Proof. First, upon using Proposition 5.2 (vi) on the entries of the matrix Aε, we obtain (7.1a)
+under the passage of limit ε → 0. Similarly, one can prove the convergence for the matrix
+At
+ε under unfolding. Next, in view of Theorem 4.3 and the fact that the triplet (uε, pε, θε)
+is an optimal solution to problem (3.1), one gets uniform estimates for the sequences {θε},
+{uε}, {pε}, {v��}, and {qε} in the spaces (L2 (O∗
+ε))n, (H1
+Γε
+0(O∗
+ε))n, L2 (O∗
+ε), (H1
+Γε
+0(O∗
+ε))n, and
+L2 (O∗
+ε), respectively.
+Using the uniform estimate of the sequence {θε} in the space (L2 (O∗
+ε))n and Proposition
+5.2 (i), we have the sequence {T ∗
+ε (θε)} to be uniformly bounded in the space (L2 (O × W ∗))n.
+Thus, by weak compactness, there exists a subsequence not relabelled and a function ˆθ in
+(L2 (O × W ∗))n, such that
+T ∗
+ε (θε) ⇀ ˆθ
+weakly in
+�
+L2 (O × W ∗)
+�n .
+(7.3)
+Now, using Proposition 5.2 (vii) in (7.3) gives
+�
+θε ⇀
+1
+|W|
+�
+W ∗
+ˆθ(x, y) dy = Θ θ0
+weakly in
+�
+L2 (O)
+�n ,
+(7.4)
+where, θ0 = MW ∗(ˆθ).
+Employing Proposition 5.2 (i), we have the uniform boundedness of the sequences {T ε(uε)},
+{T ε(∇uε)}, and {T ε(pε)} in the respective spaces (L2(O; H1 (W ∗)))n, (L2(O × W ∗))n×n,
+and L2(O × W ∗). Further, upon employing Proposition 5.3 and Proposition 5.2 (vii), there
+exist subsequences not relabelled and functions ˆu with MW ∗(ˆu) = 0, u0, and ˆp in spaces
+12
+
+(L2(O; H1
+per (W ∗)))n, (H1
+0(O))n, and L2(O × W ∗), respectively, such that
+T ∗
+ε (uε) → u0
+strongly in (L2(O; H1 (W ∗)))n,
+(7.5a)
+T ∗
+ε (∇uε) ⇀ ∇u0 + ∇y ˆu
+weakly in (L2(O × W ∗))n×n,
+(7.5b)
+�
+uε ⇀ Θ u0
+weakly in (H1
+0(O))n,
+(7.5c)
+T ∗
+ε (pε) ⇀ ˆp
+weakly in L2(O × W ∗),
+(7.5d)
+�pε ⇀ Θ MW ∗(ˆp)
+weakly in L2(O).
+(7.5e)
+Likewise, for the associated adjoint counterparts, viz., vε, and qε , one obtains that there
+exist subsequences not relabelled and functions ˆv with MW ∗(ˆv) = 0, v0, and ˆq in spaces
+(L2(O; H1
+per (W ∗)))n, (H1
+0(O))n, and L2(O × W ∗), respectively, such that
+T ∗
+ε (vε) → v0
+strongly in (L2(O; H1 (W ∗)))n,
+(7.6a)
+T ∗
+ε (∇vε) ⇀ ∇v0 + ∇yˆv
+weakly in (L2(O × W ∗))n×n,
+(7.6b)
+�
+vε ⇀ Θ v0
+weakly in (H1
+0(O))n,
+(7.6c)
+T ∗
+ε (qε) ⇀ ˆq
+weakly in L2(O × W ∗),
+(7.6d)
+�qε ⇀ MW ∗(ˆq)
+weakly in L2(O).
+(7.6e)
+The identification of the limit functions ˆu, ˆv, ˆp, ˆq, MW ∗(ˆp) and MW ∗(ˆq) is carried out in
+subsequent steps.
+Step 1: (Claim): For all ϕ ∈ (H1
+0(O))n, ψ ∈
+�
+L2 �
+O; H1
+per (W ∗)
+��n , and w ∈ L2(O), we
+claim that the ordered quadruplet (u0, ˆu, ˆp, θ0) ∈ (H1
+0(O))n ×(L2(O; H1
+per (W ∗)))n ×L2(O×
+W ∗) × (L2(O))n is a unique solution to the following limit system:
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1
+|W|
+�
+O×W ∗ A(y) (∇u0 + ∇y�u(x, y)) : (∇ϕ + ∇yψ) dx dy
+− 1
+|W|
+�
+O×W ∗ ˆp(x, y) (div(ϕ) + divy(ψ)) dx dy = Θ
+�
+O
+θ0 · ϕ dx,
+and,
+�
+O
+div(u0) w dx = 0,
+(7.7)
+and the ordered triplet (v0, ˆv, ˆq) ∈ (H1
+0(O))n×(L2(O; H1
+per (W ∗)))n×L2(O×W ∗) is a unique
+solution to the following limit adjoint system:
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1
+|W|
+�
+O×W ∗ At(y) (∇v0 + ∇y�v(x, y)) : (∇ϕ + ∇yψ) dx dy
+− 1
+|W|
+�
+O×W ∗ ˆq(x, y) (div(ϕ) + divy(ψ)) dx dy = Θ
+�
+O
+(u0 − ud) · ϕ dx,
+and,
+�
+O
+div(v0) w dx = 0.
+(7.8)
+13
+
+Proof of the Claim: Towards the proof of (7.7), let us consider a test function ϕ ∈ (D(O))n
+in (3.3) and use properties (i), (ii), and (iv) of Proposition 5.2 to get
+1
+|W|
+�
+O×W ∗ T ∗
+ε (Aε) T ∗
+ε (∇uε): T ∗
+ε (∇ϕ) dx dy +
+�
+ˆΛ∗ε
+Aε∇uε : ∇ϕ dx −
+�
+ˆΛ∗ε
+pε div(ϕ) dx
+−
+1
+|W|
+�
+O×W ∗ T ∗
+ε (pε) T ∗
+ε (div(ϕ)) dx dy =
+1
+|W|
+�
+O×W ∗ T ∗
+ε (θε) · T ∗
+ε (φε) dx dy +
+�
+ˆΛ∗ε
+θε · ϕ dx.
+(7.9)
+Using Proposition 5.2 (iii), the fact that limε→0 |ˆΛ∗
+ε| = 0, and convergences (7.3), (7.1a),
+(7.5b), (7.5d), we have under the passage of limit ε → 0 in (7.9)
+1
+|W|
+�
+O×W ∗ A(y) (∇u0 + ∇y�u(x, y)) : ∇ϕ dx dy
+− 1
+|W|
+�
+O×W ∗ ˆp(x, y) div(ϕ) dx dy = Θ
+�
+O
+θ0 · ϕ dx,
+(7.10)
+which remains valid for every ϕ ∈ (H1
+0(O))n, by density.
+Now, consider the function φε(x) = εφ(x)ξ( x
+ε), where φ ∈ D(O) and ξ ∈ (H1
+per(W ∗))n.
+Employing properties (ii), (iii), and (vi) of Proposition 5.2, one can easily obtain
+T ∗
+ε (φε) (x, y) → 0
+strongly in (L2(O × W ∗))n,
+(7.11a)
+T ∗
+ε (∇φε) (x, y) → φ(x)∇yξ(y)
+strongly in (L2(O × W ∗))n×n.
+(7.11b)
+Let us use the test function φε in (3.3) and employ properties (i), (ii), and (iv) of Proposition
+5.2 to get
+1
+|W|
+�
+O×W ∗ T ∗
+ε (Aε) T ∗
+ε (∇uε): T ∗
+ε (∇φε) dx dy +
+�
+ˆΛ∗ε
+Aε∇uε : ∇φε dx −
+�
+ˆΛ∗ε
+pε div(φε) dx
+− 1
+|W|
+�
+O×W ∗ T ∗
+ε (pε) T ∗
+ε (div(φε)) dx dy =
+1
+|W|
+�
+O×W ∗ T ∗
+ε (θε) · T ∗
+ε (φε) dx dy +
+�
+ˆΛ∗ε
+θε · φε dx.
+(7.12)
+In (7.12), the absolute value of each integral over ˆΛ∗
+ε is bounded above with a bound of
+order ε|ˆΛ∗
+ε| or |ˆΛ∗
+ε|. This with the fact that limε→0 |ˆΛ∗
+ε| = 0, and convergences (7.3), (7.1a),
+(7.5b), (7.5d), and (7.11), gives under the passage of limit ε → 0
+1
+|W|
+�
+O×W ∗ A(y) (∇u0 + ∇y�u(x, y)) : ∇yψ dx dy −
+1
+|W|
+�
+O×W ∗ ˆp(x, y) divy(ψ) dx dy = 0,
+(7.13)
+which remains valid for every φ ξ = ψ ∈ (L2(O; H1
+per(W ∗)))n, by density.
+Further, for all w ∈ L2(O), we have
+�
+O∗ε
+div(uε)w dx = 0.
+(7.14)
+14
+
+Now, upon applying unfolding on (7.14) and using properties (i), (ii), and (iii) of Proposition
+5.2 along with convergence (7.5b), we get under the passage of limit ε → 0
+1
+|W|
+�
+O×W ∗ (div(u0) + divy(ˆu)) w dx dy = 0,
+which eventually gives upon using the fact that ˆu is W ∗− periodic, for all w ∈ L2(O):
+�
+O
+div(u0)w dx = 0.
+(7.15)
+Finally, upon adding (7.10) with (7.13) and considering (7.15), we establish (7.7). Likewise,
+one can easily establish (7.8). This settles the proof of the claim.
+Step 2: First, we are going to identify the limit functions ˆu, ˆv, ˆp, and ˆq. Next, using these
+identifications, we will identify MW ∗(ˆp) and MW ∗(ˆq).
+Identification of ˆu, ˆv, ˆp, ˆq: Taking sucessively ϕ ≡ 0 and ψ ≡ 0 in (7.7), yields
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+− divy(A(y)∇y�u(x, y)) + ∇y�p(x, y) = divy(A(y))∇u0(x)
+in O × W ∗,
+− divx
+��
+W ∗ A(y)(∇u0(x) + ∇y�u(x, y))dy
+�
++ ∇�p(x, y) = |W ∗| θ0
+in O,
+div(u0) = 0
+in O,
+�u(x, ·)
+is W ∗ − periodic.
+(7.16)
+In the first line of (7.16), we have the y-independence of ∇u0(x) and the linearity of opera-
+tors, viz., divergence and gradient, which suggests �u(x, y) and �p(x, y) to be of the following
+form (see, for e.g., [17, Page 15]):
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�u(x, y) = −
+n
+�
+j,β=1
+χβ
+j (y)∂u0j
+∂xβ
++ u1(x),
+�p(x, y) =
+n
+�
+j,β=1
+Πβ
+j (y)∂u0j
+∂xβ
++ p0(x).
+(7.17)
+where the ordered pair (u1, p0) ∈ (H1(O))n × L2(O), and for 1 ≤ j, β ≤ n, the pair (χβ
+j , Πβ
+j )
+satisfy the cell problem (6.3). Likewise we obtain for the corresponding adjoint weak formu-
+lation (7.8):
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+− divy(A(y)∇y�v(x, y)) + ∇y�q(x, y) = divy(A(y))∇v0(x)
+in O × W ∗,
+− divx
+��
+W ∗ A(y)(∇v0(x) + ∇y�v(x, y))dy
+�
++ ∇�q(x, y) = |W ∗| (u0 − ud)
+in O,
+div(v0) = 0
+in O,
+�v(x, ·)
+is W ∗ − periodic,
+(7.18)
+15
+
+and,
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�v(x, y) = −
+n
+�
+j,β=1
+Hβ
+j (y)∂v0j
+∂xβ
++ v1(x),
+�q(x, y) =
+n
+�
+j,β=1
+Zβ
+j (y)∂v0j
+∂xβ
++ q0(x),
+(7.19)
+where the ordered pair (v1, q0) ∈ (H1(O))n ×L2(O), and for 1 ≤ j, β ≤ n, the pair (Hβ
+j , Zβ
+j )
+satisfy the cell problem (6.5).
+Identification of MW ∗(ˆp) and MW ∗(ˆq):
+Choosing the test function y = (y1, . . . , yn) in
+the weak formulation of (6.3), we get
+n
+�
+i,l,k,α=1
+�
+W ∗ alk
+∂
+∂yk
+�
+P β
+j − χβ
+j
+�
+· ∂P α
+i
+∂yl
+∂yi
+∂yα
+dy = n
+�
+W ∗ Πβ
+j dy.
+(7.20)
+In view of (7.5e), (7.17), and (7.20), we observe that
+MW ∗(ˆp) =
+1
+|W ∗|
+n
+�
+i,j,l,k,α,β=1
+�
+W ∗ alk
+∂
+∂yk
+�
+P β
+j − χβ
+j
+�
+· ∂P α
+i
+∂yl
+∂yi
+∂yα
+∂u0j
+∂xβ
+dy + p0,
+which upon using the definition of aαβ
+ij , gives
+MW ∗(ˆp) =
+n
+�
+i,j,α,β=1
+aαβ
+ij
+∂u0j
+∂xβ
+∂yi
+∂yα
++ p0.
+(7.21)
+Also, we re-write the equation (7.21) to get the identification of MW ∗(ˆp) as
+MW ∗(ˆp) = A0∇u0 : I + p0.
+(7.22)
+Likewise, one can obtain the identification of MW ∗(ˆq) as
+MW ∗(ˆq) = At
+0∇v0 : I + q0.
+(7.23)
+Thus, from (7.5e) and (7.22); (7.6e) and (7.23), we have the following weak convergences:
+�pε ⇀ Θ
+n A0∇u0 : I + Θ p0
+weakly in L2(O),
+(7.24a)
+�qε ⇀ Θ
+n At
+0∇v0 : I + Θ q0
+weakly in L2(O).
+(7.24b)
+Step 3: (Claim): The pairs (u0, p0) and (v0, q0) solve the systems (6.2) and (6.4), respec-
+tively.
+Proof of the Claim: We now prove that the pair (u0, p0) solves the system (6.2). The proof
+16
+
+that the pair (v0, q0) solves the system (6.4) follows analogously. Substituting the values of
+�u(x, y) and �p(x, y) from expression (7.17) into equation (7.10), we get
+1
+|W|
+n
+�
+l,k=1
+�
+O×W ∗ alk
+�
+∂u0
+∂xk
+−
+n
+�
+j,β=1
+∂χβ
+j
+∂yk
+∂u0j
+∂xβ
+�
+∂ϕ
+∂xl
+dx dy −
+1
+|W|
+n
+�
+j,β=1
+�
+O×W ∗ Πβ
+j
+∂u0j
+∂xβ
+div(ϕ) dx dy
+−Θ
+�
+O
+p0 div(ϕ) dx = Θ
+�
+O
+θ0 · ϕ dx.
+(7.25)
+With P β
+j = (0, . . . , yj, . . . , 0), we can express the terms ∂u0
+∂xk , ∂ϕ
+∂xl, and div(ϕ) as
+∂u0
+∂xk
+=
+n
+�
+j,β=1
+∂P β
+j
+∂yk
+∂u0j
+∂xβ
+,
+∂ϕ
+∂xl
+=
+n
+�
+i,α=1
+∂P α
+i
+∂yl
+∂ϕi
+∂xα
+,
+div(ϕ) =
+n
+�
+i,α=1
+divy(P α
+i ) ∂ϕi
+∂xα
+.
+Substituting these expressions in (7.25), we obtain
+n
+�
+i,j,α,β=1
+�
+O
+�
+1
+|W ∗|
+n
+�
+l,k=1
+�
+W ∗ alk
+∂
+∂yk
+�
+P β
+j − χβ
+j
+� ∂P α
+i
+∂yl
+dy
+�
+∂u0j
+∂xβ
+∂ϕi
+∂xα
+dx
+−
+n
+�
+i,j,α,β=1
+�
+O
+�
+1
+|W ∗|
+�
+W ∗ Πβ
+j divy(P α
+i ) dy
+� ∂u0j
+∂xβ
+∂ϕi
+∂xα
+dx −
+�
+O
+p0 div(ϕ) dx =
+�
+O
+θ0 · ϕ dx.
+(7.26)
+Now, choosing the test function χα
+i in the weak formulation of (6.3), we get upon using
+the fact that divy(χα
+i ) = divy(P α
+i ) = δiα, where δ denotes the Kronecker delta function, the
+following:
+�
+W ∗ A(y)∇y
+�
+P β
+j − χβ
+j
+�
+: ∇yχα
+i dy =
+�
+W ∗ Πβ
+j δiα dy.
+(7.27)
+Further, substituting (7.27) in (7.26), we obtain
+n
+�
+i,j,α,β=1
+�
+O
+�
+1
+|W ∗|
+n
+�
+l,k=1
+�
+W ∗ alk
+∂
+∂yk
+�
+P β
+j − χβ
+j
+� ∂
+∂yl
+(P α
+i − χα
+i ) dy
+�
+∂u0j
+∂xβ
+∂ϕi
+∂xα
+dx
+−
+�
+O
+p0 div(ϕ) dx =
+�
+O
+θ0 · ϕ dx.
+(7.28)
+Also, we can write equation (7.28) as
+n
+�
+i,j,α,β=1
+�
+O
+bαβ
+ij
+∂u0j
+∂xβ
+∂ϕi
+∂xα
+dx −
+�
+O
+p0 div(ϕ) dx =
+�
+O
+θ0 · ϕ dx,
+(7.29)
+17
+
+which holds true for all ϕ ∈ (H1
+0(O))n. Also, from equation (7.15), we have
+�
+O div(u)w dx =
+0, for every w ∈ L2(O). This together with equation (7.29) implies that, for θ = θ0, the
+pair (u0, p0) ∈ (H1
+0(O))n × L2(O) satisfies the variational formulation of the system (6.2).
+Therefore, we obtain the optimality system for the minimization problem (6.1). Also, in
+view of Theorem 6.1, we conclude that the triplet (u0, p0, θ0) is indeed an optimal solution to
+the problem (6.1). Finally, upon considering the optimal solution’s uniqueness, we establish
+that the subsequent pair of triplets are equal:
+(u, p, θ) = (u0, p0, θ0).
+(7.30)
+Hence, upon comparing (7.5c), (7.6c), (7.24a), (7.24b), and (7.4) with (7.30), we obtain
+convergences (7.1c), (7.1d), (7.1e), (7.1f), and (7.1b), respectively.
+Step 4: Now, we will furnish the proof of the energy convergence for the L2−cost functional.
+Choosing the test function (uε − ud) in the weak formulation of system (3.5), we get under
+unfolding upon passing ε → 0
+lim
+ε→0
+�
+O∗ε
+|uε − ud|2 dx =
+1
+|W| lim
+ε→0
+�
+O×W ∗ T ∗
+ε (At
+ε) T ∗
+ε (∇vε): T ∗
+ε (∇(uε − ud)) dx dy
++
+1
+|W| lim
+ε→0
+�
+O×W ∗ T ∗
+ε (qε) T ∗
+ε (div(ud)) dx dy,
+which gives in view of (7.30), Proposition 5.2 (iii) and convergences (7.6a), (7.5b), and (7.6d)
+lim
+ε→0
+�
+O∗ε
+|uε − ud|2 dx =
+1
+|W|
+�
+O×W ∗ At(y) (∇v + ∇y�v(x, y)) : ∇y(u − ud) dx dy
++
+1
+|W|
+�
+O×W ∗ ˆq(x, y) div(ud) dx dy.
+(7.31)
+Also, using (7.19) in (7.31) alongwith (7.30), we have upon simplification
+lim
+ε→0
+�
+O∗ε
+|uε − ud|2 dx = Θ
+�
+n
+�
+i,j,α,β=1
+�
+O
+bβα
+ji
+∂vi
+∂xα
+∂(u − ud)j
+∂xβ
+dx −
+�
+O
+q div(u − ud) dx
+�
+.
+(7.32)
+Now, using the test function (u − ud) in the weak formulation of system (6.4), we get the
+following upon comparing with the right hand side of equation (7.32)
+lim
+ε→0
+�
+O∗ε
+|uε − ud|2 dx = Θ
+�
+O
+|u − ud|2 dx.
+(7.33)
+Furthermore, in view of (3.6), (7.6a), and (7.30), we get under unfolding upon the passage
+18
+
+of limit ε → 0
+lim
+ε→0
+τ
+2
+�
+O∗ε
+|θε|2 dx = lim
+ε→0
+1
+2|W|
+�
+O×W ∗ |T ∗
+ε (θε)|2 dx dy
+= lim
+ε→0
+1
+2τ|W|
+�
+O×W ∗ |T ∗
+ε (vε)|2 dx dy
+=
+1
+2τ|W|
+�
+O×W ∗ |v|2 dx dy.
+(7.34)
+Also, since v is independent of y and comparing the right hand side of (7.34) with (6.6), we
+get
+lim
+ε→0
+τ
+2
+�
+O∗ε
+|θε|2 dx = Θτ
+2
+�
+O
+|θ|2 dx.
+(7.35)
+Thus, from equations (7.33) and (7.35), we get (7.2).
+This completes the proof of Theorem 7.1.
+8
+Conclusions
+We have addressed the limiting behavior of an interior OCP corresponding to Stokes equa-
+tions in an nD (n ≥ 2)
+periodically perforated domain
+O∗
+ε via the technique of periodic
+unfolding in perforated domains (see, [8, 11]). We employed the Neumann boundary con-
+dition on the part of the boundary of the perforated domain. Firstly, we characterized the
+optimal control in terms of the adjoint state. Secondly, we deduced the apriori optimal
+bounds for control, state, pressure, and their associated adjoint state and pressure functions.
+Thereafter, the limiting analysis for the considered OCP is carried out upon employing the
+periodic unfolding method in perforated domains. We observed the convergence between the
+optimal solution to the problem (3.1) posed on the perforated domain O∗
+ε and the optimal
+solution to that of the limit problem (6.1) governed by stationary Stokes equation posed on
+a non-perforated domain O. Finally, we established the convergence of energy corresponding
+to L2−cost functional.
+9
+Acknowledgments
+The first author would like to thank the Ministry of Education, Government of India for
+Prime Minister’s Research Fellowship (PMRF-2900953). The second author would like to
+thank the support from Science & Engineering Research Board (SERB) (SRG/2019/000997),
+Government of India.
+19
+
+References
+[1] G. Allaire, Homog´en´eisation des ´equations de Stokes dans un domaine perfor´e de petits
+trous r´epartis p´eriodiquement, C. R. Acad. Sci. Paris S´er. I Math. 309 (1989), no. 11,
+741–746.
+[2]
+, Homogenization of the Navier-Stokes equations with a slip boundary condition,
+Comm. Pure Appl. Math. 44 (1991), no. 6, 605–641.
+[3] G. Allaire and F. Murat, Homogenization of the Neumann problem with nonisolated
+holes, Asymptotic Anal. 7 (1993), no. 2, 81–95, With an appendix written jointly with
+A. K. Nandakumar.
+[4] A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic analysis for periodic struc-
+tures, Studies in Mathematics and its Applications, vol. 5, North-Holland Publishing
+Co., Amsterdam-New York, 1978.
+[5] F. Boyer and P. Fabrie, Mathematical tools for the study of the incompressible Navier-
+Stokes equations and related models, Applied Mathematical Sciences, vol. 183, Springer,
+New York, 2013.
+[6] A. Brillard, Asymptotic analysis of incompressible and viscous fluid flow through porous
+media. Brinkman’s law via epi-convergence methods, Ann. Fac. Sci. Toulouse Math. (5)
+8 (1986/87), no. 2, 225–252.
+[7] B. Cabarrubias, Homogenization of optimal control problems in perforated domains via
+periodic unfolding method, Appl. Anal. 95 (2016), no. 11, 2517–2534.
+[8] D. Cioranescu, A. Damlamian, P. Donato, G. Griso, and R. Zaki, The periodic unfolding
+method in domains with holes, SIAM J. Math. Anal. 44 (2012), no. 2, 718–760.
+[9] D. Cioranescu, P. Donato, and H. I. Ene, Homogenization of the Stokes problem with
+non-homogeneous slip boundary conditions, Math. Methods Appl. Sci. 19 (1996), no. 11,
+857–881.
+[10] D. Cioranescu, P. Donato, and R. Zaki, Periodic unfolding and Robin problems in per-
+forated domains, C. R. Math. Acad. Sci. Paris 342 (2006), no. 7, 469–474.
+[11]
+, The periodic unfolding method in perforated domains, Port. Math. (N.S.) 63
+(2006), no. 4, 467–496.
+[12] C. Conca, On the application of the homogenization theory to a class of problems arising
+in fluid mechanics, J. Math. Pures Appl. (9) 64 (1985), no. 1, 31–75.
+[13] C. Conca, P. Donato, E. C. Jose, and I. Mishra, Asymptotic analysis of optimal controls
+of a semilinear problem in a perforated domain, J. Ramanujan Math. Soc. 31 (2016),
+no. 3, 265–305.
+20
+
+[14] J. I. Diaz, A. V. Podolskiy, and T. A. Shaposhnikova, On the convergence of controls
+and cost functionals in some optimal control heterogeneous problems when the homog-
+enization process gives rise to some strange terms, J. Math. Anal. Appl. 506 (2022),
+no. 1, Paper No. 125559, 13.
+[15]
+, On the homogenization of an optimal control problem in a domain perforated
+by holes of critical size and arbitrary shape, Dokl. Math. 105 (2022), no. 1, 6–13.
+[16] H. I. Ene and E. S´anchez-Palencia,
+´Equations et ph´enom`enes de surface pour
+l′´ecoulement dans un mod`ele de milieu poreux, J. M´ecanique 14 (1975), 73–108.
+[17] S. Gu, Homogenization of Stokes Systems with Periodic Coefficients, ProQuest LLC,
+Ann Arbor, MI, 2016, Thesis (Ph.D.)–University of Kentucky.
+[18] S. Kesavan and J. Saint Jean Paulin, Homogenization of an optimal control problem,
+SIAM J. Control Optim. 35 (1997), no. 5, 1557–1573.
+[19]
+, Optimal control on perforated domains, Journal of Mathematical Analysis and
+Applications 229 (1999), no. 2, 563–586.
+[20] J.-L. Lions, Optimal control of systems governed by partial differential equations, Die
+Grundlehren der mathematischen Wissenschaften, Band 170, Springer-Verlag, New
+York-Berlin, 1971.
+[21] I. Mishra, Homogenization of boundary optimal control problem, Electron. J. Differential
+Equations 12 (2022), 1– 23.
+[22] T. Muthukumar and A. K. Nandakumaran, Darcy-type law associated to an optimal
+control problem, Electron. J. Differential Equations (2008), No. 16, 12.
+[23]
+, Homogenization of low-cost control problems on perforated domains, J. Math.
+Anal. Appl. 351 (2009), no. 1, 29–42.
+[24] J. Saint Jean Paulin and H. Zoubairi, Optimal control and “strange term” for a Stokes
+problem in perforated domains, Port. Math. (N.S.) 59 (2002), no. 2, 161–178.
+[25] R. Zaki, Homogenization of a Stokes problem in a porous medium by the periodic un-
+folding method, Asymptot. Anal. 79 (2012), no. 3-4, 229–250.
+21
+